Strategy and Game Theory: Practice Exercises with Answers (Springer Texts in Business and Economics) 3030119017, 9783030119010

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Springer Texts in Business and Economics

Felix Munoz-Garcia Daniel Toro-Gonzalez

Strategy and Game Theory Practice Exercises with Answers Second Edition

Springer Texts in Business and Economics

Springer Texts in Business and Economics (STBE) delivers high-quality instructional content for undergraduates and graduates in all areas of Business/Management Science and Economics. The series is comprised of self-contained books with a broad and comprehensive coverage that are suitable for class as well as for individual self-study. All texts are authored by established experts in their fields and offer a solid methodological background, often accompanied by problems and exercises.

More information about this series at http://www.springer.com/series/10099

Felix Munoz-Garcia • Daniel Toro-Gonzalez

Strategy and Game Theory Practice Exercises with Answers Second Edition

123

Felix Munoz-Garcia School of Economic Sciences Washington State University Pullman, WA, USA

Daniel Toro-Gonzalez School of Economics and Business Universidad Tecnológica de Bolívar Cartagena, Bolivar, Colombia

ISSN 2192-4333 ISSN 2192-4341 (electronic) Springer Texts in Business and Economics ISBN 978-3-030-11901-0 ISBN 978-3-030-11902-7 (eBook) https://doi.org/10.1007/978-3-030-11902-7 Library of Congress Control Number: 2018968092 1st edition: © Springer International Publishing Switzerland 2016 2nd edition: © Springer Nature Switzerland AG 2019, corrected publication 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This textbook presents worked-out exercises on Game Theory, with detailed step-by-step explanations, which both undergraduate and master’s students can use to further understand equilibrium behavior in strategic settings. While most textbooks on Game Theory focus on theoretical results, see, for instance, Tirole (1991), Gibbons (1992) and Osborne (2004), they offer few practice exercises. Our goal is, hence, to complement the theoretical tools in current textbooks by providing practice exercises in which students can learn to systematically apply theoretical solution concepts to different fields of Economics and Business, such as industrial economics, public policy, and regulation. The textbook provides many exercises with detailed verbal explanations (153 exercises in total), which cover the topics required by Game Theory courses at the undergraduate level, and by most courses at the master’s level. Importantly, our textbook emphasizes the economic intuition behind the main results and avoids unnecessary notation when possible, and thus is useful as a reference book regardless of the Game Theory textbook adopted by each instructor. Importantly, these points differentiate our presentation from that found in solutions manuals. Unlike these manuals, which can be rarely read in isolation, our textbook allows students to essentially read each exercise without difficulties, thanks to the detailed explanations, figures, and intuitions. Furthermore, for presentation purposes, each chapter ranks exercises according to their difficulty (with a letter A to C next to the exercise number), allowing students to first set their foundations using easy exercises (type-A), and then move on to harder applications (type-B and C exercises).

Organization of the Book We first examine games that are required in most courses at the undergraduate level, and then advance to more challenging games (which are often the content of master’s courses), both in Economics and Business programs. Specifically, Chaps. 1–6 cover complete-information games, separately analyzing simultaneous-move and sequential-move games, with applications from industrial economics and regulation, thus helping students apply Game Theory to other fields of research. Chapters 7–9 pay special attention to incomplete information games, such as auctions and v

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signaling games, while Chaps. 10 and 11 discuss cheap-talk games and equilibrium refinements. These topics have experienced a significant expansion in the last two decades, both in the theoretical and applied literature. Yet to this day most textbooks lack detailed worked-out examples that students can use as a guideline, leading them to especially struggle with this topic, which often becomes the most challenging for both undergraduate and graduate students. In contrast, our presentation emphasizes the common steps to follow when solving these types of incomplete information games and includes graphical illustrations to focus students’ attention to the most relevant payoff comparisons at each point of the analysis.

Changes in the Second Edition First, the second edition includes all the errata file identified by the authors and by several readers who generously pointed them to us. Thanks again for your nice comments, and don’t hesitate to let us know if you find any errata on the second edition! Second, we added over 40 new exercises to Chaps. 1–10, for a total of more than 200 new pages. The exercises in the first edition included many exercises at the graduate level (mainly master’s, but also some at the Ph.D. level), but relatively few exercises that undergraduate students in introductory or intermediate-level courses could use to improve their preparation on the subject. We took notice, so almost all new exercises in the second edition are appropriate for the undergraduate courses on Game Theory and Industrial Organization. Third, this edition includes a new chapter, Chap. 10, on cheap-talk games. This chapter serves as a direct application of the sequential-move games with incomplete information presented in Chap. 9. Cheap-talk games, however, assumes that messages are costless for the sender, so they can be understood as being a special type game from Chap. 9, helping us use the same approach to predict how players behave when interacting in this type of settings.

How to Use This Textbook Some instructors may use parts of the textbook in class in order to clarify how to apply certain solution concepts that are only theoretically covered in standard textbooks. Alternatively, other instructors may prefer to assign certain exercises as a required reading, since these exercises closely complement the material covered in class. This strategy could prepare students for the homework assignment on a similar topic, since our practice exercises emphasize the approach students need to follow in each class of games, and the main intuition behind each step. This strategy might be especially attractive for instructors at the graduate level, who could spend more time covering the theoretical foundations in class, asking students to go over our worked-out applications of each solution concept on their own.

Preface

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In addition, since exercises are ranked according to their difficulty, instructors at the undergraduate level can assign the reading of relatively easy exercises (type-A) and spend more time explaining the intermediate-level exercises in class (typeB questions), whereas instructors teaching a graduate-level course can assume that students are reading most type-A exercises on their own, and only use class time to explain type-C (and some type-B) exercises. In this second edition, there are 153 exercises in total: 43% are type-A, 45% are type-B, and the remaining 12% are type-C.

Acknowledgements We would first like to thank several colleagues who encouraged us in the preparation of this manuscript: Ron Mittlehammer, Jill McCluskey, and Alan Love. Ana Espinola-Arredondo reviewed several chapters on a short deadline and provided extremely valuable feedback, both in content and presentation, and we extremely thankful for her insights. Felix is especially grateful to his teachers and advisors at the University of Pittsburgh (Andreas Blume, Esther Gal-Or, John Duffy, Oliver Board, In-Uck Park, and Alexandre Matros), who taught him Game Theory and Industrial Organization, instilling a passion for the use of these topics in applied settings which hopefully transpires in the following pages. We appreciate several instructors using the first edition of the book for pointing out typos and explanations that could be improved, such as Stephan Kroll, from Colorado State University, and Ana Espinola-Arredondo, from Washington State University. We are also thankful to the “team” of teaching and research assistants, both at Washington State University and at Universidad Tecnologica de Bolivar, who helped us with this project over several years: Diem Nguyen, Gulnara Zaynutdinova, Modhurima Amin, Mianfeng Liu, Syed Badruddoza, Boris Houenou, Kiriti Kanjilal, Donald Petersen, Mursaleen Shiraj, Qingqing Wang, Jeremy Knowles, Xiaonan Liu, Ryan Bain, Eric Dunaway, Wisnu Sugiarto, Duc Vu, Tongzhe Li, Azhar Uddin, Wenxing Song, Pitchayaporn Tantihkarnchana, Roberto Fortich, Jhon Francisco Cossio Cardenas, Luis Carlos Díaz Canedo, Dayana Pinzon, and Kevin David Gomez Perez. Thanks also to Ignace Lasters, Pablo Abitbol, and Andrés Cendales for the comments to the first edition of the book. We also appreciate the support of the editors at Springer-Verlag, Rebekah McClure, Lorraine Klimowich, Jayanthi Narayanaswamy, and Dhivya Prabha. Importantly, we would like to thank our wives, Ana Espinola-Arredondo and Ericka Duncan, for supporting and inspiring us during the (long!) preparation of the manuscript. We would not have been able to do it without your encouragement and motivation. Pullman, USA Cartagena, Colombia

Felix Munoz-Garcia Daniel Toro-Gonzalez

The original version of the book was revised: Belated corrections have been incorporated throughout the book. The correction to the book is available at https://doi.org/10.1007/978-3-030-11902-7_12

Contents

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Dominance Solvable Games . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 1—From Extensive Form to Normal Form Representation-IA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 2—From Extensive Form to Normal Form Representation-IIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 3—From Extensive Form to Normal Form Representation-IIIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 4—Representing Games in Its Extensive FormA . . . . . . Exercise 5—Prisoners’ Dilemma GameA . . . . . . . . . . . . . . . . . . Exercise 6—Dominance Solvable GamesA . . . . . . . . . . . . . . . . . Exercise 7—Applying IDSDS (Iterated Deletion of Strictly Dominated Strategies)A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 8—Applying IDSDS When Players Have Five Available StrategiesA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 9—Applying IDSDS in the Battle of the Sexes GameA . Exercise 10—Applying IDSDS in Two Common GamesA . . . . . Exercise 11—Applying IDSDS in Three-Player GamesB . . . . . . . Exercise 12—IDSDS and RationalityA . . . . . . . . . . . . . . . . . . . . Exercise 13—Iterated Deletion of Strictly Dominated Strategies (IDSDS)A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 14—Unemployment BenefitsA . . . . . . . . . . . . . . . . . . . Exercise 15—Finding Dominant Strategies in Games with I  2 Players and with Continuous Strategy SpacesB . . . . . . . . . . . . . . Exercise 16—Equilibrium Predictions from IDSDS Versus IDWDSB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 17—Different Equilibrium Predictions from IDSDS and IDWDSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pure Strategy Nash Equilibrium with Complete Information . . . . Introduction . . . . . . . . . . . . . . . . Exercise 1—Prisoner’s DilemmaA

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Exercise 2—Battle of the SexesA . . . . . . . . . . . . . . . . . . . . . . . Exercise 3—Pareto Coordination GameA . . . . . . . . . . . . . . . . . Exercise 4—Finding Nash Equilibria in the Unemployment Benefits GameA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 5—Hawk-Dove GameA . . . . . . . . . . . . . . . . . . . . . . . Exercise 6—Generalized Hawk-Dove GameA . . . . . . . . . . . . . . Exercise 7—Cournot Game of Quantity CompetitionA . . . . . . . Exercise 8—Games with Positive ExternalitiesB . . . . . . . . . . . . Exercise 9—Traveler’s DilemmaB . . . . . . . . . . . . . . . . . . . . . . Exercise 10—Nash Equilibria with Three PlayersB . . . . . . . . . . Exercise 11—Simultaneous-Move Games with n  2 PlayersB Exercise 12—Political Competition (Hoteling Model)B . . . . . . . Exercise 13—TournamentsB . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 14—Lobbying GameA . . . . . . . . . . . . . . . . . . . . . . . . Exercise 15—Incentives and PunishmentB . . . . . . . . . . . . . . . . Exercise 16—Cournot Competition with Efficiency ChangesA . . Exercise 17—Cournot Mergers with Efficiency GainsB . . . . . . . 3

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Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 1—Game of ChickenA . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 2—Lobbying GameA . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 3—A Variation of the Lobbying GameB . . . . . . . . . . . . . Exercise 4—Finding a Mixed Strategy Nash Equilibrium in the Unemployment Benefits GameA . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 5—Newlyweds Buying an Apartment GameA . . . . . . . . . Exercise 6—Finding Mixed Strategies in the Hawk-Dove GameB . Exercise 7—Finding Mixed Strategies in the Generalized Hawk-Dove GameB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 8—Mixed Strategy Equilibrium with N > 2 PlayersB . . . . Exercise 9—Randomizing Over Three Available ActionsB . . . . . . Exercise 10—Rock-Paper-Scissors GameB . . . . . . . . . . . . . . . . . . Exercise 11—Penalty Kicks GameB . . . . . . . . . . . . . . . . . . . . . . . Exercise 12—Pareto Coordination GameB . . . . . . . . . . . . . . . . . . Exercise 13—Mixing Strategies in a Bargaining GameC . . . . . . . . Exercise 14—Depicting the Convex Hull of Nash Equilibrium PayoffsC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 15— Correlated EquilibriumC . . . . . . . . . . . . . . . . . . . . Exercise 16—Relationship Between Nash and Correlated Equilibrium PayoffsC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 17—Identifying Strictly Competitive GamesA . . . . . . . . . Exercise 18—Maxmin StrategiesC . . . . . . . . . . . . . . . . . . . . . . . .

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155 155 156 158 162 165 168 171 174 176 178 179 182 188 191

Sequential-Move Games with Complete Information . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 1—Ultimatum Bargaining GameB . . . . . . . . . . . . . . . . . . Exercise 2—Entry-Predation GameA . . . . . . . . . . . . . . . . . . . . . . . Exercise 3—Sequential Version of the Prisoner’s Dilemma GameA . Exercise 4—Sequential Version of the Battle of the Sexes GameA . . Exercise 5—Sequential Version of the Pareto Coordination GameA . Exercise 6—Sequential Version of the Chicken GameA . . . . . . . . . Exercise 7—Electoral CompetitionA . . . . . . . . . . . . . . . . . . . . . . . . Exercise 8—Electoral Competition with a TwistA . . . . . . . . . . . . . . Exercise 9—Centipede GameA . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 10—Trust and Reciprocity (Gift-Exchange Game)B . . . . . . Exercise 11—Stackelberg with Two FirmsA . . . . . . . . . . . . . . . . . . Exercise 12—Stackelberg Competition with 2, or 3 FirmsB . . . . . . . Exercise 13—Stackelberg Competition with Efficiency ChangesB . . Exercise 14—First- and Second-Mover Advantage in Product DifferentiationB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 15—Stackelberg Game with Three Firms Acting SequentiallyA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 16—Two-Period Bilateral Bargaining GameA . . . . . . . . . . Exercise 17—Alternating Bargaining with a TwistB . . . . . . . . . . . . Exercise 18—Backward Induction in Wage NegotiationsA . . . . . . . Exercise 19—Backward Induction-IB . . . . . . . . . . . . . . . . . . . . . . . Exercise 20—Backward Induction-IIB . . . . . . . . . . . . . . . . . . . . . . Exercise 21—Moral Hazard in the WorkplaceB . . . . . . . . . . . . . . .

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198 200 202 203 204 206 211

Applications to Industrial Organization . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 1—Bertrand Model of Price CompetitionA . . . . . . . . . Exercise 2—Bertrand Competition with Asymmetric CostsB . . . Exercise 3—Duopoly Game with A Public FirmB . . . . . . . . . . . Exercise 4—Cartel with Firms Competing a la Cournot—Unsustainable in a One-Shot gameA . . . . . . . . . . . . . Exercise 5—Cartel with firms competing a la Bertrand—Unsustainable in a one-shot gameA . . . . . . . . . . . . . Exercise 6—Cournot Competition with Asymmetric CostsA . . . Exercise 7—Cournot Competition and Cost DisadvantagesA . . . Exercise 8—Cartel with Two Asymmetric FirmsA . . . . . . . . . . Exercise 9—Strategic Advertising and Product DifferentiationC . Exercise 10—Cournot Oligopoly with CES DemandB . . . . . . . . Exercise 11—Commitment in Prices or Quantities?B . . . . . . . . . Exercise 12—Fixed Investment as a Pre-commitment StrategyB . Exercise 13—Socially Excessive Entry in an IndustryB . . . . . . .

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Exercise 14—Entry Deterring InvestmentB . . . . . . . . . . . . . . . . . . . . . 263 Exercise 15—Direct Sales or Using A Retailer?C . . . . . . . . . . . . . . . . 268 Exercise 16—Profitable and Unprofitable MergersA . . . . . . . . . . . . . . 273 6

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Repeated Games and Correlated Equilibria . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 1—Infinitely Repeated Prisoner’s Dilemma GameA . . . . . . Exercise 2—Collusion When Firms Compete in QuantitiesA . . . . . . Exercise 3—Temporary Punishments from DeviationB . . . . . . . . . . Exercise 4—Collusion When N Firms Compete in QuantitiesB . . . . Exercise 5—Collusion When N Firms Compete in PricesC . . . . . . . Exercise 6—Collusion in Donations in an Infinitely Repeated Public Good GameB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 7—Repeated Games with Three Available Strategies to Each Player in the Stage GameA . . . . . . . . . . . . . . . . . . . . . . . . Exercise 8—Infinite-Horizon Bargaining Game Between Three PlayersC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 9—Representing Feasible, Individually Rational PayoffsC . Exercise 10—Collusion and Imperfect MonitoringC . . . . . . . . . . . .

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Simultaneous-Move Games with Incomplete Information . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 1—A Simple Bayesian Game . . . . . . . . . . . . . . . . . . Exercise 2—Simple Poker GameA . . . . . . . . . . . . . . . . . . . . . Exercise 3—Incomplete Information Game, Allowing for More General ParametersB . . . . . . . . . . . . . . . . . . . . . . . . Exercise 4—More Information Might HurtB . . . . . . . . . . . . . . Exercise 5—Incomplete Information in Duopoly MarketsA . . . Exercise 6—Public Good Game with Incomplete InformationB Exercise 7—Starting a Fight Under Incomplete InformationC . Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 1—First Price Auction with Uniformly Distributed Valuations—Two PlayersA . . . . . . . . . . . . . . . . . . . . . . . . Exercise 2—First Price Auction with Uniformly Distributed Valuations—N  2 PlayersB . . . . . . . . . . . . . . . . . . . . . . Exercise 3—First Price Auction with N BiddersB . . . . . . . . Exercise 4—Second Price AuctionA . . . . . . . . . . . . . . . . . . Exercise 5—All-Pay Auctions (Easy Version)A . . . . . . . . . Exercise 6—All-Pay Auction (Complete Version)B . . . . . . . Exercise 7—Third-Price AuctionA . . . . . . . . . . . . . . . . . . . Exercise 8—FPA with Risk-Averse BiddersB . . . . . . . . . . .

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Perfect Bayesian Equilibrium and Signaling Games . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 1—Finding Separating and Pooling EquilibriaA . . . . . . . . . Exercise 2—Job-Market Signaling GameB . . . . . . . . . . . . . . . . . . . Exercise 3—Job Market Signaling with Productivity-Enhancing EducationB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 4—Job Market Signaling with Utility-Enhancing EducationB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 5—Finding Semi-Separating Equilibrium . . . . . . . . . . . . . Exercise 6—Firm Competition Under Cost UncertaintyC . . . . . . . . . Exercise 7—Signaling Game with Three Possible Types and Three MessagesC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 8—Second-Degree Price DiscriminationB . . . . . . . . . . . . . Exercise 9—Applying the Cho and Kreps’ (1987) Intuitive Criterion in the Far WestB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Cheap Talk Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 1—Cheap Talk Game with Only Two Sender’s Types, Two Messages, and Three ResponsesB . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 2—Cheap Talk Game with Only Two Sender Types, Two Messages, and a Continuum of ResponsesB . . . . . . . . . . . . . . . . . . Exercise 3—Cheap Talk with a Continuum of Types—Only Two Messages (Partitions)B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 4—Cheap Talk Game with a Continuum of Types—Three Messages (Partitions)B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 5—Cheap Talk Game with a Continuum of Types—N Messages (Partitions)C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 More Advanced Signaling Games . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 1—Poker Game with Two Uninformed PlayersB . . . Exercise 2—Incomplete Information and CertificatesB . . . . . . Exercise 3—Entry Game with Two Uninformed FirmsB . . . . Exercise 4—Labor Market Signaling Game and Equilibrium RefinementsB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 5—Entry Deterrence Through Price WarsA . . . . . . . Exercise 6—Entry Deterrence with a Sequence of Potential EntrantsC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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487 487 488 493 503

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Correction to: Strategy and Game Theory . . . . . . . . . . . . . . . . . . . . . . .

C1

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

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Dominance Solvable Games

Introduction This chapter first analyzes how to represent games in normal form (using matrices) and in extensive form (using game trees). We afterwards describe how to systematically detect strictly dominated strategies, i.e., strategies that a player would not use regardless of the action chosen by his opponents. Strictly dominated strategy. We say that player i finds strategy s
i as strictly dominated by another strategy s0i if

 ui s0i ; si [ ui s
i ; si for every strategy profile si 2 Si , where si ¼ ðs1 ; s2 ; . . .; si1 ; si þ 1 ; . . .; sN Þ represents the profile of strategies selected by player i’s opponents, i.e., a vector with N  1 components. In words, strategy s0i strictly dominates s
i if it yields a strictly higher utility than strategy s
i regardless of the strategy profile that player i’s rival choose. Since we can anticipate that strictly dominated strategies will not be selected by rational players, we apply the Iterative Deletion of Strictly Dominated Strategies (IDSDS) to predict players’ behavior. We elaborate on the application of IDSDS in games with two and more players, and in games where players are allowed to choose between two strategies, between more than two strategies, or a continuum of strategies. In some games, we will show that the application of IDSDS is powerful, as it rules out dominated strategies and leaves us with a relatively precise equilibrium prediction, i.e., only one or two strategy profiles surviving the application of IDSDS. In other games, however, we will see that IDSDS “does not have a bite” because no strategies are dominated; that is, a strategy does not provide a strictly lower payoff to player i regardless of the strategy profile selected by his opponents (it can provide a higher payoff under some of his opponents’ strategies). In this case, we won’t be able to offer an equilibrium prediction, other than to say that the © Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_1

1

2

1

Dominance Solvable Games

entire game is our most precise equilibrium prediction! In subsequent chapters, however, we explore other solution concepts that provide more precise predictions than IDSDS. Finally, we study the deletion of weakly dominated strategies does not necessarily lead to the same equilibrium outcomes as IDSDS, and its application is in fact sensible to deletion order. We can apply the above definition of strictly dominated strategies to define weakly dominated strategies. We next define what we mean when we say that player i finds that one strategy weakly dominates another strategy. Weakly dominated strategy. We say that strategy s0i weakly dominates s
i if

 ui s0i ; si  ui s
i ; si for every strategy profiles si 2 Si and

 ui s0i ; si  ui s
i ; si for at least one strategy profiles si 2 Si In words, player i finds that strategy s0i weakly dominates s
i if it yields a weakly higher utility than strategy s
i for every strategy profile that player i’s rival choose, but yields a strictly higher utility than strategy s
i for at least one of the strategy profiles that his rivals choose. To understand this definition, consider for instance a two player game between players 1 and 2, where the former chooses between A and B while the latter chooses between a and b. We can say that player 1 finds that strategy A weakly dominates B if A yields a weakly higher utility than B when player 2 chooses a (i.e., A gives player 1 the same or higher utility than B does when player 2 selects a), but yields a strictly higher utility when player 2 chooses b. This chapter presents different examples of how to apply the above definitions.

Exercise 1—From Extensive Form to Normal Form Representation-IA Represent the extensive-form game depicted in Fig. 1.1 using its normal-form (matrix) representation. Answer We start identifying the strategy sets of all players in the game. The cardinality of these sets (number of elements in each set) will determine the number of rows and columns in the normal-form representation of the game. Starting from the initial node (in the root of the game tree located on the left-hand side of the Fig. 1.1), Player 1 must select either strategy A or B, thus implying that the strategy space for player 1, S1 , is: S1 ¼ fA; Bg

Exercise 1—From Extensive Form to Normal Form Representation-IA

3

Fig. 1.1 Extensive-form game

In the next stage of the game, Player 2 conditions his strategy on player 1’s choice, since player 2 observes such a choice before selecting his own. We need to consider that the strategy profile of player 2 (S2) must be a complete plan of action (complete contingent plan). Therefore, his strategy space becomes: S2 ¼ fCE; CF; DE; DF g where the first component of every strategy describes how player 2 responds upon observing that player 1 chose A, while the second component represents player 2’s response after observing that player 1 selected B. For example, strategy CE describes that player 2 responds with C after player 1 chooses A, but with E after player 1 chooses B. Using the strategy space of player 1, with only two available strategies S1 = {A, B}, and that of player 2, with four available strategies S2 = {CE, CF, DE, DF}, we obtain the 2  4 payoff matrix represented in Fig. 1.2. For instance, the payoffs associated with the strategy profile where player 1 chooses A and player 2 chooses C if A and E if B, {A, CE}, is (0, 0). Remark: Note that, if player 2 could not observe player 1’s action before selecting his own (either C or D), then player 2’s strategy space would be S2 ¼ fC; Dg, implying that the normal form representation of the game would be a 2  2 matrix with A and B in rows for player 1, and C and D in columns for player 2.

Player 2 Player 1

CE

CF

DE

DF

A

0, 0

0, 0

1, 1

1, 1

B

2, 2

3, 4

2, 2

3, 4

Fig. 1.2 Normal-form game

4

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Dominance Solvable Games

Exercise 2—From Extensive Form to Normal Form Representation-IIA Consider the extensive form game in Fig. 1.3. Part (a) Describe player 1’s strategy space. Part (b) Describe player 2’s strategy space. Part (c) Take your results from parts (a) and (b) and construct a matrix representing the normal form game of this game tree. Answer Part (a) Player 1 has three available strategies, implying a strategy space of S1 = {H, M, L} Part (b) Since in this game players act sequentially, the second mover can condition his move on the first player’s action. Hence, player 2’s strategy set is S2 ¼ faaa; aar; arr; rrr; rra; raa; ara; rar g

Fig. 1.3 Extensive-form game

Exercise 2—From Extensive Form to Normal Form Representation-IIA

5

where each of the strategies represents a complete plan of action that specifies player 2’s response after player 1 chooses H, after player 1 selects M, and after player 1 chooses L, respectively. For instance arr indicates that player 2 responds with a after observing that player 1 chose H, with r after observing M, and with r after observing L. Remark: If player 2 could not observe player 1’s choice, the extensive-form representation of the game would depict a long dashed line connecting the three nodes at which player 2 is called on to move. (This dashed line is often referred as player 2’s “information set”) In this case, player 2 would not be able to condition his choice (since he cannot observe which action player 1 chose before him), thus implying that player 2’s set of available actions reduces to only two (accept or reject), i.e., S2 = {a, r}. Part (c) If we take the three available strategies for player 1, and the above eight strategies for player 2, we obtain the following normal form game (Fig. 1.4).

Notice that this normal form representation contains the same payoffs as the game tree. For instance, after player 1 chooses M (in the second row), payoff pairs only depend on player 2’s response after observing M (the second component of every strategy triplet in the columns). Hence, payoff pairs are either (5, 5), which arise when player 2 responds with a after M, or (0, 0), which emerge when player 2 responds instead with r after observing M.

Exercise 3—From Extensive Form to Normal Form Representation-IIIB Consider the extensive-form game in Fig. 1.5. Provide its normal form (matrix) representation. Answer Player 2. From the extensive form game, we know player 2 only plays once and has two available choices, either A or B. The dashed line connecting the two nodes at

Player 1 H M L

aaa 0, 10 5, 5 10, 0

Fig. 1.4 Normal-form game

aar 0, 10 5, 5 10, 0

arr 0, 10 0, 0 0, 0

Player 2 rrr rra 0, 0 0, 0 0, 0 0, 0 0, 0 10, 0

raa 0, 0 5, 5 10, 0

ara 0, 10 0, 0 10, 0

rar 0, 0 5, 5 0, 0

6

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Dominance Solvable Games

Fig. 1.5 Extensive-form game

which player 2 is called on to move indicates that player 2 cannot observe player 1’s choice. Hence, he cannot condition his choice on player 1’s previous action, ultimately implying that his strategy space reduces to S2 ¼ fA; Bg. Player 1. Player 1, however, plays twice (in the root of the game tree, and after player 2 responds) and has multiple choices: 1. First, he must select either U or D, at the initial node of the tree, i.e., left-hand side of the Fig. 1.5; 2. Then choose X or Y, in case that he played U at the beginning of the game. (Note that in this event he cannot condition his choice on player 2’s choice, since he cannot observe whether player 2 selected A or B); and 3. Then choose P or Q, which only becomes available to player 1 in the event that player 2 responds with B after player 1 chose D. Therefore, player 1’s strategy space is composed of triplets, as follows, S1 ¼ fUXP; UXQ; UYP; UYQ; DXP; DXQ; DYP; DYQg

Exercise 3—From Extensive Form to Normal Form Representation-IIIB

7

whereby the first component of every triplet describes player 1’s choice at the beginning of the game (the root of the game tree), the second component represents his decision (X or Y) in the event that he chose U and afterwards player 2 responded with either A or B (something player 1 cannot observe), and the third component reflects his choice in the case that he chose D at the beginning of the game and player 2 responds with B.1 As a consequence, the normal-form representation of the game is given by the following 8  2 matrix represented in Fig. 1.6.

Exercise 4—Representing Games in Its Extensive FormA Consider the standard rock-paper-scissors game, which you probably played in your childhood. If you did not play this old game before, do not worry, we will explain it next. Two players face each other with both hands on their back. Then each player simultaneously chooses rock (R), paper (P) or scissors (S) by rapidly moving one of his hands to the front, showing his fits (a symbol of a rock), his extended palm (representing a paper), or two of his fingers in form of a V (symbolizing a pair of scissors). Players seek to select an object that is ranked superior to that of his opponent, where the ranking is the following: scissors beat paper (since they cut it), paper beats rock (because it can wrap it over), and rock beats scissors (since it can smash them). For simplicity, consider that a player obtains a payoff of 1 when his object wins, −1 when it losses, and 0 if there is a tie (which only occurs when both players select the same object). Provide a figure with the extensive-form representation of this game. Answer Since the game is simultaneous, the extensive-form representation of this game will have three branches in its root ( initial node), corresponding to Player 1’s choices, as in the game tree depicted in Fig. 1.7. Since Player 2 does not observe Player 1’s choice before choosing his own, Player 2 has three available actions (Rock, Paper 1

You might be wondering why do we have to describe player 1’s choice between X and Y in triplets indicating that player 1 selected D at the beginning of the game. The reason for this detailed description is twofold: on one hand, a complete contingent plan must indicate a player’s choices at every node at which he is called on to move, even those nodes that would not emerge along the equilibrium path. This is an important description in case player 1 submits his complete contingent plan to a representative who will play on his behalf. In this context, if the representative makes a mistake and selects U, rather than D, he can later on know how to behave after player 2 responds. If player 1’s contingent plan was, instead, incomplete (not describing his choice upon player 2’s response), the representative would not know how to react afterwards. On the other hand, a players’ contingent plan can induce certain responses from a player’s opponents. For instance, if player 2 knows that player 1 will only plays Q in the last node at which he is called on to move, player 2 would have further incentives to play A. Hence, complete contingent plans can induce certain best responses from a player’s opponents, which we seek to examine. (We elaborate on this topic in the next chapters, especially when analyzing equilibrium behavior in sequential-move games.).

8

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Dominance Solvable Games

Player 2

Player 1

A

B

UXP

3, 8

1, 2

UXQ

3, 8

1, 2

UYP

8, 1

2, 1

UYQ

8, 1

2, 1

DXP

6, 6

5, 5

DXQ

6, 6

0, 0

DYP

6, 6

5, 5

DYQ

6, 6

0, 0

Fig. 1.6 Normal-form game

Fig. 1.7 Extensive-form of the Rock, Paper and Scissors game

Exercise 4—Representing Games in Its Extensive FormA

9

and Scissors) which cannot be conditioned on Player 1’s actual choice. We graphically represent Player 2’s lack of information when he is called on to move by connecting Player 2’s three nodes with an information set (dashed line in Fig. 1.7). Finally, to represent the payoffs at the terminal nodes of the tree, we just follow the ranking specified above. For instance, when player 1 chooses rock (R) and player 2 selects scissors (S), player 1 wins, obtaining a payoff of 1, while player 2 losses, accruing a payoff of −1, this set of payoffs entails the payoff pair (1, −1). If, instead, player 2 selected paper, he would become the winner (since paper wraps the rock), entailing a payoff of 1 for player 2 and −1 for player 1, that is (−1, 1). Finally, notice that in those cases in which the objects players display coincide, i.e., {R, R}, {P, P} or {S, S}, the payoff pair becomes (0, 0).

Exercise 5—Prisoners’ Dilemma GameA Two individuals have been detained for a minor offense and confined in separate cells. The investigators suspect that these individuals are involved in a major crime, and separately offer each prisoner the following deal, as depicted in Fig. 1.8: if you confess while your partner doesn’t, you will leave today without serving any time in jail; if you confess and your partner also confesses, you will have to serve 5 years in jail (since prosecutors probably can accumulate more evidence against each prisoner when they both confess); if you don’t confess and your partner does, you will have to serve 15 years in jail (since you did not cooperate with the prosecution but your partner’s confession provides the police with enough evidence against you); finally, if neither of you confess, you will have to serve one year in jail. Part (a) Draw the prisoners’ dilemma game in its extensive form representation. Part (b) Mark its initial node, its terminal nodes, and its information set. Why do we represent this information set in the prisoners’ dilemma game in its extensive form? Part (c) How many strategies player 1 has? What about player 2?

Prisoner 2 Not Confess confess Confess Prisoner 1 Not confess

-5, -5

0, -15

-15, 0

-1, -1

Fig. 1.8 Normal-form of Prisoners’ dilemma game

10

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Dominance Solvable Games

Answer Part (a) Since both players must simultaneously choose whether or not to confess, player 2 cannot condition his strategy on player 1’s decision (which he cannot observe). We depict this lack of information by connecting both of the nodes at which player 2 is called on to move with an information set (dashed line) in Fig. 1.9. Part (b) Its initial node is the “root” of the game tree, whereby player 1 is called on to move between Confess and Not confess, the terminal nodes are the nodes where the game ends (and where we represent the payoffs that are accrued to every player), and the information set is a dashed line connecting the nodes in which the second mover is called to move. We represent this information set to denote that the second mover is choosing whether to Confess or Not Confess without knowing exactly what player 1 did. Part (c) Player 1 only has two possible strategies: S1 = {Confess, Not Confess}. The second player has only two possible strategies S2 = {Confess, Not Confess} as well, since he is not able to observe what player 1 did before taking his decision. As a consequence, player 2 cannot condition his strategy on player 1’s choice.

Fig. 1.9 Prisoners’ dilemma game in its extensive-form

Exercise 6—Dominance Solvable GamesA

11

Exercise 6—Dominance Solvable GamesA Two political parties simultaneously decide how much to spend on advertising, either low, medium or high, yielding the payoffs in the following matrix (Fig. 1.10) in which the Red party chooses rows and the Blue party chooses columns. Find the strategy profile/s that survive IDSDS. Answer Let us start by analyzing the Red party (row player). First, note that High is a strictly dominant strategy for the Red party, since it yields a higher payoff than both Low and Middle, regardless of the strategy chosen by the Blue party (i.e., independently of the column the Blue party selects). Indeed, 100 > 80 > 50 when the Blue party chooses the Low column, 80 > 70 > 0 when the Blue party selects the Middle column, and 50 > 20 > 0 when the Blue party chooses the High column. As a consequence, both Low and Middle are strictly dominated strategies for the Red party (they are both strictly dominated by High in the bottom row), and we can thus delete the rows corresponding to Low and Middle from the payoff matrix, leaving us with the following reduced matrix (Fig. 1.11). We can now check if there are any strictly dominated strategies for the Blue party (in columns). Similarly as for the Red party, High strictly dominates both Low and Middle since 50 > 20 > 0; and we can thus delete the columns corresponding to Low and Middle from the payoff matrix, leaving us with a single cell, (High, High). Hence, (High, High) is the unique strategy surviving IDSDS.

Blue Party Low Red Party Middle High

Low

Middle

High

80, 80

0, 50

0, 100

50, 0

70, 70

20, 80

100, 0

80, 20

50, 50

Fig. 1.10 Political parties normal-form game

Blue Party

Red Party High

Low

Middle

High

100, 0

80, 20

50, 50

Fig. 1.11 Political parties reduced normal-form game

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Dominance Solvable Games

Exercise 7—Applying IDSDS (Iterated Deletion of Strictly Dominated Strategies)A Consider the simultaneous-move game depicted in Fig. 1.12., where two players choose between several strategies. Find which strategies survive the iterative deletion of strictly dominated strategies, IDSDS. Answer Let us start by identifying the strategies of player 1 that are strictly dominated by other of his own strategies. When player 1 chooses a, in the first row, his payoff is either 1 (when player 2 chooses x or y) or zero (when player 2 chooses z, in the third column). These payoffs are unambiguously lower than those in strategy c in the third row. In particular, when player 2 chooses x (in the first column), player 1 obtains a payoff of 3 with c but only a payoff of 1 with a; when player 2 chooses y, player 1 earns 2 with c but only 1 with a; and when player 2 selects z, player 1 obtains 1 with c but a zero payoff with a. Hence, player 1’s strategy a is strictly dominated by c, since the former yields a lower payoff than the latter regardless of the strategy that player 2 selects (i.e., regardless of the column he uses). Thus, the strategies of player 1 that survive one round of the iterative deletion of strictly dominated strategies (IDSDS) are b, c and d, as depicted in the payoff matrix in Fig. 1.13. Let us now turn to player 2 (by looking at the second payoff within every cell in the matrix). In particular, we can see that strategy z strictly dominates x, since it provides to player 2 a larger payoff than x regardless of the strategy (row) that player 1 uses Specifically, when player 1 chooses b (top row), player 2 obtains a payoff of 2 by selecting z (see the right-hand column) but only a payoff of zero from choosing x (in the left-hand column). Similarly, when player 1 chooses c (in the middle row), player 2 earns a payoff of 2 from z but only a payoff of 1 from Fig. 1.12 Normal-form game with four available strategies

x

Player 1

Player 2 y

z

a

1, 2

1, 2

0, 3

b

4, 0

1, 3

0, 2

c

3, 1

2, 1

1, 2

d

0, 2

0, 1

2, 4

Exercise 7—Applying IDSDS (Iterated Deletion of Strictly Dominated Strategies)A Fig. 1.13 Reduced normal-form game after one round of IDSDS

13

Player 2

Player 1

x

y

z

b

4, 0

1, 3

0, 2

c

3, 1

2, 1

1, 2

d

0, 2

0, 1

2, 4

x. Finally, when player 1 selects d (in the bottom row), player 2 obtains a payoff of 4 from z but only a payoff of 2 from x. Hence, strategy z yields player 2 a larger payoff independently of the strategy chosen by player 1, i.e., z strictly dominates x, which allows us to delete strategy x from the payoff matrix. Thus, the strategies of player 2 that survive one additional round of the IDSDS are y and z, which helps us further reduce the payoff matrix to that in Fig. 1.14. We can now move to player 1 again. For him, strategy c strictly dominates b, since it provides an unambiguously larger payoff than b regardless of the strategy selected by player 2 (regardless of the column). In particular, when player 2 chooses y (left-hand column), player 1 obtains a payoff of 2 from selecting strategy c but only one from strategy b. Similarly, if player 2 chooses z (in the right-hand column), player 1 obtains a payoff of one from strategy c but a payoff of zero from strategy b. As a consequence, strategy b is strictly dominated, which allows us to delete strategy b from the above matrix, obtaining the reduced matrix in Fig. 1.15. At this point, note that returning to player 2 we note that z strictly dominates y, so we can delete strategy y for player 2 and finally, considering player 2 always chooses z, for player 1strategy d strictly dominates c, since the payoff of 2 is higher than one unit derived from playing c. Therefore, our most precise equilibrium prediction after using IDSDS are the solely remaining strategy profile (d,z), indicating that player 1 will always choose d, while player 2 will always select z.

Player 2 y

z

b

1, 3

0, 2

Player 1 c

2, 1

1, 2

d

0, 1

2, 4

Fig. 1.14 Reduced normal-form game after two rounds of IDSDS

14

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Dominance Solvable Games

Player 2

Player 1

y

z

c

2, 1

1, 2

d

0, 1

2, 4

Fig. 1.15 Reduced normal-form game

Exercise 8—Applying IDSDS When Players Have Five Available StrategiesA Two students in the Game Theory course plan to take an exam tomorrow. The professor seeks to create incentives for students to study, so he tells them that the student with the highest score will receive a grade of A and the one with the lower score will receive a B. Student 1’s score equals x1 þ 1:5, where x1 denotes the amount of hours studying. (That is, he assume that the greater the effort, the higher her score is.) Student 2’s score equals x2 , where x2 is the hours she studies. Note that these score functions imply that, if both students study the same number of hours, x1 ¼ x2 , student 1 obtains a highest score, i.e., she might be the smarter of the two. Assume, for simplicity, that the hours of studying for the game theory class is an integer number, and that they cannot exceed 5 h, i.e., xi 2 f1; 2; . . .; 5g. The payoff to every student i is 10—xi if she gets an A and 8—xi if she gets a B. Part (a) Find which strategies survive the iterative deletion of strictly dominated strategies (IDSDS). Part (b) Which strategies survive the iterative deletion of weakly dominated strategies (IDWDS). Answer Part (a) Let us first show that for either player, exerting a zero effort i.e., xi ¼ 0, strictly dominates effort levels of 3, 4, and 5. If xi ¼ 0 then player i’s payoff is at least 8, which occurs when she gets a B. By choosing any other effort level xi , the highest possible payoff is 10  xi , which occurs when she gets an A. Since 8 [ 10  xi when xi [ 2, then zero effort strictly dominates efforts of 3, 4, or 5. Intuitively, we consider which is the lowest payoff that player i can obtain from exerting zero effort, and compare it with the highest payoff he could obtain from deviating to a positive effort level xi 6¼ 0. If this holds for some effort levels (as it does here for all xi [ 2), it means that, regardless of what the other student j 6¼ i does, student i is strictly better off choosing a zero effort than deviating.

Exercise 8—Applying IDSDS When Players Have Five Available StrategiesA Fig. 1.16 Reduced normal-form game

15

Player 2

Player 1

0

1

2

0

10, 8

10, 7

8, 8

1

9, 8

9, 7

9, 6

2

8, 8

8, 7

8, 6

Once we delete effort levels satisfying xi [ 2; i.e., xi ¼ 3, 4, and 5 for both player 1 (in rows) and player 2 (in columns), we obtain the 3  3 payoff matrix depicted in Fig. 1.16. As a practice of how to construct the payoffs in this matrix, note that, for instance, when player 1 chooses x1 = 1 and player 2 selects x2 = 2, player 1 still gets the highest score, i.e., player 1’s score is 1 + 1.5 = 2.5 thus exceeding player 2’s score of 2, which implies that player 1’s payoff is 10 − 1 = 9 while player 2’s payoff is 8 − 2 = 6. At this point, we can easily note that player 2 finds x2 = 1 (in the center column) to be strictly dominated by x2 = 0 (in the left-hand column). Indeed, regardless of which strategy player 1 uses (i.e., regardless of the particular row you look at) player 2 obtains the lowest score on the exam when he chooses an effort level of 0 or 1, since player 1 benefits from a 1.5 score advantage. Indeed, exerting a zero effort yields player 2 a payoff of 8, which is unambiguously higher than the payoff he obtains from exerting an effort of x2 = 1, which is 7, regardless of the particular effort level exerted by player 1. We can thus delete the column referred to x2 = 1 for player 2, which leaves us with the reduced form matrix in Fig. 1.17. Now consider player 1. Notice that an effort of x1 ¼ 1 (in the middle row) strictly dominates x1 ¼ 2 (in the bottom row), since the former yields a payoff of 9 regardless of player 2’s strategy (i.e., independently on the column), while an effort of x1 ¼ 2 only provides player 1 a payoff of 8. We can, therefore, delete the last row (corresponding to effort x1 ¼ 2) from the above matrix, which helps us further reduce the payoff matrix to the 2  2 normal-form game in Fig. 1.18.

Fig. 1.17 Reduced normal-form game

Player 2 0

2

0

10, 8

8, 8

Player 1 1

9, 8

9, 6

2

8, 8

8, 6

16

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Dominance Solvable Games

Player 2

Player 1

0

2

0

10, 8

8, 8

1

9, 8

9, 6

Fig. 1.18 Reduced normal-form game

Unfortunately, we can no longer find any strictly dominated strategy for either player. Specifically, neither of the two strategies of player 1 is dominated, since an effort of x1 ¼ 0 does not yield a weakly larger payoff than x1 = 1, i.e., it entails a strictly larger payoff when player 2 chooses x2 ¼ 0 (in the left-hand column) but lower payoff when player 2 selects x2 = 2 (in the right-hand column). Similarly, none of player 2’s strategies is strictly dominated either, since x2 ¼ 0 yields the same payoff as x2 ¼ 2 when player 1 selects x1 ¼ 0 (top row), but a larger payoff when player 1 chooses x1 ¼ 1 (in the bottom row). Therefore, the set of strategies that survive IDSDS for player 1 are {0, 1} and for player 2 are {0, 2}. Thus, the IDSDS predicts that player 1 will exert an effort of 0 or 1, and that player 2 will exert effort of 0 or 2. Part (b) Now let us repeat the analysis when we instead iteratively delete weakly dominated strategies. From part (a), we know that effort levels 3, 4, and 5 are all strictly dominated for both players, and therefore weakly dominated as well. By the same argument, we can delete all strictly dominated strategies (since they are also weakly dominated), leaving us with the reduced normal-form game we examined at the end of our discussion in part (a), which we reproduce below (Fig. 1.19). While we could not identify any further strictly dominated strategies in part (a), we can now find weakly dominated strategies in this game. In particular, notice that, for player 2, an effort level of x2 ¼ 0 (in the left-hand column) weakly dominates x2 ¼ 2 (in the right-hand column). Indeed, a zero effort yields a payoff of 8

Player 2

Player 1

0

2

0

10, 8

8, 8

1

9, 8

9, 6

Fig. 1.19 Reduced normal-form game

Exercise 8—Applying IDSDS When Players Have Five Available StrategiesA

17

Player 2 0 0

10, 8

1

9, 8

Player 1

Fig. 1.20 Reduced normal-form game

Player 2 0 Player 1

0

10, 8

Fig. 1.21 Single strategy profile surviving IDWDS

regardless of what player 1 does (i.e., in both rows), while x2 ¼ 2 yields a weakly lower payoff, i.e., 8 or 6. Thus, we can delete the column corresponding to an effort level of x2 ¼ 2 for player 2, allowing us to reduce the above payoff matrix to a single column matrix, as depicted in Fig. 1.20. We can finally notice that player 1 finds that an effort of x1 = 0 (top now) strictly dominates x1 = 1 (in the bottom row), since player 1’s payoff from a zero effort, 10, is strictly larger than that from an effort at x1 ¼ 1, 9. Hence, x1 ¼ 0 strictly dominates x1 ¼ 1, and thus x1 ¼ 0 also weakly dominates x1 ¼ 1. Intuitively, if player 2 exerts a zero effort (which is the only remaining strategy for player 2 after applying IDWDS), player 1, who starts with score advantage of 1.5, can anticipate that he will obtain the highest score in the class even if his effort level is also zero. After deleting this weakly dominated strategy, we are left with a single cell that survives the application of IDWDS, corresponding to the strategy profile (x1 = 0, x2 = 0), in which both players exert the lowest amount of effort, as Fig. 1.21 depicts. After all, it seems that the incentive scheme the instructor proposed did not induce students to study harder for the exam, but instead to not study at all!!2

2

This is, however, a product of the score advantage of the most intelligent student. If the score advantage is null, or only 0.5, the equilibrium result after applying IDWDS changes, which is left as a practice.

18

1

Dominance Solvable Games

Exercise 9—Applying IDSDS in the Battle of the Sexes GameA A husband and a wife are leaving work, and do not remember which event they are attending to tonight. Both of them, however, remember that last night’s argument was about attending either the football game (the most preferred event for the husband) or the opera (the most preferred event for the wife). To make matters worse, their cell phones broke, so they cannot call each other to confirm which event they are attending to. As a consequence, they must simultaneously and independently decide whether to attend to the football game or the opera. The payoff matrix in Fig. 1.22, describes the preference of the husband (wife) for the football game (opera, respectively). Payoffs also indicate that both players prefer to be together rather than being alone (even if they are alone at their most preferred event). Apply IDSDS. What is the most precise prediction about how this game will be played when you apply IDSDS? Answer This game does not have strictly dominated strategies (note that there is not even any weakly dominated strategy), as we separately analyze for the husband and the wife below. Husband: In particular, the husband prefers to go to the football game if his wife goes to the football game, but he would prefer to attend the opera if she goes to the opera. That is, uH ðF; F Þ ¼ 3 [ 0 ¼ uH ðO; F Þ and uH ðO; OÞ ¼ 1 [ 0 ¼ uH ðF; OÞ Intuitively, the husband would like to coordinate with his wife by selecting the same event as her, i.e., a common feature in coordination games. Therefore, there is no event that yields an unambiguous larger payoff regardless of the event his wife attends to, i.e., there is no strictly dominant strategy for the husband. Wife: A similar analysis is extensive to the wife, who would like to attend to the opera (her preferred event) obtaining a payoff of 3 only if her husband also attends the opera, i.e., uW (O, O) > uW (O, F). Otherwise, she obtains a larger payoff attending to the football game, 1, rather than attending to the opera alone, 0, i.e.,

Wife Football Opera Husband

Football

3, 1

0, 0

Opera

0, 0

1, 3

Fig. 1.22 Normal-form representation of the Battle of the Sexes game

Exercise 9—Applying IDSDS in the Battle of the Sexes GameA

19

i.e., uW (F, F) > uW (F, O). Hence, there is no event that provides her with unambiguously larger payoffs regardless of the event her husband selects, i.e., no event constitutes a strictly dominant strategy for the wife. Hence, players have neither a strictly dominant nor a strictly dominated strategy. As a consequence, the application of IDSDS does not delete any strategy whatsoever, and our equilibrium outcome prescribes that any of the four strategy profiles in the above matrix could emerge, i.e., the husband could either attend the football game or the opera, and similarly, the wife could attend either the football game or the opera. (Similarly as in previous exercises, you can return to this exercise once you are done reading Chaps. 2 and 3, and find that the Nash equilibrium solution concept yields a more precise equilibrium prediction than IDSDS.)

Exercise 10—Applying IDSDS in Two Common GamesA Show that IDSDS does not have a bite in: a. The Battle of the sexes game b. The Matching pennies game. Answer Part (a). Consider the following Battle of the Sexes game (Fig. 1.23): Starting with the Husband (the row player), we cannot find a strictly dominated strategy for him. Indeed, his payoff is higher from attending the football game (3) than from the opera (0) when his Wife attends the football game as well (see left-hand column of the matrix) but would be lower (0 vs. 1) if his Wife attends the opera (see right-hand column). In short, we cannot identify a strategy that yields strictly lower payoffs for the Husband, regardless of what his Wife does. There aren’t then any strictly dominated strategies we could delete from the Husband’s strategy space, i.e., graphically there are no rows that we could delete as never being played by the Husband. Wife

Husband

F

O

F

3, 1

0, 0

O

0, 0

1, 3

Fig. 1.23 Normal-form representation of the Battle of the Sexes game

20

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Dominance Solvable Games

A similar argument applies to the Wife (column player). Her payoff is higher from attending the football game (1) than from the opera (0) when her Husband attends the Football game (see top row of the matrix), but lower (0 vs. 3) if her Husband attends the opera (bottom row of the matrix). Therefore, there are no strategies for the Wife (that is, no columns) that we identify as strictly dominated, i.e., strategies yielding a strictly lower payoff than another strategy regardless of what her Husband chooses. In summary, we couldn’t find strictly dominated strategies for either player, and then we couldn’t delete any strategies in our application of IDSDS. As a result, our most precise equilibrium prediction after applying IDSDS is the game as a whole, that is, all four strategy profiles (one per cell): Strategies surviving IDSDS ¼ fðF; F Þ; ðF; OÞ; ðO; F Þ; ðO; OÞg

Part (b). Consider the following Matching pennies game (Fig. 1.24): Starting with the player 1 (the row player), we cannot find a strictly dominated strategy for him. Indeed, his payoff is higher (1 vs. −1) from H than from T when player 2 plays H as well (see left-hand column of the matrix) but would be lower (−1 vs. 1) if player 2 plays T (see right-hand column). In short, we cannot identify a strategy that yields strictly lower payoffs for player 1, regardless of what player 2 does. There aren’t then any strictly dominated strategies we could delete from player 1’s strategy space, i.e., graphically there are no rows that we could delete as never being played by player 1. A similar argument applies to player 2 (column player). Her payoff is higher from playing T than H when player 1 plays H (see top row of the matrix), but lower if player 1 plays T (bottom row of the matrix). Therefore, there are no strategies for player 2 (that is, no columns) that we identify as strictly dominated, i.e., strategies yielding a strictly lower payoff than another strategy regardless of what player 1 chooses.

Player 2

Player 1

H

T

H

1, -1

-1, 1

T

-1, 1

1, -1

Fig. 1.24 Normal-form representation of the matching pennies game

Exercise 10—Applying IDSDS in Two Common GamesA

21

In summary, we couldn’t find strictly dominated strategies for either player, and then we couldn’t delete any strategies in our application of IDSDS. As a result, our most precise equilibrium prediction after applying IDSDS is the game as a whole, that is, all four strategy profiles (one per cell): Strategies surviving IDSDS ¼ fðH; H Þ; ðH; T Þ; ðT; H Þ; ðT; T Þg

Exercise 11—Applying IDSDS in Three-Player GamesB Consider the following anti-coordination game in Fig. 1.25, played by three potential entrants seeking to enter into a new industry, such as the development of software applications for smartphones. Every firm (labeled as A, B, or C) has the option of entering or staying out (i.e., remain in the industry they have been traditionally operating, such as, software for personal computers). The normal form game in Fig. 1.25 depicts the market share that each firm obtains, as a function of the entering decision of its rivals. Firms simultaneously and independently choose whether or not to enter. As usual in simultaneous-move games with three players, the triplet of payoffs describes the payoff for the row player (firm A) first, for the column player (firm B) second, and for the matrix player (firm C) third. Find the set of strategy profiles that survive the iterative deletion of strictly dominated strategies (IDSDS). Is the equilibrium you found using this solution concept unique? Answer We can start by looking at the payoffs for firm C (the matrix player). [Recall that the application of IDSDS is insensitive to the deletion order. Thus, we can start deleting strictly dominated strategies for the row, column or matrix player, and still reach the same equilibrium result.] In order to test for the existence of a dominated strategy

Fig. 1.25 Normal-form representation of a three-players game

22

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Dominance Solvable Games

Fig. 1.26 Pairwise payoff comparison for firm C

for firm C, we compare the third payoff of every cell across both matrices. Figure 1.26 provides a visual illustration of this pairwise comparison across matrices. We find that for firm C (matrix player), entering strictly dominates staying out, i.e., uC (sA, sB, E) > uC (sA, sB, O) for any strategy of firm A, sA, and firm B, sB, 32 > 30, 27 > 24, 24 > 14 and 50 > 32 in the pairwise payoff comparisons depicted in Fig. 1.24 This allows us to delete the right-hand side matrix (corresponding to firm C choosing to stay out) since it is strictly dominated and thus would not be selected by firm C. We can, hence, focus on the left-hand matrix alone (where firm C chooses to enter), which we reproduce in Fig. 1.27. We can now check that entering is strictly dominated for the row player (firm A), i.e., uA (E, sB, E) < uA (O, sB, E) for any strategy of firm B, sB, once we take into account that firm C selects its strictly dominant strategy of entering. Specifically, firm A prefers to stay out both when firm B enters (in the left-hand column, since 30 > 14), and when firm B stays out (in the right-hand column, since 13 > 8). In other words, regardless of firm B’s decision, firm A prefers to stay out. This allows us to delete the top row from the above matrix, since the strategy “Enter” would never be used by firm A, which leaves us with a single row and two columns, as illustrated in Fig. 1.28.

Fig. 1.27 Reduced normal-form game

Exercise 11—Applying IDSDS in Three-Player GamesB

23

Fig. 1.28 Reduced normal-form game

Fig. 1.29 An even more reduced normal-form game

Once we deleted all but one strategy of firm C and one of firm A, the game becomes an individual-decision making problem, since only one player (firm B) must select whether to enter or stay out. Since entering yields a payoff of 16 to firm B, while staying out only entails 12, firm B chooses to enter. Firm B then regards staying out as a strictly dominated strategy, i.e., uB (O, E, E) > uB (O, O, E) where we fix the strategies of the other two firms at their strictly dominant strategies: staying out for firm A and entering for firm C. We can, thus, delete the column corresponding to staying out in the above matrix, as depicted in Fig. 1.29. As a result, the only cell (strategy profile) that survives the application of the iterative deletion of strictly dominated strategies (IDSDS) is that corresponding to (Stay Out, Enter, Enter), which predicts that firm A stays out, while both firms B and C choose to enter.

Exercise 12—IDSDS and RationalityA Consider the following 3  3 matrix (Fig. 1.30): a. Does any player have a strictly dominant strategy? b. Does any player have a strictly dominated strategy? c. Use your results from part (b) to delete players’ strictly dominated strategies assuming rationality, but not assuming common knowledge of rationality. [Hint: This means that every player can delete the strategies he finds to be strictly dominated but cannot put himself in the shoes of his opponent to delete the strategies his opponent finds to be strictly dominated.] d. Use your results from part (b) to delete players’ strictly dominated strategies assuming common knowledge of rationality. [Hint: This means that players can put themselves in the shoes of their opponents to delete strategies each other finds to be strictly dominated, as in the standard approach of IDSDS.]

24

1

Dominance Solvable Games

Player 2 F

C

B

F

0, 5

2, 4

2, 3

Player 1 C

2, 3

0, 5

3, 2

B

5, 0

2, 4

2, 3

Fig. 1.30 Normal-form representation of a three-players game

Answer Part (a). Let us start by identifying the strategies of player 1 (the row player) that are strictly dominating the other strategies. When player 1 chooses F in the first row, his payoff is 0 when player 2 chooses F or 2 when player 2 chooses C or B. However, when player 1 chooses C in the second row, his payoff is 2 when player 2 chooses F, 0 if player 2 chooses C, and 3 if player 2 chooses B. Furthermore, when player 1 chooses B in the third row, his payoff is 5 when player 2 chooses F, 2 if player 2 chooses C, and 2 if player 2 chooses B. Hence, there is no strictly dominant strategy for player 1, as there is no strategy that yields an unambiguously higher payoff for player 1 regardless of the strategy that player 2 selects. Next, we move on to identify the strategies of player 2 (the column player) that are strictly dominating the other strategies. When player 2 chooses F in the first column, his payoff is 5 when player 1 chooses F, 3 when player 1 chooses C, and 0 when player 1 chooses B. However, when player 2 chooses C in the second column, his payoff is 4 when player 1 chooses F, 5 if player 1 chooses C, and 4 if player 1 chooses B. Furthermore, when player 2 chooses B in the third column, his payoff is 3 when player 1 chooses F, 2 if player 1 chooses C, and 3 if player 1 chooses B. Hence, there is no strictly dominant strategy for player 2, as there is no strategy that yields an unambiguously higher payoff for player 2 regardless of the strategy that player 1 selects. In summary, we could not find any strictly dominant strategies for either player, and then we could not delete any strategies in our application of strict dominance. Mathematically, this is equivalent to saying that we cannot find a strategy s0i 2 Si for player i, where i = {1, 2} that satisfies
 ui s0i ; si [ ui ðsi ; si Þ for any two strategies si 6¼ s0i , and for all strategies of his opponents si 2 Si . Part (b). Starting with the player 1 (the row player), we cannot find a strictly dominated strategy for him by applying the above logic. Specifically, his payoff is higher from B that gives him a payoff of 5 than other strategies when player 2 plays

Exercise 12—IDSDS and RationalityA

25

F (see left-hand column of the matrix), but would be indifferent between F and B if player 2 plays C (see middle column) that gives him a payoff of 2, and is higher from C than other strategies when player 2 plays B (see right-hand column of the matrix) that gives him a payoff of 3. In short, we cannot identify a strategy that yields strictly lower payoffs for player 1, regardless of what player 2 does. There does not exist any such strictly dominated strategies that we could delete from player 1’s strategy space, i.e., graphically there are no rows that we could delete as never being played by player 1. A similar argument applies to player 2 (column player). Her payoff is higher from playing F than either C or B when player 1 plays F (see top row of the matrix) that gives her a payoff of 5, and from playing C than either F or B when player 1 plays C or B (see middle and bottom rows of the matrix) that also gives her a payoff of 5. In both cases, player 2 will not play B, such that there exists a strategy for player 2 (that is, column B) that we can identify as strictly dominated, i.e., strategies yielding a strictly lower payoff than another strategy regardless of what player 1 chooses. In summary, we could find strictly dominated strategies for player 2 but not player 1 for which we can delete. Mathematically, this is equivalent to saying that we can find a strategy s0i 2 Si for player i, where i = {1, 2} that satisfies
 ui s0i ; si \ui ðsi ; si Þ for any two strategies si 6¼ s0i , and for all strategy profiles of his opponents, si 2 Si . Part (c). Following part (b), after player 2 deletes strategy B, the reduced form matrix is as follows (Fig. 1.31): That yields the following strategy space assuming rationality for the players without common knowledge: Strategies surviving rationality ¼ fðF; F Þ; ðF; C Þ; ðC; F Þ; ðC; CÞ; ðB; F Þ; ðB; CÞg

Fig. 1.31 Reduced normal-form game

Player 2

Player 1

F

C

F

0, 5

2, 4

C

2, 3

0, 5

B

5, 0

2, 4

26

1

Dominance Solvable Games

Part (d). Now, player 1 can delete strategy C because his payoff is higher from B than other strategies when player 2 plays F (see left-hand column of the matrix), but would be indifferent between F and B if player 2 plays C (see middle column in the original matrix, which now became the right-hand column). In short, we can identify strategy C that yields strictly lower payoffs for player 1, regardless of what player 2 does. There is a strictly dominated strategy that we could delete from player 1’s strategy space by IDSDS, i.e., graphically there is the middle row that we could delete as never being played by player 1, rendering a reduced-form matrix as follows (Fig. 1.32): Here, starting with the player 1 (the row player), we cannot find any more strictly dominated strategy for him. Indeed, his payoff is higher from B than from F when player 2 plays F (see left-hand column of the matrix) that gives him a payoff of 5 instead of 0, but would be indifferent if player 2 plays C (see right-hand column) that gives him a payoff of 2. In short, we cannot identify additional strategies that yields strictly lower payoffs for player 1, regardless of what player 2 does. In this context, there aren’t any strictly dominated strategies we could delete from player 1’s strategy space, i.e., graphically there are no rows that we could delete as never being played by player 1. A similar argument applies to player 2 (column player). Her payoff is higher from playing F than C when player 1 plays F (see top row of the matrix) that gives her a payoff of 5 instead of 4, but lower if player 1 plays B (bottom row of the matrix) that gives her a payoff of 0 instead of 4. Therefore, there are no strategies for player 2 (that is, no columns) that we identify as strictly dominated, i.e., strategies yielding a strictly lower payoff than another strategy regardless of what player 1 chooses. In summary, we could find strictly dominated strategies for player 1 but not for player 2, and then we could delete no more strategies in our application of IDSDS. As a result, our most precise equilibrium prediction after applying IDSDS is the game without the remaining columns of the players, that is, four of above strategy profiles (one per cell): Strategies surviving IDSDS ¼ fðF; F Þ; ðF; C Þ; ðB; F Þ; ðB; CÞg

Player 2

Player 1

Fig. 1.32 Reduced normal-form game

F

C

F

0, 5

2, 4

B

5, 0

2, 4

Exercise 13—Iterated Deletion of Strictly Dominated Strategies (IDSDS)A

27

Exercise 13—Iterated Deletion of Strictly Dominated Strategies (IDSDS)A Consider the following normal form game (Fig. 1.33) (a) Find strictly dominant strategies (if any) for player 1 and for player 2. (b) Find strictly dominated strategies (if any) for player 1 and for player 2. (c) If you apply iterative deletion of strictly dominated strategies (IDSDS), what is the surviving strategy pair (or pairs)? Explain the steps you use in IDSDS, and why you use them. Answer Part (a). Player 1. Let’s start by analyzing player 1. Strategy Down yields a strictly higher payoff for player 1 than Up regardless of the strategy that player 2 chooses (i.e., independently on the column that player 2 plays). However, Down does not necessarily yield a strictly higher payoff than Middle. To see this point, notice that, when player 2 chooses Right (in the right-hand column), Middle yields a payoff of 2 while Down produces a lower payoff (1); whereas when player 2 selects Left (in the left-hand column), Middle produces a payoff of −12, which is lower than that of Down (−9). Therefore, we could not find a strictly dominant strategy for player 1. For that strategy to exist, we would need to show that Down yields a strictly higher payoff than both Middle and Up, regardless of the column that player 2 chooses. While we could show this in our payoff comparison between Down and Up, we couldn’t in our comparison between Down and Middle.

Player 2. In the case of player 2, we can identify that strategy Right yields a strictly higher payoff than Left when player 1 chooses Up and Middle (indeed, 12 > 10 and 2 > 0, respectively). However, when player 1 selects Down (bottom row of the matrix), player 2 obtains the same payoff (zero) from choosing Right and Left. This implies that strategy Right is “weakly” dominant over Left, but not strictly dominant. Hence, there are no strictly dominant strategies for player 2 either. (As a remark, if player 2’s payoff from the strategy profile (Down, Right) at the bottom

Fig. 1.33 Normal-form game

Player 2 Player 1

Left

Right

Up

-10, 10

0, 12

Middle

-12, 0

2, 2

Down

-9, 0

1, 0

28

1

Dominance Solvable Games

right-hand side of the payoff was 1 rather than 0, we could then say that player 2 finds strategy Right to be strictly dominant.) Part (b). For compactness, we denote player 1’s strategies Up, Middle and Down as U, M and D, respectively, for the rest of the exercise. Similarly, we represent player 2’s strategies Left and Right as L and R, respectively. Player 1. Starting with player 1, strategy U is strictly dominated by D, since 10 ¼ u1 ðU; LÞ\u1 ðD; LÞ ¼ 9

when player 2 chooses L;

and 0 ¼ u1 ðU; RÞ\u1 ðD; RÞ ¼ 1 when player 2 chooses R: Player 2. For player 2, neither strategy L nor R is strictly dominated. When player 1 chooses U or M, strategy R looks more attractive than L for player 2, but when player 1 chooses strategy D, player 2 obtains the same payoff from L or R (zero), not allowing us to claim that strategy R strictly dominates L. In words, strategy R does not provide a strictly higher payoff than L regardless of the strategy that player 1 chooses (i.e., regardless of the row that player 1 selects). We could say that strategy R weakly dominates L, but that was not the point of the exercise! Part (c). As shown in part (a) of the exercise, strategy U is strictly dominated. We can then remove the top row from the matrix (corresponding to strategy U for player 1), resulting in the following reduced-form matrix (Fig. 1.34): At this stage of the game, none of the strategies are strictly dominated by one another, and therefore, the strategy profile that survives IDSDS becomes IDSDS ¼ fðM; LÞ; ðM; RÞ; ðD; LÞ; ðD; RÞg

Exercise 14—Unemployment BenefitsA Consider the following simultaneous-move game between the government (row player), which decides whether to offer unemployment benefits, and an unemployed worker (column player), who chooses whether to search for a job. As you interpret from the payoff matrix below, the unemployed worker only finds it optimal to search for a job when he receives no unemployment benefit; while the government only finds it optimal to help the worker when he searches for a job (Fig. 1.35).

Exercise 14—Unemployment BenefitsA

29

Player 2 Player 1

Left

Right

Middle

-12, 0

2, 2

Down

-9, 0

1, 0

Fig. 1.34 Reduced normal-form game

Worker Government

Search

Don’t search

Benefit

3, 2

-1, 3

No benefit

-1, 1

0, 0

Fig. 1.35 Normal-form game

(a) Represent this game in its extensive form (game tree), where the government acts first and the worker responds without observing whether the government offered unemployment benefits. (b) Does the government have strictly dominant strategies? How about the worker? (c) Find which strategy profile (or profiles) survive the application of IDSDS. Answer Part (a). Representing the game tree of this game, we obtain the Fig. 1.36. Part (b). There are no strictly dominant strategies for both government and worker. Specifically, when the worker chooses to search for a job, the government will be better off (not) offering unemployment benefits. When the worker chooses to not search for a job, the government is better off not offering unemployment benefits.

A similar argument applies in the opposite direction: when the government chooses to offer unemployment benefits, the worker is better off not searching for a job; whereas when the government does not offer unemployment benefits, the worker is better off searching for a job. Part (c). In this context, every strategy profile survives IDSDS, which entails

30

1

Dominance Solvable Games

Fig. 1.36 Extensive-form (game tree) game

IDSDS ¼ fðBenefit; SearchÞ; ðBenefit; Don0 tSearchÞ; ðNoBenefit; SearchÞ; ðNoBenefit; Don0 tSearchÞg: In this type of games, we say that the application of IDSDS has “no bite” since it does not help us reduce the original set of strategy profiles.

Exercise 15—Finding Dominant Strategies in Games with I  2 Players and with Continuous Strategy SpacesB There are I firms in an industry. Each can try to convince Congress to give the industry a subsidy. Let h i denote the number of hours of effort put in by firm i, and let Ci ðhi Þ ¼ wi ðhi Þ2 be the cost of this effort to firm i, where w i is a positive constant. When the effort levels of the I firms are given by the list (h 1, …, hI), the value of the subsidy that gets approved is

Exercise 15—Finding Dominant Strategies in Games with I  2 Players...

31

XI YI
 S hi ; hj ¼ a h þ b h: i i¼1 i¼1 i where a and b are constants and a  0; and b  0. Consider a game in which the firms decide simultaneously and independently how many hours they will each devote to this lobbying effort. Show that each firm has a strictly dominant strategy if and only if b ¼ 0: What is firm i’s strictly dominant strategy when condition b ¼ 0 holds? Answer Firm i chooses the amount of lobbying h i that maximizes its profit function pi


 pi ¼ Si hi ; hj  Ci ðhi Þ h XI i YI h þ b h ¼ a  wi ðhi Þ2 i i i¼1 i¼1 Taking first-order conditions with respect to hi yields Y

aþb

! hj

 2wi hi ¼ 0

j6¼i

We can now rearrange, and solve for h i to obtain firm i’s best response function " hi ¼ a þ b

Y

!# hj

j6¼i

1 2wi

Notice that this best response function describes which is firm i’s optimal effort level in lobbying activities as a function of other firms’ effort levels, hj, for every firm j 6¼ i. Interestingly, when b = 0 firm i’s best response function becomes independent of his rival’s effort, that is, hi ¼

a : 2wi

Dominant strategy: Note that, in order for firm i to have a dominant strategy, firm i must prefer to use a given strategy regardless of the particular actions selected by the other firms. In particular, when b = 0 firm i’s best response function becomes hi ¼ 2wa i , and thus it does not depend on the action of other firms, i.e., it is not a function of hj. Therefore, firm i has a strictly dominant strategy, hi ¼ 2wa i , when b = 0 since such action is independent on the other firms’ actions.

32

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Dominance Solvable Games

Exercise 16—Equilibrium Predictions from IDSDS Versus IDWDSB In previous exercises applying IDSDS we sometimes started finding strictly dominated strategies for the row player, while in other exercises we began identifying strictly dominated strategies for the column player (or the matrix player in games with three players). While the order of deletion does not affect the equilibrium outcome when applying IDSDS, it can affect the set of equilibrium outcomes when we delete weakly (rather than strictly) dominated strategies. Use the game in Fig. 1.37 to show that the order in which weakly dominated strategies are eliminated can affect equilibrium outcomes. Answer First route: Taking the below payoff matrix, first note that, for player 1, strategy U weakly dominates D, since U yields a weakly larger payoff than D for any strategy (column) selected by player 2, i.e., u1(U, s2)  u1(D, s2) for all s2 2 {L, M, R}. In particular, U provides player 1 with the same payoff as D when player 2 selects L (a payoff of 2 for both U and D) and M (a payoff of 1 for both U and D), but a strictly higher payoff when player 2 chooses R (in the right-hand column) since 0 > − 1. Once we have deleted D because of being weakly dominated, we obtain the reduced-form matrix depicted in Fig. 1.38. We can now turn to player 2, and detect that strategy L strictly dominates R, since it yields a strictly larger payoff than R, regardless of the strategy selected by player 1 (both when he chooses U in the top row, i.e., 1 > 0, and when he chooses C in the bottom row, i.e., 2 > 1), or more compactly u2(s1, M)  u2(s2, R) for all s1 2 {U, C}. Since R is strictly dominated for player 2, it is also weakly dominated. After deleting the column corresponding to the weakly dominated strategy R, we obtain the 2  2 matrix in Fig. 1.39. At this point, notice that we are not done examining player 2, since you can easily detect that M is weakly dominated by strategy L. Indeed, when player 1 selects U (in the top row of the above matrix), player 2 obtains the same payoff from L and M, but when player 1 chooses C (in the bottom row), player 2 is better off selecting L, which yields a payoff of 2, rather than M, which only produces a payoff of 1, i.e., u2(s1, M)  u2(s1, L) for all s1 2 {U, C}. Hence, we can delete M because of being weakly dominated for player 2, leaving us with the (further reduced) payoff matrix in Figs. 1.31 and 1.40. At this point, we can turn to player 1, and identify that U strictly dominates C (and, thus, it also weakly dominates C), since the payoff that player 1 obtains from U, 2, is strictly larger than that from C, 1. Therefore, after deleting C, we are left with a single strategy profile, (U, L), as depicted in the matrix of Fig. 1.32. Hence, using this particular order in our iterative deletion of weakly dominated strategies (IDWDS) we obtain the unique equilibrium prediction (U, L) (Fig. 1.41). Second route: Let us now consider the same initial 3  3 matrix, which we reproduce in Fig. 1.33, and check if the application of IDWDS, but using a different

Exercise 16—Equilibrium Predictions from IDSDS Versus IDWDSB

Player 2 L 2, 1 1, 2 2, -2

U Player 1 C D

M 1, 1 3, 1 1, -1

R 0, 0 2, 1 -1, -1

Fig. 1.37 Normal-form game

Player 2 L 2, 1 1, 2

Player 1 U C

M 1, 1 3, 1

R 0, 0 2, 1

Fig. 1.38 Reduced normal-form game after one round of IDWDS

Player 2

Player 1

U C

L

M

2, 1 1, 2

1, 1 3, 1

Fig. 1.39 Reduced normal-form game after two rounds of IDWDS

Player 2 L Player 1

U

2, 1

C

1, 2

Fig. 1.40 Reduced normal-form game after three rounds of IDWDS

Player 2 L Player 1

U

Fig. 1.41 Strategy surviving IDWDS (first route)

2, 1

33

34

1

Dominance Solvable Games

Player 2 U Player 1 C D

L 2, 1 1, 2 2, -2

M 1, 1 3, 1 1, -1

R 0, 0 2, 1 -1, -1

Fig. 1.42 Normal-form game

Player 2

Player 1

U C D

L

M

2, 1 1, 2 2, -2

1, 1 3, 1 1, -1

Fig. 1.43 Reduced normal-form game after one round of IDWDS

deletion order (i.e., different “route”), can lead to a different equilibrium result than that found above. i.e., (U, L) (Fig. 1.42). Unlike in our first route, let us now start identifying weakly dominated strategies for player 2. In particular, note that R is weakly dominated by M, since the former yields a weakly lower payoff than the latter (i.e., it provides a strictly higher payoff when player 1 chooses U in the top row, but the same payoff otherwise). That is, u2 ðs1 ; M Þ  u2 ðs1 ; RÞ for all s1 2 {U, C, D}. Once we delete R as being weakly dominated for player 2, the remaining matrix becomes that depicted in Figs. 1.34 and 1.43. Turning to player 1, we cannot identify any other weakly dominating strategy. This equilibrium prediction using the second route of IDWDS is, hence, the six strategy profiles of Fig. 1.34, that is, {(U, L), (U, M), (C, L), (C, M), (D, L), (D,M)}. This equilibrium prediction is, of course, different (and significantly less precise) than what found when we started the application of IDWDS from player 1. Hence, equilibrium outcomes that arise from applying IDWDS are sensitive to the deletion order, while those emerging from IDSDS are not.

Exercise 17—Different Equilibrium Predictions from IDSDS and IDWDSA Consider the following 4  3 matrix with Player 1 choosing rows and Player 2 selecting columns (Fig. 1.44).

Exercise 17—Different Equilibrium Predictions from IDSDS and IDWDSA

35

Player 2 a b Player 1 c d

x

y

z

2, 3 5, 1 3, 7 4, 2

1, 4 2, 3 4, 6 2, 2

3, 2 1, 2 5, 4 6, 1

Fig. 1.44 Normal-form game

(a) Find the strategies that survive the application of IDSDS. (b) Find the strategies that survive the application of IDWDS. Answer Part (a). Player 1 choosing a is strictly dominated by c. To see this, note that when Player 2 chooses x (in the left-hand column), Player 1 obtains a higher payoff selecting row c (payoff of 3) than his payoff from choosing row a (which is only 2). Similarly, when Player 2 chooses the column corresponding to y (z), Player 1 obtains a payoff 4 (5) selecting row c which is higher than his payoff of 1 (3, respectively) from choosing row a. Since player 1 finds that strategy a is strictly dominated by strategy c, a rational player 1 will never play strategy a. As a result, we can delete row a to yield the following reduced-form matrix. (Note that strategy a is also strictly dominated by strategy d) (Fig. 1.45).

Since there are no strictly dominated strategies for player 1 in the reduced matrix, we can now move to Player 2. For him, choosing column z is strictly dominated by y. To see this point, note that when Player 1 chooses row b, selecting y gives Player 2 a payoff of 3 that is higher than his payoff from choosing z, 2. Similarly, when Player 1 chooses row c (d), choosing column y gives Player 2 a payoff of 6 (2) that is higher than the payoff he obtains from column z, 4 (1, respectively). So, we delete column z from the above matrix, which yields the following reduced-form matrix (Fig. 1.46).

Player 2 b Player 1 c d Fig. 1.45 Reduced normal-form game

x

y

z

5, 1 3, 7 4, 2

2, 3 4, 6 2, 2

1, 2 5, 4 6, 1

36

1

Dominance Solvable Games

Player 2

Player 1

b c d

x

y

5, 1 3, 7 4, 2

2, 3 4, 6 2, 2

Fig. 1.46 An even more reduced normal-form game

Let us now move to Player 1 again. When Player 2 chooses column x, selecting row b gives Player 1 a payoff of 5 which is higher than his payoff from row d, which is only 4. However, when Player 2 chooses column y, choosing b or d both gives Player 1 a payoff of 2, which is less than that of choosing row c which yields a payoff of 4. Therefore, applying IDSDS, there are no rows that can be deleted for Player 1. Next, we move to Player 2. When Player 1 chooses row b, Player 2 choosing column y gives a payoff of 3 that is higher than his payoff from choosing column x. However, when Player 1 chooses row c, the opposite occurs in which choosing column x gives Player 2 a payoff of 7 that is higher than 6 by choosing column y. Therefore, applying IDSDS, there are no columns that can be deleted for Player 2. Therefore, strategy profiles that survive IDSDS are {bx, by, cx, cy, dx, dy}. Part (b). Recall that every strictly dominated strategy is also a weakly dominated strategy. As a consequence, strategies being deleted in our application of IDSDS must also be deleted when applying IDWDS. This helps us start our analysis of IDWDS by considering the reduced-form matrix we found after applying IDSDS in part (a), which we reproduce below for illustration purposes (Fig. 1.47). We can now start applying IDWDS. We first consider Player 1 (row player). For Player 1, row d is weakly dominated by row b, since it yields a strictly higher payoff when Player 2 chooses column x but the exact same payoff when Player 2 selects column y. We can then remove row d by IDWDS to yield the following reduced-form matrix (Fig. 1.48):

Player 2

Player 1

Fig. 1.47 Reduced normal-form game

b c d

x

y

5, 1 3, 7 4, 2

2, 3 4, 6 2, 2

Exercise 17—Different Equilibrium Predictions from IDSDS and IDWDSA

37

Player 2

Player 1

b c

x

y

5, 1 3, 7

2, 3 4, 6

Fig. 1.48 An even more reduced normal-form game

Next, we move to Player 2. When Player 1 chooses row b, Player 2 choosing column y gives a payoff of 3 that is higher than his payoff from choosing column x (1). However, when Player 1 chooses row c, the opposite occurs in which choosing column x gives Player 2 a payoff of 7 that is higher than 6 by choosing column y. Therefore, applying IDWDS, there are no columns that can be deleted for Player 2. Finally, we can consider Player 1. When Player 2 chooses column x, selecting row b gives Player 1 a payoff of 5 which is higher than his payoff from row c, which is only 3. However, when Player 2 chooses column y, choosing c gives Player 1 a payoff of 4 that is higher than that of choosing row b which yields a payoff of 2. Therefore, applying IDWDS, there are no rows that can be deleted for Player 1. In summary, the strategy profiles that survive IDWDS are {bx, by, cx, cy}. Our equilibrium result, despite not being extremely precise (we are saying that four different strategy profiles can arise), are different from the six strategy profiles that survived IDSDS in part (a) of the exercise.

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games with Complete Information

Introduction This chapter analyzes behavior in relatively simple strategic settings: simultaneousmove games of complete information. Let us define the two building blocks of this chapter: best responses and Nash equilibrium. Best response. First, a strategy s
i is a best response of player i to a strategy profile si selected by other players if it provides player i with a weakly larger payoff than any of his available strategies si 2 Si . Formally, strategy s
i is a best response to si if and only if
 ui s
i ; si  ui ðsi ; si Þ

for all si 2 Si :

We then say that strategy si is a best response to si , and denote it as s
i 2 BRðsi Þ. For instance, in a two-player game, s
1 is a best response for player 1
 to strategy s2 selected by player 2 if and only if u1 s
1 ; s2  u1 ðs1 ; s2 Þ for all s1 2 S1 thus implying that s
1 2 BRðs2 Þ. Second, we define Nash equilibrium by requiring that every player uses best responses to his opponents’ strategies, i.e., players use mutual best responses in equilibrium.
 Nash equilibrium. A strategy profile s
¼ s
1 ; s
2 ; . . .; s
N is a Nash equilibrium if every player i’s strategy is a best response to his opponents’ strategies; that is, if for every player i his strategy s
i satisfies

 ui s
i ; s
i  ui si ; s
i

for all si 2 Si


 or, more compactly, strategy s
i is a best response to s
i , i.e., s
i 2 BRi s
i . In words, every player plays a best response to his opponents’ strategies, and his conjectures about his opponents’ behavior must be correct (otherwise, players could have incentives to modify their strategies and, thus, not be in equilibrium). As a © Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_2

39

40

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

consequence, players do not have incentives to deviate from their Nash equilibrium strategies; and we can understand such strategy profile as stable. We initially focus on games where two players select between two possible strategies, such as the Prisoner’s Dilemma game (where two prisoners decide to either cooperate or defect), and the Battle of the Sexes game (where a husband and a wife choose whether to attend the football game or the opera). Afterwards, we explain how to find best responses and equilibrium behavior in games where players choose among a continuum of strategies, such as in the Cournot game of quantity competition, and games where players’ actions impose externalities on other players. Furthermore, we illustrate how to find best responses in games with more than two players, and how to identify Nash Equilibria in these contexts. We finish this chapter with one application from law and economics about the incentives to commit crimes and to prosecute them through law enforcement, and a Cournot game in which the merging firms benefit from efficiency gains.1

Exercise 1—Prisoner’s DilemmaA Two individuals have been detained for a minor offense and confined in separate cells. The investigators suspect that these individuals are involved in a major crime, and separately offer each prisoner the following deal, as depicted in the matrix in Fig. 2.1: If you confess while your partner doesn’t, you will leave today without serving any time in jail; if you confess and your partner also confesses, you will serve 5 years in jail; if you don’t confess and your partner does, you have to serve 15 years in jail (since you did not cooperate with the prosecutor but your partner provided us with evidence against you); finally, if none of you confess, you will serve one year in jail (since we only have limited evidence against you). If both players must simultaneously choose whether or not to confess, and they cannot coordinate their strategies, which is the Nash Equilibrium (NE) of the game? Answer Every player i = {1, 2} has a strategy space of Si ¼ fC; NC g. In a NE, every player has complete information about all players’ strategies and maximizes his own payoff, taking the strategy of his opponents as given. That is, every player selects his best response to his opponents’ strategies. Let’s start finding the best responses of player 1, for each of the possible strategies of player 2. Player 1 If player 2 confesses (in the left-hand column), player 1’s best response is to Confess, since his payoff from doing so, −5, is larger than that from not confessing, −15.2 1

While the Nash equilibrium solution concept allows for many applications in the area of industrial organization, we only explore some basic examples in this chapter, relegating many others to Chap. 5 (Applications to Industrial Organization). 2 A common trick many students use in order to be able to focus on the fact that we are examining the case in which player 2 confesses (in the left-hand column) is to cover with their hand (or a piece of paper) the columns in which player 2 selects strategies different from Confess (in this case, that means covering Not confess, but in larger matrices it would imply covering all columns except for the one we are analyzing at that point.) Once we focus on the column corresponding to

Exercise 1—Prisoner’s DilemmaA

41 Prisoner 2 Confess

Not confess

Confess

-5, -5

0, -15

Not confess

-15, 0

-1, -1

Prisoner 1

Fig. 2.1 Prisoner’s dilemma game (normal-form) Prisoner 2 Confess

Not confess

Confess

-5, -5

0, -15

Not confess

-15, 0

-1, -1

Prisoner 1

Fig. 2.2 Prisoner’s dilemma game (normal-form)

This is indicated in Fig. 2.2 by underlining the payoff that player 1 obtains from playing this best response, −5. If, instead, player 2 does Not confess (in the right-hand column), player 1’s best response is to confess, given that his payoff from doing so, 0, is larger than that from not confessing, −1.3 This is also indicated in Fig. 2.2 with the underlined best-response payoff 0. Hence, we can compactly represent player 1’s best response (BR1 ðs2 Þ) as BR1 ðCÞ ¼ C to Confess, and BR1 ðNCÞ ¼ C to not confess. Importantly, this implies that player 1 finds confess a strictly dominant strategy, as he chooses to confess regardless of what player 2 does. Player 2 A similar argument applies for player 2. In particular, since the game is symmetric, we find that: (1) when player 1 confesses (in the top row), player 2’s best response is to Confess, since −5 > −15; and (2) when player 1 does Not confess (in the bottom row), player 2’s best response is to Confess, since 0 > −1.4 Hence, player 2’s best response can be expressed as BR2 ðCÞ ¼ C and BR2 ðNCÞ ¼ C, also Confess, player 1’s best response becomes a straightforward comparison of his payoff from Confess, −5, and that from Not confess, −15, which helps us underline the largest of the two payoffs, i.e., −5. 3 In this case, you can also focus on the column corresponding to Not confess by covering the column of Confess with your hand. This would allow you to easily compare player 1’s payoff from Confess, 0, and Not confess, −1, underlining the largest of the two, i.e., 0. 4 Similarly as for player 2, you can now focus on the row selected by player 1 by covering with your hand the row he did not select. For instance, when player 1 chooses Confess, you can cover the row corresponding to Not confess, which allows for an immediate comparison of the payoff when player 2 responds with Confess, −5, and when he does not, −15, and underline the largest of the two, i.e., −5. An analogous argument applies to the case in which player 1 selects Not confess, where you can cover the row corresponding to Confess with your hand.

42

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games... Prisoner 2 Confess

Not confess

Confess

-5, -5

0, -15

Not confess

-15, 0

-1, -1

Prisoner 1

Fig. 2.3 Prisoner’s dilemma game (normal-form)

indicating that Confess is a strictly dominant strategy for player 2, since he selects this strategy regardless of his opponent’s strategy. Payoffs associated with player 2’s best responses are underlined in Fig. 2.3 with red color. We can now see that there is a single cell in which both players are playing a mutual best response, (C, C), as indicated by the fact that both players’ payoffs are underlined (i.e., both players are playing best responses to each other’s strategies). Intuitively, since we have been underling best response payoffs, a cell that has the payoffs of all players underlined entails that every player is selecting a best response to his opponent’s strategies, as required by the definition of NE. Therefore, strategy profile (C, C) is the unique Nash Equilibrium (NE) of the game. NE ¼ ðC; CÞ Equilibrium versus Efficiency This outcome is, however, inefficient since it does not maximize social welfare (where social welfare is understood as the sum of both players’ payoffs). In particular, if players could coordinate their actions, they would both select not to confess, giving rise to outcome (NC, NC), where both players’ payoffs strictly improve relative to the payoff they obtain in the equilibrium outcome (C, C), i.e., they would only serve one year in jail rather than five years. This is a common feature in several games with intense competitive pressures, in which a conflict emerges between individual incentives (to confess in this example) and group/society incentives (not confess). Finally, notice that the NE is consistent with IDSDS. Indeed, since both players use strictly dominant strategies in the NE of the game, the equilibrium outcome according to NE coincides with that resulting from the application of IDSDS.

Exercise 2—Battle of the SexesA A husband and a wife are leaving work, and do not remember which event they are attending to tonight. Both of them, however, remember that last night’s argument was about either attending to the football game (the most preferred event for the

Exercise 2—Battle of the SexesA

43

husband) or the opera (the most preferred event for the wife). To make matters worse, their cell phones broke, so they cannot call each other to confirm which event they are attending to. As a consequence, they must simultaneously and independently decide whether to attend to the football game or the opera. The payoff matrix in Fig. 2.4 describes the preference of the husband (wife) for the football game (opera, respectively), but also indicates that both players prefer to be together rather than being alone (even if they are alone at their most preferred event). Find the set of Nash Equilibria of this game. Answer Every player i = {H, W} has strategy set Si = {F, O}. In order to find the Nash equilibrium of this game, let us separately identify the best responses of each player to his opponent’s strategies. Husband. Let’s first analyze the husband’s best responses BRH (F) and BRH (O). If his wife goes to the football game (focusing our attention in the left-hand column), the husband prefers to also attend the football game since his payoff from doing so, 3, exceeds that from attending the opera by himself, 0. If, instead, his wife attends the opera (in the right-hand column), the husband prefers to attend the opera with her, given that his payoff from doing so, 1, while low (he dislikes opera!), is still larger than that from going to the football game alone, 0. Hence we can summarize the husband’s best response as BRH (F) = F and BRH (O) = O; as indicated in the underlined payoffs in the matrix of Fig. 2.5. Intuitively, the husband’s best response is thus to attend the same event as his wife. Wife. A similar argument applies to the wife, who also best responds by attending the same event as her husband, i.e., BRW (F) = F in the top row when her husband attends the football game, and BRW (O) = O in the bottom row when he goes to the opera; as illustrated in the payoffs underlined in red color in the matrix of Fig. 2.6. Therefore, we found two strategy profiles in which both players are playing mutual best responses: (F, F) and (O, O), as indicated by the two cells in the matrix where both players’ payoffs are underlined. Hence, this game has two pure-strategy Nash equilibria (psNE): (F, F), where both players attend the football game, and (O, O), where they both attend the opera. This can be represented formally as: psNE ¼ fðF; F Þ; ðO; OÞg

Wife Football

Opera

Football

3, 1

0, 0

Opera

0, 0

1, 3

Housband

Fig. 2.4 Battle of the sexes game (Normal-form representation)

44

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games... Wife Football

Opera

Football

3, 1

0, 0

Opera

0, 0

1, 3

Housband

Fig. 2.5 Battle of the sexes game—underlining best response payoffs for husband

Wife Football

Opera

Football

3, 1

0, 0

Opera

0, 0

1, 3

Housband

Fig. 2.6 Battle of the sexes game—underlining best response payoffs for Husband and Wife

Exercise 3—Pareto Coordination GameA Consider the game in Fig. 2.7, played by two firms i = {1, 2}, each of them simultaneously and independently selecting to adopt either technology A or B. Technology A is regarded as superior by both firms, yielding a payoff of 2 to each firm if they both adopt it, while the adoption of technology B by both firms only entails a payoff of 1. Importantly, if firms do not adopt the same technology, both obtain a payoff of zero. This can be explained because, even if firm i adopts technology A, such a technology is worthless if firm i cannot exchange files, new products and practices with the other firm j 6¼ i. Find the set of Nash Equilibria (NE) in this game. Answer Firm 1. Let’s first examine firm 1’s best response. Similarly as in the battle of the sexes game, firm 1’s best response is to adopt the same technology as firm 2, i.e., BR1(A) = A when firm 2 chooses technology A, and BR1(B) = B when firm 2 selects technology B; as indicated in the payoffs underlined in blue color in the left-hand matrix of Fig. 2.8. Firm 2. A similar argument applies to firm 2, since firms’ payoffs are symmetric, i.e., BR2(A) = A and BR2(B) = B; as depicted in the payoffs underlined in red color in the right-hand matrix.

Exercise 3—Pareto Coordination GameA

45

Firm 2 Technology A Technology B Technology A

2, 2

0, 0

Technology B

0, 0

1, 1

Firm 1

Fig. 2.7 Pareto coordination game (Normal-form)

Fig. 2.8 Pareto coordination game—underlining best response payoffs for firm 1 (left matrix) and for both firms (right matrix)

Firm 2 Technology A

Technology B

Technology A

2, 2

0, 0

Technology B

0, 0

1, 1

Firm 1

Firm 2 Technology A

Technology B

Technology A

2, 2

0, 0

Technology B

0, 0

1, 1

Firm 1

Hence, we found two pure strategy Nash equilibria: (A, A) and (B, B), which are depicted in the matrix as the two cells were both players’ payoffs are underlined.5 psNE ¼ fðA; AÞ; ðB; BÞg Finally, note that, while either of the two technologies could be adopted in equilibrium, only one of them is efficient, (A, A), while the other equilibrium, (B, B), is inefficient, i.e., both firms would be better off if they could coordinate their simultaneous adoption of technology A.6

In both of these Nash equilibria, firms are playing mutual best responses, and thus no firm has incentives to unilaterally deviate. 6 However, no firm has incentives to unilaterally move from technology B to A when its competitor is selecting technology B. 5

46

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Exercise 4—Finding Nash Equilibria in the Unemployment Benefits GameA Consider the unemployment benefits game from Chap. 1. In that chapter we found that players have no strictly dominated strategies, implying that IDSDS has “no bite.” Find each player’s best response to his opponent’s strategies, to identify the Nash equilibria of this game. Answer Finding worker’s best responses. Worker’s is to choose Not search when the government offers unemployment benefits (in the top row of the matrix), that is, BRW(Benefit) = Not search; but to search when the government doesn’t offer unemployment benefits (in the bottom row), BRW(No Benefit) = Search. Finding government’s best responses. In contrast, the government’s best responses are the following: offer a benefit when the worker searches (in the left-hand column of the matrix), BRG(Search) = Benefit, but not offer it when the worker does not search (in the right-hand column), BRG(Not search) = No benefit. Underlining the payoffs associated to each of the above best responses, we obtain the following matrix (Fig. 2.9). Therefore, there is no strategy profile (no cell in the matrix) where players play a mutual best response to each other’s strategies, that is, there is no Nash equilibrium in this game. Strategically, the game resembles the “Matching pennies” game since the government seeks to offer benefits only when the worker searches, while the worker has incentives to search only when the government does not offer unemployment benefits. This holds true, however, only when players are restricted to use pure strategies, i.e., to play a strategy with 100% probability. When we allow for players to randomize their strategies, we will show that a Nash equilibrium can be sustained for this game.

Exercise 5—Hawk-Dove GameA Consider the following payoff matrix representing the Hawk-Dove game. Intuitively, Players 1 and 2 compete for a resource, each of them choosing to display an aggressive posture (hawk) or a passive attitude (dove). Assume that payoff V [ 0 denotes the value that both players assign to the resource, and C [ 0 is the cost of fighting, which only occurs if they are both aggressive by playing hawk in the top left-hand cell of the matrix (Fig. 2.10). (a) Show that if C < V, the game is strategically equivalent to a Prisoner’s Dilemma game. (b) The Hawk-Dove game commonly assumes that the value of the resource is less than the cost of a fight, i.e., C > V > 0. Find the set of pure strategy Nash equilibria.

Exercise 5—Hawk-Dove GameA

47 Worker

Government

Search

Not search

Benefit

3,2

-1,3

No benefit

-1,1

0,0

Fig. 2.9 Normal-form game

Fig. 2.10 Normal-form game

Player 2 Hawk

Dove

Hawk Player 1 Dove

Answer Part (a). Player 1’s best responses. When Player 2 (in columns) chooses Hawk (in the left-hand column), Player 1 (in rows) receives a positive payoff of VC 2 by paying Hawk, which is higher than his payoff of zero from playing Dove. Therefore, for Player 1 Hawk is a best response to Player 2 playing Hawk.

Similarly, when Player 2 chooses Dove (in the right-hand column), Player 1 receives a payoff of V by playing Hawk, which is higher than his payoff from choosing Dove, V2 ; entailing that Hawk is Player 1’s best response to Player 2 choosing Dove. Therefore, Player 1 chooses Hawk as his best response to all of Player 2’s strategies, implying that Hawk is a strictly dominant strategy for Player 1. Player 2’s best responses. Symmetrically, Player 2 chooses Hawk in response to Player 1 playing Hawk (in the top row) and to Player 1 playing Dove (in the bottom row), entailing that playing Hawk is a strictly dominant strategy for Player 2 as well. Finding pure strategy Nash equilibria. Since C  V, by assumption, the unique pure strategy Nash equilibrium is fHawk; Hawkg

48

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

. The outcome although {Dove, Dove} Pareto dominates the NE strategy as both V players can improve their payoffs from VC 2 to 2 . Since every player is choosing a strictly dominant strategy in the pure-strategy Nash equilibrium of the game, despite being Pareto dominated by another strategy profile, this game is strategically equivalent to a Prisoner’s Dilemma game. Part (b). The Hawk-Dove game commonly assumes that the value of the resource is less than the cost of a fight, i.e., C [ V [ 0. Find the set of pure strategy Nash equilibria. • Finding best responses for player 1. When Player 2 plays Hawk (in the left-hand column), Player 1 receives a negative payoff of VC 2 since C [ V [ 0. Player 1 then prefers to choose Dove (with a payoff of zero) in response to Player 2 playing Hawk. However, when Player 2 plays Dove (in the right-hand column), Player 1 receives a payoff of V by responding with Hawk, and half of this payoff, V 2 , when responding with Dove. Then, Player 1 plays Hawk in response to Player 2 playing Dove. • Finding best responses for player 2. Symmetrically, Player 2 plays Dove (Hawk) in response to Player 1 playing Hawk (Dove) to maximize his payoff. • Finding pure strategy Nash equilibria. For illustration purposes, we underline the best response payoffs in the matrix in Fig. 2.11. We can therefore identify two cells where the payoffs of both players were underlined as best response payoffs: ðHawk; DoveÞ and ðDove; HawkÞ: These two strategy profiles are the two pure strategies Nash equilibria. Our results then resemble those in the Chicken game, since every player seeks to miscoordinate by choosing the opposite strategy of his opponent.

Fig. 2.11 Normal-form game with best response payoffs underlined

Player 2 Hawk Hawk

Player 1 Dove

Dove

Exercise 6—Generalized Hawk-Dove GameA

49

Exercise 6—Generalized Hawk-Dove GameA Consider the following payoff matrix, which is often known as the “General Hawk-Dove” game. Show that this game is strategically equivalent to the Hawk– Dove game when payoff satisfy W [ T [ L  X (Fig. 2.12). Answer • Player 1’s best response. When Player 2 chooses Hawk (left-hand column), Player 1 (in rows) receives a payoff of X by playing Hawk, which is worse (or equal) than L by playing Dove, such that Player 1’s best response to Hawk is Dove. However, when Player 2 plays Dove (right-hand column), Player 1 receives a payoff of W by playing Hawk, which is larger than his payoff from Dove, T, entailing that Player 1 responds to Dove by choosing Hawk. • Player 2’s best response. Since the game is symmetric, Player 2 plays Dove (Hawk) in response to Player 1 playing Hawk (Dove, respectively) to maximize its payoff in the generalized Hawk-Dove game. • Pure-strategy Nash equilibria. For illustration purposes, we next underline the payoffs corresponding to best response strategies (Fig. 2.13). Therefore, we find two pure strategy Nash equilibria: ðHawk; DoveÞ and ðDove; HawkÞ: Fig. 2.12 Generalized normal-form game

Player 2 Hawk

Dove

Hawk Player 1 Dove

Fig. 2.13 Generalized normal-form game with best response payoffs underlined

Player 2 Hawk Hawk

Player 1 Dove

Dove

50

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Following the intuition of the basic Hawk-Dove game analyzed above, every player seeks to choose the opposite strategy than his opponent, as in the Chicken game.

Exercise 7—Cournot Game of Quantity CompetitionA Consider an industry with two firms competing in quantities, i.e., Cournot competition. For simplicity, assume that firms are symmetric in costs, c > 0, with no fixed costs and that they face a linear inverse demand p(Q) = a − bQ, where a > c, b > 0, and Q denotes aggregate output. Note that the assumption a > c implies that the highest willingness to pay for the first unit is larger than the marginal cost that firms must incur in order to produce the first unit, thus indicating that a positive production level is profitable in this industry. If firms simultaneously and independently select their output level, q1 and q2 , find the Nash Equilibrium (NE) of the Cournot game of quantity competition. Answer The profits of firm i are given by: pi ¼ pðQÞ  qi  cqi Given that Q ¼ q1 þ q2 , every firm i chooses its production level qi, taking the output level of its rival, qj, as given. That is, every firm i solves
 max a  bqi  bqj qi  cqi qi

Taking first-order conditions with respect to qi, we obtain a  2bqi  bqj  c ¼ 0 and solving for qi, we find firm i’s best response function, BRi ðqj Þ that is,
 ac 1  qj qi qj ¼ 2b 2 Figure 2.14 depicts the best response function of firm i, which originates at ac 2b , indicating the production level that firm i sells when firm j is inactive, i.e., the monopoly output level; and decreases as its rival, firm j, produces a larger amount of output. Intuitively, firm i’s and j’s output are strategic substitutes, so that firm i is forced to sell fewer units when the market becomes flooded of firm j’s products. When firm j’s production is sufficiently large, i.e., firm j produces more than ac b

Exercise 7—Cournot Game of Quantity CompetitionA Fig. 2.14 Cournot game—best response function of firm i

51

qi

qj

Fig. 2.15 Cournot game—best response function of firm j

qi

qj

units, firm i is forced to remain inactive, i.e., q
i ¼ 0.7 This property is illustrated in the figure by the fact that firm i’s best response function coincides with the horizontal axis (zero production) for all qj [ ac b . A similar argument applies to firm j, obtaining best response function 1 qj ðqi Þ ¼ ac 2b  2 qi , as depicted in Fig. 2.15. (Note that we use the same axis, in order to be able to represent both best response functions in the same figure in our ensuing discussion.) If we superimpose firm j’s best response function on top of firm i’s, we can visually see that the point where both functions cross each other represents the Nash Equilibrium of the Cournot game of quantity competition (Fig. 2.16). In order to precisely find the point in which both best response functions cross each other, let us simultaneously solve for q
i and q
j by, for instance, plugging 1 qj ðqi Þ ¼ ac 2b  2 qi into qi (qj), as we do next,

In order to obtain the output level of firm j that forces firm i to be inactive, set qi = 0 on firm i’s best response function, and solve for qj. The output you obtain should coincide with the horizontal intercept of firm i’s best response function in Fig. 2.9.

7

52

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Fig. 2.16 Cournot game— best response functions and Nash-equilibrium

qi

qj

 
 ac 1 ac 1   qi qi qj ¼ 2b 2 2b 2 which simplifies to q
i ¼

a  c þ bq
i 4b

and solving for q
i , we find the equilibrium output level for firm i, q
i ¼

ac 3b

and that of firm j, q
i ¼

a  c 1a  c a  c  ¼ : 2b 2 3b 3b

As we can see from the results, both firms produce exactly the same quantities, since they both have the same technology (they both face the same production costs). Hence, the pure strategy Nash Equilibrium is: n o na  c a  co psNE ¼ q
i ; q
j ¼ ; 3b 3b (Notice that this exercise assumes, for simplicity, that firms are symmetric in their production costs. In subsequent chapters we investigate how firms’ equilibrium production is affected when one of them exhibits a cost advantage (ci \cj ). We also examine how firms’ competition is affected when more than two firms interact in the same industry. See Chap. 5 for more details.)

Exercise 8—Games with Positive ExternalitiesB

53

Exercise 8—Games with Positive ExternalitiesB Two neighboring countries, i = 1, 2, simultaneously choose how many resources (in hours) to spend in recycling activities, ri. The average benefit (pi ) for every dollar spent on recycling is:
 rj pi ri ; rj ¼ 10  ri þ ; 2 and the (opportunity) cost per hour for each country is 4. Country i’s average benefit is increasing in the resources that neighboring country j spends on his recycling because a clean environment produces positive external effects on other countries. Part (a) Find each country’s best-response function, and compute the Equilibrium (NE), (r1*, r2*). Part (b) Graph the best-response functions and indicate the pure strategy Equilibrium on the graph. Part (c) On your previous figure, show how the equilibrium would change intercept of one of the countries’ average benefit functions fell from 10 to smaller number.

Nash Nash if the some

Answer Part (a) Since the gains of recycling are given by ðpi  ri Þ, and the costs of the activity are (4ri ), country 1’s maximization problem consists of selecting the amount of hours devoted to recycling r1 that solves: max r1

 r2  10  r1 þ r1  4ri 2

Taking the first-order condition with respect to r1 10  2r1 þ

r2 4¼0 2

Rearranging and solving for r1 yields country 1’s best-response function (BRF1 ): r1 ðr2 Þ ¼ 3 þ

r2 4

Symmetrically, Country 2’s best-response function is

54

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

r2 ðr1 Þ ¼ 3 þ

r1 4

Inserting best-response function r2(r1) into r1(r2) yields


3þ r1 ¼ 3 þ 4

r1 4



and, rearranging, we obtain an equilibrium level of recycling of r1* = 4 for country 1. Hence, country 2’s equilibrium recycling level is r2
¼ 3 þ

4 ¼4 4

Note that, alternatively, countries’ symmetry implies r1
¼ r2
. Hence, in BRF1 we can eliminate the subscript (since both countries’ recycling level coincides in equilibrium), and thus r ¼ 3 þ 4r , which, solving for r yields a symmetric equilibrium recycling of r* = 4. Hence, the psNE is given by: r1
¼ 4;

r2
¼ 4:

Part (b) Both best response functions originate at 3 and increase with a positive slope of 1/4, as depicted in Fig. 2.17. Intuitively, countries’ strategies are strategic complements, since an increase in r2 induces Country 1 to strategically increase its own level of recycling, r1, by 1/4. Part (c) A reduction in the benefits from recycling produces a fall in the intercept of one of the countries’ average benefit function, for example in Country 2. This change is indicated in Fig. 2.18 by the leftward shift (following the arrow) in

Fig. 2.17 Positive externalities—best response functions and Nash-equilibrium

Exercise 8—Games with Positive ExternalitiesB

55

Fig. 2.18 Shift in best response functions, change in Nash-equilibrium

Country 2’s best response function. In the new Nash Equilibrium, Country 2 recycles a lot less while Country 1 recycles a little less.

Exercise 9—Traveler’s DilemmaB Consider the following game, often referred to as the “ traveler’s dilemma.” An airline loses two identical suitcases that belong to two different travelers. The airline is liable for up to $100 per suitcase. The airline manager, in order to obtain an honest estimate of each suitcase, separates each traveler i in a different room and proposes the following game: “Please write an integer xi 2 ½2; 100 in this piece of paper. As a manager, I will use the following reimbursement rule: • If both of you write the same estimate, x1 ¼ x2 ¼ x, each traveler gets x. • If one of you writes a larger estimate, i.e., xi [ xj where i 6¼ j, then: – –

The traveler who wrote the lowest estimate (traveler j) receives xj þ k, where k [ 1; and The traveler who wrote the largest estimate (traveler i) only receives maxf0; xj  kg.”

Part (a) Show that asymmetric strategy profiles, in which travelers submit different estimates, cannot be sustained as Nash equilibria. Part (b) Show that symmetric strategy profiles, in which both travelers submit the same estimate, and such estimate is strictly larger than 2, cannot be sustained as Nash equilibria. Part (c) Show that the symmetric strategy profile in which both travelers submit the same estimate ðx1 ; x2 Þ ¼ ð2; 2Þ is the unique pure strategy Nash equilibrium. Part (d) Does the above result still hold when the traveler writing the largest amount receives xj  k rather than maxf0; xj  kg? Intuitively, since k [ 1 by

56

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

definition, a traveler can now receive a negative payoff if he submits the lowest estimate and xj \k. Answer Part (a) We first show that asymmetric strategy profiles, ðx1 ; x2 Þ with x1 6¼ x2 , cannot be sustained as a Nash equilibrium. Consider, without loss of generality, that player 1 submits a higher estimate than player 2, x1 [ x2 . In this setting, it is easy to see that player 1 has incentives to deviate: he now obtains a payoff of maxf0; x2  kg, and he could increase his payoff by submitting an estimate that matches that of player 2, i.e., x1 ¼ x2 , which guarantees him a payoff of x2 (as now the estimates from both travelers coincide), where maxf0; x2  kg\x2

for all x2

given that k satisfies k [ 1 by definition. Part (b) Using a similar argument, we can show symmetric strategy profiles in which both travelers submit the same estimate (but higher than two), i.e., ðx1 ; x2 Þ, with x1 ¼ x2 [ 2, cannot be supported as Nash equilibria either. To see this, note that in such strategy profile every player i obtains a payoff xi ¼ xj , but he can increase his payoff by deviating towards a lower estimate, that is, xi ¼ xj  1 since estimates must be integer numbers. With such a deviation, player i’s estimate becomes the lowest, and he thus obtains a payoff of
 xi þ k ¼ xj  1 þ k;
 where xj  1 þ k [ xj since k satisfies k [ 1 by definition. Part (c) Hence, the only remaining strategy profile is that in which both travelers submit an estimate of 2, x1 ¼ x2 ¼ 2: Let us now check if it can be sustained as a Nash equilibrium, by showing that every player i has no profitable deviation. Every traveler i obtains a payoff of 2 under the proposed strategy profile. If player i deviates towards a higher price, traveler i would be now submitting the highest estimate, and thus would obtain a payoff of  max 0; xj  k ¼ maxf0; 2  kg;

Exercise 9—Traveler’s DilemmaB

57

where maxf0; 2  kg\2 since parameter k satisfies k [ 1 by assumption. That is, submitting a higher estimate reduces player i’s payoff. Finally, note that submitting a lower estimate is not feasible since estimates must satisfy xi 2 ½2; 100 by definition. Alternative approach: While the above analysis tests whether a specific strategy profile can/cannot be sustained as Nash Equilibrium of the game, a more direct approach would identify each player’s best response function, and then find the point where player 1’s and 2’s best response functions cross each other, which constitutes the NE of the game. For a given estimate from player j, xj , if player i writes an estimate lower than xj , xi \xj , player i obtains a payoff of xi þ k, which is larger than the payoff he obtains from matching player j’s estimate, i.e., xi ¼ xj ¼ x, as long as xi þ k [ xj ;

that is; if k [ xj  xi :

Intuitively, player i profitably undercuts player j’s estimate if xi is not extremely lower than xj .8 If, instead, player i writes a larger estimate than player j, xi [ xj , his  payoff becomes max 0; xj  k , which is lower than his payoff from matching  player j’s estimate, i.e., xi ¼ xj ¼ x, since max 0; xj  k \xj . In summary, player i does not have incentives to submit a higher estimate than player j’s, but rather an estimate that is k-units lower than player j’s estimate. Hence, player i’s best response function can be written as 
 xi xj ¼ max 2; xj  k since the estimate that he writes must lie on the interval [2, 100]. This best response function is depicted in Fig. 2.19. Specifically, player i’s best response function originates at xi ¼ 2 when player j submits xj ¼ 2; remains at xi ¼ 2 when xj ¼ 3 (since k [ 1 entails that 3  k\2); and becomes xi ¼ maxf2; 4  kg when xj ¼ 4; thus increasing in xj . For instance, if k ¼ 2; player i’s best response function is xi ¼ 2 when player j’s estimate is xj ¼ 2; 3; 4; but increases to xi ¼ 3 when player j’s estimate is xj ¼ 5;
 and generally becomes xi xj ¼ xj  2 for all xj [ 4. Graphically, this function has a flat segment for low values of xj , but then increases in xj in a straight line located k-units below the 45° line. A similar argument applies to player j’s best response function. Hence, player 1’s and 2’s best response functions only cross at xi ¼ xj ¼ 2 (Fig. 2.19). Part (d) Our above argument did not rely on the property of positive payoffs for the traveler submitting the highest estimate. Hence, all the previous proof applies to this

For instance, if xj ¼ 5 and k ¼ 2, then player i has incentives to write an estimate of xi ¼ 5  2 ¼ 3, but not lower than 3 since his payoff, xi þ k, is increasing in his own estimate xi . 8

58

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Fig. 2.19 Traveler’s dilemma, best response functions

reimbursement rule as well, implying that x1 ¼ x2 ¼ 2 is the unique pure strategy Nash equilibrium of the game.

Exercise 10—Nash Equilibria with Three PlayersB Find all the Nash equilibria of the following three-player game (see Fig. 2.20), in which player 1 selects rows (a, b, or c), player 2 chooses columns (x, y, or z), and player 3 selects a matrix (either A in the left-hand matrix, or B in the right-hand matrix). Answer Player 3. Let’s start by evaluating the payoffs for player 3 when Player 2 selects x (first column). The arrows in Fig. 2.21 help us keep track of player 3’s pairwise comparison. For instance, when player 1 chooses a and player 2 selects x (in the top left-hand corner of either matrix), player 3 prefers to respond with matrix A, which gives him a payoff of 4, rather than with B, which only yields a payoff of 3. This comparison is illustrated by the top arrow in Fig. 2.21. A similar argument applies for the second arrow, which fixes the other players’ strategy profile at (b, x), and for the third arrow, which fixes their strategy profile at (c, x). The highest payoff that player 3 obtains in each of these three pairwise comparisons is circled in Fig. 2.21. Hence, we obtain that player 3’s best responses are BR3 ðx; aÞ ¼ A;

BR3 ðx; bÞ ¼ A; and BR3 ðx; cÞ ¼ A:

Exercise 10—Nash Equilibria with Three PlayersB

Fig. 2.20

59

Normal-form game with 3 players

Fig. 2.21 Normal-form game with 3 players. BR3 when player 2 chooses x

Fig. 2.22 Normal-form game with 3 players. BR3 when player 2 chooses y

If player 2 selects y (in the second column of each matrix), player 3’s pairwise comparisons are given by the three arrows in Fig. 2.22. In terms of best responses, this implies that BR3 ðy; aÞ ¼ B;

BR3 ðy; bÞ ¼ B; and BR3 ðy; cÞ ¼ fA; Bg:

The highest payoff that player 3 obtains in each pairwise comparison are also circled in Fig. 2.22. If player 2 selects z (in the third column of each matrix), player 3’s pairwise comparisons are depicted by the three arrows in Fig. 2.23 which in terms of best responses yields

60

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Fig. 2.23 Normal-form game with 3 players. BR3 when player 2 chooses z

BR3 ðz; aÞ ¼ A;

BR3 ðz; bÞ ¼ B; and BR3 ðz; cÞ ¼ A:

Hence, the payoff matrix that arises after underlying (or circling) the payoff corresponding to the best responses of player 3 is the following (see Fig. 2.24). Player 2. Let’s now identify player 2’s best responses as depicted in the circled payoffs of the matrices in Fig. 2.25. In particular, we take player 1’s strategy as given (fixing the row) and player 3’s as given (fixing the matrix) (where player 3 chooses matrix A). We obtain that player 2’s best responses are • BR2 ða; AÞ ¼ z when player 1 chooses a (in the top row), • BR2 ðb; AÞ ¼ x when player 1 selects b (in the middle row), and • BR2 ðc; AÞ ¼ z when player 1 chooses c (in the bottom row).

Fig. 2.24 Normal-form game with 3 players. Underlining player 3’s response payoffs

Fig. 2.25 Normal-form game with 3 players. Circling best responses of player 2

Exercise 10—Nash Equilibria with Three PlayersB

61

Visually, notice that we are now fixing our attention on a matrix (strategy of player 3) and on a row (strategy of player 1), and horizontally comparing the payoff that player 2 obtains from selecting the left, middle or right-hand column. Similarly, when player 3 chooses matrix B, we obtain that player 2’s best responses are • BR2 ða; BÞ ¼ fy; zg when player 1 selects a (in the top row) since both y and z yield the same payoff, $1, • BR2 ðb; BÞ ¼ x when player 1 chooses b (in the middle row), and • BR2 ðc; BÞ ¼ z when player 1 selects c (in the bottom row).9 Therefore, the matrices that arise after underlying the best response payoffs of player 2 are those in Fig. 2.26. Player 1. Let’s finally identify player 1’s best responses, given player’s 2 strategy (fixing the column) and given player 3’s strategy (fixing the matrix). When player 3 chooses A (left matrix), player 1’s best responses become: • BR1 ðx; AÞ ¼ b when player 2 chooses x (left-hand column), • BR1 ðy; AÞ ¼ a when player 2 selects y (middle column), and • BR1 ðz; AÞ ¼ c when player 2 chooses z (right-hand column). Visually, notice that we are now fixing our attention on a matrix (strategy of player 3) and on a column (strategy of player 2), and vertically comparing the payoff that player 1 obtains from choosing the top, middle or bottom row.10 Operating in an analogous fashion when player 3 chooses B (right-hand matrix), we obtain player 1’s best responses: • BR1 ðx; BÞ ¼ a when player 2 chooses x (in the left-hand column), • BR1 ðy; BÞ ¼ a when player 2 selects y (middle column), and • BR1 ðz; BÞ ¼ c when player 2 chooses z (right-hand column). We hence found three pure strategy Nash equilibria: ðb; x; AÞ; ðc; z; AÞ and ða; y; BÞ;

Visually, this implies fixing your attention on the first row of the left-hand matrix, and horizontally search for which strategy of player 2 (column) provides this player with the highest payoff. 10 For instance, in finding BR1 ðx; AÞ, we fix the matrix in which player 3 selects A (left matrix), and the column that player 2 selects x (left-hand column), and compare the payoffs that player 1 would obtain from responding with the first row (a), $2, the second row (b), $3, or with the third row (c), $1. Hence, BR1 ðx; AÞ ¼ b. A similar argument applies to other best responses of player 1. 9

62

Fig. 2.26

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Normal-form game with 3 players

Fig. 2.27 Normal-form game with 3 players

as depicted in the cells where the payoffs of all players are underlined (Fig. 2.27), as these cells correspond to outcomes where players employ mutual best responses to each others’ strategies (Fig. 2.28). In summary, the Nash equilibria of this three player game are psNE ¼ fðb; x; AÞ; ðc; z; AÞ; ða; y; BÞg

Exercise 11—Simultaneous-Move Games with n  2 PlayersB Consider a game with n  2 players. Simultaneously and independently, the players choose between two options, X and Y. These options might represent, for instance, two available technologies for the n firms operating in an industry, e.g., selling smartphones with the Android operating system or, instead, opt for the newer Windows Phone operating system from Microsoft. That is, the strategy space for each player i is Si = {X, Y}. The payoff of each player who selects X is: 2mx  m2x þ 3 where mx denotes the number of players who choose X. The payoff of each player who selects Y is

Exercise 11—Simultaneous-Move Games with n  2 PlayersB

Fig. 2.28

63

Normal-form game with 3 players. Nash equilibria

4  my where my is the number of players who choose Y. Note that mx þ my ¼ n. Part (a) For the case of only two players, n = 2, represent this game in its normal form, and find the pure-strategy Nash equilibria. Part (b) Suppose now that n = 3. How many psNE does this game have? Part (c) Consider now a game with n > 3 players. Identify an asymmetric psNE, i.e., an equilibrium in which a subset of players chooses X, while the remaining players choose Y. Answer Part (a) When both players choose X, mx = 2 and my = 0, thus implying that every player’s payoff is 2mx  m2x þ 3. Replacing mx ¼ 2, we obtain a payoff of 2ð 2Þ  ð 2Þ 2 þ 3 ¼ 3 for both players, as indicated in the cell corresponding to outcome (X, X) in the payoff matrix in Fig. 2.29. When, instead, both players choose Y, my = 2 and mx = 0, and players’ payoff becomes 4  my ¼ 4  2 ¼ 2; as depicted in outcome (Y, Y) of the payoff matrix. Finally, if only one player chooses X and another chooses Y, mx = 1 and my = 1, this yield a payoff of 2mx  m2x þ 3 ¼ 2ð1Þ  ð1Þ2 þ 3 ¼ 4 for the player who chose X, and 4  my ¼ 4  1 ¼ 3 for the player who chose Y; as represented in outcomes (X, Y) and (Y, X) in the payoff matrix (see cells away from the main diagonal in Fig. 2.29).

As usual, underlined payoffs represent a payoff corresponding to a player’s best response, as we next separately describe for each player.

64

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games... Player 2

Player 1

X Y

X 3,3 3,4

Y 4,3 2,2

Fig. 2.29 Normal-form game with n = 2 players

Player 1: In particular, player 1’s best responses are: • When player 2 chooses X (in the left-hand column), player 1’s best response is BR1 ð X Þ ¼ fX; Yg since he is indifferent between responding with X (in the top row) or Y (in the bottom row) given that they both yield a payoff of $3. As a result, we underline both payoffs of $3 for player 1 in the column in which player 2 chooses X. • If, instead, player 2 chooses Y (in the right-hand column), player 1’s best response is BR1 ðY Þ ¼ fXg, since player 1 obtains a higher payoff by selecting X ($4), than by choosing Y ($2). As a consequence, we underline the payoff of player 1 associated to his best response. Player 2: Similarly, for player 2 we find that: • When player 1 chooses X (in the top row), player 2 best responds with BR2 ð X Þ ¼ fX; Yg, since both X and Y yield a payoff of $2; • When player 1 selects Y (bottom row), player 2’s best response is BR2 ðY Þ ¼ fXg, since X yields a payoff of $4 while Y only entails a payoff of $2. Therefore, since there are three cells where payoffs of all players have been underlined as the best responses, they represent strategy profiles where players’ play mutual best responses, i.e. Nash equilibria of the game. There are, hence, three pure strategy Nash equilibrium in this game: ðX; X Þ; ðX; Y Þ; and ðY; X Þ Part (b) When introducing three players, the normal form representation of the game is depicted in the matrices of Figs. 2.30 and 2.31. (This is a standard three-player simultaneous-move game similar to that in the previous exercise). In order to identify best responses, player 1 and 2 operate as in previous exercises, i.e., taking the action of player 3 (matrix player) as given, and comparing their payoffs across rows for player 1 and across columns for player 2. However, player 3 compares his payoffs across matrices, for a given strategy profile of player 1 and 2. In particular, this pairwise payoff comparison of player 3 is analogous to that depicted in Exercises 6 and 8 of this chapter. For instance, if player’s 1 and 2 select (X, X), then player 3 obtains a payoff of only 0 if he were to select X as well

Exercise 11—Simultaneous-Move Games with n  2 PlayersB

65

Player 2 Player 1

X Y

X 0,0,0 3,3,3

Y 3,3,3 2,2,4

Fig. 2.30 Normal-form game when player 3 chooses s3 = X

Player 2 Player 1

Fig. 2.31

X Y

X 3,3,3 2,4,2

Y 4,2,2 1,1,1

Normal-form game when player 3 chooses s3 = Y

(in the upper matrix), but a higher payoff of 3 if he, instead, selects Y (in the lower matrix). For this reason, we underline 3 in the third component of the cell corresponding to (X, X, Y), in the upper left-hand corner of the lower matrix. A similar argument applies for identifying other best responses of player 3, where we compare the third component of every cell across the two matrices. For instance, when player 1 and 2 select (X, Y), player 3 obtains a payoff of 3 if he chooses X (in the upper matrix), but only a payoff of 2 if he selects Y (in the lower matrix), which leads us to underline 3 in the third component of the payoff in the cell (X, Y) of the upper matrix. Following a similar approach, we can see in Figs. 2.25 and 2.26 that there are three outcomes for which the payoffs of all players have been underlined (i.e., players are selecting mutual best responses). Specifically, the pure strategy Nash equilibria of the game with n = 3 players are: psNE ¼ fðX; Y; X Þ; ðY; X; X Þ; ðX; X; Y Þg Part (c) When n [ 3 players compete in this simultaneous-move game, the payoff from selecting strategy Y is 4  m y ¼ 4  ðn  m x Þ where the number of players selecting Y, my , is represented as those players who did not choose X, i.e., my ¼ n  mx . The payoff from selecting X is 2mx  m2x þ 3 Hence, a player selects strategy X if and only if his payoff from selecting X is weakly higher than from choosing Y, that is

66

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Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

2mx  m2x þ 3  4  ðn  mx Þ Solving for mx , yields that the number of players selecting strategy X is mx ¼

1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  4ð1  nÞ : 2

For instance, in the case of n = 5 players, the above expression becomes mx ¼ 2:56 players, which implies that three players select X. (Note that the above result for mx produces two roots, mx ¼ 2:56 and mx ¼ 1:56, but we only focus on the positive root.) For illustration purposes, Fig. 2.32 depicts the payoff from selecting strategy Y when n = 5 interact, 4  ð5  mx Þ ¼ mx  1, and that from strategy X, 2mx  m2x þ 3. Intuitively, the payoff from strategy X is decreasing in the number of players choosing it, mx (rightward movement in Fig. 2.32). Similarly, the payoff from selecting Y is also decreasing in the number of players choosing it, n  mx ; as depicted by leftward movements in Fig. 2.32. These incentives a negative network externality. For instance, settings in which a particular technology is very attractive when few other firms use it, but becomes less attractive as many other firms use it.

Exercise 12—Political Competition (Hoteling Model)B Consider two candidates competing for office: Democrat (D) and Republican (R). While they can compete along several dimensions (such as their past policies, their endorsements from labor unions, their advertising, and even their looks!), we assume for simplicity that voters compare the two candidates according to only one dimension (e.g., the budget share that each candidate promises to spend on education). Voters’ ideal policies are uniformly distributed along the interval [0, 1], and each votes for the candidate with a policy promise closest to the voter’s ideal. Candidates simultaneously and independently announce their policy positions. Fig. 2.32 Payoffs from selecting strategy X and Y

Exercise 12—Political Competition (Hoteling Model)B

67

A candidate’s payoff from winning is 1, and from losing is −1. If both candidates receive the same number of votes, then a coin toss determines the winner of the election. Part (a) Show that there exists a unique pure strategy Nash equilibrium, and that in involves both candidates proposals to promise a policy closest to the median voter. Part (b) Show that with three candidates (democrat, republican, and independent), no pure strategy Nash equilibrium exists. Answer Part (a) Let xi 2 ½0; 1 denote candidate i’s policy, where i ¼ fD; Rg. Hence, for a strategy profile ðxD ; xR Þ where, for instance, xD [ xR , voters to the right-hand side of xD vote democrat (since xD is close to their ideal policy than xR is), as well as half of the voters in the segment between xD and xR ; as depicted in Fig. 2.33. In contrast, voters to the left-hand side of xR and half of those in the segment between xD and xR vote republican. (The opposite argument applies for strategy profiles ðxD ; xR Þ satisfying xD \xR .) We can now show that there exists a unique Nash in which both candidates announce xD ¼ xR ¼ 0:5. Our proof is similar to that in the Traveler’s Dilemma game. First demonstrate that asymmetric strategy profiles where xD 6¼ xR cannot be sustained as Nash equilibria of the game; second, to show that symmetric strategy profiles where xD ¼ xR ¼ x but x 6¼ 0:5 cannot be supported as Nash equilibria either; and third, to demonstrate that symmetric strategy profile xD ¼ xR ¼ 0:5 can be sustained as Nash equilibrium of the game. Let’s first consider asymmetric strategy profiles where each candidate makes a different policy promise xi 6¼ xj , where i ¼ fD; Rg and j 6¼ i: Case 1 If xi \xj \0:5, candidate i could increase his chances to win by positioning himself e to the right of xj (where e [ 0 is assumed to be small). Thus, any strategy profile where xi \xj \0:5 cannot be supported as a Nash equilibrium.

Swing voters

1

0

Vote Republican

Fig. 2.33 Allocation of voters

Midpoint between xD and xR

Vote Democrat

68

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Case 2 If 0:5\xi \xj , candidate j could increase his chances to win by positioning himself e to the left of xi . Thus, any strategy profile where 0:5\xi \xj cannot be supported as a Nash equilibrium either. Case 3 If xi \0:5\xj , candidate i could increase his chances to win by positioning him self e to the left of xj . Thus, any case where xi \0:5\xj cannot be supported as a Nash equilibrium. (Note that the candidate j would also want to deviate to e right to the candidate i. Thus, there cannot be asymmetric Nash equilibria. Let us now consider symmetric strategy profiles where both candidates make the same policy promise, but their common policy differs from 0.5. Case 1 If xD ¼ xR \0:5, a tie occurs, and each candidate wins the election with probability 1/2. However, candidate D could win the election with certainty by positioning himself e to the right of xR . (A similar argument applies to candidate R, who would also have incentives to position himself e to the right of xD .) Thus, any strategy profile where xD ¼ xR \0:5 cannot be supported as a Nash equilibrium. Case 2 A similar argument applies if 0:5\xD ¼ xR , where a tie occurs and every candidate wins the election with probability 1/2. However, candidate R could increase his chances of winning by positioning himself e to the left of xD . Thus, any strategy profile where 0:5\xD ¼ xR cannot be supported as a Nash equilibrium either. Finally, if both candidates choose the same policy, xD ¼ xR ¼ x, and such common policy is x ¼ 1=2, each candidate receives half of the votes, and wins the election with probability 0.5. In this setting, however, neither candidate has incentives to deviate; otherwise his votes would fall from half of the electorate, guaranteeing him to lose the election. Therefore, there exists only one Nash equilibrium, in which xD ¼ xR ¼ 0:5. Part (b) Suppose that a Nash equilibrium exists with a triplet of policy proposals ðx
D ; x
R ; x
I Þ, where D denotes democrat, R republican, and I independent. We will next show that: (1) symmetric strategy profiles in which all candidates make the same that proposal, x
D ¼ x
R ¼ x
I , cannot be sustained as Nash equilibria; (2) asymmetric strategy profiles where two candidates choose the same proposal, but a third candidate differs, cannot be supported as equilibria either; and (3) asymmetric strategy profiles in which all three candidates choose different proposals cannot be sustained as equilibria; ultimately entailing that no pure strategy equilibrium exists.

Exercise 12—Political Competition (Hoteling Model)B

69

0

1 1/3

2/3

0

1

Fig. 2.34 Allocation of voters

First case. Consider, first, symmetric policy proposals x
D ¼ x
R ¼ x
I . All candidates, hence, receive the same number of votes (one third of the electorate), and each candidate wins with probability 1/3. While candidates didn’t have incentives to alter their policy promises in a setting with two candidates, with three of them we can see that candidates have incentives to deviate from such strategy profile. In particular, any candidate can win the election by moving to the right (if their common policy satisfies x
D ¼ x
R ¼ x
I \2=3) or moving to the left (if their common policy satisfies x
D ¼ x
R ¼ x
I [ 1=3); as depicted in Fig. 2.34. Similarly, when the location of the three candidates satisfies x
D ¼ x
R ¼ x
I [ 1=2 each candidate has incentives to deviate towards the left, while if x
D ¼ x
R ¼ x
I \1=2 each candidate has incentives to deviate to the right. Second case. Consider now strategy profiles in which two candidates choose the same policy, x
i ¼ x
j ¼ x
, but the third candidate differs, x
6¼ x
k , where i 6¼ j 6¼ k. If their policies satisfy x
\x
k , then candidate k has incentives to approach x
, i.e., x
þ e, as such position increases his votes. Similarly, if x
[ x
k , candidate k has incentives to approach x
, i.e., x
 e, which increases his votes. Since we found that at least one player has a profitable deviation deviation, the above strategy profile cannot be sustained as a Nash equilibrium. Third case. Finally, consider strategy profiles where all three candidates make different policy promises, x
i 6¼ x
j 6¼ x
k . The candidate that is located the farthest on the right will be able to win by moving e to the right of its closest competitor; implying that the original strategy profile cannot be equilibrium. (A similar argument applies to the other candidates, such as that located the farthest to the left, who could win by moving to the left of its closest competitor.) Therefore, there exists no pure strategy Nash Equilibrium in this game.

70

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Exercise 13—TournamentsB Several strategic settings can be modeled as a tournament, whereby the probability of winning a certain prize not only depends on how much effort you exert, but also on how much effort other participants in the tournament exert. For instance, wars between countries, or R&D competitions between different firms in order to develop a new product, not only depend on a participant’s own effort, but on the effort put by its competitors. Let’s analyze equilibrium behavior in these settings. Consider that the benefit that firm 1 obtains from being the first company to launch a new drug is $36 million. However, the probability of winning this R&D competition against its rival (i.e., being the first to launch the drug) is x1 ; x1 þ x2 which increases with this firm’s own expenditure on R&D, x1 , relative to total expenditure by both firms, x1 þ x2 . Intuitively, this suggests that, while spending more than its rival, i.e., x1 [ x2 , increases firm 1’s chances of being the winner, the fact that x1 [ x2 does not guarantee that firm 1 will be the winner. That is, there is still some randomness as to which firm will be the first to develop the new drug, e.g., a firm can spend more resources than its rival but be “unlucky” because its laboratory exploits a few weeks before being able to develop the drug. For simplicity, assume that firms’ expenditure cannot exceed 25, i.e., xi 2 ½0; 25. The cost is simply xi , so firm 1’s profit function is 

 x1 p1 ðx1 ; x2 Þ ¼ 36  x1 x1 þ x2 and there is an analogous profit function for firm 2:  p2 ðx1 ; x2 Þ ¼ 36

 x2  x2 x1 þ x2

You can easily check that these profit functions are increasing and concave in a firm’s own expenditure. Intuitively, this indicates that, while profits increase in the firm’s R&D, the first million dollar is more profitable than the 10th million dollar, e.g., the innovation process is more exhausted. Part (a) Find each firm’s best-response function. Part (b) Find a symmetric Nash equilibrium, i.e., x
1 ¼ x
2 ¼ x
.

Exercise 13—TournamentsB

71

Answer Part (a) Firm 1’s optimal expenditure is the value of x1 for which the first derivative of its profit function equals zero. That is, " # @p1 ðx1 ; x2 Þ x1 þ x2  x1 ¼ 36  1 ¼ 0: @x1 ðx1 þ x2 Þ2

Rearranging, we find " 36

x2 ðx1 þ x2 Þ2

# 1¼0

which simplifies to 36x2 ¼ ðx1 þ x2 Þ2 and further rearranging pffiffiffiffiffi 6 x2 ¼ x1 þ x2 Solving for x1 , we obtain firm 1’s best response function pffiffiffiffiffi x1 ðx2 Þ ¼ 6 x2  x2 pffiffiffiffiffi Figure 2.35 depicts firm 1’s best response function, x1 ðx2 Þ ¼ 6 x2  x2 as a function of its rival’s expenditure, x2 in the horizontal axis for the admissible set x2 2 ½0; 25.

Fig. 2.35 Tournament— firm 1’s best response function

72

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

It is straightforward to show that, for all values of x2 2 ½0; 25, firm 1’s best response also lies in the admissible set x1 2 ½0; 25. In particular, the maximum of BR1 occurs at x2 = 9 since  pffiffiffiffiffi @BR1 ðx2 Þ @ 6 x2  x2 ¼ ¼ 3ðx2 Þ1=2 1 @x2 @x2 Hence, the point at which this best response function reaches its maximum is that 1

in which its derivative is zero, i.e., 3ðx2 Þ 2 1 ¼ 0, which yields a value of x2 ¼ 9. At this point, firm 1’s best response function informs us that firm 1 optimally pffiffiffi spends 6 9  9 ¼ 9. Finally, note that the best response function is concave in its rival expenditure, x2, since 3 @BR1 ðx2 Þ 3 ¼  ðx2 Þ2 \0: 2 @x22

pffiffiffiffiffi By symmetry, firm 2’s best response function is x2 ðx1 Þ ¼ 6 x1  x1 : Part (b) In a symmetric Nash equilibrium x
1 ¼ x
2 ¼ x
. Hence, using this property in the best-response functions found in part (c), yields x
¼ 6

pffiffiffiffiffi x
1  x
1

pffiffiffiffiffi Rearranging, we obtain 2x
¼ 6 x
, and solving for x
, we find x
¼ 9: Hence, the unique symmetric Nash equilibrium has each firm spending 9. As Fig. 2.36 depicts, the points at which the best response function of player 1 and 2 cross each other occur at the 45° line (so the equilibrium is symmetric). In particular, those points are the origin, i.e., (0, 0), but this case is uninteresting since it implies that no firm spends money on R&D, and (9, 9).

Exercise 14—Lobbying GameA Consider two interest groups, A and B, seeking to influence a government policy, each with opposed interest (group A’s most preferred policy is 0, while that of group B is 1). Each group simultaneously and independently chooses a monetary contribution to government officials, si 2 ½0; 1, where i ¼ fA; Bg. The policy (x) that the government implements is a function of the contributions from both interest groups, as follows:

Exercise 14—Lobbying GameA

73

Fig. 2.36 Tournament-best response functions and Nash-equilibrium

1 xðsA ; sB Þ ¼  sA þ sB 2 Hence, if interest groups contribute zero (or if their contributions coincide, thus canceling each other), the government implements its ideal policy, 12. [In this simplified setting, the government is not a strategic player acting in the second stage of the game, since its response to contributions is exogenously described by policy function xðsA ; sB Þ.] Finally, assume that the interest groups have the following utility functions: uA ðsA ; sB Þ ¼ ½xðsA ; sB Þ2 sA uB ðsA ; sB Þ ¼ ½1  xðsA ; sB Þ2 sB which decrease in the contribution to the government, and in the squared distance between their ideal policy (0 for group A, and 1 for group B) and the implemented policy xðsA ; sB Þ. Find the Nash equilibrium of this simultaneous-move game. Answer Substituting the policy function into the utility function of every group, we obtain

 2 1  sA þ sB uA ðsA ; sB Þ ¼  1  sB 2 Taking first order conditions with respect to sA in the utility function of group A yields

2

 1  sA þ sB  1 ¼ 0 2

74

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Rearranging, we obtain 1  2sA þ 2sB  1 ¼ 0, which, solving for sA, yields the best-response function for interest group A, sA ðsB Þ ¼ sB : Similarly, taking first order conditions with respect to sB in the utility function of group B, we find



1  sA þ sB 2 1 2

 1¼0

which simplifies to sA þ sB ¼ 0; thus yielding the best-response function for interest group B, sB ðsA Þ ¼ sA : Graphically, both best response functions coincide with the 45-degree line, and completely overlap to one another. As a consequence, the set of pure strategy NEs is given by all the points in the 45° line, i.e., all points satisfying sA ¼ sB , or, more formally, the set n o ðsA ; sB Þ 2 ½0; 12 : sA ¼ sB : Furthermore, since both interest groups are contributing the same amount to the government, their contributions cancel out, and the government implements its preferred policy, 12. Finally, note that the strategic incentives in this game are similar to those in other Pareto Coordination games with symmetric NEs. While the game has multiple NEs in which both groups choose the same contribution level, the NE in which both groups choose a zero contribution Pareto dominates all other NEs with positive contributions.

Exercise 15—Incentives and PunishmentB Consider the following “law and economics” game, between a criminal and the government. The criminal selects a level of crime, y  0, and the government chooses a level of law enforcement x  0. Both choices are simultaneous and independent, and utility functions of the government (G) and the criminal (C) are, respectively, uG ¼ 

y2  xc4 x

and

uC ¼

1 pffiffiffi y 1 þ xy

Exercise 15—Incentives and PunishmentB

75

Intuitively, the government takes into account that crime, y, is harmful for society (i.e., y enters negatively into the government’s utility function uG), and that each unit of law enforcement, x, is costly to implement, at a cost of c4 per unit. In pffiffiffi contrast, the criminal enjoys y if he is not caught, which definitely occurs when pffiffiffi x = 0 (i.e., his utility becomes uc ¼ y when x = 0), while the probability of not being caught is 1 þ1 xy. Part (a) Find each player’s best-response function. Depict these best-response functions, with x on the horizontal axis and y on the vertical axis. Part (b) Compute the Nash equilibrium of this game. Part (c) Explain how the equilibrium levels of law enforcement, x and crime, y, found in part (b) change as the cost of law enforcement, c, increases. Answer Part (a) First, note that the government, G, selects the level of law enforcement, x, that solves max  x

y2  xc4 x

Taking first-order conditions with respect to x yields y2  c4 ¼ 0 x2 Rearranging and solving for x, we find the government’s (G) best response function, BRG, to be xð yÞ ¼

y : c2

Intuitively, the government’s level of law enforcement, x, increases in the amount of criminal activity, y, and decreases in the cost of every unit of law enforcement, c. Second, the criminal, C, selects the level of crime, y, that solves max y

1 pffiffiffi y 1 þ xy

Taking first-order conditions with respect to y yields 1 2y1=2 ð1 þ xyÞ



y1=2 x ð1 þ xyÞ2

¼0

76

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Rearranging and solving for y, we find the criminal’s best response function, BRC, to be yð xÞ ¼

1 x

which decreases in the level of law enforcement chosen by the government, x. The government’s best response function, BRG, and the
criminal’s best  response function, BRC, are represented in Fig. 2.37. Note that BRC i:e:; yð xÞ ¼ 1x is clearly decreasing in x but becomes flatter as x increases, that is, dyð xÞ 1 ¼  2 \0 and dx x

d 2 yð xÞ 2 ¼ 3 [ 0: 2 dx x

In order to depict the government’s best response function BRG ; xð yÞ ¼ cy2 , it is convenient to solve for y which yields y ¼ c2 x: As depicted in Fig. 2.37, BRG originates at (0, 0) and has a slope of c. Part (b) We find the values of x and y that simultaneously solve both players’ best response functions x ¼ cy2 and y ¼ 1x : For instance, you can plug in the second



expression into the first expression. This yields x
¼ 1=x c2 , which, solving for x , entails an equilibrium level of law enforcement of

1 x
¼ : c

Fig. 2.37 Incentives and punishment

Exercise 15—Incentives and PunishmentB

77

Fig. 2.38 Incentives and punishment-comparative statics

Therefore, the equilibrium level of crime is y rium is, hence, x
¼

1 c


1 c

1 ¼ 1=c ¼ c. The Nash equilib-

and y
¼ c;

as illustrated in the point where best response function BRG crosses BRC in Fig. 2.37. Part (c) As Fig. 2.38 illustrates, an increase in the cost of enforcement c pivots the government’s best response function, BRG leftward, with center at the origin (recall that c is the slope of BRG). In contrast, the criminal’s best response function is unaffected (since it is independent on c). This pivoting effect produces a new crossing point that lies to the northwest of the original Nash equilibrium, entailing a higher level of criminal activity (y) and a lower level of enforcement (x).

Exercise 16—Cournot Competition with Efficiency ChangesA Consider a Cournot duopoly model, where two firms selling a homogeneous product simultaneously and independently choose their output level q1 and q2 . Firms face a linear inverse demand curve pðQÞ ¼ 10  Q;

where Q ¼ q1 þ q2

denotes the aggregate output sold by both firms. Assume that firm 1’s cost function is given by TC1 ðq1 Þ ¼ 4q1 , so its marginal cost is 4.

78

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

(a) Assuming that firm 2 also has a cost function TC2 ðq2 Þ ¼ 4q2 , with marginal cost of 4, find each firm’s best response function, and depict them in a figure with q1 in the vertical axis and q2 in the horizontal axis. Then find the output each firm produces in equilibrium. (b) Consider now that firm 2 invests in a cost-saving technology, which transforms its cost function to TC2 ðq2 Þ ¼ 2q2 , so its marginal cost now decreases to 2. Repeat your analysis in part (a) with this new cost function for firm 2, assuming firm 1’s cost function is unaffected. (c) Compare your results from parts (a) and (b). Answer Part (a). Finding firm i’s best response function. Every firm i 2 f1; 2g, solves the following profit maximization problem: max pi ðqi Þ ¼ pðQÞqi  TCi ðqi Þ qi  0
 ¼ 10  qi  qj qi  4qi
 ¼ 6  qi  qj qi

Taking the first order condition with respect to qi , and assuming an interior solution for qi , 6  2qi  qj ¼ 0 if qi [ 0 After rearranging, the best response function of firm i in response to firm j is given by
 qj qi qj ¼ 3  2 which has a vertical intercept at 3, and a slope of −1/2. Specifically, letting firm 1 be i and firm 2 be j, the best response function of firm 1 becomes q1 ð q2 Þ ¼ 3 

q2 2

In Fig. 2.39, firm 1’s best response curve is depicted in red. Therefore, when firm 2 produces no output, that is, q2 ¼ 0, the best response of firm 1 is to produces 3 units of output, as depicted in the vertical axis of Fig. 2.39. Whereas, when firm 2 produces 6 or more units of output, since firm 1’s output

Exercise 16—Cournot Competition with Efficiency ChangesA

79

Fig. 2.39 Best response curve of firm 1

cannot be negative, the best response of firm 1 is to produce 0 units of output, as depicted in the horizontal axis. Similarly, letting firm 1 be j and firm 2 be i, the best response function of firm 2 becomes q2 ð q1 Þ ¼ 3 

q1 2

In Fig. 2.40, firm 2’s best response curve is depicted in blue. Therefore, when firm 1 produces no output, that is, q1 ¼ 0, the best response of firm 2 is to produces 3 units of output, as depicted in the horizontal axis of Fig. 2.40. Whereas, when firm 1 produces 6 or more units of output, since firm 2’s output cannot be negative, the best response of firm 2 is to produce 0 units of output, as depicted in the vertical axis. Using BRFs to find the Nash equilibrium. The intersection between the two firms’ best response functions represents the Nash Equilibrium output pair in this game. To find this crossing point, we can substitute the best response function of firm 1 into that of firm 2, q2 ¼ 3 

1 q2  3 2 2

Rearranging, we find 3q2 3 ¼ 2 4 And solving for output q2 , we obtain the equilibrium output for firm 2, q
2 ¼ 2 units. Next, substituting the equilibrium output of firm 2 into the best response function of firm 1,

80

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Fig. 2.40 Best response curves of firms 1 and 2

q
1 ¼ 3 

2 ¼2 2

As a result, both firms produce q
1 ¼ q
2 ¼ 2 units in equilibrium, graphically represented as the intersection of the best response curves in Fig. 2.40. Part (b). Finding firm 2’s best response function. While the profit maximization problem of firm 1 remains unchanged, that for firm 2 changes because of investments in cost-saving technology. In particular, firm 2 chooses q2 to solve the following profit maximization problem:

max p2 ðq2 Þ ¼ pðQÞq2  TC2 ðq2 Þ

q2  0

¼ ð10  q1  q2 Þq2  2q2 ¼ ð8  q1  q2 Þq2 Taking the first order condition with respect to q2 , and assuming an interior solution for q2 , 8  q1  2q2 ¼ 0

if q2 [ 0

After rearranging, the best response function of firm 2 in response to firm 1 is given by q2 ð q 1 Þ ¼ 4 

q1 2

Exercise 16—Cournot Competition with Efficiency ChangesA

81

which now originates at a vertical intercept of 4 (above the vertical intercept in the previous setting), and still has the same slope of −1/2. Intuitively, for a given output of firm 1, firm 2 produces a larger output due to its lower costs. The best response curves are depicted in Fig. 2.41. Using BRFs to find the Nash equilibrium. Substituting the best response function of firm 1 into the best response function of firm 2, 1 q2  3 2 2 3q2 5 ¼ ) 2 4

q2 ¼ 4 

which yields q
2 ¼ 3

1 3

Substituting the optimal output of firm 2 into the best response function of firm 1, q
1 ¼ 3 

q
2 5 1 ¼3 ¼1 3 3 2

As a result, firm 1 produces 1.33 units while firm 2 produces 3.33 units in equilibrium, graphically represented as the intersection of the best response curves in Fig. 2.41. Part (c). When both firms are symmetric in costs, each firm produces 2 units of output. Whereas, when it becomes less expensive for firm 2 to produce the output, firm 2 expands production from 2 to 3.33 units, and firm 1’s best response is to reduce production from 2 to 1.33 units.

Exercise 17—Cournot Mergers with Efficiency GainsB Consider an industry with three identical firms each selling a homogenous good and producing at a constant cost per unit c with 1 [ c [ 0. Industry demand is given by pðQÞ ¼ 1  Q, where Q ¼ q1 þ q2 þ q3 . Competition in the marketplace is in quantities. Part (a) Find the equilibrium quantities, price and profits. Part (b) Consider now a merger between two of the three firms, resulting in duopolistic structure of the market (since only two firms are left: the merged firm and the remaining firm). The merger might give rise to efficiency gains, in the sense that the firm resulting from the merger produces at a cost e  c, with e  1 (whereas the remaining firm still has a cost c):

82

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

Fig. 2.41 Best response curves when firm 2 invests in cost-saving technology

i. Find the post-merger equilibrium quantities, price and profits. ii. Under which conditions does the merger reduce prices? iii. Under which conditions is the merger beneficial to the merging firms? Answer Part (a) Finding firm i’s best response function. Each firm i = {1, 2, 3} has a profit of pi ¼ ð1  Q  cÞqi : Hence, since Q q1 þ q2 þ q3 , profits can be rewritten as:

  p i ¼ 1  qi þ qj þ qk  c qi ; The first-order conditions are given by 1  2qi  qj  qk  c ¼ 0: Using BRFs to find the Nash equilibrium. Since firms are symmetric qi ¼ qj ¼ qk ¼ q in equilibrium, that is 1  2q  q  q  c ¼ 0

Exercise 17—Cournot Mergers with Efficiency GainsB

83

or 1  4q  c ¼ 0: Solving for q at the symmetric equilibrium yields a Cournot output of, qc ¼

1c 4

Hence, equilibrium prices are   1c 1c 1c 1c 1 þ 3c pc ¼ 1    ¼13 ¼ 4 4 4 4 4 and every firm’s equilibrium profits are  pc ¼

 1 þ 3c 1  c ð1  c Þ2 c ¼ : 4 4 16

Part (b) i. After the merger, two firms are left: firm 1, with cost e  c, and firm 3, with cost c. Hence, the two profit functions are now given by: p1 ¼ ð1  Q  ecÞq1 p3 ¼ ð1  Q  cÞq3 Taking first order conditions of p1 with respect to q1 yields 1  2q1  q3  ec ¼ 0 and, solving for q1 , we obtain firm 1’s best response function q1 ð q3 Þ ¼

1  ec 1  q3 2 2

Similarly taking first-order conditions of firm 3’s profits, p3 , with respect to q3 yields 1  2q3  q1  c ¼ 0

84

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

which, solving for q3 , provides us with firm 3’s best response function q3 ð q1 Þ ¼

1c 1  q1 2 2

Plugging q3 ðq1 Þ into q1 ðq3 Þ, yields q
1 ¼

  1  ec 1 1  c 1
  q1 2 2 2 2

Rearranging and solving for q
1 , we obtain firm 1’s equilibrium output q
1 ¼

1  cð2e  1Þ 3

Plugging this output level into firm 3’s best response function yields an equilibrium output of q
3 ¼

1  cð2  eÞ 3

Note that the outsider firm can sell a positive output at equilibrium only if the merger does not give rise to strong cost savings: that is q3  0 if e 

2c  1 c

Note that if c\1=2, then the previous payoff becomes 2c1 c \0, implying that e  2c1 holds for all e  0, ultimately entailing that the outsider firm will always c sell at the equilibrium. We hence focus on values of c that satisfy c [ 1=2. Figure 2.42 illustrates cutoff e [ 2c1 c , where c [ 1=2. The region of ðe; cÞcombinations above this cutoff indicate for which the merger is not sufficiently cost saving to induce the outside firm to produce positive output levels. The opposite occurs when the cost-saving parameter, e, is lower than 2c1 c , thus indicating that the merger is so cost saving that the nonmerged firm cannot profitably compete against the merged firm. • The equilibrium price is pm ¼ 1 

1  cð2e  1Þ 1  cð2  eÞ 1 þ cð1 þ eÞ  ¼ ; 3 3 3

and equilibrium profits are given by

Exercise 17—Cournot Mergers with Efficiency GainsB

85

Fig. 2.42 Positive production after the merger

Fig. 2.43 Profitable mergers if e is low enough

p1 ¼

ð1  cð2e  1ÞÞ2 9

and p3 ¼

ð1  cð2  eÞÞ2 : 9

ii. Prices decrease after the merger only if there are sufficient efficiency gains: that is, pm  pc can be rewritten as e

5c  1 : 4c

86

2

Pure Strategy Nash Equilibrium and Simultaneous-Move Games...

5c1 Note that if c\1=5, then 5c1 4c \0, implying that e  4c cannot hold for any e  0. As a consequence, pm [ pc , and prices will never fall no matter how strong efficiency gains, e, are. iii. To see if the merger is profitable, we have to study the inequality p1  2pc ; which, solving for e, yields

e

pffiffiffi 4ð1 þ cÞ  3 2ð1  cÞ 8c

In other words, the merger is profitable only if it gives rise to enough cost savings. Figure 2.43 depicts the cutoff of e where costs are restricted to c 2 15 ; 1 . Notice that if cost savings are sufficiently strong, i.e., parameter e is sufficiently small as depicted in the region below the cutoff, the merger is profitable.

3

Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Introduction This chapter analyzes how to find equilibrium behavior when players are allowed to randomize, helping us to identify mixed strategy Nash equilibria (msNE). Finding this type of equilibrium completes our analysis in Chap. 2 where we focused on Nash equilibria involving pure strategies (not allowing for randomizations). However, as shown in Chap. 2, some games do not have a pure strategy Nash equilibrium. As this chapter explores, allowing players to randomize (mix) will help us identify a msNE in this type of games. For other games, which already have pure strategy Nash equilibria (such as the Battle of the Sexes game), we will find that allowing for randomizations gives rise to one more equilibrium outcome. Let us next define mixed strategies and msNE. Mixed strategy. Consider that every player i has a finite strategy space Si ¼ fs1 ; s2 ; . . .; sm g with its associated probability simplex DSi , which we can understand as the set of all probability distributions over Si . (For instance, if the strategy space has only two elements ðm ¼ 2Þ, then the simplex is graphically represented by a [0,1] segment in the real line; while if the strategy space has three elements ðm ¼ 3Þ the simplex can be graphically depicted by a Machina’s triangle in R2 .) Hence, a mixed strategy is an element of the simplex (a point in the previous examples for m ¼ 2 and m ¼ 3) that we denote as ri 2 DSi , ri ¼ fri ðs1 Þ; ri ðs2 Þ; . . .; ri ðsm Þg where ri ðsk Þ denotes the probability that player i plays pure strategy sk , and satisfies ri ðsk Þ  0 for all k (positive or zero probabilities for each pure strategy) and P m k¼1 ri ðsk Þ ¼ 1 (the sum of all probabilities must be equal to one). We can now use this definition of a mixed strategy to define a msNE, as follows.

© Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_3

87

88

3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

msNE. Consider a strategy profile r ¼ ðr1 ; r2 ; . . .; rN Þ where ri denotes a mixed strategy for player i. We then say that strategy profile r is a msNE if and only if ui ðri ; ri Þ  ui ðsi ; ri Þ for all si 2 Si and for every player i. Intuitively, mixed strategy ri is a best response of player i to the strategy profile ri selected by other players.1 The chapter starts with games of two players who choose among two available strategies. A natural examples is the Battle of the Sexes game, where husband and wife simultaneously and independently choose whether to attend to a football game or the opera. We then extend our analysis to settings with more than two players, and to contexts in which players are allowed to choose among more than two strategies. We illustrate how to find the convex hull of Nash equilibrium payoffs (where players use pure or mixed strategies), and afterwards explain how to find correlated equilibria when players use a publicly observable random variable. For completeness, we also examine an exercise which illustrates the difference between the equilibrium payoffs that players can reach when playing the mixed strategy Nash equilibrium of a game and when playing the correlated equilibrium, showing that they do not necessarily coincide. At the end of the chapter, we focus on strictly competitive games: first, describing how to systematically check whether a game is strictly competitive; and second, explaining how to find “maxmin strategies” (also referred to as “security strategies”) using a two-step graphical procedure.

Exercise 1—Game of ChickenA Consider the game of Chicken (depicted in Fig. 3.1), in which two players driving their cars against each other must decide whether or not to swerve. Part (a) Is there any strictly dominated strategy for Player 1? And for Player 2? Part (b) What are the best responses for Player 1? And for Player 2? Part (c) Can you find any pure strategy Nash Equilibrium (psNE) in this game? Part (d) Find the mixed strategy Nash Equilibrium (msNE) of the game. Hint: denote by p the probability that Player 1 chooses Straight and by ð1  pÞ the Note that we compare player i’s expected payoff from mixed strategy ri , ui ðri ; ri Þ, against his  expected payoff from selecting pure strategy s0i , ui s0i ; ri . We could, instead, compare ui ðri ; ri Þ   against ui r0i ; ri , where r0i 6¼ ri . However, for player i to play mixed strategy r0i , he must be indifferent between at least two pure strategies, e.g., s0i and s00i . Otherwise, player i would not be mixing, but choosing Hence, his indifference between pure strategies s0i and s00i 0  a pure  00 strategy.  entails that ui si ; ri ¼ ui si ; ri , implying that it suffices to check if mixed strategy ri yields a higher expected payoff than all pure strategies that player i could use in his randomization. (This is a convenient result, as we will not need to compare the expected payoff of mixed strategy ri against all possible mixed strategies r0i 6¼ ri ; which would entail studying the expected payoff from all feasible randomizations.) 1

Exercise 1—Game of ChickenA

89

Player 2 Straight Swerve Player 1

Straight

0,0

3,1

Swerve

1,3

2,2

Fig. 3.1 Game of Chicken

probability that he chooses to Swerve. Similarly, let q denote the probability that Player 2 chooses Straight and ð1  qÞ the probability that she chooses to Swerve. Part (e) Show your result of part (d) by graphically representing every player i’s best response function BRFi ðsj Þ, where sj ¼ fSwerve; Straightg is the strategy selected by player j 6¼ i. Answer Part (a) No, there are no strictly dominated strategies for any player. In particular, Player 1 prefers to play Straight when Player 2 plays Swerve getting a payoff of 3 instead of 2. However, he prefers to Swerve when Player 2 plays Straight, getting a payoff of 1 by swerving instead of a payoff of 0 if he chooses to go straight. Hence, there is no strictly dominated strategy for player 1, i.e., a strategy that Player 1 would never use regardless of what strategy his opponent selects. Since payoffs are symmetric, a similar argument applies to Player 2. Part (b) Player 1. If player 2 chooses straight (fixing our attention in the left-hand column), player 1’s best response is to Swerve, since his payoff from doing so, 1, is larger than that from choosing Straight. Hence, BR1 ðStraightÞ ¼ Swerve; obtaining a payoff of 1

If player 2 instead chooses to Swerve (in the right-hand column), player 1’s best response is Straight, since his payoff from Straight, 3, exceeds that from Swerve, 2. That is, BR1 ðSwerveÞ ¼ Straight; obtaining a payoff of 3 Player 2. A similar argument applies to player 2’s best responses; since players are symmetric in their payoffs. Hence, BR2 ðStraightÞ ¼ Swerve; obtaining a payoff of 1 BR2 ðSwerveÞ ¼ Straight; obtaining a payoff of 3

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

These best responses illustrate that the Game of Chicken is an Anti-Coordination game, where each player responds by choosing exactly the opposite strategy of his/her competitor. Part (c) Note that in strategy profiles (Swerve, Straight) and (Straight, Swerve), there is a mutual best response by both players. To see this more graphically, the matrix in Fig. 3.2 underlines the payoffs corresponding to each player’s selection of his/her best response. For instance, BR1 ðStraightÞ ¼ Swerve implies that, when Player 2 chooses Straight (in the left-hand column), Player 1 responds Swerving (in the bottom left-hand cell), obtaining a payoff of 1; while when Player 2 chooses Swerve (in the right-hand column) BR1 ðSwerveÞ ¼ Straight, and thus Player 1 obtains a payoff of 3 (top right-hand side cell). A similar argument applies to Player 2. There are, hence, two cells where the payoffs of all players have been underlined, i.e., mutual best responses (Swerve, Straight) and (Straight, Swerve). These are the two pure strategy Nash equilibria of the Chicken game. Part (d) Player 1. Let q denote the probability that Player 2 chooses Straight, and ð1  qÞ the probability that she chooses Swerve as depicted in Fig. 3.3. In this context, the expected value that Player 1 obtains from playing Straight (in the top row) is: EU1 ðStraightÞ ¼ 0q þ 3ð1  qÞ since, fixing our attention in the top row, Player 1 gets a payoff of zero when Player 2 chooses Straight (which occurs with probability q) or a payoff of 3 when Player 2

Player 2 Straight Swerve Player 1

Straight

0,0

3,1

Swerve

1,3

2,2

Fig. 3.2 Nash equilibria in the Game of Chicken

Player 2 Straight Swerve q 1-q Player 1

Straight

p

0,0

3,1

Swerve

1-p

1,3

2,2

Fig. 3.3 Searching for a mixed strategy equilibrium in the Game of Chicken

Exercise 1—Game of ChickenA

91

selects Swerve (which happens with the remaining probability 1 − q). If, instead, Player 1 chooses to Swerve (in the bottom row), his expected utility becomes EU1 ðSwerveÞ ¼ 1q þ 2ð1  qÞ since Player 1 obtains a payoff of 1 when Player 2 selects Straight (which occurs with the probability q) or a payoff of 2 when Player 2 chooses Swerve (which happens with probability 1 − q). Player 1 must be indifferent between choosing Straight or Swerve. Otherwise, he would not be randomizing, since one pure strategy would generate a larger expected payoff than the other, leading Player 1 to select such a pure strategy. We therefore need that EU1 ðStraightÞ ¼ EU1 ðSwerveÞ 0q þ 3ð1  qÞ ¼ 1q þ 2ð1  qÞ rearranging and solving for probability q yields 3 þ 3q ¼ q þ 2  2q ) q ¼ 1=2 That is, Player 2 must be choosing Straight with 50% probability, i.e., q ¼ 12, since otherwise Player 1 would not be indifferent between Straight and Swerve. Player 2. When he chooses Straight (in the left-hand column), he obtains an expected utility of EU2 ðStraightÞ ¼ 0p þ 3ð1  pÞ since he can get a payoff of zero when player 1 chooses Straight (which occurs with probability p, as depicted in Fig. 3.3), or a payoff of 3 when Player 1 selects to Swerve (which happens with probability 1 − p). When, instead, Player 2 chooses Swerve (directing our attention to the right-hand column), he obtains an expected utility of EU2 ðSwerveÞ ¼ 1p þ 2ð1  pÞ since his payoff is 1 if player 1 chooses Straight (which occurs with probability p) or a payoff of 2 if Player 1 selects to Swerve (which happens with probability 1 − p). Therefore, Player 2 is indifferent between choosing Straight or Swerve when EU2 ðStraightÞ ¼ EU2 ðSwerveÞ 0p þ 3ð1  pÞ ¼ 1p þ 2ð1  pÞ Solving for probability p in the above indifference condition, we obtain 1 ¼ 2p ) p ¼ 1=2

92

3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Hence, the mixed strategy Nash equilibrium of the Chicken game prescribes that every player randomizes between driving Straight and Swerve half of the time. More formally, the mixed strategy Nash equilibrium (msNE) is given by  msNE ¼

   1 1 1 1 Straight; Swerve ; Straight; Swerve 2 2 2 2

where the first parenthesis indicates the probability distribution over Straight and Swerve for Player 1, and the second parenthesis represents the analogous probability distribution for Player 2. Part (e) In Fig. 3.4, we first draw the thresholds which specify the mixed strategy Nash equilibrium (msNE): p ¼ 1=2 and q ¼ 1=2. Then, we draw the two pure strategy Nash equilibria found in question (c): • One psNE in which Player 1 plays Straight and Player 2 plays Swerve (which in probability terms means p ¼ 1 and q ¼ 0), graphically depicted at the lower right-hand corner of Fig. 3.4 (in the southeast); and • Another psNE in which Player 1 plays Swerve and Player 2 plays Straight (which in probability terms means p ¼ 0 and q ¼ 1), graphically depicted at the upper left-hand corner of Fig. 3.4 (in the northwest). Player 1’s best response function. Once we have drawn these pure strategy equilibria, notice that from Player 1’s best response function, BR1 ðqÞ, we know that, if q\ 12, then EU1 ðStraightÞ [ EU1 ðSwerveÞ, thus implying that Player 1 plays Straight using pure strategies, i.e., p ¼ 1. Intuitively, when Player 1 knows that Player 2 is rarely selecting Straight, i.e., q\ 12, his best response is to play Straight, i.e., p ¼ 1 for all q\ 12, as illustrated in the vertical segment of BR1 ðqÞ in the

Fig. 3.4 Game of Chicken— best response function of Player 1

q 1

(Swerve, Straight)

BR1 (q )

1/2

(Straight, Swerve)

1/2

1

p

Exercise 1—Game of ChickenA

93

right-hand side of Fig. 3.4. In contrast, q [ 12 entails EU1 ðStraightÞ\EU1 ðSwerveÞ, and therefore p ¼ 0. In this case, Player 2 is likely playing Straight, leading Player 1 to respond with Swerve, i.e., p = 0, as depicted in the vertical segment of BR1 ðqÞ that overlaps the vertical axis in the left-hand side of Fig. 3.4. Player 2’s best response function. A similar analysis applies to Player 2’s best response function, BR2 ðpÞ, depicted in Fig. 3.5: (1) when p\ 12, Player 2’s expected utility comparison satisfies EU2 ðStraightÞ [ EU2 ðSwerveÞ, thus implying q ¼ 1, as depicted in the horizontal segment of BR2 ðpÞ at the top left-hand side of Fig. 3.5; and (2) when p [ 12, EU2 ðStraightÞ\EU2 ðSwerveÞ, ultimately leading to q ¼ 0, as illustrated by the horizontal segment of BR2 ð pÞ that overlaps the horizontal axis in the right-hand side of the figure. Superimposing both players’ best response functions, we obtain Fig. 3.6, which depicts the two psNE of this game (southeast and northwest corners), as well as the msNE (strictly interior point, where players are randomizing their strategies). Fig. 3.5 Game of Chicken— Best response function of player 2

q 1

BR2 ( p) (Swerve, Straight)

1/2

(Straight, Swerve)

1/2

Fig. 3.6 Game of Chicken— Best response functions, psNE and msNE

1

p

1

p

q 1

BR2 ( p ) msNE

(Swerve, Straight)

BR1 (q )

1/2

(Straight, Swerve)

1/2

94

3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Exercise 2—Lobbying GameA Let us consider the following lobbying game in Fig. 3.7 where two firms simultaneously and independently decide whether to lobby Congress in favor a particular bill. When both firms (none of them) lobby, congress’ decisions are unaffected, implying that each firm earns a profit of 10 if none of them lobbies (−5 if both choose to lobby, respectively). If, instead, only one firm lobbies its payoff is 15 (since it is the only beneficiary of the policy), while that of the firm that did not lobby collapses to zero. Part (a) Find the pure strategy Nash equilibria of the lobbying game. Part (b)Find the mixed strategy Nash equilibrium of the lobbying game. Part (c) Graphically represent each player’s best response function. Answer Part (a) Let us first identify each firm’s best response function. Firm 1’s best response is BR1 ðLobbyÞ ¼ Not lobby and BR1 ðNot lobbyÞ ¼ Lobby; and similarly for Firm 2, thus implying that the lobbying game represents an anti-coordination game, such as the Game of Chicken, with two psNE: ðLobby; Not lobbyÞ and ðNot lobby; LobbyÞ: The next section analyzes the mixed strategy equilibria of this game. Part (b) Firm 1. Let q be the probability that Firm 2 chooses Lobby, and ð1  qÞ the probability that she chooses Not Lobby. The expected profit of Firm 1 playing Lobby (fixing our attention on the top row of the matrix) is EU1 ðLobbyÞ ¼ 5q þ 15ð1  qÞ ¼ 15  20q since Firm 1 obtains a payoff of −5 when Firm 2 also chooses to lobby (which happens with probability q) or a payoff of 15 when Firm 2 does not lobby (which occurs with probability 1 − q). When, instead, firm 1 chooses Not lobby (directing our attention to the bottom row of Fig. 3.7), its expected profit is:

Firm 1 Lobby Not Lobby Fig. 3.7 Lobbying game

Lobby -5,-5 0,15

Firm 2 Not Lobby 15,0 10,10

Exercise 2—Lobbying GameA

95

EU1 ðNotLobbyÞ ¼ 0q þ 10ð1  qÞ ¼ 10  10q given that Firm 1 obtains a payoff of zero when Firm 2 lobbies and a profit of 10 when Firm 2 does not lobby. If Firm 1 is mixing between Lobbying and Not lobbying, it must be indifferent between Lobbying and Not lobbying. Otherwise, it would select the pure strategy that yields the highest expected payoff. Hence, we must have that EU1 ðLobbyÞ ¼ EU1 ðNot LobbyÞ rearranging and solving for probability q yields 15  20q ¼ 10  10q ) q ¼ 1=2 Therefore, for Firm 1 to be indifferent between Lobbying and Not lobbying, it must be that Firm 2 chooses to Lobby with a probability of q ¼ 1=2. Firm 2. A similar argument applies to Firm 2. If Firm 2 chooses to lobby (fixing the right-hand column), it obtains an expected profit of EU2 ðLobbyÞ ¼ 5p þ 15ð1  pÞ since Firm 2 obtains a profit of −5 when Firm 1 also lobbies (which occurs with probability p), but a profit of 15 when Firm 1 does not lobby (which happens with probability 1 − p). When Firm 2, instead selects to Not lobby (in the right-hand column of Fig. 3.7), it obtains an expected profit of EU2 ðNot LobbyÞ ¼ 0p þ 10ð1  pÞ Hence, firm 2, must be indifferent between Lobbying and Not lobbying, that is, EU2 ðLobbyÞ ¼ EU2 ðNot LobbyÞ solving for probability p yields 5p þ 15ð1  pÞ ¼ 0p þ 10ð1  pÞ ) p ¼ 1=2 Hence, Firm 2 is indifferent between Lobbying and Not lobbying as long as Firm 1 selects to Lobby with a probability p ¼ 1=2. Then, in the lobbying game the mixed strategy Nash equilibrium (msNE) prescribes that:     1 1 1 1 Lobby; No Lobby ; Lobby; No Lobby msNE ¼ 2 2 2 2

96

3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Part (c) Figure 3.8 depicts every player’s best response function, and the crossing points of both best response functions identify the two psNE of the game ðp; qÞ ¼ ð0; 1Þ, illustrating (Not lobby, Lobby), ðp; qÞ ¼ ð1; 0Þ corresponds  which  to (Lobby, Not lobby), and the msNE of the game ðp; qÞ ¼ 12 ; 12 found above. In order to understand the construction of Fig. 3.8, let us next analyze each player’s best response function separately. Player 1: For any q [ 1=2, Firm 1 decides to play No Lobbying (in pure strategies), thus implying that p ¼ 0 for any q [ 1=2, as depicted by the vertical segment of BR1 ðqÞ (dashed line) that overlaps the vertical axis (left-hand side of Fig. 3.9). Intuitively, if Firm 2 is likely lobbying, i.e., q [ 1=2, Firm 1 prefers not to lobby. Similarly, for any q\1=2, Firm 2 is rarely lobbying, which induces firm 1 to lobby, and therefore p ¼ 1, as illustrated by the vertical segment of BR1 ðqÞ in the right-hand side of Fig. 3.9. Player 2: For any p [ 1=2, Firm 2 is better off Not lobbying than Lobbying, and hence q ¼ 0 for any p [ 1=2, as BR2 ðpÞ represents in the horizontal segment that overlaps the horizontal axis (see right-hand side of Fig. 3.10). Intuitively, if Firm 1 is likely lobbying, p [ 12, Firm 2 prefers Not to lobby. In contrast, for any p\ 12, Firm 2 is better off Lobbying than Lobbying, what implies that q ¼ 1 for any p below 1/2. Intuitively, Firm 1 is likely not lobbying, making lobbying more attractive for Firm 2. This result is graphically depicted by the horizontal segment of BR2 ðpÞ in the top left-hand side of Fig. 3.10.

q

BR2 ( p )

1

msNE

1/2

BR1 (q )

1/2

1

Fig. 3.8 Lobbying game—best response functions, psNE and msNE

p

Exercise 2—Lobbying GameA Fig. 3.9 Lobbying game— best response function , firm 1

97 q 1

1/2

BR1 ( q )

Fig. 3.10 Lobbying game— best response function, firm 2

1/2

1

p

1/2

1

p

q 1

BR2 ( p )

3/4

Superimposing BR1 ðqÞ, as depicted in Fig. 3.9, and BR2 ð pÞ, as illustrated in Fig. 3.10, we obtain the crossing points of both best response functions, as indicated in Fig. 3.8, which represent the two psNEs and one msNE in the Lobbying game.

Exercise 3—A Variation of the Lobbying GameB Consider a variation of the above lobbying game. As depicted in the payoff matrix of Fig. 3.11, if Firm 1 lobbies but Firm 2 does not, Congress decision yields a profit of x  15 for Firm 1, where x [ 25, and does not yield any profits for Firm 2.

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Firm 1

Lobby No Lobby

Lobby -5,-5 0,15

Firm 2 No Lobby -15,0 10,10

Fig. 3.11 A variation of the lobbying game

Part (a) Find the psNE of the game Part (b) Find the msNE of the game Part (c) Given the msNE you found in part (b), what is the probability that the outcome (Lobby, Not lobby) occurs? Part (d) How does your result in part (c) varies as x increases? Interpret. Answer Part (a) Firm 1’s best response function. Let us first examine Firm 1’s best response function. In particular, we write that BR1 ðLobbyÞ ¼ No Lobby; since when Firm 2 lobbies (in the left-hand column), Firm 1 obtains a larger payoff not lobbying, i.e., zero, than lobbying, −5. When, instead, Firm 2 does not lobby (in the right-hand column) Firm 1 gets a larger payoff lobbying if x − 15 > 10, or x > 25, which holds by definition. Hence, BR1 ðNot LobbyÞ ¼ Lobby: Firm 2’s best response function. Similarly, when Firm 1 chooses to lobby (in the top row), Firm 2’s best response is BR2 ðLobbyÞ ¼ Not Lobby; since 0 [  5; and when Firm 1 selects not to lobby (in the bottom row) Firm 2’s best response becomes BR2 ðNot LobbyÞ ¼ Lobby; since 15 [ 10: Therefore, there are two strategy profiles in which firms play a mutual best response to each other’s strategies: (Lobby, Not Lobby) and (Not Lobby, Lobby), i.e., only one firm lobbies in equilibrium. These are the two psNE of the game. Part (b) (in this part of the exercise, we operate in a similar fashion as in Exercise 2.) If Firm 1 is indifferent between Lobbying and Not lobbying, then it must be that

Exercise 3—A Variation of the Lobbying GameB

99

EU1 ðLobbyÞ ¼ EU1 ðNot LobbyÞ; or  5q þ ðx  15Þð1  qÞ ¼ 0q þ 10ð1  qÞ

Rearranging terms and solving for probability q, yields q ¼ 25x 20x. Similarly, Firm 2 must be indifferent between Lobbying and Not lobbying (otherwise, it would select a pure strategy). Hence, EU2 ðLobbyÞ ¼ EU2 ðNot LobbyÞ; or 5p þ 15ð1  pÞ ¼ 0p þ 10ð1  pÞ: And solving for probability p, we obtain p ¼ 12. (This comes at no surprise since the payoffs of Firm 2 coincide with those in Exercise 2, thus implying that the probability p that makes Firm 2 indifferent between Lobbying and Not lobbying also coincides with that found in Exercise 2, i.e., p ¼ 12. However, that of Firm 1 changed, since the payoffs of Firm 1 have been modified.2 Hence, the msNE is  msNE ¼

   1 1 25  x 25  x Lobby; No Lobby ; Lobby; No Lobby 2 2 20  x 20  x

Part (c) The probability that outcome (Lobby, No Lobby) occurs is given by the probability that Firm 1 lobbies, p, times the probability that Firm 2 does not lobby, 1 − q, i.e., pð1  qÞ. Since we know that p ¼ 12, and q ¼ 25x 20x, then   1 25  x 1 pð 1  qÞ ¼ ; 2 20  x or simplifying,   1 5 pð 1  qÞ ¼ 2 x  20 Part (d) As x increases, the probability of outcome (Lobby, Not Lobby) decreases, i.e.,

Nonetheless, in the specific case in which x = 30, the payoff matrix in Fig. 3.11 coincides with that in Exercise 2, and the probability q that makes Firm 1 indifferent reduces to 5 1 q ¼ 2530 2030 ¼ 10 ¼ 2, as that in Exercise 2.)

2

100

3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Fig. 3.12 Probability of outcome (Lobby, Not Lobby) as a function of x

0.20

0.15

0.10

0.05

p (1− q ) =

40

50

60

70

80

90

1 5 2 x − 20

100

x

@ ½ pð 1  qÞ  5 ¼ \0 @x 2ðx  20Þ2 which is negative given that x [ 25 by definition. Figure 3.12 depicts pð1  qÞ for different values of x (starting with x = 25, since x > 25 by definition), and illustrates that pð1  qÞ decreases in x. In addition, as x becomes larger, the aggregate payoff in outcome (Lobby, Not Lobby), i.e., ðx  15Þ þ 0, exceeds the aggregate payoff from any other strategy profile, thus becoming socially optimal. In particular, when ðx  15Þ þ 0 [ 10 þ 10, i.e., x [ 35, this outcome becomes socially efficient. Intuitively, the profit that Firm 1 obtains when it is the only firm lobbying, x − 15, is so high that aggregate profits become larger than in the case in which no firm lobbies, 10 þ 10. Otherwise, it is socially optimal that no firm lobbies.

Exercise 4—Finding a Mixed Strategy Nash Equilibrium in the Unemployment Benefits GameA Consider the unemployment benefits game from Chap. 2. We found that, when players are restricted to use pure strategies, i.e., to play a strategy with 100% probability, the game does not have a pure strategy Nash equilibrium. Allowing players to randomize, find the mixed strategy Nash equilibrium of this game. Answer Since there are no pure strategy Nash equilibrium, we provide each player with some probabilities (p and q for the Government and Worker respectively) to randomize their behavior, assuming that for a specific value of probability, each option is equivalent in terms of expected utility terms (Fig. 3.13).

Exercise 4—Finding a Mixed Strategy Nash Equilibrium …

101

Worker

Government

Search (S)

Not search (NS)

Benefit (B)

3, 2

-1, 3

No benefit (NB)

-1, 1

0, 0

Fig. 3.13 Normal-form game with probabilities and best responses

Hence, p is the probability that the Government assigns a benefit program (B), and q is the probability that the worker searches for a job (S). Government. Let’s first focus on the row player (Government). If the Government is randomizing between offering and not offering the benefit program, it must be that the Government is indifferent between these two pure strategies, that is, EUG ðBÞ ¼ EUG ðNBÞ ð3Þq þ ð1Þð1  qÞ ¼ ð1Þq þ ð0Þð1  qÞ Rearranging, we find 4q  1 ¼ q and solving for probability q, we obtain 1 q¼ : 5 Then, if the Worker searches for a job with probability q ¼ 1=5, the Government is indifferent between offering unemployment benefits and not offering them. Equivalently, if q [ 1=5, the Government prefers to offer unemployment benefits. In contrast, if q\1=5 the Government prefers not to offer unemployment benefits. We can then summarize the above best response function (BRF) for the Government as follows: 8 0). If his Wife chooses Right, however, his best response is to choose Up (indeed, in the right column of the matrix, 10 > 4). For illustration purposes, the following payoff matrix underlines best response payoff for the Husband and the Wife (Fig. 3.15). Wife’s best responses. For the Wife, her best response is to choose Right if her Husband chooses Up, in the top row of the matrix where 10 > 6; but to respond with Left if her Husband chooses Down, in the bottom of the matrix given that 6 > 0. Using best responses to find psNE. From the above payoff matrix with underlined best response payoffs, we can see that there are two cells where both players’ payoff are underlined. In these cells, players are playing best responses to each other’s strategies. In other words, players are playing mutual best responses, as

Wife Left Husband

Right

Up

0

6

10

10

Down

9

6

4

0

Fig. 3.14 Normal-form game

Wife Left

Husband

Right

Up

0

6

10

10

Down

9

6

4

0

Fig. 3.15 Normal-form game with best responses underlined

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

required by the Nash equilibrium solution concept. These two cells are then the two pure strategy Nash Equilibria (psNE) of the game: fDown; Leftg and fUp; Rightg:

Part (b). First, we assign probability p to the strategy “Up” for player 1 and probability q for the strategy “Left” for player 2 (Fig. 3.16). Husband: The row player (Husband) randomizes when he is indifferent between Up and Down, which occurs when EUH ðU Þ ¼ EUH ðDÞ 0q þ 10ð1  qÞ ¼ 9q þ 4ð1  qÞ Rearranging and solving for probability q, we find 10  10q ¼ 9q þ 4  4q 6 ¼ 15q 2=5 ¼ q Then, his Best Response Function is: 8 2 PlayersB

113

Hence, Player 1 is indifferent between choosing X and Y for values of p that satisfy EU1 ð X Þ ¼ EU1 ðY Þ: That is, pð1  pÞ6 þ 4ð1  pÞ2 ¼ 3p2 þ 4ð1  pÞp þ ð1  pÞ2 and simplifying, 2p2 þ 4p  3 ¼ 0 Solving for p, we find two roots of p: pffiffiffiffiffiffiffi • either p ¼ 1  2:5\0, which cannot be a solution to our problem, since we need p 2 ½0; 1, or pffiffiffiffiffiffiffi • p ¼ 1 þ 2:5 ¼ 0:58, which is the solution to our problem. A similar argument applies to the randomization of players 2 and 3, since their payoffs are symmetric. Hence, every player in this game randomizes between X and Y (using mixed strategies) assigning probability p ¼ 0:58 to strategy X, and 1  p ¼ 0:42 to strategy Y.

Exercise 9—Randomizing Over Three Available ActionsB Consider the following game where Player 1 has three available strategies (Top, Center, Bottom) and Player 2 has also three options (Left, Center, Right) as shown in the normal form game of Fig. 3.26. Find all Nash equilibria (NEs) of this game. Answer Deleting strictly dominated strategies. First, observe that the pure strategy C of player 2 is strictly dominated by strategy R. Indeed, regardless of the pure strategy that player 1 chooses (i.e., independently on the row he selects), player 2’s payoffs is strictly higher with R than with C, that is 3 < 4, 0 < 1 and −5 < 0. Hence, strategy C is never going to be part of a NE in pure or mixed strategies. We can then eliminate strategy C (middle column) for player 2, as depicted in the reduced game in Fig. 3.27. psNE. Underlying best response payoffs, we find that there is no psNE, i.e., there is no cell where the payoff of all players are underlined as best responses. Let’s next analyze msNE.

114

3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Fig. 3.26 A 3  3 payoff matrix

Fig. 3.27 Reduced payoff matrix (after deleting column C)

Fig. 3.28 Assigning probabilities to each pure strategy

msNE. Let us assign probability q to player 2 choosing L and (1 − q) to him selecting R; as depicted in the matrix of Fig. 3.28. Similarly, let p1 represent the probability that player 1 chooses T, p2 the probability he selects C, and 1  p1  p2 the probability of him playing B. Let us next find the expected payoffs that each player obtains from choosing each of his pure strategies, anticipating that his rival uses the above randomization profile. First, for player 1, we obtain: EU1 ðT Þ ¼ 3q þ 1ð1  qÞ ¼ 1 þ 2q EU1 ðC Þ ¼ 1q þ 2ð1  qÞ ¼ 2  q EU1 ðBÞ ¼ 2q þ 2ð1  qÞ ¼ 2

Exercise 9—Randomizing Over Three Available ActionsB

115

And for player 2 we have that: EU2 ðLÞ ¼ 2p1 þ 3p2 þ 2ð1  p1  p2 Þ ¼ 2p1 þ 3p2 þ 2  2p1  2p2 ¼ 2 þ p2 and EU2 ðRÞ ¼ 4p1 þ 1p2 þ 0ð1  p1  p2 Þ ¼ 4p1 þ p2 Hence if player 2 mixes, he must be indifferent between L and R, entailing: 2 þ p2 ¼ 4p1 þ p2 p1 ¼ 1=2 implying that player 1 mixes between C and B with a 50% probability on each. We have, thus, found the mixing strategy of player 1, and we can then turn to player 2 (which entails analyzing the expected payoffs of player 1). We next consider different mixed strategies for player 1, showing that only some of them can be sustained in equilibrium. Mixing between T and C alone. If player 1 mixes between T and C alone (assigning no probability weight to B), then he must be indifferent between T and C, as follows: EU1 ðT Þ ¼ EU1 ðC Þ 1 þ 2q ¼ 2  q; which yields q ¼ 1=3 Hence, player 1 randomizes between T and C, assigning a probability of q ¼ 13 to T and a probability 1  q ¼ 23 to C. Therefore, we found the first msNE of this game, as follows:     1 1 1 2 T; C; 0B ; L; R : mSNE 2 2 3 3 Mixing between T and B alone. If, instead, player 1 mixes between T and B (assigning no probability to C), then his indifference condition must be: EU1 ðT Þ ¼ EU1 ðBÞ 1 þ 2q ¼ 2; which yields q ¼ 1=2 implying that player 2 mixes between L and R with 50% on each. We have then identified a msNE:

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

 msNE ¼

   1 1 1 2 T; C; 0B ; L; R 2 2 3 3

In words, player 1 evenly mixes between T and B, i.e. p1 ¼ 1=2 and p2 ¼ 0, while player 2 evenly mixes between L and R. Mixing between C and B alone. Finally, if player 1 mixes between C and B, it must be that he is indifferent between their expected payoffs EU1 ðC Þ ¼ EU1 ðBÞ 1  q ¼ 2; which yields q ¼ 0 Therefore, player 2 plays R in pure strategies. Player 1 is then indifferent between C and B since both yield a payoff of $2. As a consequence, we found a third msNE which we report as follows:  msNE ¼

  1 1 0T; C; B ; R 2 2

Note that player 1 is not choosing T with probability p1 = 1/2 since this probability only holds when player 2 is indifferent between L and R. In this context, player 2 is choosing a pure strategy (R) while player 1 randomizes. Mixing among all three actions. Note that a msNE in which player 1 randomizes between all three strategies cannot be sustained since for that mixing strategy he would need: EU1 ðT Þ ¼ EU1 ðC Þ ¼ EU1 ðBÞ 1 þ 2q ¼ 2  q ¼ 2 Providing us with two equations 1 þ 2q ¼ 2 and 2  q ¼ 2, which cannot simultaneously hold, i.e., 1 þ 2q ¼ 2 entails q ¼ 1=2 while 2  q ¼ 2 yields q ¼ 0.

Exercise 10—Rock-Paper-Scissors GameB Consider the following matrix representing the Rock-Paper-Scissors game. As you probably remember from your childhood, the rules of the game are the following: Two players simultaneously and independently show their hands choosing one of

Player 1

Rock

Player 2 Paper

Rock

0, 0

-1, 1

1, -1

Paper Scissors

1, -1 -1, 1

0, 0 1, -1

-1, 1 0, 0

Fig. 3.29 Normal-form Rock-Paper-Scissors game

Scissors

Exercise 10—Rock-Paper-Scissors GameB

117

the following three options: a closed fist represents Rock, a full hand represents paper, and a V sign with the index and middle finger represents Scissors. The winner is determined by the well-known rule saying that: Rock beats Scissors, Scissors beat Paper, and Paper beats Rock (Fig. 3.29). (a) Find the pure strategy Nash equilibrium of the game. (b) Find the mixed strategy Nash equilibrium of the game. Answer Part (a). To find the pure strategy NE of the game, we first underline the individual best responses to his opponent’s actions in the following matrix (Fig. 3.30).

As we can see, there is no cell where both players’ payoffs are underlined, indicating that there is no strategy profile that constitutes a pure strategy Nash equilibrium of the game. There is, however, a mixed-strategy Nash equilibrium as we show in the next section. Part (b). Consider that every player i’s strategy is given by Si ¼ ð R i ; Pi ; 1  R i  Pi Þ where Ri denotes the probability that he plays Rock, Pi represents the probability he plays paper, and the remaining probability 1  Ri  Pi is the probability he plays Scissors. Thus, player 1’s strategy is denoted by S1 and player 2’s strategy is denoted by S2 . We now start by making player 1 indifferent between using Rock, Paper and Scissors. Thus, we need u1 ðRock; S2 Þ ¼ u1 ðPaper; S2 Þ ¼ u1 ðScissors; S2 Þ: Let us separately evaluate the expected payoff that player obtains from each of his pure strategies (Rock, Paper, and Scissors) before setting these three expected payoffs equal to each other to make him indifferent. First, if Player 1 chooses Rock (top row of the matrix), he obtains an expected payoff of

Player 1

Rock Paper Scissors

Rock 0, 0 1, -1 -1, 1

Player 2 Paper -1, 1 0, 0 1, -1

Scissors 1, -1 -1, 1 0, 0

Fig. 3.30 Normal-form Rock-Paper-Scissors game with best responses

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

u1 ðRock; S2 Þ ¼ 0R2 þ ð1ÞP2 þ 1ð1  R2  P2 Þ ¼ 1  R2  2P2 To illustrate how to obtain this expected payoff, the following payoff matrix highlights in red color the top row, where it is easier to see that player 1 receives a payoff of 0 when player 2 chooses Rock (which occurs with probability R2 in the left-hand column), a payoff of −1 when player 2 selects Paper (which happens with probability P2 in the center column), and a payoff of 1 when player 2 chooses Scissors (which occurs with probability ð1  R2  P2 Þ in the right-hand column) (Fig. 3.31). Second, if Player 1 chooses Paper (middle row of the matrix), he obtains an expected payoff of u1 ðPaper; S2 Þ ¼ 1R2 þ 0P2 þ ð1Þð1  R2  P2 Þ ¼ 2R2  1 þ P2 To illustrate how to obtain this expected payoff, the following payoff matrix highlights in red color the middle row, where it is easier to see that player 1 receives a payoff of 1 when player 2 chooses Rock (which occurs with probability R2 in the left-hand column), a payoff of 0 when player 2 selects Paper (which happens with probability P2 in the center column), and a payoff of −1 when player 2 chooses Scissors (which occurs with probability ð1  R2  P2 Þ in the right-hand column) (Fig. 3.32). Third, if Player 1 chooses Scissors (bottom row), he obtains the following expected payoff Player 2

Player 1

Rock Paper Scissors

Rock 0, 0 1, -1 -1, 1

Paper -1, 1 0, 0 1, -1

Scissors 1, -1 -1, 1 0, 0

Fig. 3.31 Normal-form Rock-Paper-Scissors game with probabilities

Player 2

Player 1

Rock Paper Scissors

Rock 0, 0 1, -1 -1, 1

Paper -1, 1 0, 0 1, -1

Scissors 1, -1 -1, 1 0, 0

Fig. 3.32 Normal-form Rock-Paper-Scissors game with probabilities

Exercise 10—Rock-Paper-Scissors GameB

119

u1 ðScissors; S2 Þ ¼ ð1ÞR2 þ 1P2 þ 0ð1  R2  P2 Þ ¼ R2 þ P2 To illustrate how to obtain this expected payoff, the following payoff matrix highlights in red color the bottom row, where it is easier to see that player 1 receives a payoff of −1 when player 2 chooses Rock (which occurs with probability R2 in the left-hand column), a payoff of 1 when player 2 selects Paper (which happens with probability P2 in the center column), and a payoff of 0 when player 2 chooses Scissors (which occurs with probability ð1  R2  P2 Þ in the right-hand column) (Fig. 3.33). We can now equate our three expected payoffs found above, u1 ðRock; S2 Þ, u1 ðPaper; S2 Þ and u1 ðScissors; S2 Þ. First, equating u1 ðPaper; S2 Þ and u1 ðScissors; S2 Þ, we get, 2R2  1 þ P2 ¼ R2 þ P2 which yields, R2 ¼

1 3

Similarly, equating u1 ðRock; S2 Þ and u1 ðScissors; S2 Þ, we get, 1  R2  2P2 ¼ R2 þ P2 which yields, P2 ¼

1 3

Finally, since 1  R2  P2 ¼ 1  13  13 ¼ 13. Therefore, the strategy that player 2 chooses in the msNE of the game is S2

 ¼

 1 1 1 ; ; ; 3 3 3

which indicates that he puts the same probability weight on each column. Remember 1 1 1that  an alternative interpretation of mixed equilibrium strategies is that  S2 ¼ 3 ; 3 ; 3 is the exact randomization that makes player 1 indifferent between Player 2

Player 1

Rock Paper Scissors

Rock 0, 0 1, -1 -1, 1

Paper -1, 1 0, 0 1, -1

Scissors 1, -1 -1, 1 0, 0

Fig. 3.33 Normal-form Rock-Paper-Scissors game with probabilities

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

playing Rock, Paper or Scissors. Otherwise, he would be playing a pure strategy (assigning 100% to one of his rows) without the need to randomize. By symmetry, the analysis of player 1 yields the same result so that  S1 ¼ 13 ; 13 ; 13 . Intuitively, player 1 assigns the same probability weight to each of his rows.

Exercise 11—Penalty Kicks GameB Consider a kicker and a goalie in a penalty kicks round, which will determine the winner of a Soccer World Cup. The following payoff matrix represents the kicker (in rows), who must choose whether to aim Left, Center, or Right; and the goalie (in columns) who must dive to the kicker’s Left, Center, or Right. The payoffs in each cell represent the probability that the kicker scores. For instance, if the kicker shoots to the right and the goalkeeper guesses the correct way (i.e., he dives to the right), the probability that the kicker will score is 0.65 while the probability that the goalkeeper will save is 0.35 (Fig. 3.34). (a) Show that the game has no pure strategy Nash equilibrium. (b) Find the mixed strategy Nash equilibrium of this game. (c) What is the probability that a goal is scored? Answer Part (a). We check for the PSNE by underlining the best responses for each action for each player (Fig. 3.35):

As we can see, there is no cell wherein two best responses match each other. This means there is always incentive for 1 player to deviate from any given outcome. Thus, there is no PSNE in this game. Part (b). We now search for a mixed strategy Nash equilibrium in which the Kicker and the Goalie randomize. Starting with the Kicker (row player), we need to make him indifferent between aiming to the Left, Center, and Right. Before we set his

Kicker

Left Center Right

Left 0.65, 0.35 0.95, 0.05 0.95, 0.05

Fig. 3.34 Normal-form penalty kicks game

Goalie Center 0.95, 0.05 0, 1 0.95, 0.05

Right 0.95, 0.05 0.95, 0.05 0.65, 0.35

Exercise 11—Penalty Kicks GameB

Kicker

Left Center Right

121

Left 0.65, 0.35 0.95, 0.05 0.95, 0.05

Goalie Center 0.95, 0.05 0, 1 0.95, 0.05

Right 0.95, 0.05 0.95, 0.05 0.65, 0.35

Fig. 3.35 Normal-form penalty kicks game with best responses

Goalie

Kicker

Left Center Right

Left

Center

Right

0.65, 0.35 0.95, 0.05 0.95, 0.05

0.95, 0.05 0, 1 0.95, 0.05

0.95, 0.05 0.95, 0.05 0.65, 0.35

Fig. 3.36 Normal-form penalty kicks game with probabilities

indifference conditions, let us find the expected utility he obtains from each of his pure strategies (Left, Center, and Right) (Fig. 3.36). In particular, when the kicker aims Left, his expected payoff becomes uk ðLÞ ¼ gl  0:65 þ gr  0:95 þ ð1  gr  gl Þ  0:95 ¼ 0:95  0:3gl

ð3:1Þ

where gl denotes the probability that the goalie dives Left, gr represents the probability that the goalie dives Right, and 1  gr  gl is the probability of the goalie diving Center. Similarly, the expected utility that the Kicker obtains from aiming Center is uk ðCÞ ¼ gl  0:95 þ gr  0:95 þ ð1  gr  gl Þ  0 ¼ 0:95ðgr þ gl Þ

ð3:2Þ

And the Kicker’s expected utility from aiming Right is uk ðRÞ ¼ gl  0:95 þ gr  0:65 þ ð1  gr  gl Þ  0:95 ¼ 0:95  0:3gr

ð3:3Þ

We know that the kicker has to be indifferent between all his strategies. Thus, we start by making him indifferent between Left and Right, uk ðLÞ ¼ uk ðRÞ, which implies setting expressions (3.1) and (3.3) equal to each other, obtaining

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

gl ¼ gr ¼ g

ð3:4Þ

Substituting this result in expression (3.2), we get 0:95ðg þ gÞ ¼ 1:9g

ð3:5Þ

Using expressions (3.1), (3.4) and (3.5), we find 0:95  0:3g ¼ 1:9g , g ¼

0:95 ¼ 0:43 2:2

Since the probability of kicking Center is gc ¼ 1  gl  gr , we obtain gc ¼ 1  0:43  0:43 ¼ 0:14: Thus, the goalkeeper’s strategy is given by Sg ¼ ð0:43; 0:14; 0:43Þ. A similar argument yields that the kicker’s probabilities are in this mixed strategy Nash equilibrium are Sk ¼ ð0:43; 0:14; 0:43Þ. Part (c). Probability a goal is scored is the probability the goalkeeper dives left and scores plus probability that the goalkeeper dives center and scores plus the probability that the goalkeeper dives right and scores. This is calculated as ½0:43  ð0:43  0:65 þ 0:14  0:95 þ 0:43  0:95Þ þ ½0:14  ð0:43  0:95 þ 0:14  0 þ 0:43  0:95Þ þ ½0:43  ð0:43  0:95 þ 0:14  0:95 þ 0:43  0:65Þ ¼ 0:82044 Thus, the kicker scores with 82% probability.

Exercise 12—Pareto Coordination GameB Consider the Pareto coordination game depicted in Fig. 3.37. Determine the set of psNE and msNE, and depict both players’ best response correspondences. Answer psNE. Let us first analyze the set of psNE of this game. Player 1’s best responses. If Player 2 chooses L (in the left-hand column), then Player 1’s best response is BR1 ðLÞ ¼ U; while if he chooses R (in the right-hand column), Player 1 responds with BR1 ðRÞ ¼ D. Player 2’s best responses. Let us now analyze Player 2’s best responses. If Player 1 chooses U (in the top row), then Player 2’s best response is BR2 ðU Þ ¼ L; while if Player 1 chooses D (in the bottom row), then Player 2 responds with BR2 ðDÞ ¼ R.

Exercise 11—Penalty Kicks GameB

123

Fig. 3.37 Pareto coordination game

Fig. 3.38 Underlining best response payoffs in the Pareto coordination game

The payoff matrix in Fig. 3.38 underlines the payoff associated to every player’s best responses. Since there are two cells in which all players payoffs are underlined (indicating that they are playing mutual best responses), there exist 2 pure strategy Nash equilibria: psNE ¼ fðU; LÞ; ðD; RÞg: msNE. Let us now examine the msNE of the game. Let p denote the probability that Player 1 chooses U and q the probability that Player 2 selects L. For Player 1 to be indifferent between U and D, we need EU1 ðU Þ ¼ EU1 ðDÞ that is, 9q þ 0ð1  qÞ ¼ 8q þ 7ð1  qÞ And solving for q yields q ¼ 78. Similarly, for Player 2 to be indifferent between L and R, we need EU2 ðLÞ ¼ EU2 ðRÞ or 9p þ 0ð1  pÞ ¼ 8p þ 7ð1  pÞ And solving for p, we obtain p ¼ 78. Hence, the msNE of this game is

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

msNE ¼

    7 1 7 1 U; D ; L; R 8 8 8 8

We can now construct the best response correspondences for each player: • For Player 1 we have that if q\ 78, then Player 1’s best response is BR1 ðqÞ ¼ D. Intuitively, in this setting, Player 2 is not likely to select L, but rather R. Hence, player 1’s expected payoff from D is larger than that of U, thus implying p ¼ 0. If, instead, q [ 78, Player 1’s best response becomes BR1 ðqÞ ¼ U, thus entailing p ¼ 1. Finally, if q is exactly q ¼ 78, then Player 1’s best response is to randomize between U and D, given that he is indifferent between both of them. This is illustrated in Fig. 3.39, where BR1 ðqÞ implies p ¼ 0 along the vertical axis of all q\ 78 (in the left-hand side of the figure), becomes flat at exactly q ¼ 78, and turns vertical again (i.e., p ¼ 1 in the right-hand side of the figure) for all q [ 78. • Similarly, for Player 2, we have that: (1) if p\ 78, his best response becomes BR2 ð pÞ ¼ R, thus implying q ¼ 0; (2) if p [ 78, his best response is BR2 ð pÞ ¼ L, thus entailing q ¼ 1; and (3) if p ¼ 78, Player 2 randomizes between U and D. Graphically, BR2 ðpÞ lies at q ¼ 0 along the horizontal axis for all p\ 78 (as illustrated in the left-hand side of Fig. 3.39), becomes vertical at exactly p ¼ 78, and turns horizontal again for all p [ 78 (i.e., q ¼ 1 at the upper right-hand part of the figure). Therefore, best response correspondences BR1 ðqÞ and BR2 ð pÞ intersect at three points; illustrating two psNE

Fig. 3.39 Pareto coordination game—best response functions

Exercise 11—Penalty Kicks GameB

125

ðp; qÞ ¼ ð0; 0Þ ) psNEðD; RÞ ðp; qÞ ¼ ð1; 1Þ ) psNEðU; LÞ and one msNE:  ðp; qÞ ¼

7 7 ; 8 8



    7 1 7 1 U; D ; L; R ) msNE 8 8 8 8

Exercise 13—Mixing Strategies in a Bargaining GameC Consider the sequential-move game in Fig. 3.40. The game tree describes a bargaining game between Player 1 (proposer) and Player 2 (responder). Player 1 makes a take-it-or-leave-it offer to Player 2, specifying an amount s ¼ 0; 2v ; v out of an initial surplus v, i.e., no share of the pie, half of the pie, or all of the pie, respectively. If Player 2 accepts such a distribution, Player 2 receives the offer s, while Player 1 keeps the remaining surplus v  s. If Player 2 rejects, both players get a zero payoff. Part Part Part Part

(a) Describe the strategy space for every player. (b) Provide the normal form representation of this bargaining game. (c) Does any player have strictly dominated pure strategies? (d) Does any player have strictly dominated mixed strategies?

Fig. 3.40 Bargaining game

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Answer Part (a) Strategy set for player 1 n v o S1 ¼ 0; ; v 2 Strategy set for player 2 S2 ¼ fAAA; AAR; ARR; RRR; RRA; RAA; ARA; RARg

For every triplet, the first component specifies Player 2’s response upon observing that Player 1 makes offer s ¼ v (in the left-hand side of Fig. 3.40), the second component is instead his response to an offer s ¼ 2v, and the third component describes Player 2’s response to an offer s ¼ 0 (in the right-hand side of Fig. 3.40). Part (b) Normal form representation: Using the three strategies for Player 1 and the eight available strategies for Player 2 found in part (a), the 3  8 matrix of Fig. 3.41 represents the normal form representation of this game. Part (c) No player has any strictly dominated pure strategy: Player 1. For Player 1, we find that s ¼ 2v yields a weakly (not strictly) higher payoff than s ¼ v, that is  v u1 s ¼ ; s2  u1 ðs ¼ v; s2 Þ for all strategies of Player 2; s2 2 S2 2 (i.e., some columns in the above matrix), which is satisfied with strict equality for some strategies of Player 2, such as ARR, RRR or RRA. Similarly, s ¼ 0 yields a weakly (but not strictly) higher payoff than s ¼ 2v. That is,  v u1 ðs ¼ v; s2 Þ  u1 s ¼ ; s2 for all s2 2 S2 ; 2

Fig. 3.41 Normal-form representation of the Bargaining game

Exercise 13—Mixing Strategies in a Bargaining GameC

127

with strict equality for some s2 2 S2 , such as ARR and RRR. Similarly, s ¼ 2v yields a weakly higher payoff than s ¼ 0 for some strategies of Player 2, such as RRR, but a strictly larger payoff for other strategies, such as RAR. Hence, there is no weakly dominated strategy for Player 1.   Player 2. Similarly, for Player 2, u2 ðs2 ; s1 Þ  u2 s02 ; s1 for any two strategies of Player 2 s2 6¼ s02 and for all s1 2 S1 with strict equality for some s1 2 S1 . Part (d) Once we have shown that there is no strictly dominated pure strategy, we focus on the existence of strictly dominated mixed strategies. We know that Player 1 is never going to mix assigning a strictly positive probability to his pure strategy s ¼ v (i.e., offering the whole pie to Player 2) given that it will reduce for sure his expected payoff, for any strategy with which Player 2 responds. Indeed, such strategy yields a strictly lower (or equal) payoff than other of his available strategies, such as s ¼ 0 or s ¼ 2v. If Player 1 mixes between s ¼ 0 and s ¼ 2v, we can see that he is going to obtain a mixed strategy r1 that yields a expected utility, u1 ðr1 ; s1 Þ, which exceeds his utility from selecting the pure strategy s ¼ v. That is, u1 ðr1 ; s1 Þ  u1 ðs ¼ v; s2 Þ for all s2 2 S2 with strict equality for s2 ¼ ARR and s2 ¼ RRR, but strict inequality (yielding a strictly higher expected payoff) for all other strategies of player 2. We can visually check this result in Fig. 3.41 by noticing that s = v, in the top row, yields a zero payoff for any strategy of player 2. However, a linear combination of strategies s ¼ 2v and s ¼ 0, in the middle and bottom rows, yields a positive expected payoff for columns AAA, AAR, RRA, RAA, ARA and RAR; since all of them contain at least one positive payoff for Player 1 in the middle or bottom row. However, in the remaining columns (ARR and RRR), Player 1’s payoff is zero both in the middle and bottom row; thus implying that his expected payoff, zero, coincides with his payoff from playing the pure strategy s ¼ v in the top row. A similar argument applies to Player 2. Therefore, there doesn’t exist any strictly dominated mixed strategy.

Exercise 14—Depicting the Convex Hull of Nash Equilibrium PayoffsC Consider the two-player normal form game depicted in Fig. 3.42. Part (a) Compute the set of Nash equilibrium payoffs (allowing for both pure and mixed strategies).

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Fig. 3.42 Normal-form game

Part (b) Draw the convex hull of such payoffs. Part (c) Is there any correlated equilibrium that yields payoffs outside this convex hull? Answer Part (a) Set of Nash equilibrium payoffs. psNE. Let us first analyze the set of psNE. Player 1’s best responses. Player 1’s best responses are BR1 ðLÞ ¼ D when player 2 chooses L (in the left-hand column), and BR1 ðRÞ ¼ U when player 2 selects R (in the right-hand column). Player 2’s best responses. Player 2’s best responses are BR2 ðU Þ ¼ R when player 1 chooses U (in the top row) and BR2 ðDÞ ¼ L when player 1 selects D (in the bottom row). Using best responses to find psNEs. Hence, there are two cells in which all players are playing mutual best responses. That is, there are two psNEs, ðU; RÞ; ðD; LÞ; with associated equilibrium payoffs of: ð2; 7Þ for psNEðU; RÞ; and ð7; 2Þ for psNEðD; LÞ: msNE. Let us now examine the msNE of this game. Player 1 randomizes with probability p, as depicted in Fig. 3.43. Hence, Player 2 is indifferent between L and R, if EU2 ðLÞ ¼ EU2 ðRÞ 6p þ 2ð1  pÞ ¼ 7p þ 0ð1  pÞ

Exercise 14—Depicting the Convex Hull …

129

Fig. 3.43 Normal-form game

And solving for probability p, we obtain p ¼ 23. Similarly, Player 2 randomizes with probability q. Therefore, Player 1 is indifferent between his two strategies, U and D, if EU1 ðU Þ ¼ EU1 ðDÞ 6q þ 2ð1  qÞ ¼ 7q þ 0ð1  qÞ And solving for probability q, we find q ¼ 23. Therefore, the msNE is given by: msNE ¼

    2 1 2 1 U; D ; L; R 3 3 3 3

The expected payoff that Player 1 obtains from this symmetric msNE is: EU1 ðp ; q Þ ¼



 2 2 1 1 2 1 38 4 14 6þ  7 þ 2þ  0 ¼ þ ¼ 3 3 3 3 3 3 9 9 3 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Player 2 plays L

Player 2 plays R

And similarly, the expected payoff of Player 2 is



 2 2 1 1 2 1 38 4 14 6þ  7 þ 2þ  0 ¼ þ ¼ EU2 ðp ; q Þ ¼ 3 3 3 3 3 3 9 9 3 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} 



Player 1 plays U

Player 1 plays D

Hence the expected payoffs for playing this msNE are given by EUi ðp ; q Þ ¼ 14=3 for both players i ¼ 1; 2. Depicting best responses. Figure 3.44 depicts the best response for each player. For Player 1, note that: (1) when q\ 23, his best response becomes BR1 ðqÞ ¼ U, thus implying p ¼ 1, as indicated in the vertical segment of BR1 ðqÞ on the right-hand side of the figure; (2) when q ¼ 23, his best response is to randomize between U and D, as illustrated in the horizontal line at exactly q ¼ 23; finally (3) when q [ 23, his best response is BR1 ðqÞ ¼ D, thus entailing p ¼ 0, as depicted

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Fig. 3.44 Best response functions

in the vertical segment of BR1 ðqÞ in the left-hand side of the figure which overlaps with the vertical axis for all q [ 23. A similar analysis applies to the best response function of player 2, BR2 ðpÞ, where BR2 ð pÞ ¼ L when p\ 23, this implying q ¼ 1 (as depicted in the horizontal segment at the top of Fig. 3.25), while his best response becomes BR2 ð pÞ ¼ R when p [ 23, which entails q ¼ 0, as indicated in the horizontal line that coincides with the p-axis for all p [ 23. Part (b) Convex hull of equilibrium payoffs . From our previous analysis, equilibrium payoffs are (2, 7) for the psNE ðU; DÞ, and (7,2) for the psNE ðD; LÞ. Figure 3.45 depicts these two payoffs pairs (see northwest and southeast corners, respectively). In addition, it also illustrates the expected payoff from the msNE, 14 3 ¼ 4:66 for each player, and the convex hull of Nash equilibrium payoffs (shaded area). Part (c) Let us now analyze the correlated equilibrium. Assume that a trusted party tells each player what to do based on the outcome of the following experiment (using a publicly observable random variable, such as a three-sided dice whose outcome all players can observe; as illustrated in Fig. 3.46). Hence, player 1’s expected payoff at the correlated equilibrium is 1 1 1 15  2þ  7þ  6 ¼ ¼5 3 3 3 3

Exercise 14—Depicting the Convex Hull …

131

Fig. 3.45 Convex hull of equilibrium payoffs

Fig. 3.46 Experiment

since player 1 obtains a payoff of 2 when players choose outcome (U, R), a payoff of 7 when they select (D, L), and a payoff of 6 in outcome (U, L). And similarly for player 2’s expected payoff 1 1 1 15  7þ  2þ  6 ¼ ¼5 3 3 3 3 Therefore, as Fig. 3.45 shows, the correlated equilibrium yields a payoff pair (5,5) outside this convex hull of Nash equilibrium payoffs. However, notice that there exists a Nash equilibrium (potentially involving mixed strategies) in which each player obtains a expected payoff of EU1 ¼ EU2 ¼ 4:5, which lies at the southwest frontier of the convex hull. The equation of the line connecting points (2, 7) and (7, 2) is u2 ¼ 9  u1 . To see this, recall that the slope of a line can be found in this context with m¼

27 ¼ 1; 72

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

while the vertical intercept is found by inserting either of the two points on the equation. For instance, using (2, 7) we find that 7 ¼ b  2 which, solving for b, yields the vertical intercept b ¼ 9. It is then easy to check that point (4.5, 4.5) lies on this line since 4:5 ¼ 9  4:5 holds with equality.

Exercise 15— Correlated EquilibriumC Consider the two games depicted in Fig. 3.47, in which player 1 chooses rows and player 2 chooses columns: Part (a) In each case, find the set of Nash equilibria (pure and mixed) and depict their equilibrium payoff pairs. Part (b) Characterize one correlated equilibrium where the correlating device is a publicly observable random variable, and illustrate the payoff pairs on the figure you developed in part (a). Part (c) Characterize the set of all correlated equilibria, and depict their payoff pairs. Answer Part (a) Game 1. Let’s first characterize the best response of each player in Game 1. If Player 1 plays U (in the top row), Player 2’s best response is BR2 ðU Þ ¼ L; while if player 1 plays D (in the bottom row), Player 2’s best response is BR2 ðDÞ ¼ R: For Player 1’s best responses, we find that if Player 2 plays L (in the left-hand column), then BR1 ðLÞ ¼ U; whereas if Player 2 plays R (in the right-hand column), BR1 ðRÞ ¼ D. These best responses yield payoffs depicted with the bold underlined numbers in the matrix of Fig. 3.48.

Fig. 3.47 Two normal-form games

Exercise 15—Correlated EquilibriumC

133

Fig. 3.48 Underlining best response payoffs of Game 1

As a consequence, we have two PSNE: fðU; LÞ; ðD; RÞg; with corresponding equilibrium payoffs pairs (2, 1) and (1, 2). In terms of mixed strategies, Player 2 randomizes by choosing a probability q that makes Player 1 indifferent between choosing Up or Down, as depicted in Fig. 3.49. Hence, if Player 1 is indifferent between U and D, we must have that EU1 ðU Þ ¼ EU1 ðDÞ which implies, 2q þ 0  ð1  qÞ ¼ 0q þ 1  ð1  qÞ rearranging, and solving for probability q, yields 2q ¼ 1  q $ q ¼

1 3

Similarly, Player 1 randomizes by choosing a probability p such that makes Player 2 indifferent between left or right, EU2 ðLÞ ¼ EU2 ðRÞ or 1  p þ 0ð1  pÞ ¼ 0p þ 2ð1  pÞ rearranging, and solving for probability p, we obtain p ¼ 2  2p $ p ¼

Fig. 3.49 Searching for msNE in Game 1

2 3

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Hence, the msNE of Game 1 is msNE ¼

    2 1 1 2 U; D ; L; R 3 3 3 3

And the associated expected payoffs in this MSNE are given by,

  2 1  2þ EU1 r1 ; r2 ¼ 3 3

  1 2  1þ EU2 r1 ; r2 ¼ 3 3

 2 0 þ 3  1 0 þ 3

1 1  0þ 3 3

2 2  0þ 3 3

 2 2 2 1 2 4 2 6 2 1 ¼  þ  ¼ þ ¼ ¼ 3 3 3 3 3 9 9 9 3  1 2 2 2 2 4 6 2 2 ¼ þ  ¼ þ ¼ ¼ 3 9 3 3 9 9 9 3

Figure 3.50 depicts the convex hull representing all Nash equilibrium payoffs of game 1. Game 2. Let’s now characterize the best responses of each player in Game 2. If Player 1 plays U (in the top row), then Player 2’s best response becomes BR2 ðU Þ ¼ R; while if player 1 plays D (in the bottom row), Player 2 responds with BR2 ðDÞ ¼ L. For Player 1’s best responses, we find that if Player 2 plays L (in the left-hand column), then BR1 ðLÞ ¼ D; whereas if Player 2 plays R (in the right-hand column), BR1 ðRÞ ¼ U. These best responses yield the payoffs depicted with the bold underlined numbers in the matrix of Fig. 3.51.

Fig. 3.50 Convex hull of equilibrium payoffs (Game 1)

Exercise 15—Correlated EquilibriumC

135

Fig. 3.51 Underlining best response payoffs in Game 2

Hence we have two cells in which all players’ payoffs were underlined, thus indicating mutual best responses in the Nash Equilibrium of the game. This game, therefore, has two psNE: fðU; RÞ; ðD; LÞg; with associated equilibrium payoff pairs of (1, 5) and (5, 1). Let us now analyze the set of msNE in this game. Similarly as in Game 1, Player 2 mixes between L and R with associated probabilities q and 1 − q to make Player 1 indifferent between his two pure strategies (U and D), as follows: EU1 ðU Þ ¼ EU1 ðDÞ or 4q þ 1  ð1  qÞ ¼ 5q þ 0  ð1  qÞ rearranging, and solving for probability q, yields 4q þ 1  q ¼ 5q $ q ¼

1 2

Similarly, Player 1 mixes to make Player 2 indifferent between choosing left or right, EU2 ðLÞ ¼ EU2 ðRÞ or 4p þ 1ð1  pÞ ¼ 5p þ 0ð1  pÞ rearranging, and solving for probability p, we obtain 4p þ 1  p ¼ 5p $ p ¼

1 2

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Therefore, the msNE of game 2 is: msNE ¼

    1 1 1 1 U; D ; L; R 2 2 2 2

And the associated payoffs of this msNE are:

  1 1  4þ EU1 r1 ; r2 ¼ 2 2

  1 1  4þ EU2 r1 ; r2 ¼ 2 2

 1 1 þ 2  1 1 þ 2

1 1 5þ 2 2

1 1 0þ 2 2

 1 1 5 1 5 10 5 0 ¼  þ  ¼ ¼ 2 2 2 2 2 4 2  1 1 5 1 5 10 5 5 ¼  þ  ¼ ¼ 2 2 2 2 2 4 2

Figure 3.52 illustrates the convex hull representing all Nash equilibrium payoffs of Game 2. Part (b) Game 1. First, note that intuitively, by using a correlating device (a publicly observable random variable) we construct convex combinations of the two PSNE of the game. For example, we can determine that the publicly observable random variable is a coin flip, where Heads makes Player 1 play U and Player 2 play L; while Tails make Player 1 play D and Player 2 play R. Hence, the probabilities assigned to every strategy profile for Game 1 are summarized in Fig. 3.53. Given the above probabilities, the expected payoffs for each player in Game 1 are:

Fig. 3.52 Convex hull of equilibrium payoffs (Game 2)

Exercise 15—Correlated EquilibriumC

137

Fig. 3.53 Probabilities assigned to each strategy profile in a correlated equilibrium (Game 1)

1 EU1 ¼  U1 ðU; LÞ þ 2 1 EU2 ¼  U2 ðU; LÞ þ 2

1 1  U1 ðD; RÞ ¼  2 þ 2 2 1 1  U2 ðD; RÞ ¼  1 þ 2 2

1 3 1¼ 2 2 1 3 2¼ 2 2

Graphically, we can see that the correlated equilibrium payoffs are located in the midpoint of the line connecting the two psNE payoffs, i.e., au2 ðU; LÞ þ ð1  aÞu2 ðD; RÞ, as depicted in Fig. 3.54, where a ¼ 1=2 since the correlating device assigns the same probability to Nash equilibria (U, L) and to (D, R). Game 2. The probabilities that the publicly observable random variable assigns to each strategy profile (in pure strategies) in Game 2 are summarized in Fig. 3.55. Yielding expected payoffs for each player of 1  U1 ðU; RÞ þ 2 1 EU2 ¼  U2 ðU; RÞ þ 2

EU1 ¼

1 1  U1 ðD; LÞ ¼  1 þ 2 2 1 1  U2 ðD; LÞ ¼  5 þ 2 2

Fig. 3.54 Nash and correlated equilibrium payoffs (Game 1)

1 6 5¼ ¼3 2 2 1 6 1¼ ¼3 2 2

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Fig. 3.55 Probabilities assigned to each strategy profile in a correlated equilibrium (Game 2)

And the graphical representation on the frontier of the convex hull is (Fig. 3.56). Similarly as for Game 1, correlated equilibrium payoffs lie at the midpoint of the NE payoffs, since the correlating device assigns the same probability to Nash equilibria (U, R) and to (D, L). Part (c) Let us now allow for the correlated equilibrium to assign a more general probability distribution given by p, q, r to strategy profiles (U, L), (U, R) and (D, L), respectively; as depicted in the table of Fig. 3.57. Player 1 Consider that Player 1 receives the order of playing U. Then, his conditional probability of being at cell (U, L) is p þp q, while his conditional probability of being at (U, R) is p þq q. (Note that, in order to find both conditional probabilities, we use the standard formula of Bayes’ rule.) If instead, Player 1 receives the order of playing D, then his conditional probr ability of being at cell (D, L) is 1pq , while his conditional probability of being at (D, R) is

1pqr 3 1pq .

Player 2 If Player 2 receives the order of playing L we have that his conditional probability of being at cell (U, L) is p þp r, while his conditional probability of being at (D, L) is p þr r. If instead, Player 2 receives the order of playing R, his conditional probability of q being at cell (U, R) is 1pr , and his conditional probability of being at (D, R) is 1pqr 4 1pr .

The denominator is given by the probability of outcomes (D, L) and (D, R), i.e., r þ ð1  p  q  r Þ ¼ 1  p  q; while the numerator reflects, respectively, the probability of outcome (D, L), r, or outcome (D, R), 1  p  q  r; as depicted in the matrix of Fig. 3.54. 4 Similarly as for Player 1, the denominator represents the probability of outcomes (U, R) and (D, R), i.e., q þ ð1  p  q  r Þ ¼ 1  p  r; while the numerator indicates, respectively, the probability of outcome (U, R), q, or (D, R), 1  q  p  r; as illustrated in the matrix of Fig. 3.54. 3

Exercise 15—Correlated EquilibriumC

139

Fig. 3.56 Correlated equilibrium payoffs (Game 2)

Fig. 3.57 Probabilities assigned to each strategy profile

We know that in a correlated equilibrium the players must prefer to stick to the rule that tells them how to play (i.e., there must be no profitable deviations from this rule), which we can represent by the following incentive compatibility conditions for Player 1 EU1 ðUjhe is told U Þ  EU1 ðDjhe is told U Þ EU1 ðDjhe is told DÞ  EU1 ðUjhe is told DÞ and for Player 2, EU2 ðLjhe is told LÞ  EU2 ðRjhe is told LÞ EU2 ðRjhe is told RÞ  EU2 ðLjhe is told RÞ Let’s now apply these conditions, and the conditional probabilities we found above, to each of the games:

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Game 1 1. EU1 ðUjhe is told U Þ  EU1 ðDjhe is told U Þ , p þ1 q  2p  p þ1 q  q, which simplifies to 2p  q. 1 2. EU1 ðDjhe is told DÞ  EU1 ðUjhe is told DÞ ⟺ 1pq  ð1  p  q  r Þ  1 1pq

 2r, which reduces to 1  p  q  3r

3. EU2 ðLjhe is told LÞ  EU2 ðRjhe is told LÞ, p þ1 r  p  p þ1 r  2r, which can be rearranged as p  2r. 2 4. EU2 ðRjhe is told RÞ  EU2 ðLjhe is told RÞ, 1pr  ð1  p  q  r Þ  1 1pr

 q, which simplifies to 1  p  r  32 q

Summarizing, the set of correlated equilibria for Game 1 can be sustained for any probability distribution (p, q, r) that satisfies: 2p  q and 1  p  q  3r for Player 1; and p  2r and 1  p  r  32 q for Player 2: As a remark, note that the numerical example we analyzed in the previous part of the exercise (where q = 0, r = 0, p = 1/2 and 1 − p − q − r = 1/2) satisfies these conditions. In particular, for Player 1, condition 2p  q reduces to 1  0, while condition 1  p  q  3r simplifies to ½  0 in this case. Similarly, for Player 2, condition p  2r reduces to 1  0, whereas condition 1  p  r  32 q simplifies to ½  0 in this numerical example. Game 2. Applying the same methodology to Game 2 yields: 1. EU1 ðUjhe is told U Þ  EU1 ðDjhe is told U Þ , p þ1 q  ð4p þ qÞ  p þ1 q  5q, which simplifies to q  p 1 1 2. EU1 ðDjhe is told DÞ  EU1 ðUjhe is told U Þ , 1pq  5r  1pq  ð4r þ 1  p  q  rÞ, which can be rearranged to 2r  1  p  q 3. EU2 ðLjhe is told LÞ  EU2 ðRjhe is told LÞ , p þ1 r  ð4p þ r Þ  p þ1 r  5p, which reduces to r  p 1 1 4. EU2 ðRjhe is told RÞ  EU2 ðLjhe is told RÞ , 1pr  5q  1pr  ð4q þ 1 p  q  rÞ, which simplifies to 2q  1  p  r. Summarizing, the set of all correlated equilibrium in Game 2 are given by probability distributions (p, q, r) that satisfy the following four constraints: q  p and 2r  1  p  q for Player 1; and r  p and 2q  1  p  r for Player 2: By definition, these probability distribution over the four outcomes of these 2  2 games are defining a polytope in D3 (the equivalent of a simplex in D2 but

Exercise 15—Correlated EquilibriumC

141

depicted in 3 dimensions) that will have associated a set of EUi which will be a superset of the convex hull of NE payoffs.

Exercise 16—Relationship Between Nash and Correlated Equilibrium PayoffsC Consider the relationship between Nash and correlated equilibrium payoffs. Can there be a correlated equilibrium where both players receive a lower payoff than their lowest symmetric Nash equilibrium payoff? Explain or give an example. Answer Yes, there is such a correlated equilibrium. In particular, it is easy to find a correlated equilibrium in an anti-coordination game (Fig. 3.58) in which both players obtain a lower payoff than in the lowest Nash equilibrium payoff of the convex hull (where each player obtains EU1 ¼ EU2 ¼ 4:5 in the MSNE of the game). Consider a correlated equilibrium with the probability distribution summarized in Fig. 3.59. Let us now check that no player has incentives to unilaterally deviate from this correlated equilibrium. If Player 1 is told to play U, it must be that the outcome arising from the above probability distribution in the correlated equilibrium is ðU; RÞ, since ðU; LÞ does not receive a positive probability. In this setting, Player 1’s expected utility from selecting U is 2, while that from unilaterally deviating towards D is only zero. Hence, Player 1 does not have incentives to deviate. Similarly, if Player 1 is told to play D, then he does not know whether the realization of the above probability distribution is outcome ðD; LÞ or ðD; RÞ. His expected payoff from agreeing to select D (in the bottom row of the payoff matrix) is EU1 ðDÞ ¼ 2 5

Fig. 3.58 Payoff matrix

2 5

þ

1 5

7þ

1 5 2 5

þ

1 5

2  0 ¼  7 ¼ 4:66 3

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Fig. 3.59 Probabilities assigned to each strategy profile

(Note that the first ratio identifies the probability of outcome ðD; LÞ, conditional on D occurring, i.e., a straightforward application of Bayes’ rule. Similarly, the second term identifies the conditional probability of outcome ðD; RÞ, given that D occurs.). If, instead, Player 1 deviates to U, his expected utility becomes EU1 ðU Þ ¼ 2 5

2 5

þ

 6þ 1 5

1 5 2 5

þ

1 5

2¼

2 1  6 þ  2 ¼ 4:66 3 3

Therefore, Player 1 does not have strict incentives to deviate in this setting either. By symmetry, we can conclude that player 2 does not have incentives to deviate from the correlated equilibrium. Let us now find the expected payoff in this correlated equilibrium P1 playsD

P1 playsU

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 2 3

z}|{ 7 2 36 2 1 EU1 ¼ 2 þ 6 7 þ 0 7 4 5 ¼ 3:6 5 5 |{z} 3 3 |{z} P2 playsL

P2 playsR

And similarly for player 2, P2 playsR

P2 playsL

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl3{ 2

z}|{ 7 2 36 2 1 7 þ 0 7 EU2 ¼  2 þ 6 4 5 ¼ 3:6 5 5 |{z} 3 3 |{z} P1 playsU

P1 playsD

Figure 3.60 depicts the payoff pair (3.6, 3.6) arising in this correlated equilibrium, and compares it with the NE payoff pairs (as illustrated in the shaded convex hull which we obtained in Exercise 8). As required, our example illustrates that players obtain a lower payoff in the correlated equilibrium, 3.6, than in any of the symmetric Nash equilibria of the game (where the lowest payoff both players can obtain is 4.5.)

Exercise 17—Identifying Strictly Competitive GamesA

143

Fig. 3.60 Nash and Correlated equilibrium payoffs

Exercise 17—Identifying Strictly Competitive GamesA For the following games, determine which of them satisfy the definition of strictly competitive games: Part (a) Matching Pennies (Anti-coordination game), Part (b) Prisoner’s Dilemma Part (c) Battle of the Sexes (Coordination game). Answer Recall the definition of strictly competitive games as those strategic situations involving players with completely opposite interests/incentives. We provide a more formal definition below. Strictly Competitive game (Definition): A two player, strictly competitive game requires that, for every two strategy profiles s and s′, if player i prefers s then his opponent, player j, must prefer s′. That is, If ui ðsÞ [ ui ðs0 Þ then uj ðsÞ [ uj ðs0 Þ for all i ¼ f1; 2g and j 6¼ i

144

3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Intuitively, if player i is better off when the outcome of the game changes from s′ to s, ui ðsÞ [ ui ðs0 Þ, player j must be worse off from such a change; a result that holds for any two strategy profiles s and s′ we compare. Alternatively, a game is not strictly competitive if we can identify at least two strategy profiles s and s′ for which players’ incentives are aliened, i.e., both players prefer s to s′, or both prefer s′ to s. Part (a) Matching Pennies game. Consider the matching pennies game depicted in Fig. 3.61, where H stands for Heads and T denotes Tails. To establish if this is a strictly competitive game, we must compare the payoff for player 1 under every pair of strategy profiles, and that of player 2 under the same two strategy profiles. When comparing strategy profiles (H, H) and (H, T), i.e., the cells in the top row of the matrix, we find that player 1’s payoffs satisfy u1 ðH; H Þ ¼ 1 [  1 ¼ u1 ðH; T Þ while that of player 2 exhibits the opposite ranking, u2 ðH; H Þ ¼ 1\1 ¼ u2 ðH; T Þ: When comparing strategy profiles (H, H) and (T, T) on the main diagonal of the matrix, we find that player 1 is indifferent between both strategy profiles, i.e., u1 ðH; H Þ ¼ 1 ¼ 1 ¼ u1 ðT; T Þ and so is player 2 since u2 ðH; H Þ ¼ 1 ¼ 1 ¼ u2 ðT; T Þ An analogous result applies for the comparison of strategy profiles (H, T) and (T, H), where player 1 is indifferent since u1 ðH; T Þ ¼ 1 ¼ 1 ¼ u1 ðT; H Þ

Fig. 3.61 Matching pennies game

Exercise 17—Identifying Strictly Competitive GamesA

145

and so is player 2 given that u2 ðH; T Þ ¼ 1 ¼ 1 ¼ u2 ðT; H Þ: Finally, when comparing (H, T) and (T, T), in the right-hand column of the matrix, we obtain that player 1 prefers (T, T) since u1 ðH; T Þ ¼ 1\1 ¼ u1 ðT; T Þ while player 2 prefers (H, T) given that u2 ðH; T Þ ¼ 1 [  1 ¼ u2 ðT; T Þ   Hence, for any two strategy profiles s ¼ ðs1 ; s2 Þ and s0 ¼ s01 ; s02 , we have shown that if u1 ðsÞ [ u1 ðs0 Þ then u2 ðs0 Þ [ u2 ðsÞ, or if u1 ðsÞ ¼ u1 ðs0 Þ then u2 ðs0 Þ ¼ u2 ðsÞ. Therefore, the Matching pennies game is strictly competitive. Part (b) Prisoner’s Dilemma game. Consider the Prisoner’s Dilemma game in Fig. 3.62, where C stands for Confess and NC denotes Not confess. For this game we also start comparing the payoffs that player 1 and player 2 obtain at every pair of strategy profiles. When comparing (C, NC) and (NC, C), we find that player 1 prefers the former since, u1 ðC; NC Þ ¼ 0 [  10\0 ¼ u1 ðNC; C Þ: while player 2 prefer the latter because u2 ðC; NC Þ ¼ 10\0 ¼ u2 ðNC; CÞ Similarly, when comparing (C, C) and (C, NC), in the left-hand column of the matrix, we find that player 1 prefers the latter given that u1 ðC; CÞ ¼ 5\0 ¼ u1 ðC; NC Þ

Fig. 3.62 Prisoner’s dilemma game

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

while player 2 prefer the former, i.e., u2 ðC; CÞ ¼ 5 [  10 ¼ u2 ðC; NC Þ: In all our comparisons of strategy profiles thus far, we have obtained that strategy profiles that are favored by one player are opposed by the other player, as required in strictly competitive games. However, when comparing (C, C) and (NC, NC), on the main diagonal of the matrix, we find that player’s preferences become aligned, since both players prefer (NC, NC) to (C, C). Indeed, player 1 finds that u1 ðNC; NC Þ ¼ 1 [  5 ¼ u1 ðC; CÞ: Similarly, player 2 also prefers (NC, NC) over (C,C) since u2 ðNC; NC Þ ¼ 1 [  5 ¼ u2 ðC; CÞ: Hence, this game is not strictly competitive because we could find a pair of strategy profiles, namely (C, C) and (NC, NC), for which both players are better off at (NC, NC) than at (C, C). That is, if s ¼ ðNC; NC Þ and s0 ¼ ðC; CÞ, then we obtain that u1 ðsÞ\u1 ðs0 Þ and u2 ðsÞ\u2 ðs0 Þ, which violates the definition of strictly competitive games. Note: As long as we can show that there are two strategy profiles for which the definition of strictly competitive games does not hold, i.e., for which players’ preferences are aligned, then we can claim that the game is not strictly competitive, without having to continue checking whether player’s preferences are aligned or misaligned over all remaining pairs of strategy profiles. Part (c) Battle of the sexes game. The payoff matrix in Fig. 3.63 describes the Battle of the Sexes game, where F denotes going to the Football game, while O represents going to the Opera. In this game, we can easily find a pair of strategy profiles for which the definition of strictly competitive games does not hold. In particular, for strategy profiles (O, F) and (F, F), both of them in the left-hand column of the matrix, we have that both the husband (row player) and the wife (column player) prefer strategy profile (F, F) over (O, F). In particular, the husband finds that

Fig. 3.63 Battle of the Sexes game

Exercise 17—Identifying Strictly Competitive GamesA

147

uH ðO; F Þ ¼ 0\3 ¼ uH ðF; FÞ and similarly for the wife, since uW ðO; F Þ ¼ 0\1 ¼ uW ðF; F Þ: Intuitively, the husband prefers that both players attend his preferred choice (football game), than being alone at the opera. Similarly, the wife prefers to attend to an event she dislikes (football game) with her husband, than being alone at the football game, as described in strategy profile (O, F). Hence, we found a pair of strategy profiles for which players’ preferences are aligned, implying that the battle of the sexes game is not a strictly competitive game.

Exercise 18—Maxmin StrategiesC Consider the game in Fig. 3.64, where player 1 chooses Top or Bottom and player 2 selects Left or Right. Part (a) Find every player’s maxmin strategy. Draw every player’s expected utility, for a given strategy of his opponent. Part (b) What is every player’s expected payoff from playing her maxmin strategy? Part (c) Find every player’s Nash equilibrium strategy, both using pure strategies (psNE) and using mixed strategies (msNE). Part (d) What is every player’s expected payoff from playing her Nash equilibrium strategy? Part (e) Compare player’s payoff when they play maxmin and Nash equilibrium strategies (from parts (b) and (d), respectively). Which is higher? Answer Part (a) Let p denote the probability with which player 1 plays Top, and thus 1  p represent the probability that he plays Bottom. Similarly, q and ð1  qÞ denote the probability that player 2 plays Left (Right, respectively); as depicted in Fig. 3.65.

Fig. 3.64 Normal-form game

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Fig. 3.65 Normal-form game with assigned probabilities

Player 1. Let us first analyze player 1. When Player 2 chooses Left (fixing our attention on the left-hand column) Player 1’s expected utility from randomizing between Top and Bottom with probabilities p and 1  p, respectively, is: EU1 ð pjLeftÞ ¼ 6p þ 3ð1  pÞ ¼ 6p þ 3  3p ¼ 3p þ 3 Figure 3.66 depicts this expected utility, with a vertical intercept at 3, and a positive slope of 3, which implies that, at p ¼ 1, EU1 becomes 6. And when Player 2 chooses Right (in the right-hand column) Player 1’s expected utility becomes EU1 ð pjRightÞ ¼ 0p þ 6ð1  pÞ ¼ 6  6p

Fig. 3.66 Expected utilities for player 1

Exercise 18—Maxmin StrategiesC

149

which, as Fig. 3.66 illustrates, originates at 6 and decreases in p, crossing the horizontal axis when p ¼ 1. In particular, EU1 ð pjRightÞ and EU1 ð pjLeftÞ cross at 3q þ 3 ¼ 6  6p , p ¼ 1=3 Recall the intuition behind maxmin strategies: player 1 first anticipates that his opponent, player 2, seeks to minimize player 1’s payoff (since by doing so, player 2 increases his own payoff in a strictly competitive game). Graphically, this implies that player 2 chooses Left for all p  13 and Right for all p [ 13, yielding a lower envelope of expected payoffs for player 1 represented by equation 3 þ 3p from p ¼ 0 to p ¼ 13, and by equation 6  6p, for all p [ 13; as depicted in Fig. 3.67. Taking such lower envelope of expected payoffs as given (i.e., assuming that Player 2 will try to minimize Player 1’s payoffs), Player 1 chooses the probability p that maximizes his expected payoff, which occurs at the point where EU1 ðpjLeftÞ crosses EU1 ðpjRightÞ. This point graphically represents the highest point of the lower envelope, as Fig. 3.67 depicts, i.e., his maxmin strategy, as it maximizes the minimum of his payoffs. Player 2. Let us now examine Player 2 using a similar procedure. When Player 1 chooses Top (in the top row), Player 2’s expected utility becomes EU2 ðqjTopÞ ¼ 0q þ 6ð1  qÞ ¼ 6  6q which originates at 6 and decreases in q, crossing the horizontal axis when q = 1; as Fig. 3.68 illustrates. And when Player 1 chooses Bottom (in the bottom row), Player 2’s expected utility is

Fig. 3.67 Lower envelope of expected payoffs (player 1)

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Fig. 3.68 Expected utilities of player 2

EU2 ðqjBottomÞ ¼ 2q þ 0ð1  qÞ ¼ 2q which originates at (0, 0) and increases with a positive slope of 2, thus reaching a height of 2 when q = 1; also depicted in Fig. 3.68. Hence, the expected utilities EU2 ðqjTopÞ and EU2 ðqjBottomÞ cross at: 6  6q ¼ 2q , q ¼ 3=4 A similar intuition as for player 1 applies to Player 2: considering that Player 1 seeks to minimize Player 2’s payoff in this strictly competitive game, Player 2 anticipates a lower envelope of expected payoffs given by 2q from q ¼ 0 to q ¼ 34, and by 6  6q for all q [ 34; as depicted in Fig. 3.69. Hence, Player 2 chooses the value of q that maximizes his expected payoff, which occurs at the point where EU2 ðqjTopÞ crosses EU2 ðqjBottomÞ, i.e., q ¼ 34 at his maxmin strategy. Summarizing, the maxmin strategy profile is: 

 1 2 Top; Bottom ; and 3 3   3 1 Left; Right Player 2 : 4 4 Player 1 :

Exercise 18—Maxmin StrategiesC

151

Fig. 3.69 Lower envelope of expected payoffs (player 2)

Part (b) In order to find player 1’s expected utility in the maxmin strategy profile, we can use either EU1 ð pjLeftÞ or EU1 ð pjRightÞ, since they both reach the some height at p ¼ 13, given that they cross at exactly this point.     1 1 EU1 p ¼ jLeft ¼ 3 þ 3p ¼ 3 þ 3 ¼ 3þ1 ¼ 4 3 3

And similarly for the expected utility of Player 2 in this maximin strategy profile     3 3 EU2 q ¼ jTop ¼ 6  6q ¼ 6  6 ¼ 1:5 4 4 Part (c) psNE. Let us first check for psNEs in this game. For player 1, we can immediately identify that his best responses are BR1 ðLÞ ¼ T and BR1 ðRÞ ¼ B, where he does not have a dominant strategy. The payoffs that player 1 obtains when selecting the above best responses are underlined in the payoff matrix of Fig. 3.70 in the (Top, Left) and (Bottom, Right) cells. Similarly, for player 2, we find that his best responses are BR2 ðT Þ ¼ R and BR2 ðBÞ ¼ L, indicated by the underlined payoff in cells (Top, Right) and (Bottom, Left). There does not exists a mutual best response by both players, i.e., in the payoff matrix there is no cell with the payoffs of all players being underlined. Hence, there is no psNE (Fig. 3.70).

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3 Mixed Strategies, Strictly Competitive Games, and Correlated Equilibria

Fig. 3.70 Normal-form game

msNE. Let us now search for msNE (recall that such equilibrium must exist given that we could not find any psNE). Player 1 must be indifferent between Top and Bottom; otherwise, he would be selecting one of these strategies in pure strategies. Hence, player 1 must face EU1 ðTopÞ ¼ EU1 ðBottomÞ, and the value of probability q that makes player 1 indifferent between T and B is EU1 ðTopÞ ¼ EU1 ðBottomÞ 6q ¼ 3q þ 6ð1  qÞ , q ¼ 2=3 Similarly, player 2 must be indifferent between Left and Right, since otherwise he would choose one of them in pure strategies. Therefore, the value of probability p that makes player 2 indifferent between L and R is EU2 ðLÞ ¼ EU2 ðRÞ 2ð1  pÞ ¼ 6p , p ¼ 1=4 Hence, the msNE of this game is     1 3 2 1 Top; Bottom ; Left; Right 4 4 3 3

Part (d) Let us now find the expected utility that player 1 obtains in the previous msNE: EU1 ¼ 6pq þ 0pð1  qÞ þ 3qð1  pÞ þ 6ð1  qÞð1  pÞ          1 2 2 3 1 3 3 1 þ3 þ6 ¼ 1þ þ ¼ 4 ¼6 4 3 3 4 3 4 2 2

Exercise 18—Maxmin StrategiesC

153

And similarly for player 2, his expected utility in the msNE becomes EU2 ¼ 0pq þ 6pð1  qÞ þ 2qð1  pÞ þ 0ð1  pÞð1  qÞ       1 1 2 3 þ2 ¼ 1:5 ¼6 4 3 3 4 Part (e) Player 1’s expected utility form playing the msNE of the game, 4, coincides with that from playing his maxmin strategy, 4. A similar argument applies to Player 2, who obtains an expected utility of 1.5 under both strategies.

4

Sequential-Move Games with Complete Information

Introduction In this chapter we move from simultaneous-move to sequential-move games, and describe how to solve these games by using backward induction, which yields the set of Subgame Perfect Nash equilibria (SPNE). Intuitively, every player anticipates the optimal actions that players acting in subsequent stages will select, and chooses his actions in the current stage accordingly. SPNE. Every player’s action is “sequentially rational,” meaning that his strategy represents an optimal action for player i at any stage of the game at which he is called on to move, and given the information he has at that point. From a practical standpoint, “sequential rationality” requires every player to put himself in the shoes of players acting in the last stage of the game and predict their optimal behavior on that last stage; then, anticipating the behavior of players acting in the last stage, study the optimal choices of players in the previous to last stage; and similarly for all previous stages until the first move. We first analyze games with two players, and then extend our analysis to settings with more than two players, such as a Stackelberg game with three firms sequentially competing in quantities. We consider several applications from industrial organization as an economic motivation of settings in which players act sequentially. For completeness, we also explore contexts in which players are allowed to choose among more than two strategies, such as bargaining games (in which proposals are a share of the pie, i.e., a percentage from 0 to 100%), or the Stackelberg game of sequential quantity competition (in which each firm chooses an output level from a continuum). In bargaining settings, we analyze equilibrium behavior not only when players are selfish, but also when they exhibit inequity aversion and thus avoid unequal payoff distributions. Finally, we study a moral hazard game in which a manager offers a menu of contracts in order to induce a high effort level from his employee. © Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_4

155

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Exercise 1—Ultimatum Bargaining GameB In the ultimatum bargaining game, a proposer is given a pie, normalized to a size $1, and he is asked to make a monetary offer, x, to the responder who, upon receiving the offer, only has the option to accept or reject it (as if he received an “ultimatum” from the proposer). If the offer x is accepted, then the responder receives it while the proposer keeps the remainder of the pie 1 − x. However, if he rejects it, both players receive a zero payoff. Operating by backward induction, the responder should accept any offer x from the proposer (even if it is low) since the alternative (reject the offer) yields an even lower payoff (zero). Anticipating such as a response, the proposer should then offer one cent (or the smallest monetary amount) to the responder, since by doing so the proposer guarantees acceptance and maximizes his own payoff. Therefore, according to the subgame perfect equilibrium prediction in the ultimatum bargaining game, the proposer should make a tiny offer (one cent or, if possible, an amount approaching zero), and the responder should accept it, since his alternative (reject the offer) would give him a zero payoff. However, in experimental tests of the ultimatum bargaining game, subjects who are assigned the role of proposer rarely make offers close to zero to the subject who plays as a responder. Furthermore, sometimes subjects in the role of the responder reject positive offers, which seems to contradict our equilibrium predictions. In order to explain this dissonance between theory and experiments, many scholars have suggested that players’ utility function is not as selfish as that specified in standard models (where players only care about the monetary payoff they receive). Instead, the utility function should also include social preferences, measured by the difference between the payoff a player obtains and that of his opponent, which gives rise to envy (when the monetary amount he receives is lower than that of his opponent) or guilt (when his monetary payoff is higher than his opponent’s). In particular, suppose that the responder’s utility is given by uR ðx; yÞ ¼ x þ aðx  yÞ; where x is the responder’s monetary payoff, y is the proposer’s monetary payoff, and a is a positive constant. That is, the responder not only cares about how much money he receives, x, but also about the payoff inequality that emerges at the end of the game, a(x − y), which gives rise to either envy, if x < y, or guilt, if x > y. For simplicity, assume that the proposer is selfish, i.e., his utility function only considers his own monetary payoffs uP ðx; yÞ ¼ y, as in the basic model. Part (a) Use a game tree to represent this game in its extensive form, writing the payoffs in terms of m, the monetary offer of the proposer, and parameter a. Part (b) Find the subgame perfect equilibrium. Describe how equilibrium payoffs are affected by changes in parameter a. Part (c) Depict the equilibrium monetary amount that the proposer keeps, and the payoff that the responder receives, as a function of parameter a.

Exercise 1—Ultimatum Bargaining GameB

157

Answer Part (a) First, player 1 offers a division of the pie, m, to player 2, who either accepts or rejects it. However, payoffs are not the same as in the standard ultimatum bargaining game. While the utility of player 1 (proposer) is just the remaining share of the pie that he does not offer to player 2, i.e., 1 − m, the utility of player 2 (responder) is m þ a½m  ð1  mÞ ¼ m þ að2m  1Þ where m is the payoff of the responder, and 1 − m is the payoff of the proposer. We depict this modified ultimatum bargaining game in Fig. 4.1. Part (b) Operating by backward induction, we first focus on the last mover (player 2, the responder). In particular, player 2 accepts any offer m from player 1 such that: m þ að2m  1Þ  0; since the payoff he obtains from rejecting the offer is zero. Solving for m, this implies that player 2 accepts any offer m that satisfies m  1 þa 2a. Anticipating such a response from player 2, player 1 offers the minimal m that generates acceptance, i.e., m  1 þa 2a, since by doing so player 1 can maximize the share of the pie he keeps. This implies that equilibrium payoffs are: 



ð1  m ; m Þ ¼




1þa a ; 1 þ 2a 1 þ 2a



The proposer’s equilibrium payoff is decreasing in a since its derivative with respect to a is @ð1  m Þ 1 ¼ @a ð1 þ 2aÞ2 which is negative for all a > 0. In contrast, the responder’s equilibrium payoff is increasing in a given that its derivative with respect to a is

Fig. 4.1 Ultimatum bargaining game (extensive-form)

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Sequential-Move Games with Complete Information

Fig. 4.2 Equilibrium payoff in ultimatum bargaining game with social preferences

@m 1 ¼ @a ð1 þ 2aÞ2 which is positive for all a [ 0. Part (c) In Fig. 4.2 we represent how the proposer’s equilibrium payoff (in red color) decreases in a, and how that of the responder (in green color) increases in a. Intuitively, when the responder does not care about the proposer’s payoff, a = 0, the proposer makes an equilibrium offer of m ¼ 0, which is accepted by the (selfish) responder, as in the SPNE of the standard ultimatum bargaining game without social preferences. However, when a increases, the minimum offer that the proposer must make to guarantee acceptance, m , increases. Intuitively, the responder will not accept very unequal offers since his concerns for envy are relatively strong. When a ¼ 0:5 the equilibrium split becomes ð1  m  ; m  Þ ¼




1 þ 0:5 0:5 ; 1 þ 2  0:5 1 þ 2  0:5




¼

1:5 0:5 ; 2 2

 ¼ ð0:75; 0:25Þ;

thus indicating that the proposer offers more to the responder as he cares more about the payoff difference, and when the proposer cares the most about the payoff difference, a ¼ 1, the equilibrium split becomes ð2=3; 1=3Þ. However, for all the values of a between zero and one, the proposer’s payoff is higher than that of the responder.

Exercise 2—Entry-Predation GameA Consider a potential entrant deciding whether to enter in an industry with only an incumbent monopolist. The game tree below depicts this scenario. First, the potential entrant chooses whether to enter the industry or stay out. If it stays out, the

Exercise 2—Entry-Predation GameA

159

game is over, the incumbent makes monopoly profits of 10, and the entrant makes no profits (zero payoff). If the entrant chooses to enter the industry, the monopolist responds either accommodating or fighting its entry (Fig. 4.3). (a) Applying backward induction, find the SPNE of this game tree. (b) Transform the above game tree into its matrix form. (c) Use your matrix representation from part (b) to find the Nash equilibria of this game [Hint: There are two of them.]. (d) Compare the SPNE you found in part (a) against the Nash equilibria you found in part (c). Which Nash equilibria is sequentially irrational? Justify your answer. Answer Part (a). To apply backward induction, we start by circling all subgames in the above game tree, as Fig. 4.4 illustrates. The smallest subgame we can find is that initiated after the entrant enters the industry (Subgame 1 in Fig. 4.4), and the next subgame we can identify is the game as a whole. Fig. 4.3 Entry-predation sequential-form game

Accommodate Entry (4, 4)

Incumbent In

Price War

Potential Entrant

(-2, -2)

Out

(0, 10) Payoff for Incumbent (2nd Mover)

Payoff for Entrant (1st Mover)

Fig. 4.4 Subgames of the entry-predation sequential-form game Incumbent In Potential Entrant

Accommodate Entry (4, 4)

Price War

(-2, -2) Subgame 1

Out

(0, 10) Game as a whole

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Subgame 1. Starting our analysis from the smallest subgame at the end of the game tree (Subgame 1), we find that the incumbent prefers to accommodate entry rather than starting a price war since his payoff of doing so satisfies 4 > −2. Figure 4.5 depicts this optimal response to entry by the incumbent, where we shaded the branch corresponding to accommodating entry with a thick red arrow. Game as a whole. We can now move on to study the potential entrant’s strategy: this firm either chooses Out in the bottom part of the game tree, with a payoff of 0, or In in the upper part of the tree, with an associated payoff of 4 since the potential entrant can anticipate how the incumbent will respond to its entry (accommodating it) in the second stage of the game. Figure 4.6 shades the branch of In for the potential entrant with a dotted thick arrow. As a result, the “equilibrium path” (the path of shaded arrows) starts with the potential entrant choosing In followed by the incumbent responding with Accommodate Entry, and equilibrium payoffs are (4, 4). The SPNE of this game (that is, the equilibrium outcome we find after applying backward induction is ðIn; AccommodateÞ . Part (b). Since the potential entrant only has two strategies (In or Out) and the incumbent also has two strategies (Accommodate or Fight), this game’s matrix representation has two rows for the entrant and two columns for the incumbent, as depicted below (Fig. 4.7).

Incumbent In Potential Entrant

Accommodate Entry (4, 4)

Price War

(-2, -2)

Out

(0, 10) Fig. 4.5 Subgame choice in the entry-predation sequential-form game

Incumbent In Potential Entrant

Accommodate Entry (4, 4)

Price War

(-2, -2)

Out

(0, 10) Fig. 4.6 Equilibrium path for the entry-predation sequential-form game

Exercise 2—Entry-Predation GameA

161

Incumbent

Entrant

Accommodate

Fight

In

4, 4

-2, -2

Out

0, 10

0, 10

Fig. 4.7 Entry-predation normal-form game

Part (c). Finding best responses for the entrant. Underlining best response payoffs for the entrant (row player), we obtain that its best response to Accommodation (in the left-hand column) is In since 4 > 0, while its best response to Fight (in the right-hand column) is Out since 0 > −2. Finding best responses for the entrant. For the incumbent (column player), we find that its best response to In (in the top row of the matrix) is Accommodate since 4 > −2 while its best response to Out (in the bottom row) is {Accommodate, Fight} since the incumbent obtains the same payoff (0) accommodating and fighting entry (Fig. 4.8). After underlining best response payoffs, we find that there are two cells with both players’ payoffs were underlined, ðIn; AccommodateÞ and ðOut; FightÞ: These two cells indicate that players are choosing best responses to each other’s strategies, i.e., players are playing mutual best responses. Therefore, these two strategy profiles are the two pure strategy Nash equilibria of the entry-predation game. Part (d). Comparing our answers from parts (a) and (c), the unique SPNE is (In, Accommodate) while we found 2 NEs: (In, Accommodate) and (Out, Fight). While the first NE coincides with the SPNE, the second does not, implying that (Out, Fight) is not sequentially rational. Indeed, it is not sequentially rational for the incumbent to respond fighting the entrant. As shown in part (a), if entry occurs the incumbent optimally responds accommodating such entry. In other words, fighting is an empty threat by the incumbent since the entrant can anticipate that the incumbent will not fight if entry occurs. Incumbent

Entrant

Accommodate

Fight

In

4, 4

-2, -2

Out

0, 10

0, 10

Fig. 4.8 Entry-predation normal-form game with best response payoffs

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Sequential-Move Games with Complete Information

Exercise 3—Sequential Version of the Prisoner’s Dilemma GameA Consider the game tree in Fig. 4.9, representing the sequential-move version of the Prisoner’s dilemma game, where player 1 acts first, and observing its move player 2 responds. (a) Find the best responses of the second mover, for each of the first mover’s actions. (b) Find the first mover’s equilibrium action and describe the subgame perfect Nash equilibrium of the game. (c) Does the equilibrium behavior in the sequential-move game differ from its simultaneous-move version? Answer Part (a). We first circle the subgames in this extensive-form game, where Subgame 1 is the whole game, and Subgames 2 and 3 are initiated after player 1 chooses Confess and Not Confess, respectively (Fig. 4.10). Using blue arrows, we depict player 2’s best responses after observing player 1’s move: • Subgame 2. After player 1 chooses Confess, player 2 responds with Confess since his payoff from doing so (−5, that is, 5 years in jail) is higher than from Not Confess (−15, implying 15 years in jail). Intuitively, player 2 observes that player 1 has already confessed, providing evidence to the police against player 2, ultimately inducing player 2 to confess as well, which saves him 10 additional years in jail. • Subgame 3. After player 1 chooses Not Confess, player 2 responds with Confess since his payoff from doing so (0, that is, no time in jail) is higher than from Not Confess (−1, which means spending one year in jail). Intuitively, player 2 observes Fig. 4.9 Prisoner’s dilemma sequential-move game

Player 1 Confess Player 2 Confess –5 –5

Not Confess

Player 2 Not Confess 0 –15

Confess –15 0

Not Confess –1 –1

Exercise 3—Sequential Version of the Prisoner’s Dilemma GameA

163

Subgame 1 Player 1 Not Confess

Confess Player 2 Confess –5 –5

Subgame 2 Not Confess 0 –15

Player 2

Subgame 3 Not Confess

Confess –15 0

–1 –1

Fig. 4.10 Prisoner’s dilemma sequential-move game with subgames

that player 1 didn’t confessed. In this setting, player 2 is better off being the only prisoner who confesses—and thus leaving the police station immediately—than remaining silent as player 1 did, since by doing so he would serve a year in jail. We can identify that, in summary, player 2 responds confessing in both Subgames 2 and 3, implying that Not Confess is a strictly dominated strategy for player 2 in the sequential-move game (as it is in its simultaneous-move version) (Fig. 4.11). Part (b). By backwards induction, player 1 anticipates that player 2 will respond with Confess regardless of his choice at the first stage of the game, i.e., player 2 chooses Confess in both Subgames 2 and 3. Therefore, the game tree can be depicted as the reduced-form version in Fig. 4.12, where the payoff pairs after player 1 chooses Confess are (−5, −5) since he anticipates player 2 responding with Fig. 4.11 Prisoner’s dilemma sequential-move game with best responses

Player 1 Confess Player 2 Confess –5 –5

Not Confess

Player 2 Not Confess 0 –15

Confess –15 0

Not Confess –1 –1

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Player 1 Confess

–5 –5

Not Confess

–15 0

Fig. 4.12 Reduced prisoner’s dilemma sequential-move game with best responses

Confess in Subgame 2. Similarly, payoff pairs after player 1 selects Not Confess are (−15, 0) since he expects player 2 responding with Confess in Subgame 3. Therefore, player 1 chooses Confess since his payoff from doing so, −5, is higher than from not confessing, −15. In words, since he anticipates that player 2 will respond with Confess regardless what action player 1 chooses in the first stage, player 1 is better off choosing Confess too. As a result, the strategy profile SPNE ¼ fConfess; ðConfess; ConfessÞg is the Subgame Perfect Nash Equilibrium (SPNE) of this sequential-move Prisoner’s dilemma game, where the first component of the triplet indicates player 1’s action in equilibrium, the second element represents player 2’s response to player 1 choosing Confess (in Subgame 2), and the third element describes player 2’s response to player 1 selecting Confess (in Subgame 3). Note that the equilibrium path is different from the SPNE of the game, as it would only say {Confess, Confess} which indicates that player 1 chooses to Confess, and player 2 responds to player 1’s confession (in equilibrium) confessing as well. In the SPNE, however, we list not only the action that player 2 chooses in equilibrium (that is, in Subgame 2) but also off-the-equilibrium (in Subgame 3, which should never be reached). Part (c). Equilibrium behavior in the simultaneous-move version of the Prisoner’s Dilemma game (where the Nash equilibrium predicts that both players choose Confess) coincides with that in its sequential-move version (where, in equilibrium, both players choose to Confess). This coincidence occurs because players find Confess to be strictly dominant both in the simultaneous and sequential version of the game, leading both players to choose Confess in all contingencies.

Exercise 4—Sequential Version of the Battle of the Sexes GameA

165

Exercise 4—Sequential Version of the Battle of the Sexes GameA Consider the game tree in Fig. 4.13, representing the sequential-move version of the Battle of the Sexes game, where wife acts first, and observing its move husband responds. (a) Find the husband’s best responses, for each of his wife’s actions in the first stage. (b) Find the wife’s equilibrium action and describe the subgame perfect Nash equilibrium of the game. (c) Does the equilibrium behavior in the sequential-move game differ from its simultaneous-move version? Answer Part (a). We first circle the subgames in this extensive-form game, where Subgame 1 is the whole game, and Subgames 2 and 3 are initiated after the wife chooses Football and Opera, respectively (Fig. 4.14). Using blue arrows, we depict the husband’s best responses after observing his wife’s move: • Subgame 2. After his wife chooses Football, the husband responds with Football since his payoff from doing so (3) is higher than from Opera (0). Intuitively, the husband observes that the wife has already chosen Football, and thus prefers to go the football game with her (that’s his most preferred event!) than going to the opera alone. • Subgame 3. After his wife chooses Opera, the husband responds with Opera since his payoff from doing so (1) is higher than from Football (0). Intuitively, the husband prefers to go to the opera with his wife than being at the football game alone. Recall that, while he likes football better than opera, he prefers to be with Fig. 4.13 Battle of the sexes sequential-move game

Wife Football Husband Football 1 3

Opera

Husband Opera 0 0

Football 0 0

Opera 3 1

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Sequential-Move Games with Complete Information

Subgame 1 Wife Football Husband Football

Opera

Husband

Subgame 2 Opera

Football

Opera

0 0

3 1

0 0

1 3

Subgame 3

Fig. 4.14 Battle of the sexes sequential-move game with subgames

his wife in either event (including the opera) than being alone at either event (including the football game). We can identify that, in summary, the husband responds by choosing the same action as his wife (Fig. 4.15). Part (b). By backwards induction, the wife anticipates that her husband will respond with Football if she goes to watch the Football game and with Opera if she goes to the Opera show. Therefore, the game tree can be depicted as the reduced-form game in Fig. 4.16, where the payoff pairs after the wife chooses Football are (1, 3) since she anticipates that her husband responds with Football in Subgame 2. Similarly, payoff pairs after the wife chooses Opera are (3, 1) since she expects that her husband responds with Opera in Subgame 3. Wife Football Husband Football 1 3

Opera

Husband Opera 0 0

Football 0 0

Opera 3 1

Fig. 4.15 Battle of the sexes sequential-move game with best responses

Exercise 4—Sequential Version of the Battle of the Sexes GameA

167

Wife Football

1 3

Opera

3 1

Fig. 4.16 Reduced battle of the sexes sequential-move game with best responses

Therefore, the wife chooses Opera at the beginning of the game since her payoff from doing so, 3, is higher than from Football, 1. In words, since the wife anticipates that her husband will follow wherever she goes, she is better off attending her most preferred event, Opera, than going to the Football game. As a result, the strategy profile SPNE ¼ fOpera; ðFootball; OperaÞg is the Subgame Perfect Nash Equilibrium (SPNE) of this sequential-move Battle of the Sexes game, where the first component of the triplet indicates the wife’s action in equilibrium, the second element represents the husband’s response to the wife choosing Football (in Subgame 2), and the third element describes the husband’s response to the wife choosing Opera (in Subgame 3). Note that the equilibrium path is different from the SPNE of the game, as it would only say {Opera, Opera} which indicates that the wife chooses Opera, and the husband responds by going to see the Opera show (in equilibrium) as well. In the SPNE, however, we list not only the action that the husband chooses in equilibrium (that is, in Subgame 3) but also off-the-equilibrium (in Subgame 2, which should never be reached). Part (c). Equilibrium behavior in the simultaneous-move version of the Battle of the Sexes game predicts two Nash equilibria in pure strategies (psNE), where both players choose Opera or both choose Football. In the sequential-move version, however, we found that only the former can be sustained in equilibrium. Intuitively, among the two strategy profiles where both players coordinate by attending the same event, the wife can pick the outcome she prefers the most (where both players attend the Opera). In other words, she exploits her first-mover advantage by choosing Opera in the first period, which is then responded by her husband choosing Opera as well. Note that, if the order of play was reversed, with the husband playing first and the wife responding to his actions, the opposite outcome would emerge in equilibrium: the husband would choose Football in the first period, and the wife would respond with Football after Football and with Opera after Opera. As a consequence, both players attend the Football game in equilibrium when the husband acts as the first mover.

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Exercise 5—Sequential Version of the Pareto Coordination GameA Consider the game tree in Fig. 4.17, representing the sequential-move version of Pareto Coordination game, where firm 1 acts first (e.g., industry leader), and observing its move firm 2 responds (e.g., industry follower). (a) Find the best responses of the second mover, for each of the first mover’s actions. (b) Find the first mover’s equilibrium action and describe the subgame perfect Nash equilibrium of the game. (c) Does the equilibrium behavior in the sequential-move game differ from its simultaneous-move version? Answer Part (a). We first circle the subgames in this extensive-form game, where Subgame 1 is the whole game, and Subgames 2 and 3 are initiated after Firm 1 chooses Technology A and Technology B, respectively (Fig. 4.18). Using blue arrows, we depict Firm 2’s best responses after observing Firm 1’s move: • Subgame 2. After Firm 1 chooses Technology A, Firm 2 responds with Technology A since its payoff from doing so (2, that is, profit of 2 million dollars) is higher than from Technology B (0, implying no profits). Intuitively, Firm 2 observes that Firm 1 has already chosen Technology A, should choose Technology A as well, which is better than choosing Technology B. • Subgame 3. After Firm 1 chooses Technology B, Firm 2 responds with Technology B since its payoff from doing so (1, that is, a profit of 1 million dollars) is Firm 1

Technology A

Technology B

Firm 2 Technology A

2 2

Firm 2 Technology B

Technology A

0 0

Fig. 4.17 Pareto coordination sequential-move game

0 0

Technology B

1 1

Exercise 5—Sequential Version of the Pareto Coordination GameA

169 Subgame 1

Firm 1

Technology A

Technology B

Subgame 2

Subgame 3

Firm 2 Technology A

Firm 2 Technology B

2 2

Technology A

0 0

Technology B

0 0

1 1

Fig. 4.18 Pareto coordination sequential-move game with subgames

higher than from Technology A (0, implying no profits). Intuitively, Firm 2 observes that Firm 1 has already chosen Technology B, should choose Technology B as well, which is better than choosing Technology A. We can see that, in summary, Firm 2 responds by choosing the same action as Firm 1 did, similarly as in the sequential-move version of the Battle of the Sexes game analyzed above (Fig. 4.19). Part (b). By backwards induction, Firm 1 anticipates that Firm 2 will respond with Technology A if it chooses Technology A and Technology B if it chooses Technology B. Therefore, the game tree can be depicted as the reduced-form game in Fig. 4.20, where the payoff pairs after Firm 1 chooses Technology A are (2, 2) since it anticipates that Firm 2 responds with Technology A in Subgame 2. Similarly,

Firm 1

Technology A

Firm 2 Technology A

2 2

Technology B

Firm 2 Technology B

0 0

Technology A

Technology B

0 0

Fig. 4.19 Pareto coordination sequential-move game with best responses

1 1

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Fig. 4.20 Reduced pareto coordination sequential-move game with best responses

Firm 1

Technology A

2 2

Technology B

1 1

payoff pairs after Firm 1 chooses Technology B are (1, 1) since it expects that Firm 2 responds with Technology B in Subgame 3. Therefore, Firm 1 chooses Technology A since its payoff from doing so, 2, is higher than that from Technology B, 1. In words, since Firm 1 anticipates that Firm 2 will follow whatever it chooses, it is better off choosing Technology A than Technology B. As a result, the strategy profile SPNE ¼ fTechnology A; ðTechnology A; Technology BÞg is the Subgame Perfect Nash Equilibrium (SPNE) of this sequential-move Pareto Coordination game, where the first component of the triplet indicates Firm 1’s action in equilibrium, the second element represents Firm 2’s response to Firm 1 choosing Technology A (in Subgame 2), and the third element describes Firm 2’s response to Firm 1 choosing Technology B (in Subgame 3). Note that the equilibrium path is different from the SPNE of the game, as it would only say {Technology A, Technology A} which indicates that Firm 1 chooses Technology A, and Firm 2 responds by choosing Technology A (in equilibrium) as well. In the SPNE, however, we list not only the action that Firm 2 chooses in equilibrium (that is, in Subgame 2) but also off-the-equilibrium (in Subgame 3, which should never be reached). Part (c). Equilibrium behavior in the simultaneous-move version of the Pareto coordination game predicts two Nash equilibria in pure strategies, where both players choose Technology A or both choose Technology B. In the sequential-move version, however, we found that only the first one can be sustained in equilibrium, where both firms choose Technology A. Intuitively, among the two strategy profiles where both players coordinate by selecting the same technology, Firm 1 can pick the outcome yielding the highest profit (where both firms choose Technology A). In other words, she exploits her first-mover advantage by choosing Technology A in the first period, which is then responded by Firm 2 with Technology A as well.

Exercise 5—Sequential Version of the Pareto Coordination GameA

171

Importantly, this is an outcome that Firm 2 (as a second mover) also prefers to Technology B, implying that the sequential-move version of the game gives rise to a Pareto optimal outcome in equilibrium; while its simultaneous-move version could give rise to both this outcome and the Pareto dominated outcome where both firms choose Technology B. As a practice, note that the equilibrium outcome would not be affected if Firm 2 acted as the first mover, since both firms have symmetric preferences over outcomes, that is, both firms prefer Technology A over Technology B.

Exercise 6—Sequential Version of the Chicken GameA Consider the game tree in Fig. 4.21, representing the sequential-move version of the Chicken game, where player 1 acts first, and observing its move player 2 responds. (a) Find the best responses of the second mover, for each of the first mover’s actions. (b) Find the first mover’s equilibrium action and describe the subgame perfect Nash equilibrium of the game. (c) Does the equilibrium behavior in the sequential-move game differ from its simultaneous-move version? Answer Part (a). We first circle the subgames in this extensive-form game, where Subgame 1 is the whole game, and Subgames 2 and 3 are initiated after Player 1 chooses Straight and Swerve, respectively (Fig. 4.22). Using blue arrows, we depict Player 2’s best responses after observing Player 1’s move: Fig. 4.21 Chicken sequential-move game

Player 1

Straight

Swerve

Player 2

Player 2

Straight

Swerve

0 0

3 1

Straight

1 3

Swerve 2 2

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Sequential-Move Games with Complete Information

Fig. 4.22 Chicken sequential-move game with subgames

Subgame 1 Player 1

Straight

Player 2

Swerve

Subgame 2

Straight

Swerve

0 0

3 1

Player 2 Straight

1 3

Subgame 3 Swerve 2 2

• Subgame 2. After Player 1 chooses Straight, Player 2 responds with Swerve since its payoff from doing so (1, that is, avoiding a heads-on collision but being humiliated by Player 1) is higher than from Straight (0, implying a heads-on collision). Intuitively, Player 2 observes that Player 1 has already chosen Straight, should choose Swerve, which is better than choosing Straight. • Subgame 3. After Player 1 chooses Swerve, Player 2 responds with Straight since its payoff from doing so (3, that is, avoiding a heads-on collision and humiliating Player 1) is higher than from Swerve (2, that is, avoiding a heads-on collision but neither humiliating nor being humiliated by Player 1). Intuitively, Player 2 observes that Player 1 has already chosen Swerve, should choose Straight, which is better than choosing Swerve. We can see that Player 2 responds by choosing the opposite action as Player 1 (Fig. 4.23). Part (b). By backwards induction, Player 1 anticipates that Player 2 will respond with Straight if it chooses Swerve and Straight if it chooses Swerve. Therefore, the game tree can be depicted as the reduced-form game in Fig. 4.24, where the payoff pairs after Player 1 chooses Straight are (3, 1) since it anticipates that Player 2 will responds with Swerve in Subgame 2. Similarly, payoff pairs after Player 1 chooses Swerve are (1, 3) since it expects that Player 2 will respond with Straight in Subgame 3. Therefore, Player 1 chooses Straight since its payoff from doing so, 3, is higher than that from Swerve, 1. In words, since Player 1 anticipates that Player 2 will anti-coordinate (choosing something different than what Player 1 selects), Player 1 is better off choosing Straight than Swerve. As a result, the strategy profile SPNE ¼ fStraight; ðSwerve; StraightÞg

Exercise 6—Sequential Version of the Chicken GameA

173

Fig. 4.23 Chicken sequential-move game with best responses

Player 1

Straight

Swerve

Player 2

Player 2

Straight

Swerve

0 0

3 1

Fig. 4.24 Reduced chicken sequential-move game with best responses

Straight

1 3

Swerve 2 2

Player 1

Straight

3 1

Swerve

1 3

is the Subgame Perfect Nash Equilibrium (SPNE) of this sequential-move Chicken game, where the first component of the triplet indicates Player 1’s action in equilibrium, the second element represents Player 2’s response to Player 1 choosing Straight (in Subgame 2), and the third element describes Player 2’s response to Player 1 choosing Swerve (in Subgame 3). Note that the “equilibrium path” is different from the SPNE of the game, as it would only say {Straight, Swerve} which indicates that Player 1 chooses Straight, and Player 2 responds by choosing Swerve (in equilibrium). In the SPNE, however, we list not only the action that Player 2 chooses in equilibrium (that is, in Subgame 2) but also off-the-equilibrium (in Subgame 3, which should never be reached). Part (c). Equilibrium behavior in the simultaneous-move version of the Chicken game predicts two Nash equilibria in pure strategies where either Player 1 chooses Straight while Player 2 Swerves, or Player 1 selects Swerve while Player 2 chooses Straight. Only the first outcome can be sustained in the sequential-move version of the game where, in equilibrium, Player 1 chooses Straight and Player 2 responds with Swerve. Intuitively, once Player 1 has chosen Straight, Player 2 is better off responding with Swerve than Straight (that is, avoiding a car crash with Player 1!).

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However, if we modified the order of play having Player 2 as the first mover, the opposite outcome emerges in equilibrium. Indeed, Player 2 chooses Straight, and Player 1 responds with Swerve.

Exercise 7—Electoral CompetitionA Consider the game tree in Fig. 4.25, describing a sequential-move game played between two political parties. The Red party acts first choosing an advertising level: Low, Middle or High. The Blue party observes the Red party’s advertising and responds with its own advertising level (Low, Middle or High as well). Find the SPNE of the game. Answer We use backwards induction to find the SPNE of the game. First we examine the optimal response of the Blue party in the last stage of the game, which is to choose High regardless of the Red party’s level of advertising in the first stage of the game. Indeed, after observing Low (see top of the tree) 100 > 80 > 50; after observing Middle (center of the game) 80 > 70 > 0; and after observing High (see bottom of the tree) 50 > 20 > 0. As a consequence, a High advertising level is a strictly dominant strategy for the Blue party. Figure 4.26 highlights the branches that the Blue party chooses as its optimal responses in each contingency. Anticipating the Blue candidate’s response of High for all advertising levels of the Red party, the Red party compares its payoffs from each advertising level, as depicted in Fig. 4.27. Specifically, the Red party’s payoffs from choosing High becomes 50, from Middle is 20, and from Low is 0. Hence, the Red party chooses High, and the unique SPNE is Fig. 4.25 Sequential electoral competition

Exercise 7—Electoral CompetitionA

175

Fig. 4.26 Sequential electoral competition— optimal responses of the last mover

Fig. 4.27 Sequential electoral competition— optimal actions

SPNE ¼ fHigh; ðHigh; High; HighÞg where the first component reflects the Red party’s choice (first mover), while the triplet (High, High, High) indicates that the Blue party (second mover) chooses a High advertising level regardless of the Red party’s advertising level in the first stage of the game.

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As a remark, note that (High, High) is the equilibrium outcome of this sequential-move game, but not the SPNE of the game. Instead, the SPNE needs to specify optimal actions both on- and off-the-equilibrium path. In this case, this means describing optimal action both after the Red party chooses Low or Middle (off-the-equilibrium).

Exercise 8—Electoral Competition with a TwistA Consider the game tree in Fig. 4.28, describing a sequential-move game played between two political parties. The game is similar to that in Exercise 7, but with different payoffs. The Red party acts first choosing an advertising level. The Blue party acts after observing the Red party’s advertising level (Low, Middle or High) and responds with its own advertising level. Find the SPNE of the game. Answer Second mover First, we examine the payoffs received by the Blue party in the lower part of the game tree. When analyzing each sub-game independently by comparing the payoffs for the Blue party (the number on the right side of each payoff pair). Starting from the subgame on the left-hand side of the tree, when the Red party chooses Low, we can see that for the Blue party is better to choose High because 100 > 80 > 50. In the case of the subgame in the center of the figure, when the Red party chooses Middle, the Blue party have incentives to choose also High, because 80 > 70 > 30. In the case the Red party chooses High, the Blue party responds with Low advertisement expenditure since 60 > 30 > 20. We shade the branches corresponding to these optimal responses (High, High, Low) in Fig. 4.29.

Fig. 4.28 Sequential electoral competition with a twist

Exercise 8—Electoral Competition with a TwistA

177

Fig. 4.29 Sequential electoral competition with a twist—optimal responses

First mover Once the best responses at the subgames for the Blue party are identified, we proceed to identify the best strategy for the Red party by comparing its payoffs in each case. In case the Red party chooses Low advertisement level, the response for the Blue party is to choose High, implying that the Red party gets 10. If the Red party chooses Middle, the payoff given the High response of Blue party is 20. Finally, if Red party’s strategy is to choose High, then its payoff is 100, since the best response by Blue is to choose Low. Hence, since 100 > 20 > 10, the backwards induction strategy to solve the game indicates that the equilibrium is SPNE ¼ fHigh; ðHigh; High; LowÞg; as depicted in Fig. 4.30. Fig. 4.30 Sequential electoral competition with a twist—optimal actions

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In this game, the Blue party (second mover) responds with Low advertising to High advertising from the Red party (first mover), and with High advertising to Low advertising, i.e., it responds choosing the opposite action than the first mover selects. Anticipating such a best response from the other party, the party acting as the first mover chooses a High level of advertising in order to induce the other party to respond with a Low level of advertisement.

Exercise 9—Centipede GameA Consider the game tree in Fig. 4.31, representing the so-called “centipede game” between depositor 1 and 2. In the first stage, depositor 1 chooses whether to take the money in the saving account, which yields a payoff of $2 for him and $0 for the other depositor, or keep the money in the account. If depositor 1 keeps the money in the account, the game continues to stage 2 where depositor 2 can take the money or keep it in the account for one more period. If he takes the money, he receives a payoff of $3 while depositor 1 receives only $1. If depositor 2 keeps the money in the account, then it’s depositor 1’s turn to choose whether to take the money or keep it for one more period. The game continues with depositors alternatively choosing whether to take the money until the 100th stage, when it is depositor 1’s turn to move. Since the game tree has 100 branches pointing down, it is often called the “centipede” game since the game tree looks like an insect with 100 legs. a. Find the best response of the last mover. b. Anticipating equilibrium behavior in the 100th stage, find the best response of the previous-to-last mover (the depositor acting in stage 99).

Fig. 4.31 Centipede sequential-form game

Exercise 9—Centipede GameA

179

c. Using your results in parts (a) and (b), find the subgame perfect Nash equilibrium of the game. Answer Part (a). The smallest subgame is highlighted in blue in the figure. In this subgame, player 2 chooses whether to stop (with a payoff of $101) or continue (with a payoff of $100). Therefore, he prefers to play stop given that u2 ðStopÞ [ u2 ðContinueÞ

since 101 [ 100:

Therefore, the last mover chooses to Stop in the last stage of the game. Importantly, this happens regardless of what player 1 and 2 did in previous rounds, so their previous actions cannot affect player 2’s decision in the last round of play. Part (b). In the previous-to-last stage, player 1 is called upon to move, choosing between stop and continue. If he stops, his payoff is $100, while if he continues he can anticipate player 2 stopping in the next stage (as shown in the part (a) of the exercise), yielding a payoff of $99 for player 1. Therefore, player 1 stops given that u1 ðStopÞ [ u1 ðContinueÞ

since 100 [ 99:

Part (c). A similar argument as in parts (a) and (b) can be extended to previous stages of the game where, the player acting at any stage t, anticipating a stopping decision by the player called on the move in the next stage t + 1, responds by choosing to stop at period t since his payoff is higher from stopping at the current period than when the game is stopped by his opponent in the next stage. This argument, of course, reaches the first node (the root of the game tree), where player 1 anticipates that player 2 will play Stop in the second stage; and generally, that every player will choose Stop in all subsequent stages of the game, and thus player 1 compares u1 ðStopÞ [ u1 ðContinueÞ

since 2 [ 1:

Summarizing, the unique subgame perfect Nash equilibrium of the game (SPNE) can be represented as ðStopt ; Stopt Þ during every period t 2 T, and for any finite length T of this centipede game.

Exercise 10—Trust and Reciprocity (Gift-Exchange Game)B Consider the game in Fig. 4.32, which has been used to experimentally test individuals’ preferences for trust and reciprocity. The experimenter starts by giving player 1 ten dollars and player 2 zero dollars. The experimenter then asks player 1 how many dollars is he willing to give back to help player 2. If he chooses to give

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Sequential-Move Games with Complete Information

Fig. 4.32 Gift-exchange game modified

x dollars, the experimenter gives player 2 that amount tripled, i.e., 3x dollars. Subsequently, player 2 has the opportunity to give any, all, or none of the money he has received from the experimenter to player 1. This game is often known as the “gift-exchange” game, since the trust that player 1 puts on player 2 (by giving him money at the beginning of the game), could be transformed into a large return to player 1, but only if player 2 reciprocates player’s trust. Part (a) Assuming that the two players are risk neutral and care only about their own payoff, find the subgame perfect equilibrium of this game (The subgame perfect equilibrium, SPNE, does not need to coincide with what happens in experiments.). Part (b) Does the game have a Nash equilibrium in which players receive higher payoffs? Part (c) Several experiments show that players do not necessarily behave as predicted in the SPNE you found in part (a). In particular, the subject who is given the role of player 1 in the experiment usually provides a non-negligible amount to player 2 (e.g., $4), and the subject acting as player 2 responds by reciprocating player 1’s kindness by giving him a positive monetary amount in return. An explanation for these experimental results is that the players may be altruistic. Show that the simplest representation of altruism—each player maximizing a weighted sum of his own dollar payoff and the other player’s dollar payoff—cannot account for this experimental regularity except for one very extreme choice of the weights. Answer Figure 4.32 depicts the extensive-form representation of the game. The amount x represents the offer that player 1 gives to player 2, and y is the amount of money player 2 gives to player 1 in return.

Exercise 10—Trust and Reciprocity (Gift-Exchange Game)B

181

Note that the utility function of the first mover is u1 ðx; yÞ ¼ 10  x þ y; since he retains 10  x dollars after giving x dollars to player 2, and afterwards receives y dollars from player 2 (and we are assuming no discounting). The utility function of the second mover is u2 ðx; yÞ ¼ 3x  y, given that for every unit given up by player 1, x, the experimenter adds up 2 units, and afterwards, player 2 gives y units to player 1. Part (a) SPNE: Working by backwards induction, we need to find the optimal amount of y that player 2 will decide to give back to player 1 at the end of the game: max u2 ðx; yÞ ¼ 3x  y y0

Obviously, since y enters negatively on player 2’s utility function, the value of y that maximizes u2 ðx; yÞ is y ð xÞ ¼ 0 for any amount of money, x, received from player 1. Hence, y ð xÞ ¼ 0 can be interpreted as player 2’s best response function (although a very simple best response function, since giving zero dollars to player 1 is a strictly dominant strategy for player 2 once he is called on to move). Player 1 can anticipate that, in the subgame originated after his decision, player 2 will respond with y ð xÞ ¼ 0. Hence, player 1 maximizes max u1 ðx; yÞ ¼ 10  x þ y x

subject to y ð xÞ ¼ 0

Inserting y ð xÞ ¼ 0 in player 1’s objective function, the above problem simplifies to max u1 ðx; yÞ ¼ 10  x x

By a similar argument, since x enters negatively into player 1’s utility function, the value of x that maximizes u1 ðx; yÞ is x ¼ 0. Therefore, the unique SPNE is ðx ; y Þ ¼ ð0; 0Þ;   which implies a SPNE payoff vector of u1 ; u2 ¼ ð10; 0Þ. Part (b) Nash equilibrium. Note that player 2 should always respond with y ð xÞ ¼ 0 along the equilibrium path for any offer x chosen by player 1 in the first period, i.e., responding with y = 0 is a strictly dominant strategy for player 2. Hence, there is no credible threat that could induce player 1 to deviate from x ¼ 0 in the first period.

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Sequential-Move Games with Complete Information

Therefore, the unique NE strategy profile (and outcome) is the same as in the SPNE described above. Part (c) Operating by backwards induction, the altruistic player 2 chooses the value of y that maximizes the weighted sum of both players’ utilities: max u2 ðx; yÞ þ a1 u1 ðx; yÞ y

where a1 denotes player 2’s concern for player 1’s utility. That is: max ð3x  yÞ þ a1 ð10  x þ yÞ ¼ 3x  a1 x  y þ a1 y þ 10a1 y

¼ 3xð1  a1 Þ þ yða1  1Þ þ 10a1 Taking first-order conditions with respect to y, we obtain a1  1  0

,

a1  1

and, in an interior solution, where y [ 0, we then have a1 ¼ 1 (Otherwise, i.e., for all a1 \1; we are at a corner solution where y ¼ 0). Therefore, the only way to justify that player 2 would ever give some positive amount of y back to player 1 is if and only if a1  1. That is, if and only if player 2 cares about player 1’s utility at least as much as he cares about his own.

Exercise 11—Stackelberg with Two FirmsA Consider a leader and a follower in a Stackelberg game of quantity competition. Firms face an inverse demand curve p(Q) = 1 − Q, where Q ¼ qL þ qF : denotes aggregate output. The leader faces a constant marginal cost cL > 0 while the follower’s constant marginal cost is cF > 0, where 1 > cF  cL, indicating that the leader has access to a more efficient technology than the follower. Part (a) Find the follower’s best response function, BRFF, i.e., qF (qL). Part (b) Determine each firm’s output strategy in the subgame perfect equilibrium (SPNE) of this sequential-move game. Part (c) Under which conditions on cL can you guarantee that both firms produce strictly positive output levels? Part (d) Assuming cost symmetry, i.e., cL = cF = c, determine the aggregate equilibrium output level in the Stackelberg game, and its associated equilibrium price. Then compare them with those arising in (1) a monopoly; in (2) the Cournot game of quantity competition; and (3) in the Bertrand game of price competition (which, under cost symmetry, is analogous to a perfectly competitive industry) [Hint: For a more direct and visual comparison, depict the inverse demand curve p

Exercise 11—Stackelberg with Two FirmsA

183

(Q) = 1 − Q, and locate each of these four aggregate output levels in the horizontal axis (with its associated prices in the vertical axis).]. Answer Part (a) The follower observes the leader’s output level, qL , and chooses its own production, qF , to solve: max ð1  qL  qF ÞqF  cF qF

qF  0

Taking first order conditions with respect to qF yields 1  qL  2qF  cF ¼ 0 and solving for qF we obtain the follower’s best response function qF ð qL Þ ¼

1  cF 1  qL 2 2

which, as usual, is decreasing in the follower’s costs, i.e., the vertical intercept decreases in cF , indicating that, graphically, the best-response function experiences a downward shift as cF increases. In addition, the follower’s best-response function decreases in the leader’s output decision (as indicated by the negative slope,  12). Part (b) The leader anticipates that the follower will respond with best response 1 F function qF ðqL Þ ¼ 1c 2  2 qL , and plugs it into the leader’s own profit maximization problem, as follows 2

3  6 7 1  cF 1 6 7  qL 7qL  cL qL max 61  qL  qL  0 4 5 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}


qF

which simplifies into 1 ½ð1 þ cF Þ  qL qL  cL qL : 2 Taking first order conditions with respect to qL yields 1 ð1 þ cF Þ  qL  cL ¼ 0: 2

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Sequential-Move Games with Complete Information

and solving for qL we find the leader’s equilibrium output level qL ¼

1 þ cF  2cL 2

which, thus, implies a follower’s equilibrium output of qF ¼ qF


 1 þ cF  2cL 1  3cF þ 2cL ¼ 2 4

Note that the equilibrium output of every firm i = L,F is decreasing in its own cost, ci, and increasing in its rival’s cost, cj, where j 6¼ i. Hence, the SPNE of the game is 

 qL ; qF ðqL Þ ¼


 1 þ cF  2cL 1  cF 1 ;  qL 2 2 2

which allows the follower to optimally respond to both the equilibrium output level from the leader, qL , but also to off-the-equilibrium production decisions qL 6¼ qL : Part (c) The follower (which operates under a cost disadvantage), produces a positive output level in equilibrium, i.e., qF [ 0, if and only if 1  3cF þ 2cL [ 0; 4 which, solving for cL , yields cL [ 

1 3 þ cF  CA : 2 2

Similarly, the leader produces a positive output level, qL [ 0, if and only if 1 þ cF  2cL [0 2 or, solving for cL , cL \

1 1 þ cF  CB : 2 2

Figure 4.33 depicts cutoffs CA (for the follower) and CB (for the leader), in the ðcF ; cL Þ—quadrant (Note that we focus on points below the 45°-line, since the leader experiences a cost advantage relative to the follower, i.e., cL  cF :). First, note that cutoff CB is not binding since it lies above the 45°-line. Intuitively, the leader produces a positive output level for all ðcF ; cL Þ—pairs in the admissible region of cost pairs (below the 45°-line). However, cutoff CA restricts the of cost

Exercise 11—Stackelberg with Two FirmsA

185

Fig. 4.33 Region of cost pairs for which both firms produce positive output

pairs below the 45°-line to only that above cutoff CA . Hence, in the shaded area of the figure, both firms produce a strictly positive output in equilibrium. Part (d) Stackelberg competition. When firms are cost symmetric, cF ¼ cL ¼ c, the leader produces qL ¼

1 þ c  2c 1  c ¼ 2 2

while the follower responds with an equilibrium output of qF ¼

1  3c þ 2c 1  c ¼ 4 4

thus producing half of the leader’s output. Hence, aggregate equilibrium output in the Stackelberg game becomes QStackel: ¼ qL þ qF ¼

1  c 1  c 3ð1  cÞ þ ¼ 2 4 4

which yields an associated price of   3ð1  cÞ 1 þ 3c ¼ P QStackel: ¼ 1  4 4

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Cournot competition. Under Cournot competition, each firm i = {1, 2} simultaneously and independently solves   max 1  qi  qj qi  cqi qi

which, taking first-order conditions with respect to qi , and solving for qi , yields a best response function of   1c 1  qj qi qj ¼ 2 2 Simultaneously solving for qi and qj , we obtain a Cournot equilibrium output of 1 qi ¼ 1c 3 for every firm i = {1, 2}. Hence, aggregate output in Cournot is QCournot ¼ q1 þ q2 ¼

2ð1  cÞ 3

which yields a market price of PðQCournot Þ ¼ 1 

2ð1  cÞ 1 þ 2c ¼ : 3 3

Bertrand competition. Under the Bertrand model of price competition and symmetric costs, the Nash equilibrium prescribes both firms to set a price that coincides with their common marginal cost, i.e., p1 ¼ p2 ¼ c (This equilibrium is analyzed in Exercise 1 of Chap. 5, where we discuss other applications to Industrial Organization. However, since this equilibrium price coincides with that in perfectly competitive industries, we can at this point consider such a market structure in our subsequent comparisons.). Hence, aggregate output in this context coincides with that under a perfectly competitive industry, c ¼ 1  Q; or QBertrand ¼ 1  c Monopoly. A monopoly chooses the output level Q that solves max ð1  QÞQ  cQ: Q0

Recall from Chap. 2, that you can find this equilibrium output by inserting best response function qj (qi) into qi (qj), and then solving for qi.

1

Exercise 11—Stackelberg with Two FirmsA

187

Fig. 4.34 Aggregate output and prices across different market structures

Taking first order conditions with respect to Q yields 1  2Q  c ¼ 0 and, solving for Q, we obtain a monopoly output of QMonop ¼

1c 2

which implies a monopoly price of   1  c 1þc ¼ : P QMonop ¼ 1  2 2 Comparison. Figure 4.34 depicts the inverse linear demand P(Q) = 1 − Q, and locates the aggregate output levels found in the four market structures described above in the horizontal axis, afterwards mapping their corresponding prices in the vertical axis. Aggregate output is the lowest at a monopolistic market structure, larger under Cournot competition (when firms simultaneously and independently select output levels), larger under Stackelberg (when firms’ competition is in quantities, but sequentially), and yet larger in a perfect competitive industry. The opposite ranking applies for equilibrium prices.

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Exercise 12—Stackelberg Competition with 2, or 3 FirmsB Consider a Stackelberg model of output competition where firm 1 is the industry leader and firm 2 is the follower. Firms face a linear inverse demand curve pðQÞ ¼ 10  Q, where Q ¼ q1 þ q2 denotes the aggregate output sold by both firms. In addition, consider that firms are cost symmetric, facing a common marginal cost of MCi ¼ 2 for every firm i ¼ f1; 2g and no fixed costs. (a) Find the equilibrium in this Stackelberg game. (b) Assume now that a new firm (firm 3) joins the industry, acting last (after firm 2 chooses its output). Find the equilibrium in this Stackelberg game. Compare your results with those in part (a). How is the output of firm 1 and 2 affected by firm 3’s entry? Answer Part (a). Firm 2 (follower). Firm 2 chooses output q2 that solves the following profit maximization problem: max p2 ðq2 Þ ¼ pðQÞq2  TC2 ðq2 Þ

q2  0

¼ ð8  q1  q2 Þq2 Taking the first order condition with respect to q2 , and assuming an interior solution for q2 , 8  q1  2q2 ¼ 0 if q1 [ 0 After rearranging, the best response function of firm 2 in response to firm 1 is given by q2 ð q 1 Þ ¼ 4 

q1 2

Firm 1 (leader). Firm 1, anticipating that firm 2 will respond with best response function q2 ðq1 Þ ¼ 4  q21 , chooses its own output level, q1 , that solves the following profit maximization problem: max p1 ðq1 Þ ¼ pðQÞq1  TC1 ðq1 Þ

q1  0

¼ ð10  q1  q2 ðq1 ÞÞq1  2q1   q1  ¼ 8  q1  4  q1 2   q1 ¼ 4 q1 2

Exercise 12—Stackelberg Competition with 2, or 3 FirmsB

189

Taking the first order condition with respect to q1 , and assuming an interior solution for q1 , 4  q1 ¼ 0

if q1 [ 0

such that the optimal output of firm 1 becomes q1 ¼ 4 Finding equilibrium output for the follower. Substituting the optimal output of firm 1 into firm 2’s best response function, the optimal output of firm 2 becomes q2 ¼ 4 

4 ¼2 2

Therefore, when both firms are symmetric in costs, the leading firm 1 produces 4 units of output while the following firm 2 produces 2 units of output. Part (b). Firm 3 (last mover). Firm 3 chooses output q3 that solves the following profit maximization problem: max p3 ðq3 Þ ¼ pðQÞq3  TC3 ðq3 Þ

q3  0

¼ ð8  q1  q2  q3 Þq3 Taking the first order condition with respect to q3 , and assuming an interior solution for q3 , 8  q1  q2  2q3 ¼ 0 if q3 [ 0 After rearranging, the best response function of firm 3 in response to firms 1 and 2 is q3 ð q1 ; q2 Þ ¼ 4 

q1 þ q2 2

Firm 2 (second mover). Firm 2, anticipating that firm 3 will respond with its best response function q3 ðq1 ; q2 Þ ¼ 4  q1 þ2 q2 , chooses its own output level, q2 , that solves the following profit maximization problem:

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max p2 ðq2 Þ ¼ pðQÞq2  TC2 ðq2 Þ

q2  0

¼ ð10  q1  q2  q3 ðq1 ; q2 ÞÞq2  2q2   q1 þ q2  ¼ 8  q1  q2  4  q2 2   q1 þ q2 ¼ 4 q2 2 Taking the first order condition with respect to q2 , and assuming an interior solution for q2 , 4

q1  q2 ¼ 0 if q1 [ 0 2

After rearranging, the best response function of firm 2 in response to firm 1 is q2 ð q1 Þ ¼ 4 

q1 2

such that the best response function of firm 3 can be rewritten as q3 ð q1 Þ ¼ 4 

q1 1  q1  q1  4 ¼2 2 2 2 4

Firm 1 (first mover). Firm 1, anticipating that firms 2 and 3 will respond with their best response functions q2 ðq1 Þ ¼ 4  q21 and q3 ðq1 Þ ¼ 2  q41 respectively, chooses its own output level, q1 , that solves the following profit maximization problem: max p1 ðq1 Þ ¼ pðQÞq1  TC1 ðq1 Þ

q1  0

¼ ð10  q1  q2 ðq1 Þ  q3 ðq1 ÞÞq1  2q1   q1   q1   ¼ 8  q1  4   2 q1 2 4   q1 ¼ 2 q1 4 Taking the first order condition with respect to q1 , and assuming an interior solution for q1 , 2

q1 ¼ 0 if q1 [ 0 2

such that the optimal output of firm 1 becomes q1 ¼ 4

Exercise 12—Stackelberg Competition with 2, or 3 FirmsB

191

Finding equilibrium output for the second and third movers. Substituting the optimal output of firm 1 into firm 2’s best response function, the optimal output of firm 2 becomes q2 ¼ 4 

4 ¼2 2

Substituting the optimal output of firm 1 into firm 3’s best response function, the optimal output of firm 3 becomes q3 ¼ 2 

4 ¼1 4

Therefore, when all firms face the same costs, firms 1 and 2 continue to produce 4 and 2 units of output respectively as in the 2-player game, while firm 3 produces 1 unit of output.

Exercise 13—Stackelberg Competition with Efficiency ChangesB Consider the industry in the above setting but assume now that firm 1 is the industry leader while firm 2 is the follower; as in the Stackelberg model of quantity competition. (a) Assuming that firm i’s cost function is TCi ðqi Þ ¼ 4qi , find the equilibrium in this Stackelberg game. (b) Assuming that firm 2’s cost function becomes TC2 ðq2 Þ ¼ 2q2 after investing in the cost-saving technology, find the equilibrium in this Stackelberg game. How are your results from part (a) affected? (c) Repeat part (b), but assuming that firm 2 is the industry leader while firm 1 is the follower. Answer Part (a). Firm 2 (follower). By backwards induction, firm 2 takes the output of firm 1, q1 , as given and chooses its own output level, q2 , that solves the following profit maximization problem: max p2 ðq2 Þ ¼ pðQÞq2  TC2 ðq2 Þ

q2  0

¼ ð6  q1  q2 Þq2 Taking the first order condition with respect to q2 , and assuming an interior solution for q2 ,

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6  q1  2q2 ¼ 0 if q2 [ 0 After rearranging, the best response function of firm 2 in response to firm 1 is given by q2 ð q 1 Þ ¼ 3 

q1 2

Firm 2 (leader). Firm 1, anticipating that firm 2 will respond with best response function q2 ðq1 Þ ¼ 3  q21 , chooses its own output level, q1 , that solves the following profit maximization problem: max p1 ðq1 Þ ¼ pðQÞq1  TC1 ðq1 Þ

q1  0

¼ ð10  q1  q2 ðq1 ÞÞq1  4q1   q1  ¼ 6  q1  3  q1 2   q1 ¼ 3 q1 2 Taking the first order condition with respect to q1 , and assuming an interior solution for q1 , 3  q1 ¼ 0 if q1 [ 0 such that the optimal output of firm 1 becomes q1 ¼ 3 Finding equilibrium output for the follower. Substituting the optimal output of firm 1 into firm 2’s best response function, the optimal output of firm 2 becomes 3 1 q2 ¼ 3  ¼ 1 2 2 Therefore, the leading firm 1 produces 3 units of output, while the following firm 2 produces 1.5 units of output, which is half of the production level of firm 1. Part (b). Firm 2 (follower). By backwards induction, firm 2 takes the output of firm 1, q1 , as given and chooses its own output level, q2 , that solves the following profit maximization problem: max p2 ðq2 Þ ¼ pðQÞq2  TC2 ðq2 Þ

q2  0

¼ ð8  q1  q2 Þq2

Exercise 13—Stackelberg Competition with Efficiency ChangesB

193

Taking the first order condition with respect to q2 , and assuming an interior solution for q2 , 8  q1  2q2 ¼ 0 if q2 [ 0 After rearranging, the best response function of firm 2 in response to firm 1 is given by q2 ð q 1 Þ ¼ 4 

q1 2

Firm 1 (leader). Firm 1, anticipating that firm 2 will respond with its best response function q2 ðq1 Þ ¼ 4  q21 , chooses its own output level, q1 , that solves the following profit maximization problem: max p1 ðq1 Þ ¼ pðQÞq1  TC1 ðq1 Þ

q1  0

¼ ð10  q1  q2 ðq1 ÞÞq1  4q1   q1  ¼ 6  q1  4  q1 2  q1  ¼ 2 q1 2 Taking the first order condition with respect to q1 , and assuming an interior solution for q1 , 2  q1 ¼ 0 if q1 [ 0 such that the optimal output of firm 1 becomes q1 ¼ 2 Finding equilibrium output for the follower. Substituting the optimal output of firm 1 into firm 2’s best response function, the optimal output of firm 2 becomes q2 ¼ 4 

2 ¼3 2

When it is less expensive for firm 2 to produce the output, firm 2 doubles production from 1.5 to 3 units, and firm 1 reduces output to 2 units which now falls below that of firm 2. Part (c). Firm 1 (last mover). By backwards induction, firm 1 takes the output of firm 2, q2 , as given and chooses its own output level, q1 , that solves the following profit maximization problem:

194

4

Sequential-Move Games with Complete Information

max p1 ðq1 Þ ¼ pðQÞq1  TC1 ðq1 Þ

q1  0

¼ ð6  q1  q2 Þq1 Taking the first order condition with respect to q1 , and assuming an interior solution for q1 , 6  2q1  q2 ¼ 0 if q1 [ 0 After rearranging, the best response function of firm 1 in response to firm 2 is given by q1 ð q 2 Þ ¼ 3 

q2 2

Firm 2 (first mover). Firm 2, anticipating that firm 1 will respond with best response function q1 ðq2 Þ ¼ 3  q22 , chooses its own output level, q2 , that solves the following profit maximization problem: max p2 ðq2 Þ ¼ pðQÞq2  TC2 ðq2 Þ

q2  0

¼ ð10  q1 ðq2 Þ  q2 Þq2  2q2    q2  ¼ 8 3  q2 q2 2  q2  ¼ 5 q2 2 Taking the first order condition with respect to q2 , and assuming an interior solution for q2 , 5  q2 ¼ 0 if q2 [ 0 such that the optimal output of firm 2 becomes q2 ¼ 5 Finding equilibrium output for the follower. Substituting the optimal output of firm 2 into firm 1’s best response function, the optimal output of firm 1 becomes q1 ¼ 3 

5 1 ¼ 2 2

Therefore, in the case that the leading firm invests in cost-saving technology, firm 2 produces 5 units of output while the following firm 1 produces ½ units of output.

Exercise 14—First- and Second-Mover Advantage in Product DifferentiationB

195

Exercise 14—First- and Second-Mover Advantage in Product DifferentiationB Consider an industry with two firms, 1 and 2, competing in prices and no production costs. Firms sell a differentiated product, with the following functions q1 ¼ 150  2p1 þ p2 and q2 ¼ 150  2p2 þ p1 Intuitively, if firm i increases its own price pi its sales decrease, while if its rival increases its price pj firm i’s sales increase. However, the own-price effect on sales is larger in absolute value than the cross-price effect, thus indicating that a marginal increase in pi has a larger negative effect on qi than the positive effect that an increase in pj has (in this particular setting, the former is twice as big as the latter). Part (a) Simultaneous-move game. If firms simultaneously and independently choose their prices (competition Bertrand), find their best response functions,  a la Sim  , the equilibrium sales by each firm the equilibrium price pair pSim ; p 1 2   Sim Sim   Sim Sim q1 ; q2 , and the equilibrium profits p1 ; p1 . Answer Firm 1 solves max ð150  2p1 þ p2 Þp1 p1

Taking first-order conditions with respect to p1 yields 150  4p1 þ p2 ¼ 0: And solving for p1 , we obtain firm 1’s best response function p1 ð p2 Þ ¼

150 1 þ p2 4 4

Similarly, firm 2 solves max ð150  2p2 þ p1 Þp2 : p2

Taking first-order conditions with respect to p2 yields 150  4p2 þ p1 ¼ 0: And solving for p2 , we obtain firm 2’s best response function

196

4

Sequential-Move Games with Complete Information

p2 ð p1 Þ ¼

150 1 þ p1 : 4 4

Note that both firms’ best response functions are positively sloped, indicating that if firm j increases its price, firm i can respond by optimally increasing its own (i.e., prices are strategic complements). Inserting p2 ðp1 Þ into p1 ðp2 Þ, we obtain the point where both best response functions cross each other, p1 ¼


 150 1 150 1 þ þ p1 4 4 4 4

which simplifies to p1 ¼

750 þ p1 : 16

Solving for price p1 , we obtain p1 = $50. Therefore, the equilibrium price pair is 

 pi ; pj ¼ ð50; 50Þ

which constitutes the Nash equilibrium of this simultaneous-move game of price competition. Note that such equilibrium is symmetric, as both firms set the same price (i.e., graphically, best response functions cross at the 45° line). Using the direct demand functions, we find that sales by each firm are q1 ¼ 150  2p1 þ p2 ¼ 100; and q2 ¼ 150  2p2 þ p1 ¼ 100: Finally, equilibrium profits are p1 ¼ 5000 and p2 ¼ 5000:

Part (b) Sequential-move game. Let us now consider the sequential version of the above game of price competition. In particular, assume that firm 1 chooses its price first (leader) and that firm 2, observing firm 1’s price,  responds with its own price

Seq (follower). Find the equilibrium price pair pSeq , the equilibrium sales by 1 ; p2     Seq Seq Seq ; q ; p each firm qSeq , and the equilibrium profits p . Which firm obtains 1 2 1 2

the highest profit, the leader or the follower?

Exercise 14—First- and Second-Mover Advantage in Product DifferentiationB

197

Answer The follower (firm 2) observes firm 1’s price, p1 , and best responds to it by using the best response function identified in part (a), i.e., p2 ðp1 Þ ¼ 1504þ p1 . Since firm 1 (the leader) can anticipate the best response function that firm 2 will subsequently use in the second stage of the game, firm 1’s profit maximization problem becomes 0

1

B 150 þ p1 C C  p1 150  2p1 þ max B p1 @ 4 ffl} A |fflfflfflfflffl{zfflfflfflffl p2

which depends on p1 alone. Taking first-order conditions with respect to p1 , we obtain 150  4p1 þ

150 þ 2p1 ¼0 4

Solving for p1 , yields an equilibrium price for the leader of p1 ¼

750 53:6: 14

Inserting this price in the direct demand functions, we find that the sales of each firm are 150 þ p1 150 þ 53:6 93:8; and ¼ 150  2ð53:6Þ þ 4 4
 150 þ 53:6 q2 ¼ 150  2p2 þ p1 ¼ 150  2 þ 53:6 101:8: 4 q1 ¼ 150  2p1 þ

And equilibrium profits in this sequential version of the game become p1 ¼ 5027:7 and p2 ¼ 5181:62: Finally, note that the leader obtains a lower profit than the follower when firms compete in prices; as opposed to the profit ranking when firms sequentially compete in quantities (standard Stackelberg competition where the leader obtains a larger profit than the follower).

198

4

Sequential-Move Games with Complete Information

Exercise 15—Stackelberg Game with Three Firms Acting SequentiallyA Consider an industry consisting of three firms. Each firm i = {1, 2, 3} has the same cost structure, given by cost function Cðqi Þ ¼ 5 þ 2qi , where qi denotes individual output. For every firm i, industry demand is given by the inverse demand function: PðQÞ ¼ 18  Q where Q denotes aggregate output, i.e., Q ¼ q1 þ q2 þ q3 . The production timing is as follows: Firm 1 produces its output first. Observing firm 1’s output, q1 , firm 2 chooses its own output, q2 : Finally, observing both firm 1’s and firm 2’s output, firm 3 then produces its own output q3 . This timing of production is common knowledge among all three firms. The industry demand and cost functions are also known to each firm. Find values of output level q1 , q2 and q3 in the SPNE of the game. Answer Firm 3. Using backward induction, we first analyze the production decision of the last mover firm 3, which maximizes profits by solving: max ½18  ðq1 þ q2 þ q3 Þq3  ð5 þ 2q3 Þ q3

Taking first order conditions with respect to q3 , we obtain 18  q1  q2  2q3  2 ¼ 0: Hence, solving for q3 we obtain firm 3’s best response function (which decreases in both q1 and q2 ) q3 ð q1 ; q2 Þ ¼

16  q1  q2 : 2

Firm 2. Given this BRF3 ðq1 ; q2 Þ, we can now examine firm 2’s production decision (the second mover in the game), which chooses an output level q2 to solve 0

0

11

B B C 16  q1  q2 C CCq2  ð5 þ 2q2 Þ: max B 18  B @ q1 þ q2 þ AA q2 @ 2 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} q3

where we inserted firm 3’s best response function, since firm 2 can anticipate firm 3’s optimal response at the subsequent stage of the game. Simplifying this profit, we find

Exercise 15—Stackelberg Game …


max q2

199

 1 1 10  q1  q2 q2  ð5 þ 2q2 Þ: 2 2

Taking first order conditions with respect to q2 , yields 1 10  q1  q2  2 ¼ 0: 2 And solving for q2 we find firm 2’s best response function (which only depends on the production that occurs before its decision, i.e., q1 ), 1 q2 ð q1 Þ ¼ 8  q 1 : 2 Firm 1. We can finally analyze firm 1’s production decision (the first mover). Taking into account the output level with which firm 2 and 3 will respond, as described by BRF2 and BRF3 , respectively, firm 1 maximizes 0

0

0

1





11

B B B C 16  q1  8  12 q1 CC B q1 þ B 8  1 q1 C þ CCq1  ð5 þ 2q1 Þ max B 18  @ @ A AA q1 @ 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} q2

q3

which simplifies to
max q1

 1 6  q1 q1  ð5 þ 2q1 Þ: 4

Taking first order conditions with respect to q1 , we obtain 1 6  q1  2 ¼ 0: 2 Solving for q1 we find the equilibrium production level for the leader (firm 1), qs1 ¼ 8. Therefore, we can plug qs1 ¼ 8 into firm 2’s and 3’s best response function to obtain 1 qs2 ¼ q2 ð8Þ ¼ 8  ð8Þ ¼ 4 units 2 for firm 2 and qs3 ¼ q3 ð8; 4Þ ¼

16  8  4 ¼ 2 units 2

200

4

Sequential-Move Games with Complete Information

for firm 3. Hence, the output levels that arise in equilibrium are (8, 4, 2). However, the SPNE of this Stackelberg game must specify optimal actions for all players, both along the equilibrium path (for instance, after firm 2 observes firm 1 producing qS1 ¼ 8 units) and off-the-equilibrium path (for instance, when firm 2 observes firm 1 producing q1 6¼ 8 units). We can accurately represent this optimal action at every point in the game in the following SPNE: ðqS1 ; q2 ðq1 Þ; q3 ðq1 ; q2 ÞÞ


¼

 1 16  q1  q2 8; 8  q1 ; : 2 2

Exercise 16—Two-Period Bilateral Bargaining GameA Alice and Bob are trying to split $100. As depicted in the game tree of Fig. 4.35, in the first round of bargaining, Alice makes an offer at cost c (to herself), proposing to keep xA and give the remaining xB ¼ 100  xA to Bob. Bob either accepts her offer (ending the game) or rejects it. In round 2, Bob makes an offer of ðyA ; yB Þ, at a cost of 10 to himself, which Alice accepts or rejects. If Alice accepts the offer, the game ends; but if she rejects it, the game proceeds to the third round, in which Alice makes an offer ðzA ; zB Þ, at a cost c to herself. If Bob accepts her offer in the third round, the game ends and payoffs are accrued to each player; whereas if he rejects it, the money is lost. Assume that players are risk-neutral (utility is equal to money obtained minus any costs), and there is no discounting. If c ¼ 0, what is the subgame-perfect equilibrium outcome? Answer Third period. Operating by backward induction, we start in the last round of the negotiation (third round). In t ¼ 3, Alice will offer herself the maximum split zA that still guarantees that it is accepted by Bob: uB ðAccept3 Þ ¼ uB ðReject3 Þ 100  zA  10 ¼ 10; i:e:; zA ¼ 100: That is, Alice offers ðzA ; zB Þ ¼ ð100; 0Þ in t ¼ 3. Second period. In t ¼ 2, Bob needs to offer Alice a split that makes her indifferent between accepting in t ¼ 2 and rejecting in order to offer herself the entire pie in t ¼ 3 (which we showed to be her equilibrium behavior in the last round of play). In particular, Alice is indifferent between accepting and rejecting if uA ðAccept2 Þ ¼ uA ðReject2 Þ 100  yB ¼ 100; that is yB ¼ 0

Exercise 16—Two-Period Bilateral Bargaining GameA

201

Fig. 4.35 Two-period bilateral bargaining game

That is, Bob offers ðyA ; yB Þ ¼ ð100; 0Þ in t ¼ 2 (This is an extreme offer, whereby Bob cannot enjoy any negotiation power from making offers, and can only offer the entire pie to Alice. Intuitively, this offer emerges as equilibrium behavior in t = 2 in this exercise because players do not discount their future payoffs and they do not incur any cost in making offers, c ¼ 0. Otherwise, the offers could not be so extreme). First period. In t ¼ 1, Alice needs to offer Bob a split that makes him indifferent between accepting in t ¼ 1 and rejecting (which entails that he will offer himself yB ¼ 0 in the subsequent period, t = 2):

202

4

Sequential-Move Games with Complete Information

uB ðAccept1 Þ ¼ uB ðReject1 Þ 100  xA  10 ¼ 0  10; that is xA ¼ 100 That is, Alice offers ðxA ; xB Þ ¼ ð100; 0Þ in t = 1. So the agreement is reached immediately and Alice gets the entire pie.

Exercise 17—Alternating Bargaining with a TwistB Consider a two-player alternating offer bargaining game with 4 periods, and where player 1 is the first one to make offers. Rather than assuming discounting of future payoffs, let us now assume that the initial surplus shrinks by one unit if players do not reach an agreement in that period, i.e., surplus at period t becomes S ¼ 6  t for all t ¼ f1; 2; 3; 4g. As in similar bargaining games, assume that every player accepts an offer when being indifferent between accept and reject. Find the SPNE of the game. Answer Let us solve this sequential-move game by applying backward induction: Period 4. In period 4, player 2 makes an offer to player 1, who accepts any offer x4  0. Player 2, anticipating such a decision rule, offers x4 ¼ 0 in the 4th period, leaving player 2 with the remaining surplus 6  4 ¼ 2. Period 3. In the previous stage (period 3), player 1 makes an offer to player 2, anticipating that, for player 2 to accept it, he must be indifferent between the offer and the payoff of $2 he will obtain in the subsequent period (when player 2 becomes the proposer). Hence, the offer in period 3 must satisfy x3  2, making player 1 an offer of exactly x3 ¼ 2, which leaves player 1 with the remaining surplus of (6 − 3) – 2 = $1. Period 2. In period 2, player 2 makes offers to player 1 anticipating that the latter will only obtain a payoff of $1 if the game were to continue one more period. Hence, player 2’s offer in period 2 must satisfy x2  1, implying that player 2 offers exactly x2 ¼ 1 to player 1 (the minimum to guarantee acceptance), yielding him a remaining surplus of (6 − 2) – 1 = $3. Period 1. In the first period, player 1 is the individual making offers and player 2 is the responder. Player 1 anticipates that player 2’s payoff will be $3 if the game progresses towards the next stage (when player 2 becomes the proposer). His offer must then satisfy x1  3, implying that player 1 offers x1 = $3 to player 2 to guarantee acceptance. Interestingly, player 1 offers a fair offer (half of the surplus) to player 2, which guarantees acceptance in the first stage of the game, implying that the surplus does not decrease over time.

Exercise 18—Backward Induction in Wage NegotiationsA

203

Exercise 18—Backward Induction in Wage NegotiationsA Consider the following bargaining game between a labor union and a firm. In the first stage, the labor union chooses the wage level, w, that all workers will receive. In the second stage, the firm responds to the wage w by choosing the number of workers it hires, h 2 ½0; 1. The labor union’s payoff function is uL ðw; hÞ ¼ w h, thus being increasing in wages and in the number of workers hired. The firm’s profit is
pðw; hÞ ¼

h2 2

h

  wh

Intuitively, revenue is increasing in the number of workers hired, h, but at a 2 decreasing rate (i.e., h  h2 is concave in h), reaching a maximum when the firm hires all workers, i.e., h ¼ 1 where revenue becomes 1/2. Part (a) Applying backward induction, analyze the optimal strategy of the last mover (firm). Find the firm’s best response function hðwÞ. Part (b) Let us now move on to the first mover in the game (labor union). Anticipating the best response function of the firm that you found in part (a), hðwÞ, determine the labor union’s optimal wage, w : Answer Part (a) The firm solves the following profit maximization problem:
max

h2½0;1

h2 h 2

  wh

Taking the first-order conditions with respect to h, we obtain: 1  h  w0 Solving for h, yields a best response function of: hðwÞ ¼ 1  w Intuitively, the firm hires all workers when the wage is zero, but hires fewer workers as w increases. If the salary is the highest, w ¼ 1, the firm responds hiring no workers [Note that the above best response function hðwÞ defines a maximum since second-order conditions yield the profit function.].

@ 2 pðw;hÞ @h2

¼ 1, thus guaranteeing concavity in

204

4

Sequential-Move Games with Complete Information

Part (b) The labor union solves: max uL ðw; hÞ ¼ w h w0

subject to hðwÞ ¼ 1  w The constraint reflects the fact that the labor union anticipates the subsequent hiring strategy of the firm. We can express this problem more compactly by inserting the (equality) constraint into the objective function, as follows: max uL ðwðhÞ; hÞ ¼ wð1  wÞ ¼ w  w2 w0

Taking first-order conditions with respect to w yields: 1  2w  0 and, solving for w, we obtain an optimal wage of w ¼ 12. Hence, the SPNE of the game is: fw ; hðwÞg ¼



1 ; 1w 2



 and, in equilibrium, salary w ¼ 12 is responded with h 12 ¼ 1  12 ¼ 12 workers hired, i.e., half of workers are hired. In addition, equilibrium payoffs become 1 1 1 uL ¼ ¼ for the labor union 2 2 4 ! 12 1 1 1 1  p ¼  2  ¼ for the firm: 2 2 2 8 2

Exercise 19—Backward Induction-IB Solve the game tree depicted in Fig. 4.36 using backward induction. Answer Starting from the terminal nodes, the smallest proper subgame we can identify is depicted in Fig. 4.37 (which initiates after player 1 chooses In). Recall that a subgame is the portion of a game tree that you can circle around without breaking any information sets. In this exercise, if we were to circle the portion of the game tree initiated after player 2 chooses A or B, we would be breaking player 3’s information set. Hence, we need to keep enlarging the circle (area of the tree) until

Exercise 19—Backward Induction-IB

205

Fig. 4.36 Extensive-form game

we do not break information sets. This happens when we circle the portion of the tree initiated after player 1 chooses In; as depicted in Fig. 4.37. In this subgame in which only two players interact (players 2 and 3), player 3 chooses his action without observing player 2’s choice. As a consequence, the interaction between players 2 and 3 in this subgame can be modeled as a simultaneous-move game. In order to find the Nash equilibrium of the subgame depicted in Fig. 4.37, we must first represent it in its normal (matrix) form, as depicted in Fig. 4.38.

Fig. 4.37 Smallest proper subgame

206

4

Sequential-Move Games with Complete Information

Fig. 4.38 Smallest proper subgame (in its normal-form) Fig. 4.39 Extensive-form game (first stage)

For completeness, Fig. 4.38 also underlines the payoffs that arise when each player selects his best responses.2 In outcome (C, X) the payoffs of player 2 and 3 are underlined, thus indicating that they are playing a mutual best response to each other’s strategies. Furthermore, we do not need to examine player 1, since only player 2 and 3 are called on to move in the subgame. Hence, the Nash equilibrium of this subgame predicts that players 2 and 3 choose strategy profile (C, X). We can now plug the payoff triple resulting from the Nash equilibrium of this subgame, (4, 3, 1), at the end of the branch indicating that player 1 chooses In (recall that this was the node initiating the smallest proper subgame depicted in Fig. 4.37), as we illustrate in Fig. 4.39. By inspecting the above game tree in which player 1 chooses In or Out, we can see that his payoff from In (4) is larger than from Out (2). Then, the SPNE of this game is (In, C, X).

Exercise 20—Backward Induction-IIB Consider the sequential-move game depicted in Fig. 4.40. The game describes Apple’s decision to develop the new iPhone (whatever the new name is, 5s, 6, ∞, etc.) with radically new software which allows for faster and better applications (apps). These apps are, however, still not developed by app developers (such as Rovio, the Finnish company that introduced Angry Birds). If Apple does not develop the new iPhone, then all companies make zero profit in this emerging market. If, instead, the new iPhone is introduced, then company 1 (the leader in the 2

See exercises in Chap. 2 for more examples of this underlining process.

Exercise 20—Backward Induction-IIB

207

Fig. 4.40 Developing iPhone and apps game

app industry) gets to decide whether to develop apps that are compatible with the new iPhone’s software. Upon observing company 1’s decision, the followers (firm 2 and 3) simultaneously decide whether to develop apps (D) or not develop (ND). Find all subgame perfect Nash equilibria in this sequential-move game. Answer Company 1 develops. Consider the subgame between firms 2 and 3 which initiates after Apple develops the new iPhone and company 1 (the industry leader) develops an application (in the left-hand side of the game tree of Fig. 4.40). Since that subgame describes that firm 2 and 3 simultaneously choose whether or not to develop apps, we must represent it using its normal form in order to find the NEs of this subgame; as we do in the payoff matrix of Fig. 4.41.

208

4

Sequential-Move Games with Complete Information

Fig. 4.41 Smallest proper subgame (after firm 1 develops)

We can identify the best responses for each player (as usual, the payoffs associated to those best responses are underlined in the payoff matrix of Fig. 4.41). In particular, BR2 ðD; DÞ ¼ D and BR2 ðD; NDÞ ¼ D for firm 2; and similarly for firm 3, BR3 ðD; DÞ ¼ D and BR3 ðD; NDÞ ¼ D: Hence, Develop is a dominant strategy for each company, so there is a unique Nash equilibrium (Develop, Develop) in this subgame. Company 1 does not develop. Next, consider the subgame associated with Apple having developed the new iPhone but company 1 not developing an application (depicted in the right-hand side of the game tree in Fig. 4.40). Since the subgame played between firms 2 and 3 is simultaneous, we represent it using its normal form in Fig. 4.42. This subgame has two Nash equilibria: ðDevelop; DevelopÞ and ðDo not develop; Do not developÞ: Note that, in our following discussion, we will have to separately analyze the case in which outcome (D, D) emerges as the NE of this subgame, and that in which (ND, ND) arises. Company 1—Case I. Let us move up the tree to the subgame initiated by Apple having developed the new iPhone. At this point, company 1 (the industry leader) has to decide whether or not to develop an application. Suppose that the Nash equilibrium for the subgame in which company 1 does not develop an application is (Develop, Develop). Replacing the two final subgames with the Nash equilibrium payoffs we found in our previous discussion, the situation is as depicted in the tree of Fig. 4.43. In particular, if firm 1 develops an application, then outcome (D, D) which entails payoffs (10, 4, 4, 4), as depicted in the terminal node at the bottom left-hand side of Fig. 4.43. If, in contrast, firm 1 does not develop an application for the new iPhone, then outcome (D, D) arises, yielding payoffs of (6, 0, 2, 2), as depicted in the bottom right-hand corner of Fig. 4.43.

Exercise 20—Backward Induction-IIB

209

Fig. 4.42 Smallest proper subgame (after firm 1 does not develop)

Fig. 4.43 Extensive-form subgame (Case I)

We can now analyze firm 1’s decision. If company 1 develops an application, then its payoff is 4, while its payoff is only 0 (since it anticipates the followers developing apps) from not doing so. Hence, company 1 chooses Develop. Company 1—Case II. Now suppose the Nash equilibrium of the game that arises after firm 1 does not develop an app has neither firm 2 or 3 developing an app. Replacing the two final subgames with the Nash equilibrium payoffs we found in our previous discussion, the situation is as depicted in Fig. 4.44. Specifically, if firm 1 develops, outcome (D, D) emerges, which entails payoffs (10, 4, 4, 4); while if firm 1 does not develop firm 2 and 3 respond not developing apps either, ultimately yielding a payoff vector of (−6, 0, 0, 0). In this setting, if firm 1 develops an application, its payoff is 4; while its payoff is only 0 from not doing so. Hence, firm 1 chooses Develop.

210

4

Sequential-Move Games with Complete Information

Fig. 4.44 Extensive-form subgame (Case II)

Thus, regardless of which Nash equilibrium is used in the subgame initiated after firm 1 chooses Do not develop (in the right-hand side of the game in Figs. 4.43 and 4.44), firm 1 (the leader) optimally chooses to Develop. First mover (Apple). Operating by backward induction, we now consider the first mover in this game (Apple). If Apple chooses to develop the new iPhone, then, as previously derived, firm 1 develops an application and this induces all followers 2 and 3 to do so as well. Hence, Apple’s payoff is 10 from introducing the new iPhone. It is then optimal for Apple to develop the new iPhone, since its payoffs from so doing, 10, is larger than from not developing it, 0. Intuitively, since Apple anticipates all app developers will react introducing new apps, it finds the initial introduction of the iPhone to be very profitable. We can then identify two subgame perfect Nash equilibria (where a strategy for firm 2, as well as for firm 3, specifies a response to firm 1 choosing Develop and a response to company 1 choosing Do not develop3): ðDevelop iPhone; Develop; Develop=Develop; Develop=DevelopÞ; and ðDevelop iPhone; Develop; Develop=Do not develop; Develop=Do not developÞ:

For instance, a strategy Develop/Develop for firm 2 reflects that this company chooses to develop apps, both after observing that firm 1 develops apps and after observing that firm 1 does not develop.

3

Exercise 20—Backward Induction-IIB

211

Note that both SPNE result in the same equilibrium path, whereby, first, Apple introduces the new iPhone, the industry leader (firm 1) subsequently chooses to develop applications for the new iPhone, and finally firms 2 and 3 (observing firm 1’s apps development) simultaneously decide to develop apps as well.

Exercise 21—Moral Hazard in the WorkplaceB Consider the following moral hazard game between a firm and a worker. A firm offers either Contract 1 to a worker, which guarantees him a wage of w = 26 regardless of the outcome of the project that he develops in the firm, or Contract 2, which gives him a salary of $36 when the outcome of the project is good (G) but only $9 if the outcome is bad (B), i.e., wG ¼ 36 and wB ¼ 9. The worker can exert two levels of effort, either high ðeH Þ or low ðeL Þ. The probability that, given a high effort level, the outcome of the project is good is f ðGjeH Þ ¼ 0:9 and, therefore, the probability that, given a high effort, the outcome is bad is only f ðBjeH Þ ¼ 0:1. If, instead, the worker exerts a low effort level, the good and bad outcome are equally likely, i.e., f ðGjeL Þ ¼ 0:5 and f ðBjeL Þ ¼ 0:5. The worker’s utility function is Uw ðw; eÞ ¼

pffiffiffiffi w  lðeÞ;

pffiffiffiffi where w reflects the utility from the salary he receives, and lðeÞ represents the disutility from effort which, in particular, is lðeH Þ ¼ 1 when he exerts a high effort but lðeL Þ ¼ 0 when his effort is low (he shirks). The payoff function for the firm is 90  w when the outcome of the project is good, but decreases to 30  w when the outcome is bad. Part (a) Depict the game tree of this sequential-move game. Be specific in your description of players’ payoffs in each of the eight terminal nodes of the game tree. Part (b) Find the subgame perfect equilibrium of the game. Part (c) Consider now the existence of a social security payment that guarantees a payoff of $x to the worker. Determine for which monetary amount of $x the worker chooses to exert a low effort level in equilibrium. Answer Part (a) Fig. 4.45 depicts the game tree of this moral hazard game: First, the firm manager offers either contract 1 or 2 to the worker. Observing the contract offer, the worker then accepts or reject the contract. If he rejects the contract, both players obtain a payoff of zero. However, if the worker accepts it, he chooses to exert a high or low effort level. Finally, for every effort level, nature randomly determines whether such effort will result in a good or bad outcome, i.e., profitable or unprofitable project for the firm. As a summary, Fig. 4.46 depicts the time structure of the game.

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Fig. 4.45 Moral hazard in the workplace

Fig. 4.46 Time structure of the moral hazard game

Exercise 21—Moral Hazard in the WorkplaceB

213

In order to better understand the construction of the payoffs in each terminal node, consider for instance the case in which the worker accepts Contract 2 (in the right-hand side of the tree), responds exerting a low effort level and, afterwards, the outcome of the project is good (surprisingly!). In this setting, his payoff becomes pffiffiffiffiffi 36  0 ¼ 6, since he receives a wage of $36 and exerts no effort, while the firm’s payoff is 90  36 ¼ 54, given that the outcome was good and, as a result of the monetary incentives in Contract 2, the firm has to pay a salary of $36 to the worker when the outcome is good. If the firm offers instead Contract 1, the worker’s salary remains constant regardless of the outcome of the project, but decreases in his effort (as indicated in the four terminal nodes in the left-hand side of the game tree). Part (b) In order to find the SPNE, we must first notice that in this game we can define 4 proper subgames plus the game as a whole (as marked by the large circles in the game tree of Fig. 4.47).

Fig. 4.47 Proper subgames in the moral hazard game

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Let us next separately analyze the Nash equilibrium of every proper subgame: Subgame (1), initiated after the worker accepts Contract 1. Let’s first find the expected utility the worker obtains from exerting a high effort level. pffiffiffiffiffi pffiffiffiffiffi EUw ðHjContract 1Þ ¼ f ðGjeH Þð 26  1Þ þ f ðBjeH Þð 26  1Þ pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi ¼ 0:9ð 26  1Þ þ 0:1ð 26  1Þ ¼ 26  1 And the expected utility of exerting a low effort level under Contract 1 is pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi EUw ðLjContract 1Þ ¼ 0:5 26 þ 0:5 26  0 ¼ 26 Hence, the worker exerts a low effort level, eL , since EUw ðHjContract 1Þ\EUw ðLjContract 1Þ $

pffiffiffiffiffi pffiffiffiffiffi 26  1\ 26

And the worker chooses to exert eL after accepting Contract 1. Intuitively, Contract 1 offers the same salary regardless of the outcome. Thus, the worker does not have incentives to exert a high effort. Subgame (2), initiated after the worker accepts Contract 2. In this case, the expected utility from exerting a high or low effort level is: EUw ðHjContract 2Þ ¼ 0:9 5 þ 0:1 2 ¼ 4:7; and EUw ðLjContract 2Þ ¼ 0:5 6 þ 0:5 3  0 ¼ 4:5 Hence, the worker exerts a high effort level, eH , after accepting Contract 2. Subgame (3), initiated after receiving Contract 1. Operating by backwards induction, the worker can anticipate that, upon accepting Contract 1, he will exert a low effort (this was, indeed, the Nash equilibrium of subgame 1, as described above). Hence, the worker faces the reduced-form game in Fig. 4.48 when analyzing subgame (3). As a consequence, the worker accepts Contract 1, since his expected payoff from pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi accepting the contract, 0:5 26 þ 0:5 26 ¼ 26, exceeds his payoff from rejecting it (zero). That is, anticipating a low effort level upon the acceptance of Contract 1, pffiffiffiffiffi pffiffiffiffiffi EUw ðAjContract 1; eL Þ  EUw ðRjContract 1; eL Þ $ 0:5 26 þ 0:5 26  0 Subgame (4), initiated after the worker receives Contract 2. Operating by backwards induction from subgame (2), the worker anticipates that he will exert a high

Exercise 21—Moral Hazard in the WorkplaceB

215

Fig. 4.48 Subgame 3 in the moral hazard game

Fig. 4.49 Subgame 4 in the moral hazard game

effort level if he accepts Contract 2. Hence, the worker faces the reduced-form game in Fig. 4.49. Therefore, the worker accepts Contract 2, given that: EUw ðAjContract 2; eH Þ  EUw ðRjContract 2; eH Þ 0:9 6 þ 0:1 3  1  0 $ 4:7 [ 0

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Fig. 4.50 Reduced-form game

Game as a whole: Operating by backward induction, we obtain the reduced-form game in Fig. 4.50. When receiving Contract 1, the worker anticipates that he will accept it to subsequently exert a low effort; while if he receives Contract 2 he will also accept it but in this case exert a high effort. Analyzing the above reduced-form game, we can conclude that the employer prefers to offer Contract 2, since it yields a larger expected payoff. In particular, EpðContract 1jA; eL Þ\EpðContract 2jA; eH Þ; that is 0:5 64 þ 0:5 4\0:9 54 þ 0:1 21 $ 34\50:7 Hence, in the SPNE the employer will offer Contract 2 to the worker paying him the wage scheme ðwG ; wB Þ ¼ ð36; 9Þ, which induces the worker to exert a high effort level in Contract 2 (in equilibrium) and a low effort in Contract 1 (off-the-equilibrium). Part (c) In part (b) we obtained that, in equilibrium, the employer offers Contract 2 and the worker exerts high effort. Since the social security payment needs to achieve that the worker exerts a low effort, the easiest way to guarantee this is by making the worker reject Contract 2. Anticipating such rejection, the employer offers Contract 1, which implies an associated low level of effort. That is, a social security payment of $x induces the worker to reject Contract 2 as long as $x satisfies

Exercise 21—Moral Hazard in the WorkplaceB

217

EUw ðAjContract 2; eH Þ\EUw ðRjContract 2; eH Þ pffiffiffiffiffi pffiffiffi 0:9 36 þ 0:1 9  1\x which simplifies to x [ 4:7: That is, the social security payment must be relatively generous. In order to guarantee that the worker still accepts Contract 1, we need that EUw ðAjContract 1; eL Þ [ EUw ðRjContract 1; eL Þ pffiffiffiffiffi pffiffiffiffiffi 0:5 26 þ 0:5 26 [ x pffiffiffiffiffi which reduces to x\ 26 ffi 5:09, thus implying that the social security payment cannot be too generous since otherwise the worker would reject both types of contracts. Hence, when the social security payment is intermediate, i.e., it lies in the pffiffiffiffiffi interval x 2 ½4:7; 26, we can guarantee that the worker rejects Contract 2, but accepts Contract 1 (where he exerts a low effort level).

5

Applications to Industrial Organization

Introduction This chapter helps us apply many of the concepts of previous chapters, dealing with simultaneous- and sequential-move games under complete information, to common industrial organization problems. In particular, we start with a systematic search for pure and mixed strategy equilibria in the Bertrand game of price competition between two symmetric firms, where we use several figures to illustrate our discussion. We then extend our explanation to settings in which firms are allowed to exhibit different costs. We also explore the effects of cost asymmetry in the Cournot game of output competition. We afterwards move to a Cournot duopoly game in which one of the firms is publicly owned and managed. While private companies seek to maximize profits, public firms seek to maximize a combination of profits and social welfare, thus affecting its incentives and equilibrium output levels. At the end of the chapter, we examine applications in which firms can choose to commit to a certain strategy (such as spending in advertising, or competing in prices or quantities) before they start their competition with other firms. Interestingly, these pre-commitment strategies affect firm’s competitiveness, and the position of their best-response functions, ultimately impacting subsequent equilibrium output levels and prices. We also analyse an incumbent’s incentives to invest in a more advanced technology that helps him to reduce its production costs in order to deter potential entrants from the industry; even if it leaves the incumbent with overcapacity (idle or unused capacity). We end the chapter with two more applications: one about a firm’s tradeoff between directly selling to its customers or using a retailer; and another in which we analyze firms’ incentives to merge, which depend on the proportion of firms in the industry that join the merger.

© Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_5

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Exercise 1—Bertrand Model of Price CompetitionA Consider a Bertrand model of price competition between two firms facing an inverse demand function pðqÞ ¼ a  bq, and symmetric marginal production costs, c1 ¼ c2 ¼ c, where c < a. Part (a) Show that in the unique Nash equilibrium (NE) of this game, both firms set a price that coincides with their common marginal cost, i.e., p1 ¼ p2 ¼ c. Part (b) Show that p1 ¼ p2 ¼ c is not only the unique NE in pure strategies, but also the unique equilibrium in mixed strategies. Answer Part (a) We will analyze that p1 ¼ p2 ¼ c is indeed a Nash equilibrium (NE). Notice that at this NE both firms make zero profits. Now let us check for profitable deviations: • If firm i tries to deviate by setting a higher price pi [ c then firm j will capture the entire market, and firm i will still make zero profits. Hence, such a deviation is not profitable. • If firm i tries to deviate by setting a lower price pi \c then it sells to the whole market but at a loss for each unit sold given that pi \c. Hence, such a deviation is not profitable either. As a consequence, once firms charge a price that coincides with their common marginal cost, that is, p1 ¼ p2 ¼ c; there does not exist any profitable deviation for any of the firms. Therefore, this pricing strategy is a NE. Let us now show that, moreover, the NE strategies p1 ¼ p2 ¼ c are the unique NE strategies in the Bertrand duopoly model with symmetric costs c1 ¼ c2 ¼ c.
 • First, let us suppose that min p1 ; p2 \c. In this case, the firm setting the lowest
  price pi ¼ min p1 ; p2 is incurring losses; as depicted in Fig. 5.1. If this firm raises its prices beyond the marginal cost, i.e., pi [ c, the worst that can happen is that its price is higher than that of its rival, pj \pi , and as a consequence firm i loses all its customers thus making zero profits (otherwise firm i will capture the entire market and make
profits,which occurs when pj > pi > c). Then, the price choices given by min p1 ; p2 \c cannot constitute a NE because there exist profitable deviations. [This argument is true for any firm i = {1, 2} which initially has the lowest price.]

Exercise 1—Bertrand Model of Price CompetitionA

221

Fig. 5.1 pi < pj < c

Fig. 5.2 Firm j has incentives to increase its price

• Second, let us suppose that pj ¼ c and pk [ c where k 6¼ j. Then, firm j captures the entire market, since it is charging the lowest price, but makes zero profits. Firm j has then incentives to raise its price from pj ¼ c to ^ pj ¼ pk  e, slightly undercutting its rival’s price; as depicted in Fig. 5.2. With this deviation, firm j still captures all the market (it is still the firm with the lowest price), but at a positive margin. Then, the initially considered price choices pj ¼ c and pk [ c where k 6¼ j cannot constitute a NE, since there are profitable deviations for firm j. • Third, let us suppose that both firms set prices above the common   marginal cost, i.e., pj [ c and pk [ c, but consider that such a price pair pj ; pk is asymmetric, so that we can identify the lowest price pj  pk , whereby firm j captures the entire market. Hence, firm k earns zero profit, and has incentives to lower its price. If it undercuts firm j’s price, i.e., pk ¼ pj  e, setting a price lower than that of its rival but higher than the common marginal cost (as illustrated in Fig. 5.3), then firm k captures all the market, raising its profits to ðpk  cÞ  qðpk Þ. Hence, we found a profitable deviation for firm k, and therefore, pk [ pj [ c cannot be sustained as a NE. Therefore, we have shown that any price configuration different from p1 ¼ ¼ c cannot constitute a NE. Since at least one firm has incentives to deviate from its strategy in such price profile. Therefore, the unique NE of the Bertrand duopoly model is p1 ¼ p2 ¼ c. Graphically representing best response functions in the Bertrand model. As a remark, we next describe how to represent firms’ best-response functions in this game. Figure 5.4 depicts, for every price firm 2 sets, p2, the profit maximizing price of firm 1. First, if p2 \c, firm 2 is capturing all the market (at a loss), and firm 1 p2

Fig. 5.3 Firm k has incentives to undercut firm j’s price

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Fig. 5.4 Firm 1’s best response function in the Bertrand game

responds setting p1 ¼ c; as illustrated in the horizontal segment for all p2 \c. Note that if firm 1 were to match firm 2’s price, despite sharing sales, it would guarantee losses for each unit sold. Hence, it is optimal to respond with a price of p1 = c. Second, if firm 2 increases its price to c\p2 \pm ; where pm denotes the monopoly price that firm 2 would set if being alone in the market, then firm 1 responds slightly undercutting its rival’s price, i.e., p1 ¼ p2  e; as depicted in the portion of firm 1’s best response function that lies e-below the 45° line. Finally, if firm 2 sets an extremely high price, i.e., p2 [ pm , then firm 1 responds with a price p1 ¼ pm . Such a monopoly price is optimal for firm 1, since it is lower than that of its rival (thus allowing firm 1 to capture all the market) and maximizes firm 1’s profits; as illustrated in the horizontal segment at the right hand of the figure. A similar analysis is applicable to firm’s 2 best response function, as depicted Fig. 5.5. Note that, in order to superimpose both firms’ best response functions in the same figure, we use the same axis as in the previous figure. We can now superimpose both firms’ best response functions, immediately obtaining that the only point where they both cross each other is p1 ¼ p2 ¼ c, as shown in our above discussion (Fig. 5.6). Part (b) Let us work by contradiction, so let us assume that there exists a mixed strategy Nash Equilibrium (MSNE). It has to be characterized by firms randomizing between at least two different prices with strictly positive probability. Firm 1 must randomize as follows:

Exercise 1—Bertrand Model of Price CompetitionA

223

Fig. 5.5 Firm 2’s best response function in the Bertrand game

Fig. 5.6 Equilibrium pricing in the Bertrand model

That is, firm 1 mixes setting a price p1,  p1 ¼

p11 p21

with with

prob prob

q21

q11 ¼ 1  q11

where, for generality, we allow the prices and their corresponding probabilities to take any values. Firm 2 must be similarly mixing between two prices, as follows:

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Applications to Industrial Organization

That is, firm 2 sets a price p2,  p2 ¼

p12 p22

with with

prob prob

q22

q12 ¼ 1  q12

For simplicity, let us first examine randomizations that cannot be profitable for either firm: • First, note that in any MSNE we cannot have both firms losing money: p1i \c and p2i \c for all firm i ¼ f1; 2g. In other words, we cannot have that both the lower and the upper bound of the randomizing interval lie below the marginal cost c, for both firms; as illustrated in Fig. 5.7. • Second, if both p1i [ c and p2i [ c for all firm i ¼ f1; 2g, both the lower and the upper bounds of the randomizing intervals of both firms lie above their common marginal cost; as depicted in Fig. 5.8. Then one of the firms, for instance firm j, can gain by deviating to a pure strategy that puts all probability weight on a single price, i.e., p1j ¼ p2j ¼ pj , which slightly undercuts its rival’s price. In particular, firm j can benefit from setting a price slightly below the lower bound of its rival’s randomization interval, i.e., pj ¼ p1k  e. Hence we cannot have p1i [ c and p2i [ c being part of a MSNE, because there are profitable deviations for at least one firm.

Fig. 5.7 Randomizing intervals below c

Fig. 5.8 Randomizing intervals above c

Exercise 1—Bertrand Model of Price CompetitionA

225

Fig. 5.9 Randomizing interval around c

• Finally, let us consider that for both firms i ¼ f1; 2g we have that p1i [ c and p2i \c, as depicted in Fig. 5.9. That is, the lower bound of randomization, p1i , lies below the common marginal cost, c, while the upper bound, p2i , lies above this cost. In this setting, every firm j could win the entire market by setting a price in which, rather than randomizing, uses a degenerate probability distribution (i.e., a pure strategy) by setting a price p1j ¼ p2j ¼ pj , which slightly undercuts its rival’s lower bound of price randomization, i.e., pj ¼ p1k  e. Therefore, having for both firms p1i \c and p2i [ c cannot be a MSNE given that there are profitable deviations for every firm j. Hence, we have reached a contradiction, since no randomizing pricing profile can be sustained as a MSNE of the Bertrand game. Thus, no MSNE exists. As a consequence, the unique NE implies that both firms play degenerated (pure) strategies, namely, p1 ¼ p2 ¼ c.

Exercise 2—Bertrand Competition with Asymmetric CostsB Consider a Bertrand model of price competition where firms 1 and 2 face different marginal production costs, i.e., c1 \c2 , and an inverse demand function pðQÞ ¼ a  bQ, where Q denotes aggregate production, and a [ c2 [ c1 . Part (a) Find all equilibria in pure strategies when you allow prices to be a continuous variable. Part (b) Find all equilibria in pure strategies when you allow prices to be a discrete variable (e.g., prices must be expressed in dollars and cents). Part (c) Repeat your analysis for mixed strategies (for simplicity, you can focus on the case of continuous pricing). Answer Part (a). We start by showing that, in a Bertrand duopoly model with heterogeneous costs c1 < c2< a and continuous pricing, there does not exist a Nash equilibrium in which both firms use pure strategies, i.e., where every firm charges a price with 100% probability.

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Proof. In order to prove this claim we will show that, for any possible pricing profile, we can find a profitable deviation. All the possible pricing profiles are: (1) pi  pj \c1 \c2 . In this case, both firms are making negative profits, since they are both setting a price below their corresponding marginal cost. Hence, they both have incentives to increase their prices. (2) pi  c1 \pj \c2 . Firm i is making negative (or zero) profits. Note that this is true regardless of its identity, either firm 1 or firm 2. Hence, this firm then has incentives to charge a price pi ¼ pj  e for a small e [ 0. When firm i’s identity is firm 1, or to a price pi = c2 when its identity is firm 2. Either way, firm i has incentives to deviate from the original configuration of prices. (3) c1 \pi  pj \c2 . Firm i serves all the market. So it is profitable for firm i to increase its price to pi ¼ pj  e, i.e., undercutting its rival’s price by e. (4) c1 \p1  c2 \p2 . Both firms are making positive (or zero) profits, and firm 1 is capturing all the market. In this setting, firm 1 has incentives to set a price marginally close to its rival’s cost, p1 ¼ c2  e, which allows firm 1 to further increase its per-unit profits. (5) c1 \p1  c2 ¼ p2 . If pi ¼ pj ¼ c2 , then firms are splitting the market. In this context, the very efficient firm 1 has an incentive to set a price p1 ¼ c2  e, which undercuts the lowest price that its (less efficient) competitor can profitably charge, helping firm 1 to capture all the market. (6) c1 \c2  pi  pj . In this context, firm i serves all the market if pi < pj, or firms share the market, if pi = pj. However, firm j has an incentive to undercut firm i’s price by setting pj ¼ pi  e. (7) p1  c1 \c2  p2 . In this setting, firm 1 serves all the market and makes negative (zero) profits if p1 \c1 ðp1 ¼ c1 , respectively). Firm 1 has incentives to set p1 ¼ c2  e for small e, which implies that this firm captures all the market and makes a positive margin on every unit sold. (8) p2  c1 \c2  p1 . This case is analog to the previous one. Since firm 1 is making zero profits (no sales), this firm has incentives to deviate to a lower price, i.e., p1 ¼ c2  e, allowing it to capture all the market. Hence, we have checked that, for any profile of prices, some (or both) firms have profitable deviations. Therefore, none of these strategy profiles constitutes a pure strategy Nash equilibria. Hence, in the Bertrand duopoly model with heterogeneous costs, c1 < c2, and continuous pricing there does not exist pure strategy Nash equilibria. Part (b). In this part of the exercise, we consider discrete pricing. In this setting, if the cost differential between firms satisfies c2  c1 [ e (that is, c2 [ c1 þ eÞ, the unique Bertrand equilibrium is for the least competitive firm (firm 2 in this case) to set a price that coincides with its marginal cost, p2 ¼ c2 , while the more competitive firm (company 1) sets a price slightly below that of its rival, i.e., p1 ¼ c2  e. Note that this pricing profile yields an equilibrium output of

Exercise 2—Bertrand Competition with Asymmetric CostsB

q2 ¼ 0 for firm 2 and q1 ¼

227

a  ðc2  eÞ for firm 1: b

In this context, firm 2 does not have incentives to deviate to a lower price. In particular, while a lower price captures some consumers, it would be at a loss for every unit sold. Similarly, firm 1 does not have incentives to lower prices, given that doing so would reduce its profit per unit. Firm 1 cannot increase prices either (further approaching them to c2) since, unlike in the previous part of the exercise (where such convergence can be done infinitely), in this case prices are discrete. Intuitively, firm 1 is charging 1 cent less than firm 2’s marginal cost c2. Part (c). We now seek to find the msNE of the Bertrand game with heterogeneous costs, c1 \c2 \a. In this setting, the following strategy constitutes a mixed strategy Nash equilibrium (MSNE) of the game: • The most competitive firm (Firm 1 in this case) sets a price that coincides with its rival’s marginal cost, i.e., p1 ¼ c2 • The least competitive firm (Firm 2) sets a price p2 by continuously randomizing over the interval ½c2 ; c2 þ e for any e [ 0, with a cumulative distribution function Fðp; eÞ, with associated density f ðp; eÞ [ 0 at all prices p; as depicted in Fig. 5.10. We now prove that the above is, indeed, a msNE of the Bertrand game with heterogeneous costs. Proof. In order to check that this strategy profile constitutes a msNE we need to check that each firm i is playing its best response, BRi . • Let us fix firm 2 randomizing over ½c2 ; c2 þ e, and check if firm 1 is playing its BR1 by setting p1 ¼ c2 (as prescribed). First, note that firm 1 (the most competitive company) sets the lowest price of all firms in equilibrium, except in the event that firm 2 sets a price exactly equal to p2 ¼ c2 . This event, however, occurs with a zero-probability measure, since firm 2 continuously randomizes over the interval ½c2 ; c2 þ e. Hence, firm 1 captures all the market demand, except in a zero-probability measure event.

Fig. 5.10 MSNE in the Bertrand game with asymmetric firms

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If firm 1 deviates towards a lower price, i.e., setting p1 \c2 , then it lowers its profits for every unit sold relative to those in p1 ¼ c2 . Hence, p1 \c2 is not a profitable deviation for firm 1. If firm 1 sets instead a higher price, p1 [ c2 , then two effects arise: on one hand, firm 1 makes a larger margin per unit (as long as its price, p1, still lies below that of firm 2) but, on the other hand, firm 1 gives rise to a negative effect, since it increases its probability of ending up with no sales (if the realization of p2 is p2 \p1 Þ at a rate of f ðp; eÞ. In order to show that the negative effect of setting a price p1 [ c2 , i.e., a price in the interval ½c2 ; c2 þ e, dominates its positive effect (and thus firm 1 does not have incentives to deviate), let us next examine firm 1’s profits Amount sold at price p zffl}|ffl{ p1 ð pÞ ¼ ð1  Fðp; eÞ Þ DðpÞ ðp  c1 Þ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} probfp1 \p2 g

per unit profits

In this context, a marginal increase in its price yields the first order condition: @p1 ð pÞ ¼ f ðp; eÞDðpÞðp  c1 Þ þ ð1  F ðp; eÞÞ½D0 ð pÞðp  c1 Þ þ DðpÞ\0 @p Hence, firm 1 doesn’t have incentives to set a price above p1 ¼ c2 . • Let us now fix firm 1’s strategy at p1 ¼ c2 , and check if firm 2 randomizing over ½c2 ; c2 þ e is a best response for firm 2. If firm 2 sets a price below c2 (or randomizing continuously with a lower bound below c2) then firm 2 makes zero (or negative) profits. If firm 2 sets a price above c2 (or randomizing continuously with an upper bound above c2) then firm 2 makes zero profits as well, since it captures no customers willing to buy the product at such a high price. Therefore, there does not exist a profitable deviation for firm 2, and the strategy profile can be sustained as a MSNE of the Bertrand game of price competition when firms exhibit asymmetric production cost.1

Exercise 3—Duopoly Game with A Public FirmB Consider a market (e.g. oil and natural gas in several countries) with one public firm (firm 0), and one private firm, (firm 1). Both firms produce a homogenous good with identical and constant marginal cost c > 0 per unit of output, and face the same

For more details on this MSNE, see Andreas Blume (2003) “Bertrand without fudge” Economic Letters, 7, pp. 167–68.

1

Exercise 3—Duopoly Game with A Public FirmB

229

inverse linear demand function p(X) = a − b X with aggregate output X = x0 + x1, and where a > c. The private firm maximizes its profit p1 ¼ pðXÞx1  cx1 ; And the public firm maximizes a combination of social welfare and profits V0 ¼ hW þ ð1  hÞp0 where social welfare (W) is given by ZX W¼

pðyÞdy  cðx0 þ x1 Þ; 0

and its profits are p0 ¼ pðXÞx0  cx0 : Intuitively, parameter h represents the weight that the manager of the public firm assigns to social welfare, while (1 − h) is the weight that he assigns to its profits. Both firms simultaneously and independently choose output (as in the Cournot model of quantity competition). Part (a) In order to better understand the incentives of these firms, and compare them with two private firms competing in a standard Cournot model, first find the best-response functions of the private firm, x1 ðx0 Þ, and of the public firm, x0 ðx1 Þ. Part (b) Depict the best response functions. For simplicity, you can restrict your analysis to h ¼ 0; h ¼ 0:5; h ¼ 1. Intuitively explain the rotation in the public firm’s best response function as h increases. Connect your results with those of a standard Cournot model where both firm’s only care about profits, i.e., h ¼ 0. Part (c) Calculate the equilibrium quantities for the private and public firms. Find the aggregate output in equilibrium. Part (d) Calculate the socially optimal output level and compare it with the equilibrium outcome you obtained in part (c). Answer Part (a) The private firm maximizes p1 ¼ ½a  bðx0 þ x1 Þx1  cx1 ; Taking the first-order conditions with respect to x1 , we find the best-response function of the private firm:

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x 1 ðx 0 Þ ¼

Applications to Industrial Organization

a  c x0  : 2b 2

The public firm maximizes V0 ¼ hW þ ð1  hÞp0 . That is, 2 x þx 3 Z0 1 V ¼ ð1  hÞð½a  bðx0 þ x1 Þx0  cx0 Þ þ h4 ða  bxÞdx  cðx0 þ x1 Þ5; 0

Taking the first-order conditions with respect to x0 , we have @V ¼ ð1  hÞða  bx1  c  2bx0 Þ þ h½a  bðx0 þ x1 Þ  cÞ ¼ 0; @x0 And solving for x0 , we find the best-response function of the public firm: x0 ¼

ac x     1 h : h 2b 1  2 2 12

Part (b) In order to better understand the effect of h on the public firm’s best response function and, as a consequence, on equilibrium behavior, let us briefly examine some comparative statistics of parameter h. In particular, when h increases from 0 to 1 the best-response function of the public firm pivots outward, with its vertical intercept being unaffected. Let us analyze some extreme cases (Fig. 5.11 depicts the public firm’s best response function evaluated at different values of h): • At h = 0, the best response function of the public firm becomes analog to that of private firm: the public firm behaves exactly as the private firm and the equilibrium is symmetric, with both firms producing the same amount of output.

Fig. 5.11 Best response function of the private firm (solid line) and public firm (dashed line)

Exercise 3—Duopoly Game with A Public FirmB

231

Intuitively, this is not surprising: when h = 0 the manager of the public company assigns no importance to social welfare, and only cares about profits.   • As h increases, the slope the best-response function of the public firm, 2 1  h2 , becomes flatter, and the public firm produces a larger share of industry output, i.e. the crossing point between both best response functions happens more to the southeast. • At h = 1, only the public company produces in equilibrium, while the private firm does not produce (corner solution). Part (c) The equilibrium quantities solve the system of two equations: a  c x0  ; and 2b 2 ac x     1 h : x0 ¼ 2b 1  h2 2 12 x1 ¼

Simultaneously solving for x0 and x1 , we find the individual output levels: x0 ¼

1 ac 1h ac ; and x1 ¼ : 3  2h b 3  2h b

Note that for h = 0 the outcome is the same as that for a Cournot duopoly, with both firms producing x0 ¼ x1 ¼ ac 3b , and for h = 1 the public firm produces the competitive outcome, x0 ¼ ac , and the private firm produces nothing, x1 ¼ 0. (This b result was already anticipated in our discussion of the pivoting effect in the best response function in the previous question.) The aggregate output as a function of h is, therefore, X ðhÞ ¼ x0 þ x1 ¼

2h ac : 3  2h b

Finally, note that differentiating the aggregate output with respect to h yields dX ðhÞ 1 ac ¼ [ 0: 2 dh ð3  2hÞ b Hence, an increase in h results in an increase in output, and an increase in social welfare. Part (d) The socially optimal output level corresponds to p(X) = c, which implies 2h ac a  bX ¼ c, i.e., X ¼ ac b :. The equilibrium output of the mixed duopoly, 32h b , is ac below the socially optimal level, b , as long as 2h \1: 3  2h

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This inequality simplifies to 2  h\3  2h, and further collapses to h\1. Intuitively, equilibrium output is socially insufficient when the manager of the public firm assigns less than 100% weight on social welfare. However, when she assigns 100% weight on social welfare, h ¼ 1, the equilibrium output in this mixed duopoly coincides with the socially optimal output.

Exercise 4—Cartel with Firms Competing a la Cournot— Unsustainable in a One-Shot gameA Consider two firms competing in output levels (a la Cournot) and facing linear inverse demand pðQÞ ¼ 100  Q, where Q ¼ q1 þ q2 denotes aggregate output. For simplicity, assume that firms face a common marginal cost of production c ¼ 10. (a) Best response function. Set up each firm’s profit-maximization problem and find its best response function. Interpret. (b) Cournot competition. Find the equilibrium output each firm produces when competing a la Cournot (that is, when they simultaneously and independently choose their output levels). In addition, find the profits that each firm earns in equilibrium. (c) Collusive agreement. Assume now that the CEOs from both companies meet for lunch and start talking about how they could increase their profits if they could increase market prices by reducing their production. That is, firms are trying to collude! Set up the maximization problem that firms solve when maximizing their joint profits (that is, the sum of profits for both firms). Find the output level that each firm should select to maximize joint profits. In addition, find the profits that each firm obtains in this collusive agreement. (d) Profit comparison. Compare the profits that firms obtain when competing a la Cournot (from part b) against their profits when they successfully collude (from part c). (e) Unsustainable cartel. Let us now show that the collusive agreement from part (c) cannot be sustained as a Nash equilibrium of the one-shot game, that is, when firms interact only once. Using the best response function found in part (a), show that when firm i chooses the collusive output found in part (c) its rival (firm j) responds with an output that is different from its collusive output. Interpret. Answer Part (a). Firm i’s profit function is given by:      pi qi ; qj ¼ 100  qi þ qj qi  10qi

Exercise 4—Cartel with Firms Competing a la Cournot …

233

Differentiating with respect to output qi yields, 100  2qi  qj  10 ¼ 0 Solving for qi , we find firm i’s best response function   1 qi qj ¼ 45  qj 2 This best response function indicates that firm i decreases its output as its rival increases its own production level, qj . Firm j has a symmetric best response function since firms are symmetric. Both best response functions have a slope of  12. They also have the same vertical intercepts as costs are symmetric. Part (b). Invoking symmetry in firms’ output, we obtain qi ¼ 45 

qi 2

Solving this equation yields qi ¼ q1 ¼ q2 ¼ 30 units. Substituting these output levels into profits p1 ðq1 ; q2 Þ and p2 ðq1 ; q2 Þ, we find equilibrium profits of

 pi qi ; qj ¼ pi ð30; 30Þ ¼ ½100  ð30 þ 30Þ30  10  30 ¼ $900 where profits coincide for every firm, that is,

equilibrium         pi qi ; qj ¼ p1 q1 ; q2 ¼ p2 q1 ; q2 ¼ $900. In words, every firm earns $900 in equilibrium, entailing a combined profit of $900 þ $900 ¼ $1800. Part (c). In this case, firms maximize their joint profits, as captured by the sum p1 ðq1 ; q2 Þ þ p2 ðq1 ; q2 Þ, which we expand as follows


  
    100  qi þ qj qi  10qi þ 100  qi þ qj qj  10qj

Differentiating joint profits with respect to qi , yields 100  2qi  qj  10  qj ¼ 0 Solving for qi , we obtain   qi qj ¼ 45  qj Invoking symmetry in output levels, we can rewrite the above result as follows qi ¼ 45  qi

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Solving for output level qi , we obtain qi ¼ q1 ¼ q2 ¼

45 ¼ 22:5 units: 2

In words, when firms maximize their joint profits, each firm only produces 22.5 units, as opposed to the 30 units every firm produces when competing a la Cournot (see part b of the exercise). This output level yields each firm a profit of

 pi qi ; qj ¼ pi ð22:5; 22:5Þ ¼ f½100  ð22:5 þ 22:5Þ22:5  ð10  22:5Þg ¼ $1012:5 Thus, each firm makes a profit of $1012.5, which yields a joint profit of $1012.5 + $1012.5 = $2025. Part (d). We can clearly see that the profits for each firm in part (b) when firms compete in quantities, $900, are lower than the profits for each firm in part (c) when firms coordinate their production decisions to maximize their joint profits, $1012.5. This naturally means that combined profits are greater in part (c) as well. Part (e). If firm i chooses the collusive outcome (producing qi ¼ 22:5 units, as found in part d), its rival (firm j) can best respond to this output level by inserting it in its best response function, qj ðqi Þ ¼ 45  12 qi , as follows 1 qj ð22:5Þ ¼ 45  22:5 ¼ 33:75 2 which yields firm j with deviation profits of pi ð33:75; 22:5Þ ¼ f½100  ð33:75 þ 22:5Þ33:75  ð10  33:75Þg ¼ $1139:06 which exceeds firm j’s collusive profit of $1012.5. Thus, every firm has incentives to unilaterally deviate from the collusive agreement. Remark: The question specifically highlights the case of a “one-shot game”. If this game were to be repeated, firms could collude, as defection would be punished by a much lower profit of $900 in future periods. However, in a one-shot game, there is no future punishments that could deter firms from unilaterally deviating from the collusive agreement.

Exercise 5—Cartel with firms competing a la Bertrand …

235

Exercise 5—Cartel with firms competing a la Bertrand— Unsustainable in a one-shot gameA Consider two firms competing in prices (a la Bertrand), and facing linear inverse demand pðQÞ ¼ 100  Q, where Q ¼ q1 þ q2 denotes aggregate output. For simplicity, assume that firms face a common marginal cost of production c ¼ 10. (a) Bertrand competition. Find the equilibrium price that each firm sets when competing a la Bertrand (that is, when they simultaneously and independently set their prices). In addition, find the profits that each firm earns in equilibrium. (b) Collusive agreement. Assume now that the CEOs from both companies meet for lunch and start talking about how they could increase their profits if they could coordinate their price setting decisions. Beware, firms are trying to collude! Set up the maximization problem that firms solve when maximizing their joint profits (that is, the sum of profits for both firms). Find the price that each firm selects when maximizing joint profits. In addition, find the profits that each firm obtains in this collusive agreement. (c) Profit comparison. Compare the profits that firms obtain when competing a la Bertrand (from part a) against their profits when they successfully collude (from part b). (d) Unsustainable cartel. Show that the collusive agreement from part (b) cannot be sustained as a Nash equilibrium of the one-shot game, that is, when firms interact only once. Interpret. Answer Part (a). The Bertrand game considers that firms’ costs are symmetric ðc1 ¼ c2 ¼ c ¼ $ 10Þ and there is no product differentiation (we can see this from the inverse demand function, where the price is equally affected by q1 and q2 Þ. As we know from Exercise#1 in this chapter, the unique Nash equilibrium of this game is for both firms to set a price equal to their common marginal cost p1 ¼ p2 ¼ c ¼ $ 10, obtaining zero profits in equilibrium. Part (b). If firms jointly maximize profits p, they maximize, p ¼ PQðPÞ  cQðPÞ ¼ Pð100  PÞ  10ð100  PÞ Differentiating with respect to price P, we find that @p ¼ 100 þ 10  2P ¼ 0 @P Rearranging, we obtain 100 þ 10  2P ¼ 0

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which yields a collusive price of PC ¼ $55. This price yields a joint profit under collusion of pC ¼ 55ð100  55Þ  10ð100  55Þ ¼ $2025: Further, this collusive profit entails a per-firm profit

$2025 2

¼ $1012:5.

Part (c). In part (a) each firm makes a profit of $0, since both firms set prices equal to their common marginal cost, while in part (b) each firm makes a larger profit of $1012.5. Part (d). Every firm has incentives to undercut its rival’s price marginally, thus claiming the entire market and making larger profits than in the collusive agreement. For example, if firm j cooperates by charging the collusive price of $55 found in part (b) of the exercise, firm i can best respond by charging a price of $54 (one dollar lower than the collusive price of $55, but a similar argument applies if firm i charges a price of $54.99). This deviation from the collusive price helps firm i capture the entire market with deviation profits of pDev ¼ 54ð100  54Þ  10ð100  54Þ ¼ $2024: which are larger than firm i’s profit in the collusive agreement, $1012.5. Therefore, every firm has incentives to unilaterally deviate from the collusive price in the one-shot Bertrand game, implying that collusion is not sustainable in the unrepeated version of the game.

Exercise 6—Cournot Competition with Asymmetric CostsA Consider a duopoly game in which firms compete a la Cournot, i.e., simultaneously and independently selecting their output levels. Assume that firm 1 and 2’s constant marginal costs of production differ, i.e., c1 [ c2 . Assume also that the inverse demand function is pðqÞ ¼ a  bq, with a [ c1 . Aggregate output is q ¼ q1 þ q2 Part (a) Find the pure strategy Nash equilibrium of this game. Under what conditions does it yield corner solutions (only one firm producing in equilibrium)? Part (b) In interior equilibria (both firms producing positive amounts), examine how do equilibrium outputs vary when firm 1’s costs change.

Exercise 6—Cournot Competition with Asymmetric CostsA

237

Answer Part (a) Finding best responses for each firm. In a Nash equilibrium ðq1 ; q2 Þ, firm 1 maximizes its profits by selecting the output level q1 that solves max 

q1  0;q2

     p1 q1 ; q2 ¼ a  b q1 þ q2 q1  c1 q1 ¼ aq1  bq21  bq2 q1  c1 q1

where firm 1 takes the equilibrium output of firm 2, q2 , as given. Similarly for firm 2, which maximizes max

q2  0;q1

     p2 q1 ; q2 ¼ a  b q1 þ q2 q2  c2 q2 ¼ aq2  bq1 q2  bq22  c2 q2

which takes the equilibrium output of firm 1, q1 , as given. Taking first order conditions with respect to q1 and q2 in the above profit-maximization problem, we obtain: @p1 ðq1 ; q2 Þ ¼ a  2bq1  bq2  c1 @q1

ð5:1Þ

@p2 ðq1 ; q2 Þ ¼ a  bq1  2bq2  c2 : @q2

ð5:2Þ

Solving for q1 in expression (5.1) we obtain firm 1’s best response function: q1 ð q2 Þ ¼

a  bq2  c1 a  c1 q2 ¼  2b 2b 2

(Note that, as usual, we rearranged the expression of the best response function 1 1 to obtain two parts: the vertical intercept, ac 2b , and the slope,  2Þ. Graphically, firm ac1 1’s best response function originates at 2b when firm 2 does not produce any units, but firm 1 decreases its output in half unit for every additional unit that firm 2 produces. Similarly, solving for q2 in expression (5.2) we find firm 2’s best response function q 2 ð q1 Þ ¼

a  bq1  c2 a  c2 q1 ¼  : 2b 2b 2

Using best responses to find the Nash equilibrium. Plugging firm 1’s best response function into that of firm 2, we obtain:

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q2 ¼

Applications to Industrial Organization

a  c2 1 a  c1 q2    2 2b 2b 2

Since this expression only depends on q2, we can now solve for q2 to obtain the equilibrium output level for firm 2, q2 ¼

a þ c1  2c2 : 3b

Plugging the output we found q2 into firm 1’s best response function q1 ¼ a2bqb 2 c2 , we obtain firm 1’s equilibrium output level: q1

¼

a  2b

a þ c

1 2c2 3b

b



 c2

¼

a þ c2  2c1 3b

2 2c1  0 we need firm 1’s costs to be sufficiently high, i.e., Hence, for q1 ¼ a þ c3b solving for c1 we obtain a þ2 c2  c1 . A symmetric condition holds for firm 2’s 1 2c2 output. In particular, q2 ¼ a þ c3b  0 arises if a þ2 c1  c2 . Therefore, we can identify three different cases (two giving rise to corner solutions, and one providing an interior solution):

aþc

(1) if ci  2 j for all firm i ¼ f1; 2g, then both firms costs are so high that no firm produces a positive output level in equilibrium; aþc (2) if ci  2 j but cj \ a þ2 ci , then only firm j produces a positive output (intuitively, firm j would be the most efficient company in this setting, thus leading it to be the only producer); and aþc (3) if ci  2 j for all firm i ¼ f1; 2g, then both firms produce positive output levels. Figure 5.12 summarizes these equilibrium results as a function of c1 and c2. In addition, the figure depicts what occurs when firms are cost symmetric, c1 = c2 = c along the 45°-line, whereby either both firms produce positive output levels, i.e., if c\ a þ2 c or c\a; or none of them does, i.e., if c [ a. For instance, when c2 [ a þ2 c1 , only firm 1 produces, as depicted in Fig. 5.13, where the best response functions of firms 1 and 2 only cross at the vertical axis where q*1 > 0 and q*2 = 0. Part (b) In order to know how the ðq1 ; q2 Þ varies when ðc1 ; c2 Þ change we separately differentiate each (interior) equilibrium output level with respect to c1 and c2 , as follows:

Exercise 6—Cournot Competition with Asymmetric CostsA

239

Fig. 5.12 Cournot game— cost pairs, and production decisions

Fig. 5.13 A corner solution in the Cournot game

@q1 2 @q1 1 [0 ¼  \0 and ¼ 3b @c1 @c2 3b Hence, firm 1’s equilibrium output decreases as its own costs go up, but increases as its rival’s costs increase. A similar intuition applies to firm 2’s equilibrium output:

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5

@q2 2 ¼  \0 and 3b @c2

Applications to Industrial Organization

@q2 1 [ 0: ¼ @c1 3b

Exercise 7—Cournot Competition and Cost DisadvantagesA Consider two firms competing in quantities (a la Cournot), facing inverse demand function pðQÞ ¼ a  Q, where Q denotes aggregate output, that is, Q ¼ q1 þ q2 . Firm 1’s marginal cost of production is c1 [ 0, while firm 2’s marginal cost is c2 [ 0, where c2  c1 indicating that firm 2 suffers a cost disadvantage relative to firm 1. Firms face no fixed costs. For compactness, let us represent firm 2’s marginal costs relative to c1 , so that c2 ¼ ac1 , where parameter a  1 indicates the cost disadvantage that firm 2 suffers relative to firm 1. Intuitively, when a approaches 1 the cost disadvantage diminishes, while when a is significantly higher than 1 firm 2’s cost disadvantage is severe. This allows us to represent both firms’ marginal costs in terms of c1 , without the need to use c2 . a. Write down every firm’s profit maximization problem and find its best response function. b. Interpret how each firm’s best response function is affected by a marginal increase in a. [Hint: Recall that an increase in parameter a indicates that firms’ marginal costs are more similar.] c. Find the Nash equilibrium of this Cournot game. d. Identify the regions of ða; aÞ-pairs for which equilibrium output levels of firm 1 and firm 2 are positive. e. How are equilibrium output levels affected by a marginal increase in parameter a? And by a marginal increase in parameter a? Answer Part (a). Finding firm 1’s best response function. Firm 1 chooses its output q1 to maximize its profits as follows max

q1  0

p1 ¼ ða  q1  q2 Þq1  c1 q1

Differentiating with respect to q1 , yields @p1 ¼ a  2q1  q2  c1 ¼ 0 @q1 Solving for q1 in the above equation gives us firm 1’s best response function q1 ð q2 Þ ¼

a  c1 1  q2 2 2

Exercise 7—Cournot Competition and Cost DisadvantagesA

241

1 which originates at ac 2 when firm 2 produces zero units of output, q2 ¼ 0, but decreases in q2 at a rate of 1/2.

Finding firm 2’s best response function. Similarly, for firm 2, its profit maximization problem is max

q2  0

p2 ¼ ða  q1  q2 Þq2  ðac1 Þq2

where we wrote firm 2’s costs as a function of firm 1’s, that is, c2 ¼ ac1 . Differentiating with respect to q2 , we obtain @p2 ¼ a  q1  2q2  ac1 ¼ 0 @q2 Solving for output q2 , we find firm 2’s best response function q2 ð q1 Þ ¼

a  ac1 1  q1 2 2

which is symmetric to that of firm 1. Part (b). Firm 1. An increase in parameter a does not affect firm 1’s best response function. It will, however, affect its equilibrium output, as we show in the next part of the exercise. Firm 2. An increase in parameter a (reflecting firm 2’s cost disadvantage relative to firm 1) means that firm 2’s best response function suffers a downward shift in its vertical intercept, without altering its slope. Intuitively, this indicates that, when firm 2’s costs are similar to those of firm 1 (i.e., a close to 1, exhibiting a small cost disadvantage) firm 2 produces a large output, for a given output of its rival q1 . However, when firm 2’s costs are significantly larger than those of firm 1 (i.e., a increases above 1, exhibiting a large cost disadvantage) firm 2 produces a smaller output, for a given output of its rival q1 . Part (c). To find the Nash equilibrium of the Cournot game, we insert one best response function into another. Note that we cannot use symmetry here, since the firms have difference costs and will hence, choose different equilibrium output. Inserting the best response function of firm 2 into that of firm 1, 1 1 q1 ðq2 Þ ¼ ac 2  2 q2 , we obtain

a  c1 1 a  ac1 1   q1 q1 ¼ 2 2 2 2

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Rearranging and solving for output q1 , we find an equilibrium output of 1 q1 ¼ ða  c1 ð2  aÞÞ 3 1 1 Inserting this result into firm 2’s best response function, q2 ðq1 Þ ¼ aac 2  2 q1 , we find

  a  ac1 1 1  q2 ¼ ð a  c 1 ð 2  aÞ Þ 2 3 2 which upon solving for output q2 yields an equilibrium output of 1 q2 ¼ ða  c1 ð2a  1ÞÞ 3 Therefore, the Nash equilibrium output pair can be summarized as follows


 q1 ; q2 ¼



1 1 ða  c1 ð2  aÞÞ; ða  c1 ð2a  1ÞÞ 3 3



Part (d). Equilibrium output for firm 1 is positive, q1 [ 0, if and only if  c1 ð2  aÞÞ [ 0. After solving for parameter a, this condition yields

1 3 ða

a [ c 1 ð 2  aÞ

ð5:3Þ

Similarly, equilibrium output for firm 2 is positive, q2 [ 0, if and only if 1 3 ða  c1 ð2a  1ÞÞ [ 0, which after solving for parameter a, yields a [ c1 ð2a  1Þ

ð5:4Þ

Since parameter a satisfies a  1 by assumption, we must have that ð2a  1Þ  ð2  aÞ. Therefore, if condition (5.4) is satisfied, condition (5.3) also holds. In summary, both firms produce a positive equilibrium output if demand is sufficiently strong, that is, a [ c1 ð2a  1Þ: To better understand this condition, note that cutoff c1 ð2a  1Þ originates at c1 since a  1 by assumption; and increases in a, indicating that condition a [ c1 ð2a  1Þ becomes more difficult to sustain as parameter a increases. Intuitively, an increase in a means that firm 2’s cost disadvantage is more severe,

Exercise 7—Cournot Competition and Cost DisadvantagesA

243

implying that firm 2 only produces a positive output level if demand becomes sufficiently large, as captured by condition a [ c1 ð2a  1Þ. Part (e). Firm 1’s equilibrium output level q1 ¼ 13 ða  c1 ð2  aÞÞ increases in both demand, as reflected by parameter a, and in the cost disadvantage that its rival (firm 2) suffers, as illustrated by parameter a. Firm 2’s equilibrium output level, q2 ¼ 13 ða  c1 ð2a  1ÞÞ also increases in demand, as captured by parameter a, but it decreases in the cost disadvantage that it suffers relative to firm 1, captured by parameter a.

Exercise 8—Cartel with Two Asymmetric FirmsA Consider two firms competing a la Cournot and facing linear inverse demand pðQÞ ¼ 100  Q, where Q ¼ q1 þ q2 denotes aggregate output. Firm 1 has more experience in the industry than firm 2, which is reflected in the fact that firm 1’s marginal cost of production is c1 ¼ 10 while that of firm 2 is c2 ¼ 16. (a) Best response function. Set up each firm’s profit-maximization problem and find its best response function. Interpret. (b) Cournot competition. Find the equilibrium output each firm produces when competing a la Cournot (that is, when they simultaneously and independently choose their output levels). In addition, find the profits that each firm earns in equilibrium. [Hint: You cannot invoke symmetry when solving for equilibrium output levels in this exercise since firms are not cost symmetric.] (c) Collusive agreement. Assume now that firms collude to increase their profits. Set up the maximization problem that firms solve when maximizing their joint profits (that is, the sum of profits for both firms). Find the output level that each firm should select to maximize joint profits. In addition, find the profits that each firm obtains in this collusive agreement. (d) Profit comparison. Compare the profits that firms obtain when competing a la Cournot (from part b) against their profits when they successfully collude (from part c). Answer Part (a). Firm 1’s profit function is given by: p1 ðq1 ; q2 Þ ¼ ½100  ðq1 þ q2 Þq1  10q1 Differentiating with respect to q1 , yields 100  2q1  q2  10 ¼ 0

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Applications to Industrial Organization

Solving for q1 , we obtain firm 1’s best response function 1 q1 ðq2 Þ ¼ 45  q2 2

ðBRF1 Þ

which originates at 45 and decreases in its rival’s output ðq2 Þ at a rate of 1/2. Similarly, firm 2’s profit function is given by: p2 ðq1 ; q2 Þ ¼ ½100  ðq1 þ q2 Þq2  16q2 Differentiating with respect to q2 yields 100  q1  2q2  16 ¼ 0 Solving for q2 , we find firm 2’s best response function 1 q2 ðq1 Þ ¼ 42  q1 2

ðBRF2 Þ

which originates at 42 and decreases in its rival’s output ðq1 Þ at a rate of 1/2. Comparing the best response functions of both firms, we find that firm 1 produces a larger output than firm 2 for a given production level of its rival. Intuitively, firm 1 enjoys a cost advantage relative to firm 2, which entails a larger output level for each of its rival’s output decisions. Part (b). Inserting firm 2’s best response function into firm 1’s, we obtain

1 1 q1 ¼ 45  42  q1 2 2 which simplifies to 1 q1  q1 ¼ 45  21 4 Solving for firm 1’s output, we find q1 ¼ 32 units. Substituting this output level in firm 2’s best response function, we obtain 1 q2 ð32Þ ¼ 42  32 ¼ 26 2 implying that firm 2 produces q2 ¼ 26 units in equilibrium. Finding equilibrium profits. Inserting these equilibrium output levels into each firm’s profit function, yields

Exercise 8—Cartel with Two Asymmetric FirmsA

245

p1 ð32; 26Þ ¼ ½100  ð32 þ 26Þ32  ð10  32Þ ¼ $1024 and p2 ð32; 26Þ ¼ ½100  ð32 þ 26Þ26  ð16  26Þ ¼ $676 The sum of these profits is then $1; 024 þ $676 ¼ $1700. Part (c). In this case, firms maximize their joint profits, that is, the sum p1 ðq1 ; q2 Þ þ p2 ðq1 ; q2 Þ, which we can expand as follows p1 þ p2 ¼ f½100  ðq1 þ q2 Þq1  10q1 g þ f½100  ðq1 þ q2 Þq2  16q2 g Differentiating joint profits with respect to q1 , yields 100  2q1  q2  10  q2 ¼ 0 Solving for q1 , we obtain q1 ðq2 Þ ¼ 45  q2 Similarly, we now differentiate joint profits with respect to q2 , yielding q1 þ 100  q1  2q2  16 ¼ 0 Solving for q2 , we find q2 ðq1 Þ ¼ 42  q1 Depicting functions q1 ðq2 Þ ¼ 45  q2 and q2 ðq1 Þ ¼ 42  q1 , we can see that they are parallel, and cross on the axes (at a corner solution), where q2 ¼ 0 and q1 ð0Þ ¼ 45  0 ¼ 45 units: Intuitively, when firms seek to maximize their joint profits, they assign all output to the firm with the lowest marginal cost (firm 1 in this case), and no output to the firm with the highest marginal cost (that is, the company with a cost disadvantage produces zero units). This leads to joint profits of p1 þ p2 ¼ f½100  ð45 þ 0Þ45  ð10  45Þg þ f½100  ð45 þ 0Þ0  ð16  0Þg ¼ 2025 þ 0 ¼ $2025: Part (d). The combined profits when firms collude ($2025) exceed those under Cournot competition ($1700).

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Exercise 9—Strategic Advertising and Product DifferentiationC Consider a two stage model. At the first stage, each firm selects how much money to spend on advertising. The amount selected by firm i = {1, 2} is denoted by Ai . After the firm selects its level of advertising, it faces an inverse demand for its product given by:  pi ðqi ; qj Þ ¼ a  bqi  di qj where i 6¼ j; b; di [ 0 and 0\di \b; where parameter di is inversely related to Ai. Hence, if firm i advertises more its product is perceived to be more differentiated relative to that of firm j (i.e., @di =@Ai < 0). At the second stage, each firm decides how much to produce. Find the SPNE of the two-stage game by solving it backwards. In particular, for given A1 and A2, derive the equilibrium quantities of the second stage (q1(A1, A2), q2 (A1, A2)). Afterwards, given the derivation of the second stage, find the equilibrium spending on advertising. (For simplicity, assume no production costs.) Answer Using backwards induction we start at the second stage: Second Stage Game: For given levels of advertising, every firm i chooses the output level qi that maximizes its profits, taking the production level of its rival, qj , as given. max½a  bqi  di qj qi qi

Taking first order conditions with respect to qi, we obtain a  2bqi  di qj ¼ 0 and solving for qi, we obtain firm i’s best response function:   a di  qj : qi qj ¼ 2b 2b Plugging firm j’s best response function into firm i’s, we find qi ¼

a  2b

di



a 2b



2b

dj qi 2b



Exercise 9—Strategic Advertising and Product DifferentiationC

247

which yields an equilibrium output level of: qi ¼

ð2b  di Þa 4b2  di dj

ð5:3Þ

First Stage Game: Given the equilibrium output levels, qi , found in expression (5.3) above, we can now move to the first stage game, where every firm independently and simultaneously selects an advertisement expenditure, Ai , that maximizes its profit function. h i max a  bqi  di qj qi  Ai where we plug the equilibrium value of qi in the second stage of the game. Taking first order conditions with respect to Ai , we find dpi @pi @qi @pi @qj @pi  ¼ þ dAi @qi @Ai @qj @Ai @Ai |{z} |{z} |fflfflffl{zfflfflffl} 0 Indirect Direct effect effect @q

where the first term on the right-hand side is zero since @Aii ¼ 0 because of the envelope theorem. The third term represents the direct effect that advertising entails on firm i’s profits (the cost of spending more dollars on advertising). The second term, however, reflects the indirect (or strategic) effect from advertising, capturing the fact that a change in a firm’s advertising expenditure affects the equilibrium production of its rival, qj , which ultimately affects firm i’s profits. Importantly, this suggest that, for Ai to be effective, it must be irreversible and perfectly observable by its rivals, since otherwise firm i would not be able to affect its rival’s output decision in the second period game. Hence, expanding the above first-order condition, we obtain: dpi @pi @qj @pi ¼  dAi @qj @Ai @Ai @pi @qj @di ¼ 1 @qj @di @Ai   @qj @di ¼ di qi 1 @di @Ai

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5

Applications to Industrial Organization

@q

where in particular, @dji is found by using the best response function, qj ¼ follows

ð2bdj Þa 4b2 dj di

as

  @qj 2b  dj adj qj dj ¼ ¼ 2 2 2 @di ð4b  di dj Þ ð4b  di dj Þ Substituting in the above expression, we obtain " #  qj dj dpi  @di   ¼ di qi  2 1 dAi 4b  di dj @Ai " # di dj   @di    1 ¼ 0: ¼ qi qj @Ai 4b2  di dj Therefore, solving for

@di @Ai ,

we find

@di 4b2  di dj 1 ¼   : qi qj @Ai di dj

ð5:4Þ

Hence, the optimal level of advertising by firm i in the first stage, Ai , is the level of Ai such that the advertising effect on firm i’s product solves the above expression. As a remark, note that if, instead, advertising was selected simultaneously with output, equilibrium advertising would be determined as follows: dpi @di ¼ qi qj 1¼0 dAi @Ai where now term

@qj @di

is absent. Intuitively, advertising Ai does not affect firm j’s

optimal output level in this case. Solving for

@di @Ai ,

@di 1 ¼ : qi qj @Ai

we find ð5:5Þ

Comparing the right-hand side of expressions (5.4) and (5.5), we can conclude that a given increase in advertising expenditures produces a larger decrease in di @di (more product differentiation), i.e. @A is larger (in absolute value), when firms i determine their level of advertising sequentially (before choosing their output level) than when they choose it simultaneously with their output level.

Exercise 10—Cournot Oligopoly with CES DemandB

249

Exercise 10—Cournot Oligopoly with CES DemandB Compute the Cournot equilibrium with N firms when the firms face the inverse demand function PðQÞ ¼ Q1=e ; where e [ 1, and have identical constant marginal cost c. Answer Every firm i simultaneously and independently chooses its output level qi in order to maximize profits

max qi

 1 Q  e  c qi :

Taking first order conditions with respect to qi , we obtain, 1 1 1 Q e  c  Q e1  qi ¼ 0: e At a symmetric equilibrium, every firm produces the same output, implying that aggregate output becomes Q ¼ N  qi . Solving for qi yields qi ¼ NQ. Inserting this result in the above first-order conditions, and rearranging, we obtain 1 1 Q 1 1 Q 1 1 Q e  c  Q e1  ¼ 0 , Q e  Q e1  ¼ c e N e   N 1 eN  1 , Q e ¼ c: eN And solving for Q we find an equilibrium output   eN  1 e : Q ¼ ceN 

In the limit when N ! 1 we obtain e 1 lim Q ¼ ; N!1 c 

which coincides with the outcome for a perfectly competitive market. To see this take the demand function PðQÞ ¼ Q1=e ; and solve for Q, Q ¼ ðPÞe ¼ ð1=PÞe In a perfectly competitive market it must be true that pðQÞ ¼ c, hence Q ¼

e 1 c

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Exercise 11—Commitment in Prices or Quantities?B This exercise is based on Singh and Vives (1984).2 Consider a market that consists of two firms offering differentiated products. The inverse demand that each firm faces is: pi ¼ a  bqi  dqj

0  d  b:

Consider the following two stage game. In the first stage each firm can commit either to supply a certain quantity qi or to set a price  pi . In the second stage, the remaining variables that were not chosen in the first stage are determined in order to clear the market (prices, if quantities were chosen in the first stage; or quantities, if prices were selected in the first stage). Show that a commitment to quantity in the first stage is a dominant strategy for each firm. Answer Operating by backward induction, let us first analyze the second stage of the game. Second Stage Part (a) If both firms commit to quantities in the first stage, then firm i’s market demand is pi ðqi ; qj Þ ¼ a  bqi  dqj : Part (b) If both firms commit to prices in the first stage, then market demand is found by solving for qi in pi ðqi ; qj Þ, as follows qi ¼

ða  pi  dqj Þ b

Similarly, solving for qj in pj ðqi ; qj Þ, we obtain qj ¼

ða  pj  dqi Þ b

Simultaneously solving for qi and qj in the above expressions, we find qi ¼

ðab  ad  bpi þ dpj Þ aðb  dÞ  bpi  dqj ¼ : ðb2  d2 Þ ðb  dÞðb þ dÞ

Singh, N., & Vives, X. (1984). “Price and Quantity Competition in a Differentiated Duopoly”. The RAND Journal of Economics, 15, 4, 546–554.

2

Exercise 11—Commitment in Prices or Quantities?B

251

Part (c) If one firm commits to prices, pi , and the other to quantities,  qj , then market demand for firm i is pi ¼ a  bqi  dqj which solving for qi yields qi ¼

a  dqj  pi b

and the demand for firm j is qj ¼

aðb  d Þ  bpi  dpj ðb  dÞðb þ dÞ

which solving for pj yields pj ¼ a  bqj  d

a  dqj   pi : b

First Stage If both firms commit to quantities, then from the demand function in point (a), we obtain the Cournot outcome. In particular, differentiating with respect to qi in max qi



 a  bqi  dqj qi

we obtain a  2bqi  dqj ¼ 0 which, solving for qi , yields a best response function of   a d  qj : qi qj ¼ 2b 2b In a symmetric equilibrium, both firms choose the same output level, qi ¼ qj ¼ q, entailing that the above best response function becomes q¼

a d  q 2b 2b

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Applications to Industrial Organization

which, solving for q; yields an equilibrium output of a q ¼ 2b þ d ; thus entailing equilibrium profits of pCi

ba2

for every firm i ¼ f1; 2g:

ð2b þ dÞ2

If both firms commit to prices, from point (b) of the exercise, we obtain the Bertrand outcome. In particular, every firm i chooses its price level pi that maximizes max pi

pi

aðb  dÞ  bpi  dqj : ðb  dÞðb þ dÞ

Taking first order conditions with respect to pi yields aðb  d Þ  2bpi  dqj ¼0 b2  d 2 and solving for pi, we obtain firm i’s best response function pi ðpj Þ ¼

að b  d Þ d þ pj 2b 2b

which increases in its rival’s price, pj. The price of firm j, pj, is symmetric. In a symmetric equilibrium, prices must coincide, pi ¼ pj ¼ p, entailing that the above best response function simplifies to p¼

að b  d Þ d þ p 2b 2b

which, solving for p, yields ab p ¼ a  2b  d : Hence, plugging these results into firm i’s profit function, we obtain firm i’s equilibrium profits

Exercise 11—Commitment in Prices or Quantities?B

pBi

bðb  dÞa2

253

for every firm i ¼ f1; 2g;

ðb þ dÞð2b  dÞ2

where profits satisfy pBi \pCi for all parameter values. If only one firm commits to prices while its rival commits to quantities, we know from part (c) of the exercise that we obtain a “hybrid” outcome. If firm i commits to prices, it solves

a  dqj   pi  pi ¼ pi b

max pi

Taking first-order conditions with respect to pi , since firm i commits to using prices, we obtain: a  dqj  2pi ¼0 b which solving for pi yields its best response   a d pi qj ¼  qj : 2 2 Similarly, firm j, which commits to using quantities, maximizes profits  max qj

a  bqj  d

a  dqj   pi b

  qj

Taking first-order conditions with respect to qj , we find a  2bqj  d

a  2dqj   pi b

¼0

which, solving for qj , yields firm j’s best response function in this setting qj ðpi Þ ¼

aðb  d Þ þ d pi : 2 2 2ðb  d Þ

  Inserting the expression we found for pi qj into  pi Þ, we obtain qj ð 

qj ¼

að b  d Þ þ d



adqj 2

2ðb2  d2 Þ

which, solving for qj , yields an equilibrium output



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5

Applications to Industrial Organization

Fig. 5.14 Payoff matrix in the first period

qj ¼

2ab  ad : 4b2  3d 2

Plugging the equilibrium quantity of firm j,  qj , into firm i’s best response   function pi qj ¼ a2  d2 qj , we obtain qj

zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{

a d 2ab  ad aðb  dÞð2b þ dÞ i ¼  p ¼ : 2 2 4b2  3d2 4b2  3d 2 We can now find qi by using the demand function firm i faces, pi ¼ a  bqi  dqj , and the expressions for pi and qj we found above. In particular, aðb  dÞð2b þ dÞ 2ab  ad ¼ a  bqi  d 2 2 2 4b ffl{zfflfflfflfflfflfflfflfflfflfflfflffl 4b ffl{zfflfflfflfflffl  3d ffl}  3dffl}2 |fflfflfflfflfflfflfflfflfflfflfflffl |fflfflfflfflffl pi

qj

Solving for qi ; we obtain an equilibrium output of qi ¼

aðb  dÞð2b þ dÞ : bð4b2  3d 2 Þ

Hence, the profits of the firm that committed to prices (firm i) are ppi ¼ pi  qi ¼

aðb  dÞð2b þ dÞ aðb  dÞð2b þ dÞ a2 ðb  d Þ2 ð2b þ dÞ2 ¼  : 4b2  3d2 bð4b2  3d2 Þ bð4b2  3d2 Þ2

Similarly, regarding firm j, we can obtain its equilibrium price, pj , by using the demand function firm j faces, pj ¼ a  bqj  dqi , and the expressions for  qj and qi found above. Specifically,

Exercise 11—Commitment in Prices or Quantities?B

255

 pj ¼ a  b

   að2b  dÞ aðb  dÞð2b þ d Þ aðb  dÞð2b  d Þðb þ dÞ d ¼ 2 2 2 2 4b  3d bð4b  3d Þ bð4b2  3d2 Þ |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} qj

qi

Thus, the profits of the firm that commits to quantities (firm j) are pQ qj ¼ j ¼ pj 

aðb  dÞð2b  dÞðb þ dÞ 2ab  ad a2 ðb  d Þð2b  d 2 Þðb þ dÞ  ¼ bð4b2  3d2 Þ 4b2  3d 2 bð4b2  3d 2 Þ2

Summarizing, the payoff matrix that firms face in the first period game is (Fig. 5.14). If firm j chooses prices (fixing our attention on the left column), firm i’s best B response is to select quantities since pQ i [ pi given that d [ 0 by assumption. Similarly, if firm j chooses quantities (in the right column), firm i’s best responds with quantities since pCi [ pPi given that d [ 0. Since payoffs are symmetric, a similar argument applies to firm j. Therefore, committing to quantities is a strictly dominant strategy for both firms, and the unique Nash equilibrium is for both firms to commit in quantities.

Exercise 12—Fixed Investment as a Pre-commitment StrategyB Consider a market that consists of two firms producing a homogeneous product. Firms face an inverse demand of pi ¼ a  bqi  dqj : The cost function facing each firm is given by TCi ðci Þ ¼ F ðci Þ þ ci qi ; where F 0 ðci Þ\0; F 00 ðci Þ  0: Hence, the fixed costs of production decline as the unit variable cost increases. Alternatively, as fixed costs increase, the unit variable costs decline. Consider the following two-stage game: in the first stage, each firm chooses its technology, which determines its unit variable cost ci (to obtain a lower unit variable cost ci the firm needs to invest in improved automation, which increases the fixed cost F(ci)). In the second stage, the firms compete as Cournot oligopolists. Part (a) Find the equilibrium of the two-stage game, and discuss the properties of the “strategic effect.” Which strategic terminology applies in this case?

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5

Applications to Industrial Organization

Part (b) Assume that F ðci Þ ¼ a  b þ 0:5c2i . Derive the closed form solution of the equilibrium of the two stage game. What restrictions on parameters a and b are necessary to support an interior equilibrium? Answer Part (a) In the first stage, firm i chooses a level of ci that maximizes its profits, given its equilibrium output in the second stage of the Cournot oligopoly game. That is, max ci

pi ðci ; qi ðci Þ; qj ðci ÞÞ

Taking first-order conditions with respect to ci, we find dpi @pi @pi @qi @pi @qj ¼ þ þ : dci @ci @qi @ci @qj @ci |{z} |fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl} ð1Þ

ð2Þ

ð3Þ

Let us separately interpret these three terms: i (1) Direct Effect: an increase in ci produces a change in costs of @p @ci : (2) This effect is zero since in the second stage qi ðci Þ is chosen in equilibrium (i.e., by the envelope theorem). i @qj (3) Strategic Effect. @p @qj @ci can be expressed as:

@pi @qj |{z} 

@qj @q |{z}i 

@qi \0 @ci |{z} 

Intuitively, the strategic effect captures the fact that an increase in ci (in the third component) reduces firm i’s production, which increases firm j’s output, ultimately reducing firms i’s profits. Hence, the overall strategic effect is negative. Therefore, the firm seeks to invest in increased automation, for unit variable costs to be lower than in the absence of competition. Firms, hence, want to become more competitive in the second period game, i.e., a “top dog” type of commitment. Let us analyze this first-order condition in more detail. Second stage. For a given ci and cj chosen in the first stage, in the second stage firm i chooses qi to maximize the following profits max qi

  pi ¼ a  bqi  bqj  ci qi  Fðci Þ

Taking first order conditions with respect to qi we obtain a  2bqi  bqj  ci ¼ 0, which yields the best response function

Exercise 12—Fixed Investment as a Pre-Commitment StrategyB

257

  ac 1  qj ; q i qj ¼ 2b 2 ultimately entailing a Nash equilibrium output level of qi ¼

a  2ci þ cj : 3b

(This is a familiar result in Cournot duopolies with asymmetric marginal costs.) Hence, first stage equilibrium profits are 

a  2ci þ cj pi ¼ 9b

2  Fðci Þ:

First stage. In the first stage, firm i chooses ci to maximize its profits, yielding first order conditions:

  4 a  2ci þ cj @pi  F 0 ðci Þ ¼ 0 ¼ 9b @ci and second order conditions: @ 2 pi 8  F 00 ðci Þ  0 ¼ 2 9b @ci 8 . In Hence, to guarantee that second order conditions hold, we need F 00 ðci Þ  9b addition, to guarantee that firms choose a positive output at the symmetric equii librium, we need that the first order condition satisfies @p > 0. Hence, @ci  c1 ¼c2 ¼0

0 0 4a  4a 9b  F ð0Þ [ 0, or F ð0Þ [ 9b. i Note that we do not solve for ci in the first order condition @p @ci since we would not be able to obtain a closed form solution for the best response function ci (cj) of firm i until we do not have more precise information about the parametric function of F (ci). We will do that in the next section of the exercise, where we are informed about the functional form of F(ci). Assessing Both Effects. To assess the strategic effect, note that in the first stage:

@pi @pi @pi @qj ¼ þ ; @ci @ci @cj @ci   where pi ¼ a  bqi  bqj  ci qi  Fðci Þ, implying that the first term (direct effect) is hence:

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5

Applications to Industrial Organization

@pi ¼ qi  F 0 ðci Þ @ci While the second term (strategic effect) is given by @pi @qj 1 ¼ ðbqi Þ \0 3b @qj @ci Part (b) We now solve the above exercise assuming that the functional relationship between fixed and variable costs is F ðci Þ ¼ a  bci þ 0:5c2i . In this context, F 0 ðci Þ ¼ b þ ci , and F 00 ðci Þ ¼ 1. Substituting them into the first order condition of the first stage, yields:

  4 a  2ci þ cj @pi þ b  ci ¼ 9b @ci While the second order condition becomes @ 2 pi 8  1  0: ¼ 2 9b @ci Therefore, in order for the second-order condition to hold, we need b  8=9. At the symmetric equilibrium ci ¼ cj ¼ c, we thus have 

4ð a  c Þ þ b  c ¼ 0: 9b

Solving for c, we obtain c ¼ Hence, we need F 0 ð0Þ [ c [ 0. 

4a 9b,

4a  9bb : 4  9b

i.e., b [

4a 9b

to guarantee an interior solution

Exercise 13—Socially Excessive Entry in an IndustryB Consider an industry with perfectly symmetric firms playing the following game. In the first stage, every firm simultaneously and independently decides whether to enter the industry. If the firm does not enter, the game is over, and its payoff is zero.

Exercise 13—Socially Excessive Entry in an IndustryB

259

If the firm enters, it incurs a fixed set-up cost F > 0, and competes a la Cournot against all entering firms (i.e., firms compete in quantities). Firms face a symmetric marginal cost of production c > 0, and inverse market demand given by p(Q) = 1 − Q, where Q denotes aggregate production by all firms that entered the industry. (a) Equilibrium entry. In the second stage of the game, find equilibrium output, price, and profits assuming that n firms enter the industry in the first stage. (b) Find the number of firms that enter the industry in the first stage of the game and describe the subgame perfect Nash equilibrium. (For simplicity, consider that the number of firms n is continuous rather than discrete.) Label the equilibrium number of firms that enter the industry as n*. (c) Socially optimal entry. Still considering the second stage of the game, assume now that in the first stage firms are not allowed to decide whether to enter the industry. Instead, a social planner (e.g., antitrust agency) decides the number of firms that should enter seeking to maximize welfare, which is defined as the sum of consumer surplus and the producer surplus minus the entry costs every firm must pay. In the second stage, consider the same quantity competition game. Find the socially optimal number of firms that the social planner selects, and label it nSO. d) Compare the number of firms that enter in equilibrium (as found in part b) against the socially optimal number of firms (found in part c). Interpret your results. Answer Part (a). In the second stage, every firm i chooses its own output qi to maximize its profits as follows max qi

pi ¼ ð1  QÞqi  cqi ¼ ð1  qi  Qi Þqi  cqi

where Qi ¼

P j6¼i

qj indicates the sum of the production from all firm i’s rivals.

Differentiating with respect to qi , yields 1  2qi  Qi  c ¼ 0 which we rearrange as 1  c  Qi ¼ 2qi

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5

Applications to Industrial Organization

Solving for firm i’s output qi , we obtain qi ¼

1c 1  Qi 2 2

which is firm i’s best response function, BRFi . Graphically, it originates at 1c 2 where firm i’s rivals do not produce any output, Qi ¼ 0, thus coinciding with the production level that firm i would choose in a monopoly market with inverse demand pðQÞ ¼ 1  Q and constant marginal cost c [ 0. However, when firm i’s rivals produce a positive output, Qi [ 0, firm i responds reducing its output at a rate of 1/2 per unit of output from its rivals. In a symmetric equilibrium, all firms produce the same output level, qi ¼ qj ¼ q , which implies that Qi ¼ ðn  1Þq . Therefore, the above best response function BRFi can be rewritten as follows q ¼

1c 1  ðn  1Þq 2 2

Rearranging this expression yields 2q ¼ 1  c  ðn  1Þq and, solving for q , we find that the equilibrium output is q ¼

1c nþ1

which is decreasing in marginal cost c, and in the number of firms competing in the industry, n. Inserting this equilibrium output q in the inverse demand function, we find that equilibrium price p is p ¼ 1  Q ¼ 1  nq ¼ 1  n ¼

n þ 1  n þ nc 1 þ nc ¼ nþ1 nþ1

1c nþ1

and equilibrium profits become

1 þ nc 1c p ðnÞ ¼ p q  cq  F ¼ ðp  cÞq  F ¼ c F nþ1 nþ1

1 þ nc  nc  c 1  c 1c 2 ¼ F ¼ F nþ1 nþ1 nþ1 

 







Exercise 13—Socially Excessive Entry in an IndustryB

261

which are decreasing in the marginal cost of production, c, the number of competing firms, n, and the fixed entry cost, F Part (b). In the first stage, every firm simultaneously and independently chooses whether to enter the industry. In particular, every firm i enters if the profits that it obtains from entering (net of entry costs), p ðnÞ, are larger than the profits from staying out (zero). That is, p ðnÞ  F  0. Entry will then occur until firms are indifferent between entering and staying out, p ðnÞ  F ¼ 0: The indifference condition p ðnÞ  F ¼ 0 can be expressed as



2

1c nþ1

¼ F.

Applying the square root on both sides, we find 1  c pffiffiffiffi ¼ F nþ1 which can be rearranged as 1c pffiffiffiffi ¼ n þ 1 F Solving for the number of firms, we obtain that the number of firms that enter in equilibrium is 1c n ¼ pffiffiffiffi  1 F which decreases in firms’ marginal cost of production, c, and in the fixed cost F that firm must incur to enter the industry. Part (c). The social planner will choose n to maximize total welfare (the sum of consumer and producer surplus). Let us first find consumer surplus, CS. Notice that the CS is given by the area of the triangle between the vertical intercept of the demand curve, 1, and the equilibrium price we find next:

p ¼ 1  Q ¼ 1  nq ¼ 1  n ¼ Hence, CS is given by

n þ 1  n þ nc 1 þ nc ¼ nþ1 nþ1

1c nþ1

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5

Applications to Industrial Organization

1 1 CS ¼ ð1  p ÞQ ¼ ð1  p Þnq 2

2 1 1 þ nc 1  c 1 ¼ n 2 nþ1 nþ1 ¼

n2 ð 1  c Þ 2 2ð n þ 1Þ 2

Producer surplus is just given by the aggregate profits of all firms entering the industry, that is, " #

1c 2 n2 ð1  cÞ2 np ðnÞ ¼ n F ¼  nF nþ1 ðn þ 1Þ2 

Therefore, the sum of consumer and producer surplus yields a total welfare of W ¼ CS þ PS ¼ ¼

n2 ð 1  c Þ 2 2ð n þ 1Þ 2

nð1  cÞ2 ðn þ 2Þ 2ð n þ 1Þ 2

þ

nð 1  c Þ 2 ð n þ 1Þ 2

!  nF

 nF

Taking first-order conditions with respect to n, we find @W ð1  cÞ2 1 ¼ ð2n þ 2Þ  F ¼ 0 þ @n 2½ðn2 þ 2nÞ ðn 2  ðn þ 1Þ2 þ 1Þ3 Simplifying this gives us ð1  cÞ2 2

½2ðn2

Fig. 5.15 Number of firms as function of entry costs (F)

2

þ 2nÞ þ ðn þ 1Þ 

ðn þ 1Þ3  F ¼ 0

Exercise 13—Socially Excessive Entry in an IndustryB

263

which further simplifies to ð1  c Þ2 ð 1 þ nÞ 3

¼F

Solving for n, we obtain that the socially optimal number of firms is. nSO

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 ð1  cÞ 1 ¼ F

which decreases in firms’ marginal cost of production, c, and in the fixed cost F that firm must incur to enter the industry. Part (d). By comparing the optimal number of firms nSO we just found with the number of firms entering at the free entry equilibrium from part (a), n , there is socially excessive entry in the industry. Intuitively, when every firm independently chooses whether to enter it only considers its profits from entering the industry against its fixed entry cost, but ignores the profit reduction that its entry produces on all firms entering the industry too. This profit-reduction effect is often referred by the literature as the “business stealing” effect of additional entry. The social planner, in contrast, internalizes the business stealing effect since he considers aggregate profits. (This result applies even if the welfare function we consider ignores consumer surplus and focuses on producer surplus alone.) For illustration purposes, the next figure depicts n and nSO as a function of F in the horizontal axis. For simplicity, both n and nSO are evaluated at a marginal cost of c = 0.5. (You can obtain similar figures using another value for firms’ marginal costs of production, c). Two important features of the figure are noteworthy. First, note that both n and nSO decrease in the entry costs, F. Second, the equilibrium number of firms entering the industry, n , lies above the socially optimal number of firms, nSO , for any given level of F; reflecting an excessive entry in the industry when entry is unregulated (Fig. 5.15).

Exercise 14—Entry Deterring InvestmentB Consider an industry with two firms, each firm i = {1, 2} with demand function qi ¼ 1  2pi þ pj . Firm 2’s (the entrant’s) marginal cost is 0. Firm 1’s (the incumbent’s) marginal cost is initially ½. By investing I = 0.205, the incumbent can buy a new technology and reduces its marginal cost to 0 as well.

264

5

Applications to Industrial Organization

Part (a) Consider the following time structure: The incumbent chooses whether to invest, then the entrant observes the incumbent’s investment decision; and subsequently the firms compete in prices. Show that in the subgame perfect equilibrium the incumbent does not invest. Part (b) Show that if the investment decision is not observed by the entrant, the incumbent’s investing becomes part of the SPNE. Comment. Answer Part (a) Second period. Recall that in the first period the incumbent decides to invest/not invest, while in the second period firms compete in a Bertrand duopoly. Operating by backward induction, let us next separately analyze firms’ pricing decisions (during the second stage) in each of the two possible investment profiles that emerge from the first period (either the incumbent invests, or it does not). The Incumbent Invests. In this setting, both firms’ costs are zero, i.e., c1 ¼ c2 ¼ 0. Then, each firm i 6¼ j chooses the price pi that maximizes   max 1  2pi þ pj pi : pi

Taking first order conditions with respect to pi , we have 1  4pi þ pj ¼ 0. Solving for pi , we obtain the firm i’s best response function pi ðpj Þ ¼

1 1 þ pj : 4 4

(Note that second order conditions also hold, since they are 4\0, guarantee that such a price level maximizes firm i’s profit function.) Graphically, this best response function originates at ¼ and increases at that rate too, implying that for every extra dollar that firm j charges for its product firm i responds increasing its own price by $0.25 (25 cents). By symmetry, firm j’s best response function is pj ðpi Þ ¼ 14 þ 14 pi . Plugging pj ðpi Þ into pi ðpj Þ, we obtain pi ¼

1þ

1 4

 þ 14 pi 5 þ pi ¼ 4 16

and thus equilibrium prices are pi ¼

1 1 and pj ¼ : 3 3

These prices yield equilibrium profits of:

Exercise 14—Entry Deterring InvestmentB

265



1 2 1 2 1 þ ¼ 3 3 3 9

for every firm i

Nonetheless, the profits that the incumbent earns after including the investment in the cost-reducing technology are pi ¼

2  0:205 ¼ 0:017: 9

The Incumbent Does Not Invest. In this case, the incumbent’s costs are higher than the entrant’s, i.e., c1 ¼ 12 and c2 ¼ 0. First, the incumbent’s profit maximization problem is max ð1  2p1 þ p2 Þðp1  0:5Þ ¼ p1  2p21 þ p2 p1  0:5 þ p1  0:5p2 p1

Taking the first order conditions with respect to p1 , we find 1  4p1 þ p2 þ 1 ¼ 0: And solving for p1 , we obtain firm 1’s (the incumbent’s) best response function p1 ðp2 Þ ¼

1 1 þ p2 2 2

ðAÞ

Second, the entrant’s profit-maximization problem is max p2

ð1  2p2 þ p1 Þp2 :

Taking first order conditions with respect to p2 , 1  4p2 þ p1 ¼ 0: And solving for p2 , we find firm 2’s (the entrant’s) best response function p2 ðp1 Þ ¼

1 1 þ p1 4 4

ðBÞ

Solving simultaneously for ðp1 ; p2 Þ in expressions (A) and (B), we obtain the equilibrium price for the incumbent

1 1 1 1 9 þ p1 þ p1 ¼ : p1 ¼ þ 2 2 4 4 16 Solving for price p1 , we find an equilibrium price of

266

5

p1 ¼

Applications to Industrial Organization

3 5

And the equilibrium price for the entrant p2 ¼

1 13 2 þ ¼ : 4 45 5

Entailing associated profits of



3 2 3 1 p1 ¼ 1  2  þ  ¼ 0:02 for the incumbent 5 5 5 2

2 3 2 ¼ 0:32 for the entrant: p2 ¼ 1  2  þ 5 5 5 First stage. After analyzing the second stage of the game, we next examine the first stage, whereby the incumbent must decide whether or not to invest in cost-reducing technologies. In particular, the incumbent decides not to invest in cost-reducing technologies since its profits when it does not invest, 0.02, are larger than when it does, 0.017. Intuitively, the incumbent prefers to compete with a more efficient entrant than having to incur a large investment cost in order to achieve the same cost efficiency as the entrant, i.e., the investment is too costly. Part (b) Now the entrant only chooses a single price p2 , i.e., a price that is not conditional on whether the incumbent invested or not. We know from part (a) that the entrant’s best response is to set a price: p2 ¼

1 3 2 5

If the incumbent chooses Invest If the incumbent chooses Not Invest

(1) If p2 ¼ 13, then the incumbent profits depends on whether it invests or not. After investment, the incumbent maximizes:

1 max 1  2p1 þ p1 p1 3 Taking first order conditions with respect to p1 , we find 1  4p1 þ 13 ¼ 0, and solving for p1 we obtain p1 ¼ 13. Hence the incumbent’s profits in this case are p1 ¼



2 1 1  0:205 ¼ 0:017: 1 þ 3 3 3

After no investment, the incumbent maximizes

Exercise 14—Entry Deterring InvestmentB

267





1 1 max 1  2p1 þ p1  p1 3 2 Taking first order conditions with respect to p1 , we find 1  4p1 þ 13 þ 1 ¼ 0, 7 and solving for p1 we obtain p1 ¼ 12 . Therefore the incumbent’s profits are



14 1 7 1 þ  p1 ¼ 1  ¼ 0:014: 12 3 12 2 [ pNotInvest if p2 ¼ 13 As a consequence, pInvest 1 1 (2) If p2 ¼ 25, then the Incumbent profits are different from above. In particular upon investment, the incumbent maximizes



2 max 1  2p1 þ p1 p1 5 Taking first order conditions with respect to p1 , we find 1  4p1 þ 25 ¼ 0, and 7 solving for p1 we obtain p1 ¼ 20 . Thus, the incumbent’s profits are p1 ¼



14 2 7 þ  0:205 ¼ 0:04: 1 20 5 20

Upon no investment, the incumbent maximizes

2 max 1 þ 2p1 þ p1 5



1 p1  : 2

Taking first order conditions with respect to p1 , we find 1  4p1 þ and solving for p1 we obtain p1 ¼ 35, thus yielding profits of p1 ¼

2 5

þ 1 ¼ 0,





6 2 3 1  1 þ ¼ 0:02: 5 5 5 2

[ pNotInvest if p2 ¼ 25 : Therefore, pInvest 1 1 Hence, incumbent invests in pure strategies, both when p2 ¼ 13 and when p2 ¼ 25, and the unique SPNE of the entry game is (Invest, p1 ¼ p2 ¼ 13Þ.

268

5

Applications to Industrial Organization

Exercise 15—Direct Sales or Using A Retailer?C Assume two firms competing in prices where the demand that firm i faces is given by qi ¼ 1  2pi þ pj for every firm i; j ¼ 1; 2 and j 6¼ i There are no costs of production. Each firm has the option between selling the product directly to the consumers, or contracting with a retailer that will sell the product of the firm. If a firm contracts with the retailer, the contract is a franchise  fee contract specifying: (a) the wholesale price piw of each unit of the product that is transferred between the firm and the retailer; and (b) the fixed franchise fee ðFi Þ for the right to sell the product. The game proceeds in the following two stages: At the first stage, each firm decides whether to contract with a retailer or to sell the product directly to the consumers. The firm that sells through a retailer, signs the contract at this stage. At the second stage, the firm or its retailer sets the retail price ðpi Þ for the product. Solve for the equilibrium of this two stage game. How does it compare with the regular Bertrand equilibrium? Explain. Answer Let us separately find which are the firm’s profits in each of the possible selling profiles that can arise in the second stage of the game: (1) both firms sell directly their good to consumers; (2) both sell their product through retailers; and (3) only one firm i sells the product through a retailer. We will afterwards compare firms’ profits in each case. Both Firms Sell Directly to Consumers: Then every firm i selects a price pi that solves max pi



 1  2pi þ pj pi

Taking first order conditions with respect to pi ; 1  4pi þ pj ¼ 0, and solving for pi , we obtain best response function   1 þ pj : pi pj ¼ 4 Similarly for firm j and pj , where firm j’s best response function is pj ð pi Þ ¼ Plugging pj ðpi Þ into pi ðpj Þ, we obtain

1 þ pi : 4

Exercise 15—Direct Sales or Using A Retailer?C

pi ¼

1þ

269

1 þ pi  4

4

which yields a equilibrium price pi ¼ 13, with associated profits of pi ¼

12

1 1 1 2 þ ¼ 3 3 3 9

Both Firms Use Retailers: Let us first analyze the retailer’s profit-maximization problem, where it chooses a price pi for the product, but has to pay a franchise fee Fi to firm i in order to become a franchisee, and a wholesale price piw to firm i for every unit sold.   max 1  2pi þ pj ðpi  piw Þ  Fi pi

Intuitively, pi  piw denotes the margin that the retailer makes per unit when it sells the product of firm i. Taking first order conditions with respect to pi ; 1  4pi þ pj þ 2piw ¼ 0, and solving for pi , we obtain the retailer’s best response function, pi ðpiw ; pj Þ ¼

1 þ pj þ 2piw : 4

By symmetry, the franchisee of firm j has a similar best-response function, i.e., pj ðpwj ; pi Þ. Hence, plugging pj ðpwj ; pi Þ into pi ðpiw ; pj Þ, we find 



pi piw ; pj ¼

1 þ 2piw þ



1 þ pi þ 2pwj 4



4

Solving for pi , we obtain the optimal price that the franchisee of firm i (retailer) charges for its products to customers pi ¼

5 þ 8piw þ 2pwj 15

for every firm i ¼ f1; 2g and j 6¼ i

Let us now study the firm’s decision of what wholesale price to charge to the franchisee, piw , and what franchisee fee to charge, Fi , in order to maximize its profits. In particular, firm i solves: max i pw ;Fi

piw qi þ Fi

270

5

Applications to Industrial Organization

And given that Fi ¼ ðpi  piw Þqi (no profits for the retailer), then firm i’s profit-maximization problem can be simplified to piw qi þ pi qi  piw qi

max i pw

(which contains a single choice variable, piw Þ. Taking first-order conditions with respect to piw , @pi @pi @pi @pj þ ¼0 @pi @piw @pj @pwj In addition, since firm i acts before the franchisee, it can anticipate the latter’s behavior, who charges a price of pi ¼

5 þ 8piw þ 2pwj : 15

8 2 This simplifies our above first order condition to ð1  4pi þ pj Þ 15 þ pi 15 ¼ 0, or 8  30pi þ 8pj ¼ 0. By symmetry 8  30pj þ 8pi ¼ 0 for firm j, which allows us to 4 simultaneously solve both equations for pi , obtaining pi ¼ 11 for all i ¼ f1; 2g. As a consequence, sales of every good at these prices are

qi ¼ 1  2 

4 4 7 þ ¼ : 11 11 11

And the associated profits that firm i obtains from selling goods through a franchisee are pi ¼

4 7 28  ¼ ’ 0:231: 11 11 121

Only one Firm Uses Retailers: If firm j sells directly to customers, it sets a price pj to maximize max pj

  pj 1  2pj þ pi

Taking first order conditions with respect to pj , we obtain 1  4pj þ pi ¼ 0, which yields firm j’s best response function pj ðpi Þ ¼

1 þ pi : 4

If instead firm i uses a retailer, then the retailer’s profit-maximization problem is

Exercise 15—Direct Sales or Using A Retailer?C



max pi

271

 1  2pi þ pj ðpi  piw Þ  Fi

Taking first order conditions with respect to pi (the only choice variable for the retailer), we obtain 1  4pi þ pj þ 2piw ¼ 0, which yields pi ðpj ; piw Þ ¼

1 þ pj þ 2piw : 4

Combining firm j’s best response function and the retailer’s best response function, we find equilibrium prices of pj ¼ pi ¼

1þ

1 þ pj þ 2piw 4

1þ

5 þ 2piw 15

4

! pj ¼

þ 2piw

4

5 þ 2piw ; and 15

! pi ¼

5 þ 8piw : 15

We can now move to firm i’s profit-maximization problem (recall that firm i uses a franchisee), max i pw ;Fi

piw qi þ Fi

And since Fi ¼ ðpi  piw Þqi , the profit maximization problem becomes max i pw

pi qi þ pi qi  piw qi

Taking first order conditions with respect to the wholesale price that firm i charges to the franchisee, piw , we obtain @pi @pi @pi @pj þ ¼ 0: i @pi @pw @pj @piw That is, 

1  4pi þ pj

 8 2 þ pi ¼ 0 15 15

And solving for pi yields pi ¼

4 4 þ pj for all i ¼ f1; 2g: 15 15

272

5

Since pi ¼ sion becomes

5 þ 8piw 15

and pj ¼

5 þ 2piw 15 ,

Applications to Industrial Organization

from our above results, the previous expres5 þ 2piw 15

5 þ 8piw 4 þ 4  ¼ 15 15

5 And solving for the wholesale price, piw , we obtain piw ¼ 112 . Therefore, the selling prices are

pi ¼

5 5 þ 8  112 5 ¼ 14 15

and pj ¼

5 5 þ 2  112 19 ¼ ; 56 15

while the amounts sold by each firm are

5 19 5 qi ¼ 1  2  þ ¼ 14 56 8

and qj ¼

19 5 þ 12 56 14

¼

19 ; 28

and their associated profits are 5 5 25 19 19 361 pi ¼  ¼ ’ 0:22 and pj ¼  ¼ ’ 0:23: 8 14 112 56 28 1568 Let us summarize the profits firms earn in each of the previous three cases with the following normal-form game (Fig. 5.16). If firm j uses a retailer (in the left column), firm i’s best response is to also use a retailer; while if firm j sells directly (in the right-hand column), firm i’s best response is to use a retailer. Hence, using a retailer is a strictly dominant strategy for firm i, i.e., it prefers to use a retailer regardless of firm j’s decision. Since firm j’s payoffs are symmetric, the unique Nash equilibrium of the game has both firms using a retailer.

Fig. 5.16 Profits matrix

Exercise 15—Direct Sales or Using A Retailer?C

273

Interestingly, in this equilibrium both firms obtain higher profits franchising their 28 ’ 0:231; than when they do not use a sales (using a retailer), i.e., pi ¼ pj ¼ 121 retailer (in which case their profits are pi ¼ pj ¼ 29 ¼ 0:22Þ.

Exercise 16—Profitable and Unprofitable MergersA Consider an industry with n identical firms competing à la Cournot. Suppose that the inverse demand function is P(Q) = a − bQ, where Q is total industry output, and a, b > 0. Each firm has a marginal costs, c, where c < a, and no fixed costs. Part (a) No merger. Find the equilibrium output that each firm produces at the symmetric Cournot equilibrium. What is the aggregate output and the equilibrium price? What are the profits that every firm obtains in the Cournot equilibrium? What is the equilibrium social welfare? Part (b) Merger. Now let m out of n firms merge. Show that the merger is profitable if and only if it involves a sufficiently large number of firms. Part (c) Are the profits of the nonmerged firms larger when m of their competitors merge than when they do not? Part (d) Show that the merger reduces consumer welfare. Answer Part (a) At the symmetric equilibrium with n firms, we have that each firm i solves max qi

where Qi

P j6¼i

ða  bQi  bqi Þqi  cqi

qj denotes the aggregate production of all other j 6¼ i firms. Taking

first-order conditions with respect to qi , we obtain a  bQi  2bqi  c ¼ 0: And at the symmetric equilibrium Qi ¼ ðn  1Þqi . Hence, the above first order conditions become a  bðn  1Þqi  2bqi  c ¼ 0 or a  c ¼ bðn þ 1Þqi . Solving for qi , we find that the individual output level in equilibrium is qi ¼

ac : ðn þ 1Þb

Hence, aggregate output in equilibrium is Q ¼ nqi ¼ n þn 1 ac b ; while the equilibrium price is

274

5

p ¼ a  bQ ¼ a  b

Applications to Industrial Organization

n a  c a  cn ¼ nþ1 b nþ1

And equilibrium profits that every firm i obtains are pi ¼ ðp  cÞqi ¼

ða  cÞ2 ð n þ 1Þ 2 b

:

The level of social welfare with n firms, Wn , is defined by RQ Wn ¼ 0 ½a  bQdQ  cQ. Calculating the integral, we obtain b 2 n ac aQ  2 ðQ Þ  cQ. Substituting for Q ¼ n þ 1 b yields Wn ¼

nðn þ 2Þða  cÞ2 2nðn þ 1Þ2

:

Part (b) Assume that m out of n firms merge. While before the merger there are n firms in this industry, after the merger there are n − m + 1. [For instance, if n = 10 and m = 7, the industry now has n − m + 1 = 4 firms, i.e., 3 nonmerged firms and the merged firm.] In order to examine whether the merger is profitable for the m merged firms, we need to show that the profit after the merger, pnm þ 1 ; satisfies pnm þ 1  mpn ; ða  cÞ2 ðn  m þ 2Þ

2

m

ða  cÞ2 ðn þ 1Þ2

Solving for n, we obtain that pffiffiffiffi n\m þ m  1; pffiffiffiffi as depicted in the (n, m)-pairs below the line m þ m  1 in Fig. 5.17. And note that this condition is compatible with the fact that the merger must involve a subset of all firms, i.e., m  n; as depicted in the points below the

Fig. 5.17 Profitable mergers pffiffiffiffi satisfy n\m þ m  1

Exercise 16—Profitable and Unprofitable MergersA

275

45-degree line. For instance, in an industry with n = 100 firms, the condition we found determines that this merger is only profitable if at least 89 firms merge. This result is due to Salant, Switzer and Reynolds (1983) and, since the function pffiffiffiffi m þ m  1 can be approximated by a straight line with a slope of 0.8, it is usually referred as the “80% rule,” intuitively indicating that the market share of the merged firms must be at least 80% for their merger to be profitable. Part (c) The output produced by the merged firms decreases (relative to their output before the merger), implying that each of the nonmerged firms earns larger profits after the merger because of pnm þ 1  pn . This condition holds for mergers pffiffiffiffi of any size, i.e., both when condition n\m þ m  1 holds and otherwise. This surprising result is often referred to as the “merger paradox.” Part (d) Since the equilibrium price p is decreasing in n, and the merger reduces the number of firms in the industry, it increases the equilibrium price. Similarly, since Wn is increasing in n, equilibrium social welfare is reduced as a consequence of the merger.

6

Repeated Games and Correlated Equilibria

Introduction In this chapter we explore agents’ incentives to cooperate when they interact in infinite repetitions of a stage game, such as the Prisoner’s Dilemma game or the Cournot oligopoly game. Repeated interactions between the same group of individuals, or repeated competition between the same group of firms in a given industry, are fairly common. As a consequence, this chapter analysis helps at characterizing strategic behavior in several real-life settings in which agents expect to interact with one another for several time periods. In these settings, we evaluate: 1. the discounted stream of payoffs that agents can obtain if they recurrently cooperate; 2. how to find a player’s optimal deviation, and the associated payoff that he would obtain from deviating; and 3. under which conditions the payoff in (1) is larger than the payoff in (2) thus motivating players to cooperate. We accompany our discussion with several figures in order to highlight the trade-off between the current benefits that players obtain from deviating versus the future losses they would sustain if they deviate today (i.e., the cooperative profits that a firm gives up if it breaks the cooperative agreement today). To emphasize these intuitions, we explore the case in which punishments from cooperation last two, three, or infinite periods. We then apply this framework to study firms’ incentives to cooperate in a cartel: first, when the industry consists of only two firms; and second, extend it to settings with more than two firms competing in either quantities or prices, with perfect or imperfect monitoring technologies (i.e., when competing firms can immediately detect a defection from other cartel participants or, instead, need a few periods before noticing such defection). For completeness, we also investigate equilibrium behavior in repeated games where players can choose among three possible strategies in the stage game, which allows for richer © Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_6

277

278

6

Repeated Games and Correlated Equilibria

deviation profiles. We then apply our analysis to bargaining games among three players, and study how equilibrium offers are affected by players’ time preferences. We end the chapter with exercises explaining how to graphically represent the set of feasible payoffs in infinitely repeated games, and how to further restrict such area to payoffs which are also individually rational for all players. This graphical representation of feasible and individually rational payoffs helps us present the so-called “Folk theorem”, as we next describe. Folk Theorem. For every Nash equilibrium in the unrepeated version of the game with expected payoff vector
v ¼ ð
v1 ;
v2 ; . . .;
vN Þ, where
vi 2 R, there is a sufficiently high discount factor d 
d, where d 2 ½0; 1 for which any feasible and individually rational payoff vector v ¼ ðv1 ; v2 ; . . .; vN Þ can be sustained as the SPNE of the infinitely repeated version of the game. Intuitively, if players care enough about their future payoffs (high discount factor), they can identify an equilibrium strategy profile in the infinitely repeated game that provides them with higher average per-period payoff than the payoff they obtain in the Nash equilibrium of the unrepeated version of the game.

Exercise 1—Infinitely Repeated Prisoner’s Dilemma GameA Consider the infinitely repeated version of the following prisoner’s dilemma game, where C denotes confess and NC represents not confess (Table 6.1): Part (a) Can players support the cooperative outcome (NC, NC) as a SPNE by using tit-for-tat strategies, whereby players punish deviations from NC in a past period by reverting to the stage-game Nash equilibrium where they both confess (C, C) for just one period, and then returning to cooperation? Part (b) Consider now that players play tit-for-tat, but reverting to the stage-game Nash equilibrium where they both confess (C, C) for two periods, and then players return to cooperation. Can the cooperative outcome (NC, NC) be supported as a SPNE? Part (c) Suppose that players use strategies that punish deviation from cooperation by reverting to the stage-game Nash equilibrium for ten periods before returning to cooperation. Compute the threshold discount factor, d, above which cooperation can be sustained as a SPNE of the infinitely repeated game.

Table 6.1 Prisoner’s dilemma game Player 1

C NC

Player 2 C

NC

2, 2 0, 6

6, 0 3, 3

Exercise 1—Infinitely Repeated Prisoner’s Dilemma GameA

279

Answer Part (a) At any period t, when cooperating in the (NC, NC) outcome, every player i = {1, 2} obtains a payoff stream of: 3 þ 3d þ 3d2 þ    where d is the discount factor. If, instead, player i deviates to C, he obtains a payoff of 6 today. However, in the following period, he is punished by the (C, C) outcome, which yields a payoff of 2, and then both players return to the competitive outcome (NC, NC), which yields a payoff of 3. Hence, the payoff stream that a deviating player obtains is: 6 þ |{z} gain

2d |{z}

punishment

þ

2 3d þ  |fflfflfflfflffl{zfflfflfflfflffl}

back to cooperation

As a remark, note that once player j 6¼ i detects that player i is defecting (i.e., playing C), he finds that NC is a best response to player i playing NC. Hence, the punishment is credible, since player j has incentives to select NC, thus giving rise to the (NC, NC) outcome. Hence, cooperation can be sustained if: 3 þ 3d þ 3d2 þ     6 þ 2d þ 3d2 þ    and rearranging, 3d  2d  6  3 which yields d  3, a condition that cannot hold since d must satisfy d 2 ½0; 1. Hence, no cooperation can be sustained with temporary reversion to the Nash Equilibrium of the stage game for only one period. Intuitively, the punishment for deviating is too small to induce players to stick to cooperation over time. Figure 6.1 represents the stream of payoffs that a given player obtains by cooperating, and compares it with his stream of payoffs from cheating. Graphically, the instantaneous gain the cheating player obtains today, 6 − 3 = 3, offsets the future loss he would suffer from being punished during only one period tomorrow, 3 − 2 = 1. Part (b) At any period t, when cooperating by choosing NC, players get the same payoff stream we identified in part (a): 3 þ 3d þ 3d2 þ 3d3 þ   

280

6

Repeated Games and Correlated Equilibria

Fig. 6.1 Payoff stream when defection is punished for one period

If, instead, a player i = {1, 2} defects to confess, C, he will now suffer a punishment during two periods, thus yielding a payoff stream of: 6 þ |fflfflfflffl 2d fflþ 2dffl}2 þ |{z} {zfflfflfflffl gain

punishment

3 3d þ  |fflfflfflfflffl{zfflfflfflfflffl}

back to cooperation

Hence, cooperation can be sustained if: 3 þ 3d þ 3d2 þ 3d3 þ     6 þ 2d þ 2d2 þ 3d3 þ    and rearranging, 3 þ 3d þ 3d2  6 þ 2d þ 2d2 d2 þ d  3  0 Solving for d we have: d¼

1 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 1 þ ð4  1  3Þ 1  13 ¼ 2 21

which yields, d1 ¼ 2:3\0

and

d2 ¼ 1:3 [ 1

Indeed, depicting d2 þ d  3 ¼ 0, we can see that the function lies in the negative quadrant, for all d 2 ½0; 1, as Fig. 6.2 illustrates. Hence, d2 þ d  3  0 cannot be satisfied for any d 2 ½0; 1. Therefore, cooperation cannot be supported in this case either, when we use a temporary punishment of only two periods. Graphically, the instantaneous gain from cheating (which increases a player’s payoff from 3 to 6 in Fig. 6.3) still offsets the future loss from being punished, since the punishment only lasts for two periods.

Exercise 1—Infinitely Repeated Prisoner’s Dilemma GameA

281

Fig. 6.2 Function d2 þ d  3

Fig. 6.3 Payoff stream when defection is punished for two periods

Part (c) At any period t, when a player i = {1, 2} chooses to cooperate, choosing NC, while his opponent cooperates, his payoff stream becomes: 3 þ 3d þ 3d2 þ 3d3 þ    þ 3d10 þ 3d11 þ    If instead, player i deviates, he obtains a payoff of 6 today, but he is subsequently punished for 10 periods, obtaining a payoff of only 2 in each period, before returning to the cooperation, where his payoff is 3. Hence, his payoff stream is: 6 þ 2d þ 2d2 þ 2d3    þ 2d10 þ 3d11 þ    Hence, cooperation can be sustained if: 3 þ 3d þ 3d2 þ 3d3 þ    þ 3d10 þ 3d11 þ     6 þ 2d þ 2d2 þ 2d3    þ 2d10 þ 3d11 þ   

282

6

Repeated Games and Correlated Equilibria

Rearranging: ð3  2Þd þ ð3  2Þd2 þ ð3  2Þd3 þ    þ ð3  2Þd10  6  3 d þ d2 þ d3 þ    þ d10 3 d 1 þ d þ d2 þ    þ d9  3 After discarding all solutions of d that are negative or larger than 1, we get: d  0:76 Hence, a long punishment of ten periods allows players to sustain cooperation if they assign a sufficiently high importance to future payoffs, i.e., d is close to 1.

Exercise 2—Collusion When Firms Compete in QuantitiesA Consider two firms competing as Cournot oligopolists in a market with demand: pðq1 ; q2 Þ ¼ a  bq1  bq2 Both firms have total costs, TC ðqi Þ ¼ cqi where c > 0 is the marginal cost of production, and a > c. Part (a) Considering that firms only interact once (playing an unrepeated Cournot game), find the equilibrium output, the market price, and the equilibrium profits for every firm. Part (b) Now assume that they could form a cartel. Which is the output that every firm should produce in order to maximize the profits of the cartel? Find the market price, and profits of every firm. Are their profits higher when they form a cartel than when they compete as Cournot oligopolists? Part (c) Can the cartel agreement be supported as the (cooperative) outcome of the infinitely repeated game? Answer Part (a) Firm 1 chooses q1 to maximize its profits max p1 ¼ ða  bq1  bq2 Þq1  cq1 q1

¼ aq1  bq21  bq2 q1  cq1

Exercise 2—Collusion When Firms Compete in QuantitiesA

283

Taking first order conditions with respect to q1 we find: @p1 ¼ a  2bq1  bq2  c ¼ 0 @q1 and solving for q1 we find firm 1’s best response function q1 ð q2 Þ ¼

ac 1  q2 2b 2

We can obtain a similar expression for firm 2’s best response function, q2(q1), since firms are symmetric q2 ð q1 Þ ¼

ac 1  q1 2b 2

Plugging q2 ðq1 Þ into q1 ðq2 Þ, we can find firm 1’s equilibrium output:   ac 1 ac 1 ac 1   q1 ) q1 ¼ þ q1 q1 ¼ 2b 2 2b 2 4b 4 ac ) q1 ¼ 3b and similarly, firm 2’s equilibrium output is q2 ¼

a  c 1 a  c a  c a  c a  c   ¼ ¼ 2b 2 3b 2b 6b 3b

Therefore, the market price is a  c  a  c b 3b 3b 3a  a þ c  a þ c a þ 2c ¼ ¼ 3 3

pðq1 ; q2 Þ ¼ a  bq1  bq2 ¼ a  b

and the equilibrium profits of every firm i = {1, 2} are pcournot ¼ pcournot ¼ pð q1 ; q2 Þ  qi  c  qi 1 2   a  c ða  cÞ2 a þ 2c a  c ¼ c ¼ 3 3b 3b 9b Part (b) Since the cartel seeks to maximize the firms’ joint profits, they simultaneously choose q1 and q2 that solve

284

6

Repeated Games and Correlated Equilibria

max p1 þ p2 ¼ ða  bq1  bq2 Þq1  cq1 þ ða  bq1  bq2 Þq2  cq2 q1 ;q2

which simplifies to maxða  bq1  bq2 Þðq1 þ q2 Þ  cðq1 þ q2 Þ q1 ;q2

Notice, however, that this profits maximization problem can be further simplified to maxða  bQÞQ  cQ Q

since Q ¼ q1 þ q2 . This maximization problem coincides with that of a regular monopolist. In other words, the overall production of the cartel of two firms which are symmetric in costs coincides with that of a standard monopoly. Indeed, taking first order conditions with respect to Q, we obtain a  2bQ  c  0 Solving for Q, we find Q ¼ ac 2b , which is an interior solution given that a > c by definition. Therefore, each firm’s output in the cartel is q1 ¼ q2 ¼

ac 2b

2

¼

ac 4b

and the market price is: p ¼ a  bQ ¼ a  b

 a  c 2b

¼

aþc 2

Therefore, each firm’s profit in the cartel is: a þ c a  c a  c ða  cÞ2 c ¼ 2 4b 4b 8b 2 ða  cÞ ¼ pcartel ¼ 2 8b

p1 ¼ pq1  TCðq1 Þ ¼ pcartel 1

Comparing the profits that every firm earns in the cartel, ðacÞ2 9b ,

ðacÞ2 8b

; against those

we can thus conclude that both firms have under Cournot competition, incentive to collude in the cartel for all parameter values. pcartel ¼ pcartel [ pcournot ¼ pcournot 1 2 1 2

Exercise 2—Collusion When Firms Compete in QuantitiesA

285

Part (c) Cooperation: If at any period t, a given firm i cooperates producing the cartel output, while all other firms produce the cartel output as well, its profit is ðacÞ2 8b .

As a consequence, the discounted sum of the infinite stream of profits from cooperating in the cartel is ða  cÞ2 ða  cÞ2 ða  cÞ2 1 ða  cÞ2 þd þ d2 þ  ¼ : 1  d 8b 8b 8b 8b (Recall that the sum 1 þ d þ d2 þ d3 þ    can be simplified to

1 1d.)

Deviation: Since firms are symmetric, we only consider one of the firms (firm 1). In particular, we need to find the optimal deviation that, conditional on firm 2 choosing the cartel output, maximizes firm 1’s profits. That is, which is the output that maximizes firm 1’s profits from deviating? Since firm 2 sticks to cooperation (i.e., produces the cartel output q2 ¼ ac 4b ), if firm 1 seeks to maximize its current profits, we only need to plug q2 ¼ ac into firm 1’s best response function, q1 ðq2 Þ ¼ 4b ac 1 2b  2 q2 ; as follows qdev 1 ¼ q1

a  c a  c 1 a  c 3ða  cÞ  ¼ ¼ 4b 2b 2 4b 8b

This result provides us with firm 1’s optimal deviation, given that firm 2 is still respecting the cartel agreement. In this context, firm 1’s profit is

    a  c 3ð a  c Þ 3ða  cÞ p1 ¼ a  b b c 8b 4b 8b   2 a  c 3ða  cÞ 9ða  cÞ ¼3 ¼ 8 8b 64b while that of firm 2 is

   a  c 3ð a  c Þ a  c p2 ¼ a  b b c 8b 4b 4b a  c a  c 3ða  cÞ2 ¼3 ¼ 8 4b 32b Hence, firm 1 has incentives to unilaterally deviate since its current profits are larger by deviating than by cooperating. That is, pdeviate ¼ 1

9ða  cÞ2 ða  cÞ2 [ pcartel ¼ 1 64b 8b

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Incentives to cooperate: We can now compare the profits that firms obtain from cooperating in the cartel agreement against those from choosing an optimal deviation plus the profits they would obtain from being punished thereafter (discounted profits in the Cournot oligopoly). In particular, for cooperation to be sustained we need 1 ða  cÞ2 9ð a  c Þ 2 d ða  cÞ2 [ þ 1  d 8b 1  d 9b 64b Solving for d in this inequality, we obtain 1 9 d [ þ ; 8ð1  dÞ 64 9ð1  dÞ which yields d[

9 17

Hence, 9 firms need to assign a sufficiently high importance to future payoffs, d 2 17 ; 1 , for the cartel agreement to be sustained. Credible punishments: Finally, notice that firm 2 has incentives to carry out the punishment. Indeed, if it doesn’t revert to the Nash equilibrium of the stage game (producing the Cournot equilibrium output), firm 2 obtains profits of qdev 1

3ðacÞ2 32b ,

since

3ðacÞ 8b

firm 1 keeps producing its optimal deviation ¼ while firm 2 produces the Cartel ac cartel output q2 ¼ 4b : If, instead, firm 2 practices the punishment, producing 2

2

ðacÞ 3ðacÞ the Cournot output ac 3b ; its profits are 9b , which exceed 32b for all parameter values. Hence, upon observing that firm 1 deviates, firm 2 prefers to revert to the production of its Cournot output level than being the only firm that produces the cartel output.

Exercise 3—Temporary Punishments from DeviationB Consider two firms competing a la Cournot, and facing linear inverse demand pðQÞ ¼ 100  Q, where Q ¼ q1 þ q2 denotes aggregate output. For simplicity, assume that firms face a common marginal cost of production c ¼ 10.

Exercise 3—Temporary Punishments from DeviationB

287

(a) Unrepeated game. Find the equilibrium output each firm produces when competing a la Cournot (that is, when they simultaneously and independently choose their output levels) in the unrepeated version of the game (that is, when firms interact only once). In addition, find the profits that each firm earns in equilibrium. (b) Repeated game—Collusion. Assume now that the CEOs from both companies meet to discuss a collusive agreement that would increase their profits. Set up the maximization problem that firms solve when maximizing their joint profits (that is, the sum of profits for both firms). Find the output level that each firm should select to maximize joint profits. In addition, find the profits that each firm obtains in this collusive agreement. (c) Repeated game—Permanent punishment. Consider a grim-trigger strategy in which every firm starts colluding in period 1, and it keeps doing so as long as both firms colluded in the past. Otherwise, every firm deviates to the Cournot equilibrium thereafter (that is, every firm produces the Nash equilibrium of the unrepeated game found in part a forever). In words, this says that the punishment of deviating from the collusive agreement is permanent, since firms never return to the collusive outcome. For which discount factors this grim-trigger strategy can be sustained as the SPNE of the infinitely-repeated game? (d) Repeated game—Temporary punishment. Consider now a “modified” grim-trigger strategy. Like in the grim-trigger strategy of part (c), every firm starts colluding in period 1, and it keeps doing so as long as both firms colluded in the past. However, if a deviation is detected by either firm, every firm deviates to the Cournot equilibrium during only 1 period, and then every firm returns to cooperation (producing the collusive output). Intuitively, this implies that the punishment of deviating from the collusive agreement is now temporary (rather than permanent) since it lasts only one period. For which discount factors this “modified” grim-trigger strategy can be sustained as the SPNE of the infinitely-repeated game? (e) Consider again the temporary punishment in part (d), but assume now that it lasts for two periods. How are your results from part (d) affected? Interpret. (f) Consider again the temporary punishment in part (d), but assume now that it lasts for three periods. How are your results from part (d) affected? Interpret. Answer Part (a). Firm i’s profit function is given by:     pt qt ; qj ¼ 100  qi þ qj qt  10qt Differentiating with respect to output qi yields, 100  2qi  qj  10 ¼ 0

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Solving for qi , we find firm i’s best response function   qj qi qj ¼ 45  2

ðBRFÞ

Since this is a symmetric game, firm j’s best response function is symmetric. Therefore, in a symmetric equilibrium both firms produce the same output level, qi ¼ qj , which helps us rewrite the above best response function as follows qi ¼ 45 

qi 2

Solving this expression yields equilibrium output of q i ¼ q 1 ¼ q 2 ¼ 30. Substituting these results into profits p1 ðq1 ; q2 Þ and p2 ðq1 ; q2 Þ, we obtain that  pi q i ; q j ¼ pi ð30; 30Þ ¼ ½100  ð30 þ 30Þ30  10  30 ¼ $900    where firm’s equilibrium profits coincide, that is, pi q t ; q j ¼ p1 q 1 ; q 2 ¼   p2 q 1 ; q 2 . In summary, equilibrium profit is $900 for each firm and combined profits are $900 þ $900 ¼ $1800 Part (b). In this case, firms maximize the sum p1 ðq1 ; q2 Þ þ p2 ðq1 ; q2 Þ, as follows 

       100  qi þ qj qi  10qi þ 100  qi þ qj qj  10qj

Differentiating with respect to output qt , yields 100  2qi  qj  10  qj ¼ 0 Solving for qt , we obtain   qi qj ¼ 45  qj and similarly when we differentiate with respect to qj . In a symmetric equilibrium both firms produce the same output level, qi ¼ qj , which helps us rewrite the above expression as follows qi ¼ 45  qi Solving for output qi we obtain equilibrium output q i ¼ q 1 ¼ q 2 ¼

45 ¼ 22:5 2

Exercise 3—Temporary Punishments from DeviationB

289

units. This yields each firm a profit of  pi q i ; q j ¼ pi ð22:5; 22:5Þ ¼ f½100  ð22:5 þ 22:5Þg22:5  ð10  22:5Þ þ f½100  ð22:5 þ 22:5Þ22:5  ð10  22:5Þg ¼ $1012:5

Thus, each firm makes a profit of $1012.5, which yields a joint profit of $2025. Part (c). For this part of the exercise, let us first list the payoffs that the firm can obtain from each of its output decisions: • Cooperation yields a payoff of $1012.5 for firm i. • We now find the payoff that firm i obtains when defecting while firm j cooperates (qj = 22.5). We need to find firm i’s optimal deviation in this context, which is found by inserting firm j’s output of qj = 22.5 units into firm i’s best response function, yielding 

22:5 qi ð22:5Þ ¼ 45  2

 ¼ 33:75

units, which generates a deviating profit for firm i of ð100  ð33:75 þ 22:5ÞÞ33:75  ð10  33:75Þ ¼ $1; 139:06: However, such defection is punished with Cournot competition in all subsequent periods, which yields a profit of only $900. Therefore, firm i cooperates as long as 1012:5 þ 1012:5d þ 1012:5d2 þ     1139:06 þ 900d þ 900d2 þ    which can be simplified to     1012:5 1 þ d þ d2 þ     1139:06 þ 900d 1 þ d þ d2 þ    and expressed more compactly as 1012:5 900d  1139:06 þ 1d 1d Multiplying both sides of the inequality by 1  d, yields 1012:5  1139:06ð1  dÞ þ 900d which rearranging and solving for discount factor d entails d  0:53

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Thus, for cooperation to be sustainable in an infinitely repeated game with permanent punishments, firms’ discount factor d has to be at least 0.53. In other words, firms must put a sufficient weight on future payoffs. Part (d). The set up is analogous to that in part (c) of the exercise, but we now write that cooperation is possible if 1012:5 þ 1012:5d þ 1012:5d2 þ     1139:06 þ 900d þ 1012:5d2 þ    Importantly, note that payoffs after the punishment period (in this case, a one period punishment of Cournot competition with $900 profits) returns to cooperation, explaining that all payoffs in the third term of the left-hand and right-hand side of the inequality coincide. We can therefore cancel them out, simplifying the above inequality to 1012:5 þ 1012:5d  1139:06 þ 900d Rearranging yields 112:5d  126:56 ultimately simplifying to d  1:125. That is, when deviations are only punished during one period cooperation cannot be sustained since discount factor d must be in [0,1] by assumption. Part (e). Following a similar approach as in part (d), we write that cooperation can be sustained if and only if 1012:5 þ 1012:5d þ 1012:5d2 þ 1012:5d3     1139:06 þ 900d þ þ 900d2 þ 1012:5d3   

The stream of payoffs after the punishment period (in this case, a two-period punishment of Cournot competition with $900 profits) returns to cooperation. Hence, the payoffs in both the left- and right-hand side of the inequality after the punishment period coincide and cancel out. The above inequality then becomes 1012:5 þ 1012:5d þ 1012:5d2  1139:06 þ 900d þ 900d2 which, after rearranging, yields d2 þ d  1:125  0 ultimately simplifying to d  0:67. In words, when deviations are punished for two periods, cooperation can be sustained if firms’ discount factor d is at least 0.67. Part (f). Following a similar approach as in part (e), we write that cooperation can be sustained if and only if

Exercise 3—Temporary Punishments from DeviationB

291

1012:5 þ 1012:5d þ 1012:5d2 þ 1012:5d3 þ 1012:5d4 . . .  1139:06 þ 900d þ 900d2 þ 900d3 þ 1012:5d4 . . .: The stream of payoffs after the punishment period (in this case, a three-period punishment of Cournot competition with $900 profits) returns to cooperation. Hence, the payoffs in both the left- and right-hand side of the inequality after the punishment period coincide and cancel out. The above inequality then becomes 1012:5 þ 1012:5d þ 1012:5d2 þ 1012:5d3  1139:06 þ 900d þ 900d2 þ 900d3 which, after rearranging, yields d3 þ d2 þ d  1:125  0 ultimately simplifying to d  0:58. Therefore, when deviations are punished for three periods, cooperation can be sustained if firms’ discount factor d is at least 0.58.

Exercise 4—Collusion When N Firms Compete in QuantitiesB Consider n firms producing homogenous goods and choosing quantities in each period for an infinite number of periods. Demand in the industry is given by p = 1 − Q, Q being the sum of individual outputs. All firms in the industry are identical: they have the same constant marginal costs c < 1, and the same discount factor d. Consider the following trigger strategy: • Each firm sets the output qm that maximizes joint profits at the beginning of the game, and continues to do so unless one or more firms deviate. • After a deviation, each firm sets the quantity q, which is the Nash equilibrium of the unrepeated Cournot game. Part (a) Find the condition on the discount factor that allows for collusion to be sustained in this industry. Part (b) Indicate whether a larger number of firms in the industry facilitates or hinders the possibility of reaching a collusive outcome. Answer Part (a) Cooperation: First, we need to find the quantities that maximize joint profits p ¼ ð1  QÞQ  cQ. This output level coincides with that under monopoly, i.e., Q ¼ 1c 2 , yielding aggregate profits of  p¼

1

 1c 1c 1  c ð1  cÞ2 c ¼ 2 2 2 4

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6

Repeated Games and Correlated Equilibria

for the cartel. Therefore, at the symmetric equilibrium, individual quantities are 2

ð1cÞ m qm ¼ 1n Q ¼ 1n 1c 2 and individual profits under the collusive strategy are p ¼ 4n .

Deviation: As for the deviation profits, the optimal deviation by firm i, qdev, is given by solving qd ðqm Þ ¼ argmaxq ½1  ðn  1Þqm  q  cq: Intuitively, the above expression represents that firm i selects the output level that maximizes its profits given that all other (n − 1) firms are still respecting the collusive agreement, thus producing qm. In particular, the value of q that maximizes the above expression is obtained through the first order conditions: ½1  ðn  1Þqm   2q  c ¼ 0 1  ðn  1Þ 1n 1c  2q  c ¼ 0 2 which, solving for q, yields, qd ¼ ð n þ 1Þ

ð1  c Þ : 4n

The profits that a firm obtains by deviating from the collusive output are hence

pd ¼ 1  ðn  1Þqm  qd ðqm Þ  qd ðqm Þ  c  qd ðqm Þ

   11  c 1c 1c 1c  c ð n þ 1Þ ;  ð n þ 1Þ ð n þ 1Þ ¼ 1  ð n  1Þ n 2 4n 4n 4n which simplifies to pd ¼

ð1  cÞ2 ðn þ 1Þ2 16n2

Incentives to collude: Given the above profits from colluding and from deviating, every firm i chooses to collude as long as 1 d pm  p d þ pc 1d 1d Multiplying both sides of the inequality by ð1  dÞ, we obtain pm  ð1  dÞpd þ pc

Exercise 4—Collusion When N Firms Compete in QuantitiesB

293

Fig. 6.4 Critical threshold of dc ðnÞ

which, solving for d, yields d

pd  pm pd  pc

Intuitively, the numerator represents the profit gain that a firm experiences when it unilaterally deviates from the cartel agreement, pd  pm . When such profit gain increases, cartel becomes more difficult to sustain, i.e., it is only supported for higher discount factors. Plugging the expressions of profits pd ; pm , and pc we previously found, yields a cutoff discount factor of d

ð1 þ nÞ2 1 þ 6n þ n2

For compactness, we denote this ratio as ð 1 þ nÞ 2

dc 1 þ 6n þ n2 Hence, under punishment strategies that involve a reversion to Cournot equilibrium forever after a deviation takes place, tacit collusion arises if and only if firms are sufficiently patient. Figure 6.4 depicts cutoff dc , as a function of the number of firms, n, and shades the region of d that exceeds such a cutoff. Differentiating the critical threshold of the discount factor, dc , with respect to the number of firms, n, we find that

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Repeated Games and Correlated Equilibria

Fig. 6.5 Critical threshold ~dðn; hÞ

@d 4ð n2  1Þ ¼ [0 @n ð1 þ 6n þ n2 Þ Intuition: Other things being equal, as the number of firms in the agreement increases, the more difficult it is to reach and sustain collusion in the cartel agreement, i.e., as the market becomes less concentrated collusion becomes less likely.

Exercise 5—Collusion When N Firms Compete in PricesC Consider a homogenous industry where n firms produce at zero cost and play the Bertrand game of price competition for an infinite number of periods. Assume that: • When firms choose the same price, they earn a per-period profit pð pÞ ¼ pa DðpÞ n , where parameter a represents the state of demand. • When a firm i charges a price of pi lower than the price of all of the other firms, it earns a profit pðpi Þ ¼ pi aDðpi Þ, and all of the other firms obtain zero profits. Imagine that in the current period demand is characterized by a = 1, but starting from the following period demand will be characterized by a = h in each of the following periods. All the players know exactly the evolution of the demand state at the beginning of the game, and firms have the same common discount factor, d. Assume that h [ 1 and consider the following trigger strategies. Each firm plays the monopoly price pm in the first period of the game, and continues to charge such a price until a profit equal to zero is observed. When this occurs, each firm charges a price equal to zero forever. Under which conditions does this trigger strategy represent a SPNE? [Hint: In particular, show how h and n affects such a condition, and give an economic intuition for this result.]

Exercise 4—Collusion When N Firms Compete in QuantitiesB

295

Answer Cooperation: Let us denote the collusive price by pc 2 ðc; pm . At time t = 0, parameter a takes the value of a = 1, whereas at any subsequent time period t = {1, 2,…}, parameter a becomes a ¼ h. Hence, by colluding, firm i obtains a discounted stream of profits of pðpc Þ pðpc Þ pðpc Þ pðpc Þ þ dh þ d2 h þ d3 h þ  n n n n pðpc Þ ¼ ð1 þ dh þ d2 h þ d3 h þ   Þ n Deviation: When deviating, firm i charges a price marginally lower than the collusive price pc and captures all the market, thus obtaining a profit of pðpc Þ. However, after that deviation, all firms revert to the Nash equilibrium of the unrepeated Bertrand game, which yields a profit of zero thereafter. Therefore, the payoff stream from deviating is pðpc Þ þ 0 þ d0 þ    ¼ pðpc Þ. Incentives to collude: Hence, every firm i colludes as long as pðpc Þ ð1 þ dh þ d2 h þ d3 h þ   Þ  pðpc Þ; n rearranging, we obtain 1 þ dh þ d2 h þ d2 h þ     n 1 þ dhð1 þ d2 h þ d2 h þ   Þ  n 1 1 þ dh 1d n n1 or equivalently, d  n1 þ h. For compactness, we hereafter denote the previous ratio n1 ~ as

dðn; hÞ. Figure 6.5 depicts this critical threshold of the discount factor, n1 þ h

evaluated at h = 1.2, i.e., demand increases 20% after the first year. Comparative statics: We can next examine how the critical discount factor ~ dðn; hÞ is affected by changes in demand, h, and in the number of firms, n. In particular, @ ~dðn; hÞ ðn  1Þ ¼ \0; @h ðn  1 þ hÞ2

whereas

@~ dðn; hÞ h ¼ [ 0: @n ðn  1 þ hÞ2

In words, the higher the increments in demand, h, the higher the present value of the stream of profits received from t = 1 onwards. That is, the opportunity cost of deviation increases as demand becomes stronger. Graphically, the critical discount factor ~dðn; hÞ shifts downwards, thus expanding the region of (d, n)-pairs for which

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Fig. 6.6 Cutoff ~ dðn; hÞ for h ¼ 1:2 and h ¼ 1:8

collusion can be sustained. Figure 6.6 provides an example of this comparative statics result, whereby h is evaluated at h = 1.2 and at h = 1.8. On the other hand, when n increases, the collusion is more difficult to sustain in equilibrium. That is, the region of (d, n)-pairs for which collusion can be sustained shrinks as n increases; as depicted by rightward movements in Fig. 6.6.

Exercise 6—Collusion in Donations in an Infinitely Repeated Public Good GameB Consider a public good game between two players, A and B. The utility function of every player i = {A, B} is   1=2 ui ðgi Þ ¼ ðw  gi Þ1=2 m gi þ gj where the first term, w  gi , represents the utility that the player receives from money (i.e., the amount of his wealth, w, not contributed to the public good). The second term indicates the utility he obtains from aggregate contributions to the public good, gi þ gj , where m  0 denotes the return from aggregate contributions. For simplicity, assume that both players have equal wealth levels w.   (a) Find every player i’s best response function, gi gj . Interpret. (b) Find the Nash Equilibrium of this game, g . (c) Find the socially optimal contribution level, gSO , that is, the donation that each player should contribute to maximize their joint utilities. Compare it with the Nash equilibrium of the game. Which of them yields the highest utility for every player i? (d) Consider the infinitely repeated version of the game. Show under which values of players’ discount factor d you can sustain cooperation, where cooperation is

Exercise 6—Collusion in Donations in an Infinitely …

297

understood as contributing the socially optimal amount (the one that maximizes joint utilities) you found in part (c). For simplicity, you can consider a “Grim-Triger strategy” where every player starts cooperating (that is, donating gSO to the public good), and continues to do so if both players cooperated in all previous periods. However, if one or both players deviate from gSO , he reverts to the Nash equilibrium of the unrepeated game, g , thereafter. Answer Part (a). Every individual i chooses his contribution to the public good, to solve the following utility maximization problem:   1=2 max ui ðgi Þ ¼ ðw  gi Þ1=2 m gi þ gj

gi [ 0

Differentiating ui ðgi Þ with respect to gi gives us qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi m gi þ gj m w  gi dui ðgi Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ¼0  ffi  2  pffiffiffiffiffiffiffiffiffiffiffiffiffi dgi w  gi 2  m gi þ gj Rearranging, we obtain   m ð w  gi Þ m gi þ gj ¼ 2 2 Solving this equation yields a best response function of   w 1 gi gj ¼  gj 2 2

ðBRFi Þ

Figure 6.7 depicts this best response function. It originates at w2 , indicating that player i contributes half of his wealth towards the public good when his opponent does not contribute anything, gj ¼ 0; see vertical intercept on the left-hand side of the figure. However, when player j makes a positive donation, player i reduces his own. Specifically, for every additional dollar that player j contributes, player i decreases his donation by half a dollar. Finally, when player j donates gj ¼ w, player i’s contribution collapses to zero. Graphically, gj ¼ w is the horizontal intercept of the best response function for player i. Intuitively, when player j contributes all his wealth to the public good, player i does not need to donate anything. Players are symmetric since they have the same wealth level and face the same utility function. Therefore, player j’s best response function is analogous to that of player i, and can be expressed as follows (note that we only switch the subscripts relative to BRFi)

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Fig. 6.7 Donor i’s best response function

gj ð gi Þ ¼

w 1  gi 2 2



BRFj



where we only switch the subscripts relative to BRFi. A similar intuition as that above applies to this best response function. Figure 6.8 depicts this best response function, using the same axis as for BRFi, which will help us superimpose both best response functions on the same figure in the next part of the exercise. Part (b). In a symmetric equilibrium, both players contribute the same amount to the public good, which entails that gi ¼ gj ¼ g. Substituting this result in the best response function of either player yields Fig. 6.8 Donor j’s best response function

Exercise 6—Collusion in Donations in an Infinitely …

299

Fig. 6.9 Superimposing both players’ best response functions

g¼

w 1  g 2 2

Solving this for g gives us the equilibrium contribution, as follows, g ¼

w 3

which represents the donation that every player i and j submit in the Nash equilibrium of the game. Figure 6.9 superimposes both players’ best response functions, illustrating that the Nash equilibrium occurs at the point where both best response functions cross each other (that is, representing a mutual best response). Part (c). The socially optimal donation maximizes players’ joint utility, as follows       1=2  1=2   1=2 max ui gi ; gj þ uj gi ; gj ¼ ðw  gi Þ1=2 m gi þ gj þ w  gj m gi þ gj

gi ;gj [ 0

Differentiating with respect to gi , we get, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m g þ g m w  gj i j dui gi ; gj m ð w  gi Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi þ  ffi þ  ffi ¼ 0 dgi 2  w  gi 2  m gi þ gj 2  m gi þ gj 



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Invoking symmetry, gi ¼ gj ¼ g, the above expression simplifies to pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi m w gþ w g 2mg pffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffi w g 2mg Rearranging, we find mðw  gÞ ¼ mg Finally, solving for g, yields a socially optimal donation of SO gSO i ¼ gj ¼

w 2

which is larger than that emerging in the Nash equilibrium of the game, g ¼ w3 . Intuition: Players free-ride off each other’s donations in the Nash equilibrium of the game, ultimately contributing a suboptimal amount to the public good. Alternatively, when every donor independently chooses his donation he ignores the positive benefit that the other donor enjoys (Recall that contributions are non-rival, thus providing the same benefit to every individual, whether he made that donation or not!) In contrast, when every player considers the benefit that his donations have on the other donor’s utility (as we did in the joint utility maximization problem), he contributes a larger amount to the public good. Comparing utility levels. When every player chooses the socially optimal w SO donation gSO i ¼ gj ¼ 2 , player i’s utility becomes 

w

w 1=2 h w w i1=2 w pffiffiffiffi þ m ¼ pffiffiffi m 2 2 2 2

while when every player chooses the donation in the Nash equilibrium of the game, g ¼ w3 , player i’s utility becomes  w 1=2 h w w i1=2 þ ðw  g Þ1=2 ½mðg þ g Þ1=2 ¼ w  m 3 3 3 2w pffiffiffiffi m ¼ 3 Comparing his utility when players coordinate choosing the socially optimal pffiffiffiffi pffiffiffiffi donation, pwffiffi2 m, and his utility in the Nash equilibrium of the game, 2w m, we 3 find that their difference

Exercise 6—Collusion in Donations in an Infinitely …

301

pffiffiffi w pffiffiffiffi 2w pffiffiffiffi 3 2  4 pffiffiffiffi pffiffiffi m  m¼ w m 3 6 2 pffiffi is positive since ratio 3 624 ffi 0:04. In words, every player obtains a higher utility when they coordinate their donations than when they independently choose their contributions to the public good. Part (d). Cooperation. As shown in part (c) of the exercise, when players SO w coordinate their donations to the public good, they donate gSO i ¼ gj ¼ 2 , which yields a per-period cooperating payoff of 

w

w 1=2 h w w i1=2 w pffiffiffiffi þ m ¼ pffiffiffi m 2 2 2 2

thus yielding a discounted stream of payoffs from cooperation of pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi w m w m 1 w m 2w m pffiffiffi þ d pffiffiffi þ d pffiffiffi    ¼ pffiffiffi 1d 2 2 2 2 Optimal deviation. When player i deviates from the cooperative donation w SO w gSO i ¼ 2 , while his opponent still cooperates by choosing gj ¼ 2 , his optimal w deviation is found by inserting gSO j ¼ 2 into player i’s best response function,   w 1 gi gj ¼ 2  2 gj , as follows gi

w 2

¼

w 1 w w   ¼ 2 2 2 4

which yields a deviating utility level for player i of 

w

w 1=2 h w w i1=2 3w pffiffiffiffi þ m m ¼ 4 4 2 4

This can be understood as his “deviating payoff” which he enjoys during only one period. w Punishment. After player i deviates from the cooperative donation gSO i ¼ 2 , his deviation is detected, and punished thereafter by both players, who choose the Nash equilibrium donation in the unrepeated version of the game, that is, g ¼ w3 , yielding a per-period payoff of  w 1=2 h w w i1=2 þ ðw  g Þ1=2 ½mðg þ g Þ1=2 ¼ w  m 3 3 3 2w pffiffiffiffi m ¼ 3

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Therefore, the stream of discounted payoffs that every player receives after player i’s deviation is detected becomes  d

     2w pffiffiffiffi 2w pffiffiffiffi d 2w pffiffiffiffi m þ d2 m þ  ¼  m 3 3 1d 3

Comparing stream of payoffs. We can now compare the discounted stream of payoffs from cooperation against that from defecting, finding that every player cooperates after a history of cooperation by both players if and only if   pffiffiffiffi 1 w m 3w pffiffiffiffi d 2w pffiffiffiffi pffiffiffi  mþ  m 1d 2 4 ð 1  dÞ 3 In words, the left-hand side of the inequality represents player i’s discounted stream of payoffs from cooperating. The first term in the right-hand side captures the payoff that player i enjoys from deviating (which only lasts one period, since his deviation is immediately detected), while the second term reflects the discounted stream of payoffs from being punished in all subsequent periods after his deviation. Multiplying both sides of the inequality by 1  d, we obtain   pffiffiffiffi w m 3w pffiffiffiffi 2w pffiffiffiffi pffiffiffi  ð1  dÞ  mþd m 4 3 2 And solving for discount factor d, we find that cooperation can be sustained in the Subgame Perfect Nash Equilibrium (SPNE) of the game if 1 3 2 pffiffiffi  ð1  dÞ þ d 4 3 2 which simplifies to pffiffiffi 6 2  ð1  dÞ0:9 þ 8d or, after solving for d, pffiffiffi d  9  6 2 0:51 Intuitively, players must care enough about their future payoffs to sustain their cooperation in the public good game.

Exercise 7—Repeated Games …

303

Exercise 7—Repeated Games with Three Available Strategies to Each Player in the Stage GameA Consider the following simultaneous-move game between player 1 (in rows) and player 2 (in columns) (Table 6.2). This stage game is played twice, with the outcome from the first stage observed before the second stage begins. There is no discounting i.e., the discount factor d is d = 1. Can payoff (4, 4) be achieved in the first stage in a pure-strategy subgame perfect equilibrium? If so, describe a strategy profile that does so and prove that it is a subgame perfect Nash equilibrium. If not, prove why not. Answer PSNE: For player 2, his set of best responses are the following: if player 1 plays T, then player 2’s best response is BR2 ðT Þ ¼ L if instead player 1 plays M, then BR2 ðM Þ ¼ C; and if player 1 plays B, BR2 ðBÞ ¼ R: For player 1, his best responses are the following: if player 2 plays L, then BR1 ðLÞ ¼ T; if he plays C, then BR1 ðCÞ ¼ M; and if player 2 plays R, then BR1 ðRÞ ¼ B: We can represent the payoffs in each best response with the bold underlined numbers in the next matrix (Table 6.3). Thus, we can identify two pure strategy Nash equilibria (PSNE): fðT; LÞ; ðM; CÞg; with corresponding equilibrium payoffs pairs uðT; LÞ ¼ ð3; 1Þ; and uðM; C Þ ¼ ð1; 2Þ: MSNE: Before starting our search of mixed strategy Nash equilibrium (MSNE) in this game, it would be convenient to eliminate those strategies that are strictly dominated for either player. Strictly dominated strategies do not receive any probability weight at the MSNE, implying that we can focus on the remaining (undominated) strategies, thus simplifying our calculations. From the above normal form game, there does not exist any pure strictly dominated strategies. Hence, we Table 6.2 Simultaneous-move game with three available actions to each player Player 1 T M B

Player 2 L 3, 1 2, 1 1, 2

C 0, 0 1, 2 0, 1

R 5, 0 3, 1 4, 4

Table 6.3 Underlining best response payoffs in the unrepeated game Player 1 T M B

Player 2 L 3, 1 2, 1 1, 2

C 0, 0 1, 2 0, 1

R 5, 0 3, 1 4, 4

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will analyze if there exists some mixed strategy that strictly dominates a pure strategy. When player 1 randomizes between  strategies  T and M with associated probabilities 23 and 13 respectively, r1 ¼ 23 T; 13 M; 0 ; we can show that he obtains an expected utility that exceeds that from selecting the pure strategy B.1 Indeed, for any possible strategy s2 2 S2 chosen by his opponents (player 2), we have that player 1’s payoffs satisfy u1 ðr1 ; s2 Þ [ u1 ðB; s2 Þ: In particular, when player 2 selects L, in the left-hand column, s2 ¼ L; we find that player 1’s expected payoff from selecting the mixed strategy r1 exceeds that from selecting B, u1 ðr1 ; LÞ [ u1 ðB; LÞ i:e:; 23  3 þ 13  2 [ 1 $ 8 [ 3 Similarly, when player 2 selects s2 ¼ C; we also obtain that player 1’s payoff satisfies u1 ðr1 ; CÞ [ u1 ðB; CÞ i:e:; 23  0 þ 13  1 [ 0 $ 13 [ 0 And finally, for the case in which player 2 selects s2 ¼ R; u1 ðr1 ; RÞ [ u1 ðB; RÞ i:e:; 23  5 þ 13  3 [ 4 $ 13 3 [4   Therefore, the payoffs obtained from the mixed strategy r1 ¼ 23 T; 13 M; 0 strictly dominate the payoffs obtained from the pure strategy B. Hence, we can eliminate the pure strategy B of player 1 because of being strictly dominated by r1 : In this context, the reduced-form matrix that emerges after deleting the row corresponding to B is shown in Table 6.4. Given this reduced-form matrix, let us check if, similarly as we did for player 1, we can find a mixed strategy of player 2 that strictly dominates one of his pure strategies. In particular, if player 2 randomizes between s2 ¼ L and s2 ¼ C with equal probability 12, his resulting expected utility is larger than that under pure strategy R, regardless of the specific strategy selected by player 1 (either T or M). In particular, if player 1 chooses s1 ¼ T (in the top row), player 2’s expected utility from randomizing with strategy r2 yields an expected payoff above the payoff he would obtain from selecting R.2 u2 ðr2 ; TÞ [ u2 ðR; TÞ; i:e:;

1 1 1  1þ  0[0 $ [0 2 2 2

You can easily find many other probability weights on T and M that yield an expected utility exceeding the utility from playing strategy B, for any strategy selected by player 2 (i.e., for any column), ultimately allowing you to delete B as being strictly dominated. 2 Note that we do not analyze the case of u2 ðr2 ; BÞ [ u2 ðR; BÞ given that strategy s1 ¼ B was already deleted from the strategy set of player 1. 1

Exercise 7—Repeated Games …

305

Table 6.4 Surviving strategies in the unrepeated game Player 2 L 3, 1 2, 1

Player 1 T M

C 0, 0 1, 2

R 5, 0 3, 1

And similarly, if player 1 chooses s1 = M (in the bottom row) u2 ðr2 ; MÞ [ u2 ðR; MÞ;

i:e:;

1 1 3  1þ  2[1 $ [1 2 2 2

Hence, we can conclude that player pure 1 2’s  strategy s2 ¼ R is strictly domi1 nated by the randomization r2 ¼ 2 L; 2 C; 0 . Therefore, after deleting s2 ¼ R from the payoff matrix, we end up with the following reduced-form game (Table 6.5). At this point, however, we cannot delete any further strategies for player 1 or 2. Hence, when developing our analysis of the MSNE of the game, we will only consider S1 ¼ fT; M g and S2 ¼ fL; C g as the strategy sets of players 1 and 2, respectively. We can now start to find the MSNE of the remaining 2  2 matrix. Player 2 wants to randomize over L and C with a probability q that makes player 1 indifferent between his pure strategies s1 ¼ fT; M g. That is, EU1 ðT Þ ¼ EU1 ðMÞ 3q þ 0ð1  qÞ ¼ 2q þ 1  ð1  qÞ; which yields q ¼ 12 And similarly, player 1 randomizes with a probability p that makes player 2 indifferent between her pure strategies S2 ¼ fL; Cg: That is, EU2 ðLÞ ¼ EU2 ðCÞ 1  p þ 1  ð1  pÞ ¼ 0  p þ 2ð1  pÞ; yielding p ¼ 12. Hence, the MSNE strategy profile is given by: r ¼

    1 1 1 1 T; M ; L; C 2 2 2 2

Table 6.5 Surviving strategies in the unrepeated game (after two rounds of deletion) Player 1 T M

Player 2 L 3, 1 2, 1

C 0, 0 1, 2

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And the expected payoffs associated to this MSNE are:   1 1 1 3  þ0  þ 2 2 2   1 1 1 1  þ1  EU2 ¼ þ 2 2 2

EU1 ¼

  1 1 1 3 3 6 3 2  þ1  ¼ þ ¼ ¼ 2 2 2 4 4 4 2   1 1 1 1 1 0  þ2  ¼ þ ¼1 2 2 2 2 2

  Hence, the expected utility pair in this MSNE is 32 ; 1 . Summarizing, if the game is played only once, we have three Nash-equilibria (two PSNEs and one MSNE), with the following associated payoffs: PSNEs : uðT; LÞ ¼ ð3; 1Þ and uðM; C  Þ ¼ ð1; 2Þ MSNE : EU ðr1 ; r2 Þ ¼ 32 ; 1 Repeated game: But the efficient payoff (4, 4) is not attainable in these equilibria of the unrepeated game.3 However, in the two-stage game the following strategy profile (punishment scheme) constitutes a SPNE: • Play (B, R) in the first stage. • If the first-stage outcome is (B, R), then play (T, L) in the second stage. • If the first-stage outcome is not (B, R), then play the MSNE r in the second stage. Proof Let’s analyze if the former punishment scheme played by every agent induces both players to not deviate from outcome (B,R), with associated payoff (4, 4), in the first stage of the game. To clarify our discussion, we separately examine the incentives to deviate by players 1 and 2, Player 1. Let’s take as given that player 2 adopts the previous punishment scheme: • If player 2 sticks to (B, R), then player 1 obtains u1 ¼ 4 þ 3 ¼ 7; where 4 reflects his payoffs in the first stage of the game, when both cooperate in (B, R); and 3 represents his payoffs in the second stage, where outcome (T, L) arises according to the above punishment scheme. • If, instead, player 1 deviates from (B, R), then his optimal deviation is to play (T, R) which yields a utility level of 5 in the first period, but a payoff of 32 in the second period (the punishment implies the play of the MSNE in the second period). Therefore, player 1 does not have incentives to deviate since his payoff from selecting the cooperative outcome (B, R), 4 + 3 = 7, exceed his payoff from deviating 5 þ 32 ¼ 13 2. 3

Note that payoff pair (4, 4) is efficient since a movement to another payoff pair, while it benefit one player, reduces the utility level of the other player.

Exercise 7—Repeated Games …

307

Player 2. Taking as given that player 1 sticks to the punishment scheme: • If player 2 plays (B, R) in the first stage, then she obtains an overall payoff of u2 ðB; RÞ ¼ 4 þ 1 ¼ 5. As we can see, player 2 doesn’t have incentives to deviate, because his best response function is in fact BR2 ðBÞ ¼ R when we take as given that player 1 is playing B. Therefore, no player has incentives to deviate from (B, R) in the first stage of the game. As a consequence, the efficient payoff (4, 4) can be sustained in the first stage of the game in a pure strategy SPNE strategy profile.

Exercise 8—Infinite-Horizon Bargaining Game Between Three PlayersC Consider a three-player variant of Rubinstein’s infinite-horizon bargaining game, where players divide a surplus (pie) of size 1, and the set of possible agreements is X ¼ fðx1 ; x2 ; x3 Þjxi  0 for i ¼ 1; 2; 3 and x1 þ x2 þ x3 ¼ 1g: Players take turns to offer agreements, starting with player 1 in round t = 0, and the game ends when both responders accept an offer. If agreement x is reached in round t, player i′s utility is given by dt xi ; where 12 \d\1: Show that, if each player is restricted to using stationary strategies (in which she makes the same proposal every time she is the proposer, and uses the same acceptance rule whenever she is a responder), there is a unique subgame perfect equilibrium outcome. How much does each player receive in equilibrium? Answer First, note that every proposal is voted using the unanimity rule. For instance, if player 1 offers x2 to player 2 and x3 to player 3, then players 2 and 3 independently and simultaneously decide if they accept or reject player 1’s proposal. If they both accept, players get x = (x1, x2, x3), while if either player rejects, player 1’s offer is rejected (because of unanimity rule) and player 2 becomes the proposer in the following period, making an offer to players 1 and 3, that they choose whether to accept or reject. Consider any player i, and let xprop denote the offer the proposer at this time period makes to himself, xnext the offer that he makes to the player who will become the proposer in the next period, and xtwo the offer he makes to the player who will become the proposer two periods from now. In addition, we know that the sum of all offers must be equal to the size of the pie, xprop + xnext + xtwo = 1.

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We know that the offer the proposer makes must satisfy two main conditions: 1. First, his offer to the player who will become the proposer in the next period, xnext, must be weakly higher than the discounted value of the offer that such player would make to himself during the next period (when he becomes the proposer), dxprop. That is, xnext  dxprop. Moreover, since the proposer seeks to minimize the offers he makes, he wants to reduce xnext as much as possible, but still guarantee that his offer is accepted, i.e., xnext = dxprop. 2. Second, the offer he makes to the player who will be making proposals two periods from now, xtwo, must be weakly higher than the discounted value of the offer that such a player will make to himself two periods from now as a proposer, d2xprop. That is, xtwo  d2xprop, and since the proposer seeks to minimize the offer xtwo, he will choose a proposal that satisfies xtwo = d2xprop. Using the above three equations, xprop + xnext + xtwo = 1, xnext = dxprop, and x = d2xprop, we can simultaneously solve for xprop, xnext and xtwo. In particular, plugging xnext = dxprop and xtwo = d2xprop into the constraint xprop + xnext + xtwo = 1, we obtain two

xprop þ dxprop þ d2 xprop ¼ 1; and solving for xprop yields xprop ¼

1 1 þ d þ d2

Using this result into the other two equations, we obtain xnext ¼ dxprop ¼

d 1 þ d þ d2

and xtwo ¼ d2 xprop ¼

d2 ; 1 þ d þ d2

which constitute the equilibrium payoffs in this game. Intuitively, this implies that player 1 offers xprop to himself in the first round of play, i.e., in period t = 0, xnext to player 2, and xtwo to player 3, his offers are then accepted by players 2 and 3, and the game ends. Comparative statics: Let us next check how these payoffs are affected by the common discount factor, d, as depicted in Fig. 6.10. When players are very impatient, i.e., d is close to zero, the player who makes the first proposal fares better than do others, and retains most of the pie, i.e., xprop is close to one, while xnext and xtwo are close to zero. However, when players become more patient, the proposer is forced to make more equal divisions of the pie, since otherwise his proposal would be rejected. In the extreme case of perfectly patient players, when d = 1, all players obtain a third of the pie.

Exercise 9—Representing Feasible, Individually Rational PayoffsC

309

Fig. 6.10 Equilibrium payoffs in the 3-player bargaining game

Exercise 9—Representing Feasible, Individually Rational PayoffsC Consider an infinitely-repeated game where the stage game is depicted Table 6.6. Player 1 chooses rows and player 2 chooses columns. Players discount the future using a common discount factor d. Part (a) What outcomes in the stage-game are consistent with Nash equilibrium play? Part (b) Let v1 and v2 be the repeated game payoffs to player 1 and player 2 respectively. Draw the set of feasible payoffs from the repeated game, explaining any normalization you use. Part (c) Find the set of individually rational feasible set of payoffs. Part (d) Find a Nash equilibrium in which the players obtain (9, 9) each period. What restrictions on d are necessary? Answer We first need to find the set of Nash equilibrium strategy profiles for the stage game represented in Table 6.6: Let us start identifying best responses for each player. In particular, if player 1 plays U (in the top row), player 2’s best response is BR2 ðU Þ ¼ R; while if player 1 plays D (in the bottom row), BR2 ðDÞ ¼ R, thus indicating that R is a strictly dominant strategy for player 2. Regarding player 1’s Table 6.6 Simultaneous-move game Player 1 U D

Player 2 L 9, 9 10, 1

R 1, 10 7, 7

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Fig. 6.11 Expected utilities for players 1 and 2

best responses, we find that BR1 ðLÞ ¼ D and BR1 ðRÞ ¼ D, showing that D is a strictly dominant strategy for player 1. Hence, (D, R) is the unique Nash equilibrium (NE) outcome of the stage game, in which players use strictly dominant strategies, and which yields an equilibrium payoff pair of (7, 7). Figure 6.11 depicts the utility of player 1 and 2. For player 1, for instance, u1 ðD; qÞ represents his utility from selecting D, for any probability q with which his opponent selects Left, i.e., 10q þ 7ð1  qÞ ¼ 7 þ 3q; and, similarly, u1 ðU; qÞ reflects player 1’s utility from playing U, i.e., 9q þ 1ð1  qÞ ¼ 1 þ 8q. A similar argument applies for player 2: when he selects L, his expected utility is u2 ðL; PÞ ¼ 9p þ 1ð1  pÞ ¼ 7 þ 3p. From Fig. 6.7 it becomes obvious that strategy U is strictly dominated for player 1, since it provides a strictly lower utility than D regardless of the precise value of q with which player 2 randomizes. Similarly, strategy L is strictly dominated for player 2. From previous chapters, we know that every Nash equilibrium strategy profile must put weight only on strategies that are not strictly dominated. This explains why in this case the unique Nash equilibrium of the stage game does not put any weight on strategy U for player 1 and strategy L for player 2, since they are both strictly dominated. Anticipating that player 1 will choose D, since it gives an unambiguously larger payoff then U (as depicted in the figure), player 2 can minimize his opponent’s payoff by selecting q = 0, which entails a minimax payoff of v1 ¼ 7 for player 1. Similarly, anticipating player 2 will choose R, since it provides an unambiguously larger payoff than L (as illustrated in the figure), player 1 can minimize his opponent’s payoff by selecting p = 0, which yields a minimax payoff of v2 ¼ 7 for player 2. Therefore, the minimax payoff ðv1 ; v2 Þ ¼ ð7; 7Þ is consistent with the Nash equilibrium payoff of (7, 7). Part (b) We know that the set of feasible payoffs V is represented by the convex hull of all possible payoffs obtained in the stage game, as depicted in Fig. 6.12. (This set of feasible payoffs is often referred to in the literature as FP set.)

Exercise 9—Representing Feasible, Individually Rational PayoffsC

311

Fig. 6.12 Set of feasible payoffs

But obviously the above set of feasible payoffs in the repeated game is not convex. In order to address this non-convexity, we can convexify the set of feasible payoffs of the game by assuming that all players observe the outcome of a public randomization device at the start of each period. Using this normalization, the set of feasible payoffs can be depicted as the shaded area in Fig. 6.13.

Fig. 6.13 Set of feasible payoff after the normalization

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Part (c) The minimax values for player i are given by

 vi ¼ min max ui ðai ; aj Þ where ai ; aj 2 ½0; 1 j 6¼ i: aj

ai

Player 1. She maximizes her expected payoffs by playing D, a strictly dominant strategy, regardless of the probability used by player 2, q. Player 2 minimizes the payoff obtained from the previous maximization problem by choosing the minimax profile, implying that the minimax payoff for player 1 is v1 ¼ 7: Player 2. A similar argument applies to player 2, who maximizes her expected payoffs by playing R, a strictly dominant strategy, regardless of the probabilities used by player 1, p. Player 2 minimizes the payoff obtained from the previous maximization problem by choosing the minimax profile p ¼ 0; thus entailing that the minimax payoff for player 2 is v2 ¼ 7: The feasible and individually rational set of payoffs (often referred to as set of FIR payoffs) is given by all those feasible payoffs vi 2 V such that vi [ vi ; that is, all feasible payoffs in which every player obtains strictly more than his payoffs from playing the minimax profiles, i.e., vi [ 7 for every player i in this game, as Fig. 6.14 depicts in the shaded area. Part (d) For a sufficiently high discount factor, i.e., d 2 ðd; 1Þ; we can design an appropriate punishment scheme that introduces incentives for every player not to deviate from payoff (9, 9). In fact, this result is just a particular application of the Folk theorem, given that the proposed outcome (9, 9) is feasible and satisfies individual rationality for every player; i.e., vi [ vi for all i:

Fig. 6.14 Set of feasible, individually rational payoffs

Exercise 9—Representing Feasible, Individually Rational PayoffsC

313

Proof Let’s take the payoff from outcome (U, L), 9, for every player i ¼ 1; 2: Given symmetry, we can proceed all this proof by focusing on one of the players. Hence, without loss of generality let’s analyze player 1’s strategies. In particular, consider that player 1 uses the following grim-trigger strategy: • Play U in round t = 0 and continue to do so as long as the realized action profile in previous periods was (U, L). • Play D as soon as an outcome different from (U, L) is observed. Hence, in order to show that strategy profile (U, L) is a Nash equilibrium of the infinitely repeated game we need to demonstrate that player 1 does not want to deviate from selecting U at every time period. To show that, we separately evaluate the payoff stream that player 1 obtains when cooperating and defecting. Cooperation: If at any given time period t player 1 decides to play U forever after, his discounted stream of payoffs becomes 9 þ d9 þ d2 9 þ    ¼

9 1d

Deviation: If, instead, player 1 deviates to D, his current payoff increases (from 9 to 10), but his deviation is observed (both by player 2 and by himself), triggering as a consequence an infinite punishment whereby both players revert to the Nash equilibrium of the stage game, namely (D, R), with associated payoff pair (7, 7). Hence, player 1 obtains the following stream of payoffs from deviating 10 þ d7 þ d2 7 þ    ¼ 10 þ

d 7 1d

Incentives to cooperate: Thus, player 1 prefers to select U, i.e., cooperate, if and only if 9 d  10 þ 7 1d 1d which, solving for d, yields d  13 : That is, player 1 cooperates as long as he assigns a sufficiently high value to future payoffs. Incentives to punish: In addition, note that player 2 has incentives to implement the punishment scheme upon observing player 1’s deviation. In particular, given that player 1 has deviated to D, player 2’s best response is BR2 ðDÞ ¼ R; thus giving rise to outcome (D, R), the Nash equilibrium of the stage game, as prescribed by the punishment scheme. Summary. Hence, for any d [ 13 we can design an appropriate punishment scheme that supports the above specified strategy for player 1 as his optimal strategy in this infinitely repeated game. We can show the same results for player 2, given the symmetry of their payoffs. Hence, the outcome (U, L) can be supported as a subgame perfect equilibrium strategy profile in this infinitely repeated game.

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Exercise 10—Collusion and Imperfect MonitoringC Two firms produce a differentiated product, and every firm i = {1, 2} faces the following linear demand: pi ¼ a  bqi  dqj ;

where a; b; d [ 0 and 0\d\b

Intuitively, when d ! 0 firm j’s output decisions do not affect firm i’s sales (i.e., products are heterogeneous), while when d ! b, firm j’s output decisions have a significant impact on firm i’s sales, i.e., products are relatively homogeneous. The firms expect to compete against each other for infinite periods, that is, there is no entry or exit in the industry. [Hint: Assume that when a firm detects deviations from collusive behavior, it retaliates by reverting to the non-cooperative Cournot (Bertrand) production level (pricing, respectively), in all subsequent periods. Prior to detection, each firm assumes that its rival follows the collusive outcome.] Part (a) Assume that firms compete as Cournot oligopolists. What is the minimum discount factor necessary in order to sustain the collusive outcome assuming that (i) Deviations are detected immediately, and (ii) Deviations are detected with a lag of one period. Part (b) Repeat part (a) of the exercise under the assumption that firms compete as Bertrand oligopolists. (You can assume immediate detection) Answer Part (a) (i)

Deviations are immediately detected. Cournot outcome. Every firm i selects output level qi by solving max pi ¼ ða  bqi  dqj Þqi qi

i Taking first-order conditions with respect to qi , we obtain @p @qi ¼ a  2bqi  dqj ¼   adqj 0; which yields the best response function qi qj ¼ 2b for every firm i and j 6¼ i. Note that, as products become more heterogeneous, d close to zero indicating very differentiated products, firm i’s best response function approaches the a . monopoly output qi ¼ 2b i Plugging firm j’s best response function qj ðqi Þ ¼ adq 2b ; into firm i’s best response function yields

  i a  d adq a dða  dqi Þ 2b  ¼ qi ¼ 2b 4b2 2b

Exercise 9—Representing Feasible, Individually Rational PayoffsC

315

Solving for qi in this equality, we obtain the (interior) Cournot equilibrium output when firms sell a differentiated product qci ¼

a 2b þ d

Therefore, firm i’s equilibrium profits become (recall that production is costless): pci ¼

 ab

 a a a ba2 d ¼ 2b þ d 2b þ d 2b þ d ð2b þ dÞ2

Cooperative outcome (Monopoly). The merged firm selects the individual output of firm i and j, qi and qj, that maximizes the joint profits, as follows     max p ¼ pi þ pj ¼ a  bqi  dqj qi þ a  bqj  dqi qj : qi ;qj

Differentiating with respect to qi, we obtain   @p ¼ a  2 bqi þ dqj ¼ 0 @qi and differentiating with respect to qj, we find a symmetric expression, as follows   @p ¼ a  2 bqi þ dqj ¼ 0 @qj In a symmetric output profile, we have that qi ¼ qj : Inserting this property in any of the above first-order conditions yields a  2ðb þ dÞqi ¼ 0 Solving for qi, we find qi ¼ qj ¼

a : 2ð b þ d Þ

Thus, the joint profits that firms earn if the collusive agreement is respected are   a a a p ¼ ab d 2ð b þ d Þ 2ð b þ d Þ 2ð b þ d Þ   a a a d þ ab 2ð b þ d Þ 2ð b þ d Þ 2ð b þ d Þ m

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which simplifies to pm ¼

a2 2ð b þ d Þ

implying that every firm i earns half of these profits, that is, pm i ¼

1 a2 a2 ¼ : 2 2ð b þ d Þ 4ð b þ d Þ

Best deviation for firm i. Let us assume that firm j 6¼ i produces the cooperative a output level qm j ¼ 2ðb þ dÞ : In order to determine the optimal deviation for firm i, we just need to plug firm j’s output qm j into firm i’s best response function, as follows 



qi qm ¼ j

ad





a 2ðb þ d Þ

2b

¼

að2b þ d Þ : 4bðb þ d Þ

Therefore, the profits that firm i earns from deviating are pD i ¼

  að2b þ dÞ a að2b þ dÞ a2 ð2b þ dÞ2 d ¼ ab : 4bðb þ dÞ 2ðb þ dÞ 4bðb þ dÞ 16bðb þ d Þ2

Incentives to collude: Thus, for firms to collude, we need that   2 2 D C pm i 1 þ d þ d þ     pi þ pi ðd þ d þ   Þ That is, pm i

1 d C  pD i þ pi 1d 1d

D C Plugging the expression of profits pm i ; pi and pi we found in previous steps yields

1 a2 a2 ð2b þ dÞ2 d ba2  þ : 1  d 4ðb þ d Þ 16bðb þ dÞ2 1  d ð2b þ dÞ2 Multiplying by (1 − d) on both sides of the inequality, we obtain a2 a2 ð2b þ dÞ2 ba2  ð 1  d Þ þ d 4ðb þ dÞ 16bðb þ d Þ2 ð2b þ dÞ2

Exercise 9—Representing Feasible, Individually Rational PayoffsC

317

and solving for d yields the minimal discount factor that supports collusion in this industry, d

ð2b þ d Þ2

d1 : 8b2 þ 8bd þ d 2

Importantly, cutoff d1 increases in the parameter reflecting product differentiation, d, because the derivative @d1 4bdð2b þ d Þ ¼ @d ð8b2 þ 8bd þ d 2 Þ2 is positive for all parameter values b; d [ 0: Intuitively, cooperation is more difficult to sustain when the product both firms sell is relatively homogeneous (i.e., d increases approaching b); but becomes easier to support as firms sell more differentiated products (i.e., d is close to zero), as if each firm operated as an independent monopolist. (ii) Deviations are detected with a lag of one period. In this case, collusion can be supported if     2 3 D D C 2 pm i 1 þ d þ d þ     pi þ dpi þ pi d þ d þ    where note that, if firm i deviates, it obtains the deviating profits of pD i for two consecutive periods. The above inequality can be alternatively expressed as pm i

2 1 C d  ð1 þ dÞ pD þ p i i 1d 1d

D C Plugging the expression for profits pm i ; pi and pi we obtained in previous steps of this exercise yields

1 a2 a2 ð2b þ dÞ2 d2 ba2  ð 1 þ dÞ þ : 1  d 4ð b þ d Þ 16bðb þ dÞ2 1  d ð2b þ dÞ2 Multiplying by (1 − d) on both sides of the inequality, we obtain  a2 a2 ð2b þ dÞ2  ba2  1  d2 þ d2 2 4ðb þ dÞ 16bðb þ dÞ ð2b þ dÞ2

318

6

Repeated Games and Correlated Equilibria

Fig. 6.15 Minimal discount factors d1 and d2

where, in the first term on the right-hand side of the inequality, we used the property ð1 þ dÞð1  dÞ ¼ ð1  d2 Þ. Solving for d, we obtain the minimal discount factor sustained cooperation d

2b þ d ð8b2

þ 8bd þ d2 Þ1=2

d2 :

where @d2 2bd ¼ [ 0: 2 @d ð8b þ 8bd þ d2 Þ3=2 Note that, similarly as for cutoff d1 (when deviations where immediately observed and punished), d2 is also increasing in d. However, d2 [ d1 implying that, when deviations are only detected (and punished) after one-period lag, cooperation can be sustained under more restricting conditions than when deviations are immediately detected. Figure 6.15 depicts cutoffs d2 and d1 as a function of parameter d, both of them evaluated at b = 1/2. Graphically, the region of discount factors sustaining cooperation shrinks as firms detect deviations with a lag one one period. Part (b) Let us now analyze cooperative outcomes if firms compete a la Bertrand. The inverse demand function every firm i = {1, 2} faces is pi ¼ a  bqi  dqj Hence, solving for qi and qj we find the direct demands,

Exercise 9—Representing Feasible, Individually Rational PayoffsC

a dpj  bpi þ 2 bþd b  d2 a dpi  bpj þ 2 qj ¼ bþd b  d2

qi ¼

319

for firm i for firm j

Since pi ¼ pj ¼ p in the collusive outcome, firms choose a common price p that maximizes their joint profits    a pð d  bÞ a p max þ 2  p p¼ p bþd b  d2 bþd bþd 

Taking first-order conditions with respect to the common price p, we obtain @p a 2p ¼  ¼ 0; @p b þ d b þ d which, solving for p, yields the monopolistic (collusive) price pm ¼ a2, and entails collusive profits of pm ¼



  a  a pm a a a2   2 ¼ pm ¼ bþd bþd b þ d b þ d 2 4ðb þ dÞ

Hence, the profits that every firm obtains when colluding in prices are half of this monopoly profit pm i ¼

a2 8ðb þ dÞ

Bertrand outcome. In this case, every firm i independently and simultaneously selects a price pi in order to maximize its individual profits 

 a d b þ max pj  2 pi pi pi b þ d b2  d 2 b  d2 Taking first-order conditions with respect to pi , we find @pi a d 2b þ ¼ pj  2 pi ¼ 0 b  d2 @pi b þ d b2  d 2   aðbdÞ þ dpj and solving for pi we obtain firm i′s best response function pi pj ¼ . 2b A similar argument applies for firm j’s best response function pj ðpi Þ. Plugging firm j’s best response function,

320

6

pj ðpi Þ ¼

Repeated Games and Correlated Equilibria

aðb  dÞ þ dpi ; 2b

into firm i’s best response function yields pi ¼

að b  d Þ þ d

h

aðbdÞ þ dpi 2b

i

2b

and rearranging pi ¼

aðb  d Þð2b þ dÞ þ d2 pi 4b2

Solving for pi, we obtain equilibrium prices pi ¼

aðb  dÞ ¼ pj 2b  d

Therefore, the equilibrium output level for firm i becomes a d b þ pj  2 pi b þ d b2  d 2 b  d2 a d aðb  dÞ b aðb  dÞ þ 2  2 ¼   2 2 bþd b  d 2b  d b d 2b  d ba ¼ ð2b  dÞðb þ dÞ

qi ¼

And Bertrand equilibrium profits are thus aðb  dÞ ba 2b  d ð2b  dÞðb þ dÞ bðb  dÞa2

pBi ¼ pi qi ¼ ¼

ðb þ d Þð2b  dÞ2

a Best deviation by firm i. Assuming that firm j selects the collusive price pm j ¼ 2; m firm i can obtain its most profitable deviation by plugging the collusive price pj ¼ a2 into its best response function, as follows

 aðb  d Þ þ da að2b  dÞ 2 ¼ pi pm ¼ j 4b 2b

Exercise 9—Representing Feasible, Individually Rational PayoffsC

321

Therefore, the profits that firm i obtains from such a deviation are pD i ¼ ¼



 a d a b að2b  dÞ að2b  dÞ þ 2    b þ d b  d 2 2 b2  d 2 4b 4b

a2 ð2b  dÞ2 16bðb  dÞðb þ dÞ

Incentives to collude. Hence, collusion can be sustained if   2 2 D B pm i 1 þ d þ d þ     pi þ pi ðd þ d þ   Þ m 1 D B d pi 1d  pi þ pi 1d D B Plugging the expression of profits pm i ; pi and pi we found in previous steps yields

a2 1 a2 ð2b  dÞ2 d bðb  dÞa2    þ 2ðb þ dÞ 1  d 16bðb  dÞðb þ dÞ 1  d ðb þ d Þð2b  dÞ2 Multiply both sides of the inequality by (1 − d), we obtain a2 a2 ð2b  dÞ2 bðb  dÞa2  ð1  dÞ þ d 2ðb þ dÞ 16bðb  dÞðb þ dÞ ðb þ dÞð2b  dÞ2 Solving for d, we find the minimal discount factor supporting cooperation when firms sell a heterogeneous product and compete a la Bertrand, d

ðd  2bÞ2 ð2b2  2bd þ d2 Þ a2 ð8b2  8bd þ d2 Þ

7

Simultaneous-Move Games with Incomplete Information

Introduction This chapter introduces incomplete information in simultaneous-move games, by allowing one player to be perfectly informed about some relevant characteristic, such as the state of market demand, or its production costs; while other players cannot observe this information. In this setting, we still identify players’ best responses, but we need to condition them on the available information that every player observes when formulating its optimal strategy. Once we find the (conditional) best responses for each player, we are able to describe the Nash equilibria arising under incomplete information (the so-called Bayesian Nash equilibria, BNE) of the game; as the vector of strategies simultaneously satisfying all best responses. We next define BNE. Consider a game with N
2 players, each with discrete strategy space Si and observing information hi 2 Hi , where hi 2 R. Parameter hi describes a characteristic that only player i can privately observe, such as its production costs, the market demand, or its willingness to pay for a product. A strategy for player i in this setting, is a function of hi , i.e., si ðhi Þ, entailing that the strategy profile of all other N − 1 players is also a function of hi ¼ ðh1 ; h2 ; . . .; hi1 ; hi þ 1 ; . . .; hN Þ, i.e., si ðhi Þ. We are now ready to use this notation to define a Bayesian Nash Equilibrium. Bayesian Nash Equilibrium (BNE): Strategy profile ðs1 ðh1 Þ; . . .; sN ðhN ÞÞ is a BNE if and only if

 EUi si ðhi Þ; si ðhi Þ
EUi si ðhi Þ; si ðhi Þ for every strategy si ðhi Þ 2 Si ; every hi 2 Hi ; and every player i. In words, the expected utility that player i obtains from selecting strategy si ðhi Þ is larger than from deviating to any other strategy si ðhi Þ; a condition that holds for all players i 2 N for all realizations of parameter hi . © Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_7

323

324

7

Simultaneous-Move Games with Incomplete Information

As initial motivation, we explore a simple poker game in which we study the construction of the “Bayesian normal form representation” of incomplete information games, which later on will allow us to solve for the set of BNEs in the same fashion as we found Nash equilibria in Chap. 2 for complete information games, that is, underlining best response payoffs. Afterwards, we apply this solution concept to a Cournot game in which one of the firms (e.g., a newcomer) is uninformed about the state of the demand, while its competitor (e.g., an incumbent with a long history in the industry) has accurate information about such demand. In this context, we first analyze the best response function of the privately informed firm, and afterwards characterize the best response function of the uninformed firm. We then study settings in which both agents are uninformed about each other’s strengths, and must choose whether to start a fight, as if they were operating “in the dark.”

Exercise 1—A Simple Bayesian Game Consider the payoff matrix in Fig. 7.1, where player 1 (in rows) chooses between A and B, and player 2 (in columns) selects C or D. Player 1 observes the realization of random variable x, which takes value x = 15 with probability 1/2 or x = 4 with probability 1/2. Player 2, however, doesn’t observe the realization of x, but knows its probability distribution. Therefore, player 1 is perfectly informed about the payoff matrix they face, while player 2 does not know the payoff that player 1 earns in strategy profile (A, C). While this payoff only affects player 1, player 2’s behavior is affected by this uncertainty since he could anticipate that player 1 best responds to column C with A when x = 15, but player 1 would best respond with B when x = 4. (a) Complete information. As a benchmark, let us assume in this part of the exercise that both players are informed about the realization of x. Which are the pure strategy Nash equilibria of the game when x = 15? And what are the pure strategy Nash equilibria when x = 4? (b) Incomplete information. Let us now return to the game where only player 1 privately observes the realization of x. Find the Bayesian Nash Equilibrium of this incomplete information game by following the next three steps: 1. Identify each player’s strategy space in this incomplete information setting, 2. Construct the Bayesian normal form representation, and 3. Underline each player’s best response payoffs to find the BNE of the game.

Exercise 1—A Simple Bayesian Game

Player 1

325

A B

Player 2 C D x, 18 6, 7 8, 0 8, 11

Fig. 7.1 Simultaneous-move game under asymmetric information about x

Answer Part (a). When x takes the value x = 15, the above payoff matrix becomes that depicted in Fig. 7.2a. Player 1’s best responses. When player 2 chooses C (in the left column), player 1’s best response is A, since 15 > 8, which we more compactly write as BR1 ðC Þ ¼ A. When player 2 chooses D (in the right column), player 1’s best response is B, since 8 > 6, which we write as BR1 ðDÞ ¼ B. Player 2’s best responses. When player 1 chooses A (in the top row), player 2’s best response is C, since 18 > 7, which we more compactly write as BR2 ð AÞ ¼ C. When player 1 chooses B (in the bottom row), player 2’s best response is D, since 11 > 0, which we write as BR2 ðBÞ ¼ D. The matrix in Fig. 7.2b underlines best response payoff for each player. Therefore, the pure strategy Nash equilibria when x takes the value x = 15 are ðA; C Þ and ðB; DÞ: When x takes the value x = 4, the above payoff matrix is that depicted in Fig. 7.3a. Player 1’s best responses. When player 2 chooses C (in the left column), player 1’s best response is B, since 8 > 4, which we more compactly write as BR1 ðC Þ ¼ B. When player 2 chooses D (in the right column), player 1’s best response is B, since 8 > 6, which we write as BR1 ðDÞ ¼ B.

(a) Player 1

A B

Player 2 C D 15, 18 6, 7 8, 0 8, 11

A B

Player 2 C D 6, 7 15, 18 8, 0 8, 11

(b) Player 1

Fig. 7.2 a Simultaneous-move game—all players observe x = 15. b Simultaneous-move game— all players observe x = 15, underlining best response payoffs

326

7

Simultaneous-Move Games with Incomplete Information

(a) Player 1

A

Player 2 C D 4, 18 6, 7

B

8, 0

A

Player 2 C D 6, 7 4, 18

B

8, 0

(b) Player 1

8, 11

8, 11

Fig. 7.3 a Simultaneous-move game—all players observe x = 4. b Simultaneous-move game— all players observe x = 4, underlining best response payoffs

Player 2’s best responses. When player 1 chooses A (in the top row), player 2’s best response is C, since 18 > 7, which we more compactly write as BR2 ð AÞ ¼ C. When player 1 chooses B (in the bottom row), player 2’s best response is D, since 11 > 0, which we write as BR2 ðBÞ ¼ D. Figure 7.3(b) underlines best response payoff for each player. Therefore, the only pure strategy Nash equilibrium when x takes the value x = 4 is: ðB; DÞ. Part (b). To find the BNEs of the incomplete information version of the game, we follow the next three steps. 1st step: Let us first identify player 2’s strategy space S2 ¼ fC; Dg and player 1’s strategy space is   S1 ¼ A15 A4 ; A15 B4 ; B15 A4 ; B15 B4 ; where the superscript 15 means that x takes the value x = 15, while the superscript 4 means that x = 4. Intuitively, player 2 cannot condition his strategy on player 1’s type (either x = 15 or x = 4) since player 2 cannot observe the realization of x. In contrast, player 1 can condition his strategy on whether he observes x = 15 or x = 4. For instance, strategy A15 A4 represents that player 1 chooses row A regardless of whether x takes the value x = 15 or x = 4, strategy A15 B4 denotes that player 1 selects row A after observing x = 15 but B otherwise, strategy B15 A4 indicates that player 1 only chooses row B after observing x = 15 but A otherwise, and finally strategy B15 B4 says that player 1 selects row B regardless of the value that x takes.

Exercise 1—A Simple Bayesian Game

327

Player 2

Fig. 7.4 Simultaneous-move game—Bayesian normal form representation (no expected payoffs yet)

15 4 15

Player 1

C

D

4

15 4 15

4

The above strategy spaces imply that player 2 only has two available strategies while player 1 has 4, as depicted in the following Bayesian normal form matrix with 2 columns and 4 rows (Fig. 7.4). Let’s find out the expected payoffs we must insert in the cells. 2nd step: Let us now find the expected payoffs arising in each strategy profile and place them in the appropriate cell. In strategy profile ðA15 A4 ; CÞ, at the top left corner of the Bayesian normal form representation, each player earns the following expected payoffs
 1 EU1 A15 A4 ; C ¼  15 þ 2
15 4  1 EU2 A A ; C ¼  18 þ 2

1  4 ¼ 9:5 2 1  18 ¼ 18 2

In strategy profile ðA15 A4 ; DÞ, in the top right corner of the Bayesian normal form representation, their expected payoffs are
 1 EU1 A15 A4 ; D ¼  6 þ 2
15 4  1 EU2 A A ; D ¼  7 þ 2

1 6¼6 2 1 7¼7 2

In strategy profile ðA15 B4 ; CÞ players’ expected payoffs become
 1 EU1 A15 B4 ; C ¼  15 þ 2
15 4  1 EU2 A B ; C ¼  18 þ 2

1  8 ¼ 11:5 2 1 0¼9 2

In strategy profile ðA15 B4 ; DÞ, we obtain
 1 EU1 A15 B4 ; D ¼  6 þ 2
15 4  1 EU2 A B ; D ¼  7 þ 2

1 8¼7 2 1  11 ¼ 9 2

328

7

Simultaneous-Move Games with Incomplete Information

In strategy profile ðB15 A4 ; CÞ, expected payoffs are
 1 EU1 B15 A4 ; C ¼  8 þ 2
15 4  1 EU2 B A ; C ¼  0 þ 2

1 4¼6 2 1  18 ¼ 9 2

In strategy profile ðB15 A4 ; DÞ, each player’s expected payoff is
 1 1 EU1 B15 A4 ; D ¼  8 þ  6 ¼ 7 2 2
15 4  1 1 EU2 B A ; D ¼  11 þ  7 ¼ 9 2 2 In strategy profile ðB15 B4 ; CÞ, each player earns an expected payoff of
 1 EU1 B15 B4 ; C ¼  8 þ 2
15 4  1 EU2 B B ; C ¼  0 þ 2

1 8¼8 2 1 0¼0 2

And, finally, in strategy profile ðB15 B4 ; DÞ, located at the bottom left-hand corner of the above Bayesian normal form representation of the game, players earn expected payoffs
 1 1 EU1 B15 B4 ; D ¼  8 þ  8 ¼ 8 2 2
15 4  1 1 EU2 B B ; D ¼  11 þ  11 ¼ 11 2 2 Inserting these expected payoffs in the cells of the matrix representing the Bayesian normal form representation of the game, we obtain Fig. 7.5. 3rd step: We can now underline best response payoffs in the above matrix, to find the BNEs of this incomplete information game.

Player 2

Player 1

C 9.5, 18 11.5, 9 6, 9 8, 0

D 6, 7 7, 9 7, 9 8, 11

Fig. 7.5 Simultaneous-move game—Bayesian-normal form representation including expected payoffs

Exercise 1—A Simple Bayesian Game

329

Player 2 C 9.5, 18 11.5, 9 6, 9 8, 0

Player 1

D 6, 7 7, 9 7, 9 8, 11

Fig. 7.6 Simultaneous-move game—Bayesian-normal form representation, underlining best responses

Player 1’s best responses. When player 2 chooses C (in the left column), player 1’s best response is A15 B4 , since this strategy yields the highest payoff, 11.5, than any other row, which we more compactly write as BR1 ðCÞ ¼ A15 B4 . This best response payoff, 11.5, is underlined in the payoff matrix in Fig. 7.6. When player 2 chooses D (in the right column), player 1’s best response is B15 B4 , since it yields 8, A higher payoff than any other row, which we write as BR1 ðDÞ ¼ B. This best response payoff is also underlined in matrix 7.6 (see right column). Player 2’s best responses. When player 1 chooses A15 A4 (in the top row), player 2’s
best response is C, since 18 > 7, which we more compactly write as  BR1 A15 A4 ¼ C. When player 1 chooses A15 B4 (in the second row), player 2’s best
response  is C or D, since both yield the same payoff of 9, which we write as BR1 A15 B4 ¼ fC; Dg. When player 1 chooses B15 A4 (in the third row), player 2’s best
response  is C or D, since both yield the same payoff of 9, which we write as BR1 B15 A4 ¼ fC; Dg. Finally, when player 1 chooses B15 B4 (in the bottom row), player
2’s best response is D, since 11 > 0, which we more compactly write as BR1 B15 B4 ¼ D. Hence, there are two cells where both players’ payoffs are underlined. These are strategy profiles where players play mutual best responses to each other, implying that they are the two BNEs of the game: ðA15 B4 ; CÞ

and ðB15 B4 ; DÞ:

Exercise 2—Simple Poker GameA Here is a description of the simplest poker game. There are two players and only two types of cards in the deck, an Ace and a King. First, the deck is shuffled and one of the two cards in dealt to player 1. That is, nature chooses the card for player 1: being the Ace with probability 2/3 and the King with probability 1/3. Player 2 has previously received a card, which both players had a chance to observe. Hence, the only uninformed player is player 2, who does not know whether his opponent

330

7

Simultaneous-Move Games with Incomplete Information

Fig. 7.7 A simple poker game

has received a King or an Ace. Player 1 observes his card and then chooses whether to bet (B) or fold (F). If he folds, the game ends, with player 1 obtaining a payoff of −1 and player 2 getting a payoff of 1 (that is, player 1 loses his ante to player 2). If player 1 bets, then player 2 must decide whether to respond betting or folding. When player 2 makes this decision, she knows that player 1 bets, but she has not observed player 1’s card. The game ends after player 2’s action. If player 2 folds, then the payoff vector is (1, −1), meaning player 1 gets 1 and player 2 gets −1, regardless of player 1’s hand. If player 2, instead, responds betting, then the payoff depends on player 1’s card: if player 1 holds the Ace then the payoff vector is (2, −2), thus indicating that player 1 wins; if player 1 holds the King, then the payoff vector is (−2, 2), reflecting that player 1 looses. Represent this game in the extensive form and in the Bayesian normal form, and find the Bayesian Nash Equilibrium (BNE) of the game. Answer Figure 7.7 depicts the game tree of this incomplete information game. Player 2 has only two available strategies S2 ¼ fBet; Fold g, but player 1 has four available strategies S1 ¼ fBb; Bf ; Fb; Ff g. In particular, for each of his strategies, the first component represents player 1’s action when the card he receives is the Ace while the second component indicates his action after receiving the King. This implies that the Bayesian normal form representation of the game is given by the following 4  2 matrix (Table 7.1).

Table 7.1 Strategies in the Bayesian normal form representation of the poker game

Player 2 Bet Player 1

Bb Bf Fb Ff

Fold

Exercise 2—Simple Poker GameA

331

In order to find the expected payoffs for strategy profile (Bb, Bet), i.e., in the top left-hand side cell of the matrix, we proceed as follows: EU1 ¼

2 1 2 2 1 2  2 þ  ð2Þ ¼ and EU2 ¼  ð2Þ þ  ð2Þ ¼  3 3 3 3 3 3

Similarly for strategy profile (Bf, Bet), 2 1 2 1 EU1 ¼  ð2Þ þ  ð1Þ ¼ 1 and EU2 ¼  ð2Þ þ  1 ¼ 1 3 3 3 3 For strategy profile (Fb, Bet), 2 1 4 2 1 4 EU1 ¼ ð1Þ þ ð2Þ ¼  and EU2 ¼  1 þ  2 ¼ 3 3 3 3 3 3 For strategy profile (Ff, Bet), in the bottom left-hand side cell of the matrix, we obtain 2 1 2 1 EU1 ¼ ð1Þ þ ð1Þ ¼ 1 and EU2 ¼  1 þ  1 ¼ 1 3 3 3 3 In strategy profile (Bb, Fold), located in the top right-hand side cell of the matrix, we have 2 1 2 1 EU1 ¼  1 þ  1 ¼ 1 and EU2 ¼ ð1Þ þ ð1Þ ¼ 1 3 3 3 3 For strategy profile (Bf, Fold), EU1 ¼

2 1 1 2 1 1  1 þ ð1Þ ¼ and EU2 ¼ ð1Þ þ  1 ¼  3 3 3 3 3 3

Similarly for (Fb, Fold), 2 1 1 2 1 1 EU1 ¼  ð1Þ þ  1 ¼  and EU2 ¼  1 þ ð1Þ ¼ 3 3 3 3 3 3 Finally, for (Ff, Fold), in the bottom right-hand side cell of the matrix EU1 ¼

2 1 2 1  ð1Þ þ ð1Þ ¼ 1 and EU2 ¼  1 þ  1 ¼ 1 3 3 3 3

We can now insert these expected payoffs into the Bayesian normal form representation (Table 7.2).

332 Table 7.2 Bayesian form representation of the poker game after inserting expected payoffs

7

Simultaneous-Move Games with Incomplete Information

Player 1

Bb Bf Fb Ff

Table 7.3 Underlined best response payoffs for the poker game Player 1

Bb Bf Fb Ff

Player 2 Bet

Fold

2/3, −2/3 1, −1 −4/3, 4/3 −1, 1

1, −1 1/3, −1/3 −1/3, 1/3 −1, 1

Player 2 Bet

Fold

2/3, −2/3 1, −1 −4/3, 4/3 −1, 1

1, −1 1/3, −1/3 −1/3, 1/3 −1, 1

We can now identify the best response to each player. • For player 1, his best response when player 2 bets (in the left-hand column) is to play Bf since it yields a higher payoff, i.e., 1, than any other strategy, that is BR1(Bet) = Bf. Similarly, when player 2 chooses to fold (in the right-hand column), player 1’s best response is to play Bb, given that its associated payoff, $1, exceeds that of all other strategies, i.e., BR1 ðFold Þ ¼ Bb. Hence, the best responses yield an expected payoff of 23 and 1, respectively; as depicted in the underlined payoffs of Table 7.3. • Similarly, for player 2, when player 1 chooses Bb (in the top row), his best response is to bet, since his payoff, 2 3 , is larger than that from folding, −1, i.e., BR2 ðBbÞ ¼ Bet. If, instead, player 1 chooses Bf (in the second row), player 2’s best response is to fold, BR2(Bf) = Fold, since his payoff from folding, −1/3, is larger than from betting, −1. A similar argument applies to the case in which player 1 chooses Fb (in the third row), where player 2 responds betting, BR2 ðFbÞ ¼ Bet, obtaining a payoff of 43. Finally, when player 1 chooses Ff in the bottom row, player 2 is indifferent between responding with bet or fold since they both yield the same payoff ($1), i.e., BR2 ðFf Þ ¼ fBet; Fold g. The payoffs that player 2 obtains from selecting these best responses are underlined in the matrix of Table 7.3. Since there is no cell in Table 7.3 where both players’ payoffs are underlined, we can claim that there is no mutual best response when players use pure strategies, implying that there is no pure strategy BNE in this game. Intuitively, player 1 wants to be unpredictable, as otherwise his action (bet or fold) would provide player 2 with information about player 1’s hand. In Chap. 9 we revisit this poker game again, finding an equilibrium where players 1 and 2 randomize.

Exercise 3—Incomplete Information Game …

333

Exercise 3—Incomplete Information Game, Allowing for More General ParametersB Consider the Bayesian game in Fig. 7.8. First, nature selects player 1’s type, either high or low with probabilities p and 1 − p, respectively. Observing his type, player 1 chooses between x or y (when his type is high) and between x′ and y′ (when his type is low). Finally, player 2, neither observing player 1’s type nor player 1’s chosen action, responds with a or b. Note that this game can be interpreted as players 1 and 2 acting simultaneously, with player 2 being uninformed about player 1’s type. Part (a) Assume that p = 0.75. Find a Bayesian-Nash equilibrium. Part (b) For each value of p, find all Bayesian-Nash equilibria. Answer Part (a). In order to represent the game tree of Fig. 7.8 into its Bayesian normal form representation, we first need to identify the available strategies for each player. In particular, player 2 has only two strategies S2 ¼ fa; bg. In contrast, player 1 has four available strategies S1 ¼ fxx0 ; xy0 ; yx0 ; yy0 g where the first component of every strategy pair denotes what player 1 chooses when his type is H and the second component reflects what he selects when his type is L. Hence, the Bayesian normal form representation of the game is given by the following 4  2 matrix (Table 7.4).

Fig. 7.8 A Bayesian game where player 2 is uninformed

334

7

Simultaneous-Move Games with Incomplete Information

Table 7.4 Bayesian normal form representation of the game

Player 2 a

b

xx′ xy′ yx′ yy′

Player 1

Let’s find the expected utilities that each player obtains from strategy profile (xx′, a), i.e., top left corner of the above matrix, EU1 ¼ p  3 þ ð1  pÞ  2 ¼ 2 þ p; and EU2 ¼ p  1 þ ð1  pÞ  3 ¼ 3  2  p Similarly, strategy profile (xy′, a) yields expected utilities of EU1 ¼ p  3 þ ð1  pÞ  3 ¼ 3 and EU2 ¼ p  1 þ ð1  pÞ  1 ¼ 1 Proceeding in this fashion for all remaining cells, we find the Bayesian normal form representation of the game, as illustrated in the following matrix (Table 7.5): When p = 0.75 (as assumed in part (a) of the exercise) the above matrix becomes (Table 7.6): We can now identify player 1’s best responses: when player 2 chooses a (in the left-hand column), BR1 ðaÞ ¼ xy0 , yielding a payoff of 3; while when player 2 selects b (in the right-hand column), BR1 ðbÞ ¼ yy0 , yielding a payoff of 4.75. Similarly operating for player 2, we obtain that his best responses are BR2 ðxx0 Þ ¼ b, with a payoff of 2.75 when player 1 selects xx′ the first row; BR2 ðxy0 Þ ¼ b, with a payoff of 2.25 when player 1 chooses xy′ in the second row; BR2 ðyx0 Þ ¼ b, where player 2’s payoff is 2 when player 1 selects yx′ in the third row; and BR2 ðyy0 Þ ¼ b, with a payoff of 1.5 when player 1 chooses yy′ in the bottom row. Underlining the payoff associated with the best response of each player, we find that there is a unique BNE in this incomplete information game: (yy′, b).

Table 7.5 Inserting expected payoffs in the Bayesian normal form representation of the game

Player 1

xx′ xy′ yx′ yy′

Player 2 a

b

2 þ p; 3  2p 3; 1 2; 3  2p 3  p; 1

1; 2 þ p 4  3p; 3p 1 þ 4p; 2 4 þ p; 2p

Exercise 3—Incomplete Information Game …

335

Table 7.6 Underlined best response payoffs Player 1

Table 7.7 Bayesian normal form representation of the game, for a general probability p

Player 1

xx′ xy′ yx′ yy′

xx′ xy′ yx′ yy′

Player 2 a

b

2.75, 1.5 3, 1 2, 1.5 2:25; 1

1, 2.75 1.75,2.25 4; 2 4:75; 1:5

Player 2 a

b

2 þ p; 3  2p 3; 1 2; 3  2p 3  p; 1

1; 2 þ p 4  3p; 3p 1 þ 4p; 2 4 þ p; 2p

Part (b). We can now approach this exercise without assuming a particular value for the probability p. Let us first analyze player 1’s best responses. We reproduce the Bayesian normal form representation of the game for any probability p (Table 7.7). Player 1’s best responses: When player 2 selects the left column (strategy a), player 1 compares the payoff he obtains from xy′, 3, against the payoff arising in his other strategies, i.e., 3 > 2 + p, 3 > 2, and 3 > 3 − p, which hold for all values of p. Hence, player 1’s best response to a is xy′. Similarly, when player 2 chooses the right-hand column (strategy b), player 1 compares the payoff from yy′, 4 + p, against that in his other available strategies, i.e., 4 + p > 1, 4 + p > 4 − 3p, and 4 + p > 1 + 4p, which hold for all values of p. Hence, player 1’s best response to b is yy′. Summarizing, player 1’s best responses are BR1 ðaÞ ¼ xy0 and BR1 ðbÞ ¼ yy0 . Player 2’s best responses: Let us know examine player 2’s to each of the four possible strategies of player 1:. When player 1 selects xx′, he responds with a if 3 − 2p > 2 + p. This condition simplifies to 1 > 3p that is, he responds with a if 13 [ p: When player 1 selects xy′, he responds with a if 1 [ 3p or 13 [ p (otherwise he chooses b). When player 1 selects yx′, he responds with a if 3  2p [ 2; or 1 [ 2p that is, he chooses a if 12 [ p. When player 1 selects yy′, he responds with a if 1 [ 2p, or if 12 [ p (otherwise he chooses b). Figure 7.9 summarizes player 2’s best responses (either a or b), as a function of the probability p.

336

7

Simultaneous-Move Games with Incomplete Information

Fig. 7.9 Player 2’s best responses as a function of p

We can then divide our analysis into three different matrices, depending on the specific value of the probability p: • One matrix for p\ 13   • A second matrix for p 2 13 ; 12 , and • A third matrix for p [ 12 First case: p\ 13 In this case, player 2 responds with a regardless of the action (row) selected by player 1 (see the range of Fig. 7.3 where p\ 13). This is indicated by player 2’s payoffs in the matrix of Table 7.8, which are all underlined in the column where player 1 selects a. Hence, strategy a becomes a strictly dominant strategy for player 2 in this context, while player 1’s best responses are insensitive to p, namely, BR1 ðaÞ ¼ xy0 and BR1 ðbÞ ¼ yy0 (Table 7.8). In this case, the unique BNE when p\ 13 is (xy′, a)   Second case: p 2 13 ; 12 When p 2 ½13 ; 12, the intermediate range of p in Fig. 7.3 indicates that player 2 still responds with a when player 1 chooses yx′ and yy′, but with b when player 1 selects xx′ and xy′. This is indicated in player 2’s payoffs in the matrix of Table 7.9, which are underlined in the column corresponding to a when player 1 chooses yx′ and yy′ (last two rows), but are underlined in the right-hand column (corresponding to strategy b) otherwise (Table 7.9). Table 7.8 Bayesian normal form game with underlined best responses when p < 1/3

Player 1

Player 2 a

b

xx′

2 þ p; 3  2p

1; 2 þ p

xy′ yx′

3, 1 2; 3  2p

4  3p; 3p 1 þ 4p; 2

yy′

3 – p, 1

4 + p, 2p

Exercise 3—Incomplete Information Game … Table 7.9 Bayesian normal form game with underlined best responses if 1/3  p  1/2

Table 7.10 Bayesian normal form game with underlined best responses when p > 1/2

Player 1

Player 1

337 Player 2 a

b

xx′

2 þ p; 3  2p

1; 2 þ p

xy′

3, 1

4  3p; 3p

yx′

2; 3  2p

1 þ 4p; 2

yy′

3  p; 1

4 þ p ; 2p

Player 2 a

b

xx′

2 þ p; 3  2p

1; 2 þ p

xy′

3; 1

4  3p; 3p

yx′ yy′

2; 3  2p 3  p; 1

1 þ 4p; 2 4 þ p ; 2p

In this case there is no BNE when we restrict players to only use pure strategies. Third case: p [ 12 Finally, when p [ 12, Fig. 7.3 reminds us that player 2 best responds with strategy b regardless of the action (row) selected by player 1, i.e., b becomes a strictly dominant strategy. This result is depicted in the matrix of Table 7.10, where player 2’s payoffs corresponding to b are all underlined (see right-hand column) (Table 7.10). In this case the unique BNE is (yy′, b) (Note that this BNE is consistent with part (a) of the exercise, where p was assumed to be 0.75, and thus corresponds to the third case analyzed here p [ 12. Needless to say, we found the same BNE as in the third case).

Exercise 4—More Information Might HurtB Show that more information may hurt a player by constructing a two-player game with the following features: Player 1 is fully informed while player 2 is not; the game has a unique Bayesian Nash equilibrium, in which player 2’s expected payoff is higher than his payoff in the unique equilibrium of any of the related games in which he knows player 1’s type. Answer Player 2 is fully informed about the types of both players, while player 1 is only informed about his own type. The following payoff matrices describe this incomplete information game (Table 7.11).

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7

Simultaneous-Move Games with Incomplete Information

Table 7.11 “More information might hurt” game

Full Information: When both players know player 2’s type, we have a unique Nash Equilibrium (NE), both when player 2’s type is low and when its type is high. We next separately analyze each case. Player 2 is low type. As depicted in the left-hand matrix, player 1 finds strategy U strictly dominated by strategy D. Similarly, for player 2, strategy L is strictly dominated by strategy R. Hence, the unique strategy pair surviving the iterative deletion of strictly dominated strategies (IDSDS) is (D, R). In addition, we know that when the set of strategies surviving IDSDS is a singleton, then such strategy profile is also the unique NE in pure strategies. Hence, the equilibrium payoffs in this case are u1 ¼ u2 ¼ 25: Player 2 is high type. As depicted in the right-hand matrix, we cannot delete any strategy because of being strictly dominated. In addition, there doesn’t exist any NE in pure strategies, since BR1 ðLÞ ¼ U and BR1 ðRÞ ¼ D for player 1, and BR2 ðU Þ ¼ R and BR2 ðDÞ ¼ L for player 2, i.e., there is no strategy profile that constitutes a mutual best response for both players. We, hence, need to find the MSNE for this game. Assume that player 2 randomizes between L and R with probability y and 1 − y, respectively. Thus, player 1 is indifferent between playing U and D if:

EU 1 ðUjyÞ ¼ EU 1 ðDjyÞ or y  100 þ ð1  yÞ  1 ¼ y  99 þ ð1  yÞ  2 and solving for y, we find that y ¼ 12 which entails that player 2 chooses L with 12 probability. We can similarly operate for EU2 . Assuming that player 1 randomizes between L and R with probability x and 1 − x, respectively, player 2 is made indifferent between L and R if EU 2 ðLjxÞ ¼ EU 2 ðRjxÞ or x  1 þ ð1  xÞ  4 ¼ 2x þ ð1  xÞ  1

Exercise 4—More Information Might HurtB

339

and solving for x, we obtain that x ¼ 34. This is the probability of player 1 playing U when he knows that P2 is high type. Hence, the unique MSNE for the case of player 2 being high type (right-hand matrix) is:  3 1 U þ D; MSNE ¼ 4 4

1 1 Lþ R 2 2



And the associated expected payoffs for each player are:   3 1 1 1 1 1 EU 1 ¼   100 þ  1 þ   99 þ  2 4 2 2 4 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} when P1 plays U

when P2 plays D

3 101 1 101 404 101  þ  ¼ ¼ ¼ 50:5 4 2 4 2 8  2 1 3 1 1 3 1 1 7 1 7 7  1þ  4 þ   2 þ  1 ¼  þ  ¼ ¼ 1:75 EU 2 ¼  2 4 4 2 4 4 2 4 2 4 4 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼

when P2 plays L

when P2 plays R

Therefore, EU 1 ¼ 50:5 and EU 2 ¼ 1:75 in the MSNE complete-information game in which player 2 is of a high type.

of

the

Imperfect Information Case: In this setting, we consider that player 1 is not informed about player 2’s type. However, player 2 is informed about his own type, and so he has perfect information. Let p denote the probability (exogenously determined by nature) that player 2 is of low type i.e., that players interact in the left-hand matrix. Player 1. We know that player 1 will take his decision between U and D “in the dark” since he does not observe player 2’s type. Hence, player 1 will be comparing EU 1 ðUjyÞ against EU 1 ðDjyÞ. In particular, if player 1 selects U, his expected payoff becomes: EU 1 ðUjyÞ ¼

p  ð1000Þ |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} because P2

þ ð1  pÞ  ½y  100 þ ð1  yÞ  1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} when P2 is hightype

always plays R when being low  type ¼ 1000 p þ ð1  pÞ½100y þ ð1  yÞ Note that player 2 plays R when being a low-type because R is a strictly dominant strategy in that case. When his type is high, however, player 1 cannot anticipate whether player 2 will be playing L or R, since there is no strictly dominant strategy

340

7

Simultaneous-Move Games with Incomplete Information

for player 2 in this case. We can operate similarly in order to find the expected utility that player 1 obtains from selecting D, as follows. EU 1 ðDjyÞ ¼

þ ð1  pÞ  ½y  99 þ ð1  yÞ  2 p  25 |fflffl{zffl ffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} when P2 is hightype because P2 always plays R when being lowtype

1 For simplicity, assume that this probability is, in particular, p ¼ 25 , and therefore 24 1  p ¼ 25. Hence, comparing both expected utilities, we have

EU 1 ðUjyÞ\EU1 ðDjyÞ 1000 24 24 þ ½99y þ 1\1 þ ½97y þ 2  25 25 25 and solving for y yields 1049 48 \ y ! y\21:85 25 25 which is always true, since y 2 ½0; 1, thus implying that EU1 ðUÞ\EU1 ðDÞ holds 1 . Hence, the uninformed player 1 chooses D. for all values of y when p ¼ 25 More generally, for any value of probability p, we can easily find the difference EU1 ðUjyÞ  EU1 ðDjyÞ ¼ 1 þ 2y  2pð512 þ yÞ Hence, solving for p, we obtain that EU1 ðUjyÞ\EU1 ðDjyÞ for all p[

1 þ 2y ^; p 2ð512 þ yÞ

^ becomes negative for all y\ 12, thus implying which holds for all y\ 12, i.e., cutoff p ^ is satisfied since p 2 ½0; 1. For all y [ 12, the condition p [ p ^ is not very that p [ p ^ increases from p ^ ¼ 0 at y ¼ 12 to p ^ ¼ 0:0009 when restrictive either since cutoff p y ¼ 1, thus implying that, for most combinations of probabilities p and y, player 1 prefers to choose D. Figure 7.10 depicts this area of ðp; yÞ-pairs, namely that above ^ cutoff, thus spanning most admissible values. the p Player 2. Since player 2 (the informed player) can anticipate player 1’s selection of D (for any value of y), player 2’s best responses become:

Exercise 4—More Information Might HurtB

341

Fig. 7.10 Admissible probability pairs

BR2 ðP1 always plays DjP2 is low typeÞ ¼ R BR2 ðP1 always plays D; jP2 is high typeÞ ¼ L Therefore, the unique Bayesian Nash equilibrium (BNE) of the game is given by: ðD; RLÞ, where R indicates that the informed player 2 plays R when he is a low-type, and the second component (L) indicates what he plays when he is a high-type. The associated payoff for the uninformed player 1 in this BNE is EU1 ¼ p  EU1 ðDjP2 is low typeÞ þ ð1  pÞEU1 ðDjP2 is high typeÞ 1 24  25  99 ¼ þ ¼ 96:04 25 25 |fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl} when ðD; RÞ if P2

when ðD; LÞ if P2

is low  type

is hightype

Payoff comparison: Therefore, we can show that the uninformed player’s payoff is higher when he is uninformed (in the BNE) than perfectly informed (in the NE in pure strategies), both when his opponent’s type is low: EU1 ðP2 is low typeÞ \ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} informed case

EU |{z}1

! 25\96:04

uninformed case

and similarly for the case in which his opponent’s type is high:

342

7

Simultaneous-Move Games with Incomplete Information

EU1 ðP2 is high typeÞ \ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} informed case

EU |{z}1

! 50\96:04

uninformed case

As a consequence, “more information may hurt” player 1. Alternatively, he would prefer to play the incomplete information game in which he does not know player 2’s type before selecting between U and D, than interacting with player 2 in the complete information game in which he can observe player 2’s type before choosing U or D.

Exercise 5—Incomplete Information in Duopoly MarketsA Consider a differentiated duopoly market in which firms compete by selecting prices. Let p1 be the price chosen by firm 1 and let p2 be the price of firm 2. Let q1 and q2 denote the quantities demanded (and produced) by the two firms. Suppose that the demand for firm 1 is given by q1 ¼ 22  2p1 þ p2 , and the demand for firm 2 is given by q2 ¼ 22  2p2 þ p1 . Firm 1 produces at a constant marginal cost of 10, while firm 2’s constant marginal cost is c > 0. Firms face no fixed costs. Part (a) Complete information. Consider that firms compete in prices. Represent the normal form by writing the firms’ payoff functions. Part (b) Calculate the firms’ best-response functions. Part (c) Suppose that c = 10 so the firms are identical (the game is symmetric). Calculate the Nash equilibrium prices in this complete information game. Part (d) Incomplete information. Now suppose that firm 1 does not know firm 2’s marginal cost c. With probability ½ nature picks c = 14 and with probability ½ nature picks c = 6. Firm 2 knows its own cost (that is, it observes nature’s move), but firm 1 only knows that firm 2’s marginal cost is either 6 or 14 (with equal probabilities). Calculate the best-response function of player 1 and those for player 2 (one for each of its two types, c = 6 and c = 14), and find the Bayesian Nash equilibrium quantities. Answer Part (a) For firm 1, with constant marginal costs of production of c, we have that its profit function is p1 ðp1 ; p2 Þ ¼ p1  q1  10q1 ¼ ðp1  10Þq1 ¼ ðp1  10Þð22  2p1 þ p2 Þ ¼ ð22 þ 2cÞp1 þ p1 p2  2ðp2 Þ2  22c  cp2 Similarly, for firm 2, with marginal costs given by c > 0, we have a profit function of

Exercise 5—Incomplete Information in Duopoly MarketsA

343

p2 ðp1 ; p2 Þ ¼ p2  q2  c  q2 ¼ ðp2  cÞq2 ¼ ðp2  cÞð22  2p2 þ p1 Þ ¼ ð22 þ 2cÞp2 þ p1 p2  2p22  22c  cp1 Part (b) Taking first-order conditions of firm 1’s profit with respect to p1 we obtain ð22 þ 2cÞ þ p2  4p1 ¼ 0 and, solving for p1 ; yields firm 10 s best response function 22 þ 2c þ p2 p1 ðp2 Þ¼ ðBRF1 Þ 4 1 Thus, the best response function of firm 1 originates at 42 4 and has a vertical slope of 4. Similarly, taking first-order conditions of p2 ðp1 ; p2 Þ with respect to p2 yields,

22 þ 2c þ p1  4p2 ¼ 0 and solving for p2 , we obtain firm 2’s best response function p2 ð p1 Þ ¼

22 þ 2c þ p1 4

BRF2

which originates at 22 þ4 2c and has a vertical slope of 14 (Note that if firm 2’s marginal costs were 10, then 22 þ4 2c would reduce to 42 4 , yielding a BRF2 symmetric to that of firm 1.). Part (c) In this part, the exercise assumes a marginal cost of c = 10. Plugging BRF1 into BRF2 , we obtain p2 ¼

42 þ

42 þ p2  4

4

And solving for p2 yields an equilibrium price of p2 = $14. We can finally plug p2 = $14 into BRF1 , to find firm 1’s equilibrium price p1 ð14Þ ¼

42 þ 14 ¼ $14 4


Hence, the Nash equilibrium of this complete information game is the price pair p1 ; p2 = ($14, $14). Therefore, firm i’s profits in the equilibrium of this complete information game are
 pi pi ; pj ¼ ð14  10Þð22  2  14 þ 14Þ ¼ $32

344

7

Simultaneous-Move Games with Incomplete Information

Part (d) Incomplete information game. Starting with the informed player (Player 2), we have that: • Firm 2’s best-response function evaluated at high costs, i.e., c = 14, BRF2H , is pH 2 ð p1 Þ ¼

22 þ 2  14 þ p1 50 þ p1 ¼ 4 4

• Firm 2’s best-response function when having low costs, i.e., c = 6, BRF2L , is pL2 ðp1 Þ ¼

22 þ 2  6 þ p1 34 þ p1 ¼ 4 4

• The uninformed firm 1 chooses the price p1 that maximizes its expected profits max p1

 1  1 2 H 42p1 þ p1 pH 42p1 þ p1 pL2  2p21  10pL2 2  2p1  10p2 þ 2 2

where the first term represents the case in which firm 2 has high costs, and thus sets a price pH 2 ; while the second term denotes the case in which firm 2’s costs are low, setting a price pL2 . Importantly, firm 1 cannot condition its price, p1, on the production costs of firm 2 since, as opposed to firm 2, firm 1 does not observe this information. Taking first order conditions with respect to p1 we obtain  1  1 42 þ pH 42 þ pL2  4p1 2  4p1 þ 2 2 or 42 þ

1 H 1 L p þ p2  4p1 ¼ 0 2 2 2

L And solving with respect to p1 we obtain firm 1’s BRF1 ðpH 2 ; p2 Þ, which depends on H L p2 and p2 , as follows. 1 L
 42 þ 12 pH 2 þ 2 p2 L p1 pH 2 ; p2 ¼ 4 L H L Plugging pH 2 ðp1 Þ and p2 ðp1 Þ into BRF1 ðp2 ; p1 Þ, we obtain




L p1 pH 2 ; p2



¼

42 þ



1 50 þ p1 2 4



þ



1 34 þ p1 2 4



4

And solving for p1 , we find firm 1’s equilibrium price, p1 = $14.

Exercise 5—Incomplete Information in Duopoly MarketsA

345

Finally, plugging this equilibrium price p1 = $14 into firm 2’s BRF2H and BRF2L yields equilibrium prices of pH 2 ð14Þ ¼

50 þ 14 ¼ $16 4

when firm 2’s costs are high, and pL2 ð14Þ ¼

34 þ 14 ¼ $12 4

when its costs are low. Thus, BNE of this Bertrand game under incomplete information is L ðp1 ; pH 2 ; p2 Þ ¼ ð14; 16; 12Þ

In this setting, the informed player (firm 1) makes an expected equilibrium profits of 
 1  L p 1 p1 ; pH 42 14 þ 14 16  2 142  10  16 2 ; p2 ¼ 2  1 þ 42  14 þ 14  12  2  142  10  12 2 ¼ $252 while the uninformed player (firm 2) obtains an equilibrium profit of
 L  p 2 p1 ; pH 2 ; p2 ¼ ð16  14Þð22  2 16 þ 16Þ ¼ $12 when its costs are high, i.e., c = 14; and
 L  p 2 p1 ; pH 2 ; p2 ¼ ð12  6Þð22  2 12 þ 12Þ ¼ $60 when its costs are low, i.e., c = 6.

Exercise 6—Public Good Game with Incomplete InformationB Consider a public good game between two players, A and B. The utility function of every player i is 
1=2 ui ðgi Þ ¼ ðwi  gi Þ1=2 m gi þ gj

346

7

Simultaneous-Move Games with Incomplete Information

where the first term, wi  gi , represents the utility that the player receives from money (i.e., the amount of his wealth not contributed to the public good which he can dedicate to private uses). The second term indicates the utility he obtains from aggregate contributions to the public good, gi þ gj , where m
0 denotes the return from aggregate contributions. Since public goods are non-rival in consumption, player i’s utility from this good originates from both his own donations, gi , and in those of player j, gj . Consider that the wealth of player A, wA [ 0, is common knowledge among the players; but that of player B, wB , is privately observed by player B but not observed A. However, player A knows that player B’s wealth level wB can be high
by Hplayer 
 L wB or low wLB with equal probabilities, where wH B [ wB [ 0. Let us next find the Bayesian Nash Equilibrium (BNE) of this public good game where player A is uninformed about the exact realization of parameter wB . (a) Starting with the privately informed player B, set up his utility maximization problem, and separately find his best response function, first, when wB ¼ wH B and, second, when wB ¼ wLB . (b) Find the best response function of the uninformed player A. (c) Using your results in parts (a) and (b), find the equilibrium contributions to the public good by each player. For simplicity, you can assume that player A’s wealth is wA = $14, and those of player B are wH B = $20 when his wealth is high, and wLB = $10 when his wealth is low [Hint: You will find one equilibrium contribution for player A, but two equilibrium contributions for player B as his contribution is dependent on his wealth level wB ]. Answer Part (a). Player B with high wealth. When wB ¼ wH B , player B chooses his con[ 0 to solve the following utility maximization tribution to the public good gH B problem:
H   1=2 H 1=2 uB ðgH mðgA þ gH max B Þ ¼ w B  gB BÞ H

gB
0

where the first term represents his utility from private uses (that is, the amount of wealth not contributed to the public good), while the second term reflects his utility from total donations to the public good from him and player A. Differentiating with respect to gH B , yields ffi pffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H m gA þ gH m wH @EUB ðgH B B  gB BÞ ¼ þ H H H H 2 2 @gB wB  gB gA þ gB

Exercise 6—Public Good Game with Incomplete InformationB

347


 Assuming interior solutions gH B [ 0 , we set the above first order condition equal to zero, which helps us rearrange the above expression as follows H H gA þ gH B ¼ wB  gB

Solving for player B’s contribution to the public good, gH B , yields the following best response function, wH 1 B  gA 2 2

gH B ð gA Þ ¼ wH

which originates at half of his wealth, 2B , and decreases in player A’s donation by 1/2. In words, when player A does not contribute to the public good (gA ¼ 0), wH

player B donates half of his wealth, 2B , but for every dollar that player A contributes to the public good, player B reduces his donation by 50 cents. Player B with low wealth. When wB ¼ wLB , player B chooses his contribution to the public good gLB [ 0 to solve the following utility maximization problem


1=2  1=2 uB ðgLB Þ ¼ wLB  gLB mðgA þ gLB Þ max L

gB
0

Differentiating with respect to gLB , yields ffi pffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m gA þ gLB m wLB  gLB @uB ðgLB Þ ¼  þ 2 2 @gLB wLB  gLB gA þ gLB
 Assuming interior solutions gLB [ 0 , we set the above first order condition equal to zero, which helps us simplify the expression as follows gA þ gLB ¼ wLB  gLB After solving for player B’s contribution to the public good, gLB , we find the following best response function gLB ðgA Þ ¼ which originates at half of his wealth, the public good by 1/2.

wLB 2,

wLB 1  gA 2 2 and decreases in player A’s contribution to

348

7

Simultaneous-Move Games with Incomplete Information

Part (b). Player A chooses his contribution to the public good gA
0 to solve the following expected utility maximization problem,  1=2 1  1=2 1 max EUA ðgA Þ ¼ ðwA  gA Þ1=2 mðgA þ gLB Þ þ ðwA  gA Þ1=2 mðgA þ gH BÞ gA
0 2 2 since he does not observe player B’s wealth level, he does not know whether player B contributes gLB (which occurs when his wealth is low) or gH B (which happens when his wealth is high). Differentiating with respect to gA , yields ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# pffiffiffiffi "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m @EUA ðgA Þ wA  gA w A  gA gA þ gLB gA þ gH B  ¼ þ  L H 4 @gA gA þ gB gA þ gB w A  gA wA  gA equal Assuming interior solutions ðgA [ 0Þ, we set the above first order condition
 to zero. Rearranging, we find that player A’s best response function, gA gLB ; gH B , is implicitly defined by the following expression sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gA þ gLB gA þ gH wA  gA w A  gA B þ ¼ þ L w A  gA wA  gA gA þ gB gA þ gH B Part (c). At a Bayesian Nash equilibrium (BNE), the contribution of each player constitutes a best response to one another (mutual best response). Hence, we can substitute the best responses of the two types of player B into the best response of player A, to obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wL g wH g gA þ B 2 A gA þ B 2 A wA  gA wA  gA þ ¼ L g þ w wH g A w A  gA wA  gA gA þ B gA þ B A 2

2

While the expression is rather large, you can notice that it now depends on player A’s contribution to the public good, gA , alone. We can then start to simplify the above expression, obtaining qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 gA þ wLB þ gA þ wH þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B ¼ 2ðwA  gA Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L gA þ wB gA þ wH B which is further simplified to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðgA þ wLB ÞðgA þ wH B Þ ¼ 2ðwA  gA Þ

!

Exercise 6—Public Good Game with Incomplete InformationB

349

Squaring both sides yields



 2 gA þ wLB gA þ wH B ¼ 4ðwA  gA Þ
 2 L H ) 3 g2A  8wA þ wLB þ wH B gA þ 4wA  wB wB ¼ 0

L Substituting wealth levels wA = $14, wH B = $20, and wB = $10 into the above quadratic equation, we obtain

3g2A  142gA þ 584 ¼ 0 The quadratic equation returns two roots, namely gA ¼ 4:55 or gA ¼ 42:78. However, since player A’s contribution to public good cannot exceed his wealth, wA = $14, we take gA ¼ 4:55 as the only feasible solution. Therefore, the contribution made by player 2 when his wealth is high or low are gH B ¼

20  4:55 ¼ 7:73 2

gL B ¼

14  4:55 ¼ 4:73 2

and

As a result, the Bayesian Nash equilibrium (BNE) of this public good game is the triplet of contributions 

 L gA ; gH ¼ f4:55; 7:73; 4:73g B ; gB

Exercise 7—Starting a Fight Under Incomplete InformationC Consider two students looking for trouble on a Saturday night (hopefully, they are not game theory students). After arguing about some silly topic, they are in the verge of a fight. Each individual privately observes its own ability as a fighter (its type), which is summarized as the probability of winning the fight if both attack each other: either high, pH, or low, pL, where 0 < pL < pH < 1, with corresponding probabilities q and 1 − q, respectively. If a student chooses to attack and its rival does not, then he wins with probability apK , where K = {H, L}, and a takes a value such that pK \apK \1. If both students attack each other, then the probability that a student with type-k wins is pK. If a type-k fighter does not attack but the other fighter attacks, then the probability of victory for the student who did not attack decreases to only bpK , where b takes a value such that 0\bpK \pK . Finally, if neither student attacks, there is no fight. A student is then more likely to win the

350

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Simultaneous-Move Games with Incomplete Information

fight the higher is his type. A student’s payoff when there is no fight is 0, the benefit from winning a fight is B (e.g., being the guy in the gang), and from losing a fight is L (shame, red face). Assume that B > 0 > L. a. Under which conditions there is a symmetric Bayesian-Nash equilibrium in this simultaneous-move game where every student attacks regardless of his type. b. Find the conditions to support a symmetric BNE in which every student attacks only if his type is high. Answer Part (a) If one student expects the other to attack for sure (that is, both when his type is high and low), then it is optimal to attack regardless of his type. Doing so results in an expected payoff of pB þ ð1  pÞL, where p is the probability of wining the fight when both students attack, and not doing so results in an expected payoff of bpB þ ð1  bpÞL, where bp is the probability of wining the fight when the student does not attack but his rival does. Comparing these expected payoffs, we obtain that every student prefers to attack if: pB þ ð1  pÞL [ bpB þ ð1  bpÞL rearranging yields: Bðp  bpÞ [ Lðp  bpÞ or, further simplifying, B > L, which holds by definition since B > 0 > L. Therefore, attacking regardless of one’s type is a symmetric Bayesian-Nash equilibrium (BNE). Part (b) Consider a symmetric strategy profile in which a student attacks only if his type is high. High-type student. For this strategy profile to be an equilibrium, we need that the expected utility from fighting when being a strong fighter must exceed that from not fighting. In particular, 
  
  q pH B þ 1  pH L þ ð1  qÞ apH B þ 1  apH L
q½bpH B þ ð1  bpH ÞL þ ð1  qÞ0 Let’s start analyzing the left-hand side of the inequality: • The first term represents the expected payoff that a type-pH student obtains when facing a student who is also type pH (which occurs with probability q). In such a case, this equilibrium prescribes that both students attack (since they are both pH ),

Exercise 7—Starting a Fight Under Incomplete InformationC

351

and the student we analyze wins the fight with probability pH (and alternatively losses with probability 1  pH ). • The second term represents the expected payoff that a type pH student obtains when facing a student who is, instead, type pL (which occurs with probability 1  q). In this case, his rival doesn’t attack in equilibrium, increasing the probability of wining the fight for the student we analyze to apH , where apH [ pH . Let us now examine the right-hand side of the inequality, which illustrates the expected payoff that student pH obtains when he deviates from his equilibrium strategy, i.e., he does not attack despite having a type pH . • The first term represents the expected payoff that a type-pH student obtains when facing a student who is also pH (which occurs with probability q). In such a case, the student we analyze doesn’t attack while his rival (also of type H) attacks, lowering the chances that the former wins the fight to bpH , where bpH \pH . • If, in contrast, the type of his rival is pL , then no individual attacks and their payoffs are both zero. Rearranging the above inequality, we obtain: 
  q pH B þ 1  pH L þ ð1  qÞ½apH B þ ð1  apH ÞL

q½bpH B þ 1  bpH L þ ð1  qÞ0 Factoring out B and L, we have B½qpH þ apH  qapH  qbpH 
L½qpH þ apH  qapH  qbpH  ð1  qÞ and solving for B, yields  B
L 1 

1q pH ½qð1  bÞ þ ð1  qÞa



Since ð1  qÞ [ 0 and pH ½qð1  bÞ þ ð1  qÞa [ 0, the term in parenthesis ð1qÞ satisfies 1  pH ½qð1b Þ þ ð1qÞa \1, and this condition holds by the initial condition B [ L. For instance, if pH ¼ 0:8, a ¼ 1:3, and b ¼ 0:8, the above inequality becomes 

1 þ 3q B
L : 22q  26 If, for instance, losing the fight yields a payoff of L = − 2, then Fig. 7.11 depicts the (B, q)-pairs for which the high-type student starts a fight, i.e., shaded region. Intuitively, the benefit from winning, B, must be relatively high. This is likely when

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7

Simultaneous-Move Games with Incomplete Information

Fig. 7.11 (B, p)-pairs for which the high-type student fights (shaded region)

the probability of the other student being strong, q, is relatively low. For instance, when q = 0, B only needs be larger than 1/13 in this numerical example for the student to start a fight. However, when his opponent is likely strong, i.e., q approaches 1, the benefit from winning the fight must be much higher (large value of B) for the high-type student to start a fight. Low-type student. Let us now examine the student of type pL . Recall that, according to the equilibrium we analyze, this student prefers to not attack. That is, his expected utility must be larger from not attacking than attacking, as follows. 
  q bpL B þ 1  bpL L þ ð1  qÞ0
q½pL B þ ð1  pL ÞL þ ð1  qÞ½apL B þ ð1  apL ÞL: An opposite intuition as above can be constructed for this inequality, i.e., a student prefers not to attack when it type is pL . Factoring out B and L, and solving for B, yields  B
L 1 

1q pL ½qb  a þ qa  q



Since ð1  qÞ [ 0 and pL ½qb  a þ qa  q [ 0, the term in parenthesis satisfies  1

1q \1 pL ½qb  a þ qa  q

As a consequence, the above inequality is also implied by the initial condition B [ L.

Exercise 7—Starting a Fight Under Incomplete InformationC

353

Following a similar numerical example as above, where pL ¼ 0:4; a ¼ 1:3; b ¼ 0:8, we obtain B
2Lð18q19Þ 11q13 . Using the same numerical example as for the 7672q high-type player, this yields a cutoff B
11q13 , which is negative for all values of q, and thus satisfied for any positive benefit from winning the fight, B > 0.

8

Auctions

Introduction In this chapter we examine different auction formats, such as first-, second-, thirdand all-pay auctions. Auctions are a perfect setting in which to apply the Bayesian Nash Equilibrium (BNE) solution concept learned in Chap. 7, since competing bidders are informed about their private valuation for the object for sale, but are uninformed about each other’s valuations. Since, in addition, bidders are asked to simultaneously and independently submit their bids under an incomplete information environment; we can use BNE to identify equilibrium behavior, namely, equilibrium bidding strategies. In order to provide numerical examples, the following exercises often depict equilibrium bids for the case in which valuations are distributed according to a uniform distribution. We start analyzing of the first-price auction (whereby the bidder submitting the highest bid wins the auction and must pay his bid), and show how to find equilibrium bids. For simplicity, we start with settings in which bidders’ valuations are uniformly distributed (which are mathematically easier to manipulate), first with only two bidders, and then with more than two bidders. Exercise 3 then generalizes our analysis to contexts where bidders’ valuations are distributed according to any other distribution (not only uniform). For completeness, we use two approaches (first order conditions, and the envelope theorem). We also investigate the effect of a larger number of competing bidders on players’ equilibrium bids, showing that players become more aggressive as more bidders compete for the object. Exercise 4 then examines the second-price auction, where the winner of the object is still the bidder submitting the highest bid. However, the winner does not pay his own bid, but the second highest bid. Exercises 5 and 6 move to equilibrium bids in the so-called “all pay auction,” whereby all bidders, irrespective of whether they win the object for sale, must pay the bid they submitted.

© Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_8

355

356

8

Auctions

We afterwards compare equilibrium bidding functions in the first-, second- and all-pay auction. Exercise 7 explores third-price auctions, where the winner is the bidder who submits the highest bid (as in all previous auction formats), but he/she only pays the third highest bid. Finally, we introduce the possibility that bidders are risk averse, and examine how equilibrium bidding functions in the first- and second-price auctions are affected by players’ risk aversion.

Exercise 1—First Price Auction with Uniformly Distributed Valuations—Two PlayersA Consider a first-price auction with two players, each of them drawing his valuation for the object vi from a uniform distribution vi
U½0; 1. Every player i simultaneously and independently submits his bid for the object, bi , and the seller chooses the player who submitted the highest bid as the winning bidder, who pays the bid he submitted. The other bidder loses the auction and receives a zero payoff. (a) Set up every bidder i’s expected utility maximization problem. Interpret. (b) Find the equilibrium bidding function that every bidder i uses in the Bayesian Nash Equilibrium (BNE) of the first-price auction. Answer Part (a). Every player i 2 f1; 2g receives: 1. a net payoff of ðvi  bi Þ if he wins the auction (that is, the difference between his private valuation for the object and the bid he pays for it), and 2. a payoff of 0 if he loses the auction. Then, his expected payoff from participating in the auction is given by EUi ðbi jvi Þ ¼ probðwinÞ  ðvi  bi Þ þ probðloseÞ  0

ð1Þ

Player i’s expected payoff of bidding above his own valuation, bi [ vi , is negative. Indeed, if he were to win the auction, his margin would be negative! Hence, bidding above his valuation is a dominated strategy (since bidding any amount less than vi guarantees at least a payoff of 0 if he loses, and a positive expected payoff if he wins). Thus, in equilibrium, we must have that every bidder submits a bid below his valuation, bi \vi . That is, bidders must practice what is usually referred to as “bid shading.” In particular, if bidder i’s valuation is vi , his bid must be a share of his true valuation, bi ðvi Þ ¼ avi ; where parameter a 2 ð0; 1Þ represents the extent of bid shading.

Exercise 1—First Price Auction with Uniformly Distributed …

357

Consider now, that the bidder submits a bid x so that bi ðvi Þ ¼ x ¼ avi Solving for valuation vi yields vi ¼

x a

Now, in order to find the optimal bidding strategy, we need to maximize expected payoffs as described in expression (1) above. For this, we first need to find player i’s probability of winning the auction, i.e., probðwinÞ in expression (1). For a bid x, bidder j can use the symmetric bidding function avi to recover bidder i’s valuation, x. Hence, the probability that player i wins the auction is given by x 

 prob bi [ bj ¼ prob x [ bj ¼ prob [ vj a We depict this probability in Fig. 8.1. Starting from the vertical axis, the probability that bidder i wins is captured by all bids satisfying bi [ bj , in the shaded segment of the vertical axis. In words, player j’s bid must be lower than player i’s for player i to win the auction. These bids originate from valuations in the shaded segment of the horizontal axis, where player j’s valuations for the object, vj , are relatively low. Indeed, player j’s valuation must satisfy vj \ ax, as indicated in the figure and in our above probability expression. Because distribution on valuations, we find that probability
x of the uniform x prob a [ vj becomes a, as depicted on Fig. 8.1. Substituting this probability into bidder i’s expected utility from participating in the auction (expression 1), we obtain EUi ðxjvi Þ ¼ probðwinÞ  ðvi  bi Þ þ probðloseÞ  0 x ¼  ðvi  xÞ þ 0 a vi x  x2 ¼ a Part (b). Differentiating the above expected utility from participating in the auction, with respect to player i’s bid, x yields

vi xx2 a ,

@EUi ðxjvi Þ vi  2x ¼ ¼0 @x a

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Auctions

Fig. 8.1 Understanding the probability of winning a FPA using bids and valuations

Solving for bid x, we find that bidder i’s equilibrium bidding function is 1 xðvi Þ ¼ vi 2 Thus, the optimal bidding strategy in a first-price auction with two players where valuations are uniformly distributed is to bid half of one’s valuation vi . For instance, if player i’s valuation is vi ¼ 23, his optimal bid should be xðvi Þ ¼ 12 23 ¼ 26 ¼ 13.

Exercise 2—First Price Auction with Uniformly Distributed Valuations—N  2 PlayersB Consider a first-price auction with N  2 players, each of them drawing his valuation for the object vi from a uniform distribution vi
U½0; 1. Every player i simultaneously and independently submits his bid for the object, bi , and the seller chooses the player who submitted the highest bid as the winning bidder, who pays the bid he submitted. All other bidders lose the auction and receive a zero payoff.

Exercise 2—First Price Auction with Uniformly Distributed Valuations …

359

(a) Set up every bidder i’s expected utility maximization problem. Interpret. (b) Find the equilibrium bidding function that every bidder i uses in the Bayesian Nash Equilibrium (BNE) of the first-price auction. c) How is the equilibrium bidding function you found in part (b) affected when the number of bidders, N, grows? Answer Part (a). Like in the previous exercise, where we analyzed a first-price auction with only two players, every bidder i’s expected payoff from participating in the auction is EUi ðbi jvi Þ ¼ probðwinÞ  ðvi  bi Þ þ probðloseÞ  0 However, the probability of winning the auction, probðwinÞ, now implies that player i’s bid to be higher than every other bidder. That is, probðwinÞ ¼ probðbi [ b1 Þ  . . .  probðbi [ bi1 Þ  probðbi [ bi þ 1 Þ  . . .  probðbi [ bN Þ By the same argument as in the previous problem, the probability that player i’s bid is higher than any other bidder j can be written as x 
 prob bi [ bj ¼ prob [ vj a which helps us express the above probability of winning the auction as follows  x  x  [ v1  . . .  prob [ vi1  prob [ vi þ 1  . . . a a ax   prob [ vN a x x x x ¼  ...    ...  a a a a x N1 ¼ a

probðwinÞ ¼ prob

x

Therefore, the expected utility that bidder i obtains from participating in the auction can now be expressed as EUi ðxjvi Þ ¼

xN1 a

ðvi  xÞ ¼

1 aN1

½xN1 vi  xN 

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Auctions

Part (b). Differentiating the above expected utility from participating in the auction with respect to player i’s bid, x, we obtain @EUi ðxjvi Þ 1 ¼ N1 ½ðN  1ÞxN2 vi  NxN1  ¼ 0 @x a which is zero when the term in brackets is nil. This implies that ðN  1ÞxN2 vi ¼ NxN1 which further simplifies to xN1 N  1 vi ¼ N xN2 Solving for bid x we find that every bidder i’s optimal bidding function is x ðvi Þ ¼

N1 vi N

In words, the optimal bidding strategy in a first-price auction with N players where valuations are uniformly distributed is to bid N1 N times one’s valuation. Part (c). When N = 2 bidders compete in the auction, the above bid collapses to ¼ 12 vi , which coincides with the equilibrium bidding strategy we found in the previous exercise. When N = 3 bidders compete for the object, the above bidding 2 function increases to 31 3 vi ¼ 3 vi , thus indicating that every bidder shades his bid less significantly when we increase the number of bidders he competes with, i.e., more aggressive bidding. A similar argument applies when the number of bidders increases, for instance to N = 100, where the above bidding function becomes 1001 99 100 vi ¼ 100 vi , indicating that every bidder only shades his bid by 1%, i.e., he submits a bid that represents 99% of his true valuation for the object. More formally, we can differentiate the equilibrium bidding function we found in part (b), x ðvi Þ ¼ N1 N vi , with respect to the number of bidders, N, to obtain 21 2 vi

@x ðvi Þ N  ðN  1Þ 1 ¼ vi ¼ 2 vi 2 @N N N which is positive since the number of players is positive, N > 0 and their valuation is weakly positive, vi  0. Therefore, equilibrium bidding function x ðvi Þ ¼ N1 N vi increases in the number of bidders competing for the object, N.

Exercise 3—First Price Auction with N BiddersB

361

Exercise 3—First Price Auction with N BiddersB Consider a first-price sealed-bid auction in which bidders simultaneously and independently submit bids and the object goes to the highest bidder at a price equal to his/her bid. Suppose that there are N  2 bidders and that their values for the object are independently drawn from a uniform distribution over [0, 1]. Player i’s payoff is v  bi when she wins the object by bidding bi and her value is v; while her payoff is 0 if she does not win the object. Part (a) Formulate this auction as a Bayesian game. Part (b) Let bi ðvÞ denote the bid made by player i when his valuation for the object is v. Show that there is a Bayesian Nash Equilibrium in which bidders submit a bid which is a linear function of their valuation, i.e., bi ðvÞ ¼ a þ bv for all i and v, where parameters a and b satisfy a; b [ 0. Determine the values of a and b. Answer First, note that the expected payoff of every bidder i with valuation vi is given by:

ui ðbi ; bi ; vi Þ ¼

8 v i  bi > > >
> > :

vi bi 2

if

bi ¼ max bj

0

if

bi \ max bj

j6¼i

j6¼i

j6¼i

where, in the top line, bidder i is the winner of the auction, since his bid, bi , exceeds that of the highest competing bidder, maxj6¼i bj . In this case, his net payoff after paying bi for the object is vi − bi. The bottom line, bidder i loses the auction, since his bid is lower than that of the highest competing bidder. In this case, his payoff is zero. Finally, in the middle line, there is a tie, since bidder i’s bid coincides with that of the highest competing bidder. In this case, the object is assigned with equal probability among the two winning bidders. Part (a) To formulate this first price auction (FPA) as a Bayesian game we need to identify the following elements: • Set of N players (bidders). • Set of actions available to bidder i, Ai , which in this case is the space of admissible bids bi 2 ½0; 1Þ, i.e., a positive real number. • Set of types of every bidder i, Ti , which in this case is just his set of valuations for the object on sale, i.e., vi 2 ½0; 1. • A payoff function ui , as described above in ui ðbi ; bi ; vi Þ: • The probability that bidder i wins the auction is, intuitively, the probability that his bid exceeds that of the highest competing bidder, i.e., pi ¼ probi ðwinÞ ¼ probi ðbi  maxj6¼i bj Þ.

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Auctions

Hence, the Bayesian game is compactly defined as: G ¼ \N; ðAi Þi2N ; Ti ; ðui Þi2N ; pi [ : Part (b) Let bi ðvi Þ denote the bid submitted by bidder i with type (valuation) given by vi . First, note that bidder i will not bid above his own valuation, given that it constitutes a strictly dominated strategy: if he wins he would obtain a negative payoff (since he has to pay for the object a price above his own valuation), while if he loses he would obtain the same payoff submitting a lower bid. Therefore, there must be some “bid shading” in equilibrium, i.e., bidder i’s bid lies below his own valuation, bi \v, or alternatively, bi ðvÞ ¼ a þ bv\v: Hence, we need to show that in a FPA there exists a BNE in which the optimal bidding function is linear and given by bi ðvÞ ¼ a þ bv for every bidder i, and for any valuation v. Proof First, we will find the optimal bidding function using the Envelope Theorem. Afterwards, we will show that the same bidding function can be found by differentiating the bidder’s expected utility function with respect to his bid i.e., the so-called first-order approach (or direct approach). For generality, the proof does not make use of the uniform distribution U
½0; 1, i.e., FðvÞ ¼ v, until the end. (This would allow you to use other distribution functions different from the uniform distribution.) Envelope Theorem Approach Let’s take a bidder i whose valuation is given by vi and who is considering to bid according to another valuation zi 6¼ vi in order to obtain a higher expected payoff. His expected payoff of participating in the auction will then be given by:

EUi ðv; zÞ ¼ probi ðwinÞ½vi  bi ðzi Þ where the probability of winning is given by:    
 probi ðwinÞ ¼ probi max bj vj \bi ðzi Þ ¼ probi max vj \zi j6¼i

j6¼i

given that we assume that the bidding function, bi ðÞ, is monotonically increasing, and all bidders use the same bidding function. [Note that this does not imply that bidders submit the same bid, but just that they use the same function in order to determine their optimal bid.] In addition, the probability that
 valuation zi lies above of the other bidder’s valuation, vj, is F ðzi Þ ¼ prob vj \zi : Therefore, the probability that zi exceeds all other N − 1 competing bidders’ valuations is

Exercise 3—First Price Auction with N BiddersB

363

F ðzi Þ  F ðzi Þ. . .F ðzi Þ ¼ Fðzi ÞN1 , thus implying that the expected utility from participating in the auction can be rewritten as EUi ðvi ; zi Þ ¼ Fðzi ÞN1 ½vi  bi ðzi Þ: Bidder i then chooses to bid according to a valuation zi that maximizes his expected utility  dEUi ðv; zÞ ¼ ðN  1ÞF ðzi ÞN2 f ðzi Þ vi  ^bðzi Þ  F ðzi ÞN1 ^ b0 ðzi Þ ¼ 0 dzi For ^bi ðvÞ to be an optimal bidding function, it should be optimal for the bidder not to pretend to have a valuation zi different from his real one, vi . Hence zi ¼ vi and bðzi Þ ¼ ^bðvÞ, implying that the above first-order condition becomes  b0 ðvÞ ¼ 0 ðN  1ÞFðvÞN2 f ðzi Þ vi  ^bðvÞ  FðvÞN1 ^ And rearranging ðN  1ÞFðvÞN2 f ðvÞ^bðvÞ þ FðvÞN1 ^b0 ðvÞ ¼ ðN  1ÞFðvÞN2 f ðvÞv Note that the left-hand side of the above expression can be alternative represented as

dFðvÞN1 ^ bðvÞ . dv

Hence we can rewrite the above expression as dFðvÞN1 ^bðvÞ ¼ ðN  1ÞFðvÞN2 f ðvÞv dv

Applying integrals to both sides of the equality, we obtain:

FðvÞ

Zv

N1 ^

ðN  1ÞFðxÞN2 f ðxÞxdx

bðvÞ ¼ 0

Hence, solving for the optimal bidding function ^ bðvÞ, we find ^bðvi Þ ¼

1 FðvÞ

Zv ðN  1ÞFðxÞN2 f ðxÞx dx

N1 0

Finally, since dFðxÞN1 ¼ ðN  1ÞFðxÞN2 f ðxÞ, we can more completely express the optimal bidding function in the FPA as:

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8

^bðvÞ ¼

1 FðvÞ

Auctions

Zv xdFðxÞN1

N1 0

Direct Approach We can confirm our previous result by finding the optimal bidding function without using the Envelope Theorem. Similarly as in our above approach, bidder i wins the object when his bid, bi , exceeds that of the highest competing bidder that is probðwinÞ ¼ probðbðY1 Þ  bi Þ where Y1 is the highest valuation among the N  1 remaining bidders (the highest order statistic), and bðÞ is the symmetric bidding function that all bidders use. We can, hence, invert this bidding function, to express bidder i’s probability of winning in terms of valuations (rather than in terms of bids),
 probðbðY1 Þ  bi Þ ¼ prob Y1  b1 ðbi Þ ¼ Gðb1 ðbi ÞÞ

ð8:1Þ

where in the first equality we apply the inverse of the bidding function b1 ðÞ on both sides of the inequality inside the parenthesis. Intuitively, we move from expressing the probability of winning in terms of bids to expressing it in terms of valuations, where Y1 is the valuation of the highest competing bidder and b1 ðbi Þ is bidder i’s valuation. Figure 8.2 depicts the bidding function of bidder i: when he has a valuation vi for the object on sale, his bid becomes bðvi Þ. If, observing such a bid, we seek to infer his initial valuation, we would move from the vertical to the horizontal axes on Fig. 8.2 by inverting the bidding function, i.e., b1 ðvi Þ ¼ vi . The second equality of expression (8.1) evaluates the probability that the valuation of all competing bidders is lower than the valuation of bidder i (given by b1 ðbi Þ ¼ vi ). Hence, the expected payoff function for a bidder with valuation vi can be rewritten as: EUi ðvi Þ ¼ Gðb1 ðbi ÞÞ ½vi  bi  We can now take first-order conditions with respect to bidder i’s bid, bi , in order to find which is his optimal bid,

 dEUi ðvi Þ g b1 ðbi Þ  ½vi  bi   G b1 ðbi Þ ¼ 0 ¼ 0
1 dbi b b ð bi Þ

Exercise 3—First Price Auction with N BiddersB

365

Fig. 8.2 Bidding function

and rearranging,
 gðb1 ðbi ÞÞ ½vi  bi   b0 ðb1 ðbi ÞÞG b1 ðbi Þ ¼ 0 Given that in equilibrium we have that bðvÞ ¼ bi we can invert this function to obtain v ¼ b1 ðbi Þ; as depicted in Fig. 8.2. As a consequence, we can rewrite the above expression as: gðvi Þðvi  bðvi ÞÞ  bðvi ÞGðvi Þ ¼ 0; Or more compactly as, gðvi Þbðvi Þ þ bðvi ÞGðvi Þ ¼ gðvi Þvi Similarly as in our proof using the Envelope Theorem, note that the left-hand side i ÞGðvi Þ , implying that the equality can be can be alternatively expressed as d½bðvdv i rewritten as: d½bðvi ÞGðvi Þ ¼ gðvi Þvi dvi Integrating on both sides, yields Zv bðvi ÞGðvi Þ ¼

gðyÞydy þ C; 0

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where C is the integration constant and it is zero given that bð0Þ ¼ 0, i.e., a bidder with a zero valuation for the object can be assumed to submit a zero bid. Hence, solving for the optimal bid bðvi Þ we find that the optimal bidding function in a FPA is: 1 bðvi Þ ¼ Gðvi Þ

Zv gðyÞydy 0

Equivalent Bidding Functions. Let us now check that the two optimal bidding functions that we found (the first one with the Envelope Theorem approach, and the second with the so-called Direct approach) are in fact equivalent. First note that GðxÞ ¼ FðxÞN1 , and differentiating with respect to x yields G0 ðxÞ ¼ ðN  1ÞFðxÞN2 f ðxÞ. Hence, substituting into the expression of the optimal bidding function that we found using the Direct approach, we obtain: 1 bðvi Þ ¼ Gðvi Þ

Zv gðyÞydy ¼ 0

1 FðvÞ

Zv ðN  1ÞFðxÞN2 f ðxÞxdx

N1 0

N1

And noting that ðN  1ÞFðxÞN2 f ðxÞ ¼ dFðxÞ dx , we can express this bidding v R N1 function as bðvi Þ ¼ FðvÞ1N1 xdFðxÞ , which exactly coincides with the optimal 0

bðvi Þ, and bidding function we found using the Envelope Theorem. Hence, bðvi Þ ¼ ^ both methods are equivalent. Uniformly Distributed Valuation. Given that in this case we know that the valuations are drawn from a uniform cumulative distribution FðxÞ ¼ x with density f ðxÞ ¼ 1 and support [0, 1], we obtain that: FðvÞN1 ¼ vN1 ;

and dFðvÞN1 ¼ ðN  1ÞxN2 dx

Therefore, the optimal bidding function ^bðvi Þ in this case becomes ^bðvi Þ ¼ 1 vN1

Zv xðN  1Þx

N2

0

dx ¼

1

Zv ðN  1ÞxN1 dx

vN1 0

Exercise 3—First Price Auction with N BiddersB

367

rearranging, ð N  1Þ ¼ N1 v ¼

Zv x 0 N

N1

v ð N  1Þ x N dx ¼ N1 v N 0

ðN  1Þ v N 1 v ¼ vN1 N N

Comparative Statics: More Bidders. Hence, the optimal bidding function in a FPA with N bidders with uniformly distributed valuations is ^bðvi Þ ¼ N  1 vi N Therefore, when we only have N = 2 bidders, their symmetric bidding function Interestingly, as N increases, bidders also increase their bid, i.e., from 12 v is 99 when N = 2, to 23 v when N = 3, and to 100 v when N = 100; as depicted in Fig. 8.3. Intuitively, their bids get closer to their actual valuation for the object i.e., the equilibrium bidding functions approach the 45°-line where bi ¼ vi : Hence, bidders reduce their “bid shading” as more bidders compete for the same object. Hence, the optimal bidding function corresponds to the linear function bðvi Þ ¼ a þ bv, where: 1 2 v.

• a ¼ 0 (vertical intercept is zero) because we have assumed that the optimal bid for a bidder with a valuation of zero is zero, b(0) = 0.

Fig. 8.3 Equilibrium bidding function in FPA with more bidders

368

• b ¼ FðvÞ1N1

8

Rv 0

Auctions

xdFðxÞN1 , which is the slope of the bidding function, e.g., under

uniformly distributed valuations b ¼ N1 N . It represents to what extent the bidder shades his own valuation. In the case with two bidders, every player just bids half of his valuation, whereas with more competing bidders, bidder i would submit bids closer to his own valuation; graphically pivoting the bidding function towards the 45°-line where no bid shading occurs.

Exercise 4—Second Price AuctionA Consider a setting where every bidder privately observes his/her own valuation for the object. In addition, assume that N bidders interact in an auction in which they simultaneously and independently submit their own bids, the highest bidder wins the object, but he/she pays the second-highest bid (this is the so-called second-price auction, SPA). All other bidders pay zero. a. Argue that it is a weakly dominant strategy for a bidder to bid her value in the SPA. b. Are optimal bids affected if the joint distribution of bidders’ values exhibits correlation, i.e., if bidder i observes that he has a high valuation for the object he can infer that other bidders must also have a high valuation (in the case of positively correlated values) or that, instead, they must have a low valuation for the object (in the case of negatively correlated values)? c. Are optimal bids affected by the number of bidders participating in the auction? Are optimal bids affected by bidders’ risk aversion? Answer Part (a) Let ri ¼ maxj6¼i bj be the maximum bid submitted by the other players (different from bidder i). Suppose first that player i is considering bidding above his own valuation, i.e., bi > vi. We have the following possibilities: • If ri  bi [ vi , bidder i does not win the object (or he does so in a tie with other players). However, his expected payoff is zero (if he loses the auction) or negative (if he wins in a tie), thus implying that he obtains the same (or lower) payoff than when bids his own valuation, vi. • If bi > ri > vi, player i wins the object but gets a payoff of vi − ri < 0; he would be strictly better off by bidding vi and not getting the object (i.e. obtaining a payoff of zero).

Exercise 4—Second Price AuctionA

369

• If bi > vi > ri, player i wins the object and gets a payoff of vi − ri  0. However, his payoff would be exactly the same if he were to bid his own valuation vi. We can apply a similar argument to the case in which bidder i considers bidding below his valuation, bi < vi. • When ri [ vi [ bi , bidder i loses the object since the highest competing bid, ri , is higher than his bid, bi . In this setting, bidder i’s expected payoff would be unchanged if he were to submit a bid equal to his own valuation, bi = vi, instead of bi < vi. • If, however, the highest competing bid, ri , lies between bidder i’s valuation and his bid, i.e., vi [ ri [ bi , the bidder would be forgoing a positive payoff by underbidding. He currently loses the object (and gets a zero payoff) whereas if he bids bi = vi, he would win the object and obtain a positive payoff of vi − ri > 0. Therefore, bidder i does not have incentives to submit a bid below his valuation for the object. This argument, hence, establishes that bidding one’s valuation is a weakly dominant strategy, i.e., submitting bids that lie above or below his valuation will provide him with exactly the same utility level as he obtains when submitting his valuation, or a strictly lower utility; thus implying that the bidder does not have strict incentives to deviate from bi = vi. Part (b) By the definition of weak dominance, this means that bidding one’s valuation is weakly optimal no matter what the bidding strategies of the other players are. It is, therefore, irrelevant how the other players’ strategies are related to their own valuations, or how their valuations are related to bidder i’s own valuation. In other words, bidding one’s valuation is weakly optimal irrespectively of the presence of correlation (positive or negative) in the joint distribution of player’s values. Part (c) The above equilibrium bidding strategy in the SPA is, importantly, unaffected by the number of bidders who participate in the auction, N, or their risk-aversion preferences. In terms of Fig. 8.3 in Exercise 1, this result implies that the equilibrium bidding function in the SPA, bi ðvi Þ ¼ vi , coincides with the 45°-line where players do not shade their valuation for the object. Intuitively, since bidders anticipate that they will not have to pay the bid they submitted, but that of the second-highest bidder, every bidder becomes more aggressive in his bidding behavior. In particular, our above discussion considered the presence of N bidders, and an increase in their number does not emphasize or ameliorate the incentives that every bidder has to submit a bid that coincides with his own valuation for the object, bi ðvi Þ ¼ vi . Furthermore, the above results remain valid when bidders evaluate their net payoff according to a concave utility function, such as uðwÞ ¼ wa ; where a 2 ð1; 0Þ, thus, exhibiting risk aversion. Specifically, for a given value of the highest competing bid, ri , bidder i’s expected payoff from

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submitting a bid bi ðvi Þ ¼ vi would still be weakly larger than from deviating to a bidding strategy strictly above his own valuation for the object, bi ðvi Þ [ vi , or strictly below it, bi ðvi Þ\vi .

Exercise 5—All-Pay Auctions (Easy Version)A Consider the following all-pay auction with two bidders privately observing their valuation for the object. Valuations are uniformly distributed, i.e., vi
U½0; 1. The player submitting the highest bid wins the object, but all players must pay the bid they submitted. Find the optimal bidding strategy, taking into account that it is of the form bi ðvi Þ ¼ m  vi2 ; where m denotes a positive constant. Answer • Bidder i’s expected utility from submitting a bid of x dollars in an all-pay auction is EUi ðxjvi Þ ¼ probðwinÞ  vi  x where, if winning, bidder i gets the object (which he values at vi ), but he pays his bid, x, both when winning and losing the auction • Let us now specify the probability of winning, probðwinÞ. If bidder i submits a bid $x using a bidding function x ¼ m  v2i , we can recover the valuation vi that pffiffiffi generated such a bid, i.e., solving for vi in x ¼ m  v2i we obtain mx ¼ vi . Hence, since valuations are distributed according to vi
U½0; 1, the probability of winning is given by rffiffiffiffi rffiffiffiffi 
 x x prob vj \vi ¼ prob vj \ ¼ : m m • Therefore, the above expected utility becomes EUi ðxjvi Þ ¼

rffiffiffiffi x  vi  x m

Taking first-order conditions with respect to the bid, x, we obtain 1 þ

pffiffiffi vi mx ¼0 2x

Exercise 5—All-Pay Auctions (Easy Version)A

371

Fig. 8.4 Optimal bidding function in the APA

and solving for x, we find bidder i’s optimal bidding function in the all-pay auction bi ðvi Þ ¼

1  v2 4m i

• If, for instance, m = 0.5, this bidding function becomes bi ðvi Þ ¼ 12  v2i . Figure 8.4 depicts this bidding function, comparing it with bids that coincide with players’ valuation for the object, i.e., bi ¼ vi in the 45°-line. (For a more general identification of optimal bidding functions in APAs, see Exercise 6.)

Exercise 6—All-Pay Auction (Complete Version)B Modify the first-price, sealed-bid auction in Exercise 3 so that the loser also pays his bid (but does not win the object). This modified auction is the so-called the all-pay auction (APA), since all bidders must pay their bids, regardless whether they win the object or not. Part (a) Show that, in the case of two bidders, there is a Bayesian Nash Equilibrium in which every bidder i’s optimal bidding function is given by bi ðvÞ ¼ c þ dv þ /v2 for all i and v where c; d; /  0, i.e., every bidder submits a bid which is a convex function of his private valuation v for the object. Part (b) Analyze how the optimal bidding function varies in the number of bidders, N.

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Part (c) How do players’ bids in the APA compare to those in the first price auction? What is the intuition behind this difference in bids? Answer In this case, the payoff function for bidder i with valuation vi will be:

ui ðbi ; bi ; vi Þ ¼

8 v i  bi > > > < vi bi 2

> > > : bi

if bi [ max bj j6¼i

if bi ¼ max bj j6¼i

if bi \ max bj j6¼i

This payoff function coincides with that of the FPA, except for the case in which bidder i’s bid is lower than that of the highest competing bidder, i.e., bi \ maxj6¼i bj , and thus loses the auction (see bottom row), since in the APA bidder i must still pay his bid, implying that his payoff in this event is bi ; as opposed to a payoff zero in the FPA. We need to show that there exists a BNE in which the optimal bidding function in the APA is given by the convex function bi ðvÞ ¼ c þ dv þ /v2 , for every bidder i with valuation v. [Similarly as in Exercise 1, we only consider that bidders’ valuations are distributed according to a uniform distribution function U
½0; 1 after finding the optimal bidding function for any cumulative distribution function FðxÞ.] Proof The expected payoff of a bidder i with a real valuation of vi but bidding according to a different valuation of zi 6¼ vi is: EUi ðv; zÞ ¼ probi ðwinÞðvi  bi ðzi ÞÞ þ ð1  probi ðwinÞÞ ðbi ðzi ÞÞ where, if winning, bidder i obtains a net payoff vi  bi ðzi Þ; but, if losing, which occurs with probability 1  probi ðwinÞ, he must pay the bid he submitted. Similarly to the FPA, his probability of wining is also given by     probi ðwinÞ ¼ probi max bj ðvj Þ\bi ðzi Þ ¼ probi max vj \zi ¼ Fðzi ÞN1 ; j6¼i

j6¼i

since we assume a symmetric and monotonic bidding function. Hence, the expected utility from participating in a APA can be rewritten as h i EUi ðv; zÞ ¼ Fðzi ÞN1 ðvi  bi ðzi ÞÞ þ 1  F ðzi ÞN1 ðbi ðzi ÞÞ

Exercise 6—All-Pay Auction (Complete Version)B

373

or, rearranging, as: EUi ðv; zÞ ¼ Fðzi ÞN1 vi  bi ðzi Þ Intuitively, if winning, bidder i enjoys his valuation for the object, vi , but he must pay the bid he submitted both when winning and losing the auction. For completeness, we next show that bidding function bi ðvÞ ¼ c þ dv þ /v2 is optimal in the APA using first the Envelope Theorem approach and afterwards using the so-called Direct approach. Envelope Theorem Approach Since the bidder is supposed to be utility maximizing, we can find the following first-order condition using the Envelope Theorem: dEUi ðv; zÞ dbi ðzi Þ ¼ ðN  1ÞF ðzi ÞN2 f ðzi Þvi  ¼0 dzi dzi Hence, for bi ðvÞ to be an optimal bidding function, it should not be optimal for the bidder to pretend to have a valuation zi different from his real one, vi. Hence zi ¼ vi , which entails that the above first-order condition becomes ðN  1ÞF ðvi ÞN2 f ðvi Þvi ¼

d^ bi ðvi Þ dvi

Given that this equality holds for every valuation, v, we can apply integrals on both sides of the equality to obtain, ^bðvi Þ ¼

Zv ðN  1ÞFðxÞN2 f ðxÞxdx þ C; 0

where C is the constant of integration, which equals 0 given that ^ bð0Þ ¼ 0. Hence, using the property that dFðxÞN1 ¼ ðN  1ÞFðxÞN2 f ðxÞ, we obtain that the optimal bidding function for this APA is: ^bðvi Þ ¼

Zv xdFðxÞN1 0

Direct Approach We can alternatively find the above bidding function by using a methodology that does not require the Envelope Theorem. We know that bidder i wins the auction if his bid, bi , exceeds that of the highest competing bidder, bðYi Þ, where Y1 represents

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the highest valuation among the N  1 remaining bidders (the first order statistic). Hence, the probability of winning is given by:
 probðwinÞ ¼ probðbðY1 Þ  bi Þ ¼ prob Y1  b1 ðbi Þ ¼ Gðb1 ðbi ÞÞ (for a discussion of this expression, see Exercise 3). Thus, the expected payoff function for a bidder with valuation vi is given by:  EUi ðvi Þ ¼ Gðb1 ðbi ÞÞðvi  bi Þ þ 1  Gðb1 ðbi ÞÞ ðbi Þ or, rearranging, EUi ðvi Þ ¼ Gðb1 ðbi ÞÞvi  bi We can now take first-order conditions with respect to the optimal bid of this player, bi , obtaining
 dEUi ðvi Þ g b1 ðbi Þ  vi  1 ¼ 0 ¼ 0
1 dbi b b ð bi Þ
 Multiplying both sides by b0 b1 ðbi Þ , and rearranging,

 g b1 ðbi Þ vi  b0 b1 ðbi Þ ¼ 0 Given that in equilibrium we have that bðvi Þ ¼ bi , we can invert this bidding function to obtain vi ¼ b1 ðbi Þ, Therefore, we can rewrite the above expression as: gðvi Þvi  b0 ðvi Þ ¼ 0;

or as b0 ðvi Þ ¼ gðvi Þvi

Given that this equality holds for R v any value of v, we can integrate on both sides of the equality, to obtain bðvÞ ¼ 0 gðyÞydy þ C, where C is the integration constant which is zero given that bð0Þ ¼ 0. Thus, the optimal bidding function in an APA is: Zv bðvÞ ¼

gðyÞy dy 0

Equivalent Bidding Functions. Let us now check that the two optimal bidding functions that we found (the first using the Envelope Theorem approach, and the second with the Direct approach) are in fact equivalent. First, note that GðxÞ ¼ FðxÞN1 and, hence, its derivative is gðxÞ ¼ ðN  1ÞFðxÞN2 f ðxÞ.

Exercise 6—All-Pay Auction (Complete Version)B

375

Substituting g(x) into the expression of the optimal bidding function that we found using the Direct approach, yields Zv bðvi Þ ¼

Zv xðN  1ÞFðxÞ

N2

xdFðxÞN1

f ðxÞdx ¼

0

0

^ i Þ and both methods since ðN  1ÞFðxÞN2 f ðxÞ ¼ dFðxÞN1 . Hence, bðvi Þ ¼ bðv produce equivalent bidding functions. Uniform Distribution. Given that in this case we know that the valuations are drawn from a uniform distribution F(x) = x with density f(x) = 1 and support [0, 1], FðxÞN1 ¼ xN1 and its derivate is dFðxÞN1 ¼ ðN  1ÞxN2 dx, implying that the optimal bidding function becomes ^bðvi Þ ¼

Zv



Zv xðN  1Þx

N2

dx ¼ ðN  1Þ

0

x 0

N1

xN dx ¼ ðN  1Þ N

v ¼ 0

N1 N v N

Therefore, in the case of N = 2 bidders, the optimal bidding function for the APA will be ^bðvi Þ ¼ 12 v2i , as depicted in Fig. 8.5. Hence, this (convex) optimal bidding function corresponds to bi ðvi Þ ¼ c þ dv þ /v2 where c ¼ 0 (vertical intercept is zero) because we have assumed that the optimal bid for a bidder with a valuation of zero is zero, i.e., ^ bi ð0Þ ¼ 0; d ¼ 0 and / ¼ 1=2, which implies that the optimal bidding function is non linear.

Fig. 8.5 Optimal bidding function in the APA

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Part (b). When increasing the number of bidders, N, we obtain a more convex function. Figure 8.6 depicts the optimal bidding function in the APA with two 2 9 10 v . Intuitively, as more bidders bidders, v2 , three bidders, 23 v3 , and ten bidders, 10 compete for the object, bid shading becomes substantial when bidder i has a relatively low valuation, but induces him to bid more aggressively when his valuation is likely the highest among all other players, i.e., when vi ! 1: Part (c). Figure 8.7 compares the optimal bidding function (for the case of uniformly distributed valuations and two bidders) in the FPA against that in the APA.

Fig. 8.6 Comparative statics of the optimal bidding function in the APA

Fig. 8.7 Optimal bidding functions in FPA versus APA

Exercise 6—All-Pay Auction (Complete Version)B

377

As we can see in Fig. 8.7, every bidder in the APA bids less aggressively than in the FPA, since he has to pay the bid he submits, i.e., the optimal bidding function for the APA lies below that of the FPA. More formally, the optimal bidding APA N function in the FPA, bFPA ðvÞ ¼ N1 ðvÞ ¼ N1 N v, lies above that in the APA, b N v , since the difference bFPA ðvÞ  bAPA ðvÞ ¼

ðN  1Þðv  vN Þ N

is positive, since N  2 and v [ vN , graphically, the 45°-line lies above any convex function such as vN .

Exercise 7—Third-Price AuctionA Consider a third-price auction, where the winner is the bidder who submits the highest bid, but he/she only pays the third highest bid. Assume that you compete against two other bidders, whose valuations you are unable to observe, and that your valuation for the object is $10. Show that bidding above your valuation (with a bid of, for instance, $15) can be a best response to the other bidders’ bid, while submitting a bid that coincides with your valuation ($10) might not be a best response to your opponents’ bids. Solution • If you believe that the other players’ bids are $5 and $12, then submitting a bid that coincides with your valuation, $10, will only lead you to lose the auction, yielding a payoff of zero. If, instead, you submit a bid of $15, your bid becomes the highest of the three, and you win the auction. Since, upon winning, you pay the third highest bid, which in this case is only $5, your net payoff is vi  p ¼ 10  5 ¼ 5. • Therefore, submitting a bid above your own valuation for the object can be a best response in the third-price auction. Intuitively, this occurs if one of your competitors submits a bid above your own valuation, while the other submits a bid below your valuation.

Exercise 8—FPA with Risk-Averse BiddersB In previous exercises, we assumed that both the seller and all bidders are risk neutral. Let us next analyze how our equilibrium results would be affected if bidders are risk averse, i.e., their utility function is concave in income, w, and given by

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uðwÞ ¼ wa , where 0\a  1 denotes bidder i’s risk-aversion parameter. In particular, when a ¼ 1 he is risk neutral, while when a decreases, he becomes risk averse.1 Part (a) Find the optimal bidding function, bðvi Þ, of every bidder i in a FPA where N = 2 bidders compete for the object and where valuations are uniformly distributed U
½0; 1. Part (b) Explain how this bidding function is affected when bidders become more risk averse. Answer Part (a). First, note that the probability of winning the object is unaffected, since, for a symmetric bidding function bi ðvi Þ ¼ a  vi for bidder i, where a 2 ð0; 1Þ, the probability that bidder i wins the auction against another bidder j is











bi prob bi [ bj ¼ prob bi [ a  vj ¼ prob [ vj a

 ¼

bi a

where the first equality is due to the fact that bj ¼ a  vj , the second equality rearranges the terms in the parenthesis, and the last equality makes use of the uniform distribution of bidders’ valuations. Therefore, bidder i’s expected utility from participating in this auction by submitting a bid bi when his valuation is vi is given by EUi ðbi jvi Þ ¼

bi ðvi  bi Þa a

where, relative to the case of risk-neutral bidders analyzed in Exercise 1, the only difference arises in the evaluation of the net payoff from winning, vi  bi , which it is now evaluated with the concave function ðvi  bi Þa . Taking first-order conditions with respect to his bid, bi , yields 1 bi ðvi  bi Þa  aðvi  bi Þa1 ¼ 0; a a and solving for bi , we find the optimal bidding function, bi ðvi Þ ¼

vi : 1þa

Importantly, this optimal bidding function embodies as a special case the context in which bidders are risk neutral (as in Exercises 1–3). Specifically, when a ¼ 1, 1

An example you have probably encountered in intermediate microeconomics courses includes the pffiffiffi pffiffiffi concave utility function uðxÞ ¼ x since x ¼ x1=2 . As a practice, note that the Arrow-Pratt 00 coefficient of absolute risk aversion rA ðxÞ ¼  uu0 ððxxÞÞ for this utility function yields 1a x , confirming that, when a ¼ 1, the coefficient of risk aversion becomes zero, but when 0\a\1, the coefficient is positive. That is, as a approaches zero, the function becomes more concave.

Exercise 8—FPA with Risk-Averse BiddersB

379

Fig. 8.8 Risk aversion and optimal bidding strategies in the FPA

bidder i’s optimal bidding function becomes bi ðvi Þ ¼ v2i . However, when his risk aversion increases, i.e., a decreases, bidder i’s optimal bidding function increases. Specifically, @xðvi Þ vi ¼ @a ð1  aÞ2 which is negative for all parameter values. In the extreme case in which a decreases to a ! 0, the optimal bidding function becomes xðvi Þ ¼ vi , and players do not practice bid shading. Figure 8.8 illustrates the increasing pattern in players’ bidding function, starting from v2i when bidders are risk neutral, a ¼ 1, and approaching the 45-degree line (no bid shading) as players become more risk averse. Intuitively, a risk-averse bidder submits more aggressive bids than a risk-neutral bidder in order to minimize the probability of losing the auction. In particular, consider that bidder i reduces his bid from bi to bi  e. In this context, if he wins the auction, he obtains an additional profit of e, since he has to pay a lower price for the object he acquires. However, by lowering his bid, he increases the probability of losing the auction. Importantly, for a risk-averse bidder, the positive effect of slightly lowering his bid, arising from getting the object at a cheaper price, is offset by the negative effect of increasing the probability that he loses the auction. In other words, since the possible loss from losing the auction dominates the benefit from acquiring the object at a cheaper price, the risk-averse bidder does not have incentives to reduce his bid, but rather to increase it, relative to risk-neutral bidders.

9

Perfect Bayesian Equilibrium and Signaling Games

Introduction This chapter examines again contexts of incomplete information but in sequential move games. Unlike simultaneous-move settings, sequential moves allow for players’ actions to convey or conceal the information they privately observe to players acting in subsequent of the game and who did not have access to such information (uninformed players). That is, we explore the possibility that players’ actions may signal certain information to other players acting latter on in the game. In order to analyze this possibility, we first examine how uninformed players form beliefs upon observing certain actions, by the use of the so-called Bayes’ rule of conditional probability. Once uninformed players update their beliefs, they optimally respond to other players’ actions. If all players’ actions are in equilibrium, then such strategy profile constitutes a Perfect Bayesian Equilibrium (PBE) of the sequential-move game under incomplete information; as we next define. Perfect Bayesian Equilibrium (PBE). A strategy profile for N players ðs1 ðh1 Þ; . . .; sN ðhN ÞÞ and a system of beliefs ðl1 ; l2 ; . . .; lN Þ are a PBE if: (a) the strategy of every player i, si ðhi Þ, is optimal given the strategy profile selected by other players si ðhi Þ and given player i’s beliefs, li ; and (b) the beliefs of every player i; li , are consistent with Bayes’ rule (i.e., they follow Bayesian updating) wherever possible. In particular, if the first-mover’s actions differ depending on the (private) information he observes, then his actions “speak louder than words,” thus giving rise to a separating PBE. If, instead, the first-mover behaves in the same manner regardless of the information he observes, we say that such strategy profile forms a pooling PBE.

© Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_9

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Perfect Bayesian Equilibrium and Signaling Games

We first study games in which both first and second movers have only two available actions at their disposal, thus allowing for a straightforward specification of the second mover’s beliefs. To facilitate our presentation, we accompany each of the possible strategy profiles (two separating and two pooling) with their corresponding graphical representation, which helps focus the reader’s attention on specific nodes of the game tree. We then examine a simplified version of Spence’s (1973) job market signaling game where workers can only choose two education levels and firms can only respond with two possible actions: hiring the worker as a manager or as a cashier. Afterwards, we study more elaborate signaling games in which the uninformed agent can respond with more than two actions, and then we move to more general settings in which both the informed and the uninformed player can choose among a continuum of strategies. In particular, we consider an industry where the incumbent firm observes its production costs, while the potential entrant does not, but uses the incumbent’s pricing decision as a signal to infer this incumbent’s competitiveness. We finally examine signaling games where the privately informed player has three possible types, and three possible messages, and investigate under which conditions a fully separating strategy profile can be sustained as a PBE of the game (where every type of informed player chooses a different message), and under which circumstances only a partially separating profile can be supported (where two types choose the same message and the third type selects a different message). We end the chapter with an exercise that studies equilibrium refinements in incomplete information games, applying the Cho and Kreps’ (1987) Intuitive Criterion to a standard signaling game where the informed player has only two types and two available messages. In this context, we make extensive use of figures illustrating equilibrium and off-the-equilibrium behavior, in order to facilitate the presentation of off-the-equilibrium beliefs, and its subsequent restriction as specified by the Intuitive Criterion.

Exercise 1—Finding Separating and Pooling EquilibriaA Consider the following game of incomplete information: nature first determines player 1’s type, either high or low, with equal probabilities. Then player 1, observing his own type, decides whether to choose left (L) or Right (R). If he chooses left, the game ends and players’ payoffs become (u1, u2) = (2, 0), regardless of player 1’s type. Figure 9.1 depicts these payoffs in the left-hand side of the tree. If, instead, player 1 chooses Right, player 2 is called on to move. In particular, without observing whether player 1’s type is high or low, player 2 must respond by either selecting Up (U) or Down (D). As indicated by the payoffs in the right-hand side of the tree, when player 1’s type is high, player 2 prefers to play Up. The opposite preference ranking applies when player 1’s type is low.

Exercise 1—Finding Separating and Pooling EquilibriaA

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Fig. 9.1 Game tree where player 2 is uninformed

Part (a) Does this game have a Separating Perfect Bayesian Equilibrium (PBE)? Part (b) Does this game have a Pooling PBE? Answer: Part (a) Separating PBE First type of separating equilibrium: RL′ Figure 9.2 shades the branches corresponding to the separating strategy profile RL′, whereby player 1 chooses R when his type is high (in the upper part of the game tree), but selects L′ when his type is Low (see lower part of the game tree). First step (use Bayes’ rule to specify the second mover’s beliefs): After observing that player 1 chose right, the second-mover’s beliefs of dealing with a high-type player 1 (upper node of his information set) are q ¼ pðHjRÞ ¼

1 1 pðHÞ  pðRjHÞ ¼1 2 1 ¼1 pðRÞ 2 1þ 2 0

Intuitively, this implies that the second mover, after observing that player 1 chose Right, assigns full probability to such a message originating from an H-type of first mover. In other words, only a H-type would be selecting Right in this separating strategy profile. Second step (focus on the second mover): After observing that player 1 chose Right, and given the second-mover’s beliefs specified above, the second mover is essentially convinced on being on the upper node of the information set. At this point, he obtains a higher payoff responding with U, which provides him a payoff of 2, than with D, which only yields a payoff

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Fig. 9.2 Separating strategy profile RL′

of 0. Hence, the second mover responds with U. To keep track of player 2’s optimal response in this strategy profile, Fig. 9.3 shades the branches corresponding to U.1 Third step (we now analyze the first mover): When the first mover is H-type (in the upper part of Fig. 9.3), he prefers to select R (as prescribed in this separating strategy profile) than deviate towards L, given that he anticipates that player 2 will respond with U. Indeed, player 1’s payoff from choosing R, 3, is larger than that from deviating to L′, 2. When the first mover is L-type (in the lower part of Fig. 9.3), he prefers to select L′ (as prescribed in this separating strategy profile) than deviate towards R′, since he can also anticipate that player 2 will respond with U. Indeed, player 1’s payoff from choosing L′, 2, is larger than that from his deviation towards R′, 1. Then, this separating strategy profile can be sustained as a PBE where (RL′, U) and beliefs are q = 1. Second type of separating equilibrium, LR′ Figure 9.4 depicts this separating strategy profile, in which the high-type of player 1 now chooses L while the low-type selects R′, as indicated by the thick shaded arrows of the figure.

1

Note that player 2 responds with U after observing that player 1 chooses Right, and that such response cannot be made conditional on player 1’s type (since player 2 does not observe his opponent’s type). In other words, player 2 responds with U both when player 1’s type is high and when it is low (for this reason, both arrows labeled with U are shaded in Fig. 9.3 in the upper and lower part of the game tree).

Exercise 1—Finding Separating and Pooling EquilibriaA

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Fig. 9.3 Optimal response by player 2 (U) in strategy profile RL′

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Fig. 9.4 Separating strategy profile LR′

First step (use Bayes’ rule to determine the second mover’s beliefs): After observing that the first mover chose Right, the second-mover’s beliefs of dealing with a high-type of player 1 are q¼

1 0 pðHÞ  pðRjHÞ ¼1 2 1 ¼0 pðRÞ  1 þ 2 20

Intuitively, this implies that the second mover, after observing that the first mover chose Right, assigns full probability to such a message originating from a low-type. Alternatively, a message of “Right” can never originate from an H-type in this separating strategy profile, i.e., q = 0.

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Second step (examine the second mover): After observing that the first mover chose Right, and given that the second-mover’s beliefs specify that he is convinced of being in the lower node of his information set, he responds with D, which provides him a payoff of 1, rather than with U, which provides him a payoff of 0 (conditional on being in the lower node); see the lower right-hand corner of Fig. 9.4 for a visual reference. To keep track of the optimal response of D from player 2, Fig. 9.5 shades the D branches in thick arrows. Third step (analyze the first mover): When the first mover is H-type (upper part of the game tree in Fig. 9.5), he prefers to select L (as prescribed in this separating strategy profile) since L yields him a payoff of 2 while deviating towards R would only provide a payoff of 1, as he anticipates player 2 will respond with D. When the first mover is L-type (lower part of the game tree), however, he prefers to deviate towards L′ than selecting R′ (the strategy prescribed for him in this separating strategy profile) since his payoff from R′ is only 1 whereas that of deviating towards L′ is 2. Then, this separating strategy profile cannot be sustained as a PBE, since one of the players (namely, the low type) has incentives to deviate. Part (b) Does this game have a pooling PBE? First type of pooling equilibrium: LL′ Figure 9.6 depicts this pooling strategy profile in which both types of player 1 choose Left, i.e., L for the high type in the upper part of the game tree and L′ for the low type in the lower part of the tree.

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Fig. 9.5 Optimal response of player 2 (D) in strategy profile LR′

Exercise 1—Finding Separating and Pooling EquilibriaA

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Fig. 9.6 Pooling strategy profile LL′

First step (use Bayes’ rule to determine the second mover’s beliefs): Upon observing that the first mover chose Right (which occurs off-the-equilibrium, as indicated in the figure), the second-mover’s beliefs are q¼

0:5  0 0 ¼ 0:5  0 þ 0:5  0 0

Thus, beliefs cannot be updated using Bayes’ rule, and must be left undefined in the entire range of probabilities, q 2 ½0; 1. Second step (examine the second mover): Let us analyze the second-mover’s optimal response. Note that the second mover is only called on to move if the first mover chooses Right, which occurs off-the-equilibrium path. In such an event, the second mover must compare his expected utility from responding with U versus that of selecting D, as follows EU2 ðUjRÞ ¼ 2  q þ 0  ð1  qÞ ¼ 2q EU2 ðDjRÞ ¼ 0  q þ 1  ð1  qÞ ¼ 1  q Hence, EU2 ðUjRÞ [ EU2 ðDjRÞ holds if 2q [ 1  q, or simply q [ 13 : Third step (analyze the first mover): Let us next divide our following analysis of the first mover (since we have already examined the second mover) into two cases depending on the optimal response of the second mover: Case 1: q [ 13, which implies that the second mover responds selecting U; and Case 2: q\ 13, leading the second mover to respond with D.

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CASE 1: q [ 13. Figure 9.7 depicts the optimal response of the second mover (U) in this case where q [ 13, where we shaded the arrows corresponding to U. When the first mover is H-type (in the upper part of the tree), he prefers to deviate towards R rather than selecting L (as prescribed in this strategy profile) since his payoff from deviating to R, 3, is larger than that from L, 2, given that he anticipates that the second mover will respond with U since q [ 13. This is already sufficient to conclude that the pooling strategy profile LL′ cannot be sustained as a PBE of the game. When off-the-equilibrium beliefs are relatively high, i.e., q [ 13 (As a practice, you can check that the L-type of first mover does not have incentives to deviate, since his payoff from L′, 2, is larger than that from R′, 1. However, since we have found a type of player with incentives to deviate, we conclude that the pooling strategy profile LL′ cannot be supported as a PBE). CASE 2: q\ 13. In this setting the second mover responds with D, as depicted in Fig. 9.8, where the arrows corresponding to D are thick and shaded in blue color. When the first mover is H-type, he prefers to select L (as prescribed in this strategy profile), which provides him with a payoff of 2, rather than deviating towards R, which only yields a payoff of 1, as indicated in the upper part of the figure (given that player 1 anticipates that player 2 will respond with D). Similarly, when the first mover is L-type, he prefers to select L′ (as prescribed in this strategy profile), which yields a payoff of 2, rather than deviating towards R′, which only yields a payoff of 1, as indicated in the lower part of the figure. Hence, the pooling strategy profile LL′ can be sustained as a PBE of the game when off-the-equilibrium beliefs are relatively low, i.e., q\ 13.

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Fig. 9.7 Optimal response of player 2 (U) in strategy profile LL′

1,1

Exercise 1—Finding Separating and Pooling EquilibriaA

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Fig. 9.8 Optimal response of player 2(D) in strategy profile LL′

Second type of pooling equilibrium: RR′ Figure 9.9 illustrates that pooling strategy profile in which both types of player 1 choose to move Right; as indicated by the thick arrows. First step (use Bayes’ rule to define he second mover’s beliefs): After observing that the first mover chose Right (which occurs in-equilibrium, as indicated in the shaded branches of the figure), the second-mover’s beliefs are q¼1 2

1 2

þ

1 2

¼

1 2

Therefore, the second-mover’s updated beliefs after observing Right coincide with the prior probability distribution over types. Intuitively, this indicates that, as both types of player 1 choose to play Right, then the fact of observing Right does not

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Fig. 9.9 Pooling strategy profile RR′

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provide player 2 more precise information about player 1’s type than the probability with which each type occurs in nature, i.e., 12. In this case, we formally say that the prior probability of types (due to nature) coincides with the posterior probability of types (after applying Bayes’ rule) or, more compactly, that priors and posteriors coincide. Second step (examine the second mover): Let us analyze the second-mover’s optimal response after observing that the first mover chose Right. The second mover must compare his expected utility from responding with U versus that of selecting D, as follows
 1 1 EU2 ðUjRÞ ¼ 2  þ 0  1  ¼1 2 2
 1 1 1 ¼ EU2 ðDjRÞ ¼ 0  þ 1  1  2 2 2 Hence EU2 ðUjRÞ [ EU2 ðDjRÞ since 1 > 12, and the second mover responds with U. Third step (analyze the first mover): Given that the first mover can anticipate that the second mover will respond with U (as analyzed in our previous step), we can shade this branch for player 2, as depicted in Fig. 9.10. When the first mover is H-type, he prefers to select R (as prescribed in this pooling strategy profile), which provides him a payoff of 3, rather than deviating towards L, which only yields a payoff of 2. However, when the first mover is L-type,

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Fig. 9.10 Optimal response for player 2 (U) in strategy profile RR′

1,1

Exercise 1—Finding Separating and Pooling EquilibriaA

391

he prefers to deviate towards L′, which yields a payoff of 2, rather than selecting R′ (as prescribed in this strategy profile), which only provides a payoff of 1.2 Hence, the pooling strategy profile in which both types of player 1 choose Right cannot be sustained as a PBE of the game since the L-type of first mover has incentives to deviate (from R′ to L′).

Exercise 2—Job-Market Signaling GameB Let us consider the sequential game with incomplete information depicted in Fig. 9.11. A worker privately observes whether he has a High productivity or a Low productivity, and then decides whether to acquire some education, such as a college degree, which he will subsequently be able to use as a signal about his innate productivity to potential employers. For simplicity, assume that education is not productivity enhancing. A firm that considers hiring the candidate, however, does not observe the real productivity of the worker, but only knows whether the worker decided to acquire a college education or not (the firm can observe whether the candidate has a valid degree). In this setting, the firm must respond hiring the worker as a manager (M) or as a cashier (C).3 Part (a) Does this game have an equilibrium using the strategy profile [NE’] Part (b) Does this game have an equilibrium using the opposite separating strategy profile [EN’] Part (c) Does this game have an equilibrium using the Pooling strategy profile [EE’] in which both types of workers acquire education. Part (d) Does this game have an equilibrium using the pooling strategy profile [NN’], in which neither type of worker acquires education. Answer: Part (a) Consider the separating strategy profile [NE′]

2 Importantly, the low-type of player 1 finds R′ to be strictly dominated by L′, since R′ yields a lower payoff than L′ regardless of player 2’s response. This is an important result, since it allows us to discard all strategy profiles in which R′ is involved, thus leaving us with only two potential candidates for PBE: the separating strategy profile RL′ and the pooling strategy profile LL′. Nonetheless, and as a practice, this exercise examined each of the four possible strategy profiles in this game. 3 This game presents a simplified version of Spence’s (1973) labor market signaling game, where the worker can be of only two types (either high or low productivity), and the firm manager can only respond with two possible actions (either hiring the candidates as a manager, thus offering him a high salary, or as a cashier, at a lower salary). A richer version of this model, with a continuum of education levels and a continuum of salaries, is presented in the next chapter.

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Fig. 9.11 The job-market signaling game

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Fig. 9.12 Separating strategy profile NE′

Figure 9.12 depicts the separating strategy profile in which only the low productivity worker chooses to acquire education (We know it’s a rather crazy strategy profile, but we have to check for all possible profiles!). 1. Responder’s beliefs: Firm’s updated beliefs about the worker’s type are p = 1 after observing No education, since such a message must only originate from a high-productivity worker in this strategy profile; and q = 0 after observing Education, given that such a message is only sent by low-productivity workers in this strategy profile. Graphically, p = 1 implies that the firm is convinced to be in the upper node after observing no education (in the left-hand side of the tree); whereas q = 0 entails that the firm believes to be in the lower node upon observing an educated worker (in the right-hand side of the tree). 2. Firms’ optimal response given their updated beliefs: After observing “No Education” the firm responds with M′ since, conditional on being in the upper left-hand node of Fig. 9.13, the firm’s profit from M′, 10,

Exercise 2—Job-Market Signaling GameB 10,10

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Fig. 9.13 Optimal responses of the firm in strategy profile NE′

exceeds that from C′, 4; while after observing “Education” the firm chooses C, receiving 4, instead of M, which yields a zero payoff. Figure 9.13 summarizes the optimal responses of the firm, M′ and C. 3. Given steps 1 and 2, the worker’s optimal actions are: When the worker is a high-productivity type, he does not deviate from “No Education” (behaving as prescribed) since his payoff from no education and be recognized as a high productivity worker by the firm (thus being hired as a manager), 10, exceeds that from acquiring education and be identified as a low-productivity worker (and thus be hired as a cashier), 0; as indicated in the upper part of Fig. 9.13. When the worker is of low type, he deviates from “Education”, which would only provide him a payoff of −3, to “No Education”, which gives him a larger payoff of 10; as indicated in the lower part of Fig. 9.13. Intuitively, by acquiring education, the low-productivity worker is only identifying himself as a 10,10

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Fig. 9.14 Separating strategy profile EN′

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Perfect Bayesian Equilibrium and Signaling Games

low-productivity candidate, which induces the firm to hire him as a cashier. By, instead, deviating towards No Education this worker “mimics” the high-productivity worker. Therefore, this separating strategy profile cannot be supported as a PBE, since we found that at least one type of worker had incentives to deviate. Part (b) Consider now the opposite separating strategy profile [EN′] Figure 9.14 depicts this (more natural!) separating strategy profile, whereby only the high productivity worker acquires education, as indicated in the thick arrows E and N′. 1. Responder’s beliefs: Firm’s beliefs about the worker’s type can be updated using Bayes’ rule, as follows: q ¼ pðHjEÞ ¼

1 pðHÞ  pðEjHÞ 2 1  ¼1 ¼1 1 pðEÞ 2 1þ 12  0

Intuitively, upon observing that the worker acquires education, the firm infers that the worker must be of high productivity, i.e., q = 1, since only this type of worker acquires education in this separating strategy profile; while No education conveys the opposite information, i.e., p = 0, thus implying that the worker is not of high productivity but instead of low productivity. Graphically, the firm restricts its attention to the upper right-hand corner, i.e., q = 1, and to the lower left-hand corner, i.e., p = 0. 2. Firms’ optimal response given the firm’s updated beliefs: After observing “Education” the firm responds with M. Graphically, the firm is convinced to be located in the upper right-hand corner of the game tree since q = 1. In this corner, the best response of the firm is M, which provides a payoff of 10, rather than C, which only yields a payoff of 4. After observing “No Education” the firm responds with C′. In this case the firm is convinced to be located in the lower left-hand corner of the game tree given that p = 0. In such a corner, the firm’s best response is C′, providing a payoff of 4, rather than M′, which yields a zero payoff. Figure 9.15 illustrates these optimal responses for the firm. 3. Given the previous steps 1 and 2, let us now find the worker’s optimal actions: When he is a high-productivity worker, he acquires education (as prescribed in this strategy profile) since his payoff from E (6) is higher than that from deviating to N (4); as indicated in the upper part of Fig. 9.15.

Exercise 2—Job-Market Signaling GameB 10,10

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Fig. 9.15 Optimal responses of the firm in strategy profile EN′

When he is a low-productivity worker, he does not deviate from “No Education”, i.e., N′, since his payoff from No Education, 4, is larger than that from deviating to Education, 3; as indicated in the lower part of the game tree. Intuitively, even if acquiring no education reveals his low productivity to the firm, and ultimately leads him to be hired as a cashier, his payoff is larger than what he would receive when acquiring education (even if such education helped him be hired as a manager). In short, the low-productivity worker finds too costly to acquire education. Then, the separating strategy profile [N′E, C′M] can be supported as a PBE of this signaling game, where firm’s beliefs are q = 1 and p = 0. Part (c) Pooling strategy profile [EE′] in which both types of workers acquire education. Figure 9.16 describes the pooling strategy profile in which all types of workers acquire education by graphically shading branches E and E′ of the tree.

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Fig. 9.16 Pooling strategy profile EE′

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1. Responder’s beliefs: Upon observing the equilibrium message of Education, the firm cannot further update its beliefs about the worker’s type. That is, its beliefs are q ¼ pðHjEÞ ¼

1 1 pðHÞ  pðEjHÞ 1 ¼1 2 1 ¼ pðEÞ 2 2  1þ 2  1

which coincide with the prior probability of the worker being of high productivity, i.e., q ¼ 12. Intuitively, since both types of workers are acquiring education, observing a worker with education does not help the firm to further restrict its beliefs. After observing the off-the-equilibrium message of No Education, the firm’s beliefs are p ¼ pðHjNEÞ ¼

1 0 pðHÞ  pðNEjHÞ 0 ¼1 2 1 ¼ pðNEÞ 0 2  0þ 2  0

and must then be left unrestricted, i.e., p 2 ½0; 1. 2. Firms’ optimal response given its updated beliefs: Given the previous beliefs, after observing “Education” (in equilibrium): if the firm responds hiring the worker as a manager (M), it obtains an expected payoff of 1 1 EUF ðMÞ ¼  10 þ  0 ¼ 5 2 2 if, instead, the firm hires him as a cashier (C), its expected payoff is only EUF ðCÞ ¼

1 1 4þ  4 ¼ 4 2 2

Thus inducing the firm to hire the worker as a Manager (M). After observing “No Education” (off-the-equilibrium): the firm obtains a expected payoffs of EUF ðM 0 Þ ¼ p  10 þ ð1  pÞ  0 ¼ 10p; and when it hires the worker as a manager (M′), and EUF ðC0 Þ ¼ p  4 þ ð1  pÞ  4 ¼ 4 when it hires him as a cashier (C′) (Note that, since firm’s beliefs upon observing the off-the-equilibrium message of No Education, p, had to be left unrestricted, we

Exercise 2—Job-Market Signaling GameB 10,10

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Fig. 9.17 Optimal responses to strategy profile EE′—Case 1

must express the above expected utilities as a function of p.). Hence, the firm prefers to hire him as a manager (M′) after observing no education if and only if 10p [ 4, or p [ 2=5: Otherwise, the firm hires the worker as a cashier (C′). 3. Given the previous steps 1 and 2, the worker’s optimal actions must be divided into two cases (one where the firm’s off-the-equilibrium beliefs satisfy p [ 2=5, thus implying that the firm responds hiring the worker as a manager when he does not acquire education, and another case in which p  2=5 entailing that the firm hires the worker as a cashier upon observing that he did not acquire education): Case 1: When p [ 2=5 the firm responds hiring him as a manager when he does not acquire education (M′), as a depicted in Fig. 9.17 (see left-hand side of the figure). In this setting, if the worker is a high-productivity type, he deviates from “Education,” where he only obtains a payoff of 6, to “No Education,” where his payoff increases to 10; as indicated in the upper part of the game tree. As a consequence, the pooling strategy profile in which both workers acquire

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Firm

C

0,4

M

3,0

(1-q)

(1-p)

4,4

6,10

q

p C’

M Firm

N’

Worker_L

E’

Fig. 9.18 Optimal responses in strategy profile EE′—Case 2

Firm C

-3,4

398

9

Perfect Bayesian Equilibrium and Signaling Games

education (EE′) cannot be sustained as a PBE when off-the-equilibrium beliefs satisfy p [ 2=5: Case 2: When p  2=5 the firm responds hiring the worker as a cashier (C′) after observing the off-the-equilibrium message of No education, as a depicted in the shaded branches on the left-hand side of Fig. 9.18. In this context, if the worker is a low-productivity type, he deviates from “Education,” where his payoff is only 3, to “No Education,” where his payoff increases to 4; as indicated in the lower part of the game tree. Therefore, the pooling strategy profile in which both types of workers acquire education (EE′) cannot be sustained when off-the-equilibrium beliefs satisfy p  2=5: In summary, this strategy profile cannot be sustained as a PBE for any off-the-equilibrium beliefs that the firm sustains about the worker’s type. Part (d) Let us finally consider the pooling strategy profile [NN′], in which neither type of worker acquires education. Figure 9.19 depicts the pooling strategy profile in which both types of worker choose No Education by shading branches N and N′ (see thick arrows). 1. Responder’s beliefs: Analogously to the previous pooling strategy profile, the firm’s equilibrium beliefs (after observing No Education) coincide with the prior probability of a high type, p ¼ 12; while its off-the-equilibrium beliefs (after observing Education) are left unrestricted, i.e., q 2 ½0; 1: 2. Firms’ optimal response given its updated beliefs: Given the previous beliefs, after observing “No Education” (in equilibrium): if the firm hires the worker as a manager (M′), it obtains an expected payoff of 1 1 EUF ðM 0 Þ ¼  10 þ  0 ¼ 5 2 2 10,10

M’ Firm

N

Worker_H

E

4,4

10,0

M’

H

1/2

L

1/2

4,4

Firm

C

0,4

M

3,0

(1-q)

(1-p)

C’

6,10

q

p C’

M Firm

N’

Fig. 9.19 Pooling strategy profile NN′

Worker_L

E’

Firm C

-3,4

Exercise 2—Job-Market Signaling GameB

399

and if it hires him as a cashier (C′) its expected payoff is only EUF ðC0 Þ ¼

1 1  4 þ  4 ¼ 4; 2 2

leading the firm to hire the worker as a manager (M′) after observing the equilibrium message of No Education. After observing “Education” (off-the-equilibrium): the expected payoff the firm obtains from hiring the worker as a manager (M) or cashier (C) are, respectively, EUF ðMÞ ¼ q  10 þ ð1  qÞ  0 ¼ 10q; and EUF ðCÞ ¼ q  4 þ ð1  qÞ  4 ¼ 4 Hence, the firm responds hiring the worker as a manager (M) after observing the off-the-equilibrium message of Education if and only if 10q [ 4, or q [ 25. Otherwise, the firm hires the worker as a cashier. 3. Given the previous steps 1 and 2, let us find the worker’s optimal actions. In this case, we will also need to split our analysis into two cases (one in which q [ 25 and thus the firm hires the worker as a manager upon observing the off-the-equilibrium message of Education, and the case in which q  25, in which the firm responds hiring him as a cashier): Case 1: When q [ 25, the firm responds hiring the worker as a manager (M) when he acquires education, as depicted in Fig. 9.20 (see thick shaded arrows on the right-hand side of the figure). If the worker is a high-productivity type, he plays “No Education” (as prescribed in this strategy profile) since his payoff from doing so, 10, exceeds that from deviating to E, 6; as indicated in the shaded branches of upper part of the game tree. Similarly, if he is a low-productivity type, he plays “No Education” (as

10,10

M’ Firm

N

Worker_H

E

4,4

10,0

M’

H

1/2

L

1/2

4,4

Firm

C

0,4

M

3,0

(1-q)

(1-p)

C’

6,10

q

p C’

M Firm

N’

Worker_L

E’

Fig. 9.20 Optimal responses in strategy profile NN′—Case 1

Firm C

-3,4

400

9

10,10

Perfect Bayesian Equilibrium and Signaling Games

M’ Firm

N

Worker_H

E

C’

10,0

M’

H

1/2

L

1/2

4,4

Firm

C

0,4

M

3,0

Firm

(1-q)

(1-p)

C’

6,10

q

p

4,4

M

N’

Worker_L

E’

Firm C

-3,4

Fig. 9.21 Optimal responses in strategy profile NN′—Case 2

prescribed) since his payoff from this strategy, 10, is higher than from deviating towards E′, 3; as indicated in the lower part of the game tree. Hence, the pooling strategy profile in which no worker acquires education, [NN′, M′M], can be supported as a PBE when off-the-equilibrium beliefs satisfy q [ 25. Case 2: When q  25 ; the firm responds hiring the worker as a cashier (C) upon observing the off-the-equilibrium message of Education, as depicted in the thick shaded branches at the right-hand side of Fig. 9.21. If the worker is a high-productivity type, he does not deviate from No Education since his payoff from N, 10, exceeds the zero payoff he would obtain by deviating to E; as indicated in the shaded branches at the upper part of the game tree. Similarly, if he is a low-productivity worker, he does not deviate from No Education since his (positive) payoff from N′ (10) is larger than the negative payoff he would obtain by deviating to E′, −3; as indicated in the lower part of the game tree. Therefore, the pooling strategy profile in with no type of worker acquires education, [NN′, M′C], can also be supported when off-the-equilibrium beliefs satisfy q  25.

Exercise 3—Job Market Signaling with Productivity-Enhancing EducationB Consider a variation of the Job-market signaling game (Exercise 2 of this chapter). Still assume two types of worker (high or low productivity), two messages he can send (education or no education), and two responses by the firm (hiring the worker as a manager or cashier). However, let us now allow for education to have a productivity-enhancing effect. In particular, when the worker acquires education (graphically represented in the right-hand side of the game tree), the firm’s payoff from hiring him as a manager is 10 þ a when the worker is of high productivity, and

Exercise 3—Job Market Signaling …

(10,10)

M’

Firm

N

401

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

Firm

H 1/2

L

M’

1/2

M

(1-p)

(4,4)

(6,10+α)

q

C’

Firm

M

(3,β)

(1-q)

N’

Worker L

E’

Firm

C’

C

(-3,4)

M

(6,10+α)

C

(0,4)

Fig. 9.22 Job-market signaling game with productivity-enhancing education

(10,10)

M’

Firm

N

Worker H

E

p

(4,4)

(10,0)

q

C’

H 1/2

L

M’

1/2

M

(1-p)

Firm (4,4)

Firm

(3,β)

(1-q)

N’

C’

Worker L

E’

Firm C

(-3,4)

Fig. 9.23 Separating strategy profile EN′

b when the worker is of low productivity. In words, parameter aðbÞ represents the productivity-enhancing effect of education for the high (low, respectively) productivity worker respectively, where we assume that a  b  0. For simplicity, we assume that education does not increase the firm’s payoff when the worker is hired as a cashier, providing a payoff of 4 regardless of the worker’s productivity. All other payoffs for both the firm and the worker are unaffected in Fig. 9.22 relative to Fig. 9.11. (a) Consider the separating strategy profile EN′. Under which conditions of parameters a and b can this strategy profile be supported as a PBE? Figure 9.23 depicts the separating strategy profile, in which only the high productivity worker acquires education, as indicated in the thick arrows E and N′.

402

9

Perfect Bayesian Equilibrium and Signaling Games

1. Responder’s beliefs: Firm’s beliefs about the worker’s type can be updated using Bayes’ rule, as follows: q ¼ pðHjEÞ ¼

1 pðHÞpðEjHÞ 2 1  ¼1 ¼1 pðHÞpðEjHÞ þ pðLÞpðEjLÞ 2  1 þ 1  12  0

Intuitively, upon observing that the worker acquires education, the firm infers that the worker must be of high productivity, i.e., q = 1, since only this type of worker acquires education in this separating strategy profile; while no education conveys the opposite information, i.e., p = 0, thus implying that the worker is not of high productivity but instead of low productivity. Graphically, the firm restricts its attention to the upper right-hand corner, i.e., q = 1, and to the lower left-hand corner, i.e., p = 0. 2. The firm’s optimal response given its updated beliefs: After observing “Education” the firm responds with M. Graphically, the firm is convinced to be located in the upper right-hand corner of the game tree since q = 1. In this corner, the best response of the firm is M, which provides a payoff of 10 + a, rather than C, which only yields a payoff of 4. After observing “No education” the firm responds with C′. In particular, in this case the firm is convinced to be located in the lower left-hand corner of the game tree given that p = 0. In such a corner, the firm’s best response is C′, providing a payoff of 4, rather than M′, which yields a zero payoff. Figure 9.24 illustrates these optimal responses for the firm. 3. Given the previous steps 1 and 2, let us now find the worker’s optimal actions:

(10,10)

M’

Firm

N

Worker H

E

(10,0)

C’

L

M’

1/2

(0,4)

M

(1-p)

(4,4)

C

Firm

H 1/2

Firm

(6,10+α)

q

p

(4,4)

M

(3,β)

(1-q)

N’

Worker L

E’

C’

Fig. 9.24 Optimal responses of the firm in strategy profile EN′

Firm C

(-3,4)

Exercise 3—Job Market Signaling …

403

When he is a high-productivity worker, he acquires education (as prescribed in this strategy profile) since his payoff from E (6) is higher than that from N (4); as indicated in the upper part of Fig. 9.24. When he is a low-productivity worker, he does not deviate from “No education”, i.e., N′, since his payoff from “No education”, 4, is larger than that from “Education”, 3; as indicated in the lower part of the game tree. Intuitively, even if acquiring no education reveals his low productivity to the firm, and ultimately leads him to be hired as a cashier, his payoff is larger than what he would receive when acquiring education (even if such education helped him to be hired as a manager). In short, the low-productivity worker finds it too costly to acquire education. Then, the separating strategy profile [N′E, C′M] can be supported as a PBE of this signaling game, where firm’s beliefs are q = 1 and p = 0 for all values of a  b  0. (b) Consider the separating strategy profile E′N. Under which conditions of parameters a and b can this strategy profile be supported as a PBE? Figure 9.25 depicts the separating strategy profile in which only the low productivity worker chooses to acquire education. 1. Responder’s beliefs: Firm’s beliefs about the worker’s type can be updated using Bayes’ rule, as follows: p ¼ pðHjNÞ ¼

(10,10)

M’

1 pðHÞpðNjHÞ 2 1  ¼1 ¼1 pðHÞpðNjHÞ þ pðLÞpðNjLÞ 2  1 þ 1  12  0

Firm

N

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

Firm

H 1/2

L

M’

1/2

M

(1-p)

(4,4)

(6,10+α)

q

C’

Firm

M

(3,β)

(1-q)

N’

C’

Fig. 9.25 Separating strategy profile NE′

Worker L

E’

Firm C

(-3,4)

404

9

Perfect Bayesian Equilibrium and Signaling Games

Firm’s updated beliefs about the worker’s type are p = 1 after observing “No education”, since such a message must only originate from a high-productivity worker in this strategy profile; and q = 0 after observing “Education”, given that such a message is only sent by low-productivity workers in this strategy profile. Graphically, p = 1 implies that the firm is convinced to be in the upper node after observing no education (in the left-hand side of the tree); whereas q = 0 entails that the firm believes to be in the lower node upon observing an educated worker (in the right side of the tree). In the following paragraphs, we first assume that b < 4. 2. Firm’s optimal response given update beliefs: After observing “No education” the firm responds with M′ since, conditional on being in the upper left-hand node of Fig. 9.25, the firm’s profit from M′, 10, exceeds that from C′, 4; while after observing “Education” the firm chooses C, receiving 4, instead of b. Figure 9.26 summarizes the optimal responses of the firm, M′ and C when b\4, which implies that the firm responds with C after observing “Education”. Intuitively, the productivity-enhancing effect of education for the low-productivity worker is low ðb\4Þ, thus not compensating the firm to hire the worker as a manger. 3. Given steps 1 and 2, the worker’s optimal actions are: When the worker is a high-productivity type, he does not deviate from “No education” (behaving as prescribed) since his payoff from no education and be recognized as a high productivity worker by the firm (thus being hired as a manager), 10, exceeds that from acquiring education and be identified as a low-productivity worker (and thus be hired as a cashier), 0; as indicated in the upper part of Fig. 9.26. When the worker is of low type, he deviates from “Education”, which would only provides him a payoff of −3, to “No education”, which gives him a larger payoff of 10; as indicated in the lower part of Fig. 9.26. Intuitively, by acquiring education, the low-productivity worker is only identifying himself as a (10,10)

M’

Firm

N

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

Firm

H 1/2

L

M’

1/2

M

(1-p)

(4,4)

(6,10+α)

q

C’

Firm

M

(3,β)

(1-q)

N’

Worker L

E’

Firm

C’

Fig. 9.26 Optimal responses of the firm in strategy profile NE′ (when b\4)

C

(-3,4)

Exercise 3—Job Market Signaling …

(10,10)

M’

Firm

N

405

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

Firm

H 1/2

L

M’

1/2

M

(1-p)

(4,4)

(6,10+α)

q

C’

Firm

M

(3,β)

(1-q)

N’

Worker L

E’

Firm

C’

C

(-3,4)

Fig. 9.27 Optimal responses of the firm in strategy profile NE′ (when b  4)

low-productivity candidate, which induces the firm to hire him as a cashier. But, instead, by deviating towards “No education”, this worker “mimics” the high-productivity worker. Therefore, this separating strategy profile cannot be supported as a PBE for b\4, since we found that at least one type of worker had incentives to deviate. In the following paragraphs, we assume that the productivity-enhancing effect of education for the low-productivity worker is strong, that is, b  4. 2. Firm’s optimal response given update beliefs: After observing “No education” the firm responds with M′ since, conditional on being in the upper left-hand node of Fig. 9.25, the firm’s profit from M′, 10, exceeds that from C′, 4; while after observing “Education” the firm also chooses M, receiving b instead of 4. Intuitively, the productivity-enhancing effect of education for the low-productivity worker is strong ðb  4Þ, thus compensating the firm to hire the worker as a manager. Figure 9.27 summarizes the optimal responses of the firm, M′ and M. 3. Given steps 1 and 2, the worker’s optimal actions are: When the worker is a high-productivity type, he does not deviate from “No education” (behaving as prescribed) since his payoff from no education and be recognized as a high productivity worker by the firm (thus being hired as a manager), 10, exceeds that from acquiring education and still be hired as a manager with a payoff of 6 (from incurring an education cost of 4); as indicated in the upper part of Fig. 9.27. When the worker is of low type, he deviates from “Education”, which would only provide him a payoff of 3, to “No education”, which gives him a larger payoff of 10; as indicated in the lower part of Fig. 9.27. Intuitively, by acquiring education, the low-productivity worker incurs an education cost of 7 and is hired as a manager. But, instead, by deviating towards “No education”, this worker can obtain a higher payoff of 10 and still be hired as a manager.

406

9

(10,10)

M’

Firm

N

Perfect Bayesian Equilibrium and Signaling Games

Worker H

E

p

(4,4)

(10,0)

(6,10+α)

C

(0,4)

Firm q

C’

H 1/2

L

M’

1/2

M

(1-p)

(3,β)

(1-q)

N’

Firm (4,4)

M

C’

Worker L

E’

Firm C

(-3,4)

Fig. 9.28 Pooling strategy profile EE′

Therefore, this separating strategy profile cannot be supported as a PBE for b  4, since we found that at least one type of worker had incentives to deviate. In summary, the separating strategy profile NE′ cannot be supported as a PBE for any values of b. (c) Consider the pooling strategy profile EE′. Under which conditions of parameters a and b can this strategy profile be supported as a PBE? Figure 9.28 describes the pooling strategy profile in which all types of workers acquire education by graphically shading branches E and E′ of the tree. 1. Responder’s beliefs: Upon observing the equilibrium message of “Education”, the firm cannot further update its beliefs about the worker’s type. That is, its beliefs are q ¼ pðHjEÞ ¼

1 1 pðHÞpðEjHÞ 1 ¼1 2 1 ¼ pðHÞpðEjHÞ þ pðLÞpðEjLÞ 2  1 þ 2  1 2

which coincide with the prior probability of the worker being of high productivity, i.e., q ¼ 12. Intuitively, since both types of workers are acquiring education, observing a worker with education does not help the firm to further restrict its beliefs. After observing the off-the-equilibrium message of “No education”, the firm’s beliefs are p ¼ pðHjNEÞ ¼

1 0 pðHÞpðNEjHÞ 0 ¼1 2 1 ¼ pðHÞpðNEjHÞ þ pðLÞpðNEjLÞ 2  0 þ 2  0 0

Exercise 3—Job Market Signaling …

407

and must then be left unrestricted, i.e., p 2 ½0; 1. 2. Firms’ optimal response given its updated beliefs: Given the previous beliefs, after observing “Education” (in equilibrium): if the firm responds by hiring the worker as a manager (M), it obtains an expected payoff of EUF ðMÞ ¼

1 1 aþb  ð10 þ aÞ þ  b ¼ 5 þ 2 2 2

if, instead, the firm hires him as a cashier (C), its expected payoff is only EUF ðCÞ ¼

1 1 4þ  4 ¼ 4 2 2

which induces the firm to hire the worker as a Manager (M). After observing “No education” (off-the-equilibrium): the firm obtains expected payoffs of EUF ðM 0 Þ ¼ p  10 þ ð1  pÞ  0 ¼ 10p; and when it hires the worker as a manager (M′), EUF ðC0 Þ ¼ p  4 þ ð1  pÞ  4 ¼ 4 when it hires him as a cashier (C′) (Note that, since firm’s beliefs upon observing the off-the-equilibrium message of no education, p, had to be left unrestricted, we must express the above expected utilities as a function of p.). Hence, the firm prefers to hire him as a manager (M′) after observing no education if and only if 10p [ 4, or p [ 25, Otherwise, the firm hires the worker as a cashier (C′). 3. Given the previous steps 1 and 2, the worker’s optimal actions must be divided into two cases (one where the firm’s off-the-equilibrium beliefs satisfy p [ 25, thus implying that the firm responds hiring the worker as a manager when he does not acquire education, and another case in which p  25, entailing that the firm hires the worker as a cashier upon observing that he did not acquire education): Case 1: When p [ 25, the firm responds by hiring him as a manager when he does not acquire education (M′), as a depicted in Fig. 9.29 (left-hand side). In this setting, if the worker is a high-productivity type, he deviates from “Education,” where he only obtains a payoff of 6, to “No education,” where his payoff increases to 10; as indicated in the upper part of the game tree. As a consequence, the pooling strategy profile in which both workers acquire education (EE′) cannot be sustained as a PBE when off-the-equilibrium beliefs satisfy p [ 25.

408

9

(10,10)

M’

Firm

N

Perfect Bayesian Equilibrium and Signaling Games

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

Firm

H 1/2

L

M’

1/2

M

(1-p)

(4,4)

(6,10+α)

q

C’

Firm

M

(3,β)

(1-q)

N’

Worker L

E’

Firm

C’

C

(-3,4)

Fig. 9.29 Optimal responses to strategy profile EE′—Case 1

Case 2: When p  25 the firm responds by hiring the worker as a cashier (C′) after observing the off-the-equilibrium message of “No education”, as a depicted in the shaded branches on the left-hand side of Fig. 9.30. In this context, if the worker is a low-productivity type, he deviates from “Education,” where his payoff is only 3, to “No education,” where his payoff increases to 4; as indicated in the lower part of the game tree. Therefore, the pooling strategy profile in which both types of workers acquire education (EE′) cannot be sustained when off-the-equilibrium beliefs satisfy p  25. In summary, this strategy profile cannot be sustained as a PBE for all values of a  b  0 and for any off-the-equilibrium beliefs that the firm sustains about the worker’s type. (d) Consider the pooling strategy profile NN′. Under which conditions of parameters a and b can this strategy profile be supported as a PBE?

(10,10)

M’

Firm

N

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

Firm

H 1/2

L

M’

1/2

M

(1-p)

(4,4)

(6,10+α)

q

C’

Firm

M

(3,β)

(1-q)

N’

Worker L

E’

C’

Fig. 9.30 Optimal responses to strategy profile EE′—Case 2

Firm C

(-3,4)

Exercise 3—Job Market Signaling …

(10,10)

M’

Firm

N

409

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

Firm

H 1/2

L

M’

1/2

M

(1-p)

(4,4)

(6,10+α)

q

C’

Firm

M

(3,β)

(1-q)

N’

Worker L

E’

C’

Firm C

(-3,4)

Fig. 9.31 Pooling strategy profile NN′

Figure 9.31 depicts the pooling strategy profile in which both types of worker choose “No education” by shading branches N and N′ (see thick arrows). 1. Responder’s beliefs: Analogously as the previous pooling strategy profile, the firm’s equilibrium beliefs (after observing “No education”) coincide with the prior probability of a high type, p ¼ 12; while its off-the-equilibrium beliefs (after observing “Education”) are left unrestricted, i.e., q 2 ½0; 1. 2. Firms’ optimal response given its updated beliefs: Given the previous beliefs, after observing “No education” (in equilibrium): if the firm hires the worker as a manager (M′), it obtains an expected payoff of 1 1 EUF ðM 0 Þ ¼  10 þ  0 ¼ 5 2 2 and if it hires him as a cashier (C′), its expected payoff only becomes EUF ðC0 Þ ¼

1 1  4 þ  4 ¼ 4; 2 2

leading the firm to hire the worker as a manager (M′) after observing the equilibrium message of “No education”. After observing “Education” (off-the-equilibrium): the expected payoff the firm obtains from hiring the worker as a manager (M) or cashier (C) are, respectively, EUF ðMÞ ¼ q  ð10 þ aÞ þ ð1  qÞ  b ¼ ð10 þ a  bÞq þ b EUF ðCÞ ¼ q  4 þ ð1  qÞ  4 ¼ 4

410

9

(10,10)

M’

Firm

N

Perfect Bayesian Equilibrium and Signaling Games

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

Firm

H 1/2

L

M’

1/2

M

(1-p)

(4,4)

(6,10+α)

q

C’

Firm

M

(3,β)

(1-q)

N’

Worker L

E’

C’

Firm C

(-3,4)

Fig. 9.32 Optimal responses in strategy profile NN′—Case 1

Hence, the firm responds hiring the worker as a manager (M) after observing the off-the-equilibrium message of Education if and only if ð10 þ a  bÞq þ b [ 4, or q [ 10 4b þ ab. Otherwise, the firm hires the worker as a cashier (C). 3. Given the previous steps 1 and 2, let us find the worker’s optimal actions. In this case, we will also need to split our analysis into two cases (one in which q  10 4b þ ab and thus the firm respond by hiring him as a cashier upon observing the off-the-equilibrium message of “Education”, and the case in which q  10 4b þ ab, in which the firm hires the worker as a manager): Case 1: When q  10 4b þ ab, the firm responds by hiring the worker as a cashier (C) upon observing the off-the-equilibrium message of “Education”, as depicted in the right-hand shaded branches of Fig. 9.32. If the worker is a high-productivity type, he does not deviate from “No education” since his payoff from N, 10, exceeds the zero payoff he would obtain by deviating to E; as indicated in the shaded branches upper part of the game tree. Similarly, if he is a low-productivity worker, he does not deviate from “No education” since his (positive) payoff from N′ (10) is larger than the negative payoff he would obtain by deviating to E′, −3; as indicated in the lower part of the game tree. Therefore, the pooling strategy profile in with no type of worker acquires education, [NN′, M′C], can be supported when off-the-equilibrium beliefs satisfy q  10 4b þ ab for all values of a  b  0. Case 2: When q [ 10 4b þ ab the firm responds by hiring the worker as a manager (M) when he acquires education, as depicted in Fig. 9.33. If the worker is a high-productivity type, he plays “No education” (as prescribed in this strategy profile) since his payoff from doing so, 10, exceeds that from deviating to E, 6; as indicated in the shaded branches of upper part of the game tree. Similarly, if he is a low-productivity type, he plays “No education” (as

Exercise 3—Job Market Signaling …

(10,10)

M’

Firm

N

411

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

Firm

H 1/2

L

M’

1/2

M

(1-p)

(4,4)

(6,10+α)

q

C’

Firm

M

(3,β)

(1-q)

N’

Worker L

E’

C’

Firm C

(-3,4)

Fig. 9.33 Optimal responses in strategy profile NN′—Case 2

prescribed) since his payoff from this strategy, 10, is higher than from deviating towards E′, 3; as indicated in the lower part of the game tree. Hence, the pooling strategy profile in which no worker acquires education, [NN′, M′M], can be supported as a PBE when off-the-equilibrium beliefs satisfy q [ 10 4b þ ab, for all values of a  b  0 (that is, when b [ 4, all off-the-equilibrium beliefs q 2 ½0; 1 support the poling strategy profile NN′ as a PBE). (e) How are your results affected by an increase in parameter a? And an increase in parameter b? Summarizing, the following strategy profiles can be sustained as PBEs: – [N′E, C′M], with q = 1 and p = 0. 1 – [NN′, M′C] with q  q ¼ 10 4b þ ab and p ¼ 2 1 – [NN′, M′M] with q [ q ¼ 10 4b þ ab and p ¼ 2

Therefore, increasing parameter a reduces cutoff probability  q, that is, @q 4b ¼ \0: @a ð10 þ a  bÞ2 Intuitively, as education improves the productivity of the high-productivity worker, the expected profit from hiring an educated worker as a manager increases, leading the firm to become more willing to respond hiring an educated worker as a manager. A similar argument applies to an increase in parameter b since

412

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Perfect Bayesian Equilibrium and Signaling Games

@q 6þa ¼ \0: @b ð10 þ a  bÞ2 This result indicates that, as education improves the productivity of the low-productivity worker, the expected profit from hiring an educated worker as a manager increases, and the firm responds hiring him in this position for a larger range of priors (i.e., a larger range of probability q). Finally, note that when both parameters are zero, a ¼ b ¼ 0, probability cutoff  q 2 becomes q ¼ 10 40 ¼ , and we return to the same payoffs as in Exercise 2 of this þ 00 5 chapter, and the same set of PBEs.

Exercise 4—Job Market Signaling with Utility-Enhancing EducationB Consider a variation of the labor market signaling game (Exercise 2 of this chapter). Still assume two types of worker (high or low productivity), two messages he can send (education or no education), and two responses by the firm (hiring the worker as a manager or cashier). However, let us now allow for education to have a utility-enhancing effect. In particular, when the worker acquires education (graphically represented in the right-hand side of the game tree), the worker’s payoff from being hired as a manager is 6 þ uH when the worker is of high productivity, and 3 þ uL when the worker is of low productivity. In words, parameter uH ; uL  0 represents the utility-enhancing effect of education for the high (low) productivity worker respectively. For simplicity, we assume that education does not increase the firm’s payoff, that is, productivity-enhancing effects are absent. All other payoffs for both the firm and the worker are unaffected in Fig. 9.34 relative to Fig. 9.11. (a) Consider the separating strategy profile EN′. Under which conditions of parameters uH and uL can this strategy profile be supported as a PBE? • Fig. 9.35 depicts the separating strategy profile, in which only the high productivity worker acquires education, as indicated in the thick arrows E and N′. 1. Responder’s beliefs: • Firm’s beliefs about the worker’s type can be updated using Bayes’ rule, as follows: qðHjEÞ ¼

1 pðHÞpðEjHÞ 2 1  ¼1 ¼1 pðHÞpðEjHÞ þ pðLÞpðEjLÞ 2  1 þ 1  12  0

Exercise 4—Job Market Signaling with Utility-Enhancing EducationB

(10,10)

M’

Firm

N

Worker H

(10,0)

H 1/2

L

M’

1/2

(1-p)

(4,4)

(6 + uH,10)

C

(0,4)

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(-3,4)

q

C’

Firm

M Firm

E

p

(4,4)

413

(1-q)

N’

Worker L

E’

Firm

C’

Fig. 9.34 Job-market signaling game with utility-enhancing education

• Intuitively, upon observing that the worker acquires education, the firm infers that the worker must be of high productivity, i.e., q = 1, since only this type of worker acquires education in this separating strategy profile; while “No education” conveys the opposite information, i.e., p = 0, thus implying that the worker is not of high but instead of low productivity. Graphically, the firm restricts its attention to the upper right-hand corner, i.e., q = 1, and to the lower left-hand corner, i.e., p = 0. 2. The firm’s optimal response given its updated beliefs: • After observing “Education” the firm responds with M. Graphically, the firm is convinced to be located in the upper right-hand corner of the game tree since q = 1. In this corner, the best response of the firm is M, which provides a payoff of 10, rather than C, which only yields a payoff of 4. (10,10)

M’

Firm

N

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

M

(3 + uL,0)

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(-3,4)

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H 1/2

L

M’

1/2

(1-p)

(4,4)

(6 + uH,10)

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C’

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M

(1-q)

N’

C’

Fig. 9.35 Separating strategy profile EN′

Worker L

E’

Firm

414

9

(10,10)

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Perfect Bayesian Equilibrium and Signaling Games

M N

Worker H

E

p

(4,4)

(10,0)

C

H 1/2

L

M’

1/2

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(1-p)

(4,4)

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(6 + uH,10)

(0,4)

(3 + uL,0)

(1-q)

N’

Worker L

E’

C’

Firm C

(-3,4)

Fig. 9.36 Optimal responses of the firm in strategy profile EN′

• After observing “No education” the firm responds with C′. In particular, in this case the firm is convinced to be located in the lower left-hand corner of the game tree given that p = 0. In such a corner, the firm’s best response is C′, providing a payoff of 4, rather than M′, which yields a zero payoff. Figure 9.36 illustrates these optimal responses for the firm. 3. Given the previous steps 1 and 2, let us now find the worker’s optimal actions: • When he is a high-productivity worker, he acquires education (as prescribed in this strategy profile) since his payoff from E ð6 þ uH Þ is higher than that from N (4); as indicated in the upper part of Fig. 9.36. • When uL \1, the low-productivity worker does not deviate from “No education”, i.e., N′, since his payoff from “No education”, 4, is larger than that from “Education”, 3 þ uL ; as indicated in the lower part of the game tree. Intuitively, even if acquiring education conceals his low productivity from the firm, and ultimately leads him to be hired as a manager, his payoff is lower than what he would receive when not acquiring education. In short, the low-productivity worker finds too costly to acquire education. Then, the separating strategy profile [N′E, C′M] can be supported as a PBE, where firm’s beliefs are q = 1 and p = 0, and for uH  0 and uL \1. • However, when uL  1, the low-productivity worker would find education worth pursuing, because the productivity-enhancing effect is so strong that being hired as a manager would yield a payoff of 3 þ uL  4 that is higher than that of taking no education and being hired as a cashier. Then, the low-productivity worker has an incentive to deviate from no education to education, such that the separating strategy profile [EN′, MC′] cannot be supported as a PBE of this signaling game, where firm’s beliefs are q = 1 and p = 0, and for uH  0 and uL  1.

Exercise 4—Job Market Signaling with Utility-Enhancing EducationB

(10,10)

M’

Firm

N

Worker H

E

p

(4,4)

(10,0)

M

(6 + uH,10)

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(0,4)

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L

M’

1/2

(1-p)

(1-q)

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Firm (4,4)

415

Worker L

E’

Firm

C’

Fig. 9.37 Separating strategy profile NE′

(b) Consider the separating strategy profile NE′. Under which conditions of parameters uH and uL can this strategy profile be supported as a PBE? • Fig. 9.37 depicts the separating strategy profile in which only the low productivity worker chooses to acquire education. 1. Responder’s beliefs: • Firm’s beliefs about the worker’s type can be updated using Bayes’ rule, as follows: p ¼ pðHjNÞ ¼

(10,10)

M’

Firm

1 pðNjHÞpðHÞ 2 1  ¼1 ¼1 pðNjHÞpðHÞ þ pðNjLÞpðLÞ 2  1 þ 1  12  0

N

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

M

(3 + uL,0)

C

(-3,4)

Firm

H 1/2

L

M’

1/2

(1-p)

(4,4)

(6 + uH,10)

q

C’

Firm

M

(1-q)

N’

Worker L

E’

C’

Fig. 9.38 Optimal responses of the firm in strategy profile NE′

Firm

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Perfect Bayesian Equilibrium and Signaling Games

• Firm’s updated beliefs about the worker’s type are p = 1 after observing “No education”, since such a message must only originate from a high-productivity worker in this strategy profile; and q = 0 after observing “Education”, given that such a message is only sent by low-productivity workers in this strategy profile. Graphically, p = 1 implies that the firm is convinced to be in the upper node after observing no education (in the left-hand side of the tree); whereas q = 0 entails that the firm believes to be in the lower node upon observing an educated worker (in the right side of the tree). 2. Firm’s optimal response given update beliefs: • After observing “No education” the firm responds with M′ since, conditional on being in the upper left-hand node of Fig. 9.37, the firm’s profit from M′, 10, exceeds that from C′, 4; while after observing “Education” the firm chooses C, receiving 4 instead of 0. Figure 9.38 summarizes the optimal responses of the firm, M′ and C. 3. Given steps 1 and 2, the worker’s optimal actions are: • When the worker is a high-productivity type, he does not deviate from “No education” (behaving as prescribed) since his payoff from no education and be recognized as a high productivity worker by the firm (thus being hired as a manager), 10, exceeds that from acquiring education and be identified as a low-productivity worker (and thus be hired as a cashier), 0; as indicated in the upper part of Fig. 9.38. Intuitively, he is in the best of the worlds: avoids having to invest in education, but he reveals his type, and is hired as a manager. • When the worker is of low type, he deviates from “Education”, which would only provide him a payoff of −3, to “No education”, which gives him a larger payoff of 10; as indicated in the lower part of Fig. 9.38. Intuitively, by acquiring education, the low-productivity worker is only identifying himself as a low-productivity candidate, which induces the firm to hire him as a cashier. But, instead, by deviating towards “No education”, this worker “mimics” the high-productivity worker. • Therefore, this separating strategy profile cannot be supported as a PBE of this signaling game, where firm’s beliefs are q = 0 and p = 1, and for uH ; uL  0, since we found that at least one type of worker had incentives to deviate. (c) Consider the pooling strategy profile EE′. Under which conditions of parameters uH and uL can this strategy profile be supported as a PBE? • Fig. 9.39 describes the pooling strategy profile in which all types of workers acquire education by graphically shading branches E and E′ of the tree.

Exercise 4—Job Market Signaling with Utility-Enhancing EducationB

(10,10)

M’

Firm

N

Worker H

E

p

(4,4)

(10,0)

M

(6 + uH,10)

C

(0,4)

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L

M’

1/2

(1-p)

(1-q)

N’

Firm (4,4)

417

Worker L

E’

C’

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Fig. 9.39 Pooling strategy profile EE′

1. Responder’s beliefs: • Upon observing the equilibrium message of “Education”, the firm cannot further update its beliefs about the worker’s type. That is, its beliefs are q ¼ pðHjEÞ ¼

1 1 pðEjHÞpðHÞ 1 ¼1 2 1 ¼ pðEjHÞpðHÞ þ pðEjLÞpðLÞ 2  1 þ 2  1 2

which coincide with the prior probability of the worker being of high productivity, i.e., q ¼ 12. Intuitively, since both types of workers are acquiring education, observing a worker with education does not help the firm to further restrict its beliefs. • After observing the off-the-equilibrium message of “No education”, the firm’s beliefs are p ¼ pðHjNEÞ ¼

1 0 pðNEjHÞpðHÞ 0 ¼1 2 1 ¼ pðNEjHÞpðHÞ þ pðNEjLÞpðLÞ 2  0 þ 2  0 0

and must then be left unrestricted, i.e., p 2 ½0; 1. 2. Firms’ optimal response given its updated beliefs: • Given the previous beliefs, after observing “Education” (in equilibrium): if the firm responds by hiring the worker as a manager (M), it obtains an expected payoff of

418

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Perfect Bayesian Equilibrium and Signaling Games

1 1 EUF ðMÞ ¼  10 þ  0 ¼ 5 2 2 if, instead, the firm hires him as a cashier (C), its expected payoff is only EUF ðCÞ ¼

1 1 4þ  4 ¼ 4 2 2

Thus inducing the firm to hire the worker as a Manager (M). • After observing “No education” (off-the-equilibrium): the firm obtains expected payoffs of EUF ðM 0 Þ ¼ p  10 þ ð1  pÞ  0 ¼ 10p; and when it hires the worker as a manager (M′), EUF ðC0 Þ ¼ p  4 þ ð1  pÞ  4 ¼ 4 when it hires him as a cashier (C′) (Note that, since firm’s beliefs upon observing the off-the-equilibrium message of No education, p, had to be left unrestricted, we must express the above expected utilities as a function of p.). Hence, the firm prefers to hire him as a manager (M′) after observing no education if and only if 10p  4, or p  25. Otherwise, the firm hires the worker as a cashier (C′). 3. Given the previous steps 1 and 2, the worker’s optimal actions must be divided into two cases (one where the firm’s off-the-equilibrium beliefs satisfy p  25, thus implying that the firm responds hiring the worker as a manager when he does not acquire education, and another case in which p\ 25 entailing that the firm hires the worker as a cashier upon observing that he did not acquire education): (10,10)

M’

Firm

M N

Worker H

E

p

(4,4)

(10,0)

C

H 1/2

L

M’

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M

(1-p)

(4,4)

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C’

Firm

(6 + uH,10)

(0,4)

(3 + uL,0)

(1-q)

N’

Worker L

E’

C’

Fig. 9.40 Optimal responses to strategy profile EE′—Case 1

Firm C

(-3,4)

Exercise 4—Job Market Signaling with Utility-Enhancing EducationB

419

Case 1: When p  25 the firm responds by hiring him as a manager when he does not acquire education (M′), as a depicted in Fig. 9.40 (left-hand side). • When uH \4, the high-productivity type worker deviates from “Education”, where he only obtains a payoff of 6 þ uH , to “No education”, where his payoff increases to 10; as indicated in the upper part of the game tree. As a consequence, the pooling strategy profile in which both workers acquire education (EE′) cannot be sustained as a PBE when off-the-equilibrium beliefs satisfy p  25, and for uH \4 and uL  0. However, when uH  4, the high-productivity type worker does not deviate from “Education” to “No education” since doing so yields him a lower payoff. However, when uL \7, the low-productivity type would deviate from “Education” to “No education”, since this yields him a higher payoff of 10 than 3 þ uL . In other words, the pooling strategy profile in which both workers acquire education (EE′) can be sustained as a PBE when off-the-equilibrium beliefs satisfy p  25 for uH  4 and uL  7. Case 2: When p\ 25 the firm responds by hiring the worker as a cashier (C′) after observing the off-the-equilibrium message of “No education”, as a depicted in the shaded branches on the left-hand side of Fig. 9.41. • When uL \1, then the low-productivity type worker deviates from “Education,” where his payoff is 3 þ uL , to “No education,” where his payoff increases to 4; as indicated in the lower part of the game tree. Therefore, the pooling strategy profile in which both types of workers acquire education (EE′) cannot be sustained when off-the-equilibrium beliefs satisfy p\ 25, and for of uH  0 and uL \1. Whereas, when uL  1, the low-productivity type would not deviate from

(10,10)

M’

Firm

N

Worker H

E

p

(4,4)

(10,0)

C

(0,4)

M

(3 + uL,0)

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(-3,4)

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H 1/2

L

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(1-p)

(4,4)

(6 + uH,10)

q

C’

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(1-q)

N’

Worker L

E’

C’

Fig. 9.41 Optimal responses to strategy profile EE′—Case 2

Firm

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Perfect Bayesian Equilibrium and Signaling Games

“Education” to “No education”, since this yields him a lower payoff of 4 than 3 þ uL . In other words, the pooling strategy profile that both workers acquire education (EE′) can be sustained as a PBE, where off-the-equilibrium beliefs satisfy p\ 25, and for uH  0 and uL  1. (d) Apply Cho and Kreps’ (1987) Intuitive Criterion to the pooling strategy profile EE′. Under which conditions of parameters uH and uL will this strategy profile violate Cho and Kreps’ Intuitive Criterion? In the PBE, (EE′, MC′), where the firm’s off-the-equilibrium beliefs satisfy 0  p\ 25 with uH  0 and uL  1, we consider the following four cases regarding the parameter values. • First case. 0  uH \4 and 1  uL \7 First Step: Check the maximum payoff that each type of worker can obtain by deviation. maxpH ðNÞ ¼ 10 [ 6 þ uH ¼ pH ðEÞ

maxpL ðN 0 Þ ¼ 10 [ 3 þ uL ¼ pL ðE0 Þ

Therefore, both high and low productivity type workers have incentives to deviate towards no education, entailing that H ðNÞ ¼ fH; Lg, such that the firm is unable to update its off-the-equilibrium beliefs at the left-hand side of the game tree by Bayes’ rule. Second Step: Given off-the-equilibrium beliefs unchanged at 0  p\ 25, the firm continues to hire any uneducated worker as a cashier, such that min

pH ðaÞ ¼ 4k6 þ uH ¼ pH ðEÞ

min

pL ðaÞ ¼ 4k3 þ uL ¼ pL ðE0 Þ

a2A ðH ðNÞ;NÞ a2A ðH ðNÞ;NÞ

where A ðÞ ¼ fM 0 ; C 0 g represents the action space of the firm given its updated beliefs H ðNÞ of observing an off-the-equilibrium message N, and the action that minimizes the payoff of the workers is C 0 that gives each type of worker a payoff of 4 instead of 10. Therefore, no types of worker could benefit by unilateral deviating to no education. As a result, the PBE (EE′, MC′), where 0  p\ 25, 0  uH \4, and 1  uL \7, survives Cho and Kreps’ Intuitive Criterion.

Exercise 4—Job Market Signaling with Utility-Enhancing EducationB

421

• Second case. 0  uH \4 and uL  7 First Step: Check the maximum payoff that each type of worker can obtain by deviation. maxpH ðNÞ ¼ 10 [ 6 þ uH ¼ pH ðEÞ

maxpL ðN 0 Þ ¼ 10  3 þ uL ¼ pL ðE0 Þ

Therefore, only the high productivity type worker has an incentive to deviate towards no education, entailing that H ðNÞ ¼ fHg, such that the firm updates its off-the-equilibrium beliefs at the left-hand side of the game tree to p ¼ 1 by Bayes’ rule. Second Step: Given the firm’s updated belief that any worker without an education is of the high type, the firm responds by hiring a H-type worker as a manager which increases its payoff from 4 to 10, such that the H-type worker would deviate to no education because min

a2A ðH ðNÞ;NÞ

pH ðaÞ ¼ 10 [ pH ðEÞ ¼ 6 þ uH

where a ¼ fM 0 g representing the best response of the firm after observing N. As a result, the PBE (EE′, MC′), where 0  p\ 25, 0  uH \4, and uL  7, violates Cho and Kreps’ Intuitive Criterion. • Third case. uH  4 and 1  uL \7 First Step: Check the maximum payoff that each type of worker can obtain by deviation. maxpH ðNÞ ¼ 10  6 þ uH ¼ pH ðEÞ

maxpL ðN 0 Þ ¼ 10 [ 3 þ uL ¼ pL ðE0 Þ Therefore, only the low productivity type worker has an incentive to deviate towards no education, entailing that H ðNÞ ¼ fLg, such that the firm updates its off-the-equilibrium beliefs at the left-hand side of the game tree by to p ¼ 0 Bayes’ rule. Second Step: Given the firm’s updated belief that any worker without an education is of the low type, the firm responds by hiring a L-type worker as a cashier, such that the L-type worker would not deviate to no education because min

a2A ðH ðNÞ;NÞ

pL ðaÞ ¼ 4kpL ðEÞ ¼ 3 þ uL

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Perfect Bayesian Equilibrium and Signaling Games

where a ¼ fC 0 g representing the best response of the firm after observing N. As a result, the PBE (EE′, MC′), where 0  p\ 25, uH [ 4, and 1  uL \7, survives Cho and Kreps’ Intuitive Criterion. • Fourth case. uH  4 and uL  7 First Step: Check the maximum payoff that each type of worker can obtain by deviation. maxpH ðNÞ ¼ 10  6 þ uH ¼ pH ðEÞ

maxpL ðN 0 Þ ¼ 10  3 þ uL ¼ pL ðE0 Þ

Therefore, both high and low productivity type workers have no incentives to deviate towards no education, entailing that H ðNÞ ¼ ;, such that the firm is unable to update its off-the-equilibrium beliefs at the left-hand side of the game tree by Bayes’ rule. As a result, the PBE (EE′, MC′), where 0  p\ 25, uH  4, and uL  7, survives Cho and Kreps’ Intuitive Criterion. (e) Consider the pooling strategy profile NN′. Under which conditions of parameters uH and uL can this strategy profile be supported as a PBE? • Fig. 9.42 depicts the pooling strategy profile in which both types of worker choose “No education” by shading branches N and N′ (see thick arrows). 1. Responder’s beliefs: • Analogously as the previous pooling strategy profile, the firm’s equilibrium beliefs (after observing “No education”) coincide with the prior probability of a high type, p ¼ 12; while its off-the-equilibrium beliefs (after observing “Education”) are left unrestricted, i.e., q 2 ½0; 1 2. Firms’ optimal response given its updated beliefs: • Given the previous beliefs, after observing “No education” (in equilibrium): if the firm hires the worker as a manager (M′), it obtains an expected payoff of 1 1 EUF ðM 0 Þ ¼  10 þ  0 ¼ 5 2 2 and if it hires him as a cashier (C′), its expected payoff only becomes EUF ðC0 Þ ¼

1 1  4 þ  4 ¼ 4; 2 2

Exercise 4—Job Market Signaling with Utility-Enhancing EducationB

(10,10)

M’

Firm

N

Worker H

E

p

(4,4)

(10,0)

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(6 + uH,10)

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423

(1-q)

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C’

Fig. 9.42 Pooling strategy profile NN′

leading the firm to hire the worker as a manager (M′) after observing the equilibrium message of “No education”. • After observing “Education” (off-the-equilibrium): the expected payoff the firm obtains from hiring the worker as a manager (M) or cashier (C) are, respectively, EUF ðMÞ ¼ q  10 þ ð1  qÞ  0 ¼ 10q; EUF ðCÞ ¼ q  4 þ ð1  qÞ  4 ¼ 4; Hence, the firm responds hiring the worker as a manager (M) after observing the off-the-equilibrium message of Education if and only if q  25. Otherwise, the firm hires the worker as a cashier (C). 3. Given the previous steps 1 and 2, let us find the worker’s optimal actions. In this case, we will also need to split our analysis into two cases (one in which q\ 25 and thus the firm respond by hiring him as a cashier upon observing the off-the-equilibrium message of “Education”, and the case in which q  25, in which the firm hires the worker as a manager): Case 1: When q\ 25, the firm responds by hiring the worker as a cashier (C) upon observing the off-the-equilibrium message of “Education”, as depicted in the right-hand shaded branches of Fig. 9.43. • If the worker is a high-productivity type, he does not deviate from “No education” since his payoff from N, 10, exceeds the zero payoff he would obtain by deviating to E; as indicated in the shaded branches upper part of the game tree. Similarly, if he is a low-productivity worker, he does not deviate from “No education” since his (positive) payoff from N′ (10) is larger than the negative payoff he would obtain by deviating to E′, −3; as indicated in the lower part of

424

(10,10)

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Perfect Bayesian Equilibrium and Signaling Games

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Fig. 9.43 Optimal responses in strategy profile NN′—Case 1

the game tree. Therefore, the pooling strategy profile in with no type of worker acquires education, [NN′, CM′], can be supported when off-the-equilibrium beliefs satisfy q\ 25 for all values of uH ; uL  0. Case 2: When q  25 the firm responds by hiring the worker as a manager (M) when he acquires education, as depicted in Fig. 9.44. • When uH \4, the high-productivity type worker plays “No education” (as prescribed in this strategy profile) since his payoff from doing so, 10, exceeds that from deviating to E, 6 þ uH ; as indicated in the shaded branches of upper part of the game tree. Similarly, uL \7, the low-productivity worker plays “No education” (as prescribed) since his payoff from this strategy, 10, is higher than from deviating towards E′, 3 þ uL ; as indicated in the lower part of the game tree. Hence, the pooling strategy profile where no worker acquires education [NN′, MM′], can be supported as a PBE when off-the-equilibrium beliefs satisfy q  25, for uH \4 and uL \7. Note that when uH  4ðuL  7Þ, the high- (low-) productivity worker has an incentive to deviate to “Education” since the (10,10)

M’

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Fig. 9.44 Optimal responses in strategy profile NN′—Case 2

Firm

Exercise 4—Job Market Signaling with Utility-Enhancing EducationB

425

utility-enhancing effect of education is so strong, invalidating the pooling strategy profile [NN′, MM′] as a PBE given off equilibrium beliefs q  25. (f) Apply the Cho and Kreps’ (1987) Intuitive Criterion to the pooling strategy profile NN′. Under which conditions of parameters uH and uL will this strategy profile violate Cho and Kreps’ Intuitive Criterion? In the PBE, (NN′, CM′), where the firm’s off-the-equilibrium beliefs satisfy 0  q\ 25 with uL ; uH  0, we consider the following four cases regarding the parameter values. • First case. 0  uH \4 and 0  uL \7 First Step: Check the maximum payoff that each type of worker can obtain by deviation. maxpH ðEÞ ¼ 6 þ uH \10 ¼ pH ðNÞ maxpL ðE0 Þ ¼ 3 þ uL \10 ¼ pL ðN 0 Þ Therefore, both high and low productivity type workers have no incentives to deviate towards education, entailing that H ðNÞ ¼ ;, such that the firm is unable to update its off-the-equilibrium beliefs at the right-hand side of the game tree by Bayes’ rule. As a result, the PBE (NN′, CM′), where 0  q\ 25, 0  uH \4, and 0  uL \7, survives Cho and Kreps’ Intuitive Criterion. • Second case. 0  uH \4 and uL  7 First Step: Check the maximum payoff that each type of worker can obtain by deviation. maxpH ðEÞ ¼ 6 þ uH \10 ¼ pH ðNÞ maxpL ðE0 Þ ¼ 3 þ uL  10 ¼ pL ðN 0 Þ Therefore, only the low productivity type worker has an incentive to deviate towards education, entailing that H ðNÞ ¼ fLg, such that the firm updates its off-the-equilibrium beliefs at the right-hand side of the game tree by to q ¼ 0 Bayes’ rule. Second Step: Given the firm’s updated belief that any worker with an education is of the low type, the firm responds by hiring a L-type worker as a cashier, such that the L-type worker would not deviate to education because

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9

min

a2A ðH ðNÞ;NÞ

Perfect Bayesian Equilibrium and Signaling Games

pL ðaÞ ¼ 3kpL ðEÞ ¼ 10

where a ¼ fCg representing the best response of the firm after observing N. As a result, the PBE (NN′, CM′), where 0  q\ 25, 0  uH \4, and uL  7, survives Cho and Kreps’ Intuitive Criterion. • Third case. uH  4 and 1  uL \7 First Step: Check the maximum payoff that each type of worker can obtain by deviation. maxpH ðEÞ ¼ 6 þ uH  10 ¼ pH ðNÞ

maxpL ðE0 Þ ¼ 3 þ uL \10 ¼ pL ðN 0 Þ

Therefore, only high productivity type worker has an incentive to deviate towards education, entailing that H ðNÞ ¼ fHg, such that the firm updates its off-the-equilibrium beliefs at the right-hand side of the game tree to q ¼ 1 by Bayes’ rule. Second Step: Given the firm’s updated belief that any worker with an education is of the high type, the firm responds by hiring a H-type worker as a manager, such that the H-type worker would deviate to education because min

a2A ðH ðNÞ;NÞ

pH ðaÞ ¼ 6 þ uH [ 10 ¼ pH ðEÞ

where a ¼ fMg representing the best response of the firm after observing N. As a result, the PBE (NN′, CM′), where 0  q\ 25, uH  4, and 1  uL \7, violates Cho and Kreps’ Intuitive Criterion. • Fourth case. uH  4 and uL  7 First Step: Check the maximum payoff that each type of worker can obtain by deviation. maxpH ðEÞ ¼ 6 þ uH  10 ¼ pH ðNÞ

maxpL ðE0 Þ ¼ 3 þ uL  10 ¼ pL ðN 0 Þ

Therefore, both high and low productivity type workers have incentives to deviate towards education, entailing that H ðNÞ ¼ fH; Lg, such that the firm is unable to update its off-the-equilibrium beliefs at the right-hand side of the game tree by Bayes’ rule.

Exercise 4—Job Market Signaling with Utility-Enhancing EducationB

427

Second Step: Given off-the-equilibrium beliefs maintained at 0  q\ 25, the firm continues to hire an educated worker as a cashier, such that min

a2A ðH ðNÞ;NÞ

min

a2A ðH ðNÞ;NÞ

pH ðaÞ ¼ 0k6 þ uH ¼ pH ðEÞ

pL ðaÞ ¼ 3k3 þ uL ¼ pL ðE0 Þ

where A ðÞ ¼ fM; Cg represents the action space of the firm given its updated beliefs H ðNÞ of observing an off-the-equilibrium message N, and the action that minimizes the payoff of the workers is C that gives high (low) productivity worker a payoff of 0 (−3) instead of 10. Therefore, no worker could benefit by unilateral deviating to education. As a result, the PBE (NN′, MC′), where 0  q\ 25, uH  4, and uL  7, survives Cho and Kreps’ Intuitive Criterion. Whereas, in the PBE, (NN′, MM′), where the firm’s off-the-equilibrium beliefs satisfy 25  q  1 with uL \4 and uH \7, it is obvious that neither high or low productivity type of worker would be better off by deviating to education, as maxpH ðEÞ ¼ 6 þ uH \10 ¼ pH ðNÞ

maxpL ðE0 Þ ¼ 3 þ uL \10 ¼ pL ðN 0 Þ

This implies that both high and low productivity type workers have no incentives to deviate towards education, entailing that H ðNÞ ¼ ;, such that the firm is unable to update its off-the-equilibrium beliefs at the right-hand side of the game tree by Bayes’ rule. As a result, the PBE (NN′, MM′), where 25  q  1 with uL \4, and uH \7, survives Cho and Kreps’ Intuitive Criterion. (g) How are your results affected by an increase in parameter uH or uL ? • If uH ¼ uL ¼ 0, we return to the standard labor-market signaling game as in Exercise 2. We next depict the PBEs that emerge in the ðuH ; uL Þ plane, where uL  0 (on the horizontal axis) and uH  0 (on the vertical axis), as in Fig. 9.45. • The yellow-shared region corresponds to the pooling PBE [NN′, CM′], given equilibrium beliefs of p ¼ 12 and off-the-equilibrium beliefs of 0  q\ 25, that is supported by all non-negative values of uL ; uH  0; such that any increase in the values of uL and uH has no impact on this pooling strategy being a PBE. • The blue-shared region corresponds to the pooling PBE [NN′, MM′], given equilibrium beliefs of p ¼ 12 and off-the-equilibrium beliefs of 25  q  1, that is supported by uL \7 and uH \4; such that if the value of uL or uH increase beyond 7 and 4 respectively, this pooling strategy can no longer be supported as a PBE.

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Perfect Bayesian Equilibrium and Signaling Games

(EE’, MC’) for 0 ≤ p < 2/5 (EN’,MC’) for p = 0 and q = 1 (EE’, MM’) for 2/5 ≤ p ≤ 1 and q = ½

uH

4

1

(NN’, MM’) for p = ½ and 2/5 ≤ q ≤ 1

7

uL

(NN’, CM’) for p = ½ and 0 ≤ q < 2/5 Fig. 9.45 PBEs on different values of uL and uH

• The red-shared region corresponds to the pooling PBE [EE′, MM′], given off-the-equilibrium beliefs of 25  q  1 and equilibrium beliefs of q ¼ 12, that is supported by uL  7 and uH  4; such that only when the value of uL and uH both increase beyond 7 and 4 respectively can this pooling strategy be supported as a PBE. • Lastly, the green-shared region corresponds to the pooling PBE [EE′, MC′], given off-the-equilibrium beliefs of 0  p\ 25 and equilibrium beliefs of q ¼ 12, that is supported by uL  1 and uH  0. Also, the separating strategy PBE [EN′, MC′], given equilibrium beliefs of p ¼ 0 and q ¼ 1, is supported by uL  1 and uH  0. Therefore, when the value of uL increases beyond 1, and for all values of uH , these separating and pooling strategy profiles can be supported as PBEs.

Exercise 5—Finding Semi-Separating Equilibrium Consider the sequential game with incomplete information depicted in Fig. 9.46. Two players are playing the following super-simple poker game. The first player gets his cards, and obviously he is the only one who can observe them. He can either get a High hand or a Low hand, with equal probabilities. After observing his

Exercise 5—Finding Semi-Separating Equilibrium

429

Fig. 9.46 A simple poker game

hand, he must decide whether to Bet (let us imagine that he is betting a fixed amount of a dollar) or Resign from the game. If he resigns his payoff is −1, independent of whether he has a High or a Low hand, and player 2 gets a dollar. If he bets, player 2 must decide whether to Call or Fold from the game. Obviously, player makes his choice without being able to observe player 1’s hands, but only observing that player 1 Bets. (a) Show that no separating PBE can be sustained where player 1 chooses a pure strategy (i.e., betting or resigning, given his cards, with 100% probability). (b) Show that no pooling PBE can be sustained where player 1 chooses a pure strategy (i.e., betting or resigning, given his cards, with 100% probability). (c) Find a semi-separating PBE where player 1 chooses Bet or Resign in pure strategies when having a high hand, but randomizes between bet and resign when having a low hand. Answer: (a) Show that no separating PBE can be sustained where player 1 chooses a pure strategy (i.e., betting or resigning, given his cards, with 100% probability). 1. Separating PBE where player 1 Bet when having a High hand, and Resign when having a Low hand: (Bet, Resign); depicted in Fig. 9.47a. 1st step. We first find player 2’s beliefs (responder beliefs) in this separating strategy profile. After observing Bet, player 2’s beliefs are l = 1, intuitively indicating that player 2 believes that a Bet can only originate from a player 1 with a High hand. Graphically, this implies that player 2 believes to be in the upper node of the information set.

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9

Perfect Bayesian Equilibrium and Signaling Games

Fig. 9.47 a Separating strategy profile (Bet, Resign) in the poker game. b Separating strategy profile (Bet, Resign) in the poker game, including player 2’s responses

2nd step. Given player 2’s beliefs, player 2’s optimal response is to Fold since −1 > −2. Figure 9.47b shades the Fold branch for player 2, for both types of player 1 (since player 2 cannot condition his response on player 1’s true hand). 3rd step. Given the previous steps, we need to find what is player 1’s optimal action (whether to Resign or Bet) when he has a High hand, and what is his optimal action when having a Low hand. • When player 1 has a High hand, he prefers to Bet (as prescribed in this strategy profile) since his payoff from doing so (1) exceeds that from Resigning (−1); see shaded branches at the top right-hand side of Fig. 2b. • When player 1 has a Low hand, he prefers to Bet (deviating from the prescribed strategy profile) since his payoff from doing so (1) exceeds that from Resigning (−1). We can then conclude that this separating strategy profile (Bet, Resign) cannot be supported as a PBE of the poker game. In particular, we found that player 1 prefers to deviate from this strategy profile when having a Low hand, as he anticipates that player 2 will respond folding.

Exercise 5—Finding Semi-Separating Equilibrium

431

Fig. 9.48 a Separating strategy profile (Resign, Bet) in the poker game. b Separating strategy profile (Resign, Bet) in the poker game, including player 2’s responses

2. Separating PBE where player 1 Resigns when having a High hand, and Bets when having a Low hand: (Resign, Bet); as illustrated in Fig. 9.48a. 1st step. We first find player 2’s beliefs (responder beliefs) in this separating strategy profile. After observing Bet, player 2’s beliefs are l = 0, intuitively indicating that player 2 believes that a Bet can only originate from a player 1 with a Low hand. Graphically, this implies that player 2 believes to be in the lower node of the information set. 2nd step. Given player 2’s beliefs, which is player 2’s optimal response is to Call, since 2 > −1. Figure 9.48b shades the Call branch for player 2, for both types of player 1 (since player 2 cannot condition his response on player 1’s true hand). 3rd step. Given the previous steps, we need to find what is player 1’s optimal action (whether to Resign or Bet) when he has a High hand, and what is his optimal action when having a Low hand.

432

9

Perfect Bayesian Equilibrium and Signaling Games

• When player 1 has a High hand, he prefers to Bet (deviating from the prescribed strategy profile) since his payoff from doing so (2) exceeds his payoff from Resigning (−1). • When player 1 has a Low hand, he prefers to Resign (deviating from the prescribed strategy profile) since his payoff from doing so (−1) exceeds his payoff from Betting (−2). We can then conclude that this separating strategy profile (Resign, Bet) cannot be supported as a PBE of the poker game. In particular, we found that player 1 prefers to deviate from this strategy profile when having a High hand, (he prefers to Bet) and when he has a Low hand (he prefers to Resign). (b) Show that no pooling PBE can be sustained where player 1 chooses a pure strategy (i.e., betting or resigning, given his cards, with 100% probability). 1. Pooling PBE with both types of player 1 playing Bet: (Bet, Bet); as depicted in Fig. 9.49a.

Fig. 9.49 a Pooling strategy profile (Bet, Bet) in the poker game. b Pooling strategy profile (Bet, Bet) in the poker game, including player 2’s responses

Exercise 5—Finding Semi-Separating Equilibrium

433

1st step. We first find player 2’s beliefs (responder beliefs) in this pooling strategy profile. After observing Bet, player 2’s beliefs are: 1 1 1 2 ¼ ¼ 1 2 1 2  1þ 2  1

l¼1

1 2

Intuitively, player 2 beliefs coincide with the prior probability distribution overtypes, i.e., l ¼ 12. 2nd step. Given player 2’s beliefs, his optimal response is to Call. In order to examine player 2’s response to Bet, he must separately find the expected utility derived from both strategies (Call and Fold), as follows: 1 EU2 ðCallÞ ¼ ð2Þ þ 2 1 EU2 ðFold Þ ¼ ð1Þ þ 2

1 ð 2Þ ¼ 0 2 1 ð1Þ ¼ 1 2

The evaluation shows that player 2’s optimal response is to Call since his expected utility is higher (0 > −1). Figure 9.49b shades the Call branch for player 2, for both types of player 1 (since player 2 cannot condition his response on player 1’s true hand). 3rd step. Given the previous steps, we need to find what is player 1’s optimal action (whether to Resign or Bet) when he has a High hand, and what is his optimal action when having a Low hand. • When player 1 has a High hand, he prefers to Bet (as prescribed in this strategy profile) since his payoff from doing so (2) exceeds his payoff from Resigning (−1); see shaded branches at the right-hand side of Fig. 2f. • When player 1 has a Low hand, he prefers to Resign (deviating from the prescribed strategy profile) since his payoff from doing so (−1) exceeds his payoff from Betting (−2). We can then conclude that this pooling strategy profile with both types of player 1 playing Bet cannot be supported as a PBE, because player 1 prefers to Resign when he has a Low hand, contradicting the prescribed pooling strategy profile where both types of player 1 Bet. 2. Pooling PBE with both types of player 1 playing Resign: (Resign, Resign); as illustrated in Fig. 9.50a.

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9

Perfect Bayesian Equilibrium and Signaling Games

Fig. 9.50 a Pooling strategy profile (Resign, Resign) in the poker game. b Pooling strategy profile (Resign, Resign) in the poker game, including player 2’s responses when Calling (l \ 3=4). c Pooling strategy profile (Resign, Resign) in the poker game, including player 2’s responses when Folding (l [ 3=4)

Exercise 5—Finding Semi-Separating Equilibrium

435

1st step. We first find player 2’s beliefs (responder beliefs) in this pooling strategy profile. After observing Bet (something that occurs off-the-equilibrium path, as indicated in the figure), player 2’s beliefs are: 0 0 ¼ 1 0 2  0þ 2  0

l¼1

1 2

and thus, beliefs must be left unrestricted, that is, µ takes any number µ 2 [0, 1]. 2nd step. Given player 2’s beliefs, in order to identify which is player 2’s response to Bet we must separately find his expected utility from Call and Fold as follows: EU2 ðCallÞ ¼ lð2Þ þ ð1  lÞð2Þ ¼ 2  4l EU2 ðFoldÞ ¼ lð1Þ þ ð1  lÞð1Þ ¼ 1 implying that player 2 prefers to Call if and only if 2  4l [  1, or l \ 34. Hence, we need to divide our subsequent analysis into two cases: • Case 1: l \ 34, implying that player 2 responds Calling if he observes player 1 betting. • Case 2: l \ 34, implying that player 2 responds Folding if he observes player 1 betting. CASE 1: l \ 34, as depicted in Fig. 9.50b. 3rd step. Let us now check  if this pooling strategy profile can be sustained as a PBE in this case l \ 34 : • When player 1 has a High hand, he prefers to Bet, since his payoff from doing so (2) is larger than that from Resigning (−1). • We don’t even need to check the case in which player 1 has a Low hand, since the above argument already shows that the pooling strategy profile cannot be sustained as a PBE when l \ 34. CASE 2: l \ 34, as illustrated in Fig. 9.50c. 3rd step. Let us now  check  if this pooling strategy profile can be sustained as a 3 PBE in this case l [ 4 : • When player 1 has a High hand, he prefers to Bet, since his payoff from doing so (1) is larger than that from Rosining (−1).

436

9

Perfect Bayesian Equilibrium and Signaling Games

• We don’t even need to check the case in which P1 has a Low hand, since the above argument already shows that the pooling strategy profile cannot be sustained as a PBE when l [ 34 : Then, the pooling strategy profile where player 1 Bets both when his hand is High and Low cannot be sustained as a PBE, regardless of player 2’s off-the-equilibrium beliefs. Overall, we have ruled out all pure strategy PBEs (either separating or pooling) and we must now move to check for semi-separating equilibria where at least one player randomizes. (c) Find a semi-separating PBE where player 1 chooses Bet or Resign in pure strategies when having a high hand, but randomizes between bet and resign when having a low hand. • First, note that Bet is a strictly dominant strategy for player 1 when he has a High hand. Indeed, his payoff from doing so (either 2 or 1, depending on whether player 2 calls or folds) is strictly higher than his payoff from Resigning (−1). • However, Bet is not a dominant strategy for player 1 when he has a Low hand. He prefers to Bet if he anticipates that player 2 will Fold, but prefers to Resign if player 2 responds with Calls. • Intuitively, player 2 has incentives to call a Bet originating from a player 1 with a Low hand. As a consequence, player 1 doesn’t want to convey his type to player 2, but instead to conceal it so that player 2 Folds. The way in which player 1 can conceal his type is by randomizing his Betting strategy. This semi-separating strategy profile is depicted in Fig. 9.51.

Fig. 9.51 Semi-separating strategy profile (Bet, Resign/Bet) in the poker game

Exercise 5—Finding Semi-Separating Equilibrium

437

i. First, let us find player 2’s beliefs (l) that support the fact that player 2 responds mixing. That is, what is the value of l that makes player 2 indifferent between Call and Fold? Player 2 must be mixing. If he wasn’t, player 1 could anticipate his action and play pure strategies as in any of the above strategy profiles (which are not PBE of the game, as we just showed). Hence, player 2 must be indifferent between Call and Fold, as follows lð1Þ þ ð1  lÞ  2 ¼ l  ð1Þ þ ð1  lÞð1Þ which simplifies to l ¼ 34. Hence, player 2’s beliefs in this semi-separating PBE must satisfy l ¼ 34. ii. Given player 2’s beliefs, l ¼ 34, let us write Bayes’ rule, considering that player 1 always Bets when having a High hand, but that he mixes (with probability pL ) when having a Low hand. l¼

1 H 3 2p ¼1 4 2  pH þ 12  pL

where pH denotes the probability with which player 1 Bets when he has a High hand, and similarly pL represents the probability that player 1 Bets when he has a Low hand. Since we know that pH ¼ 1 given that player 1 always Bets when he has a High hand (he Bets using pure strategies), then the above ratio becomes 1 3 ¼ 1 12 4 2 þ 2  pL

Solving for probability pL , we obtain pL ¼ 13. At this stage of our solution we know everything about player 1: he Bets with probability pL ¼ 13 when he has a Low hand and Bets using pure strategies (with 100% probability) when he has a High hand, i.e., pH ¼ 1. iii. The probability that player 2’s Calls (denoted by q) that makes player 1 indifferent between Bet and Resign when he has a Low hand, satisfies EU1 ðBetjLowÞ ¼ EU1 ðResignjLowÞ qð2Þ þ ð1  qÞ  1 ¼ 1 Solving for probability q, we obtain that player 2 randomizes with probability q ¼ 23, that is, he Calls with probability 66.67%. iv. Summarizing, all the previous results, the Semi-Separating PBE of this poker game is:

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Perfect Bayesian Equilibrium and Signaling Games

• Player 1 Bets when he has a High hand with full probability, pH ¼ 1; whereas he Bets when he has a Low hand with probability pL ¼ 13 : • Player 2 Calls with probability q ¼ 23, and his beliefs are l ¼ 34.

Exercise 6—Firm Competition Under Cost UncertaintyC Consider two firms competing in prices. Each firm faces a direct demand function of: qi ¼ 1  pi þ pj

for every firm i; j ¼ 1; 2; where i 6¼ j

The firms compete as Bertrand oligopolists over two periods of production. Assume that, while the unit cost of firm 2 is deterministically set at zero, i.e., c2 ¼ 0, the unit cost of firm 1, c1 , is stochastically determined at the beginning of the first period. The distribution function of c1 is defined over the support ½c0 ; c0  with expected value Eðc1 Þ ¼ 0. At the beginning of period 1, firm 1 can observe the realization of its unit cost, c1 , while its rival cannot. Firms simultaneously and independently select first-period prices after firm 1’s unit cost has been realized. At the beginning of period 2, both firms can observe the prices set in the first period of production. In particular, firm 2 observes the price selected by firm 1 in period 1 and draw some conclusions about firm 1’s unit cost, c1 . After observing first-period prices, the firms simultaneously select second-period prices. Find equilibrium prices in each of the two periods [Hint: Assume that firm 1’s first period price is linear in its private information, c1, i.e., p11 ¼ f ðc1 Þ ¼ A0 þ A1 C1 , where A0 and A1 are positive constants.]. Answer: Let us start with a summary of the time structure of the game: t = 0: Firm 1 privately observes c0. t = 1: Both firms i and j simultaneously select their own price ðp1i ; p1j Þ, where the superscript indicates the first period. t = 2: Observing first-period prices ðp1i ; p1j Þ, firms i and j simultaneously choose second-period prices ðp2i ; p2j Þ: We start analyzing the second-period game Firm 1. Let us first examine the informed Firm 1. This, firm chooses the second-period price, p21 , that solves

Exercise 6—Firm Competition Under Cost UncertaintyC

439

max ð1  p21 þ p22 Þðp21  c1 Þ p21

Taking first order conditions with respect to p21 , yields 1  2p21 þ p22 þ c1 ¼ 0 and solving for p21 we obtain firm 1’s best response function in the second-period game (which is positively sloped in its rival’s price, p22 ), p21 ¼

1 þ c1 þ p22 2

BRF12

Firm 2. Let us now analyze the uninformed Firm 2. This firm chooses price p22 that solves ð1 þ p21  p22 Þðp22 Þ max 2 p2

We recall that c2 = 0, i.e., this firm faces no unit costs. Taking first order conditions with respect to p22 , we obtain 1  2p22 þ p21 ¼ 0 and solving for p22 we find firm 2’s best response function in the second-period game p22 ¼

1 þ p21 2

BRF22

(which is also positively sloped in its rival’s price). Plugging one best response function into another, we can find the price that the informed Firm 1 sets: p21 ¼

1 þ c1 þ 2

1 þ p21 2

which solving for p21 yields a price of p21 ¼ 1 þ

2 c1 3

ðAÞ

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9

Perfect Bayesian Equilibrium and Signaling Games

Therefore, the uninformed Firm 2 sets a price p22

  1 þ 1 þ 23 f 1 ðp11 Þ ¼ 2

which, rearranging, yields p22 ¼ 1 þ

1 1 1 f ðp1 Þ 3

ðBÞ

Explanation: The price that firm 1 sets is a function of its privately observed cost, p11 ¼ f ðc1 Þ. Then, upon observing a price from firm 1 in the first period, the uninformed firm 2 can infer firm 1’s costs by inverting the above function, as follows: f 1 ðp11 Þ ¼ c1 . This helps firm 2 set a second-period price p22 . Let’s now analyze the first period: Firm 2. The uninformed Firm 2 chooses a price p12 that maximizes maxð1  p12 þ p11 Þp12 p12

Taking first order conditions with respect to p12 yields: 1  2p12 þ p11 ¼ 0 and solving for p12 , we obtain firm 2’s best response function in the first-period game p12 ¼

1 þ p11 2

BRF21

Firm 1. The informed Firm 1 solves   maxðp11  c1 Þ  1  p11 þ p12 þ ðp21  c1 Þ  ð1  p21 þ p22 Þ p11

where the first term indicates first-period profits, while the second term reflects second-period profits. Note that this firm takes into account the effect of its first-period price on its second-period profits because a lower price today could lead firm 2 to believe that firm 1’s costs are low (that is, firm 1 would be seen as a “tougher” competitor); ultimately improving firm 1’s competitiveness in the second-period game. Firm 2’s profit-maximization problem, in contrast, does not consider the effect of first-period prices on second-period profits since firm 1 is perfectly informed about firm 2’s costs.

Exercise 6—Firm Competition Under Cost UncertaintyC

441

Firm 1 anticipates second-period prices, p21 ¼ 1 þ 23 c1 for firm 1 and p22 ¼ 1 þ 13 f 1 ðp11 Þ for firm 2. Hence, his maximization problem can be rewritten as

  2 max ðp11  c1 Þ  ð1  p11 þ p12 Þ þ 1 þ c1  c1 3 p11


2 1 1 1  1  1 þ c1 þ 1 þ f ðp1 Þ 3 3 Taking first order condition with respect to p11 , we obtain
 1 1 1  1  ¼ 0: 1  2p11 þ p12 þ c1 þ 2 1  c1 0 1 3 3 f ðf p1 Simplifying and solving for p11 , we find firm 1’s best response function in the first-period game p11

           1  6 þ c1 ð9 f 0 f 1 p11  2 þ 9½f 0 f 1 p11 1 þ p12    p2 ¼ 18½f 0 f 1 p11

Inserting this expression of p11 into p12 ¼ second-period price for Firm 1 p12 ¼

1 þ p11 2 ,

we obtain the optimal

3 þ c1 2ð 3  c 1 Þ    þ 3 27½f 0 f 1 p11

Plugging this result into the best response function p11 ðp12 Þ, yields the optimal first-period price for firm 1 p11 ¼ 1 þ

2c1 4ð 3  c 1 Þ    þ 3 27f 0 f 1 p11

As suggested in the exercise, let us now assume that there exists a linear function p11 ¼ f ðc1 Þ, that generates the previous strategy profile. That is, p11 ¼ A0 þ A1 c1 ¼ f ðc1 Þ where A0 and A1 are positive constants. Intuitively, a firm with zero unit costs, c1 ¼ 0, would charge a first-period price of A0, while a marginal increase in its unit costs, c1, would entail a corresponding increase in prices of A1. Figure 9.52 illustrates this pricing function for firm 1 (Note that this is a separating strategy profile, as firm 1 charges a different first-period price depending on its unit cost as long as A1 > 0. A pooling strategy profile would exist if A1 = 0.).

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Perfect Bayesian Equilibrium and Signaling Games

Fig. 9.52 Firm 1’s pricing function

Setting it equal to the expression for p11 we found above and letting, A1 ¼ f 0 f 11 p1 , we obtain ð ð 1 ÞÞ A 0 þ A 1  c1 ¼ 1 þ

2c1 4ð 3  c 1 Þ þ A1 27 3

since A1 measures the slope of the pricing function (see Fig. 9.52), thus implying A1 ¼ f 0 f 11 p1 . Rearranging the above expression, we find ð ð 1 ÞÞ 27A0 þ 31A1 c1 ¼ 3ð9 þ 4A1 þ 6c1 Þ or, after solving for A1 , we obtain A1 ¼

9ð3  3A0 þ 2c1 Þ 31c1  12

In addition, when firm 1’s costs are nil, c1 ¼ 0, the above expression becomes, 4 A0 ¼ 1 þ A1 : 9 Inserting this equation O þ 2c1 Þ , yields A1 ¼ 9ð33A 31c1 12

into

the

expression

for

A1

found

above,

Exercise 6—Firm Competition Under Cost UncertaintyC

443

    9 3  3 1 þ 49 A1 þ 2c1 A1 ¼ 31c1  12 Solving for A1 , we obtain A1 ¼ 18 31 0:58. Therefore, the intercept of the pricing function, A0 , becomes: A0 ¼

39 1:26: 31

Hence, the pricing function p11 of firm 1, p11 ¼ A0 þ A1 c1 , becomes: p11 ¼

18 39 þ c1 : 31 31

Exercise 7—Signaling Game with Three Possible Types and Three MessagesC Consider the signaling game depicted in Fig. 9.53. Nature first determines the production costs of firm 1. Observing this information, firm 1 decides whether to set low (L), medium (M) or High (H) prices. Afterwards, observing firm 1’s prices, but not observing its costs, firm 2 responds with either high (h) or low (l) prices. Find a PBE in which firm 1 chooses intermediate prices both when its costs are low and medium, but selects high prices when its costs are high. Answer: Figure 9.54 shades the branches in which firm 1 chooses intermediate prices both when its costs are low and medium, M and M′ respectively, but a high price otherwise (H″). This type of strategy profile is often referred to as “semi-separating” since it has two types of sender pooling into the same message (medium prices in this example), but has another type of sender choosing a different (“separating”) message (i.e., H″ by the high-cost firm in our example). Receiver beliefs:

After observing medium prices, the probability that such action originates from a firm 1 with low costs can be computed using Bayes’ rule, as follows

444

9

Perfect Bayesian Equilibrium and Signaling Games Nature

Low costs Prob=1/4

High costs Medium costs

Prob=1/4 Prob=1/2

Firm 1

M

L

H

L’

M’

L’’

H’

M’’

H’’

Firm 2

Firm 1 Firm 2

l

h

7 1

2 0

l’

4 1

h’

l’’

2 0

2 1

h’’

3 0

l

b

1 1

4 1

l’

6 1

h’

l’’

1 0

3 1

h’’

5 0

l

h

2 0

4 1

l’

3 0

h’

l’’

h’’

1 1

5 0

5 1

Fig. 9.53 Signaling game with 3 types, each with 3 possible messages

Nature

Low costs Prob=1/4

High costs Medium costs

Prob=1/4 Prob=1/2

Firm 1

M

L

H

L’

M’

L’’

H’

M’’

H’’

Firm 2

Firm 1 Firm 2

l

h

7 1

2 0

l’

4 1

h’

l’’

2 0

2 1

h’’

3 0

l

4 1

b

1 1

l’

6 1

h’

l’’

1 0

3 1

Fig. 9.54 Semi-separating strategy profile (M, M′, H″)

h’’

5 0

l

h

2 0

4 1

l’

3 0

h’

l’’

h’’

1 1

5 0

5 1

Exercise 7—Signaling Game with Three Possible Types …

445

probðlow costsÞ  probðMjlow costsÞ probðMÞ 1 1 1 41 ¼1 ¼ 43 ¼ 1 1 3  1 þ  1 þ  0 4 2 4 4

probðlow costsjMÞ ¼

Intuitively, since low- and medium-cost firms choose medium prices in this strategy profile, the conditional probability that, upon observing medium prices, firm 1’s costs are low is just the relative frequency of low cost firms within the subsample of low- and medium-cost firms. A similar argument applies to medium-cost firms. Specifically, upon observing a medium price, the probability that such price originates from a firm with medium costs is 1 1 2 2 ¼ ¼ 1 3 3  1 þ  1 þ  0 4 4 4 1 2 1 2

probðmedium costsjMÞ ¼ 1

which is the relative frequency of medium-cost firms within the subsample of lowand medium-costs firms. Finally, high-cost firms should never select a medium price in this strategy profile, implying that firm 2 assigns no probability to the fact that the medium price originates from this type of firm, that is probðhigh costsjMÞ ¼ 0 Similarly, after observing a high price, the probability that such a price originates from a firm 1 with low costs can be computed using Bayes’ rule, as follows 0 ¼0  0 þ  0 þ 14  1 4 1 4 1 2

probðlow costsjHÞ ¼ 1

since a high price should never originate from a low-cost firm in this strategy profile, 0 ¼0 1 4  0þ  0þ 4  1 1 2 1 2

probðmedium costsjHÞ ¼ 1

since a high price should not originate from a medium-cost firm either, and 1 1 ¼ 41 ¼ 1 1 4  0þ  0þ 4  1 4

probðhigh costsjHÞ ¼ 1

1 4 1 2

given that only high-cost firms choose high prices in this strategy profile. Finally regarding low prices, we know that these can only occur off-the-equilibrium path, since no type of firm 1 selects this price level in the strategy profile we are testing. Hence, firm 2’s off-the-equilibrium beliefs are

446

9

Perfect Bayesian Equilibrium and Signaling Games

probðlow costsjLÞ ¼ c1 2 ½0; 1 (Recall that, as shown in previous exercises, the use of Bayes’ rule doesn’t provide a precise value for off-the-equilibrium beliefs such as c1 , and we must leave the receiver’s beliefs unrestricted, i.e., c1 2 ½0; 1). Similarly, the conditional probability that such low prices originate from a medium-cost firm is probðmedium costsjLÞ ¼ c2 ½0; 1; And, as a consequence, the conditional probability that such low price originates from a high-cost firm is probðHighjLÞ ¼ 1  c1  c2 Receiver (firm 2): Given the above beliefs, after observing a medium price (M), firm 2 responds with either low (l) or high (h) prices depending on which action yields the highest expected utility. In particular, these expected utilities are 1 EU2 ðljMÞ ¼  1 þ 3 1 EU2 ðhjMÞ ¼  0 þ 3

2  1 ¼ 1; and 3 2 0¼0 3

Therefore, after observing a medium price, firm 2 responds selecting a low price, l. After observing high prices (H), firm 2 similarly compares its payoff from selecting l and h (Note that in this case, the receiver does not need to compute expected utilities, since it is convinced to be dealing with a high-cost firm, i.e., firm 2 is convinced of being at the right-most node of the game tree.). In particular, responding with l″ at this node yields a payoff of zero, while responding with h″ entails a payoff of 1. Thus, firm 2 responds with high prices (h) after observing that firm 1 sets high prices. Finally, after observing low prices (L) (off-the-equilibrium path), firm 2 compares its expected payoff from responding with l and h, as follows EU2 ðljLÞ ¼ c1  1 þ c2  1 þ ð1 þ c1  c2 Þ  0 ¼ c1 þ c2 ; and EU2 ðhjLÞ ¼ c1  0 þ c2  1 þ ð1 þ c1  c2 Þ  1 ¼ 1 þ c1 Hence, after observing low prices (L), firm 2 responds choosing low prices (l) if and only if c1 þ c2 [ 1 þ c1 , or c2 [ 1  2c1 . In our following discussion of the sender’s optimal messages, we will have to consider two cases: one in which c2 [ 1  2c1 holds, thus implying that firm 2 responds with low prices after observing the off-the-equilibrium message of low prices from firm 1; and that in

Exercise 7—Signaling Game with Three Possible Types …

447

Nature

Low costs Prob=1/4

High costs Medium costs

Prob=1/4 Prob=1/2

Firm 1

M

L

H

L’

M’

L’’

H’

M’’

H’’

Firm 2

Firm 1 Firm 2

l

h

7 1

2 0

l’

4 1

h’

l’’

2 0

2 1

h’’

3 0

l

4 1

b

1 1

l’

6 1

h’

l’’

1 0

3 1

h’’

5 0

l

h

2 0

4 1

l’

3 0

h’

l’’

h’’

1 1

5 0

5 1

Fig. 9.55 Optimal responses in the semi-separating strategy profile

which c2  1  2c1 , whereby firm 2 responds with high prices to such an off-the-equilibrium message. Figure 9.55 summarizes the optimal responses of firm 2 by shading l′ and h″. Sender Low costs. If its costs are low (left-hand side of the game tree) and firm 1 selects medium prices (as prescribed in this strategy profile), it obtains a payoff of 4, since firm 1 can anticipate that its message will be responded with l′ (see shaded branches in Fig. 9.55). If instead, it deviates towards a high price, it is responded with h″ yielding a payoff of only 3. If instead, it deviates towards a low price, firm 1 obtains a payoff of 7 if firm 2’s off-the-equilibrium beliefs satisfy c2 [ 1  2c1 (which lead firm 2 to respond with l), but a payoff of only 2 if c2  1  2c1 (case in which firm 2 responds with h). Hence, in order for firm 1 to prefer medium prices (obtaining a payoff of 4) we need that off-the-equilibrium beliefs satisfy c2  1  2c1 (otherwise firm 1 would have incentives to deviate towards a low price). Medium costs. If firm 1’s costs are medium and it sets a medium price (as prescribed), firm 1 obtains a payoff of 6 (see shaded arrows at the center of Fig. 9.55). Deviating towards a high price would only provide a payoff of 5. Similarly, deviating towards a low price gives firm 1 a payoff of 4 (when c2 [ 1  2c1 , and firm 2 responds with l) or a payoff of 1 (when c2  1  2c1 , and firm 2 responds with h). Hence, regardless of firm 2’s off-the-equilibrium beliefs,

448

9

Perfect Bayesian Equilibrium and Signaling Games

1

Not supported

Supported

1/2

Fig. 9.56 Off-the-equilibrium beliefs of firm 2

firm 1 does not have incentives to deviate from setting a medium price, as prescribed in this semi-separating strategy profile. High costs. A similar argument applies to firm 1 when its costs are high (see right-hand side of Fig. 9.55). If it sets a high price, as prescribed, its profit is 5. Deviating towards a medium price would reduce its profits to 3, and so would a deviation to low prices (where firm 1 obtains a payoff of 2 if firm 2 responds with l or a payoff of 4 if firm 2 responds with h). Thus, independently on firm 2’s off-the-equilibrium beliefs, firm 1 does not have incentives to deviate from high prices. Summarizing, the only condition that we found for this separating strategy profile to be sustained as a PBE is that off-the-equilibrium beliefs must satisfy c2  1  2c1 . Figure 9.56 represents all combinations of c1 and c2 for which the above strategy profile can be sustained as a PBE of this game.4

Exercise 8—Second-Degree Price DiscriminationB Assume that Alaska Airlines is the only airline flying between the Seattle and Pullman. There are two types of passengers, tourist and businessmen, with the latter being willing to pay more than tourists. The airline, however, cannot directly tell whether a ticket purchaser is a tourist or a business traveler. Nevertheless, the two types differ in how much they are willing to pay to avoid having to purchase their 4

This condition allows for different combinations of off-the-equilibrium beliefs. For instance, when c1 < 1/2 but c2 > 1/2, the uninformed firm is essentially placing a large probability weight on the off-the-equilibrium message of low price originating from a medium-cost firm, and not from a lowor high-cost firm. If, instead, c1 = 1/2 but c2 = 0, the uninformed firm believes that the off-the-equilibrium message of low price is equally likely to originate from a low- or a high-cost firm.

Exercise 8—Second-Degree Price DiscriminationB

449

tickets in advance (Passengers do not like to commit themselves in advance to traveling at a particular time.). More specifically, the utility functions of each traveler net of the price of the ticket, p, are: Business traveler : v  hB p  w Tourist traveler : v  hT p  w where w is the disutility of buying the ticket w days prior to the flight and the price coefficients are such that 0\hB \hT . The proportion of travelers who are tourists is k. Assume that the cost that the airline incurs when transporting a passenger is c [ 0, regardless of the passenger’s type. Part (a) Describe the profit-maximizing price discrimination scheme (second degree price discrimination) under the assumption that Alaska Airlines sells tickets to both types of travelers. How does it depend on the underlying parameters k, hB, hT, and c. Part (b) Under what circumstances does Alaska Airlines choose to serve only business travelers? Answer: Part (a) In order to find the profit-maximizing discrimination scheme, we must find pairs of prices and waiting time, ðpB ; wB Þ and ðpT ; wT Þ, that solve the following constrained maximization problem max kpT þ ð1  kÞpB  2c subject to v  hB pB  wB  v  hB pT  wT v  hT pT  wT  v  hT pB  wB v  hB pB  wB  0 v  hT pT  wT  0 pB ; w B ; pT ; w T ;  0

ðICB Þ ðICT Þ ðIRB Þ ðIRT Þ

Intuitively, incentive compatibility condition ICB (ICT) represents that business (tourist, respectively) travelers prefer the pair of price and waiting time offered to their type then that of the other type. Individual rationality constraint IRB (IRT) describe that business (tourist) travelers obtain a positive utility level from accepting the pair of ticket price and waiting time from Alaska Airlines (we normalize the utility from not traveling, in the right-hand side of the individual rationality constraints, to be zero).

450

9

Perfect Bayesian Equilibrium and Signaling Games

In addition, constraints ICB and IRT must be binding. Otherwise, the firm could extract more surplus from either type of traveler, thus increasing profits. Therefore, IRT can be expressed as v  hT pT  wT ¼ 0 , wT ¼ v  hT pT while ICB can be written as v  hB pB  wB ¼ v  hB pT  v þ hT pT , wB ¼ v  hB pB  ðhT  hB ÞpT Hence, the firm’s maximization problem becomes selecting ticket prices pB and pT (which reduces our number of choice variables from four in the previous problem to only two now) as follows max kpT þ ð1  kÞpB  2c pT ;pB

subject to

ðpT ; wT Þ ¼ ðpT ; v  hT pT Þ ðpB ; wB Þ ¼ ðpB ; v  hB pB  ðhT  hB ÞpT Þ

Before continuing any further, let us show that the waiting time for business travelers, wB , does not affect Alaska Airlines profits. Proof: Assume, by contradiction, that wB [ 0. Then wB can be reduced by e [ 0 units, and increase the price charged to this type of passengers, pB , by heB , so that business travelers’ utility level does not change, and the firm earns higher profits. We now need to check that tourists will not choose this new package offered to business travelers. That is, v  hT p T  w T  v  h B p B  w B rearranging, hT pT þ wT  hB pB þ wB , which implies,

 e e hT pB þ þ ðwB  eÞ\hT pB þ þ ðwB  eÞ hT hB This result, however, contradicts that wB [ 0 can be part of an optimal contract. Hence, only wB ¼ 0 can be part of an optimal incentive contract (Q.E.D.). Hence, from the above claim wB ¼ 0, which yields v  hB pB  ðhT  hB ÞpT ¼ 0

Exercise 8—Second-Degree Price DiscriminationB

451

and solving for price pB , we obtain pB ¼

1 ½v  ðhT  hB ÞpT  hB

Plugging this price into the firm’s maximization problem, we further reduce the set of choice variables to only one price, pT , as follows: max kpT þ ð1  kÞ pT

1 ½v  ðhT  hB ÞpT   2c hB

which can be alternatively expressed as  hT  hB 1 max pT k  ð1  kÞ þ ð1  kÞ v  2c pT hB hB Taking first-order conditions with respect to pT , we find k  ð1  kÞ

hT  hB ¼0 hB

and rearranging k hT  hB ¼ 1k hB At this point we can identify three solutions: one interior and two corners, as we next separately describe. k B (1) If 1k ¼ hThh , any contract ðpB ; wB Þ and ðpT ; wT Þ satisfying the following B conditions is optimal:

1 ½v  ðhT  hB ÞpT   0 hB Waiting times : wB ¼ 0 and wT ¼ 0

Prices : pT  0 and pB ¼

k B (2) If 1k [ hThh , the willingness to pay of tourist travelers is relatively high B (recall that a low value of parameter hT indicates that a higher price does not reduce the traveler’s utility significantly. Since hT is relatively low, a marginal increase in prices does not substantially reduce the tourists’ utility. Then, it is optimal to raise price pT (which reduces the waiting time of this type of traveler wT ). In fact, it is optimal to do it until wT is reduced to wT ¼ 0. Hence,

wT ¼ v  hT pT ¼ 0;

452

9

Perfect Bayesian Equilibrium and Signaling Games

which yields a price of pT ¼ hvT and wB ¼ v  hB pB  ðhT  hB Þ

v ¼ 0; hT

which entails a price of pB ¼ hvT : Therefore, the optimal contract in this case

 coincides for both type of travelers, ðpT ; wT Þ ¼ ðpB ; wB Þ ¼ hvT ; 0 , i.e., a pooling scheme. k B (3) 1k \ hThh . In this case, tourists’ willingness to pay is relatively low, i.e., the B disutility they experience from a higher price, hT , is very high. It is, hence, optimal to reduce the price pT as much as possible ðpT ¼ 0Þ, which implies that its corresponding waiting time wT becomes wT ¼ v  ht pT ¼ v (because ht pT ¼ 0). For business travelers, this implies a waiting time of wB ¼ v  hB pB  ðhT  hB ÞpT ¼ 0; which entails a price of pB ¼ hvB . Therefore, in case 3 optimal contracts satisfy

 ðpT ; wT Þ ¼ ð0; vÞ and ðpB ; wB Þ ¼ hvB ; 0 . Intuitively, case 3 (in which only business travelers are serviced) is more likely to emerge when a) the proportion of business travelers is large enough (small k); and/or b) business travelers suffer less from higher prices (small hB ). Case 2 (serving both types of travelers) is more likely to arise when: a) the proportion of business travelers is small (large k); and/or b) tourists suffer less from higher prices (small hT ). Figure 9.57 summarizes under which conditions of the unit cost, c, the discriminating monopolist chooses to serve all types of travelers, only business travelers, or no traveler at all. Serving everyone: The (pooling) price of serving every type of traveler described in case 2 is pT ¼ pB ¼ hvT . Hence, when costs are below this price, the firm serves all types of travelers. Only Business travelers: When c [ hvB the above pooling pricing scheme would imply a sure loss of money for the firm. In this case, the firm chooses to serve only business travelers since pB ¼ hvB [ c (as in case 3 above). No operation: When pB ¼ hvB \c, costs are so high that it is not profitable to serve business travelers. Hence the firm decides to not operate at all.

Fig. 9.57 Summary of price discrimination results

Exercise 9—Applying the Cho …

453

Exercise 9—Applying the Cho and Kreps’ (1987) Intuitive Criterion in the Far WestB Consider the signaling game depicted in Fig. 9.58, usually referred as “Breakfast in the Far West” or “Beer-Quiche game.” The game describes a saloon in the Far West, in which a newcomer (player 1) enters. His shooting ability (he is either a wimpy or a surely type) is his private information. Observing his own type, he orders breakfast: either quiche or beer (no more options today; this is a saloon in the Far West!). The troublemaker in this town (player 2) shows up who, observing the choice for breakfast of the newcomer but not observing the newcomer’s type, decides whether to duel or not duel him. Despite being a troublemaker, he would prefer a duel with a wimpy type and avoid the duel otherwise. This signaling game has several PBEs but, importantly, it has two pooling equilibria: one in which both types of player 1 choose to have quiche for breakfast, and another in which both types have beer for breakfast (At this point of the chapter you should be able to confirm this result. You can do it as practice.). Part (a) Check if the pooling equilibrium in which both types of player 1 have beer for breakfast survives the Cho and Kreps’ (1987) Intuitive Criterion. Figure 9.59 depicts the pooling PBE where both types of player 1 have beer for breakfast, shading the branches that each player selects. Part (b) Check if the pooling equilibrium in which both types of player 1 have quiche for breakfast survives the Cho and Kreps’ (1987) Intuitive Criterion. Figure 9.60 illustrates the pooling PBE in which both types of player 1 have quiche for breakfast. Answer: Part (a) First step: In the first step of Cho and Kreps’ (1987) Intuitive Criterion, we want to eliminate those off-the-equilibrium messages that are equilibrium dominated. For

(1,1)

Duel Player 2

(3,0)

Quiche

Player 1

Wimpy

Not Duel

Beer

Duel

(0,1)

Not Duel

(2,0)

Duel

(1,-1)

Not Duel

(3,0)

Player 2

0.1

Nature

(0,-1)

(2,0)

Duel

Not Duel

Surely

Player 2

Quiche

0.9

Player 1

Beer

Fig. 9.58 Breakfast in the far west (“Beer-Quicke” game)

Player 2

454

9 (1,1)

Duel Player 2

(3,0)

Perfect Bayesian Equilibrium and Signaling Games

Player 1

Quiche

Wimpy

Not Duel

Beer

Duel

(0,1)

Not Duel

(2,0)

Duel

(1,-1)

Not Duel

(3,0)

Duel

(0,1)

Not Duel

(2,0)

Duel

(1,-1)

Not Duel

(3,0)

Player 2

0.1 Nature

(0,-1)

(2,0)

Duel

Not Duel

Surely

Player 2

Quiche

0.9

Player 1

Beer

Player 2

Fig. 9.59 Pooling equilibrium (Beer, Beer)

(1,1)

Duel Player 2

(3,0)

Quiche

Player 1

Wimpy

Not Duel

Beer

Player 2

0.1 Nature

(0,-1)

(2,0)

Duel

Not Duel

Surely

Player 2

Quiche

0.9

Player 1

Beer

Player 2

Fig. 9.60 Pooling equilibrium (Quiche, Quiche)

the case of a wimpy player 1, we need to check if having quiche can improve his equilibrium utility level (from having beer). That is, if u1 ðBeerjWimpyÞ \ |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Equil:Payoff;2

Max u1 ðQuichejWimpyÞ a2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Highest payoff from deviating towards Quiche; 3

This condition can be indeed satisfied if player 2 responds to the off-the-equilibrium message of quiche not dueling: the wimpy player 1 would obtain a payoff of 3 instead of 2 in this pooling equilibrium. Hence, the wimpy player 1 has incentives to deviate from this separating PBE. Intuitively, this would occur if, by deviating towards quiche, the wimpy type of player 1 can still convince player 2 not to duel, i.e., player 1 would be having his preferred breakfast while still avoiding the duel.

Exercise 9—Applying the Cho …

455

Let us now check the surely type of player 1. We need to check if an (off-the-equilibrium) message of quiche could ever be convenient for the surely type. That is, if u1 ðBeerjSurelyÞ \ |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Equil:Payoff;3

max u1 ðQuichejSurelyÞ a2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Highest payoff from deviating towards Quiche; 2

But this condition is not satisfied: the surely player 1 obtains in equilibrium a payoff of 3 (from having beer for breakfast), and the highest payoff he could obtain from deviating to quiche is only 2 (as a surely type, he dislikes quiche but loves to have a beer for breakfast). Hence, the surely type would never deviate from beer. Therefore, Player 2’s beliefs, after observing the (off-the-equilibrium) message of quiche, can be restricted to H ðQuicheÞ ¼ fWimpyg. That is, if player 2 were to observe player 1 choosing quiche, he would believe that player 1 must be wimpy, since this is the only type of player 1 who could benefit from deviating from the pooling equilibrium of beer. Second step: After restricting the receiver’s beliefs to H ðQuicheÞ ¼ fWimpyg, player 2 plays duel every time that he observes the (off-the-equilibrium) message of quiche, since he is sure that player 1 must be wimpy. The second step of the Intuitive Criterion analyzes if there is any type of sender (wimpy or surely) and any type of message he would send that satisfies: min

a2A ½H ðmÞ;m

ui ðm; a; hÞ [ ui ðhÞ

In this context, this condition does not hold for the wimpy type: the minimal payoff he can obtain by deviating (having quiche for breakfast) once the receiver’s beliefs have been restricted to H ðQuicheÞ ¼ fWimpyg, is 1 (given that when observing that player 1 had quiche for breakfast, player 2 infers that he must be wimpy, and responds dueling). In contrast, his equilibrium payoff from having beer for breakfast in this pooling equilibrium is 2. A similar argument is applicable to the surely type: in equilibrium he obtains a payoff of 3, and by deviating towards quiche he will be dueled by player 2, obtaining a payoff of 0 (since he does not like quiche, and in addition, he has to fight). As a consequence, no type of player deviates towards quiche, and the pooling equilibrium in which both types of player 1 have beer for breakfast survives the Cho and Kreps’ (1987) Intuitive Criterion.

456

9

Perfect Bayesian Equilibrium and Signaling Games

Part (b) First step: Let us now analyze the pooling PBE in which both types of player 1 choose quiche for breakfast. In the first step of the Intuitive Criterion, we seek to eliminate those off-the-equilibrium messages that are equilibrium dominated. For the case of a wimpy player 1, we need to check if having beer can improve his equilibrium utility level (of having quiche in this pooling PBE). That is, if u1 ðQuichejWimpyÞ \ |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Equil:Payoff ;3

max u1 ðBeerjWimpyÞ a2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Highest payoff from deviating towards Beer; 2

This condition is not satisfied: the wimpy player 1 obtains a payoff of 3 in this equilibrium (look at the lower left-hand corner of Fig. 9.54: the wimpy type loves to have quiche for breakfast and avoid the duel!). By deviating towards beer, the highest payoff that he could obtain is only 2 (if he were to still avoid the duel). Let us now check if the equivalent condition holds for the surely type of player 1, u1 ðQuichejSurelyÞ \ |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Equil:Payoff;2

max u1 ðBeerjSurelyÞ a2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Highest payoff from deviatingtowards Beer;3

This condition is satisfied in this case: the surely player 1 obtains in equilibrium a payoff of 2 (from having quiche for breakfast, which he dislikes), and the highest payoff he could obtain from deviating towards beer is 3. Notice that this would happen if, upon observing the off-the-equilibrium message of beer, player 2 infers that such a message can only originate from a surely type, and thus refrains from dueling. In summary, the surely type is the only type of player 1 who has incentives to deviate towards beer. Player 2’s beliefs, after observing the (off-the-equilibrium) message of beer, can then be restricted to H ðBeerÞ ¼ fSurelyg. Second step: After restricting the receiver’s beliefs to H ðBeerÞ ¼ fSurelyg, player 2 responds by not dueling player 1 as he assigns full probability to player 1 being a surely type. The second step of the Intuitive Criterion analyzes if there is any type of sender (wimpy or surely) and any type of message he would send that satisfies: min

a2A ½H ðmÞ;m

ui ðm; a; hÞ [ ui ðhÞ

This condition holds for the surely type: the minimal payoff he can obtain by deviating (having beer for breakfast) after the receiver’s beliefs have been restricted to H ðBeerÞ ¼ fSurelyg, is 3 (given that player 2 responds by not dueling any

References

457

player who drinks beer). In contrast, his equilibrium payoff from having quiche for breakfast in this pooling equilibrium is only 2. As a consequence, the surely type has incentives to deviate to beer, and the pooling equilibrium in which both types of player 1 have quiche for breakfast does not survive the Cho and Kreps’ (1987) Intuitive Criterion.5

References Cho, I. and Kreps, D. (1987). Signaling games and stable equilibrium. Quarterly Journal of Economics, 102, 179–222. Spence, M. (1973). Job market signaling. Quarterly Journal of Economics, 87, 355–374.

5 For applications of the Cho and Kreps’ (1987) Intuitive Criterion to games in which the sender has a continuum of possible messages and the receiver a continuum of available responses, see the Exercise 4 in Chap. 10 about the labor market signaling game in the next chapter.

Cheap Talk Games

10

Introduction This chapter focuses on a particular type of signaling games where all messages are equally costly for the sender and, in addition, this common cost is normalized to zero. Intuitively, all messages are costless for the sender. A typical example is that of a lobbyist, who privately observes information about some industry (or a NGO privately observing precise information about a natural park), sending a message to a politician who, despite having some information about the industry, does not have access to the precise information of the lobbyist. The receiver (politician) knows that all messages from the sender (lobbyist) are costless, and that the sender may have a bias towards certain outcomes, e.g., more generous subsidies. This type of games are often known as “cheap talk” games since the sender’s messages are all cheap (costless), just as talking is cheap (e.g., when speaking to another person, you can say one word or another, both being equally costly for you). In this setting, we seek to identify under which conditions “communication” can take place between the sender and receiver, understood as a separating PBE where the sender chooses a different message depending on his privately observed type, so the receiver can infer the sender’s type by just looking at the message he sent. At first glance, we may expect that a separating PBE cannot be sustained, since all types of sender will have incentives to lie about their type since they do not experience higher costs when sending other messages; and only pooling PBE can be supported where all types of sender choose the same message. However, as our next exercises show, a separating PBE can be sustained if, essentially, the preferences of the sender and receiver are not extremely different, i.e., if the sender’s bias towards a particular outcome is not substantial. In other words, when sender’s and receiver’s preferences are sufficiently aligned, meaningful communication can emerge in equilibrium as part of a separating PBE. For simplicity, Exercise 1 examines a cheap talk game where the sender can only have two possible types (e.g., high or low). The advantage of starting with this setting is that we can graphically represent the game as the labor-market signaling © Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_10

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game in Chap. 9, and readily apply the same approach to test for separating and pooling PBEs. In Exercise 1, the receiver (politician) can only choose among three possible responses, while Exercise 2 extends this setting to one in which the receiver has a continuum of policies to respond with. Exercise 3 then provides a further extension, where both the sender and receiver have a continuous action space, i.e., the sender has a continuum of message to choose from and the receiver has a continuum of potential responses to choose from. However, the exercise still restricts sender’s types to be only two. Exercise 4 relaxes this assumption too, by allowing for the sender to have N possible types. This is the canonical cheap-talk model presented in the literature.1

Exercise 1—Cheap Talk Game with Only Two Sender’s Types, Two Messages, and Three ResponsesB Figure 10.1 depicts a cheap talk game. In particular, the sender’s payoff coincides when he sends message m1 or m2, and only depends on the receiver’s response (either a, b or c) and the nature’s type. You can interpret this strategic setting as a lobbyist (Sender) informing a Congressman (Receiver) about the situation of the industry he represents: messages “good situation” or “bad situation” are equally costly for him, but the reaction of the politician to these message (and the actual state of the industry) determine the lobbyist payoff. A similar argument applies for the payoffs of the receiver (Congressman), which do not depend on the particular message he receives in his conversation with the lobbyist, but are only a function of the specific state of the industry (something he cannot observe) and the action he chooses (e.g., the policy that he designs for the industry after his conversation with the lobbyist). For instance, when the sender is type t1, in the left-hand side of the tree, payoff pairs only depend on the receiver’s response (a, b or c) but do not depend on the sender’s message, e.g., when the receiver responds with a players obtain (4, 3), both when the original message was m1 and m2. This cheap talk game can be alternatively represented with Fig. 10.2. Part (a) Check if a separating strategy profile in which only the sender with type t1 sends message m1 can be sustained as a PBE. Part (b) Check if a pooling strategy profile in which both types of the sender choose message m1 can be supported as a PBE. Part (c) Assume now that the probability that the sender is type t1 is p and, hence, the probability that the sender is type t2 is 1 − p. Find the values of p such that the pooling PBE you found in part (b) of the exercise can be sustained and predict that the receiver responds using action b.

See the original article by Crawford, V. and Sobel, J. 1982. “Strategic information transmission.” Econometrica, 50, pp. 1431–51.

1

Exercise 1—Cheap Talk Game with Only Two Sender’s Types …

461

Fig. 10.1 Cheap talk game

Fig. 10.2 An alternative representation of the cheap talk game

Answer Part (a) Separating strategy profile (m1,m2′): Figure 10.3 depicts the separating strategy profile (m1, m2′), where message m1 only originates from a sender of type t1 (see shaded branch in the upper part of the tree) while message m2 only stems from a sender with type t2 (see lower part of the tree).

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Fig. 10.3 Separating strategy profile

Receiver’s
beliefs: Let l ti jmj represent the conditional probability that the receiver assigns to the sender being of type ti given that message mj is observed, where i, j 2 {1, 2}. Thus, after observing message m1 , the receiver assigns full probability to such a message originating only from t1-type of sender, lðt1 jm1 Þ ¼ 1;

and lðt2 jm1 Þ ¼ 0

while after observing message m2 , the receiver infers that it must originate from t2type of sender, lðt1 jm2 Þ ¼ 0;

and lðt2 jm2 Þ ¼ 1

Receiver’s optimal response: • After observing m1 , the receiver believes that such a message can only originate from a t1 -type of sender. Graphically, the receiver is convinced to be in the upper left-hand corner of Fig. 10.3. In this setting, the receiver’s optimal response is b given that it yields a payoff of 4 (higher than what he gets from a, 3, and from c, 0.) • After observing message m2 , the receiver believes that such a message can only originate from a t2 -type of sender. That is, he is convinced to be located in the lower right-hand corner of Fig. 10.3. Hence, his optimal response is c, which yields a payoff of 5, which exceeds his payoff from a, 4; or b, which entails a zero payoff. Figure 10.4 summarizes these optimal responses of the receiver, by shading branch b upon observing message m1 in the left-hand side of the tree, and c upon observing m2 in the right-hand side of the tree.

Exercise 1—Cheap Talk Game with Only Two Sender’s Types …

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Fig. 10.4 Optimal responses in the separating strategy profile

Sender’s optimal messages: • If his type is t1 , by sending m1 he obtains a payoff of 2 (since m1 is responded with b), but a lower payoff of 1 if he deviates towards message m2 (since such a message is responded with c). Hence, the sender doesn’t want to deviate from m1 . • If his type is t2 , he obtains a payoff of 3 by sending message m2 (which is responded with c) and a payoff of 1 if he deviates to message m1 (which is responded with b). Hence, he doesn’t have incentives to deviate from m2 . Hence, the initially prescribed separating strategy profile can be supported as a PBE. As a curiosity, note that the opposite separating strategy profile ðm2 ; m01 Þ can also sustained as a PBE (you can check that as a practice). Part (b) Pooling strategy profile ðm1 ; m01 Þ. Figure 10.5 depicts the pooling strategy profile ðm1 ; m01 Þ in which both types of senders choose the same message m1. Receiver’s beliefs: After observing message m1 (which occurs in equilibrium), beliefs coincide with the prior probability distribution over types. Indeed, applying Bayes’ rule we find that lðt1 jm1 Þ ¼

pðt1 Þ  pðm1 jt1 Þ 0:6  1 ¼ 0:6 ¼ pðm1 Þ 0:6  1 þ 0:4  1

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Fig. 10.5 Pooling strategy profile

lðt2 jm1 Þ ¼

pðt2 Þ  pðm1 jt2 Þ 0:4  1 ¼ 0:4 ¼ pðm1 Þ 0:6  1 þ 0:4  1

After receiving message m2 (what happens off-the-equilibrium path), beliefs cannot be updated using Bayes’ rule since lðt1 jm2 Þ ¼

pðt1 Þ  pðm2 jt1 Þ 0:6  0 0 ¼ ¼ pðm2 Þ 0:6  0 þ 0:4  0 0

and hence off-the-equilibrium beliefs must be arbitrarily specified, i.e. l ¼ lðt1 jm2 Þ 2 ½0; 1. Receiver’s optimal response: • After receiving message m1 (in equilibrium), the receiver’s expected utility from responding with actions a, b, and c is, respectively Action a: 0:6  3 þ 0:4  4 ¼ 3:4 Action b: 0:6  4 þ 0:4  0 ¼ 2:4; and Action c: 0:6  0 þ 0:4  5 ¼ 2:0 Hence, the receiver’s optimal strategy is to choose action a in response to the equilibrium message of m1 . • After receiving message m2 (off-the-equilibrium), the receiver’s expected utility from each of his three possible responses are

Exercise 1—Cheap Talk Game with Only Two Sender’s Types …

465

EUReceiver ðajm2 Þ ¼ l  3 þ ð1  lÞ  4 ¼ 4  l; EUReceiver ðbjm2 Þ ¼ l  4 þ ð1  lÞ  0 ¼ 4l; and EUReceiver ðcjm2 Þ ¼ l  0 þ ð1  lÞ  5 ¼ 5  5l where the receiver’s response critically depends on the particular value of his off-the-equilibrium belief l, i.e., 4l > 4 − l if and only if l [ 45, 4 – l > 5 – 5l if and only if l [ 14, and 4l > 5 − 5l if and only if l [ 59. Hence, if   off-the-equilibrium beliefs lie within the interval l 2 0; 14 , 5 − 5l is the highest expected payoff, thus inducing the receiver to respond with c; if they lie on the interval l 2 ð14 ; 45, 4 − l becomes the highest expected payoff and the receiver chooses a; finally, if off-the-equilibrium beliefs lie on the interval l 2 ð45 ; 1, 4l is the highest expected payoff and the receiver responds with b. For simplicity, we only focus on the case in which off-the-equilibrium beliefs satisfy l = 1 (and thus the responder chooses b upon observing message m2). Figure 10.6 summarizes the optimal responses of the receiver (a after m1, and b after m2) in this pooling strategy profile. Sender’s optimal message: • If his type is t1 , the sender obtains a payoff of 4 from sending m1 (since it is responded with a), but a payoff of only 2 when deviating towards m2 (since it is responded with b). Hence, he doesn’t have incentives to deviate from m1 . • If his type is t2 , the sender obtains a payoff of 2 by sending m1 ’ (since it is responded with a), but a payoff of only 1 by deviating towards m2 ’ (since it is responded with b). Hence, he doesn’t have incentives to deviate from m1 ’.

Fig. 10.6 Optimal responses in the pooling strategy profile

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Fig. 10.7 Pooling strategy profile

Therefore, the initially prescribed pooling strategy profile where both types of sender select m1 can be sustained as a PBE of the game. As a practice, check that this equilibrium can be supported for any off-the-equilibrium beliefs, lðt1 jm2 Þ, the receiver sustains upon observing message m2. You can easily check that the opposite pooling strategy profile ðm2 ; m02 Þ can also be sustained as a PBE. However, this PBE is only supported for a precise set of off-the-equilibrium beliefs, and the receiver does not respond with action b in equilibrium. Part (c) Fig. 10.7 depicts the pooling strategy profile ðm1 ; m01 Þ, where note that the prior probability of type t1 is now, for generality, p, rather than 0.6 in the previous parts of the exercise. Receiver’s beliefs: After observing message m1 (which occurs in equilibrium), his beliefs coincide with the prior probabilities over types, lðt1 jm1 Þ ¼

p1 ¼ p; and p  1 þ ð1  pÞ  1

lðt2 jm1 Þ ¼

ð1  pÞ  1 ¼1p p  1 þ ð1  pÞ  1

After receiving message m2 (off-the-equilibrium path), his beliefs cannot be updated using Bayes’ rule since lðt1 jm2 Þ ¼

p0 0 ¼ p  0 þ ð1  pÞ  0 0

Exercise 1—Cheap Talk Game with Only Two Sender’s Types …

467

Hence, off-the-equilibrium beliefs must be left arbitrarily specified, i.e., l ¼ lðt1 jm2 Þ 2 ½0; 1. Receiver’s optimal response: • After receiving a message m1 (in equilibrium), the receiver’s expected utility from responding with actions a, b, and c is, respectively, Action a: p  3 þ ð1  pÞ  4 ¼ 4  p Action b: p  4 þ ð1  pÞ  0 ¼ 4p; and Action c: p  0 þ ð1  pÞ  5 ¼ 5  5p For the receiver to be optimal to respond choosing action b (as required in the exercise) it must be the case that 4 ; and 5 5 4p [ 5  5p; i:e:; p [ 9 4p [ 4  p; i:e:; p [

Thus, since p [ 45 is more restrictive than p [ 59, we can claim that if p [ 45 the receiver’s optimal strategy is to choose b in response to the equilibrium message m1 . • After receiving message m2 (off-the-equilibrium), the receiver’s expected utility from each of his three possible responses is, respectively, EUReceiver ðajm2 Þ ¼ l  3 þ ð1  lÞ  4 ¼ 4  l; EUReceiver ðbjm2 Þ ¼ l  4 þ ð1  lÞ  0 ¼ 4l; and EUReceiver ðcjm2 Þ ¼ l  0 þ ð1  lÞ  5 ¼ 5  5l which coincides with the receiver’s expected payoffs in part (b). From our discussion in part (b), the receiver’s response depends on his off-the-equilibrium belief l. However, for simplicity, we consider that l ¼ 1, which implies that the receiver responds with b after observing the off-the-equilibrium message m2. Figure 10.8 summarizes the optimal responses of the receiver in this pooling strategy profile. Sender’s optimal message: • If his type is t1 (in the upper part of Fig. 10.8), the sender obtains a payoff of 2 from sending m1 (since it is responded with b), and the same exact payoff when

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Fig. 10.8 Optimal responses in the pooling strategy profile

deviating towards m2 (since it is responded with b). Hence, he does not have strict incentives to deviate from m1 . • If his type is t2 (in the lower part of Fig. 10.8), the sender obtains a payoff of 1 by sending m1 ’ (since it is responded with b), and the same payoff by deviating towards m2 ’ (since it is responded with b). Hence, he does not have strict incentives to deviate from m1 . Therefore, the initially prescribed pooling strategy profile, where both types of sender select m1 and the receiver responds with b, can be sustained as a PBE of the game when p [ 45.

Exercise 2—Cheap Talk Game with Only Two Sender Types, Two Messages, and a Continuum of ResponsesB Let us consider a cheap talk game, where a lobbyist tries to persuade a politician through the information that the lobby transmits with its messages. The lobbyist initially observes the true state of the world, either hH and hL (both equally likely), which can be interpreted as the current situation of the industry or constituents the lobbyist represents. Afterwards, the lobbyist decides which message to send to the politician. For simplicity, let us assume that the message space coincides with the type space. That is, the lobbyist can only send either hH or hL. After observing the lobbyist’s message, the government (or a politician) decides which optimal action to take as a response. The government’s and the lobbyist’s utility functions (G and U below) are quadratic in the distance between their ideal point and the final policy implemented by the politician: Gðp; hÞ ¼ ðp  hÞ2

and

U ðp; hÞ ¼ ðp  h  bÞ2 ;

Exercise 2—Cheap Talk Game with Only Two Sender Types …

469

where b represents a parameter reflecting the degree of divergence between the lobbyist and the politician preferences. Alternatively, b represents the bids that the lobbyist exhibits in favor of more intense policies. (a) Find a Separating PBE where the sender sends hH when hH is the true state of the world, and hL when hL is the true state of the world. (b) Find a Pooling PBE where the sender sends hH when hH is the true state of the world, and also hH when hL is the true state of the world. Answer (a) Let us follow the usual steps to test for the existence of a separating PBE. In order to facilitate a visual reference, Fig. 10.9 illustrates a cheap talk game with only two types (hH or hL), only two messages (µ = hH or µ = hL), but a continuum of possible responses from the receiver (i.e., a continuum of policies from the politician). The figure shades the branches of messages µ = hH for the hH-lobbyist and µ = hL for the hL-lobbyist in the separating strategy profile we analyze. (1) We first define the receiver’s beliefs (the politician’s beliefs) at this Separating PBE. • When observing hH, µ(hH|hH) = 1 since this message can only originate from a lobbyist who observed hH as the state of the world. Hence µ (hH|hL) = 0. • And when observing hL,, µ(hL|hL) = 1 given that message µ = hL only originates from a lobbyist who observed hL. Hence µ(hL|hH) = 0.

Fig. 10.9 Separating strategy profile in the cheap talk game

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(2) We can now describe the politician’s optimal actions: first after observing hH; and second, after observing hL. • If hH is observed, then hH is believed, and this exact policy p = hH is implemented, since hH minimizes the difference (p − hH) in the politician’s utility function, G(p, h) = −(p − hH)2. (Formally, you can take first order conditions with respect to p, which yield −2(p − hH) = 0. Hence, p ¼ hH is the utility-maximizing policy for the politician.) • Similarly, if hL is observed, then hL is believed, and this policy p = hL is implemented, since hL minimizes the difference (p- hL) in the politician’s utility function, G(p, h) = −(p − hL)2. (3) Given the previous steps, we can describe the sender’s (lobbyist’s) optimal message when hH is the true state of the world. • If hH is the true state of the world, then a message hH is sent. Sending a message hL is not utility-maximizing in this case since it would be believed as hL (and as a consequence a policy p = hL would be implemented). This is clearly not a profitable deviation, given that the lobbyist seeks to induce a policy hH from the politician. (4) We can now describe the sender’s (lobbyist) optimal message when hL is the true state of the world. (Remember that the need to check what condition must be satisfied in order to induce the lobbyist to truthfully reveal that hL is the true state of the world, instead of lying by sending a message hH.) • If hL is the true state of the world, then a message hL is sent if and only if the sender (lobbyist) is better off by sending hL (and having hL implemented as a policy) rather than by sending a false message hH (and having hH believed and implemented by the politician). The last statement is true if and only if ðhL  hL  bÞ2   ðhH  hL  bÞ2 rearranging, and solving for b, we obtain b

hH  hL 2

Then, the lobbyist sends hL when hL is the true state of the world if and L only if b  hH h holds. 2

Exercise 2—Cheap Talk Game with Only Two Sender Types …

471

Fig. 10.10 Pooling strategy profile in the cheap talk game

(5) Let us finally describe this fully informative (separating) PBE we just found. Sender (lobbyist): send hH when hH is the true state of the world (under all parameter values), and send hL when hL is the true state of the world if and L only if b\ hH h holds. 2 Receiver (politician): implement a policy p = hH if a message hH is received, and a policy p = hL if a message hL is received. Receiver’s beliefs: When observing hH, µ(hH|hH) = 1 and hence µ (hH|hL) = 0; and when observing hL, µ(hL|hL) = 1 and hence µ(hL|hH) = 0. (b) Fig. 10.10 illustrates the pooling strategy profile in which both types of lobbyists send message µ = hH. (1) Let us first define the receiver’s beliefs (the politician beliefs) at this Pooling PBE. • When observing hH, µ(hH|hH) = µ(hH|hL) = 1/2. Since both types of lobbyists are sending hH, the receiver cannot infer any additional information from the sender’s actions, and his posterior probability distribution coincides with the prior probability of having a true state of the world of hH. • And when observing hL,, µ(hL|hL) and µ(hL|hH) can be arbitrarily specified, since beliefs are on the off-the-equilibrium path. That is, if the sender sends hH regardless of the true state of the world, the receiver will only observe hL as a surprise (since it is not happening on-the-equilibrium path that we specified above). Remember that in this

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case using Bayes’ rule is not helpful, since it provides us with a ratio of 0/0, which is undetermined. (2) We can now describe the politician’s optimal actions: first, after observing hH; and second, after observing hL. • If hH is received, then the true state of the world can be anything. The result from maximizing the receiver’s expected utility gives us an optimal policy of p = 0.5hH + 0.5hL. (In particular, the politician now maximizes an expected utility  12 ðp  hH Þ2  12 ðp  hL Þ2 , which yields first order condition of ðp  hH Þ  ðp  hL Þ ¼ 0, and solving for p, we find p ¼ 12 hH þ 12 hL .) • If hL is received, then we are on the off-the-equilibrium path. In this situation, in order to determine the optimal policy of the receiver, we need to figure out the politican’s policy choice p that maximizes his expected utility, as follows: max lðhL jhH Þðp  hH Þ2  lðhL jhL Þðp  hL Þ2 p0

where the first (second) component measures the politician’s utility when the off-the-equilibrium message is sent from the H-type (L-type) lobbyists, weighted by the politician’s off-the-equilibrium belief of receiving such a message. Since lðhL jhL Þ ¼ 1  lðhL jhH Þ, and the negative of a maximization problem is the same as a minimization problem, we can simplify to yield min p0

lðhL jhH Þðp  hH Þ2 þ ½1  lðhL jhH Þðp  hL Þ2

Taking the first order condition with respect to the policy choice p, and setting it equal to zero by assuming an interior solution for p, that is, p > 0, 2p  2lðhL jhH ÞhH  2½1  lðhL jhH ÞhL ¼ 0 which, after rearranging, yields p ¼ lðhL jhH ÞhH þ lðhL jhL ÞhL that says the optimal policy choice p on the off-the-equilibrium path should be the realization of the senders’ types weighted by the probability of deviation. • Let us assume, for example, that the politician believes that only a sender with a true state of the world of hL could ever make such a

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473

surprising message of hL. That is, when observing hL, the receiver beliefs are updated using Bayes’s rule, yielding lðhL jhL Þ ¼ 1

and hence lðhL jhH Þ ¼ 0

Then, his optimal policy (the one that minimizes the distance between p and hL in his utility function, is p ¼ 0  hH þ 1  hL ¼ hL :

(3) Let us now describe the sender’s (lobbyist’s) optimal message when hH is the true state of the world; and second, when hL is the true state of the world. • If hH is the true state of the world, then a message hH is sent. • If hL is the true state of the world, then a message hH is also sent if the sender prefers to lie rather than to tell the truth. In point (4) of the Separating PBE we found a condition for truth telling when hL is the true state of the world. Reversing the inequality sign of this condition, L we obtain that if b [ hH h holds then the sender would prefer to send a 2 message of hH when the true state of the world is hL rather than sending a true message of hL. (4) Let us finally describe this uninformative (pooling) PBE. Sender (lobbyist): sends hH when hH is the true state of the world (under all parameter values), and sends hH when hL is the true state of the world if and L only if b [ hH h holds. 2 Receiver (politician): implements a policy p = 0.5hH + 0.5hL if message hH is received, but implements a policy p ¼ lðhL jhH ÞhH þ lðhL jhL ÞhL if message hL is received based on his off-the-equilibrium beliefs at the information set on the right-hand side of the game tree. Receiver’s beliefs: When observing message hH, beliefs µ(hH|hH) = µ (hL|hH) = 1/2; and when observing message hL, beliefs µ(hL|hL) and µ (hL|hL) are unrestricted between [0, 1].

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Exercise 3—Cheap Talk with a Continuum of Types—Only Two Messages (Partitions)B Let us now extend the previous exercise about cheap talk messages, to a setting in which the sender (lobbyist) can have a continuum of types (rather than only two), uniformly distributed in the interval [hmin, hmax]. Answer the following questions, assuming the same utility functions as the ones in the previous exercise. (a) Consider the partially informative strategy profile in Fig. 10.11. In particular, all lobbyists with a type h\h1 send the same message m1 , while those above this cutoff send a different message, m2 . What are the receiver’s beliefs in this partially informative equilibrium? (b) What is the receiver’s optimal action (or policy), given the beliefs you specified in the previous question? (c) What is the value of h1 that makes a sender with type h1 to be indifferent between sending message m1 (inducing an action of p ¼ hmin2þ h1 ) and sending a message of m2 (inducing an action of p ¼ h1 þ2hmax )? [Hint: your expression of h1 should depend on hmax, hmin, and the parameter reflecting the divergence between the receiver and the sender’s preferences, b.] (d) Numerical example. Let us now consider a numerical example. Assume that hmin = 0 and hmax = 1, and rewrite the expression of h1 you found in part (c). We know that h1 must be below 1, since otherwise it would be beyond the upper bound hmax , i.e., h1  1. Plug in this inequality the value for h1 that you found in the previous questions, and solve for parameter b. What is range of values of parameter b that supports this partially informative PBE with two steps (two partitions)? Answer (a) If message m1 is received, the receiver’s beliefs are totally concentrated on the fact that the true state of the world must be within some of the values of interval 1, where h 2 ½hmin ; h1 , and never from interval 2, where h 2 ½h1 ; hmax :

min

Fig. 10.11 A partially informative PBE with two partitions

max

Exercise 3—Cheap Talk with a Continuum of Types …

lðInterval 1jm1 Þ ¼ 1

475

and lðInterval 2jm1 Þ ¼ 0

Similarly if message m2 is received, the receiver is convinced that the true state of the world must be within some of the values of interval 2, and never from interval 1: lðInterval 2jm2 Þ ¼ 1

and lðInterval 1jm2 Þ ¼ 0

(b) If a message on the first interval is received, then beliefs are updated (concentrated on the fact that the true state must be in this first interval). Hence, the optimal action of the receiver is p ¼ hmin2þ h1 . This action minimizes the distance between p and the expected value of h in the first interval, which is hmin2þ h1 . If a message on the second interval is received, then beliefs are updated (concentrated on the fact that the true state must be in this second interval), and hence, the optimal action of the receiver is p ¼ h1 þ2hmax . This action minimizes the distance between p and the expected value of h in the second interval, which is h1 þ2hmax . (c) If h1 is the true state, then the sender sends any message in the first interval. We must check that the sender does not want to deviate to send l2 . The sender observing a true state of the world in the first interval who has the highest incentives to deviate towards a message on the second interval is precisely h1. In fact, he is indifferent between sending a message on the first and on the second interval:  2  2 hmin þ h1 h1 þ hmax   ðh1 þ bÞ ¼   ðh1 þ bÞ 2 2 Rearranging the expression yields, ðh1 þ bÞ 

hmin þ h1 h1 þ hmax ¼  ðh1 þ bÞ 2 2

Further simplifying, 4ðh1 þ bÞ ¼ hmax þ hmin þ 2h1 Solving for h1, we obtain. 1 h1 ¼ ðhmax þ hmin Þ  2b 2

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(d) When hmax ¼ 1 and hmin ¼ 0, the above expression becomes 1 1 h1 ¼ ðhmax þ hmin Þ  2b ¼  2b 2 2 In addition, if h1 ¼ 12  2b must be smaller than 1, we obtain that 1  \2b; 2

and hence b\

1 4

Intuitively, if the preferences between the receiver (politician) and the sender (lobbyist) are no very divergent (b < 1/4), we can obtain a 2-step equilibrium which is partially informative about the true state of the world. That is, some (although not full) information is transmitted from the sender to the receiver. Otherwise (if b > 1/4) we cannot support such a partially informative equilibrium in two steps, and we would only have a pooling equilibrium in which all type of lobbyists in ½hmin ; hmax  send the same message to the politician.

Exercise 4—Cheap Talk Game with a Continuum of Types— Three Messages (Partitions)B Consider the partially informative strategy profile in Fig. 10.12 which shows three partitions of the state space, and where the sender (e.g., lobbyist) sends the following messages: 8 < m1 if h 2 ½hmin ; h1  m if h 2 ½h1 ; h2  mðhÞ ¼ : 2 m3 if h 2 ½h2 ; hmax 

m1

θmin

m2

θ1

m3

θmax

θ2

min

Fig. 10.12 A partially informative PBE with three partitions

max

Exercise 4—Cheap Talk Game with a Continuum of Types …

477

(a) What are the receiver’s beliefs in this partially informative equilibrium? (b) What is the receiver’s optimal action (or policy), given the beliefs you specified in part (a)? (c) What is the value of h1 that makes a sender with type h1 to be indifferent between sending message m1 (inducing an action of p ¼ hmin2þ h1 ) and a message of m2 (inducing an action of p ¼ h1 þ2 h2 )? Similarly, what is the value of h2 that makes a sender with type h2 to be indifferent between sending message m2 (inducing an action of p ¼ h1 þ2 h2 ) and a message of m3 (inducing an action of p ¼ h2 þ2hmax )? (d) Numerical example. Let us now consider a numerical example. Assume that hmin = 0 and hmax = 1, rewrite the expressions of h1 and h2 you found in part (c). What is range of values of parameter d that supports this partially informative PBE with three partitions? Answer (a) If message m1 is received, the receiver’s beliefs are totally concentrated on the fact that the true state of the world must be within some values of interval 1, where h 2 ½hmin ; h1 , and neither from interval 2, where h 2 ½h1 ; h2 , nor interval 3, where h 2 ½h2 ; hmax , such that lðInterval 1jm1 Þ ¼ 1

and

lðInterval 2jm1 Þ ¼ 0

and

lðInterval 3jm1 Þ ¼ 0

If message m2 is received, the receiver’s beliefs are totally concentrated on the fact that the true state of the world must be within some values of interval 2, where h 2 ½h1 ; h2 , and neither from interval 1, where h 2 ½hmin ; h1 , nor interval 3, where h 2 ½h2 ; hmax , such that lðInterval 1jm2 Þ ¼ 0

and

lðInterval 2jm2 Þ ¼ 1

and

lðInterval 3jm2 Þ ¼ 0

Lastly, if message m3 is received, the receiver’s beliefs are totally concentrated on the fact that the true state of the world must be within some values of interval 3, where h 2 ½h2 ; hmax , and neither from interval 1, where h 2 ½hmin ; h1 , nor interval 2, where h 2 ½h1 ; h2 , such that lðInterval 1jm3 Þ ¼ 0

and

lðInterval 2jm3 Þ ¼ 0

and

lðInterval 3jm3 Þ ¼ 1

(b) If m1 is received, then beliefs are updated (concentrated on the fact that the true state must be in the first interval). Hence, the receiver’s optimal action is p¼

hmin þ h1 ; 2

478

10

Cheap Talk Games

which minimizes the distance between p and the expected value of h in the first interval. If m2 is received, then beliefs are updated (concentrated on the fact that the true state must be in the second interval). Hence, the receiver’s optimal action is p¼

h1 þ h2 ; 2

which minimizes the distance between p and the expected value of h in the second interval. If m3 is received, then beliefs are updated (concentrated on the fact that the true state must be in the third interval). Hence, the receiver’s optimal action is p¼

h2 þ hmax ; 2

which minimizes the distance between p and the expected value of h in the third interval. (c) If h1 is the true state, then the sender sends any message in the first interval. We must check that the sender does not want to deviate to send m2 . The sender observing a true state of the world in the first interval who has the highest incentives to deviate towards a message on the second interval is precisely h1. In fact, he is indifferent between sending a message on the first and on the second interval:



 2  2 hmin þ h1 h1 þ h2  ðh1 þ dÞ ¼   ðh1 þ dÞ 2 2

Rearranging the above expression yields, hmin þ h1  2ðh1 þ dÞ ¼ h1  h2 þ 2ðh1 þ dÞ Further simplifying, we obtain h1 ¼

hmin þ h2  2d 2

Similarly, if h2 is the true state, then the sender sends any message in the second interval. We must check that the sender does not want to deviate to send m3 . The sender observing a true state of the world in the second interval who has the highest incentives to deviate towards a message on the third interval is precisely h2. In fact, he is indifferent between sending a message on the second and on the third interval:

Exercise 4—Cheap Talk Game with a Continuum of Types …

479

 2  2 h1 þ h2 h2 þ hmax  ðh2 þ dÞ ¼   ðh2 þ dÞ  2 2 Rearranging the above expression yields, h1 þ h2  2ðh2 þ dÞ ¼ h2  hmax þ 2ðh2 þ dÞ Further simplifying, yields h2 ¼

h1 þ hmax  2d 2

The above calculations imply that if the bias parameter is not too large, the three partitions can be sustained as a partially informative Perfect Bayesian equilibrium. (d) When hmax ¼ 1 and hmin ¼ 0, the above expressions become

h1 ¼ h2 ¼

h2  2d 2

1 þ h1  2d 2

Substituting the first expression into the second expression, and rearranging, we obtain h1 ¼

1  4d 3

h2 ¼

2  4d 3

Since d  0, we have h2  h1 . Therefore, we need h1 [ 0 (which is sufficient for h2 [ 0) as the limiting condition to support a three-partition partially informative equilibrium: 1  4d [ 0; 3

and hence d\

1 12

Intuitively, if the preferences between the receiver (politician) and the sender (lobbyist) are not very divergent ðd\1=12Þ, we can obtain a three-partition partially informative equilibrium. That is, some (although not full) information is transmitted from the sender to the receiver. Otherwise (if d [ 1=12) we cannot

480

10

m1

θ0

m2

θ1

Cheap Talk Games

m3

θ2

mN

θ3

θN-1

θN

Fig. 10.13 A partially informative PBE with N partitions (not necessarily drawn to scale)

support such a partially informative equilibrium, and we may end up with at least two types of lobbyists sending the same message to the politician.

Exercise 5—Cheap Talk Game with a Continuum of Types— N Messages (Partitions)C Consider the cheap-talk model in Fig. 10.13, which depicts a state space divided in N partitions. This generalizes our previous exercise, which only divided the state space into 3 partitions, to a setting where it is divided into finer partitions. Intuitively, we seek to understand the maximal number of partitions that we can sustain as a Perfect Bayesian Equilibrium (PBE) of the cheap-talk game, since a larger number of partitions allows for more information transmission between the informed sender (e.g., lobbyist) and the uniformed receiver of messages (e.g., politician). In this context, consider that the sender sends the following messages for each partition:

mðhÞ ¼

8 m1 > > > > m > 2 > > >
> > > .. > > > . > : mN if h 2 ½hN1 ; hN 

(a) What are the receiver’s beliefs in this partially informative equilibrium? (b) What is the receiver’s optimal action (or policy), given the beliefs you specified in part (a)? (c) What is the value of hk that makes a sender with type hk sending message mk?

Exercise 5—Cheap Talk Game with a Continuum of Types …

481

(d) Numerical example. Let us now consider a numerical example. Assume that the upper and lower bounds of the sender’s type are h0 = 0 and hN = 1. Rewrite the expressions of hk you found in part (c). What is range of values of parameter d that supports this partially informative PBE with N partitions? Answer (a) If message mk is received, the receiver’s beliefs are totally concentrated on the fact that the true state of the world must be within some values of interval k, where h 2 ½hk1 ; hk , such that lðInterval kjmk Þ ¼ 1

and lðInterval jjmk Þ ¼ 0 for all intervals j 6¼ k:

(b) If message mk is received, then beliefs are updated (concentrated on the fact that the true state must be in the kth interval). Hence, the receiver’s optimal action is p¼

hk1 þ hk ; 2

which minimizes the distance between the policy p and the expected value of h in the kth interval. (c) If hk is the true state, then the sender sends any message in the kth interval. We must check that the sender does not have incentives to deviate to send any message mj , where j 6¼ k. It suffices our analysis to show that the sender observing the true state of the world in interval k sends neither message mk1 or mk þ 1 . Specifically, if sending message mk1 ðmk þ 1 Þ is dominated by message mk ,
then sending messages corresponding to even more distant intervals  mkj mk þ j where j  2 would also be dominated (because his utility function is the negative quadratic distance between his type and the message to be sent). First, we check that the kth sender has no incentive to overreport by sending the message corresponding to the interval immediately above k, mk þ 1 , that is 

 2  2 hk1 þ hk hk þ 1 þ hk  ðhk þ dÞ   ðhk þ dÞ 2 2

Rearranging the above expression, we obtain hk1 þ hk  2ðhk þ dÞ   hk þ 1  hk þ 2ðhk þ dÞ Further simplifying, yields hk þ 1  2hk  hk1 þ 4d

482

10

Cheap Talk Games

As an illustration, let us check the initial condition, for which the sender of type h1 has no incentives to overreport by sending a message hk þ 1 ¼ h2 . 

h0 þ h1  ðh1 þ dÞ  2

2

 2 h2 þ h1  ðh1 þ dÞ  2

Rearranging the above expression, h0 þ h1  2ðh1 þ dÞ   h2  h1 þ 2ðh1 þ dÞ Further simplifying, h2  2h1  h0 þ 4d Next, we check that the k + 1th sender has no incentive to underreport by sending the message corresponding to the interval immediately below k, mk ; that is,  2  2 hk þ 1 þ hk hk þ hk1   ðhk þ 1 þ dÞ    ðhk þ 1 þ dÞ 2 2 Rearranging the above expression, we find hk þ 1 þ d 

hk þ 1 þ hk hk þ hk1   ðhk þ 1 þ dÞ 2 2

Further simplifying, yields hk þ 1 

1 ð2hk þ hk1  4dÞ 3

Since hk þ 1 [ hk by construction, if the kth sender has no incentives to overreport, it suffices to say that the sender also has no incentives to underreport.2 Therefore, in general, the condition for the kth sender to send the message corresponding to the interval where the state of the world lies, mk, is hk þ 1  2hk  hk1 þ 4d Check that condition hk þ 1 [ hk [ 23 hk þ 13 hk1 ¼ 13 ð2hk þ hk1 Þ [ 13 ð2hk þ hk1  4dÞ holds for all values of the bias parameter d, which means that the incentive compatibility constraint for no underreporting becomes slack.

2

Exercise 5—Cheap Talk Game with a Continuum of Types …

483

(d) Rearranging the incentive compatibility condition for no overreporting that we obtained at the end of part (c), we find hk þ 1  hk  ðhk  hk1 Þ þ 4d which means that the interval to the right must be weakly larger than the interval to the left to induce the sender not to overreport its type. Further assuming that the above inequality binds, then hk þ 1  hk ¼ ðhk  hk1 Þ þ 4d Graphically, this says that every interval must be at least 4d larger than its predecessor. Assuming that the distance of the first interval, h1  h0 , is d, then h2  h1 ¼ ðh1  h0 Þ þ 4d ¼ d þ 4d h3  h2 ¼ ðh2  h1 Þ þ 4d ¼ ðd þ 4dÞ þ 4d ¼ d þ 2  4d And by recursion, the length of the kth interval is given by hk  hk1 ¼ ðhk1  hk2 Þ þ 4d ¼ ðhk2  hk3 Þ þ 2  4d ¼ ðhk3  hk4 Þ þ 3  4d ¼  ¼ ðh1  h0 Þ þ ðk  1Þ  4d ¼ d þ ðk  1Þ  4d As an illustration, the length of the final interval is hN  hN1 ¼ d þ ðN  1Þ  4d Given that the sender’s type is in between [0, 1], in particular, h0 ¼ hmin ¼ 0 and hN ¼ hmax ¼ 1, we can express the support of the sender’s type as the sum length of N partitions. hN  h0 ¼ ðhN  hN1 Þ þ ðhN1  hN2 Þ þ ðhN2  hN3 Þ þ    þ ðh1  h0 Þ ¼ d þ ðd þ 4dÞ þ ðd þ 2  4dÞ þ    þ ðd þ ðN  1Þ  4dÞ ¼ Nd þ 4dð1 þ 2 þ    þ ðN  1ÞÞ ¼ Nd þ 2NðN  1Þd

484

10

Cheap Talk Games

Since the left-hand side of the above expression is just hN  h0 ¼ 1, we can write 1 ¼ Nd þ 2NðN  1Þd and solve for d to obtain d¼

1  Nd 2NðN  1Þ

Hence, the maximum value of d occurs when the initial partition is of length zero, d = 0, that is d  d ¼

1 2NðN  1Þ

Therefore, to sustain a partially informative PBE with more partitions (higher N), we need a smaller preference divergence parameter d between the sender (e.g., lobbyist) and the receiver (e.g., politician). Alternatively, we can solve for the number of partitions N in the above expression, to find the maximum number of partitions, NðdÞ, as a function of d, NðN  1Þ 

1 2d

Further rearranging, N2  N 

1 0 2d

Factorizing the above inequality, we obtain 0 @N 

1þ

qffiffiffiffiffiffiffiffiffiffiffi10 qffiffiffiffiffiffiffiffiffiffiffi1 1 þ 2d 1  1 þ 2d A@N  A0 2 2

Since N  1 (that is, there must be at least one partition), we can rule out the negative root such that

N

1þ

qffiffiffiffiffiffiffiffiffiffiffi 1 þ 2d 2

Exercise 5—Cheap Talk Game with a Continuum of Types …

485

Number of partitions, N 3.0 2.5 2.0 1.5 1.0 0.5

1

2

3

4

5

Preference divergence, δ

 in a partially informative PBE with N partitions Fig. 10.14 Cutoff N

Furthermore, as N 2 N (that is, N must take on positive integer values), we have qffiffiffiffiffiffiffiffiffiffiffi7 6 61 þ 1 þ 27 6 d7 5  ¼4 N N 2  from below (e.g., numbers where the symbol b:c indicates that we round cutoff N  represents the such as 4.2 or 4.9 are both rounded to 4). Intuitively, cutoff N maximum number of partitions that can be supported by d As the divergence  decreases. parameter, d, increases, the fraction 2d becomes smaller, so that cutoff N Intuitively, as the lobbyist and the politician become more divergent in their preferences, the lobbyist has more incentives to overreport its type, so that the message becomes less informative. In other words, more accurate information between the lobbyist and the politician (larger number of partitions) can only be sustained when the bias becomes extremely small, that is, d ! 0. As Fig. 10.14 illustrates, the partially informative PBE yields a smaller number of partitions as the preference divergence parameter d increases.

More Advanced Signaling Games

11

Introduction The final chapter presents extensions and variations of signaling games, thus providing more practice about how to find the set of PBEs in incomplete information settings. We first study a poker game where, rather than having only one player being privately informed about his cards (as in Chap. 8), both players are privately informed. In this context, the first mover’s actions can reveal information about his cards to the second mover, thus affecting the latter’s incentives to bet or fold relative to a context of complete information. We then explore a game in which one player is privately informed about his benefits from participating in a project, and evaluate his incentives to acquire a costly certificate that publicly discloses such benefit to other players. In the same vein, we then examine an entry game where firms are privately informed about their own type (e.g., production costs) before choosing to simultaneously and independently enter a certain industry. In this setting, firms can spend on advertising in order to convey their benefits from entering the industry to other potential entrants, where we identify symmetric and asymmetric PBEs. We extend the Spence’s (1974) job market signaling game examined in Chap. 9 to a context in which workers can choose an education level from a continuum (rather than only two education levels), and the firm can respond with a continuum of salaries. We first study under which conditions separating and pooling PBEs can be sustained in this version of the game when the worker has only two types, and then extend it to contexts in which the worker might have three or an infinite number of types. In order to provide further practice of the application of Cho and Kreps’ (1987) Intuitive Criterion in games where players have a continuum of available strategies, we offer a step-by-step explanation of this criterion. Afterwards we consider two types of entry-deterrence games. In the first game, the incumbent might be of a “sane” type (if he prefers to avoid price wars when entry occurs) or a “crazy” type (if he enjoys such price wars); following an incomplete information setting similar to that in Kreps et al. (1982). After © Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_11

487

488

11

More Advanced Signaling Games

simplifying the game, we systematically describe how to test for the presence of separating and pooling equilibria. In the second game, which follows Kreps et al. and Wilson (1982), an incumbent faces entry threats from a sequence of potential entrants, and must choose whether to start a price war with the first entrant in order to build a reputation that would protect him from further entry. While this game seems at first glance more complicated than standard signaling games with a single potential entrant, we show that it can be simplified to a point where we easily apply a similar methodology to that developed in most games of Chap. 9.

Exercise 1—Poker Game with Two Uninformed PlayersB Consider a simple poker game in which each player receives a card which only he can observe. 1. There is an ante of $A in the pot. Player 1 has either a good hand (G) or a bad hand (B) chosen by Nature with probability p and 1 − p, respectively. The hand is observed by player 1 but not by player 2. Player 1 then chooses whether to bet or resign: If player 1 resigns, player 2 wins the ante. If player 1 bets, he puts $B in the pot. 2. If player 1 bets then player 2, observing his own card (either good or bad), must decide whether to call or fold: if player 2 responds folding, player 1 gets the pot, winning the ante. If, instead, player 2 responds calling, he puts $1 in the pot. Player 1 wins the pot if he has a high hand; player 2 wins the pot if player 1’s hand is low. Part (a) Draw the extensive form game. Part (b) Fully characterize the set of equilibria for all sets of parameter values p, B, and A. Answer: Part (a) Fig. 11.1 depicts the game tree of this poker game: Nature determines whether both players receive a good hand (GG), only player 1 does (GB), etc. Only observing his good hand (left-hand side of the tree), player 1 decides whether to bet (B) or fold (F). Note that player 1’s lack of information about player 2’s hand is graphically represented by the fact that player 1 chooses to bet or fold on an information set. For instance, when player 1 receives a good hand, he does not observe player 2’s hand, i.e., is uncertain about being located in node GG or GB. If player 1 bets, player 2 observes such an action and must respond betting or folding when his own hand is good (but without knowing whether player 1’s hand is good, in l1 , or bad, in 1
l1 ), and similarly when his own hand is bad (and he does not know whether his rival’s hand is good, in l3 , or bad, in 1
l3 ). A similar argument applies to the case in which player 2 observes player 1 folding,

Exercise 1—Poker Game with Two Uninformed PlayersB

489

Fig. 11.1 A poker game with two uninformed players

leaving player 2 in information set ðl2 ; 1
l2 Þ when his own hand is good, and ðl4 ; 1
l4 Þ when his hand is bad. First, note that betting is a strictly dominant strategy for every player who receives a good hand, irrespective of the strategy his opponent selects. You can see this result by starting, for instance, to examine player 1’s decision to fold or bet when he receives a good card (in the GG and GB nodes). In particular, betting would yield an expected payoff of pðA=2Þ þ ð1
pÞðA þ BÞ; but a lower expected payoff of pðA=2Þ þ ð1
pÞðA=2Þ from folding. A similar argument applies to player 2 when his cards are good. However, when players receive a bad hand, they do not have a strictly dominant strategy. For compactness, let a denote the probability that player 1 bets when he receives a bad hand, and b the probability that player 2 bets when he receives a bad hand, as indicated in Fig. 11.1. Part (b) Fully characterize the set of equilibrium for all parameter values p, B, and A. As a roadmap, note that we need to first examine player 2’s optimal responses in all four information sets in which he is called on to move: those in which he

490

11

More Advanced Signaling Games

observes player 1 betting, ðl1 ; 1
l1 Þ and ðl3 ; 1
l3 Þ, and those in which he responds to player 1 folding, ðl2 ; 1
l2 Þ and ðl4 ; 1
l4 Þ. Player 1 bets. After observing that player 1 bets, player 2 will respond betting when he receives a bad hand, as depicted in information set ðl3 ; 1
l3 Þ, if EU2 ðBetjBad hand; P1 BetsÞ  EU2 ðFoldjBad hand; P1 BetsÞ; that is ð
BÞ  prob ðp1 Betsjp1 has a good handÞ þ

A  prob ðp1 Betsjp1 has a bad handÞ  0 2

where, graphically, note that player 2 obtains a payoff of −B when he responds betting and player 1’s hand is good, but a higher payoff of A2 when player 1’s hand is bad. Using conditional probabilities, we can rewrite the above inequality as ð
BÞ

p A að1
pÞ þ 0 p þ ð1
pÞa 2 p þ ð1
pÞa

Note that the two large ratios use Bayes’ rule in order to determine the probability that, upon observing player 1 betting, such a player received a good hand, að1
pÞ p p þ ð1
pÞa, or a bad hand, p þ ð1
pÞa. Intuitively, player 1 bets with probability 1 when he has a good had (which occurs with probability p), but with probability a when he has a bad hand (which happens with probability 1 − p). Solving for probability a in the above inequality, we obtain that player 2 bets when having a bad hand and observing player 1 betting if and only if a

2Bp : Að1
pÞ

Note that, in order for this ratio to be a probability we need that: 0  p\ 2BAþ A.

2Bp Að1
pÞ

 1,

i.e., the prior probability p must be relatively low, In contrast, when player 2 observes that player 1 bets but player 2’s hand is good, he responds betting [as indicated in information set ðl1 ; 1
l1 Þ], since EU2 ðBetjGood hand; BetÞ  EU2 ðFoldjGood hand; BetÞ or A l þ ðA þ BÞð1
l1 Þ  0l1 þ 0ð1
l1 Þ 2 1 which reduces to

Exercise 1—Poker Game with Two Uninformed PlayersB


ð A þ BÞ 

491

 A þ B l1 2

which holds since A, B > 0, and l1  1. Thus, player 2 responds to player 1’s bet with a bet when his hand is good, under all parameter values. Player 1 folds. Let us now move to the case in which, upon receiving a good hand, player 2 observes that player 1 folded; as depicted in information set ðl2 ; 1
l2 Þ. In this case, player 2 responds betting if EU2 ðBetjGood hand; FoldÞ  EU2 ðFoldjGood hand; FoldÞ or Al2 þ Að1
l2 Þ 

A A l 2 þ ð 1
l2 Þ 2 2

which simplifies to A  A2 , a condition which is always true. Hence, player 2 bets when having a good hand, both after observing that player 1 bets and that he folds. Let us finally analyze the case in which player 2 receives a bad hand and observes that player 1 folds; as depicted in information set ðl4 ; 1
l4 Þ. In this setting, player 2 responds betting if EU2 ðBetjBad hand; FoldsÞ  EU2 ðFoldjBad hand; FoldsÞ which implies A

0p ð1
aÞð1
pÞ þA 0p þ ð1
aÞð1
pÞ 0p þ ð1
aÞð1
pÞ

 A 0p A ð1
aÞð1
pÞ  þ 2 0p þ ð1
aÞð1
pÞ 2 0p þ ð1
aÞð1
pÞ

which simplifies to 2A  A. Hence, player 2 will respond betting if, despite having a bad hand, he observes that player 1 folded. Player 1. Let us now move to the first mover, player 1. When having a good hand, betting is strictly dominant, as discussed above. However, when his hand is bad (as indicated in the two branches in the right-hand side of the tree), he finds profitable to bet if EU1 ðBetjBad handÞ  EU1 ðFoldjBad handÞ that is,

492

11

More Advanced Signaling Games

      A A A p½1  ð
BÞ þ 0  A þ ð1
pÞ b þ ð1
bÞA  p 1  0 þ 0  þ ð1
pÞ b  0 þ ð1
bÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} if P0 s hand is good 2

if P02 s hand is bad he bets with prob b

he responds betting

if P02 s hand is good he responds betting

if P02 s hand is bad he bets with prob b

and solving for probability p we find that player 1 bets after receiving a bad hand if and only if p\

A 
p 2B þ A

(Note that this condition only depends on the prior probability of nature, p, and on the size of the payoffs A and B). Therefore, player 1 bets, despite receiving a bad hand, when the chances that player 2 has a better hand are relatively low, i.e., p\
p.1 (For a numerical example, note that if the ante contains $1 and the amount of money that each player must add to the ante when betting is also $1, i.e., A = B = $1, then we obtain that the cutoff probability is
p ¼ 1=3:). Hence, we can summarize the PBE of this game as follows: Player 1: Bet after receiving a good hand for all p, but bet after receiving a bad hand if and only if p\
p. Player 1’s posterior beliefs coincide with his prior p. Player 2: Bet after receiving a good hand for all p. After receiving a bad hand, and 2Bp observing that player 1 bets, player 2 responds betting if and only if a  Að1
pÞ ; while after observing that player 1 folds, player 2 responds betting under all parameter values. Player 2’s beliefs are given by Bayesian updating, as follows:

p 0 p ;l ¼ ¼ 0; p þ ð1
pÞa 2 1  p þ ð1
pÞð1
aÞ p 0 p ; and l4 ¼ ¼0 l3 ¼ p þ ð1
pÞa 0  p þ ð1
pÞð1
aÞ l1 ¼

As a remark, note that l3 ¼ 0 ðl4 ¼ 0Þ means that player 2, in equilibrium, anticipates that the bet (fold, respectively) decision from player 1 he observes must indicate that player 1’s cards are bad, both when player 2’s hand is good (in l3 ) and bad (in l4 ).

Importantly, this condition holds both when a satisfies a 

1

2Bp Að1
pÞ,

which implies that player 2

2Bp responds betting when his hand is bad, thus implying b ¼ 1, and when a  ðAð1
pÞÞ , implying that player 2 responds folding when his hand is bad, i.e., b ¼ 0, since the cutoff strategy p\
p is independent on a.

Exercise 2—Incomplete Information and CertificatesB

493

Exercise 2—Incomplete Information and CertificatesB Consider the following game with complete information. Two agents, A and B, simultaneously and independently decide whether to join, J, or not join, NJ, a common enterprise. Matrix, Fig. 11.2 represents the payoffs accruing to agents A and B under each strategy profile: Note that every agent i = {A, B} obtains a payoff of 1 if the other agent j 6¼ i joins, but a payoff of −1 if the other agent does not join (while the agent who did not join obtains a payoff of zero). Finally, if no agent joins, both agents obtain a payoff of zero. Part (a) Find the set of pure and mixed strategy Nash equilibria. Part (b) Let us now consider the following extension of the previous game in which we introduce incomplete information: before players are called to move (deciding whether they join or not join), nature selects a type t 2 ½0; 1, uniformly distributed between 0 and 1, i.e., t  U½0; 1, which affects both players’ payoff when all players join. Intuitively, the benefit from having both players joining the common enterprise is weakly larger than under complete information, $1, but such additional benefit t is player B’s private information. Thus, after such choice by nature, the payoff matrix becomes (Fig. 11.3) Find the set of Bayesian Nash equilibria (BNE) of this game. Part (c) Consider now a variation of the above incomplete information game: First, nature determines the specific value of t, either t = 0 or t = 1, which only player B observes. Player B proposes to the uninformed player A that the latter will make a transfer p to the former. If player A accepts the offer, the transfer is automatically implemented. Upon observing the value of t, player B chooses whether or not to acquire a certificate at a cost c [ 0: Player A can only observe whether player B acquired a certificate, and knows that both types (t = 0 and t = 1) are equally likely. Find under which conditions can a pooling equilibrium be sustained in which player B acquires a certificate, both when his type is t = 0 and when it is t = 1. Part (d) In the version of the game described in part (c), identify under which conditions can a separating PBE be sustained, in which player B acquires a certificate only when his type is t = 1.

Fig. 11.2 Payoff matrix under complete information

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More Advanced Signaling Games

Fig. 11.3 Payoff matrix under incomplete information

Answer: Part (a) Complete information version of the game (Fig. 11.4): PSNE. Let us first examine each player’s best response function. If player A joins (top row), then player B’s best response, BRB , is to join; while if player A does not join (bottom row), then BRB is not join. By symmetry, if player B joins, then player A’s best response, BRA is to join; while if player B does not join, then BRA is not join. Intuitively, both players have incentives to mimic the action selected by the other player, as in similar coordination games we analyzed in previous chapters (e.g., the Battle of the Sexes game). Thus, the game has the following two pure (and symmetric) strategy Nash equilibrium: PSNE : fðJ; JÞ; ðNJ; NJÞg Hence, the game reflects the same strategic incentives as in a Pareto coordination game with two PSNE, where one equilibrium yields unambiguously larger payoffs to both players than the other equilibrium, i.e., (J, J) Pareto dominates (NJ, NJ). MSNE. Let us now find the MSNE of this game, denoting p ðqÞ the probability with which player A (player B, respectively) joins. Making player A indifferent between joining and not joining, we obtain

Fig. 11.4 Payoff matrix under complete information—Underlining best response payoffs

Exercise 2—Incomplete Information and CertificatesB

495

Fig. 11.5 Payoff matrix under incomplete information—underlining best response payoffs

EU A ðJÞ ¼ EU A ðNJÞ; that is q  1 þ ð1
qÞð
1Þ ¼ 0 $ q ¼ 12 Similarly p ¼ 12 given that agents are symmetric. Hence, the mixed strategy Nash equilibrium of the game is 
MSNE :


 1 1 1 1 J; NJ ; J; NJ 2 2 2 2

Part (b) Let us now consider the incomplete information extension of the previous game, where t  U ½0; 1, as depicted in the payoff matrix, Fig. 11.5. Let us first examine the informed player (player B). Denoting by l the probability with which player A joins, then player B’s expected utility from joining and not joining are EU B ðJ Þ ¼ lð1 þ tÞ þ ð1
lÞð
1Þ ¼ l þ lt
1 þ l ¼ lð2 þ tÞ
1; and EU B ðNJ Þ ¼ 0 where l denotes the probability with which his opponent joins the enterprise. Hence, player B will decide to join if and only if: EU B ðJ Þ  EU B ðNJ Þ $ lð2 þ tÞ
1  0 $ l 

1 2þt

The decision for the uninformed player A is analogous in this case since payoffs are symmetric. However, player A doesn’t consider the realization of parameter t but its expected value, t = 1/2. Using l again to denote the probability with which his opponent (player B) joins the enterprise, we can say the following: (1) when 1 l  2þ t, player B joins inducing player A to also join since his expected payoff from doing so, 1 + 1/2 = 3/2, is higher than that from not joining, 0; and (2) when 1 l\ 2 þ t, player B does not join the enterprise leading player A to not join since his

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More Advanced Signaling Games

expected payoff from doing so, −1 (in this case not affected by parameter t), is lower than that from not joining, 0. This implies the existence of two BNEs, as we next describe: 1st BNE: If l 

1 2þt

n then both players join, and the BNE is ðrA ; rB ; lÞ ¼ J; J; l 

1 2þt

o

2nd BNE:

n o 1 A B 1 then no player joins, and the BNE is ð r ; r ; l Þ ¼ NJ; NJ; l\ If l\ 2 þ t 2þt Note that both of these BNEs involve structurally consistent beliefs: first, when 1 both players join, l ¼ 1, which satisfies l [ 2 þ t since t  U½0; 1; second, when 1 no player joins, l ¼ 0, which satisfies l\ 2 þ t. Intuitively, if every player believes that his opponent will likely join, he responds joining; thus giving rise to the first type of BNE where both players join the enterprise. Otherwise, neither of them joins, and the second type of BNE emerges. Part (c) Pooling PBE Figure 11.6 depicts the pooling strategy profile in which the privately informed player, player B, acquires a certificate, both when his type is t = 0 and t = 1. After observing that player B acquired a certificate. When player A is called on to move on information set ðl1 ; 1
l1 Þ, he accepts the transfer p after observing that player B acquired a certificate (in equilibrium) if and only if2: l1 ð1
pÞ þ ð1
l1 Þð2
pÞ  0 $ 2
p  l1 Since we are in a pooling strategy profile in which both types of player B acquire a certificate, then l1 ¼ lðt ¼ 0jcertÞ ¼ 1=2 (i.e., posterior = prior). This allows us to further reduce the above inequality to 2
p  1=2 $ p  3=2: that is, any transfer p in the range p 2 ½0; 3=2 will be accepted by the uninformed player A after observing that player B acquired a certificate. After observing that player B does not acquire a certificate. When player A is called on to move on information set ðl2 ; 1
l2 Þ, i.e., he observes the off-the-equilibrium action of no certificate from player B, he rejects the transfer p if and only if

Notice that here p represents a monetary transfer or payment not necessarily contained in the unitary range, different to the probability p used in part (a) of this exercise.

2

Exercise 2—Incomplete Information and CertificatesB

497

Fig. 11.6 Pooling strategy profile—certificates

EU A ðAcceptjNC; l2 Þ  EU A ðRejectjNC; l2 Þ; that is l2 ð1
pÞ þ ð1
l2 Þð2
pÞ  0 $ 2
p  l2 At this point, note that any transfer p satisfying 2
p  l2 ; or 2
l2  p, will be rejected. Several off-the-equilibrium beliefs l2 support such a behavior. For instance, if l2 ¼ 1, we obtain that 2
1  p, or 1  p, leading player A to reject the transfer proposed by player B if no certificate is offered and the transfer A needs to give to B is weakly larger than $1. Intuitively, note that l2 ¼ 1 implies that the uninformed player A assigns the worst possible beliefs about player B’s type upon observing no certificate, i.e., player B’s type must be low (t = 0). If, instead, his beliefs assign full probability to agent B’s type being good, i.e., l2 ¼ 0, then the above incentive compatibility condition becomes 2
0  p, or 2  p, thus implying that player A is willing to make transfers to player B as long as the transfer does not exceed $2. Our subsequent analysis focuses on the case in which conditions p  3=2 and 2
l2  p hold because the pooling strategy profile where both types of player B acquire a certificate cannot be supported otherwise. To see this point, consider a player with type t (either t = 0 or t = 1).

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More Advanced Signaling Games

• First, if p  3=2 does not hold but 2
l2  p is satisfied, then player A responds rejecting player B’s offer regardless of whether B acquired a certificate or not. Anticipating this response profile, player B does not acquire a certificate since 0 
c, which holds by definition. As a consequence, the pooling strategy profile where both types of player B acquire a certificate cannot be supported in this setting. • A similar argument applies if p  3=2 holds but 2
l2  p does not. In this case, player A responds accepting player B’s offer regardless of whether B acquired a certificate or not. Anticipating this response profile, player B does not acquire a certificate since 1 þ t  1 þ t
c, which holds by definition. • Finally, a similar argument applies if neither condition p  3=2 or 2
l2  p hold. In this setting, player A responds rejecting player B’s offer after observing a certificate, but accepting it otherwise. Anticipating this response profile, player B does not acquire a certificate since 1 þ t 
c, which holds by definition. Informed player B. Let us now move to the informed player B (the first mover in this sequential move game with incomplete information). In order to keep track of player A’s optimal responses in this game, i.e., accepting after observing that player B acquired a certificate but rejecting otherwise, Fig. 11.7 shades the acceptance (rejection) branch of player A after observing that player B acquires (does not acquire, respectively) a certificate. • Player B’s type is t = 0. In this case, player B anticipates that a certificate will be responded with acceptance, yielding for him a payoff of 1 + p − c; but not acquiring a certificate will be subsequently rejected, entailing a payoff of 0. Hence, he offers the certificate if 1 þ p
c  0 and since the transfer enters positively, he will choose the highest offer p that guarantees acceptance, i.e., p = 3/2, implying that he will acquire a certificate as long as its cost is sufficiently low, i.e., 1 þ 3=2
c  0 or, solving for c; c  5=2. • Player B’s type is t = 1. Similarly, when his type is t = 1, player B offers the certificate if 2 + p > 0. By the same argument as above, any transfer p  1 is accepted upon offering no certificate (note that this occurs off-the-equilibrium path), leading player B to select the highest offer that guarantees rejection, i.e., p = 1. This inequality thus becomes 2 + 1 > 0, which holds for all parameter conditions. Hence, a Pooling strategy profile can be sustained as a PBE, and it can be summarized as follows: • Player B acquires a certificate, both when his type is t = 0 and when t = 1, as long as the certificate costs are sufficiently low, i.e., c\5=2, and asks for a transfer of p ¼ 32 after acquiring a certificate (in equilibrium), and a transfer of p = 1 after not acquiring a certificate (off-the-equilibrium).

Exercise 2—Incomplete Information and CertificatesB

499

Fig. 11.7 Pooling strategy profile (including responses)

• Player A responds accepting to make any transfer p  32 to player B after observing a certificate, and any transfer p  1 after observing no certificate. His equilibrium beliefs are l1 ¼ 12 (after observing a certificate, since both types of player B acquire such a certificate in equilibrium and both types are equally likely), while his off-the-equilibrium beliefs are l2 ¼ 1.3 Part (d) Separating PBE Let us now check if the separating strategy profile in which only t = 1 acquires certificate (while t = 0 does not) can be sustained as a PBE. In order to facilitate our visual presentation, Fig. 11.8 shades the arrow corresponding to certificate by player B when his type is t = 1, and that corresponding to no certificate when t = 0. Beliefs. In a separating equilibrium, we must have that, upon observing the certificate, player A’s posterior beliefs are,

Note that if, instead, off-the-equilibrium beliefs are l2 ¼ 0, then the highest transfer that player B can request from player A increases from p = 1 to p = 2 when he does not acquire a certificate. All other results of the pooling PBE would remain unaffected.

3

500

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More Advanced Signaling Games

Fig. 11.8 Separating strategy profile

l1 ¼ lðt ¼ 0jC Þ ¼ 0; and l2 ¼ lðt ¼ 0jNC Þ ¼ 1; since player B does not acquire a certificate in this strategy profile when his type is t ¼ 0. Let us next examine player A’s optimal responses after observing a certificate or no certificate from player B. Player A after observing that player B acquired a certificate. Given the above beliefs, if player A observes a certificate (in the node marked with 1
l1 in Fig. 11.7) he accepts if and only if EU A ðAcceptjNC; l1 Þ ¼ 2
p  EU A ðRejectjNC; l1 Þ ¼ 0; that is, if p  2: Intuitively, player A is convinced to be in the right-hand node (marked with 1
l1 in Fig. 11.7), thus making his optimal response straightforward: he accepts if the transfer he must pay to B is sufficiently small (p  2) but rejects it otherwise.

Exercise 2—Incomplete Information and CertificatesB

501

Player A after observing that player B did not acquire a certificate. If, instead, player A observes that player B did not acquire a certificate (in the node marked with l2 in Fig. 11.8), he responds accepting if and only if EU A ðAcceptjNC; l2 Þ ¼ 1
p  EU A ðRejectjNC; l2 Þ ¼ 0 that is, if p  1: This response is analogous to that after observing a certificate: player A responds accepting if the transfer he must pay to B is sufficiently small ðp  1Þ but rejects otherwise. Overall, we can summarize the above two responses according to the transfer p: 1. When p  1, player B responds accepting player A’s offer, both after player A acquires a certificate and when he does not. 2. When 2  p [ 1, player B responds accepting player A’s offer after player A acquires a certificate, but rejecting it otherwise. 3. When p [ 2, player B responds rejecting player A’s offer, regardless of whether A acquires a certificate or not. Player B. Let us now analyze player B, who anticipates under which conditions a certificate will be accepted and rejected by player A as described in our above decisions. • Player B’s type is t = 1. In particular, when his type is t = 1, player B acquire a certificate (as prescribed in this separating equilibrium) if and only if:

EU B ðCjt ¼ 1Þ  EU B ðNCjt ¼ 1Þ Depending on the value of transfer p, this inequality becomes: (a) When p  1, player B acquires a certificate if and only if 2 þ p
c  2 þ p, which simplifies to
c  0, and thus cannot hold. Therefore, when p  1, player B does not acquire a certificate. (b) When 2  p [ 1, player B acquires a certificate if and only if 2 þ p
c  0, which reduces to 2 þ p  c. (c) When p [ 2, player B acquires a certificate if and only if 0
c  0, which cannot hold since the certificate is costly. Therefore, when p [ 2, player B does not acquire a certificate. Hence, since the transfer p enters positively into player B’s utility, he will acquire a certificate and select the highest value of p that still guarantees acceptance, i.e., p = 2. In this case, the cost of acquiring the certificate must satisfy 2 þ 2
c  0, i.e., c  4:

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More Advanced Signaling Games

• Player B’s type is t = 0. In contrast, when player B’s type is t = 0, he doesn’t acquire a certificate (as prescribed) if and only if EU B ðNCjt ¼ 0Þ  EU B ðCjt ¼ 0Þ Similarly as in the case of t = 1, this inequality can be separately examined according to the value of transfer p: (d) When p  1, player B does not acquire a certificate if and only if 1 þ p  1 þ p
c, which holds since the certificate is costly. Therefore, when p  1, player B does not acquire a certificate; as prescribed in this separating strategy profile. (e) When 2  p [ 1, player B does not acquire a certificate if and only if 0  1 þ p
c, which simplifies to c  1 þ p. (f) When p [ 2, player B does not acquire a certificate if and only if 0  0
c, which holds by definition. Therefore, when p [ 2, player B does not acquire a certificate, as prescribed. In summary, for the separating equilibrium where player B only acquires a certificate when his type is t = 1, we need the following conditions to simultaneously hold: (1) 2  p [ 1, (2) 2 þ p  c, and (3) c  1 þ p; which originate from points (b) and (e) in the previous lists. Figure 11.9 depicts these three conditions in the shaded area, where condition (1) is defined by intermediate values of transfer p in the horizontal axes, condition (2) by values of c below line 2 + p, and condition (3) by values of c above line 1 + p. Intuitively, the certificate must be sufficiently costly for player B to not have incentives to acquire it when his type is t = 0, i.e., 1 þ p  c, but sufficiently inexpensive for player B to have incentives to acquire it when t = 1, i.e., 2 þ p  c. In addition, the transfer that player B requests from player A, p, cannot be too high, since otherwise player A would not be induced to accept the proposal (and, as a consequence, the common enterprise will not be started), i.e., 1  p  2: Once we identified the conditions under which a separating PBE can be sustained where player B only acquires a certificate when his type is t = 1, we can find which offer p he requests from player A. Since player B receives the offer (i.e., p enters positively in his payoff function) he will increase p as much as possible in the set of (c, p)-pairs for which this separating PBE can be sustained; as depicted in right-hand border of the shaded region in Fig. 11.9. That is, the separating PBE prescribes p = 2 and 2  c  4: This condition on c can be intuitively understood as saying that the certificate must be sufficiently inexpensive for player B to acquire it when its type is t = 1, but sufficiently expensive for him to not acquire it when its type is t = 0. In other words, when his type is t = 0 player B cannot find it profitable to mimic the other type, t = 1, by acquiring a certificate. We can, then, summarize this separating PBE as follows:

Exercise 2—Incomplete Information and CertificatesB

503

Fig. 11.9 (p, c)-pairs for which a separating PBE can be sustained

• Player B: – When his type is t = 1, he acquires a certificate and offers a transfer p = 2 if and only if 4  c. – When his type is t = 0, he does not acquire a certificate if 1 þ p  c. • Player A accepts player B’s offer p according to the following rule: – When p  1, player B responds accepting, both after player A acquired a certificate and when he did not. – When 2  p [ 1, player B responds accepting player A’s offer after player A acquires a certificate, but rejecting it otherwise. – When p [ 2, player B responds rejecting player A’s offer, regardless of whether A acquires a certificate or not. – Player A’s beliefs are l1 ¼ lðt ¼ 0jC Þ ¼ 0 after observing a certificate and l2 ¼ lðt ¼ 0jNC Þ ¼ 1 after observing no certificate.

Exercise 3—Entry Game with Two Uninformed FirmsB Suppose there are two symmetric firms, A and B, each privately observing its type (t), either g (good) or b (bad) with probabilities p and 1
p, respectively, where 0\p\1. For instance, “good” can represent that a firm is cost efficient, while “bad” can indicate that the firm is cost inefficient. They play an entry game in which the two firms simultaneous choose to enter or stay out of the industry. Denote the outcome of the entry game by the set of types that entered, i.e., (g, g) means two good firms entered, (b) means only one bad firm entered, and so on. If a firm does not enter, its payoff is 0. When it enters, its payoff is determined by its own type and the set of types that entered, with the following properties:

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More Advanced Signaling Games

ug ðgÞ [ ug ðg; bÞ [ ug ðg; gÞ [ ub ðbÞ [ 0 [ ub ðb; bÞ [ ub ðg; bÞ where the superscript is the firm’s type and the argument is the outcome (profile of firms that entered the industry). Find a symmetric separating Perfect Bayesian equilibrium (PBE) in which every firm i enters only when its type is good assuming that the following condition holds: pub ðg; bÞ þ ð1
pÞ ub ðbÞ\0: Answer: Since this is a simultaneous-move game of incomplete information (both firms are uninformed about each other’s types), and firms are ex-ante symmetric (before each observes its type), we next analyze the entry decision of every firm i. First for the case in which its own type is good, and then when its type is bad. Firm i’s type is good. When firm i’s type is good, ti ¼ g, the expected utility that firm i obtains from entering is EUi ðEnterjti ¼ gÞ ¼

p½r0 ug ðg; gÞ þ ð1
r0 Þug ðgÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

if firm j is good; j can decide either to enter or not

þ ð1
pÞ½rug ðg; bÞ þ ð1
rÞug ðgÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} if firm j is bad; j can decide either to enter or not

where r′ denotes the probability that firm j enters when its type is good, ti ¼ g; while r represents the probability that firm j enters when its type is bad, ti ¼ b. But we know that when a firm’s type is good it always enters given that its payoff is strictly positive by definition, i.e., entering is a strictly dominant strategy for a good firm since staying out yields a payoff of zero. That is, a good firm i is better off entering when firm j does not enter, ug ðgÞ [ 0, and when firm j enters, both when firm j’s type is good, ug ðg; gÞ [ 0, and bad ug ðg; bÞ [ 0. This implies that firm j when its type is good, tj ¼ g, will decide to enter, i.e., r′ = 1. Therefore the above EUi can be simplified to: EUi ðEnterjti ¼ gÞ ¼ pug ðg; gÞ þ ð1
pÞ ½rug ðg; bÞ þ ð1
rÞug ðgÞ On the other hand, firm i obtains a payoff of zero from staying out regardless of firm j’s type and irrespective of whether firm j enters or not. Indeed, EUi ðNotEnterjti ¼ gÞ ¼ p½r 0  0 þ ð1
r 0 Þ  0 þ ð1
pÞ½r  0 þ ð1
r Þ  0 ¼ 0 Therefore, when deciding whether to join or not, a good firm i will compare these two expected utilities, inducing it to enter if and only if

Exercise 3—Entry Game with Two Uninformed FirmsB

505

EUi ðEnterjti ¼2gÞ [ EUi ðNot Enterjti ¼ gÞ 3 p  ug ðg; gÞ þ ð1
pÞ4r  ug ðg; bÞ þ ð1
rÞ ug ðgÞ 5 [ 0 |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |ffl{zffl} þ

þ

þ

which holds for all values of p and r, since ug ðg; gÞ; ug ðg; bÞ, and ug ðgÞ are all positive by definition. Hence, a good firm i will enter, i.e., we don’t have to impose additional conditions in order to guarantee that this type of firm enters. Firm i’s type is bad. Let us now consider the case in which firm i’s type is bad, i.e., ti ¼ b. The expected utility that this firm obtains from entering is EUi ðEnterjti ¼ bÞ if firm j is good

if firm j is bad

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{

¼ p r 0 ub ðg; bÞ þ ð1
r 0 Þub ðbÞ þ ð1
pÞ rub ðb; bÞ þ ð1
rÞub ðbÞ But given that firm j enters when its type is good, then r 0 ¼ 1, and the previous expression reduces to:

EUi ðEnterjti ¼ bÞ ¼ pub ðg; bÞ þ ð1
pÞ rub ðb; bÞ þ ð1
rÞub ðbÞ This firm’s expected utility from not entering is zero, i.e., EUi ðNot Enterjti ¼ bÞ ¼ 0, since firm i attains a zero payoff from staying out regardless of firm j’s decision and type. Hence, for firm i to stay out when its type is bad (as prescribed in the separating strategy profile) we need that

pub ðg; bÞ þ ð1
pÞ rub ðb; bÞ þ ð1
r Þub ðbÞ  0 But, since the strategy profile we are testing requires firms to enter when their type is good, we have that r = 1, which further simplifies the above inequality to p ub ðg; bÞ þ ð1
pÞ ub ðb; bÞ  0 |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl}





which holds since ub ðg; bÞ\0 and ub ðb; bÞ\0 by definition. Hence, firm i will not enter when being of bad type. Therefore, we have found a symmetric PBE in which, every firm i = {A, B} enters (stays out) when its type is good (bad, respectively), and each firm believes the other firm will only enter when its type is good.

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More Advanced Signaling Games

Exercise 4—Labor Market Signaling Game and Equilibrium RefinementsB Consider the following labor market signaling model. A worker with privately known productivity h chooses an education level e. Upon observing e, each firm responds with a wage offer that corresponds to the worker’s expected productivity (That is, we assume that the labor market is perfectly competitive, so firms pay the value of the expected marginal productivity to every worker. Otherwise, competing firms would have incentives to increase their salaries in order to attract workers with higher productivities.). The worker’s payoff from receiving a wage w and acquiring 3 an education level e, given an innate productivity h, is uðw; ejhÞ ¼ w
eh , i.e., the 3 cost of acquiring education, eh , is convex, and decreases in the worker’s innate productivity, h (Note that, for simplicity, we assume that the worker’s innate productivity, h, is unaffected by the education level he acquires.). Part (a) Assuming that there are two equally likely types h1 ¼ 1 and h2 ¼ 2, characterize the set of all separating equilibria. Part (b) Which of these equilibria survive the Cho and Kreps’ (1987) Intuitive Criterion? Answer: Part (a) Separating PBE when h1 ¼ 1 and h2 ¼ 2. Figure 11.10 depicts ðw; eÞpairs, and two indifference curves: one for the high-productivity worker, ICH , i.e., 3 u ¼ w
e2 since h2 ¼ 2, in which this type of worker reaches the same utility level at all points of ICH ; and other for the low-productivity worker, ICL , whereby this worker reaches the same utility level among all points satisfying w ¼ u þ e3 . Intuitively, note that an increase in education must be associated with an increase in wages in order to guarantee that the worker’s utility level is unaffected, which explains why the worker’s indifference curve is increasing in education. Furthermore, an increase in education must be accompanied be a more-than-proportional increase in wages in order to guarantee that the worker’s utility level remains unaltered, which explains why indifference curves are convex in e. In addition, note that indifference curves closer to the northeast of the figure represent a higher utility level, since they are associated with a higher wage. The set of all separating PBE satisfies: • low productivity worker chooses no education, i.e., e L ðhL Þ ¼ 0 • high productivity

worker chooses an Heducation level in 3 the range H e H ðhH Þ 2 eH ; e 1 2 , where the lower bound e1 solves uL ¼ wH
e , i.e., wH ¼ 3 uL þ e as represented in the figure. A similar argument applies for the upper bound eH 2 but, for compactness, its exact expression is presented below.

Exercise 4—Labor Market Signaling Game and Equilibrium RefinementsB

507

Fig. 11.10 Separating PBE in the labor market signaling game

• Firms, after observing equilibrium

H e H ðhH Þ 2 eH 1 ; e2 , offer wages:

education

levels,

e L ðhL Þ ¼ 0

and

 w e L ðhL Þ ¼ hL ¼ 1; and  w e H ðhH Þ ¼ hH ¼ 2 while for any off-the-equilibrium education level e 6¼ e L ðhL Þ 6¼ e H ðhH Þ firms offer a wage schedule w ðeÞ which lies weakly below the indifference curve of both types of workers (as represented in the thick black line in the Fig. 11.10).4 H In order to determine the exact value of eH 1 and e2 , we need that the following incentive compatibility conditions are satisfied:   EU e L ; w ðeÞjhL  EU e H ; w ðeÞjhL so the low-productivity worker does not have incentive to deviate from his education level, e L , to that of the high-productivity worker, e H . Similarly,   EU e H ; w ðeÞjhH  EU e L ; w ðeÞjhH

The wage schedule then lies below ICH for all off-the-equilibrium education levels, i.e., for all H e\eH 1 and e [ e2 . When the firm observes, instead, an equilibrium education level of zero, the firm manager offers a wage of wð0Þ ¼ 0, as depicted in the vertical intercept

of wage schedule ; eH wðeÞ. Similarly, upon receiving an equilibrium education level of e 2 eH , the firm manager 1 2

H offers salary wðeÞ ¼ 2, as depicted in Fig. 11.10 for all e 2 eH 1 ; e2 . 4

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implying that the high-productivity worker does not have incentives to reduce his education level from e H to e L . For the utility functions and parameters given in the exercise, the above two incentive compatibility conditions reduce to, 03 ðeH Þ3 2
whichimplies eH  1 1 1 ffiffiffi p ðeH Þ3 0 3 whichimplies 2  eH 1
2
2 2 1


Therefore, the education level that the high-productivity worker selects in ; eH must lie in the interval equilibrium, e ðh Þ 2 eH ffiffiffi 1 2 ffiffiffi H H p p 3 3 eH ðhH Þ 2 1; 2 ; where 2 ’ 1:26 Note that the above strategy profile is indeed a PBE: • firms are optimizing given their benefits about workers’ types and given the equilibrium strategies for the workers. • workers are optimizing given firms’ equilibrium strategies (wages) and given firms’ beliefs about the workers’ types. • firms’ beliefs are computed by Bayes’ rule when possible. Note that the hL -worker will never acquire more education than e L ðhL Þ ¼ 0: by H acquiring more education, such as any eH 1 satisfying e1 \1, he is still identified as a low-productivity worker and offered a wage of hL , and he is incurring a large costs in acquiring this additional education. Furthermore, even if he acquires eH 1 ¼ 1 (or higher) and he is identified as a high-productivity worker (fooling the firm), the increase in his wage will not offset the increase in his education costs (which are especially high for him), ultimately indicating that he does not have incentives to deviate from his equilibrium education level e L ¼ 0. A similar argument applies to the high-productivity worker: he is already identified as such by the firm manager, who offers him a salary of $2. Acquiring an even larger education level would only reduce his utility (since acquiring education is costly) without increasing his salary. Lowering his education level is not profitable either, since any education level below eH 1 would imply that he is identified as a low-productivity worker, and thus receives a wage of $1. Since the savings in education costs he experiences from lowering his education level do not offset the reduction in wages, the high-productivity worker does not have incentives to deviate from his equilibrium education level. Part (b) Intuitive criterion Figure 11.11 depicts indifference curve ICL which crosses the (w, e) equilibrium pair the low-productivity worker, i.e., an education level of eL* ðh2 Þ ¼ 0 responded with a wage of wL= 1. Similarly, ICH crosses the (w, e) equilibrium pair for the pffiffiffi high-productivity worker, namely, an education level of e H ðhH Þ ¼ 3 2 responded

Exercise 4—Labor Market Signaling Game and Equilibrium RefinementsB

509

Fig. 11.11 Applying the intuitive criterion

with a wage of wH = $2. This education level is the upper bound of the equilibrium pffiffiffi

interval e H ðhH Þ 2 1; 3 2 of admissible education levels in equilibrium. We next seek to show that all education levels in this interval, other than the least-costly education level in lower bound of the interval, e H ðhH Þ ¼ 1, violate the Cho and Kreps’ (1987) Intuitive Criterion. First Step: Firms’ beliefs after observing off-the-equilibrium messages, such as e′ in Fig. 11.10 need to be restricted to H ðe0 Þ 2 H, which represents all those types of workers for whom sending e′ is never equilibrium dominated: n o H ðe0 Þ ¼ h 2 HjU ðhi Þ  max U ðs; e0 ; hi Þ ; s

where s 2 S ðH; e0 Þ

In words, upon observing effort e′, the firm restricts its beliefs to those types of workers h 2 H for whom their maximal deviating payoff exceeds their equilibrium payoff we can see this inequality is only satisfied for the hH -type, but is never satisfied for the low type. That is, even if firms believe that hL is in fact a hH -worker and pay w(e′) = 2, it is not convenient for a hL -worker to send such a message because of the high education costs that he would need to incur, that is, 1


02 ðe 0 Þ3 2
$ e0  1: 1 1

Hence, e′ is equilibrium dominated for the hL -worker. Graphically, this argument can be checked by noticing that the indifference curve of the low-productivity worker that passes through point A (guaranteeing a salary of 2 after acquiring an education level of e′) would lie to the southeast of the indifference curve that this worker reaches in equilibrium (crossing the vertical axis when he acquires a zero

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education and is offered a wage of 1). Hence, this type of worker does not have incentives to deviate from his equilibrium education level of zero, even if by doing so he could alter the firm’s equilibrium beliefs. However, sending message e′ is not equilibrium dominated for the hH -worker. That is, if firms believe that he is a hH -worker when he sends an education level e′ rather than the (higher) eH 2 , his salary would be unaffected while his education costs would go down. This is graphically illustrated by the fact that the indifference curve of the hH -worker passing through point A in the figure (corresponding to wage hH ¼ 2 and education level e′) would lie to the northwest of the indifference curve that he reaches when acquiring the education level eH 2 , ICH , thus indicating that this worker would be able to reach a higher utility level by deviating to e′. Therefore, the only type of worker for whom sending e′ is never equilibrium dominated is Hðe0 Þ ¼ fhH g, i.e., only the hH -worker could benefit by deviating from his equilibrium message. Second Step: Once we have restricted firms’ beliefs to H ðe0 Þ ¼ fhH g we need to find whether there exists a message that can be sent by hH that implies a higher level of utility than sending his equilibrium message. That is, we need to find a pair ðhH ; eÞ for which the following inequality holds: U ðhH Þ\ min U ðs; e; hH Þ; s 2 S ðH ðeÞ; eÞ s

As we can see, once we have restricted firms’ beliefs to H ðe0 Þ ¼ fhH g firms will simply offer wages of wðeÞ ¼ hH for all those education levels that the hL worker finds unprofitable, i.e., any e  eH 1. As a consequence, the hH -worker will be better-off by acquiring an education level lower than the one he acquires in any of the separating PBEs where e [ eH 1. Indeed,   hH
c e H ; hH \hH
cðe; hH Þ $ c e H ; hH [ cðe; hH Þ which is true given that eH [ e and that costs are increasing in education. Therefore, all the inefficient separating PBEs can be eliminated by the Intuitive Criterion. However, the so-called “Riley outcome” (the efficient separating PBE where the high-productivity worker only needs to acquire the lowest possible amount of education that separates him from the low-productivity worker, i.e., e ¼ eH 1 ) cannot be eliminated by the Intuitive Criterion. Intuitively, lowering his education level below eH 1 would identify him as a low-productivity worker,

Exercise 4—Labor Market Signaling Game and Equilibrium RefinementsB

511

receiving a wage offer of hL ¼ 1, which is not profitable for the high-productivity worker. Hence, e ¼ eH 1 is the so-called least-costly separating equilibrium, since the hH -worker acquires the minimum amount of education that helps him convey his type to the firms, separating himself from the low-productivity worker.5

Exercise 5—Entry Deterrence Through Price WarsA Consider two firms competing in an industry, an incumbent and an entrant. The entrant has already entered the market, and in period 1 the incumbent must decide whether to fight or accommodate; if she fights, the entrant loses fE [ 0 (i.e., profit ¼
fE ) and if she accommodates the entrant makes a profit of aE [ 0. The incumbent may be either “sane”, in which case he loses fI [ 0 from fighting, and gains aI [ 0 from accommodating; or “crazy”, in which case his only available action is to fight. In period 2, the entrant must choose whether to stay in the market or exit; if she exits, the incumbent makes a profit of m; if she stays, the incumbent gets to fight or accommodate again and the payoffs are as described before. Assume that there is no discounting, and that the prior probability of a “sane” incumbent is 0 < p < 1 (the actual type of the incumbent is private knowledge to the incumbent). Part (a) Draw the game in its extensive form. Part (b) Show that if payoffs satisfy m
fI  2aI , then there is a separating equilibrium. Part (c) Show that if m
fI  2aI does not hold there is a pooling equilibrium when p  fE þfE aE : Answer: Part (a) Figure 11.12 depicts the game tree. Here is a summary of the entrant’s payoff:
fE are his losses from fighting, while aE is the benefit from accommodating. For the incumbent we have that
fI represents the cost of fighting for the sane incumbent, and aI is the benefit of accommodating for the sane incumbent. This game tree can be alternatively represented as in Fig. 11.13. We can apply backwards induction in the proper subgames of the above game tree. In particular, after Fight1 (in the right-hand side of the tree) and the entrant responding with Stay, we encounter a proper subgame in which the sane incumbent operates under complete information. In particular, the sane incumbent prefers to choose Acc2 , which yields a payoff of aI
fI , than Fight2 , which entails a lower payoff of
2fI . In addition, after Acc1 (in the left-hand side of the tree) and the entrant responding with Stay, we also find a proper subgame in which the sane 5

For more details on the application of the Cho and Kreps’ (1987) Intuitive Criterion and other equilibrium requirements, see Espinola-Arredondo and Munoz-Garcia (2010) “The Intuitive and Divinity Criterion: Interpretation and Step-by-Step examples,” Journal of Industrial Organization Education, vol. 5(1), article 7.

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Fig. 11.12 Entry deterrence game—I

incumbent is called on to move under complete information. In this case, the sane incumbent prefers Acc2 , which yields a payoff of 2aI , than Fight2 , which only entails aI
fI . Anticipating this response from the sane incumbent after Acc1 , the entrant chooses Stay, since its payoff from doing so, 2aE , exceed that from exiting, aE . These optimal responses help us reduce the game tree as depicted in Fig. 11.14. Part (b) Once we have reduced the game tree, we can apply our standard approach to search for PBEs, first to check for the existence of separating PBEs, and then for pooling PBEs.

Exercise 5—Entry Deterrence Through Price WarsA

Fig. 11.13 Entry deterrence game—II

Fig. 11.14 Reduced Entry deterrence game

Fig. 11.15 Separating strategy profile

513

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Fig. 11.16 Separating strategy profile (with responses)

Fig. 11.17 Pooling strategy profile

Separating PBE. Figure 11.15 shades the branches corresponding to separating strategy profile in which only the sane incumbent accommodates can be sustained (note that the crazy incumbent can only fight by definition). In this strategy profile, the entrant’s beliefs are l ¼ 0, thus considering that a decision to fight can never originate from a sane incumbent. Given these beliefs, the entrant focuses on the node at the lower right-hand corner of the game tree, and responds exiting, since
fE \
2fE . We depict this response of the entrant in Fig. 11.16, with the blue shaded branch corresponding to exit (Note that the entrant chooses exit irrespective of the incumbent’s type, since the entrant cannot condition its response on the unobserved incumbent’s type.). Once we found such a response from the uninformed entrant, we can show that the sane incumbent prefers Acc1 (as prescribed), obtaining 2aI , rather than deviate

Exercise 5—Entry Deterrence Through Price WarsA

515

Fig. 11.18 Pooling strategy profile (with responses)—Case 1

towards Fight1 , where he would only obtain a payoff of m
fI ; where 2aI [ m
fI holds by definition. Hence, the sane incumbent sticks to the prescribed strategy of accommodation, and this separating strategy profile can be sustained as a PBE.6 Part (c) Pooling PBE. Let us now examine if a pooling strategy profile, where both types of incumbent fight, can be supported as a PBE. Figure 11.17 illustrates this strategy profile. First, in this strategy profile the entrant’s beliefs cannot be updated using Bayes’ rule, and must coincide with his priors, i.e., p ¼ p. Intuitively, since both types of incumbent are choosing to fight, the entrant cannot refine its beliefs about the incumbent’s type upon observing that the incumbent fought. In this case, the entrant chooses to stay if and only if pðaE
fE Þ þ ð1
pÞð
2fE Þ  pð
fE Þ þ ð1
pÞð
fE Þ ¼
fE or alternatively, if p

fE 
p: f E þ aE


Þ and We separately consider the case in which the entrant responds staying ðp  p the case in which it responds exiting ðp\
pÞ. Case 1 p 
p. Figure 11.18 depicts the case in which the entrant responds staying (blue shaded arrows). In this setting, the sane incumbent would choose to Fight1 (as prescribed) if aI
fI  2aI , or alternatively
fI  aI , which cannot hold, since aI [ 0and
fI \0 by definition. Hence, the pooling strategy profile in which both types of incumbent fight cannot be sustained as a PBE when p 
p: 6

Note that this is the only separating equilibrium of the game, since the crazy incumbent cannot choose to accommodate (his only available action is to fight), as depicted in the lower part of Fig. 11.16.

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Fig. 11.19 Pooling strategy profile (with responses)—Case 2

Case 2, p\
p. Let us next check the case in which p\
p, which indicates that the uninformed entrant responds exiting, as Fig. 11.19 depicts (see blue shaded branches). In this context, the sane incumbent chooses Fight1 (as prescribed) if and only if m
fI  2aI . Hence, as long as this condition holds, the pooling strategy profile in which both types of incumbents fight can be sustained as a PBE if the prior probability of the incumbent being sane, p, is sufficiently low, i.e., p\
p.

Exercise 6—Entry Deterrence with a Sequence of Potential EntrantsC The following entry model is inspired on the original paper of Kreps et al. (1982).7 Consider an incumbent monopolist building a reputation as a tough competitor who does not allow entry without a fight. The entrant first decides whether to enter the market, and, if he does, the monopolist chooses whether to fight or acquiesce. If the entrant stays out, the monopolist obtains a profit of a > 1, and the entrant gets 0. If the entrant enters, the monopolist gets 0 from fighting and −1 from acquiescing if he is a “tough” monopolist, and −1 from fighting and 0 from acquiescing if he is a “normal” monopolist. The entrant obtains a profit of b if the monopolist acquiesces and b − 1 if he fights, where 0\b\1. Suppose the entrant believes the monopolist to be tough (normal) with probability p (1
p, respectively), while the monopolist observes his own type. Part (a) Depict a game tree representing this incomplete information game. Part (b) Solve for the PBEs game tree, and solve for the PBE of this game.

Kreps et al. (1982) “Reputation and Imperfect Information,” Journal of Economic Theory, 27, pp. 253–279.

7

Exercise 6—Entry Deterrence …

517

Fig. 11.20 Entry deterrence with only one entrant

Part (c) Suppose the monopolist faces two entrants in sequence, and the second entrant observes the outcome of the first game (there is no discounting). Depict the game tree, and solve for the PBE [Hint: You can use backward induction to reduce the game tree as much as possible before checking for the existence of separating or pooling PBEs. For simplicity, focus on the case in which prior beliefs satisfy p  b:]. Answer Part (a) See Fig. 11.20. Part (b) The tough monopolist fights with probability 1, since fight is a dominant strategy for him; while the normal monopolist accommodates with probability 1, since accommodation constitutes a dominant strategy for this type of incumbent. Indeed, at the node labeled with MonopT on the left-hand side of the game tree in Fig. 11.17, the monopolist’s payoff from fighting, 0, is larger than from accommodating, −1. In contrast, at the node labeled with MonopN for the normal monopolist (see right-hand side of Fig. 11.19), the monopolist’s payoff from

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More Advanced Signaling Games

Fig. 11.21 Entry deterrence with two entrants in sequence

fighting, −1, is strictly lower than from accommodating, 0. Hence, the entrant’s decision on whether or not to enter will be based on: EUE ðEnterjpÞ ¼ pðb
1Þ þ ð1
pÞb ¼ b
p; and EUE ðNot EnterjpÞ ¼ 0 Therefore, the entrant enters if and only if b
p [ 0, or alternately, b [ p. We can, hence, summarize the equilibrium as follows: • The entrant enters if b [ p, but doesn’t enter if b  p: • The incumbent fights if tough, but accommodates if normal.

Exercise 6—Entry Deterrence …

519

Fig. 11.22 Reduced-form game

Part (c) The following game tree depicts an entry game in which the incumbent faces two entrants in sequence (see Fig. 11.21). Hence, applying backward induction on the proper subgames (those labeled with MT and MN in the last stages of the game tree) we can reduce the previous game tree to that in Fig. 11.22. For instance, after the second entrant chooses to enter despite observing a fight with the first entrant (left side of game tree), the tough monopolist chooses between fighting and accommodating in the node labeled MT. In this case, the tough monopolist prefers to fight, which yields a payoff of zero, rather than accommodate, which entails a payoff of −1. A similar analysis applies to the other node labeled with MT (where the second entrant has entered after observing that the incumbent accommodated the first entrant). However, an opposite argument applies for the nodes marked with MN in the right-hand side of the tree, where the normal monopolist prefers to accommodate the second entrant, regardless of whether he fought or accommodated the first entrant, since his payoff from accommodating (zero) is larger than from fighting (−1). In addition, note that the first entrant behaves in exactly the same way as in exercise (a): entering if and only if b [ p. Hence, when p  b, the first entrant

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More Advanced Signaling Games

Fig. 11.23 Entry of the first potential entrant

Fig. 11.24 A further reduction of the game

enters, as shown in exercise (a). Figure 11.23 shades this choice of the first entrant (green shaded branches). Therefore, upon entry, the first entrant gives rise to a beer-quiche type of signaling game, which can be more compactly represented as the game tree in Fig. 11.24. Intuitively, all elements before MonopT and MonopN can be predicted

Exercise 6—Entry Deterrence …

521

Fig. 11.25 Pooling strategy profile—Acc

(i.e., the first entrant enters as long as p  b), while the subsequent stages characterize a signaling game between the monopolist (privately informed about its type) and the second entrant. In this context, the monopolist uses his decision to fight or accommodate the first entrant as a message to the second entrant, in order to convey or conceal his type. (For every triplet of payoffs, the first corresponds to the monopolist, the second to the first entrant, and the third to the second entrant.) Let us now check if a pooling strategy profile in which both types of monopolists accommodate can be sustained as a PBE. Pooling PBE with Acc. Figure 11.25 shades the braches corresponding to such a pooling strategy. In this setting, posterior beliefs cannot be updated using Bayes’ rule, which entails r ¼ p. As in similar exercises, the observation of accommodation by the uninformed entrant does not allow him to further refine his beliefs about the monopolist’s type. Hence, the second entrant responds entering (e) after observing that the monopolist accommodates (in equilibrium), since pðb
1Þ þ ð1
pÞb [ p  0 þ ð1
pÞ  0 $ b [ p

Fig. 11.26 Pooling strategy profile—Acc (with responses)

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More Advanced Signaling Games

which holds in this case. If, in contrast, the second entrant observes the off-the-equilibrium message of fight, then this player also enters if qðb
1Þ þ ð1
qÞb [ q  0 þ ð1
qÞ  0 $ b [ q Hence, the second entrant enters regardless of the incumbent’s action if off-the-equilibrium beliefs, q, satisfy q\b, as depicted in Fig. 11.26 (see blue shaded branches in the right-hand side of the figure). Otherwise, the entrant only enters after observing the equilibrium message of Acc. The tough monopolist, MT , is hence indifferent between Acc (as prescribed) which yields a payoff of 0, and deviate to Fight, which also yields a payoff of zero. A similar argument applies to the normal monopolist, MN , in the lower part of the game tree. Hence, the pooling strategy profile where both types of incumbents accommodate can be sustained as a PBE. A remark on the Intuitive Criterion. Let us next show that the above pooling equilibrium, despite constituting a PBE, violates the Cho and Kreps’ (1987) Intuitive Criterion. In particular, the tough monopolist has incentives to deviate towards Fight if, by doing so, he is identified as a tough player, q ¼ 1, which induces the entrant to respond not entering. In this case, the tough monopolist obtains a payoff of a, which exceeds his equilibrium payoff of 0. In contrast, the normal monopolist doesn’t have incentives to deviate since, even if his deviation to Fight deters entry, his payoff from doing so, −1 + a, would still be lower than his equilibrium payoff of 0, given that −1 + a < 0 or a < 1. Hence, only the tough monopolist has incentives to deviate, and the entrant’s off-the-equilibrium beliefs can thus be restricted to q = 1 upon observing that the monopolist fights. Intuitively, the entrant infers that the observation of Fight can only originate from the tough monopolist. In this case, the tough incumbent indeed prefers to select Fight, thus implying that the above pooling PBE violates Cho Kreps’ (1983) Intuitive Criterion (Q.E.D). Let us finally check if this pooling strategy profile can be sustained when off-the-equilibrium beliefs satisfy, instead, q  b, thus inducing the entrant to

Fig. 11.27 Pooling strategy profile—Acc (with responses)

Exercise 6—Entry Deterrence …

523

Fig. 11.28 Pooling strategy profile—Fight

respond not entering upon observing the off-the-equilibrium message of Fight, as illustrated in the game tree of Fig. 11.27 (see blue shaded branches in the right-hand side of the tree). In this case, the MT has incentives to deviate from Acc, and thus the pooling strategy profile where both MT and MN select to Acc cannot be sustained as a PBE. Pooling PBE with Fight. Let us now examine the opposite pooling strategy profile (Fight, Fight), in which both types of monopolists fight, as depicted in Fig. 11.28. Hence, equilibrium beliefs after observing Fight cannot be updated, and thus satisfy q = p; while off-the-equilibrium beliefs are arbitrary r 2 ½0; 1 after observing the off-the-equilibrium message of accommodation. Given these beliefs, upon observing the equilibrium message of Fight, the entrant responds entering since pð b
1Þ þ p  b [ p  0 þ ð 1
pÞ  0 $ b [ p which holds by definition. If, in contrast, the entrant off-the-equilibrium message of Acc, then it responds entering if

Fig. 11.29 Pooling strategy profile—Fight (with responses)

observes the

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Fig. 11.30 Pooling strategy profile—Fight (with responses)

r ð b
1Þ þ r  b [ p  0 þ ð 1
r Þ  0 $ b [ r Hence, if b > r, the entrant enters both after observing Fight (in equilibrium) and Acc (off-the-equilibrium path). If, instead, b  r, then the entrant only responds entering after observing Fight, but is deterred from the industry otherwise. We next separately analyze each case. Case 1. Figure 11.29 illustrates the entrant’s responses when b > r, and thus the second entrant enters after Acc. In this context, the tough monopolist is indifferent between Fight, obtaining zero profits, and accommodating, which also yields zero profits. A similar argument applies to the normal monopolist in the lower part of the game tree. Hence, in this case the pooling strategy profile (Fight, Fight) can be supported as a PBE if off-the-equilibrium beliefs, r, satisfy r < b. Case 2. If, instead, r  b, then the entrant is deterred upon observing the off-the-equilibrium message of Acc, as Fig. 11.30 depicts. The pooling strategy profile cannot be sustained in this context, since MN has incentives to deviate towards Acc, obtaining a payoff of a, which exceeds its payoff of zero when he Fights. Hence, the pooling strategy profile (Fight, Fight) cannot be supported as a PBE when off-the-equilibrium beliefs satisfy r  b:

Fig. 11.31 Separating strategy profile (Fight, Acc)

Exercise 6—Entry Deterrence …

525

Fig. 11.32 Separating strategy profile (Fight, Acc), with responses

Separating PBE (Fight, Acc). Let us next examine if the separating strategy profile in which only the tough monopolist fights can be sustained as a PBE. Figure 11.31 depicts this strategy profile. In this case, entrant’s beliefs are updated to q = 1 and r = 0 using Bayes’ rule, implying that, upon observing Fight, the entrant is deterred from the market since b − 1 < 0, given that b < 1 by definition. Upon observing Acc, the entrant is instead attracted to the market since b > 0. Figure 11.32 illustrates the entrant’s responses (see blue shaded arrows). In this setting, no type of monopolist has incentives to deviate: (1) the tough monopolist obtains a payoff of a by fighting (as prescribed) but only 0 from deviating towards Acc; and similarly (2) the normal monopolist obtains 0 by accommodating (as prescribed) but a negative payoff, −1 + a, by deviating towards Fight, given that a < 1 by definition. Hence, this separating strategy profile can be sustained as a PBE.

Fig. 11.33 Separating strategy profile (Acc, Fight)

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Fig. 11.34 Separating strategy profile (Acc, Fight), with responses

Separating PBE (Acc, Fight). Let us now check if the alternative separating strategy profile, in which only the normal monopolist fights, can be supported as a PBE (We know that this strategy profile sounds crazy, but we want to formally show that it cannot be sustained as a PBE.). Figure 11.33 illustrates this strategy profile. In this setting, the entrant’s beliefs can be updated to r = 1 and q = 0, inducing the entrant to respond not entering after observing Acc, but entering after observing Fight, as depicted in Fig. 11.34 (see blue shaded branches). Given these responses by the entrant, the tough monopolist has incentives to deviate from Acc, which yields a negative payoff of −1 + a, to Fight, which yields a higher payoff of zero. Therefore, this separating strategy cannot be supported as a PBE.

References Cho, I., and Kreps, D. (1987). Signaling games and stable equilibrium. Quarterly Journal of Economics, 102, 179–222. Kreps, D. M., Milbrom, P., Roberts, J., and Wilson, R. (1982). Rational cooperation in the finitely repeated prisoners’ dilemma. Journal of Economic Theory, 27(2), 245–252. Spence, M. (1974). Job market signaling. Quarterly Journal of Economics, 87, 355–374.

Correction to: Strategy and Game Theory

12

Correction to: F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7 In the original version of the book, belated corrections provided by the author in the chapters 1–6, 9 and 10 have now been incorporated.

The updated version of the book can be found at https://doi.org/10.1007/978-981-13-0463-7 © Springer Nature Switzerland AG 2019, corrected publication 2020 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7_12

C1

Index

A All-pay auction, 355, 370, 371 Anti-coordination game, 21, 90, 94, 141, 143 Auctions, 355, 361–363, 368–373, 377–379 B Backward induction, 155–157, 198, 202–204, 210, 264, 517, 519 Bargaining game, 125, 126, 156, 157, 200, 202, 307, 309 Battle of the sexes game, 18, 19, 40, 43, 44, 88, 146, 147 Bayesian Nash equilibria (BNE), 323 Bayes’ rule, 138, 142, 381, 387, 389, 445, 463, 466, 490, 515, 525 Beliefs, 381–385, 387–389, 392, 394, 396, 398, 447, 455, 456, 462, 465–467, 497, 508, 514, 521, 526 Belief updating, 381, 389, 392, 394, 396, 398, 525, 526 Bertrand game, 67, 182, 219, 222, 223, 225, 227, 228, 294, 295, 345 Best response function, 31, 51, 55, 57, 58, 72, 73, 76, 83, 93, 94, 97, 130, 182, 195, 196, 199, 203, 222, 229, 231, 237, 246, 257, 269, 314, 319, 439, 494 Best responses, 39, 40, 43, 59, 60, 64, 65, 122, 128, 132, 177, 208, 303, 309, 332, 334, 335, 337 Blume, 228 Branch (of a game tree), 7, 206, 395, 398–400, 491, 523 C Cartel, 277, 282–284, 286, 292–294 Cheating, 279, 280 Cho and Kreps’ (1987) Intuitive Criterion, 382, 453, 455, 457, 487, 506, 509, 522 Collusion, 282, 291, 293–295, 314, 317, 321

Competition in prices, 67, 186, 196, 219, 220, 225, 228, 294 Competition in quantities, 50, 81, 155, 187, 197, 219, 234, 240, 250, 259, 277, 282, 291 Complete information, 39, 40, 219, 324, 342, 343, 487, 493, 494, 512 Concavity, 203 Conditional probability, 138, 446, 462 Constitutes, 196 Continuous action, 460 Convex hull of equilibrium payoffs, 130, 131, 136 Convexity, 311 Cooperation, 277–281, 285, 317, 318 Coordination games, 18, 74, 122, 123, 494 Correlated equilibrium, 88, 128, 130, 132, 137, 139, 141–143 Cournot game, 40, 50, 51, 182, 219, 239, 282, 291, 324 Credible punishment, 286 D Defect (in repeated games), 277, 279–281, 313 Deviation, 56, 220, 221, 224, 226, 228, 277, 278, 285, 286, 292, 295, 314, 318, 321, 448 Direct approach, 57, 362, 364, 366, 373, 374 Direct demand, 195–197, 318, 438 Discrete action, 225 Dominance solvable games, 1, 11 Duopolists, 81 Duopoly, 219–221, 226, 231, 264, 342 E Efficiency gains, 40, 81, 85 Entrants, 488 Entry deterrence, 511–513, 516–518

© Springer Nature Switzerland AG 2019 F. Munoz-Garcia and D. Toro-Gonzalez, Strategy and Game Theory, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-11902-7

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528 Envelope theorem, 247, 256, 355, 362, 364–366, 373, 374 Expected utility, 91, 93, 112, 141, 147–149, 151, 153, 214, 304, 310, 340, 352, 363, 370, 387, 446, 464, 467, 504, 505 Extensive-form game, 2, 5 F Finitely-repeated games, 87, 277 First order statistic, 374 Folk theorem, 278, 312 G Game of Chicken, 88, 90, 94 Game tree, 1, 2, 5–7, 125, 174, 204, 206, 211, 213, 333, 383, 386, 394, 399, 447, 512, 517, 519, 523 Gift-Exchange game, 179, 180 Grim-trigger strategy, 313 H Heterogeneous products, 225, 226 Homogeneous products, 255, 314, 317 Hoteling, 66 I Incentives and punishment, 74, 77 Incomplete Information, 323, 330, 334, 342, 344, 355, 382, 487, 493, 498, 516 Inequity aversion, 155 Infinitely-repeated games, 278, 282, 309, 313 Information set, 5, 9, 10, 383, 386, 487–489, 491, 496 Initial node, 2, 6, 7, 9, 10 Inverse demand, 50, 182, 198, 220, 225, 236, 246, 249, 250, 255, 273, 318 Iterative Deletion of Strictly Dominated Strategies (IDSDS), 1, 12, 14, 21, 23, 338 Iterative Deletion of Weakly Dominated Strategies (IDWDS), 14, 32 K Kreps, 382, 453, 455, 457, 487, 488, 506, 509, 511, 516, 522 L Labor market signaling, 391, 457, 506 Lobbying, 31, 72, 94–99 M Matching pennies game, 144, 145 Maxmin strategies, 88, 147, 149 Mergers, 81, 85, 273–275

Index Mixed strategy, 87, 88, 92, 94, 95, 127, 219, 222, 227, 304, 495 Mixed strategy Nash Equilibrium (msNE), 88, 92, 95, 222, 227, 303 Monopolist, 284, 317, 452, 516, 519, 521, 522, 524–526 Monopolistic, 319 Monopoly, 50, 182, 186, 187, 222, 284, 315, 318, 319 Moral hazard, 155, 211, 213, 215 N Nash equilibrium, 19, 39, 43, 52, 57, 64, 67–69, 77, 88, 92, 127, 131, 136, 147, 196, 208, 209, 220, 227, 237, 278, 286, 303, 310, 313, 337, 343, 361, 494 Nash reversion (in repeated games), 279 Node (of a game tree), 5, 9, 10, 208, 213, 383, 386, 392, 446, 514, 519 Normal-form game, 3, 5, 8, 11–17, 23, 25, 26, 33, 34, 59, 60, 62, 63, 65, 129, 133, 148 O Oligopolists, 255, 282, 314, 438 Oligopoly, 256, 277, 286 Outcomes of the game, 3 P Pareto coordination game, 74, 122, 123, 494 Pareto optimality, 171 Payoffs, 3, 9, 11, 19, 42, 43, 45, 61, 64, 66, 90, 113, 115, 127, 128, 130, 131, 134, 136, 141, 149, 156, 176, 202, 208, 210, 277, 282, 303, 306, 308, 309, 311, 313, 332, 339, 460 Perfect Bayesian Equilibrium (PBE), 381, 383, 504 Pooling equilibria, 382, 453, 488 Posterior beliefs, 492, 499, 521 Price discrimination, 448, 449, 452 Prior beliefs, 517 Prisoners’ Dilemma game, 9, 10 Private firms, 229 Private information, 438, 453, 493 Probability, 68, 75, 88, 90, 91, 94, 95, 99, 112, 114, 138, 141, 152, 225, 305, 329, 338, 349, 350, 361, 364, 370, 379, 389, 445, 460, 490, 492, 504, 511, 517 Product differentiation, 248, 317 Profitable deviation, 56, 69, 139, 220, 221, 224, 226, 228, 320 Profit maximization, 183, 197, 203, 265, 271 Public firms, 219, 228–231

Index Pure strategies, 87, 92, 96, 114, 135, 152, 304, 338 Pure strategy Nash equilibrium (psNE), 39, 53, 55, 58, 64, 67, 74, 87, 88, 90–92, 123, 226, 236, 303, 307 Q Quantity competition, 40, 50, 51, 155, 182, 229 R Repeated games, 277, 279, 282, 303, 309, 313 Reputation, 488, 516 Risk, 180, 200, 356, 369, 377–379 Risk aversion, 356, 368, 369, 378, 379 Rock-paper-scissors game, 7 S Semi-separating equilibrium, 443, 444, 447, 448 Separating equilibria, 383, 384, 499, 501, 506, 511, 514 Sequence of entrants, 516 Sequential-move game, 125, 174, 182, 202, 211, 381 Sequential rationality, 155 Signaling, 381, 382, 392, 443, 453, 487, 506, 520 Simultaneous-move games, 21, 39, 62 Singh, 250 Social welfare, 42, 219, 229, 231, 274, 275 Spence, 382, 391, 487

529 Spence’s (1974) job market signaling game, 382, 487 Stackelberg game, 155, 182, 185, 198, 200 Strategic complements, 54, 196 Strategic substitutes, 50 Strictly competitive game, 87, 88, 143, 144, 146, 147, 149, 150 Strictly dominated strategies, 1, 2, 12, 14, 18, 21, 32, 303, 338 Subgame Perfect Nash equilibria (SPNE), 155, 158, 174, 180, 181, 184, 200, 204, 213, 278, 307 Symmetric strategy profiles, 55, 67, 68 T Terminal nodes, 9, 10, 204, 208, 211, 213 Traveler’s Dilemma game, 45, 55 U Ultimatum bargaining game, 156–158 Uncertainty, 438 Utility function, 73, 74, 156, 181, 211, 362, 369, 377, 449 V Vives, 250 W Weakly dominated strategies, 2, 14, 16, 32, 34 Wilson, 488