Game Theory - Successful Negotiation in Purchasing: Requirements, Incentives and Award 3658408677, 9783658408671

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Christoph Pfeiffer

Game Theory —Successful Negotiation in Purchasing Requirements, Incentives and Award

Game Theory—Successful Negotiation in Purchasing

Christoph Pfeiffer

Game Theory—Successful Negotiation in Purchasing Requirements, Incentives and Award

Christoph Pfeiffer Competitio Consulting GmbH Hamburg, Germany

ISBN 978-3-658-40867-1 ISBN 978-3-658-40868-8 (eBook) https://doi.org/10.1007/978-3-658-40868-8 This book is a translation of the original German „Spieltheorie – Erfolgreich verhandeln im Einkauf“ by Pfeiffer, Christoph, published by Springer Fachmedien Wiesbaden GmbH in 2021. The translation was done with the help of an artificial intelligence machine translation tool. A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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 Fachmedien Wiesbaden GmbH, part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Preface and Acknowledgments

Construction projects are not the only ones that tend to run over their original schedules. Books, too, usually require more energy and time than expected. This book was no exception. A book is necessarily a collaborative project. First and foremost, I would like to thank Sarah Pfeiffer for her patience and loving support. Without her, I could not have completed this book. Professionally, I am most indebted to Gero von Grawert. His well-founded and precise comments have greatly improved the content and technical aspects of this text. Any remaining errors are mine. Hamburg, Germany

Christoph Pfeiffer

v

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 5

2 Basic Game Theory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Randomized Strategies and Second Mover Advantage . . . . . . . . 2.5 Extensive Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Backward Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Evolutionary Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Mechanism Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 8 8 10 12 16 18 23 26 29

3 Examples of Applied Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Price Guarantees—A Case for the Cartel Office? . . . . . . . . . . . . 3.2 Location Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Number Selection and Level n Rationality . . . . . . . . . . . . . . . . . . 3.4 The Tragedy of the Commons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Price Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Three Goals and a Car . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 From Duel to Truel and the Advantages of Weakness . . . . . . . . 3.8 Focal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 The Prisoner’s Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 The Repeated Prisoner’s Dilemma and the Evolution of Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 You can win with stubbornness . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 35 39 40 45 48 53 56 59 61 65

vii

viii

Contents

3.12 The Ultimatum Game and the Perception of Fairness . . . . . . . . . 3.13 The 100-Euro Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 67 70

4 Purchasing Negotiations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Basic Model of a Negotiation . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Central Auction Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 English and Japanese Auction . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Dutch Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 First-price Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Second-price Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Ausubel Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Average Price Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.7 Multi-good Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Further Aspects of Auction Design . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Winner’s Curse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Monetization of the Advantages and Disadvantages of an Offer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Credibility of the Mechanism . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 (Stochastic) Reserve Prices . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Duration of the Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Ambiguity in the English Auction . . . . . . . . . . . . . . . . . . . 4.3.7 The English-Dutch Auction . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 73 76 80 84 85 89 91 94 100 100 103 104 104 105 106 108 109

5 Data-based Identification of Cooperation between Suppliers . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 120

6 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 A Multi-good Auction with Partial Monopoly Markets . . . . . . . 6.2 Chilean School Meals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Building a Factory in Mexico . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 123 124 126

7 Application and Limits of Game Theoretical Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 129

8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

9 Appendix—Derivation of Optimal Bid in a First Price Auction . . . .

133

1

Introduction

A Tier 1 automotive supplier increases annual profits by more than 20% with a single sourcing project. A 15% savings is achieved in another project that was expected to result in price increases. A school district achieves significant cost reductions in its statewide purchasing of school lunches, while improving the quality of the meals and increasing margins for its suppliers. These remarkable results, described in more detail in a later chapter, are based on a logical structure that can be applied systematically: game theory. Game theory is the mathematical modeling of interactions in which the utility of one party depends on the behavior of the other. In principle, all human interactions can be understood as games. However, game theory does not only aim at understanding interdependent decisions and deriving the optimal behavior of the actors involved with the help of analytical methods. It also provides answers to the question of how to optimally design incentives in the context of such strategic interdependencies. As a result, game theory has become the analytical foundation for many areas of auction and negotiation theory. Over the past few decades, purchasing has evolved from an “order taker” to a strategically important function with a wide range of responsibilities. In many companies, the ratio of purchasing volume to revenue exceeds 50%, making it clear that efficient purchasing plays a significant role in achieving both short- and long-term goals. Purchasing is no longer limited to improving the bottom line, but is also responsible for the top line as suppliers increasingly contribute to the overall attractiveness of the final product through their own innovations. Purchasing objectives are derived from the business strategy. These objectives may be to reduce costs, reduce the number of suppliers, or improve product quality. The objectives are translated into a purchasing strategy and further broken down into commodity group strategies. The tactical purchasing process includes the specification of the service to be purchased, the selection of the best suppliers, © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 C. Pfeiffer, Game Theory—Successful Negotiation in Purchasing, https://doi.org/10.1007/978-3-658-40868-8_1

1

2

1

Introduction

Table 1.1 The Savings-Revenue Equivalence Matrix shows what increase in sales would be required to achieve the same EBIT effect as the respective savings Savings [EUR million] Margin

0.1

0.2

0.5

1

10

2%

5

10

25

50

500

5%

2

4

10

20

200

6%

1

3

7

14

143

10%

1

2

5

10

100

20%

1

1

3

5

50

and the preparation and conduct of negotiations (cf. van Weele 2018, pp. 66– 69). Suppliers can be selected either through traditional bilateral negotiations or through a competitive bidding process in which suppliers compete on a level playing field. The most common application of game theory within tactical sourcing processes is the rule-based design of negotiations with pre-qualified suppliers. Mechanism design, a subfield of game theory, provides recommendations for the optimal design of a negotiation with defined goals under various conditions. Applied to procurement, game theory has become an important tool to systematically ensure efficient allocations. However, there are significant differences in the degree of its diffusion. While some automotive OEMs or technology companies already have their own game theory departments, game theory is often still new territory in the SME sector, in retail, in public procurement or in procurement under the sector regulation. An interesting consideration is how much sales would have to increase to match the EBIT effect of the savings achieved through sourcing. The result depends on the average margin achieved. With a 5% margin, sales would have to increase by EUR 2 million to match EUR 100,000 of savings (Table 1.1). Let us begin, however, with an example that illustrates the mutual interdependence of decision-making. In the short story The Final Problem (Doyle 1993, pp. 30–48; last filmed in the BBC series Sherlock) towards the end of the plot, an interesting situation arises (cf. von Neumann and Morgenstern 1947, pp. 176– 178): Sherlock Holmes and Dr. Watson have collected enough evidence to convict the mathematics professor (!) and villain James Moriarty. Moriarty is aware of this and will therefore try to kill Sherlock and Watson. The two of them flee on a train to Dover, from where they intend to cross over to Continental Europe by ferry. As the train leaves Victoria Station, they see Moriarty on the platform,

1

Introduction

3

Victoria Station Sherlock

Canterbury Dover

Moriarty

Canterbury

Dover

Sherlock:

Sherlock:

Morriarty:

Morriarty:

Sherlock:

Sherlock:

Morriarty:

Morriarty:

Fig. 1.1 Sherlock Holmes and Dr. Watson consider at which station to get off the train. Sherlock’s decision is based on how Professor Moriarty is likely to behave, and Moriarty’s behavior is based on how he expects Sherlock and Dr. Watson to behave

who unsuccessfully tries to stop the train. Apparently, Moriarty is aware of their escape plan. Sherlock and Watson have two options: get off at the next stop, Canterbury, or continue on to Dover. Which decision is better for Sherlock depends on what Moriarty decides. If Sherlock and Watson get off at the stop where Moriarty is waiting for them, he will probably kill them. To make an optimal decision, Sherlock will ask himself what he would do if he were Moriarty, and Moriarty will ask himself how he would behave in this situation if he were Sherlock (Fig. 1.1).

4

1

Introduction

Assuming another person’s point of view implies one’s own point of view, which implies the other person’s point of view. But the problem does not end here; it can continue indefinitely. This creates an infinite regress, that is, an infinite sequence of successive considerations. This mental tangle can be untangled with the help of game theory. Without going into a detailed analysis of the strategic situation at this point, suffice it to say that Sherlock and Dr. Watson decide not to continue to Dover, but to get off the train at Canterbury. This allows them to avoid Professor Moriarty, who is taking the express train to Dover without any stops. The game-theoretic analysis and design of auctions has been one of the most productive areas of economics in recent years (Klemperer 2018). Although theoretical insights have not always been confirmed, the possibility and necessity of applying theoretical insights to economically highly relevant situations has accelerated the growth of practically applicable knowledge. Although most theoretical, experimental, or empirical studies of auctions focus on forward auctions, many of the findings are also applicable to reverse auctions. In particular, the auctioning of third-generation mobile frequencies and the resulting revenues (e.g. EUR 50 billion in Germany) raised awareness of auctions. The results of UMTS frequency auctions vary widely across Europe. While Germany and the United Kingdom achieved high per capita revenues, revenues in other countries, such as the Netherlands and Switzerland, were significantly lower (Fig. 1.2). The auction designs used in the UMTS auctions had certain weaknesses, which auction participants adapted to over time. The first auctions in the United Kingdom and Germany worked well in terms of revenues, but administrators had to accept much lower revenues in later auctions. Purchasing auctions are less transparent to outsiders and do not attract the same media attention as public spectrum auctions. Nevertheless, auction rules used have a significant impact on the acquisition cost, which contradicts the theoretical statement of the revenue equivalence theorem. This theorem states (under restrictive assumptions) that the outcome of the auction, i.e. the price level achieved, is independent of the auction format chosen. A basic knowledge of auctions and game theory is therefore especially important for buyers. But every purchase needs a seller. As (complex) buying auctions have become more common, sellers are increasingly confronted with this problem, so it is also worthwhile for them to study game and auction theory.

References

5

-98% 3,690 3,150

1,050

850

618 76

GER

UK

IT

NL

AU

CH

Result in € / population / license

Fig. 1.2 Overview of UMTS Auction Results in Europe 2000 (cf. Grimm et al. 2002, p. 224)

References Doyle AC (1993) The final problem, the memoirs of Sherlock Holmes, Roden C (ed). Oxford University Press, Oxford Grimm V, Riedel F, Wolfstetter E (2002) The third generation (UMTS) spectrum auction in Germany. ifo Studien 48:123–143 Klemperer P (2018) Auctions: theory and practice. The Toulouse lectures in economics. Princeton University Press, Princeton v. Neumann J, Morgenstern O (1947) Theory of games and economic behavior. Princeton University Press, Princeton van Weele AJ (2018) Purchasing and supply chain management. Cengage, Andover

2

Basic Game Theory Concepts

Abstract

This chapter gives a brief overview of the basic concepts of game theory. Without claiming completeness, it introduces elementary methods that are applied in the following chapters. For a detailed, formal presentation, I would recommend Berninghaus (2010) or—largely without mathematics— Binmore (2007). If you have prior knowledge of game theory, or are simply impatient, you could skip directly to Chap. 4. Game theory was developed by John von Neumann as a way to mathematically model and better understand the logic of strategic interaction. In 1928, von Neumann introduced the foundations of game theory in his article Zur Theorie der Gesellschaftsspiele. Through his collaboration with Oscar Morgenstern, the foundation for the further development of game theory was laid in 1944 in Theory of Games and Economic Behavior. The analysis was still primarily limited to zero-sum games and two-player games. In zero-sum games, the advantage of one player is the disadvantage of the other, and there is no way to increase total utility. Games with more than two players, on the other hand, were only possible by forming coalitions, so that the games logically corresponded to a two-player game again. John Nash (1950) developed the tools necessary to generalize to non-zero-sum games and non-cooperative games with more than two players. The Nash equilibrium has been further refined, for example, by the introduction of sequential games (Reinhard Selten) or imperfect information (John Harsanyi). Experimental studies in which subjects (usually with real money) play theoretical models are now an integral part of game theoretic analysis.

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 C. Pfeiffer, Game Theory—Successful Negotiation in Purchasing, https://doi.org/10.1007/978-3-658-40868-8_2

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Basic Game Theory Concepts

In game theory, a game consists of at least two players. Each player has one or more strategies to choose from. Depending on which combination of strategies the players pick, there is an outcome that results in a payoff for each player. Payoff is often equated with monetary gain, but non-monetary payoffs are just as common.1

2.1

Normal Form

The most common representation of a game is the normal form (Fig. 2.1). The normal form is a compact representation of all strategies and possible outcomes. It assumes that there is a single interaction in which all players decide simultaneously. Another common assumption is that all players are rational. In this sense, each player is rational if she maximizes her payoff as defined. In addition, common knowledge of rationality is often assumed: all players know their own rationality, the rationality of the other players, and the knowledge of the other players about the rationality of all players. Figure 2.1 shows a two-player game in normal form, consisting of two rows and two columns. Player 1 can choose a row and player 2 can choose a column. The players may also be referred to as row players or column players. Depending on which combination of rows and columns (also called strategy combination) is chosen, the payoffs for each player are realized in one of the four squares. However, it is not possible to solve the game without knowing the payoffs. No statement can be made about the expected behavior of the players until at least the payoff relationships are known.

2.2

Dominance

Figure 2.2 shows specific payoffs. We can see that player 1 always has a better payoff when he chooses strategy 1. Strategy 1 is therefore the dominant option for player 1. Player 2 can infer that player 1 will not choose the dominated alternative. The decision situation is simplified for player 2 by eliminating one of player 1’s options. 1

Deviations are also plausible because there is usually a nonlinear relationship between monetary gain and utility. Money tends to have diminishing marginal utility. That is, each additional unit of money returns less value as the total amount increases. Similarly, a loss is valued more intensely in absolute terms than a gain of the same amount.

2.2 Dominance

9

Player 2

Option 1 Option 2

Player 1

Option 1

Option 2

Payoff player 1

Payoff player 1

Payoff player 2

Payoff player 2

Payoff player 1

Payoff player 1

Payoff player 2

Payoff player 2

Fig. 2.1 A general two-player game in normal form

In the remaining first row, player 2 gets a better result if he chooses the second option. As a result, after a simple iteration (cf. Sect. 2.6), the strategy combination where player 1 chooses option 1 and player 2 chooses option 2 emerges. Is an equilibrium a realistic prediction if it results from a dominance derived from multiple steps? It has been shown that only a simple iteration can be reliably reproduced in experiments. Equilibria resulting analytically from multi-step eliminations due to nested dominant alternatives are no longer reliably reproduced in experimental interactions (cf. Kreps 2019, p. 22). However, it can be argued that firms tend to have better analytical capabilities than individuals and that they use them when faced with sufficiently relevant decision problems. As a result, interactions between firms may lead to equilibria that are derived from a higher number of iterations. A weakly dominant strategy always achieves a payoff that is better or equal to a payoff gained by any other strategy. Experimental studies show that iteration in the case of weak dominance can only be reproduced unreliably (Ibid., pp. 24–25).

10

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Basic Game Theory Concepts

Player 2

Option 1 Option 2

Player 1

Option 1

Option 2

P1: 3

P1: 2

P2: 0

P2: 1

P1: 2

P1: 0

P2: 4

P2: 0

Fig. 2.2 A two-player game with defined payoffs shown in normal form. P1 denotes player 1’s payoff, P2 denotes player 2’s payoff

2.3

Nash Equilibrium

A Nash equilibrium (NE) is a combination of strategies in which no player can improve her payoff by unilaterally changing her strategy.2 The Nash equilibrium and John Nash, who gave the Nash equilibrium its name, were popularized through the movie A Beautiful Mind. In one scene, John Nash, played by Russell Crowe, is in a bar with friends. There is also a group of five women present, one of whom, from the perspective of John Nash and his group, is a particularly attractive blonde. The other four women are brunettes. John is now considering what strategy the group should use: If everyone competes for the blond, we block each other, and no one gets her. So, then we all go for her friends. But they give us the cold shoulder, because no one likes to be second choice. Again, no winner. But what if none of us goes for the blonde? We 2

In a strict NE, a unilateral change in strategy results in a worse payoff for any player who makes the change.

2.3 Nash Equilibrium

11

don’t get in each other’s way; we don’t insult the other girls. That’s the only way we win.

Unfortunately, the strategy described in the movie is not a Nash equilibrium, because each of the men could improve his payoff by unilaterally deviating from the strategy proposed by John and turning to the blonde woman. Varian (2002) notes that, atypically for game theory, the consideration of the other side was largely neglected in the example. A game with two true Nash equilibria is shown in the simple coordination game in Fig. 2.3. Two players can individually decide whether to go to a movie theater or a coffee shop. Both players are social, i.e. if they both visit the places separately, they both get a payoff of zero. The two asymmetric solutions do not yet represent a Nash equilibrium, since both players can improve their payoff by unilaterally changing their strategy. Player 1 prefers going to a coffee shop to going to a movie theater. For player 2, it is the other way around. The strategy combinations of both players going to the

Player 2

Café Cine

Player 1

Café

Cinema

P1: 10

P1: 0

P2: 6

P2: 0

P1: 0

P1: 6

P2: 0

P2: 10

Fig. 2.3 There are two Nash equilibria in this simple coordination game

12

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Basic Game Theory Concepts

coffee shop or both players going to the movies each represent a Nash equilibrium. If there is an opportunity for interaction before the decision is made, there is a good chance that a Nash equilibrium will be realized. In repeated interactions, coordination can also occur through learning processes. One interpretation of the Nash equilibrium is that of an agreement reached over time through adaptation.

2.4

Randomized Strategies and Second Mover Advantage

Under certain circumstances, it is advantageous for players not to follow a deterministic strategy, but to make their own decision somewhat unpredictable. This is obviously the case in many sports. In a penalty kick from the 11-meter spot in soccer, the ball is so fast, at least when advanced players perform the kick, that the goalkeeper has no chance of preventing a goal if he decides which side to jump to after the kick. Therefore, he must decide to jump to one side or stay in the middle before or during the shot. This gives the kicker the opportunity to make a reactive shot, as practiced by Wynton Rufer, Rodolfo Esteban Cardoso and Hans-Jörg Butt. “This is a deliberate delay in the last steps of the approach (as a temporal deception), which is used to provoke or wait for a reaction from the goalkeeper to one side, in order to then shoot the penalty kick into the other corner” (Kibele 2013). Let’s stay with the case where the goalkeeper and the soccer player choose an area of the soccer goal at the same time. For simplicity, let’s assume that the areas of the soccer goal to choose from are left, center, and right. The scoring probabilities are shown in Fig. 2.4. The penalty kick can be viewed as an anti-coordination game in which the shooter tries to choose the area that the goalkeeper does not choose. To further simplify the analysis, we assume that if the kicker shoots into an area not chosen by the goalkeeper, a goal is certain. But even if the goalkeeper chooses the same area as the shooter, there is a certain probability that a goal will be scored. In the example, we assume that the shooter has a particularly strong shot into the left corner, so there is a 60% probability of a goal, even if the goalkeeper chooses the same corner. In the middle and right corners, the remaining probabilities are 30% and 40%, respectively. So what area should the shooter target? With a goalkeeper who makes random decisions, shooting to the left side would be a dominant strategy. In this case, the scoring probability can be calculated as shown in Fig. 2.6. As the probability of scoring is 87% when shooting to the left, this choice would always be preferred.

2.4 Randomized Strategies and Second Mover Advantage

13

Goalkeeper Right

100 %

100 %

40 %

Center

Center

100 %

30 %

100 %

Left

Shooter

Right

Left

60 %

100 %

100 %

Fig. 2.4 The penalty kick game between shooter and goalkeeper. The scoring probabilities represent the shooter’s expected payoff and the goalkeeper’s negative payoff

However, if we assume that all the probabilities are known to both players and that the goalkeeper considers his choice of strategy rationally, then the goalkeeper could simply counter the shooter’s strategy of always aiming to the left by jumping to the left himself. The probability of scoring would then drop to only 60%. However, viewing the shooter deterministically and ignoring the reasoning of the other side may not lead to an optimal result. What is the alternative? Another possibility would be to shoot randomly. This would already lead to a better result than the deterministic choice of strategy, because it is more difficult for the other side to adapt to it. A deterministic strategy cannot be the optimal solution for either player, as it can be easily exploited by the other side. The pure random strategy is an improvement over the deterministic strategy, but it does not seem to be optimal either, as it does not take

14

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Basic Game Theory Concepts

into account the different probabilities of scoring. The pure random strategy can also be exploited, for example by the goalkeeper only choosing the centre. This would reduce the score probability to 77%. If the shooter considers the different hit probabilities, this should mean that he shoots less to the center and more to the left. However, it still needs to be determined what the exact probabilities should be. How can this question be answered? The optimal strategy of the players should not be exploitable by the other team. Since the shooter’s advantage is always the goalkeeper’s disadvantage (and vice versa), the goalkeeper will try to achieve the worst case for the shooter. A non-vulnerable strategy is the maximin strategy. As the name suggests, it maximizes the minimum utility, i.e. it optimizes the worst-case scenario. This makes the other side indifferent between the alternatives. In Fig. 2.6 the probabilities from Fig. 2.5 are supplemented by a worstcase column, in which the line minimum, i.e., the outcome least favorable to the shooter’s chosen strategy, is entered. The three deterministic strategies (left, middle, right) are extended by two random strategies for the shooter, which are added in the two rows below. For simplicity, we restrict ourselves to deterministic strategies for the goalkeeper. The payoffs for the randomized strategies result as expected values, analogous to the calculation in Fig. 2.5. This results in the following optimization problem: Which probabilities for each shooting direction (P(L), P(M), P(R)) maximize the minimum payoff? The following constraints apply: Each probability may take a value between 0% and 100%. In addition, the sum of the three probabilities must be 100%.

Static probability analysis Probability goalkeeper left = 1/3 Probability goalkeeper center = 1/3

The goalkeeper behaves purely randomly (0-rational)

Probability goalkeeper right = 1/3

Hit probability shooter shoots left = 1/3 x 100% + 1/3 x 100% + 1/3 x 60%. = 2/3 + 1/5 = 13/15 = 87 % Hit probability shooter shoots center = 1/3 x 100% + 1/3 x 30% + 1/3 x 100% = 2/3 + 1/10 = 23/30 = 77 % Hit probability shooter shoots right = 1/3 x 40% + 1/3 x 100% + 1/3 x 100%. = 4/30 + 2/3 = 24/30 = 80 %

Fig. 2.5 If we assume a purely random behavior of the goalkeeper, a shot to the left is the optimal choice. However, if the goalkeeper anticipates this, the probability of success drops to 60%

2.4 Randomized Strategies and Second Mover Advantage

15

Right

Right

100 %

100 %

40 %

40 %

100 %

30 %

100 %

30 %

60 %

100 %

100 %

60 %

87 %

77 %

80 %

77 %

82 %

82 %

82 %

82 %

1/3-Pr All options

Left

Center

Center

Worst Case Shooter

Left

Optimal probability

Deterministic strategies Random strategies

Shooter

Goalkeeper

Fig. 2.6 In this zero-sum game, the only Nash equilibrium is a randomized strategy

A good approximation can be obtained by trial and error. For example, if we decrease the probability of aiming at the center by 5 percentage points and increase the probability of aiming at the right by 5 percentage points, the worst case for the shooter improves to a hit probability of 80%. This value is already 3 percentage points higher than the randomly chosen value. A more precise optimal value can be determined using a mathematical optimization method, linear programming. It uses algorithms to solve an optimization problem under constraints. This can be done in Microsoft Excel. The following optimal values are obtained from optimization: The shooter shoots to the left with 44.7%, to the center with 25.5%, and to the right with 29.8% according to the maximin optimization. The worst case for the

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shooter improves by another 2 percentage points to 82% compared to the first approximation solution (Fig. 2.6). What is interesting about this solution is that it makes the choice of strategy by the goalkeeper irrelevant to the outcome. Regardless of the goalkeeper’s choice, both players always have the same expected payoff. This is true even if, for example, the goalkeeper always stays in the middle. The maximin strategy creates immunity to any attempt by the opponent to gain an advantage.3

2.5

Extensive Form

At the beginning of the last section, we saw that some shooters have developed a shooting technique in which they simulate the shot and delay it for a short moment. This allows them to observe the goalkeeper’s decision before taking their own shot. In effect, they turn a simultaneous game into a sequential one. You can anticipate the goalkeeper’s decision and then shoot where the goalkeeper is not. This way, almost 100% accuracy is possible, regardless of the goalkeeper’s decision.4 A sequential game in which players make their decisions one at a time can be represented in the extensive form (Fig. 2.7). In the sequential game, the shooter has an advantage because he can observe the goalkeeper’s decision and then make an optimal decision for himself. This is the second mover advantage. In the same way, however, a first mover advantage is also possible under different circumstances. This can be seen, for example, in the coordination game in Sect. 2.3, where two players have the choice of going to a movie theater or a coffee shop. Player 1, who prefers the coffee shop, can now go to the coffee shop first, call player 2 from there and let him know what he has decided. Since player 1 has a preference for going to a coffee shop together rathan than being alone at the cinema, he will go to the coffee shop. Each individual decision situation can be considered a subgame. The equilibrium that emerges in this game is therefore a subgame perfect equilibrium, which is a refinement of the Nash equilibrium. This implies that the players behave foresightedly and anticipate the behavior of the other player. The optimal choice for player 1 is derived by backward induction. 3

Levitt et al. (2002) statistically evaluated penalty kicks in the Italian and French soccer leagues over a three-year period. They conclude that the interaction between the kicker and the goalkeeper is essentially simultaneous. Furthermore, they show that both randomize their strategies during the penalty kick, in line with the theoretical analysis. 4 To make the results of penalty kicks more open and exciting, the delay in execution could be prohibited.

2.5 Extensive Form

17 100 %

100 %

Left

gh t

er

er

nt Ce

40 %

Left

Ri g

ft Le

ht

Right Center

60 %

nt

100 %

Ri

t

Ce

ft Le

Righ

100 %

30 %

100 %

Center

100 %

Goalkeeper Shooter ...% hit probability

Fig. 2.7 The subgame perfect equilibrium can be determined by backward induction

In the upper decision situation for player 2, he can choose between a payoff of 6 (café) and a payoff of 0 (cinema). In this situation, he will obviously choose the café. In the lower decision node, player 2 can choose between a payoff of 0 (café) and a payoff of 10 (cinema). The choice would obviously be the movie theater. This leaves only two possible outcomes, namely (café, café) or (cinema, cinema), i.e. the two Nash equilibria of the simultaneously played game. Player 1 will choose the café because he knows that player 2 will follow him and the café generates the higher payoff for him (Fig. 2.8). Backward induction is only possible with complete information. As soon as there are information asymmetries, backward induction may not be feasible. So if player 2 does not know where player 1 is in the second phase of the game, for example because there is no communication between them, then the game is sequential, but strategically it is equivalent to the simultaneous game (Fig. 2.9).

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Basic Game Theory Concepts

Player 1 Player 2

Player 1: 10 Player 2: 6

Ci

Ci

ne

m

a

Player 1: 0 Player 2: 0 Player 1: 0 Player 2: 0

ne

m

a

Ci

ne

m

a

Player 1: 6 Player 2: 10

Fig. 2.8 The subgame-perfect equilibrium can be determined by backward induction under perfect information

2.6

Backward Induction

In the centipede game5 (Fig. 2.10) players take turns for a predetermined number of rounds, either keeping an envelope filled with a certain amount of money for themselves or passing it on to another player. If the envelope is passed, the amount of money in the envelope is increased by a certain amount. Let’s make the conditions more concrete: Player 1 starts in round 1 with an amount of EUR 0. If a player chooses to pass, the total amount increases by EUR 100 in each subsequent round6 , so that in round 2, player 2 would be faced with the decision of either keeping EUR 100 or passing the envelope back to 5

The centipede game goes back to Rosenthal (1980). The name of the game is derived from the many ways to act, which make it look like a centipede. 6 In this case, it is a constant increase in the amount of money. Other interesting scenarios might include increasing the amount itself with each round.

2.6 Backward Induction

19

Player 1 Player 2

Player 1: 10 Player 2: 6

Ci

ne

m

a

Information asymmetry

Player 1: 0 Player 2: 0

Ci

ne

m

Player 1: 0 Player 2: 0

a

Ci

ne

m

a

Player 1: 6 Player 2: 10

Fig. 2.9 With information asymmetry at the second node, the sequential game is strategically equivalent to the simultaneous game

player 1 with EUR 200. The number of rounds is limited to 10. At the start of round 10, the total amount in the envelope has increased to EUR 900. Since player 1 starts in round 1, player 2 would be up in round 10. After round 10, no more rounds are scheduled, so player 2 can only choose between EUR 900 (keep) and EUR 0 (pass). A self-interested, rational player will choose the keep strategy. However, in round 9, player 1 can decide whether to give the envelope to player 2, and thus whether there will be a round 10 at all. Since player 2 is expected to keep the envelope in round 10, it is also advantageous for player 1 to keep the envelope in round 9. This logic can be applied step by step back to round 1. The analysis performed step by step from the end of a game back to the logically derived first step is the backward induction.

Keep

Keep

Keep

Pass

S1: 600 S2: 0

Pass

S1: 0 S2: 700

Pass

S1: 800 S2: 0

Pass

S1: 0 S2: 900

Pass

S1: 1000 S2: 0

2

Fig. 2.10 The centipede game with 10 rounds

Player 2

Keep

Player 1

Keep

S1: 0 S2: 500

Keep

S1: 400 S2: 0

Pass

Keep

S1: 0 S2: 300

Pass

Keep

S1: 200 S2: 0

Pass

Keep

S1: 0 S2: 100

Pass

Keep

S1: 0 S2: 0

Pass

20 Basic Game Theory Concepts

2.6 Backward Induction

21

By backward induction, player 1 can deduce that player 2 will keep the envelope in round 2. So how will player 1 behave in the first round? Based on the previous considerations, player 1 would be indifferent between the two possible actions, since in a deterministic-rational world they both imply a payoff of zero. However, it is clear that in practice most players will not choose an empty envelope as long as there is some chance of winning in subsequent rounds. What is the explanation for this discrepancy between the analytical prediction and the observed behavior? García-Pola et al. (2020, p. 2) list the following approaches: 1. The players are boundedly rational and make mistakes with a certain probability in the sense of a deviation from the Nash equilibrium. When making decisions, they anticipate possible errors by themselves or others. The Quantal Response Equilibrium (QRE) is a statistical solution concept in game theory developed by McKelvey and Palfrey (1995). In the QRE, players’ rational expectations are complemented by possible errors. In this way, rational strategies are still played more often than bad strategies, but not always (Ibid. p. 22). In a sequential game, players assume that both they and the other players will make mistakes in later rounds, in the sense of deviating from the Nash equilibrium with some probability.7 The empirically observed behavior of not ending the game at the first opportunity in the centipede game or similar interactions can thus be interpreted in part8 as a rational anticipation of one’s own or others’ irrational behavior. The sum of possible future errors decreases with each additional round, so the probability of passing the envelope on also decreases. In Fig. 2.11 the centipede game is calculated as a QRE. For the model it is still necessary to specify the parameter lambda, which indicates the degree of irrationality. Even with very low values, it can be seen that the results obtained

7

In extensive games, the Agent-Quantal-Response-Equilibrium (AQRE) is also used (cf. McKelvey and Palfrey 1998). The player sees his future self as another person over whose behavior he has no control, and who will deviate from the analytic optimum with some probability. 8 In addition to the expectation of an error, the deviation may also be the result of a previous error.

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Probabilities envelope forwarding AQRE Probability

Round

Player 1

Player 2

Fig. 2.11 If a certain probability of error is built into the calculation of both players, the probability of passing the envelope is initially higher and decreases as the game progresses. The QRE model is reproduced here with three different rationality values for both players. The QRE was calculated using the Gambit software package (McKelvey et al. 2014)

are closer to what would be expected in reality compared to the model with full rationality.9 If player 1 assumes in the first round that there is a certain probability of errors (in the sense of deviations from the Nash equilibrium) in later rounds by both her and the other side, then a certain payoff of 0 Euro is an inferior outcome to a possible higher payoff in later rounds. Thus, handing over the envelope is the rational choice. 2. The second approach is based on level-k reasoning. Players expect the other player to use a certain iteration depth to analyze the game. If this is known, it

9

It should be noted, however, that these values are based on a statistical model that gains its explanatory power by being fitted to existing data, for which data from comparable situations must be available.

2.7 Evolutionary Game Theory

23

is optimal to choose an iteration depth that is exactly one level higher than that used by the other side. In the logic of level-k reasoning, the optimal behavior of a player results from the expected depth of iterations of the other side. With a fully rational opponent, player 1 must assume that player 2 will claim the contents of the envelope in round 2. However, if player 2 is a so-called level-0 player, this means that he makes his decisions purely randomly. In this case, it would make more sense to pass on the envelope in round 1. If player 2 also gives up the envelope in round 2, player 1 can choose between 200 euros for sure or 400 euros with a 50% probability in round 3 (assuming that he would definitely keep the envelope in round 5 if he got that far). The calculation of a player who assumes that the other side is purely random is that of a level 1 player. A player who anticipates the reasoning of a level 1 player is a level 2 player, and so on. Since the level of the other side cannot be predicted exactly, statistical evaluations can also be used to make statements about how often a certain iteration depth is encountered. However, these are highly context-dependent, and it should be noted that the existence of these statistical measures can itself cause a change in behavior. Nevertheless, k=2 and k=3 are among the most commonly observed levels of rationality. Note that Nash equilibria determined by multiple iterations have relatively little predictive power in reality. The more iterations required, the weaker a statement. The centipede game takes the predictive weakness of Nash equilibria to the extreme, as it is determined by a particularly large number of iterations. Models that also take into account the possibility of error or limited rationality generally have better predictive power when the required reasoning involves multiple steps.

2.7

Evolutionary Game Theory

The games considered so far are aimed at rational agents who make conscious and (largely) rational decisions according to their strategic calculus. In evolutionary game theory, however, players have fixed, genetically determined strategies. In this perspective, members of a population compete for scarce resources using different fixed strategies. Successful interaction secures resources, while failure results in fewer or no resources. Secured resources lead to population growth, while lack of resources leads to population contraction. In this context, the payoff can be understood as a measure of the evolutionary fitness of a strategy or a species (Cowden 2012).

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Both aggressive and defensive strategies are possible when competing for resources. The best-known game in evolutionary game theory is the doves-hawks game, which, in addition to its importance for understanding evolutionary processes, also shows the role of aggression in social interactions. In this game, hawks pursue an aggressive strategy, while doves rely on cooperation and conflict avoidance. If a hawk meets a dove, the hawk can claim the entire resource for itself, because the dove will retreat as soon as it sees a hawk. If two pigeons meet, the resource is shared equally. A hawk that is surrounded by doves always benefits from its aggressive strategy, because it can always claim the resource that is being fought over. The disadvantage of the aggressive strategy arises as soon as two conflict-oriented strategies meet and a conflict starts to occur. The higher the conflict costs and the higher the number of other aggressive players, the more disadvantageous the conflict-oriented strategy becomes. Species that fight each other for resources or status tend to have traits that reduce conflict costs. For example, a lion’s mane is not only impressive, but also serves to reduce the number of neck and throat injuries in fights between male lions. The mane is a brilliant evolutionary development that only reduces the conflict costs of the lion in fights with other lions, but in no way impairs its fighting power against other species. The doves-hawks game is shown in Fig. 2.12. The value of the resource to be fought over is assumed to be 20 for all players. The amount of conflict cost is represented by the variable C and is not specified so that different magnitudes of C can be compared. A key concept in evolutionary game theory is evolutionary stability. An evolutionarily stable strategy (ESS) is a strategy or strategy component in a population that cannot be invaded by other strategies. Let’s imagine a population consisting entirely of pigeons. If a mutation occurs and one or more individuals begin to adopt a hawkish strategy, they would initially have an easy time (as long as the cost of conflict is not very high) and would spread quickly. A population consisting entirely of pigeons can therefore be easily invaded and would not represent an ESS. For a strategy not to be uninvadable, the expected payoff of a mutation must be less than the expected payoff of the existing strategies. The concept of ESS is closely related to the Nash equilibrium; a strict Nash equilibrium (see Sect. 2.3) always represents an ESS. Under what circumstances is the hawk strategy strongly dominant? A strategy is strongly dominant if it is always advantageous to the player, regardless of the strategy chosen by the other player. The hawk’s payoff is always higher than the dove’s. The dove strategy results in a zero payoff if the other player is a hawk.

Dove

Fig. 2.12 The hawk-dove game in normal form (Pastine and Pastine 2017). The higher the cost of conflict, the less advantageous an aggressive strategy

25

Hawk

2.7 Evolutionary Game Theory

Hawk

Dove

(20 - C) / 2

20

(20 – C) / 2

0

0

10

20

10

For the hawk strategy to become a strongly dominant strategy, it must produce a payoff greater than zero when confronted with another hawk strategy. Solving the inequality (20 − C)/2 > 0 results in a conflict cost, C, that is less than 20. As long as the conflict cost is below this threshold, the hawk strategy is a strongly dominant option, a strict Nash equilibrium, and therefore an evolutionarily stable strategy. The evolution of the proportions of a strategy in a finite population can be simulated using a Moran process. The genetic fitness, which results from the respective payoffs, determines the probability of reproduction and extinction. If the hawk strategy is dominant at low conflict costs, the entire population will eventually become hawks (Fig. 2.13), even though the sum of payoffs in a population would be higher if it consisted only of doves. At higher conflict costs, the situation changes, but the dove strategy can never be dominant at any conflict cost value, since the hawk strategy is always advantageous if the other player behaves in a conflict-avoidant manner. As conflict costs increase, the evolutionarily stable fraction of hawks decreases. However, even with very high conflict costs, a single hawk can still generate a high payoff if it encounters only doves. The only limit to the growth of hawks comes from encounters with other hawks. When conflict costs are high, a small fraction of an aggressive strategy is the equilibrium. Consider, for example, the office tyrant who is surrounded

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Basic Game Theory Concepts

High conflict costs (C=40)

Low conflict costs (C