40 Years of Research on Rent Seeking 1: Theory of Rent Seeking 3540791817, 9783540791812

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40 Years of Research on Rent Seeking 1

Roger D. Congleton • Arye L. Hillman Kai A. Konrad (Eds.)

40 Years of Research on Rent Seeking 1 Theory of Rent Seeking

Springer

Prof. Roger D. Congleton George Mason University Center for the Study of Public Choice MSN 1D3 Fairfax, Virginia 22030 USA [email protected]

Prof. Arye L. Hillman Bar-Ilan University Department of Economics 52900 Ramat-Gan Israel [email protected]

Prof. Kai A. Konrad Wissenschaftszentrum Berlin fUr Sozialforschung Reichpietschufer 50 10785 Berlin Germany [email protected]

ISBN: 978-3-540-79181-2 Library of Congress Control Number: 2008926320 © 2008 springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of die German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper 987654321 springer.com

Preface

The last survey of the rent-seeking literature took place more than a decade ago. Since that time a great deal of new research has been published in a wide variety of journals, covering a wide variety of topics. The scope of that research is such that very few researchers will be familiar with more than a small part of contemporary research, and very few libraries will be able to provide access to the full breadth of that research. This two-volume collection provides an extensive overview of 40 years of rent-seeking research. The volumes include the foundational papers, many of which have not been in print for two decades. They include recent game-theoretic analyses of rent-seeking contests and also appHcations of the rent-seeking concepts and methodology to economic regulation, international trade policy, economic history, poUtical competition, and other social phenomena. The new collection is more than twice as large as any previous collection and both updates and extends the earlier surveys. Volume I contains previously published research on the theory of rent-seeking contests, which is an important strand of contemporary game theory. Volume II contains previously pubHshed research that uses the theory of rent-seeking to analyze a broad range of public policy and social science topics. The editors spent more than a year assembling possible papers and, although the selectionsfilltwo large volumes, many more papers could have been included. Our aim has been to include the most important contributions in the literature and give a broad overview of secondary contributions. The end result is afinecollection that shows theflexibiUtyand power of the rent-seeking methodology, and the Ught shed on a broad range of political, social, and institutional research issues. Each volume begins with an extensive survey of the Uterature written by the editors and an overview of the contributions included in the two volumes. Although responsibility for the papers included and organization of the volumes rests entirely with the editors, a number of debts must be acknowledged. The Center for Study of Public Choice provided a grant that made the pubUcation of these volumes possible. Professor Congleton's contribution to the project was supported by the Center for Study of PubHc Choice at George Mason University and the Department of Political Science at the University of Southern Denmark. Professor Hillman's contribution was supported by the Department of Economics at Bar Ilan University, Professor Konrad's contribution was supported by the Social Science Research Center (WZB) and the Department of Economics at the Free University of Berlin. Thanks are due to Nina Bonge for her administrative

Preface

assistance and to Martina Bihn of Springer for her encouragement and oversight of the production of the two volumes. Numerous colleagues provided suggestions for the contents and commented on our selection. Roger D. Congleton Arye L. Hillman Kai A. Konrad

VI

Contents Volume I

Forty Years of Research on Rent Seeking: An Overview Roger D. Congleton, Arye L. Hillman andKaiA. Konrad

1

Part 1 Rents The welfare costs of tariffs, monopoUes, and theft Gordon Tullock

45

Rent seeking and profit seeking James M. Buchanan

55

Competitive process, competitive waste, and institutions Roger D. Congleton

69

Risk-averse rent seekers and the social cost of monopoly power Arye L. Hillman and Eliakim Katz

97

Efficient rent seeking Gordon Tullock

105

Free entry and efficient rent seeking Richard S. Higgins, William E Shughart II, and Robert D. Tollison

121

A general analysis of rent-seeking games /. David Perez-Castrillo and Thierry Verdier

133

Rent-seeking with asymmetric valuations Kofi O, Nti

149

Dissipation of contestable rents by small numbers of contenders Arye L. Hillman and Dov Samet

165

Politically contestable rents and transfers Arye L. Hillman and John G. Riley

185

The all-pay auction with complete information Michael R. Baye, Dan Kovenock, and Casper G. de Vries

209

vu

Contents Rent seeking with bounded rationality: An analysis of the all-pay auction Simon P. Anderson, Jacob K. Goeree, and Charles A. Holt

225

Conflict and rent-seeking success functions: Ratio vs. difference models of relative success Jack Hirshleifer

251

Contest success functions Stergios Skaperdas On the existence and uniqueness of pure Nash equilibrium in rent-seeking games Ferenc Szidarovszky and Koji Okuguchi Part 2

263

271

Collective Dimensions

Committees and rent-seeking effort Roger D. Congleton

279

Risk-averse rent seeking with shared rents Ngo Van Long and Neil Vousden

293

Collective rent dissipation Shmuel Nitzan

309

The equivalence of rent-seeking outcomes for competitive-share and strategic groups KyungHwan Baik, Bouwe R. Dijkstra, Sanghack Lee, and Shi Young Lee

323

PubUc goods, rent dissipation, and candidate competition Heinrich W. Ursprung

329

Effort levels in contests. The pubUc-good prize case Kyung Hwan Baik

347

Rent seeking and the provision of public goods Mark Gradstein

353

A general model of rent seeking for public goods Khalid Riaz, Jason E Shogren, and Stanley JR. Johnson

361

Collective action and the group size paradox Joan Esteban and Debraj Ray

379

Part 3

Extensions

Transfer seeking and avoidance: On the full social costs of rent seeking Elie Appelbaum and Eliakim Katz

vui

391

Contents Strategic buyers and the social cost of monopoly Tore Ellingsen

399

Sabotage in rent-seeking contests KaiA. Konrad

409

Strategic restraint in contests Gil S. Epstein and Shmuel Nitzan

421

Strategic behavior in contests Avinash K. Dixit

431

Strategic behavior in contests: Comment Kyung Hwan Baik and Jason E Shogren

439

The social cost of rent seeking when victories are potentially transient and losses final Joerg Stephan and Heinrich W. Urspmng Uncertain preassigned non-contestable and contestable rents Nava Kahana and Shmuel Nitzan Evolutionary equilibrium in TuUock contests: Spite and overdissipation Burkhard Hehenkamp, Wolfgang Leininger, and Alex Possajennikov

443 455

473

Information in conflicts Karl Wdmeryd

487

Rent seeking with private values David A, Malueg and Andrew J. Yates

503

Part 4

Structure of Contests

Hierarchical structure and the social costs of bribes and transfers Arye L. Hillman and Eliakim Katz

523

Group competition for rents Eliakim Katz and Julia Tokatlidu

537

Bidding in hierarchies KaiA. Konrad

547

Seeking rents by setting rents: The political economy of rent seeking Elie Appelbaum and Eliakim Katz Orchestrating rent seeking contests Mark Gradstein and KaiA. Konrad

555 571

IX

Contents Maximum efforts in contests with asymmetric valuations Kofi O. Nti

581

Optimal contests Amihai Glazer and Refael Hassin

589

Competition over more than one prize Derek J. Clark and Christian Riis

601

The optimal allocation of prizes in contests Benny Moldovanu andAner Sela

615

Incentive effects of second prizes Stefan Szymanski and Tommaso M. Valletti

633

Part 5

Experiments

Reexamining efficient rent-seeking in laboratory markets Jason E Shogren and Kyung Hwan Baik

651

An experimental examination of rational rent-seeking Jan Potters, Casper G. de Vries, and Frans van Winden

663

Efficient rent-seeking in experiment Carsten Vogt, Joachim Weimann, and Chun-Lei Yang

681

Acknowledgements

693

Contents

Contents Volume II

Forty Years of Research on Rent Seeking: An Overview Roger D. Congleton, Arye L, Hillman and KaiA. Konrad Part 1

1

Regulation and Protection

The social costs of monopoly and regulation Richard A. Posner

45

The social costs of monopoly power Keith Cowling and Dennis C. Mueller

67

Misleading calculations of the social costs of monopoly power Stephen C. Littlechild

89

Declining industries and political-support protectionist motives Arye L. Hillman

105

Domestic politics, foreign interests, and international trade policy Arye L. Hillman and Heinrich W. Ursprung

113

Protection for sale Gene M. Grossman and Elhanan Helpman

131

Part 2

Economic Development and Growth

The political economy of the rent-seeking society Anne O, Krueger

151

Foreign aid and rent seeking Jakob Svensson

165

The political economy of coffee, dictatorship, and genocide Philip Verwimp

191

Why is rent seeking so costly to growth? Kevin M. Murphy, Andrei Shleifer, and Robert W. Vishny

213

Political culture and economic decline Arye L, Hillman and Heinrich W, Ursprung

219

XI

Contents Institutions and the resource curse Halvor Mehlum, Karl Moene, and Ragnar Torvik

245

The king never emigrates Gil S. Epstein, Arye L. Hillman, and Heinrich W. Ursprung

265

Immigration as a challenge to the Danish welfare state? Peter Nannestad

281

Part 3

Political and Legal Institutions

Rent-seeking aspects ofpolitical advertising Roger D. Congleton

297

Rent extraction and rent creation in the economic theory of regulation Fred 5. McChesney

313

Rigging the lobbying process: An application of the all-pay auction Michael R. Baye, Dan Kovenock, and Casper G. de Vries

331

Caps on political lobbying Yeon-Koo Che and Ian L. Gale

337

Inverse campaigning KaiA. Konrad

347

On the efficient organization of trials Gordon Tullock

361

Legal expenditure as a rent-seeking game Amy Farmer and Paul Pecorino

379

Rent-seeking through litigation: Adversarial and inquisitorial systems compared Francesco Parisi

397

Comparative analysis of litigation systems: An auction-theoretic approach Michael R. Baye, Dan Kovenock, and Casper G. de Vries

421

Part 4

Institutions and History

Rent seeking, noncompensated transfers, and laws of succession James M. Buchanan A model of institutional formation within a rent seeking environment Kevin Sylwester

Xll

443 . . . .

459

Contents The 2002 Winter Olympics scandal: Rent-seeking and committees J. Atsu Amegashie

467

Mercantilism as a rent-seeking society Barry Baysinger, Robert B, EkelundJn, and Robert D. Tollison

475

Efficient transactors or rent-seeking monopolists? The rationale for early chartered trading companies S. R. H. Jones and Simon R Ville

509

The open constitution and its enemies: Competition, rent seeking, and the rise of the modem state Oliver Volckart

527

Illegal economic activities and purges in a Soviet-type economy: A rent-seeking perspective Arye L, Hillman andAdi Schnytzer

545

Rent seeking and taxation in the Ancient Roman Empire Charles D. DeLormeJr,, Stacey Isom, and David R. Kamerschen Parts

559

The Firm

"Hard" and "soft" budget constraint J, Komai Workers as insurance: Anticipated government assistance and factor demand Arye L. Hillman, Eliakim Katz, and Jacob Rosenberg Rent seeking and rent dissipation in state enterprises Steven T. Buccola and James E. McCandlish

569

585 593

Discouraging rivals: Managerial rent-seeking and economic inefficiencies Aaron S. Edlin and Joseph E. Stiglitz

609

The dark side of internal capital markets: Divisional rent-seeking and inefficient investment David S. Scharfstein and Jeremy C. Stein

621

Allies as rivals: Internal and external rent seeking Amihai Glazer

649

Efficiency wages versus insiders and outsiders Assar Lindbeck and Dennis J. Snower

657

Monitoring rent-seeking managers: Advantages of diffuse ownership Roger D. Congleton

667

xui

Contents Inside versus outside ownership: A political theory of the Holger M. MiXller and Karl Wdmeryd Part 6

firm

679

Societal Relations

Efficient status seeking: Externalities, and the evolution of status games Roger D. Congleton

697

A signaling explanation for charity Amihai Glazer and Kai A. Konrad

713

Competition for sainthood and the millennial church Mario Ferrero

723

Publishing as prostitution? - Choosing between one's own ideas and academic success Bruno S. Frey

749

Ideological conviction and persuasion in the rent-seeking society Roger D. Congleton

769

Political economy and political correctness Arye L. Hillman

797

Acknowledgements

813

XIV

Forty Years of Research on Rent Seeking: An Overview Roger D. Congleton, Arye L. Hillman and Kai A. Konrad

The quest for rents has always been part of human behavior. People have long fought and contended over possessions, rather than directing abilities and resources to productive activity. The great empires and conquests were the consequences of successful rent seeking. Resources were also expended in defending the rents that the empires provided. The unproductive use of resources to contest, rather than create wealth, also occurred within societies in attempts to replace incumbent rulers and in seeking the favor of rulers who dispensed rewards and indeed often determined life and death. Sacrifices made by early peoples to their deities were instances of rent seeking; valuable possessions were given up with the intent of seeking to influence assignment of other rewards. In contemporary times, rent seeking takes place within democratic institutions and also under conditions of autocracy that are akin to the circumstances of the earlier rent-dispensing despots. Incentives for rent seeking are present whenever decisions of others influence personal outcomes or more broadly when resources can be used to affect distributional outcomes. The search for rents, defined as rewards and prizes not earned or not consistent with competitive market returns, is, thus, clearly ancient. Efforts to understand how wealth, status, and other rewards can be acquired, and how contests for such prizes can be designed to reduce losses associated with unproductive conflict and encourage productive forms of competition, are also likely to have begun at the dawn of social Ufe. The academic rent-seeking literature, however, is relatively new and emerged from papers by Gordon TuUock, Anne Krueger, and Richard Posner published during the course of some 10 years in the 1960s and 1970s (reprinted in these volumes). The early rent-seeking analyses sought accurate measures of social losses from pubUc poUcies and monopoly. TuUock, Krueger, and Posner argued that the resources used to establish, maintain, or eliminate trade restrictions and monopoUes are part of the social cost of those policies, but had previously been neglected. The idea that resources are unproductively used in rent-seeking contests has much broader application than the initial rent-seeking papers suggested. The rentseeking logic has been applied to issues in history, sociology, anthropology, biology, and philosophy. The core idea has also been formalized and analyzed more rigorously, using the tools of modern game theory. The modern rent-seeking Uterature describes the rational decision to invest in contesting pre-existing wealth or income, rather than undertaking productive activity.

Roger D. Congleton, Arye L. Hillman and Kai A. Konrad The starting point of this Hterature is often considered to be Gordon TuUock's paper on the "Welfare Costs of Tariffs, MonopoHes, and Theft" in 1967. TuUock focused on the efficiency consequences of income transfers and observed that "Transfers themselves cost society nothing, but for the people engaging in them they are just like any other activity, and this means large resources may be invested in attempting to make or prevent transfers. These largely offsetting commitments of resources are totally wasted from the standpoint of society as a whole." TuUock's observations implied that there was more to inefficiency than deadweight losses. Beneficiaries of inefficient policies have personal incentives to influence creation and assignment of income and wealth created by political decisions. Tullock reasoned that the resources used in activities of persuasion should be counted as a cost to society. The quests for income and wealth redistribution through public policy are comparable to the activities of thieves, who also use personal resources and initiative in unproductive endeavors to redistribute, rather than create wealth. The act of theft results in an income transfer that does not change total national income, but social losses do arise before a theft takes place, because the aspiring thieves' contest who in the end will be successful in the act of theft, and prospective victims' invest in various means for resisting the thieves' efforts. These resources could have been used to produce goods and services with a positive value, rather than devoted to distributional conflict. The social loss from rent seeking similarly occurs ex ante, through unproductively used resources and initiative before policy decisions are made. PubUc policies often directly or indirectly transfer income or wealth among people. Of course, there is no suggestion that all public poUcies are akin to theft. The theory of rent seeking, however, is based on the possibiUty of influencing public poUcies for personal gain. The quest for personal advantage may be masked with the rhetoric of social advantage. A focal question of the rent-seeking hterature is the computation of social loss through the value of the resources unproductively used because of the presence of the rents or prizes that are assigned by the personal discretion of others, as when political decision makers determine public policies. After four decades of research following the pubUcation of TuUock's paper, the literature expanding on the rent-seeking idea is substantial. The JStor data base of academic journals reports that 74 papers include the term "rent seeking" in their titles. The Scopus on-line search reports 170 papers. The more representative EconLit data base of academic journals and books reports 401. The broader Google Scholar search engine reports that the titles of more than 1,500 papers on the Web include the term "rent seeking." Moreover, not every paper on rent seeking includes those words in its title. EconLifs data base reports that more than 8,000 published papers and books use the terms "rent seeking" or "rent seeker" somewhere within their pages. The quite different backgrounds of the editors of these two volumes provide a balance in perspectives on rent seeking and indeed on public choice. Roger Congleton was present from the beginning in the Center for the Study of Public Choice as a member of the Virginia School when the rent-seeking concept developed, and

Forty Years of Research on Rent Seeking: An Overview was an editor of a previous collection and contributed to the influential first compendium on rent seeking published in 1980. Arye Hillman, a former president of the European Public Choice Society, has pursued political economy research, but not as a member of the Virginia School. Kai Konrad provides a perspective that includes rent seeking in the more general study of contests. It was, of course, a difficult task to choose papers for inclusion in these volumes. The decision rule for inclusion of papers was consensus among the editors. The prediction of the theoretical literature (Buchanan and TuUock 1962) that decision costs increase with a consensus rule for collective decision making was borne out in the natural experiment of selecting papers for these volumes, which continued for more than a year. Other scholars were also consulted and recommendations heeded. The two volumes include classics and major extensions of the rent-seeking literature. Papers were valued that expressed informative novel ideas and that directed attention to applications that expanded the scope of the rent-seeking concept. Many were initially published in journals and books to which few readers have easy access. Books that contain seminal contributions are out of print, including the now classic 1980 edited volume. Many papers that have proven to be significant were not published in "leading" journals. Our choice of papers confirms the more generalfindingof Andrew Oswald (2007) that the most significant papers are not always published in the most prominent economic journals. The papers are organized to provide a sense of the development of the rentseeking literature by topic rather than by date of publication. The papers in volume I are analytical developments on the rent-seeking theme. The papers in volume II are applications of the rent-seeking concept. Our introduction provides an overview of the literature and summarizes the contributions of the papers in the two volumes.

Origins of the Literature In the second half of the twentieth century, there were a variety of efforts to place the normative analysis of public poUcy on firmer analytical ground. The first efforts, like the positive theories of that time, attempted to rank order allocations of real resources without considering how or why particular allocations might have arisen (Bergson 1938, Samuelson 1947, Harsanyi 1955). Governments and therefore political decision makers were described as seeking social optimality. Yet individuals andfirmsin the private sector were at the same time viewed as having self-interested objectives. The argument was that, if governments were making decisions, the decisions were in the best interest of society, because government is socially benevolent. The public choice view proposed consistency in application of the principle of rationality and self-interest. If utility- and profit-maximizing models explain a good deal of private sector behavior, they are also Ukely to explain a good deal of the behavior of political and bureaucratic decision makers. In appUed research, the public choice view could provide an answer as to why governments often adopt economically inefficient policies. In contrast, the mainstream

Roger D. Congleton, Arye L. Hillman and Kai A. Konrad literature used the classic analysis of monopoly by Arnold Harberger (1954) to measure "deadweight" losses of such pubhc policies, ignoring how they might have come to be adopted. The evaluations of deadweight losses were influential and soon found their way into textbooks as the core of modern welfare economics. The perceived benevolence of government made policies in these studies exogenous, rather than endogenous, consequences of choices made under specific legal and poUtical institutions. Public policies that result in deadweight losses do not come into existence spontaneously. Yet the welfare implications of interest group activities and poUtically endogenous poUcies were either not fully appreciated in the mainstream Uterature or else were simply neglected to avoid confronting the question as to why poUtical and bureaucratic decision makers created and assigned rents ~ and extracted poUtical rents for themselves. A prime case was the literature on international economics: here the deadweight efficiency losses from protectionist policies were studied, but the mainstream literature of the time did not address the question as to why Pareto-inefficient departures from free trade took place. Yet, the protectionist policies clearly reduced national income while benefiting some people at the expense of others. In his 1967 paper inaugurating the modern literature on rent seeking, TuUock made the fundamental observation that, if inefficient pubUc poUcies, such as trade poUcies, were poUtically endogenous, part of the social cost of those policies was the use of scarce personal abilities and resources in efforts to influence policy decisions. TuUock thus pointed to a source of social loss beyond Harberger triangles. The next important step in TuUock's analysis - and for the rent-seeking literature that later emerged - was to investigate the extent to which resources are attracted to rent-seeking activities. TuUock reasoned that, if government could be induced to redistribute wealth, the rate of return from political wealth-enhancing activities would equal the return from other investments in long-run competitive equilibrium. This implies that social losses from contesting rents are equal to the values of observed contested rents. Although profits and other rents could often be measured, the value of the resources used in rent seeking is usually not observable. TuUock's logic suggests that the value of rents generated by public poUcies can be used as a proxy for the resources used in rent seeking. That is, rent dissipation could be viewed as complete. Given this, the mainstream accounts of losses from monopoly, tariffs, and other pubUc policies that had been carefully worked out in the previous two decades substantially understated the true extent of the losses that a society incurred from inefficient pubUc policies. TuUock's insight was slow to find its way into print (see Brady and TolUson 1994) and slow to be integrated into new research. The rent-seeking idea was not totally neglected after publication in 1967; however, wider recognition of the worth of TuUock's idea required re-publication and re-expression in a more prominent place and the accompaniment of an appropriate pithy phrase. Anne Krueger (1974: reprinted in volume II) provided the descriptive term "rent seeking" and thereafter the Uterature could refer to "rent seekers." Krueger set out a general

Forty Years of Research on Rent Seeking: An Overview equilibrium model of social loss from contesting quota rents and presented estimates of losses in Turkey and India. She based her measures on the complete dissipation presumption that rents would attract resources of equal value. Richard Posner (1975: reprinted in volume II) used the complete-dissipation presumption to estimate losses from monopoly in U.S. industries. Keith Cowling and Dennis Mueller (1978: reprinted in volume II) followed with estimates of the cost of monopoly for the U.S. and U.K. economies. These and other empirical studies, for example, David Laband and John Sophocleus (1992) and Martin Paldam (1997), suggest social losses considerably greater than the rather small losses that had been reported from measurement of Harberger triangles. With its new appellation and evidence of its importance, and also the expression of the idea outside of the public choice school, the literature on rent seeking began to expand, although not very rapidly at first. In the beginning, the rent-seeking concept was, in a sense, proprietary to the Virginia School that TuUock had been instrumental in founding. Much of the early research was by faculty and students associated with the Center for Study of Public Choice, where the new research on rent seeking stimulated several papers that were presented in an evening seminar series in 1978. It was in that seminar series that TuUock noted that the complete rent dissipation presumption that Krueger, Posner, and others had also adopted was not necessarily appropriate. TuUock (1980: reprinted in volume I) set out a rent-seeking game and, with assistance from his colleagues Nicolas Tideman and Joseph Greenberg and his graduate student assistant, characterized the Nash equilibrium for a contest success function that probabUistically designated the winner of a contest according to how much individual contenders spend. The mathematical and simulation results confirmed that complete dissipation was a special case. Actual rent dissipation depended on the number of players and on a returns-toscale parameter in the contest success function. The seminar inspired many of the papers that appeared in thefirstrent-seeking volume edited by Buchanan, ToUison, and TuUock (1980), in which TuUock's paper with the probabilistic contest success function first appeared. The explanatory power of the rent-seeking idea arises from its linkage of neoclassical economics to modem game theory and rational choice politics. TuUock's lottery-based characterization of the contest success function of a rent-seeking game was an important advance in demonstrating that the "rules of the game" matter. The institutions and technologies that determine the parameters of rentseeking contests affect society's losses from rent seeking. TuUock's 1980 paper was also important for the development of the literature, because his characterization of rent-seeking contests as lotteries was relatively easy to generalize and extend. Intuition and a preference for a simple and elegant mapping that transforms competing players' efforts into probabiUties of success guided TuUock in his choice of a contest success function. Later axiomatic work by Skaperdas (1996: reprinted in volume I), Kooreman and Schoonbeek (1997), and Clark and Riis (1998: reprinted in volume I) confirmed that only the functional form that TuUock chose is com-

Roger D. Congleton, Arye L. Hillman and Kai A. Konrad patible with a number of desirable and plausible properties of a contest success function. The first collection of rent-seeking papers that appeared in the 1980 volume launched the broader literature that emerged during the next 25 years. The 1980 collection included 22 papers, only 10 of which had been previously published. The 12 new papers covered a broad range of topics. The papers in thatfirstcollection were for the most part by colleagues and students of TuUock, including James Buchanan and an editor (Congleton) of the present collection. The inclusion of TuUock's efficient rent seeking paper, which had now been provided with a natural publication outlet, allowed the research on rent dissipation to begin in earnest. The 1980 volume influenced scholars in societies outside of the United States. Far away from Virginia, in Israel, where economic liberalization was yet a decade away, another editor of the present collection (Hillman) observed the substantial presence of non-market allocation and non-market-determined personal rewards in the self-managed worker sector of the economy. As in Virginia, the Bar-Ilan School set out to investigate efficiency consequences of the presence of poUtically assigned rents. The scholarly divisions were amazingly similar to those in the United States where rent seeking was not a topic favored outside of the Virginia School. In Israel, only scholars at Bar-Ilan University or graduates of Bar-Ilan undertook research on rent seeking. The first contribution of the Bar-Ilan School to the study of rent-seeking contests, by Hillman and Katz (1984: reprinted in volume I), investigated rent dissipation in winner-take-all contests, such as those for monopoly. They demonstrated that complete rent dissipation emerges if rent seekers are risk neutral and rent-seeking contests can be freely entered, but not if rent seekers are risk averse or there are barriers to entry into contests. Hillman and Samet (1987: reprinted in volume I) solved the rent-seeking game for the case of an all-pay auction in which the highest spending individual or group wins the prize. Their results provide another justification for the complete-dissipation presumption. Subsequently Shmuel Nitzan and others at Bar-Ilan made contributions to the rent-seeking literature. The analysis of rent seeking also attracted the attention of scholars in Korea, including in particular Kyung Hwan Baik. In Korea substantial rent-creating nonmarket allocation occurred through the interaction between government and vertically integrated conglomerates. Scholars from the Philippines and sub-Saharan Africa where corruption and non-benevolent governments created and assigned significant rents also contributed to the rent-seeking literature. However, in many autocratic societies where rent seeking has been endemic, there appear to have been impediments to analyses and discussion of rent seeking by local scholars. The post-1980 literature was often theoretical and positive, rather than empirical or normative as the first papers had been. The extent of rent dissipation was studied, often in abstract terms without description of the types of policies that gave rise to the rents and to rent seeking. Other contest success functions were considered and changes were made to incorporate more general assumptions. The analysis gained in precision and increased in complexity. The Uterature

Forty Years of Research on Rent Seeking: An Overview has explored hov^ contest structures affect resource use in rent-seeking activities and has investigated the efficiency properties of alternative methods of allocating "prizes." Collective goods, free-riding incentives in contests, hold-up problems from repeated rent seeking, issues of endogenous timing, budget constraints, nested contests, and the role of incomplete information have been analyzed in this context. The papers in volume I are largely from this theoretical strand of the rent-seeking literature.^ The idea of rent seeking has been acknowledged as important for understanding a broad range of long-standing applied economic topics encompassing regulation, international trade policy, economic development, the transition from socialism, and communal property. An approach based on political economy of protection, which gradually became part of the mainstream economics literature, clearly pointed out the link between trade policies and rents for protected groups (Hillman 1989). The rent-seeking concept has also been applied to deepen our understanding of topics in economic history, law, sociology, and biology. If rentseeking contests can be created and conditions of contests revised, many aspects of institutional design and evolution are also part of the applied rent-seeking research program. A broad selection of applications of the rent-seeking concept is included in volume II.

Criteria for inclusion A general principle for inclusion of papers in these volumes is consistency with the political-economy and institutional origins of the rent-seeking concept. Rent seeking is a political economy concept. The intent of Gordon TuUock, the Virginiabased Public Choice School, and the Bar-Ilan School was to show that societies incur efficiency losses beyond the traditional economic deadweight losses when personal benefits and costs are politically assigned rather than market determined. Their research generally implies that reducing rent-seeking losses requires institutional reform. Not all contests involve rent-seeking activities, although most involve decisions about how much to invest in a given contest. A decision was therefore confronted whether to include literature in which contests similar in structure to rent seeking are studied outside of the domain of political economy. One such literature involves the theory of the firm. This literature is extensive and revolves around internal principal-agent problems. Another question was whether to include literature that was in principle about rent creation and rent seeking but did not use or acknowledge the rent-seeking concept. Prominent among such cases is again the firm-related literature in which private-sector internal-firm tournaments or ^ A strand of more technically oriented research analyzed a large set of formal structures that can be applied to analyze rent-seeking contests. Konrad (2007) surveys the literature focusing on strategic aspects of contests.

Roger D. Congleton, Aiye L. Hillman and Kai A. Konrad contests for promotion have structures similar to rent seeking.^ In a literature on research and development (R&D) contests, winners provide productive outputs rather than contests being purely distributional. In sports contests also, effort is a source of benefit for spectators. We have excluded the literature on R&D contests and the Uterature exphcitly on sports contests.^ We have, however, included representative papers on the rents created and contested within the firm. Contests are also described in a large body of literature known as conflict theory. We have excluded most of this Uterature. Conflict theory emerged from the recognition that, without the rule of law and without possibilities for contractual enforcement, property rights are estabHshed endogenously by efforts to defend own wealth or efforts to acquire the wealth of others. In the 1990s conflict theory developed for the most part parallel and apart from the theory of rent-seeking literature. Models of rent seeking and models of conflict share the common element of contestability of wealth or income, although conflict models do not in general focus on the core rent-seeking issue of dissipation. There is also a difference in institutional setting: most rent-seeking models are motivated by the presence of government that can be influenced to create and assign rents, whereas most models of conflict are motivated by the absence of government and thereby absence of the rule of law."^ Questions were faced about how to categorize papers in which authors did not relate their analyses and conclusions to the prior insights of the rent-seeking literature, even though their papers described circumstances in which politically created and assigned rents are contested. A general rule was to exclude papers if they are not described by authors as being about rents or rent seeking. However, exceptions were made in the applications volume when the behavior being described clearly constitutes rent creation by government and rent seeking by interest groups. The principal sources of papers in the present collection are the American Economic Review, Public Choice, the Economic Journal, and the European Journal of Political Economy, which in the mid-1990s adopted a political economy focus that attracted authors of papers on rent seeking. Attitudes to the concept and terminology of rent seeking are of interest for understanding academic economics. As we have observed, in the beginning, rent seeking as a descriptive and explanatory concept was not accepted into the "main^ The literature on tournaments began with seminal contributions of Lazear and Rosen (1981) and Rosen (1986). ^ Early influential contributions to the R&D contest literature were Loury (1979), Dasgupta and Stiglitz (1980), and Nalebuff and Stiglitz (1983). FuUerton and McAfee (1999) and Baye and Hoppe (2003) provide microfoundations of TuUock's contest success function in applications to R&D contests. On sports contests, see Szymanski (2003). ^ For elaboration on the distinction between rent seeking due to the response of government and conflict theory based on anarchy and the absence of government, see Hillman (2003, chapter 6). See also TuUock (1974) on conflict in anarchy. Skaperdas (2003) and Garfinkel and Skaperdas (2007) survey the conflict literature. See Fearon (1995) on military conflict and Hillman (2004) on conflict between strong and weak under Nietzschean conditions in anarchy.

Forty Years of Research on Rent Seeking: An Overview

Stream" of academic economics. TuUock's 1967 paper appeared in a relatively new regional journal and his 1980 paper had to await the edited volume for publication. In the 1990s rational political behavior and political economy concepts came to be more broadly recognized as descriptive of realities of government decision making. However, there seemed to be a reluctance to acknowledge the antecedents of the public-choice school. For example, a "new" political economy literature emerged in the 1990s that often failed to acknowledge that their research addressed questions that had previously been addressed by public choice scholars. When asked (by one of the present editors) in 1990s why the contributions of public choice scholars were not being acknowledged, a prominent contributor to the "new" political economy Uterature replied that "we cannot cite everyone since Adam Smith". The answer was not "what is public choice?" Or "what is rent seeking?" The answer suggests that the concept of rent seeking has had wider influence than indicated by the rent-seeking literature per se. We proceed now to describe the structure of the two volumes and to link and summarize the papers.

Roger D. Congleton, Arye L. Hillman and Kai A. Konrad

Volume I - Theory of Rent Seeking The focus of volume I is on conceptual and theoretical developments.

Theory Part 1 - Rents 1.1 The social cost of rent seeking The early papers followed TuUock's original exposition in further considering the social cost of contestable rents. Gordon TuUock, 1967. The welfare costs of tariffs, monopolies, and theft. Westem Economic Journal 5, 224-32. James M. Buchanan, 1980. Rent seeking and profit seeking. In James M. Buchanan, Robert D. ToUison, and Gordon TuUock (eds.). Toward a Theory of the Rent-Seeking Society. Texas A&M University Press, College Station, pp. 3-15. Roger D. Congleton, 1980. Competitive process, competitive waste, and institutions. In James M. Buchanan, Robert D. TolHson, and Gordon Tullock (eds.). Toward a Theory of the Rent-Seeking Society. Texas A&M University Press, College Station, pp. 153-79. Arye L. Hillman and Eliakim Katz, 1984. Risk-averse rent seekers and the social cost of monopoly power. Economic Journal 94,104-10. Tullock (1967) is the seminal paper that points out the social losses from unproductive use of resources in quests to influence political decisions about income distribution. Buchanan (1980) distinguishes socially productive and unproductive competition. Congleton (1980) explicitly analyzes the role of institutions, or "the rules of the game," using deterministic models that would later be called all-pay auctions. His analysis suggests that rent dissipation can be reduced by majorityrule allocation of prizes and by rules that distribute the "prize" in proportion to effort, rather than through winner-take-all contests. Hillman and Katz (1984) use a general contest success function in which the probabihty of winning increases with own rent-seeking effort and decreases with opponents' effort to show that the complete dissipation presumption (that the value of a contested rent is equal to the value of the resources used in contesting the rent) is valid in contests with large numbers of risk neutral players; however, rent dissipation is incomplete when rent seekers are risk averse. The early Hterature thus established that contestable rents create social losses, that we should distinguish between socially productive and unproductive forms of competition, that institutions matter, and that complete dissipation in competitive contests is the predicted outcome when rent seekers are risk neutral and when the rent-seeking game is fully competitive but not otherwise.

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Forty Years of Research on Rent Seeking: An Overview 1.2 Tullock contests Tullock (1980) introduced the probabilistic contest success function for which personal expenditures on rent seeking are like buying lottery tickets. Other studies subsequently amended and extended the Tullock contest success function. A primary issue was whether over-dissipation of a rent could ever occur. Gordon Tullock 1980. Efficient rent seeking. In James M. Buchanan, Robert D. ToUison, and Gordon Tullock (eds.), Towards a Theory of the Rent-Seeking Society, Texas A&M University Press, College Station, pp. 97-112. Richard S. Higgins, William F. Shughart II, and Robert D. ToUison, 1985. Free entry and efficient rent seeking. Public Choice 46, 247-58. J. David Perez-Castrillo and Thierry Verdier, 1992. A general analysis of rentseeking games. Public Choice 73, 335-50. Kofi O. Nti, 1999. Rent seeking with asymmetric valuations. Public Choice 98, 415-30. A scale parameter was included in TuUock's lottery model of rent seeking, which (partly) determines the return from purchasing lottery tickets and implicitly represents institutional aspects of rent-seeking contests, Tullock showed that with constant returns from rent-seeking expenditures two contenders dissipate half of the rent in the unique Nash equilibrium. Rent-seeking losses increase as the number of contenders increase and with increases in the scale parameter. TuUock's results suggest that over-dissipation occurs if there are large economies of scale in rent-seeking. Higgins, Shughart, and ToUison (1985) point out that the value of the scale parameter determines whether the Tullock contest success function is consistent with existence of Nash equilibria. In a generalization that that does not rely on the TuUock contest success function, Higgins et al consider a case in which effort of risk-neutral rent seekers is observed subject to error and rent seekers choose a probability with which to participate in a contest. In a symmetric zero-profit mixedstrategy equUibrium, rents are on average completely dissipated, although ex post, under-, or over-dissipation may be observed (depending on the realizations of the mixed strategies). Over-dissipation is inconsistent with Nash equilibrium when participation in rent-seeking contests is voluntary. Perez-Castrillo and Verdier (1992) reformulate the Tullock contest using reaction curves and consider consequences of free entry and Stackelberg equUibrium. Nti (1999) extends TuUock's contest to asymmetric valuations of a prize. Equilibria depend on the different valuations of rent seekers as weU as on the scale parameter of the contest success function. The asymmetric valuations determine a favorite and an "underdog." A player with a higher valuation expends more effort to win. The "underdog" values the prize less and in consequence chooses to spend less to obtain the prize. Asymmetry in practice requires either a discriminatory contest design or disagreements about

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Roger D. Congleton, Arye L. Hillman and Kai A. Konrad the estimated value of the prize, such as might arise in contests for a mate, for ego rents from poUtical office, or for a non-pecuniary honor bestowed on a winner. 1.3 Contests as all-pay auctions In the TuUock contest, random elements through the lottery nature of the contest success function determine the identity of the winner. An alternative contest success function designates the participant exerting the highest effort as the winner with certainty. Arye L. Hillman and Dov Samet, 1987. Dissipation of contestable rents by small numbers of contenders. Public Choice 54, 63-82. Arye L. Hillman and John G. Riley, 1989. Politically contestable rents and transfers. Economics and Politics 1,17-39. Michael R. Baye, Dan Kovenock, and Casper G. de Vries, 1996. The all-pay auction with complete information. Economic Theory 8, 291-305. Simon R Anderson, Jacob K. Goeree, and Charles A. Holt, 1998. Rent seeking with bounded rationaUty: An analysis of the all-pay auction. Journal of Political Economy 106, 828-53. The possibility of a contest success function in which the highest effort wins was noted in Congleton (1980) above. Hillman and Riley (1989) introduce the terminology of a discriminating contest to describe such a contest success function. The TuUock lottery implies an inability to discern with precision the efforts of different contestants and accounts for sources of noise. The discriminating rent-seeking contest is an all-pay auction: contenders bid for the rent, the highest bid wins, and all contenders lose the value of their bids whether they win or not. Hillman and Samet (1987) derive the mixed strategy solution for such success functions. A Nash equilibrium in pure strategies does not exist in such contests, as noted in Congleton (1980).^ They show that in a mixed-strategy equilibrium, with risk neutrality and equal valuations of the rent, complete dissipation on average holds, in that the expected value of resources used in rent seeking by all contenders is equal to the value of the rent. They conclude that the rent is on average fully dissipated for any number of contestants larger than one. A justification other than competitive rent seeking by risk-neutral rent seekers is thereby provided for the descriptions of social losses from rent seeking by Tullock, Krueger, and Posner, and others, in which it is taken for granted that rent dissipation is complete. Hillman and Riley (1989) also extend the all-pay auction to cases in which contestants have different valuations of the prize. Only the two highest valuation ^ If everybody makes the same bid (less than the value of the prize), there is an incentive to bid a little more. If someone bids the value of the prize, others bid zero, in which case the contender who bid the value of the prize can reduce his or her bid to a little above zero, but then others will not bid zero - and so on.

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Forty Years of Research on Rent Seeking: An Overview contenders contest the prize. Others are deterred by the high valuations of competitors for the rent. In this case, the outcome can be less than complete rent dissipation, and an inefficient allocation of the prize (through the low-value contender winning) can occur. Hillman and Riley distinguish between contests for pre-existing rents and contests for transfers (in which one person's gain is another person's loss). Baye, Kovenock, and de Vries (1996) provide a complete characterization of the various types of equilibria that can emerge in an all-pay auction in the case of many players with different valuations of the prize, and when the equilibrium is unique. The all-pay contest is used as a building block in more complex contests and in many applications. Anderson, Goeree, and Holt (1998) study all-pay auctions when there is bounded rationality. Rational behavior is inconsistent with systematic over-dissipation of rents, yet over-dissipation is observed in experiments. Bounded rationality, in which decisions with higher expected payoffs are more likely to be made, but not with probability one, is proposed as an explanation for the overdissipation observed in some laboratory experiments. A generalization of Nash equilibrium obtained by incorporating bounded rationaUty into the determination of equilibrium yields a prediction of over-dissipation similar to that of TuUock's (1980) analysis. The extent of rent dissipation increases with the number of players

1.4 Contest success functions reconsidered The contest success function was reconsidered in different contexts. Jack Hirshleifer, 1989. Conflict and rent-seeking success functions: Ratio vs. difference models of relative success. Public Choice 63,101-12. Stergios Skaperdas, 1996. Contest success functions. Economic Theory 7,28390. Ferenc Szidarovszky and Koji Okuguchi, 1997. On the existence and uniqueness of pure Nash equilibrium in rent-seeking games. Games and Economic Behavior 18,135-40. Hirshleifer (1989) compared the TuUock ratio (lottery) contest success function with a specification based on differences in effort. In the TuUock case, non-conflict cannot be an equilibrium, and there is never an equilibrium in which one side just gives up. Hirshleifer's difference-based specification for the contest success function is consistent with equilibrium outcomes of mutual non-conflict and submission, the latter occurring when there are sufficiently large differences in the valuation of the prize. Skaperdas (1996) shows that the TuUock lottery is the only contest success function that is consistent with seven reasonable axioms about the relationship between efforts and win probabiUties. The most important axiom needed is an independence of irrelevant alternatives property. Szidarovszky and Okuguchi (1997) use a clever transformation to provide sufficient conditions for

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Roger D. Congleton, Arye L. Hillman and Kai A. Konrad the existence and uniqueness of equilibrium for contests with contest success functions that are considerably more general than the TuUock contest success function.

Theory Part 2 - Collective Dimensions 2.1 Collective decisions, collective effort, and shared rents Rent seeking often involves collective choices of various kinds. Collective decisions are often made about who receives rents. There might also be collective effort. Indeed, rent-seeking contests often involve groups, rather than single individuals, and collective action issues can arise, as noted by Mancur Olson (1965). In such cases, the prize might be shared by members of successful rent-seeking teams. Outcomes depend on whether the prize is allocated by casting votes, is a groupspecific public good that can be enjoyed in a non-rival manner by all members of the winning group, or is a private good that needs to be allocated among the members of the winning group. Roger D. Congleton, 1984. Committees and rent-seeking effort. Journal of Public Economics 25,197-209. Ngo Van Long and Neil Vousden, 1987. Risk-averse rent seeking with shared rents. Economic Journal 97, 971-85. Shmuel Nitzan, 1991. Collective rent dissipation. Economic Journal 101,152234. Joan Esteban and Debraj Ray, 2001. Collective action and the group size paradox, ylmencan Political Science Review 95, 663-72. Kyung Hwan Baik, Bouwe R. Dijkstra, Sanghack Lee, and Shi Young Lee, 2006. The equivalence of rent-seeking outcomes for competitive-share and strategic groups. European Journal of Political Economy 22, 337-42. Congleton (1984) extends his 1980 analysis of the effects of institutions on rent dissipation in deterministic contests between two groups. In two-party contests, he shows that investments in deterministic rent-seeking contests tend to be lower when decisions are made by committees through majority rule rather than by a single person, because investing resources to form majority coalitions tends to deescalate, rather than escalate. This conclusion provides an explanation for the widespread use of committees to make decisions within both democratic and nondemocratic organizations. Investments in such contests also tend to be smaller under deterministic proportional sharing rules than under winner-take-all rules. Long and Vousden (1987) analyze the case in which rents are shared and shares are not deterministic and demonstrate that dissipation falls with the extent of risk aversion of rent seekers and with uncertainty about the shares received by individual participants. They also explore the case in which the prize distributed is increased by the total effort of rent seekers. In such cases, individual efforts

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Forty Years of Research on Rent Seeking: An Overview produce both positive (larger prize) and negative (reduced probability of winning) externaUties for fellow players. As a consequence, individual investments increase, because investments provide a higher rate of return by increasing the value of the rent to be distributed to winners. Total resources invested in rentseeking games thus increase, although overall dissipation rates do not necessarily increase. Nitzan (1991) considered shared rents and showed that rules specifying the division of collectively sought rents within a winning group determine the magnitude of the free-riding problem among group members and determine thereby a group's effectiveness in an inter-group contest. Assignment rules that make a group member's share in the prize an appropriately chosen function of personal and other group members' efforts countervail free-riding incentives and increase equilibrium rent-seeking effort and also overall rent dissipation. A critical role is demonstrated for the rule by which the prize is shared among the members of the winning group, as a function of their number, type, or contest effort. Baik, Dijkstra, Lee, and Lee (2006) synthesize previous portrayals of contests by showing equivalence between contests in which groups compete for shared rents that are assigned to members of the group through distribution rules and contests in which members of a group compete individually for a rent and a winner is obliged to share with group members. 2.2 Rent seeking for public goods The above papers are about sharing of benefits when the benefits are private. For example, money might simply be shared. In other cases, however, the rent that is contested provides a public-good benefit to a group. PubUc expenditure may, for example, provide local public goods that benefit regional populations. The following papers describe rent seeking for public goods. Heinrich W. Ursprung, 1990. Public goods, rent dissipation, and candidate competition. Economics and Politics 2,115-32. Kyung Hwan Baik, 1993. Effort levels in contests: The public-good prize case. Economics Letters 41, 363-67. Mark Gradstein, 1993. Rent seeking and the provision of public goods. Economic Journal 103,1236-43. KhaUd Riaz, Jason F. Shogren, and Stanley R. Johnson, 1995. A general model of rent seeking for pubUc goods. Public Choice 82, 243-59. Joan Esteban and Debraj Ray, 2001. Collective action and the group size paradox, ^menca^ Political Science Review 95, 663-72. Ursprung (1990) embeds rent seeking for a public good in a model of political competition. Individual utility is additively separable so there are only substitution effects when the size of a group supporting a political candidate increases. Freeriding incentives through substitution effects between own-spending and spending

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Roger D. Congleton, Aiye L. Hillman and Kai A. Konrad by others reduce a group's total rent-seeking effort. In the Nash equihbrium, the total effort of a group is independent of the size of the group, and there is substantial under-dissipation of the pubHc-good rent that the groups contest.^ Riaz, Shogren, and Johnson (1995) point out that an income effect would increase total group effort when group size increases. Convex contribution costs have a similar effect of making total contributions to group rent-seeking effort increase in the group size: Esteban and Ray (2001) show this in a model that allows also for a mix between private and public components of the prize. Gradstein (1993) draws attention to the choice between inefficient private, uncoordinated provision of a public good, and governmental provision. Rent seeking uses resources unproductively, but the private supply of public goods is also generally inefficient because of free-riding incentives. Although the government can overcome the free-rider problem by compelling payment, the government may also be lobbied, which is costly in using resources and also does not ensure first-best provision. Baik (1993) describes groups composed of people with different valuations of a pubUc good. The groups compete for the public good. Free riding tends to be complete within each group, with one high valuation contender active on behalf of the group. In effect, the contest becomes one of high-value individuals representing their group. Rent dissipation is clearly low.

Theory Part 3 - Extensions 3.1 Opposition The losers from public policies, such as assigning monopoly rights, protectionist international trade pohcies, and privileged budgetary allocations, and other cases in which income is transferred, have incentives to resist the transfer. Hillman and Riley (1989) call such cases transfer contests to distinguish them from contests in which a pre-existing prize or rent exists and the providers of the rent are not identified or do not resist. There is a social cost associated with rent seeking, even if the rent seekers are not successful in persuading political decision makers to create rents. Those who have been successful in blocking such policies have nonetheless used resources in their opposition to the rent seekers. These losses highlight the point that the source of the social cost of rent seeking is the ex ante contest, rather than the ex post policy outcome. Elie Appelbaum and Eliakim Katz, 1986. Transfer seeking and avoidance: On the full social costs of rent seeking. Public Choice 48,175-81. Tore EUingsen, 1991. Strategic buyers and the social cost of monopoly. y4mencan Economic Review 81, 648-57. ^ Katz, Nitzan, and Rosenberg (1990, reprinted in Lockard and Tullock 2001) likewise study rent seeking for a public good when there are no income effects, and demonstrate precisely the same result.

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Forty Years of Research on Rent Seeking: An Overview Kai A. Konrad, 2000. Sabotage in rent-seeking contests. Journal of Law, Economics, and Organization 16,155-65. Gil S. Epstein and Shmuel Nitzan, 2004. Strategic restraint in contests. European Economic Review 48, 201-10. Appelbaum and Katz (1986) point out that the size of the stakes and the group of rent seekers need to be considered, as v^ell as whether people can abstain from the contest. EUingsen (1991) considers the consequences of active transfer-avoidance behavior by consumers opposing the creation of monopoly rents. He shoves that pre-existing rent seekers will change their behavior if new players enter and that this will generally prevent the players from over-dissipating the rent. Konrad (2000) explores contests in which there are two types of rent-seeking effort. He distinguishes between efforts that improve one's own competitive position with respect to all other contestants, and efforts that disadvantage a subset of the other competitors. The latter, sabotaging a subset of the other contenders, has the characteristics of a public good, because it benefits all but the contestants who are sabotaged. For this reason, sabotage as a form of opposition is a phenomenon that is more likely to occur in games with a small number of players. Epstein and Nitzan (2004) show that competition over policy alternatives induces strategic restraint in policy proposals (the prizes sought), which reduces resources used in rent seeking.^

3.2 Choice of timing Nash equilibrium is often based on simultaneity in choice of strategies, but one player may move first and commit to a strategy, which introduces issues of timing. Avinash K. Dixit, 1987. Strategic behavior in coniQSi^. American Economic Review 11, 891-98. Kjomg Hwan Baik and Jason F. Shogren, 1992. Strategic behavior in contests: Commtni. American Economic Review 82, 359-62. Dixit (1987) investigated the incentives of players to commit to choose a level of effort other than their Nash equilibrium effort if a player can act as a Stackelberg leader, and asked how the choice of commitment depends on the valuations of the prize. Baik and Shogren (1992) showed that Stackelberg-leader-follower behavior emerges endogenously in a non-discriminating probabilistic contest, with the weaker player or "underdog" moving first. The weaker player has a lower valuation of the prize than the stronger player, or has a probability of winning the contest of

^ See also Leidy (1994) on reduced rent dissipation when monopoly is threatened by regulatory policy.

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Roger D. Congleton, Arye L. Hillman and Kai A. Konrad less than 50 percent in the simultaneous-move Nash equilibrium. Rent dissipation is less than in the simultaneous-move Nash equilibrium.^ 3.3 Time Rent seeking can take place sequentially, in the course of time, and in repeated contests. Two important time-related aspects of rent-seeking contests concern asymmetries in living on to compete again and the properties of evolutionary equilibria. Joerg Stephan and Heinrich W. Ursprung, 1998. The social cost of rent seeking when victories are potentially transient and losses final. In Karl-Josef Koch and Klaus Jaeger (eds.), TradCy Growth, and Economic Policy in Open Economies: Essays in Honour of Hans-JUrgen Vosgerau, Springer, Berlin, pp. 369-80. Nava Kahana and Shmuel Nitzan, 1999. Uncertain preassigned non-contestable and contestable rents. European Economic Review 43,1705-21. Burkhard Hehenkamp, Wolfgang Leininger, and Alex Possajennikov, 2004. Evolutionary equilibrium in Tullock contests: Spite and overdissipation. European Journal of Political Economy 20,1045-57. Stephan and Ursprung (1998) describe rent seeking in sequential contests over time, with the important asymmetry that one side can lose in a contest and nonetheless return to contest the rent in a future contest, whereas for the other side any loss is permanent. For example, an incumbent once ousted may not be able to return to office or a new policy, once adopted, may not be easily reversed. The possibility that one side wins once and forever does not necessarily reduce the social cost of rent seeking. Kahana and Nitzan (1999) describe a government bureaucracy that procrastinates and does not deliver assigned payments or rents in a timely way, or may never deliver. The uncertainty about timing of payment affects the value of the rent. They examine circumstances in which the rent if delivered has been pre-assigned and the purpose of rent seeking is to eUcit the payment from the bureaucracy, and also in which the rent is contestable. They evaluate rent dissipation in each case. When there is no uncertainty regarding timing of delivery of a contestable rent, the model reduces to the standard Tullock contest. Hehenkamp, Leininger, and Possajennikov (2004) provide a dynamic analysis of the Tullock contest using the concept of evolutionary stable strategies (ESS).^ An evolutionary contest is a contest for survival and has a natural rent-seeking connotation. Hehenkamp et al derive the equilibrium in ESS for Tullock contests and show that more resources are used in rent seeking than in the unique Nash ^ An equivalent result is derived by Leininger (1993, reprinted in Lockard and Tullock 2001). For observations on the two formulations, see Nitzan (1994, reprinted in Lockard and Tullock 2001). ^ An ESS has the property that, if generally adopted by the group, there is no alternative strategy that can give a higher payoff to a member of the group. For finite populations, an ESS can differ from Nash equilibrium.

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Forty Years of Research on Rent Seeking: An Overview equilibrium. Total use of resources in rent seeking does not depend on the number of players and is solely determined by the contest success function and the value of the rent. Whether there is over-dissipation depends on the scale parameter of the contest function. Over-dissipation is interpreted as the consequence of "spite". Given the nature of ESS, it pays to reduce one's own prospects of success by increasing rent-seeking outlays, if this reduces the other strategy's success even more, which is what a person using ESS will do. An equilibrium in ESS is consistent with over-dissipation. Vindication is therefore provided for TuUock's observations on over-dissipation, although not in the usual context of Nash equilibrium.

3.4 Information Information is expected to affect the outcome of rent-seeking contests. Most of the literature on contests assumes that players are completely informed, or at least symmetrically informed. As in many other games, the information of players may not always be symmetric. Information asymmetries can exist with respect to a number of aspects of a contest. Karl Warneryd, 2003. Information in convicts Journal ofEconomic Theory 110, 121-36. David A. Malueg and Andrew J. Yates, 2004. Rent seeking with private values. Public Choice 119,161-78. Warneryd (2003) considers a contest for a prize, the value of which is the same for both players, but known only by one of the players. He finds that the less informed player may win with a higher probability. Malueg and Yates (2004) consider the case in which each player's valuation of the prize is private information and drawn from the same binary probability distribution.

Theory Part 4 - Structure of Contests 4.1 Hierarchies and nested contests Contests can take place in hierarchies, or similarly can be nested in that the results of one contest give rise to another contest. Arye L. Hillman and EUakim Katz, 1987. Hierarchical structure and the social costs of bribes and transfers. Journal of Public Economics 34,129-42. Eliakim Katz and JuUa Tokatlidu, 1996. Group competition for rents. European Journal of Political Economy 12, 599-607. Kai A. Konrad, 2004. Bidding in hierarchies. European Economic Review 48, 1301-08.

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Roger D. Congleton, Aiye L. Hillman and Kai A. Konrad Hillman and Katz (1987) described rent seeking in bureaucratic hierarchies in which bribes are transferred up the hierarchy (for example, from the corrupt pohce officer to the senior officer to the minister of poUce, to the president). Bribes are transfers and do not in themselves indicate social losses through rent seeking. Contests to occupy the positions to which bribes accrue at each level of the hierarchy, however, attract resources into rent seeking. Katz and Tokatlidu (1996) describe a nested contest in which initially members of a group compete for a rent, and in a second stage the members of the group that has won the rent compete for the rent among themselves. In the absence of risk aversion, whether or not the rent is divisible is of no significance. The rent is a collective benefit in the stage at which groups compete and is a private benefit when members of the winning group compete among themselves. Rent dissipation depends on the relative sizes of the groups. The model describes cases in which, first, a coalition is formed to contest or create a rent and, second, if the rent is won or made available, the personal division of the rent becomes the issue of contention. Konrad (2004) shows that group composition effects become important in nested contests if the group members are asymmetric. For some group compositions, players who value the prize very little may win the prize with high probabiUty and with very Uttle effort.

4,2 Contest design The structures of contests affect the effort that contenders exert. The consequences are similar to different distributional rules for contenders. The Uterature demonstrates that fine-grained rules and the structure of contests have a wide variety of subtle effects on the investments in effort made by participants. Under slightly different rules for entry, sequencing, or dividing the prize, the social losses associated with rent-seeking contests can differ substantially. These are important conclusions because structures of rent-seeking contests are not entirely historical accidents. Rather, contests are often contrived with various ends in mind. The rules of the game are often affected by the gains of those who contrive the contests. Although effort used in rent seeking is a cost for rent seekers, and so potentially a source of social loss, those efforts can be a source of benefit for government officials.^^ In principle, the rules of the game can be revised to reduce or to increase social losses by inducing changes in both the extent and kind of competitive effort (Congleton 1980). Elie Appelbaum and Eliakim Katz, 1987. Seeking rents by setting rents: The poUtical economy of rent seeking. Economic Journal 97, 685-99. Mark Gradstein and Kai A. Konrad, 1999. Orchestrating rent seeking contests. Economic Journal 109, 536-45. ^^ See Congleton (1988) for a discussion of the extent to which rent-seeking efforts may be regarded as completely wasteful.

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Forty Years of Research on Rent Seeking: An Overview Kofi O. Nti, 2004. Maximum efforts in contests with asymmetric valuations. European Journal of Political Economy 20,1059-66. Appelbaum and Katz (1987) point out the active role that the "rent setter" may have in devising rent-seeking contests, Gradstein and Konrad (1999) show that organizing a contest in a structure with multiple rounds, in which there are period contests among subgroups of players and only the winners advance to the next round, may induce higher total rent-seeking effort, in particular, if the discriminatory power of the contest at each round is low. Nti (2004) considers the choice of the contest success function that maximizes effort when contestants have asymmetric valuations of the prize. Although the TuUock function with constant returns is optimal in circumstances in which valuations are symmetric and contest success functions are restricted, in the unconstrained case the optimal contest success function is equivalent to an all-pay auction with a reserve price. The optimal design internalizes the incentives to exert effort that derive from different valuations of the prize and discounts the incentive of a high-valuation contestant to evoke more effort. 4.3 The structure of prizes The early studies proposed a single prize for the winner of a contest. The structure of prizes is, however, an important determinant of effort in contests. There can be more than one prize, and the prizes can have different values. Amihai Glazer and Refael Hassin, 1988. Optimal contests. Economic Inquiry 26,133-43. Derek J. Clark and Christian Riis, 1998. Competition over more than one prize. American Economic Review 88, 276-89. Benny Moldovanu and Aner Sela, 2001. The optimal allocation of prizes in conit^i^, American Economic Review 91, 542-58. Stefan Szymanski and Tommaso M. Valletti, 2005. Incentive effects of second prizes. European Journal of Political Economy 21, 467-81. Glazer and Hassin (1988) examined the structure of prizes as incentive mechanisms. With prizes allocated in accord with individuals' output or effort rankings, they derived properties of a structure of prizes that maximizes the output (or effort) of contestants and found different optimal structures of prizes in different circumstances. Clark and Riis (1998) and Moldovanu and Sela (2001) investigate the structure of prizes in optimal design of contests under conditions of both complete and incomplete information. They find that splitting the prize into several smaller prizes is typically not a good strategy for inducing higher overall effort. Convex cost of effort is one of the cases for which multiple prizes may, however, generate higher overall effort. Szymanski and Valletti (2005) investigate the effect of introducing a second place prize in contests in which contestants have asym-

21

Roger D. Congleton, Arye L. Hillman and Kai A. Konrad metric abilities. In a three-person contest, a second prize increases total effort if one contestant is favored to win thefirstprize. The model directly applies to sports contests in which the efforts of contestants provide utility for spectators; however, other applications are proposed in which there is asymmetry in the abilities of contenders. The second prize is an alternative to an exclusion rule that would deny participation to a contestant whose likelihood of winning is so high as to make the outcome of a contest almost a foregone conclusion. Theory Part 5 - Experiments Many of the predictions of the theory have been tested in experiments. Jason F. Shogren and Kyung Hwan Baik, 1991. Reexamining efficient rent seeking in laboratory markets. Public Choice 69, 69-79. Jan Potters, Casper G. de Vries, and Frans van Winden, 1998. An experimental examination of rational rent seeking. European Journal ofPolitical Economy 14, 783-800. Carsten Vogt, Joachim Weimann, and Chun-Lei Yang, 2002. Efficient rent seeking in experiment. Public Choice 110, 61-lS, Shogren and Baik (1991) report on experimental behavior in TuUock's efficient rent-seeking game and find outcomes consistent with predicted behavior and rent dissipation. Potters, de Vries, and van Winden (1998) report on experiments using both the TuUock probabilistic and highest-bid (discriminating or all-pay auction) contest success functions. In the Tullock contests, rent dissipation was initially greater than the predicted 50 percent for two contenders but declined toward the predicted outcome as further games were played. In the contests in which the highest bidder won, ex post rent dissipation fluctuated around the Hillman-Samet predicted on-average complete dissipation. Some participants showed learning from experience and changed their behavior, while others in both types of contests did not approach the consistent rational behavior predicted by the models. Vogt, Weimann, and Yang (2002) report rational behavior in variants of the Tullock contest.

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Forty Years of Research on Rent Seeking: An Overview

Volume II - Applications: Rent Seeking in Practice Volume II focuses on applications of the rent-seeking approach. The earUest applied papers demonstrate that the rent-seeking approach can be used to shed light on the behavior of politically active individuals and interest groups, and to provide a rational choice - based explanation for the wide range of unproductive economic regulations observed in the present and past. The rent-seeking approach has also been used to investigate a variety of other contest-like settings in which the resources invested by participants may be socially unproductive. For example, rent-seeking models have been used to analyze electoral contests, court proceedings, status seeking, terrorism, war, and revolution. Again, the literature is large and selection was required. Our decision was to focus on classics, significant contributions, and to sample the breadth of the appHed work. Many more papers could have been included. In the second volume we have sorted papers into areas of application by institutional setting and sector of the economy: regulation of industry, protectionist rent seeking, soft budgets and moral hazard, rent seeking in the context of economic development, the relationship between rent seeking and economic growth, rent seeking inside thefirm,rent seeking between insiders and outsiders, office seeking and rent creation in democratic politics, Htigation, history, and the civil society.

Applications Part 1 - Regulation and Protection 1.1 Monopoly and regulation of industry Tullock's (1967) observations of the social cost of rent seeking included monopoly. Subsequent studies focused on measuring the social costs of monopoly due to rent seeking. Richard A. Posner, 1975. The social costs of monopoly and regulation. Journal of Political Economy 83, 807-27. Keith Cowling and Dennis C. Mueller, 1978. The social costs of monopoly power. Economic Journal 88, 727-48. Stephen C. Littlechild, 1981. Misleading calculations of the social costs of monopoly power. Economic Journal 91, 348-63. Posner (1975) set out assumptions consistent with complete dissipation of monopoly rents and computed a formula for the relation between deadweight losses and social costs of full-dissipation rent seeking. The elasticity of demand is critical for evaluating social costs. Estimates of elasticities and social loss were computed for a number of U.S. industries. Posner's calculations suggested that between 1.7 and 3.5 percent of GNP may have been lost through monopolization. He also noted that, in price-regulated industries, rent dissipation occurs through non-price com-

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Roger D. Congleton, Arye L. Hillman and Kai A. Konrad petition. ^^ Posner extended his observations to tax policy, the effects of monopoly power on the distribution of income, and the internal practices of labor unions. For example, he argued that taxes that increase government revenue also provide greater incentives for taxpayers to seek means of avoiding the tax payments, and so, when rent avoidance costs are taken into account, broader tax bases can be socially costly. Cowling and Mueller (1978) took the full-dissipation assumption to its logical conclusion and assigned social loss to all profits and all expenditures on advertising. They estimate social losses for relatively large firms and for the economies as a whole in the U.S. and the U.K. Estimated welfare losses ranged from 3.0 to 7.2 percent of GNP. Littlechild (1981) re-evaluated the Cowling-Mueller study and suggested that their estimates overstate the true social cost of monopoly. 1.2 Protectionist international trade policies TuUock (1967) also referred to tariffs. Beyond monopoly and regulation, policies that restrict international trade have been a significant source of rents. The following papers view protectionist policies as means of rent creation. Rent seeking is introduced through the issue of who is to be protected. Arye L. Hillman, 1982. Declining industries and political-support protectionist motiwQS. American Economic Review 72,1180-87. Arye L. Hillman and Heinrich W. Ursprung, 1988. Domestic politics, foreign interests, and international trade poUcy. American Economic Review 78, 72945. Gene M. Grossman and Elhanan Helpman, 1994. Protection for salt. American Economic Review 84, 833-50. In normative international trade models, protectionism was shown to be socially optimal under various second-best circumstances; in particular, subsidizing domestic firms was proposed as socially optimal when international markets are imperfectly competitive. Hillman (1982) pointed out that protection to industries in dechne because of changing comparative advantage could be explained by political-support motives. Protection increased industry-specific rents and benefited an identifiable group. When industries are in decline, new entry does not occur and the beneficiaries of protection can readily identify themselves and express political gratitude, whereas costs of protection of any industry are widely dispersed over the population. The view of protection as politically motivated rent creation and protection contrasted with the prior views of protection as reflecting social welfare objectives in second-best situations. Hillman and Ursprung (1988) describe trade policy as the outcome of a contest between competing poUtical candidates who have committed to implement the ^^ Posner's (1975) description of non-price competition as a form of rent seeking was preceded by studies of structurally similar processes in markets with promotional competition in the form of marketing and advertising effort (e.g., Friedman 1958).

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Forty Years of Research on Rent Seeking: An Overview preferred policies of domestic import-competing and foreign exporting producers. When tariffs are the means of protection, the candidates announce polarized poUcies that benefit their respective political supporters. Tariff revenue is assumed to be without political value. Voluntary export restraints that replace tariffs transform the tariff revenue to quota rents that are transferred to foreign exporters. Domestic producers gain from protection and foreign producers gain from the quota rents. The rents to foreign producers are compensation - and indeed overcompensation - for the protectionist policies. When voluntary export restraints restrict international trade, political candidates choose identical Hotelling-type poUcies, so ending the poUtical tensions when tariffs are the means of protection. Grossman and Helpman (1994) describe a policy maker who stands ready to accept offers for "sale of protection" to industry interests. The industry producer groups are perfectly organized and consumers are not organized at all, which is the source of the industry's poUtical advantage. The model solves a common agency problem in which the poUtician selling protection secures all rents (at the margin). Market characteristics determine the structure of protection chosen to maximize political rents. Each of these papers describes rent creation and rent assignment through poUtical discretion over international trade policy.

Applications Part 2 - Economic Development and Growth 2.1 Economic development Anne O. Krueger, 1974. The poUtical economy of the rent-seeking society. American Economic Review 64, 291-303. Jakob Svensson, 2000. Foreign aid and rent seeking. Journal of International Economics 51, 437-61. Philip Verwimp, 2003. The political economy of coffee, dictatorship, and genocide. European/owma/o/Poto'ca/£^conomy 19,161-81. Krueger (1974) observed that rents from import quotas attracted resources to rent seeking and computed estimates of the social cost of rent seeking for quota rents in India and Turkey. She set out a general equilibrium model using the completedissipation assumption. The rent dissipation arose in the course of the government assigning import quotas based on firms' productive capacities. The method for assigning quotas provided incentives for excessive productive capacity. Resources were also used in seeking to influence government officials' decisions regarding quota assignments. She estimated that rent dissipation accounted for 7.3 percent of national income for India and 15 percent for Turkey. Svensson (2000) points to the evidence that foreign aid has been ineffective in increasing incomes in poor countries and notes that foreign aid made up more than half of the government budgets of the 50 most aid-dependent countries in the 1975-1995 period. He describes a repeated game among competing domestic groups in which aid is provided, and considers an aid policy that takes account of losses from rent seeking. In an empir-

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Roger D. Congleton, Arye L. Hillman and Kai A. Konrad ical section, ethnic diversity is used as a proxy for the number of competing groups and an index of corruption is a proxy for rent seeking. Foreign aid is positively associated with rent seeking (proxied by corruption). Verwimp (2003) describes how the antecedents to the Rwanda genocide centered on government responses to the value of rents that were tied to the price of coffee.

22 Property rights and corruption Conditions are favorable for rent seeking when the rule of law is not present to protect property rights and where there is corruption. Kevin M. Murphy, Andrei Shleifer, and Robert W. Vishny, 1993. Why is rent seeking so costly to growth? American Economic Review 83, 409-14. Arye L. Hillman and Heinrich W. Ursprung, 2000. Political culture and economic decline, ^^wropean/owma/c/Poft'rica/Economy 16,189-213. Halvor Mehlum, Karl Moene, and Ragnar Torvik, 2006. Institutions and the resource curse. Economic Journal 116,1-20. Murphy, Shleifer, and Vishney (1993) observe that effectiveness of protection of property rights determines returns from rent seeking and propose that rent seeking inhibits economic growth for two principal reasons: because of increasing returns to rent seeking relative to productive activity and because bureaucratic rent seeking deters innovation more so than ongoing productive activity. In their model with increasing returns, a "bad" equilibrium exists that is stable and is not affected by minor improvement in property rights protection. Countries can also slide into this equilibrium as the consequence of civil turmoil. With respect to innovation, they note that rent seeking by government officials impedes growth because of the need for licenses, etc. to start new business activities. Also, whereas large pre-existing firms have political influence and can protect themselves from a rent-seeking bureaucracy, innovators are least protected from the rent seeking by government officials and can least afford bribes. Innovative projects are also long-term risky investments that are most vulnerable to rent extraction by government officials. Hillman and Ursprung (2000) use the background of the transition from socialism to address the question as to why rent seeking appears to increase with political liberalization. The transition from socialism offered substantial rents through the processes of privatization. The privatization occurred with initial property rights not defined, and after rounds of privatization property rights could at times remain not well protected. The processes of privatization often involved corruption by political insiders who had the authority and means to designate owners of property and natural resources. Hillman and Ursprung use a nested model to describe the privileged insiders competing for rents, while at the same time outsiders compete to become insiders. Political liberalization gives outsiders direct access to contests for rents and thereby increases social costs of rent seeking, as long as a political

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Forty Years of Research on Rent Seeking: An Overview culture of rent seeking persists, ^^ Mehlum, Moene, and Torvik (2006) confirm that institutions conducive to rent seeking underUe failures of societies to realize benefits from natural resource v^ealth. Natural-resource v^ealth is a "curse," rather than a source of social benefit when property rights are not defined or respected and the wealth becomes a rent-seeking prize. ^^ 2.3 Migration Rents and rent-seeking losses are associated with migration and migration policies. When people emigrate, they may be escaping a rent-seeking society or they may be attracted by rents available in new locations. Gil S. Epstein, Arye L. Hillman, and Heinrich W. Ursprung, 1999. The king never emigrates. Review of Development Economics 3,107-21. Peter Nannestad, 2004. Immigration as a challenge to the Danish welfare state? European Journal of Political Economy 20, 755-67. Epstein, Hillman, and Ursprung (1999) describe a king or ruler who creates and assigns rents by taxing part of the population for both own personal benefit and for the benefit of other privileged parts of the population. Whether people in the population gain or lose depends on the outcome of a contest that determines proximity to the king. People differ in personal comparative advantage in productive and rent-seeking activities. The contest success function determines whether the most productive people or the superior rent seekers are closest to the king. Those furthest from the king have the greatest incentives to emigrate. Rents provided by welfare systems make immigration a form of rent seeking. Nannestad (2004) describes the creation of rents for immigrants through the welfare budget of Denmark. Applications Part 3 - Political and Legal Institutions 3.1 Electoral politics Rent-seeking contests within political systems take place at several levels, as noted in volume I. Analysis of the efforts of would-be monopolists, transfer recipients, and beneficiaries of entry barriers to change the policies of standing governments and regulatory agencies is the focus of most of the research described in volume II. In other cases, however, the government and the rent at issue are determined simultaneously. Contests to become the government - through electoral competition in democracies - exhibit some of the properties of rent-seeking games. Moreover, incumbent politicians and poUtical parties may create or threaten to create ^^ Gelb, Hillman, and Ursprung (1998) describe the institutional background of rent seeking in the transition from socialism. ^^ OUson (2007) considers the case of diamonds.

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Roger D. Congleton, Arye L. Hillman and Kai A. Konrad rent-seeking contests to attract campaign "contributions," insofar as campaign resources increase their prospects for electoral success. Roger D. Congleton, 1986. Rent-seeking aspects of political advertising. PwW/c Choice 49, 249-63. Fred S. McChesney, 1987. Rent extraction and rent creation in the economic theory of TQgul2ition. Journal of Legal Studies 16,101-18. Michael R. Baye, Dan Kovenock, and Casper G. de Vries, 1993. Rigging the lobbying process: An application of the all-pay auction. American Economic Review 83, 289-94. Yeon-Koo Che and Ian L. Gale, 1998. Caps on political lobbying. American Economic Review 88, 643-51. Kai A. Konrad, 2004. Inverse campaigning. Economic Journal 114, 69-82. Congleton (1986) notes that competition among candidates (and parties) for the votes of their electorates often resembles a rent-seeking contest. Advertising is often used to affect voter expectations about the relative merits of the policies and candidates. To the extent that poHtical advertising is effective, but provides biased information, the quality of voter information may be eroded by persuasive campaigns, at least at the margin. This may occur even in cases in which the efforts of proponents and opponents of a given policy exactly offset each other, because such persuasive campaigns tend to increase the variance of voter estimates of policy consequences. When the informational value of poHtical advertising to voters is less than the expenditures of opposing candidates, at least some political advertising is wasteful in the sense of a rent-seeking contest. McChesney (1987) suggests that the demand for campaign contributions can induce competing candidates and political parties to create new rent-seeking games. Incumbent politicians, may for example, threaten to eliminate existing rents, or threaten firms with new taxation, to obtain additional poHtical support. Such contests increase rent-seeking losses by creating new contests for political influence with costs greater than benefits. They may also create conventional deadweight losses by affecting the allocation of investment resources and the flow of indirect payments to politicians. Baye, Kovenock, and de Vries (1993) demonstrate how a rent-maximizing official can benefit by creating a two-stage lobbying game when participants disagree about the value of the prize to be awarded. Lobbyists in the second stage actively compete for favor by providing services or campaign contributions, and the closer the lobbyists in the second round are in valuing the prize, the higher the total lobbying expenditure tends to be. Consequently, it is clear that conditions exist in which officials will exclude the highest bidder from the final group of participants. Similarly, Che and Gale (1998) demonstrate that caps on the amounts that may be given to political candidates can increase total expenditures in cases in which substantial valuation disagreements exist.

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Forty Years of Research on Rent Seeking: An Overview Konrad (2004) explores a setting in which campaign expenditures are informative, but nonetheless give rise to a deadweight cost through the electoral process. Uninformed voters have sufficient information in the absence of campaign expenditures to make the correct (welfare-enhancing) choice. Voters initially know that a majority benefits from the program of one of the two parties, but not who actually benefits. In a process termed "inverse campaigning", parties each diminish poUtical support for political opponents by informing uninformed voters about the beneficiaries of the opponent's programs. The expected benefits of the uninformed voters from the opponent's program are reduced (the voters realize that they are less likely to be members of the favored subset of voters). In equilibrium, voters become informed, but are no better off because they knew enough in the first place to make the correct decision in the election. Campaign expenditures have been wasted in the political contest. 3.2 The courts, the judiciary, and litigation A good deal of mainstream economics rests on the assumption that property rights are both secure and clearly understood by one and all. This allows trade to take place and contracts to be negotiated with participating parties all expecting to benefit from exchange. Although this is a useful first approximation of legal systems, disagreements can exist about the nature of a contract and about property rights. The result is then litigation that aims at clarifying or establishing property rights. The outcome of a civil suit redistributes wealth between defendants and plaintiffs. Civil law proceedings are thus rent-seeking contests in which the "prize" is dissipated through conflict. Gordon TuUock, 1975. On the efficient organization of trials. Kyklos 28,745-62. Amy Farmer and Paul Pecorino, 1999. Legal expenditure as a rent-seeking game. Public Choice 100, 271-88. Francesco Parisi, 2002. Rent-seeking through Utigation: Adversarial and inquisitorial systems compared. International Review of Law and Economics 22, 193-216. Michael R. Baye, Dan Kovenock, and Casper G. de Vries, 2005. Comparative analysis of litigation systems: An auction-theoretic approach. Economic /owma/115, 583-601. TuUock (1975) developed an early version of his contest success function to describe a legal contest between two sides of a civil law suit. In related research, TuUock used the resources committed to Utigation as an index of the cost-effectiveness of legal systems. By that measure, he argues that the judge-run continental system is superior to the adversarial proceedings of the Anglo-Saxon system. The technique of using rent dissipation as an index of court performance was developed further by Farmer and Peccorino (1999), Parisi (2002), and Baye, Kovenock, and de Vries (2005), who highlight different aspects of court procedures and outcomes. Farmer

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Roger D. Congleton, Arye L. Hillman and Kai A. Konrad and Peccorino and Baye, Kovenock and de Vries show how the design of the legal system, in particular fee shifting rules, that is, the allocation of litigation fees as a function of the court decision, influences the efficiency of the Utigation system. Parisi notes the existence of a continuum of court procedures, rather than the dichotomous Anglo-Continental Europe choice.

Applications Part 4 - Institutions and History 4.1 Institutions The rules of a rent-seeking contest determine both the feasible range of rentseeking methods and the net returns from private investments in rent-seeking contests. The "rules of the game" are simply another name for the array of formal and informal institutions under which the rent-seeking contest takes place. To the extent that existing formal and informal rules can be modified or new formal rules introduced, rent-seeking expenditures can be reduced (or increased) through institutional design (Congleton, 1980,1984: reprinted in volume I). Since institutions can both induce and curtail rent-seeking activities, normative research on institutional design attempts to identify rules that reduce or increase unproductive conflict and to suggest reforms that can improve on existing rules. For example, the research on court systems and alternative ownership structures for firms noted above falls into this general category of research. The rent-seeking approach can also be appUed to understand the law itself and other civil institutions. James M. Buchanan, 1983. Rent seeking, noncompensated transfers, and laws of succession. Journal of Law and Economics 26, 71-85. Kevin Sylwester, 2001. A model of institutional formation within a rent-seeking environment. Journal of Economic Behavior and Organization 44,169-76. J. Atsu Amegashie, 2006. The 2002 winter Olympics scandal: Rent seeking and committees. Social Choice and Welfare 26,183-89. Buchanan (1983) explores an area of civil law that is one of the oldest and most important, namely inheritance law. Many models assume that agents live forever, although this is not a reasonable assumption for long-term analysis. Even if intragenerational law is efficient in the sense that it minimizes rent-seeking losses by channeUng conflict into productive activities, resources may still be dissipated in intergenerational conflict. Such large-scale conflict is most evident in great dynastic conflicts for power and wealth, but may also occur in any household that has wealth that may be passed on to the next generation, or even within governments or bureaucracies insofar as an office may be said to be created and passed on to the next office holder. Buchanan notes that some legal institutions clearly tend to reduce conflict levels, as with primogeniture and requirements for equal division, while others clearly increase conflict over what he terms uncompensated transfers.

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Forty Years of Research on Rent Seeking: An Overview The logic of rent seeking can also be used to explain the emergence of the law and state enforcement of the law itself as a means of avoiding losses from wasteful conflict. This approach to political theory was first clearly stated by Thomas Hobbes in 1651, and rational choice-based analysis of the state as a device for reducing conflict in a setting of anarchy has existed since the early 1970s (see for example TuUock 1974).^'^ Sylwester (2001) analyzes an intermediate setting in which a group of producers confronts rent seeking by a large group of pragmatists who can choose either to be productive or to rent seek (steal) from the producers. Additional protection through the rule of law can be provided collectively by the producers to reduce losses from rent seeking, although this is not always consistent with individual producers' incentives, because additional security is a public good for the producers. Sylwester suggests that the larger the initial productive group is relative to the group of rent seekers, and the more productive are the producers, the more likely it is that additional law enforcement will be provided. In a relatively simple model, he demonstrates a clear interdependence between production technology (income), rent-seeking, and effective legal institutions. Societies with effective legal institutions are more prosperous because of reduced rent-seeking activity, although more costly legal institutions can be adopted only by societies that are relatively prosperous. In effect, the results suggest that some societies "boot strap" themselves out of poverty by adopting successively more effective legal institutions that curtail rent seeking and enhance productivity. Determining the outcome of rent-seeking contests requires the choice of judges. When measures of performance are not entirely objective, but involve subjective aesthetics, the choice of winners can be influenced by extraneous considerations or corruption. Amegashie (2006) uses the case of the skating judgment scandal at the 2002 winter Olympic Games as background to investigate the consequence of change in the rules of committee decisions. ^^ Because of the scandal, not all judges' evaluations were included in determination of winners. Rather, there was random selection of which judges' evaluations were used. Although the identity of which judges' opinions will count is then not known, Amegashie finds no systematic effects that reduce the incentives for rent seeking through influence on judges. If the intent was to diminish rent seeking, this was a case of institutional design that failed.

^^ As noted, we have elected to exclude the theoretical literature on anarchy from the present volume for space considerations and in order to focus on settings in which a political and legal system of some sort already exists. ^^ This was a case of corruption. If the decisions of an international committee and international organizations reflect the values or behavioral norms of individual national members, we expect corruption to emerge, and rent-seeking contests to replace objective decisions.

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Roger D. Congleton, Arye L. Hillman and Kai A. Konrad 4,2 Mercantilism Mercantilism has been studied as an example of a rent-seeking society. ^^ Barry Baysinger, Robert B. Ekelund Jr., and Robert D. ToUison, 1980. Mercantilism as a rent-seeking society. In James M. Buchanan, Robert D. ToUison, and Gordon TuUock (eds.), Towards a Theory of the Rent-Seeking Society. Texas A&M University Press, College Station, pp. 235-68. S. R. H. Jones and Simon E Ville, 1996. Efficient transactors or rent-seeking monopoUsts? The rationale for early chartered trading companies. Journal of Economic History 56, 898-915. Oliver Volckart, 2000. The open constitution and its enemies: Competition, rent seeking, and the rise of the modern state. Journal of Economic Behavior and Organization 42,1-17. Baysinger, Ekelund, and ToUison (1980) quote Adam Smith that "mercantilism is nothing but a tissue of protectionist fallacies foisted upon a venal parUament by our merchants and manufacturers" based on the idea that "wealth consists in money." They describe mercantUism using a model of the state as a source of private rents, with appUcations to the different political and legal institutions of England and France. MercantiUsm in France persisted into the nineteenth century, whereas mercantilism in England was compromised by a competitive judiciary that created uncertainty about whether monopoly rights could be sustained, and also by the intellectual arguments of economists and philosophers.^'^ Volckart (2000) analyzes the emergence of the early mercantiUst state in the late Middle Ages as an exercise in rent extraction by lords and vassals providing mUitary protection for peasants in exchange for other services. The rent-extracting ability arose because of reductions in information and transaction costs, along with increasing population, which together shifted bargaining power and miUtary authority to regional lords. In the early Middle Ages, labor had been scarce and competition between large landowners and fortified towns for labor resulted in contracts that were relatively favorable to peasants. As population increased, regional political authorities were able to create and enforce new laws that generated new rents for those controUing large blocks of land and supplying military services. Rents were created for towns, for example, by requiring farmers to sell their grain to the nearest grain dealer, reducing competition among resellers for grain, who in turn would obtain this profitable privUege by accepting the regional lord's provision of military services. Jones and Ville (1996) propose that joint stock companies that emerged in the seventeenth and eighteenth centuries held exclusive trading rights in particular goods and/or regions of the world, not because they reduced transactions costs associated with ^^ Prior to mercantilism, Ekelund et al (1996) describe the medieval church as an economic firm. ^^ For an extended study, see Ekelund and ToUison (1997).

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Forty Years of Research on Rent Seeking: An Overview long-distance trade, but because they maximized monopoly rents. This, in turn, maximized the fees that the crown could charge for issuing such charters. For example, in 1687, private traders to West Africa paid a premium of 40 percent of the value of their cargoes for the right to trade. Very few of these companies survived in the more competitive environment that emerged in the late eighteenth century.

4.3 Authoritarian regimes Rent seeking has historically been prevalent under authoritarian regimes. The following papers describe communism and the Roman Empire. Arye L. Hillman and Adi Schnytzer, 1986. Illegal economic activities and purges in a Soviet-type economy: A rent-seeking perspective. International Review of Law and Economics 6, 87-99. Charles D. DeLorme Jr., Stacey Isom, and David R. Kamershen, 2005. Rent seeking and taxation in the Ancient Roman Empire. Applied Economics 37, 705-11. Hillman and Schnytzer (1986) describe the role of rents and rent seeking under communism, under which rewards were non-market determined and market transactions constituted economic crimes. Purges were means of protecting the incumbent ruler from rent seekers. Data from the prosecution of economic crimes reveals the large magnitudes of rents from personal transactions within the planned system. A puzzle is that large rents were contested and secured when there were limited opportunities for spending wealth because of the Hmited presence of markets. Data is also presented on the value of payments made to obtain positions in the official hierarchy. The payments are indicative of the rents that could be extracted. DeLorme, Isom, and Kamershen (2005) describe the role of rent seeking in the demise of the Roman Empire, The change in political institutions from republic to rule by an emperor changed the behavior of the ruling classes. Rent seeking resulted in use of tax revenue for privileged benefits. Military control by the emperor prevented popular expression of discontent with the privileged assignment of tax revenue.

Applications Part 5 - The Firm 5.7 Soft budgets and moral hazard The interface between firms and benefits through pubHc policy introduces the idea of the soft budget, which is closely related to rent seeking. "Soft" budgets are budgets that are not binding and that can thereby be manipulated to create rents. Government subsidies that cover producers' costs provide soft budgets for the subsidy recipients. Soft budgets are associated with moral hazard.

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Roger D. Congleton, Aiye L. Hillman and Kai A. Konrad Janos Kornai, 1980. 'Hard' and 'soft' budget constraint. Acta Oeconomica 25, 231-46. Arye L. Hillman, Eliakim Katz, and Jacob Rosenberg, 1987. Workers as insurance: Anticipated government intervention and factor demand. Oxford Economic Papers 39, 813-20. Steven T. Buccola and James E. McCandish, 1999. Rent seeking and rent dissipation in state enterprises. Review ofAgricultural Economics 21, 358-73. Kornai (1980) described soft budgets in a context in which state-ownedfirmsfunction in markets but cannot become bankrupt, because of the political unacceptabiUty of unemployment or the closure of a state-ownedfirm.A state guarantee to cover all losses allows rent creation and rent extraction by managers and workers in the state-ownedfirms.Kornai's soft budgets also apply in market economies to government departments and bureaucracies, which similarly are protected from bankruptcy. Hillman, Katz, and Rosenberg (1987) describe a firm whose owners are aware that the political disutility of unemployment and the likeUhood of protectionist pohcies increase with the number of workers who would lose their jobs if the firm were to confront low-cost import competition. Rents in the form of returns to industry-specific capital are protected by employing more than the profit-maximizing number of workers. Firms producing output under conditions of market risk have incentives to produce in peripheral locations where the political disutility of unemployment is greater. Beyond moral hazard, there is therefore adverse selection. Buccola and McCandish (1999) provide a case study from Africa in which a private firm competes against a privatized former state enterprise that retains its ties to government officials and thereby its privileges. The case study is the background for a description of how state-owned firms seek to maximize costs subject to the aid that is provided by international donors. In this case, the source of the soft budget is development assistance.

5,2 Rent seeking within the firm When rent seeking occurs within the firm, government need not be involved and so political economy issues need not arise. The firm is an institution of economic organization based on the incentives of markets and private property. In principle, the competitivefirmis devoid of rents. The logic of the rent-seeking approach suggests, however, that afirm'slabor force, management, and owners have incentives to invest resources in socially (and organizationally) fruitless disputes over their firm's profits. Aaron S. Edlin and Joseph E. Stiglitz, 1995. Discouraging rivals: Managerial rent seeking and economic inefficiencies, ^mencan Economic Review 85,130112.

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Forty Years of Research on Rent Seeking: An Overview David S. Scharfstein and Jeremy C. Stein, 2000. The dark side of internal capital markets: Divisional rent-seeking and inefficient investment. The Journal of Finance 55, 2527-64. Amihai Glazer, 2002. AlUes as rivals: Internal and external rent stoking. Journal of Economic Behavior and Organization 48,155-62. Edlin and Stiglitz (1995) present a model in which managers entrench themselves in their positions by making investment decisions that discourage rivals from applying for or contesting their positions. The rents of entrenchment are achieved by creating asymmetric information and by making acquisitions that require the personal information of the incumbent managers for the realization of potential synergies. Incumbent managers, thus, increase uncertainty about the firm's prospects to reduce competition for their positions, given the reservation rewards of other prospective applicants. Edlin and StigUtz thereby suggest that acquisitions and mergers reflect rent seeking by incumbent managers. Scharfstein and Stein (2000) note the empirical evidence that diversified firms or conglomerates trade on the stock market at a discount compared with firms that are more specialized in their activities, A CEO personally gains from empire building or from increasing total investment beyond levels that maximize the value of the firm. Nonetheless the CEO has an incentive to allocate capital efficiently within the firm. Scharfstein and Stein (2000) suggest, however, that the internal allocation of capital within the firm is inefficient because rent seeking by divisional managers results in value-reducing cross-subsidization among the divisions of conglomerates. Rent seeking increases the bargaining power of the weaker divisional managers. Weaker divisions of the firm are subsidized by stronger divisions, because the opportunity cost of allocating time to rent seeking, rather than productive activities, is lower for the managers of the weaker divisions. With accountability constraining the CEO from increasing managerial incomes directly, managerial incomes are increased by increasing investment and thereby managerial responsibility, which can be used to justify increases in manager incomes. Glazer (2002) notes that employees have the option of using their rent-seeking abilities on behalf of the firm in confronting external competitors, or in rent seeking for personal benefit within the firm, which reduces the firm's profits. The ability or means to use rent seeking in either way reduces the incentives of the firm to hire proficient rent seekers.

5.3 Firm ownership, outsiders, and rents Rent seeking within the firm provides an explanation for the existence of outside ownership and for unemployment. Assar Lindbeck and Dennis J. Snower, 1987. Efficiency wages versus insiders and outsiders. European Economic Review 31, 407-16.

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Roger D. Congleton, Arye L. Hillman and Kai A. Konrad Roger D. Congleton, 1989. Monitoring rent-seeking managers: Advantages of diffuse ownership. Canadian Journal of Economics 22, 662-72. Holger M. MixUer and Karl Warneryd, 2001. Inside versus outside ownership: A political theory of thefirm.Rand Journal of Economics 32, 527-41. Intra-firm rent-seeking opportunities arise because the institutional structure of thefirmis unable to aUgn perfectly the interests of the parties participating in joint production. This may be a consequence of informational asymmetries within the firm, contract and institutional imperfections, and/or a firm's market power. It is clear that intra-firm conflict over profits tends to reduce a firm's efficiency and thereby its prospects for survival in competitive markets. And, it is equally clear thatfirmswith an organizational structure that reduces such losses will be relatively more efficient and more likely to survive. Thus, the rent-seeking approach predicts the emergence of organizational structures that reduce intra-firm rent-seeking activities. Congleton (1989) and Miiller and Warneryd (2001) suggest that a firm's ownership structure is such a device. Congleton (1989) notes that owner efforts to monitor shirking employees create a rent seeking-like contest in which owners may over-monitor their employees. Owners may sacrifice totalfirmprofits, as long as their own share of the profits can be increases sufficiently through monitoring. In some cases, diffuse ownership can increase afirm'sprofits by reducing monitoring by owners, because productivity increases as employee profit shares increase. The more profit seeking are employees, the more diffuse ownership should be, if a firm's profits are to be maximized. Miiller and Warneryd explore distinctions between partnerships (inside ownership) and outside owners. They demonstrate that adding outside owners has the effect of creating a hierarchical game in which investments in rent seeking tend to fall relative to the single-level game among partners (insiders). In effect, insiders free ride in the contest, with outside owners leaving less on the insider's table to contest. They explore incentives for insider sell-outs to outsiders and suggest that the common evolution of firm organizational structures from partnerships to corporations reflects diminishing returns to investments infirm-specifichuman capital and increasing intra-firm distributional conflict. Lindbeck and Snower (1987) use the firm insider-outsider relation to propose an explanation for unemployment. The "efficiency wage" hypothesis suggests that unemployment is the outcome of a Nash equilibrium in which higher than marketclearing wages are paid to employees to provide incentives for workers not to shirk. Efficiency wages therefore create rents, at the same time that they increase the opportunity cost of shirking. Such rents produce unemployment. Snower and Lindbeck describe rents as created by insiders within thefirm,who decide on the number of workers who are hired.

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Forty Years of Research on Rent Seeking: An Overview Applications Part 6 ~ Societal Relations Societal relations involve forms of rent seeking, as expounded by Thorstein Veblen (1899) in his classic book, The Theory of the Leisure Class, Veblen described the quest for social status as a rent-seeking contest that involved conspicuously refraining from engaging in productive activity, or if more expedient, conspicuously engaging in consumption or having unwarranted servants for the purpose of conspicuous display. Resources were wastefully used in the display of status. More recent literature extends Veblen's observations.

6.1 Status A great many social settings resemble rent-seeking contests, in that a prize of one kind or another is to be awarded in a manner that depends on the relative efforts of the persons seeking the prize. In some cases the prize is distributed among all those seeking it, as might be said of status in a status game. In other cases, the prize tends to be of the winner-take-all variety, as might be said of the quest for sainthood. Whether the resources used in attempting to secure the prize are socially wasted depends upon the nature of the activities that influence the distribution of the prize or probability of obtaining the prize. If status is conferred by good works or public goods, the game may consume resources but external benefits may exceed the cost of seeking the status, in which case the game may be efficiency enhancing. On the other hand, if the resources used produce no positive externalities outside the game, the game is wasteful in the usual sense of a rent-seeking contest. All contestants would benefit from a proportionate reduction in their expenditures, because this would not diminish their relative position (which determines their share of honor or probability of winning), but would free resources for other purposes. The following papers consider the quest for status. Roger D. Congleton, 1989. Efficient status seeking: Externalities and the evolution of status gdivat^. Journal ofEconomic Behavior and Organization 11,175-90. Amihai Glazer and Kai A. Konrad, 1996. A signaling explanation for charity. American Economic Review 86,1019-28. Mario Ferrero, 2002. Competing for sainthood and the millennial church. Kyklos 55, 335-60. Bruno S. Frey, 2003. Publishing as prostitution? Choosing between one's own ideas and academic success. Public Choice 116, 205-23. Congleton (1989) suggests that there is a tendency for status games to evolve toward more productive contests in which games with negative externalities (private duals and criminal competition) are replaced by games producing no or positive externalities for non-participants, as in gift-giving contests. Good deeds and knowledge accumulate as a consequence of the latter contests. Yet this process of social

37

Roger D. Congleton, Arye L. Hillman and Kai A. Konrad evolution is slow and imperfect. Glazer and Konrad (1996) describe charitable giving as a contest for status. Ferrero (2002) describes contests for the status of sainthood, in which proponents of candidates use resources in post-mortem contests to attract attention to the case for status. Frey (2003) considers status conferred by academic publishing. Rent seeking occurs insofar as personal honor and higher personal income are sought through socially unproductive uses of time and ability. Frey observes that authors seeking pubHcations exhibit a willingness to make any and all changes that an editor or reviewers demand in order to ensure publication of their papers. Frey points out that there is no honor in such quests for honor. Frey also proposes changes in responsibilities of editors to make the quest for publication less like prostitution. 6,2 Civil society and rent seeking Twofinalpapers consider the role of rent seeking in the context of civil society. Roger D. Congleton, 1991. Ideological conviction and persuasion in the rentseeking society. Journal of Public Economics 44, 65-86. Arye L. Hillman, 1998. Political economy and poHtical correctness. Public Choice 96, 219-39. Rent-seeking activity in democracies involves persuasion: directly of voters, and indirectly of poUtical decision makers who exercise some policy discretion, because of specialization and the rational ignorance of uniformed voters. Inefficient poHcies that create rents can be adopted at Uttle cost to responsible politicians or bureaucrats, if few voters know about the specific policies at issues. To understand why voters often support policies favoring more wealthy interest groups, however, requires a morefinelygrained representation of voter interests than provided by models that focus exclusively on economic wealth. Relatively well-informed voters often favor such programs, and it bears noting that the public arguments of economic interest groups rarely directly mention their own economic stakes or those of voters. Rather, political campaigns tend to use arguments based on the interests that voters have in a more attractive society, which usually reflects implications of broadly shared norms and ideology. Congleton (1991) explores how economic and ideological groups conduct advertising and lobbying campaigns to persuade voters and bureaucrats of the merits of particular poUcies. He demonstrates that persuasive contests among ideological groups are more likely to escalate than are contests among economic interest groups, and so rent-seeking losses tend to be higher for ideological than economic persuasive campaigns. Persuasive campaigns of rent seekers are more likely to be successful during times of ideological confusion or uncertainty, because at such times, voters are more open to persuasion. Hillman (1998) considers the slow acceptance of the rent-seeking concept in its first two decades. He argues that contemporary ideology requires that democratic government be perceived as acting in the pubhc interest. Consequently, the idea

38

Forty Years of Research on Rent Seeking: An Overview that rent seekers might be able to persuade others that their personal interests are actually the pubUc's interest was simply rejected by the "mainstream orthodoxy," as impossible or at least politically incorrect. The poUcies that create and assign rents are then left unexplained, as simply part of the error term of democratic theory. It might be argued that the intent of a democratic theory that did not countenance rent seeking v^as pedagogical. Perhaps, education in the v^ays of normatively desirable behavior required not exposing students to the possibility of undeserved rewards obtained through unproductive activities. Yet assuming rent seeking out of existence, or assuming that all assignments of income and wealth affected by democratic governance are meritorious, also teaches students that all pohtically assigned rewards reflect intrinsic merit. Students are thereby not taught to be wary of pohtical assignments of personal rewards. Eventually, as these volumes demonstrate, the social costs associated with rent creation, rent assignment, and rent extraction by political decision makers have come to be widely acknowledged among social scientists and in the public domain, where the term "rent seeking" has emerged in an increasingly wide range of academic publications and in newspaper editorial pages around the world. The literature on corruption (for example, Tanzi 1998, Aidt 2003) and the willingness of international organizations, such as the World Bank and the International Monetary Fund, to ascribe ineffectiveness of aid to rent seeking and other political problems has also increased the awareness and application of the rent-seeking approach (for example. Easterly 2001, Abed and Gupta 2002). The fear of those who acknowledge that rent seeking takes place, but oppose the academic research program, is that democratic institutions will be undermined by that research. In contrast, the hope of those who engage in that research is that, by raising awareness of the problems of political decision making, voters will exercise better oversight of their elected representatives. Democratic governments, although imperfect, sustain more attractive societies than other systems that we are aware of, and their policies are likely to be improved by more informed monitoring of the activities of rent seekers. And, moreover, as suggested in Congleton (1980: reprinted in volume 1,2000,2003a, 2003b), the research may lead to improvements in our institutions for developing and implementing public policies. Without acknowledging the problems, improvements are unlikely to be forthcoming.

Forty Years of Rent-Seeking Research: A Progress Report The importance of a theory can be judged in different ways. Within academia itself, the importance of a new idea can be gauged by its ability to capture the attention and imagination of other academics. The breadth and depth of the academic research on rent seeking undertaken in the past 40 years clearly suggests that this test has been passed. The analytical literature on rent-seeking contests represented in volume I demonstrates that the properties of rent-seeking contests are widely regarded to be interesting, subtle, and important. The applied Uterature represented in volume II demonstrates that the rent-seeking model provides a powerful and

39

Roger D. Congleton, Arye L. Hillman and Kai A. Konrad versatile tool for understanding a wide variety of social, economic, and political phenomena. The phenomenon identified by Gordon TuUock in 1967 has clearly proven to be both subtle and general. The collection of papers in these two volumes provides an overview of important contributions of the literature. The result is an especially interesting and broad subset of the Uterature as a whole that should be of interest to economists, poUtical scientists, and policy makers, whether for their own research or to better understand the world.

References (papers referred to in the introduction, but not included in the volumes) Abed, George T. and Sanyeev Gupta (eds.), 2002. Govemancey Corruption, and Economic Performance. International Monetary Fund, Washington DC. Aidt, Toke S., 2003. Economic analysis of corruption: A survey. Economic Journal 113, 632-52. Baye, Michael R. and Heidrun C. Hoppe, 2003. The strategic equivalence of rent-seeking, innovation, and patent-race games. Games and Economic Behavior 44, 217-26. Bergson, Abram, 1938. A reformulation of certain aspects of welfare economics. Quarterly Journal of Economics 52,310-34. Brady, Gordon and Robert D. Tollison, 1994. On The Trail of Homo Economicus: Essays by Gordon Tullock. George Mason University Press, Fairfax. Buchanan, James M. and Gordon Tullock, 1962. The Calculus of Consent: Logical Foundations of Constitutional Democracy. University of Michigan Press, Ann Arbor. Buchanan, James M., Robert D. Tollison, and Gordon Tullock (eds.), 1980. Toward a Theory of the Rent-Seeking Society. Texas A&M University Press, College Station. Congleton, Roger D., 1988. Evaluating rent-seeking losses. Public Choice 56,181-84. Congleton, Roger D., 2000. Apolitical-efficiency case for federalism in multinational states: Controlling ethnic rent seeking. In Gianluigi Galeotti, Pierre Salmon, and Ronald Wintrobe (eds.). Competition and Structure: The Political Economy of Collective Decisions: Essays in Honor ofAlbert Breton. Cambridge University Press, New York, pp. 284-308. Congleton, Roger D., 2003a. Rent seeking and political institutions. In Charles K. Rowley and Friedrich Schneider (eds.), The Encyclopedia of Public Choice. Kluwer Academic PubHshers, Dordrecht, pp. 499-501. Congleton, Roger D., 2003b. Improving Democracy through Constitutional Reform: Some Swedish Lessons. Kluwer Academic Publishers, Dordrecht. Dasgupta, Partha and Joseph E. Stiglitz, 1980. Uncertainty, industrial structure, and the speed of R&D. Bell Journal of Economics 11,1-28. Easterly, William, 2001. The Elusive Quest for Growth: Economists'Adventures and Misadventures in the Tropics. MIT Press, Cambridge MA. Ekelund, Robert B. Jr. and Robert D. Tollison, 1997. Politicized Economies: Monarchy, Monopoly, and Mercantilism. Texas A&M University Press, College Station. Ekelund, Robert B, Jr., Robert D. Tollison, Gary M. Anderson, Robert F. Hebert, and Audrey B. Davidson, 1996. Sacred Trust: The Medieval Church as an Economic Firm. Oxford University Press, Oxford. Fearon, James D., 1995. Rationalist explanations for war. International Organization 49, 379-414.

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Forty Years of Research on Rent Seeking: An Overview Friedman, Lawrence, 1958. Game-theory models in the allocation of advertising expenditures. Operations Research 6, 699-709. FuUerton, Richard L. and R. Preston McAfee, 1999. Auctioning entry into tournaments. Journal of Political Economy 107, 573-605. Garfinkel, Michelle R. and Stergios Skaperdas, 2007. Economics of conflict: An overview. In Todd Sandler and Keith Hartley (Qds.),Handbook ofDefense Economics, Vol. 2. Elsevier, Amsterdam, pp. 649-709. Gelb, Alan, Arye L. Hillman, and Heinrich W. Ursprung, 1998. Rents as distractions: Why the exit from transition is prolonged. In Nicolas C. Baltas, George Demopoulos, and Joseph Hassid (eds.), Economic Interdependence and Cooperation in Europe. Springer, Berlin, 1998, pp. 21-38. Harberger, Arnold C, 1954. Monopoly and resource dXloodXion.American Economic Review 44, 77-87. Harsanyi, John C., 1955. Cardinal welfare, individualistic ethics, and interpersonal comparisons of utiUty. Journal of Political Economy 63, 309-21. Hillman, Arye L., 1989. The Political Economy ofProtection, Harwood Academic Publishers, Chur. Reprinted 2001 by Routledge, London. Hillman, Arye L., 2003. Public Finance and Public Policy: Responsibilities and Limitations of Government, Cambridge University Press, New York. Hillman, Arye L., 2004. Nietzschean development failures. Public Choice 119, 263-80. Katz, EUakim, Shmuel Nitzan, and Jacob Rosenberg, 1990. Rent seeking for pure public goods. Public Choice 65,49-60. Reprinted in: Alan Lockard and Gordon TuUock (eds.), Efficient Rent Seeking: Chronicle of an Intellectual Quagmire, Kluwer Academic Publishers, Dordrecht, 2001, pp. 137-48. Konrad, Kai A., 2007. Strategy in contests: An introduction. Wissenschaftszentrum Berlin, Discussion Paper SPII 2007-01. Kooreman, Peter and Lambert Schoonbeek, 1997. The specification of the probabihty functions in Tullock's rent-seeking contest. Economics Letters 56,59-61. Laband, David N. and John P. Sophocleus, 1992. An estimate of resource expenditure on transfer activity in the United States. Quarterly Journal of Economics 107, 959-83. Lazear, Edward P., and Sherwin Rosen, 1981. Rank-order tournaments as optimum labor conixdiOXs. Journal of Political Economy 89, 841-64. Leidy, Michael P., 1994. Rent dissipation through self-regulation: The social cost of monopoly under threat of reform. Public Choice 80,105-28. Leininger, Wolfgang, 1993. More efficient rent-seeking - A Munchhausen solution. Public Choice 75, 43-62. Reprinted in Alan Lockard and Gordon Tullock (eds.). Efficient Rent Seeking: Chronicle of an Intellectual Quagmire. Kluwer Academic Publishers, Dordrecht, 2001, pp. 187-206. Lockard, Alan, and Gordon Tullock, 20^1. Efficient Rent-seeking: Chronicle of an Intellectual Quagmire. Kluwer Academic Publishers, Dordrecht. Loury, Greg C, 1979. Market structure and innovation. Quarterly Journal of Economics 93, 395-410. Nalebuff, Barry J. and Joseph E. Stiglitz, 1983. Prizes and incentives: Towards a general theory of compensation and competition. Bell Journal of Economics 14, 21-43. Nitzan, Shmuel, 1994. More on more efficient rent seeking and strategic behavior in contests: Comment. Public Choice 79, 355-56. Reprinted in Alan Lockard and Gordon Tullock (eds.). Efficient Rent Seeking: Chronicle of an Intellectual Quagmire, Kluwer Academic Publishers, Dordrecht, 2001, pp. 239-40.

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Roger D. Congleton, Arye L. Hillman and Kai A. Konrad Olson, Mancur, 1965. The Logic of Collective Action, Public Goods and the Theory of Groups, Harvard University Press, Cambridge MA. Olsson, Ola, 2007. Conflict diamonds. Journal of Development Economics 82, 267-86. Oswald, Andrew J., 2007. An examination of the reliability of prestigious scholarly journals: Evidence and implications for decision-makers. Economica lA, 21-31. Paldam, Martin, 1997. Dutch disease and rent seeking: The Greenland model. European Journal of Political Economy 13,591-614. Rosen, Sherwin, 1986. Prizes and incentives in eUmination tournaments. American Economic Review 76, 701-15. Samuelson, Paul A., 1947. Foundations of Economic Analysis, Harvard University Press, Cambridge MA. Skaperdas, Stergios, 2003. Restraining the genuine homo economicus: Why the economy cannot be divorced from its governance. Economics and Politics 15,135-62. Szymanski, Stefan, 2003. The economic design of sporting contests. Journal of Economic Literature 41,1137-87. Tanzi, Vito, 1998. Corruption around the world: Causes, consequences, scope, and cures. IMF Staff Papers 45, 559-94. Tullock, Gordon, 1974. The Social Dilemma: The Economics of War and Revolution. University Publications, Blacksburg. Veblen, Thorstein, 1934 (1899). The Theory of the Leisure Class. Modern Library, New York.

A Prince had some Monkeys trained to dance. Being naturally great mimics of men's actions, they showed themselves most apt pupils, and when arrayed in their rich clothes and masks, they danced as well as any of the courtiers. The spectacle was often repeated with great applause, till on one occasion a courtier, bent on mischief, took from his pocket a handful of nuts and threw them upon the stage. The Monkeys at the sight of the nuts forgot their dancing and became (as indeed they were) monkeys instead of actors. Pulling off their masks and tearing their robes, they fought with one another for the nuts. The dancing spectacle thus came to an end amidst the laughter and ridicule of the audience. (Aesop, circa 600 BCE, "Fable of the Dancing Monkeys")

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Parti Rents

1.1.1.1 The welfare costs of tariffs, monopolies, and theft THE W E L F A R E C O S T S O F TARIFFS, A N D THEFT

MONOPOLIES,

GORDON TULLOCK RICE

UNIVERSITY

In recent years a considerable number of studies have been published that purport to measure the welfare costs of monopolies and tariffs/ The results have uniformly shown very small costs for practices that economists normally deplore. This led Mundell to comment in 1962 that **Unless there is a thorough theoretical re-examination of the validity of the tools upon which these studies are founded . . . someone will inevitably draw the conclusion that economics has ceased to be important."^ Judging from conversations with graduate students, a number of younger economists are in fact drawing the conclusion that tariffs and monopolies are not of much importance. This view is now beginning to appear in the literature. On the basis of these measurements Professor Harvey Leibenstein has argued *'Microeconomic theory focuses on allocative efficiency to the exclusion of other types of efficiencies that, in fact, are much more significant in many instances."^ It is my purpose to take the other route suggested by Mundell and demonstrate that the ''tools on which these studies are founded" produce an underestimation of the welfare costs of tariffs and monopolies. The classical economists were not concerning themselves with trifles when they argued against tariffs, and the Department of Justice is not dealing with a miniscule problem in its attacks on monopoly. STATICS

The present method for measuring these costs was pioneered by Professor Harberger.'* Let us, therefore, begin with a very simple use of his diagram to analyze a tariff. Figure 1 shows a commodity that can be produced ^The.se studies are conveniently listed with a useful table of the welfare losses computed in each in Haivey Leibenstein, "Allocative EfHciency vs. 'X-Efiiciency'/' Am, Ecotj. Rev., June 1966, >6, 392-415. 'R. A. Mundell, Review of L. H. Jansscn, Free Trade, Protecthu and Customs Uniofi, Am, Ecoit. Rev., June 1962, ^2, 622. ^Olf. cif., p. 392. ia this article Leibenstein consistently uses the phrase *'allocative efBcicncy" to refer solely to the absence of tariffs and monopolies. ^A, C. Harberger, *'Using the Resources at Hand More Effectively," Am, Ecou, Rev., May 1959, 49, 134-46. It should be noted that Harberger suggested the method for the measurement of the welfare costs of monopoly, but its extension to cover tariffs was the work of other scholars. The more careful scholars who have measxired the welfare costs of tariffs have not all used this very simple application of Hacberger's method, but a method such as illustrated in Figure 2. I have chosen to begin with this method of measurement partly because it simplifies the exposition and partly because this procedure is the "conventional wisdom" on the matter. (Cf. Leibenstein, op* at,) 224

Western Economic Journal 5, 224—232

45

Gordon Tullock TULLOCK: WELFARE COSTS

'. Such analysis does, however, turn much of modern economics inside out. The-latter tends to commence with the presumed stmcture of an ordered market, and its analysis tends to be concentrated on spinning out even more elegant and rigorous "proofs" or "theorems" about the ideahzed model of the competitive process. But let us be honest. How much more do we know about market process than Adam Smith knew that is of practical relevance? The analysis of rent seeking, as the contributions in this book indicate, shifts attention to interactions and to institutions outside of and beyond the confined competitive market process, while applying essentially the same tools as those applied to interactions within the process. The analysis of rent seeking is, therefore, properly designated as institutional economics in a very real sense. The analysis also falls within public choice, especially ff the latter is defined methodologically as the extension of the basic tools of economics to nonmarket interaction. Indeed, the previously used rubric, "theory of nonmarket de66

1.1.1.2 Rent seeking and profit seeking Rent Seeking and Profit Seeking

15

cision making," allows rent seeking to be included directly under its umbrella. As many critics, both friendly and unfriendly, have noted, public choice theory and the economic theory of property rights have several aiBSnities. Rent seeking analysis can readily be incorporated within the property-rights approach, and, as with public choice, the theory of rent seeking can be interpreted as an appropriate extension. The primary purpose of this book is to collect the most relevant contributions to the analysis of rent seeking and by so doing to call more attention to the opportunities for further inquiry As the contents of this book suggest, the subject remains new, and opportuiiities for productive and relevant research seem almost unlimited. The book contains the early bits and pieces of a line of inquiry that can be, should be, and will be extensively expanded. In the process, additional institutional and historical detail will be elaborated; additional empirical tests will be conducted; additional rigor wrill characterize the formal analysis. We shall come to know much more about rent seeking. As, when, and if we do, we may hope that some contribution may be made in shifting public attitudes toward constitutional reform that will reduce rather than continue to expand rent-seeking opportunities in our society.

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1.1.1.3 Competitive process, competitive waste, and institutions

In: James M. Buchanan, Robert D. Tollison, and Gordon TuUock (Eds.), Toward a Theory of the Rent-Seeking Society. Texas A&M University Press, College Station, 153-179

9 Competitive Process^ Competitive Wasteland Institutions by ROGER CONGLETON "The activity which we call economic, whether of production or of consumption or of the two together, is also, if we look below the surface, to be interpreted largely by the motives of the competitive contest or game, rather than those of mechanical utility functions to be maximized.*'* ECONOMIC models have by and large focused on the cooperative aspect of economic activity: that of mutually beneficial exchange in a world of scarcity. In so doing, economic theory has sought to illuminate the principle of "spontaneous coordination" by which the multifarious ends of individuals are woven into a network of transactions that benefit everyone involved. In the world normally modeled by economists, there is no explicit conflict or resource devoted to games of conflict. Property rights are enforced without cost and clearly defined areas of individual autonomy (opportunity sets) specify the range of individual endeavor. The ingredients that determine an individual's opportunity set are essentially unalterable features of the world: human and nonhuman wealth legitimately possessed and exchange possibilities defined by externally provided prices. The clear definition of property rights is such that within them no attempt to transfer another's wealth can be successful unless it is the result of voluntary exNoTE: I wish to thank Charles Breedon who made several helpful comments on an earlier drafl of this paper* R H. Knight, The Ethics of Competition and Other Essays (New York: Harper and Brothers, 1935), p. 301.

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Roger Congleton

154 Subsequent Contributions

to Theory and

Measurement

change. In such a world the cost of conflict is effectively infinite, and thus no resources are devoted specifically to the conflicts or competitive processes of normal economic activity. No resources are devoted to bargaining, to monopolizing, toward increasing one s market share, or to political wheeling and dealing. It is a world of complete rule of law, natural and social. Ones opportunities in society are rigorously defined and clearly understood by all in such economic models. However, if the world is not clearly understood by all, or if one's opportunity set is not entirely determined externally by forces beyond the influence of an individual actor, situations are very likely to arise in which an economically rational individual will use the resources at his disposal to influence his range of options at the expense of others. Resources will be devoted to activities that are purely redistributional, at best, and voluntary only in the sense that affected individuals (may) have voluntarily accepted the rules of the competitive arena, rules that may be very Hmited indeed. The mutual advantage of voluntary exchange is often tied to the conflict of bargaining, to real efforts to protect one s property, and to attempts at misleading, if not defrauding, one s potential trading "partner.*' Property rights, as they exist, are social products that can be and are influenced by the actions of individuals to the benefit or detriment of other members of the affected society. Indeed, the right to alter rights seems to be one of the most enduring, though the methods that must be used vary greatly from place to place and time to time. Attempts to model the disposition of economic resources that ignore these important uses of economic wealth will miss important aspects of the process of resource allocation and distribution. It is within this world that the rent-seeking literature attempts to shed Ught. The focus of this Hterature has been primarily instances of what might be called bureaucratic rent seeking: seeking favors of or positions in important bureaus. A general theme of this approach has been that monopoly power is socially inefiicient, not only because of its static welfare properties, but also because individuals will waste resources attempting to establish positions where monopoly rents can be earned (or avoided). A paradox of this line of reasoning seems to have gone largely unnoticed. The dominant source of the waste associated with rentseeking activity arises because not all of the would-be rent acquirers will be successful in their attempts to acquire monopoly power—that

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1.1.1.3 Competitive process, competitive waste, and institutions

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is t9 say, because of the competitive nature of rent seeking itself. The essential difference between rent-seeking activity and competition in ordinary markets is not whether resources are "wasted" in the process, but rather that successful monopolization ends competition (or at least lessens it to a large extent) and the competition that takes place in competitive markets does not. Thus, one cannot arrive at antitrust policy simply by pointing to the costs associated vdth monopolization; the costs of monopolization will tend to be less than the costs of competitive markets if they are measured strictly in terms of the resources utilized for strictly competitive purposes. For in the case of successful monopolization, one has these costs only until the monopoly position is successfully established, while in a competitive market these costs continue to be borne ad infinitum. With the creation of a monopoly position, it would seem, the costs of competing are avoided, and thus the strictly competitive use of resources may be greatly reduced. This is not to say that the general line of argument offered in this literature is incorrect, but rather that this particular inference should not be made in the language used above. A proper statement of the general thrust of the rent-seeking hterature would be that S07ne kinds of competition are preferable to others. As this wording suggests, the importance of what might be called the rent-seeking approach extends far beyond the narrow, but important, confines of bureaucratic rent seeking. The rent-seeking approach seems to be a natural device to examine and compare competitive processes that differ in the nature of the rivalry and/or the institutional setting under which the competitive process takes place. In this paper, two general types of rivalry are examined under three institutional arrangements for determining the dispensation of the rewards of competition.- It is hoped that analysis of highly simplified models will add to the theory of competitive process in general and that some hght v^ll be shed upon features of competitive incentives that affect the extent of competitive efforts and the degree to which such efforts may be wasteful* The settings examined, though ad^This use of the word competition is more like its use in biologx' than in ordinar\' sports or games, for it includes arenas of conflict in which the strategic possibilities of competing parties are not approximately the same. Games often include both aggressive (oflPensive) and defensive strategies, though in practice a competing part>' may be circumstantially constrained to use one or the other of these possibihties. In the normal use of the word competition^ there is an assumed parity that may be lacking in the cases

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mittedly a caricature of real-world settings, seem to be relevant to many issues of interest to both political scientists and economists. Competitive activities are, for our purposes, characterized by the possibility of using resources to alter one s share of some sought reward in a way that is at least partly at the expense of other competitors. There are thus two necessary preconditions of competitive activities: (1) the reward must be scarce, and (2) there must be the possibility of affecting one's share of that reward by the use of resources at one s disposal. Scarcity alone creates the incentive and environment necessary for competition to take place. In order to focus clearly on the competitive use of resources, our primary concern will be a world that has only a single, homogeneous, all-purpose commodity that we shall call "wealth.** This restriction will later be relaxed to allow consideration of competitive games in which strategies take the form of indirect bribes to third parties controlling the dispensation game rewards. Our analysis will use much of the jargon of game theory and focus on the types of pure strategies that individuals may be expected to adopt in a variety of institutional circumstances. Although single-good worlds have been used before in economic analysis, there has not been, to my knowledge, an exploration of the conflict that must naturally follow from such circumstances. It may well be due to the conceptual subtlety of mutually beneficial exchange that economists spend their efforts there, for people without economic training seem intuitively aware of the great potential for conflict. So much so that in the absence of economic training, most individuals seem hard pressed to grasp the very idea of a mutually advantageous transaction; the jargon and concepts of force and conflict seem much more natural to them. "Winner Take All That Remains'' Games in a State of Nature The first type of rivalry to be examined will be games that are characterized by a "winner take all that remains" distribution of the game considered here. It is the use of resources In an arena of conflict that demarcates competitive activities in the sense used here from noncompetitive acti\ities» games in which there are scarce rewards, struggled for by those directly involved in the game. Relative success comes at the expense of other competitors in the game settings of interest here. Note that this does not in principle Hmit our discussion to negative-sum or zero-sum games, since positive-sum games can also be competitive in the sense used here.

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rewards. This game form loosely characterizes all games of establishing monopoly power, including the preponderance of work done in the rent-seeking literature. The game is, as any competitive game must be, characterized by a scarce (hence finite) reward to be distributed to the "winners" of the game while alloting nothing to the "losers." The activity of competing in this game is itself not a costless activity. The winner of the game is he who devotes the greater resources to the competitive activity. To illustrate this kind of game in the simplest terms, consider the famihar island of Crusoe and Friday, where a finite quantity of "wealth" exists, insuflScient to satisfy both Friday and Crusoe simultaneously. Imagine Friday and Crusoe engaged in a game of conquest for the privilege of winning all the wealth that exists on the island. Since wealth is an all-purpose good, it may be used to attack or defend wealth as well as for consumption, consumption being the only direct source of utihty that wealth provides. If neither Friday nor Crusoe has a particular advantage in the initial distribution of wealth or in ability to deploy wealth for attack or defense, then the victor will be that individual who devotes most of his wealth to "attack" (or who vanquishes the attack through superior defense). If the game limited strategies to nearly all-out attack (or defense), one would have a familiar "prisoner s dilemma" situation with the equally familiar Nash equilibrium of fiill attack-full attack shown in figure 9.1. EflFectively all resources would be wasted in competitive activity with essentially no wealth left: for consumption. A somewhat more interesting game, and a more "realistic" one, would be the similar game in which each competitor can choose to employ as much wealth for attack as he wishes, saving the remainder to be consumed along with the hoped-for spoils of victory. The consumption payoJBFto Crusoe of devoting X units of wealth to attack-defense in this game would b e R — Y — X i f X i s greater than Y, and zero if X is less than Y; where Y is the extent of the resources that Friday devotes to attack-defense and R is the total level of wealth present on the island at the disposal of the two game players. Friday s payofiF is zero if Crusoe s level of investment is greater than his (X > Y) and equals R — Y —X if his level of attack-defense is greater than that of Crusoe (Y > X). In the event that X = Y, each player receives (R — Y — X)/2, the remainder of his own resources after the standoff. To maxi-

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FIGURE 9.1

Friday

Crusoe

Full Attack-Defense (less one unit)

Inaction

Full Attack-Defense (lessone unit)

I naction

I.I

16,0

0,16

15, 15

mize one's consumption payofiF in this game, a player should devote "one more" unit of wealth to attack-defense than does his competitor. By minimizing the cost of victory, one maximizes the amount of wealth left to be consumed by the victor. Of course, the problem is that each competitor will be attempting to maximize the size of his consumption payoff according to this rule, and in order to use the rule one must know the other s strategy. If a single choice of strategy must be made simultaneously by both parties, the selections can be based only upon judgment and not upon assuredness of mathematical optimization. There exist no dominant strategies, no level of competitive investment that will prove optimal regardless of ones competitors decision. Ones best course of action depends ultimately upon the course of action chosen by one s competitor. However, if strategies may be revised in light of evidence about ones competitors intent, use of the payoff-maximizing rule leads each competitor to revise continuously upward the level of resources that should be devoted to the competitive process. Friday may choose to use half his resources, Crusoe would rationally counter writh a bit more

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than half his own, and Friday with a bit more than that. In the absence of scarcity this process can continue ad infinitum, but in the presence of resource constraints the process of escalation is limited by the resources at the disposal of the competitors. Since we have assumed that Friday and Crusoe initially each command half the islands wealth, this process would continue until each invests all the resources at his disposal, efiPectively duphcating the game solution of the simpler game developed above. The possibihty of escalation may, in this way, tend to generate outcomes in which all the competitors' resources are wasted on competitive activity. ''Winner Take All That Remains'' Games and Arbitrators to Influence

Open

Not all games that seem totally wastefiil from the competitors' perspective are necessarily socially wasteful in the Faretian sense. Often third parties benefit from the competitive efforts of those directly involved in rivalrous conduct, lessening the waste associated with competitive processes. Such third parties may be divided into two categories: those who can affect the game outcome and those who cannot. The first group will be, of course, the primary interest of competitors and is the first to be included in our analysis. The extreme case of third-party influence is the case where some individual outside the competitive arena can actually determine the outcome. If the opinion of an arbitrator, judge, superior, or other authority is beyond the influence of the would-be competitors, or if the division is in a sense already determined by some rule that uses criteria beyond the influence of the affected parties—such as race, consanguinity, or the occurrence of some truly random event—then there can be no fiuitfiil competitive use of resources. The reward may still be scarce, but the possibility of efficacious competitive action is ruled out. However, this situation is unlikely to be the case. If even the facts of the situation matter, there will be an opportunity for each affected party to attempt to provide "facts" in the quantity and detail that seem most likely to add effectively to his expected share of the reward. Matters other than the "facts'* often enter in as well. Demonstrations of loyalty, trustworthiness, or other influencing merits often affect decisions of binding arbitrators. More direct means of influence are also

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sometimes available. For our purposes, influences can be divided into two categories: direct and indirect transfers of wealth. A direct transfer involves a simple transfer of ownership from one of the competitors to the arbitrator. An indirect transfer is one in which a competitor uses some of his resources to provide some "service" to the arbitrator. The possibihty of either or both kinds of influence being used by two or more parties returns the allocation of resources to the competitive arena. The competitive means have changed from guns to buttering up the relevant authority, but if the prize is to be awarded to some winning party at the expense of the losing party, the game format remains. However, preserving the game format does not mean that the game will necessarily be as wasteftil as it was under different institutional arrangements. To illustrate the effect of institutions on the extent of social waste, even in cases where the game format is not or cannot be changed, let us modify the circumstances facing Crusoe and Friday on the island. Suppose that the chief of the neighboring island hears that both Friday and Crusoe have arrived and are about to be involved in the game of conquest outlined above. Suppose farther that, being an imaginative and industrious politician, he sees that the game of conquest will be disastrously wastefal and sets sail for the island to impose some kind of settlement on the new islanders. Assume that the chief is willing and able to accept bribes and will reward the high bidder with all the wealth that has not been turned over to him in the form of bribes. This is, of course, one famihar version of the rent-seeking game. Note that, given this decision rule, Friday and Crusoe face the same game format as before the chief s arrival, with the one difference that competitive efforts are now devoted to bribing the chief rather than to battle in the literal sense. The strategic use of resources is the same as before so far as Crusoe and Friday are concerned. The optimal strategy is to invest just a bit more than the other invests in competitive activity. In this case, each should try just barely to outbribe the other. If a period of escalation is allowed (as at an open auction), there will be a tendency for each competitor to devote all his resources to the competitive bidding process. As far as the individual competitors are concerned, there will once again be complete waste of the island s resources. However, because wealth is not exhausted in batde but merely transferred to the

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chief, there is no waste in the usual economic sense. The result would be Pareto-optimal, since any distribution of wealth in a single-good world is Pareto-eflBcient. A comparison between the earher game and this game of competitive bribing reveals that this shift of institutions is a Pareto-superior move (assuming that the wealth of the island is valued over the chiefs opportunity cost at home). The chief is better oflF and Crusoe and Friday are no worse off under the new arrangements. (Of course, competitive efforts to become chief may be intensified, that is to say, this competitive game may be imbedded in others, but these other levels of competitive activity are beyond the scope of interest here.) There are many situations in which the arbitrator is prevented from taking direct bribes: perhaps the factors of concern are not merely wealth; perhaps it is feared that there would be more wastefol competition to become arbitrator tf his income were to be increased or partially dependent upon direct bribes; or perhaps allowing bribes would lead to a change in the role of arbitration itself. Thus it is common to find competitive games similar to the one faced above in which the chief cannot accept direct bribes (at least as an ordinary aspect of his entitlements) but may be indirectly influenced. If, in our example, the chief were constrained by law, custom, or fear of the gods to accept only indirect emoluments, the payoffs to being the arbitrator are changed. But tf these indirect payments utilize wealth and the quality or quantity of the influence remains an increasing function of the level of resource investment, the game will not be changed markedly from the viewpoint of the competitors so long as the reward is to be allocated in the same manner as before, that is, to the victor go the spoils. The game remains a game of judgment in the absence of escalation, and again the optimal strategy would be to employ just the barest minimum of resources beyond that of ones competitor. The same tendency for risk aversion and escalation to lead to the total employment of the wealth for competitive purposes would be present. The primary difference is that the value of the transfers to the chief may, in his eyes, be below that of the wealth used to effect the transfers. The competitive activity remains wasteftil in the eyes of the competitors and a transfer of wealth is made to the chief, but the value of the resources or performance actually received by the chief will tend to be less than before. It cannot be greater, since he could have used

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the all-purpose commodity, wealth, to recreate his indirect payments. Thus, in his eyes, resources will be wasted relative to the case of direct bribes, although this case would still be Pareto-superior to the case of direct conquest in a state of nature. The ineflBciency of indirect transfers is a primary source of rent-seeking waste. There are two senses of waste involved here, waste to the society of three, including the arbitrator, and waste in so far as the competitors are concerned. In some cases, the second of these wastes may be more relevant. A person is not usually considered obligated to purchase unproductive services from another in the normal course of economic transactions, although, strictly speaking, under the Pareto criteria such transactions may be required if no net harm was incurred by the "purchasers.*' The inefiBciency of indirect transfers to an arbitrator or group with such powers may be lessened (or enlarged) if the preferred method generates positive benefits to individuals not directly involved in the competitive activity, third parties who may be called "innocent bystanders.*' The judge rhay, for example, choose to make his decision based upon evidence of greater altruism, or he may organize the competition so that it has some pubhc-good character deemed useful, such as a competitive foot race or the display of some other socially appreciated skill or process. These considerations can increase the social efficiency of indirect payments. Indirect influence may in this way be Pareto-superior to direct payments if the recipients collectively would be willing to oflFer greater wealth for their treatment than the chief otherwise would have received. Of course, increased efficiency is not guaranteed simply by the existence of "bystander" eflfects. The chief could have created competitive incentives leading to competitive activity with effects outside the competitive game but imposing costs rather than benefits on nonparticipants. "Scalps" or their conceptual equivalents are often used as evidence of merit, particularly the "scalps" of the chief s own competitors at a different game level. Thus indirect transfer schemes could be even less efficient than the usual rentseeking analysis of situations would seem to indicate. The efficiency of indirect transfers depends entirely upon the nature of competitive incentives and the indirect instruments used in the process of competition. The normal conclusion of the rent-seeking Hterature is thus by no means guaranteed simply because of the indirect nature of trans-

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fers. It is true that in the cases usually examined, the uninvolved "bystander" effects are negative or small and as a first approximation may be ignored. However, generally in discussions of competitive institutions, they cannot be so easily discounted. Competitive Games in a Majority-Rule

State

A simple modification of the institutional setting developed above is the case where the game outcome is determined by the majority of some arbitrating group of individuals. In modern democratic states this is often the case for legislative matters, to some extent for trials by jury, and common when a ruling board of one sort or another vdll be the deciding factor. For most questions of policy, promotion, or the dispensation of badges of merit, only a limited number of the wouldbe candidates will be successful. In many cases there will be but a single winner, particularly in the case of policy. The "winner takes all that remains" format thus remains an appropriate game form under majority-rule institutions. To continue with our illustration, suppose that the neighboring tribe is ruled by a triumvirate of a form where two of the three leaders (triumvirs) have by custom or law the power to decide, in more or less the same manner as the chief in our previous examples, the fate of the distribution of wealth between Crusoe and Friday. Once again, if the decisions of the triumvirate are beyond the possible influence of both Friday and Crusoe, no resources would be devoted to competitive activities. However, for reasons touched on before, this is unlikely to be the case. To the extent that the game format remains "winner take all that remains," Friday and Crusoe will continue to devote wealth to the competitive activity of influence. However, in this institutional setting there is no parallel tendency for all of the island resources to be devoted to the competitive activities. Let us again begin with the case of direct bribes. If the island resources are to be distributed between the competitors according to a single majority vote so that the competitor who gets a majority gets all the reward and the loser(s) none, then it will clearly be in the interests of the competing parties to influence the way voters cast their ballots. In the case of direct bribery, one should attempt to offer larger bribes than one's competitor(s) to a majority of the voting agents. However,

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one need not always increase the level of ones own bribes in response to increases by one*s competitor. One can gain the same advantage by changing the distribution of bribes across voters, and do so at lesser expense. In fact, there is no pattern of bribes that can be dominated only with the fiill use of ones resources, if competitors begin on more or less even ground. Suppose that Crusoe were reckless enough to devote his full share of the island s wealth to competitive bribery and then announce his intentions to Friday. Would Friday be forced to reciprocate with a similar level of competitive effort? Let us consider two possible bribing patterns for Crusoe under these circumstances. First, suppose that Crusoe had announced that he was going to distribute his wealth equally across the voters, that is, a bribing pattern of (5, 5, 5). It is clear that Friday could successfully respond to this extreme effort by devoting just shghtly more than the amount spent by Crusoe to a bare majority of the voters. For example, he might use a pattern of (5.5,5.5, 0) which yields a two-to-one majority among the triumvirs and requires only eleven units of wealth to oppose Crusoe's use of fifteen. One might ask whether Crusoe's strategy was a reasonable one under the circumstances. A bettjer strategy might appear to have been allocating his wealth equally between a bare majority of the voters, for example (7.5, 7.5, 0). Were such the announced pattern of Crusoe, Friday could have responded with bribes of (0, 8, 1) which yields an even more economical majority for Friday requiring only nine units of wealth. If Crusoe were similarly able to revise his bribing pattern in light of Friday's intentions, he may counter any pattern of bribes offered or intended with a distribution of bribes requiring fewer resources. For example, Crusoe might respond to Friday's pattern of (5.5, 5.5, 0) with a pattern of (6, 0, 1), and Friday to this new pattern with (0, 1, 2), and so forth. Each iteration uses fewer resources for competitive purposes than the round before it.^ Thus we have the rather surprising result ^To determine the least-cost majority: (1) order the bribes of ones competitor from lowest to highest bribe; (2) offer a bit more, rf, than is currently being received to a bare majority drawnfromthe low end of the spectrum determined under (1); (3) offer nothing to all others. If the sum of these bits, d, can be less than the sum of the bribes received by the new minorit>', then the new majority can be formed using fewer resources than those of one*s competitor (if this amount is greater than zero). If d can be arbitrarily small, this will always be possible.

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that under majority-rule arbitration a series of response and counterresponse de-escalates. In the Hmit virtually no resources will be devoted to the competitive activity. However, since the strateg)' (0, 0, 0) is never a winning pattern of bribes, no equilibrium level of competitive activity will be reached. Although there is no dominant distribution of bribes or even level of bribes that will ultimately be received by the triumvirs, there is a clear tendency for the level of bribes to be small (indeed vanishingly small) if time is available for competitor interaction to occur. (If e is the minimum effective bribe, there is a tendency for equihbrium strategies to converge toward patterns resembling (e,e 4- (i,0), where d is the smallest amount that can be added to e so that a triumvir perceives e -^ d to be greater than e.) Majority-rule arbitration changes the nature of the game faced by competitors. Although wealth is still to be allocated on a "winner take all that remains** basis, the winner is no longer simply the one who devotes the larger amount of resources to the game of influence. Instead, he is the one who successfully brings a majority coalition to vote, an activity that requires resources, yet is not determined by sheer level of competitive effort. Had competitors been constrained to offer equal bribes to all triumvirs, or had the triumvirate used a unanimous-decision rule instead of majority rule, the original format could have been preserved, and so would the tendency for competitors to exhaust their resources in the process of competition. However, the tendency for majority-rule arbitration to defuse competitive games of influence seems to accord well with the observations of Gordon Tullock and Anne Krueger regarding the relatively larger amounts of rent seeking that occurs in the third world vis-a-vis the western democracies,^ and so remains an important variant of "winner take all that remains" games. The efficiency characteristics of majority-rule arbitration parallel those of single-man arbitration. Because of the single-good character of the situation analyzed, any outcome will be Pareto-optimal given the existing institutional arrangements. Comparing the majority-rule outcomes with those under anarchy reveals that majority-rule arbitration is Pareto-superior to the game of dominion under anarchy. Both competitors are better off since comparative efforts tend to de-esca^ Gordon Tullock, "Rent Seeking as a Negative-Sum Game*'; Anne O. Krueger, "The Political Economy of the Rent-Seeking Society/' chapters 2 and 4 in this volume.

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166 Subsequent Contributions to Theory and Measurement late, which allows more wealth to be used for consumptive purposes. The triumvirs may be assumed better oflP, although this inference depends, upon their unmentioned alternatives at home. A comparison between single-man arbitration and majority-rule arbitration cannot be made without much additional detail, except to point out the obvious, that the competitors are better ofiF under majority-rule arbitration. Analysis of indirect means of influence and the existence of external benefits and/or costs of the process of competition is as before. If sufficient external benefits are generated to more than offset the inherent losses of indirect transfers, indirect means of competition may be Pareto-superior to direct forms of influence. If not, indirect means of influence will be Pareto-inferior to competition utilizing direct means of influence. "Proportional Share'' Games under Nature At this point we would like to move fi-om changing the institutional setting of the game to a change in the format of the game itself Clearly not all competitive games are of the "winner take all that remains" variety, nor even all important games, though clearly they are an accurate description of a great many important areas of competitive activity. A second general type of game, what might be called the "proportional share" game, is equally important and often an alternative game form to the "winner take all" variety. In a "proportional share" game, competition takes place over the division of a scarce reward, where the share of the prize received depends upon the relative size of individual competitive efforts. One who has expended more resources than another wins not all the reward, as in the previous game format, but a proportionately larger share according to the extent of his efforts relative to others. The particular form of interest here is the case where an individual who devotes twice as much "effort" as another receives a share of the reward twice as large as the others. In an N-person version of such games, the jth person s reward can be written as (Ej/ X £f)R, where R is the total reward available and Ejis the level of resource investment devoted to this competitive activity. The important new feature of this format is that a non-zero investment in competitive resources guarantees one a non-zero share of the reward.

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FIGURE 9.2

Friday

Crusoe

Full Attack

Half Attack

Inaction

Full Attack

0,0

7.5,0

15,0

Half Attack

0,7.5

7.5,75

22.5,0

Inaction

0,15

i 0,22.5

i 15,15

No level of competitive investment would assure this in the "winner take all that remains" format developed above. Consider the changed circumstances that Crusoe and Friday would face in the original island setting if the game is changed from one of all-or-nothing dominance to a "proportional share*' version. Suppose that the wealth may be used as before to attack or defend but that this time the circumstances allow one to influence the size of ones share of the islands wealth vis-a-vis the other's rather than to estabhsh island-wide dominance. The game form analogous to the two-by-two matrix of the game of conquest is a three-by-three matrix. Such a matrix is shown in figure 9.2, Note that in this game there is 2L dominant strategy for each and hence a Nash equiUbrium. The dominant strategy for each is to employ half the resources at his disposal to competitive activity. Thus, in this discrete form of the "proportional share" game, only half the island's resources are wasted by strictly competi-

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Roger Congleton 168 Subsequent Contribtitions to Theory and Measurement tive uses. The game outcome in eflFect ratifies the individual s decision to save half his resources for consumption. Because few^er resources are wasted in this game than in the anarchistic version of the "winner take all that remains'' games discussed previously, a move to the former from the latter would be a Pareto-superior move. The three-strategy game, however, is not as natural a game as one in which the competitors can vary their competitive efforts nearly continuously between no effort and complete commitment. A player's task in such games is, of course, to maximize his return to competitive investment P(X), which in this game format may be written as:

(1)

PW = Y T Y (« - ^ - ^)

where X is the competitive effort of player X, Y is the level of effort of other players, and R is the resource base of the game. In our illustration X may be thought of as Crusoe's competitive use of wealth, Y as Friday's, and R as the total wealth of the island to be allocated by the game outcomes and strategies. Differentiating with respect to X and setting the partial derivative obtained equal to zero yields: -X- - 2XY - Y^ -f Yfl . (X + Y)^ • ^•'^ Solving this for X yields: X = - Y ± VYR, (3) Since X, R, and Y must be real and greater than or equal to zero, one of the possible solutions is eliminated, leaving as our result X = -Y + V Y R . (4) To determine whether or not this function represents maxima rather than minima of the payoff function, one must establish that the second derivative with respect to X over the relevant range is negative. f R ^ - 2 X - 2Y _ (-X" - 2YX - P -f YR) . .

ax^

(x-f Y)^

(x + Yf

'

^^

Placing over the common denominator and collecting terms yields: ^ = -2YR . . dX^ {X + Yf ^^ Since X, Y, and R must be positive, - 2YR/(X + Yf will be negative. Hence X = ~Y + VYR"represents maxima of the payoff function over the range of game strategies allowed. It specifies Crusoe's optimal

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level of competitive investment for a level of Friday's competitive effort, Y. There is no dominant strategy for Crusoe, since his best strategy depends upon the level of Y chosen by Friday. Hov^^ever, the lack of a dominant strategy does not preclude the existence of equilibrium strategies, strategies that, once chosen, cannot be improved upon without a change in strategy on the part of one s competitor(s). To determine whether such strategies exist, one must see if there is a strategy pair that simultaneously maximizes the competitive revi^ard to each. Since the games reward structure is sraimetiical with respect to Friday and Crusoe, Friday s reward-maximizing strategy can be determined, as was Crusoes, and would be described by the foUowdng equation: Y = - X + VXR, (7) Substituting this representation of Friday s strategy into Crusoe's optimal strategy function yields: X = - ( ~ X + VXR) + V{-X + VmjiR . (8) Solving this for X yields: X = R/4 , (9) Crusoe's half of the equilibrium strategy pain Substituting this value into Friday's optimal strategy function yields Y = R/4, the same strategy investment in competitive activity chosen by Crusoe. Recall that R in this game refers to the total amount of resources available. If R were distributed equally at the start of the game, as assumed in our illustration, an R/4 investment in competitive effort means that each individual will invest half the resources at his disposal. This was of course the dominant strategy and equihbrium in the discrete three-bythree version of the game developed above. However, the lack of a dominant strategy means that in a one-time run of this game, strategy must be a matter of judgment rather than optimization alone.. The proper strategy depends upon both the optimal strategy function and one's judgment of the hkely strategic choice of one's competitor(s). However, if there is a chance for this game to escalate—that is, for one to gather information about one's competitor and he about one's reaction to this information and so on—this game has a tendency to escalate and to converge toward the equilibrium developed above. The illustration of the optimal strategy functions for Crusoe and Friday

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Roger Congleton 170 Subsequent Contributions to Theory and Measurement FIGURE 9.3 o

Fridoy*s Optimal Strategy Function CD

CL

E o o

> o

Crusoe's Optimal Strotegy Function

Q

5

_i

Level Of Wealth Devoted To Competition By Friday

m figure 9,1 allows us to make this point clear with the help of an illustrating converging series. The general shapes of these curves will be the same regardless of the size of R, rising from zero to R/4 and falling thereafter back to zero at R. In our example, R is thirt>', and feasible strategies for each participant are limited to half the resources of the game, though this does not really matter so long as each competitor has at least one-quarter of the resource total. Selection of any strategy will, if naively reacted to by adopting the optimal strategy called for, lead to a series of strategic adjustments that converge to the strategy where each invests half his own resources (one-quarter of the game total).^ Referring to figure 9.3, or to the two optimal strategy ftmctions, one should notice that an optimal strategy never calls for a level of competitive investment greater than 7.5 (R/4 in general). Suppose Friday begins or plans to begin with a single unit level of competitive investment. Upon learning Fridays intention, Crusoe would respond * Notice that if Y = 0, the optimal strateg\' function is not defined. However, it is apparent that some non-zero level of competitive eflfort by X will, according to the payofi* function (equation [1]), be better than similar inaction. Note that the payoffs to mutual inaction is H/2. Given inaction on the part of ones competition (Y = 0), X > 0 would yield R - X, which is greater than R/2 if X is less than R/2.

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with an investment of approximately 4.5 units* Friday, learning of Crusoe's intentions, would respond with approximately 7.2 units, whereupon Crusoe would counter vnth about 7.5 units. Friday would reciprocate vdth 7.5 units. This being an equilibrium strategy pair, no further movement would occur. Both Friday and Crusoe invest half their initial wealth in the purely competitive activity. Regardless of the initial choice (so long as it is less than R), the strategies will converge smoothly to the equilibrium strategies of R/4, R/4 (7.5, 7,5 in this case). Thus, while the "proportional share" game is one whose properties are not as intuitively obvious as those of the other game format, the two formats do have some similar properties. They are both games ofjudgment that, in the simple recurring adjustment to one's competitors' competitive play or intentions, lead one to an equiHbrium situation captured by their respective simpler discrete versions. The important diflFerence is the extent to which the island's resources are wasted in competitive activity. The wasted resources of the "proportional share" game amount to only half those of the "winner take all" game (in the two person cases examined). Clearly Friday and Crusoe, oflFered a choice of games, would prefer the second wdth its less competitive environment. Proportional Sharing with an Arbitrator Open to Influence As in the "winner take all that remains" games developed earlier, "proportional share" games are often influenced by third parties and/or generate effects affecting third parties who are not directiy involved in the competitive arena. In this section of the paper we are primarily concerned with the effects of "influencers," since other third-part\' effects will tend to have properties mirroring those of our earlier discussion. The first case examined will once again be the extreme case in which a single individual has the power and/or authority to allocate completely the resources of the competing parties. As before, our interest lies in games with formats similar to the one developed under the anarchy setting. Suppose, once again, that a neighboring chief becomes aware of the wastefiil nature of Crusoe and Friday's competitive system of allocation and comes to the island to distribute them by other means.

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172 Subsequent Contribtitions to Theory and Measurement Once again, if his judgment is beyond the influence of both Crusoe and Friday, there will be httle value gained by attempting to affect the chiefs decision. For example, a commitment to distribute the wealth of the island equally might nearly accomplish this, though efforts to obscure the extent of the island's wealth (in effect competing with the chief) would, of course, follow to some extent. However, most societies recognize that some individuals merit a greater share of the pie than others, though there is a wide range of variation in the nature of merits considered relevant for the allocative decision. If the chief decides to allocate wealth proportionate to demonstrable merit, the game format faced by Friday and Crusoe at the onset remains unchanged, though the competitive means change from weaponry to demonstrations of the relevant category of merit. If merit is simply a matter of "gifts'^* rendered to the chief, the competitive game becomes a game of transferring wealth to the chief. The resources devoted to strictly competitive activity would be simply transferred to the chief, who, in view of the resulting demonstration of equal "merit** by Friday and Crusoe, would distribute the remaining wealth equally. The result so far as Crusoe and Friday are concerned is unchanged; each has wasted half his resources on competitive activity. However, since these resources have been transferred to the chief intact, there is no social waste to this competitive process. The result is Fareto-optimal and Pareto-superior to the results under anarchy if the wealth received by the chief exceeds the value of his opportunity cost. However, considerations of merit are rarely such that direct transfers are allowed to carry all the weight of evidence. Indirect transfers, as before, may create value for the chief below that possible by direct transfers. Hence they tend, other things being equal, to generate a social loss when viewed from the direct transfer alternative. As before, the ingredients of merit may demand that Crusoe and Friday bestow benefits to individuals other than the chief, which may in total moderate or overcome the inherent inefBciencies of indirect means of influencing the chief. One might require that wealth be used effectively to generate more wealth (inventiveness) for the chief s tribe or to demonstrate benevolence or altruism, which might lead to an alternative Pareto-efficient outpome, or, in the extreme case, even one that is Pareto-superior to the case of direct bribes (for example, a successful innovation may dramatically improve techniques of creating or using

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wealth). As developed before, indirect means of payment or influence may generate outcomes worse or better than the apparent inefficiency found by focusing upon the chiefs loss of weFare. It should be noted that, whereas the competitors are each better off in this competitive format, the chief was much better off under the other. His transfers within a "proportional share" game between two people are only haff what they were under the "winner take all'' format. Thus, if allowed to pick the game format played by those under his authority, he would naturally prefer the "winner take all that remains" format over "proportional share" games, other things being equal. ''Proportional Share" Gaines and Majority Rule "Proportional share" games are not easily adapted to majority-rule allocation. As was true of "winner take all" games, this format cannot be maintained intact in a shift to majority-rule arbitration. However, this feature of majority-rule institutions is itsetf of interest and will be examined next. The most straightforward way of estabHshing something approximating a majority-rule version of the "proportional share" game is to allow voting on the same issue of the proportion of the reward due each competitor. In the two-person case of Friday and Crusoe on the island, this amounts to determining the ratio of rewards received by Friday to those received by Crusoe, a single number greater than zero. When a single number is to be chosen from a continuum by using a majority-rule decision-making procedure, the number chosen will be that preferred by the median voter (tf the preferences of the voters are single peaked over the domain voted upon). For this reason, the attentions of the competitors will be directed toward generating a favorable median voter "opinion" on the matter of their relative merit. If the distribution of voters is stable, so that the same individual would continue to be the median voter during any election count, this voter would receive virtually all the attention of the competitors. His position would be very much like that of the single arbitrator developed above. However, if influence is possible at all, it is likely that more than one person s vote can be influenced. In the extreme case where voter preferences are, in effect, determined by the relative "bribes" received from Crusoe and Friday, every distribiition of bribes will have a voter distribution associated with it» The median voter iSy in

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Roger Congleton 174 Subsequent Contributions to Theory and Measurement such instances, entirely determined by the strategic use of wealth employed by the competing parties. In this game, the payoff received by Crusoe if he allocates M^ units of wealth to the median voter and S, to the other voters is: ^^^') = i r ^ V (fl - M. - M, - S, - SJ

(10)

where M^ is Friday's expenditure on the median voter, R is the total resource base of the game, and Sy is the total ejqpenditure by Friday on members other than the median voter. The distribution of M,,, M^, Sj,, S, across voters estabhshes a particular median voter. Since transfers to voters other than the median voter subtract from the size of the reward to be distributed but do not add to the size of ones share (apart from helping to determine a particular median voter), both competitors will make these auxiliary payments as small as possible consistent with generating the sought median voter. As a first step, these auxiliary payments should be limited to just half the voters (the half receiving the smallest sum from one s competitor). If this strategy is adopted by both parties, then S^ and Sy can be very small indeed, approaching zero. All but the median voter will, under this scheme, receive nothing from one competitor and some vanishingly small bribe from the other. Their votes would thus tend to favor granting all the remaining reward to the competitor offering the non-zero bribe. The proper allocation of effort toward the median voter can be determined by maximizing one's payoff with respect to M,. Taking the partial derivative of the payoff ftmction given above with respect to M, and setting the result equal to zero yields: 0 = fl -- 2M. - M. - S. - S„ _ MXR-M^-M^-S^-'S.) (M^-^My) {M^-^Mf

. '^ ^

Solving for M, yields: M, = -M, ± VMyiR-S^-Sy).

(12)

Since M, must be no less than zero, this ehminates one of the possible descriptions of M^, leaving: M, = -My 4- VMyiR-

S, - Sy).

(13)

Notice that this formula is very similar to the optimal strategy fimction developed in the original "proportional share'* games (equation [4]). However, this strategy function estabhshes the proper level of re90

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sources to devote to a median voter, given (1) a particular voter distribution resulting in a particular median voter and (2) values for S„ S^, and My. Like the earlier function, this optimal median voter bribe formula depends critically upon the level oflfered by one s competitor, here Friday. Like the earlier fimction, this function has a maximum, a maximum that approaches R/4 as S, and Sy approach zero. Thus the largest bribe that it is rational to oflFer the median voter will tend to be less than half Crusoe's initial endovmient. Hov^ever, there is no tendency for strategic revision to generate convergence to that maximum, as was the case in the other two institutional settings. In this setting a competitor can often achieve better results and use fewer resources by generating a new voter distribution and a new median voter. Although it is not possible to determine any particular equilibrium distribution of bribes between the voting arbitrators, since every such distribution can be beaten by an alternative, it is possible to characterize the equilibrium levels of competitive effort. An optimal strategy will entail giving the minimum effective bribe, e, to half the arbitrators, zero to the other half, and some amount to the median voter dictated by equation (13). In the case of a triumvirate, one triumvir will get 0, one will get M^ and one will get e. Given such a sb-ategy announcement by Crusoe, (0, M^, e), Friday could respond with a strategy of (^, 0, M^), which makes the bribing ratio of the third triumvir the median apportionment of the remaining wealth, namely MJe (the others being all to Crusoe or all to Friday). Crusoe could respond with (M« e, 0), which makes the first triumvir the median voter, Friday with (0, Myy e\ and so on. If M^ and M, eventually reach stable levels, then there will be a constant distribution of effort or intended effort across triumvirs. It should be clear from this discussion that such an equihbrium will depend upon the resource base of the game, R, and the minimum resource level required to influence a voters decision, e. Notice that the series of bribing strategy adjustments entails choosing the other competitors e level of effort as the most promising candidate for median voter and relegating the former median voter to an extreme. Since eis the median voter effort of Y, and Myy the level paid to the former median, becomes S^y and S, is e, the new median voter bribe for Crusoe becomes: M, = ~ e + Ve{R - M,j - e). (14)

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Because this game is symmetric, a similar function describes the median voter strategies of Y, namely: M, =- -e + Ve{R - M, - e) . (15) Stable levels of competitive effort are implied if both equations (14) and (15) are simultaneously satisfied, as is the case in M, = -e + VeiR - [-e + Ve(R ~ M, -"!)"] - e) . (16) Squaring, collecting terms, squaring again, and collecting terms yields: 0 = M/ + 4eM/ -f (6e- - 2eR)M/ + {Se" - 4em)M, -h 2e' - 3e^R + e^R-. (17) Equation (17) can be factored into a product of quadraticis and the four roots then established using the quadratic formula: 0 = (M/ + 3eM, + 2e^ - eR) (M/ + eU, -he'- eR), (18) the roots of which are: r — •X

r rr

-3e ± Ve^ + 4eR 2

y

(19a)

-e ± V4eR - 3 g

2 . ''*' Although all these roots wiU satisfy the conditions of equation (17), not all will satisfy those of equation (16), nor will those roots that satisfy (16) necessarily satisfy the constraint that M^ be greater than or equal to zero. The single root that does satisfy these conditions characterizes equihbrium strategies in terms of e and R.

M^=-^e + V7Tl^

(20)

This equation indicates that as the minimal effective bribe approaches zero, so does the equihbrium level of competitive investment. Contrariwise, as it gets relatively large, so does the total resource conimitment to the competitive process. A similar equation can be found for My impl)dng that My will equal M, at the resource equilibrium. Given a period for iterative adjustment of status, equation (20) imphes that the competitive effort generated under majority-rule arbitration may be substantially larger or smaller than that which occurs under a single arbitrator. However, our tacit assumption that e is relatively small suggests that the total competitive efforts will be smaller

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than those which would occur under a single arbitrator. For example, if R is 30 units of wealth and g is 0.1 unit of wealth, then M, and M,j are each approximately 1.58 units of wealth. In addition to expenditures on the median voter, there are the resources devoted to establishing a particular median voter, in this case 0.1 units of wealth each. The total competitive effort of both competitors is twice 1.68 units of wealth, or 3.38 units of wealth, which is, of course, far less than the 15 units of wealth used in our illustration under a single arbitrator. (Had e been 4 units instead, this total would have been approximately 18.27 units of wealth.) Recall that in* similar circumstances, a "winner take all that remains" format under majority-rule arbitration tended toward greater de-escalation. That is, levels of competitive effort tended toward a smaller hmit. Under a triumvirate government and where the minimum effective bribe is e, the bribes tended in the limit toward 4e, 0.4 units of wealth, if e is 0.1. Thus majority-rule arbitrators may tend to receive, on average, higher transfers if the game format is "proportional share*' rather than "winner take all that remains" (if a period is allowed for interaction and adjustment), and arbitrators may be expected to prefer the former game format to the latter, even in the complete absence of any concerns for equity. Conclusion and

Summary

This paper has explored some of the effects that institutions and rules governing the disposition of competitive rewards can have on both the quantity of resources used for competitive purposes and the extent to which such uses may be wastefal or beneficial. While it is not news that institutions matter, it is useful to know how institutions may affect levels of competitive effort and, more importantly, the land of competitive efforts engaged in. Further, the extent to which competitive effort may be socially wasteful is clearly another important concern. Light has been shed on this issue by our efforts to trace the course of competitive resources: are resources merely transferred, are they transformed to less valuable assets, and, tf so, does the process of transformation generate any spillover benefits or costs? Answers to these questions will, by and large, determine whether some particular competitive instrument is efiScient or not. It has been shown that in-

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stitutions can, at least in highly simplified circumstances, create incentives for individuals to use less v^asteful or even productive competitive means. Under both game formats, it proved possible to shift from competitive methods that consumed resources to ones that merely transferred them (e.g., the shift from anarchy to a single arbitrator changed the competitive means from warfare to bribery and generated a Pareto-superior state). A second possible institutional device was also illustrated: changing the dispensation of the game reward. One can change the level of competitive effort that vidll be called forth by changing the way in which that effort increases one s share of the reward. A shift from a "winner take all that remains** rule to a "proportional share" rule decreased the level of competitive effort under both anarchy and singleman arbitration, and, though the results of such a shift under democratic arbitration are less clear, there seems to be a tendency for such a shift to increase the level of competition effort. Although changes in distributive rules are not always possible—that is, some game outcomes are mutually exclusive and so necessarily of the "winner take all that remains" variety—it is clear from the cases developed that, when possible, such changes can have substantial results on the level of competitive effort called forth. Thus one can affect both the level and Idnd of competitive effort that will take place through institutional measures. However, it has not been shown that any "refinement" will be to the advantage of either the competitors or society as a whole. In general the interests of arbitrators and those of the competitors are juxtaposed. In each of the circumstances examined arbitrators benefit from additional competition whereas competitors emerge poorer from additional competitive requirements. Thus arbitrators have an incentive to increase the level of competition, even when such competition may be judged ineflBcient from a Paretian standpoint. In the case of a single arbitrator it proves to his advantage to shift from the proportional-sharing rule to a "winner take all that remains" rule of distributing game rewards, regardless of whether the competitive means being used are socially eflBcient or not. Greater levels of indirect transfers surely are to such an arbitrator s advantage. It seems apparent that contriving circumstances to promote competition, even where none is inherent in the problem at hand, would be advantageous to wpuld-be

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arbitrators. Policies that artificially limit the number of possible "winners" clearly will be in the interest of those with the power to decide who will be among the victorious. Further, arbitrators have an interest in avoiding situations that prevent aflFected parties from influencing their judgment. While it is clear that institutions play an important role in generating the land and level of competitive eJBPort that will exist, it should also be clear that transforming institutions is itself a competitive game. Both scarcity (only one institution can be adopted) and eflBcacy (the ability to transform or adopt institutions) are clearly present. No particular facet of behavior can be both iDegal and legal in the same sense at the same instant. Nor can there be but one collective decision-making rule, however complex, embodied in explicit or implicit constitutions. This is not to say that Pareto-superior changes in institutions are impossible, but that such changes, even if possible, will tend to be accompanied by at least the normal conflict of any bargaining process. Our analysis of such games suggests that there may be less competitive waste under democratic institutions than under anarchy or single-man rule, though there may be on similar grounds more "improvements*^ forthcoming. Constitutional revision represents but one extreme of the competitive arena. In principle, competitive incentives exist at all levels from the household to the firm to the political and judicial levels of action, and competitive efforts should be expected to accompany these incentives. Because the process of competition may be expected to utilize resources at every level, economic welfare analyses clearly require an examiuation of these processes as well as the end states generated. The costs (or benefits) of "getting there" are bound to have an effect upon the desirability of the destination. The possibility of adopting rules or customs at each level that affect both the costs and the directions taken clearly represents one essential way of improving the voyage.

95

1.1.1.4 Risk-averse rent seekers and the social cost of monopoly power Tlie Economic Journal, 94 (March 1984), 104-110 Printed in Great Britain

RISK-AVERSE RENT SEEKERS AND THE SOCIAL COST OF MONOPOLY POWER Arye L. Hillman and Eliakim Katz There is now quite general recognition from beginnings by Tullock (1967) and developments by Posner (1975) and Cowling and Mueller (1978) that the social cost of monopoly power encompasses the resources expended by individuals seeking to become the beneficiaries of monopoly rents. However, since the activity of rent seeking is generally not observable, direct estimates of resources expended in quests to acquire monopoly power are usually impossible to come by.^ As a consequence, the indirect approach of taking the observed value of monopoly rents as indicative of the unobserved value of the resources expended in rent seeking is quite generally adopted in approaches to evaluation of the social cost of such activity. When a particular rent is biddable and the resources used in rent seeking have positive shadow prices, competitive rent seeking by risk-neutral rent seekers results in complete rent dissipation and the full amount of rent reflects a social loss. In some instances rents are not biddable but are preassigned, as for example where the premium-carrying import licences and revenues analysed by Krueger (1974) and Bhagwati and Srinivasan (1980) are preallocated. Then the incentive to expend resources in rent seeking is clearly absent. Also, the general theory of directly unproductive profit-seeking activities as expounded by Bhagwati (1982) makes clear that because of intrinsic second-best considerations resources used in rent seeking may not have positive shadow prices, implying that individuals' quests to secure biddable rents need not always entail socially wasteful activity. When neither of these qualifications are pertinent, so that biddable rents are sought using positively-valued resources, rent dissipation will still nevertheless be less than complete if rent seeking is not perfectly competitive: Tullock (1980) has shown how, in small-numbers cases, substantial portions of biddable rents may remain undissipated as a consequence of game-theoretic considerations. Suppose that perfectly competitive rent seeking takes place in quest of a biddable rent, with socially positively valued resources being expended. Then there remains the question of the effect of attitudes to risk on rent dissipation. Uncertainty is intrinsic in a competitive quest to attain a biddable monopoly rent, since the rent by its nature accrues indivisibly to the one rent seeker who is to be ultimate monopolist. In face of this uncertainty rent seekers may quite reasonably be risk averse, and, if this is so, individual rent seekers will allocate

• We thank Jagdish Bhagwati, Peter Neary, and Gordon Tullock for helpful comments. * As Tullock (1967) pointed out: *The problem of identifying and measuring these resources is a difficult one, partly because the activity of monopolizing is illegal/ (p. 49, in Buchanan et al, 1980}. [ 104 ]

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less to a particular rent seeking quest than the expected value of the gain from the activity. However, although individual allocations to rent seeking are reduced as a consequence of risk aversion, in a competitive environment a large number of individuals is nevertheless each allocating some resources (perhaps e) in the quest to obtain a biddable rent. So to what extent, then, will a biddable rent be dissipated by competitive rent seeking when rent seekers are risk averse? Should competitive rent dissipation be revealed to remain more or less complete notwithstanding risk aversion on the part of rent seekers, then the procedure of equating the observed value of monopoly profits to the value of the resources expended in the quest for monopoly power would be provided with some justification. But the procedure of taking observed monopoly profits as a guide to the social cost of monopoly power due to rent seeking would be called into question if as a consequence of risk aversion rent dissipation were to be substantially reduced. To address this issue of the effect of risk aversion on the appropriateness of taking the value of a particular rent as indicative of the value of the resources expended in seeking that rent, we derive in Section I a limiting expression describing competitive rent dissipation when rent seekers are risk averse and proceed to show how rent dissipation is sensitive to both the degree of risk aversion and to initial wealth. The limiting expression is valid only for small rents. In Section II, by introducing a particular form for rent seekers' utility functions (logarithmic utility), we are able via direct computation to provide illustrative values for rent dissipation when monopoly rents are large. A brief concluding summary is then presented in Section III. The focus of our analysis will be on competitive rent seeking. However, an appendix demonstrates how risk aversion can be introduced into the strategic rent seeking settings investigated by Tullock. I. COMPETITION FOR SMALL RENTS

We shall consider a market which is monopolised and which yields, as a result of monopolistic profit maximisation, a monopoly profit of value X.^ The right to monopoly power which allows this monopoly rent to be secured is taken to be biddable, and we suppose that any ofn firms might potentially secure the rent. Of the nfirms,only one can be the ultimately successful monopolist. No firm is assured of success, and hence anyfirmseeking to acquire the monopoly power in question confronts uncertainty. Given this uncertainty, let eachfirm'sbehaviour be as described by the Von Neumann Morgenstem axioms,^ withfirmshaving a common utility function U{,) and common initial wealth A, * This rent, given as the outcome of monopolistic optimisation, is assumed to be independent in size of the resources expended in seeking to acquire monopoly power. It should be noted that this does not imply that the ultimate monopolist will not attempt to increase profits by rent-seeking activity after monopoly power is attained. What is implied is that the value of the monopoly profit X already encompasses the anticipated results of such further rent-seeking activities. * For a description of these axioms, see for example Hey (i979)» Chapter 4.

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The sole means available to any firm of increasing the likelihood of its success in the quest for the monopoly rent is an increase in its allocation to rent-seeking activity relative to its rivals. There is no favouritism in the rent-seeking quest; an increase in the allocation to rent seeking by any particular firm increases its likelihood of success to the same extent as does a corresponding increase in the allocation to rent seeking by any other firm. Denoting by Uj the probability that firm J will be the ultimate monopolist and by P^ the value of the resources allocated to rent seeking activity by firm i, we posit n,.=^.(/>,A,...,/^,...,i>j,

(I)

where^.(.) is concave, differentable, and dfJdP^ > o, ^ / ^ / ^ < o {j i^j). W i t h ^ ( . ) identical for all firms, in equilibrium each firm allocates the same sum in the quest to attain the monopoly rent. If rent seeking is a competitive activity, then, regardless of the particular form which the function^(.) might have, the value of the resources expended by each firm in rent seeking is implicitly given by P = i^. (j = i,...,n) in E(f/) = [ ( n - i ) / « ] f / ( ^ - P ) + ( i A ) f / ( ^ - P + Z) = [/(^).

(2)

This is the basic rent seeking equilibrium condition, and it states that the expected utility of the marginal firm entering rent seeking activity is equal to the utility to be derived by that firm's maintaining its initial wealth with certainty by refraining from rent seeking. Now, taking a Taylor Expansion (with a remainder), we obtain the expressions U{A-^P) = U{A)-PU'{A)

^^U\A)

^^U"\A),

(3)

and

U[A^-{X^P)] =

U{A)^{X^P)U'{A)^'^^^P^U''{A)-\-^^''J^''u'"{A) (4)

where A = {A-aP) and A = [ ^ + / ? ( Z - P ) ] , with o < a < i and o < /? < i. Substituting (3) and (4) into (2) yields

(Z^P)t/- + ^^^^^V--H-^^'^^^V^^(J) •-(«--i)P^^ 4-^"""^^^^V^ -^^^V'''(l) = o

(5)

where U' and U" in (5) are defined at A. From (5) one can then readily establish that limP=o. (6) That is, as the number of firms becomes very large, the amount which an individual firm is prepared to expend in the quest for the rent approaches zero.^ However, although the small likelihood of any particular firm's success means * Divide through (5) by n and then take the limit as n approaches infinity. Of the roots of the resuhing equation in P, the relevant solution is P = o.

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that rent dissipation by any one firm is small, this cannot be taken to imply that overall the total rent dissipated will also necessarily be smalL Denote by S the total rent dissipated by all rent-seekers (i.e. S = nP). Then, using (5) and (6) we have

Hence, if X is relatively small, we obtain the limiting expression for rent dis^^P^^^^^'

,. S kR lim^=i—--,

,«, (oj

where R is the coefficient of relative risk aversion and k = X/A. Not surprisingly, rent dissipation declines as risk aversion increases. However, rent dissipation also decreases as the value of the rent declines relative to initial wealth. Suppose for example that rent seekers exhibited unitary relative risk aversion.^ Then, if the monopoly rent as a percentage of initial wealth were i %, (8) indicates the limiting percentage of monopoly rent dissipated to be 99-5 %. On the other hand, i![X/A = 5 %, 97-5 % of the rent is dissipated; when X/A = 1 0 % , 95 % is dissipated; when X/A = 20 %, 90 % is dissipated. These outcomes suggest that, at least for relatively small rents, the value of a rent observed to accrue as a consequence of monopoly power approximates the value of the resources expended in the course of competitive rent seeking in quest of that rent. II. COMPETITION FOR LARGE RENTS

The limiting expression (8) is restricted in its application to instances where rents are small. When rents are large, the evaluation of rent dissipation requires solving the equilibrium rent seeking condition (2) directly for a particular specification of rent seekers' utilities. Let us assume that utility is logarithmic. The coefficient of relative risk aversion is accordingly unity, and is equal then to that in the small-rent examples based on (8). Resource dissipation outcomes for logarithmic utility are indicated in Table i for rents ranging from 10 % to ten times initial wealth,^ and with the number of rent seekers ranging from 2 to 1,000. We observe that in the small-rent cases where the rent is 10-20 % of initial wealth, rent disspation as indicated by the limiting approximation (8) is approached by competition among 50 rent seekers. Since increasing the number of rent seekers beyond 50 does not affect rent dissipation in other cases, this number of rent seekers also appears to suffice to approximate the competitive limit for larger rents with respect to which direct comparisons with (8) cannot be made. Table i points to a revealing relationship between the size of the rent sought and the extent to which a rent is dissipated: as the size of the rent increases, smaller » A choice of unity for the coefficient of relative risk aversion is not entirely arbitrary. Arrow suggested (1970, p. 98) that one would expect relative risk aversion *to hover around* unity. * Since equation (2) is homogeneous of degree zero in A, P, and X when the utility function is logarithmic, what matters is not the absolute sizes of ^ and X but only the ratio X/A,

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proportions of the rent are dissipated. For example, rent dissipation is around 70 % when the rent is equal to initial wealth, but dissipation falls to just over a third at five times initial wealth; and at ten times initial wealth only a quarter of the rent is dissipated. This falling away in rent dissipation as the size of the rent increases is not to be attributed to limits imposed by the nonavailability of resources to be dissipated in rent seeking. In particular, in the competitive limit, there are infinitely many individuals each with strictly positive initial wealth which can in part be allocated to rent-seeking activity. Table i Competitive rent dissipation, logarithmic utility, A = loo n A

X/A

2

3

5

O'lO 0-20 0*50

98 95 88 76

97 94 85 74 34

96 93 83

21

I-OO

500 lO'OO

32 18

10

50

100

1,000

96

95

95

95

91 81 70

91 81

91 81

72

92 82 70

35

36

36

69 36

69 36

22

23

24

24

24

III. CONCLUDING SUMMARY

Evaluations of the social cost of monopoly which encompass the resources expended by individuals seeking to acquire monopoly power yield measures of social loss which are high relative to outcomes when rent seeking is not accounted for. Indeed, Littlechild (1981) has suggested that the Cowling and Mueller (1978) measures of the social cost of monopoly power which make allowance for the role of rent seeking are inordinately high.^ A central question relating to the indicated high social cost of monopoly power concerns the extent to which it is appropriate to take observed monopoly profits to reflect the value of resources expended in seeking to acquire monopoly power. Presupposing that resources expended in rent seeking have positive marginal social value, competition among rent seekers for a biddable rent dissipates that rent in its entirety if rent seekers are risk neutral; but one cannot presuppose that, when confronted with the intrinsic uncertainty associated with competing to acquire monopoly power, rent seekers will be risk-neutral. Acknowledging that rent seekers may be risk averse, we find that competition among risk-averse rent seekers appears to result in substantial rent dissipation for small rents; but, on the other hand, when rents are large, competitive rent dissipation by risk averse rent seekers may be far from complete. Accordingly, for small rents, risk aversion does not appear to compromise the presumption that the observed value of a monopoly rent can be taken as an approximate guide to the unobserved value of the resources expended by individuals seeking to become the beneficiary of the rent; but not so for large * See also Cowling and Mueller's reply (1981) to Littlechild.

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rents, where the value of the resources expended in competitive rent seeking may deviate substantially from the value of the rent to be obtained by the rent seeker who is ultimately successful in securing the rights to monopoly power. Bar-Ilan University^ Israel, Date of receipt offinaltypescript: August igSj APPENDIX

When rent seeking is not competitive, dissipation of a biddable rent is incomplete even if rent seekers are risk neutral. The non-competitive or small-numbers case has been analysed by Tullock (1980) under the assumption of risk neutrality. In this appendix we provide some illustrative values of the extent of rent dissipation which might be expected to occur when in small-numbers cases rent seekers are risk averse. Following Tullock, we assume non-cooperative behaviour with rent-seeking outcomes established by Cournot-Nash equilibria. To consider non-competitive or strategic rent-seeking when rent seekers are risk averse, denote the probability thatfirm7 will be the ultimate monopolist by n , = p,/i^p,

= /5./[(«-1) Q+z^.],

(^ ^^

where Q is the average outlay by firm/'s rivals in rent-seeking activity. If firms adopt Cournot conjectural variations with respect to rivals' behaviour, then each firm chooses its allocation to rent-seeking /^ to solve max[UjU^iA+X^P^

-h(i - Uj) U^iA-P^l

(A 2)

Pi

where 11^ is given by (A i). In the subsequent Nash-Cournot equilibrium, assuming an interior solution (and dropping the subscript/) ^„.,^Q+pi^'(^-P)

- U'{A+X-P)]

- U'{A-P)

+ i(^;^^^^dUiA+X-P)-U{A-P)]

= o.

(A3)

Because of symmetry between firms, in equilibrium P = Q, so l[U'{A-P)

- U'(A +X-P}]

- U'{A -P) + ^

[U{A+X-Pn

= o. (A 4)

Noting that, as «->oo, P->-o and letting nP = S-we then obtain nmm±^Zm.U'iA)=o. n->oo

(A 5)

"

Now expanding U{A +X) and substituting in (13) yields Urr.^

102

,^^^ZMl^^^Mli3

fA6)

1.1.1.4 Risk-averse rent seekers and the social cost of monopoly power no

THE ECONOMIC JOURNAL

[MARCH I 9 8 4 ]

Hence, if X is relatively small, lim-^=i-—.

(A 7)

(A 7) is equivalent to (8), so confirming that in the limit strategic and competitive rent dissipation under risk aversion coincide. Since (A 6) is equivalent to (7), this coincidence is not restricted to small rents. A logarithmic utility function yields the strategic rent dissipation results in Table 2. For large numbers of rent seekers {n ^ 50), which we previously indicated to approximate competitive outcomes, there is no substantial difference between rent dissipated in the strategic and competitive cases. However, for small numbers of rent seekers (« == 2,3,5,10), more rent is revealed to be dissipated by competitive than strategic rent seeking, where of course in these latter small-numbers cases the strategic outcomes are the pertinent ones. Table 2 Strategic rent dissipation^ logarithmic utility^ A = 100 n

X/A

2

3

5

10

50

100

1,000

o*io

50 50

66 64

94

95

60

87 83 75 65

94

48 45

78 75 69

90 80

90 81

91 81

21 12

25

32

68 35

69 35

69 36

21

23

24

24

0*20

0-50 POO

500 lO-OO

53 «5

60 29 18

REFERENCES

Arrow, K. H. (1970). Essays in the Theory of Risk-Bearing. Amsterdam: North Holland. Bhagwati, J. N . (1982). * Directly unproductive, profit-seeking (DUP) activities.* Journal of Political Economyy vol. 90 (August), pp. 988-1002. and Srinivasan, T . N . (1980). 'Revenue seeking: a generalization of the theory of tariffs.* Journal of Political Economyy vol. 88 (December), pp. 1069-87. Buchanan, J, M., Tollison, R. D . and Tullock, G. (1980), editors. Toward a Theory of the Rent-Seeking Society, College Station: Texas A & M Press. Cowling, K. and Mueller, D . C. (1978). 'The social cost of monopoloy power.* ECONOMIC JOURNAL, vol. 88 (December), pp. 727-48. Reprinted in Buchanan et al, (1980), pp. 125-52. and (1981). 'The social costs of monopoly power revisited.* ECONOMIC JOURNAL, vol. 91 (September), pp. 721-25. Hey, J. D . (1979). Uncertainty in Microeconomics. New York: New York University Press. Krueger, A. O. (1974). *The political economy of the rent-seeking society.' American Economic Review, vol. 64 (Jime), pp. 291-303. Reprinted in Buchanan et al, (1980), pp. 51-70. Littlechild, S. C. (1981). 'Misleading calculations of the social cost of monopoly power.* ECONOMIC JOURNAL, vol. 91 (June), pp. 348-63. Posner, R. A. (1975). *The social costs of monopoly and regulation.* Journal of Political Economy, vol. 83 (August), pp. 807-27. Reprinted in Buchanan et al (1980), pp. 71-94. Tullock, G. (1967). 'The welfare costs of tariffi, monopolies and theft.* Western Economic Journal, vol. 5 (June), pp. 224-32. Reprinted in Buchanan et al. (1980), pp. 39-50. (1980). 'Efficient rent-seeking.* In Buchanan et al. (1980), pp. 97-112.

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1.1.2.1 Efficient rent seeking In: James M. Buchanan, Robert D. Tollison, and Gordon Tullock (Eds.), Toward a Theory of the Rent-Seeking Society. Texas A&M University Press, College Station, 97-112

6 Efficient Rent Seeking by GORDON TULLOCK MOST of the papers in this volume impHcitly or exphcitly assume that rent-seeking activity discounts the entire rent to be derived. Unfortunately, this is not necessarily true; the reality is much more complicated. The problem here is that the average cost and marginal cost are not necessarily identical. This is surprising because in competitive equilibrium the average cost and marginal cost are equal and rent seeking is usually a competitive industry. If marginal cost is continuously rising, then marginal and average cost will be different.^ In the ordinary industry the average cost curve of an individual enterprise is usually U-shaped, with economies of scale in the early range and diseconomies of scale in the latter range. In equilibrium, the companies will be operating at the bottom of this cost curve, and therefore average and marginal costs will be equated. A second and much more important reason for the equality of marginal and average cost is that if there is some resource used in production of anything produced under continuously rising costs, then the owners of that resource will charge the marginal cost. People engaged in manufacturing (or whatever activity with which we are dealing) will face a cost that incorporates these rents of the original factor owners. Thus, the assumption that the costs are constant over scale is suitable for practical use. Unfortunately, both these reasons are of dubious validity in the case of rent seeking. First, there seem to be no particular economies of scale. As far as we can see, for example, such monster industries as big * This is obviously also true if marginal cost is continuously falling.

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oil and the natural gas producers do not do as well in dealing with the government as do little oil or, in the gas case, householders. In general, it would appear that there is no range of increasing returns in rent seeking. However, this is admittedly an empirical problem and one for which, at the moment, we have little data. It is, in any event, dangerous to assume that the curves are all U-shaped and competition will adjust us to the minimum point of these curves. This is particularly so, since there is no obvious reason why all rent seekers should have identical efficiencies. The second and more important reason why we can normally assume that supply curves are, in the long run, flat is that if they are continuously rising, factory owners can generally achieve the full rent by selling their factors at their marginal value; hence, the enterprises face essentially flat supply prices. Unfortunately, this has only a limited application in rent seeking. Suppose, for example, that we organize a lobby in Washington for the purpose of raising the price of milk and are unsuccessful. We cannot simply transfer our collection of contacts, influences, past bribes, and so forth to the steel manufacturers' lobby. In general, our investments are too specialized, and, in many cases, they are matters of very particular and detailed good will to a specific organization. It is true that we could sell the steel lobby our lobbyists with their connections and perhaps our mailing list. But presumably all these things have been bought by us at their proper cost. Our investment has not paid, but there is nothing left to transfer. Similarly, the individual lobbyist spends much time cultivating congressmen and government officials and learning the ins and outs of government regulations. There is no way he can simply transfer these contacts, connections, and knowledge to a younger colleague if he wishes to change his line of business. The younger colleague must start at the bottom and work his way up. Thus, it seems likely that in most rent-seeking cases, the supply curve slants up and to the right fi-om its very beginning. This means that rent-seeking activities are very likely to have diflFerent marginal and average costs, even if we can find an equilibrium. It might seem that with continuously upward sloping supply curves and a competitive industry, there would be no equilibrium. This turns out not to be true, although the equilibrium is of a some-

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what unusual nature. The analytical tools required to deal with it are drawn more from game theory than from classical economics. In my article, "On the Efficient Organization of Trials,"' I introduced a game that I thought had much resemblance to a court trial or, indeed, to any other two-party conflict. In its simplest form, we assume two parties who are participating in a lottery under somewhat unusual rules. Each is permitted to buy as many lottery tickets as he wishes at one dollar each, the lottery tickets are put in a drum, one is pulled out, and whoever owns that ticket wins the prize. Thus, the probability of success for A is shown in equation (1), because the number of lottery tickets he holds is amount A and the total number in the drum is A -f B.'

In the previously cited article, I pointed out that this model could be generalized by making various modifications in it, and it is my puipose now to generalize it radically.^ Let us assume, then, that a wealthy eccentric has put up $100 as a prize for the special lottery between A and B. Note that the amount spent on lottery tickets is retained by the lottery, not added onto the prize. This makes the game equivalent to rent seeking, where resources are also wasted. How much should each invest? It is obvious that the answer to this question, from the standpoint of each party, depends on what he thinks the other will do. Here, and throughout the rest of this paper, I am going to use a rather special assumption about individual knowledge. I am going to assume that if there is a correct solution for individual strategy, then each player will assume that the other parties can also figure out what that correct solution is. In other words, if the correct strategy in this game were to play $50, each party would assume that the other was playing $50 and would only buy fifty tickets for himself, if that were the optimal amount under those circumstances. 'Gordon Tullock, "On the EfiRcient Organization of Trials,'^ Kijklos 28 (1975): 745-762. For a previous generalization of the model and an application to arms races, see Gordon Tullock, The Social Dilemma: The Economics of War and Revolution (Blacksburg, Va.: Center for Study of Public Choice, 1974), pp. 87-125.

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As a matter of fact, the optimal strategy in this game is not to buy $50.00 worth of tickets but to buy $25.00. As a very simple explanation, suppose that I have bought $25.00 and you have bought $50.00. I have a one in three chance of getting the $100.00 and you have a two in three chance. Thus, the present value of my investment is $33.33 and the present value of yours is $66.66, or, for this particular case, an equal percentage gain. Suppose, however, that you decided to reduce your purchases to $40.00 and I stayed at $25.00. This saves you $10.00 on your investment, but it lowers your present value of expectancy to only $61.53 and you are about $5.00 better off Of course, I have gained from your reduction, too. You could continue reducing your bet with profit until you also reached $25.00. For example, if you lowered your purchase from $26.00 to $25,00, the present value of your investment would fall from $50.98 to $50.00, and you would save $1.00 in investment. Going beyond $25.00, however, would cost you money. If you lowered it to $24.00, you would reduce the value of your investment by $1.02 and only save $1.00. It is assumed, of course, that I keep my purchase at $25.00. I suppose it is obvious from what I have said already that $25.00 is equilibrium for both, that is, departure from it costs either one something. It is not true, however, that if the other party has made a mistake, I maximize my returns by paying $25.00. For example, if the other party has put up $50.00 and I pay $24,00 instead of $25.00, I save $1.00 in my investment but reduce my expectancy by only $0.90. My optimal investment, in fact, is $17.00. However, if we assume a game in which each party knows what the other party has invested and then adjusts his investment accordingly, the ultimate outcome must be at approximately $25.00 for each party."* The game is clearly a profitable one to play, and, in fact, it will impress the average economist as rather improbable. However, it is a case in which inframarginal profits are made, although we are in inarginal balance. At first glance, most people feel that the appropriate bet is $50.00, but that is bringing the "^It would make no difference in the reasoning here, or in any of the following work, if there were an insurance company always willing to buy a bid at its true actuarial value. For example, if you had put in $25,00 and the other party had also put in $25.00, it would give you $50.00 for it, and if you had put in $26.00 and the other party $25.00, it would give you $50.98. But rent seeking normally involves risk, and hence I have kept the examples in the risky form.

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total return into equality with the total cost rather than equating the margins. To repeat, this line of reasoning depends on the assumption that the individuals can figure out the correct strategy, if there is a correct strategy, and that they assume that the other people will be able to figure it out, also. It is similar to the problem that started John von Neumann on the invention of game theory, and I think it is not too irrational a set of assumptions if we assume the kind of problem that rent seeking raises. But there is no reason why the odds in our game should be a simple linear function of contributions. For example, they could be an exponential function, as in equation (2): A'' There are, of course, many other functions that could be substituted, but in this paper we will stick to exponentials. It is also possible for more than two people to play, in which case we would have equation (3): ^- = I T T ^ f :• • (3) A' + B', . . . , n' The individuals need not receive the same return on their investment. Indeed, in many cases we would hope that the situation is biased. For example, we hope that the likelihood of passing a civil service examination is not simply a function of the amount of time spent cramming, but that other types of merit are also important. This would be shown in our equations by some kind of bias in which one party receives more lottery tickets for his money than another. We will begin by changing the shape of the marginal cost curve and the number of people playing, and leave bias until later. Table 6.1 shows the individual equilibrium payments by players of the game, with varying exponents (which means varying marginal cost structures) and varying numbers of players. Table 6.2 shows the total amount paid by all of the players, if they all play the equilibrium strategy. I have drawn lines dividing these two tables into zones I, II, and III. Let us temporarily confine ourselves to discussing zone I. This is the zone in which the equilibrium price summed over all players leads 109

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TABLE 6.1

Individual Investments (N-person, No Bias, with Exponent) Number of Players Exponent

2

1/3 1/2 1 2 3 5 8 12

8.33 12.50 25.00 50.00 75.00 125.00 200.00 300.00

4

6.25 9.37 18.75 37.50 56.25

j

1

15

10

93.75 150.00 225.00

J

3.00 4.50 9.00 18.00 27.00 45.00 72.00 108.00

I

II 1

2.07 3.11 6.22 12.44 18.67 31.11 49.78 74.67

L

TABLE 6.2

Sum of Investments (N-Person, No Bias, with Exponent) Number of Players Exponent

1/3 1/2 1 2 3 5 8 12

2

4

10

30.00 16.66 25.00 45.00 25.00 37.40 50.00 75.00 90.00 100.00 J 150.00 180.00 150.00 270.00 225.00 450.00 250.00 1 375.00 400.00 600.00 L 720.00 600.00 900.00 1,080.00

jjj

15

Limit

31.05 46.65 93.30 186.60 280.05 466.65 746.70 1,120.05

33.30 50.00 100.00 200.00 300.00 500,00 800.00 1,200.00

II

to a payment equal to or less than the total price. In other v^ords, these are the games in which expectancy of the players, if they all play, v^ould be positive. Although we vdll start with these games, as we shall see below there are cases in which we may be compelled to play games in zones II and III where the expectancy is negative. If we look at zone I, it is immediately obvious that the individual payments go down as the number of players rises, but the total amount paid rises. In a way, what is happening here is that a monopoly profit is 110

1.1.2.1 Efficient rent seeking Efficient R e n t Seeking

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being eompeted away. Note, however, when the exponent is one-third or one-half, even in the Hmit there is profit of $66.66 or $50.00 to the players taken as a whole. Thus, some profit remains. With the cost curve slanting steeply upward, these results are to some extent counterintuitive. One might assume that with a positive return on investment, it will always be sensible for more players to enter, thereby driving down the profits. In this case, however, each additional player lowers the payments of all the preceding players and his own, and the limit as the number of players goes to infinity turns out to be one where that infinity of players has, at least in expectancy terms, sizable profits. Throughout the table, in zones I, II, and III, individual payments go down as we move from left to right, and total payments rise. We can deduce a policy implication from this, although it is a policy implication to which many people may object on moral grounds. It would appear that if one is going to distribute rents, nepotism is a good thing because it reduces the number of players and, therefore, the total investment. This is one of the classical arguments for hereditary monarchies. By reducing the number of candidates for an extremely rentrich job to one, you eliminate such rent-seeking activities as civil war, assassination, and so forth. Of course, there are costs here. If we reduce the number of people who may compete for a given job, you may eliminate the best candidate or even the best two thousand candidates. This cost must be offset against the reduction in rent-seeking costs. On the other hand, many cases of rent seeking are not ones in which we care particularly who gets the rent. In such matters as government appointments where there are large incomes from illegal sources, pressure groups obtaining special aid from the government, and so on, we would prefer that there be no rent at all, and, if there must be rent, it does not make much difference to whom it goes. In these cases, clearly measures to reduce rent seeking are unambiguous gains. Thus, if Mayor Richard Daley had confined all of the more lucrative appointments to his close relatives, the social savings might have been considerable. If we go dovvni the table, the numbers also steadily rise. Looking at two players, for example, from an exponent of one-third, which represents an extremely steeply rising cost curve, to an exponent of two. 111

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which is much flatter, we get a sixfold increase in the individual and total payments. This also suggests a policy conclusion. On the whole, it would be desirable to estabHsh institutions so that the marginal cost is very steeply rising. For example, civil servants' examinations should be, as far as possible, designed so that the return on cramming is low, or, putting it another way, so that the marginal cost of improving ones grade is rapidly rising. Similarly, it is better if the political appointments of the corrupt governments are made quickly and rather arbitrarily, so that not so many resources are invested in rent seeking. Once again, however, there is a cost. It may be hard to design civil service examinations so that they are difficult to prepare for and yet make efficient selections.^ Here again, if we are dealing with appointments to jobs that we would rather not have exist, the achievement of profits through political manipulations and the like, there is no particular loss in moving down our table. Thus, laws that make it more expensive or more difficult to influence the government—such as the campaign contribution laws—may have considerable net gain by making the rise in marginal cost steeper. There is a considerable expense involved, however. The actual restrictions placed on campaign contributions are designed in a highly asymmetrical manner, so that they increase the cost for some potential lobbyists and not for others. Whether there is a net social gain from this process is hard to say. So much for zone I; let us now turn to zones II and III. In zone II, the sum of the payments made by the individual players is greater than the prize; in other words, it is a negative-sum game instead of a positive-sum game as in zone I. In zone III, the individual players make payments that are higher than the prize. It might seem obvious that no one would play games of this sort, but, unfortunately, this is not true. Before von Neumann began his work on the theory of games, students of probability divided gambling situations into two categories: pure chance and games of strategy. We may take two simple examples. If Smith flips a coin and Jones calls the outcome, we have a game of pure chance, provided only that Smith does not have enough skill ac^ There is another solution, which is to put the civil service salary at the same level as equivalent private salaries. Under these circumstances, there would be no rent seeking. Given the political power of civil servants, however, I doubt that this would be possible.

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tually to control the coin. This is so even if the coin is not a fair one, although Jones might not properly calculate the odds under those circumstances. In this game, the properly calculated, but mathematical, odds are fifty-fifty, and there is no great problem. Consider, however, a very similar game, in which Smith chooses which side of the coin will be up and covers it with his hand until Jones calls either heads or tails. The coin is then uncovered, and if Jones has properly called the bet, Smith pays him; if he has not, Jones pays Smith. This is a game of strategy. The early writers in this case reasoned that there was no proper solution to the game, because if there were a proper solution, both parties could figure it out. Thus, for example, if the proper thing for Smith to do was to play heads, he would know that Jones would know that this was the proper thing to do; hence, the proper thing for Smith to do would be to play tails. Of course, if the proper thing is to play tails, then Jones will also know that; therefore, the proper thing to do is to play heads. It will be seen that this is an example of the paradox of the liar. The early students of probability argued that in circumstances like these there was no proper solution and referred to it as a game of strategy, which was roughly equivalent to throwing up their hands. In games of this sort, von Neumann discovered that there might be (not necessarily was, but might be) a solution. In the particular case of coin matching, there is no simple solution, but in many real-world situations there could be a strategy for Smith that he would still retain even though Jones could figure it out and make the best reply. If there was such a strategy, it was called a saddle point. Von Neumann also pointed out that one should consider not only pure strategies but also mixed strategies. Further, in zero-sum games there is always some mixed strategy that has a saddle point. This proof can also be extended to differential games, which are the kind of games we are now discussing, but, unfortunately, it applies only to zero-sum games, and our games are not zero-sum.® A broader concept of equilibrium was developed by Nash, but unfortunately the games in zones II and III have a very pronounced discontinuity at 0. In consequence, there is no Nash equifibrium. These games have neither dominant pure strategies, saddle points, nor domi® Except, of course, for those games which lie along the boundary between zone I and zone XL

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nant mixed strategies. They are games of strategy in the older sense of the word, games for which we can offer no solution. Let us here reexamine the idea of a solution in order to make this clear. If there is such a solution, anyone can compute it. Thus, Smith must choose his strategy knowing that Jones will know what he is going to do. Similarly, Jones must choose knowing that Smith will be able to predict accurately what he will do. There is no law of nature that says all games will have solutions of this sort, and these, unfortunately, are in a category that do not. For a simple example, consider the game shown on table 6.1 in which there are two players. Smith and Jones again, and assume that the exponent on the cost function is 3. The individual payment is shown as $75, and the result of the two players putting up $75 is that they will jointly pay $150 for $100. Each is paying $75 for a fifty-fifty chance on $50, which appears to be stupid. However, let us run through the line of reasoning that may lead the two parties to a $75 investment. Suppose, for example, that we start with both parties at $50. Smith raises to $51. With the exponent of three, the increase in the probability that he will win is worth more than $1—in fact, considerably more. If Jones counters, he also gains more than $1 by his investment. By a series of small steps of this sort, each one of whic^ is a profitable investment, the two parties will eventually reach $75, at which point there is no motive for either one to raise or lower his bid by any small amount. They are in marginal adjustment, even though the total conditions are very obviously not satisfied. But what of the total conditions? For example, suppose that Jones decides not to play. Obviously, his withdrawal means that Smith is guaranteed success, and, indeed, he will probably regret that he has $75 down rather than $1, but, still, he is going to make a fairly good profit on his investment. Here we are back in the trap of the coin-matching games. If the best thing to do, the rational strategy, in this game is not to play, then obviously the sensible thing to do is to put in $1. On the other hand, if the rational strategy is to play, and one can anticipate the other party will figure that out, too, so that he will invest, then the rational thing to do is to stay out, because you are going to end up with parties investing at $75. There is no stable solution. 114

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Games like this occur many times in the real world. Poker, as it is actually played, is an example, and most real-world negotiations are also examples of this sort of thing; in the case of poker, there is no social waste, because the parties are presumably deriving entertainment from the game. Negotiations, although they always involve at least some waste, may involve fairly small amounts because the waste involved in strategic maneuvering may be more than compensated by the transfer of information that may permit achievement of a superior outcome. But in our game this is not possible. In the real world there may be some such ejffect that partially oflFsets the waste of the rent seeking. In most rent-seeking cases, however, it is clear that this offset is only partial, and in many cases of rent seeking the activity from which^the rent will be derived is, in and of itself, of negative social value. Under these circumstances, not only do we have the waste of rent seeking, we also have the net social waste imposed by the rent itself In the real world, the solution to rent seeking is rather apt to end up at $75 in our particular case instead of at zero, because normally the game does not permit bets, once placed, to be withdrawn. In other words, the sunk costs are truly sunk; you cannot withdraw your bid. For example, if I decide to cram for an examination or invest a certain amount of money in a lobby in Washington that is intended to increase the salaries of people studying public choice, once the money is spent, I cannot get it back. If it turns out that I am in this kind of competitive game, the sunk-cost aspect of the existing investment means that I will continue making further investments in competition with other people studying for the examination or in hiring lobbyists. In a way, the fact that there is an optimal amount—that even with the previous costs all sunk we will not go beyond $75 in the particular example we are now using—is encouraging. Although sunk costs are truly sunk, there is still a limit to the amount that will be invested in the game. Note that this game has a possible precommitment strategy." If one of the parties can get his $75 in first and make it clear that it will not be withdrawn, the sensible poHcy for the second party is to play zero; hence, the party who precommits makes, on this particular game, a profit of $25. ^Thomas C. Schelling, The Strategy of Conflict (Cambridge, Mass.: Harvard University Press, 1960).

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Unfortunately, this analysis, although true, is not very helpful. It simply means that there is another precommitment game played. We would have to investigate t h e parameters of that game, as well as the parameters of the game shown in tables 6.1 and 6.2, and determine the sum of the resources invested in both. Offhand, it would appear that most precommitment games would be extremely expensive because it is necessary to make large investments on very little information. You must be wilhng to move before other people, and this means moving when you are badly informed.'* But, in any event, this precommitment game would have some set of parameters, and, if we investigate them and then combine them with the parameters of the game that you precommit, we would obtain the total cost. I doubt that this would turn out to be a low amount of social waste. The situation is even more bizarre in zone III. Here the equilibrium involves each of the players' investing more than the total prize offered. It is perhaps sensible to reemphasize the meaning of the payments shown in table 6.1. They are the payments that would be reached if all parties, properly calculating what the others would do, made minor adjustments in their bids and finally reached the situation where they stopped in proper marginal adjustment. They are not in total equilibrium, of course. Once again, the simple rule—do not play such games—is not correct, because if it were the correct rule, then anyone who violated it could make large profits. Consider a particular game invented by GeoJBFrey Brennan, which is the limit of table 6.1 as the exponent is raised to infinity. In this game, $100 is put up and will be sold to the highest bidder, but all the bids are retained, that is, when you put in a bid, you cannot reduce it. Under these circumstances, no one would put in an initial bid of more than $100, but it is not at all obvious what one should put in. Further, assume that the bids, once made, cannot be withdrawn but can be raised. Under these circumstances, there is no equilibrium maximum bid. In other words, it is always sensible to increase your bid above its present level if less than $100 will make ** As an amusing sidelight on this problem, a referee of an earlier draft of this paper objected to my above paragraph on the ground that the first party should not put in $75 but some smaller number closer to $55 that would be enough to bar the other party. Note, however, that if one paused to figure out the actual optimal number, the other party would get in first with his $75.

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1.1.2.1 Efficient rent seeking Efficient R e n t Seeking

109

you the highest bidder. The dangers are obvious, but it is also obvious that refusal to play the game is not an equilibrium strategy, because of the paradox of the Har mentioned above. In games in zones II and III, formal theory can say little. Clearly, these are areas where the ability to guess what other people will do, interpret facial expressions, and so on, pays ofiFvery highly. They are also areas where it is particularly likely that very large wastes will be incurred by society as a whole. Unfortunately, it seems likely that rent seeking is apt to lead to these areas in some cases. Obviously, as a good social policy, we should try to avoid having games that are likely to lead to this kind of waste. Again, we should try to arrange that the payoif to further investment in resources is comparatively low, or, in other words, that the cost curve points sharply upward. One way to lower the social costs is to introduce bias into the selection process. Note that we normally refer to bias as a bad thing, but one could be biased in the direction of the correct decision. For example, a civil service exam might be so designed that it is very likely to pick out people who have the necessaiy natural traits and is very hard to prepare for. This would be bias in favor of the appropriate traits, but it would be a desirable thing. Similarly, we would like t o have court proceedings biased in such a way that whoever is on the right side need not make very large investments in order to win, and if this is true, the people on the wrong side will not make very large investments either, because they do not pay. On the other hand, bias can be something which, at least morally, is incorrect. We referred above to Mayor Daley s appointments of his relatives, and this would be a kind of bias. In that particular case, presumably bias would reduce total rent seeking and not lower the functional efficiency of the government of Chicago, but there are many cases where this kind of bias would lower efficiency. Bias, it will be seen, is rather similar to the restriction on the number of players we have discussed above. Instead of totally cutting oflFsome players, we differentially weigh the players. For example, assume that player A is given five times as many coupons for his onedollar investment as are the other players. This would bias the game in his favor, although not to the extreme of prohibiting others from buying tickets. This kind of bias, once again, is rather similar to designing 117

Gordon Tullock

110 Subsequent Contributions to Theory aiid Measu rement TABLE 6.3

Individual Investments (2-Party, Bias, Exponent) Bias Exponent

2

1 2 3 5 8 12

22.22 44.44 66.67 111.11 177.78 266.67

4

1

16.00 32.00 48.00 80.00 1 128.00 1 192.00 III

10

15

8.30 16.53 24.79 41.32 66.12 99.17

5.90 11.72 17.58 29.30 46.88 70.31

II L

your examination to select natural traits. If player A can, with one hour of cramming, increase his probable score on a civil service exam as much as can player B vsdth five hours of cramming, then the system is biased in favor of A, and v^e would anticipate that the total cost of rent seeking would go down. Let us now turn to table 6.3. In this table, we have only two parties competing because the situation is mathematically complex and, in any event, having more than two parties would require a threedimensional diagram. Along the top is the degree of bias toward one player, which is measured here simply in the number of tickets he gets per dollar, it being assumed that the less-advantaged player gets one ticket per dollar. We have omitted the lower exponents of table 6.1, because it is immediately obvious that bias very sharply reduces total rent seeking. Table 6.4 is the sum over both players of all the payments shown in table 6.3, and, in this case, they always just double the figures in table 6.3. It turns out that, using our simple mathematical apparatus, both players—the one who is favored by the bias and the one who is not— make the same investment. This is a little counterintuitive, but not very, since most of us do not have very strong intuitions on these matters. In any event, it may simply be an artifact of the particular mathematical formahsm we have chosen. It will be noted immediately that zone I is much larger in this case than in the unbiased cases of tables 6.1 and 6.2. Indeed, even with an 118

1.1.2.1 Efficient rent seeking Efficient R e n t Seeking

111

TABLE 6.4

Sum of Investments (2-Party, Bias, Exponent) Bias

Exponent

1 2 3 5 8 12

10 44.44 88.88 133.34 222.22 355.56 533.34

32.00 64.00 96.00 160.00

I

256.00 L 384.00

III

16.60 30.06 49.58 82.64 132.24 198.34 II

15

11.80 23.44 35.16 58.60 93.76 140.62

exponent of 8—which means an extremely flat cost curve—a bias of 15 leads to the game still being in zone I. Thus, such bias does pay off heavily in reducing rent seeking. It is also true that this kind of bias, in general, is easier to arrange by socially desirable techniques than the earlier suggestions made to reduce rent seeking. Once again, designing personnel selection procedures so that they select the best man at relatively low cost to him is an example. Another would be some kind of policy selection process that was heavily biased in favor of efficient, or "right," policies. Both these techniques, if we could design them, would have large payoffs, not only in reducing rent-seeking activity but also in increasing eflBciency of government in general. Thus, it seems to me that introducing this rather special kind of bias into rent seeking would be desirable in many areas, even if we ignore the rent-seeking savings. However, for many rent-seeking activities, it is admittedly very hard to find a way to introduce bias at all or to introduce bias in a way that leads to better outcomes. Once again, if we assume that Mayor Daley does not restrict his appointments to his relatives but simply gives relatives a differential advantage, depending on how close they are to him, we have a bias system that will reduce rent seeking. However, it will not lead to outcomes in any way superior. Similarly, the restrictions placed on campaign contributions and other methods of attempting to influence government pohcy are biased in the sense that they are heavier burdens for some people than for others, and it is not clear whether this bias will lead to policy choices superior to those ob119

Gordon Tullock

112 Subsequent

Contributions

to Theory and

Measurement

tained without it. Thus, the only gain is the possibiHty of reduction in total rent seeking. Thus ends our preliminary investigation of rent seeking and ways to reduce its social cost. When I have discussed the problem with colleagues, I have found that the intellectually fascinating problem of zones II and III tends to dominate the discussion. This is, indeed, intellectually very interesting, but the real problem we face is the attempt to lower the cost of rent seeking, and this will normally move us into zone I. Thus, I hope .that the result of this paper is not mathematical examination of the admittedly fascinating intellectual problems of zones II and III, but practical investigation of methods to lower the cost of rent seeking.

APPENDIX T O CHAPTER 6

Mathematical Appendix, or Labor Saving Calcidation Methods When I first began working on this paper, I discovered that the equations that would have to be solved were higher-order equations, and therefore simply assigned to my graduate assistant, William J. Hunter, the job of approximating the results by using a pocket calculator. H e promptly discovered the rather astonishing regularity of column 1, which implied that it would not be all that difficult to solve the equations even if they were higher order. Before I had had time to do anything other than shudder vaguely about the problem, however, I went to lunch with my colleague, Nicolaus Tideman, told him the problem, and he solved it on a napkin. This gave us the equation for tables 6.1 and 6.2. Having discovered this simple algorithm, when we wanted to prepare tables 6.3 and 6.4, once again we asked Tideman, and he obliged with equal speed. The equations used are: P, = « ^ ~ ^

(Tables 6.1, 6.2)

(b + 1)where F^ R N b

120

= equilibrium investment, = exponent, or the determinant of steepness of the supply curve, — number of players, and = bias weight.

1.1.2.2 Free entry and efficient rent seeking Public Choice 46: 247-258 (1985). © 1985 Martinus Nijhoff Publishers, Dordrecht. Printed in the Netherlands. Efficient rents 2

Free entry and efficient rent seeking

RICHARD S, HIGGINS Federal Trade Commission WILLIAM F. SHUGHART II ROBERT D. TOLLISON* Center for Study of Public Choice, George Mason University, Fairfax, VA 22030

1. Introduction This paper concerns rent seeking and the extent to which rents are dissipated under various circumstances. Gordon Tullock's (1967) insight that expenditures made to capture an artificially created transfer represent a social waste suggested that the cost to the economy of monopoly and regulation is greater than the simple Harberger (1954) deadweight loss. Indeed, under Tullock's original formulation and in the extensions of his work by Krueger (1974) and Posner (1975), rents are exactly dissipated at the social level ($1 is spent to capture $1), so that the total welfare loss from such activities is equal to the Harberger triangle plus the rectangle of monopoly profits. More recent contributions have identified instances of imperfect rent dissipation. Rogerson (1982) finds that rent dissipation will be incomplete when rent seekers face differential start-up costs or in situations where a monopoly franchise is reassigned periodically in such a way as to give advantage to the incumbent monopolist. Hillman and Katz (1984) obtain similar results when rent seekers exhibit risk aversion and the value of the transfer is *large.' Lastly, using a model in which rent seekers know the correct strategy (if it exists), but where the probability of obtaining a transfer is not a linear function of rent-seeking expenditures, TuUock (1980) finds situations of either underbidding or overbidding. We shall have more to say about Tullock's analysis in Section 2 below. While the importance of these results to the measurement of the social cost of monopoly and regulation is transparent, there is a curious aspect to most of the models in the literature on rent seeking. Both the value of the transfer and the number of rent seekers are normally fixed. The latter assumption removes from consideration the method by which rents are dissipated where entry is possible. Indeed, since monopoly rights are assigned in these models either by a pure lottery or by award to the highest bidder • We have benefitted from comments by James Buchanan, Gerard Butters, Robert Mackay, Gordon Tullock, and two anonymous referees. Remaining errors are our own.

121

Richard S. Higgins, William E Shughart II, and Robert D. Tollison 248

among a predetermined number of contestants, an important antecedent question is left unanswered: What determines the equilibrium number of rent seekers? In this paper, we investigate the degree of rent dissipation when the number of active rent seekers is endogenous. Building upon Tullock's (1980) model, we show that in a market where the award is based on ^effort,' entry fees are nonrefundable, and contestants are risk neutral, there exists a symmetric mixed-strategy zero-profit equilibrium. This result implies that all rents are dissipated (expectationally), and that the expected number of active rent seekers depends on the value of the rents to be appropriated, among other things. Moreover, we are able to demonstrate that under-, over-, or exact dissipation of the monopoly rents is possible depending on the actual number of active rent seekers. On average, however, rents are completely dissipated. This paper is organized as follows. Section 2 provides a summary of Tullock's model, introducing along the way the key elements in our analysis. Our model is presented in Section 3, and Section 4 contains some concluding remarks.

2. A critique of the Tullock model Consider the static monopoly depicted in Figure 1. A competitive industry with constant long-run marginal costs (LMC) equal to Pc is costlessly transformed into monopoly. Output falls from Qc to Qm as a result, and market price rises to Pm- In the standard analysis, the welfare loss is represented by the triangle ABC^ and the rectangle of monopoly profits, PmABPc, is treated as a pure transfer from consumers to the monopolist. Tullock's (1967) insight was that any resources spent to capture PmABPc were also part of the social cost of monopoly. Posner's (1975) example is instructive on this point. Suppose that 10 bidders vie for a transfer worth $100,000. If the bids are nonrefundable and all contestants are risk neutral, each will offer $10,000 for the right to be the monopolist. At the social level, the monopoly returns are exactly dissipated - $100,000 is spent to capture $100,000. The welfare cost of the monopoly is thus equal to the area of the trapezoid, PmACPc. Tullock (1980) subsequently proposed a model in which results other than exact dissipation of rents are possible.* In the model, N participants seek a prize, L. The probabiHty of winning is assumed to be Pi = Xl/ E 7=1

122

Xl

1.1.2.2 Free entry and efficient rent seeking 249

A

s

\ c

D

Figure J.

where Xi is the amount bid by individual / and the value of the exponent, R, denotes possible marginal cost structures confronting the participants. (The higher is /?, the easier it is for contestants to raise their probability of winning at the margin.) TuUock assumes further that if a correct individual strategy exists, each participant knows it and plays with the presumption that the other parties can discover the same solution. Given these circumstances, each individual chooses the bid, X-^ that maximizes LPf-Xi. The joint solution of these N marginal conditions is Xf = R{N- \)L/N^, Thus, Tullock shows that, depending on AT and R, the players will spend in the aggregate more or less than L, Actually, however, the value of R drives the entire result. In the case of two participants, for example, overbidding occurs whenever R exceeds two. Indeed, with two players and R > 2, each of the bids will be greater than the value of the prize. We can illuminate the relationship between R and A/^ further by extending Tullock*s model to allow entry of participants until there is zero profit from playing the game, that is, until all rents are dissipated. Zero profit occurs when NX-" = X. Solving this expression for A^ yields

N* =

R/(R-l),

123

Richard S. Higgins, William F. Shughart II, and Robert D. ToUison 250

Figure 2.

Note, first, that N* is independent of JL. This suggests that for a given R, the same number of people will play the game when the prize is $1 as will participate when the prize is $100,000. Second, when i? = 1, underbidding obtains with any finite N\ it requires an infinite number of players to dissipate all rents. Third, as R increases from 1, N falls asymptotically to unity. (These relations are depicted in Figure 2.) In short, R must lie on the interval (1,2) for the zero-profit result to make sense. Only if we are willing to assert 1 < i? < 2 is there an optimal N that dissipates all rents. Extendmg Tullock's model by allowing for entry, we find that all rents are dissipated, but only under such restrictive conditions that we are led to ask, why does the probability of winning the game have the particular form assumed by TuUock? Moreover, within that structure, what determines Rl In what follows, we offer a more general specification of the competition for rents.

3. Free entry and rent dissipation In our model there are TV potential rent seekers.^ Each of them has two decisions to make. Each first decides whether or not to compete for the monopoly right. Each of the m{< N) active rent seekers then chooses a level of effort to maximize the expected value of securing rents, given the efforts

124

1.1.2.2 Free entry and efficient rent seeking 251 of other active participants.^ In contrast to existing models in the literature, we let the expected number of active rent seekers be defined by a zero-profit condition ^ The actual number of active players may under- or overdissipate rents, but ex ante and on average, rents are exactly dissipated when the players exhibit risk neutrality. We first describe how the rents are assigned among the active rent seekers, and then we analyze potential rent seekers' decisions concerning whether to participate actively. It is useful to think of our model of rent assignment in terms of the *comparative hearings* conducted by the Federal Communications Commission (FCC) in granting a radio or television licensdv We suppose the license to be worth L, and individuals expend effort Ui to qualify for the franchise.^ They invest in market research, program development, lobbying activities, and so forth. The effort costs C({//), assumed to be identical for all contestants. We also suppose, however, that the FCC*s decision-making process is subject to error. Perhaps there is some room for (unbiased) discretion, the effort of applicants can be observed only imperfectly, or in the case that several applicants supply equal effort, the winner is chosen at random. Thus, the FCC looks at the vector of efforts (C/i, C/2, . • •, ^m) and grants the license to the applicant supplying the highest level of effort, subject to the error element. Each contestant's measured effort (relevant effort) is thus Wi - Ui + Ely the true effort plus a random variable. We assume that the Ei are independently distributed with zero mean and constant variance. First, consider the case of two contestants. The license will be awarded to applicant 1 when C/i + £"1 > C/2 + £2, or when Ui - U2 -^ Ei > E2, Thus, the probability of winning for applicant 1 is given by KiUu

Ui) ^ p

e{Ex) J^' d\E2)dE2dEu

— 00

(1)

~oo

where Vi = Ui -U2+E1, We now use the Nash condition to define equilibrium. Applicant 1 chooses the level of effort, Ui, that maximizes LPUUU

U2)'-C{Ui)

holding 1/2 constant, and applicant 2 optimizes over the symmetric problem. The marginal conditions are L{dPl/dUi)-'C'(Ui) L{dPl/dU2)

= 0, and

- C'{U2) = 0,

(2a) (2b)

where the C'(0 represent each contestant's marginal cost of effort. Using the definition in (1), the two first-order conditions become 125

Richard S. Higgins, William F. Shughart II, and Robert D. Tollison 252 L Y

e{Ei)eiUi'-U2-^Ei)dEi-C'(Ui)

= 0, and

(3a)

= 0.

(3b)

— 00

L J°° eiEz) e(U2 - Ui +E2)dEt -C'iUi) — 00

These relations give reaction functions that in equilibrium define equal efforts Uf and U* described by

L 1"° e\E)dE--C\U*) = 0.

(4)

— 00

Equation (4) is obtained by exploiting the symmetry properties of the problem, noting that ${Ui - Uz +Ei) and BiUi - Vi +£"2) are in equilibrium equal to ^(£'1) and ^(£'2), respectively, since (7* = C/*. Equation (4) suggests that equilibrium effort depends on the value of the license, L, on the steepness of marginal costs, and on the nature of the error distribution, specifically the *mean density* (see, e.g., Nalebuff and Stiglitz, 1983). In particular, the applicants will each expend a greater level of effort the more valuable the prize and the less rapidly marginal costs rise with effort. (If marginal costs are constant, there is no optimal effort.) In order to examine the relationship between total rents and total rentseeking effort, we must specify the functions more completely. Suppose that E follows a uniform distribution on the interval [-D/2, D/2] and that C(U) = a + (6/2)C/^ Equation (4) is now L{\/D)-bU*

= 0,

which implies that U* = L/bD, Total rents are, of course, L, and from the cost function aggregate rent-seeking expenditures are 2[a -f {b/2)(L/bDf]. The net gains are therefore L - [2a-^(L^/blf)]. Note that the more error there is in the decision-making process, the lower the effort and hence the lower the cost. In the extreme case where effort does not count, the competition reduces to a pure lottery with fixed entry cost, a. Here the net gain is L-2a. We next generalize to m applicants. In the special case outlined above where each of m applicants incurs costs of C{U) = 3, + (b/2)lf' and the distribution of decision-making errors is uniform on [-£)/2, D/2], the optimal individual effort is U* = L/bD.^ Effort is, therefore, independent of m when the Ei are uniformly distributed. Although this is clearly a special case, we retain the uniform distribution because it greatly facilitates the analysis of rent seekers' prior decisions whether to be active participants. For each number of active participants greater than one there is an expected profit given by L/m - la-\-(b/2){L/bDf] ^ Eirim), When m = 0, profit is zero, and when m - \, there is really no contest so profit is L - a. 126

1.1.2.2 Free entry and efficient rent seeking 253 As m varies from 2 to N, Eirim) declines at a constant rate equal to C(C/*).^ Depending on the parameters of the model, Eir may be negative beyond some particular value of m. And finally, since m assumes only integer values, there may be no value of m for which ET is exactly zero. In the latter case there will be some particular value of m, say m^, at which ETc{m^)>0 and£'7r(w^ + l ) < 0 . To model the A^ potential rent seekers* participation decisions, we make some simplifying assumptions. First, we assume that all rent seekers announce simultaneously their decisions to play or not to play. The fixed cost component, a, is incurred when a rent seeker decides to be active, and a decision to be active is binding. That is, the rent seeker must pay an entry fee equal to a to be an active participant, which would be forfeited if the entrant were to withdraw. The rent seekers do not decide on their optimal levels of effort at the time they commit to active participation. They choose the appropriate levels of effort only after they know how many other active participants there are.^ As Tullock (1980: 105-109) has emphasized, because expected profit is in general negative for some number of active participants less than A^, there is no symmetric pure strategy equilibrium in our entry game. He refers to this problem as the ^paradox of the liar.' We resolve the paradox by allowing our rent seekers to choose mixed strategies. Furthermore, since effort is independent of /w in our example based on the uniform error distribution, expected net return over the variable cost of effort may be negative beyond some number of active players. Under these circumstances, players minimize losses by dropping out according to an optimal mixed strategy, forfeiting the fixed entry fee. To avoid complicating the analysis of the initial entry decision we assume that expected profit (net of variable cost) from the contest is nonnegative for all w < N.^ Each of the A/'rent seekers chooses to participate actively with probability Pi, The expected value of actively seeking rents is found by weighting Eirim) by the probability that only m - 1 other rent seekers will actively participate, summing these terms, and multiplying by/?/. In the symmetric mixed-strategy equilibrium the expected value of being an active rent seeker must be zero for each rent seeker. And, each player's choice of/7i, given the probabilities of the others, must simultaneously maxhnize the expected values for each player, which equal zero. In the present case, there is a straightforward way to find equilibrium p . We know that the individual expected gain must equal the individual expected cost. Expected cost is jc>{ C*[l-(l~/?)^~M + a(l-/;)^*M = pla + (b/2)U^[l-(l-p)^'^]]^^ Expected gain for the individual is L[l - ( 1 -pf^yN, That is, expected gain is total gains, L, times the probability that at least one rent seeker becomes active, all divided by the total number of rent seekers. Thus, 127

Richard S. Higgins, William E Shughart II, and Robert D. ToUison 254 L[l-(\-pf]/N

= p{a^{b/2)U^[l~{\--pf^']].

(5)

When N is large, (5) is simply L/N = pC*y and /?* = L/NC*, provided L/NC* is not greater than 1. When L/NC* > 1, the equilibrium strategy is for all N rent seekers to contest actively for the monopoly rents. When the N rent seekers choose probability p* so that there is zero ex~ pected profit from engaging in rent seeking, rents are fully dissipated ex ante. Ex post - that is, after a realization from the binomial distribution of active participants - profit from engaging in the contest may be positive or negative depending on the particular realization. On average, the actual number of active contestants equals the expected number, Np*, and expected return over the cost of effort in the contest will just equal the fixed cost. For other realizations of active contestants, expected net return from the contest may exceed sunk cost (underdissipation of rents) or be less than sunk cost (overdissipation). We conclude by noting the relationship between the average number of active rent seekers and the other parameters of the model. Specifically, AT/?* = NiL/NC*) = L/la + {b/2HL/bDf].

(6)

From (6) we determine that dNp*/da < 0 and dNp*/dD > 0. Thus, assuming that/7* < 1 to begin with (i.e., L-a > 0 and L/m-C* < 0 for all m beyond m^ < N), there are fewer active rent seekers the higher the entry fee, and more active rent seekers the greater the role of chance in assigning rents. Moreover, we see from (6) that the expected number of active rent seekers is independent of the size of the pool of potential rent seekers. Of course, in general when effort varies with the number of active rent seekers, this result is unlikely. The relationship between the expected number of active rent seekers and the size of the prize is surprisingly ambiguous. A higher prize induces two opposing responses. For a given level of effort, a higher L increases the expected gain from being an active rent seekers. However, since a higher L induces greater effort the cost of active participation is also raised, and the overall effect of increasing L on the probability of actively seeking rents is ambiguous. Specifically, when the variable cost of effort exceeds the fixed entry fee, an increase in the size of the prize induces less active participation. Finally, when the error in the decision-making process is very large, that is, D ^ 00, the ^hearing' reduces to a pure lottery, and effort becomes irrelevant. In this case Np* = L/a (when L/a < N), and an increase in the size of the prize unambiguously increases active participation.

128

1.1.2.2 Free entry and efficient rent seeking 255 4. Concluding remarks In the competition for a monopoly right in which the number of bidders is fixed, Tullock and others have found the value of the resources spent in the aggregate to capture the transfer to be sometimes less than and sometimes greater than the value of the monopoly. We think this approach to be incomplete since it leaves unanswered the question of what determines the number of individuals who will vie for the right to be the monopolist. It is unsatisfactory to imagine, for example, that the franchisor sets the number of contestants. One could then foresee that rent seeking would arise to influence the permissible number of bidders, and this merely moves the rent-seeking dissipation question one step back. Our approach has been to extend these models in two ways. First, for a given number of active rent seekers, the monopoly right is granted according to the contest model developed by Nalebuff and Stiglitz (1983). This model clearly reveals that overdissipation of monopoly rents generally occurs only when there is some fixed cost of effort - or what amounts to the same thing, when active participation requires a nonrefundable entry fee. According to the contest model of granting rents, the extent to which rents are dissipated depends positively on the number of active rent seekers. Second, since expected profit in the contest is generally negative beyond some number of contestants less then the potential number of contestants, we construct an economic model of the entry decision. To avoid Tullock's 'paradox of the liar' - the absence of a symmetric pure-strategy equilibrium - our potential rent seekers adopt mixed entry strategies. We show that there is a symmetric mixed-strategy zero-profit equilibrium in which each of A^ potential rent seekers actively engages in the rent-seeking contest with probability/?. Thus, the actual number of active rent seekers is a draw from the binomial distribution with parameters N and p. For the expected number of contestants, Np, rents are exactly and fully dissipated. Over- and underdissipation of monopoly rents are possible, but only ex post. The implications of our analysis are straightforward. First, when there are no restrictions on the number of individuals who may vie for the right to capture an artificially created transfer, entry will occur, and resources will be spent up to the point where the expected net value of the transfer is zero. Such competition leads to exact dissipation of the present value of the flow of rents associated with the transfer, and in static terms, makes the social cost of the monopoly equal to the value of the Tullock trapezoid. Second, even if entry is limited, overbidding for the franchise will in general not occur, the value of the Tullock trapezoid sets an upper limit on the social cost of monopoly. 129

Richard S. Higgins, William R Shughart II, and Robert D. ToUison 256 The result that rents are fully dissipated depends critically on the assumption of risk neutrality. While we have not analyzed the case of risk aversion completely, several predictions about the characteristics of equilibrium appear straightforward. First, if the marginal contestant is risk averse, then setting net expected utility equal to zero implies that in the limit the monetary value of the rents will not be fully dissipated. Moreover, the extent to which rents are dissipated will be less the greater the degree of risk aversion, the smaller the value of the appropriable rents relative to initial wealth, and the higher the fixed cost of entry (see Hillman and Katz, 1984: 107). Second, the extent of rent dissipation will also depend on the assumptions made concerning the supply of rent seekers and their risk aversion distribution. For example, there may be a large enough pool of potential rent seekers with zero risk aversion that the equilibrium number of active rent seekers will all be risk neutral. In this case all rent will be dissipated expectationally. Third, and most importantly, with risk aversion as with risk neutrality, overdissipation will not be observed ex ante. Finally, the theory of rent seeking, as exposited here and elsewhere, puts considerable pressure on the argument that monopoly promotes a transfer of wealth from consumers to owners of monopoly firms (Comanor and Smiley, 1975). As Posner (1975: 821) observed, rent seeking implies that monopoly profits are dissipated, not transferred. This argument is correct as far as it goes. Only it does not go far enough, and it would carry us well beyond the scope of this paper to present a careful analysis of the impact of rent seeking on the level and distribution of wealth. Suffice it to say here that the effect of rent seeking on the level and distribution of wealth will be a function of the mechanism used to assign rents in a society, attitudes toward risk, comparative advantages in rent seeking, and so on (Higgins and ToUison, 1984).

NOTES 1. In a recent paper, Corcoran (1984) raises several of the same points about Tullock*s analysis which we independently noted, and which are set out in the remainder of Section 2. He suggests, for example, that bidders will enter until in the long run the expected payoff from playing the game is just equal to the return realizable from alternative investments, and that in equilibrium, the number of contestants depends only upon the value of the parameter, i?, which in TuUock's model represents the structure of marginal costs facing the participants. Corcoran goes on to state, correctly, that aggregate rent-seeking expenditures will in the long run be invariant with respect to R. What Corcoran fails to note (and what Tullock, 1984, shows suspicion of, but does not fully demonstrate in his comment) is that for the zero-profit result to make sense, R must be restricted to the interval (1, 2). Where we differ from Corcoran is in our development of a more general approach to the question of rent dissipation under conditions of free entry (see Section 3).

130

1.1.2.2 Free entry and efficient rent seeking 257 2. The number of potential rent seekers is fixed in our model. We do not analyze the decision to be among the pool of potential rent seekers. 3. Thus, our rent-seeking model is analogous to Shubik*s (1959) game-theoretic model of oligopoly in which firms in-being are distinguished from active competitors. 4. The major exception is Corcoran (1984); see note 1. One may question whether equilibrium is the appropriate restriction to apply to the problem of rent seeking for a known monopoly right. Consider Frank Knight's example of the California gold rush. Overinvestment occurred in that case because no one entrant could possibly determine how many others would attempt to stake claims. Admittedly, our model requires that all players have this type of information, but we think that the assumption is justifiable on the ground that it permits us to derive testable implications about the determinants of rent-seeking activity. 5. If the license is perfectly durable, L reflects the discounted value of the flow of rents in perpetuity. On the other hand, if there is some positive probability that the rents will be expropriated in the future, this will reduce the present value of the license. The exact nature of the license right in this regard is inunaterial to our results. 6. When there are m rent seekers, individual / wins when Wi > Wj for ally 5^ i, that is, when Ui - Uj •¥ Ei > Ej for ally 5^ /. If we assume that the errors are distributed independently, the probability that player / wins given Ei is cVi n

giXj)dXj,

(a)

where Vi = Ui - Uj + £ i . To get the probability that / is successful, we integrate {a) over Ei and obtain

r'" r

giXj)dXj]g{Xi)dXi

- P C t / i , . . . , ( / / , . . . , U„),

(b)

The partial derivative of P{U} with respect to Ui evaluated at the symmetric solution is

(m-l)j

g'{X)G"'-\X)dX,

(c)

— 00

Thus, the marginal condition in the symmetric case when g is the uniform density over [ - P / 2 , D/2] is L{\/D)-bU*

= 0.

(d)

7. Obviously, if U* were not independent of m, fir would depend on m in a more complex way. 8. Alternatively, we might have supposed the N rent seekers to make a joint effort and entry decision. We have not worked through the implications of such a model. 9. The drop-out decision would have to be modeled in the same way as the initial decision to enter. The drop-out game would be played repeatedly until the expected return to effort in the contest was nonnegative for the remaining players. Thus, for all m for which expected net return over variable cost would be negative, Ev will be limited to the loss of fixed cost. 10. Expected cost is not simply pC* because when there is only one active participant, no effort needs to be expended. The probability that a particular individual incurs only cost a is (1 -p)^~*. The probability that a particular individual is not the only active rent seeker and thereby incurs cost C* is 1 - ( ! -p)^~^.

131

Richard S. Higgins, William F. Shughart II, and Robert D. ToUison 258 REFERENCES Corcoran, W.J. (1984). Long-run equilibrium and total expenditures in rent-seeking. Public Choice 43: 89-94. Comanor, W.S., and Smiley, R.H. (1975). Monopoly and the distribution of wealth. Quarterly Journal of Economics 89 (May): 177-194. Harberger, A. (1954). Monopoly and resource allocation. American Economic Review 44 (May): 77-87. Higgins, R.S., and Tollison, R.D. (1984). Notes on the theory of rent seeking. Unpublished manuscript. Hillman, A.L., and Katz, E. (1984). Risk-averse rent seekers and the social cost of monopoly power. Economic Journal 94 (March): 104-110. Krueger, A.O. (1974). The political economy of the rent-seeking society. American Economic Review 64 (June): 291-303. Nalebuff, B.J., and Stiglitz, J.E. (1983). Prizes and incentives: Towards a general theory of compensation and competition. Bell Journal of Economics 14 (Spring): 21-43. Posner, R.A. (1974), The social costs of monopoly and regulation. Journal of Political Economy 83 (August): 807-827. Roger son, W.P. (1982). The social costs of monopoly and regulation: A game-theoretic analysis. Bell Journal of Economics 13 (Autumn): 391-401. Shubik, M. (1959). Strategy and market structure: Competition, oligopoly, and the theory of games. New York: Wiley. Tullock, G. (1%7). The welfare costs of tariffs, monopolies, and theft. Western Economic Journal 5 (June): 224-232. Tullock, G. (1980). Efficient rent seeking. In J.M. Buchanan, R.D. Tollison and G. Tullock (Eds.), Toward a theory of the rent-seeking society, 97-112. College Station: Texas A&M University Press. Tullock, G. (1984). Long-run equilibrium and total expenditures in rent-seeking: A comment. Public Choice 43: 95-97.

132

1.1.2.3 A general analysis of rent-seeking games Public Choice 73: 335-350. 1992. © 1992 Kluwer Academic PubOshers, Printed in the Netherlands.

A general analysis of rent-seeking games'^ J. David Perez-Castrillo Thierry Verdier DELTA Joint Research Unit (CNRS EHESS ENS), 48 Bd Jourdan, F'75014 Paris Received 18 July 1990; accepted 31 October 1990

Abstract. In this paper we reconsider the basic model of "efficient rent seeking." We stress the importance of the shape of the players* reaction curve in order to understand the impact of the technology of rent-seeking on the structure of the outcome of the game. We give a complete characterization of the pure strategy equilibria. Moreover, the possibility of preemption by a Stakelberg leader is discussed according to the nature of the technology of rent-seeking available to the agents.

1. Introduction The literature on efficient rent-seeking has expanded quite rapidly since the seminal work of TuUock (1980) and there has been quite a lot of debate and controversies around the subject. Typically, the basic discussion raised two interrelated points of interest. First was the question of the structure of the equilibria that should emerge in the rent-seeking process and notably whether the equilibria should be symmetric or not. Secondly was the point that complete dissipation of rents in rent-seeking processes does not necessarily occur even when there was competition and free entry among the participants, A certain number of works have tried to investigate both points. Corcoran (1984) and Corcoran-Karels (1985) extended Tullock's model to a long-run setting and tried to endogenize the number of participants in the rent-seeking process through free entry. It was then found that under some circumstances there was total dissipation of rents. Tullock (1984) and (1985) pointed out the difficulties with the long-run solution depending on whether the technology of rent-seeking was with decreasing or increasing returns of scale; and secondly they commented that in a-priori symmetric rent-seeking models, there was a strong presumption of the possibility of asymmetric behavior of the participants. Hillman-Samet (1987a) have touched the problem through the use of mbced• We would like to thank an anonymous referee for his useful comments

133

J. David Perez-Castrillo and Thierry Verdier 336 strategies. In their model, rent-seeking is a contest in which the rent prize is allocated to the greatest outlay. They found that if no initial bet was imposed in order to participate in rent-seeking, then on average one would observe total dissipation of rents. Here again Tullock (1987) questioned the reasonability of mixed-strategies in the context of rent-seeking. As Hillman-Samet (1987b) rightly replied, the critic of Tullock was in fact more fundamental by raising the difficult question of the use and interpretation of mixed-strategies in game theory in general. As it comes out from the preceding discussion, it seems that we are still **in the swamp*' inspite of the many efforts undertaken. However, a potential way to **drier lands** is perhaps to go back to *Hhe roots", reconsider more systematically the basic model of rent-seeking and analyse explicit its properties. In this respect, one point has not probably received the attention that it deserves. It is the rigorous determination of the optimal behavior of a participant given its strategic environment, namely the exact derivation of the **reaction** curve is important because one can reasonably think that much of what can be said about the rent-seeking game is certainly connected with the shape of this reaction curve. The main purpose of this paper is to try to fill this gap by investigating systematically the properties of the reaction function of a particular participant to a rent-seeking game. As we will show, this will provide us with some insights why the case with decreasing returns of scale in the technology of rent-seeking is very different from the case with increasing returns. Moreover, with the explicit specification of this reaction curve in mind, we will analyse rigorously the nature of the equilibria of the rent-seeking game in the short run as well as in the long run. We will discuss the problem of symmetry or asymmetry of the equilibria and reconsider the basic problem of rent-seeking dissipation in this kind of game. Our analysis also enables us to consider the importance of the technology of the shape of rent-seeking's shape in rent-seeking games where there is an asymmetry between the agents: one agent being in a dominant position over the others and acting as a Stackelberg leader. Here again, according to the type of technology used by the players, the properties of the reaction curve help us identify the conditions where the Stackelberg leader effectively preempts the other agents from entering actively in the race for political favors. The plan of the paper is the following. In Section 2 we describe the basic model and derive explicitly the reaction curve of a particular agent given his strategic environment. Section 3 derives necessary conditions for an equilibrium to exist. Section 3 derives necessary conditions for an equilibrium to exist. Section 4 analyses the structure of the Nash equilibria of the rent-seeking game when the number of the potential participants is exogenously fixed. In Sectoin 5 we consider the case of the long-run competition with an endogenous

134

1.1.2.3 A general analysis of rent-seeking games 337

number of agents. Section 6 considers the problem of preemption in a leaderfollower framework. Finally Section 7 concludes.

2. The model We consider a rent-seeking process involving N participants. For example, one may think of N firms in a particular industry and bidding for the attribution of some monopoly right. A particular agent is noted by the indice i (i € ( 1 , . . ,N}). The rents which are the object of the competition may be represented by a prize X. Each agent i makes a bet a^ in the rent-seeking process and following Tullock (1980) and Corcoran (1984), we suppose that his probability of winning this prize X is given: Pr(ai,a_,) =

^ a[+

(1) E

J5^i

where r > 0 caracterizes the returns of scale of the technology of rent-seeking. When r < 1 the technology of rent-seeking may be considered with decreasing returns of scale while when r > 1 the technology is with increasing returns. For simplicity, as we will consider initially only the decision problem of a single N

agent i, we will omit the indice i and note B =

D aj. Now the expected j=i

profit of our agent may be written as: En(a) = ^ - — X ~ a a

(2)

+ D

and his decision program is simply to maximize his expected profit Ell(a), taking as given his '^strategic environment B " and under the constraint the a > 0. The first order condition of this program yields for an interior solution: a^-^(a^ + B) = r B X

(3)

Let note a = a(B) the **reaction curve"' of our agent given an **outside*' competition B from the other agents, that is the best response to B. We pose also the useful notation A(B) = (a(B))^. Then we have the following proposition characterizing this best response function:

135

J. David Perez-Castrillo and Thierry Verdier 338

Proposition 1 1) If r < 1; then the best response a(B) is strictly positive for all B > 0 and is determined by the first order condition (3). 2) If r > 1; two cases are possible: For B < M(r,X), then a(B) is strictly positive for B > 0 and determined by the first order condition (3), and a(0) = 0. For B > M{r,X), then a(B) = 0. where M(r,X) = ^''"'^^'

^'

Proof: See the Appendix 1. The preceding proposition strikingly separates the case where the technology of rent'seeking is with decreasing or constant returns of scale (r< 1) from the. case where the technology has some increasing returns of scale (r > 1). In the case of r < 1, the agent will always find in its interest to invest strictly positively in the rent-seeking process. On the contrary, in the case r > 1, he does invest in rent-seeking only if the *'outside" competition (represented by the term B) is not too important. In fact, the outside competition has to be lower than a certain level M(r,X) which is increasing in the amount of the rent. For levels of B higher than that amount, the agent simply does not participate actively in rent-seeking. The next step is to analyse how does the agent react to changes of his strategic environment B. The following proposition characterizes the shape of the best response function a(B). Proposition 2 1) a'(B) < 0 (equivalently A'(B) > 0) if and only if: A(B) > B. 2) VB, A'(B) > - 1 . Proof: 1) Simple differentiation of equation (3) yields the result. 2) See Appendix 2 Propositions 1) and 2) define three possibilities for the shape of the function A(B). They are represented in Figures 1 to 3. Typically, when we are in the case r < 1, the function is increasing at first when the outside competition is weak and decreases continuously as B increases. On the other hand, for the case r > 1, the function A(B) jumps discontinuously from a positive level to zero as B passes through M(r,X). This discontinuity is essentially due to the nonconvexity of the profit function of the agent in the case r > 1. When r > 2 the reaction curve in its positive part is always increasing while in the case 1 < r < 2 it is first increasing and then decreasing in Figure 3).

136

1.1.2.3 A general analysis of rent-seeking games 339 A(B)

Figure I, Reaction curve in the case r < 1. A(B)

Figure 2. Reaction curve in the case r ^ 2. A(B)

M(r^X)

B

Figure 3. Reaction curve in the case 1 < r < 2.

Having completely examined the behavior of our single agent, we are now in a position to consider the characterization of the equilibrium of the rentseeking process. In a first step we give some results about the equilibrium in outlays for a given number of active agents. This will be useful to analyse subsequently the structure of equilibria when the number N of agents involved is fixed and then to discuss the problem of entry and endogenization of N.

137

J. David Perez-Castrillo and Thierry Verdier 340 3. Non-cooperative rent-seeking equilibria In this section we show that in a Nash equilibrium with a given number of active agents, those active agents necessarily make the same bet. While this is always taken for granted in the literature, it is not however as obvious as it seems. There are numbers of symmetric models in economic theory which in fact admit asymmetric equilibria. We give then a characterization of this symmetric equilibrium in outlays. We first have the following lemma: Lemma 7: In a rent-seeking Nash equihbrium with K active agents, necessarily those agents invest all the same amount of resources. Proof: Let us consider a rent-seeking Nash equilibrium with K active agents and suppose that two active agents i and j devote respectively in this equilibrium the amounts a* and a] with aj 5«^ a]. Let us note H* = E a^^ in which a^ is the investment made by another agent active in this rent-seeking equilibrium. Then from proposition 2) we have: a]r - ay = A(a]r + H*) - A(a^ + H*) < (a^ + H*) - (a]^ + H*) = a^ - ay. which is absurd. Hence the lemma results. Q.E.D. We may now characterize a Nash equilibrium with K active agents (K < N) by the following proposition which gives the usual symmetric characterization of the rent-seeking equilibrium in bets. Proposition 3: A Nash equilibrium with K active agents is such that all agents invest the same amoung a* (K) of money in rent-seeking with aMK) =

^ r X .

A necessary condition for the existence of such an equilibrium in the case of r > 1 is K
0

This is equivalent to the condition that r < 1 or r > 1 and K
2). While a useful step of analysis for the general situation of endogenous N, this case is in itself already interesting. For example, if the rents to be distributed ared monopoly rents specific to a particular industry, we may reasonably suppose that in the short run the firms already in the market have some informational advantage over the potential entrants about those rents and the best way to lobby for them. Therefore the rent-seeking game will be played with a fixed number of firms. We have the following proposition characterizing completely the Nash equilibria of rent-seeking for a fixed number of agents N > 2. (N = 1 is trivial case) Proposition 4: a) if r < 1, then there is a unique Nash equilibrium in which all the N agents N- 1 are active players and each of them invests a* = — z - r X. Moreover, the N^ equilibrium expected profit for each agent is equal to:

139

J. David Perez-Castrillo and Thierry Verdier 342 En* = ( l - r ) N -fr b) If r > 1 and N
0

, then a symmetric equilibrium with N active players r- 1

exists with the same characteristics as in case a). (EH* = 0 only if N =

) r-1

c) If r > 1 and N >

, let N* the highest number such that N* < . r-1 r-1 Then if N* > 1, the Nash equilibria in pure strategies are asymmetric. Moreover, there exists an equilibrium with N* be active agents and N - N * N*-l non-active agents in which each of the active players devotes a* = =- r X of resources to rent-seeking and receives a profit EH* = resources to rent(l-r)N* + r seeking and receives a profit EH* = ^ X positive. Each of the non-active players invests 0 and has a null profit.

Proof: See Appendix 3. As it was already mentioned in the literature, we observe a strong difference between the case r < 1 and the case r > 1. In the case where the technology of rent-seeking is with constant or decreasing returns of scale, there is for a fixed number of agents a unique Nash equiHbrium which is symmetric. In the case of increasing returns of scale, if the number of possible participants is not too high, here again there is a unique symmetric equilibrium with all agents participating actively in the rent-seeking activity. However, when the number of agents potentially interested in rent-seeking is higher than a certain level, we find a multiplicity of equilibria which are asymmetric and in which there are N* devoting the same amount of resources to rent-seeking and N - N * staying inactive. The active agents derive a perhaps small but strictly positive profit while the inactive agents receive nothing. The reason for the mujltiplicity of equilibria comes two reasons. The first one is the obvious indeterminacy of who will be the active agents or the inactive agents. The second one is more intricate and comes from the fact that there may be an asymmetric Nash equilibrium with N* active agents and an asymmetric Nash equilibrium with N* - 1 active participants. (It is not difficult to show that an equilibrium with N* - 2 active agents cannot exist.) As a matter of fact, this happens when we are in the following configuration for N*: (N* - 1) a*(N* - 1) > M(r) > (N* - 1) a*(N*).

(5)

Reconsidering proposition 1), those inequalities simply say that if a potential

140

1.1.2.3 A general analysis of rent-seeking games 343

participant anticipates that N* - 1 agents play a* (N* - 1 ) then he does not bet anything and thus we observe effectively a Nash equilibrium with N* - 1 active agents. But, similarly, if the agent anticipates that the N* - 1 play a*(N*) then he has some incentive to make a positive bet which in this case will be his best response a*(N*) and consequently we may also observe a Nash equilibrium with N* active agents. It is interesting to see that the number N* does not depend on the number of potential players N. This will have some importance in the following section when we consider a rent-seeking process with free entry. Note finally that we have characterized all the equilibria in pure strategies of this game of rentseeking.

S. Rent-seeking with free entry In this section we discuss the case where the number of players in the gaem is not a-priori fixed. The resuhs of proposition 4) show us that in the case r < 1, for any number of agents N, there is always a unique equilibrium in which the expected payoffs are strictly positive. There is therefore always some incentive for a potential entrant to decide to participate in rent-seeking. Here, in the ideal theoretical case, as was pointed out rightly by Corcoran and Karels (1985), the equilibrium entails a very large number of agents, each of them undertaking an infinitesimal bet and competing for some expected profit which tends from above to zero (TuUock 1984). Note however that in this kind of setting we do observe a convergence of the total amount of outlays to the rent prize when the number of participants N tends to infinity only when r = 1 Lim N — 00

N a'^ (N)

= r

(6)

A

In this sense, the rents prize cannot be necessarily a good proxy for the amount of resources spent in rent-seeking. One can easily solve the indeterminacy of the number of participants by introducing some little fixed fee to pay at the beginning of the rent-seeking process or equivalently some small range of increasing returns of scale in the technology. In that case profits need not be zero but would remain relatively small. If r > 1, then as long as N < N*, any potential entrant has some incentive to enter in the game. As we observed in the previous section, when N > N*, an agent contemplating entry will not engage in the process of rent-seeking because by doing this he will receive a negative expected payoff. Therefore the only possible solution with free entry is N* agents undertaking rent-seeking and

141

J. David Perez-Castrillo and Thieny Verdier 344

obtaining a strictly positive but generally small expected profit. Rents are not completely dissipated.

6. Stackelberg leader behavior Until now we have supposed that all the agents were identical and therefore a natural way to model their behavior was to view them all as Nash players. However, in many cases we find situations where firms or agents are not placed symmetrically in the rent-seeking game. Some of them may possess superior knowledge about the rent-seeking environment and may have better connections with the politicians before the beginning of the race to political favors. This kind of situation calls for a different type of modelization of the game and a leader-follower framework seems to be appropriate in this case. In order to analyse the consequences of such an asymmetry, let us consider the case where one of the agents plays as a Stackelberg leader compared to the other agents. In such a case he may use strategically his capacity of commitment of playing the first **shot'' in order to obtain a better outcome in the rent-seeking game. A particular interesting point to analyse in this context is how the shape of the technology of rent-seeking is related to the capacity of preemption of the lader of the game. As a matter of fact, we will see below that a technology of rent-seeking with increasing returns to scale (r > 1) allows the agent playing as a Stackelberg leader to preempt successfully the other players and forbid them the possibility of active entry in the rent-seeking game. This is most simply seen by considering the case of a K-person natural rentseeking game with free entry. In that case we know from proposition 4) that only K agents can be active players in a Nash equilibrium of the rent-seeking game. Now suppose that player 1 plays as a Stackelberg leader compared to the other players. It is easy to see that by playing M(r,X) as a bet, he is able to preempt the other players by preventing them from any possible active entry in the rent-seeking game. Is it in his interest to do that? In order to give an answer to this question we need some few more notations. Let us note the profit function of agent 1 as: En(a,B) =

^' '' a^ + B

Let assume that player 1, as a Stackelberg leader, bets first a > 0. The other agents play then between themselves as Nash players and there is an resulting Nash equilibrium with free entry between them. Let denote E as the set of the active agents in this equilibrium and let b | (a) be the bet of agent i, i € E, and

142

1.1.2.3 A general analysis of rent-seeking games 345 EII* (a) the profit he receives (which both depend of course on the amount a previously chosen by the Stackelberg leader). Thus we should have the following inequality: a +

E bf (a) > M(r,X) i€E

This inequality ensures that any agent jgE is not interested in participating in rent-seeking. But then we have: En(M(r,X),0) < En (a +

E br(a),0)

(7)

i€E

where En(. ,0) is the profit function of player 1 when he is the only one to play actively in the rent-seeking process. Moreover, as the technology of rentseeking is with increasing returns of scale (r > 1), it is easy to see that: v a > 0 such that b* (a) > 0 En(a + (a-f E biXa),0)>En(a,{a+ E bJ(a)) + (a4- E bpEnj(a) > i€E

i6E

> En(a, (a -h E br(a))

i€E

(8)

i€E

From equations (7) and (8) we see that agent 1 has an interest, as a leader of Stackelberg, to bet M(r,X) and preempt all the other potential players. The interesing point to note here is the fact that his possibility of total preemption of the Stackelberg leader is only feasible because of the non-convexity associated with the technology of rent-seeking. In the case of r < 1, it is easy to see that a leader of Stackelberg cannot preempt the other players from entering actively in the game. In that case the Stackelberg leader will still have an advantage over the other players but his power will be obviously more limited than in the case with increasing returns of scale.

7. Conclusion In this paper, we have tried to investigate more closely the implications of the rent-seeking technology on the shape of the typical reaction curve of an agent wishing to participate in rent-seeking. We saw that in the case of decreasing return of scale, this reaction curve is continuous in the bets of the other agents. On the contrary the case with increasing returns of scale shows a sharp discontinuity in the reaction curve. This systematic analysis of the reaction curve allowed us to discuss, in a consistent framework, the problem of ^'efficient rent-

143

J. David Perez-Castrillo and Thierry Verdier 346 seeking'* and the natural structure of competition that is supposed to appear in such games. In this respect, according to the type of the technology of rentseeking, we characterize completely the type of pure strategies Nash equilibria that may result in the game. Moreover, we have been able to investigate also the consequences of forms of strategic behavior that depart from the traditional Nash behavior. We considered the case where one of the agents has some kind of superior position compared to the other agents, namely he acts as a Stackelberg leader with respect to the other participants of the rent-seeking. We show that a technology with increasing returns of scale enables the leader to preempt the other players from entering actively into the race for rents. Evidently several extensions have not been explored in this paper. An interesting one may be to consider different alternative assumptions on the conjectures that one agent forms about the others* reaction to his behavior and to study the implications of those assumptions on the nature of the resulting equilibria of the rent-seeking game. Another interesting question would also be to consider rent-seeking processes where participants acquire explicitly some form of experience in the way to efficiently rent-seek bureaucrats. In that way we would have a fully dynamic rent-seeking process where memory effects take place (see Cairns,, 1989, for a first analysis of dynamic rent-seeking games). While all those extensions are obviously beyond the scope of this paper, we believe that the basic results that we get there are nevertheless a useful step in our understanding of rent-seeking processes. References Cairns, R.D. (1989). Dynamic rent-seeking. Journal of Public Economics 39: 315-334. Corcoran, W.J. (1984). Long-run equilibrium and total expenditures in rent-seeking. Public Choice 43: S9-94, Corcoran, W.J. and Karels, G.V. (1985). Rent-seeking behavior in the long-run. Public Choice 43: 227-246. Hillman, A.L. (1988). The political economy of protection. Murray Kemp: University of New South Wales, Australia. Hillman, A.L. and Samet, D. (1987a). Dissipation of contestable rents by small numbers of contenders. Public Choice 54: 63-82. Hillman, A.L. and Samet, D (1987b). Characterizing equilibrium rent-seeking behavior: A reply to Tullock. Public Choice 54: 85-87. Paul, C. and Wilhite, A. (1990). Efficient rent-seeking under varying cost structures. Public Choice 64: 279-290. Tullock, G. (1980). Efficient rent-seeking. In J.M. Buchanan, R.D. Tollison and G. Tullock (Eds.), Toward a Theory of the rent-seeking society. College Station: Texas A&M Press. Tullock, G. (1984). Long-run equilibrium and total expenditures in rent-seeking: Comment. Public Choice 43: 95-97. Tullock, G. (1985). Back to the bog. Public Choice 46: 256-263. Tullock, G. (1987). Another part of the swamp. Public Choice 54: 83-84.

144

1.1.2.3 A general analysis of rent-seeking games 347

Appendix 1 Let ^ - ( r - l ) a - J + Ira'-^ (a^ + B)-' > 0 r-1 r+1

^ af ^ — - B

1

. r-1 and thus 4* is decreasing for a < [ — T B

and increasing thereafter. Condition (4.A) is

then verified if and only if:

1 (r-1)^

^ B
7A^ (A4-B)2 + 2 A ^ (A + B ) - A ^ (A + B)2 > 0

I -1

A-J(A + B) + 2 > 0

0

(A2.3)

thus the denominator of the right hand side of equation (A2.2) is positive if and only if we have inequality (A2.3). When r < 1 this inequality is automatically satisfied. If r > 1, on the decreasing part of A(B) we have that A + B > 2 A, hence ( A + B ) ( l - r ) + 2r A > A > 2 A(l~r4-r) > 0 and thus the sign of the denominator of (A2.2). dA •• Let us show now that — > - 1. This is equivalent to showing that dB 1

r orAX>

-BX + rBX

r-l « A > -y-B r-l Let B* the point where the curve A(B) intersect the line A = —f~ B dA Then the property "TT > - 1 is verified for all point B if and only if B* > M(r). We may calculate uB

B* easily. l2

r^i7B*H^B.+B-

r ~ 4 B*2 X :^ 0

which gives:

"•°

(r-l)r-»r2r+i (2,-.)^ ""

hence B* > M(r) *>

(r-l)f-J r^r^^J {r-iy-^ -^—-rX^ > ^^ 7 — X^ (2r — I )•*•*

146

r^

1.1.2.3 A general analysis of rent-seeking games 349 o r3r+i > (2r-l)2r 3 3 inequality which is always verified for r € [1, x)- (When r > r the reaction curve is always increasand thu trivially the property is verified). OED.

Appendix 3 a) For the case r < 1. From proposition 3) we know that for any K s N there is a unique rentseeking equilibrium with K active agents. In each equilibrium the active agents derive a strictly positive profit. Suppose now that K < N and let us consider the stability of the equilibrium associated with this K. In order to do this we have to examine the behavior of one of the non-active agents. Has he got any incentive to stay inactive? By becoming active and anticipating that his expected profit will be the one of an active agent in a the new resulting equilibrium with K + 1 agents, he obtains a strictly positive expected payoff which is higher than staying inactive and contemplating an equilibrium with K active agents in which he receives 0. Hence in any equilibrium with K active agents the non-active agents always have an incentive to enter actively in the game. Moreover, it is not difficult to show that if agent K + 1 takes as completely given the bets of the other K agents, he has also an incentive to participate actively in the rent-seeking process. Therefore the unique Nash equilibrium and the only stable solution is the equilibrium with all the N agents active in rent-seeking. The conclusions on the equilibrium amount of resources and profits then follow immediately from proposition 3). b) In the case where r > I and N ^ TTT* ^^^^ proposition 3) there exists an equilibrium with N active agents. For the same reasons as in a) it constitutes the only stable equilibrium. c) In the case r > 1 and N > -—r» ^^ consider N* the number such that N*
0. For the same reasons as previously, all equilibria with K.active agents for K < N* are not stable. Moreover, again from proposition 3), no equilibria with K active agents exist for K > N*. We want to show now that any equilibrium with N* is stable. In order to do this, we have to show that EII* (N* -1-1) < 0. In that case, no inactive agent will have any incentive to move from the position of no investment. Showing that EII* (N* + 1) < 0 is equivalent, after some manipulation to show that; N*

N*-I N»2 r X

^

(r-ir-» X^

which is equivalent to |S|*I-2r (N* _ i)r > ( r - l ) r - l r'^^

(A3.2)

But then it is easy to sec that for all N < — - , the function N - N'-2r (N-1)^ is increasing in

147

J. David Perez-Castrillo and Thierry Verdier

350 N and therefore recalling equation (A2.1) we find that:

N*i-2r(N*~i)r >

—J

Try-I

= (r-ir-M2-r)r>(r-l)f-ir2-r.

The last inequality follows from {I-TY > r~^ which is always true for r € [!,»]. This demonstrates inequality (A2,2) and finally proves that any equilibrium with N* is stable.

148

OED

1.1.2.4 Rent-seeking with asymmetric valuations Public Choice 98: 415-430,1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.

415

Rent-seeking with asymmetric valuations KOFIO.NTI* Smeal College of Business Administration, Penn State University, 308 Beam BAB, University Park, PA 16802, U.S.A. Accepted 11 March 1997 Abstract. This paper analyzes TuIIock's rent-seeking game with asymmetric valuations for a variable range of the returns to scale parameter. A necessary and sufficient condition for a unique pure strategy Nash equilibrium is established. Equilibrium effort and expected profits are determined and subjected-to comparative statics analysis. Increasing the underdog's valuation induces both players to increase their efforts. Increasing the favorite's valuation increases his effort but decreases the effort of the underdog. Expected profits increase with a player's valuation but decreases with the valuation of the competitor. The impact of the returns to scale parameter is also analyzed.

L Introduction The theory of rent-seeking has advanced considerably since Tullock (1980) proposed his well-known game theory model to explore the issue of rent dissipation. Various researchers have clarified the conditions under which partial or full rent dissipation occurs. (See Corcoran and Karels 1985, Hilhnan and Samet 1987, Higgins et al. 1987, Shogren and Baik 1991, Perez-Castrillo and Verdier 1992, Baye et al. 1994; also see the recent survey by Nitzan 1994a for additional references on rent-seeking contests.) A striking feature of the majority of the existing theoretical and experimental research is the assumption that the contestants are identical or symmetrically situated. The rent-seeking paradigm has tremendous possibilities. It can be used to illuminate issues in procurement (Rogerson 1989), lobbying for trade protection (Hillman 1989), public goods (Katz et al. 1990), market entry (Nti 1995), and others. In many of these situations, the contestants may value the prize differently. In government procurement, a previous contract holder may value the prize more because it has sunk resources or gained experience (Baik 1994); in lobbying, the imposition or removal of various taxes and subsidies may be valued differently by different pressure groups because these policies will typically impact them differently (Becker 1983); * I thank an anonymous referee for helpful comments and suggestions.

149

Kofi O. Nti 416 pollution removal may be valued differently by different localities (Katz et aL 1990); and an incumbent may value a patent for a new substitute product more highly than a potential entrant because the incumbent has a monopoly position to protect (Baik 1994). These examples clearly show that asymmetric player valuations are natural and quite common in many applications of rent-seeking paradigm. Most of the extensive literature on rent-seeking have assumed that the contestants assign a common valuation to the prize. The few exceptions include Hillman (1988), Hilhnan and Riley (1989), Katz et al. (1990), Ellingsen (1991), Leininger (1993), Baik (1994), and Baye et al. (1993, 1996).^ Hillman (1988) and Hillman and Riley (1989) study a perfectly discriminating contest (i.e., a first price all-pay auction) and noted that asymmetric valuations may act as a barrier to entry and reduce rent-dissipation. They also extended their results to the Tullock contest with constant returns to scale rent-seeking technology. Ellingsen (1991) applies the results of Hillman and Riley (1989) to study the welfare imphcations of rent-seeking by prospective monopolists and strategic buyers. Baye et al. (1993) show that the designer of an all-pay auction may have an incentive to exclude certain high valuation lobbyist from the game. Baye et al. (1996) provide a comprehensive solution to the all-pay auction with asymmetric valuations, noting that the auctioneer's expected revenue may differ across the multiple equilibria. Clearly, the all-pay auction with asymmetric valuations has been wellstudied. In contrast, imperfectly discriminating rent-seeking contests (i.e., the lottery form) with asymmetric valuations have not been analyzed as comprehensively. Most of the analyses study asymmetric valuations for the special case of the Tullock game with constant returns to scale rent-seeking technology. Within this framework, Hillman and Riley (1989) relate total rent-seeking efforts to the harmonic mean of the players' valuations. In a study of public goods, Katz et al. (1990) discuss the impact of asymmetric valuations and population size on rent-seeking efforts in a group contest; and Leininger (1993) contains explicit expressions for the Nash equilibrium efforts and payoffs for constant returns to scale Tullock game with asymmetric valuations. Baik (1994) by-passes the Tullock game and uses a general rentseeking technology to study the qualitative properties of player efforts in a contest with asymmetric valuations and differential abilities. But his analysis does not readily apply to the generalized Tullock game, which is the subject of this paper. The purpose of this paper is to contribute to the theory of imperfectly discriminating rent-seeking contests with asymmetric valuations. We analyze the Tullock (1980) rent-seeking game with asymmetric valuations for a variable range of the returns to scale parameter. A necessary and sufficient condition

150

1.1.2.4 Rent-seeking with asymmetric valuations 417 for the existence of a unique pure strategy Nash equihbrium is estabhshed. We show that the player with the higher valuation expends more effort but both players allocate the same fraction of their valuations to rent-seeking activities. Thus the higher valuation player is favored to win the contest and the lower valuation player is the underdog. We comment on issues relating to rent dissipation; however, our primary focus is on the qualitative structure of equilibrium efforts and expected profits. We develop explicit expressions for equilibrium efforts and expected profits and subject them to comparative statics analyses. We investigate how changes in valuations affect the efforts and payoffs of the contestants. Increasing the underdog's valuation induces both players to increase their efforts. In contrast, an increase in the favorite's valuation leads him to increases his effort but induces the underdog to decrease her effort. Expected profits increase with a player's valuation but decreases with the valuation of the competitor. The impact of the returns to scale parameter is also analyzed. This paper extends and generalizes certain results on imperfectly discriminating rent-seeking contests with asymmetric valuations. Because we consider the Tullock rent-seeking technology for a variable range of the returns to scale parameter, our analysis generalizes certain aspects of Katz et al. (1990) and Leininger (1993), both of which involve the constant returns case. The focus of our analysis is closer to Baik (1994), but he employs a more general rent-seeking technology. However, we exploit the relative simplicity of the rent-seeking technology to produce more definitive results. We should mention that we study the qualitative properties of both effort and payoffs in the generalized Tullock model; how payoffs vary with valuations are not studied in Baik (1994). Other interesting relationships and differences between Baik (1994) and this paper will be discussed after we have developed the main results. A strength of this paper is that it contains simple expressions for the efforts and payoffs of the players in the general Tullock contest with asymmetric valuations. In addition, all the results in this paper are stated explicitly in terms of the parameters of the model, namely the valuations and the returns to scale parameter. The rest of the paper is organized as follows. The model is presented in Section 2. Section 3 provides a closed form solution and discusses conditions for the existence of a pure strategy equilibrium. Section 4 analyzes the effect of changes in player valuations on efforts and expected profits. Section 5 concludes the paper.

151

Kofi O. Nti 418 2. The model We consider Tullock's (1980) model of rent-seeking where two risk neutral players compete to win a prize but we assume that players valuation of the prize may be different. Let Vi be player I's valuation of the prize and let V2 be player 2's valuation, where Vi < V2. Suppose player i, i = 1 and 2, expends effort Xi in hopes of winning the prize, where the effort is measured in the same units as the prize. Given a profile of effort levels (xi,X2), we assume, as in Tullock (1980), that the probability that player i wins the prize takes the symmetric, logit form

where r > 0, and we may define Pi to equal 1/2 if xi = X2 = 0. In order words, the rent-seeking technology is P(x) = x*", where r is the returns to scale parameter. There is decreasing returns to scale when 0 < r < 1 and increasing returns to scale when r > 1. As noted earlier, a great deal of the rent-seeking literature has been devoted to the constant returns to scale case where r == 1. This paper considers returns to scale of all types. Given the above specification of the winning probabilities, the expected profit of player i is 7ri(xi,X2) = Pi(xi,X2)Vi-Xi, Vix[ ^r The first order conditions for an interior Nash equilibrium of the model are

The second order sufficiency conditions are

Later we will provide a remarkably simple condition that ensures that the second order sufficiency conditions are satisfied. We may rewrite the first order condition (3) as rVix|-'x[ , , ,,\ = 1 X + x'2 2 152

(5)

1.1.2.4 Rent-seeking with asymmetric valuations 419 and take the ratio of the two first order conditions to get the following relationship between equilibrium effort levels and a player's valuation of the prize:

Since V2 > Vi, the above relationship also implies that X2 > xi. These relationships are summarized in the next proposition. Proposition 1 The player who values the prize more expends more effort in equilibrium but both players allocate the same fraction of their valuations to the contest, regardless of the magnitude of the returns to scale parameter r. Proposition 1 is quite remarkable. Regardless of whether there is increasing, constant, or decreasing returns to scale, the higher valuation player expends more effort in equilibrium; however, the two players dissipate the same fraction of their values in rent-seeking activities. Katz et al. (1990) obtained Proposition 1 for the constant returns case but we have shown that it holds for all positive r that are compatible with the existence of a pure strategy equilibrium. Proposition 1 is also relevant for the two stage group contest of Katz and Tokatlidu (1996), which also employs a constant returns to scale rent-seeking technology. Under the endogenous sharing rule they employed, the value of the prize expected at the end of stage one is inversely proportional to the square of the group size. Thus the smaller group has a higher valuation for the prize. Proposition 1 implies that the smaller group should supply more aggregate effort than the larger group, which agrees with the results in Katz and Tokatlidu (1996). Proposition 1 raises interesting questions about the structure of relative advantages and profits of the players. Obviously, the results may depend on the valuations as well as the returns to scale parameter r. To obtain insightful answers to these questions, we now proceed to provide a detailed analysis of the model.

3. Closed form solution A closed form solution is available for the Tullock rent-seeking game with asymmetric valuations. The idea is to use the effort to value relationship (6) to eliminate Xj from the first order condition for player i and then solve the

153

Kofi O. Nti 420

resulting equation for xi. Substituting X2 = (V2/Vi)xi into the first order condition for player 1 and solving forxj yields 1

^

^t = (V^+V^)2 nn\.rr\2-

(?)

Hence M _

Vi

^2

V2

""^^

(Vi+v92

(8)

and

It is interesting to observe from (7) and (8) that when the players entertain a common valuation V for the prize (Vi =: V2 = V) we obtain the familiar results that Xj = X2 = rV/4. That is, each player dissipates a fraction r/4 of his or her valuation in rent-seeking. Thus, in the symmetric case, the fraction of the value dissipated per player increases (linearly) in r but is independent of the common valuation. When Vi and V2 are different but r = 1, the effort levels coincide with those obtained by Leininger (1993) for the simultaneous move game with equal rent-seeking abilities. Now let us examine the issue of rent dissipation under asymmetric valuations. Let v = V1/V2, 0 < v < 1, then

xr Vi

(l+vO^

and a ( x r / V O ^ r^v-^ dY (i+vo^^

^

Thus the fraction of a player's valuation expended in rent seeking is increasing in the valuation ratio V1/V2. That is, the more skewed the valuations, the smaller the fraction of the value dissipated in rent-seeking. It can also be seen from Figure 1 that the fraction of the value dissipated per player may or may not increase with r when the valuation ratio is fixed. Consider the three values of r used in Figure 1. When v = V1/V2 is small (v < 0.1), the fraction corresponding to r = 0.5 exceeds that for r = 1.0, which in turn exceeds thefi*actionfor r = 1.5. The relationship is reversed when v is large (v > 0.3). The relationship is not monotonic if v Hes between 0.1 and 0.3.

154

1.1.2.4 Rent-seeking with asymmetric valuations 421

f 0.4

r « 1.5 0.35 4-

0.3

+ r « 1.0

0.25

0.2

0.15

0.1

+

0.05

+

Figure 1. Fraction of player's valuation dissipated.

These insights may be usefully exploited by a contest designer to control the degree of rent-dissipation.

Proposition 2 The fraction of player valuations expended in rent-seeking increases with the valuation ratio v = V1/V2 but is not monotonic in the returns to scale parameter r. We may also calculate the winning probabilities and expected payoffs for the players. The equilibrium w^inning probability for player i is

155

Kofi O. Nti 422

p r - ^ .

(10)

And the expected payoffs are V\+' TT,

rVj+'V^

=

yr+l I [V^+V^-rV^] (V^+V^) 2

(11)

and V^+'

rV^+'V^

^2

^'^ [V^V^-rV,]. (V^+V92

(12)

Two observations are in order. First, equilibrium winning probabilities P* in (10) are computed as if the players fully dissipate their values in the rent-seeking contest. Of course, the players only dissipate a fraction of their values in equilibrium, but the winning probabilities are the same as in the non-equilibrium outcome where X] = Vj. This insight can be used to develop intuition on the structure of the model. For example, the higher valuation player is favored to win the contest. Thus we may follow the terminology of Dixit (1987) and refer to the higher valuation player as the favored player or favorite and the lower valuation player as the underdog, whenever it is convenient (or less verbose). We should note that if Vi < V2 then the equilibrium probability ratio Pj /P2 = (Vi /N2Y is increasing m the valuation ratio Vi /V2 but decreasing in r. In the symmetric case (Vi = V2 = V) the odds are equal and independent of r or the common valuation V. The second observation is that the equilibrium expected profits in 11 and 12 are positive if and only if V^ + V2 - rV[ > 0 for i =: 1 and 2. Since Vi < V2, the following assumption is sufficient for the analysis. Assumption The returns to scale parameter r and the players' valuations N\ and Yj satisfy the relationship Vj + V2 > rV2 > rV^j.

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1.1.2.4 Rent-seeking with asymmetric valuations 423 Proposition 3 Given Vi < V2, the Tullock rent-seeking game with asymmetric valuations has a unique pure strategy Nash equilibrium if and only ifY\ + V2 > rV2. Proof Since the prize is valued positively by both players, any pure strategy equilibrium involves positive effort levels by both players. The analysis has shown that there is a unique solution (xj, X2) to the first order conditions. In addition, the expected profit of both players is positive under the assumption. Evaluating the second order suflBciency condition for player 1, for example, and making use of xi /x2 = V1/V2, shows that

< 0, since rV2 — V2 — Vj < 0 from the assumption. Therefore player 1 is maximizing against X2, and similarly for player 2. Thus the sufficiency condition implies a unique pure strategy equilibrium. And if the sufficiency assimiption is violated, at least one player will receive a negative payoff; that player can improve by reducing his or her effort to zero, which will destroy the equilibrium. D Let V = V1/V2, where 0 < v < 1. Then we may write the sufficiency condition as r < 1 + v^. Given v, let R(v) be the maximum value of r that solves the inequality r < 1 + v^ Then for any v, 0 < v < 1, the Tullock rent-seeking game has a pure strategy equilibrium for 0 < r < R(v). It is easy to see that R(v) > 1 for all v > 0 and that R(l) = 2. This shows that, in general, R depends on the valuation ratio, is greater than 1, and equals 2 only when the contest is symmetric. (See Shogren and Baik 1991 and Baye et al. 1994 for a discussion on the crucial role of R = 2 in the analysis of symmetrically valued contest; also see Chung 1996 for a related result where R is not a constant but depends on the players common marginal valuation.) Figure 2 illustrates the general relationship between R and v. It is interesting to observe that R is increasing from 1 to 2 within the feasible range. Thus we may conclude that the more skewed the players valuations, the smaller the range of the returns to scale parameter supporting pure strategy equilibria.

157

Kofi O. Nti 424

0.6

0.4

0.2

H0

0.1

-h-—\ 0.2

0.3

0.4

\

h

H

0.5

0.6

0.7

1

1

0.8

0.9

1V 1

Figure 2. Dependence of R on valuation ratio v.

Evidently, the symmetric game has pure strategy equihbria for the largest possible range of the returns to scale parameter. 4. Comparative statics The need to obtain the comparative statics properties of contests and rentseeking games arises naturally in a variety of applications. Recently, Nti (1997) has developed comprehensive comparative statics results for symmetric contests. His analysis applies to the symmetric Tullock game with decreasing returns to scale. But his results do not readily extend to contest with asymmetric valuations or with increasing returns technology. It is also

158

1.1.2.4 Rent-seeking with asymmetric valuations 425 not straightforward to apply the results from the asymmetric contest model of Baik (1994), as we will explain later. Fortunately, we have obtained a closed form solution to the Tullock game. Therefore, we can use elementary methods to determine how efforts and payoffs respond to variations in the parameters defining the contest. First, we examine the response of effort levels. Performing the necessary differentiation, simplifying, and using the relationship V'j + V^ > rV^ > rV^ yields^ xV\N\ [V^+V^-rV^+rV^] > 0, ( V i + V 9 3 •[V^4 dx] dV2

{v\ + w\y

dxl

rV^-''V^+'

dWx

xV\+'' V - '

{v\+\\f

[rV^ - rV^] < 0,

[rV^-rV^] > 0 ,

and Fhi*

r V V

^ - ^^^^ aV2 ( V j + V ^ [W\ + Y\ - rV] - rVy > 0. This proves the next proposition. Proposition 4 Effort of the favored player increases with his own valuation and with the valuation of the underdog; effort of the underdog increases with her own valuation but decreases with the valuation of the favoredplayer. Proposition 4 shows that an increase in the valuation of the underdog raises the effort of both players. That is, competition becomes more intense as the valuation of the underdog increases. On the other hand, an increase in the valuation of the favored player dampens the effort of the underdog and stimulates the effort of the favorite. We may say that competition becomes less intense as the valuation of the favored player increases. This is quite interesting and seems to be in agreement with casual observations. Competition becomes keener when the gap between the contestants is not too large. Sports and political elections generally attract more attention and effort when the contestants are about equally matched.

159

Kofi O. Nti

426 Finally, we examine the response of expected profits to parametric variations. Performing the necessary differentiation, simplifying, and making use of the relationship V^ + V^ > rV^ > rV'j leads to d^

_ [Vf + (r+l)V-,Vy

aV2 -

;9 * av,

, v^ _ , v n +

(V^ + V93 1^1 + ^2

rV2j -

Y^-lyr+l

'"^'^^^

>0

( y r + y r ) 3 < «'

^Y^~^V^^+^

(V^+V93 L 1 •

^

iJ

(V^+V93

(V^ + V^)3

+ V2 - rV,J + ^^^ ^ ^^^3 > 0.

and aV2 ~

LV,

This proves the next proposition. Proposition 5 Expected profit of a player increases with his or her own valuation but decreases with the valuation of the other player. Proposition 5 shows that an increase in valuations have symmetrical effects on the players' expected profits. An increase in a player's valuation is beneficial to him or her but has an adverse effect on the competitor. A player benefits when he or she competes against others who have a lower valuation for the prize. This suggests that there may be incentives for sophisticated rent seekers to contest in arenas where the competition is likely to have a lower valuation. The reverse is that shrewd contest organizers may want to separate contestants with highly skewed valuations. Finally, it is instructive to comment on the relationship between the results in this paper and those of Baik (1994). Baik formulates a two player asymmetric contest to study the response of effort levels to variations in relative valuations and relative abilities. His results for the simultaneous move game with equal abilities is relevant to this paper. Using the notation of this paper, Baik's model involves a general rent-seeking technology P(x) and player valuations

160

1.1.2.4 Rent-seeking with asymmetric valuations 427 Vi = vV and V2 = V, where v > 0. Clearly, player 1 has a lower valuation for the prize when 0 < v < 1, and has a higher valuation when v > 1. Baik (1994) shows that player 1 is the favorite if the Nash equilibrium effort profile (x|,X2) satisfies vP'(x^) > P'(x^); player 1 is the underdog if the inequality is reversed. (See Lemma 2 of Baik 1994.) If P is linear, Baik's condition clearly implies that player 1 is the underdog if 0 < v < 1; player 1 is the favorite if v > 1. But if we consider P(x) =x\x ^ 1, then it is difficult to apply Baik's result to determine who the underdog is. To do so requires an explicit knowledge of the Nash equilibrium profile. Baik acknowledges that his result is conditional. We have shown in this paper (Proposition 1) that for P(x) = x"* the higher valuation player is unconditionally the favorite. Applying these results to Baik's model, we can conclude that player 1 is the underdog if 0 < v < 1; player 1 is the favorite if v > 1. This may also be verified by substituting our solution for Xj and X2 into Baik's condition.^ Baik (1994) also investigates how effort levels respond to changes in v. He showed that both players increase their efforts as v increases from zero to one, but as v increases beyond one player 1 expends more effort while player 2 expends less. (See Proposition 1 of Baik 1994, noting that with equal abilities the contest is "even" when v = 1 . ) For P(x) — x^ Proposition 4 in this paper gives the same qualitative results as Baik. To see this, note that increasing v from zero to one is equivalent to increasing the valuation of the underdog; increasing v beyond one is equivalent to increasing the valuation of the favorite. We established in Proposition 4 that effort of both players increase their effort when the underdog's valuation is increased; effort of the favorite increases as his valuation is increased but effort of the underdog decreases. Although we have the same qualitative results as Baik (1994), we should mention that Baik's conclusions are conditional. Baik requires that the conditions P"(x|)(P(x|) +P(x^)) < 2{V'{x\)f andP"(x^)(P(x|) + P(x^)) < 2(P'(x^))2 hold at the Nash equilibrium (x|,x^). For P(x) == x^ these conditions are obviously satisfied when 0 < r < 1, but when r > 1 it is difficult to determine from Baik (1994) alone whether these conditions are satisfied or not. Again, we may use our explicit expression for xf and X2 to check his conditions but that would be laborious. We also mentioned eariier that Baik (1994) does not include a discussion of how expected profits vary with valuations. Proposition 5 in this paper addresses these issues. On the other hand, Baik (1994) contains interesting discussions on how variations in relative abilities impact effort levels. Thus this paper and Baik (1994) complement each other substantively and technically.

161

Kofi O. Nti 428 5. Conclusion This paper extended the Tullock rent-seeking game to consider a situation where the players may have different valuations for the prize and for a variable range of the returns to scale parameter. It was shown that the player with the higher valuation expends more effort but both players allocate the same fraction of their valuations to rent-seeking activities. Thus the higher valuation player is favored to wm the contest and the lower valuation player is the underdog. A necessary and sufficient condition for a unique pure strategy Nash equilibrium was established. Equilibrium effort levels and expected profits were subjected to various parametric variations. An increase in the valuation of the underdog induces both players to increase their efforts. In contrast, an increase in the valuation of the favored player increases his own effort but decreases the effort of the underdog. Expected profits of a player increases with his or her own valuation but decreases with the valuation of the competitor. The impact of the returns to scale parameter was also studied. Similarities and differences between the results obtained under asymmetric versus symmetric valuations were noted throughout the paper. We also discussed the relationship between the results obtained here and those in Baik (1994), which also studies a two player contest with asymmetric valuations. This paper suggests several directions for future research. First, it contains a rich set of hypotheses that may be subjected to empirical or experimental testing. Asymmetric player valuations introduces an additional degree of freedom that can be exploited to improve experimental design and analysis of rent-seeking in actual or simulated markets. Secondly, the results and insights obtained here may be applied to illuminate issues in lobbying, public goods, procurement, and market entry where asymmetric player valuations are natural and prevalent. Finally, it would be interesting to generalize the analysis to derive analogous results for other specifications of the winning probability and also for games with sequential or dynamic structure.

Notes 1. Asymmetries in rent-seeking abilities or costs are discussed in Dixit (1987), Allard (1988), Baik and Shogren (1992), Leininger (1993), Baik (1994), and Gradstein (1995); strategic implications of introducing asymmetries in the order of moves are examined in Dixit (1987), Baik and Shogren (1992), and Leininger (1993). Group contests with asymmetric population sizes are discussed in Katz et al. (1990). Nitzan (1991) studies a group contest with asymmetric sharing rules; a two stage group contest of Katz and Tokatlidu (1996) with asymmetric population sizes also generates a contest with an asymmetrically valued prize.

162

1.1.2.4 Rent-seeking with asymmetric valuations 429 2. These variations are of significant interest when V2 > case where Vi = V2 = V has effort levels xj = Xj = V and r. Expected profits are TTJ* = TTJ = (1/4) (2 decreasing in r. 3. Nitzan (1994b) uses this approach to reconcile Baik (1993).

Vi. As noted eariier, the symmetric (r/4) V, which is increasing in both r)V, which is increasing in V and and Shogren (1992) and Leininger

References Allard, R.J. (1988). Rent-seeking with non-identical players. Public Choice 57: 3-14. Baik, K.H. (1994). Effort levels in contests with two asymmetric players. Southern Economic Journal 6\\ 361-31%. Baik, K.H. and Shogren, J.F. (1992). Strategic behavior in contests: Comment. American Economic Review 82: 359-362. Baye, M., Kovenock, D. and de Vries, C.G. (1996). The all-pay auction with complete information. Economic Theory 8: 291-305. Baye, M., Kovenock, D. and de Vries, C.G. (1994). The solution to the Tullock rent-seeking game when R > 2: Mixed-strategy equilibria and mean dissipation rates. Public Choice 81:363-380. Baye, M., Kovenock, D. and de Vries, C.G. (1993). Rigging the lobbying process: An application of the all-pay auction. American Economic Review 83: 289-294. Becker, G.S. (1983). A theory of competition among pressure groups for political influence. Quarterly Journal of Economics 98: 371-400. Chung, T-Y. (1996). Rent-seeking contest when the prize increases with aggregate efforts. Public Choice %1: 55-66. Corcoran, W.J. and Karels, G.V. (1985). Rent-seeking behavior in the long-run. Public Choice 46: 227-246. Dixit, A. (1987). Strategic behavior in contests. American Economic Review 11: 891-898. Ellingsen, T. (1991). Strategic buyers and the social cost of monopoly. American Economic Review %\'. 64^-651. Gradstein, M. (1995). Intensity of competition, entry and entry deterrence in rent seeking contests. Economics & Politics 7: 79-91. Higgins, R.S., Shughart, W.F. and Tollison, R.D. (1985). Free entry and efficient rent-seeking. Public Choice 46: 247-258. Hillman, A.L. (1989). The political economy of protection. London: Harwood Academic Publishers. Hillman, A.L. and Riley, J.C. (1989). Politically contestable rents and transfers. Economics and Politics 1: 17-39. Hillman, A.L. and Samet, D. (1987). Dissipation of rents and revenues in small number contests. Public Choice 54: 63-82. Katz, E., Nitzan, S. and Rosenberg, J. (1990). Rent-seeking for pure public goods. Public Choice 65: 49-60. Katz, E. and Tokatlidu, J. (1996). Group competition for rents. European Journal of Political Economy 12: 599-607. Leininger, W. (1993). More efficient rent-seeking - A Miinchhausen solution. Public Choice 75: 43-62. Nitzan, S. (1991). Collective rent dissipation. Economic Journal 101:1522-1534. Nitzan, S. (1994a). ModelHng rent-seeking contests. European Journal of Political Economy 10:41-60. Nitzan, S. (1994b). More on efficient rent seeking and strategic behavior in contests: Comment. Public Choice 79: 355-356.

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Kofi O. Nti 430 Nti, K.O. (1997). Comparative statics of contests and rent-seeking games. International Economic Review 38: 43-59. Nti, K.O. (1995). Potential competition and coordination in a market entry game. MSIS working paper 95-9, Penn State University. Perez-Castrillo, J. and Verdier, R.D. (1992). A general analysis of rent-seeking games. Public Choice 73: 335-350. Rogerson, W.R (1989). Profit regulation of defense contractors and prizes for innovation. Journal ofPolitical Economy 97: 1284-1305. Shogren, J.R and Baik, K.Y. (1991). Reexamining efficient rent-seeking in laboratory markets. Public Choice 69: 69-79. Tullock, G. (1980). Efficient rent seeking. In J. Buchanan, R. Tollison, and G. Tullock (Eds.), Toward a theory of the rent-seeking society. College Station: Texas A&M University Press, 97-112,

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1.1.3.1 Dissipation of contestable rents by small numbers of contenders Pubiic Choice 54: 63-82 (1987). © 1987 Martinus Nijhoff Publishers (Kluwer), Dordrecht, Printed in the Netherlands.

Dissipation of contestable rents by small numbers of contenders

ARYE L. HILLMAN Bar-Ilan University, Ramat Gan, 52100, Israel and University of California, Los Angeles, Los Angeles, CA 90024, USA DOV SAMET* Bar-Ilan University, Ramat Gan, 52100, Israel and Northwestern University, Evanston, IL 60201, USA

1. Introduction 1.1 Rent dissipation The theory of rent seeking with its origins in the observations of Gordon Tullock (1967) - or to use Jagdish Bhagwati's (1982) proposed term, the theory of directly unproductive profit-seeking activities - is concerned with the potentially adverse effects on resource allocation of incentives to capture and defend artificially-contrived rents and transfers. The scope for social loss proposed by the theory derives from the relation between the value of a contestable prize and the value of the resources attracted into the contest to determine the beneficiary of the prize. Underlying this social loss is a specification of how rational behavior by optimizing agents links the value of the prize sought to the resources expended. It has been traditional to assume competitive behavior in describing the activities of lobbying and influence seeking. Then, if some further conditions are satisfied,* the total value of the resources expended precisely equals the value of the prize sought, so dissipation is complete.^ Consequently, the social cost associated with contestability of a rent can be inferred from the value of the rent itself, and the detailed and hard-to-comeby information on individual outlays made in the course of the contest becomes unnecessary. By basing their analyses on competitive dissipation, contributors to the rent seeking literature (see the review by Robert Tollison, 1982) have been able to presume that the observed value of a contested rent is an exact measure of the associated social cost of monopoly power or regulation. Similarly, in the trade-theoretic literature where the rights contested are to quota premia or revenues from trade taxes (Krueger, 1974; Bhagwati *We thank without impHcating Harold Demsetz, Franklin Fisher, Jack Hirshleifer, John Riley, Al Roth and Gordon Tullock for comments on a previous draft of this paper.

165

Arye L. Hillman and Dov Samet 64 and Srinivasan, 1980, 1982, 1983) rent and revenue seeking have been portrayed as competitive activities.^ Competition may however well be absent from contests to secure rights to rents and transfers, just as it is absent from the regulated, monopolized and protected markets where the rents arise. Special knowledge, connections, prior positioning in relationships within bureaucratic structures, political advantage, or a general historically-based advantage of incumbent placement, may all restrict participation in a contest to only a small number of participants. The assumption of a perfectly competitive environment therefore makes the claim of social loss due to contestability of rents or transfers vulnerable to the observation that contests may be restricted to small numbers of participants. Absence of conditions assuring competitive rent seeking is for example an element in Franklin Fisher's critique (1985) of the Tullock/Krueger/Posner presumption that an observed rent reflects a social cost of equal value."^ In this paper we present a theoretical basis for a presumption of complete dissipation of indivisible artificially-contrived contestable rents or transfers which does not rely on a competitive environment. Rational equilibrium behavior is investigated in small-numbers contests wherein the successful contender will have made the greatest outlay in seeking to influence the outcome in his or her favor.^ Our basic result is that with risk-neutrality and no minimum outlay required for participation, equilibrium behavior results in the sum of all outlays made being equal to, in an expected sense, the value of the prize secured by the successful participant. Such expectationally complete dissipation arises for any number of participants in a contest. Hence, the number of rival contenders does not influence the expected magnitude of social loss due to contestability of rents or transfers. Dissipation is quite simply complete on average independently of the number of individuals placed to participate in a contest. A theoretical foundation other than limiting competitive behavior is therefore provided for basing estimates of the social costs of contestability of rents and transfers on observed values of the prizes contested. With appropriate measurement (see Franklin Fisher and John McGowan, 1983; Harold Demsetz, 1985), averaging on an economy-wide basis, any number of potential monopolists is consistent with an association of the full value of a contested rent with an equivalent social cost. Or any number of individuals competing for a position in a bureaucratic hierarchy is consistent with outlays by contenders equal on average to the rent associated with incumbency. In the trade-theoretic settings, there may not be free entry into contests where the objective is to influence the transfer policy of governments, but dissipation of rents arising from trade restrictions can nevertheless be proposed to be expectationally complete even if the number of contenders is small. 166

1.1.3.1 Dissipation of contestable rents by small numbers of contenders 65 1.2. Equilibrium behavior In the contests with which we are concerned - where all outlays made are irretrievably lost and the largest outlay determines the successful contender - it is well known that there can exist no pure strategy equilibrium. An equilibrium requires that no contender have an incentive to change his behavior, given the behavior of rivals. However, if the contender making the greatest outlay wins, the incentive is ever present to expend marginally more than the previous highest outlay. This had led to conjectures that no equilibrium exists. We shall show that there does however exist a rational pattern of equilibrium behavior for rival contenders characterized by mixed strategies. Let the value of the indivisible prize being contested be given by V, and let there be n risk-neutral agents with the requisite information and positioning to compete symmetrically as potential beneficiaries. Denote by x > 0 the outlay made by a potential beneficiary, where there is no minimum outlay required. Then there is a unique symmetric Nash equilibrium which is described by contenders' randomizing choice of outlays according to the continuous function F*(x) = (—)i/(n-i),

0 < X< V

where F*(x) is the probability that a contender's outlay does not exceed x. Thus, when there are but two contenders, equilibrium behavior entails the choice of an outlay from the uniform distribution over (0, V). That the equilibrium is the uniform distribution for the two-contender case has been previously noted by Hirshleifer and Riley (1978). When the number of contenders increases to three, equilibrium behavior entails randomization of outlays according to (x/V)^^^; and so on. As the number of contenders further increases, the equilibrium distribution places increasingly greater weight on smaller outlays, to compensate for the reduced Hkelihood of any particular contender winning. When individuals adopt the above equilibrium pattern of behavior, the expected value of an outlay is given by

Ex = J^xd{—)^/ 0 < X < V 1

V^ X

171

Aiye L. Hillman and Dov Samet 70 (ii) The associated equilibrium payoff ir* is zero. Proof: We begin by showing that F* in (6) is an equilibrium: When player i outlays x, his probability of winning is given by F*(x)"~* (i.e., the joint probability that the other (n - 1) contenders outlay less than x). Taking participant 1 as representative, the expected payoff from outlaying x is Tj(x, F*, . . . , F*) = F*(x)n-»V-x.

(7)

Thus, substituting for F*(x) from (6) in the range [0,V], p*(x)n-iv_x == (x/V)V-x = Tj(x, F*, . . . , F*) = 0.

(7')

That is, each pure strategy in [0,V] yields an expected return of zero, and therefore so does the equilibrium strategy F* which mixes these pure strategies. TT* is therefore zero. Clearly outlays greater than V yield less than 0. This with (7') guarantees by Proposition 1 that F* is an equilibrium. We now proceed to demonstrate that F* in (6) is the unique symmetric equilibrium. As a first step to showing that F* must be given by (6), substitute X = 0 into (4) and, noting that Tj (0, F*, . . . , F*) ^ 0, it follows that TT* ^ 0. That is, the equilibrium payoff cannot be negative. The option always exists of outlaying zero and receiving a return (with certainty) of zero, making zero a lower bound to ir*. Next, we observe that the equilibrium distribution F* must be continuous.^ Since F* does not place positive weight on points, the probability of ties is zero. An individuaFs expected gain x* is then given by the probability F*(x)"~^ that x exceeds the maximal bid of the other ( n - 1) contenders, multiplied by the rent V, minus the outlay x. T^ (F*, . . . , x, . . . , F*) is continuous in X and hence, by Proposition (2), for each increasing point x of F*, F*(x)n-l V - X = TT*.

(10)

In particular x = 0 is an increasing point^ and so yields TT*. Hence the expected return at x = 0 can be used to establish TT*. Substituting x = 0 into (10), we obtain F*(0)"-^V-0 = TT* which reveals that x* = 0. It therefore follows from (10) that at each increasing point of F*, F*(x)"-^ = x/V.

(11)

Since F* is continuous and increasing, (10) holds for all x in the interval [0,V]. That is, the set of all increasing points coincides with this interval. (6) follows directly from (11). The unique symmetric equilibrium is therefore given by F* in (6).

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1.1.3.1 Dissipation of contestable rents by small numbers of contenders 71

2.3 Rent dissipation Now consider rent dissipation. In the symmetric equilibrium described by F*, the expected outlay of a participant is Ex = [ X dF*(x)

(12)

0

= j

Xd(^)l/(n-l)

0

= [ xF*'(x)dx ! y-\/{n-\) (n-1)

[ x^/ 0 since he would always do better outlaying (a - e). Hence, there are at least two active individuals. Proposition 4: A set of strategies in which ( n - m ) players are inactive

173

Aiye L. Hillman and Dov Samet 72

(2 ^ m ^ n) and the remaining players choose the mixed strategy given by 0

X ^0

F*(x) = (—)J/(ni-i) 0 < x ^ V 1

V A, ^^ ^ U(V-x)~U(-x) V

(23) ^ ^

Thus Ex = V - f F*(x)dx = V - [ (x/V)i/ 0 such that agent7 will bid on the interval [jS —e, /3] with zero probability, for all 7V/. But then agent / is better off spending {fi — e) rather than jS since his probability of winning is the same, contradicting the hypothesis that X/ = /? is an equilibrium strategy. Given this result, it follows immediately that if there are just two agents, they must have the same maximum spending level. For if jc is agent I's maximum spending level, agent 2 wins with probability 1 by spending x and vice versa. A similar argument establishes that the minimum spending level is zero for each agent. To see this, suppose to the contrary that agent / spends less than ^ with zero probability, where i3>0. Then any spending level between zero and jS yields a negative payoff since the probability of winning is zero. Since other agents can always spend zero it follows that no other agent will spend in the interval (0,i5). But then agent / could reduce his spending level below jS without altering the probability of winning, contradicting the hypothesis that agent / could, in equilibrium, do no better than take ^ as his minimum spending level. Given these results, if we define 1 — G/(JC/) to be the probability that agent i spends more than jC/, then G/(x/) is continuous over (0,oo). If 00 for a l l x , > 0 . (b) With only 2 active agents the maximum spending levels are the same. (c) At most one agent spends zero with strictly positive probability. 3.

TRANSFERS

We now consider the simplest case of a contest where a transfer is to take place between two agents and provide a complete characterization of the equilibrium. If agent 1 spends x i his expected payoff is j^r _

j probability j [value as J _

(probability ) jpayment )

I of winning j i winner j

I ^^ losing j 1 as loser J

= Pl^l =

where

-Li

+

(l-Pl)^l VLPI

-

_

,^x

--^1

xi,

Vi = Wi -h L^.

The amount V/ is the gross value to agent / of winning the competition relative to the option of remaining inactive. Henceforth we shall assume that agents are ranked according to their gross values, that is Vj exceeds V2. Since, by Proposition 1, the probability of a tie is zero for any jcj > 0 , agent I's win probability is G2(xi). Agent I's expected payoff is therefore U,(xO - - L , + G2(xi)v, - xi.

(3)

Arguing symmetrically, agent 2's expected payoff is U2(X2) =

-Lj

+ Gi(X2)V2 -

X2.

(4)

By remaining inactive, agent / loses L,. Therefore agent / will enter with probability 1 whenever his equilibrium payoff exceeds — L/. To analyze the equilibrium, we first note that, since agent 2 has the option of remaining out of the contest, he will never spend more than his valuation and so earn a return below — L2. Then agent 1 will always enter since, for any small e, he can guarantee himself a return o f v i - V 2 - L i — e b y spending V2+e. It follows that f/1 > - L1. Setting xx = 0 in (3) we conclude that G2(0)>0. We now show that the equilibrium expected payoff for agent 2 must be — L2. For, if not, setting jc2 = 0 in (4) we obtain G,(0)>0. Moreover, with U2> —L2, agent 2 also enters the contest with probability 1. With both Gi(0) and G2(0) strictly positive and both agents always competing,

191

Arye L. Hillman and John G. Riley 24

HILLMAN AND RILEY

both agents must spend zero with strictly positive probability. But this contradicts Proposition 1. Then U2 = —L2 and so, from (4) ^1(^1) = - ^ ^ , X, € [0,V2].

(5)

V2

That \s, agent 1 's equilibrium mixed strategy is to spend according to the unifonn distribution over [OjVj. Since both agents have the same maximum spending level V2, we know that V2 is in the support of agent I's bid distribution. Moreover G2(v2) = i. Setting JC{ = V2 in (3) we obtain ^1 = G2(x2)v, - X2-I.1 = vj - V2 - L i -

(6)

Rearranging, it follows that agent 2's equilibrium mixed strategy is

\

V| /

Vi

\

Vj /

\

Vi

/

^.(7) V2

Note that agent 2 makes a strictly positive bid with probability 1 - G,(0) = V2/V1 < 1. The most natural interpretation of this result is that agent 2 stays out of the contest with probability (1 — V2/F1) and enters with probability V2/V1. From (7), conditional upon entering the contest, agent 2 also adopts a uniform mixed strategy over the interval [0,V2]. To sunmiarize, we have derived the following result.

Proposition 2: Characterizionon of the Equilibrium With perfect discrimination and two agents whose gross valuations are V| and V2 (v2:^ Vj), agent 1 always enters the contest while agent 2 enters with probability V2/V|. Conditional upon entry each agent spends according to a unifonn mixed strategy over the interval [0, V2]. For the special case of pure rent seeking, the gross valuation v, is simply agent f s value of the prize. As we have seen, the value to agent 2 is completely dissipated in the contest while agent 1 has an expected net gain equal to Vj —V2, the difference in valuations. While the model is rather different, there is a close parallel here widi the ouUx>me under Bertrand price competition when firm 1 has a lower marginal cost than firm 2. Price competition eliminates any profits for firm 2 and firm 1 's unit profit is the difference in marginal costs. Appealing to Proposition 2, we can easily compute &e expected spending levels of the two agents. Agent Ts spending is uniformly distributed on [0,V2] and so

192

1.1.3.2 Politically contestable rents and transfers POLITICALLY CONTESTABLE RENTS AND TRANSFERS

25

his expected spending is (1/2)V2. Conditional upon entering, agent 2's spending is also (1/2)V2. Multiplying the latter by the probability of entry, V2/V1, expected total spending is therefore

.,„..,,=i,.±,(^).^(„^)

(8)

We therefore have Corollary 1. With perfect discrimination and two agents whose gross valuations are v, and V2 (v2^V|), expected total spending is E[Xx+X2]

-" [^]-

For the symmetric case this, reduces to the common gross valuation v. See also Hirshleifer and Riley (1978) or Hillman and Samet (1987) for a discussion of this case. However, under asymmetry, note that expected spending is lower than either of the gross valuations. We now introduce a third agent with gross valuation V3 where V3 V3 > . . . > v„. Then if agents 1 and 2 act as if there were no other agents, the other agents have no incentive to compete. This result is most easily understood for the case of pure rent seeking. That is, Vi = Wf and L/ = 0. Then, even with only two agents, rent seeking expenditures are sufficiently high to eliminate any net surplus to the lower valued agent. That is, for any x V2 > V3 . . . > v„ and assume that all n agents actively participate in the contest. Hence x, > 0, / = 1, . . ., n. The harmonic mean of the n agents' valuations is «

1

(21)

From (20) =

2/1

199

Arye L. Hillman and John G. Riley 32

HILLMAN AND RILEY

Summing over n ^n -

Z xj = 4 ( 1 : — ) .

From the definitions of 5„ and v it follows that the total value of the outlays made in the contest is

'^ = ( ^ )

(22)

'-

Thus, when valuations of the political prize differ, total outlays closely approximate the harmonic mean of individuals' valuations as the number of participants increases. TuUock investigated the symmetric case where V/ = v, 1 = 1 , . . . , n. In that case (22) reduces to

s„ = ( - ^ )

V,

(22')

with the common valuation v replacing the harmonic mean of valuations. The same limiting rent-dissipation result emerges now with respect to the common valuation v. When evaluations differ, we cannot however presume that all agents will choose to participate actively in a contest. Consider the appearance of an (n + l)th individual whose valuation of the prize is v„+i v^, there is a unique ''symmetric equilibrium" (symmetric in the sense that ail agents with identical values use the same strategy), as well as a continuum of asymmetric equilibria. The expected sum of the bids (revenue to the auctioneer) varies across the continuum of equihbria; there is not "revenue equivalence." The case where v^>V2>v^>'">v„is known to have a unique equihbrium (Hillman and Riley, 1989).^ Our theoretical results have important imphcations for economic applications of the all-pay auction. To highlight these imphcations, Section III reconsiders the regulation game analyzed by Wenders (1987) and ElHngsen (1991). II Characterization of equiUbria The all-pay auction with complete information does not have a Nash equilibrium in pure strategies, but does have a Nash equilibrium in mixed-strategies. Accordingly, let G^(Xj.) denote the cumulative distribution function (cdf) representing the equilibrium mixed-strategy of player i. Player z is said to randomize continuously onA^Rif he plays a mixed strategy that is atomless (i.e., contains no mass points) and has a strictly increasing cdf almost everywhere on A. Our first theorem characterizes equilibrium for the case when m > 2 players have the highest valuation of the prize. For this case, Hillman and Samet (1987) have shown that there exists a symmetric equilibrium and a finite number of asymmetric equilibria where some agents with the highest valuation bid zero with probability one, and claim this exhausts all equilibria. Our Theorem 1 shows, however, that there actually exists a continuum of asymmetric equilibria when three or more players have the highest valuation of the prize. Nonetheless, we show that all of the equilibria imply the same expected payoif (zero) for each player, and yield the auctioneer the same expected revenue. Theorem 1: When i^i = •••== i^^ > t)^+j >•••> u„ and m > 2: (A) Ifm = 2, the Nash equilibrium is unique and symmetric. If3 0.^ When two or more players randomize continuously on a common interval, their corresponding cdfs are identical over that interval.^ (B) In any equilibrium, the expected payoff to each player is zero. (C) All equilibria are revenue equivalent: the expected sum of the bids in any equilibrium equals v^.

"^ In this unique equilibrium, only players one and two actively bid (players 3 through n bid zero with probabiHty one). ^ If 6; > u,, player i bids 0 with probability one. ^ Equation 2 below summarizes the algebraic form of the complete set of equilibria.

211

Michael R. Baye, Dan Kovenock, and Casper G. de Vries 294

M.R.Bayeetal.

The formal proof of Theorem 1 is similar to the proof contained in the Appendix for our Theorem 2 below, and is thus omitted (our 1990 working paper contains a complete proof). However, it is useful to highhght some of the features of equilibrium, as well as some intuition for the existence of a continuum of equilibria. The basic issues can be illustrated in the case where m = n = 3, so that Vj^=^V2=v^{ = v, say). Theorem 1 implies, in this case, that in every equilibrium two players randomize continuously on the interval [0,t;], while the third player randomizes continuously on the interval [b, v] and concentrates all remaining mass at zero (this mass is (b/v)^'^, and is thus zero if & = 0). (Note that ^ > 0 is an arbitrary constant). Since two players randomize continuously on [0, u], and any atoms in the third player's mixed strategy (player 3's, say) are isolated at 0, the highest bid is positive and unique with probability one. Furthermore, since zero is contained in the support of all three players' mixed strategies and at least two players use mixed strategies that do not put mass at zero, each player earns an expected payoff of zero. Given the characterization of the support of each player's mixed strategy, we know that all three players randomize continuously on [b, u], and hence, all three are capable of generating a winning bid in the interval [b,v^. Equihbrium requires that, for any bid in [^, y], each player earns an expected payoff of zero, given the mixed strategies used by the other two players. Three non-degenerate mixed strategies over [b, u] are uniquely determined as the solution to three equations that set the expected payoff of each player i to be zero for bids in [b, i;]: For i ^ j,fc Uiix) = Gj{x)Gf,{x)[t? - x] - [1 - Gj{x)G^(x)']x = 0

\fxelb,vl

The solution to this system of equations is symmetric and given by G, = G2 = G3 - (x/i;)^/2 for xe[5,t;]. The probability player 3 submits a winning bid in the interval [0,5] is zero, since the characterization of player 3's support requires that (remaining) mass of G^ib) = {blvf^ be isolated at 0 if b > 0. Given G^jibX and the fact that only players 1 and 2 can submit a winning bid in the interval [0, b'] with positive probability, the mixed strategies for players 1 and 2 must satisfy For / ^ i, 3: u^x) = Gj{x)G^{b)[p - x] - [1 - Gjix)G^{b)^x - 0 Vx6[b, vl For a given b, the solution to this system of equations is symmetric: G, = G,=^ (x/v)lG,{b)r'

= (x/v)(b/v)- 'f^ for xe [0, b].

These mixed>strategies for players 1 and 2 are sufficiently aggressive on the interval [0, b'] to ensure that player 3 will not find it profitable to deviate by submitting a bid in the open interval (0, b). Thus, for a given b, we have constructed Nash equilibrium mixed-strategies for the three players. On the interval [b, v\, all players randomize continuously according to the three-player symmetric equilibrium. On the interval [0, b], player 3's mixed strategy concentrates all mass at zero (unless b — 0), while players 1 and 2 randomize continuously according to mixed-strategies that are proportional to the two-player symmetric equilibrium. But since b is arbitrary, by varying b from 0 to

212

1.1.3.3 The all-pay auction with complete information The all-pay auction

295

V one generates a continuum of equilibria, ranging from the unique symmetric equilibrium (when b = 0)to the extremely asymmetric one in which only players 1 and 2 actively compete (when b = v, player 3 bids zero with probability one).*^ More generally, Theorem 1 allows us to explicitly characterize the algebraic form of the family of equilibrium mixed strategies for the case where v^=V2='V^ — ,>u^+i>'••>!;„. Let v = v^=v2 • v . = •• v^. By the theorem, players m + 1 through n bid zero with probability one, so suppose without loss of generality that players i=l,2,,.,,h, m>h>2, randomize continuously over [0,i;] with players i = /i + l,...,m randomizing continuously over [^,-,1;], with bf^+i< bf^+2^'" ^b^v^^^>

••- > v„, and 3 v^+^> ••• > t>„. By the theorem, players m + 1 through n bid zero with probabihty one, so suppose without loss of generality that of the players {2,...,m} players i = 2,...,h,h>2 randomize continuously over (0,172], with players i = /i 4-1,,.., m randomizing continuously over {bi, V2}, (where b^ = V2 implies ai(0) = 1) with ^ft+i i

^ This serves as a caveat to the claim by Magee, Brock and Young(1989, p. 217) that two-ness is a general property of political contests, ^^ If^>i;2,a,-(0)=1. ^^ Equation 4 below summarizes the algebraic form of the complete set of equihbria.

214

1.1.3.3 The all-pay auction with complete information The all-pay auction

297

;e{/i-f l , , . . , m - l }

-Hih-i)

^21

^1

J

lk>H

J

(4)

In addition to the multiplicity of equilibria, the key imphcation of Theorem 2 is part C: expected revenue varies across the continuum of equilibria. Note that the theorem states that expected revenue is maximized in the equilibrium that maximizes the expected bid of the player with the highest valuation. Given the form of the mixed strategies in equation (4), this occurs in the asymmetric equilibrium where player 1 and exactly one other player submit a positive bid with positive probabiHty.^^ To complete the characterization, we need the following result originally formulated by Hillman (1988) and Hillman and Riley (1989) (a rigorous proof is contained in our 1990 CentER working paper): Theorem 3 (Hillman and Riley): If Vi>V2>v^>'" >v„, the Nash equilibrium is unique. In equilibrium, player 1 randomizes continuously on [0,y2]- Player 2 randomizes continuously on (0,^2], placing an atom of size 0C2{0) == (v^ — V2)/vx at zero. Players 3 through n bid zero with probability one. Player Vs equilibrium payoff is uf = v^--V2, while players 2 through n earn payoffs of zero. The algebraic form of the equilibrium mixed strategies for the case when i;i > i;2 > 1^3 > ••' > y^ are as follows. Players 3 through n bid zero with probabihty one. Players one and two randomize according to G^(x) = x/v2 and ^iM

— (^1 — ^2 + ^)/^i ^or XG[0,I;2].

Ill A concluding example We conclude with an example that highHghts our results in the context of the regulatory contest discussed by Wenders (1987) and Ellingsen (1991).^^ Suppose ^^ Milgrom (1981) and Bikhchandani and Riley (1991) examine a similar issue in standard (winner pay) auctions, and find the opposite result often holds for classes of standard auctions: symmetric strategies may yield higher expected revenues. ^^ Of course, there are numerous other applications, as noted in the introduction. For example, the following analysis is analogous to the case of an incumbent versus a number of potential entrants as discussed in Rogerson (1982) and Dasgupta (1986).

215

Michael R. Baye, Dan Kovenock, and Casper G. de Vries 298

M. R. Baye et al.

M > 2 potential producers compete for the monopoly right to run a public utility. They face opposition from a consumer organization. The regulatory body decides to reward the organization which exerts the highest effort in the lobbying process. If this turns out to be one of the producers, the monopoly solution is implemented. If the consumer organization wins, the marginal cost pricing solution is implemented. If the consumer group wins, it earns a payoff equal to the sum of the would-be monopoly profits (call this amount "T" for "Tullock square") and the would-be deadweight loss (if, for "Harberger triangle"). If one of the producers wins, it earns the monopoly profits, T. Thus v^ = T-hH and y^ = ^3 = ••• = % + i = T. By Theorem 2A, there exists a continuum of equihbria to this game, and by 2C the equihbria are not revenue equivalent. In particular, the expected revenue to the regulator is EXX.. = % , + ( I - J ) E X , = ^ + ^ £ X , .

(5)

Since Ex^ varies depending upon which equihbrium is played, when the regulator receives the lobbying expenditures as "bribes" she is not indifferent to the equilibrium that is played. By Theorem 2C, Ex^ is maximized when only one of the firms participates in the lobbying process. ^"^ The selfish regulator does best in the equilibrium where only the consumer group and one of the M firms engage in lobbying. It turns out that the expected social waste due to lobbying also depends on which equihbrium is played. Suppose that only a proportion 1,0 < A < 1, of the lobbying expenditures is socially wasteful (see, e.g., Fudenberg and Tirole, 1977; Brooks and Heydra, 1990; and Dougan, 1991). Expected social waste, W, equals the expected deadweight loss plus a fraction X of the expected lobbying expenditures. If P^ is the probability the consumer group wins, then the expected social waste is W = il-P,)H

+ XEY^x,

(6)

Using equation (5) and the fact^^ that Ex^ = P^v^ + DJ — v^, this can be written as W=^XT + {l-X)j^{T-ElxJ).

(7)

If A = 1 (all of the lobbying is socially wasteful) the expected social waste is T. Notice that this result is independent of which equilibrium is played.^ ^ In contrast, when ^* As an example, consider the three player case with v^ = 2, and yj ~ Ug = 1. In the most asymmetric equihbrium, player 3 bids zero with probability one, while G^ — x and Gj = (1 + x)/2 on [0,1]. In this equihbrium, Exj^ = 1/2, and by equation (5), ZExi — 3/4. In the "symmetric equilibrium'*, player one randomizes with Gi(x) = x[(l + x)/2]~ ^^^, while players two and three use 02 = 0^ = [(I + x)/2]*^^. In this case,

X;£:x,= [5-2v^]/3, which is less than 3/4, as of course it must be by Theorem 2C. ^' Theorem 2B implies that, in equihbrium, Eu^ = P^v^—Ex^==v^—V2> ^^ Ellingsen (Proposition 1) considers the case where A = 1 and a finite number of possible equilibria. Equation 7, however, reveals that EUingsen's result is valid across the entire continuum of other equilibria.

216

1.1.3.3 The all-pay auction with complete information The all-pay auction

299

0 < /I < 1 the expected social waste is a decreasing function of Ex^, which in turn depends on which equilibrium is played. By Theorem 2C, it follows that the more symmetric the bidding strategies of the producers, the greater the expected social waste, W. This holds irrespective of A, except when lobbying is completely wasteful (in which case, A = 1 and hence W=Ty When AG[0, 1), different equilibria imply different expected social wastes, and society prefers fewer firms lobbying for monopoly rights to more. Appendix: Proof of Theorem 2 Proof of 2A and 2B: The proof of parts A and B of Theorem 2 are contained in the following lemmas. Before proceeding, note that if 5,. and 5,. are the upper and lower bounds of the support of player fs mixed strategy, then Vi, v^ > s^ > 5,- > 0. Also, recall that oL^ix) is the mass placed at x by player fs mixed strategy. The first lemma is used in Lemma 2 to show that the lower bound of the support of each player's mixed strategy is zero. Lemma 1: If 3i such that Si>Sj and af(sj) = 0, then Sj = 0 and Gj(0) = lim,^,^G/x). If, in addition, a,(s,) = 0, then Gj{0) = Gj{s,), Proof: Let tij{xpG_j) denote / s payoff to bidding Xj when strategies G_j are employed by the other n—1 players. Now Uj(_Sp G_j) = —Sj < 0 for^^. > 0. Since the same holds for Uj{xj,G_j) for XJKS^, and also for Xj-^i if a,(s,') = 0, the claim follows. D Lemma 2: 5^ = 0 Vz. Proof: Clearly, Vi>Si>0 Vf, so it is sufficient to show that no player employs a mixed strategy that has a support with a strictly positive lower bound. By way of contradiction, suppose S = {i\Si > 0} is nonempty, i.e., 5^ > 0 for at least one L If S consists of a single player i, tben s^ >Sj = 0 Vj ^ I In this case, if OLJ^S^^ = 0, Lemma 1 implies that Gj(0)=Gj(5f) V/^i, which in turn implies that Wj(5j.,G_,) EQIX"]. (a) By Lemma 10, in any equilibrium at least one of the players 2,3,..., m, randomizes continuously on the interval (0, V2']. Suppose player i is such a player. By Lemma 6, B^ix) = 0 Vxe(0,172]. Isolating the cdf of player 1, G^, in the expression for Ai yields Gi(x) = [x/(t;2iIj^i^fGy(x))]. Hence, across all equilibria, G^{x) is minimized for each x 6 (0,^2] when the denominator is maximized (note that every equihbrium must have an f e (2,..., m} randomizing continuously over (0, V2']), This imphes that G^(x) is minimized when 77^.^^ ^.G^(x)= 1 (that is, in the equilibrium where only players 1 and i actively bid.) But this means that G^ in this asymmetric equilibrium stochastically dominates the corresponding G^'s that arise in the other equilibria, which implies Ex^^ is maximized in this equilibrium. (b) Similarly, suppose player ie{2,...,m} randomizes continuously on (0,^2]Then G^{x) is maximized for each xe[0,V2'] across equilibria when IJ^^^fipa) is minimized. By Lemma 12, in any equilibrium player 1 randomizes continuously

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over (0, V2], This implies by Lemma 6 that in any equilibrium A^{x) = {v^ — v2-i- x)/ Uj,Vxe(0,V2]. Sinceyli(x) = IIj^^Gpc)is constant across equilibria, Uj^^fijix)is minimized in an equilibrium in which Gi(x) is maximized. But by Lemmas 5,9, and 10, in any equilibrium and for every ;e{2,..., m}, j V i, G,(x) < Gj{x) Vx6[0, V2]. Hence maximizing G^{x) across equilibria requires maximizing the minimum of the Gfc(x)'s, /ce{2,...,m}. Since for each xe(0,V2], ^i(x) is constant across equilibria, this is done by setting G^Cx) = G/x)for all kje{2,...,m} on [0,172]- D References Baye, M. R., de Vries, C. G.: Mixed strategy trade equilibria. Canadian Journal of Economics 25,281-293 (1992) Baye, M. R., Kovenock, D., de Vries, C. G.: The all-pay auction with complete information. CentER Discussion Paper 9051,1990 Baye, M. R., Kovenock, D., de Vries, C. G.: It takes two to tango: equihbria in a model of sales. Games and Economic Behavior 4, 493-510 (1992) Baye, M. R., Kovenock, D., de Vries, C. G,: Rigging the lobbying process: an appHcation of the all-pay auction. American Economic Review 83, 289-294 (1993) Bikhchandani, S., Riley, J. G.: Equilibria in open common value auctions. Journal of Economic Theory 101-130(1991) Broecker, T.: Credit-worthiness tests and interbank competition. Econometrica 58,429-452 (1990) Brooks, M. A., Heydra, B. J.: Rent-seeking and the privatization of the commons. European Journal of Pohtical Economy 6, 41-59 (1990) Bull, C, Schotter, A., Weigelt, K.: Tournaments and price rates: an experimental study. Journal of Political Economy 95, 1-33 (1987) Dasgupta, P.: The theory of technological competition. In: J. E. Stiglitz and G. F. Mathewson (eds.) New developments in the analysis of market structure, pp. 519-547. Cambridge: MIT Press 1986 Deneckere, R., Kovenock, D., Lee, R. E.: A model of price leadership based on consumer loyalty. Journal of Industrial Economics 40,147-156 (1992) Dennert, T.: Price competition between market makers. Review of Economic Studies 60,735-752 (1993) Oixit, A.: Strategic behavior in contests. American Economic Review, 77, 891-898 (1987) Dougan, W. R.: The cost of rent seeking: is GNP negative? Journal of Political Economy 99, 660-664 (1991) EUingsen, T.: Strategic buyers and the social cost of monopoly. American Economic Review 81,660-664 (1991) Fundenberg, D., Tirole, J.: Understanding rent dissipation: on the use of game theory in industrial organization. American Economic Review 77,176-183 (1987) Hendricks, K., Weiss, A., Wilson, C: The war of attrition in continuous time with complete information. International Economic Review 29, 663-680 (1988) Hillman, A. L.: The pohtical economy of protectionism. Harwood: New York 1988 Hillman, A. L., Riley, J. G.: Pohtically contestable rents and transfers. Economics and Pohtics 1,17-39 (1989) Hillman, A. L., Samet, D.: Dissipation of contestable rents by small numbers of contenders. Public Choice 54,63-82 (1987) Lazear, E. P., Rosen, S.: Rank-order tournaments as optimum labor contracts. Journal of Political Economy 89,341-364 (1981) Magee, S, P., Brock, W. A., Young, L.: Black hole tariffs and endogenous policy theory, political economy in general equilibrium, Cambridge: Cambridge University Press 1989 Milgrom, P.: Rational expectations, information acquisition, and competitive bidding. Econometrica 49, 921-944 (1981) Moulin, H.: Game theory of the social sciences, 2nd ed. New York: New York University Press 1986a Moulin H.: Eighty-nine exercises with solutions from game theory for the social sciences, 2nd edn. New York: New York University Press 1986b

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Naiebuff, B. X, Stiglitz, J. E.: Prizes and incentives: towards a general theory of compensation and competition. Bell Journal of Economics 13,21-43 (1982) Narasimhan, C: Competitive promotional strategies. Journal of Business 61,427-449 (1988) Raju, J. S., Scrinvasan, V., Lai, R.: The effects of brand loyalty on competitive price promotional strategies. Management Science 36,276-304 (1990) Rogerson, W.: The social costs of monopoly and regulation: a game theoretical analysis. Bell Journal of Economics 13, 391-401 (1982) Rosen, S.: Prizes and incentives in elimination tournaments. American Economic Review 76, 701-715 (1986) Snyder, J. M.: Election goals and the allocation of campaign resources. Econometrica57,637-660(1989) Tullock, G.: Efficient rent seeking. In: J, Buchanan et al. (eds.) Toward a theory of the rent seeking society. College Station: Texas A&M University Press 1980 Varian, H.: A model of sales. American Economic Review 70,651-659 (1980) Weber, R. J.: Auctions and competitive bidding. In: H. P. Young (ed.) Fair allocation, pp. 143-170. American Mathematical Society 1985 Wenders, J. T.: On perfect rent dissipation. American Economic Review 77,456-459 (1987)

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1.1.3.4 Rent seeking with bounded rationality: An analysis of the all-pay auction

Rent Seeking with Bounded Rationality: An Analysis of the All-Pay Auction

Simon P. Anderson, Jacob K. Goeree, and Charles A. Holt University of Virginia

The winner-take-all nature of all-pay auctions makes the outcome sensitive to decision errors, which we introduce with a logit formulation. The equilibrium bid distribution is a fixed point: the belief distributions that determine expected payoffs equal the choice distributions determined by expected payoffs. We prove existence, uniqueness, and symmetry properties. In contrast to the Nash equilibrium, the comparative statics of the logit equilibrium are intuitive: rent dissipation increases with the number of players and the bid cost. Overdissipation of rents is impossible under full rationality but is observed in laboratory experiments. Our model predicts this property.

I.

Introduction

Many economic allocations are decided by competition for a prize on the basis of costly activities. For example, monopoly licenses may be awarded to the person (or group) that lobbies the hardest (Tullock 1967), or tickets may be given to those who wait in line the longest (Holt and Sherman 1982). In such contests, losers' efforts are costly and are generally n o t compensated. These situations, This research was funded in part by the University of Virginia Center for Advanced Studies and the National Science Foundation (SBR-9617784). We should like to thank Doug Davis, Maxim Engers, Glenn Harrison, Dan Kovenock, Jan Potters, the participants at the Conference on Contests in Rotterdam, and an anonymous referee for helpful suggestions. Section V of this paper is dedicated to enlightened university administrators everywhere. [Journal of Political Economy, 1998, vol. 106, no. 4] © 1998 by The University of Chicago. All rights reserved. 0022-3808/98/0604-0001$02.50

828

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which are especially c o m m o n in nonmarket allocations, are of concern to economists precisely because competition involves the expenditure of real resources, o r 'Vent-seeking'* behavior. Krueger (1974) estimated the annual welfare costs of r e n t seeking induced by price and quantity controls to be 7 percent of gross national product in India and somewhat higher in Turkey. Mohammad and Whalley (1984) reconsidered the cost of rent seeking in India and came u p with much larger estimates, on the order of 3 0 - 4 5 percent of GNP. They conclude that ** these numbers put rent seeking in India into an entirely different category from more traditional policy issues such as trade liberalization, tax reform, a n d the like'' (pp. 3 8 7 88). In the United States, Posner (1975) estimated the social cost of regulation to be u p to 30 percent of sales in some industries (motor carriers, oil, and physicians' services)} Following Tullock (1980), the literature on rent seeking is based on the assumption that the probability of obtaining the prize is an increasing function of one's own effort. T h e hmiting case in which the prize is always awarded to the competitor who exerts the highest effort is called an ''all-pay auction." T h e auction formulation apphes when efforts are like monetary bids that can be ranked easily, as is the case with awarding tickets to those who wait the longest in line or choosing a weapon system on the basis of easily measured performance criteria. T h e all-pay assumption is commonly used in the literature on lobbying (Hillman and Samet 1987; Hillman 1988) since expenditures incurred in the competition for a government grant, license, or contract are usually n o t reimbursed. O t h e r applications of the all-pay auction include research and development races, political contests, and promotion tournaments. T h e prize goes to the highest bidder in an all-pay auction, so each bidder has an incentive to bid just above the highest of the others, as long as this allows a positive payoff. Therefore, there is typically n o equilibrium in p u r e strategies. In symmetric all-pay auctions, the mixed-strategy equilibria involve full dissipation of the rent; that is, the sum of the expected bids equals the value of the prize (Baye, Kovenock, and de Vries 1996). In particular, rents could never be overdissipated with rational players who can always ensure a zero payoff by bidding zero. However, overdissipation might occur when people are not perfectly rational. Davis and Reilly (1994) report a pervasive pattern of overdissipation for all-pay auctions in laboratory experiments with financially motivated subjects. ^ All these estimates are based on the assumption that rents are fully dissipated. The analysis can also be criticized on other grounds, but the magnitude of these figures suggests the importance of reducing rent seeking.

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This p a p e r develops a theoretical model in which bidding behavior is subject to error. T h e introduction of errors is motivated by the observation that behavior in laboratory experiments can b e **noisy'* (e.g., Bull, Schotter, and Weigelt 1987; Smith a n d Walker 1993*, 1997) and can systematically deviate from Nash predictions.^ T o p u t this into perspective, recall that a Nash analysis has two components: perfectly rational decision making and consistency of beliefs a n d decisions. H e r e , we relax the assumption of perfect rationality, while keeping the consistency of beliefs a n d decisions.^ O u r a p p r o a c h should be t h o u g h t of as an equilibrium analysis with boundedly rational players.^ Bid decisions are assumed to be d e t e r m i n e d by expected payoffs via a logit probabilistic choice rule, where decisions with higher expected payoffs are m o r e likely to b e chosen, although n o t with probability one. T h e sensitivity of decisions to payoff differences is d e t e r m i n e d by an error p a r a m e t e r that allows perfect radonality as a limiting case. T h e equilibrium is a fixed point in probability distributions: the bid distributions d e t e r m i n e ^ Deviations from Nash predictions are summarized in chaps. 2, 5, and 6 of Davis and Holt (1993). ^ Another possibility is to relax the consistency of beliefs and decisions. This raises the related issue of learning and adjustment to equilibrium, as proposed by Sargent (1993, p. 3). In Anderson, Goeree, and Holt (1997), we specify a stochastic evolutionary model, for which the steady state is a logit equilibrium. In particular, players are assumed to adjust their decisions in the direction of increasing payoffs, subject to some randomness. The variance of the noise determines the error parameter in the logit equilibrium. Alternatively, Chen, Friedman, and Thisse (1997) show convergence to a logit-type equilibrium in a model of naive learning (fictitious play) when players make decision errors that are determined by a probabilistic choice rule. Brandts and Holt (1995) and Offerman, Schram, and Sonnemans (in press) show that naive Bayesian learning, together with logit decision error, provides a good explanation of the patterns of adjustment in data from laboratory experiments with step-level public goods and signaling games. McKelvey and Palfrey (1996) provide a theoredcal logit analysis of a step-level public goods game. ^ Smith and Walker (1993a, 1997) model decision error as noise around a target decision level. This approach could be thought of as "implementation error** in that players know that they (or their agents) will imprecisely implement desired actions. In the Smith and Walker model, players can decrease the variance of errors at a cost. The model predicts that scaling up payoffs should shift the average outcome toward that of rational play (with higher payoffs, players increase effort to reduce the error variance) and decrease the variance of outcomes. (Similar properties hold under our approach.) Smith and Walker (1993a) then show that these predictions are broadly consistent with results from a survey of 31 experimental studies. Smith and Walker (1993^) provide further evidence for these hypotheses (in the context of a first-price auction). One difference between their approach and ours is that errors in their model are assumed to be centered around the Nash equilibrium (with no error): the actual decision is equal to the Nash equilibrium decision plus a random error with zero mean and a variance that is determined by a costly effort. In contrast, the equilibrium distribution of decisions in our model is not necessarily centered around the Nash equilibrium, and systematic biases can occur even when the Nash equilibrium is not at a boundary of the set of feasible actions.

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the expected payoffs for each bid, which in turn determine the probability distributions of actual bids. T h e model is closed by requiring that the belief distributions correspond to the decision distributions. This ''Nash plus logit'' approach has been termed a logit equilibrium by McKelvey and Palfrey (1996). T h e logit equilibrium is a stochastic generalization of the Nash equilibrium and a special case of the quantal response equiHbrium proposed by McKelvey a n d Palfrey (1995).^ O n e especially appealing feature of the incorporation of errors into the equilibrium analysis is that comparative statics properties are, in some cases, more intuitive than for the standard Nash analysis (with n o error). If two bidders' prize valuations are known and different, for example, then an increase in value for the player with the higher value will stochastically increase that player's bids. In contrast, this increase in the high value will n o t affect the equilibrium bid density of the high-value bidder in the mixed-strategy Nash equilibrium. In other games, the logit equilibrium also provides a plausible explanation of data patterns that are consistent with economic intuition but are n o t predicted by a Nash equilibrium.^ For the all-pay auction, Lopez (1995, 1996) identifies conditions u n d e r which the logit and Nash equilibria are identical and there is exact dissipation of rents in both cases. This equivalence holds only with two bidders, identical prize values, and a maximum allowed bid that equals the c o m m o n value. We show that overdissipation is possible in the logit equilibrium when these assumptions are relaxed, for example, when there are m o r e than two players. In particular, the logit equilibrium model provides an explanation of overdissipation of rents observed by Davis a n d Reilly (1994) in laboratory experiments. Moreover, the model allows us to analyze the trade-

^ Rosenthal (1989) pioneered the use of probabilistic choice in an equilibrium framework: he essentially used a linear probability model instead of a logit formulation. Our approach is closer to that of Lopez (1995), who considers Bertrand and auction games with continuous choice variables. Lopez uses both the logit rule and a ratio choice function that was first proposed by Luce (1959). ^ In public goods games in which free-riding is a dominant strategy, the level of voluntary contributions in laboratory experiments is increasing in the marginal value of the public good. The logit equilibrium model explains both this and other anomalous patterns in the data (Anderson et al., iii press). Palfrey and Prisbrey (1996) use an ordered probit analysis of the data from a public goods experiment, and they conclude that errors are significant. The logit model also explains data patterns in continuous coordination games; i.e., an increase in the cost of "effort" or in the number of players reduces observed effort levels (Anderson et al. 1996). This pattern is consistent with economic intuition, but not with Nash predictions (since any common effort level is a Nash equilibrium). McKelvey and Palfirey (1995) evaluate data from several normal-form games and reject the Nash equilibrium in favor of the logit equilibrium.

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off between the costs of r e n t seeking and the benefit from allowing the high-value bidder to compete m o r e aggressively for the prize. T h e model is described in Section II, and the equilibrium for the two-player case is analyzed in Section III. Some key properties of the n-player all-pay auction (existence, uniqueness, symmetry, and comparative statics) are derived in Section IV. O n a first reading, o n e may wish to skip the proofs in Section IV and go on to the analysis of rent seeking a n d efficiency in Section V. Section VI presents a conclusion. II.

The Model

In an n-player all-pay auction, player i bids bi for a prize that is worth Vi dollars to player L Bids are m a d e simultaneously. T h e prize goes to the highest bidder, but each player incurs the cost of bidding, cbi, c > 0? In the event of a tie for the highest bid, the prize is either split equally or randomly allocated to o n e of the high bidders. This model can be interpreted as a lobbying game in which cis the cost of lobbying effort that must be b o r n e whether or not the effort is successful. In many contexts, a maximum allowable bid, B, is specified by the rules of the auction or is implied by resource constraints. For instance, subjects in a laboratory experiment may n o t be allowed to bid m o r e than their initial cash endowments given o u t at the beginning of the experiment. In a Nash equilibrium (without errors), the cost associated with the maximum observed bid will never exceed the prize value since higher bids are dominated by a bid of zero. Therefore, in a Nash analysis of a symmetric model, there would be n o loss of generality in assuming that the maximal allowable bid is equal to V/c. However, in some laboratory experiments with c= 1, bids above value have been observed (Davis and Reilly 1994). To permit this kind of error, we allow the cost of the maximum bid to exceed the prize value (i.e., cB ^ V), although o u r comparative static and characterization results apply also to the case cB < V, In particular, we show that overdissipation can occur even when the cost of the maximum allowable bid is less than the prize value,^ ^ In most theoretical work, as in laboratory experiments, c = 1. We chose not to normalize in order to consider the effects of independent changes in the bid cost and of differences in individuals' bid costs. Moreover, normalizations are not innocuous in a logit analysis. A multiplicative change in payoffs could be used to normalize c to one. This would not affect the Nash equilibrium (with no error) since any payoff difference, no matter how small, mil determine which decision is made. However, doubling payoffs will double all payoff differences, which reduces the impact of errors in the logit model specified below (by halving the error parameter \i). ^ See the discussion follov^ing proposition 6 below. Note, however, that overdissipation is precluded by assumption when the maximum bid, B, is less than V/cn, since then the maximum total effort is ncB < V.

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T h e first step in the analysis is to establish the connection between a player's expected payoffs and the others' bid distributions. Let Fj(b) denote the cumulative bid distribution for player7, so the probability of winning with a bid of b is the probability that all other bids are below ft, that is, the product of the others' distribution functions evaluated at b. Thus the expected payoff for player i, for a bid b, :c9 is"^

nib) = V;n^;(*) - cb, i= \,

(1)

J^X

T h e second step is to introduce decision error, by specifying the bid density as an increasing function of expected payoff, b u t without having all of the probability located at the bid that maximizes the expected payoff. T h e logit form is one particularly useful parametric model of such probabilistic choice: it specifies decision probabilities to be proportional to an exponential function of expected payoffs. ^^ In the continuous version of the logit model, the bid density is exponential in expected payoffs: fi{b) = ft, exp[7C-(ft)/|x], where \i is an error parameter and ki is a constant that ensures that the density integrates to one. Since expected payoff in (1) is zero at ft = 0, it follows that ki = fi{0). T h e expected payoff in (1) is finite for all possible bids, so the logit density is finite for all nonzero values of |Ll. Therefore, the resulting distribution functions are continuous, and the probability of ties is zero in a logit equilibrium. By substituting the expected payoff from (1) into the logit choice density, we obtain

H'

Vi[[Fj{b)-cb

fdb)=MO)exp

,

ftG[0,£],

i=l,...,n.

(2)

T h e density in (2) is greatest at the bid that yields the highest expected profit, b u t nonoptimal bids have densities that are nonzero and increasing in the expected payoffs for those bids. T h e parameter |Ll reflects the degree of irrationality: as \l tends to infinity, the density function in (2) becomes flat over its whole support and behavior ^The right side of (1) would have to be modified if ties occurred with positive probability. We show below that the logit equilibrium density is continuous, so ties will occur with probability zero. ^^ For example, if there are two decisions, Dx and A , with associated expected payoffs of K{ and %\, then the logit probability of choosing A is an increasing (exponential) function of its expected payoff: Pr(A)=

'-^^^^^ . exp(7c;/^) + exp(7i^/p,)

i=l,2.

where the denominator ensures that the probabilities sum to one.

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becomes random. As the error rate becomes smaller, decisions with higher payoffs are chosen with increasingly higher probability. In the limit as the error rate goes to zero, only optimal decisions are made, and we shall show in the next section that the logit equilibrium converges to a Nash equilibrium. The logit equilibrium condition is that the distribution functions that determine expected payoffs in (1) correspond to the choice densities determined in (2). It follows from differentiation of (2) that the equilibrium densities satisfy the logit differential equations: , fi< fi=^—^.

i = l , ...,7i,

(3)

where the primes denote derivatives and the b arguments of the functions have been suppressed. Equations (2) and (3) are used in Section IV to establish some general properties of the logit equilibrium: existence, symmetry (for the symmetric model), uniqueness when all prize values are equal, and comparative statics effects. These proofs are somewhat technical, and it is instructive to begin by considering the special case of two bidders.

ni.

The All-Pay Auction with Two Players

In this section we derive closed-form solutions for the logit equilibrium, for both identical and asymmetric values. In the symmetric two-player case with V\ = V/'^] + X* where K* is a constant that forces the density to integrate to one. The solution for the other player's density is obtained by replacing V; by ^2- These formulas could be useful in evaluating data from laboratory experiments.

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results in an increase in that player's bids (in the sense of first-degree stochastic dominance). Proof, Without loss of generality, let player 1 be the one whose value will increase. Equation (9) can be integrated from zero to b to obtain VxF^ib) = V^F.ib) +

VA^

1 — exp

-cb\

Evaluating this equation at b — B, we can determine the value of the integration constant: 1

•cB^

exp

Using these expressions, we can write the expected payoff for player 1, V1F2 — cb, in terms of i^i, which allows us to write the density condition in (2):

Mb) =M0) X exp]

V2F, + {V, - V2)

1 -

exp{-cb/li)

1 -

exp{-cB/ii)

cb

(10)

This is the equation that determines the density of player 1 in the asymmetric model. When player Ts value is increased to Vf, the corresponding formula for the d e n s i t y , / f , becomes

ffib) =/r(0) X exp

V,Ff + {Vr-

V,)

1 -

exp(-cV^)

1 -

exp{-cB/\i)

- cb

•

(11)

T h e structure of the proof is to show that there can be only two crossings of the distribution functions, Fi and Ff, a n d since the distributions are equal at zero and B, these are the only crossings. We subsequendy show that J^i starts out above Ff at low bids, so that bids are stochastically higher u n d e r Ff, At any crossing, Fi = Ff, a n d it follows from (10) a n d (11) that the ratio of the slopes at a crossing IS

ffjb) Mb)

_/*(0) /i(0)

1 exp

JLI

^

1 -

exp{-cb/\i)

1 -

exp{-cB/\i)

(12)

T h e ratio on the right side of (12) is strictly increasing in b since Vf > Vi, If there were m o r e than two crossings, the ratio of slopes

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at successive crossings would either decrease and then increase or the reverse, a contradiction. Since the ratio in (12) is increasing, it must be less than one at i = 0 and greater than one 3tb = B, Therefore, Fi > Ff for all interior values of b. The proof for the other player is analogous. Q.E.D. In particular, notice that a value increase for the player with the higher value will stochastically raise that player's bids. As noted above, this intuitive result is not a property of the mixed-strategy Nash equilibrium. Similar results hold when players have identical prize values but different bid costs, c,. In particular, if Ci > ^2, then player 2's Nash equilibrium bid density is independent of C2, whereas the logit equilibrium predicts that player 2 will bid stochastically less in response to a higher bid cost. The effects of cost and value asymmetries on rent dissipation are considered in Section V below. IV. The All-Pay Auction with n Players Although closed-form solutions are not available for the general asymmetric n-player case, we can prove some existence, symmetry, and uniqueness results that are useful in the subsequent analysis of comparative statics effects and rent dissipation. PROPOSITION 2. A logit equilibrium exists for the n-player all-pay auction. The general existence proof for the n-player, eisymmetric-value case is given in Appendix A. The proof is based on Schauder's fixedpoint theorem, which is a generalization of Brouwer's theorem to function spaces (which are not compact).^® When all the players have the same prize value, we are able to derive the closed-form solution, which is useful for the analysis of rent dissipation (see App. B). The next issue is symmetry. The equilibrium will not be symmetric across players if their values differ. In particular, the player with the higher value will have stochastically higher bids. Nevertheless, we can show that those with identical values have the same bid distributions, even if others' values are different. PROPOSITION 3. In any logit equilibrium for the all-pay auction, players with identical values have identical bid distributions. When players have different values, those with higher values bid more (in the sense of first-degree stochastic dominance). Proof, We start by proving the final statement in the proposition. Let JFI and F2 denote the distributions corresponding to Vi and V2 '^ Other applications of Schauder*s theorem are given in Stokey and Lucas (1989, chap. 17).

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(Vi > V2). Suppose that F^ = F2 on some interval of bids. Then the derivatives of these distributions must also be the same on this interval, which is impossible when one considers (2). At any crossing of the distribution functions, Fi = F2 — F, and it follows from (2) that the ratio of the slopes at all such crossings is Mb)

MO)

(13) exp J Mb) MO) ^ which is strictiy increasing in F and hence in b. If there were three crossings, the ratio of slopes at successive crossings would either decrease and then increase or the reverse, a contradiction. Since the ratio in (13) is increasing, it must be less than one at 6 = 0 and greater than one at 6 = J5. Therefore, F2 > Fi for all interior values of b. Next, consider the case of equal values, V, = ^2 = V, for which we must show that the bid densities for players 1 and 2 are idendcal. Consider a particular value of b. Since F2{b) > Fi{b) for all Vi > I4 and F^ib) < Fi{b) for all Vj < V2, it follows from a continuity argument that the distributions are equal at b when Vi = 14- Obviously, this argument holds for all values of b, Q.E.D. It is readily verified that when players have identical prize values but different bid costs, ^„ those with lower bid costs bid (stochastically) higher. The proposition implies that the equilibrium will be symmetric when all players' costs and values are identical. This symmetry result is interesting because Baye et al. (1996) have shovm that there can be asymmetric mixed-strategy Nash equilibria (with no errors), even in a symmetric model, as long as there are more than two players. The effect of errors is to ''smooth out" the best response functions in a way that precludes asymmetric equilibria in the symmetric model.^^ In addition, we can show that the symmetric equilibrium is unique. PROPOSITION 4. The logit equilibrium is unique when values are identical. Proof, In light of the symmetry result in proposition 3, it suffices to show that there is at most one symmetric equilibrium. Suppose in contradiction that there are two symmetric equilibria, distinguished by I and II subscripts. By dropping the player-specific subscripts from (3) and using the derivative of the payoff in (1), we obtain the following differential equations for the two candidate solutions: =

'^ In the limit in which \i goes to zero, the logit equilibrium ^'selects'* the symmetric mixed-strategy equilibrium. 237

Simon P. Anderson, Jacob K. Goeree, and Charles A. Holt RENT SEEKING

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,^__fAVf,Fr\n-\) f\

- c]

n-2,. _ - . . _ . ,

^, __h{Vf,,Fl'\n-\)

(14)

- c]

jn

•

Without loss of generality, suppose that/,(&) =/ii(6) for ft less than some bid bi, possibly zero, and that/i( ft) > /„(ft) forftjust above h. Since the densities must integrate to one, they must cross again at some higher bid,ft^/,with/i crossing from above. Thus, at ft = bu, it must be the case t h a t / [ < / „ . However, at buy we also have /, = /„ and Fi > Fu, which together with (14) imply/[ > /fi, a contradiction. Q.E.D. It is more difficult to establish uniqueness of the logit equilibrium in asymmetric models, but the special case of two players is tractable. Recall that (10) determines the bid density for player 1 in the asymmetric model for n = 2. Suppose that there were two solutions, /i a n d / f , to this equation. Taking the ratio of (10) to the analogous equation for/f, we obtain /i(*)

ffib)

V^JFi -

/i(0)

~

/f(0)

exp

Ff)

(15)

If the initial conditions for the two candidate solutions were equal, then the differential equation in (10) would trace out the same density in each case. So without loss of generality, let/i(0) > / f (0). Since both densities integrate to one, they must cross at some interior point, ft,. At the crossing,/i (ft,) = / f (ft,) and Fi{b,) > Ff (ft,), which together with (15) contradict the initial assumption that /i(0) > / f (0). Hence the bid distribution for player 1 is unique. An analogous argument establishes uniqueness for the other player.^^ Next we consider the effects of changes in the exogenous parameters on the equilibrium bid distributions. As would be expected, bids are stochastically decreasing in the cost parameter, c, and stochastically increasing in both the value parameter, F, and the uppermost bid, B, PROPOSITION 5. In the logit equilibrium for a symmetric all-pay auction, bids are raised in the sense of first-degree stochastic dominance by a decrease in the cost parameter, c, an increase in the common value, Vy or an increase in the maximum feasible bid, 5 . ^^ We have not been able to use this method to prove uniqueness in the case of value asymmetries with more than two players. Possible nonuniqueness would not affect any of the propositions. In particular, the result of proposition 3, which pertains to the n-player asymmetric case, holds for any logit equilibrium.

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Proof, Let Ci and ^n be the common cost parameter for all players {ci> cii)y and leti^i and Fn denote the corresponding distributions. As before, the structure of the proof is to show that the distribution functions can cross only twice, at the boundaries. At any crossing, Fi = Fiu and it follows from (2), applied to the symmetric case, that the ratio of the slopes at a crossing is

Mb)

MO)

(ci -

Cv)b\

(16) Mb) - /i(0) exp 1^ The right side of (16) is stricdy increasing in b. By the argument used in proposition 4, this impUes that the only two crossings occur at the boundaries, and Fi > Fu for all interior values. The proof for an increase in the players' common value is analogous. The effect of an increase in the maximum feasible bid, B, is also proved along the same lines. As before, leti^i and J^n denote the distribution functions corresponding to Bi and B^ (Bi > J52). For all be [0, £2], equation (2) implies that/i(6)//„(&) ==/i(0)//„(0) at any crossing of Fi and Fill that is, the ratio of slopes of the distribution functions at any crossing on [0, ^2] must equal the ratio of their slopes at * = 0. Therefore, the distribution functions can cross only once on this interval (i.e., at ft = 0). Since F2 reaches its maximum value of one at B2, it lies everywhere above Fi, which reaches its maximum value of one at Bi > B^. Q.E.D. In the benchmark case in which c= 1 and B = V/it is natural to consider what happens when B and Vare raised by the same amount. A simultaneous increase in B and V can be decomposed into an increase of V with B held constant, followed by an increase in £ with V held constant It follows from proposition 5 that the combined effect will raise bids since each effect alone raises bids. The method of proof for proposition 5 cannot be used to determine the comparative static effect of the number of bidders, n, on the bid distributions. In the next section, we evaluate the effect of n on net rents, using the closed-form solution for the equilibrium distribution derived in Appendix B. V. Rent Seeking Allocations based on lobbying effort are likely to be used when ethical or equity considerations preclude selling the prize outright as in a market transaction.^^ For example, it is not reasonable to expect ^^ In addition, the person awarding the prize may care about more than just the net value of the prize to the contestants. The person making the award may have an independent preference in favor of one of the contestants. This could make a significant difference if the rent-seeking efforts are closely balanced. In other cases, the person making the award decision may enjoy the attention that comes with rent-

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that the benefits that academic deans dispense could be sold in this manner. Effort-based competitions, however, have the undesirable feature of using up real resources, and this rent dissipation can be considerable. In fact, when prize values are equal, rents are fully dissipated in a Nash equilibrium and are overdissipated for a wide range of parameter values in the presence of errors, as shown in proposition 6 below. Here an equal division of the prize, if this is feasible, is more efficient since there is no unnecessary expenditure of real resources. (Many prizes such as grant money, funding for computers, or teaching reductions are fairly divisible, despite the fact that they are usually allocated in discrete lumps.) The counterargument, in favor of using effort-based competitions, is that contestants with higher values will exert more effort and, hence, have a higher probability of winning. This raises the issue ofjust how much value asymmetry is required for the all-pay auction to achieve a higher net rent than a simple random allocation or equal division. Before dealing with value asymmetries, we consider rent dissipation in a symmetric model. The pattern of rent dissipation, shown in figure 1 for the symmetric, two-player model, shows overdissipation for high costs, that is, when c > V/B = V2. We now show that this result is true more generally. PROPOSITION 6. In a logit equilibrium, there is overdissipation of rent in the symmetric-value model with more than two players and

cB^V. Proof, Recall that the logit equilibrium corresponds to the Nash equilibrium in the limit as \i goes to zero. For the Nash mixedstrategy equilibrium, the lower bound of the support is zero, and therefore expected payoffs are zero. Hence, the expected net rent is zero, so there is exact dissipation in the absence of errors. In addition, the symmetric mixed-strategy Nash equilibrium bid density,/^(Z^) = ( n - l)-M^/y)^/6/^-Ms clearly decreasing in ft when n > 2 (Baye et al. 1996). Dividing both sides of the formula for the logit equilibrium density in (2) by the Nash density/^(ft) yields

fib) /(O) wiby-' ~ exp Mb) Mb)

- cb

(17)

Recall that VF^'^b) - eft = 0 for all ft by construction of the mixedstrategy Nash equilibrium, so we know that the term in brackets in seeking efforts. In this section, we restrict attention to the total net rents for the contestants. These qualifications should be kept in mind when we use the term overdissipation.

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(17) is zero when the Nash distribution,/^,,, crosses the logit distribution, F. It follows th?it f{b)/f^{b) = f{0)/f^{b) at crossings. Since /{c (6) is decreasing, the ratio on the left side of (17) is increasing at successive crossings. Thus this ratio has to be less than one at the lower bound and greater than one at the upper bound, and hence, F{b) < iSjc(*) at all interior points. It follows that the logit equilibrium distribution lies below the Nash mixed-strategy distribution that fully dissipates the rent. A lower distribution function implies higher expected bids, which in turn imply that rent is overdissipated. Q.E.D. Proposition 6 gives a sufficient condition for overdissipation, which includes the benchmark case in which c= I and B = 7. However, overdissipation can also occur for cB < V, To see this, recall that as the error parameter goes to infinity, bidding behavior becomes completely random, that is, uniform on the interval [0, B], Therefore, the expected bid converges to B/2 for each player, and the total effort cost converges to ncB/2, Thus the total effort cost can exceed the prize value Vfor sufficiently large values of n and |LI, even when cB < V. Davis and Reilly (1994) conducted a series of all-pay auction experiments with financially motivated human subjects. In their treatments, ^ = 1 and the maximum possible bid was not restricted to be less than the prize value, so cB > V, There were four bidders, so proposition 6 implies that rent will be overdissipated. They report that the social costs of rent-seeking activities consistently exceeded the prize value, so subjects lost money on average. This tendency for losses was handled by providing each subject with a relatively large initial cash beilance.^^ Many annoying requirements that are imposed in all-pay competitions can be understood as attempts to limit the number of contestants, where direct exclusion may be perceived as being unfair. Indirect exclusion is probably intended to make the task of comparing bids easier for the person awarding the prize, but the effect of such limits may be to reduce rent dissipation as well. We can use the closed-form solution for the equilibrium distribution in the n-player model, equation (B3) in Appendix B, to determine the effects of changing the number of bidders on net rents. Figure 2 shows the relationship between net rents and n for selected values of the error parameter \i. Notice that numbers restrictions raise net rents, but ^° In these experiments, the worst errors (bidding above value) tended to occur in early rounds. We would expect error rates to decline over time in stationary environments, but not to disappear altogether. In fact, the error rates estimated by McKelvey and Palfrey (1995) for simple matrix games decline in successive periods of laboratory experiments, although some residual noise remains in most cases.

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Net Rent sN.

-0.2 '

4

6

8

10

'-^ \

-0.4 -0.6 -0.8

-1 s

-1.2 s N

-1 4 FIG. 2.—The effect of the number of players, w, on net rent for \i = 0.05 (solid line), \i = 0.075 (long dashes), and \i = 0.1 (short dashes), given c= B - V= 1.

the effect is small when the error rate is low; indeed, rent is fully dissipated in the Nash equilibrium case of |X = 0 for all n. Finally, we investigate the effects of asymmetries on rent dissipation, using an equal division or random allocation of the prize as a basis of comparison. For simplicity, we restrict attention to two players. We start with different values, Vi > V^, and identical bid costs, c = 1. Recall that net efficiency in an all-pay auction depends on the trade-off between the costs of rent seeking and the increased probability that the prize is awarded to the person who values it the most. This trade-oflf is very simple to evaluate with full rationality (|Li = 0). In a mixed Nash equilibrium with Vi> V^, Baye et a l (1996) show that the expected payoff of the high-value player is VI - V^, whereas the expected payoff of the low-value player is zero. Thus total net rent in the Nash equilibrium is Vi - 14- In comparison, an equal division or random allocation involves no effort cost and produces a net rent of (T^ + V2)/2, It follows from these observations that the all-pay auction with fully rational bidders is less efficient than a random allocation unless Vi — V2 exceeds {Vi -f T^)/? or, equivalently, Vi > 3^2- This three-to-one ratio seems like a relatively large value asymmetry, considering that the all-pay auction is likely to be replaced by a unilateral, dictatorial allocation when value asymmetries are so great as to be obvious to all concerned ex ante. 242

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Net Rent

vl FIG. 3.—The effect of value asymmetry on net rent, given c= V2 = 1, B= 6, and n = 2. The upper (lower) solid line shows net rent in the logit model with |i = 0.1 {\i — 0.3). Net rent in a Nash equilibrium is given by Fi — 1 (short dashes), and in an equal division by (Vi + l ) / 2 (long dashes).

In the presence of decision error, we can calculate net rents using the formulas for the bid densities in note 15. Figure 3 shows the relationship between net rent and the high value, Vi, when the low value is set equal to one. The solid lines show net rent for the logit equilibrium for \i = 0.1 and \i — 0.3. The line with short dashes shows net rent for the Nash equilibrium. For comparison, the net rent line for the equal-division allocation is shown as a line with long dashes, which is higher when Vi < 3. The effect of adding errors is to further increase the inefficiency of the all-pay auction. Another aspect of the inefficiency of the all-pay auction is apparent when different players have different bid costs, Q. Suppose for illustration that player Ts value exceeds that of player 2 but the latter's bid cost is much lower. The optimal allocation is to give the prize to player 1. In a Nash equilibrium, player 2 will bid more aggressively when Vi/ci < V2/C2, and therefore the player with the lower value will win more often. Cost differences are a major determinant of the final allocation in the auction but are irrelevant to

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the optimum. T h e all-pay auction is inefficient n o t only because of the rent dissipation b u t also because it relies on a sorting criterion that is inappropriate for choosing the contestant who would benefit the most. VI.

Conclusion

T h e all-pay auction has been widely studied because it is an allocation mechanism in which competition for a prize involves the expenditure of real resources, for example, lobbying. Since losers also incur costs, economists have considered the extent to which rents associated with the prize are dissipated by the competitive process. In theory, full dissipation is possible, but overdissipation is impossible in a Nash equilibrium because a zero effort ensures a zero payoff. T h e approach taken h e r e is to introduce the possibility that players are n o t perfectiy rational: bid choices are probabilistic, with an error parameter that allows perfect rationality in the limit. T h e resulting lo^t equilibrium yields a continuous relationship between the extent of r e n t dissipation and the cost of bidding. T h e overdissipation observed in the Davis and Reilly (1994) experiment is consistent with the predictions of the logit equilibrium. T h e logit equilibrium provides a n u m b e r of intuitive comparative statics predictions, which can be tested in laboratory experiments. In the symmetric-value case, for example, the extent of r e n t dissipation is increasing in the n u m b e r of players, which might explain eligibility restrictions that are sometimes imposed. (In contrast, rent is fully dissipated in a Nash equilibrium.) With asymmetric values, the all-pay competition provides the high-value player with a higher probabilit)^ of obtaining the prize, but the added cost of competitive efforts m o r e than offsets this benefit unless value differences are relatively large. Value asymmetries suggest o t h e r cases in which the Nash and logit equilibria differ. With two bidders, the high-value player's bid distribution is i n d e p e n d e n t of the player's own value in the Nash equilibrium (with n o error). In contrast, increases in values result in stochastic increases in bids for the logit equilibrium. Economists have long suspected that some of the most glaring inefficiencies in an economy arise in n o n m a r k e t allocations. T h e standard way of analyzing behavior in n o n m a r k e t situations is to apply the notion of a Nash equilibrium or some refinement thereof. Although mathematically appealing, game theory is difficult to evaluate empirically because the predictions often d e p e n d o n subde, difficult to measure effects of informational a n d preference asymmetries. T o date, the most direct evaluations of game theory have come from the laboratory. Many experimental studies report system244

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atic differences between Nash predictions and data patterns. Even where behavior appears to be converging to a Nash equilibrium, there is almost always some residual noise in the data. Statistical tests are generally based on adding symmetric noise to the Nash prediction (in an ad hoc manner), which becomes the baseline from which the significance of deviations can be assessed. The disparities between Nash and logit predictions in the all-pay auction, summarized above, indicate that modeling endogenous decision errors can be quite different from adding symmetric, exogenous noise to the Nash prediction. Moreover, the logit equilibrium is convenient for empirical work because it specifies a likelihood function and because it nests the Nash equilibrium as a limiting case; that is, it allows arbitrarily small deviations from perfect rationality and provides a natural null hypothesis (p, = 0). It is important to point out that the Nash equilibrium does provide reasonably good predictions in many contexts. The logit equilibrium should be viewed as a generalization of Nash that preserves its usefulness in organizing the data but offers a new perspective in explaining anomalies. Although the logit choice function can be derived from basic axioms, we recognize that it is a specific parameterization that can be generalized in a number of ways. Other parameterizations may generate even better predictions in specific contexts. Nevertheless, in terms of qualitative predictions, the logit equilibrium seems clearly better than the Nash equilibrium. Appendix A Proof of Proposition 2 (Existence of Equilibrium) Let ¥(b) denote the vector of bid distributions, whose ith entry, Fi{b), is the bid distribution of player e, for f = 1, . . . , n. Integrating the left- and right-hand sides of (2) yields an operator T that maps a vector F(^) into a vector TF(^), with components

'iUpjiy) _ I ^' ^ pf m(b) I '"p{ ViUfjiy) j*i

-

A}

- Ciy\

dy (Al)

Ady

J

The vector of logit equilibrium distributions is a fixed point of this operator, diat is, TFiib) = Fi{b) for all b e [0, B] and f = 1, . . . , n. Since the right side of (Al) is continuous in b even when the distributions Fj are not, the equilibrium distributions are necessarily continuous. So there is no loss of generalit)^ in restricting attention to C[0, B], the set of continuous functions

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on [0, B]An particular, consider t h e set S = {FG C [ 0 , B]\\\F]\ < 1}, where | | | | denotes the sup n o r m . T h e set S, which includes all continuous cumulative distributions, is an infinite-dimensional u n i t ball a n d is thus closed a n d convex. Hence, the n-fold (Cartesian) product 5 " = S X - X 5 i s a closed and convex subset of C[0, B] X - - X C[0, J5], the set of all continuous n-vector valued functions o n [0, B]. This latter space is endowed with the n o r m ||F||„ = max^^i,.. „ ||7^||. T h e operator T m a p s elements from S" to itself, b u t since 5" is n o t compact, we cannot rely on Brouwer's fixed-point theorem. Instead, we use the following fixed-point t h e o r e m d u e to Schauder (see, e.g., Griffel 1985). SCHAUDER'S SECOND THEOREM. If S"* is a closed convex subset of a n o r m e d

space and H"" is a relatively compact subset of S", then every continuous mapping of S" to if" has a fixed point. To apply the theorem, we need to prove (i) that / / " = {TF|F E S") is relatively compact a n d (ii) that T is a continuous mapping from S" to / / " . T h e proof of part i requires showing that elements of if" are uniformly b o u n d e d a n d equicontinuous o n [0, B]. From (Al) it is clear that the mapping TFiib) is nondecreasing. So \TFi{b)\ < TFi{B) = 1 for all x€ [0, B], Fi e Sy and 2 = 1, . . . , n, a n d elements of H"" are uniformly b o u n d e d . T o prove equicontinuity of if", we must show that for every € > 0 there exists a 6 > 0 such that |77^(^i) - 77^(^2)1 < € whenever \bi - ^2! < 8, for all FiG Sy i = 1, , , , y n. Consider the difference

I'expj un^iC^) - ^^.

\i\db

|7i^(^.,) - 7F,(^2)I =

[ expjUn^yC^) - ^i^ /^lU We can b o u n d the right side by replacing the distribution functions with 1 in the n u m e r a t o r a n d with 0 in the denominator to obtain

\TF,{b,) - 7FKWI

f

exp[(y,-

c,b)/li]db

exp{ — Cib/ii)db This inequality is maintained if b is replaced by 0 in the integrand of the n u m e r a t o r a n d by i5in the integrand of the denominator. T h e n integration yields \m{bi)

-

TF,{b,)\

1^2-

b,\exp{VJii)

Bexpi-dB/ii)

T h u s the difference in the values of TFi is ensured to b e less than e for all FiG Sy i = ly . , . y riy by setting \bi — ^2! < 8, where 8 = eB m i n exp

•Vi-

Therefore, T F is equicontinuous for all F € S".

246

CiB

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JOURNAL OF POLITICAL ECONOMY

Finally, we prove continuity of T. The mapping T is continuous if, for all FS F2 G S" and for all e > 0, there exists a 5 > 0 such that HTF^ - TF^H „ < € when ||F^ - F^H „ < 6. In order to get a bound on ||TF^ - TF^H „, let us write F\{b) = F^iib) + hiib), with - 6 < h,(b) < 5 for all be [0, B], i = 1, . , , , n. Using the upper bound, we derive

n-f"](*) K(8y^ TF]{b). Thus we conclude that K,(d)-'TF',{b) < TF]{b) < K,{S)TF]{b),

f = 1, . . . , n.

Notice that i;(5) is strictly increasing for 6 > 0, with Ki{0) = 1, for all z = 1, . . . , n. The final step is to obtain a bound on HTF^ - TF^||„ = max,=!,...„ \\TF] - 7F?||. Suppose without loss of generality that the (maximum) supremum is attained for f = 1, ^ = b*, at which TF\ > TF], We also have TF\ < K^iS) TF] at b*. So

247

Simon P. Anderson, Jacob K. Goeree, and Charles A. Holt RENT SEEKING

851 ||TF^ - TF2||„ = |7Fl(^*) 1), can only hold up to some Cj < C^* Postulating that contest success depends upon the difference between the resource commitments, the required conditions — that Cj = 0 need not imply Pj = 0, and that the inflection point occurs at Cj = C2 - are met by the logistic family of curves: Pi =

(4) l+exp{k(C2-Ci)}

where P2 is defined correspondingly. (As is logically required Pj -f p2 = 1.) In particular, when Cj = 0 player #1 still retains a share of success Pj = 1/(1 + exp{kC2}). Figure 2 shows several CSF curves for varying k, where k h the **mass effect parameter** applicable to the logistic function. In a military context we might expect the ratio form of the Contest Success Function to be applicable when clashes take place under close to "idealized" conditions such as: an undifferentiated battlefield, full information, and unflagging weapons effectiveness. In contrast, the difference form tends to apply where there are sanctuaries and refuges, where information is imperfect, and where the victorious player is subject to fatigue and distraction. Given such "imperfections of the combat market,*' the defeated side need not lose absolutely everything. (For the sake of concreteness I have been using military metaphors and examples, but analogous statements can evidently be made about non-military struggles - e.g., lawsuits or political campaigns or rentseeking competitions.) The generahzation of equation (2) for any number of players N was provided in TuUock's initial paper. For the i^^ contestant, the probabihty of success becomes: p =

\

C^4-C^+...+CJ5

=

L_

(5)

E.C]"

Of course, the Pj's sum to unity. Employing the difference (logistic) form instead, the corresponding generalization of equation (4) is: explkCj} p. =

(6) EjexplkCj}

It is evident from the form of the last fraction on the right that, as required, the sum of these Pj's will also be unity.^

254

1.1.4.1 Conflict and rent-seeking success functions: Ratio vs. difference models of relative success 105 r

•

1 1

k.0.02 ~k-0.04

.

^

/

- ^

1 -^^ \

/ /

t

0

20

40

60

80

1

1

r

1

1

100 120 t40 160 180 200 220

IC,

240

Figure 2. Contest success function: Difference (logistic) form

To illustrate, if N = 3 and i= 1, equation (6) becomes: 1

(6a)

l-hexp{k(C2-C,)) + exp{k(C3-Cj)) Both (5) and (6) fall within the more general category of logit functions.^

3. Symmetrical Nash-Cournot equilibrium As has been mentioned, when the ratio form of the CSF applies each side will surely always commit some resources to the contest. If peace is defined by the condition C, = C2 = 0, then peace can never occur as a Cournot equilibrium under the traditional ratio model! The demonstration is simple. Side #1 will be seeking to maximize its '^profit*': Yj = Vp, - C,

(7)

where V is the given value of the prize and Pj is determined as in equation (2), A similar equation holds of course for player #2. Suppose momentarily it were the case that Cj = C2 = 0, the parties sharing the prize equally without fighting. Then, assuming only that V > 0, under the Cournot assumption either player would be motivated to defect, since even the smallest finite commitment of resources makes the defector's relative success jump from 50^o to lOO^o. In effect, the marginal profitability of i*s contest contribution is infinite when Cj = 0. In contrast, when the logistic Contest Success Function applies, two-sided

255

Jack Hirshleifer 106 peace may easily hold as a stable Cournot solution. Since the player who defects from Cj = C2 = 0 does not get the benefit of a discrete jump from 50% to 100% success, there is a finite marginal gain to be balanced against the marginal cost of contest effort.^ Numerical Example I Player #1 seeks to maximize his profit as in equation (7), with pj defined by the logistic CSF equation (4) above. If Cj = 0, then finding the derivative in the usual way leads to: kexp{-kCj)

1

(l+exp(-kC,})2 For Cj = 0 to be a solution, we must have V = 4/k. By symmetry, an analogous equation will hold for player #2. So if, for example, k = .04 and V = 100, then (as claimed) C, = C2 = 0 will indeed be a Cournot equilibrium. In this equihbrium pj = P2 = .5 so that the parties each have profit of 50.

4. Asymmetrical equilibrium What about the possibility of one-sided submission rather than two-sided peace? This means that player #1 (say) chooses Cj > 0 while player #2 sets C2 = 0. For such an outcome, some kind of asymmetry must be introduced - in the parties' valuations of the prize, in the effectiveness of their respective contest efforts, or possibly in the costs of such efforts. But regardless of any such asymmetries, under the ratio model one-sided submission as a Cournot equilibrium can no more occur than could two-sided peace! We need look only at asymmetries due to inequalities in valuations of the prize. Specifically, suppose V, > V2, suggesting that there might be a Cournot equilibrium with Cj > 0 while C2 = 0. Using the profit equation (7) for player #1, and equation (2) for the Contest Success Function in ratio form, the firstorder condition is:

ac,

(cf+c^)2

(8)

Evidently, whenever C2 = 0 the marginal profit of contest effort to player #1 is always negative. So under the ratio form of the CSF, it will never be possible to have an asymmetrical contest outcome with one party having zero and the other having positive commitment of resources.

256

1.1.4.1 Conflict and rent-seeking success functions: Ratio vs. difference models of relative success 107 For the difference form (logistic CSF), however, the asymmetrical outcomes are quite different. First of all, taking the partial derivatives of the respective Contest Success Functions leads to: dY. kexp {k(C,-hC.)) ^Y^ —i = 1 ±= —i aC, (exp{kCj) + exp {kC2))^ aC2

(9)

This possibly surprising proposition states that at any pair of Cj, C2 choices, the partial derivatives (the '^marginal products" of the respective contest efforts) are always the same for both sides. It follows immediately that, if the valuations Vj and V2 are unequal, it is impossible to simultaneously satisfy (as the respective first-order conditions for a profit maximum would require): dY. — - = l/Vj aCj

dY. and —^ = I/V2 ^^2

(10)

Thus, when the difference form of the CSF apphes, there cannot be an interior asymmetrical Nash-Cournot solution. (Whereas, we have just seen, using the ratio form there cannot be a corner asymmetrical solution.) This impossibihty theorem for the difference form is somewhat too strong, since it is an artifact of the assumption implicit in equation (7) that the Marginal Cost of contest effort is constant. If the Marginal Cost of contest effort is rising, equations (10) might be satisfied so as to permit an interior NashCournot equilibrium. More generally, with rising Marginal Cost there could be either a corner or an interior asymmetrical solution, depending upon the numerical parameters and the exact functional form.^ Assuming for simplicity that Marginal Cost is constant as in equation (7), the Reaction Curves RCj and RC2 associated with the logistic CSF are parallel straight lines of 45^ slope, up to a point of discontinuity. More specifically, in the continuous range the Reaction Curve equations are:^^ Cj = C2 + A,, where A, = (2/k)cosh-* {.5sqrt(kV,)) C2 = Cj + A2, where A2 = (2/k)cosh-i {.5sqrt(kV2)}

(11)

The discontinuities fall into three distinct patterns - depending upon the relative positions of the points Cf, Cj, and C." as sketched in Figures 3 through 5 — each leading to a particular class of Cournot solution. The pattern of Reaction Curves RC, and RC2 pictured in Figure 3 represents the *'strong asymmetry" case, which stems from a relatively large difference Vj - V2 between the parties* valuations of the prize. Here RC2, the Reaction Curve for the lower-valuing player, rises as Cj increases - but only

257

Jack Hirshleifer 108

Figure 3, Logistic reaction curves: Strong asymmetry

2

i

i

c? J

^ • • • "

\y/^

Q

05 2

-/RC,

1

RCg

1 C|

L

D E>

q

: ^

• —

~c

e^

Figure 4, Logistic reaction curves: Moderate asymmetry

up to point G where the opponent's effort has reached a certain critical value C\. At G, player #2*s optimum drops off discretely to C2 = 0 (point F), and of course remains at zero for all higher values of Cj. (The explanation is that, given a logistic CSF, the lower-valuing player can always take home some profit by investing zero effort. Hence doing so always remains a viable alternative, and eventually becomes more advantageous than trying to keep up with very large contest efforts on the part of his higher-valuing opponent.) If, as in Figure 3, Cj < C? - that is, point F is to the left of pomt E, the latter being the point where the higher valuing player's Reaction Curve RCj intercepts the horizontal axis - then the Nash-Cournot equilibrium is at E, where (C,, C2)

258

1.1.4.1 Conflict and rent-seeking success functions: Ratio vs. difference models of relative success 109

Figure 5. Logistic reaction curves: Symmetry or near-symmetry

= (C^, 0). It is easy to verify that, at point E, each player's effort is a best response to the opponent's choice. This solution represents one-sided submission: the lower-valuing player has abandoned the struggle. Numerical Example 2 Once again each player seeks to maximize his profit Yj Vpj - Cp where Pj is defined by equation (4) above. Let the required asymmetry be in the valuations of the prize, where specifically VJ = 400 and V2 = 100. Assuming k = .04, the Reaction Curves are as pictured in Figure 3, with C^ = Aj = 65.848 and C^ = A^ = 0. If the higher-valuing player #1 takes €2 = 0 as given, his profit-maximizing solution Cj equals Aj = 65.848 (point E). Turning to player #2, with C, = 65.848 taken as given the profitmaximizing'^ C2 is indeed C2 0. The expectations on each side as to the other party's behavior being mutually consistent, this is a Cournot equilibrium. The associated shares are Pj = .933, and P2 = .067, and the profits are Y, = 307.4 and Y2 = 6.699. Note that the higher-valuing player does disproportionately better: not only is his prize worth more, but he fights harder for it. Figure 4 illustrates a "moderate asymmetry" pattern. Here, the difference between Vj and V2 being smaller, point E (the horizontal intercept of RCj) lies to the left of point F (at the discontinuity along RC2). In consequence, point

259

Jack Hirshleifer 110 E, where player #2 unilaterally submits, is no longer a Cournot equilibrium. (That is, player §2 will no longer choose C2 = 0 as his best response to player ^Vs choice of Cj = Cj = Aj.) As the prize valuations Vj and V2 approach equality, finally, the ^'symmetrical or near-symmetrical" pattern of Figure 5 is obtained. Here also, it will be evident, unilateral submission will not occur. The actual solutions for both the Figure 4 and the Figure 5 patterns involve mixed strategies on one or both sides,^^ but the specifics of these solutions are not of immediate concern to us. As the next step, it would be natural to ask whether the ratio versus the difference forms of the Contest Succes Function lead to correspondingly different outcomes in terms of the Stackelberg or other asymmetrical solution concepts. I will not, however, be pursuing these implications here.

5. Conclusion In analyzing rent-seeking or other conflict competitions, models allowing relative success to respond continuously to changes in contest commitments have heretofore assumed that success must be a function of the ratio of the parties' resource commitments. However, this assumption is inconsistent with the observation that two-sided peace or one-sided submission do sometimes occur in the world. When relative success is postulated to stem instead from the numerical difference between the respective contest inputs, a Contest Success Function taking the form of a logistic equation is derived. Two-sided peaceful outcomes emerge in Cournot equilibrium when the '*mass effect parameter'' of the logistic CSF curve is sufficiently low. One-sided submission can also occur when there is a large disparity between the parties' valuation of the prize. As these valuations approach equality, the logistic CSF leads to mixed-strategy Cournot equilibria.

Notes 1. See, e.g., Hillman and Katz (1984), Corcoran and Karels (1985), Higgins, Shughart and Tollison (1985), Appelbaum and Katz (1986), AUard (1988), Hillman and Samet (1987). 2. A recent paper of Hillman and Riley (1988) makes use of still another family of contest payoff functions, in which - in contrast with the sharing rules analyzed here - the entire prize, as in an auction, goes to the high bidder. Their paper also allows for differing prize valuations. 3. In the standard Lanchester equations of military combat (Lanchester, 1916 (1956); Brackney, 1959), the outcome is also assumed to depend upon the ratio of the forces commiUed. But for Lanchester the battle result is always fully deterministic, in the sense that the side with larger forces (adjusted for fighting effectiveness) is 10007o certain to win. This makes the CSF a step function, which jumps from pi = 0 to p, = 1 when Cj - C2. So Lanchester's formula can be regarded as the limiting case of equation (2) as the mass effect parameter m goes to infinity.

260

1.1.4.1 Conflict and rent-seeking success functions: Ratio vs. difference models of relative success 111 The same holds also for the auction-style payoffs in Hillman and Riley (1988). 4. As seen in the previous footnote, the Lanchester equations of combat take this to the extreme. The larger force is 100' = (>'i,>'2»"->y«) denote a vector of efforts for the n players. Each player I's winning probability is denoted by p\y){p': [0,7]" -> U where Y > y^ for all ieN), The following properties are maintained. ^ Another way to ground a CSF is its derivation from primitive distributional assumptions on the underlying contest. Whereas the logit CSF can be derived in such a way (see McFadden, 1984), the general additive form and the power functional form axiomatized in this paper do not appear to have been derived that way. The two approaches - the axiomatic and the one based on distributions - are clearly complementary in understanding the limitations and advantages of a CSF. ^ As, for example, axiomatizations of utiUty functions do not depend on an agent's particular choice or on the model the utility function is embedded.

264

1.1.4.2 Contest success functions Contest success functions

285

(Al) ZieNP\y) = 1 and p\y) > 0 for all ieN and all;;; if 3;^ > 0 then p\y) > 0. (A2) For all ieN p'{y) is increasing in j,. and decreasing in yj for all; ^ i. (A3) For any permutation nof N (i.e., a bijection TT: N -> AT) we have P"%) = p(y.,,y..,...,3^JVf6Ar (Al) says that the contest success function satisfies the conditions of a probability distribution function and, in addition, when the effort of a player is positive that player's probability of success is also positive. The positivity of each player's probabiHty of success, although not needed for the respresentation in (1) below, becomes necessary in the proof of Theorem L"*" (A2) states that a player's probabihty of success is increasing in the player's own effort but decreasing in every other player's effort. (A3) is an anonymity property stating that each player's probabiHty of success should not depend on his or her identity or on the identities of their opponents, but just on the efforts of the players. This property also impHes that if in some vector of efforts two players have identical efforts then their probabilities of success must be equal and if all players were to exert identical efforts, then each one of them would have a probability of success equal to 1/n. For any/ceiV, let j.fc = ( j i , . . . , y^. 1, jfc^ 1,...,yJ Let/?\p^...,p" be functions satisfying (A1)-(A3). Then it can be shown that there exists a function p: [0, 7]" -> U which is increasing in its first argument, decreasing in the remainder n — 1 arguments, and such that for any given vector of efforts y^ = (y J, y 2» - • •' >^n) ^^ ^^^^ p\y') = Piyly'-,)

for ^WkeN

(1)

That is, (A1)-(A3) guarantee that each player's probabihty of success is governed by the same function as every other player's probability. Examples of CSF's satisfying (A1)-(A3) include: (i) e^y*/(ZjeNe^'') wherefe> 0, (ii) yT/iZjeNy?) where m > 0 , (iii) yT-(Zj^iyJ)/(n-l)-^l/n where yj w -f z) [ = 1 — P2(w 4- z, x)]. Examples (iii) and (iv), mentioned at the end of section 2.1, do not satisfy (A5). Examples (i) and (ii), however, do satisfy (A5) and (A4); both of these examples belong to the following class of GSFs that satisfies (A1)-(A5): (A5') m^fiy^l^^^Jiy^)) for all ieN and pL(y) =/(y.')/(Z,eM/(y;)) for all ieM((=:N) where /(•) is a positive increasing function of its argument. It is evident that /(•) is unique up to positive multiplicative transformations. The following result establishes that the class described in (A5') is also the only class of CSFs satisfying (A1)-(A5). Theorem 1: {A1)-{A5) are satisfied ifand only if the Contest Success Function satisfies (A5y

266

1.1.4.2 Contest success functions 287

Contest success functions

Proof: It is straightforward to show that a CSF satisfying (A5') satisfies {A1)~{A5). Thus, we only show that (A1)-(A5) imply a CSF of the form described in (A5'). By (A4) and (A5) we have

Ay)

(2)

Pi(yJ

for all subsets M of N. By (A1)-(A3) (which imply the representation in (1)), along with (A4) and (A5), for every McN there is a function of |M| variables such that pln{yJ=Pmiyi>y'-i) where y_f represents the vector efforts of the other players who belong to M and p^{') is the projection of the function p(-) in |M| dimensions. In addition, by (Al) and (A4) "ZPJyi^y-d^^ ieM

Thus, we have lp2{'y') denotes the two-player CSF] 1 =

Plniym) / Z Piiym)

\lpi{ym)/pUym)l\

Z

Piiym)/Pm{ym)

= lPmiyi^y-i)/i^ - Pmiyi^y-iWLPziyk^yiVPziyiyyk)']TjPiiypykVPziy} :'yj)] JeM

= [Pm(y.>>'-,)/(l -Pm(j',-.y-,))3[(l -P2(3'.-.>'*))/P2(j',>>'»)]

X

(3)

LPiiypykVi^-Piiypyk)) jeM

Fix y^ = a>0 and let /(j,) = P2{yi,«)/(! — Pziyh °^}) which is positive (since, by (Al), 0 < P2(3'i. °f) < 1) and, by (A2), increasing in j?,-. Then (3) becomes

1 = LpJyi,y-i)/ii -Pjyi,y-imLyfiyi)lil

'^.•^^yA JeM

which impb'es Pmiyiyy~i)=fiyr

4i-M

Therefore, for all M czN the CSF has the additive representation described in (A5'). • 23, Two Functional Forms: Power and Logit The next task is to axiomatize the two functional forms that have been employeed in applications of contests; both are special cases of (A5') and therefore satisfy (A1HA5).

267

Stergios Skaperdas 288

S. Skaperdas

First, let ly = (Xy^^, >^-y2> • • • > ^-J^n) ^^^ consider the following homogeneity axiom. p\Xy) = p\y) for all A > 0 and for all ieN,

(A6)

According to this property an equiproportionate change in the efforts of all players would leave the winning probability of every player unaffected. The implication is that the ratio of winning probabiHties of any two players depends on the ratio of their efforts. The following functional form, mentioned earlier and which is widely employed in the rent seeking literature, satisfies (A6) [in addition to (A1HA5)]: fiyd = ^y? f^^ some a > 0 and m > 0.

(A6')

Moreoever, as shown below this is the only continuous functional form satisfying all six axioms. Theorem 2: {A6') is the only continuous functional form satisfying (A1)-{A6). Proof: By Theorem 1, (A1)-(A5) imply the additive representation in (A5') which along with (A6) yields f{Xx)/f{x) = /(A2)//(z) for any x, z, A > 0

(4)

By setting z = 1, (4) implies:

mx)^mm

...

Let F(z) ^ /(z)//(l). Then (4') becomes: F(Ax)==F(l)F(x)(A>Oandx>0)

(4")

The rest of the proof consists of transforming (4") to a Cauchy equation (see (5') below).^ Let X = e\ x = ^^ and denote F{e^) by g{t). By substitution into (4'') we obtain: g{t-i-s) = g{t)g{s) (5) Finally, letting h{u) = In g{u), (5) implies: h(t-hs) = hit) + h{s)

(50

The only continuous solution of (50 [see, e.g., Aczel (1969)] is h{u) = mu for some m, or that g{u) = e'"". Substituting back into ¥(-) we have F(e") = ( e T and z = e" yields F(z) = z"*. Finally, we have f{z) = af^ where /(I) = a > 0 and, given (A2), m > 0 which satisfies (A60. D To axiomatize the second functional form, let ceK" denote the constant vector with all its components equal to c and consider the following axiom which implies that the winning probability of each player depends only on the difference in the efforts among all players. p\y) = p\y -f c) ^ceR" such that y,- 4- c > 0, VieiV; in addition, assume /(•) is defined for y^ = 0. (A7) See Aczel (1969) which the rest of the proof closely follows (pp. 51-53).

268

1.1.4.2 Contest success functions Contest success functions

289

Clearly, this is a strong property since it requires the winning probabilities of, say, yx = 1 and ^2 = 2 to be the same as when y^ = 10001 and 3^2 = 10002. The main objective of an axiomatization, however, is a better understanding of the representation that is being axiomatized, not necessarily the advocacy of a certain axiom. And, as we will mention at the end of this section, the companion functional form to (A 7) is not completely devoid of merit. This companion functional form is the logit function: f{y.) = e^y' for some k>0. (AT) Suppose (A7'). Clearly such a functional form satisfies (A1)-(A5). To show that it satisfies (A7) as well, divide both the numerator and the denominator of pXy + c) by ^^^••^^>, so that p'iy + c) = 1/[1 + S^^/^^^"^'^] = ppl Thus, (A70 imphes (A1)-(A5) and (A 7). As shown below, the direction of implication essentially goes the other way as well. Theorem 3: (AV) is the only continuousfunctionalform satisfying {A1)-{A5) and (A?), Proof: Suppose (A1)-(A6) and (A7) are satisfied. By Theorem 1, (A1)-(A5) imply the additive representation in (A5'). Then, (A5') along with (A7) imphes that for all X, z > 0 and for all c such that x + c>0 and z + c ^ 0 we have /W//(^) = / ( ^ + c)//(z + c) By setting z = 0 (and thus c > 0) and letting F{w) = /(w)//(0) (> 0 since, by (A6') /(•) is positive), the equation above imphes Fix-]-c) = F{x)F(c) which has the same form as (5) in the proof of Theorem 2. Since F(-) is positive, its only continuous solution is F{w) = e^^ for some k>0 and for all w > 0. Consequently, /(w) = f{0)e^^ = ^^ with /(O) = 1, which satisfies (AT). D Although, as indicated earlier, (A7) can be criticized for its treatment of the outcome of contests when c is large, its associated logit function in (A7') has some merits. First, it can be derived from primitive distributional assumptions as in McFadden (1984); the same cannot be said of the power or "ratio" form in (A6'). Second, as Plirshleifer (1989) has suggested using data from battles in Dupuy (1987), the outcomes of at least some conflict situations appear to fit the logit function well, especially in the neighborhood where the players have equal winning probabilities. Although this evidence is far from being conclusive and it concerns only battles, not necessarily other kinds of contests, this is the only empirical evidence on CSFs that we are aware of. Thus, the criticisms that can be levelled against (A7) represent one side of the coin and should not be the only criterion in judging the relevance of the logit function as a CSF - perhaps it holds locally in some contest situations when c cannot vary much. 3. Concluding comment The axiomatizations in this paper are meant to provide a better understanding of the properties and a grounding of the functional forms of Contest Success Functions

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Stergios Skaperdas 290

S. Skaperdas

used in different applications of contests. An independence from irrelevant alternatives axiom is mainly responsible for generating the class of additive functional forms. Within that class, the power or "ratio" form, used in the rent-seeking literature, and the logit function were also axiomatized. Although helpful, axiomatizations by themselves are unlikely to settle the issue of appropriateness of a CSF for any particular contest situation. As with production functions (and to a lesser degree, with utility functions), finding ways to discriminate among functional forms empirically would be a complentary and welcome endeavor. References Aczel, J.: On applications and theory of functional equations. New York: Academic Press 1969 Baye, M., Kovenock, D., de Vries, C G.: Rigging the lobbying process: An application of the all-pay auction. Am. Econ. Rev. 83, 289-94 (1993) Coughlin, P.: Probabilistic voting models. In: Kotz-Johnson (ed.) Encyclopedia of Statistical Sciences, Vol. 7. New York: John Wiley & Sons 1986 Dixit, A.: Strategic behavior in contests. Am. Econ. Rev. 77, 891-898 (1987) Dupuy, T. N.: Understanding war. New York: Paragon House 1987 Hirshleifer, J.: Conflict and rent-seeking success functions: Ratio vs. difference models of relative success.Publ. Choice 63, 101-12 (1989) Hirshleifer, J.: The paradox of power. Econ. Polit. 3,177-200 (1991) Luce, R. D., Suppes, P.: Preferences, utihty, and subjective probability. In: Luce, R. D., Bush, R. R., Galanter, E. (eds.) Handbook of mathematical psychology, Vol. III. New York: Wiley 1965 McFadden, D. L.: Econometric analysis of quahtative response models. In: Griliches, Intriligator (eds.) Handbook of econometrics, Vol. 2. New York: North-Holland 1984 Nitzan, S.: Collective rent dissipation. Econ. J. 101,1522-34 (1991) Rosen, S.: Prizes and incentives in elimination tournaments. Am. Econ. Rev. 76,701-714 (1986) Samuelson, L.: On the independence from irrelevant alternatives in probabilistic choice models. J. Econ. Theory 35, 376-389 (1985) Skaperdas, S.: Cooperation, conflict, and power in the absence of property rights. Am. Econ. Rev. 82, 720-39 (1992) Skaperdas, S., Grofman, B.: Modeling negative campaigning. Am. Polit. Sci. Rev. 89, 49-61 (1995) Suppes, P., Krantz, D. H., Luce, R. D., Tversky, A.: Foundations of measurement. Vol. II. San Diego: Academic Press 1989 Tullock, G.: Efficient rent seeking. In: Buchanan, J. M., Tollison, R. D., Tullock, G. (eds.) Toward a theory of the rent-seeking society. College Station: Texas A&M University Press 1980

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1.1.4.3 On the existence and uniqueness of pure Nash equihbrium in rent-seeking games GAMES AND ECONOMIC BEHAVIOR 18, 135-140 (1997) ARTICLE NO. GA970517

On the Existence and Uniqueness of Pure Nash Equilibrium in Rent-Seeking Games Ferenc Szidarovszky Systems and Industrial Engineering Department, University of Arizona, Tucson, Arizona 85721

and Koji Okuguchi^ Department of Economics, Nanzan University, Yamazoto-cho, Showa-ku, Nagoya 466, Japan Received December 14, 1995

The existence of a unique non-symmetric pure Nash equilibrium is proved for rent-seeking games under the assumption that the production functions of the agents for lotteries are twice differentiable, strictly increasing, and concave. It is also proved that at the equilibrium each agent has nonnegative expected net rent. Journal of Economic Literature Classification Numbers: C72, D43, LIS. © 1997 Academic Press

L INTRODUCTION Many researchers have studied rent-seeking games based on the introductory paper of Tullock (1980). One of the most important questions is the existence and uniqueness of pure Nash equilibrium. Perez-Castrillo and Verdier (1992) have analyzed the case of decreasing returns in rent-seeking technology as well as that of increasing returns and discussed the possibility of the agent's reaction function to be upward sloping. In addition, they have offered a systematic analysis of the existence of equilibrium. As was pointed out in Okuguchi (1995), a rent-seeking game is formally equivalent to a special profit-maximizing Coumot oligopoly without product differentiation. However, the classical results on Cournot oligopolies (see, for example, Okuguchi, 1976; Okuguchi and Szidarovszky, 1990) cannot be used directly, since different convexity assumptions are made and the sets of strategies are unbounded. Okuguchi (1995) has Copyright © 1997 by Academic Press All rights of reproduction in any form reserved.

^^^

271

Ferenc Szidarovszky and Koji Okuguchi 136

SZIDAROVSZKY AND OKUGUCHI

proved the existence and uniqueness of a symmetric pure Nash equilibrium in rent-seeking games under quite general conditions, assuming that the agents' production functions for lotteries are identical. In practical situations, agents may have different experiences in rentseeking activity; furthermore, agents may have different connections with politicians, which will influence the effectivenesss of rent-seeking activity. Hence, in this paper, we will keep the monotonicity and convexity assumptions of Okuguchi (1995), but drop the assumption that agents' production functions for the lotteries are identical. The method used by Szidarovszky and Yakowitz (1977) and Okuguchi (1993; 1995) will be used to show the existence and uniqueness of the pure Nash equilibrium. 2. THE MODEL Let n be the number of agents. If x^ is agent fs ( / = 1,2,... ,n) expenditure on the rent-seeking activity and fi(x^) is his production function for lotteries, then the probability for winning the rent is given as

Assume that for all / and x^ > 0, (A) /; is twice differentiate,

f;(Xi) > 0, f/ix^)

< 0, /;(0) == 0.

Notice that under this assumption, each agent's increase in expenditure on rent-seeking activity yields increased lotteries at a decreasing rate. The expected net rent of agent / is (P,(xi,...,xj = p . - X , =

, -X,,

(2)

where we have normalized rent to be equal to 1. If all x^ = 0, then cp^ is defined to be zero. Introducing the notation y^ = ftixi) (yi e [0,/^(oo)), the net rent of agent / can be rewritten as ^iiyi

y. yn) = T^;r—r - Siiyi)'

(3)

^j-iyj

where g, = /T' • Assumption (A) implies that g;(y/)>o. 272

g';{yd>o.

(4)

1.1.4.3 On the existence and uniqueness of pure Nash equiUbrium in rent-seeking games NASH EQUILIBRIUM IN RENT-SEEKING GAMES

137

The net-rent function (3) is formally identical to the profit function of Cournot oligopolies without product differentiation when the inverse demand function, the output, and the cost of firm / are given as 1

respectively. The classical results on Cournot oligopoly cannot be applied in this case, since the feasible set for yj is usually unbounded and the inverse demand function is convex. However, using the slight modification of the methodology used earlier for Cournot oligopoly, the existence and uniqueness of the pure Nash equilibrium can be established.

3. EXISTENCE ANALYSIS First, we note that yi^yi^ ••• y„ == 0 is not an equilibrium, since if any one of the agents changes his strategy selection to a small positive y^, then his payoff becomes positive: i-^,(y,)>o. Notice next that function (3) is concave in y^; therefore, with fixed y/y ^ 0, the best response yf of agent / is given as follows: If

then yf = 0: otherwise, yf is the unique positive solution of

iZj^jj+yf) It is well known that a vector ( y f , . . . , y*) is an equilibrium if and only if for all i, yf is the best response with fiixed values of yf (j ^ i). Introduce

273

Ferenc Szidarovszky and Koji Okuguchi 138

SZIDAROVSZKY AND OKUGUCHI

the notation s = E"= i y^ and for all s > 0, define

[0 y.(s)

if^g;(0)>l

= < unique positive solution

(5)

I of equation ^^g-(y/) = ^ — y^,

otherwise.

First, we show that y^(s) is well defined. If ^g-CO) < 1, then at y / = 0, s^g'i(yi) < s — y^, and with y^ = 5, s^g^iy^) > s - y^. Furthermore, s^g'^iyi) is increasing in y^, while ^ - y ^ decreases. Hence, the unique positive solution is guaranteed, and it is in interval (0, s). From the above observation, it follows that ( y * , . . . , y*) is a pure Nash equilibrium if and only if 5* = X"=iyf satisfies n

> ' ( ^ * ) = E}^/(^*) - ^ * = 0,

(6)

and then >;*=y,(5*),

/=l,2,...,n.

From Eq. (5), it is easy to see that y^is) is continuous in s, and if yiis) = 0 and s < s', then y^C^O = 0. Next we show that Eq. (6) has a unique solution by proving that there is a pair 5* > 5^ > 0 such that no solution exists with s < S^ and s > 5*; furthermore, in the interval [5^,5*], Y strictly decreases, and y ( 5 ^ ) > 0, y(5*) < 0. Assume therefore that y^is) > 0. Differentiate equation s^g^iy^is)) = s — y^(s) with respect to s to have

y'M

l+s'g';{y,)

l+s'g'liy,)

2j,. - s

If y, < 5/2, then y^ decreases. Notice that y^ decreases even if y^ = 0. If y^ > s/Z, then y^ strictly increases in s. From Eq. (5), we see that the second case occurs if and only if

*'g;(^)-^ + ^ < o . 2

which is equivalent to 2ss;

'(i) S2>

-•

>S^.

If ss/2, and y2(s)>s/2\ hence, Eq. (6) has no solution. Select S^ = 5 2 . A t 5 = 5 ^ , Y(s) > 0. Assume next that ^ > 5 ^ . Then y2(s),..., y^is) all decrease. If y^(s) is also decreasing, then Y(s) obviously decreases. If y\(s) > 0, then assumption (A) implies that

2

=

-{yi{s)-s) S*, then the first case of Eq. (5) implies that for all /, y^is) = 0; therefore, for 5 > 5*, Y(s) < 0, so no solution exists here. Finally, we show that 5^ < 5*: 1

1

1

1

5

^* ^ 2 g ' 2 ( 5 , / 2 ) - 2g'2(0) - "2 * min,g,(0) ^

* Y'

Hence, we proved the following result: THEOREM

1. Under condition (A), there is a unique pure Nash equilib-

rium. Notice that for all agents / and any fixed values of yj (j ^ i), the solution y^ = 0 always gives zero payoff value for this agent. Therefore, at the best response, his/her payoff must not be negative. Hence, we proved the following result: THEOREM 2. Under condition (A), each agent enjoys nonnegative expected net rent at the equilibrium.

4. CONCLUSIONS The existence of the unique equilibrium in rent-seeking games was proved under realistic conditions in this paper. We also showed that at the equilibrium, each agent's expected net rent is nonnegative. The proof was

275

Ferenc Szidarovszky and Koji Okuguchi 140

SZIDAROVSZKY AND OKUGUCHI

constructive, giving a computational method to find the unique equihbrium. One has to solve the monotonic single variable equation (6), where functions y^ are defined by relation (5). For solving monotone equations, routine methods are available (see, for example, Szidarovszky and Yakowitz, 1978). ACKNOWLEDGMENT We are grateful to an associate editor for very helpful comments.

REFERENCES Okuguchi, K. (1976). Expectations and Stability in Oligopoly Models. Berlin/Heidelberg/New York: Springer-Verlag. Okuguchi, K. (1993). ^'Unified Approach to Cournot Models: Oligopoly, Taxation and Aggregate Provision of a Pure Public Good," Eur. J. Polit. Econ., 9, 233-245. Okuguchi, K. (1995). "Decreasing Returns and Existence of Nash Equilibrium in Rent-Seeking Games," Mimeo. Department of Economics, Nanzan University, Nagoya, Japan. Okuguchi, K., and Szidarovszky, F. (1990). The Theory of Oligopoly with Multi-Product Firms. Berlin/Heidelberg/New York: Springer-Verlag. Perez-Castrillo, J. D., and Verdier, T. (1992). "A General Analysis of Rent-Seeking Games," Public Choice 73, 335-350. Szidarovszky, F., and Yakowitz, S. (1977). "A New Proof of the Existence and Uniqueness of the Cournot Equilibrium." Int. Econ. Rev. 18, 787-789. Szidarovszky, F., and Yakowitz, S. (1978). Principles and Procedures of Numerical Analysis. New York/London: Plenum Press. Tullock, G. (1980). "Efficient Rent-Seeking," in Toward a Theory of the Rent-Seeking Society J. M. Buchanan, R. D. Tollison and G. Tullock, Eds., College Station: Texas A & M Press.

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Part 2 Collective Dimensions

1.2.1.1 Committees and rent-seeking effort

Journal of Public Economics 25 (1984) 197-209. North-Holland

COMMITTEES AND RENT-SEEKING EFFORT Roger D. CONGLETON* Clarkson College, Potsdam, New York, NY 13676, USA Received August 1981, revised version received December 1983 This paper explores the extent to which administrative structures may influence the extent of rent-seeking effort. Within the environments explored, relatively smaller efforts tend to be invested to influence the deliberations of committees than those of single administrators. To the extent that these efforts may be considered unproductive rent-seeking, the widespread use of committees may reflect the fact that savings from reduced rent-seeking more than offset the greater decision costs of committees.

1. Introduction Committees are widely used by nearly all social organizations as a means of developing and implementing policies that often have clear distributive implications. Applications range from academic committees responsible for conferring minor student awards to corporate, Congressional, and Party committees responsible for decisions that have substantial effects upon the allocation of national resources. Such a state of affairs must be more than a little puzzling to those familiar with the modern literature on committee deliberations. Committee deliberations fall prone to all the problems of majority rule decision-making: the obvious diseconomies of multi-person decision-making, the possible absence of unique equiUbria, the potential for intransitive rankings of alternatives and the implied arbitrariness of decisions noted by Arrow (1951), Black (1958), Buchanan and Tullock (1962) and Usher (1981). These weaknesses would seem to suggest that allocative decisions would be better made by single individuals than by committees. There are many possible defenses of committees, from democractic values and economies of sampHng to differences in information possessed by participating individuals. Unfortunately, few, if any, of these ideas seem to account for the widespread use of committees by cultures and institutions that apparently place little value of majority rule procedures, per se.^ This *The author wouJd like to thank Fred Menz, Paul Downing and the anonymous referees for helpful suggestions on earlier drafts of this paper. ^Buchanan and Tullock (1962) provide an individualistic rationale for the adoption of majority rule institutions that requires no ethical commitment to democratic principles. They suggest that individuals may prefer majority decision rules over rules requiring unanimity or minority approval because of a trade-off between achieving successful authorization of policies © 1984, Elsevier Science Publishers B.V. (North-Holland)

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paper provides an explanation that may be more broadly applicable. Namely, the use of committees may reduce rent-seeking waste. The work of Tullock (1967) and others suggest that efforts to influence the decisions of resource allocating agencies tend to be wasteful in so far they are redistributive rather the productive. This paper suggests that fewer resources may be used to affect the deliberations of committees than those of single individuals with equivalent authority. To the extent that such efforts at influence constitute wasteful rent-seeking, committees may be expected to out perform and out survive the institution of one-person allocation wherever the economies of reduced rent-seeking waste more than offset the additional costs of majority-rule decision-making. The paper thus provides a rare, if limited, efficiency based defense of majority rule institutions. In order to demonstrate the potential economies of committees, the methodology of rent-seeking as developed by Tullock (1967), Krueger (1974) and in Buchanan, Tullock, and Tollison (1980) is applied to the task of analysing competitive games of influence under committee and one-man decision-making regimes. Our intent is to contrast the resource commitments made to influence committee decisions with those which would have occurred under one-person administration. Because our interest is essentially a comparison of institutions, all other sources of variation are to be rendered as small as possible. Administrators are assumed to be drawn from the same homogeneous pool of individuals, and so have identical tastes and skills at decision-making. The agenda for both committee deliberations and one-man administration is presumed to be given at the outset. A fixed award or grant is to be distributed between two possible recipients according to the dictates of a particular allocative rule. Potential recipients are assumed to have equal skills at influencing the various decisions of the award allocating body. And since our interest is the level of resources utilized in the award-seeking process, we shall ignore the particular methods by which award-seekers influence the decisions of award administrators. All such mechanisms are to be included under the broad umbrella of what might be called the influence production function or the 'application process'. For our purposes, it is enough that influence varies directly with the value of resources committed to the 'application process*. thai benefit particular individuals and avoiding collective actions that damage those same interests. This line of reasoning can be used to explain the adoption of majority rule procedures by a committee of the whole, even if individuals have no allegiance to majoritarian norms, per se. Representative committees similarly may be defended as a transactions-costs-reducing means of accompUshing the same ends. However, the existence of committees that are not intended to be representative or are too small to represent a statistically useful sub-sample of the population cannot easily be explained by the Buchanan and Tullock theory of constitutional choice. An alternative explanation of the use of small representative committees is that such committees reduce incentives for individuals outside the deliberating body to invest in awardseeking effort, since those efforts may tend to needlessly duplicate the efforts of the committee members themselves.

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The analysis focuses on cases where there are but two award-seekers in order to facilitate analysis and because it is often possible to characterize the award-seeking efforts of more than two individuals as competition between two coalitions. Each of two decision-making methods (one-man and committee) will be paired with two allocative rules (all-or-nothing and proportionate-share rules) which allows comparison of four administrative structures. We first examine the effects of committee versus one-man administration of all-or-nothing rules and then proceed to a similar analysis of proportionate-share rules. 2. Award-seeking effort under all-or-nothing allocative rules Administration of all-or-nothing rules requires the selection of deserving grantees from a pool of appHcants. Under such rules, some applicants will be conferred awards and so find themselves among the 'most worthy', while others will receive no award at all. As a consequence, there is a clear incentive for potential grantees to attempt to influence the allocative decision of the administrative body so that they might be included among those 'honored'. Our analysis suggests that there is a tendency for fewer resources to be employed in the resulting competitive game of influence if selections are made by committees rather than by single administrators. 2, J, One-man administration of all-or-nothing rules When resource grants are to be made by a single administrator, the task of influencing one's future award is straightforward. Would-be recipients simply devote effort to influence the administrator's evaluation of their own merit vis-a-vis others. Because worthiness is assumed to be an increasing function of the resources employed demonstrating or manufacturing the properties of interest to the administrator, successful award-seekers will be those who have invested the greatest award-seeking effort. (In the unlikely event of a tie, the award will be granted arbitrarily to one of the applicants.) Potential recipients will undertake this effort — play the rent-seeking game — if the anticipated gain exceeds the sacrifice incurred by actively participating in the game. If e is the opportunity cost of the minimum recognizable effort, then the smallest prize that might attract individual award-seeking effort is P>e. The optimal strategy for an award-seeker in this game is to devote just a bit more effort to the process of lobbying than other applicants. By minimizing the cost of success, the net advantage of winning is maximized. The problem with this ideal strategy is that it requires the award-seeker to know the level of resources that will be devoted by other competitiors. Unfortunately, it is not possible for each supplicant to devote more resources than all others. As a result, there are no dominant pure strategies. Awardseekers will choose strategies based upon their judgement of the eff'ort levels 281

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forthcoming from their competitiors rather than an objective determination of the net-payoff maximizing commitment. Thus, little can be said regarding the level of resources that will be employed in a single round of the 'appHcation process' beyond noting that it will never pay to devote resources as valuable as the sought award.^ This ambiguity is an important feature of games of influence in all of the institutional settings explored by this paper. Deterministic allocation rules do not necessarily imply deterministic strategies. Nor are mixed or random strategies particularly attractive alternatives in settings where a game is to be played only a few times. A variety of methods for choosing strategies have been suggested for such circumstances, although none is universally accepted. One possibility which is widely used, but less widely accepted, is to choose strategies that maximize one's payoff given some expectation about the play of other game-players. This paper explores the implications of the simplist of the conjectural variation schemes, attributed to Antoine Augustin Cournot. Under the Cournot rule, a player chooses a strategy based on knowledge of the previous round of the game and the expectation that other players will not alter their strategies. In the small number setting of interest here, it is assumed that strategic planning is based upon information obtained about competitor intentions rather than previous rounds of the game of influence. In the two-party game of influence characterized above, Cournot interaction generates a pattern of adjustment often associated with competitive settings, namely escalation. If player A anticipates some level of effort from player B, A will devote a bit more effort than that in order to obtain a favorable decision from the administrator. If player B learns of this intention, a strategy involving a bit more effort than planned by A becomes B's optimal strategy. A's reaction to B's new strategy will again require efforts beyond those planned by B. The deterministic nature of the grant-allocating regime, together with information concerning competitor intentions, allows each competitor to believe that his own commitment will be a winning ^The process of judgement, itself, may be open to analysis. For example, Foster (1981) suggests that judgement may be modeled as an exercise in subjective probability calculus. Note that the presumed deterministic nature of the allocative rule implies that the only probabilistic aspect to be modeled is the behavior of one's competitors. Thus, under this model of judgement, a typical competitor forms priors concerning the efforts of his opposition and, if risk neutral, invests the level of effort that maximizes the expected net payofT. If a competitor were risk averse, he would invest somewhat more than this amount or zero (and not play the game). Unfortunately, the range of plausible subjective probability distributions allow 'optimal' investment levels over the entire 0-A interval. Another possible representation of judgement is to consider it a mechanism for implementing optimal mixed strategies. Shiffman (1953) established that games with mathematical structures similar to the ones of interest here have equilibrium mixed strategies, although the optimal density functions to be employed by the players cannot always be determined. Such models do not eliminate the role of judgement, but rather reduce it to the determination of prior probabilities and the probability-event space. For our purposes, the lack of a deterministic choice is the matter of interest rather than analysis of particular representations of the process of judgement itself.

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Strategy. Cournot interaction causes each competitor's effort at influence to move toward upper bound F, the maximum effort that can ever be a reasonable strategy. As this limit is approached, the total effort committed to the game of influence approaches 2P,^ At this limit, resources approaching twice the award granted are consumed in the competitive game of influence.* 2.2. Committee administration of all-or-nothing rules Under committee administration, grants are awarded to individuals considered to be the most deserving by the majority of committee members. Because the number of awards is assumed to be smaller than the pool of potential recipients, there will again be clear incentives for would-be grantees to attempt to influence the decisions made by the administrative body. If a particular committee member may be influenced only to the extent that specific resources are devoted to that end, then successful award-seekers will be those who have devoted the greatest effort to a majority of individual committee members. A committee having three members will reward the efforts of those successfully targeting two of the three administrators.^ The smallest prize capable of attracting award-seeking effort under committee administration will be somewhat larger than that required under oneman administration. If e is the opportunity cost of minimal efforts targeted at a single committee member, then the minimum commitment required to influence a committee having three members is the amount required to be noticed by a majority of committee members, or 2e. Thus, the minimum grant sufficient to attract award-seeking effort under committee administration will exceed 2e, rather than e, the minimum under a single administrator. Fewer award-seeking games will be played under committee administration since some prizes will be sufficient to attract the active participation of award-seekers under one-man administration but not under committees. However, many awards will be sufficient to overcome the participation thresholds for both committees and single administrators. Under committee administration, active award-seekers have the dual problems of determining appropriate investment levels and properly distributing that effort among committee members in a manner which gains ^The F,F strategic pair is not a stable equilibrium, merely the upper bound toward which Cournot adjustment tends to converge. ^Other games in which players jointly *over-invest' relative to prospective rewards are, of course, not uncommon. State-run lotteries and private gambling establishments are, for the most part, profitable ventures. ^Personalized apphcations are often the only means of influence available to would-be recipients in the informal setting typical of games of influence. Without the assumed private character of applications, homogeneous committee members would have always acted in unison and so been equivalent to one-man rule. In cases where apphcations efforts are limited to the production of public goods or their equivalents, the escalating tendencies of one-man administration are also implied for committee administration of all-or-nothing games.

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majority approval for their preferred resource allocation. To maximize net advantage, an av^ard-seeker should identify the minimally sized majority of administrators who will receive the least attention by the competition and submit applications to that majority which embody just a bit more effort. Unfortunately, this ideal strategy suffers from the same sort of practical defect posed under one-man administration. In this case, the award-seeker must know both the extent and the distribution of the efforts of other prospective grantees. Because it is always possible for a competitor to be successful by altering his applicative efforts given this information, there are no pure strategies that dominate all others. Award-seekers are once again left to uncertainties of judgement in any single play of the grant-seeking game. However, in contrast to the situation under one-man arbitration, there is no clear tendency for pre-application Cournot adjustment to increase overall effort levels in two-party games of influence. Mr. A's ideal strategy given Mr. B's intent never requires more resources than those committed by B. For example, suppose that A has learned of B's intention to distribute relatively large efforts across all committee members, as in a pattern of (100,100,100). A can dominate B's strategy by concentrating a smaller overall effort on a simple majority, as in a pattern of (101,101,0). Concentrating one's efforts on a minimally sized majority coalition will generally prove to be a cost-effective strategy. Unfortunately for A, B can respond to this plan by focusing still smaller total efforts on the neglected minority and a small portion of A's majority coalition, as with (0,102,1). This strategy requires just a bit more than half of the resources put forth by A. A could respond to B's new intent with a similar strategy, as with a pattern of (1,0,2) which establishes a very cost-effective majority. In this game, a process of Cournot adjustment leads to a de-escalation, rather than escalation, of competitive efforts. As this process continues, patterns of effort converge toward {e,Q,e^d\ where e is the resource cost of the smallest effective application, and d is the smallest increment of effort above e that allows a committee member to distinguish an application based upon an effort level of e + d from one fashioned from effort level e. Once at this limit, further adjustment will cause potential majorities to change in cyclic pattern. Other strategies are possible, but none is consistent with the Cournot method of choosing strategies, that is, with maximizing the net pay-off of successful application given competitor strategies. While convergence of the adjustment process does not allow us to predict a particular winner or loser, it does allow us to predict the total resource commitments that will be made at the limit, namely Ae + 2d, To the extent that minimal efforts are small relative to the prospective award, the investment limit for committees, 4^ +2d, will be less than the limiting case for one-man administration, IP. Perhaps more important than these limiting cases is the contrast between the escalation of competitive efforts under oneman administration and the potential for de-escalation under committees.

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These tendencies suggest that fewer games of influence will be played under committee administration, and the games that are played tend to be played with relatively fewer resources. Committee administration itself may reduce the extent that resources are utilized in competitive games of influence.

3. Award-seeking effort under proportionate-share rules All-or-nothing distributive rules are by no means the only means of allocating a pool of resources between competing parties. The explore the possibility that the results obtained above are peculiarities of the all-ornothing rule, we now modify that administrative structure by changing the allocative rule to be implemented. Rather than restrict awards to a specific subset of all potential grantees, awards will now be distributed across all eligible parties according to some criteria of relative merit or 'grantworthiness'. Merit raises and governmental grants are often apportioned among applicants according to some criteria of relative performance rather than restricted to a select few. The alternative rule will be called a proportionate-share rule. Under a porportionate-share rule, some pool of wards is to be distributed across all appHcants according to relative merit. If A is twice as deserving as B, A will receive a grant twice as large as that given B. If P is the total value of the pool of awards to be apportioned, Ej is the level of effort put forth by the yth individual, and demonstrated merit is produced via such efforts with constant returns to scale, then the award received by the yth individual can be written as:

-[:it]As might have been expected, the adoption of this more equitable rule changes the incentives for competitive efforts. Before we proceed, a few differences between all-or-nothing and proportionate-share rules should be noted. First, under poportionate-share rules any level of appHcative effort will guarantee the applicant some nonzero award, whereas there is no similar guarantee under all-or-nothing rules. Second, proportionate-share rules require a divisible pool of rewards, whereas all-or-nothing rules do not. Third, administration of proportionateshare rules requires a cardinal measure of worthiness, whereas all-or-nothing rules require only ordinal measures. As a consequence of the last two characteristics, it. will not always be possible to replace all-or-nothing allocative rules with proportionate-share rules. Some awards are intrinsically all-or-nothing, as there can be only one ambassador to the United Nations

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or but few Nobel laureates. In some settings, the required cardinal measure of worthiness may not be available, or possible. Thus, comparisons across allocating rules must be limited to those areas in which a change can be readily made. 3.1. One-man administration of proportionate-share rules Under one-man administration, awards are determined by the administrator's perception of relative merit. If we assume that demonstrated merit is produced by a process exhibiting constant returns to scale, then relative merit amounts to the relative value of the resources invested by each competitor. In this setting, the net advantage of application is the award received less the cost of applicative efforts, or Nj=^A^^Ey

(2)

If we return to our two-applicant world and let individual A's effort level be Ea and let E^, represent the effort level of B, eq. (2) becomes:

The first-order conditions for maximizing A's net return is satisfied in:

dE,

\{E,^E,y

Solving this condition for £« in terms of Ej, gives us A's optimal strategy function:

This function specifies the pay-off maximizing effort for A given B's effort and the size of the award pool, P. Once again, the best strategy for A depends on the level of effort chosen by B. Symmetry implies that B's strategy will be based upon a similar rule. Because no single effort level proves optimal independent of the actions of one's competitors, the resource level embodied in any single application will again be the result of active judgement rather than an objective process of optimization. However, if it is possible for each competitor to learn the intended effort levels of the other and to adjust accordingly, there is a tendency for effort levels to converge to equilibrium levels. Fig. 1 depicts the optimal strategy functions (or reaction functions) of the two players. Note that selection of

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1.2.1.1 Committees and rent-seeking effort R,D. Congleton, Committees and rent-seeking effort

P/4

205

L,

Fig. 1

any strategy requiring fewer than P resources leads to a series of adjustments that converge to P/4, P/4, which constitute equihbrium levels of effort. Thus, one-man administration of proportionate-share rules differs from the previous examples in that equilibrium strategies do exist. At this equilibrium, fewer resources are devoted to the competitive game of influence than under a similar administration of all-or-nothing rules at or near their escalation limit. P/2 will always be less than 2P. 32. Committee administration of proportionate-share rules Committees charged with administering proportionate-share rules will tend to allocate the pool of resource grants in accordance with the median administrator's evaluation of relative merit. Because the administrators are assumed to be homogeneous, both the median voter and his inclinations regarding relative merit will be determined by the distribution of efforts

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across members of the decision-making body. For convenience, it will again be assumed that the allocating committee has three members. Grant-seekers facing this administrative structure have the threefold task of (1) determining optimal levels of grant-influencing effort, (2) distributing that effort across members of the decision-making body so as to generate a particular median voter, and (3) demonstrating the optimal level of merit to that median voter. In our two-applicant world, Mr. A's net gain for an effort level of Ma focused on the median voter is:

where P is the size of the award fund, M^ is B's effort level toward the committee member that A determines to be the best median voter, and S^ is the level of resources required to assure a particular median voter. Needless to say, Sa should be as small as possible since it does not directly affect the grant ultimately received by A. Allocating no effort to one of the committee members efficiently assures that he will be an extreme and not the median. Similarly, allocating some very small amount of effort to members neglected by one's competitor assures that they will be the other extreme (in this case, of the view that essentially all of the award should go to A). Thus, a typical pattern of applicative effort will look like (0,S^, M^). An optimal award-seeking effort function can be derived from eq. (6) in a manner similar to that used above for proportional share rules under oneman administration. The resulting function is: £.= -£/, + V ^ ^

(7)

This equation does not dictate a total level of effort, but rather the optimal level of effort to direct at the median voter given the efforts of B. Note that the cost of assuring this particular median, S^, is irrelevent at this point, it being a fixed cost. A similar function can be determined for B:

E,^-E,^JJE^,

(8)

Eqs. (7) and (8) indicate that the optimal level of resources to be directed toward the median voter varies with the efforts of one's competitor. Again, there are no dominant pure strategies in the game of influence. If the competitors confront a stable median voter, then eqs. (7) and (8) represent reaction functions similar to those developed above for one-man administration of similar rules. However, the presumed homogeneous character of the administrators implies that the median committee member is endogenously determined by the relative effort levels of the competitors.

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Instead of following the dictates of eq. (7) or (8), competitors will generally find it cost-effective to alter the distribution of their efforts and generate a new median voter. For example, were Mr. B to learn of Mr. A's intent to distribute his efforts in a pattern like (0,S«,M^), B's best response would be to ignore committee member 3 (A's intended median) and attempt a strategy like (S^jAffejO), This strategy makes administrator number 2 the median voter. Unfortunately, B's strategy renders A's plan sub-optimal. Were A to learn of B's new plan, he would be inclined to adopt a pattern like (Ma,0,S„). This revision in turn renders B's strategy sub-optimal. In contrast to one-man administration of proportionate-share rules, Cournot adjustments do not converge to stable strategy pairs. There are no equilibrium pure strategies under committee administration. However, a series of Cournot adjustments will converge to stable levels of resource commitment. As was mentioned above, both S^ and S^ should be as small as possible in order to minimize the cost of establishing a particular median administrator. If e is the smallest effective effort level, then eq. (7) implies that optimal strategies will involve patterns of effort like (e,0, — e-h^^P), where e is the minimum discernible effort and —e + y/eP is the level of effort focused on a median voter who is the target * of e efforts from one's competitor. Strategic revision at this point requires changes in the distribution of efforts rather than the level of resources committed. To the extent that minimal effort, e, is 'small', fewer resources will tend to be used in the competitive game of influence under committee administration of proportionate-share rules than under one-man administration. This parallels our result under all-or-nothing rules. However, since -e-\-y/ep must be at least as large as e-\-d, the resource investments made under committee administration of proportionate-share rules will be at least as large as those committed under a similar administration of all-or-nothing rules. 4. Conclusion and summary In admittedly abstract and highly simplified circumstances, committee administration has been shown to reduce the extent to which scarce resources are utilized by would-be recipients to influence the deliberations of administrators. Two properties of committee administration are responsible for this result. First, the minimum award necessary to induce participation in a game of influence is greater for committees than for single administrators. There are simply more people to be influenced under committee administration than under one-man administration and thus the participation threshold tends to be greater. Second, two-party games of influence have a tendency to de-escalate under committee administration, whereas such games tend to escalate under one-man administration within the environments explored. Games of influence played under majority rule institutions are not simply a matter of resource commitments but of coalition building. The

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usually undesirable absence of stable majority coalitions allows awardseekers to economize on efforts devoted to influencing committee deliberations by targeting alternative majority coalitions. Table 1 summarizes the resource commitments made at the respective Cournot hmit points under the four allocative regimes examined. Both decision-making method and allocative rules have substantial effects on the extent to which resources are utilized in award-seeking games. Fewer resources were committed to games of influence under committee administration than under one-man administration of either allocative rule. Casual empiricism tends to support these theoretical results. Lobbying efforts directed toward committees do seem to be somewhat smaller than those directed at single individuals with similar allocative authority. It should be noted that the effects of allocative rule changes were not decision-method invariant. Under one-man administration, a shift from all-or-nothing rules to the more equitable proportionate share rules tended to greatly reduce competitive efforts, whereas a similar shift under committee administration tended to increase effort levels.^ Table 1 Administrative structure

Limiting case or equilibrium strategy

Grant-seeking effort rank

1. All-or-Nothing (a) One administrator (b) Majority rule administration

P (0, e, e + 5)

1 4

2. Proportionate-share (a) One administrator (b) Majority rule administration

P/4 {0, e, - e-^^ ^/eP)

2 3

The question of whether all competitive efforts at influence should be classified as counter-productive rent-seeking has not been explored and is beyond the scope of this paper. A reasonably thorough characterization of the sorts of competitive efforts that do tend to be wasteful can be found in the previous work of TuUock (1967) and Congleton (1980). It is presumed here that minor efforts to influence decisions may be productive in so far as more informed choices might be made, but that greater efforts tend to have little, if any, positive effect on the decision reached. These supra-productive efforts may properly be regarded as wasteful rent-seeking if they neither ^It may also be worth noting that to the extent that administrators are beneficiaries of competitor efforts, one expects committees to prefer the more equitable proportionate>share rules because of the above competitive propensities alone. Thus, to the extent that equity, itself, is of general interest, our analysis suggests that committees may be a preferred method of decision-making because they may tend to adopt more equitable distributive procedures.

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generate positive externalities nor represent the simple transfer of resources to others. In this context, the above analysis implies that committees themselves tend to reduce rent-seeking waste. To the extent that these economies more than offset the generally higher decision costs of multi-party deliberations, committees may serve as efficiency enhancing instruments despite their cumbersomeness. The scope of this conclusion is limited by the institutional environment explored and by the mechanism used to generate competitor strategies. The Cournot method of modeling individual choice and interdependency has a limited appeal in settings where decision-makers are considered to be wellinformed and interaction extensive. Although even here, the Cournot assumption provides substantial insight into the operation of auctions and other important market processes. On the other hand, if individuals are presumed to be less than completely informed and social institutions largely the result of evolutionary tendencies, then the convergence properties of relatively simple choice mechanisms may be of greater relevance. To the extent that Cournot adjustments are accepted as a reasonable first approximation of actual decision-making procedures, committees might be used because of a conscious awareness of the economies developed above. Or, more likely, they may, in Hayek's (1973) terms, be examples of kosmos or social evolution; institutions created by chance and perhaps maintained for incorrect reasons, but nonetheless reasonably efficient social mechanisms.

References Arrow, K„ 1951, Social choice and individual values (John Wiley and Sons Inc., New York). Black, D., 1958, The theory of committees and elections (Cambridge University Press, Cambridge). Buchanan, J.M. and G. TuUock, 1962, The calculus of consent (University of Michigan Press, Ann Arbor). Buchanan, J.M., R.D, Tollison and G. TuUock, eds., 1980, Towards a theory of the rent-seeking society (Texas A&M Press, College Station). Congleton, R.D., 1980, Competitive process, competitive waste, and institutions, in: J.M, Buchanan, R.D. Tollison and G. Tullock, eds.. Towards a theory of the rent-seeking society (Texas A&M Press, College Station). Foster, E., 1981, Competitively awarded government grants. Journal of Public Economics 13, 105-111. Hayek, F.A., 1973, Law legislation and liberty (University of Chicago Press, Chicago). Krueger, A.O., 1974, The political economy of the rent seeking society, American Economic Review 64, 291-303, Shiffman, M., 1953, Games of timing, in: H.W. Kuhn and A.W. Tucker, eds., Contributions to the theory of games II, Annals of mathematical studies, 28 (Princeton University Press, Princeton) 97-123. Tullock, G., 1967, The welfare costs of tariffs, monopoly, and theft. Western Economic Journal 5 (June), 224-232. Usher, D., 1981, The economic prerequisites to democracy (Columbia University Press, New York).

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1.2.1.2 Risk-averse rent seeking with shared rents The Economic Joumaly 97 {December 1987), 971-985 Printed in Great Britain

RISK-AVERSE RENT SEEKING WITH SHARED RENTS* Ngo Van Long and Neil Vousden

The theory of rent-seeking behaviour has advanced in many directions since the pioneering contributions by Tullock (1967), Krueger (1974), and Posner (1975). Perhaps because of the early preoccupation with the social costs of monopoly, the literature has tended to emphasise contests in which the ' winner takes all' (e.g. Tullock (1980, 1984), Corcoran (1984), Hillman and Katz (1984) and Hillman and Samet (1987)). This approach is appropriate when agents compete for a monopoly rent, a government contract or any other indivisible transfer. There is however another important case, relatively neglected in the literature,^ in which agents expend resources competing for a share of a divisible rent rather than for the whole of an indivisible rent. It is this type of rent seeking which is the subject of the present paper. It is not difficult to find examples of contests in which a rent is shared. Indeed, the original Krueger (1974) paper considered one such c a s e - t h e allocation of import quota licences among competing importers.^ Competition by groups or individuals for a share of the government's tax/tariff revenue falls into the same category. Other examples are: (i) allocation of rents among the members of a cartel; (ii) lobbying by factor owners for a share of GNP in economies where wages are wholly or partly determined by a central authority; {Hi) lobbying by intermediate and final good producers in an industry for/ against protection of intermediate goods (i.e. competition for the rents which would accrue to final good producers in the absence of such protection); and (iv) resources expended by Deans of faculties preparing submissions etc. to increase a faculty's share of a fixed university budget! In all such instances, an issue of vital concern is the measurement of the social waste associated with the rent seeking process. In particular, under what circumstances is the value of rents a good measure of the value of resources expended in contesting those rents? This question has been a recurring theme in the literature on ' winner-takes-all' games cited above. One reason why the value of a rent may overstate the true social cost of rent seeking is risk aversion on the part of rent seekers. The role of risk aversion in the lobbying process has been analysed by Hillman and Katz (1984) and Hillman and Samet (1987) for the case of indivisible rents. In particular, Hillman and Katz derive a simple expression showing how risk aversion reduces the proportion of rents dissipated by lobbying below unity. They also show that for * The authors are indebted to Jurgen Eichberger, Arye L. Hillman, Tracy Lewis and two anonymous referees for helpful comments on an earlier draft. The usual caveat applies. ^ There is some discussion of'proportional share' games in Congleton (1980). * Bhagwati and Srinivasan (1980, 1982) also consider rent seeking for a share of import quota rents or tariff revenue. [971 ]

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a relatively small rent contested, the value of the rent is a good approximation to the resource cost of the associated rent seeking. In this paper these and other issues are explored using a model in which n risk averse players expend resources/effort lobbying for a share of a 'cake' (total rents). Initially we assume that the size of the cake is fixed and known with certainty, but that each player's share is uncertain, its probability distribution depending on the player's lobbying effort relative to that of his opponents.^ Comparative static results for the symmetric Nash equilibrium are derived, in particular the effects of changes in the cake size and the number of players on the equilibrium level of lobbying effort. Of course, the assumption that the total rent is fixed is not always appropriate. Frequently the aggregate non-cooperative efforts of lobbyists for a share of the cake, will affect the size of the cake. For example, lobbying by pressure groups for a share of tax revenue may lead to the government increasing taxes, so that the 'subsidised' group's non-cooperative efforts collectively result in a larger cake. To capture such possibiUties we allow for cake size to be uncertain (subject to a known upper Hmit) and for its expected value to depend on the aggregate non-cooperative efforts of the players. An important feature of this case is that a player's lobbying effort is simultaneously directed towards a private return (his share) and a public good (the size of the rent to be shared). Finally, the model with fixed total rents is extended to allow for free entry/ exit to/from the lobbying activity. We are then able to derive a simple formula for the proportion of rents dissipated in the lobbying process in the long run. The formula obtained is a natural extension of that obtained by Hillman and Katz, involving the coefficient of relative risk aversion and the variance of the individual share. The proportion of rents dissipated is seen to be reduced further below unity the more risk-averse are the rent-seekers and the greater the risk faced. The paper is set out as follows. Section I presents the basic model with a fixed number of rent seekers and a fixed, known cake size. Section II extends the model to allow the expected cake size to be influenced by the aggregate noncooperative efforts of the rent seekers. In Section III we relax the assumption that the number of players is fixed and allow free entry/exit to/from the lobbying activity. Section IV summarises the main conclusions of the paper. I. FIXED TOTAL RENTS, FIXED NUMBER OF RENT SEEKERS

n identical players (lobbyists) are assumed to expend resources competing for a share of a cake (rents) of fixed, known size, K, The share of the cake going to any one player is uncertain, with its probability depending on that lobbyist's efforts relative to those of his opponents. We assume that if all players increase * In this respect our paper is more in the tradition of the * efficient rent seeking* models of Tullock (1980) where a Nash equilibrium yields an agent's probability of success in a contest. This is in contrast to contests such as the 'Brennan game' where the highest ouriay wins and all ouriays are irretrievably l o s t - i n such cases a Nash equilibrium in pure strategies does not exist. Hillman and Samet (1987) however show that a mixed-strategy equilibrium exists for such contests.

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their efforts in the same proportion, then the probability distribution for each player*s share is unaffected. Moreover, in wrhat foUov^s v^e confine our attention to a special case of this homogeneity - specifically, we assume that the probability distribution associated with player z's share is unaffected if player i and his aggregate opposition both increase their efforts in the same proportion.^ Thus each player only considers the total amount of pressure applied by his opponents when deciding on his own optimal degree of pressure. The model may now be stated formally as follows. Let the share of the cake secured by player i be denoted by 5^ (f = i, 2,..., n;o ^Si^ i). Player i can increase the probability that his share exceeds any given value ye{Oyi) by increasing his lobbying effort (x^) relative to the aggregate effort (ZJ of the other players; specifically we define the distribution function: G^(y,;c„Z,) = Pr(5,O{>O

for

\

(i)

^6(0,1));

iiin^-.^ G*2 (y, I, ^) = o,

where

q = Zjx^;

the signs on the partial derivatives indicating that: (a) increased lobbying effort by player i increases the probability that his share exceeds y; (b) increased aggregate lobbying effort by players other than i reduces the probability that Si exceeds y; {c) increased aggregate lobbying effort by players other than i dampens the marginal effect of fs effort; and (rf) the marginal effect of f s effort becomes insignificant as fs relative effort (i/q) becomes small. Let g^yyXiyZf) denote the density function derived from G^(i.e. / = dG^/dy = G{). It follows that / will also be homogeneous of degree zero in (:Vi,ZJ, which implies, in turn, t h a t ^ ( = d^/dx^) v^ll be homogeneous of degree minus one in (^i,Z^), a property which will be useful when analysing the symmetric Nash equilibrium. Each player is assumed to be risk averse, with the utility of rents obtained represented by an increasing, stricdy concave function, u{yK)y which is assumed to be bounded in the relevant range. The cost of lobbying effort is represented by an increasing, convex function, C{Xi), Since x^ is not a random variable, we define relative risk aversion with respect to the uncertain rent as the elasticity of the marginal utility u\ i.e. u' d{yK)

u'

^^ '

* This amounts to assuming that player I's probability distribution does not depend on the composition of opposition effort across players or, equivalcntly, that the players* 'political influence functions* exhibit constant returns to scale with respect to lobbying effort.

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Player i is then assumed to solve the following optimisation problem for given Z,: MaxF,= u{yK).g\y,Xi,Zi)dy-C{Xi), (z = i,2,...,n). (2) Xi

Jo

The first order condition for a solution to this problem is:

£'

^{y^)-gliyy^iy^i)dy = (^i^i) (^*= 1,2,...,«).

(3)

It is convenient to work with the following alternative form of (3), obtained by integrating by parts: G'^{y,x,,Z,)u{yK)\' -- (" Ku'{yK)(?,{y,x,,Z,)dy lo Jo

= C{x,)

(z= I,2,...,;^). (3a)

Since G*(o,',-) = o and C*'(i,-,-) = i the derivative G{ is identically zero when evaluated at y = o and y = i. Hence the first term on the left hand side of (3 a) is zero and the condition reduces to

KJ'u'{yK)[-G[{y,x,,Z,)]dy

= C{x,),

(f = 1,2,...,«).

(3^)

The left hand side of (3 b) is the marginal return to effort and the right hand side is the marginal cost of effort. The solution (xi,^2> •••>'^n) of the system (3^) represents a Nash equilibrium. Given that all players are identical, it is natural to consider the symmetric Nash equiHbrium (SNE)^ in which x^ = x^^ ,^. = x^ = X. (3^) then becomes (omitting the 2-superscript from G): KJ'^ u'(yK){-G^{y,x,{n--i)x]}dy-C{x)

= o.

(4)

Now, since g^ is homogeneous of degree — i in [x^.Z^), G^ will also be homogeneous of degree — i. Thus (4) becomes A{x,K,n)^K?

u\yK)[-G^{y,i,n-i)]dy--xC{x)^o.

[^a)

We shall now proceed to derive comparative static results for the effects of changes in the value of total rents, iC, and the number of rent seekers, «, on the SNE (4a). {a) Effect of Changes in K The following proposition reveals how the degree of risk aversion influences the use of resources in lobbying when the rents sought are of different magnitudes: PROPOSITION I. When the number of rent seekers isfixed,an increase in the value of total rents, K, will cause the SNE level of lobbying effort by each player (x) to increase * Note that (o, o,..., o) may also be a SNE, but if C{o) = o then it is not an equilibrium because it always pays for each player to increase his effort. Alternatively, we could rule out zero outlays by assuming a minimum required level of eifort for each player participating in the contest.

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{decrease) if the relative risk aversion for u is less than {greater than) unity in the interval Proof From (4 a), dx/dK = —A^/A^, where A^ = -C{x)-xCr{x) A, = - [

u^{yK)G,dy--K[

JO

yu\yK)G,dy= Jo

o otherwise). A may also be interpreted as a 'perception parameter': o < A < i if individuals imperfectly perceive the contribution of their effort to cake size, A = i for perfect perception and A = o for zero perception.' H is assumed to have the following properties: H{K,AX,K) = I >fX,K^o;

H{o,AX,K) = oVZ,X^o;

//j ^ o ( > o for some se(o,K)); H^ < o ( < o for se{o,K));

H2^o{ 00, we obtain (13).

•

REFERENCES

Bhagwati, J. N. and Srinivasan, T. N. (1980). * Revenue seeking: a generalization of the theory of tariffs.* Journal of Political Economy ^ vol. 88, December, pp. 1069-87. and (1982). 'The welfare consequences of directly unproductive profit seeking lobbying activities: Price vs. quantity distortions.* Journal of International EconomicSy vol. 13, August, pp. 33-44. Buchanan, J. M., Tollison, R. D. and TuUock, G. (eds.). (1980). Toward a Theory of the Rent-Seeking Society. College Station: Texas A&M Press. Congleton, R. (1980). 'Competitive process, competitive waste, and institutions.* In Buchanan et al. (1980), PP- 153-79Corcoran, W. J. (1984). 'Long run equilibrium and total expenditures in rent seeking.* Public Choice^ vol. 43, no. I, pp. 89-94. Comes, R. and Sandler, T. (1986). The Theory of Externalities, Public Goods and Club Goods. Cambridge: Cambridge University Press. Hillman, A. L. and Katz, E. (1984). 'Risk-averse rent seekers and the social cost of monopoly power.* ECONOMIC JOURNAL, vol. 94, March, pp. 104-10. and Samet, D. (1987). 'Dissipation of rents and revenues in small-numbers contests.* Public Choice (forthcoming).

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Krueger, A. O. (1974). *The political economy of the rent-seeking society.' American Economic Reviewy vol. 64, June, pp. 291-303. Reprinted in Buchanan et al, (1980), pp. 51-70. Posner, R. A. (1975). *The social costs of monopoly and regulation.' Journal of Political Economyy vol. 83 August, pp. 807-27. Reprinted in Buchanan el al. (1980), pp. 71-94. TuUock, G. (1967). 'The welfare costs of tariffs, monopolies and theft.' IVestern Economic Joumaly vol 5, June, pp. 224-32. Reprinted in Buchanan et al. (1980), pp. 39-50. (1980). 'Efficient rent seeking.' In Buchanan et d.y (1980), pp. 97-112. (1984). 'Long-run equilibrium and total expenditures in rent-seeking: a comment.' Public Choice, vol. 43, no. I, pp. 95-7.

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1.2.1.3 Collective rent dissipation The Economic Journalf l o i {November 1991), 1522-1534 Printed in Great Britain

COLLECTIVE RENT DISSIPATION* Shmuel Nitzan The theory of rent seeking was initiated by Tullock (1967, 1980), followed by Krueger (1974) and Posner (1975) and, more recently, extended in various directions by AUard (1988); Appelbaum and Katz (1986 a, A, r); Gradstein and Nitzan (1989); Hillman and Katz (1984, 1987), Hillman and Samet (1987); Hillman and Riley (1989); Katz et aL (1990); Long and Vousden (1987); Ursprung (1990) and Varian (1989). The purpose of this paper is to extend this theory by introducing the possibility of collective-group rent seeking with voluntary individual decisions regarding their extent of participation in the collective rent-seeking efforts. The type of rent seeking I consider has the following characteristics: (i) a number of groups of individuals compete for a single rent, (ii) the rent is indivisibly allocated in the sense that a single group wins the entire rent, (iii) the rent exhibits private good characteristics and therefore it can be divided among the members of the winning group and (iv) group members decide voluntarily on the extent of their rent-seeking efforts. Common examples of this type of rent seeking are the struggles for the budgets at the discretion of politicians by various interest groups (parties, localities, industries, etc.). In imperfectly discriminating^ collective rent-seeking contests substantial dissipation of the contested rents is usually not the case. The reduced dissipation rate is obtained in small or large-number contests, with risk neutral or risk averse potential rent seekers and with homogeneous or heterogeneous agents. Reduced dissipation in our extended group rent-seeking model is basically due to two factors: the free-riding incentives within the groups competing on the rent and the deterrent group-size effects - since rents are subtractable, an increase in group size reduces the individual members' shares in the rent. Both of these factors depend on the rules applied by the groups to distribute rents among their members. In this study I focus on mixed rules that distribute part of the rent on an egalitarian basis and the rest on the basis of the principle 'to each according to his relative effort'. The main results provide the relationship between the proportion of the rent dissipated in equilibrium and the following parameters: the number of competing groups, the size of the groups, the distribution rule applied by the groups and the characteristics of the individual players: their endowed wealth and attitudes towards risk. It is shown that, in general, the extent of rent dissipation is positively related to the * I am very much indebted to John Hey, Arye L. Hillman, Nava Kahana and two anonymous referees for their useful comments and suggestions. I also wish to express my appreciation to Moshe Glazman for his help in data processing. ^ In an imperfecdy discriminating contest an ultimate winner is not designated, but rather, each contending group is assigned a probability of winning the rent. For an analysis of perfectiy discriminating individual rent-seeking contests see Hilhnan and Samet (1987) and Hillman and Riley (1989).

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number of contesting groups, and is inversely related to the degree of ^ egalitarianism' in distributing the rents and to the degree of individual risk aversion. The effect of the total number of potential rent seekers is ambiguous. Thus, in contrast to earlier studies that deal with individual rent seeking, in our setting an increase in population may reduce the extent of rent dissipation. Finally, wealth variability may also set a bound on the possible outlays on the rent seeking activities. The basic elements of the extended model of collective rent seeking are introduced in Section I. In Section II, I study the special case where all groups distribute rents according to relative effort. The diametrically opposed case where all groups distribute rents equally among their members is analysed in Section III. In Section IV I consider the general case of groups applying the same mixed distribution rule. Individuals are initially assumed to be risk neutral and identical in their endowed wealth. The only asymmetry allowed in the basic extended model is the variability in group-size. The analysis of the extended collective rent-seeking model with heterogeneous players (agents are endowed with different wealth levels and are characterised by different attitudes towards risk) is presented in Section V. The empirical applicability of the extended approach is discussed and illustrated in Section VI. Section VII contains a brief summary of the main conclusions.

I. VOLUNTARY PARTICIPATION IN COLLECTIVE RENT SEEKING

Consider n interest groups confronting the opportunity of winning a prespecified prize S, This prize is referred to as a contestable rent. We focus on Tullock's (1980) special case of an imperfectly discriminating contest where the political process cannot discriminate among the competing groups to designate a winner with certainty, but rather the outcome of the contest is the assignment to each group of a probability that it wins the contest. Specifically, group fs probability of success in the contest, TT^, is given by the value of its outlay X^, X^ = S^l*i Xj^i, relative to the total oudays Z, Z = S^.j Z}, made by all groups engaged in the rent seeking activities. That is, 7r,(Z„...,ZJ=-/^ = | .

(I)

SZ, i"!

Let us denote by n{i) the number of individuals in group i. The size of the groups is given exogenously and is held constant, i.e. inter-group mobility is not allowed in our model. Initially individuals are assumed to be identical risk neutrals endowed with wealth y. The expected payoff of a representative individual in group e, individual ki^ from being engaged in the rent seeking activity is assumed to be given by: V^, = n,{y + Sf^,[X,,,,,,,X^,,,.,,X,,,,,]-^XJ^-{i--n,){y^X^,^

(2)

where/^^[Zi ^n(i)i] ~

"y

^ '^TK'

v3)

That is, a proportion a of the rent is distributed on egalitarian grounds and the rest is distributed according to relative effort. When a—i, each member in group i receives i/n{i) of the rent regardless of his personal effort. When a = o the member ki receives Xj^JX^ of the rent. Substituting (i) and (3) into (2) we get:

»'-.-f{'+*[('-)f*;i]-'..)+T^ Y^ for k-= 1,2. From equation (8) it follows immediately that this condition is equivalent to Zi > 0 and Z2 ^ 0. 4J

(10)

The equilibrium in the symmetric case

Consider now the symmetric case where the two groups are of equal size, i.e. m = n. In equilibrium, the condition 62-1 -61 is then satisfied. Three types of equilibria can be distinguished: (1) convergent equilibria in the sense of Hotelling and Downs at the midpoint of the political spectrum [^i = (?2= ^Z^], (2) completely polarized equilibria [61 = 1,62 = 0], and (3) partially polarized equilibria [0 < ^2= 1 ~