An Introduction to Game Theory

An Introduction to Game Theory

Please send comments to Martin J. Osborne Department of Economics 150 St. George Street University of Toronto Toronto, Canada M5S 3G7 email: [email protected]

This version: 2002/7/25

Note: The categorization of exercises according to the availability of their solutions is tentative. No solutions are currently publicly available. Note: The Index currently covers only Chapters 1 and 2 (and a few other bits and pieces). Note: Bad line breaks and page breaks have not yet been corrected.

An Introduction to Game Theory Martin J. Osborne

New York

Oxford OXFORD UNIVERSITY PRESS 2002

OXFORD UNIVERSITY PRESS Oxford New York Athens Auckland Bangkok Bombay Calcutta Cape Town Dar es Salaam Deli Florence Hong Kong Istanbul Karachi Kuala Lumpur Madras Madrid Melbourne Mexico City Nairobi Paris Singapore Taipei Tokyo Toronto and associated companies in Berlin Ibadan

Copyright © 1995–2002 by Martin J. Osborne Published by Oxford University Press, Inc., 198 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, by any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. This book was typeset by the author, who is greatly indebted to Donald Knuth (TEX), Leslie Lamport (LATEX), Diego Puga (mathpazo), Christian Schenk (MiKTEX), Ed Sznyter (ppctr), Timothy van Zandt (PSTricks), and others, for generously making superlative software freely available. The main font is 10pt Palatino. Library of Congress Cataloging-in-Publication Data Osborne, Martin John. An Introduction to Game Theory / Martin J. Osborne p. cm. Includes bibliographical references and index. ISBN ????? 1. Game Theory. 2. Economics, Mathematical. I. II. Title. HB144.O???? 2002

Contents

Preface 1

I 2

xi

Introduction 1 1.1 What is game theory? 1 An outline of the history of game theory John von Neumann 3 1.2 The theory of rational choice 4 1.3 Coming attractions 7 Notes 8

Games with Perfect Information

3

9

Nash Equilibrium: Theory 11 2.1 Strategic games 11 2.2 Example: the Prisoner’s Dilemma 12 2.3 Example: Bach or Stravinsky? 16 2.4 Example: Matching Pennies 17 2.5 Example: the Stag Hunt 18 2.6 Nash equilibrium 19 John F. Nash, Jr. 21 Studying Nash equilibrium experimentally 22 2.7 Examples of Nash equilibrium 24 Experimental evidence on the Prisoner’s Dilemma 26 Focal points 30 2.8 Best response functions 33 2.9 Dominated actions 43 2.10 Equilibrium in a single population: symmetric games and symmetric equilibria 49 Notes 51

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Nash Equilibrium: Illustrations 53 3.1 Cournot’s model of oligopoly 53 3.2 Bertrand’s model of oligopoly 61 Cournot, Bertrand, and Nash: some historical notes 67 3.3 Electoral competition 68 3.4 The War of Attrition 75 3.5 Auctions 78 Auctions from Babylonia to eBay 79 3.6 Accident law 89 Notes 94

4

Mixed Strategy Equilibrium 97 4.1 Introduction 97 Some evidence on expected payoff functions 102 4.2 Strategic games in which players may randomize 103 4.3 Mixed strategy Nash equilibrium 105 4.4 Dominated actions 117 4.5 Pure equilibria when randomization is allowed 120 4.6 Illustration: expert diagnosis 121 4.7 Equilibrium in a single population 126 4.8 Illustration: reporting a crime 128 Reporting a crime: social psychology and game theory 131 4.9 The formation of players’ beliefs 132 4.10 Extension: Finding all mixed strategy Nash equilibria 135 4.11 Extension: Mixed strategy Nash equilibria of games in which each player has a continuum of actions 140 4.12 Appendix: Representing preferences over lotteries by the expected value of a payoff function 144 Notes 148

5

Extensive Games with Perfect Information: Theory 151 5.1 Extensive games with perfect information 151 5.2 Strategies and outcomes 157 5.3 Nash equilibrium 159 5.4 Subgame perfect equilibrium 162 5.5 Finding subgame perfect equilibria of finite horizon games: backward induction 168 Ticktacktoe, chess, and related games 176 Notes 177

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6

Extensive Games with Perfect Information: Illustrations 179 6.1 The ultimatum game, the holdup game, and agenda control 179 Experiments on the ultimatum game 181 6.2 Stackelberg’s model of duopoly 185 6.3 Buying votes 190 6.4 A race 195 Notes 201

7

Extensive Games with Perfect Information: Extensions and Discussion 203 7.1 Allowing for simultaneous moves 203 More experimental evidence on subgame perfect equilibrium 209 7.2 Illustration: entry into a monopolized industry 211 7.3 Illustration: electoral competition with strategic voters 213 7.4 Illustration: committee decision-making 215 7.5 Illustration: exit from a declining industry 219 7.6 Allowing for exogenous uncertainty 224 7.7 Discussion: subgame perfect equilibrium and backward induction 228 Experimental evidence on the centipede game 232 Notes 234

8

Coalitional Games and the Core 237 8.1 Coalitional games 237 8.2 The core 241 8.3 Illustration: ownership and the distribution of wealth 245 8.4 Illustration: exchanging homogeneous horses 249 8.5 Illustration: exchanging heterogeneous houses 254 8.6 Illustration: voting 258 8.7 Illustration: matching 261 Matching doctors with hospitals 266 8.8 Discussion: other solution concepts 267 Notes 268

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II

Games with Imperfect Information

271

9

Bayesian Games 273 9.1 Motivational examples 273 9.2 General definitions 278 9.3 Two examples concerning information 283 9.4 Illustration: Cournot’s duopoly game with imperfect information 9.5 Illustration: providing a public good 289 9.6 Illustration: auctions 292 Auctions of the radio spectrum 300 9.7 Illustration: juries 301 9.8 Appendix: Analysis of auctions for an arbitrary distribution of valuations 308 Notes 311

285

10 Extensive Games with Imperfect Information 313 10.1 Extensive games with imperfect information 313 10.2 Strategies 318 10.3 Nash equilibrium 318 10.4 Beliefs and sequential equilibrium 323 10.5 Signaling games 332 10.6 Illustration: conspicuous expenditure as a signal of quality 336 10.7 Illustration: education as a signal of ability 341 10.8 Illustration: strategic information transmission 343 10.9 Illustration: agenda control with imperfect information 351 Notes 358

III

Variants and Extensions

359

11 Strictly Competitive Games and Maxminimization 361 11.1 Maxminimization 361 11.2 Maxminimization and Nash equilibrium 364 11.3 Strictly competitive games 365 11.4 Maxminimization and Nash equilibrium in strictly competitive games 367 Maxminimization: some history 370 Empirical tests: experiments, tennis, and soccer 373 Notes 375 12 Rationalizability 377 12.1 Rationalizability 377 12.2 Iterated elimination of strictly dominated actions 385 12.3 Iterated elimination of weakly dominated actions 389 12.4 Dominance solvability 391 Notes 392

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13 Evolutionary Equilibrium 395 13.1 Monomorphic pure strategy equilibrium 396 Evolutionary game theory: some history 401 13.2 Mixed strategies and polymorphic equilibrium 402 13.3 Asymmetric contests 408 Side-blotched lizards 410 Explaining the outcomes of contests in nature 412 13.4 Variation on a theme: sibling behavior 413 13.5 Variation on a theme: the nesting behavior of wasps 417 13.6 Variation on a theme: the evolution of the sex ratio 419 Notes 420 14 Repeated Games: The Prisoner’s Dilemma 421 14.1 The main idea 421 14.2 Preferences 423 14.3 Repeated games 425 14.4 Finitely repeated Prisoner’s Dilemma 426 14.5 Infinitely repeated Prisoner’s Dilemma 428 14.6 Strategies in an infinitely repeated Prisoner’s Dilemma 428 14.7 Some Nash equilibria of an infinitely repeated Prisoner’s Dilemma 430 14.8 Nash equilibrium payoffs of an infinitely repeated Prisoner’s Dilemma 433 Experimental evidence 438 14.9 Subgame perfect equilibria and the one-deviation property 439 Axelrod’s tournaments 442 14.10 Some subgame perfect equilibria of an infinitely repeated Prisoner’s Dilemma 443 Reciprocal altruism among sticklebacks 447 14.11 Subgame perfect equilibrium payoffs of an infinitely repeated Prisoner’s Dilemma 448 Medieval trade fairs 450 14.12 Concluding remarks 451 Notes 452 15 Repeated Games: General Results 453 15.1 Nash equilibria of general infinitely repeated games 453 15.2 Subgame perfect equilibria of general infinitely repeated games 457 15.3 Finitely repeated games 462 15.4 Variation on a theme: imperfect observability 463 Notes 465

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16 Bargaining 467 16.1 Bargaining as an extensive game 467 16.2 Illustration: trade in a market 479 16.3 Nash’s axiomatic model 482 16.4 Relation between strategic and axiomatic models 490 Notes 493 17 Appendix: Mathematics 17.1 Numbers 495 17.2 Sets 496 17.3 Functions 497 17.4 Profiles 500 17.5 Sequences 501 17.6 Probability 501 17.7 Proofs 506 References Index 527

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Preface

pervades economic theory and is used widely in other social and behavioral sciences. This book presents the main ideas of game theory and shows how they can be used to understand economic, social, political, and biological phenomena. It assumes no knowledge of economics, political science, or any other social or behavioral science. It emphasizes the ideas behind the theory rather than their mathematical expression, and assumes no specific mathematical knowledge beyond that typically taught in US and Canadian high schools. (Chapter 17 reviews the mathematical concepts used in the book.) In particular, calculus is not used, except in the appendix of Chapter 9 (Section 9.8). Nevertheless, all concepts are defined precisely, and logical reasoning is used throughout. My aim is to explain the main ideas of game theory as simply as possible while maintaining complete precision; the more comfortable you are with tight logical analysis, the easier you will find the arguments. The only way to appreciate the theory is to see it in action, or better still to put it into action. So the book includes a wide variety of illustrations from the social and behavioral sciences, and over 280 exercises. The structure of the book is illustrated in the figure on the next page. The gray boxes indicate core chapters (the darker gray, the more important). A black arrow from Chapter i to Chapter j means that Chapter j depends on Chapter i. The gray arrow from Chapter 4 to Chapter 9 means that the latter depends weakly on the former; for all but Section 9.7 only an understanding of expected payoffs (Section 4.1.3) is required, not a knowledge of mixed strategy Nash equilibrium. (Two chapters are not included in this figure: Chapter 1 reviews the theory of a single rational decision-maker, and Chapter 17 reviews the mathematical concepts used in the book.) Each topic is presented with the aid of “Examples”, which highlight theoretical points, and “Illustrations”, which demonstrate how the theory may be used to understand social, economic, political, and biological phenomena. The “Illustrations” introduce no new theoretical points, and any or all of them may be skipped without loss of continuity. The “Illustrations” for the key models of strategic and extensive games are grouped in separate chapters (3 and 6). The limited dependencies between chapters mean that several routes may be taken through the book.

G

AME - THEORETIC REASONING

• At a minimum, you should study Chapters 2 (Nash Equilibrium: Theory) and 5 (Extensive Games with Perfect Information: Theory). • Optionally you may sample some sections of Chapters 3 (Nash Equilibrium: Illustrations) and 6 (Extensive Games with Perfect Information: Illustrations). xi

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Strategic games 3: Illustrations (53)

4: Mixed strategies (97) Imperfect information 2: Theory (11)

9: Bayesian games (273) Topics 11: Maxminimization (361) 12: Rationalizability (377) 13: Evolutionary equilibrium (395)

Extensive games 6: Illustrations (179) 7: Extensions (203) Imperfect information 5: Theory (151)

10: Signaling games (313)

Topics 14, 15: Repeated games (421, 453)

Coalitional games

16: Bargaining (467)

8: Core (237) The structure of the book. Each chapter is represented by a box whose area is proportional to the length of the chapter; the number in parentheses is the page on which the chapter begins. Boxes shaded gray correspond to core chapters; dark gray chapters are more central that light gray ones. An arrow from Chapter i to Chapter j means that Chapter i is a prerequisite for Chapter j. The gray arrow from Chapter 4 to Chapter 9 means that the latter depends only weakly on the former.

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• You may add to this plan any combination of Chapters 4 (Mixed Strategy Equilibrium), 9 (Bayesian Games, except Section 9.7, which requires Chapter 4), 7 (Extensive Games with Perfect Information: Extensions and Discussion), 8 (Coalitional Games and the Core), and 16 (Bargaining). • If you read Chapter 4 (Mixed Strategy Equilibrium) then you may in addition study any combination of the remaining chapters covering strategic games, and if you study Chapter 7 (Extensive Games with Perfect Information: Extensions and Discussion) then you are ready to tackle Chapters 14 and 15 (Repeated Games). All the material is intended to be accessible to undergraduate students. A onesemester course for third or fourth year North American economics majors (who have been exposed to a few of the main ideas in first and second year courses) could cover up to about half the material in the book in moderate detail. Osborne and Rubsintein (1994), a graduate text, offers a more advanced treatment of the field. With few exceptions, the two books use the same notation and terminology. The exceptions are noted on the website for this book (see the end of the preface). Examples modeling economic, political, and biological phenomena The main examples that involve economic, political, or biological phenomena are listed below. (Examples used to make a primarily game-theoretic point are omitted.) Economic phenomena Accident law: Section 3.6 Adverse selection: Exercise 282.4 Auctions: Section 3.5 (perfect information), Example 142.1 (all-pay), Section 9.6 (imperfect information), Section 9.8 (imperfect information) Bertrand’s model of oligopoly: Section 3.2, Exercise 134.2, Exercise 143.2, Exercise 190.1, Exercise 213.1, Exercise 392.4 (dominance solvability), Exercise 457.1 (repeated game), Exercise 461.1, Exercise 461.2, Section 15.4 (repeated game with imperfectly observable actions) Chain-store game: Example 230.1 Collective decision-making: Section 2.9.4 Common property: Section 3.1.5 Competition in product characteristics: Exercise 74.2 Cournot’s model of oligopoly: Section 3.1, Exercise 134.1, Section 9.4 (imperfect information), Exercise 388.3 (rationalizability) Entry into an industry by a financially-constrained challenger: Exercise 173.1 Entry into a monopolized industry: Section 7.2 Exit from a declining industry: Section 7.5

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Expert diagnosis: Section 4.6 Firm–union bargaining: Exercise 175.1, Exercise 225.1, Exercise 490.2 Holdup game: Section 6.1.2 Market games: Exercise 209.3, Example 244.1, Section 8.4, Section 8.5, Section 16.2 Matching: Example 240.3, Section 8.7 Ownership and the distribution of wealth: Example 238.2, Example 243.1, Section 8.3 Price competition between sellers: Exercise 126.1, Exercise 210.1 Provision of a public good: Exercise 31.1, Section 2.8.4, Exercise 43.1, Exercise 130.3, Section 9.5, Exercise 388.2 (rationalizability) Reporting a crime (private provision of a public good): Section 4.8 “Rotten kid theorem”: Exercise 175.2 Signaling ability with education: Section 10.7 Signaling quality with conspicuous expenditure: Section 10.6 Stackelberg’s model of duopoly: Section 6.2 Strategic information transmission: Section 10.8 Timing product release: Exercise 78.1

Political phenomena Agenda control: Section 6.1.3 (perfect information), Section 10.9 (imperfect information) Allocating resources in election campaigns: Exercise 139.2 Approval voting: Exercise 47.3 Buying votes in a legislature: Section 6.3 Cohesion of governing coalitions in legislatures: Exercise 226.2 Collective decision-making: Section 2.9.4 Committee decision-making: Section 7.4 Electoral competition between citizen-candidates: Exercise 73.2 Electoral competition between policy-motivated candidates: Exercise 73.1 Hotelling’s model of electoral competition: Section 3.3, Exercise 194.3, Exercise 194.4, Section 7.3 (strategic voters), Exercise 388.1 (rationalizability), Exercise 456.2 (repeated game) Juries: Section 9.7 Lobbying as an auction: Exercise 88.3 Majority game: Example 239.1, Exercise 243.2 Vote trading: Exercise 245.2 Voter participation: Exercise 32.2, Exercise 116.1 Voting: Section 2.9.3 (strategic game), Section 8.6 (coalitional game), Exercise 307.1 (swing voter’s curse), Exercise 392.3 (dominance solvability) Voting by alternating veto: Exercise 172.1

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Biological phenomena Evolution of sex ratio: Section 13.6 Hawk–Dove: Example 400.2, Example 406.2, Section 13.3.2 Hermaphroditic fish: Exercise 16.1 Nesting behavior of wasps: Section 13.5 Reciprocal altruism: Box on page 447 Sibling competition: Section 13.4 Signaling hunger (Sir Philip Sydney game): Exercise 335.2 War of attrition: Section 3.4

Personal pronouns The English language lacks a third person singular pronoun widely interpreted to be sex-neutral. In particular, many experiments have shown that “he” in not neutral1 , a finding consistent with the observation that whereas people may say “when an airplane pilot is working, he needs to concentrate”, they do not usually say “when a flight attendant is working, he needs to concentrate” or “when a secretary is working, he needs to concentrate”. To quote the American Heritage Dictionary (third edition, page 831), “Thus he is not really a gender-neutral pronoun; rather it refers to a male who is to be taken as the representative member of the group referred to by its antecedent. The traditional usage, then, is not simply a grammatical convention; it also suggests a particular pattern of thought.” Like many writers, I regard as unacceptable the bias implicit in the use of “he” for individuals of unspecified sex. The New Oxford Dictionary of English states the case clearly: “[the use of he to refer to a person of unspecified sex] has become . . . a hallmark of old-fashioned language or sexism in language.” Writers have become sensitive to this issue in the last fifty years, but the lack of a sex-neutral pronoun “has been felt since at least as far back as Middle English” (Webster’s Dictionary of English Usage, Merriam-Webster Inc., 1989, 499). A common solution has been to use “they”, 2 a usage that the New Oxford Dictionary of English endorses (and employs). In some contexts this usage sounds natural, but in others it does not; it can also create ambiguity when the pronoun follows references to more than one person. I choose a different solution: I use “she” exclusively. Obviously this usage, like that of “he”, is not sex-neutral, but it may help to counterbalance the widespread use of “he”, and seems unlikely to do any harm. 1 See, for example, Janice Moulton, George M. Robinson, and Cherin Elias, “Sex bias in language use: ‘neutral’ pronouns that aren’t”, American Psychologist 33 (1978), 1032–1036; Janet Shibley Hyde, “Children’s understanding of sexist language”, Development Psychology 20 (1984), 697–706; and John Gastil, “Generic pronouns and sexist language: the oxymoronic character of masculine generics”, Sex Roles 23 (1990), 629–643. 2 For a discussion of the history of the use of “they”, see Ann Bodine, “Androcentrism in prescriptive grammar: singular ‘they’, sex-indefinite ‘he’, and ‘he or she’ ”, Language in Society 4 (1975), 129–146.

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References The “Notes” section at the end of each chapter attempts to assign credit for the ideas discussed. Several cases present difficulties. In some cases, ideas evolved over a long period of time, with contributions by many people, making their origins hard to summarize in a sentence or two. In a few cases, my research has led to a conclusion about the origins of an idea different from the standard one. In all cases, I cite the relevant papers without regard to their difficulty. Over the years, I have taken exercises from many sources. I have attempted to remember the origins of the ones I use in this book, and give credit appropriately. Conventions, numbering, and nomenclature I use the term “dollar” for a unit of money because it is probably recognizable as such to a majority of readers of this book, even if they pay their bills in rupees, yuan, or pa’anga. In formal definitions, the terms being defined are set in boldface. Terms set in italics are informal definitions. Definitions, propositions, examples, and exercises are numbered according to the page on which they appear. If the first such object on page n is an exercise, for example, it is called Exercise n.1; if the next object on that page is a definition, it is called Definition n.2. For example, the definition of a strategic game with ordinal preferences on page 11 is Definition 11.1. This scheme allows numbered items to found rapidly, and also facilitates precise index entries. Symbol/term

Meaning

?

Exercise, with solution on website

??

Hard exercise, with solution on website

?

Exercise, with solution available only to instructors (see website for conditions)

??

Hard exercise, with solution available only to instructors (see website for conditions)

I

Definition Result Example: a game that illustrates a game-theoretic point

Illustration

A game, or family of games, that shows how the theory can illuminate observed phenomena

Acknowledgements I owe a huge debt to Ariel Rubinstein. I have learned, and continue to learn, vastly from him about game theory. His influence on this book will be clear to anyone

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familiar with our jointly-authored book A course in game theory. Had we not written that book and our previous book Bargaining and markets, I doubt that I would have embarked on this project. I was privileged as a graduate student to learn game theory from Robert Aumann, Sergiu Hart, Mordecai Kurz, Al Roth, and Robert Wilson at Stanford University. Discussions over the years with Jean-Pierre Benoît, Haruo Imai, Vijay Krishna, Michael Peters, and Carolyn Pitchik have improved my understanding of many game-theoretic topics. Many people have generously commented on all or parts of drafts of the book. I am particularly grateful to Gian Luigi Albano, Jeffrey Banks, Nikolaos Benos, Ted Bergstrom, Tilman Börgers, Randy Calvert, Vu Cao, In-Koo Cho, Rachel Croson, Eddie Dekel, Marina De Vos, Laurie Duke, Patrick Elias, Mukesh Eswaran, Xinhua Gu, Costas Halatsis, Joe Harrington, Hiroyuki Kawakatsu, Lewis Kornhauser, Jack Leach, Simon Link, Bart Lipman, Kin Chung Lo, Massimo Marinacci, Peter McCabe, Massimo Morelli, Nathan Nunn, Barry O’Neill, Robin G. Osborne, Marco Ottaviani, Carolyn Pitchik, Marie Rekkas, Bob Rosenthal, Al Roth, Matthew Shum, Giora Slutzki, Michael Smart, Nick Vriend, Charles A. Wilson, and Luís Zemborain. I thank also the anonymous reviewers consulted by Oxford University Press and several other presses; the suggestions in their reviews greatly improved the book. The book has its origins in a course I taught at Columbia University in the early 1980s. My experience in that course, and in courses at McMaster University and the University of Toronto, brought the book to its current form. The Kyoto Institute of Economic Research at Kyoto University and the School of Economics at the Australian National University provided me with splendid environments in which to work on the book during visits in 1999 and 2001. I maintain a website for the book. The current URL is http://www.economics.utoronto.ca/osborne/igt/ M ARTIN J. O SBORNE [email protected] http://www.economics.utoronto.ca/osborne/ Department of Economics, University of Toronto 150 St. George Street, Toronto, Canada, M5S 3G7 July 2002