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Copyright c 1995–2000 by Martin J. Osborne All rights reserved. No part of this book may be reproduced by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from Oxford University Press.

Contents Preface xiii 1 Introduction 1 1.1 What is game theory? 1 An outline of the history of game theory John von Neumann 3 3 1.2 The theory of rational choice 4 1.3 Coming attractions 7 Notes 8 I Games with Perfect Information 9 2 Nash Equilibrium: Theory 11 11 2.1 Strategic games 2.2 Example: the Prisoner’s Dilemma 2.3 Example: Bach or Stravinsky? 16 2.4 Example: Matching Pennies 17 2.5 Example: the Stag Hunt 2.6 Nash equilibrium 19 18 12 John F. Nash, Jr. Studying Nash equilibrium experimentally 20 22 2.7 Examples of Nash equilibrium 24 Experimental evidence on the Prisoner’s Dilemma Focal points 30 26 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 v

vi Contents 3 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 79 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 119 4.6 4.7 Equilibrium in a single population 125 4.8 Illustration: expert diagnosis 120 Illustration: reporting a crime 128 Reporting a crime: social psychology and game theory 130 4.9 The formation of players’ beliefs 4.10 Extension: Finding all mixed strategy Nash equilibria 135 4.11 Extension: Mixed strategy Nash equilibria of games in which each player 131 has a continuum of actions 139 4.12 Appendix: Representing preferences over lotteries by the expected value of a payoﬀ function 143 Notes 148 5 Introduction 151 Extensive Games with Perfect Information: Theory 5.1 5.2 Extensive games with perfect information 151 5.3 Strategies and outcomes 157 5.4 Nash equilibrium 159 5.5 Subgame perfect equilibrium 162 5.6 Finding subgame perfect equilibria of ﬁnite horizon games: backward 151 induction 167 Ticktacktoe, chess, and related games 176 Notes 177

Contents vii 6 Introduction 179 Extensive Games with Perfect Information: Illustrations 179 6.1 6.2 The ultimatum game and the holdup game 179 181 Experiments on the ultimatum game 6.3 Stackelberg’s model of duopoly 184 6.4 Buying votes 189 6.5 A race 194 Notes 200 7 Extensive Games with Perfect Information: Extensions and Discussion 201 7.1 Allowing for simultaneous moves 201 More experimental evidence on subgame perfect equilibrium 207 Illustration: entry into a monopolized industry 209 Illustration: electoral competition with strategic voters Illustration: committee decision-making 213 Illustration: exit from a declining industry 217 7.2 7.3 7.4 7.5 7.6 Allowing for exogenous uncertainty 222 7.7 Discussion: subgame perfect equilibrium and backward induction 226 211 Experimental evidence on the centipede game 230 Notes 232 8 239 235 Coalitional Games and the Core 8.1 Coalitional games 235 8.2 The core 8.3 8.4 8.5 8.6 8.7 Illustration: ownership and the distribution of wealth 243 Illustration: exchanging homogeneous horses 247 Illustration: exchanging heterogeneous houses 252 Illustration: voting 256 Illustration: matching 259 Matching doctors with hospitals 264 8.8 Discussion: other solution concepts 265 Notes 266

viii Contents II Games with Imperfect Information 269 9 Bayesian Games 271 9.1 Introduction 271 9.2 Motivational examples 9.3 General deﬁnitions 276 9.4 Two examples concerning information 281 9.5 9.6 9.7 271 Illustration: Cournot’s duopoly game with imperfect information 283 Illustration: providing a public good 287 Illustration: auctions 290 Auctions of the radio spectrum 298 Illustration: juries 9.8 9.9 Appendix: Analysis of auctions for an arbitrary distribution of 299 valuations 306 Notes 309 10 Extensive games with imperfect information 311 10.1 To be written 311 Notes 312 III Variants and Extensions 333 11 Strictly Competitive Games and Maxminimization 335 11.1 Introduction 335 11.2 Deﬁnitions and examples 11.3 Strictly competitive games 338 335 Maxminimization: some history Testing the theory of Nash equilibrium in strictly competitive 344 Notes 348 games 347 12 Rationalizability 349 12.1 Introduction 349 12.2 Iterated elimination of strictly dominated actions 355 12.3 Iterated elimination of weakly dominated actions 359 Notes 361

Contents ix 13 Evolutionary Equilibrium 363 13.1 Introduction 363 13.2 Monomorphic pure strategy equilibrium 364 Evolutionary game theory: some history 369 13.3 Mixed strategies and polymorphic equilibrium 370 13.4 Asymmetric equilibria 377 Explaining the outcomes of contests in nature 379 13.5 Variation on a theme: sibling behavior 380 13.6 Variation on a theme: nesting behavior of wasps 386 Notes 388 14 Repeated games: The Prisoner’s Dilemma 389 391 14.1 The main idea 389 14.2 Preferences 14.3 Inﬁnitely repeated games 393 14.4 Strategies 14.5 Some Nash equilibria of the inﬁnitely repeated Prisoner’s Dilemma 14.6 Nash equilibrium payoﬀs of the inﬁnitely repeated Prisoner’s Dilemma when 394 396 the players are patient 398 14.7 Subgame perfect equilibria and the one-deviation property 402 14.8 Some subgame perfect equilibria of the inﬁnitely repeated Prisoner’s Dilemma Notes 409 404 15 Repeated games: General Results 411 15.1 Nash equilibria of general inﬁnitely repeated games 411 15.2 Subgame perfect equilibria of general inﬁnitely repeated games 414 Axelrod’s experiments Reciprocal altruism among sticklebacks 418 419 15.3 Finitely repeated games 420 Notes 420 16 Bargaining 421 16.1 To be written 421 16.2 Repeated ultimatum game 421 16.3 Holdup game 421

x Contents 17 Appendix: Mathematics 443 443 444 17.1 Introduction 443 17.2 Numbers 17.3 Sets 17.4 Functions 445 17.5 Proﬁles 448 17.6 Sequences 449 17.7 Probability 449 17.8 Proofs 454 References 457

Preface Game theoretic reasoning 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 bi- ological 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. (Chap- ter 17 reviews the mathematical concepts used in the book.) In particular, calculus is not used, except in the appendix of Chapter 9 (Section 9.7). Nevertheless, all concepts are deﬁned precisely, and logical reasoning is used extensively. The more comfortable you are with tight logical analysis, the easier you will ﬁnd the argu- ments. In brief, my aim is to explain the main ideas of game theory as simply as possible while maintaining complete precision. 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 200 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). An 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 9.8 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 theoreti- cal points, and “Illustrations”, which demonstrate how the theory may be used to understand social, economic, political, and biological phenomena. The “Illustra- tions” for the key models of strategic and extensive games are grouped in separate chapters (3 and 6), whereas those for the other models occupy the same chapters as the theory. The “Illustrations” introduce no new theoretical points, and any or all of them may be skipped without loss of continuity. The limited dependencies between chapters mean that several routes may be taken through the book. 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:

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