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Copyright C© 1995–2000 by Martin J. Osborne All Rights Reserved. No Part 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 3 John von Neumann 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 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. 20 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 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 Illustration:expert diagnosis 120 4.7 Equilibrium in a single population 125 4.8 Illustration:reporting a crime 128 Reporting a crime: social psychology and game theory 130 4.9 The formation of players’ beliefs 131 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 139 4.12 Appendix:Representing preferences over lotteries by the expected value of a payoff function 143 Notes 148 5 Extensive Games with Perfect Information: Theory 151 5.1 Introduction 151 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 induction 167 Ticktacktoe, chess, and related games 176 Notes 177 Contents vii 6 Extensive Games with Perfect Information: Illustrations 179 6.1 Introduction 179 6.2 The ultimatum game and the holdup game 179 Experiments on the ultimatum game 181 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 7.2 Illustration:entry into a monopolized industry 209 7.3 Illustration:electoral competition with strategic voters 211 7.4 Illustration:committee decision-making 213 7.5 Illustration:exit from a declining industry 217 7.6 Allowing for exogenous uncertainty 222 7.7 Discussion:subgame perfect equilibrium and backward induction 226 Experimental evidence on the centipede game 230 Notes 232 8 Coalitional Games and the Core 235 8.1 Coalitional games 235 8.2 The core 239 8.3 Illustration:ownership and the distribution of wealth 243 8.4 Illustration:exchanging homogeneous horses 247 8.5 Illustration:exchanging heterogeneous houses 252 8.6 Illustration:voting 256 8.7 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 271 9.3 General definitions 276 9.4 Two examples concerning information 281 9.5 Illustration:Cournot’s duopoly game with imperfect information 283 9.6 Illustration:providing a public good 287 9.7 Illustration:auctions 290 Auctions of the radio spectrum 298 9.8 Illustration:juries 299 9.9 Appendix:Analysis of auctions for an arbitrary distribution of 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 335 11.3 Strictly competitive games 338 Maxminimization: some history 344 Testing the theory of Nash equilibrium in strictly competitive games 347 Notes 348 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 14.1 The main idea 389 14.2 Preferences 391 14.3 Infinitely repeated games 393 14.4 Strategies 394 14.5 Some Nash equilibria of the infinitely repeated Prisoner’s Dilemma 396 14.6 Nash equilibrium payoffs of the infinitely repeated Prisoner’s Dilemma when 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 404 Notes 409 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 418 Reciprocal altruism among sticklebacks 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 17.1 Introduction 443 17.2 Numbers 443 17.3 Sets 444 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 grayboxesindicatecorechapters(thedarkergray,themoreimportant).Anblack 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: 14 Preface Strategic games 3: Illustrations 4: Mixed strategies Imperfect information 2: Theory 9: Bayesian games Topics 11: Maxminimization 12: Rationalizability 13: Evolutionary equilibrium Extensive games 6: Illustrations 7: Extensions Imperfect information 5: Theory 10: Signaling games Topics 14, 15: Repeated games (I, II) 16: Bargaining Coalitional games 8: Core xivFigure 0.1 The structure of the book. The area of each box is proportional to the length of the chapter the box represents. The boxes corresponding to the core chapters are shaded gray; the ones shaded dark gray are more central that the ones shaded light gray.
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