An Introduction to Game Theory by Martin J. Osborne

Please send comments to Martin J. Osborne Department of Economics 150 St. George Street University of Toronto Toronto, Canada M5S 3G7 email: martin.osborne@utoronto.ca

This version:

2000/11/6

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 1

xiii

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

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I

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Games with Perfect Information

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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

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Contents

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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 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

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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 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

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Contents

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Extensive Games with Perfect Information: Illustrations 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

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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 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...