An Introduction to Game Theory by Martin J. Osborne

685 Pages·2001·6.28 MB·English

Selected chapters from draft of
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: [email protected]
This version: 2000/11/6
Copyright(cid:1)c 1995–2000byMartinJ.Osborne
Allrightsreserved.Nopartofthisbookmaybereproducedbyanyelectronicormechanicalmeans
(includingphotocopying,recording,orinformationstorageandretrieval)withoutpermissionin
writingfromOxfordUniversityPress.
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 ExtensiveGameswithPerfectInformation: ExtensionsandDiscussion 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 reasoningpervadeseconomic theory and is used widely in other
socialandbehavioralsciences. Thisbookpresentsthemainideasofgametheory
andshowshowtheycanbeusedtounderstandeconomic,social,political,andbi-
ologicalphenomena. Itassumesnoknowledge ofeconomics, politicalscience, or
anyother socialor behavioralscience. Itemphasizesthe ideasbehind the theory
ratherthantheirmathematicalexpression,andassumesnospecificmathematical
knowledgebeyondthattypicallytaughtinUSandCanadianhighschools. (Chap-
ter17reviewsthemathematicalconceptsusedinthebook.) Inparticular,calculus
is not used, except in the appendix of Chapter 9 (Section 9.7). Nevertheless, all
conceptsaredefinedprecisely,andlogicalreasoningisusedextensively. Themore
comfortable you arewith tight logical analysis, the easier you will find the argu-
ments. Inbrief, my aimistoexplainthe mainideasof gametheory assimply as
possiblewhilemaintainingcompleteprecision.
Theonlywaytoappreciatethetheoryistoseeitinaction,orbetterstilltoput
it into action. Sothe book includesa wide varietyof illustrations fromthe social
andbehavioralsciences,andover200exercises.
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.
ThegrayarrowfromChapter4toChapter9meansthatthelatterdependsweakly
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
singlerationaldecision-maker,andChapter17reviewsthemathematicalconcepts
usedinthebook.)
Eachtopic is presented with the aid of “Examples”, which highlight theoreti-
calpoints,and“Illustrations”,whichdemonstratehowthetheorymaybeusedto
understand social, economic, political, and biological phenomena. The “Illustra-
tions”forthekeymodelsofstrategicandextensivegamesaregroupedinseparate
chapters(3and 6),whereasthose for the other modelsoccupythe same chapters
asthetheory. The“Illustrations”introducenonewtheoreticalpoints, andanyor
allofthemmaybeskippedwithoutlossofcontinuity.
The limited dependencies between chapters mean that severalroutes may be
takenthroughthebook.
(cid:127) At a minimum, you should study Chapters 2 (Nash Equilibrium: Theory)
and5(ExtensiveGameswithPerfectInformation: Theory).
(cid:127) OptionallyyoumaysamplesomesectionsofChapters3(NashEquilibrium:
14 Preface
Strategicgames
3: Illustrations
4: Mixedstrategies
Imperfectinformation
2: Theory 9: Bayesiangames
Topics
11: Maxminimization
12:Rationalizability
13:Evolutionaryequilibrium
Extensivegames
6: Illustrations
7: Extensions
Imperfectinformation
5: Theory
10: Signalinggames
Topics
14,15: Repeatedgames(I,II)
16:Bargaining
Coalitionalgames
8: Core
xivFigure0.1 Thestructureofthebook.Theareaofeachboxisproportionaltothelengthofthechapter
theboxrepresents.Theboxescorrespondingtothecorechaptersareshadedgray;theonesshadeddark
grayaremorecentralthattheonesshadedlightgray.AnarrowfromChapteritoChapterjmeansthat
ChapteriisaprerequisiteforChapterj. ThegrayarrowfromChapter4toChapter9meansthatthe
latterdependsonlyweaklyontheformer.