Game Theory in Action: An Introduction to Classical and Evolutionary Models

289 Pages·2016·1.54 MB·English

by  Stephen Schecter

Game Theory in Action

Game Theory in Action
An Introduction to Classical
and Evolutionary Models
Stephen Schecter
Herbert Gintis
PrincetonUniversityPress
PrincetonandOxford
Copyright©2016 byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,
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IntheUnitedKingdom:PrincetonUniversityPress,
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press.princeton.edu
AllRightsReserved
ISBN978-0-691-16764-0
ISBN(pbk.)978-0-691-16765-7
LibraryofCongressControlNumber:2015956193
BritishLibraryCataloging-in-PublicationDataisavailable
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TypesetbyT&TProductionsLtd,London
Printedonacid-freepaper(cid:2)∞
PrintedintheUnitedStatesofAmerica
1 3 5 7 9 10 8 6 4 2
ForNancy,Erica,andJeff.
S.S.
Tothemathteacherswhoinspiredme:
Mr.Gettyinjuniorhighschool,
PincusSchubandWalterGottschalkattheUniversityofPennsylvania,
andLynnLoomisandOscarZariskiatHarvardUniversity.
H.G.

Contents
Prefaceandacknowledgments xi
Chapter 1. Backward induction 1
1.1 Tony’sAccident 1
1.2 Gamesinextensiveformwithcompleteinformation 3
1.3 Strategies 5
1.4 Backwardinduction 6
1.5 BigMonkeyandLittleMonkey1 9
1.6 Threats,promises,commitments 10
1.7 UltimatumGame 12
1.8 Rosenthal’sCentipedeGame 13
1.9 Continuousgames 15
1.10 Stackelberg’smodelofduopoly1 16
1.11 Stackelberg’smodelofduopoly2 20
1.12 Backwardinductionforfinitehorizongames 24
1.13 Critiqueofbackwardinduction 25
1.14 Problems 27
Chapter 2. Eliminating dominated strategies 37
2.1 Prisoner’sDilemma 37
2.2 Gamesinnormalform 39
2.3 Dominatedstrategies 40
2.4 IsraelisandPalestinians 41
2.5 GlobalWarming 44
2.6 Hagar’sBattles 45
2.7 Second-priceauctions 47
2.8 Iteratedeliminationofdominatedstrategies 49
2.9 TheBattleoftheBismarckSea 49
2.10 Normalformofagameinextensiveformwithcomplete
information 51
2.11 BigMonkeyandLittleMonkey2 51
2.12 Backwardinduction 53
2.13 Critiqueofeliminationofdominatedstrategies 55
2.14 Problems 55
viii • Contents
Chapter 3. Nash equilibria 61
3.1 BigMonkeyandLittleMonkey3andthedefinitionofNash
equilibria 61
3.2 FindingNashequilibriabyinspection:Importantexamples 63
3.3 WaterPollution1 66
3.4 ArguingoverMarbles 67
3.5 TobaccoMarket 69
3.6 Iteratedeliminationofdominatedstrategies 71
3.7 BigMonkeyandLittleMonkey4 74
3.8 FindingNashequilibriausingbestresponse 75
3.9 BigMonkeyandLittleMonkey5 76
3.10 WaterPollution2 77
3.11 Cournot’smodelofduopoly 77
3.12 Problems 79
Chapter 4. Games in extensive form with incomplete information 88
4.1 Utilityfunctionsandlotteries 88
4.2 BuyingFireInsurance 89
4.3 Gamesinextensiveformwithincompleteinformation 90
4.4 BuyingaUsedCar 91
4.5 TheTravailsofBossGorilla1 95
4.6 CubanMissileCrisis 98
4.7 Problems 104
Chapter 5. Mixed strategy Nash equilibria 114
5.1 MixedstrategyNashequilibria 114
5.2 Tennis 120
5.3 OtherwaystofindmixedstrategyNashequilibria 122
5.4 One-cardTwo-roundPoker 123
5.5 Two-playerzero-sumgames 128
5.6 TheUltimatumMinigame 132
5.7 ColonelBlottovs.thePeople’sMilitia 134
5.8 WaterPollution3 140
5.9 Equivalentgames 141
5.10 SoftwareforcomputingNashequilibria 142
5.11 CritiqueofNashequilibrium 143
5.12 Problems 144
Chapter 6. More about games in extensive form with
complete information 151
6.1 SubgameperfectNashequilibria 152
6.2 BigMonkeyandLittleMonkey6 152
Contents • ix
6.3 Subgameperfectequilibriaandbackwardinduction 153
6.4 DuelsandTruels 155
6.5 TheRubinsteinbargainingmodel 160
6.6 Discountfactorandrepeatedgames 163
6.7 TheWineMerchantandtheConnoisseur 165
6.8 TheFolkTheorem 168
6.9 Maximumvalueofafunction 172
6.10 TheSamaritan’sDilemma 173
6.11 TheRottenKidTheorem 177
6.12 Problems 180
Chapter 7. Symmetries of games 186
7.1 Interchangeableplayers 186
7.2 ReportingaCrime 189
7.3 SexRatio1 191
7.4 Othersymmetriesofgames 193
7.5 Problems 198
Chapter 8. Alternatives to the Nash equilibrium 203
8.1 Correlatedequilibrium 203
8.2 Epistemicgametheory 205
8.3 Evolutionarystability 206
8.4 Evolutionarystabilitywithtwopurestrategies 209
8.5 SexRatio2 213
8.6 Problems 214
Chapter 9. Differential equations 217
9.1 Differentialequationsandscientificlaws 217
9.2 Thephaseline 219
9.3 Vectorfields 220
9.4 Functionsanddifferentialequations 222
9.5 Lineardifferentialequations 225
9.6 Linearization 228
Chapter 10. Evolutionary dynamics 232
10.1 Replicatorsystem 232
10.2 Microsoftvs.Apple 235
10.3 Evolutionarydynamicswithtwopurestrategies 237
10.4 HawksandDovesrevisited 238
10.5 Side-blotchedLizards 240
10.6 Equilibriaofthereplicatorsystem 244
10.7 Cooperators,Defectors,andTit-for-Tatters 246