Table Of Contentcress-79032 cres79032˙fm January23,2003 14:51
EvolutionaryDynamics
andExtensiveFormGames
i
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EconomicLearningandSocialEvolution
GeneralEditor
Ken Binmore, Director of the Economic Learning and Social
EvolutionCentre,UniversityCollegeLondon.
1. EvolutionaryGamesandEquilibriumSelection,LarrySamuelson,
1997
2. The Theory of Learning in Games, Drew Fudenberg and David
K.Levine,1998
3. GameTheoryandtheSocialContract,Volume2:JustPlaying,Ken
Binmore,1998
4. Social Dynamics, Steven N. Durlauf and H. Peyton Young,
editors,2001
5. Evolutionary Dynamics and Extensive Form Games, Ross
Cressman,2003
ii
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EvolutionaryDynamics
andExtensiveFormGames
RossCressman
TheMITPress
Cambridge,Massachusetts
London,England
iii
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(cid:1)c 2003MassachusettsInstituteofTechnology
Allrightsreserved.Nopartofthisbookmaybereproducedinanyformbyanyelectronic
or mechanical means (including photocopying, recording, or information storage and
retrieval)withoutpermissioninwritingfromthepublisher.
ThisbookwassetinPalatinobyInteractiveCompositionCorporation(inLATEX)andwas
printedandboundintheUnitedStatesofAmerica.
LibraryofCongressCataloging-in-PublicationData
Cressman,Ross.
Evolutionarydynamicsandextensiveformgames/RossCressman.
p. cm.—(Economiclearningandsocialevolution;5)
Includesbibliographicalreferencesandindex.
ISBN0-262-03305-4(hc.:alk.paper)
1. Gametheory. 2. Evolution—Mathematicalmodels. I. Title. II. MITPress
seriesoneconomiclearningandsocialevolution;5.
QA269.C69 2003
519.3—dc21 2002038682
iv
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Contents
SeriesForeword ix
Preface xi
1 Introduction 1
1.1 ExtensiveFormversusNormalForm 2
1.2 BiologyversusEconomics 8
1.3 Imitation 12
1.4 OrganizationalMatters 15
1.5 Notes 16
2 SymmetricNormalFormGames 19
2.1 TheReplicatorDynamic 19
2.2 DynamicsforTwo-StrategyGames 23
2.3 MonotoneSelectionDynamics 27
2.4 FictitiousPlayandBestResponseDynamic 31
2.5 ConvergenceandStability:NEandESS 34
2.6 Three-StrategyGameDynamics 37
2.6.1 Rock–Scissors–PaperGames 37
2.6.2 ESSetsandNEComponents 42
2.6.3 MoreThree-StrategyGames 43
2.7 DynamicStabilityforGeneralGames 46
2.8 NaturalSelectionataSingleLocus 53
2.8.1 Discrete-TimeViabilitySelection 53
2.8.2 Continuous-TimeNaturalSelection 54
2.9 One-StageSimultaneityGames 56
2.10 Multi-armedBandits 58
2.11 Appendix 64
2.12 Notes 66
v
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vi Contents
3 BimatrixGames 69
3.1 NashEquilibriaandStrictEquilibriumSets 70
3.2 BimatrixReplicatorandBestResponseDynamics 71
3.2.1 TheOwner-IntruderGame 74
3.3 DynamicsforTwo-StrategyBimatrixGames 75
3.3.1 NondegenerateBimatrixGames 76
3.3.2 DegenerateBimatrixGames 79
3.4 SymmetrizedBimatrixGames 85
3.4.1 TheSymmetrizedBimatrixReplicatorDynamic 88
3.4.2 TheSymmetrizedBestResponseDynamic 93
3.5 BimatrixMonotoneSelectionDynamics 96
3.6 Notes 101
4 AsymmetricGames 103
4.1 TheNormalForm 104
4.2 TheExtensiveForm:NEandESSets 106
4.2.1 AnAge-StructuredOwner-IntruderGame 108
4.3 SESetsandAgentNormalForms 110
4.4 DynamicsandtheWrightManifold 113
4.4.1 TheReplicatorDynamicandSubgames 114
4.4.2 BestResponseDynamics 116
4.5 TrulyAsymmetricTwo-PlayerGames 117
4.5.1 TheAge-StructuredOwner-Intruder
GameDynamic 121
4.6 TrulySymmetricTwo-PlayerGames 123
4.6.1 ATrulySymmetricGameDynamic
Counterexample 125
4.6.2 ParallelBandits 128
4.7 AsymmetricGameswithTwoRoles 137
4.7.1 AFamilyofAsymmetricGames 137
4.7.2 Two-SpeciesEvolutionarilyStableStrategies 140
4.8 AHierarchicalHawk-DoveGame 146
4.9 AppendixA 150
4.10 AppendixB 152
4.11 Notes 153
5 NaturalSelectionwithMultipleLoci 155
5.1 Continuous-TimeSelection-Recombination 155
5.2 SymmetricExtensiveFormwithAdditiveFitness 157
5.3 Recombination 160
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Contents vii
5.4 SelectionandRecombination 162
5.5 Notes 163
6 ExtensiveFormGames 165
6.1 N-PlayerExtensiveFormGames 166
6.1.1 StrategiesandPayoffs 168
6.1.2 NashEquilibria,Subgames,and
BackwardInduction 171
6.2 NormalFormsandtheReplicatorDynamic 173
6.3 TheWrightManifoldandReplicatorDynamic 175
6.4 SymmetricExtensiveFormGames 180
6.5 Appendix 184
6.6 Notes 185
7 SimultaneityGames 187
7.1 ElementaryTwo-StageSimultaneityGames 188
7.2 Two-StageTwo-StrategyGames 192
7.2.1 Two-StageTwo-StrategyRepeatedGames 195
7.2.2 SymmetricSignalingGames 197
7.2.3 CheapTalkGames 200
7.3 AsymptoticStabilityofPervasiveNE 201
7.3.1 SimultaneityGameswithNo
AsymmetricSubgames 201
7.3.2 Simultaneity Games with Asymmetric Subgames 204
7.3.3 SimultaneityGameswithMovesbyNature 206
7.4 TheWarofAttrition 207
7.4.1 TheDiscreteWarofAttrition 208
7.4.2 TheContinuousWarofAttrition 213
7.4.3 TheDiscreteWarofAggression 215
7.5 TheFinitelyRepeatedPrisoner’sDilemmaGame 217
7.5.1 TheReplicatorandMonotone
SelectionDynamics 220
7.5.2 TheBestResponseDynamicandFictitiousPlay 222
7.6 AppendixA:ProofofTheorem7.5.3 228
7.7 AppendixB:MaximalAttractor 232
7.8 Notes 233
8 PerfectInformationGames 235
8.1 ElementaryPerfectInformationGames 237
8.2 EquilibriumSelection:DynamicApproach 240
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viii Contents
8.2.1 TheReplicatorandMonotone
SelectionDynamics 242
8.2.2 FictitiousPlayandBestResponseDynamic 248
8.2.3 BehaviorStrategyFictitiousPlay 252
8.3 TheCentipedeGame 255
8.3.1 CentipedeGamesofLengthsTwoandThree 256
8.3.2 CentipedeGamesofLength N≥4 258
8.4 ExtensiveFormBandits 260
8.4.1 TheCentipedeBandit 268
8.5 AppendixA 276
8.6 AppendixB 283
8.7 Notes 287
9 SubgameMonotonicity 289
9.1 MonotoneTrajectories 289
9.2 SubgameMonotoneTrajectories 292
9.3 AnImitationExample 301
9.4 Discussion 303
9.5 Notes 304
Bibliography 307
Index 313
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SeriesForeword
TheMITPressseriesonEconomicLearningandSocialEvolutionreflects
thecontinuinginterestinthedynamicsofhumaninteraction.Thisissue
has provided a broad community of economists, psychologists, biolo-
gists,anthropologists,mathematicians,philosophers,andothers,with
asenseofcommonpurposesostrongthattraditionalinterdisciplinary
boundarieshavemeltedaway.Werejecttheoutmodednotionthatwhat
happensawayfromequilibriumcansafellybeignored,butthinkitno
longer adequate to speak in vague terms of bounded rationality and
spontaneousorder.Webelievethetimehascometoputsomebeefon
thetable.
Thebooksintheseriessofarare:
• EvolutionaryGamesandEquilibriumSelection,byLarrySamuelson(1997).
Traditional economic models have only one equilibrium, and so fail
tocometogripswithsocialnormswhosefunctionistoselectanequi-
libriumwhentherearemultiplealternatives.Thisbookstudieshow
suchnormsmayevolve.
• TheTheoryofLearninginGames,byDrewFudenbergandDavidLevine
(1998).VonNeumannintroduced“fictitiousplay”asawayoffinding
equilibriainzero-sumgames.Inthisbooktheideaisreinterpretedas
alearningprocedure,anddevelopedforuseingeneralgames.
• JustPlaying, byKenBinmore(1998).Thisbookappliesevolutionary
gametheorytomoralphilosophy.Howandwhydowemakefairness
judgments?
• Social Dynamics, edited by Steve Durlauf and Peyton Young (2001).
Theessaysinthiscollectionprovideanoverviewofthefieldofsocial
dynamics,inwhichsomeofthecreatorsofthefielddiscussavarietyof
ix
