Population Games and Evolutionary Dynamics (Economic Learning and Social Evolution)

618 Pages·2010·26.727 MB·English

by  William H. Sandholm

PopulationGamesandEvolutionaryDynamics
EconomicLearningandSocialEvolution
GeneralEditor
KenBinmore,EmeritusProfessorofEconomicsatUniversityCollegeLondon
EvolutionaryGamesandEquilibriumSelection,LarrySamuelson,1997
TheTheoryofLearninginGames,DrewFudenbergandDavidK.Levine,1998
GameTheoryandtheSocialContract,Volume2:JustPlaying,KenBinmore,1998
SocialDynamics,StevenN.DurlaufandH.PeytonYoung,editors,2001
EvolutionaryDynamicsandExtensiveFormGames,RossCressman,2003
Moral Sentiments and Material Interests: The Foundations of Cooperation in Economic
Life, Herbert Gintis, Samuel Bowles, Robert Boyd, and Ernst Fehr, editors, 2004
DoesGameTheoryWork?TheBargainingChallenge,KenBinmore,2006
PopulationGamesandEvolutionaryDynamics,WilliamH.Sandholm,2010
PopulationGamesandEvolutionaryDynamics
WilliamH.Sandholm
TheMITPress
Cambridge,Massachusetts
London,England
©2010MassachusettsInstituteofTechnology
Allrightsreserved.Nopartofthisbookmaybereproducedinanyformbyanyelectronicormechanical
means(includingphotocopying,recording,orinformationstorageandretrieval)withoutpermission
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ThisbookwassetinPalatinobyWestchesterBookComposition.PrintedandboundintheUnitedStates
ofAmerica.
LibraryofCongressCataloging-in-PublicationData
Sandholm,WilliamH.,1970–
Populationgamesandevolutionarydynamics/WilliamH.Sandholm.
p. cm.—(Economiclearningandsocialevolution)
Includesbibliographicalreferencesandindex.
ISBN978-0-262-19587-4(hbk.:alk.paper)
1.Gametheory. 2.Evolution—Mathematicalmodels. I.Title.
HB144.S27 2011
303.4—dc22
2010011973
10 9 8 7 6 5 4 3 2 1
ContentsinBrief
SeriesForeword xvii
Preface xix
1 Introduction 1
I PopulationGames
2 PopulationGames 21
3 PotentialGames,StableGames,andSupermodularGames 53
II DeterministicEvolutionaryDynamics
4 RevisionProtocolsandEvolutionaryDynamics 119
5 DeterministicDynamics:FamiliesandProperties 139
6 BestResponseandProjectionDynamics 177
III ConvergenceandNonconvergenceofDeterministicDynamics
7 GlobalConvergenceofEvolutionaryDynamics 221
8 LocalStabilityunderEvolutionaryDynamics 271
9 NonconvergenceofEvolutionaryDynamics 319
IV StochasticEvolutionaryModels
10 StochasticEvolutionandDeterministicApproximation 367
11 StationaryDistributionsandInfinite-HorizonBehavior 397
12 LimitingStationaryDistributionsandStochasticStability 451
References 541
NotationIndex 565
Index 575

Contents
SeriesForeword xvii
Preface xix
1 Introduction 1
1.1 PopulationGames 2
1.1.1 ModelingInteractionsinLargePopulations 2
1.1.2 DefinitionsandClassesofPopulationGames 5
1.2 EvolutionaryDynamics 6
1.2.1 Knowledge,Rationality,andLargeGames 7
1.2.2 FoundationsforEvolutionaryDynamics 8
1.2.3 DeterministicEvolutionaryDynamics 9
1.2.4 OrdersofLimitsforStochasticEvolutionaryModels 11
1.2.5 StationaryDistributionsandStochasticStability 12
1.3 RemarksonHistory,Motivation,andInterpretation 13
Notes 15
I PopulationGames
2 PopulationGames 21
2.1 PopulationGames 22
2.1.1 Populations,Strategies,andStates 22
2.1.2 Payoffs 23
2.1.3 BestResponsesandNashEquilibria 23
2.1.4 PreludetoEvolutionaryDynamics 24
2.2 Examples 25
2.2.1 MatchinginNormalFormGames 25
2.2.2 CongestionGames 27
2.2.3 TwoSimpleExternalityModels 28
2.3 TheGeometryofPopulationGamesandNashEquilibria 29
2.3.1 DrawingTwo-StrategyGames 29
2.3.2 DisplacementVectorsandTangentSpaces 31
viii Contents
2.3.3 OrthogonalProjections 34
2.3.4 DrawingThree-StrategyGames 36
2.3.5 TangentConesandNormalCones 38
2.3.6 NormalConesandNashEquilibria 41
2.A Appendix:AffineSpaces,TangentSpaces,andProjections 44
2.A.1 AffineSpaces 44
2.A.2 AffineHullsofConvexSets 45
2.A.3 OrthogonalProjections 46
2.A.4 TheMoreauDecompositionTheorem 50
Notes 51
3 PotentialGames,StableGames,andSupermodularGames 53
3.1 FullPotentialGames 53
3.1.1 FullPopulationGames 54
3.1.2 DefinitionandCharacterization 54
3.1.3 Examples 55
3.1.4 NashEquilibriaofFullPotentialGames 58
3.1.5 TheGeometryofNashEquilibriuminFullPotentialGames 64
3.1.6 EfficiencyinHomogeneousFullPotentialGames 66
3.1.7 InefficiencyBoundsforCongestionGames 68
3.2 PotentialGames 71
3.2.1 MotivatingExamples 71
3.2.2 Definition,Characterizations,andExamples 72
3.2.3 PotentialGamesandFullPotentialGames 76
3.2.4 PassiveGamesandConstantGames 77
3.3 StableGames 79
3.3.1 Definition 79
3.3.2 Examples 81
3.3.3 Invasion 85
3.3.4 GlobalNeutralStabilityandGlobalEvolutionaryStability 87
3.3.5 NashEquilibriumandGlobalNeutralStabilityinStableGames 90
3.4 SupermodularGames 94
3.4.1 Definition 94
3.4.2 Examples 97
3.4.3 BestResponseMonotonicityinSupermodularGames 98
3.4.4 NashEquilibriaofSupermodularGames 99
3.A Appendix:MultivariateCalculus 100
3.A.1 UnivariateCalculus 100
3.A.2 TheDerivativeasaLinearMap 101
3.A.3 DifferentiationasaLinearOperation 103
3.A.4 TheProductRuleandtheChainRule 103
Contents ix
3.A.5 HomogeneityandEuler’sTheorem 104
3.A.6 Higher-OrderDerivatives 105
3.A.7 TheWhitneyExtensionTheorem 106
3.A.8 VectorIntegrationandtheFundamentalTheoremofCalculus 107
3.A.9 PotentialFunctionsandIntegrability 108
3.B Appendix:AffineCalculus 108
3.B.1 LinearFormsandtheRieszRepresentationTheorem 109
3.B.2 DualCharacterizationsofMultiplesofLinearForms 110
3.B.3 DerivativesofFunctionsonAffineSpaces 111
3.B.4 AffineIntegrability 113
Notes 115
II DeterministicEvolutionaryDynamics
4 RevisionProtocolsandEvolutionaryDynamics 119
4.1 TheBasicStochasticEvolutionaryModel 120
4.1.1 InertiaandMyopia 120
4.1.2 RevisionProtocols 121
4.2 MeanDynamics 122
4.2.1 Derivation 122
4.2.2 TargetProtocolsandTargetDynamics 124
4.3 Examples 125
4.3.1 ImitativeProtocolsandDynamics 126
4.3.2 DirectProtocolsandDynamics 128
4.4 DeterministicEvolutionaryDynamics 129
4.A Appendix:OrdinaryDifferentialEquations 131
4.A.1 BasicDefinitions 131
4.A.2 Existence,Uniqueness,andContinuityofSolutions 133
4.A.3 OrdinaryDifferentialEquationsonCompactConvexSets 135
Notes 137
5 DeterministicDynamics:FamiliesandProperties 139
5.1 InformationRequirementsforRevisionProtocols 141
5.2 IncentivesandAggregateBehavior 144
5.3 FamiliesofEvolutionaryDynamics 148
5.4 ImitativeDynamics 153
5.4.1 Definition 153
5.4.2 Examples 154
5.4.3 BiologicalDerivationsoftheReplicatorDynamic 158
5.4.4 ExtinctionandInvariance 160
5.4.5 MonotonePercentageGrowthRatesandPositiveCorrelation 162
5.4.6 RestPointsandRestrictedEquilibria 164