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1、DYNAMICSFORINFINITEDIMENSIONALGAMES1byR.M.Seymour∗DepartmentofMathematicsUniversityCollegeLondonGowerStreetLondonWC1E6BT,UKe-mail:r.seymour@ucl.ac.ukRevisedversion;January2000∗ThesupportoftheEconomicandSocialResearchCouncil(ESRC)isgratefullyacknowledged.Theworkwa
2、spartoftheprogrammeoftheESRCResearchCentreforEconomicLearningandSocialEvolution.IntroductionTheexplosionofinterestamongstgame-theoristsinrecentyearsinthe‘evolutionary’(learn-ing)dynamicsofrepeatedgames,hasgenerallybeenconcernedwith2-player(usuallysymmet-ric)games
3、inwhicheachplayerhasavailableafinitenumberofpurestrategies.Thelearningdynamicsofplayerschosenfromlarge,usuallyinfinite,populationsofsuchplayers,arethentakentodescribetheevolutionoftheprobabilitywithwhicharandomlychosenplayerwillplayagivenpurestrategy.Inasuitablecon
4、tinuous-timelimit,inwhichthegameisrepeatedcontin-uously,thesedynamicstaketheformofafinite-dimensionalsystemofdifferentialequations.Variousversionsofthesedynamicshavebeenextensivelystudiedinvariousgame-theoreticcontexts,andup-to-dateaccountsofmuchoftherecentresearc
5、hinthisareacanbefoundinWeibull(1996),Samuelson(1997)andHofbauerandSigmund(1999).1Anevolutionarydynamicsoncontinuousstrategyspaces,whichhasfeaturesincommonwiththeapproachpresentedhere,hasalsorecentlybeendevelopedbyOechsslerandRiedel(1998).SeealsoPontiandSeymour(19
6、98),appendixA,andOeschlerandRiedel(1999)forasomewhatdifferentapproaches.1However,inmanygamesthestrategysetsopentoplayersarecontinua,ratherthanfinitesets.Forexample,inbargaininggamesovercontinuouslydivisiblecommodities.Littleworkhasbeendoneinexploringevolutionarydy
7、namicsinthismoregeneralcontext,notleastbecauseoftheoftenformidabletechnicaldifficultiesinvolved.RecentexceptionsareHopkinsandSeymour(1999),Seymour(1999),FriedmanandYellin(1997),PontiandSeymour(1998),andOeschlerandRiedel(1998,1999).However,moreadventurousexcursions
8、intotheevolutionarydynamicsofinfinitedimensionalgamesareundoubtedlyhinderedbythelackofaneasilyaccessibleandrigorousdevelopmentofageneraltheoryappropriatetothisc
