博弈论英文版lecture

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1、LectureIX:EvolutionMarkusM.MÄobiusMarch10,2004Learningandevolutionarethesecondsetoftopicswhicharenotdiscussedinthetwomaintexts.Itriedtomakethelecturenotesself-contained.²FudenbergandLevine(1998),TheTheoryofLearninginGames,Chap-ter1and21IntroductionFormode

2、lsoflearningwetypicallyassumea¯xednumberofplayerswho¯ndoutabouteachother'sintentionsovertime.Inevolutionarymodelstheprocessoflearningisnotexplicitlymodeled.Instead,weassumethatstrate-gieswhichdobetteronaverageareplayedmoreofteninthepopulationovertime.Theb

3、iologicalexplanationforthisisthatindividualsaregeneticallyprogrammedtoplayonestrategyandtheirreproductionratedependsontheir¯tness,i.e.theaveragepayo®theyobtaininthegame.Theeconomicexplanationisthatthereissociallearninggoingoninthebackground-people¯ndoutgr

4、aduallywhichstrategiesdobetterandadjustaccordingly.However,thatadjustmentprocessisslowerthantherateatwhichagentsplaythegame.Wewillfocusinitiallyonmodelswithrandommatching:thereareNagentswhoarerandomlymatchedagainsteachotherovertimetoplayacertaingame.Frequ

5、ently,weassumethatNasin¯nite.Wehavediscussedinthelastlecturethatrandommatchinggivesrisetomyopicplaybecausetherearenorepeatedgameconcerns(I'munlikelytoeverencountermycurrentopponentagain).Wewillfocusonsymmetricnbyngamesforthepurposeofthiscourse.Inasymmetri

6、cgameeachplayerhasthesamestrategysetandthepayo®1matrixsatis¯esui(si;sj)=uj(sj;si)foreachplayeriandjandstrategiessi;sj2Si=Sj=fs1;::;sng.ManygamesweencounteredsofarinthecoursearesymmetricsuchasthePrisoner'sDilemma,BattleoftheSexes,Chickenandallcoordinationg

7、ames.Insymmetricgamesbothplayersfaceexactlythesameproblemandtheiroptimalstrategiesdonotdependonwhethertheyplaytheroleofplayer1orplayer2.Animportantassumptioninevolutionarymodelsisthateachagentplaysa¯xedpurestrategyuntilshedies,orhasanopportunitytolearnand

8、aboutherbelief.Thegameisfullyspeci¯edifweknowthefractionofagentswhoplaystrategyPs1;s2;::;snwhichwedenotewithx1;x2;::;xnsuchthatni=1xi=1.2MutationsandSelectionEverymodelofevolutionreliesontwokeyconcepts-amutationmech