复旦微观经济学repeated game

AdvancedMicroeconomicsJunjiXiaoLecture3JunjiXiao(Lecture3)Micro1/73RepeatedgamesAnalyzewhetherthreatsandpromisesaboutfuturebehaviorcanin‡uencecurrentbehaviorinrepeatedrelationships.JunjiXiao(Lecture3)Micro2/73RoadmapRoadmapRepeatedgamesFinite-stagerepeatedgamesIn…nite-stagerepeatedgamesJunjiXiao(Lecture3)Micro3/73Finite-stagerepeatedgamesTwo-stagePrisoners’DilemmaConsidertwoplayersplaythePrisoners’Dilemmatwice,observingtheoutcomeofthe…rstplaybeforethesecondplaybegins.Supposethepayo¤fortheentiregameissimplythesumofthepayo¤sfromthetwostages.Player2Player1L2R2L11,15,0R10,54,4JunjiXiao(Lecture3)Micro4/73Finite-stagerepeatedgamesTwo-stagePrisoners’Dilemma(cont’d)Theuniqueequilibriumofthesecond-stagegameis(L1,L2),regardlessofthe…rst-stageoutcome.Weanalyzethe…rststageofthetwostagePrisoners’Dilemmabytakingintoaccountthattheoutcomeofthegameremaininginthesecondstage,(L1,L2).Theplayers’…rst-stageinteractioninthegameamountstotheone-shotgameasfollows:Player2Player1L2R2L12,26,1R11,65,5.ThegamehasauniqueNashequilibrium:(L1,L2).Theuniquesubgame-perfectoutcomeofthetwo-stagePrisoners’Dilemmais(L1,L2)inthe…rststage,followedby(L1,L2)inthesecondstage.JunjiXiao(Lecture3)Micro5/73Finite-stagerepeatedgamesTwo-stagePrisoners’Dilemma(cont’d)Theuniqueequilibriumofthesecond-stagegameis(L1,L2),regardlessofthe…rst-stageoutcome.Weanalyzethe…rststageofthetwostagePrisoners’Dilemmabytakingintoaccountthattheoutcomeofthegameremaininginthesecondstage,(L1,L2).Theplayers’…rst-stageinteractioninthegameamountstotheone-shotgameasfollows:Player2Player1L2R2L12,26,1R11,65,5.ThegamehasauniqueNashequilibrium:(L1,L2).Theuniquesubgame-perfectoutcomeofthetwo-stagePrisoners’Dilemmais(L1,L2)inthe…rststage,followedby(L1,L2)inthesecondstage.JunjiXiao(Lecture3)Micro5/73Finite-stagerepeatedgamesTwo-stagePrisoners’Dilemma(cont’d)Theuniqueequilibriumofthesecond-stagegameis(L1,L2),regardlessofthe…rst-stageoutcome.Weanalyzethe…rststageofthetwostagePrisoners’Dilemmabytakingintoaccountthattheoutcomeofthegameremaininginthesecondstage,(L1,L2).Theplayers’…rst-stageinteractioninthegameamountstotheone-shotgameasfollows:Player2Player1L2R2L12,26,1R11,65,5.ThegamehasauniqueNashequilibrium:(L1,L2).Theuniquesubgame-perfectoutcomeofthetwo-stagePrisoners’Dilemmais(L1,L2)inthe…rststage,followedby(L1,L2)inthesecondstage.JunjiXiao(Lecture3)Micro5/73Finite-stagerepeatedgamesTwo-stagePrisoners’Dilemma(cont’d)Theuniqueequilibriumofthesecond-stagegameis(L1,L2),regardlessofthe…rst-stageoutcome.Weanalyzethe…rststageofthetwostagePrisoners’Dilemmabytakingintoaccountthattheoutcomeofthegameremaininginthesecondstage,(L1,L2).Theplayers’…rst-stageinteractioninthegameamountstotheone-shotgameasfollows:Player2Player1L2R2L12,26,1R11,65,5.ThegamehasauniqueNashequilibrium:(L1,L2).Theuniquesubgame-perfectoutcomeofthetwo-stagePrisoners’Dilemmais(L1,L2)inthe…rststage,followedby(L1,L2)inthesecondstage.JunjiXiao(Lecture3)Micro5/73Finite-stagerepeatedgamesTwo-stagePrisoners’Dilemma(cont’d)Theuniqueequilibriumofthesecond-stagegameis(L1,L2),regardlessofthe…rst-stageoutcome.Weanalyzethe…rststageofthetwostagePrisoners’Dilemmabytakingintoaccountthattheoutcomeofthegameremaininginthesecondstage,(L1,L2).Theplayers’…rst-stageinteractioninthegameamountstotheone-shotgameasfollows:Player2Player1L2R2L12,26,1R11,65,5.ThegamehasauniqueNashequilibrium:(L1,L2).Theuniquesubgame-perfectoutcomeofthetwo-stagePrisoners’Dilemmais(L1,L2)inthe…rststage,followedby(L1,L2)inthesecondstage.JunjiXiao(Lecture3)Micro5/73Finite-stagerepeatedgamesDe…nitionAny…nitenumberofrepetitions,T.LetG=fA1,...,An;u1,...,ungdenoteastaticgameofcompleteinformationinwhichplayers1throughnsimultaneouslychooseactionsa1throughanfromtheactionspacesA1throughAn,respectively,andthepayo¤sareu1(a1,...,an)throughun(a1,...,an).ThegameGwillbecalledthestagegameoftherepeatedgame.JunjiXiao(Lecture3)Micro6/73Finite-stagerepeatedgamesDe…nition(cont’d)De…nitionGivenastagegameG,letG(T)denotethe…nitelyrepeatedgameinwhichGisplayedTtimes,withtheoutcomesofallprecedingplaysobservedbeforethenextplaybegins.Thepayo¤sforG(T)aresimplythesumofthepayo¤sfromtheTstagegames.JunjiXiao(Lecture3)Micro7/73Finite-stagerepeatedgamesFinite-stagerepeatedgamesPropositionIfthestagegameGhasauniqueNashequilibriumthen,forany…niteT,therepeatedgameG(T)hasauniquesubgame-perfectoutcome:theNashequilibriumofGisplayedineverystage.JunjiXiao(Lecture3)Micro8/73Finite-stagerepeatedgamesTwo-stagerepeatedgames-stagegamewithmultipleNashequilibriaL2M2R2L11,15,00,0M10,54,40,0R10,00,03,3Supposethestagegameisplayedtwice,withthe…rst-stageoutcomeobservedbeforethesecondstagebegins.Twopure-strategyNashequilibria:(L1,L2),(R1,R2).JunjiXiao(Lecture3)Micro9/73Finite-stagerepeatedgamesTwo-stagerepeatedgames-stagegamewithmultipleNashequilibriaL2M2R2L11,15,00,0M10,54,40,0R10,00,03,3Supposethestagegameisplayedtwice,withthe…rst-stageoutcomeobservedbeforethesecondstagebegins.Twopure-strategyNashequilibria:(L1,L2),(R1,R2).JunjiXiao(Lecture3)Micro9/73Finite-stagerepeatedgamesTwo-stagerepeatedgames-stagegamewithmultipleNashequilibria(cont