博弈论(复旦大学中国经济研究中心).pptx
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博弈论(复旦大学中国经济研究中心).pptx
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GameTheory(MicroeconomicTheory(IV),Instructor:
YongqinWangEmail:
yongqin_SchoolofEconomics,FudanUniversityDecember,2004MainReference:
RobertGibbons,1992:
GameTheoryforAppliedEconomists,PrincetonUniversityPressFudenbergandTirole,1991:
GameTheory,MITPress,1.StaticGameofCompleteInformation,1.3FurtherDiscussiononNashEquilibrium(NE)1.3.1NEversusIteratedEliminationofStrictDominanceStrategiesPropositionAInthe-playernormalformgameifiteratedeliminationofstrictlydominatedstrategieseliminatesallbutthestrategies,thenthesestrategiesaretheuniqueNEofthegame.,AFormalDefinitionofNE,Inthen-playernormalformthestrategiesareaNE,ifforeachplayeri,is(atleasttiedfor)playerisbestresponsetothestrategiesspecifiedforthen-1otherplayers,Contd,PropositionBInthe-playernormalformgameifthestrategiesareaNE,thentheysurviveiteratedeliminationofstrictlydominatedstrategies.,1.3.2ExistenceofNE,Theorem(Nash,1950):
Inthe-playernormalformgameifisfiniteandisfiniteforevery,thenthereexistatleastoneNE,possiblyinvolvingmixedstrategies.SeeFudenbergandTirole(1991)forarigorousproof.,1.4Applications1.4.1CournotModel,TwofirmsAandBquantitycompete.InversedemandfunctionTheyhavethesameconstantmarginalcost,andthereisnofixedcost.,Contd,FirmAsproblem:
Contd,Bysymmetry,firmBsproblem.FigureIllustration:
ResponseFunction,TatonnementProcessExercise:
whatwillhappensiftherearenidenticalCournotcompetingfirms?
(ConvergencetoCompetitiveEquilibrium),1.4.2TheproblemofCommons,DavidHume(1739):
ifpeoplerespondonlytoprivateincentives,publicgoodswillbeunderprovidedandpublicresourcesover-utilized.Hardin(1968):
TheTragedyofCommons,Contd,Therearefarmersinavillage.Theyallgrazetheirgoatonthevillagegreen.Denotethenumberofgoatsthefarmerownsby,andthetotalnumberofgoatsinthevillagebyBuyingandcaringeachgoatcostandvaluetoafarmerofgrazingeachgoatis.,Contd,Amaximumnumberofgoats:
forbutforAlsoThevillagersproblemissimultaneouslychoosinghowmanygoatstoown(tochoose).,Contd,Hispayoffis
(1)InNE,foreach,mustmaximize
(1),giventhatotherfarmerschoose,Contd,Firstordercondition(FOC):
(2)(where)SummingupallfarmersFOCandthendividingbyyields(3),Contd,Incontrast,thesocialoptimumshouldresolveFOC:
(4)Comparing(3)and(4),wecanseethatImplicationsforsocialandeconomicsystems(CoaseTheorem),2.DynamicGamesofCompleteInformation,2.1DynamicGamesofCompleteandPerfectInformation2.1.ATheory:
BackwardInductionExample:
TheTrustGameGeneralfeatures:
(1)Player1choosesanactionfromthefeasibleset.
(2)Player2observesandthenchoosesanactionfromthefeasibleset.(3)Payoffsareand.,Contd,BackwardInduction:
Then“Peoplethinkbackwards”,2.1.BAnexample:
StackelbergModelofDuopoly,Twofirmsquantitycompetesequentially.Timing:
(1)Firm1choosesaquantity;
(2)Firm2observesandthenchoosesaquantity;(3)Thepayofftofirmisgivenbytheprofitfunctionistheinversedemandfunction,andistheconstantmarginalcostofproduction(fixedcostbeingzero).,Contd,Wesolvethisgamewithbackwardinduction(providedthat).,Contd,Now,firm1sproblemso,.,Contd,ComparewiththeCournotmodel.HavingmoreinformationmaybeabadthingExercise:
Extendtheanalysistofirmcase.,2.2Twostagegamesofcompletebutimperfectinformation2.2.ATheory:
Sub-GamePerfection,Heretheinformationsetisnotasingleton.Considerfollowinggames
(1)Players1and2simultaneouslychooseactionsandfromfeasiblesetsand,respectively.
(2)Players3and4observetheoutcomeofthefirststage(,)andthensimultaneouslychooseactionsandfromfeasiblesetsand,respectively.(3)Payoffsare,AnapproachsimilartoBackwardInduction,1and2anticipatethesecondbehaviorof3and4willbegivenbythenthefirststageinteractionbetween1and2amountstothefollowingsimultaneous-movegame:
(1)Players1and2simultaneouslychooseactionsandfromfeasiblesetsandrespectively.
(2)PayoffsareSub-gameperfectNashEquilibriumis,2.2BAnExample:
BanksRuns,Twodepositors:
eachdepositsDinabank,whichinvestthesedepositsinalong-termproject.Earlyliquidationbeforetheprojectmatures,2rcanberecovered,whereDrD/2.Ifthebankallowstheinvestmenttoreachmaturity,theprojectwillpayoutatotalof2R,whereRD.Assumethereisnodiscounting.InsertMatrixesInterpretationofThemodel,goodversusbadequilibrium.,Contd,Date1Date2,Contd,InEquilibriumInterpretationoftheModelandtheRoleoflawandotherinstitutions,2.3RepeatedGame,2.3ATheory:
Two-StageRepeatedGameRepeatedPrisonersDilemmaStageGame,Contd,DefinitionGivenastagegameG,letthefinitelyrepeatedgameinwhichGisplayedTtimes,withtheoutcomesofallprecedingplaysobservedbeforethenextplaybegins.ThepayoffforG(T)aresimplythesumofthepayoffsfromthestagegames.PropositionIfthestagegameGhasauniqueNE,thenforanyfiniteT,therepeatedgameG(T)hasauniquesub-gameperfectoutcome:
theNashequilibriumofGisplayedineverystage.(Theparadoxofbackwardinduction),SomeWaysoutoftheParadox,BoundedRationality(Tremblesmaymatter)MultipleNashEquilibrium(AnTwo-PeriodExample)UncertaintyaboutotherplayersUncertaintyaboutthefutures,2.3BTheory:
InfinitelyRepeatedGames,Definition1Giventhediscountfactor,thepresentvalueoftheinfinitelyrepeatedsequenceofpayoffsisDefinition2(Selten,1965)ANashEquilibriumissubgameperfectiftheplayersstrategiesconstituteaNashequilibriumineverysubgame.,Contd,Definition3:
Giventhediscountedfactor,theaveragepayoffoftheinfinitesequenceofpayoffsisFolkTheorem(Friedman,1971):
SeeGibbons(p97).DiscussReputationModel,2.4DynamicGameswithCompletebutImperfectInformation,Informationsetisnotasingleton.JustificationforSub-GamePerfectArgument.Commitment,Reputation,SunkCostandCheaptalk.,
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