博弈论(复旦大学中国经济研究中心).pptx

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.,