博弈论 第五章(Chapter 5)

Lecture5SequentialGamesandBackwardInductionZhenfaXieDepartmentofPE,SE,XMUAGame:CashinAHat(ToyVersionofLenderandBorrower)Player1canput$0,$1,or$3inahatThehatispassedtoplayer2Player2caneither“match”(i.e.addthesameamount)ortakethecashPayoffs:Player1:0→0;1→doubleifmatch,-1ifnot;3→doubleifmatch,-3ifnot.Player2:1.5ifmatch1;2ifmatch2;the$inthehatiftakes.2SequentialMoveGamePlayer2knowsplayer1’schoicebefore2chooses;Player1knowsthatthiswillbethecase.3AGame:CashinAHat40,01,1.5-1,13,2-3,30131-13-3122BackwardInduction5Lookforward,workback.0,01,1.5-1,13,2-3,30131-13-3122MoralHazard6Moralhazard:agenthasincentivestodothingsthatarebadfortheprincipal.Examplein“cashinahat”:keptthesizeofloan/projectsmalltoreducethetemptationtocheat.IncentiveDesign7Changecontracttogiveincentivesnottoshirk“Asmallshareofalargepie”canbebiggerthan“alargeshareofasmallpie”.IncentiveDesign80,01,1.5-1,11.9,3.1-3,30131-13-3122IncentiveContracts9PieceratesSharecroppingCollateralSubtracthousefromrunawaypayoffsLowerspayoffstoborroweratsometreepoint,yetmaketheborrowerbetterofchangesthechoicesofothersinawaythathelpsyouCollateral100,01,1.5-1,1(-house)3,2-3,3(-house)0131-13-3122CommitmentStrategy11GettingridofchoicescanmakemebetteroffCommitment:tohavefeweroptionsanditchangesthebehaviorofothersKnowledge:theotherplayersmustknowthesituationchanged破釜沉舟12CommitmentStrategy130,02,11,20,02,1FRA项章章项项项项NotburnburnFFFFFRARARA1,2Liongame14Stackelberg’sDuopolyGameConstantUnitCostandLinearInverseDemandPlayers：thetwofirms(aleaderandafollower)；Timing:Firm1choosesaquantity；Firm2observesandthenchoosesaquantity;Payoff：Eachfirm’spayoffisrepresentedbyitsprofit.10q1q20q15BasicAssumptionsConstantunitcost：Linearinversedemandfunction：Assume：foralliiiiCqcqqif,0ifiQQPQQqQc16BackwardinductionFirstcomputefirm2’sreactiontoanarbitraryquantitybyfirm1:Nextfirm1’sprobleminthefirststageofthegameamountsto:Thustheoutcomeofthegameis:222122120011211max(,)max[]if()20ifqqqqqqqcqcqcbqqc1111121112100110max(,())max()1max()2qqqqbqqqbqcqcq**1221and()24ccqqbq17ConclusionTheoutcomeoftheequilibriumoutputis:Firm1’sprofitis,andfirm2’sprofitis.Bycontrast,intheuniqueNashequilibriumofCournot’s(simultaneous-move)gameunderthesameassumptions,eachfirmproducesunitsofoutputandobtainstheprofit.Thusfirm1producesmoreoutputandobtainsmoreprofitinthesubgameperfectequilibriumofthesequentialgame,andfirm2produceslessoutputandobtainslessprofit.**12and24ccqq218()c2116()c219()c13()c18Lessons19Commitment:sunkcostscanhelpSpyorhavingmoreinformationcanhurtyouKey:theotherplayersknewyouhadmoreinformationReason:itcanleadotherplayerstotakeactionsthathurtyouFirst-moveradvantageFirst-moveradvantage20Yessometimes:StackelbergButnotalways—SecondmoveradvantageRock,paper,scissorsInformationhereishelpfulSometimesneitherfirstnorsecondmoveradvantageDivideacakewithyoursibling:Isplit,youchoose.ItcanbefirstorsecondmoveradvantagewithinthesamegamedependingonsetupThegameofNimTheGameofNimPilesequal→secondmoveradvantagePilesunequal→firstmoveradvantage21Definitions22PerfectinformationAgameofperfectinformationisoneinwhichateachnode,theplayerwhoseturnitistomoveknowswhichnodesheisat(andhowshegotthere).PurestrategyApurestrategyforplayeriinagameofperfectinformationisacompleteplanofactions:Itspecifieswhichactioniwilltakeateachofitsdecisionnodes.EntryGameChallengerOut0，3InIncumbentAcquiesceFight1，1-1，023EntryGameInentrygame，thestrategiesis：Challenger：In，OutIncumbent：Strategy1(s1):PlayAcquiesceifChallengerplaysIn;Strategy2(s2):PlayFightifChallengerplaysIn.24Example11DC2EF2，13，00，21，3GH225Example1Play1hastwostrategies：CandD；Play2hasfourstrategies：EG，EH，FG，FH.ActionassignedtohistoryCActionassignedtohistoryDStrategy1EGStrategy2EHStrategy3FGStrategy4FH26Example21DC22，0EF3，110，01，2GH27Example2Player1hasfourstrategies：CG,CH,DG,DH.Inparticular,eachstrategyspecifiesanactionafterthehistory(C,E)evenifitspecifiestheactionDatthebeginningofthegame,inwhichcasethehistory(C,E)doesnotoccur!Player2hastwostrategies：EandF.28ThestrategicformoftheentrygameanditsNashequilibriumsIncumbentAcquiesceFightChallengerIn1*，1*-1，0Out0，3*0*，3*NE:(In,Acquiesce)←BackwardInduction(Out,Fight)ItisaNEthatreliesonbelievinganincrediblethreat.29Chain-storeParadox2020/1/26厦门大学财政系谢贞发30试想一个博弈：在每个省会城市，都有麦当劳连锁店，每个城市都有一个挑战者想进入该行业。请利用backwardinduction寻找该博弈的均衡。如果你是麦当劳在中国的总代理，你会怎么做？如何解释“连锁店悖论”？EntryGameinMarketkChallengerkOut0，3InIncumbentAcquiesceFight1，1-1，031Chain-storeParadox2020/1/26厦门大学财政系谢贞发32Youcanfindauniqueoutcomebybackwardinduction,inwhicheverychallengerentersandthechain-storealwaysacquiescestoentry.Butinrealityyoumayobservethateverypreviouschallengerenteredandthatthechain-storefoughteachone,thenfuturechallengersstayout.Chain-storeParadox2020/1/26厦门大学财政系谢贞发33TwopointsSmallprobabilityofcrazychangesthingsIfɛ-chance(1%)thatthechain-storeiscraze,thenhecandeterentrybyfighting:seemingcraze.Reputationmatters兵临城下34DuelGame2020/1/26厦门大学财政系谢贞发35Players:2pla