國外博弈論課件lecture18

DynamicGamesofCompleteInformationDynamicGamesofCompleteandImperfectInformationJune13,20031OutlineofdynamicgamesofcompleteinformationDynamicgamesofcompleteinformationExtensive-formrepresentationDynamicgamesofcompleteandperfectinformationGametreeSubgame-perfectNashequilibriumBackwardinductionApplicationsDynamicgamesofcompleteandimperfectinformationMoreapplicationsRepeatedgamesJune13,20032Today’sAgendaReviewofpreviousclassTariffsandimperfectinternationalcompetition(2.2.CofGibbons)RepeatedgamesJune13,20033InformationsetGibbons’definition:Aninformationsetforaplayerisacollectionofnodessatisfying:theplayerhasthemoveateverynodeintheinformationset,andwhentheplayofthegamereachesanodeintheinformationset,theplayerwiththemovedoesnotknowwhichnodeintheinformationsethas(orhasnot)beenreached.AllthenodesinaninformationsetbelongtothesameplayerTheplayermusthavethesamesetoffeasibleactionsateachnodeintheinformationset.June13,20034Subgame-perfectNashequilibriumANashequilibriumofadynamicgameissubgame-perfectifthestrategiesoftheNashequilibriumconstituteorinduceaNashequilibriumineverysubgameofthegame.Asubgameofagametreebeginsatasingletoninformationset(aninformationsetcontainingasinglenode),andincludesallthenodesandedgesfollowingthesingletoninformationset,anddoesnotcutanyinformationset;thatis,ifanodeofaninformationsetbelongstothissubgamethenallthenodesoftheinformationsetalsobelongtothesubgame.June13,20035ExamplePlayer1LRPlayer2L’R’2,2,0Player2L’R’3L〞R〞3L〞R〞3L〞R〞3L〞R〞1,2,33,1,22,2,12,2,10,0,11,1,21,1,1June13,20036Tariffsandimperfectinternationalcompetition(2.2.CofGibbons)Twoidenticalcountries,1and2,simultaneouslychoosetheirtariffrates,denotedt1,t2,respectively.Firm1fromcountry1andfirm2fromcountry2produceahomogeneousproductforbothhomeconsumptionandexport.Afterobservingthetariffrateschosenbythetwocountries,firm1and2simultaneouslychoosesquantitiesforhomeconsumptionandforexport,denotedby(h1,e1)and(h2,e2),respectively.MarketpriceintwocountriesPi(Qi)=a–Qi,

fori=1,2.Q1=h1+e2,Q2=h2+e1.Bothfirmshaveaconstantmarginalcostc.Eachfirmpaystariffonexporttotheothercountry.June13,20037TariffsandimperfectinternationalcompetitionJune13,20038TariffsandimperfectinternationalcompetitionJune13,20039Backwardinduction:

subgamebetweenthetwofirmsJune13,200310Backwardinduction:

subgamebetweenthetwofirmsJune13,200311Backwardinduction:wholegameJune13,200312TariffsandimperfectinternationalcompetitionJune13,200313RepeatedgameArepeatedgameisadynamicgameofcompleteinformationinwhicha(simultaneous-move)gameisplayedatleasttwice,andthepreviousplaysareobservedbeforethenextplay.Wewillfindoutthebehavioroftheplayersinarepeatedgame.June13,200314Two-stagerepeatedgameTwo-stageprisoners’dilemmaTwoplayersplaythefollowingsimultaneousmovegametwiceTheoutcomeofthefirstplayisobservedbeforethesecondplaybeginsThepayofffortheentiregameissimplythesumofthepayoffsfromthetwostages.Thatis,thediscountfactoris1.Player2L2R2Player1L11,15,0R10,54,4June13,200315Gametreeofthetwo-stageprisoners’dilemma1L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R21+1

1+11+5

1+01+0

1+51+4

1+411115+1

0+15+5

0+05+0

0+55+4

0+40+1

5+10+5

5+00+0

5+50+4

5+44+1

4+14+5

4+04+0

4+54+4

4+4June13,200316Informalgametreeofthetwo-stageprisoners’dilemma1L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R21

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41111(1,1)(5,0)(0,5)(4,4)1

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4June13,200317Informalgametreeofthetwo-stageprisoners’dilemma1L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R21

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41111(2,2)(6,1)(1,6)(5,5)1

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4June13,200318two-stageprisoners’dilemmaThesubgame-perfectNashequilibrium

(L1L1L1L1L1,L2L2L2L2L2)

Player1playsL1atstage1,andplaysL1atstage2foranyoutcomeofstage1.

Player2playsL2atstage1,andplaysL2atstage2foranyoutcomeofstage1.Player2L2R2Player1L11,15,0R10,54,4June13,200319FinitelyrepeatedgameAfinitely

repeatedgameisadynamicgameofcompleteinformationinwhicha(simultaneous-move)gameisplayedafinitenumberoftimes,andthepreviousplaysareobservedbeforethenextplay.ThefinitelyrepeatedgamehasauniquesubgameperfectNashequilibriumifthestagegame(thesimultaneous-movegame)hasauniqueNashequilibrium.TheNashequilibriumofthestagegameisplayedineverystage.June13,200320WhathappensifthestagegamehasmorethanoneNashequilibrium?TwoplayersplaythefollowingsimultaneousmovegametwiceTheoutcomeofthefirstplayisobservedbeforethesecondplaybeginsThepayofffortheentiregameissimplythesumofthepayoffsfromthetwostages.Thatis,thediscountfactoris1.Question:canwefindasubgameperfectNashequilibriuminwhichM1,M2areplayed?Or,canthetwoplayerscooperateinasubgameperfectNashequilibrium?Player2L2M2R2Player1L11,15,00,0M10,54,40,0R10,00,03,3June13,200321Informalgametree1L1R122L2R2M2L2R2M2L2R2M22L1R122L2R2M2L2R2M2L2R2M22M1(1,1)(5,0)(0,5)(4,4)(0,0)M1(0,0)(0,0)(0,0)(3,3)1(1,1)(5,0)(0,5)(0,0)(0,0)(0,0)(0,0)(3,3)(4,4)June13,200322Informalgametreeandbackwardinduction1L1R122L2R2M2L2R2M2L2R2M22L1R122L2R2M2L2R2M2L2R2M22M1(1,1)(5,0)(0,5)(4,4)(0,0)M1(0,0)(0,0)(0,0)(3,3)1(1,1)(5,0)(0,5)(0,0)(0,0)(0,0)(0,0)(3,3)(4,4)(1,1)(1,1)(1,1)(3,3)(1,1)(1,1)(1,1)(1,1)(