国外博弈论课件lecture(19)

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1、Exercise112.1,138.1ofOsborneSolutionJune6,2003173-347GameTheory--Lecture14Mixedstrategyequilibrium:2-playereachwithtwostrategiesTheorem2Let((r*,1-r*),(q*,1-q*))beapairofmixedstrategies,where0

2、(q*,1-q*))EU2(s21,(r*,1-r*))=EU2(s22,(r*,1-r*))Remark:whenyouuseTheorem2tosolveagame,ifthesolutiondoesnotsatisfy0

3、)u1(s11,s22),u2(s11,s22)s12(1-r)u1(s12,s21),u2(s12,s21)u1(s12,s22),u2(s12,s22)June6,2003273-347GameTheory--Lecture14Exercise112.1ofOsborne(rightgame)Player1’sexpectedpayoffofplayingTEU1(T,(q,1–q))=q×0+(1–q)×0=0Player1’sexpectedpayoffofplayingBEU1(B,(q,1–q))=q×2+(1–q)×0=2qPlayer1isindifferentbetwee

4、nplayingTandBEU1(T,(q,1–q))=EU1(B,(q,1–q))0=2qSoq=0Hence,wecanconcludethatthereisnoNashequilibriuminwhicheachplayerassignspositiveprobabilitytoeachofhertwopurestrategiesPlayer2L(q)R(1–q)Player1T(r)0,10,2B(1–r)2,20,1June6,2003373-347GameTheory--Lecture14Exercise112.1ofOsborne(rightgame)Wecansolveth

5、isgamebyappealingtoTheorem4.SincewehavefoundtwopurestrategyNashequilibriaandcheckedonecaseinthepreviousslide,soweneedtoconsiderfourmorecasesPlayer2L(q)R(1–q)Player1T(r)0,10,2B(1–r)2,20,1June6,2003473-347GameTheory--Lecture14Exercise112.1ofOsborne(rightgame)Case1:checkwhetherthereisamixedstrategyin

6、whichr=1,0

7、ereisamixedstrategyinwhichr=0,0