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1、lecture11MixedStrategyNashEquilibrium(博弈论,CarnegieMellonUniversity)Static(orSimultaneous-Move)GamesofCompleteInformationMixedStrategyNashEquilibrium73-347GameTheory--Lecture11OutlineofStaticGamesofCompleteInformationIntroductiontogamesNormal-form(orstrategic-form)representationIterate
2、deliminationofstrictlydominatedstrategiesNashequilibriumReviewofconcavefunctions,optimizationApplicationsofNashequilibriumMixedstrategyNashequilibrium73-347GameTheory--Lecture11Today’sAgendaReviewofpreviousclassExercise137><8.1ofOsborneReviewHW173-347GameTheory--Lecture11Exercise13<8.
3、1ofOsborneWefirstconsiderpure-strategyNashequilibria.Howmanycanyoufind?InordertofindallNashequilibria,weneedtoconsider15morecasesbyTheorem4ofLecture10!Wefirstconsidercomplicatedcases.Somecasesareveryeasy.Player20,21,3R(p23)M(p22)1,13,2B(p12)Player10,32,2T(p11)L(p21)73-347GameTheory--L
4、ecture11Exercise13<8.1ofOsborneCase1:checkwhetherthereisamixedstrategyinwhichp11>0,p12>0,p21>0,p22>0,p23>0Bytheorem4,weshouldhave2?8?p11+2?8?p12=3?8?p11+1?8?p12=3?8?p11+2?8?p12andp11+p12=1.Weshouldhave2?8?p21+0?8?p22+1?8?p23=3?8?p21+1?8?p22+0?8?p23andp21+p22+p23=1Solvet
5、heseequations.Ifwecangetasolutionthatsatisfiesp11>0,p12>0,p21>0,p22>0,p23>0thenwehaveamixedstrategyNashequilibrium.Otherwise,ifwecannotfindasolutionorwefindasolutionthatdoesnotsatisfiesp11>0,p12>0,p21>0,p22>0,p23>0,thenwedonothavesuchamixedstrategyNashequ
6、ilibrium.Player20,21,3R(p23)M(p22)1,13,2B(p12)Player10,32,2T(p11)L(p21)73-347GameTheory--Lecture11Exercise13<8.1ofOsborneCase2:checkwhetherthereisamixedstrategyinwhichp11>0,p12>0,p21>0,p22>0,p23=0Bytheorem4,weshouldhave2?8?p11+2?8?p12=3?8?p11+1?8?p12?8?3?8?p11+2?8?p12andp1
7、1+p12=1.Weshouldhave2?8?p21+0?8?p22+1?8?p23=3?8?p21+1?8?p22+0?8?p23andp21+p22+p23=1Solvethese.Ifwecangetasolutionthatsatisfiesp11>0,p12>0,p21>0,p22>0,p23=0thenwehaveamixedstrategyNashequilibrium.Otherwise,ifwecannotfindasolutionorwefindasolutionthatdoesnotsatisfiesp11>0
8、,p12&
