04 - Nash equilibrium and mixed strategies.pdf

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1、GameTheory4:NashequilibriumindifferentgamesandmixedstrategiesNASHEQUILIBRIUMANDMIXEDSTRATEGIESReviewoflecturethree•Agamewithnodominatedstrategy:“Thebattleofthesexes”•TheconceptofNashequilibrium•TheformaldefinitionofNE•HowtofindNEinmatrixgames2NASHEQUILIBRIU

2、MANDMIXEDSTRATEGIESWhichsideoftheroad?Mr.RedLEFTRIGHTMr.LEFT(1,1)(-1,-1)GreenRIGHT(-1,-1)(1,1)•YouhavetwoNE:(LEFT,LEFT)and(RIGHT,RIGHT)•Thisgameisacoordinationone•Inacoordinationgametheproblemisnottocheatorliebuttofindoutthewaytogetthemutualbenefitbycoordin

3、atingtheiractions(i.e.,choosingthesamestrategy)•Howtodothat?3NASHEQUILIBRIUMANDMIXEDSTRATEGIESFocalpoints•Forgamesofcoordinationthiscanbedonebyfindingsomeelementsofinteractionthatappearsinsomewayasprominenttoplayers•Theseaspectsarecalledfocalpoints4NASHEQUI

4、LIBRIUMANDMIXEDSTRATEGIESExamples(T.Schelling1960)•NameacityofUK.Ifyouallnamethesame,youwinaprize•YouaretomeetsomebodyinMilanforareasonthatyoubothconsiderveryimportant.Youwerenotbeentoldwheretomeetandyoucannotcommunicate.Guesswheretogo…•AsktoPeppino!!!5NASH

5、EQUILIBRIUMANDMIXEDSTRATEGIESTotò,PeppinoelaMalafemmina:http://www.youtube.com/watch?v=im2GpqWk8uk6NASHEQUILIBRIUMANDMIXEDSTRATEGIESExamples(T.Schelling1960)•“peoplecanoftenconcerttheirintentionsorexpectationswithothersifeachknowsthattheotheristryingtodothe

6、same”•Focalpointsimposethemselvesontheplayers’attentionforreasonsthattheformaltheoryoverlooks7NASHEQUILIBRIUMANDMIXEDSTRATEGIESApplicationtoBS:asymmetriccoordinationgameAnnfightballetfight(2,1)(0,0)Bertballet(0,0)(1,2)1.TodayisAnnbirthday2.LasteveningoutAnn

7、andBertwenttoaballet•Stillitmayhappenthatplayersdonotfindthewaytoreachoneoftheirpreferredresultscoordinationfailure8NASHEQUILIBRIUMANDMIXEDSTRATEGIESAnapplicationtoPoliticsNucleararmsrace(firstcase)•BothUSandUSSRhavetwostrategies:“ARM”and“REFRAIN”•Supposeb

8、othUSandUSSRcareonlyformilitarysupremacy(Avs.R)>(Rvs.R)>(Avs.A)>(Rvs.A)9NASHEQUILIBRIUMANDMIXEDSTRATEGIESToughsuperpowersarmsraceUSSRREFRAINARMREFRAIN(3,3)(1,4)USARM(4,1)(2,2)•BychoosingARM(leadingto(