Advanced Micro (2) 09v3 Proof of Nash Equilibrium (1)英文文献资料

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1、SolutionConcepts:EquilibriaofStrategicFormGame(V)11.DominationandRationalisability2.NashEquilibrium2.1Purestrategiesandrandomisedstrategies2.2Randomisedstrategiesindetail3.ComputingNashEquilibrium4.Minimax,Maximin,SecurityStrategy,andZeroSumGame4.1MinimaxandMa

2、ximin4.2SecurityStrategy4.3Zero-SumGames25.ProofofExistenceofNashTheorem31.1ReviewofContinuityConcept1.1.1What’sdefinitionofafunction?4Afunction:onetoonerelationship.LetXandYbesubsetsofRNandRMrespectively.Afunctionf:X→YassignsauniqueelementyofthesetYtoeachelem

3、entxoftheX.TheelementyinYassignedbyftoxisdenotedbyf(x).51.1.2AlighttouchofBrouwerfixedpointtheoremBrouwerTheorem:SupposethatXisanon-empty,compact,convexsetinRn.Ifthefunctionf:X→Xiscontinuous,thenafixedpointxexistsatisfyingx=f(x).6Indoingthis,onemaythinkofXa

4、satankfullofwater.Acontinuousfunctionf:X→Xcanthenbeenvisagedasastirringofthewater.Afterthewaterhasbeenstirred,adropletthatwaslocatedatpointxwillhavebeenshiftedtoanewlocationf(x)inthetank.Brouwertheoremcanthenbeseenasthefarfromobviouspropositionthat,howeverthew

5、aterisstirred,therewillalwaysbeatleastonedropletthatisreturnedtoitsinitiallocation.71.1.3DefinitionofContinuity8Afunctionf:X→Yiscontinuous,ifitisalwaystruethatx→xkask→∞impliesthatf(xk)→f(x)ask→∞.Afunctionf:X→Yiscontinuousatapointx0if,forallε>0,thereexistsaδ>0s

6、uchthatd(x,x0)<εimpliesthatd(f(x),f(x0))<δ.Afunctioniscalledacontinuousfunctionifitiscontinuousateverypointinitsdomain.Theflowofariverisabitlikeacontinuousprocess,setsofdropletsthatareneighboursattheoriginoftheriverwillstillbeneighboursbeforetheymergeinthesea,

7、providingthequalityofwaterareeven,nobodypoolsorstopstheriver,andnobodycatcheswaterfromtheriverbeforetheriverarriveattheexittothesea.9Thedefinitionofcontinuityfocusesonapointx,whichisassumedtobeaneighbouroforgluedtothesetS={x,x,12x3,…}.Beforetherivermergesinthe

8、seaorevenwhentheriverismerginginthesea,therequirementforcontinuitycanthenbeinterpretedassayingthatthedropletofwaterthatstartedatxshouldstillbeaneighbouroforgluedtothesetofdropletso