gath16 - Stat 155 Game theory Yuval Peres Fall 2004...

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Stat 155 Game theory, Yuval PeresFall 2004Lectures 16,17The minimax theorem:Suppose given a payoff matrixAm×n= (aij)1im,1jn,withaijequal to the payment ofIItoIisIpicksiandIIpicksj. PlayerIcan assure an expected payoff of maxxΔmminyΔnxTAy. PlayerIIcanbe assured of not paying more than minyΔnmaxxΔmxTAy. The notationis:Δm={xRm:xi0,mXi=1xi= 1}.Having prepared some required tools, we will now prove:Theorem 1 (Von Neumann minimax)maxxΔmminyΔnxTAy= minyΔnmaxxΔmxTAy.Proof:ThatmaxxΔmminyΔnxTAyminyΔnmaxxΔmxTAy.is easy, and has the same proof as that given for pure strategies. For theother inequality, we firstly assume thatminyΔnmaxxΔmxTAy >0.(1)LetKdenote the set of payoff vectors that playerIIcan achieve, or thatare better than some such vector. That is,K=ny1A(1)+y2A(2)+. . .+ynA(n)+v:y= (y1, . . . , yn)TΔn, v= (v1, . . . , vm)T, vi0o,whereA(i)denotes thei-th column ofA. That is,yis a strategy for playerII, with thei-th component of a member ofKat least as favourable to playerIas the payoff to playerIof playing moveiagainst the mixed strategyyfor playerII. It is easy to show thatKis convex and closed: this uses thefact that Δnis closed and bounded. Note also that06∈K.(2)
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Lectures 16,172To see this, note that (1) means that, for everyy, there existsxsuch thatplayerIhas a uniformly positive expected payoffxTAy(> δ >0). If 0K,this means that, for somey, we have thatAy=y1A(1)+y2A(2)+. . .+ynA(n)0,where by0, we mean in each of the coordinates. However, this contradicts

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