复杂度:零和博弈,最小最大定理以及LP对偶
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Complexity of 2-Player Zero-sum Game
lecturer: Constantinos Daskalakis
Games and Equilibria
Penaliy Shot Game
Drive/Kick Left Right Left 1,-1 -1,1 Right -1,1 1,-1这个零和博弈存在混合策略纳什均衡,我们考虑支付期望
∑i,jci,jxiyj ,(x1 x2)T∗[1 −1;−1,1] 。这里的均衡是1/2.1/2[von Neumann ‘28]: An equilibrium exists in every two-player zero-sum game (R+C=0)
[Dantzig’ 40s] in fact, this follows from strong LP duality
[Khachivan ‘79’] in P time
[B. 56++] dynamics converges
Penaliy Shot Game - not zero-sum game
Drive/Kick Left Right Left 2,-1 -1,1 Right -1,1 1,-1这里的纳什均衡是2/5,3/5
[Nash ‘50/’51]: An equilibrium exists in every finite game.
- proof used Kakutani/Brouswer’s fixed point theorem, and no constructive proof has been found in 70+ years.
- same is true for economic equilibria: supply different goods max utility no good is over demanded
Equlibrium:
A pair
Minimax Theory
Minimax Theorem [von Newmann’28]
Suppose
Proof: Zero-sum Game Two player game has nash equilibrum
(R,C)n×m
R+C=0
X=Δn= {X:Ex=xi≥0 }
Y=Δm
In a zero-sum game, take
f(x,y)=XTCY - how much row plays colum
Then
(x∗,y∗) is an equilibria, wherex∗∈argminx∈Xmaxy∈Y f(x,y) andy∗∈argminy∈Ymaxx∈X f(x,y) xTCy∗≥minx xTCy∗=maxx xTCy∗=maxy minxxTCy=minxmaxyxTCy=maxy x∗Tcy
Existence of Equilibrium in Zero-Sum Game [von Neumann’28]
In two-player zero-sum
Proof:
Let
Then,
Presidential Elections
Clin/Tru Morality Tax Cuts Economy +3,-3 -1, +1 Society -2, +2 1, -1Suppose Clinton commits to strategy
(x1,x2)
E["Morility"]=−3x1+2x2
E["TaxCuts"]=x1−x2 Tru: max
(−3x1+2x2,x1−x2) Clin:
max (-3x_1+2x_2, x_1-x_2),(x1,x2)∈argmax min(−3x1+2x2,x1−x2) , which is a maximin problemIf Clinton is forced to commit to
(x1,x2) ,argmax(x1,x2) min(−3x1+2x2,x1−x2) ,argmaxX min(XTR)
max z - s.t.
3x1−2x2≥z −x1+x2≥z x1+x2=1
x1,x2>0 No matter what Clin does Trump can guarantee 1/7 to himself by playing (3/7, 4/7)
No matter what Clin does Trump can guarantee -1/7 to himself by playing (2/7, 5/7)
i.e. (3/7, 4/7) is best response to (2/7, 5/7) and vise versa
两边的LP问题其实是对偶问题 strong linear programming duality,这也可以从minimax theory这个角度来看
方法二: 从minimax问题直接切入
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