复杂度：零和博弈，最小最大定理以及LP对偶

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 (x,y) of randomized strategies so that no player has incentive to deviate if the others does not.

xTRy≥x′TRy, ∀x′xTCy≥xTCy′, ∀y′

Minimax Theory

Minimax Theorem [von Newmann’28]

Suppose X and Y are compact (closed and bounded) convex sets, and f:X×Y→ is a continuous function that is convex-concave, i.e., f(.,y) is convex for all fixed y , and f(x,.) is concave for all fixed x , then:

minx∈Xmaxy∈Yf(x,y)=maxy∈Yminx∈Xf(x,y)

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, where

  • x∗∈argminx∈Xmaxy∈Y f(x,y) and y∗∈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 (R+C=0) games, equilibrium always exists.

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Proof:

Let f(x,y)=xTCy (the payoff of column player), then f(x,y) satisfies Minimax theorem. Assume

f(x∗,y∗)=x∗TCy∗minx∈Xmaxy∈Yf(x,y)=maxy∈Yminx∈Xf(x,y)

Then,

x∗TCy∗=maxy∈Yf(x∗,y)≥x∗TCy′,  ∀y′x∗TCy∗=−x∗TRy∗⇒x∗TRy∗=−minx∈Xf(x,y∗)≥x′TRy∗,  ∀x′

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Presidential Elections

Clin/Tru	Morality	Tax Cuts
Economy	+3,-3	-1, +1
Society	-2, +2	1, -1

Suppose 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 problem

If 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