复杂度:零和博弈,最小最大定理以及LP对偶

这篇博客探讨了两玩家零和博弈的复杂度,阐述了纳什均衡的存在性和计算性。内容涵盖游戏与均衡理论、最小最大定理以及线性规划对偶在寻找均衡策略中的应用。文中通过实例解释了如何在博弈中找到最优策略,并指出在某些情况下,这些问题是可以在多项式时间内解决的。

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.

xTRyxTRy, xxTCyxTCy, 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:

minxXmaxyYf(x,y)=maxy
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