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$\sum_{i,j}c_{i,j}x_{i}y_j$，$(x_1\ x_2)^T*[1\ -1; -1, 1]$。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.

$$x^T R y \geq x'^T R y, ~\forall x'\\ x^T C y \geq x^TC y', ~\forall y'$$

Minimax Theory

Minimax Theorem [von Newmann’28]

Suppose $X$ and $Y$ are compact (closed and bounded) convex sets, and $f: X \times Y \rightarrow \mathcal{R}$ 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