矩阵对策的两个注记

Two Notes on a Matrix Game

设 (x*,y*)是以 A=[aij]m× n为赢得矩阵 G的对策解 ,则当局中人 1,2各自独立地使用其最优策略 x* =(x*1,x*2 ,… ,x*m) ,y* =(y*1,y*2 ,… ,y*n )时 ,局中人 1的赢得期望为对策值 v* =x*Ay* T。若局中人双方使用使得方差 D(x*,y*) = mi=1 nj=1(aij- v* ) 2 x*i y*j 达最小的对策解 (x*,y* ) ,则其赢得靠近 v*的概率达到最大。以 O记使方差达到最小的对策解的集合。若 O满足 (x( 1) ,y( 1) ) ,(x( 2 ) ,y( 2 ) )∈ O蕴涵 (x( 1) ,y( 2 ) ) ,(x( 2 ) ,y( 1) )∈ O,则说 O是可换的。本文首先证明了 :若矩阵对策 G有纯解 ,则 O是可换的。然后证明了如果限定局中人 1在其混合扩充策略集的一个非空紧凸子集 X-中选取策略 ,那么存在 X-的一个非空紧子集 O(X-) ,它是有限个非空互不相交紧凸集之并 ,使得只要局中人 1使用 O(X-)中的策略 。

Let (x *,y *) is a game solution to a matrix game G with the payoff matrix A= m×n. When the two players 1 and 2 use independently their optimal strategies x *=(x * 1,x * 2,...,x * m),y *=(y * 1,y * 2,...,y * n), respectively, the player 1's payoff expectation is v *=x *Ay *T. If they use the game solution (x *,y *) whose variance D(x *,y *)=mi=1nj=1(a ij-v *) 2x * iy * j is the smallest, the probability that the payment closes to v * is the greatest. The optimal solution set O is defined as the set of all the game solutions whose variance D(x *,y *) is the smallest. O is said to be commutative if (x (1),y (1)), (x (2),y (2))∈O implies (x (1),y (2)), (x (2),y (1))∈O. In this paper, the author proves, first, that O is commutative if the matrix game has a pure solution. Secondly, it is proved that if the player 1 has to use strategies in X-, a compact convex subset of mixed strategy set, then there exists a non-empty compact subset O(X-) of X-, which is a union of finite number of mutually disjoint compact non-empty convex sets, such that the player 1 can obtain the best wining in the most unfavorable situation if he uses strategies in O(X-).