On the optimal strategy in a random game

Consider a two-person zero-sum game played on a random n x n-matrix where the entries are iid normal random variables. Let Z be the number of rows ins the support of the optimal strategy for player I given the realization of the matrix. (The optimal strategy is a.s. unique and Z a.s. coincides wit the number of columns of the support of the optimal strategy for player II.) Faris an Maier [4] make simulations that suggest that as n gets large Z has a distribution close to binomial with parameters n and 1/2 and prove that P(Z = n) less than or equal to 2(-(k-1)). In this paper a few more theoretically rigorous steps are taken towards the limiting distribution of Z: It is shown that there exists a < 1/2 (indeed a < 0.4) such that P((1/2 - a) n < Z < (1/2 + a)n) --> 1 as n --> infinity. It is also shown that EZ = (1/2 + o(1))n. We also prove that the value of the game with probability 1 - o(1) is at most C-n(-1/2) for some C < infinity independent of n. The proof suggests that an upper bound is in fact given by f(n)n(-1), where f(n) is any sequence such that f(n) --> infinity, and it is pointed out that if this is true, then the variance of Z is o(n(2)) so that any a > 0 will do in the bound on Z above.