On the optimal strategy in a random game

Artikel i vetenskaplig tidskrift, 2004

Consider a two-person zero-sum game played on a random n by n matrix where the entries are iid normal random variables. Let Z be the number of rows in 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 with the number of columns of the support of the optimal strategy for player II.) Faris an Maier (see the references) 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) < 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) tends to 1 as n increases. It is also shown that the expectation of Z is (1/2+o(1))n. We also prove that the value of the game with probability 1-o(1) is at most Cn-1/2 for some finite C independent of n. The proof suggests that an upper bound is in fact given by f(n)/n, where f(n) is any sequence tending to infinity as n increases, 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.