Abstract: "This paper presents a parallel randomized algorithm which computes a pair of [epsilon]-optimal strategies for a given (m, n)- matrix game A = [a[subscript i, subscript j]] [is in the set of] [-1,1] in O([epsilon][superscript-2] log² (n + m)) expected time on an (n + m) / log (n + m)-processor EREW PRAM. This algorithm is a natural extension of the classical method of fictitious play by Brown and Robinson. For any fixed accuracy [epsilon]> 0, the expected sequential running time of the suggested algorithm is O((n + m) log (n + m)), which is sublinear in mn, the number of input elements of A. On the other hand, simple arguments are given to show that for [epsilon] <1/2, any sequential deterministic algorithm for computing a pair of [epsilon]-optimal strategies of an (m, n)- matrix game A with [plus or minus] 1 elements examines [omega] (m, n) of its elements. In particular, for m = n the randomized algorithm achieves an almost quadratic expected speedup relative to any deterministic method."