Patience of matrix games

For matrix games we study how small nonzero probability must be used in optimal strategies. We show that for n×nn×n win–lose–draw games (i.e.  (−1,0,1)(−1,0,1) matrix games) nonzero probabilities smaller than n−O(n)n−O(n) are never needed. We also construct an explicit n×nn×n win–lose game such that the unique optimal strategy uses a nonzero probability as small as n−Ω(n)n−Ω(n). This is done by constructing an explicit (−1,1)(−1,1) nonsingular n×nn×n matrix, for which the inverse has only nonnegative entries and where some of the entries are of value nΩ(n)nΩ(n).