Patience of Matrix Games

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