Patience of matrix games

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