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 $n^{\Omega(n)}$.