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Patience of matrix games. (English) Zbl 1285.91005
Summary: 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 \(n^{\Omega (n)}\).
MSC:
91A05 2-person games
91A60 Probabilistic games; gambling
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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