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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.)
##### Keywords:
matrix games; ill-conditioned matrices; nonnegative inverse
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##### References:
 [1] Alon, N.; Vũ, V. H., Anti-Hadamard matrices, coin weighing, threshold gates, and indecomposable hypergraphs, Journal of Combinatorial Theory, Series A, 79, 133-160, (1997) · Zbl 0890.05011 [2] Babai, L.; Hansen, K. A.; Podolskii, V. V.; Sun, X., Weights of exact threshold functions, (Proceedings of the 35th International Symposium on Mathematical Foundations of Computer Science, MFCS 2010, Lecture Notes in Computer Science, vol. 6281, (2010), Springer), 66-77 · Zbl 1287.94129 [3] Ching, L., The maximum determinant of an $$n \times n$$ lower Hessenberg $$(0, 1)$$ matrix, Linear Algebra and its Applications, 183, 147-153, (1993) · Zbl 0769.15008 [4] Everett, H., Recursive games, (Contributions to the Theory of Games Vol. III, Ann. Math. Studies, vol. 39, (1957), Princeton University Press), 67-78 · Zbl 0078.32802 [5] Faddeev, D. K.; Sominskii, I. S., Problems in higher algebra, (1965), W. H. Freeman, translated by J.L. Brenner · Zbl 0125.00702 [6] Ferguson, T. S., Game theory, (2011) [7] Graham, R. L.; Knuth, D. E.; Patashnik, O., Concrete mathematics: A foundation for computer science, (1994), Addison-Wesley Longman Publishing Co., Inc. Boston, MA, USA · Zbl 0836.00001 [8] Graham, R. L.; Sloane, N. J.A., Anti-Hadamard matrices, Linear Algebra and its Applications, 62, 113-137, (1984) · Zbl 0552.15010 [9] Hansen, K. A.; Ibsen-Jensen, R.; Miltersen, P. B., The complexity of solving reachability games using value and strategy iteration, (6th Int. Comp. Sci. Symp. in Russia, CSR, LNCS, (2011), Springer), 77-90 · Zbl 1330.68112 [10] Hansen, K. A.; Koucký, M.; Lauritzen, N.; Miltersen, P. B.; Tsigaridas, E. P., Exact algorithms for solving stochastic games: extended abstract, (Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, (2011), ACM), 205-214 · Zbl 1288.68120 [11] K.A. Hansen, M. Koucký, P.B. Miltersen, Winning concurrent reachability games requires doubly exponential patience, in: Proc. of IEEE Symp. on Logic in Comp. Sci., LICS, pp. 332-341. [12] Håstad, J., On the size of weights for threshold gates, SIAM Journal on Discrete Mathematics, 7, 484-492, (1994) · Zbl 0811.68100 [13] Johnson, D. S.; Papadimitriou, C. H.; Yannakakis, M., How easy is local search?, Journal of Computer and System Sciences, 37, 79-100, (1988) · Zbl 0655.68074 [14] Krentel, M. W., Structure in locally optimal solutions (extended abstract), (30th Annual Symposium on Foundations of Computer Science, FOCS’89, (1989), IEEE Computer Society), 216-221 [15] Lipton, R. J.; Young, N. E., Simple strategies for large zero-sum games with applications to complexity theory, (Proceedings of the 26th Annual Symposium on the Theory of Computing, (1994), ACM Press), 734-740 · Zbl 1345.68175 [16] McCormick, S. T.; Rao, M.; Rinaldi, G., Easy and difficult objective functions for MAX cut, Mathematical Programming, 94, 459-466, (2003) · Zbl 1030.90130 [17] Podolskii, V. V., Perceptrons of large weight, Problems of Information Transmission, 45, 46-53, (2009) · Zbl 1171.68585 [18] Roth, R. M.; Viswanathan, K., On the hardness of decoding the gale-berlekamp code, IEEE Transactions on Information Theory, 54, 1050-1060, (2008) · Zbl 1311.94121 [19] Schäffer, A. A.; Yannakakis, M., Simple local search problems that are hard to solve, SIAM Journal on Computing, 20, 56-87, (1991) · Zbl 0716.68048 [20] Shapley, L. S.; Snow, R. N., Basic solutions of discrete games, (Contributions to the Theory of Games, Annals of Mathematics Studies, vol. 24, (1950), Princeton University Press), 27-35 · Zbl 0041.25403 [21] Sloane, N. J.A., Unsolved problems related to the covering radius of codes, (Cover, T. M.; Gopinath, B., Open Problems in Communication and Computation, (1987), Springer-Verlag), 51-56 [22] Spencer, J., Ten lectures on the probabilistic method, (1987), SIAM · Zbl 0703.05046 [23] von Neumann, J., Zur theorie der gesellschaftsspiele, Mathematische Annalen, 100, 295-320, (1928) · JFM 54.0543.02
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