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September  2015, 35(9): 3901-3931. doi: 10.3934/dcds.2015.35.3901

## Ergodicity conditions for zero-sum games

 1 INRIA and CMAP, Ecole polytechnique, CNRS, CMAP, Ecole polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France, France, France

Received  May 2014 Revised  December 2014 Published  April 2015

A basic question for zero-sum repeated games consists in determining whether the mean payoff per time unit is independent of the initial state. In the special case of zero-player'' games, i.e., of Markov chains equipped with additive functionals, the answer is provided by the mean ergodic theorem. We generalize this result to repeated games. We show that the mean payoff is independent of the initial state for all state-dependent perturbations of the rewards if and only if an ergodicity condition is verified. The latter is characterized by the uniqueness modulo constants of nonlinear harmonic functions (fixed points of the recession function associated to the Shapley operator), or, in the special case of stochastic games with finite action spaces and perfect information, by a reachability condition involving conjugate subsets of states in directed hypergraphs. We show that the ergodicity condition for games only depends on the support of the transition probability, and that it can be checked in polynomial time when the number of states is fixed.
Citation: Marianne Akian, Stéphane Gaubert, Antoine Hochart. Ergodicity conditions for zero-sum games. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3901-3931. doi: 10.3934/dcds.2015.35.3901
##### References:
 [1] M. Akian and S. Gaubert, Spectral theorem for convex monotone homogeneous maps and ergodic control, Nonlinear Analysis, T.M.A., 52 (2003), 637-679. doi: 10.1016/S0362-546X(02)00170-0. [2] M. Akian, S. Gaubert and R. Nussbaum, Uniqueness of the fixed point of nonexpansive semidifferentiable maps,, Trans. Amer. Math. Soc., ().  doi: 10.1090/S0002-9947-2015-06413-7. [3] M. Akian, S. Gaubert and C. Walsh, The max-plus Martin boundary, Documenta Mathematica, 14 (2009), 195-240, URL http://www.math.uni-bielefeld.de/documenta/vol-14/09.html. [4] M. Akian, S. Gaubert and A. Hochart, Fixed point sets of payment-free shapley operators and structural properties of mean payoff games, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, July 7-11, Groningen, The Netherlands, (2014), 1438-1441. [5] X. Allamigeon, On the complexity of strongly connected components in directed hypergraphs, Algorithmica, 69 (2014), 335-369. doi: 10.1007/s00453-012-9729-0. [6] F. Baccelli, G. Cohen, G. Olsder and J.-P. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, Wiley Series in Probability and Mathematical Statistics, John Wiley, 1992. [7] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, vol. 9 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994, Revised reprint of the 1979 original. doi: 10.1137/1.9781611971262. [8] G. Birkhoff, Lattice Theory, vol. 25 of Colloquium publications, American Mathematical Society, Providence, 1995 (first edition, 1940). [9] J. Bolte, S. Gaubert and G. Vigeral, Definable zero-sum stochastic games, Mathematics of Operations Research, 40 (2014), 171-191, URL http://arxiv.org/abs/1301.1967, Published online. doi: 10.1287/moor.2014.0666. [10] R. Cavazos-Cadena and D. Hernández-Hernández, Poisson equations associated with a homogeneous and monotone function: necessary and sufficient conditions for a solution in a weakly convex case, Nonlinear Anal., 72 (2010), 3303-3313. doi: 10.1016/j.na.2009.12.010. [11] H. Everett, Recursive games, in Contributions to the theory of games Vol. III, vol. 39 of Ann. Math. Studies, Princeton University Press, (1957), 47-78. [12] A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228. doi: 10.1007/s00526-004-0271-z. [13] A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics, 2014, To appear. [14] G. Gallo, G. Longo, S. Nguyen and S. Pallottino, Directed hypergraphs and applications, Discrete Appl. Math., 42 (1993), 177-201. doi: 10.1016/0166-218X(93)90045-P. [15] S. Gaubert and J. Gunawardena, The Perron-Frobenius theorem for homogeneous, monotone functions, Trans. Amer. Math. Soc., 356 (2004), 4931-4950. doi: 10.1090/S0002-9947-04-03470-1. [16] O. Hernández-Lerma and J. B. Lasserre, Further Topics on Discrete-Time Markov Control Processes, vol. 42 of Applications of Mathematics (New York), Springer-Verlag, New York, 1999, URL http://dx.doi.org/10.1007/978-1-4612-0561-6. doi: 10.1007/978-1-4612-0561-6. [17] J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer-Verlag, New York-Heidelberg, 1976, Reprinting of the 1960 original, Undergraduate Texts in Mathematics. [18] E. Kohlberg and A. Neyman, Asymptotic behavior of nonexpansive mappings in normed linear spaces, Israel J. Math., 38 (1981), 269-275. doi: 10.1007/BF02762772. [19] B. Lemmens and R. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge university Press, 2012. doi: 10.1017/CBO9781139026079. [20] J.-F. Mertens and A. Neyman, Stochastic games, Internat. J. Game Theory, 10 (1981), 53-66. doi: 10.1007/BF01769259. [21] A. Neyman and S. Sorin, Stochastic Games and Applications, vol. 570 of Nato Science Series C, Kluwer Academic Publishers, 2003. doi: 10.1007/978-94-010-0189-2. [22] R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc., 75 (1988), iv+137pp. doi: 10.1090/memo/0391. [23] R. D. Nussbaum, Iterated nonlinear maps and Hilbert's projective metric. II, Mem. Amer. Math. Soc., 79 (1989), iv+118pp. doi: 10.1090/memo/0401. [24] O. Ore, Galois connexions, Trans. Amer. Math. Soc., 55 (1944), 493-513. doi: 10.1090/S0002-9947-1944-0010555-7. [25] J. Renault, Uniform value in dynamic programming, Journal of the European Mathematical Society, 13 (2011), 309-330. doi: 10.4171/JEMS/254. [26] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, 1998. doi: 10.1007/978-3-642-02431-3. [27] D. Rosenberg and S. Sorin, An operator approach to zero-sum repeated games, Israel J. Math., 121 (2001), 221-246. doi: 10.1007/BF02802505. [28] S. Sorin, Asymptotic properties of monotonic nonexpansive mappings, Discrete Event Dynamic Systems, 14 (2004), 109-122. doi: 10.1023/B:DISC.0000005011.93152.d8. [29] G. Vigeral, A zero-zum stochastic game with compact action sets and no asymptotic value, Dyn. Games Appl., 3 (2013), 172-186. doi: 10.1007/s13235-013-0073-z. [30] P. Whittle, Optimization Over Time. Vol. II, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1983, Dynamic programming and stochastic control. [31] K. Yang and Q. Zhao, The balance problem of min-max systems is co-NP hard, Systems & Control Letters, 53 (2004), 303-310. doi: 10.1016/j.sysconle.2004.05.009.

show all references

##### References:
 [1] M. Akian and S. Gaubert, Spectral theorem for convex monotone homogeneous maps and ergodic control, Nonlinear Analysis, T.M.A., 52 (2003), 637-679. doi: 10.1016/S0362-546X(02)00170-0. [2] M. Akian, S. Gaubert and R. Nussbaum, Uniqueness of the fixed point of nonexpansive semidifferentiable maps,, Trans. Amer. Math. Soc., ().  doi: 10.1090/S0002-9947-2015-06413-7. [3] M. Akian, S. Gaubert and C. Walsh, The max-plus Martin boundary, Documenta Mathematica, 14 (2009), 195-240, URL http://www.math.uni-bielefeld.de/documenta/vol-14/09.html. [4] M. Akian, S. Gaubert and A. Hochart, Fixed point sets of payment-free shapley operators and structural properties of mean payoff games, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, July 7-11, Groningen, The Netherlands, (2014), 1438-1441. [5] X. Allamigeon, On the complexity of strongly connected components in directed hypergraphs, Algorithmica, 69 (2014), 335-369. doi: 10.1007/s00453-012-9729-0. [6] F. Baccelli, G. Cohen, G. Olsder and J.-P. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, Wiley Series in Probability and Mathematical Statistics, John Wiley, 1992. [7] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, vol. 9 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994, Revised reprint of the 1979 original. doi: 10.1137/1.9781611971262. [8] G. Birkhoff, Lattice Theory, vol. 25 of Colloquium publications, American Mathematical Society, Providence, 1995 (first edition, 1940). [9] J. Bolte, S. Gaubert and G. Vigeral, Definable zero-sum stochastic games, Mathematics of Operations Research, 40 (2014), 171-191, URL http://arxiv.org/abs/1301.1967, Published online. doi: 10.1287/moor.2014.0666. [10] R. Cavazos-Cadena and D. Hernández-Hernández, Poisson equations associated with a homogeneous and monotone function: necessary and sufficient conditions for a solution in a weakly convex case, Nonlinear Anal., 72 (2010), 3303-3313. doi: 10.1016/j.na.2009.12.010. [11] H. Everett, Recursive games, in Contributions to the theory of games Vol. III, vol. 39 of Ann. Math. Studies, Princeton University Press, (1957), 47-78. [12] A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228. doi: 10.1007/s00526-004-0271-z. [13] A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics, 2014, To appear. [14] G. Gallo, G. Longo, S. Nguyen and S. Pallottino, Directed hypergraphs and applications, Discrete Appl. Math., 42 (1993), 177-201. doi: 10.1016/0166-218X(93)90045-P. [15] S. Gaubert and J. Gunawardena, The Perron-Frobenius theorem for homogeneous, monotone functions, Trans. Amer. Math. Soc., 356 (2004), 4931-4950. doi: 10.1090/S0002-9947-04-03470-1. [16] O. Hernández-Lerma and J. B. Lasserre, Further Topics on Discrete-Time Markov Control Processes, vol. 42 of Applications of Mathematics (New York), Springer-Verlag, New York, 1999, URL http://dx.doi.org/10.1007/978-1-4612-0561-6. doi: 10.1007/978-1-4612-0561-6. [17] J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer-Verlag, New York-Heidelberg, 1976, Reprinting of the 1960 original, Undergraduate Texts in Mathematics. [18] E. Kohlberg and A. Neyman, Asymptotic behavior of nonexpansive mappings in normed linear spaces, Israel J. Math., 38 (1981), 269-275. doi: 10.1007/BF02762772. [19] B. Lemmens and R. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge university Press, 2012. doi: 10.1017/CBO9781139026079. [20] J.-F. Mertens and A. Neyman, Stochastic games, Internat. J. Game Theory, 10 (1981), 53-66. doi: 10.1007/BF01769259. [21] A. Neyman and S. Sorin, Stochastic Games and Applications, vol. 570 of Nato Science Series C, Kluwer Academic Publishers, 2003. doi: 10.1007/978-94-010-0189-2. [22] R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc., 75 (1988), iv+137pp. doi: 10.1090/memo/0391. [23] R. D. Nussbaum, Iterated nonlinear maps and Hilbert's projective metric. II, Mem. Amer. Math. Soc., 79 (1989), iv+118pp. doi: 10.1090/memo/0401. [24] O. Ore, Galois connexions, Trans. Amer. Math. Soc., 55 (1944), 493-513. doi: 10.1090/S0002-9947-1944-0010555-7. [25] J. Renault, Uniform value in dynamic programming, Journal of the European Mathematical Society, 13 (2011), 309-330. doi: 10.4171/JEMS/254. [26] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, 1998. doi: 10.1007/978-3-642-02431-3. [27] D. Rosenberg and S. Sorin, An operator approach to zero-sum repeated games, Israel J. Math., 121 (2001), 221-246. doi: 10.1007/BF02802505. [28] S. Sorin, Asymptotic properties of monotonic nonexpansive mappings, Discrete Event Dynamic Systems, 14 (2004), 109-122. doi: 10.1023/B:DISC.0000005011.93152.d8. [29] G. Vigeral, A zero-zum stochastic game with compact action sets and no asymptotic value, Dyn. Games Appl., 3 (2013), 172-186. doi: 10.1007/s13235-013-0073-z. [30] P. Whittle, Optimization Over Time. Vol. II, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1983, Dynamic programming and stochastic control. [31] K. Yang and Q. Zhao, The balance problem of min-max systems is co-NP hard, Systems & Control Letters, 53 (2004), 303-310. doi: 10.1016/j.sysconle.2004.05.009.
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