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

Ergodicity conditions for zero-sum games

Marianne Akian 1, , Stéphane Gaubert 1, and  Antoine Hochart 1,

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.
Keywords: fixed point, mean ergodic theorem, Shapley operators, dynamic programming, nonexpansive mappings, zero-sum games, mean payoff, directed hypergraphs., Nonlinear ergodic theorem, Galois connections, semigroups of contractions, Poisson equation.
Mathematics Subject Classification: Primary: 47H25; Secondary: 91A20, 05C65, 06A1.
Citation: Marianne Akian, Stéphane Gaubert, Antoine Hochart. Ergodicity conditions for zero-sum games. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. Birkhoff, Lattice Theory, vol. 25 of Colloquium publications, American Mathematical Society, Providence, 1995 (first edition, 1940). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics, 2014, To appear. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

B. Lemmens and R. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge university Press, 2012. doi: 10.1017/CBO9781139026079.  Google Scholar

[20]

J.-F. Mertens and A. Neyman, Stochastic games, Internat. J. Game Theory, 10 (1981), 53-66. doi: 10.1007/BF01769259.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

O. Ore, Galois connexions, Trans. Amer. Math. Soc., 55 (1944), 493-513. doi: 10.1090/S0002-9947-1944-0010555-7.  Google Scholar

[25]

J. Renault, Uniform value in dynamic programming, Journal of the European Mathematical Society, 13 (2011), 309-330. doi: 10.4171/JEMS/254.  Google Scholar

[26]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. Birkhoff, Lattice Theory, vol. 25 of Colloquium publications, American Mathematical Society, Providence, 1995 (first edition, 1940). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics, 2014, To appear. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

B. Lemmens and R. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge university Press, 2012. doi: 10.1017/CBO9781139026079.  Google Scholar

[20]

J.-F. Mertens and A. Neyman, Stochastic games, Internat. J. Game Theory, 10 (1981), 53-66. doi: 10.1007/BF01769259.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

O. Ore, Galois connexions, Trans. Amer. Math. Soc., 55 (1944), 493-513. doi: 10.1090/S0002-9947-1944-0010555-7.  Google Scholar

[25]

J. Renault, Uniform value in dynamic programming, Journal of the European Mathematical Society, 13 (2011), 309-330. doi: 10.4171/JEMS/254.  Google Scholar

[26]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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