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January  2019, 6(1): 27-51. doi: 10.3934/jdg.2019003

An accretive operator approach to ergodic zero-sum stochastic games

Universidad Adolfo Ibáñez, Santiago, Chile

Received  August 2017 Revised  July 2018 Published  January 2019

Fund Project: The author is funded by FONDECYT grant 3180662.

We study some ergodicity property of zero-sum stochastic games with a finite state space and possibly unbounded payoffs. We formulate this property in operator-theoretical terms, involving the solvability of an optimality equation for the Shapley operators (i.e., the dynamic programming operators) of a family of perturbed games. The solvability of this equation entails the existence of the uniform value, and its solutions yield uniform optimal stationary strategies. We first provide an analytical characterization of this ergodicity property, and address the generic uniqueness, up to an additive constant, of the solutions of the optimality equation. Our analysis relies on the theory of accretive mappings, which we apply to maps of the form $Id - T$ where $T$ is nonexpansive. Then, we use the results of a companion work to characterize the ergodicity of stochastic games by a geometrical condition imposed on the transition probabilities. This condition generalizes classical notion of ergodicity for finite Markov chains and Markov decision processes.

Citation: Antoine Hochart. An accretive operator approach to ergodic zero-sum stochastic games. Journal of Dynamics & Games, 2019, 6 (1) : 27-51. doi: 10.3934/jdg.2019003
References:
[1]

M. AkianS. Gaubert and A. Hochart, Ergodicity conditions for zero-sum games, Discrete Contin. Dyn. Syst., 35 (2015), 3901-3931.  doi: 10.3934/dcds.2015.35.3901.  Google Scholar

[2]

M. Akian, S. Gaubert and A. Hochart, Hypergraph conditions for the solvability of the ergodic equation for zero-sum games, in 54th IEEE Conference on Decision and Control, Osaka, Japan, 2015, 5845-5850, arXiv: 1510.05396. doi: 10.3934/dcds.2015.35.3901.  Google Scholar

[3]

M. Akian, S. Gaubert and A. Hochart, A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors, 2018, Preprint, arXiv: 1812.09871. Google Scholar

[4]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edition, Springer, Berlin, 2006.  Google Scholar

[5]

E. AltmanA. Hordijk and F. M. Spieksma, Contraction conditions for average and $α$-discount optimality in countable state Markov games with unbounded rewards, Math. Oper. Res., 22 (1997), 588-618.  doi: 10.1287/moor.22.3.588.  Google Scholar

[6]

E. Asplund, Positivity of duality mappings, Bull. Amer. Math. Soc., 73 (1967), 200-203.  doi: 10.1090/S0002-9904-1967-11678-1.  Google Scholar

[7]

J.-P. Aubin and H. Frankowska, Set-valued Analysis, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009, Reprint of the 1990 edition. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[8]

J. Bather, Optimal decision procedures for finite Markov chains. Ⅱ. Communicating systems, Advances in Appl. Probability, 5 (1973), 521-540.  doi: 10.2307/1425832.  Google Scholar

[9]

T. Bewley and E. Kohlberg, The asymptotic theory of stochastic games, Math. Oper. Res., 1 (1976), 197-208.  doi: 10.1287/moor.1.3.197.  Google Scholar

[10]

J. BolteS. Gaubert and G. Vigeral, Definable zero-sum stochastic games, Math. Oper. Res., 40 (2015), 171-191.  doi: 10.1287/moor.2014.0666.  Google Scholar

[11]

V. S. Borkar and M. K. Ghosh, Denumerable state stochastic games with limiting average payoff, J. Optim. Theory Appl., 76 (1993), 539-560.  doi: 10.1007/BF00939382.  Google Scholar

[12]

E. Boros, K. Elbassioni, V. Gurvich and K. Makino, A pumping algorithm for ergodic stochastic mean payoff games with perfect information, in Integer Programming and Combinatorial Optimization, vol. 6080 of Lecture Notes in Comput. Sci., Springer, Berlin, 2010, 341-354. doi: 10.1007/978-3-642-13036-6_26.  Google Scholar

[13]

F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), Amer. Math. Soc., Providence, R. I., 1976, 1-308.  Google Scholar

[14]

I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1990. doi: 10.1007/978-94-009-2121-4.  Google Scholar

[15]

M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), 385-390.  doi: 10.1090/S0002-9939-1980-0553381-X.  Google Scholar

[16]

H. Everett, Recursive games, in Contributions to the Theory of Games, Annals of Mathematics Studies, no. 39, Princeton University Press, Princeton, N. J., 3 (1957), 47-78.  Google Scholar

[17]

P. M. FitzpatrickP. Hess and T. Kato, Local boundedness of monotone-type operators, Proc. Japan Acad., 48 (1972), 275-277.  doi: 10.3792/pja/1195519662.  Google Scholar

[18]

S. Gaubert and J. Gunawardena, The Perron-Frobenius theorem for homogeneous, monotone functions, Trans. Amer. Math. Soc., 356 (2004), 4931-4950 (electronic).  doi: 10.1090/S0002-9947-04-03470-1.  Google Scholar

[19]

M. K. Ghosh and A. Bagchi, Stochastic games with average payoff criterion, Appl. Math. Optim., 38 (1998), 283-301.  doi: 10.1007/s002459900092.  Google Scholar

[20]

V. A. Gurvich and V. N. Lebedev, A criterion and verification of the ergodicity of cyclic game forms, Uspekhi Mat. Nauk, 44 (1989), 193-194.  doi: 10.1070/RM1989v044n01ABEH002010.  Google Scholar

[21]

O. Hernández-Lerma and J. B. Lasserre, Zero-sum stochastic games in Borel spaces: Average payoff criteria, SIAM J. Control Optim., 39 (2000), 1520-1539.  doi: 10.1137/S0363012999361962.  Google Scholar

[22]

A. Hochart, An accretive operator approach to ergodic problems for zero-sum games, in 22nd International Symposium on Mathematical Theory of Networks and Systems, Minneapolis, Minnesota, USA, 2016,315-318, arXiv: 1605.04520, hdl: 11299/181518. Google Scholar

[23]

A. J. Hoffman and R. M. Karp, On nonterminating stochastic games, Management Sci., 12 (1966), 359-370.  doi: 10.1287/mnsc.12.5.359.  Google Scholar

[24]

A. Jaśkiewicz and A. S. Nowak, On the optimality equation for zero-sum ergodic stochastic games, Math. Methods Oper. Res., 54 (2001), 291-301.  doi: 10.1007/s001860100144.  Google Scholar

[25]

M. JurdzińskiM. Paterson and U. Zwick, A deterministic subexponential algorithm for solving parity games, SIAM J. Comput., 38 (2008), 1519-1532.  doi: 10.1137/070686652.  Google Scholar

[26]

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

[27]

W. A. Kirk and R. Schöneberg, Zeros of $m$-accretive operators in Banach spaces, Israel J. Math., 35 (1980), 1-8.  doi: 10.1007/BF02760935.  Google Scholar

[28]

E. Kohlberg, Repeated games with absorbing states, Ann. Statist., 2 (1974), 724-738.  doi: 10.1214/aos/1176342760.  Google Scholar

[29]

H.-U. Küenle, On multichain Markov games, in Advances in Dynamic Games and Applications (Maastricht, 1998), vol. 6 of Ann. Internat. Soc. Dynam. Games, Birkhäuser Boston, Boston, MA, 2001,147-163.  Google Scholar

[30]

B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, vol. 189 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139026079.  Google Scholar

[31]

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

[32]

J.-F. MertensA. Neyman and D. Rosenberg, Absorbing games with compact action spaces, Math. Oper. Res., 34 (2009), 257-262.  doi: 10.1287/moor.1080.0372.  Google Scholar

[33]

J.-F. Mertens, S. Sorin and S. Zamir, Repeated Games, Econometric Society Monographs, Cambridge University Press, New York, 2015, With a foreword by Robert J. Aumann. doi: 10.1017/CBO9781139343275.  Google Scholar

[34]

A. Neyman, Stochastic games and nonexpansive maps, in Stochastic Games and Applications (Stony Brook, NY, 1999), vol. 570 of NATO Sci. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 2003, 397-415. doi: 10.1007/978-94-010-0189-2_26.  Google Scholar

[35]

A. Neyman and S. Sorin, Stochastic Games and Applications, vol. 570 of Nato Science Series C, Springer Netherlands, 2003. Google Scholar

[36]

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

[37]

W. V. Petryshyn, A characterization of strict convexity of Banach spaces and other uses of duality mappings, J. Functional Analysis, 6 (1970), 282-291.  doi: 10.1016/0022-1236(70)90061-3.  Google Scholar

[38]

S. Reich, Approximating fixed points of nonexpansive mappings, Panamer. Math. J., 4 (1994), 23-28.   Google Scholar

[39]

J. Renault, Uniform value in dynamic programming, J. Eur. Math. Soc. (JEMS), 13 (2011), 309-330.  doi: 10.4171/JEMS/254.  Google Scholar

[40]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[41]

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

[42]

L. I. Sennott, Zero-sum stochastic games with unbounded costs: Discounted and average cost cases, Z. Oper. Res., 39 (1993), 209-225.  doi: 10.1007/BF01415582.  Google Scholar

[43]

L. S. Shapley, Stochastic games, Proc. Nat. Acad. Sci. U. S. A., 39 (1953), 1095-1100.  doi: 10.1073/pnas.39.10.1953.  Google Scholar

[44]

S. Sorin, The operator approach to zero-sum stochastic games, in Stochastic Games and Applications (Stony Brook, NY, 1999), vol. 570 of NATO Sci. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 2003, 417-426.  Google Scholar

[45]

S. Sorin, Asymptotic properties of monotonic nonexpansive mappings, Discrete Event Dyn. Syst., 14 (2004), 109-122.  doi: 10.1023/B:DISC.0000005011.93152.d8.  Google Scholar

[46]

S. Sorin and G. Vigeral, Operator approach to values of stochastic games with varying stage duration, Internat. J. Game Theory, 45 (2016), 389-410.  doi: 10.1007/s00182-015-0512-8.  Google Scholar

[47]

G. Vigeral, Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces, ESAIM Control Optim. Calc. Var., 16 (2010), 809-832.  doi: 10.1051/cocv/2009026.  Google Scholar

[48]

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

[49]

O. J. Vrieze, Stochastic games and stationary strategies, in Stochastic Games and Applications (Stony Brook, NY, 1999), vol. 570 of NATO Sci. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 2003, 37-50.  Google Scholar

[50]

B. Ziliotto, A tauberian theorem for nonexpansive operators and applications to zero-sum stochastic games, Math. Oper. Res., 41 (2016), 1522-1534.  doi: 10.1287/moor.2016.0788.  Google Scholar

[51]

B. Ziliotto, Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture ${\text{maxmin}}=\lim \ {{v}_{n}}$, Ann. Probab., 44 (2016), 1107-1133.  doi: 10.1214/14-AOP997.  Google Scholar

show all references

References:
[1]

M. AkianS. Gaubert and A. Hochart, Ergodicity conditions for zero-sum games, Discrete Contin. Dyn. Syst., 35 (2015), 3901-3931.  doi: 10.3934/dcds.2015.35.3901.  Google Scholar

[2]

M. Akian, S. Gaubert and A. Hochart, Hypergraph conditions for the solvability of the ergodic equation for zero-sum games, in 54th IEEE Conference on Decision and Control, Osaka, Japan, 2015, 5845-5850, arXiv: 1510.05396. doi: 10.3934/dcds.2015.35.3901.  Google Scholar

[3]

M. Akian, S. Gaubert and A. Hochart, A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors, 2018, Preprint, arXiv: 1812.09871. Google Scholar

[4]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edition, Springer, Berlin, 2006.  Google Scholar

[5]

E. AltmanA. Hordijk and F. M. Spieksma, Contraction conditions for average and $α$-discount optimality in countable state Markov games with unbounded rewards, Math. Oper. Res., 22 (1997), 588-618.  doi: 10.1287/moor.22.3.588.  Google Scholar

[6]

E. Asplund, Positivity of duality mappings, Bull. Amer. Math. Soc., 73 (1967), 200-203.  doi: 10.1090/S0002-9904-1967-11678-1.  Google Scholar

[7]

J.-P. Aubin and H. Frankowska, Set-valued Analysis, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009, Reprint of the 1990 edition. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[8]

J. Bather, Optimal decision procedures for finite Markov chains. Ⅱ. Communicating systems, Advances in Appl. Probability, 5 (1973), 521-540.  doi: 10.2307/1425832.  Google Scholar

[9]

T. Bewley and E. Kohlberg, The asymptotic theory of stochastic games, Math. Oper. Res., 1 (1976), 197-208.  doi: 10.1287/moor.1.3.197.  Google Scholar

[10]

J. BolteS. Gaubert and G. Vigeral, Definable zero-sum stochastic games, Math. Oper. Res., 40 (2015), 171-191.  doi: 10.1287/moor.2014.0666.  Google Scholar

[11]

V. S. Borkar and M. K. Ghosh, Denumerable state stochastic games with limiting average payoff, J. Optim. Theory Appl., 76 (1993), 539-560.  doi: 10.1007/BF00939382.  Google Scholar

[12]

E. Boros, K. Elbassioni, V. Gurvich and K. Makino, A pumping algorithm for ergodic stochastic mean payoff games with perfect information, in Integer Programming and Combinatorial Optimization, vol. 6080 of Lecture Notes in Comput. Sci., Springer, Berlin, 2010, 341-354. doi: 10.1007/978-3-642-13036-6_26.  Google Scholar

[13]

F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), Amer. Math. Soc., Providence, R. I., 1976, 1-308.  Google Scholar

[14]

I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1990. doi: 10.1007/978-94-009-2121-4.  Google Scholar

[15]

M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), 385-390.  doi: 10.1090/S0002-9939-1980-0553381-X.  Google Scholar

[16]

H. Everett, Recursive games, in Contributions to the Theory of Games, Annals of Mathematics Studies, no. 39, Princeton University Press, Princeton, N. J., 3 (1957), 47-78.  Google Scholar

[17]

P. M. FitzpatrickP. Hess and T. Kato, Local boundedness of monotone-type operators, Proc. Japan Acad., 48 (1972), 275-277.  doi: 10.3792/pja/1195519662.  Google Scholar

[18]

S. Gaubert and J. Gunawardena, The Perron-Frobenius theorem for homogeneous, monotone functions, Trans. Amer. Math. Soc., 356 (2004), 4931-4950 (electronic).  doi: 10.1090/S0002-9947-04-03470-1.  Google Scholar

[19]

M. K. Ghosh and A. Bagchi, Stochastic games with average payoff criterion, Appl. Math. Optim., 38 (1998), 283-301.  doi: 10.1007/s002459900092.  Google Scholar

[20]

V. A. Gurvich and V. N. Lebedev, A criterion and verification of the ergodicity of cyclic game forms, Uspekhi Mat. Nauk, 44 (1989), 193-194.  doi: 10.1070/RM1989v044n01ABEH002010.  Google Scholar

[21]

O. Hernández-Lerma and J. B. Lasserre, Zero-sum stochastic games in Borel spaces: Average payoff criteria, SIAM J. Control Optim., 39 (2000), 1520-1539.  doi: 10.1137/S0363012999361962.  Google Scholar

[22]

A. Hochart, An accretive operator approach to ergodic problems for zero-sum games, in 22nd International Symposium on Mathematical Theory of Networks and Systems, Minneapolis, Minnesota, USA, 2016,315-318, arXiv: 1605.04520, hdl: 11299/181518. Google Scholar

[23]

A. J. Hoffman and R. M. Karp, On nonterminating stochastic games, Management Sci., 12 (1966), 359-370.  doi: 10.1287/mnsc.12.5.359.  Google Scholar

[24]

A. Jaśkiewicz and A. S. Nowak, On the optimality equation for zero-sum ergodic stochastic games, Math. Methods Oper. Res., 54 (2001), 291-301.  doi: 10.1007/s001860100144.  Google Scholar

[25]

M. JurdzińskiM. Paterson and U. Zwick, A deterministic subexponential algorithm for solving parity games, SIAM J. Comput., 38 (2008), 1519-1532.  doi: 10.1137/070686652.  Google Scholar

[26]

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

[27]

W. A. Kirk and R. Schöneberg, Zeros of $m$-accretive operators in Banach spaces, Israel J. Math., 35 (1980), 1-8.  doi: 10.1007/BF02760935.  Google Scholar

[28]

E. Kohlberg, Repeated games with absorbing states, Ann. Statist., 2 (1974), 724-738.  doi: 10.1214/aos/1176342760.  Google Scholar

[29]

H.-U. Küenle, On multichain Markov games, in Advances in Dynamic Games and Applications (Maastricht, 1998), vol. 6 of Ann. Internat. Soc. Dynam. Games, Birkhäuser Boston, Boston, MA, 2001,147-163.  Google Scholar

[30]

B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, vol. 189 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139026079.  Google Scholar

[31]

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

[32]

J.-F. MertensA. Neyman and D. Rosenberg, Absorbing games with compact action spaces, Math. Oper. Res., 34 (2009), 257-262.  doi: 10.1287/moor.1080.0372.  Google Scholar

[33]

J.-F. Mertens, S. Sorin and S. Zamir, Repeated Games, Econometric Society Monographs, Cambridge University Press, New York, 2015, With a foreword by Robert J. Aumann. doi: 10.1017/CBO9781139343275.  Google Scholar

[34]

A. Neyman, Stochastic games and nonexpansive maps, in Stochastic Games and Applications (Stony Brook, NY, 1999), vol. 570 of NATO Sci. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 2003, 397-415. doi: 10.1007/978-94-010-0189-2_26.  Google Scholar

[35]

A. Neyman and S. Sorin, Stochastic Games and Applications, vol. 570 of Nato Science Series C, Springer Netherlands, 2003. Google Scholar

[36]

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

[37]

W. V. Petryshyn, A characterization of strict convexity of Banach spaces and other uses of duality mappings, J. Functional Analysis, 6 (1970), 282-291.  doi: 10.1016/0022-1236(70)90061-3.  Google Scholar

[38]

S. Reich, Approximating fixed points of nonexpansive mappings, Panamer. Math. J., 4 (1994), 23-28.   Google Scholar

[39]

J. Renault, Uniform value in dynamic programming, J. Eur. Math. Soc. (JEMS), 13 (2011), 309-330.  doi: 10.4171/JEMS/254.  Google Scholar

[40]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[41]

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

[42]

L. I. Sennott, Zero-sum stochastic games with unbounded costs: Discounted and average cost cases, Z. Oper. Res., 39 (1993), 209-225.  doi: 10.1007/BF01415582.  Google Scholar

[43]

L. S. Shapley, Stochastic games, Proc. Nat. Acad. Sci. U. S. A., 39 (1953), 1095-1100.  doi: 10.1073/pnas.39.10.1953.  Google Scholar

[44]

S. Sorin, The operator approach to zero-sum stochastic games, in Stochastic Games and Applications (Stony Brook, NY, 1999), vol. 570 of NATO Sci. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 2003, 417-426.  Google Scholar

[45]

S. Sorin, Asymptotic properties of monotonic nonexpansive mappings, Discrete Event Dyn. Syst., 14 (2004), 109-122.  doi: 10.1023/B:DISC.0000005011.93152.d8.  Google Scholar

[46]

S. Sorin and G. Vigeral, Operator approach to values of stochastic games with varying stage duration, Internat. J. Game Theory, 45 (2016), 389-410.  doi: 10.1007/s00182-015-0512-8.  Google Scholar

[47]

G. Vigeral, Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces, ESAIM Control Optim. Calc. Var., 16 (2010), 809-832.  doi: 10.1051/cocv/2009026.  Google Scholar

[48]

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

[49]

O. J. Vrieze, Stochastic games and stationary strategies, in Stochastic Games and Applications (Stony Brook, NY, 1999), vol. 570 of NATO Sci. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 2003, 37-50.  Google Scholar

[50]

B. Ziliotto, A tauberian theorem for nonexpansive operators and applications to zero-sum stochastic games, Math. Oper. Res., 41 (2016), 1522-1534.  doi: 10.1287/moor.2016.0788.  Google Scholar

[51]

B. Ziliotto, Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture ${\text{maxmin}}=\lim \ {{v}_{n}}$, Ann. Probab., 44 (2016), 1107-1133.  doi: 10.1214/14-AOP997.  Google Scholar

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