Reversibility and oscillations in zero-sum discounted stochastic games

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January 2015, 2(1): 103-115. doi: 10.3934/jdg.2015.2.103

Reversibility and oscillations in zero-sum discounted stochastic games

Sylvain Sorin ^1, and Guillaume Vigeral ^2,

1.	Sorbonne Universités, UPMC Univ Paris 06, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Univ Paris Diderot, Sorbonne Paris Cité, F-75005, Paris, France
2.	Université Paris-Dauphine, CEREMADE, Place du Maréchal De Lattre de Tassigny, 75775 Paris cedex 16, France

Received November 2014 Revised January 2015 Published June 2015

Abstract
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We show that by coupling two well-behaved exit-time problems one can construct two-person zero-sum dynamic games having oscillating discounted values. This unifies and generalizes recent examples of stochastic games with finite state space, due to Vigeral (2013) and Ziliotto (2013).

Keywords: oscillations, reversibility., dynamic games, Zero-sum games, stochastic games.

Mathematics Subject Classification: Primary: 91A15, 91A05; Secondary: 91A2.

Citation: Sylvain Sorin, Guillaume Vigeral. Reversibility and oscillations in zero-sum discounted stochastic games. Journal of Dynamics and Games, 2015, 2 (1) : 103-115. doi: 10.3934/jdg.2015.2.103

References:

[1]	T. Bewley and E. Kohlberg, On stochastic games with stationary optimal strategies, Mathematics of Operations Research, 3 (1978), 104-125. doi: 10.1287/moor.3.2.104.
[2]	J. Bolte, S. Gaubert and G. Vigeral, Definable zero-sum stochastic games, Mathematics of Operation Research, 40 (2015), 171-191. doi: 10.1287/moor.2014.0666.
[3]	G. Grimmett and D. Stirzaker, Probability and Random Processes, Oxford University Press, 2001.
[4]	R. Laraki, Explicit formulas for repeated games with absorbing states, International Journal of Game Theory, 39 (2010), 53-69. doi: 10.1007/s00182-009-0193-2.
[5]	J.-F. Mertens, A. Neyman and D. Rosenberg, Absorbing games with compact action spaces, Mathematics of Operation Research, 34 (2009), 257-262. doi: 10.1287/moor.1080.0372.
[6]	J.-F. Mertens, S. Sorin and S. Zamir, Repeated Games, Cambridge University Press, 2015.
[7]	A. Neyman, Stochastic games and nonexpansive maps, in Stochastic Games and Applications (eds. A. Neyman and S. Sorin), NATO Sci. Ser. C Math. Phys. Sci., 570, Kluwer Academic Publishers, Dordrecht, 2003, 397-415. doi: 10.1007/978-94-010-0189-2_26.
[8]	D. Rosenberg and S. Sorin, An operator approach to zero-sum repeated games, Israel Journal of Mathematics,121 (2001), 221-246. doi: 10.1007/BF02802505.
[9]	S. Sorin, A First Course on Zero-SumRepeated Games, Springer-Verlag, 2002.
[10]	S. Sorin, The operator approach to zero-sum stochastic games, in Stochastic Games and Applications (eds. A. Neyman and S. Sorin), NATO Sci. Ser. C Math. Phys. Sci., 570, Kluwer Academic Publishers, Dordrecht, 2003, 417-426.
[11]	S. Sorin and G. Vigeral, Existence of the limit value of two person zero-sum discounted repeated games via comparison theorems, Journal of Opimization Theory and Applications, 157 (2013), 564-576. doi: 10.1007/s10957-012-0193-4.
[12]	G. Vigeral, A zero-sum stochastic game with compact action sets and no asymptotic value, Dynamic Games and Applications, 3 (2013), 172-186. doi: 10.1007/s13235-013-0073-z.
[13]	B. Ziliotto, Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture maxmin=$\lim v_n$, to appear in Annals of Probability, 2013. Available from: https://hal.archives-ouvertes.fr/hal-00824039.

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References:

[1]	T. Bewley and E. Kohlberg, On stochastic games with stationary optimal strategies, Mathematics of Operations Research, 3 (1978), 104-125. doi: 10.1287/moor.3.2.104.
[2]	J. Bolte, S. Gaubert and G. Vigeral, Definable zero-sum stochastic games, Mathematics of Operation Research, 40 (2015), 171-191. doi: 10.1287/moor.2014.0666.
[3]	G. Grimmett and D. Stirzaker, Probability and Random Processes, Oxford University Press, 2001.
[4]	R. Laraki, Explicit formulas for repeated games with absorbing states, International Journal of Game Theory, 39 (2010), 53-69. doi: 10.1007/s00182-009-0193-2.
[5]	J.-F. Mertens, A. Neyman and D. Rosenberg, Absorbing games with compact action spaces, Mathematics of Operation Research, 34 (2009), 257-262. doi: 10.1287/moor.1080.0372.
[6]	J.-F. Mertens, S. Sorin and S. Zamir, Repeated Games, Cambridge University Press, 2015.
[7]	A. Neyman, Stochastic games and nonexpansive maps, in Stochastic Games and Applications (eds. A. Neyman and S. Sorin), NATO Sci. Ser. C Math. Phys. Sci., 570, Kluwer Academic Publishers, Dordrecht, 2003, 397-415. doi: 10.1007/978-94-010-0189-2_26.
[8]	D. Rosenberg and S. Sorin, An operator approach to zero-sum repeated games, Israel Journal of Mathematics,121 (2001), 221-246. doi: 10.1007/BF02802505.
[9]	S. Sorin, A First Course on Zero-SumRepeated Games, Springer-Verlag, 2002.
[10]	S. Sorin, The operator approach to zero-sum stochastic games, in Stochastic Games and Applications (eds. A. Neyman and S. Sorin), NATO Sci. Ser. C Math. Phys. Sci., 570, Kluwer Academic Publishers, Dordrecht, 2003, 417-426.
[11]	S. Sorin and G. Vigeral, Existence of the limit value of two person zero-sum discounted repeated games via comparison theorems, Journal of Opimization Theory and Applications, 157 (2013), 564-576. doi: 10.1007/s10957-012-0193-4.
[12]	G. Vigeral, A zero-sum stochastic game with compact action sets and no asymptotic value, Dynamic Games and Applications, 3 (2013), 172-186. doi: 10.1007/s13235-013-0073-z.
[13]	B. Ziliotto, Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture maxmin=$\lim v_n$, to appear in Annals of Probability, 2013. Available from: https://hal.archives-ouvertes.fr/hal-00824039.

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