# American Institute of Mathematical Sciences

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

## Reversibility and oscillations in zero-sum discounted stochastic games

 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

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).
Citation: Sylvain Sorin, Guillaume Vigeral. Reversibility and oscillations in zero-sum discounted stochastic games. Journal of Dynamics & 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. 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. 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. 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. 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), (2003), 397. 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. 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), (2003), 417. [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. 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. 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).

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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. 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. 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. 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. 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), (2003), 397. 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. 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), (2003), 417. [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. 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. 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).
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