American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

G. Grimmett and D. Stirzaker, Probability and Random Processes, Oxford University Press, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J.-F. Mertens, S. Sorin and S. Zamir, Repeated Games, Cambridge University Press, 2015. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Sorin, A First Course on Zero-SumRepeated Games, Springer-Verlag, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

G. Grimmett and D. Stirzaker, Probability and Random Processes, Oxford University Press, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J.-F. Mertens, S. Sorin and S. Zamir, Repeated Games, Cambridge University Press, 2015. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Sorin, A First Course on Zero-SumRepeated Games, Springer-Verlag, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[1]

Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics & Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020
[2]

Alexander J. Zaslavski. Structure of approximate solutions of dynamic continuous time zero-sum games. Journal of Dynamics & Games, 2014, 1 (1) : 153-179. doi: 10.3934/jdg.2014.1.153
[3]

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
[4]

Qingmeng Wei, Zhiyong Yu. Time-inconsistent recursive zero-sum stochastic differential games. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1051-1079. doi: 10.3934/mcrf.2018045
[5]

Beatris A. Escobedo-Trujillo. Discount-sensitive equilibria in zero-sum stochastic differential games. Journal of Dynamics & Games, 2016, 3 (1) : 25-50. doi: 10.3934/jdg.2016002
[6]

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
[7]

Alexander J. Zaslavski. Turnpike properties of approximate solutions of dynamic discrete time zero-sum games. Journal of Dynamics & Games, 2014, 1 (2) : 299-330. doi: 10.3934/jdg.2014.1.299
[8]

Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial & Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95
[9]

Fernando Luque-Vásquez, J. Adolfo Minjárez-Sosa. Average optimal strategies for zero-sum Markov games with poorly known payoff function on one side. Journal of Dynamics & Games, 2014, 1 (1) : 105-119. doi: 10.3934/jdg.2014.1.105
[10]

Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control & Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026
[11]

Fabien Gensbittel, Miquel Oliu-Barton, Xavier Venel. Existence of the uniform value in zero-sum repeated games with a more informed controller. Journal of Dynamics & Games, 2014, 1 (3) : 411-445. doi: 10.3934/jdg.2014.1.411
[12]

Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial & Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27
[13]

Josef Hofbauer, Sylvain Sorin. Best response dynamics for continuous zero--sum games. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 215-224. doi: 10.3934/dcdsb.2006.6.215
[14]

Beatris Adriana Escobedo-Trujillo, José Daniel López-Barrientos. Nonzero-sum stochastic differential games with additive structure and average payoffs. Journal of Dynamics & Games, 2014, 1 (4) : 555-578. doi: 10.3934/jdg.2014.1.555
[15]

Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control & Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010
[16]

Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics & Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013
[17]

Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889
[18]

Alain Bensoussan, Jens Frehse, Christine Grün. Stochastic differential games with a varying number of players. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1719-1736. doi: 10.3934/cpaa.2014.13.1719
[19]

Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics & Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117
[20]

Konstantin Avrachenkov, Giovanni Neglia, Vikas Vikram Singh. Network formation games with teams. Journal of Dynamics & Games, 2016, 3 (4) : 303-318. doi: 10.3934/jdg.2016016

 Impact Factor: 

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences