The Annals of Applied Probability

On values of repeated games with signals

Hugo Gimbert, Jérôme Renault, Sylvain Sorin, Xavier Venel, and Wiesław Zielonka

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Abstract

We study the existence of different notions of value in two-person zero-sum repeated games where the state evolves and players receive signals. We provide some examples showing that the limsup value (and the uniform value) may not exist in general. Then we show the existence of the value for any Borel payoff function if the players observe a public signal including the actions played. We also prove two other positive results without assumptions on the signaling structure: the existence of the $\sup$ value in any game and the existence of the uniform value in recursive games with nonnegative payoffs.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 1 (2016), 402-424.

Dates
Received: July 2014
Revised: December 2014
First available in Project Euclid: 5 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1452003243

Digital Object Identifier
doi:10.1214/14-AAP1095

Mathematical Reviews number (MathSciNet)
MR3449322

Zentralblatt MATH identifier
1332.91023

Subjects
Primary: 91A20: Multistage and repeated games
Secondary: 91A05: 2-person games 91A15: Stochastic games 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

Keywords
Multistage game repeated games with signals repeated games with symmetric information Borelian evaluation limsup value uniform value

Citation

Gimbert, Hugo; Renault, Jérôme; Sorin, Sylvain; Venel, Xavier; Zielonka, Wiesław. On values of repeated games with signals. Ann. Appl. Probab. 26 (2016), no. 1, 402--424. doi:10.1214/14-AAP1095. https://projecteuclid.org/euclid.aoap/1452003243


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References

  • [1] Blackwell, D. (1969). Infinite $G_{\delta }$-games with imperfect information. Zastos. Mat. 10 99–101.
  • [2] Blackwell, D. and Ferguson, T. S. (1968). The big match. Ann. Math. Statist 39 159–163.
  • [3] Gale, D. and Stewart, F. M. (1953). Infinite games with perfect information. In Contributions to the Theory of Games, Vol. 2. 245–266. Princeton Univ. Press, Princeton, NJ.
  • [4] Gensbittel, F., Oliu-Barton, M. and Venel, X. (2014). Existence of the uniform value in repeated games with a more informed controller. Journal of Dynamics and Games 1 411–445.
  • [5] Ghosh, M. K., McDonald, D. and Sinha, S. (2004). Zero-sum stochastic games with partial information. J. Optim. Theory Appl. 121 99–118.
  • [6] Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
  • [7] Kohlberg, E. and Zamir, S. (1974). Repeated games of incomplete information: The symmetric case. Ann. Statist. 2 1040–1041.
  • [8] Kuhn, H. W. (1953). Extensive games and the problem of information. In Contributions to the Theory of Games, Vol. 2. 193–216. Princeton Univ. Press, Princeton, NJ.
  • [9] Lehrer, E. and Sorin, S. (1992). A uniform Tauberian theorem in dynamic programming. Math. Oper. Res. 17 303–307.
  • [10] Maitra, A. and Sudderth, W. (1992). An operator solution of stochastic games. Israel J. Math. 78 33–49.
  • [11] Maitra, A. and Sudderth, W. (1993). Borel stochastic games with $\lim\sup$ payoff. Ann. Probab. 21 861–885.
  • [12] Maitra, A. and Sudderth, W. (1993). Finitely additive and measurable stochastic games. Internat. J. Game Theory 22 201–223.
  • [13] Maitra, A. and Sudderth, W. (1998). Finitely additive stochastic games with Borel measurable payoffs. Internat. J. Game Theory 27 257–267.
  • [14] Maitra, A. and Sudderth, W. (2003). Stochastic games with Borel payoffs. In Stochastic Games and Applications (Stony Brook, NY, 1999) (A. Neyman and S. Sorin, eds.). NATO Sci. Ser. C Math. Phys. Sci. 570 367–373. Kluwer Academic, Dordrecht.
  • [15] Martin, D. A. (1975). Borel determinacy. Ann. of Math. (2) 102 363–371.
  • [16] Martin, D. A. (1998). The determinacy of Blackwell games. J. Symbolic Logic 63 1565–1581.
  • [17] Mertens, J.-F. (1987). Repeated games. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) 1528–1577. Amer. Math. Soc., Providence, RI.
  • [18] Mertens, J.-F. and Neyman, A. (1981). Stochastic games. Internat. J. Game Theory 10 53–66.
  • [19] Mertens, I.-F., Sorin, S. and Zamir, S. (2014). Repeated Games. Cambridge Univ. Press, Cambridge.
  • [20] Neyman, A. and Sorin, S., eds. (2003). Stochastic games and applications. In Proceedings of the Nato Advanced Study Institute Held in Stony Brook, NY, July 717, 1999. NATO Science Series C: Mathematical and Physical Sciences 570. Kluwer Academic, Dordrecht.
  • [21] Renault, J. (2011). Uniform value in dynamic programming. J. Eur. Math. Soc. (JEMS) 13 309–330.
  • [22] Renault, J. (2012). The value of repeated games with an informed controller. Math. Oper. Res. 37 154–179.
  • [23] Rosenberg, D., Solan, E. and Vieille, N. (2002). Blackwell optimality in Markov decision processes with partial observation. Ann. Statist. 30 1178–1193.
  • [24] Rosenberg, D., Solan, E. and Vieille, N. (2004). Stochastic games with a single controller and incomplete information. SIAM J. Control Optim. 43 86–110 (electronic).
  • [25] Rosenberg, D., Solan, E. and Vieille, N. (2009). Protocols with no acknowledgment. Oper. Res. 57 905–915.
  • [26] Shapley, L. S. (1953). Stochastic games. Proc. Natl. Acad. Sci. USA 39 1095–1100.
  • [27] Solan, E. and Vieille, N. (2002). Uniform value in recursive games. Ann. Appl. Probab. 12 1185–1201.
  • [28] Sorin, S. (2002). A First Course on Zero-Sum Repeated Games. Mathématiques & Applications (Berlin) [Mathematics & Applications] 37. Springer, Berlin.
  • [29] Sorin, S. (2003). Symmetric incomplete information games as stochastic games. In Stochastic Games and Applications (Stony Brook, NY, 1999) (A. Neyman and S. Sorin, eds.). NATO Sci. Ser. C Math. Phys. Sci. 570 323–334. Kluwer Academic, Dordrecht.
  • [30] Ziliotto, B. (2013). Zero-sum repeated games: Counterexamples to the existence of the asymptotic value and the conjecture $\max\min=\lim v(n)$, pages 1–20. Available at arXiv:1305.4778.