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Recursive games: Uniform value, Tauberian theorem and the Mertens conjecture " M axmin = lim v n = lim v λ "

Author

Listed:
  • Xiaoxi Li

    (Department of Computer Science - Zhejiang University - Zhejiang University)

  • Xavier Venel

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

Abstract

We study two-player zero-sum recursive games with a countable state space and finite action spaces at each state. When the family of n-stage values {v_n;n >0} is totally bounded for the uniform norm, we prove the existence of the uniform value. Together with a result in Rosenberg and Vieille [12], we obtain a uniform Tauberian theorem for recursive game: (v_n) converges uniformly if and only if (v_λ) converges uniformly. We apply our main result to finite recursive games with signals (where players observe only signals on the state and on past actions). When the maximizer is more informed than the minimizer, we prove the Mertens conjecture Maxmin = lim v_n = lim v_λ. Finally, we deduce the existence of the uniform value in finite recursive games with symmetric information.

Suggested Citation

  • Xiaoxi Li & Xavier Venel, 2016. "Recursive games: Uniform value, Tauberian theorem and the Mertens conjecture " M axmin = lim v n = lim v λ "," PSE-Ecole d'économie de Paris (Postprint) hal-01302553, HAL.
    • Handle: RePEc:hal:pseptp:hal-01302553
    DOI: 10.1007/s00182-015-0496-4
    Note: View the original document on HAL open archive server: https://paris1.hal.science/hal-01302553
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    References listed on IDEAS

    as
    1. Lehrer Ehud & Monderer Dov, 1994. "Discounting versus Averaging in Dynamic Programming," Games and Economic Behavior, Elsevier, vol. 6(1), pages 97-113, January.
    2. MERTENS, Jean-François, 1987. "Repeated games. Proceedings of the International Congress of Mathematicians," LIDAM Reprints CORE 788, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Monderer, Dov & Sorin, Sylvain, 1993. "Asymptotic Properties in Dynamic Programming," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(1), pages 1-11.
    4. SORIN, Sylvain, 1984. "'Big match' with lack of information on one side (part 1)," LIDAM Reprints CORE 601, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. VIEILLE, Nicolas & ROSENBERG, Dinah & SOLAN, Eilon, 2002. "Stochastic games with a single controller and incomplete information," HEC Research Papers Series 754, HEC Paris.
    6. Ehud Lehrer & Sylvain Sorin, 1992. "A Uniform Tauberian Theorem in Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 17(2), pages 303-307, May.
    7. Dinah Rosenberg & Nicolas Vieille, 2000. "The Maxmin of Recursive Games with Incomplete Information on one Side," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 23-35, February.
    8. repec:dau:papers:123456789/6231 is not listed on IDEAS
    9. Nicolas Vieille & Dinah Rosenberg, 2000. "The Maxmin of Recursive Games with Incomplete Information on one Side," Post-Print hal-00481429, HAL.
    10. Truman Bewley & Elon Kohlberg, 1976. "The Asymptotic Theory of Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 197-208, August.
    11. Eilon Solan & Nicolas Vieille, 2000. "Uniform Value in Recursive Games," Discussion Papers 1293, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    12. Jérôme Renault, 2012. "The Value of Repeated Games with an Informed Controller," Mathematics of Operations Research, INFORMS, vol. 37(1), pages 154-179, February.
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    Cited by:

    1. Dhruva Kartik & Ashutosh Nayyar, 2021. "Upper and Lower Values in Zero-Sum Stochastic Games with Asymmetric Information," Dynamic Games and Applications, Springer, vol. 11(2), pages 363-388, June.
    2. Ziliotto, Bruno, 2018. "Tauberian theorems for general iterations of operators: Applications to zero-sum stochastic games," Games and Economic Behavior, Elsevier, vol. 108(C), pages 486-503.
    3. Dmitry Khlopin, 2018. "Tauberian Theorem for Value Functions," Dynamic Games and Applications, Springer, vol. 8(2), pages 401-422, June.

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