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General limit value in zero-sum stochastic games

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  • Bruno Ziliotto

    (Université Paris Dauphine)

Abstract

Bewley and Kohlberg (Math Oper Res 1(3):197–208, 1976) and Mertens and Neyman (Int J Game Theory 10(2):53–66, 1981) have respectively proved the existence of the asymptotic value and the uniform value in zero-sum stochastic games with finite state space and finite action sets. In their work, the total payoff in a stochastic game is defined either as a Cesaro mean or an Abel mean of the stage payoffs. The contribution of this paper is twofold: first, it generalizes the result of Bewley and Kohlberg (1976) to a more general class of payoff evaluations, and it proves with an example that this new result is tight. It also investigates the particular case of absorbing games. Second, for the uniform approach of Mertens and Neyman, this paper provides an example of absorbing game to demonstrate that there is no natural way to generalize their result to a wider class of payoff evaluations.

Suggested Citation

  • Bruno Ziliotto, 2016. "General limit value in zero-sum stochastic games," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(1), pages 353-374, March.
    • Handle: RePEc:spr:jogath:v:45:y:2016:i:1:d:10.1007_s00182-015-0509-3
    DOI: 10.1007/s00182-015-0509-3
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    References listed on IDEAS

    as
    1. Jean-François Mertens & Abraham Neyman & Dinah Rosenberg, 2009. "Absorbing Games with Compact Action Spaces," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 257-262, May.
    2. Renault, Jérôme & Venel, Xavier, 2017. "A distance for probability spaces, and long-term values in Markov Decision Processes and Repeated Games," TSE Working Papers 17-748, Toulouse School of Economics (TSE).
    3. Truman Bewley & Elon Kohlberg, 1976. "The Asymptotic Theory of Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 197-208, August.
    4. Miquel Oliu-Barton, 2014. "The Asymptotic Value in Finite Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 712-721, August.
    5. Pierre Cardaliaguet & Rida Laraki & Sylvain Sorin, 2012. "A Continuous Time Approach for the Asymptotic Value in Two-Person Zero-Sum Repeated Games," Post-Print hal-00609476, HAL.
    6. repec:dau:papers:123456789/6775 is not listed on IDEAS
    7. Guillaume Vigeral, 2013. "A Zero-Sum Stochastic Game with Compact Action Sets and no Asymptotic Value," Dynamic Games and Applications, Springer, vol. 3(2), pages 172-186, June.
    8. repec:dau:papers:123456789/10880 is not listed on IDEAS
    9. Abraham Neyman & Sylvain Sorin, 2010. "Repeated games with public uncertain duration process," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 29-52, March.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. 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.
    2. Dmitry Khlopin, 2018. "Tauberian Theorem for Value Functions," Dynamic Games and Applications, Springer, vol. 8(2), pages 401-422, June.

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