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A Zero-Sum Stochastic Game with Compact Action Sets and no Asymptotic Value

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  • Guillaume Vigeral

Abstract

We give an example of a zero-sum stochastic game with four states, compact action sets for each player, and continuous payoff and transition functions, such that the discounted value does not converge as the discount factor tends to 0, and the value of the n-stage game does not converge as n goes to infinity. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • 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.
    • Handle: RePEc:spr:dyngam:v:3:y:2013:i:2:p:172-186
    DOI: 10.1007/s13235-013-0073-z
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    References listed on IDEAS

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    8. Jérôme Bolte & Stéphane Gaubert & Guillaume Vigeral, 2015. "Definable Zero-Sum Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 40(1), pages 171-191, February.
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    15. 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.
    16. Sylvain Sorin & Guillaume Vigeral, 2013. "Existence of the Limit Value of Two Person Zero-Sum Discounted Repeated Games via Comparison Theorems," Journal of Optimization Theory and Applications, Springer, vol. 157(2), pages 564-576, May.
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    Cited by:

    1. Laraki, Rida & Sorin, Sylvain, 2015. "Advances in Zero-Sum Dynamic Games," Handbook of Game Theory with Economic Applications,, Elsevier.
    2. Laraki, Rida & Renault, Jérôme, 2017. "Acyclic Gambling Games," TSE Working Papers 17-768, Toulouse School of Economics (TSE).
    3. Sylvain Sorin, 2018. "Limit Value of Dynamic Zero-Sum Games with Vanishing Stage Duration," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 51-63, February.
    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. Flesch, János & Laraki, Rida & Perchet, Vianney, 2018. "Approachability of convex sets in generalized quitting games," Games and Economic Behavior, Elsevier, vol. 108(C), pages 411-431.
    6. Levy, Yehuda John, 2022. "Uniformly supported approximate equilibria in families of games," Journal of Mathematical Economics, Elsevier, vol. 98(C).
    7. Sylvain Sorin & Guillaume Vigeral, 2020. "Limit Optimal Trajectories in Zero-Sum Stochastic Games," Dynamic Games and Applications, Springer, vol. 10(2), pages 555-572, June.
    8. Xavier Venel, 2015. "Commutative Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 40(2), pages 403-428, February.
    9. 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.
    10. Sylvain Sorin & Guillaume Vigeral, 2016. "Operator approach to values of stochastic games with varying stage duration," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(1), pages 389-410, March.
    11. Dmitry Khlopin, 2018. "Tauberian Theorem for Value Functions," Dynamic Games and Applications, Springer, vol. 8(2), pages 401-422, June.
    12. Jérôme Bolte & Stéphane Gaubert & Guillaume Vigeral, 2015. "Definable Zero-Sum Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 40(1), pages 171-191, February.
    13. Antoine Hochart, 2021. "Unique Ergodicity of Deterministic Zero-Sum Differential Games," Dynamic Games and Applications, Springer, vol. 11(1), pages 109-136, March.
    14. 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.
    15. Jérôme Renault & Bruno Ziliotto, 2020. "Limit Equilibrium Payoffs in Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 889-895, August.
    16. Bruno Ziliotto, 2016. "A Tauberian Theorem for Nonexpansive Operators and Applications to Zero-Sum Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1522-1534, November.
    17. Rida Laraki & Jérôme Renault, 2020. "Acyclic Gambling Games," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1237-1257, November.

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