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Stochastic Games with a Single Controller and Incomplete Information

Author

Listed:
  • Dinah Rosenberg

    (GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique)

  • Eilon Solan

    (TAU - School of Mathematical Sciences [Tel Aviv] - TAU - Raymond and Beverly Sackler Faculty of Exact Sciences [Tel Aviv] - TAU - Tel Aviv University)

  • Nicolas Vieille

    (GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique)

Abstract

We study stochastic games with incomplete information on one side, in which the transition is controlled by one of the players. We prove that if the informed player also controls the transitions, the game has a value, whereas if the uninformed player controls the transitions, the max-min value as well as the min-max value exist, but they may differ. We discuss the structure of the optimal strategies, and provide extensions to the case of incomplete information on both sides.

Suggested Citation

  • Dinah Rosenberg & Eilon Solan & Nicolas Vieille, 2002. "Stochastic Games with a Single Controller and Incomplete Information," Working Papers hal-00593394, HAL.
    • Handle: RePEc:hal:wpaper:hal-00593394
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    References listed on IDEAS

    as
    1. Bernard De Meyer, 1996. "Repeated Games, Duality and the Central Limit Theorem," Mathematics of Operations Research, INFORMS, vol. 21(1), pages 237-251, February.
    2. Robert J. Aumann, 1995. "Repeated Games with Incomplete Information," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262011476, April.
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    5. Mertens, Jean-Francois, 2002. "Stochastic games," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 47, pages 1809-1832, Elsevier.
    6. repec:dau:papers:123456789/6231 is not listed on IDEAS
    7. Nicolas Vieille & Dinah Rosenberg, 2000. "The Maxmin of Recursive Games with Incomplete Information on one Side," Post-Print hal-00481429, HAL.
    8. Mertens, J.-F. & Zamir, S., 1980. "Minmax and maxmin of repeated games with incomplete information," LIDAM Reprints CORE 433, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    13. MERTENS, Jean-François & ZAMIR, Shmuel, 1971. "The value of two-person zero-sum repeated games with lack of information on both sides," LIDAM Reprints CORE 154, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    16. 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.
    17. Bernard de Meyer, 1996. "Repeated games, Duality, and the Central Limit Theorem," Post-Print hal-00259714, HAL.
    Full references (including those not matched with items on IDEAS)

    Citations

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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. Hugo Gimbert & Jérôme Renault & Sylvain Sorin & Xavier Venel & Wieslaw Zielonka, 2016. "On the values of repeated games with signals," PSE-Ecole d'économie de Paris (Postprint) hal-01006951, HAL.
    3. Laraki, Rida & Sorin, Sylvain, 2015. "Advances in Zero-Sum Dynamic Games," Handbook of Game Theory with Economic Applications,, Elsevier.
    4. Dinah Rosenberg & Eilon Solan & Nicolas Vieille, 2002. "Stochastic Games with Imperfect Monitoring," Discussion Papers 1341, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    5. Jérôme Renault, 2006. "The Value of Markov Chain Games with Lack of Information on One Side," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 490-512, August.
    6. Jérôme Renault & Xavier Venel, 2017. "Long-Term Values in Markov Decision Processes and Repeated Games, and a New Distance for Probability Spaces," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 349-376, May.
    7. 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.
    8. Sylvain Sorin, 2011. "Zero-Sum Repeated Games: Recent Advances and New Links with Differential Games," Dynamic Games and Applications, Springer, vol. 1(1), pages 172-207, March.
    9. Erim Kardeş & Fernando Ordóñez & Randolph W. Hall, 2011. "Discounted Robust Stochastic Games and an Application to Queueing Control," Operations Research, INFORMS, vol. 59(2), pages 365-382, April.
    10. Abraham Neyman, 2002. "Stochastic games: Existence of the MinMax," Discussion Paper Series dp295, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    11. Banas, Lubomir & Ferrari, Giorgio & Randrianasolo, Tsiry Avisoa, 2020. "Numerical Appromixation of the Value of a Stochastic Differential Game with Asymmetric Information," Center for Mathematical Economics Working Papers 630, Center for Mathematical Economics, Bielefeld University.
    12. 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.
    13. Jérôme Renault, 2012. "The Value of Repeated Games with an Informed Controller," Mathematics of Operations Research, INFORMS, vol. 37(1), pages 154-179, February.
    14. Xiaoxi Li & Xavier Venel, 2016. "Recursive games: Uniform value, Tauberian theorem and the Mertens conjecture " M axmin = lim v n = lim v λ "," Post-Print hal-01302553, HAL.
    15. Xiaoxi Li & Xavier Venel, 2016. "Recursive games: Uniform value, Tauberian theorem and the Mertens conjecture " M axmin = lim v n = lim v λ "," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01302553, HAL.
    16. P. Cardaliaguet, 2008. "Representations Formulas for Some Differential Games with Asymmetric Information," Journal of Optimization Theory and Applications, Springer, vol. 138(1), pages 1-16, July.

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    More about this item

    Keywords

    Minimax method; Game value; Stochastic game; Incomplete information; Optimal strategy; Uncertain system;
    All these keywords.

    JEL classification:

    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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