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Continuous-Time Markov Games with Asymmetric Information

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  • Fabien Gensbittel

    (University Toulouse 1 Capitole)

Abstract

We study a two-player zero-sum stochastic differential game with asymmetric information where the payoff depends on a controlled continuous-time Markov chain X with finite state space which is only observed by player 1. This model was already studied in Cardaliaguet et al (Math Oper Res 41(1):49–71, 2016) through an approximating sequence of discrete-time games. Our first contribution is the proof of the existence of the value in the continuous-time model based on duality techniques. This value is shown to be the unique solution of the same Hamilton–Jacobi equation with convexity constraints which characterized the limit value obtained in Cardaliaguet et al. (2016). Our second main contribution is to provide a simpler equivalent formulation for this Hamilton–Jacobi equation using directional derivatives and exposed points, which we think is interesting for its own sake as the associated comparison principle has a very simple proof which avoids all the technical machinery of viscosity solutions.

Suggested Citation

  • Fabien Gensbittel, 2019. "Continuous-Time Markov Games with Asymmetric Information," Dynamic Games and Applications, Springer, vol. 9(3), pages 671-699, September.
    • Handle: RePEc:spr:dyngam:v:9:y:2019:i:3:d:10.1007_s13235-018-0273-7
    DOI: 10.1007/s13235-018-0273-7
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    References listed on IDEAS

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    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. Miquel Oliu-Barton, 2015. "Differential Games with Asymmetric and Correlated Information," Dynamic Games and Applications, Springer, vol. 5(3), pages 378-396, September.
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    6. Fabien Gensbittel & Catherine Rainer, 2018. "A Two-Player Zero-sum Game Where Only One Player Observes a Brownian Motion," Dynamic Games and Applications, Springer, vol. 8(2), pages 280-314, June.
    7. Pierre Cardaliaguet & Catherine Rainer, 2012. "Games with Incomplete Information in Continuous Time and for Continuous Types," Dynamic Games and Applications, Springer, vol. 2(2), pages 206-227, June.
    8. Chloe Jimenez & Marc Quincampoix & Yuhong Xu, 2016. "Differential Games with Incomplete Information on a Continuum of Initial Positions and without Isaacs Condition," Dynamic Games and Applications, Springer, vol. 6(1), pages 82-96, March.
    9. Fabien Gensbittel & Jérôme Renault, 2015. "The Value of Markov Chain Games with Incomplete Information on Both Sides," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 820-841, October.
    10. Abraham Neyman, 2008. "Existence of optimal strategies in Markov games with incomplete information," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(4), pages 581-596, December.
    11. Fabien Gensbittel & Christine Grün, 2019. "Zero-Sum Stopping Games with Asymmetric Information," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 277-302, February.
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    13. Bernard de Meyer, 1996. "Repeated games, Duality, and the Central Limit Theorem," Post-Print hal-00259714, HAL.
    14. Rainer Buckdahn & Marc Quincampoix & Catherine Rainer & Yuhong Xu, 2016. "Differential games with asymmetric information and without Isaacs’ condition," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(4), pages 795-816, November.
    15. Pierre Cardaliaguet & Catherine Rainer & Dinah Rosenberg & Nicolas Vieille, 2016. "Markov Games with Frequent Actions and Incomplete Information—The Limit Case," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 49-71, February.
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    17. 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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    Cited by:

    1. Ashkenazi-Golan, Galit & Rainer, Catherine & Solan, Eilon, 2020. "Solving two-state Markov games with incomplete information on one side," Games and Economic Behavior, Elsevier, vol. 122(C), pages 83-104.
    2. Ashkenazi-Golan, Galit & Hernández, Penélope & Neeman, Zvika & Solan, Eilon, 2023. "Markovian persuasion with two states," LSE Research Online Documents on Economics 119970, London School of Economics and Political Science, LSE Library.

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