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Differential Games with Incomplete Information on a Continuum of Initial Positions and without Isaacs Condition

Author

Listed:
  • Chloe Jimenez

    (Université de Brest)

  • Marc Quincampoix

    (Université de Brest)

  • Yuhong Xu

    (Université de Brest
    Soochow University)

Abstract

This article deals with a two-player zero-sum differential game with infinitely many initial positions and without Isaacs condition. The structure of information is asymmetric: The first player has a private information on the initial position, while the second player knows only a probability distribution on the initial position. In the present model, we face two difficulties: First, the incomplete information structure does not reduce to a finite set (as in the famous Aumann–Maschler model for repeated games). Second, the game does not satisfy the Isaacs condition (crucially used in classical approaches to differential games). Therefore, we use tools from optimal transportation theory and stochastic control. Our main result shows that with a suitable concept of mixed strategies, there exists a value of the game with such random strategies. As a byproduct of our approach, we obtain the Lipschitz continuity of the random value with respect to the Wasserstein distance and we show the existence of a value in pure strategies in the specific case of an initial distribution without atoms. We also discuss an extension of our model when the asymmetric information concerns continuous scenarios.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:dyngam:v:6:y:2016:i:1:d:10.1007_s13235-014-0134-y
    DOI: 10.1007/s13235-014-0134-y
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    References listed on IDEAS

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    1. Robert J. Aumann, 1995. "Repeated Games with Incomplete Information," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262011476, April.
    2. Rainer Buckdahn & Juan Li & Marc Quincampoix, 2013. "Value function of differential games without Isaacs conditions. An approach with nonanticipative mixed strategies," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(4), pages 989-1020, November.
    3. SCHMEIDLER, David, 1973. "Equilibrium points of nonatomic games," LIDAM Reprints CORE 146, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. P. Cardaliaguet & M. Quincampoix, 2008. "Deterministic Differential Games Under Probability Knowledge Of Initial Condition," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 10(01), pages 1-16.
    5. 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.
    6. 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).
    7. R. Buckdahn & P. Cardaliaguet & M. Quincampoix, 2011. "Some Recent Aspects of Differential Game Theory," Dynamic Games and Applications, Springer, vol. 1(1), pages 74-114, March.
    8. repec:dau:papers:123456789/6046 is not listed on IDEAS
    9. Ma, Jinpeng, 1995. "An infinitely repeated rental model with incomplete information," Economics Letters, Elsevier, vol. 49(3), pages 261-266, September.
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    Cited by:

    1. Fabien Gensbittel, 2019. "Continuous-Time Markov Games with Asymmetric Information," Dynamic Games and Applications, Springer, vol. 9(3), pages 671-699, September.
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    3. Xiaochi Wu, 2021. "Differential Games with Incomplete Information and with Signal Revealing: The Symmetric Case," Dynamic Games and Applications, Springer, vol. 11(4), pages 863-891, December.
    4. Xiaochi Wu, 2019. "Infinite Horizon Differential Games with Asymmetric Information," Dynamic Games and Applications, Springer, vol. 9(3), pages 858-880, September.

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