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Differential games with asymmetric information and without Isaacs’ condition

Author

Listed:
  • Rainer Buckdahn

    (Université de Brest
    Shandong University)

  • Marc Quincampoix

    (Université de Brest)

  • Catherine Rainer

    (Université de Brest)

  • Yuhong Xu

    (Université de Brest
    Soochow University)

Abstract

We investigate a two-player zero-sum differential game with asymmetric information on the payoff and without Isaacs’ condition. The dynamics is an ordinary differential equation parametrized by two controls chosen by the players. Each player has a private information on the payoff of the game, while his opponent knows only the probability distribution on the information of the other player. We show that a suitable definition of random strategies allows to prove the existence of a value in mixed strategies. This value is taken in the sense of the limit of any time discretization, as the mesh of the time partition tends to zero. We characterize it in terms of the unique viscosity solution in some dual sense of a Hamilton–Jacobi–Isaacs equation. Here we do not suppose the Isaacs’ condition, which is usually assumed in differential games.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jogath:v:45:y:2016:i:4:d:10.1007_s00182-015-0482-x
    DOI: 10.1007/s00182-015-0482-x
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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. 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. Back, Kerry, 1993. "Asymmetric Information and Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(3), pages 435-472.
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    5. De Meyer, B., 1996. "Repeated games, duality and the central limit theorem," LIDAM Reprints CORE 1210, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Bebczuk,Ricardo N., 2003. "Asymmetric Information in Financial Markets," Cambridge Books, Cambridge University Press, number 9780521793421.
    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. 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.
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    11. Bernard de Meyer, 1996. "Repeated games, Duality, and the Central Limit Theorem," Post-Print hal-00259714, HAL.
    12. Robert J. Aumann, 1995. "Repeated Games with Incomplete Information," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262011476, December.
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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.
    2. 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.
    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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