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Deterministic Differential Games Under Probability Knowledge Of Initial Condition

Author

Listed:
  • P. CARDALIAGUET

    (Laboratoire de Mathématiques, CNRS UMR 6205 Université de Bretagne Occidentale, 6 Avenue Le Gorgeu, 29200 Brest, France)

  • M. QUINCAMPOIX

    (Laboratoire de Mathématiques, CNRS UMR 6205 Université de Bretagne Occidentale, 6 Avenue Le Gorgeu, 29200 Brest, France)

Abstract

We study a zero-sum differential game where the players have only an unperfect information on the state of the system. In the beginning of the game only a random distribution on the initial state is available. The main result of the paper is the existence of the value obtained through an uniqueness result for Hamilton-Jacobi-Isaacs equations stated on the space of measure inℝn. This result is the first step for future work on differential games with lack of information.

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:igtrxx:v:10:y:2008:i:01:n:s021919890800173x
    DOI: 10.1142/S021919890800173X
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    Citations

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    Cited by:

    1. Miquel Oliu-Barton, 2015. "Differential Games with Asymmetric and Correlated Information," Dynamic Games and Applications, Springer, vol. 5(3), pages 378-396, September.
    2. Laraki, Rida & Sorin, Sylvain, 2015. "Advances in Zero-Sum Dynamic Games," Handbook of Game Theory with Economic Applications,, Elsevier.
    3. 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.
    4. Xiaochi Wu, 2022. "Existence of value for a differential game with asymmetric information and signal revealing," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(1), pages 213-247, March.
    5. Jun Moon, 2022. "State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations," Mathematics, MDPI, vol. 10(10), pages 1-32, May.
    6. 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.
    7. 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.
    8. Valerii Patsko & Marc Quincampoix, 2019. "Preface: DGAA Special Issue on Pursuit-Evasion Games and Differential Games with Incomplete Information," Dynamic Games and Applications, Springer, vol. 9(3), pages 569-572, September.
    9. 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.
    10. Vianney Perchet & Marc Quincampoix, 2015. "On a Unified Framework for Approachability with Full or Partial Monitoring," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 596-610, March.
    11. 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.
    12. Yurii Averboukh, 2019. "Krasovskii–Subbotin Approach to Mean Field Type Differential Games," Dynamic Games and Applications, Springer, vol. 9(3), pages 573-593, September.
    13. Xiaochi Wu, 2019. "Infinite Horizon Differential Games with Asymmetric Information," Dynamic Games and Applications, Springer, vol. 9(3), pages 858-880, September.
    14. Alberto Bressan & Deling Wei, 2013. "Stackelberg Solutions of Feedback Type for Differential Games with Random Initial Data," Dynamic Games and Applications, Springer, vol. 3(3), pages 341-358, September.
    15. Yurii Averboukh & Antonio Marigonda & Marc Quincampoix, 2021. "Extremal Shift Rule and Viability Property for Mean Field-Type Control Systems," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 244-270, April.
    16. Alós-Ferrer, Carlos & Kern, Johannes, 2015. "Repeated games in continuous time as extensive form games," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 34-57.
    17. 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.

    More about this item

    Keywords

    Differential games; incomplete information; Hamilton-Jacobi-Isaacs equation; Subject Classification: 22E46; Subject Classification: 53C35; Subject Classification: 57S20;
    All these keywords.

    JEL classification:

    • B4 - Schools of Economic Thought and Methodology - - Economic Methodology
    • C0 - Mathematical and Quantitative Methods - - General
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D5 - Microeconomics - - General Equilibrium and Disequilibrium
    • D7 - Microeconomics - - Analysis of Collective Decision-Making
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics

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