Differential Games with Incomplete Information and with Signal Revealing: The Symmetric Case

Abstract

In this paper, we investigate the existence of value for a two-person zero-sum differential game with symmetric incomplete information and with signal revealing. Before the game begins, the initial state of the dynamic is chosen randomly among a finite number of points in \(\mathbb {R}^n\), while both players have only a probabilistic knowledge of the chosen initial state. During the game, if the system reaches a fixed closed target set K, the current state of the system at the hitting time is revealed to both players. We prove in this paper that this game has a value and its value function is the unique bounded continuous viscosity solution of a suitable Hamilton–Jacobi–Isaacs equation.

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References

  1. 1.

    Aumann RJ, Maschler MB (1995) Repeated games with incomplete information. With the collaboration of Richard E. Stearns. MIT Press, Cambridge

    Google Scholar 

  2. 2.

    Bardi M, Capuzzo-Dolcetta I (1996) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, Basel

    Google Scholar 

  3. 3.

    Bardi M, Koike S, Soravia P (2000) Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations. Discrete Contin Dyn Syst 6(2):361–380

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bernhard P, Rapaport A (1995) Étude d’un jeu de poursuite plane avec connaissance imparfaite d’une coordonnée. Automatique-productique informatique industrielle 29:575–601

    Google Scholar 

  5. 5.

    Buckdahn R, Cardaliaguet P, Quincampoix M (2011) Some recent aspects of differential game theory. Dyn Game Application 1(1):74–114

    MathSciNet  Article  Google Scholar 

  6. 6.

    Buckdahn R, Quincampoix M, Rainer C, Xu Y (2016) Differential games with asymmetric information and without Isaacs’ condition. Int J Game Theory 45:795–816

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cardaliaguet P (2007) Differential games with asymmetric information. SIAM J Control Optim 46(3):816–838

    MathSciNet  Article  Google Scholar 

  8. 8.

    Cardaliaguet P (2009) A double obstacle problem arising in differential game theory. J Math Anal Appl 360(1):95–107

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cardaliaguet P (2010) Introduction to differential games. Université de Bretagne Occidentale, Lecture notes

  10. 10.

    Cardaliaguet P, Jimenez C, Quincampoix M (2014) Pure and random strategies in differential game with incomplete informations. J Dyn Games 1(3):363–375

    MathSciNet  Article  Google Scholar 

  11. 11.

    Cardaliaguet P, Quincampoix M (2008) Deterministic differential games under probability knowledge of initial condition. Int Game Theory Rev 10(01):1–16

    MathSciNet  Article  Google Scholar 

  12. 12.

    Cardaliaguet P, Quincampoix M, Saint-Pierre P (2001) Pursuit differential games with state constraints. SIAM J Control Optim 39(5):1615–1632

    MathSciNet  Article  Google Scholar 

  13. 13.

    Cardaliaguet P, Rainer C (2009) Stochastic differential games with asymmetric information. Appl Math Optim 59(1):1–36

    MathSciNet  Article  Google Scholar 

  14. 14.

    Crandall MG, Ishii H, Lions P-L (1992) User’s guide to viscosity solutions of second order partial differential equations. Bull Am Soc 27:1–67

    MathSciNet  Article  Google Scholar 

  15. 15.

    Crandall MG, Lions P-L (1983) Viscosity solutions of Hamilton–Jacobi equations. Trans Am Math Soc 277:1–42

    MathSciNet  Article  Google Scholar 

  16. 16.

    Elliott RJ, Kalton NJ (1972) The existence of value in differential games of pursuit and evasion. J Differ Equ 12(3):504–523

    MathSciNet  Article  Google Scholar 

  17. 17.

    Evans LC, Souganidis PE (1984) Differential games and representation formulas for solutions of Hamilton–Jacobi–Isaacs equations. Indiana Univ Math J 33(5):773–797

    MathSciNet  Article  Google Scholar 

  18. 18.

    Forges F (1982) Infinitely repeated games of incomplete information: symmetric case with random signals. Int J Game Theory 11:203–213

    MathSciNet  Article  Google Scholar 

  19. 19.

    Isaacs R (1967) Differential games. Wiley, London

    Google Scholar 

  20. 20.

    Jimenez C, Quincampoix M, Xu Y (2016) Differential games with incomplete information on a continuum of initial positions and without Isaacs condition. Dyn Games Appl 6:82–96

    MathSciNet  Article  Google Scholar 

  21. 21.

    Kohlberg E, Zamir S (1974) Repeated games of incomplete information: the symmetric case. Anal Stat 2:1040–1041

    MathSciNet  Article  Google Scholar 

  22. 22.

    Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New York

    Google Scholar 

  23. 23.

    Neyman A, Sorin S (1997) Equilibria in repeated games of incomplete information: the deterministic symmetric case. In: Parthasaraty T (ed) Game theoretic applications to economics and operations research. Springer, US

    Google Scholar 

  24. 24.

    Neyman A, Sorin S (1998) Equilibria in repeated games of incomplete information: the general symmetric case. Int J Game Theory 27:201–210

    MathSciNet  Article  Google Scholar 

  25. 25.

    Oliu-Barton M (2015) Differential games with asymmetric and correlated information. Dyn Games Appl 5(3):378–396

    MathSciNet  Article  Google Scholar 

  26. 26.

    Petrosjan LA (1993) Differential games of pursuit, volume 2 of series on optimization. World Scientific Publishing co Ltd, Singapore

    Google Scholar 

  27. 27.

    Roxin E (1979) Feedback strategies with finite memory in differential games. J Optim Theory Appl 27(1):127–134

    MathSciNet  Article  Google Scholar 

  28. 28.

    Soravia P (1993) Pursuit-evasion problems and viscosity solutions of Isaacs’ equations. SIAM J Control Optim 31(3):604–623

    MathSciNet  Article  Google Scholar 

  29. 29.

    Varaiya P (1967) On the existence of solutions to a differential game. SIAM J Control 5(1):153–162

    MathSciNet  Article  Google Scholar 

  30. 30.

    Veliov VM (1997) Lipschitz continuity of the value function in optimal control. J Optim Theory Appl 94(2):335–363

    MathSciNet  Article  Google Scholar 

  31. 31.

    Wu X (2017) Existence of value for differential games with incomplete information and signals on initial states and payoffs. J Math Anal Appl 446(2):1196–1218

    MathSciNet  Article  Google Scholar 

  32. 32.

    Wu X (2018) Existence of value for a differential game with incomplete information and revealing. SIAM J Control Optim 56(4):2536–2562

    MathSciNet  Article  Google Scholar 

  33. 33.

    Yong JM (1988) On differential pursuit games. SIAM J Control Optim 26(2):478–495

    MathSciNet  Article  Google Scholar 

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Funding

This study was supported by China Postdoctoral Science Foundation (Grant No. 2020M672037).

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Correspondence to Xiaochi Wu.

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Wu, X. Differential Games with Incomplete Information and with Signal Revealing: The Symmetric Case. Dyn Games Appl (2021). https://doi.org/10.1007/s13235-021-00376-1

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Keywords

  • Differential games
  • Incomplete information
  • Hamilton–Jacobi–Isaacs equation
  • Signal

JEL Classification

  • 91A05
  • 91A10
  • 91A23