Asymptotic Value in Frequency-Dependent Games with Separable Payoffs: A Differential Approach


We study the asymptotic value of a frequency-dependent zero-sum game with separable payoff following a differential approach. The stage payoffs in such games depend on the current actions and on a linear function of the frequency of actions played so far. We associate with the repeated game, in a natural way, a differential game, and although the latter presents an irregularity at the origin, we prove that it has a value. We conclude, using appropriate approximations, that the asymptotic value of the original game exists in both the n-stage and the \(\lambda \)-discounted games and that it coincides with the value of the continuous time game.

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  1. 1.

    As far as we know, in general stochastic games, payoffs with separable structure first appeared in [16]. The authors study games with transitions independent of the current state.

  2. 2.

    Existence of the value follows from the standard comparison and uniqueness theorems for viscosity solutions presented in [15].

  3. 3.

    The authors prove that under some regularity conditions on the payoff and dynamics functions, the discrete values converge to the values of the continuous time game as the mesh of the discretization tends to 0. These approximations do not converge in general if the value function is discontinuous.

  4. 4.

    In the literature, the lower and upper values of a differential game have been first characterized by means of DPP in [8].


  1. 1.

    Bardi M, Capuzzo-Dolcetta I (2008) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Springer, New York

    MATH  Google Scholar 

  2. 2.

    Barles G, Souganidis PE (1991) Convergence of approximation schemes for fully nonlinear second order equations. Asymptot Anal 4(3):271–283

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Brenner T, Witt U (2003) Melioration learning in games with constant and frequency-dependent payoffs. J Econ Behav Org 50(4):429–448

    Article  Google Scholar 

  4. 4.

    Cardaliaguet P (2000) Introduction à la théorie des jeux différentiels. Lecture Notes. Université Paris-Dauphine

  5. 5.

    Cardaliaguet P, Laraki R, Sorin S (2012) A continuous time approach for the asymptotic value in two-person zero-sum repeated games. SIAM J Control Optim 50(3):1573–1596

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Contou-Carrère P (2011) Contributions aux jeux répétés. Ph.D. thesis, Paris 1

  7. 7.

    Elliott RJ, Kalton NJ (1972) Values in differential games. Bull Am Math Soc 78(3):427–431

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Elliott RJ, Kalton NJ (1974) Cauchy problems for certain Isaacs–Bellman equations and games of survival. Trans Am Math Soc 198:45–72

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

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

  10. 10.

    Friedman A (1970) On the definition of differential games and the existence of value and of saddle points. J Differ Equ 7(1):69–91

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Jean-François M, Sylvain S, Shmuel Z (2015) Repeated games, vol 55. Cambridge University Press, New York

    MATH  Google Scholar 

  12. 12.

    Joosten R (2004) Strategic interaction and externalities: FD-games and pollution. Pap Econ Evol 2004–1:1–26

    Google Scholar 

  13. 13.

    Joosten R, Brenner T, Witt U (2003) Games with frequency-dependent stage payoffs. Int J Game Theory 31(4):609–620

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Laraki R (2002) Repeated games with lack of information on one side: the dual differential approach. Math Oper Res 27(2):419–440

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Michael G (1983) Crandall and Pierre-Louis Lions. Viscosity solutions of Hamilton–Jacobi equations. Trans Am Math Soc 277(1):1–42

    Article  Google Scholar 

  16. 16.

    Parthasarathy T, Tijs SH, Vrieze OJ (1984) Stochastic games with state independent transitions and separable rewards. Selected Topics in Operations Research and Mathematical Economics. Springer, New York, pp 262–271

    MATH  Google Scholar 

  17. 17.

    Roxin E (1969) Axiomatic approach in differential games. J Optim Theory Appl 3(3):153–163

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Smale S (1980) The prisoner’s dilemma and dynamical systems associated to non-cooperative games. Econometrica 48:1617–1634

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Souganidis PE (1999) Two-player, zero-sum differential games and viscosity solutions, pp 69–104. Stochastic and differential games, Springer

  20. 20.

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

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Vieille N (1992) Weak approachability. Math Oper Res 17(4):781–791

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    von Neumann J (1928) Zur Theorie der Gesellschaftsspiele. Math Ann 100(1):295–320

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Ziliotto B (2016) A Tauberian theorem for nonexpansive operators and applications to zero-sum stochastic games. Math Oper Res 41(4):1522–1534

    MathSciNet  Article  MATH  Google Scholar 

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Corresponding author

Correspondence to Nikolaos Pnevmatikos.

Additional information

This author’s research was supported by Labex MME-DII. Part of this research was carried out when the author was working at GERAD of HEC Montréal.

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Abdou, J.M., Pnevmatikos, N. Asymptotic Value in Frequency-Dependent Games with Separable Payoffs: A Differential Approach. Dyn Games Appl 9, 295–313 (2019).

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  • Stochastic game
  • Frequency-dependent payoffs
  • Continuous time game
  • Discretization
  • Hamilton–Jacobi–Bellman–Isaacs equation

Mathematics Subject Classification:

  • 91A15
  • 91A23
  • 91A25