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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As far as we know, in general stochastic games, payoffs with separable structure first appeared in . The authors study games with transitions independent of the current state.
Existence of the value follows from the standard comparison and uniqueness theorems for viscosity solutions presented in .
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.
In the literature, the lower and upper values of a differential game have been first characterized by means of DPP in .
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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). https://doi.org/10.1007/s13235-018-0278-2
- Stochastic game
- Frequency-dependent payoffs
- Continuous time game
- Hamilton–Jacobi–Bellman–Isaacs equation
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