From Nash to Cournot-Nash via $\Gamma$-convergence
Résumé
In this work we study the question of when will Nash equilibria of N -player games converge to a suitable notion of equilibrium of a game with a continuum of players. This is a central question in Mean Field Games theory, but instead we consider the notion of Cournot-Nash equilibrium. When such a game has a so-called potential structure, we give a complete characterization of equilibria as the stationary points of a functional, which we call a potential function. In particular, its minimizers are equilibria. In the sequel, we focus on the case that the players interact pair-wise; in this case we show that the original sequence of N -player games also admit a potential structure and prove that their corresponding potential functions converge in the sense of Γ-convergence to the potential function of the game with a continuum of players. This allows us to answer positively to the question of convergence of equilibria.
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