Discrete Mean Field Games: Existence of Equilibria and Convergence

Josu Doncel 1, 2 Nicolas Gast 3, 4 Bruno Gaujal 3, 4
2 DYOGENE - Dynamics of Geometric Networks
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique : UMR 8548, Inria de Paris
4 POLARIS - Performance analysis and optimization of LARge Infrastructures and Systems
Inria Grenoble - Rhône-Alpes, LIG - Laboratoire d'Informatique de Grenoble
Abstract : We consider mean field games with discrete state spaces (called discrete mean field games in the following) and we analyze these games in continuous and discrete time, over finite as well as infinite time horizons. We prove the existence of a mean field equilibrium assuming continuity of the cost and of the drift. These conditions are more general than the existing papers studying finite state space mean field games. Besides, we also study the convergence of the equilibria of N -player games to mean field equilibria in our four settings. On the one hand, we define a class of strategies in which any sequence of equilibria of the finite games converges weakly to a mean field equilibrium when the number of players goes to infinity. On the other hand, we exhibit equilibria outside this class that do not converge to mean field equilibria and for which the value of the game does not converge. In discrete time this non- convergence phenomenon implies that the Folk theorem does not scale to the mean field limit.
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Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete Mean Field Games: Existence of Equilibria and Convergence. Journal of Dynamics & Games, 2019, pp.1-19. ⟨10.3934/jdg.2019016⟩. ⟨hal-01277098v2⟩

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