Mean Field Game systems describe equilibrium configurations in differential games with infinitely many infinitesimal interacting agents. We introduce a learning procedure (similar to the Fictitious Play) for these games and show its convergence when the Mean Field Game is potential.

In this article we consider finite Mean Field Games (MFGs), i.e. with finite time and finite states. We adopt the framework introduced in [15] and study two seemly unexplored subjects. In the first one, we analyze the convergence of the fictitious play learning procedure, inspired by the results in continuous MFGs (see [12] and [19]). In the second one, we consider the relation of some finite MFGs and continuous first order MFGs. Namely, given a continuous first order MFG problem and a sequence of refined space/time grids, we construct a sequence finite MFGs whose solutions admit limits points and every such limit point solves the continuous first order MFG problem.in [12], for a particular class of MFGs called potential MFGs. This analysis has then been extended in [19], by assuming that the MFG is monotone, which means that agents have aversion to imitate the strategies of other players. Under an analogous monotonicity assumption, we prove in Theorem 4 that the fictitious play procedure converges also in the case of finite MFGs.Our second contribution concerns the relation between continuous and finite MFGs. We consider here a first order continuous MFG and we associate to it a family of finite MFGs defined on finite space/time grids. By applying the results in [15], we know that for any fixed space/time grid the associated finite MFG admits at least one solution. Moreover, any such solution induces a probability measure on the space of strategies. Letting the grid length tend to zero, we prove that the aforementioned sequence of probability measures is precompact and, hence, has at least one limit point. The main result of this article is given in Theorem 4.1 and asserts that any such limit point is an equilibrium of the continuous MFG problem. To the best of our knowledge, this is the first result relating the equilibria for continuous MFGs, introduced in [23], with the equilibria for finite MFGs, introduced in [15].The article is organized as follows. In Section 2 we recall the finite MFG introduced in [15] and we state our first assumption that ensures the existence of at least one equilibrium. In Section 3 we describe the fictitious play procedure for the finite MFG and prove its convergence under a monotonicity assumption on the data. In Section 4 we introduce the first order continuous MFG under study, as well as the corresponding space/time discretization and the associated finite MFGs. As the length of the space/time grid tends to zero, we prove several asymptotic properties of the finite MFGs equilibria and we also prove our main result showing their convergence to a solution of the continuous MFG problem. Acknowledgements: The second author acknowledges financial support by the ANR project MFG ANR-16-CE40-0015-01 and the PEPS-INSMI Jeunes project "Some open problems in Mean Field Games" for the years 2016 and 2017. Both authors acknowledge financial support by the PGMO project VarPDEMFG. 2 The finite state and discrete time Mean Field Game problem We begin this section by presenting the MFG problem i...