Discrete mean field games

In this paper we study a mean field model for discrete time, finite number of states, dynamic games. These models arise in situations that involve a very large number of agents moving from state to state according to certain optimality criteria. The mean field approach for optimal control and differential games (continuous state and time) was introduced by Lasry and Lions. Here we consider a discrete version of the problem. Our setting is the following: we assume that there is a very large number of identical agents which can be in a finite number of states. Because the number of agents is very large, we assume the mean field hypothesis, that is, that the only relevant information for the global evolution is the fraction $\pi^n_i$ of players in each state $i$ at time $n$. The agents look for minimizing a running cost, which depends on $\pi$, plus a terminal cost $V^N$. In contrast with optimal control, where usually only the terminal cost $V^N$ is necessary to solve the problem, in mean-field games both the initial distribution of agents $\pi^0$ and the terminal cost $V^N$ are necessary to determine the solutions, that is, the distribution of players $\pi^n$ and value function $V^n$, for $0\leq n\leq N$. Because both initial and terminal data needs to be specified, we call this problem the initial-terminal value problem. Existence of solutions is non-trivial. Nevertheless, following the ideas of Lasry and Lions, we were able to establish existence and uniqueness, both for the stationary and for the initial-terminal value problems. In the last part of the paper we prove the main result of the paper, namely the exponential convergence to a stationary solution of $(\pi^0, V^0)$, as $N\to \infty$, for the initial-terminal value problem with (fixed) data $\pi^{-N}$ and $V^N$.