Discrete time,finite state space 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 was introduced by Lasry and Lions (2006, 2007) [3], [4], [5]. The discrete time, finite state space setting is motivated both by its independent interest as well as by numerical analysis questions which appear in the discretization of the problems introduced by Lasry and Lions. The main contribution of this paper is the exponential convergence to equilibrium of the initial-terminal value problem.