Mean field limit of a continuous time finite state game

Mean field games is a recent area of study introduced by Lions and Lasry in a series of seminal papers in 2006. Mean field games model situations of competition between large number of rational agents that play non-cooperative dynamic games under certain symmetry assumptions. They key step is to develop a mean field model, in a similar way that what is done in statistical physics in order to construct a mathematically tractable model. A main question that arises in the study of such mean field problems is the rigorous justification of the mean field models by a limiting procedure. In this paper we consider the mean field limit of two-state Markov decision problem as the number of players $N\to \infty$. First we establish the existence and uniqueness of a symmetric partial information Markov perfect equilibrium. Then we derive a mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. Our main result is the convergence as $N\to \infty$ of the $N$ player game to the mean field model and an estimate of the rate of convergence.