  In this paper we consider symmetric games where a large number of players can
be in any one of d states. We derive a limiting mean field model and
characterize its main properties. This mean field limit is a system of coupled
ordinary differential equations with initial-terminal data. For this mean field
problem we prove a trend to equilibrium theorem, that is convergence, in an
appropriate limit, to stationary solutions. Then we study the $N+1$-player
problem, which the mean field model attempts to approximate. Our main result is
the convergence as $N\to \infty$ of the mean field model and an estimate of the
rate of convergence. We end the paper with some further examples for potential
mean field games.
