In this paper we study the long time convergence (trend to equilibrium problem) for finite state mean-field games using Gamma-convergence. Our
techniques are based upon the observation that an important class of mean-field games can be seen as the Euler-Lagrange equation of a suitable
functional. Therefore, by a scaling argument, one can convert the long time convergence problem into a Gamma-convergence problem. Our results
generalize previous results on long-time convergence for finite state problems.
