  We show the convergence of finite state symmetric N-player differential
games, where players control their transition rates from state to state, to a
limiting dynamics given by a finite state Mean Field Game system made of two
coupled forward-backward ODEs. We exploit the so-called Master Equation, which
in this finite-dimensional framework is a first order PDE in the simplex of
probability measures, obtaining the convergence of the feedback Nash
equilibria, the value functions and the optimal trajectories. The convergence
argument requires only the regularity of a solution to the Master equation.
Moreover, we employ the convergence method to prove a Central Limit Theorem and
a Large Deviation Principle for the evolution of the N-player empirical
measures. The well-posedness and regularity of solution to the Master Equation
are also studied.
