  Mean field games (MFGs) model equilibria in games with a continuum of weakly
interacting players as limiting systems of symmetric $n$-player games. We
consider the finite-state, infinite-horizon problem with ergodic cost. Assuming
Markovian strategies, we first prove that any solution to the MFG system gives
rise to a $(C/\sqrt{n})$-Nash equilibrium in the $n$-player game. We follow
this result by proving the same is true for the strategy profile derived from
the master equation. We conclude the main theoretical portion of the paper by
establishing a large deviation principle for empirical measures associated with
the asymptotic Nash equilibria. Then, we contrast the asymptotic Nash
equilibria using an example. We solve the MFG system directly and numerically
solve the ergodic master equation by adapting the deep Galerkin method of
Sirignano and Spiliopoulos. We use these results to derive the strategies of
the asymptotic Nash equilibria and compare them. Finally, we derive an explicit
form for the rate functions in dimension two.
