  Mean field games model equilibria in games with a continuum of players as
limiting systems of symmetric $n$-player games with weak interaction between
the players. We consider a finite-state, infinite-horizon problem with two cost
criteria: discounted and ergodic. Under the Lasry--Lions monotonicity condition
we characterize the stationary ergodic mean field game equilibrium by a mean
field game system of two coupled equations: one for the value and the other for
the stationary measure. This system is linked with the ergodic master equation.
Several discounted mean field game systems are utilized in order to set up the
relevant discounted master equations. We show that the discounted master
equations are smooth, uniformly in the discount factor. Taking the discount
factor to zero, we achieve the smoothness of the ergodic master equation.
