  Mean Field Game (MFG) systems describe equilibrium configurations in games
with infinitely many interacting controllers. We are interested in the behavior
of this system as the horizon becomes large, or as the discount factor tends to
$0$. We show that, in the two cases, the asymptotic behavior of the Mean Field
Game system is strongly related with the long time behavior of the so-called
master equation and with the vanishing discount limit of the discounted master
equation, respectively. Both equations are nonlinear transport equations in the
space of measures. We prove the existence of a solution to an ergodic master
equation, towards which the time-dependent master equation converges as the
horizon becomes large, and towards which the discounted master equation
converges as the discount factor tends to $0$. The whole analysis is based on
the obtention of new estimates for the exponential rates of convergence of the
time-dependent MFG system and the discounted MFG system.
