  We consider learning approximate Nash equilibria for discrete-time mean-field
games with nonlinear stochastic state dynamics subject to both average and
discounted costs. To this end, we introduce a mean-field equilibrium (MFE)
operator, whose fixed point is a mean-field equilibrium (i.e. equilibrium in
the infinite population limit). We first prove that this operator is a
contraction, and propose a learning algorithm to compute an approximate
mean-field equilibrium by approximating the MFE operator with a random one.
Moreover, using the contraction property of the MFE operator, we establish the
error analysis of the proposed learning algorithm. We then show that the
learned mean-field equilibrium constitutes an approximate Nash equilibrium for
finite-agent games.
