  In a probabilistic mean-field game driven by a linear diffusion an individual
player aims to minimize an ergodic long-run cost by controlling the diffusion
through a pair of -- increasing and decreasing -- c\`adl\`ag processes, while
he is interacting with an aggregate of players through the expectation of a
similar diffusion controlled by another pair of c\`adl\`ag processes. In order
to find equilibrium points in this game, we first consider the control problem,
in which the individual player has no interaction with the aggregate of
players. In this case, we prove that the best policy is to reflect the
diffusion process within two thresholds. Based on these results, we obtain
criteria for the existence of equilibrium points in the mean-field game in the
case when the controls of the aggregate of players are of reflection type, and
give a pair of nonlinear equations to find these equilibrium points. In
addition, we present an approximation result for Nash equilibria of erdogic
games with finitely many players to the mean-field game equilibria considered
above when the number of players tends to infinity. These results are
illustrated by several examples where the existence and uniqueness of the
equilibrium points depend on the coefficients of the underlying diffusion.
