Stationary Discounted and Ergodic Mean Field Games with Singular Controls

We study stationary mean field games with singular controls in which the representative player interacts with a long-time weighted average of the population through a discounted and ergodic performance criteria. This class of games finds natural applications in the context of optimal productivity expansion in dynamic oligopolies. We prove the existence and uniqueness of the mean field equilibria, which are completely characterized through nonlinear equations. Furthermore, we relate the mean field equilibria for the discounted and ergodic games by showing the validity of an Abelian limit. The latter also allows us to approximate Nash equilibria of—so far unexplored—symmetric N-player ergodic singular control games through the mean field equilibrium of the discounted game. Numerical examples finally illustrate in a case study the dependency of the mean field equilibria with respect to the parameters of the games.