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We study stationary mean field games with singular controls in which therepresentative player interacts with a long-time weighted average of thepopulation through a discounted and an ergodic performance criterion. Thisclass of games finds natural applications in the context of optimalproductivity expansion in dynamic oligopolies. We prove existence anduniqueness of the mean field equilibria, which are completely characterizedthrough nonlinear equations. Furthermore, we relate the mean field equilibriafor the discounted and the ergodic games by showing the validity of an Abelianlimit. The latter allows also to approximate Nash equilibria of - so farunexplored - symmetric N-player ergodic singular control games through the meanfield equilibrium of the discounted game. Numerical examples finally illustratein a case study the dependency of the mean field equilibria with respect to theparameters of the games.

Stationary Discounted and Ergodic Mean Field Games of Singular Control