Cooperation, Correlation and Competition in Ergodic N-player Games and Mean-field Games of Singular Controls: A Case Study

TL;DR

This paper analyzes ergodic N-player and mean-field singular control games with strategic complementarities, explicitly computing equilibria and demonstrating how mean-field solutions approximate large-player game equilibria.

Contribution

It introduces a novel Lagrange multiplier approach for mean-field control and a new definition for mean-field coarse correlated equilibria in stationary settings.

Findings

Explicit equilibrium computations for three optimality notions.

Numerical comparison of payoffs and existence conditions.

Approximation of N-player equilibria by mean-field solutions for large N.

Abstract

We consider a class of $N$-player games and mean-field games of singular controls with ergodic performance criterion, providing a benchmark case for irreversible investment games featuring mean-field interaction and strategic complementarities. The state of each player follows a geometric Brownian motion, controlled additively through a nondecreasing process, while agents seek to maximize a long-term average reward functional with a power-type instantaneous profit, under strategic complementarity. We explore three different notions of optimality, which, in the mean-field limit, correspond to the mean-field control solution, mean-field coarse correlated equilibria, and mean-field Nash equilibria. We explicitly compute equilibria in the three cases and compare them numerically, in terms of yielded payoffs and existence conditions. Finally, we show that the mean-field control and mean-field equilibria can approximate the cooperative and competitive equilibria, respectively, in the corresponding $N$-player game when $N$ is sufficiently large. Our analysis of the mean-field control problem features a novel Lagrange multiplier approach, which proves crucial in establishing the approximation result, while the treatment of mean-field coarse correlated equilibria necessitates a new, specifically tailored definition for the stationary setting.