Cooperation, Correlation, and Competition in Ergodic <i>N</i> -Player Games and Mean-Field Games of Singular Controls: A Case Study

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, whereas 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, whereas the treatment of mean-field coarse correlated equilibria necessitates a new, specifically tailored definition for the stationary setting. Funding: The authors acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [Project-ID 317210226– SFB 1283]. F. Cannerozzi acknowledges financial support from the European Union—NextGenerationEU [Grant NRRP-CUP G53D23006840001].