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

2024-04-23

AI Summary

This paper studies an N-player stochastic game with irreversible investment and its corresponding mean-field game: each player controls a geometric Brownian motion state variable via nondecreasing singular controls to maximize long-run average power-type utility. Within an ergodic singular control framework, we pioneer the application of Lagrange multiplier methods to solve the mean-field control problem. We introduce the novel concept of “mean-field coarse correlated equilibrium” tailored to stationary settings, capturing strategic complementarities and interdependence among agents. Explicit constructions are provided for three types of equilibria—mean-field control, coarse correlated equilibrium, and Nash equilibrium—and we rigorously establish that, as (N o infty), both the coarse correlated and Nash equilibria converge to the mean-field solution. Numerical experiments further compare existence conditions and payoff differences across these equilibria.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Study N-player and mean-field games with ergodic singular controls
• Compare three optimality notions in mean-field game settings
• Approximate N-player equilibria using mean-field solutions for large N

Innovation

• Lagrange multiplier approach for mean-field control
• Novel definition for coarse correlated equilibria
• Geometric Brownian motion with additive control