Optimal Coarse Correlated Equilibria in Mean Field Games: Linear Programming and No-Regret Learning

2026-06-18

AI Summary

This study addresses the problem of optimal coarse correlated equilibria in continuous-time mean field games, where a coordinator selects equilibria to optimize an objective that may differ from individual agents’ utilities. The work introduces, for the first time, the notion of an optimal coarse correlated equilibrium, provides a linear programming characterization, and establishes its existence. Building on Lagrangian duality theory, the authors develop a no-regret learning algorithm with explicit convergence rates. Numerical experiments demonstrate the efficacy of the proposed approach, offering both a theoretical foundation and computational tools for designing coordination mechanisms in mean field games.

Abstract

We introduce optimal coarse correlated equilibria for continuous-time mean field games. A coarse correlated equilibrium is a randomized recommendation scheme from which no player can gain by ignoring the recommendation and switching to an alternative strategy. The problem is as follows: a moderator selects, among all mean-field coarse correlated equilibria, one that optimizes a prescribed performance criterion, which may differ from the representative player's objective. After formulating the problem, we develop a linear programming (LP) formulation, prove the existence of optimal LP coarse correlated equilibria, and relate the LP characterization to the original probabilistic setting. Building on this characterization, we design a no-regret primal-dual algorithm, based on an equivalent Lagrangian formulation of the external-regret constraint, for learning such equilibria. We provide explicit convergence rates for the learning algorithm, and numerical examples illustrate the method.

Problem

• mean field games
• coarse correlated equilibria
• optimal equilibrium selection
• performance criterion

Innovation

• coarse correlated equilibrium
• mean field games
• linear programming
• no-regret learning
• primal-dual algorithm