OPTIMAL COARSE CORRELATED EQUILIBRIA IN MEAN FIELD GAMES: LINEAR PROGRAMMING AND NO-REGRET LEARNING

Mean field games (MFGs) are typically solved using the Nash equilibrium concept, where players optimize against a population flow. However, in many settings, a moderator can influence outcomes by sending recommendations to agents. This paper addresses the selection problem for coarse correlated equilibria (CCEs) in continuous-time MFGs, where a moderator aims to select a CCE that optimizes a specific performance criterion, which may differ from the individual players' objectives.

The authors propose a two-part solution. First, they develop a linear programming (LP) formulation that relaxes the equilibrium conditions into linear constraints on occupation measures. This approach allows the moderator to optimize over the set of all possible mean-field CCEs, which is shown to be convex and compact under the paper's assumptions. Second, they translate the CCE optimality condition into an external-regret constraint. By using a Lagrangian formulation, they design a primal-dual learning algorithm that alternates between updating the primal variable (the probability measure over martingale flows) and the dual variable (the Lagrange multiplier for the regret constraint).

The study establishes that the LP formulation is a tractable framework for both proving the existence of optimal mean-field CCEs and computing them. The proposed no-regret primal-dual algorithm is shown to converge to an optimal LP-CCE at a rate of O(N^-1/2) for averaged iterates. This provides a constructive method for equilibrium selection that does not rely on the restrictive Lasry–Lions monotonicity condition, making it applicable to a wider range of games. Numerical experiments on flocking and emission abatement models demonstrate that the method successfully interpolates between Nash-type behavior and more coordinated outcomes.

This work bridges the gap between classical MFG theory and modern algorithmic game theory. By providing a systematic way to select among the often-abundant set of CCEs, the paper offers a powerful tool for incentive design and policy-making in large-scale systems. The use of no-regret learning ensures that the resulting equilibria are robust to deviations, providing a practical computational path for complex, continuous-time strategic interactions.