Ergodic McKean-Vlasov Games: Verification Theorems and Linear-Quadratic Applications

TL;DR

This paper develops verification theorems for ergodic McKean-Vlasov games, linking solutions of coupled HJB Master equations to Nash equilibria, and applies the theory to explicit LQG models.

Contribution

It introduces a verification framework for ergodic McKean-Vlasov games and derives explicit solutions in LQG settings using polynomial measure structures.

Findings

Unique determination of value functions by invariant measures

Verification theorem connecting HJB Master equations to Nash equilibria

Explicit solutions for LQG models

Abstract

This paper investigates two-player ergodic nonzero-sum stochastic differential games with McKean-Vlasov dynamics. We establish a verification theorem connecting solutions of coupled Hamilton-Jacobi-Bellman (HJB) Master equations to Nash equilibria, characterized through an auxiliary control problem defined on the measure space. A key contribution is showing that the value functions are uniquely determined (up to an additive constant) by the uniqueness of the invariant measure of the optimal state process. The theory is applied to Linear-Quadratic-Gaussian (LQG) settings, where explicit solutions to the Master equations are derived by exploiting their polynomial structure in measure variables.