Linear-Quadratic Mean Field Games of Controls with Non-Monotone Data

TL;DR

This paper analyzes linear-quadratic mean field games of controls with common noise, establishing existence, uniqueness, and convergence results without requiring monotonicity conditions, and providing explicit solutions via the master equation.

Contribution

It introduces a framework for solving LQ mean field games of controls with common noise without monotonicity assumptions, including explicit solutions and convergence analysis.

Findings

Unique classical solution to the master equation without monotonicity.

Convergence of N-player games to the mean field limit.

Propagation of chaos for optimal trajectories.

Abstract

In this paper, we study a class of linear-quadratic (LQ) mean field games of controls with common noises and their corresponding $N$-player games. The theory of mean field game of controls considers a class of mean field games where the interaction is via the joint law of both the state and control. By the stochastic maximum principle, we first analyze the limiting behavior of the representative player and obtain his/her optimal control in a feedback form with the given distributional flow of the population and its control. The mean field equilibrium is determined by the Nash certainty equivalence (NCE) system. Thanks to the common noise, we do not require any monotonicity conditions for the solvability of the NCE system. We also study the master equation arising from LQ mean field games of controls, which is a finite-dimensional second-order parabolic equation. It can be shown that the master equation admits a unique classical solution over an arbitrary time horizon without any monotonicity conditions. Beyond that, we can solve the $N$-player games directly by further assuming the non-degeneracy of the idiosyncratic noises. As byproducts, we prove the quantitative convergence results from the $N$-player game to the mean field game and the propagation of chaos property for the related optimal trajectories.