New Uniqueness Results For A Mean Field Game Of Controls

TL;DR

This paper introduces a novel method for establishing the uniqueness of solutions in a class of mean field games of controls, focusing on the relationship between aggregate quantities and agent trajectories, applicable to economic models.

Contribution

It presents a new approach that relaxes traditional monotonicity conditions to prove uniqueness in mean field games of controls, expanding applicability to economic models.

Findings

New conditions for uniqueness that differ from Lasry-Lions monotonicity

Applicable to existing economic models with aggregate controls

Provides a framework for analyzing equilibrium uniqueness

Abstract

We propose a new approach to proving the uniqueness of solutions to a certain class of mean field games of controls. In this class, the equilibrium is determined by an aggregate quantity $Q(t)$, e.g. the market price or production, which then determines optimal trajectories for agents. Our approach consists in analyzing the relationship between $Q(t)$ and corresponding optimal trajectories to find conditions under which there is at most one equilibrium. We show that our conditions do not match those prescribed by the Lasry-Lions monotonicity condition, nor even displacement monotonicity, but they do apply to economic models that have been proposed in the literature.