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New Uniqueness Results For A Mean Field Game Of Controls

Jameson Graber, Elizabeth Matter

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.

New Uniqueness Results For A Mean Field Game Of Controls

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 , e.g. the market price or production, which then determines optimal trajectories for agents. Our approach consists in analyzing the relationship between 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.

Paper Structure

This paper contains 7 sections, 10 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Description of the Model
  3. Monotonicity of the Error Map
  4. When the Lagrangian does not depend on $x$
  5. When the Lagrangian depends on $x$
  6. Existence and Uniqueness
  7. Examples and Counterexamples

Key Result

Lemma 2.2

If $L$ satisfies Lassum, then there exists a unique solution of the Euler-Lagrange equations E-L. Its velocity is given by

Theorems & Definitions (29)

  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • ...and 19 more