A Lagrangian approach for aggregative mean field games of controls with mixed and final constraints

TL;DR

This paper investigates the existence and uniqueness of equilibria in deterministic mean field games of controls with mixed and final constraints, using a Lagrangian approach and fixed point theorems.

Contribution

It introduces a Lagrangian framework for analyzing mean field games with complex constraints and nonlinear dynamics, extending existing methods to more general settings.

Findings

Existence of equilibria established via Kakutani's fixed point theorem.

Uniqueness of equilibria shown under monotonicity conditions.

Framework accommodates congestion, price functions, and mixed constraints.

Abstract

The objective of this paper is to analyze the existence of equilibria for a class of deterministic mean field games of controls. The interaction between players is due to both a congestion term and a price function which depends on the distributions of the optimal strategies. Moreover, final state and mixed state-control constraints are considered, the dynamics being nonlinear and affine with respect to the control. The existence of equilibria is obtained by Kakutani's theorem, applied to a fixed point formulation of the problem. Finally, uniqueness results are shown under monotonicity assumptions.