Mean Field Games of Controls with Dirichlet boundary conditions

TL;DR

This paper develops a mathematical framework for mean-field games with control influences and Dirichlet boundary conditions, establishing existence of Nash equilibria and providing examples and numerical methods.

Contribution

It introduces a novel mean-field control model with Dirichlet boundary conditions and proves the existence of Nash equilibria under regularity assumptions.

Findings

Existence of Nash equilibria in the proposed model.

Development of a priori estimates for non-conservative mass systems.

Numerical implementations demonstrating the model's applicability.

Abstract

In this paper we study a mean-field games system with Dirichlet boundary conditions in a closed domain and in a mean-field of control setting, that is in which the dynamics of each agent is affected not only by the average position of the rest of the agents but also by their average optimal choice. This setting allows the modeling of more realistic real-life scenarios in which agents not only will leave the domain at a certain point in time (like during the evacuation of pedestrians or in debt refinancing dynamics) but also act competitively to anticipate the strategies of the other agents. We shall establish the existence of Nash Equilibria for such class of mean-field of controls systems under certain regularity assumptions on the dynamics and the Lagrangian cost. Much of the paper is devoted to establishing several a priori estimates which are needed to circumvent the fact that the mass is not conserved (as we are in a Dirichlet boundary condition setting). In the conclusive sections, we provide examples of systems falling into our framework as well as numerical implementations.