Existence and uniqueness for Mean Field Games with state constraints

TL;DR

This paper investigates the existence and uniqueness of Nash equilibria in deterministic mean field games with agents confined to a bounded domain, using measure-based approaches and fixed point arguments.

Contribution

It introduces a measure-based framework for analyzing mean field games with state constraints and establishes existence and uniqueness results under classical assumptions.

Findings

Existence of solutions via fixed point methods

Uniqueness under monotonicity assumptions

Framework applicable to bounded domain scenarios

Abstract

In this paper, we study deterministic mean field games for agents who operate in a bounded domain. In this case, the existence and uniqueness of Nash equilibria cannot be deduced as for unrestricted state space because, for a large set of initial conditions, the uniqueness of the solution to the associated minimization problem is no longer guaranteed. We attack the problem by interpreting equilibria as measures in a space of arcs. In such a relaxed environment the existence of solutions follows by set-valued fixed point arguments. Then, we give a uniqueness result for such equilibria under a classical monotonicity assumption.