The Convergence Problem in Mean Field Games with Neumann Boundary Conditions

TL;DR

This paper investigates how Nash Equilibria in N-player differential games with reflection boundary conditions converge to mean field game solutions, using the Master Equation to establish convergence of finite-dimensional projections.

Contribution

It introduces a framework for analyzing convergence in mean field games with Neumann boundary conditions, leveraging the well-posedness of the Master Equation.

Findings

Nash Equilibria converge to mean field game solutions

Finite-dimensional projections approximate the Nash system

The Master Equation solution facilitates convergence analysis

Abstract

In this article we study the convergence of the Nash Equilibria in a N-player differential game towards the optimal strategies in the Mean Field Games, when the dynamic of the generic player includes a reflection process which guarantees the invariance of the state space. The well-posedness of the Master Equation allows us to use its solution U in order to construct finite dimensional projections, which will converge, in some suitable spaces, to the solution of the Nash system.