# On the asymptotic nature of first order mean field games

## TL;DR

This paper proves that symmetric N-player Nash equilibria in finite horizon first order mean field games converge to solutions of the mean field game, linking Lagrangian and PDE formulations, and establishing convergence in Markov strategies.

## Contribution

It provides a simple proof of convergence of Nash equilibria to mean field game solutions and connects Lagrangian and PDE approaches.

## Key findings

- Nash equilibria converge to mean field game solutions
- Lagrangian solutions relate to PDE formulations
- Convergence established for distributed Markov strategies

## Abstract

For a class of finite horizon first order mean field games and associated N-player games, we give a simple proof of convergence of symmetric N-player Nash equilibria in distributed open-loop strategies to solutions of the mean field game in Lagrangian form. Lagrangian solutions are then connected with those determined by the usual mean field game system of two coupled first order PDEs, and convergence of Nash equilibria in distributed Markov strategies is established.