# An example of multiple mean field limits in ergodic differential games

## TL;DR

This paper presents a symmetric ergodic N-players differential game where the Nash equilibrium payoffs' limit set expands as N grows, despite having a unique mean field game equilibrium, contrasting finite horizon results.

## Contribution

It provides a novel example demonstrating multiple mean field limits in ergodic differential games, highlighting differences from finite horizon cases.

## Key findings

- Limit set of Nash payoffs becomes large as N increases.
- Single mean field game equilibrium exists despite multiple limits.
- Contrasts with finite horizon game results.

## Abstract

We present an example of symmetric ergodic $N$-players differential games, played in memory strategies on the position of the players, for which the limit set, as $N\to +\infty$, of Nash equilibrium payoffs is large, although the game has a single mean field game equilibrium. This example is in sharp contrast with a result by Lacker [23] for finite horizon problems.