An example of multiple mean field limits in ergodic differential games

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 $$N\rightarrow +\infty $$ , 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 (On the convergence of closed-loop Nash equilibria to the mean field game limit, 2018. arXiv:1808.02745 ) for finite horizon problems.