On (in)efficiency of mean field equilibria

Mean field games (MFG) are dynamic games with infinitely many infinitesimal agents. In this context, we study the efficiency of Nash equilibria: Namely we compare the social cost of a MFG equilibrium with the terminal cost a global planner can achieve. In addition to its intrinsic interest, the result is used to provide an example of a N-players differential game for which the limit set, as N tends to infinity, of Nash equilibrium payoffs is large, although the game has a single MFG equilibrium.