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On efficiency in mean field differential games

We investigate the efficiency of Nash equilibria of a class of Mean Field Games. We focus on the stationary case with entry and exit of players, and derive an expression for the social cost at a Nash equilibrium, based on value function and agent density. We propose a model for a Mean Field Congestion Game, in which the agents' control cost depends (locally) on the agent density. We present numerical results that show that the Nash equilibria of these games are inefficient in general. Also, we point out an interesting paradox, which can be seen as a continuous analogue of Braess's paradox known from selfish routing games. Finally, we cast the welfare maximization problem as a PDE-constrained optimization problem.