Generalized conditional gradient and learning in potential mean field games

Understand

We apply the generalized conditional gradient algorithm to potential mean field games and we show its well-posedeness.

• It turns out that this method can be interpreted as a learning method called fictitious play.
• More precisely, each step of the generalized conditional gradient method amounts to compute the best-response of the representative agent, for a predicted value of the coupling terms of the game.
• We show that for the learning sequence $\delta_k = 2/(k+2)$, the potential cost converges in $O(1/k)$, the exploitability and the variables of the problem (distribution, congestion, price, value function and control terms) converge in $O(1/\sqrt{k})$, for specific norms.