Generalized conditional gradient and learning in potential mean field games

TL;DR

This paper demonstrates the application of the generalized conditional gradient algorithm to potential mean field games, showing convergence properties and interpreting it as a fictitious play learning method.

Contribution

It introduces a novel application of the generalized conditional gradient method to potential mean field games and analyzes its convergence as a learning process.

Findings

Potential cost converges at a rate of O(1/k).

Variables such as distribution, congestion, and value function converge at O(1/√k).

The method is well-posed and interpretable as fictitious play.

Abstract

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.