Metric gradient flows with state dependent functionals: the Nash-MFG equilibrium flows and their numerical schemes

Gabriel Turinici (1, 2)

1 IUF - Institut universitaire de France
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We investigate the convergence of a relaxed version of the best reply numerical schemes (also known as best response or fictitious play) used to find Nash-mean field games equilibriums. This leads us to consider evolution equations in metric spaces similar to gradient flows except that the functional to be differentiated depends on the current point; these are called equilibrium flows. We give two definitions of solutions and prove that as the time step tends to zero the interpolated (`a la de Giorgi) numerical curves converge to equilibrium flows. As a by-product we obtain a sufficient condition for the uniqueness of a mean field games equilibrium. We close with applications to congestion and vaccination mean field games.

Mots clés: gradient flows; mean field games; vaccination games

Domaines: Analyse numérique [math.NA]; Probabilités [math.PR]; Statistiques [math.ST]; Analyse fonctionnelle [math.FA]; Géométrie métrique [math.MG]; Santé publique et épidémiologie

Gabriel Turinici. Metric gradient flows with state dependent functionals: the Nash-MFG equilibrium flows and their numerical schemes. Nonlinear Analysis: Theory, Methods and Applications, 2017, 165 (December 2017), pp.163-181. ⟨10.1016/j.na.2017.10.002⟩. ⟨hal-01528480v2⟩