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

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Date
2017
Link to item file
https://hal.archives-ouvertes.fr/hal-01528480
Dewey
Probabilités et mathématiques appliquées
Sujet
gradient flows; mean field games; vaccination games
Journal issue
Nonlinear Analysis
Volume
165
Publication date
2017
Article pages
163-181
Publisher
Elsevier
DOI
http://dx.doi.org/10.1016/j.na.2017.10.002
URI
https://basepub.dauphine.fr/handle/123456789/16864
Collections
  • CEREMADE : Publications
Metadata
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Author
Turinici, Gabriel
Type
Article accepté pour publication ou publié
Abstract (EN)
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.

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