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Applied Mathematics & Optimization

, Volume 72, Issue 2, pp 291–303 | Cite as

Existence and Uniqueness Result for Mean Field Games with Congestion Effect on Graphs

  • Olivier Guéant
Article

Abstract

This paper presents a general existence and uniqueness result for mean field games equations on graphs (\(\mathcal {G}\)-MFG). In particular, our setting allows to take into account congestion effects of almost any form. These general congestion effects are particularly relevant in graphs in which the cost to move from one node to another may for instance depend on the proportion of players in both the source node and the target node. Existence is proved using a priori estimates and a fixed point argument à la Schauder. We propose a new criterion to ensure uniqueness in the case of Hamiltonian functions with a complex (non-local) structure. This result generalizes the discrete counterpart of uniqueness results obtained in Lasry and Lions (C. R. Acad. Sci. Paris 343(10):679–684, 2006). Lions (http://www.college-de-france.fr/default/EN/all/equ_der/audio_video.jsp, 2014).

Keywords

Mean field games Discrete state space Congestion effect 

Notes

Acknowledgments

The author wishes to acknowledge the helpful conversations with Yves Achdou (Université Paris-Diderot), François Delarue (Université Nice Sophia Antipolis), Jean-Michel Lasry (Université Paris-Dauphine) and Pierre-Louis Lions (Collège de France).

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.UFR de Mathématiques, Laboratoire Jacques-Louis LionsUniversité Paris-DiderotParisFrance

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