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

Existence and Uniqueness Result for Mean Field Games with Congestion Effect on Graphs This paper presents a general existence and uniqueness result for mean field games equations on graphs ( $$\mathcal {G}$$ 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 ). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

, Volume 72 (2) – Oct 1, 2015
13 pages

/lp/springer_journal/existence-and-uniqueness-result-for-mean-field-games-with-congestion-lj0ZoaRZbp
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-014-9280-2
Publisher site
See Article on Publisher Site

### Abstract

This paper presents a general existence and uniqueness result for mean field games equations on graphs ( $$\mathcal {G}$$ 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 ).

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 2015

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