On some singular mean-field games

Marco Cirant; Diogo A. Gomes; Edgard A. Pimentel; Héctor Sánchez-Morgado

doi:10.3934/jdg.2021006

Article Contents

2021, Volume 8, Issue 4: 445-465. Doi: 10.3934/jdg.2021006

This issue Previous Article Linear-quadratic zero-sum mean-field type games: Optimality conditions and policy optimization Next Article Splitting methods for a class of non-potential mean field games

On some singular mean-field games

1.
Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121, Padova, Italy
2.
King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia
3.
Department of Mathematics, Pontifícia Universidade Católica do Rio de Janeiro, 22451-900, Rio de Janeiro, Brazil
4.
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cd. México 04510, México

Received: May 2020

Early access: March 2021

Published: October 2021

M. Cirant is partially supported by the Fondazione CaRiPaRo Project "Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games" and the INdAM-GNAMPA project "Fenomeni di segregazione in sistemi stazionari di tipo Mean Field Games a più popolazioni".
D. Gomes was partially supported by KAUST baseline and start-up funds.
E. Pimentel was partially supported by FAPESP (Grant 2015/13011-6) and PUC-Rio baseline funds.

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Abstract

Here, we prove the existence of smooth solutions for mean-field games with a singular mean-field coupling; that is, a coupling in the Hamilton-Jacobi equation of the form $ g(m) = -m^{- \alpha} $ with $ \alpha>0 $. We consider stationary and time-dependent settings. The function $ g $ is monotone, but it is not bounded from below. With the exception of the logarithmic coupling, this is the first time that MFGs whose coupling is not bounded from below is examined in the literature. This coupling arises in models where agents have a strong preference for low-density regions. Paradoxically, this causes the agents move towards low-density regions and, thus, prevents the creation of those regions. To prove the existence of solutions, we consider an approximate problem for which the existence of smooth solutions is known. Then, we prove new a priori bounds for the solutions that show that $ \frac 1 m $ is bounded. Finally, using a limiting argument, we obtain the existence of solutions. The proof in the stationary case relies on a blow-up argument and in the time-dependent case on new bounds for $ m^{-1} $.

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Mathematics Subject Classification: 35A01, 35K61, 35B45.

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