# Singular mean-filed games

Handle URI:
http://hdl.handle.net/10754/626543
Title:
Singular mean-filed games
Authors:
Cirant, Marco; Gomes, Diogo A. ( 0000-0002-3129-3956 ) ; Pimentel, Edgard A.; Sánchez-Morgado, Héctor
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 $g(m)=-m^{-\alpha}$. 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 to spread and prevents the creation of solutions with a very-low density. 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 $m^{-1}$.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
arXiv
Issue Date:
22-Nov-2016
ARXIV:
arXiv:1611.07187
Type:
Preprint
http://arxiv.org/abs/1611.07187v1; http://arxiv.org/pdf/1611.07187v1
Appears in Collections:
Other/General Submission; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

# Full metadata record

DC FieldValue Language
dc.contributor.authorCirant, Marcoen
dc.contributor.authorGomes, Diogo A.en
dc.contributor.authorPimentel, Edgard A.en
dc.date.accessioned2017-12-28T07:32:15Z-
dc.date.available2017-12-28T07:32:15Z-
dc.date.issued2016-11-22en
dc.identifier.urihttp://hdl.handle.net/10754/626543-
dc.description.abstractHere, 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 $g(m)=-m^{-\alpha}$. 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 to spread and prevents the creation of solutions with a very-low density. 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 $m^{-1}$.en
dc.publisherarXiven
dc.relation.urlhttp://arxiv.org/abs/1611.07187v1en
dc.relation.urlhttp://arxiv.org/pdf/1611.07187v1en
dc.rightsArchived with thanks to arXiven
dc.titleSingular mean-filed gamesen
dc.typePreprinten
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.eprint.versionPre-printen
dc.contributor.institutionDipartimento di Matematica, Universita di Padova, Via Trieste 63, 35121, Padova, Italy.en
dc.contributor.institutionDepartment of Mathematics, Universidade Federal de Sao Carlos, 13506-905, Sao Carlos, Brazil.en
dc.contributor.institutionInstituto de Matematicas, Universidad Nacional Autonoma de Mexico. Cd. Mexico 04510. Mexico.en
dc.identifier.arxividarXiv:1611.07187en
kaust.authorGomes, Diogo A.en
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