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Variational Mean Field Games for Market Competition

  • Philip Jameson GraberEmail author
  • Charafeddine Mouzouni
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 28)

Abstract

In this paper, we explore Bertrand and Cournot Mean Field Games models for market competition with reflection boundary conditions. We prove existence, uniqueness and regularity of solutions to the system of equations, and show that this system can be written as an optimality condition of a convex minimization problem. We also provide a short proof of uniqueness to the system addressed in Graber and Bensoussan (Appl Math Optim 77:47–71, 2018), where uniqueness was only proved for small parameters 𝜖. Finally, we prove existence and uniqueness of weak solutions to the corresponding first order system at the deterministic limit.

Keywords

Cournot competition Extended mean field games Optimal control Forward-backward systems of PDE 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Baylor UniversityDepartment of MathematicsWacoUSA
  2. 2.Univ LyonÉcole centrale de Lyon, CNRS UMR 5208, Institut Camille JordanEcully CedexFrance

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