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Chinese Annals of Mathematics, Series B

, Volume 34, Issue 2, pp 161–192 | Cite as

Control and nash games with mean field effect

  • Alain Bensoussan
  • Jens Frehse
Article

Abstract

Mean field theory has raised a lot of interest in the recent years (see in particular the results of Lasry-Lions in 2006 and 2007, of Gueant-Lasry-Lions in 2011, of Huang-Caines-Malham in 2007 and many others). There are a lot of applications. In general, the applications concern approximating an infinite number of players with common behavior by a representative agent. This agent has to solve a control problem perturbed by a field equation, representing in some way the behavior of the average infinite number of agents. This approach does not lead easily to the problems of Nash equilibrium for a finite number of players, perturbed by field equations, unless one considers averaging within different groups, which has not been done in the literature, and seems quite challenging. In this paper, the authors approach similar problems with a different motivation which makes sense for control and also for differential games. Thus the systems of nonlinear partial differential equations with mean field terms, which have not been addressed in the literature so far, are considered here.

Keywords

Mean field Dynamic programming Nash games Equilibrium Calculus of variations 

2000 MR Subject Classification

49L20 

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Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.International Center for Decision and Risk Analysis, School of ManagementUniversity of Texas-DallasDallasFrance
  2. 2.School of Businessthe Hong Kong Polytechnic UniversityHong KongChina
  3. 3.Graduate Department of Financial EngineeringAjou UniversityAjouKorea
  4. 4.Institute for Applied MathematicsUniversity of BonnBonnGermany

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