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Control and Nash Games with Mean Field Effect

In: Partial Differential Equations: Theory, Control and Approximation

Author

Listed:
  • Alain Bensoussan

    (University of Texas-Dallas, International Center for Decision and Risk Analysis, School of Management
    The Hong Kong Polytechnic University, School of Business
    Ajou University, Graduate Department of Financial Engineering)

  • Jens Frehse

    (University of Bonn, Institute for Applied Mathematics)

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.

Suggested Citation

  • Alain Bensoussan & Jens Frehse, 2014. "Control and Nash Games with Mean Field Effect," Springer Books, in: Philippe G. Ciarlet & Tatsien Li & Yvon Maday (ed.), Partial Differential Equations: Theory, Control and Approximation, edition 127, pages 1-39, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-41401-5_1
    DOI: 10.1007/978-3-642-41401-5_1
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