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Existence and Uniqueness for Mean Field Games with State Constraints

  • Piermarco Cannarsa
  • Rossana Capuani
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 28)

Abstract

In this paper, we study deterministic mean field games for agents who operate in a bounded domain. In this case, the existence and uniqueness of Nash equilibria cannot be deduced as for unrestricted state space because, for a large set of initial conditions, the uniqueness of the solution to the associated minimization problem is no longer guaranteed. We attack the problem by interpreting equilibria as measures in a space of arcs. In such a relaxed environment the existence of solutions follows by set-valued fixed point arguments. Then, we give a uniqueness result for such equilibria under a classical monotonicity assumption.

Keywords

Mean field games Nash equilibrium State constraints Hamilton-Jacobi-Bellman equations 

MSC Subject Classifications

49J15 49J30 49J53 49N90 

Notes

Acknowledgements

This work was partly supported by the University of Rome “Tor Vergata” (Consolidate the Foundations 2015) and by the Istituto Nazionale di Alta Matematica “F. Severi” (GNAMPA 2016 Research Projects). The second author is grateful to the Universitá Italo Francese (Vinci Project 2015).

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversitá di Roma “Tor Vergata”RomeItaly
  2. 2.CEREMADEUniversité Paris-DauphineParisFrance

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