Existence and Uniqueness for Mean Field Games with State Constraints

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


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.


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

MSC Subject Classifications

49J15 49J30 49J53 49N90 



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).


  1. 1.
    Ambrosio, L., Gigli, N., Savare, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures in Mathematics (ETH Zürich. Birkhäuser Verlag, Basel, 2008)Google Scholar
  2. 2.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis (Birkhäuser, Boston, 1990)zbMATHGoogle Scholar
  3. 3.
    Benamou, J.D., Carlier, G., Santambrogio, F., Variational Mean Field Games (Birkhäuser, Cham, 2017), pp. 141–171CrossRefGoogle Scholar
  4. 4.
    Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Progress in Nonlinear Differential Equations and their Applications, vol. 58 (Birkhäuser Boston Inc., Boston, 2004)Google Scholar
  5. 5.
    Cardaliaguet, P.: Notes on mean field games from P.-L. Lions’ lectures at Collège de France (2012).
  6. 6.
    Cardaliaguet, P., Marchi, C.: Regularity of the Eikonal equation with Neumann boundary conditions in the plane: application to fronts with nonlocal terms. SIAM J. Control Optim. 45, 1017–1038 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advance Mathematics (CRC Press, Ann Arbor, 1992)Google Scholar
  8. 8.
    Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainly equivalence principle. Commun. Inf. Syst. 6(3), 221–252 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Huang, M., Caines, P.E., Malhamé, R.P.: Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized 𝜖-Nash equilibria. IEEE Trans. Autom. Control 52(9), 1560–1571 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math J. 8(3), 457–459 (1941)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lasry, J.-M., Lions, P.-L.: Jeux á champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619–625 (2006)CrossRefGoogle Scholar
  12. 12.
    Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(9), 679–684 (2006)CrossRefGoogle Scholar
  13. 13.
    Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)MathSciNetCrossRefGoogle Scholar

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© 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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