Existence and Uniqueness for Mean Field Games with State Constraints
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.
KeywordsMean field games Nash equilibrium State constraints Hamilton-Jacobi-Bellman equations
MSC Subject Classifications49J15 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.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
- 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.Cardaliaguet, P.: Notes on mean field games from P.-L. Lions’ lectures at Collège de France (2012). https://www.ceremade.dauphine.fr/~cardalia/MFG100629.pdf
- 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