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.
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Notes
- 1.
We say that {μ y}y ∈ Y is a Borel family (of probability measures) if \(y\in Y\longmapsto \mu _y(B)\in \mathbb {R}\) is Borel for any Borel set B ⊂ X.
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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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Cannarsa, P., Capuani, R. (2018). Existence and Uniqueness for Mean Field Games with State Constraints. In: Cardaliaguet, P., Porretta, A., Salvarani, F. (eds) PDE Models for Multi-Agent Phenomena. Springer INdAM Series, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-030-01947-1_3
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DOI: https://doi.org/10.1007/978-3-030-01947-1_3
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