Calculus of Variations and Geometric Measure Theory
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M. Bardi - M. Fischer

On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games

created by bardi on 31 Jul 2017
modified on 18 Jun 2018


Accepted Paper

Inserted: 31 jul 2017
Last Updated: 18 jun 2018

Journal: ESAIM Control Optim. Calc. Var.
Year: 2017

ArXiv: 1707.00628 PDF


This paper presents a class of evolutive Mean Field Games with multiple solutions for all time horizons $T$ and convex but non-smooth Hamiltonian $H$, as well as for smooth $H$ and $T$ large enough. The phenomenon is analysed in both the PDE and the probabilistic setting. The examples are compared with the current theory about uniqueness of solutions. In particular, a new result on uniqueness for the MFG PDEs with small data, e.g., small $T$, is proved. Some results are also extended to MFGs with two populations.

Keywords: uniqueness of solutions, Mean Field Games, finite horizon, non-uniqueness of solutions, multipopulation MFG


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