Mean-field games (MFGs) are models for large populations of competing rational agents that seek to optimize a suitable functional. In the case of congestion, this functional takes into account the difficulty of moving in high-density areas. Here, we study stationary MFGs with congestion with quadratic or power-like Hamiltonians. First, using explicit examples, we illustrate two main difficulties: the lack of classical solutions and the existence of areas with vanishing densities. Our main contribution is a new variational formulation for MFGs with congestion. With this formulation, we prove the existence and uniqueness of solutions. Finally, we consider applications to numerical methods.
|Original language||English (US)|
|Number of pages||38|
|State||Published - Apr 30 2018|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): OSR-CRG2017-3452
Acknowledgements: The authors were partially supported by King Abdullah University of Science and Technology (KAUST) baseline and start-up funds, and grant OSR-CRG2017-3452.