Meanfield games (MFG) are models of large populations of rational agents who seek to optimize an objective function that takes into account their state variables and the distribution of the state variable of the remaining agents. MFG with congestion model problems where the agentsâ€™ motion is hampered in highdensity regions.
First, we study radial solutions for first and secondorder stationary MFG with congestion on Rd. The radial case, which is one of the simplest non onedimensional MFG, is relatively tractable. As we observe, the FokkerPlanck equation is integrable with respect to one of the unknowns. Consequently, we obtain a single equation substituting this solution into the HamiltonJacobi equation. For the firstorder case, we derive explicit formulas; for the elliptic case, we study a variational formulation of the resulting equation. For the first case, we use our approach to compute numerical approximations to the solutions of the corresponding MFG systems.
Next, we consider secondorder stationary MFG with congestion and prove the existence of stationary solutions. Because moving in congested areas is difficult, agents prefer to move in noncongested areas. As a consequence, the model becomes singular near the zero density. The existence of stationary solutions was previously obtained for MFG with quadratic Hamiltonians thanks to a very particular identity. Here, we develop robust estimates that give the existence of a solution for general subquadratic Hamiltonians.
Additionally, we study firstorder stationary MFG with congestion with quadratic or powerlike Hamiltonians. Using explicit examples, we illustrate two key difficulties: the lack of classical solutions and the existence of areas with vanishing densities. Our
main contribution is a new variational formulation for MFG with congestion. With this formulation, we prove the existence and uniqueness of solutions. Finally, we devise a discretization that is combined with optimization algorithms to numerically solve various MFG with congestion.
Date of Award  Jun 23 2019 

Original language  English (US) 

Awarding Institution   Computer, Electrical and Mathematical Science and Engineering


Supervisor  Diogo Gomes (Supervisor) 

 meanfield games
 congestion problems
 stationary problems
 calculus f variations