This was part of Mathematical Advances in Mean Field Games
MFG Systems with Interactions though the Law of Controls: Existence, Uniqueness and Numerical Simulations
Ziad Kobeissi, Institut Louis Bachelier
Tuesday, December 14, 2021
Abstract: This talk is dedicated to a class of games in which agents may interact through their law of states and controls; we use the terminology mean field games of controls (MFGC for short) to refer to this class of games. We first give existence and uniqueness results under various sets of assumptions. We introduce a new structural condition, namely that the optimal dynamics depends upon the law of controls in a Lipschitz way, with a Lipschitz constant smaller than one. In this case, we give several existence results on the solutions of the MFGC system, and one uniqueness result under a short-time horizon assumption. Then, under a monotonicity assumption on the interactions through the law of controls (which can be interpreted as the adaptation of the Lasry-Lions monotonicity condition to the MFGC system), we prove existence and uniqueness of the solution of the MFGC system. Finally, numerical simulations of a model of crowd motion are presented, in which an agent is more likely to go in the mainstream direction (which is in contradiction with the above-mentioned monotonicity condition). Original behaviors of the agents are observed leading to non-uniqueness or queue formation.