A Machine Learning Method for Stackelberg Mean Field Games

We propose a single-level numerical approach to solve Stackelberg mean field game (MFG) problems. In the Stackelberg MFG, an infinite population of agents plays a noncooperative game and chooses their controls to optimize their individual objectives while interacting with the principal and other agents through the population distribution. The principal can influence the mean field Nash equilibrium at the population level through policies, and she optimizes her own objective, which depends on the population distribution. This leads to a bilevel problem between the principal and mean field of agents that cannot be solved using traditional methods for MFGs. We propose a reformulation of this problem as a single-level mean field optimal control problem through a penalization approach. We prove convergence of the reformulated problem to the original problem. We propose a machine learning method based on (feed-forward and recurrent) neural networks and illustrate it on several examples from the literature. We also give a modified example to show the scalability of the proposed method and show its efficiency by comparing its performance with an alternative bilevel approach.