Crowd-Averse Robust Mean-Field Games: Approximation via State Space Extension

We consider a population of dynamic agents, also referred to as players. The state of each player evolves according to a linear stochastic differential equation driven by a Brownian motion and under the influence of a control and an adversarial disturbance. Every player minimizes a cost functional which involves quadratic terms on state and control plus a cross-coupling mean-field term measuring the congestion resulting from the collective behavior, which motivates the term "crowd-averse." Motivations for this model are analyzed and discussed in three main contexts: a stock market application, a production engineering example, and a dynamic demand management problem in power systems. For the problem in its abstract formulation, we illustrate the paradigm of robust mean-field games. Main contributions involve first the formulation of the problem as a robust mean-field game; second, the development of a new approximate solution approach based on the extension of the state space; third, a relaxation method to minimize the approximation error. Further results are provided for the scalar case, for which we establish performance bounds, and analyze stochastic stability of both the microscopic and the macroscopic dynamics.