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

Abstract

Abstract-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 crosscoupling mean-field term measuring the congestion resulting from the collective behavior, which motivates the term "crowdaverse". 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.

Strategic thinking under social influence: Scalability, stability and robustness of allocations

“…The problem has connections to recent research on robust dynamic coalitional TU games [12] and robust mean-field games [9,13]. A quantitative analysis of the approximation error of the solution presented is left as future work.…”

“…In other words, the allocation strategies are built on aggregate information and therefore their structures do not depend on the number of players. Such policies are obtained by carrying out a robust control design based on augmentation and regularization of the state space [13].…”

“…This section describes a simple heuristic approach toward solving the set of equations (8), based on state space augmentation [13]. The augmented state space includes the mean distribution, and thus the augmented state variables evolve according to the equations…”

Crowd-Averse Cyber-Physical Systems: The Paradigm of Robust Mean-Field Games

“…The method was first developed in [3] and it is here adapted to the problem at hand. The heuristic approach reframes the problem in an extended state space involving both the state of the player and the average state distribution.…”

“…Using the above expression of the Hamiltonian in the HJI equation in (19), we obtain the HJI in (3).…”