Mean field control hierarchy

TL;DR

This paper models government influence on large populations as a mean field optimal control problem, deriving conditions for optimality and introducing a computationally efficient hierarchy of sub-optimal controls with numerical validation.

Contribution

It introduces a novel hierarchy of sub-optimal controls based on a Boltzmann approach for mean field control problems, with rigorous derivation and numerical experiments.

Findings

Existence of mean field optimal controls in stochastic and deterministic settings

Development of a Boltzmann-based hierarchy of sub-optimal controls

Numerical experiments demonstrating the control hierarchy in opinion formation models

Abstract

In this paper we model the role of a government of a large population as a mean field optimal control problem. Such control problems are constrainted by a PDE of continuity-type, governing the dynamics of the probability distribution of the agent population. We show the existence of mean field optimal controls both in the stochastic and deterministic setting. We derive rigorously the first order optimality conditions useful for numerical computation of mean field optimal controls. We introduce a novel approximating hierarchy of sub-optimal controls based on a Boltzmann approach, whose computation requires a very moderate numerical complexity with respect to the one of the optimal control. We provide numerical experiments for models in opinion formation comparing the behavior of the control hierarchy.