Mean field social optimization: feedback person-by-person optimality and the dynamic programming equation

TL;DR

This paper develops a new Hamilton-Jacobi-Bellman equation for mean field social optimization in nonlinear diffusion models, establishing approximate person-by-person optimality and providing explicit solutions for the linear-quadratic case.

Contribution

It introduces the master equation for the value function in mean field social optimization and proves its near-optimality under certain regularity conditions.

Findings

Derived a new master equation for social optimization.

Proved $ ext{epsilon}$-person-by-person optimality of control laws.

Provided explicit solutions for the linear-quadratic case.

Abstract

We consider mean field social optimization in nonlinear diffusion models. By dynamic programming with a representative agent employing cooperative optimizer selection, we derive a new Hamilton--Jacobi--Bellman (HJB) equation to be called the master equation of the value function. Under some regularity conditions, we establish $\epsilon$-person-by-person optimality of the master equation-based control laws, which may be viewed as a necessary condition for nearly attaining the social optimum. A major challenge in the analysis is to obtain tight estimates, within an error of $O(1/N)$, of the social cost having order $O(N)$. This will be accomplished by multi-scale analysis via constructing two auxiliary master equations. We illustrate explicit solutions of the master equations for the linear-quadratic (LQ) case, and give an application to systemic risk.