Mean Field Social Optimization: Feedback Person-by-Person Optimality and the Dynamic Programming Equation

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 [Formula: see text]-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 [Formula: see text], of the social cost having order [Formula: see text]. This will be accomplished by multiscale 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.