Linear quadratic mean field social optimization: Asymptotic solvability and decentralized control

TL;DR

This paper investigates the asymptotic solvability of linear quadratic mean field social optimization problems with controlled diffusions, deriving conditions via Riccati equations and analyzing decentralized control efficiency.

Contribution

It introduces a low-dimensional Riccati ODE system to characterize asymptotic solvability and quantifies the efficiency gain of social optimization over mean field games.

Findings

Derived a Riccati ODE system for asymptotic solvability

Decentralized control achieves bounded optimality loss of O(1/N)

Quantified efficiency gain of social optimum over mean field game

Abstract

This paper studies asymptotic solvability of a linear quadratic (LQ) mean field social optimization problem with controlled diffusions and indefinite state and control weights. Starting with an $N$-agent model, we employ a rescaling approach to derive a low-dimensional Riccati ordinary differential equation (ODE) system, which characterizes a necessary and sufficient condition for asymptotic solvability. The decentralized control obtained from the mean field limit ensures a bounded optimality loss in minimizing the social cost having magnitude $O(N)$, which implies an optimality loss of $O(1/N)$ per agent. We further quantify the efficiency gain of the social optimum with respect to the solution of the mean field game.