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

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.