Social optima in mean field LQG control: Centralized and decentralized strategies

We study a class of LQG control problems with N decision makers possessing different dynamics. The basic objective is to minimize a social cost function as the sum of N individual costs containing mean field coupling. For large N, the exact socially optimal solution (which also has the Pareto optimality property) requires centralized information and this leads to high computational complexity. In this paper, we develop decentralized cooperative optimization so that the strategy of each agent only uses its own state and a function which can be computed off-line; the resulting set of strategies asymptotically achieves the social optimum as N ¿ ¿. Our solution relies on a mean field approximation and the calculation of the social cost variation due to individual strategy perturbations. We also discuss the relationship between the socially optimal solution and the so-called Nash Certainty Equivalence based solution presented in previous work (see e.g. Huang, Caines and Malhame¿, IEEE Trans. Auto. Control, Sept. 2007) on mean field LQG games. This leads to insights into connections and differences between cooperative and noncooperative solutions to mean field LQG problems.