Social Optima in Mean Field LQG Control: Centralized and Decentralized Strategies

We study a class of linear-quadratic-Gaussian (LQG) control problems with N decision makers, where the basic objective is to minimize a social cost as the sum of N individual costs containing mean field coupling. The exact socially optimal solution (determining a particular Pareto optimum) requires centralized information for each agent and has high implementational complexity. As an alternative we subsequently exploit a mean field structure in the centralized optimal control problem to develop decentralized cooperative optimization so that each agent only uses its own state and a function which may be computed offline; the resulting set of strategies asymptotically achieves the social optimum as N → ∞. A key feature in this scheme is to let each agent optimize a new cost as the sum of its own cost and another component capturing its social impact on all other agents. We also discuss the relationship between the decentralized cooperative solution and the so-called Nash Certainty Equivalence based solution presented in previous work on mean field LQG games.