Social Certainty Equivalence in Mean Field LQG Control: Social, Nash and Centralized Strategies

We study social decision problems and Nash games for a class of linear-quadratic-Gaussian (LQG) models with N decision makers possessing different dynamics. For the social decision case, the basic objective is to minimize a social cost as the sum of N individual costs containing mean field coupling, and the exact social optimum requires centralized information. Continuing from the previous work (Huang, Caines, and Malhamé, 2009 Allerton Conference), we develop decentralized cooperative optimization so that each agent only uses its own state and a function which can be computed off-line. We prove asymptotic social optimality results with general vector individual states and continuum dynamic parameters. In finding the asymptotic social optimum, a key step 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 socially optimal solution and the so-called Nash Certainty Equivalence (NCE) based solution presented in previous work on mean field LQG games, and for the NCE case we illustrate a cost blow-up effect due to the strength of interaction exceeding a certain threshold.