Discrete‐time mean field social control with a major agent

This paper considers a linear‐quadratic mean field control problem involving a major agent and N minor agents. We aim to optimize a social cost as a weighted sum of the individual costs under decentralized information. Firstly, the forward‐backward stochastic difference equations (FBSDEs) are obtained for this problem by variational analysis. Then, by decoupling the FBSDEs, we design the decentralized control laws, which are further shown to be asymptotically optimal.