Mixed Nash games and social optima for linear-quadratic forward-backward mean-field systems

We consider a new class of mixed linear-quadratic Nash games and social optimization for two types of interactive agents. One is called a major agent and the others are minor agents. By 'mixed', we mean that all minor agents team up with each other to compete against this major agent for their contradictory cost functions. Different from the standard setup, this major's state is governed by some linear stochastic differential equation where the diffusion term and drift term both contain a control process, while the states of these minors are all weakly-coupled and driven by some linear backward stochastic differential equations because their terminal conditions are specified. To construct decentralized strategies for these two types of agents respectively, the backward person-by-person optimization method, combining some variational method and mean-field approximation are applied. Under some suitable conditions, we also verify the asymptotic optimality of these decentralized strategies.