  In this article, we consider mean field games between a dominating player and
a group of representative agents, each of which acts similarly and also
interacts with each other through a mean field term being substantially
influenced by the dominating player. We first provide the general theory and
discuss the necessary condition for the optimal controls and game condition by
adopting adjoint equation approach. We then present a special case in the
context of linear-quadratic framework, in which a necessary and sufficient
condition can be asserted by stochastic maximum principle; we finally establish
the sufficient condition that guarantees the unique existence of the
equilibrium control. The proof of the convergence result of finite player game
to mean field counterpart is provided in Appendix.
