  In this article, we provide a comprehensive study of the linear-quadratic
mean field games via the adjoint equation approach; although the problem has
been considered in the literature by Huang, Caines and Malhame (HCM, 2007a),
their method is based on Dynamic Programming. It turns out that two methods are
not equivalent, as far as giving sufficient condition for the existence of a
solution is concerned. Due to the linearity of the adjoint equations, the
optimal mean field term satisfies a linear forward-backward ordinary
differential equation. For the one dimensional case, we show that the
equilibrium strategy always exists uniquely. For dimension greater than one, by
choosing a suitable norm and then applying the Banach Fixed Point Theorem, a
sufficient condition, which is independent of the solution of the standard
Riccati differential equation, for the unique existence of the equilibrium
strategy is provided. As a by-product, we also establish a neat and instructive
sufficient condition for the unique existence of the solution for a class of
non-trivial nonsymmetric Riccati equations. Numerical examples of non-existence
of the equilibrium strategy and the comparison of HCM's approach will also be
provided.
