  Mean field game theory has been developed largely following two routes. One
of them, called the direct approach, starts by solving a large-scale game and
next derives a set of limiting equations as the population size tends to
infinity. The second route is to apply mean field approximations and formalize
a fixed point problem by analyzing the best response of a representative
player. This paper addresses the connection and difference of the two
approaches in a linear quadratic (LQ) setting. We first introduce an asymptotic
solvability notion for the direct approach, which means for all sufficiently
large population sizes, the corresponding game has a set of feedback Nash
strategies in addition to a mild regularity requirement. We provide a necessary
and sufficient condition for asymptotic solvability and show that in this case
the solution converges to a mean field limit. This is accomplished by
developing a re-scaling method to derive a low dimensional ordinary
differential equation (ODE) system, where a non-symmetric Riccati ODE has a
central role. We next compare with the fixed point approach which determines a
two point boundary value (TPBV) problem, and show that asymptotic solvability
implies feasibility of the fixed point approach, but the converse is not true.
We further address non-uniqueness in the fixed point approach and examine the
long time behavior of the non-symmetric Riccati ODE in the asymptotic
solvability problem.
