Mean field game theory has been developed largely following two routes by different researchers. One route, called the direct approach, starts by solving a large-scale game and next derives a set of limiting equations. The second route is to apply mean field approximations and formalize a fixed point problem by obtaining the best response of a representative player. A systematic comparison of the two approaches is generally difficult since in the literature very often each approach is applied under various sufficient conditions.
This work addresses the connection and difference of the two approaches in a linear quadratic (LQ) game setting, where an asymptotic solvability notion is adopted for the direct approach. By using a re-scaling technique, a necessary and sufficient condition for asymptotic solvability is derived in terms of a low dimensional non-symmetric Riccati ODE, which provides a foundation for comparison with the fixed point approach. Next we show that asymptotic solvability implies feasibility of the fixed point approach, but the converse is not true. We further examine the long time behavior of the non-symmetric Riccati ODE and address non-uniqueness in the fixed point problem. (This is a joint work with Mengjie Zhou)