  This paper investigates the linear-quadratic-Gaussian (LQG) mean-field game
(MFG) for a class of stochastic delay systems. We consider a large population
system in which the dynamics of each player satisfies some forward stochastic
differential delay equation (SDDE). The consistency condition or Nash certainty
equivalence (NCE) principle is established through an auxiliary mean-field
system of anticipated forward-backward stochastic differential equation with
delay (AFBSDDE). The wellposedness of such consistency condition system can be
further established by some continuation method instead the classical
fixed-point analysis. Thus, the consistency condition maybe given on arbitrary
time horizon. The decentralized strategies are derived which are shown to
satisfy the $\epsilon$-Nash equilibrium property. Two special cases of our MFG
for delayed system are further investigated.
