  In this paper we discuss a class of mean field linear-quadratic-Gaussian
(LQG) games for large population system which has never been addressed by
existing literature. The features of our works are sketched as follows. First
of all, our state is modeled by stochastic Volterra-type equation which leads
to some new study on stochastic "integral" system. This feature makes our setup
significantly different from the previous mean field games where the states
always follow some stochastic "differential" equations. Actually, our
stochastic integral system is rather general and can be viewed as natural
generalization of stochastic differential equations. In addition, it also
includes some types of stochastic delayed systems as its special cases. Second,
some new techniques are explored to tackle our mean-field LQG games due to the
special structure of integral system. For example, unlike the Riccati equation
in linear controlled differential system, some Fredholm-type equations are
introduced to characterize the consistency condition of our integral system via
the resolvent kernels. Third, based on the state aggregation technique, the
Nash certainty equivalence (NCE) equation is derived and the set of associated
decentralized controls are verified to satisfy the $\epsilon$-Nash equilibrium
property. To this end, some new estimates of stochastic Volterra equations are
developed which also have their own interests.
