  The paper is concerned with two-person zero-sum mean-field linear-quadratic
stochastic differential games over finite horizons. By a Hilbert space method,
a necessary condition and a sufficient condition are derived for the existence
of an open-loop saddle point. It is shown that under the sufficient condition,
the associated two Riccati equations admit unique strongly regular solutions,
in terms of which the open-loop saddle point can be represented as a linear
feedback of the current state. When the game only satisfies the necessary
condition, an approximate sequence is constructed by solving a family of
Riccati equations and closed-loop systems.The convergence of the approximate
sequence turns out to be equivalent to the open-loop solvability of the game,
and the limit is exactly an open-loop saddle point, provided that the game is
open-loop solvable.
