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.

该论文涉及有限范围内的两人零和平均场线性二次随机微分博弈。通过希尔伯特空间方法，推导出开环鞍点存在的必要条件和充分条件。结果表明，在充分条件下，关联的两个 Riccati 方程允许唯一的强正则解，由此开环鞍点可以表示为当前状态的线性反馈。当博弈只满足必要条件时，通过求解一族Riccati方程和闭环系统来构造近似序列。近似序列的收敛结果等价于博弈的开环可解性，而极限恰好是一个开环鞍点，