This paper discusses the linear quadratic (LQ) mean-field type stochastic differential game with non-homogeneous terms in infinite horizon. Employing the equivalent description of Pareto efficiency, necessary conditions for the existence of Pareto solutions are presented under an assumption on the Lagrange multiplier set. Two conditions are introduced to guarantee that the assumption is established. Further, sufficient conditions for a control to be Pareto efficient are put forward in terms of the necessary conditions, the convexity condition on the homogeneous weighted sum cost functional and a transversality condition. The characterization of Pareto solutions is also studied for the homogeneous case. If the system is MF-$L^2$-stabilizable, then the solvability of the related generalized algebraic Riccati equations (GAREs) provides a sufficient condition under which the cost functionals are convex and all Pareto efficient strategies can be obtained by the weighted sum minimization method. In addition, by introducing two algebraic Lyapunov equations (ALEs), we derive all Pareto solutions.

本文讨论了无限范围内具有非齐次项的线性二次(LQ)平均场型随机微分博弈。采用帕累托效率的等价描述，在拉格朗日乘子集的假设下，提出了帕累托解存在的必要条件。引入两个条件来保证假设成立。进一步从必要条件、齐次加权和代价函数的凸性条件和横向条件等方面提出了控制是帕累托有效的充分条件。还研究了齐次情况下帕累托解的表征。如果系统是 MF-${升}^2$-stabilizable，那么相关的广义代数Riccati方程（GARE）的可解性提供了一个充分条件，在该条件下成本函数是凸的，并且所有帕累托有效策略都可以通过加权和最小化方法获得。此外，通过引入两个代数李雅普诺夫方程 (ALE)，我们推导出所有帕累托解。