Pareto efficiency in the infinite horizon mean-field type cooperative stochastic differential game

Abstract

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-L2-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.