Linear-quadratic mean field stochastic zero-sum differential games

Abstract

In this paper, we consider the two-player linear-quadratic mean field stochastic differential game (LQ-MF-SZSDG), where the expected values of state and players' control variables are included in the corresponding mean field stochastic system and the objective functional. We characterize the explicit condition under which the feedback saddle-point equilibrium for the LQ-MF-SZSDG exists. The results are obtained by using the "completion of squares" method and the "ordered interchangeability" property of multiple saddle points. We show that the corresponding saddle-point equilibrium is linear in state, which is characterized by the two coupled Riccati differential equations (CRDEs). We also discuss the solvability conditions of the CRDEs. Numerical results are presented to verify the theoretical results of the paper.