Pareto Optimality in Infinite Horizon Mean-Field Stochastic Cooperative Linear-Quadratic Difference Games

This article is concerned with the mean-field stochastic cooperative linear-quadratic dynamic difference game in an infinite time horizon. First, the necessary and sufficient conditions for the stability in the mean-square sense and the stochastic Popov-Belevitch-Hautus eigenvector tests for the exact observability and exact detectability of mean-field stochastic linear difference systems are derived by the $\mathscr H-representation technique. Second, the relation between the solvability of the cross-coupled generalized Lyapunov equations and the exact observability, exact detectability, and stability of the mean-field dynamic system is well characterized. It is then shown that the cross-coupled algebraic Riccati equations (CC-AREs) admit a unique positive-definite (positive-semidefinite, respectively) solution under exact observability (exact detectability, respectively), which is also a feedback stabilizing solution. Furthermore, all the Pareto optimal strategies and solutions can be, respectively, derived via the solutions to the weighted CC-AREs and the weighted cross-coupled algebraic Lyapunov equations. Finally, a practical application on the computation offloading in the multiaccess edge computing network is presented to illustrate the proposed theoretical results.