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

Abstract

This paper is concerned with the mean-field stochastic cooperative linear-quadratic (LQ) 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-Belevith-Hautus (PBH) eigenvector tests for exact observability and exact detectability of mean-field stochastic linear difference systems are derived by the <inline-formula><tex-math notation="LaTeX">$\mathscr {H}$</tex-math></inline-formula>-representation technique. Second, the relation between the solvability of the cross-coupled generalized Lyapunov equations&#x00A0;(CC-GLEs) and 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&#x00A0;(CC-AREs) admit a unique positive definite (positive semi-definite, respectively) solution under exact observability (exact detectability, respectively), which is also a feedback stabilizing solution. Furthermore, all Pareto optimal strategies and solutions can be respectively derived via the solutions to the weighted CC-AREs (WCC-AREs) and the weighted cross-coupled algebraic Lyapunov equations&#x00A0;(WCC-ALEs). Finally, a practical application on the computation offloading in the multi-access edge computing network (MECN) is presented to illustrate the proposed theoretical results.