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In this paper, the Nash games are investigated for weakly coupled large-scale stochastic discrete-time systems with state and control dependent noise. Under the assumption that each subsystem is stabilizable in the mean square sense, a state feedback strategy set is constructively designed by solving the cross-coupled stochastic algebraic Riccati equations (CSAREs). After establishing the asymptotic structure for the solutions of CSAREs, we show that there exists a unique solution of the equations in a small neighborhood of ε = 0. The main contribution of this paper is to design a new parameter independent strategy set. Moreover, a computational method for solving the CSAREs is also developed if the information on the small parameter is available. Finally, a numerical example for a practical aircraft control problem is solved to show the effectiveness of the proposed method.