A discrete-time nonsymmetric algebraic Riccati system which incorporates as special cases of various discrete-time nonsymmetric algebraic Riccati equations is introduced and studied without any restrictive assumptions on the matrix coefficients. Necessary and sufficient existence conditions together with computable formulas for the stabilizing solution are given in terms of proper deflating subspaces of an associated matrix pencil. The theory is applied in the framework of game theory with an open-loop information structure to design Nash strategy without the classical assumptions on the invertibility of some matrix coefficients.