In order to obtain a closed-loop strategies in Nash differential game with infinite horizon, one needs to solve a system of coupled algebraic Riccati equations. Under standard conditions it is not yet known if solutions for such equations exist. One way to achieve that goal is to consider discrete dynamical systems, whose fixed points (if they exist) are solutions of the problem under study. These discrete dynamical systems of coupled algebraic Riccati equations can also serve as numerical algorithms to compute possible solutions. In this paper, we propose a new discrete dynamical system. Through the study of pertinent examples, we show numerically that this algorithm behaves better than the existing ones, both in terms of convergence speed and detection of a stabilizable solution (when it exists).