The solution set of the N-player scalar feedback Nash algebraic Riccati equations

We analyze the set of scalar algebraic Riccati equations (ARE) that play an important role in finding feedback Nash equilibria of the scalar N-player linear-quadratic differential game. We show that in general there exist at most 2^N-1 solutions of the ARE that give rise to a Nash equilibrium. In particular we analyze the number of equilibria as a function of the autonomous growth parameter and present both necessary and sufficient conditions for the existence of a unique solution of the ARE