A Numerical Algorithm to Find All Feedback Nash Equilibria in Scalar Affine Quadratic Differential Games

Abstract

This note deals with solving scalar coupled algebraic Riccati equations. These equations arise in finding linear feedback Nash equilibria of the scalar $N$ -player affine quadratic differential game. A numerical procedure is provided to compute all the stabilizing solutions. The main idea is to reformulate the Riccati equations into an extended eigenvalue-eigenvector problem for a specific parametrized matrix $U\in I\!\!R^{2^{N}\times 2^{N}}$ . Since the size of $U$ increases exponentially on $N$ , the algorithm only applies for games where the number of players is not too large.