On the convergence of the best-response algorithm in routing games

We investigate the convergence of sequential best-response dynamics in a routing game over parallel links. Each player controls a nonnegligible portion of the total traffic, and seeks to split its flow over the links of the network so as to minimize its own cost. We prove that best-response operators are lipschitz continuous, which implies that a sufficient condition for the convergence of the best-response dynamics is that the joint spectral radius of Jacobian matrices of best-response operators be strictly less than unity. We establish the specific structure of these Jacobian matrices for our game, and show that this condition is met in two cases: (a) two-player game for an arbitrary number of links and for a wide class of cost functions; and (b) for arbitrary numbers of players and links in the case of linear latency functions. For latency functions satisfying reasonable convexity assumptions, we conjecture that the proposed sufficient condition is met for arbitrary numbers of players and links.