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

Abstract

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.

A sequential procurement model for a PPP project pipeline

“…Once none of the bidders can improve their response given the opponents' actions, there is evidence that one might have arrived in an equilibrium. Academic literature only offers theoretical proofs for a limited number of circumstances for which the best response algorithm converges [6,47]. Also in the context of the PPP procurement setting, we did not succeed in theoretically guaranteeing the convergence of the heuristic, while the computational results indicated convergence for the majority of the investigated cases.…”

“…Moreover, in contrast to models that study infinite timeframes (e.g., [16,33,38,52]), the PPP pipeline has a finite nature which is a logical consequence of the magnitude of the projects and the limited budget horizon of governments. 6 The experimental set-up relies on a heuristic approach to derive the equilibrium. Algorithmic game theory attempts to deal with the complexity of real-life models [32].…”

A penalized best‐response algorithm for nonlinear single‐path routing problems

“…For the unsplittable routing games considered in this article, it has been proven only (a) for linear latency functions , and (b) when all flows have the same traffic demands (i.e., when ), in which case the game is a congestion game . It is also worth mentioning that for splittable routing games, the convergence of the standard best‐response algorithm has been proven only in some special cases for networks of parallel links .…”

Analysis and Implementation of an Hourly Billing Mechanism for Demand Response Management

“…In general games, CBRD might not converge [30] or might take an exponential time to converge [31]. In atomic splittable congestion games on a parallel network, as in our case, the convergence and the speed of Algo 1 has been studied previously in [12] and [13], where the authors show by different methods that there is a geometric convergence in the case of N = 2 players and convex and strictly increasing price functions (Assumption 1). However, to the best of our knowledge, the convergence in this setting and for more players N > 2 is still an open question.…”

“…In that case, convergence is known but to our knowledge, no bound on the rate has ever been given. The convergence has been conjectured more generally for any convex prices [12,13]. We introduce a different algorithm based on a simultaneous projected gradient descent (Algo 2), and show its geometric convergence (Thm.…”