Cost-Balancing Tolls for Atomic Network Congestion Games

Abstract

We investigate the existence of optimal tolls for atomic symmetric network congestion games with unsplittable traffic and arbitrary nondecreasing latency functions. We focus on pure Nash equilibria, and consider a natural toll mechanism, which we call cost-balancing tolls. A set of cost-balancing tolls turns every path with positive traffic on its edges into a minimum-cost path. Hence any given configuration is induced as a pure Nash equilibrium of the modified game with the corresponding cost-balancing tolls. We show how to compute in linear time a set of cost-balancing tolls for the optimal solution such that the total amount of tolls paid by any player in any pure Nash equilibrium of the modified game does not exceed the latency on the maximum-latency path in the optimal solution. Our main result is that for congestion games on series-parallel networks with strictly increasing latency functions, the optimal solution is induced as the unique pure Nash equilibrium of the game with the corresponding cost-balancing tolls. To the best of our knowledge, only linear congestion games on parallel links were known to admit optimal tolls prior to this work. To demonstrate the difficulty of computing a better set of optimal tolls, we show that even for two-player linear congestion games on series-parallel networks, it is NP-hard to decide whether the optimal solution is the unique pure Nash equilibrium or there is another pure Nash equilibrium of total cost at least $6/5$ times the optimal cost.