Pricing network edges for heterogeneous selfish users

We study the negative consequences of selfish behavior in a congested network and economic means of influencing such behavior. We consider a model of selfish routing in which the latency experienced by network traffic on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route traffic on minimum-latency paths. The quality of a routing of traffic is measured by the sum of travel times (the total latency). It is well known that the outcome of selfish routing (a Nash equilibrium) does not minimize the total latency. An ancient strategy for improving the selfish solution is the principle of marginal cost pricing, which asserts that on each edge of the network, each network user on the edge should pay a tax offsetting the congestion effects caused by its presence. By pricing network edges according to this principle, the inefficiency of selfish routing can always be eradicated. This result, while fundamental, assumes a very strong homogeneity property: all network users are assumed to trade off time and money in an identical way. The guarantee also ignores both the algorithmic aspects of edge pricing and the unfortunate possibility that an efficient routing of traffic might only be achieved with exorbitant taxes. Motivated by these shortcomings, we extend this classical work on edge pricing in several different directions and prove the following results. We prove that the edges of a single-commodity network can always be priced so that an optimal routing of traffic arises as a Nash equilibrium, even for very general heterogeneous populations of network users. When there are only finitely many different types of network users and all edge latency functions are convex, we show how to compute such edge prices efficiently. We prove that an easy-to-check mathematical condition on the population of heterogeneous network users is both necessary and sufficient for the existence of edge prices that induce an optimal routing while requiring only moderate taxes.