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.