It is well known that the outcome of selfish routing (a flow at Nash equilibrium) does not minimize the total latency, and that marginal cost pricing—charging each network user for the congestion effects caused by its presence—eliminates the inefficiency of selfish routing. However, the principle of marginal cost pricing assumes that taxes cause no disutility to network users; this is appropriate only when collected taxes can be feasibly returned (directly or indirectly) to the users. If this assumption does not hold and we wish to minimize the total user disutility (latency plus taxes paid)—the total cost—how should we price the network edges? Intuition may suggest that taxes can never improve the cost of a Nash equilibrium, but the famous Braess's Paradox shows this intuition to be incorrect.

We consider strategies for pricing network edges to reduce the cost of a Nash equilibrium. Since levying a sufficiently large tax on an edge effectively removes it from the network, our study generalizes previous work on designing networks for selfish users [T. Roughgarden, Designing networks for selfish users is hard, in: Proceedings of the 42nd Annual Symposium on Foundations of Computer Science (FOCS), 2001, pp. 472–481 (full version to appear in Journal of Computer and System Sciences)]. In this paper, we prove the following results.

• In a large class of networks—including all networks with linear latency functions—marginal cost taxes do not improve the cost of a Nash equilibrium.

• The largest-possible benefit from taxes does not exceed that from edge removals. In every network with linear latency functions, the benefit of taxes cannot exceed that of removing edges. There are networks with nonlinear latency functions, however, in which taxes are radically more powerful than edge removals.

• For every ε>0, there is no -approximation algorithm for computing optimal taxes, even in networks with linear latency functions (assuming P≠NP).