  We consider a resource allocation problem where individual users wish to send
data across a network to maximize their utility, and a cost is incurred at each
link that depends on the total rate sent through the link. It is known that as
long as users do not anticipate the effect of their actions on prices, a simple
proportional pricing mechanism can maximize the sum of users' utilities minus
the cost (called aggregate surplus). Continuing previous efforts to quantify
the effects of selfish behavior in network pricing mechanisms, we consider the
possibility that users anticipate the effect of their actions on link prices.
Under the assumption that the links' marginal cost functions are convex, we
establish existence of a Nash equilibrium. We show that the aggregate surplus
at a Nash equilibrium is no worse than a factor of 4*sqrt{2} - 5 times the
optimal aggregate surplus; thus, the efficiency loss when users are selfish is
no more than approximately 34%.
