Efficiency Loss in a Network Resource Allocation Game: The Case of   Elastic Supply
  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%. 
Ramesh Johari (Stanford University), Shie Mannor (McGill University),   John N. Tsitsiklis (MIT)