Stackelberg strategies for selfish routing in general multicommodity networks

A natural generalization of the selfish routing setting arises when some of the users obey a central coordinating authority, while the rest act selfishly. Such behavior can be modeled by dividing the users into an a fraction that are routed according to the central coordinator's routing strategy (Stackelberg strategy), and the remaining 1-a that determine their strategy selfishly, given the routing of the coordinated users. One seeks to quantify the resulting price of anarchy, i.e., the ratio of the cost of the worst traffic equilibrium to the system optimum, as a function of a. It is well-known that for a=0 and linear latency functions the price of anarchy is at most 4/3 (2002). If a tends to 1, the price of anarchy should also tend to 1 for any reasonable algorithm used by the coordinator.