The price of anarchy of the proportional allocation mechanism revisited

We consider the proportional allocation mechanism first studied by Kelly (1997) in the context of congestion control algorithms for communication networks. A single infinitely divisible resource is to be allocated efficiently to competing players whose individual utility functions are unknown to the resource manager. If players anticipate the effect of their bids on the price of the resource and their utility functions are concave, strictly increasing and continuously differentiable, Johari and Tsitsiklis (2004) proved that the price of anarchy is 4/3. The question was raised whether there is a relationship between this result and that of Roughgarden and Tardos (2002), who had earlier shown exactly the same bound for nonatomic selfish routing with affine-linear congestion functions. We establish such a relationship and show, in particular, that the efficiency loss can be characterized by precisely the same geometric quantity. We also present a new variational inequality characterization of Nash equilibria in this setting, which enables us to extend the price-of-anarchy analysis to important classes of utility functions that are not necessarily concave.