Price of Anarchy in Networks with Heterogeneous Latency Functions

We address the performance of selfish network routing in multicommodity flows where the latency or delay function on edges is dependent on the flow of individual commodities, rather than on the aggregate flow. An application of this study is the analysis of a network with differentiated traffic, that is, in transportation networks where there are multiple types of traffic and in networks where traffic is prioritized according to type classification. We consider the inefficiency of equilibrium in this model and provide price of anarchy bounds for networks with k (types of) commodities, where each link is associated with heterogeneous polynomial delays, that is, commodity i on edge e faces delay specified by a multivariate polynomial dependent on the individual flow of each commodity on the edge. We consider both atomic and nonatomic flows and show bounds on the price of anarchy that depend on the relative impact of each type of traffic on the edge delay when the delay functions are polynomials of degree θ , for example, ∑ i a i f i ( e ) θ . The price of anarchy is unbounded for arbitrary polynomials. For networks with decomposable delay functions where the delay is the same for all commodities using the edge, we show improved bounds on the price of anarchy, for both nonatomic and atomic flows. The results illustrate that the inefficiency of selfish routing worsens in the case of heterogeneous delays compared with the standard delay functions that do not consider type differentiation.