The Price of Anarchy for Restricted Parallel Links

In the model of restricted parallel links, n users must be routed on m parallel links under the restriction that the link for each user be chosen from a certain set of allowed links for the user. In a (pure) Nash equilibrium, no user may improve its own Individual Cost (latency) by unilaterally switching to another link from its set of allowed links. The Price of Anarchy is a widely adopted measure of the worst-case loss (relative to optimum) in system performance (maximum latency) incurred in a Nash equilibrium. In this work, we present a comprehensive collection of bounds on Price of Anarchy for the model of restricted parallel links and for the special case of pure Nash equilibria. Specifically, we prove: • For the case of identical users and identical links, the Price of Anarchy is Ω “ lg m lg lg m ” . • For the case of identical users, the Price of Anarchy is O “ lg n lg lg n ” . • For the case of identical links, the Price of Anarchy is O “ lg m lg lg m ” , which is asymptotically tight. • For the most general case of arbitrary users and related links, the Price of Anarchy is at least m− 1 and less than m. The shown bounds reveal the dependence of the Price of Anarchy on n and m for all possible assumptions on users and links.