Worst-Case Nash Equilibria in Restricted Routing

Abstract

We study a restricted related model of the network routing problem. There are m parallel links with possibly different speeds, between a source and a sink. And there are n users, and each user i has a traffic of weight w _i to assign to one of the links from a subset of all the links, named his/her allowable set. We analyze the Price of Anarchy (denoted by PoA) of the system, which is the ratio of the maximum delay in the worst-case Nash equilibrium and in an optimal solution. In order to better understand this model, we introduce a parameter λ for the system, and define an instance to be λ-good if for every user, there exist a link with speed at least $\frac{s_{max}}{\lambda}$ in his/her allowable set. In this paper, we prove that for λ-good instances, the Price of Anarchy is $\Theta \big( \min\{\frac{\log \lambda m}{\log \log \lambda m}, m\}\big)$. We also show an important application of our result in coordination mechanism design for task scheduling game. We propose a new coordination mechanism, Group-Makespan, for unrelated selfish task scheduling game. Our new mechanism ensures the existence of pure Nash equilibrium and its PoA is $O \big(\frac{\log^2 m}{\log \log m}\big)$. This result improves the best known result of O(log² m) by Azar, Jain and Mirrokni in [2].

Supported by the National Natural Science Foundation of China Grant 60553001 and the National Basic Research Program of China Grant 2007CB807900, 2007CB807901.