Worst-Case Nash Equilibria in Restricted Routing

We study the network routing problem with restricted and related links. There are parallel links with possibly different speeds, between a source and a sink. Also there are users, and each user has a traffic of some weight to assign to one of the links from a subset of all the links, named his/her allowable set. The users choosing the same link suffer the same delay, which is equal to the total weight assigned to that link over its speed. A state of the system is called a Nash equilibrium if no user can decrease his/her delay by unilaterally changing his/her link. To measure the performance degradation of the system due to the selfish behavior of all the users, Koutsoupias and Papadimitriou proposed the notion Price of Anarchy (denoted by PoA), which is the ratio of the maximum delay in the worst-case Nash equilibrium and in an optimal solution. The PoA for this restricted related model has been studied, and a linear lower bound was obtained. However in their bad instance, some users can only use extremely slow links. This is a little artificial and unlikely to appear in a real world. So in order to better understand this model, we introduce a parameter for the system, and prove a better Price of Anarchy in terms of the parameter. 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 with improved price of anarchy.