Fast Convergence of Selfish Rerouting
Eyal Even-Dar and Yishay Mansour
In: Sixteenth ACM-SIAM Symposium on Discrete Algorithms (SODA), 23-25 Jan 2005, Van Couver, Canada.

## Abstract

We consider $n$ identical selfish users that route their communication using $m$ parallel links. The users are allowed to reroute, concurrently, from overloaded links to underloaded links. The different rerouting decisions are concurrent, randomized and independent. The rerouting process terminates when the system reaches a Nash equilibrium, in which no user can improve its state. We study a simple migration policy, which balances the expected load on the links, and show that if all the users apply it, the rerouting terminates in expected $\tilde{O}(\log \log n + \log^{1.5} m)$ stages. We extend our study by considering Nash rerouting policies, where {\em every} rerouting phase is a Nash equilibrium, and show similar termination bounds in this setting. We study the structural properties of such Nash rerouting policies, and derive both existence result and an efficient algorithm for the case that the number of links is small.