Fast Convergence of Selfish Routing
Eyal Even-Dar and Yishay Mansour
In: ACM-SIAM Symposium on Discrete Algorithms, Jan. 23-25, 2005, Vancouver, Canada.

## Abstract

We consider $n$ anonymous selfish users that route their communication through $m$ 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 the convergence rate of several migration policies. The first is a very natural policy, which balances the expected load on the links, for the case that all users are identical and apply it, we show that the rerouting terminates in expected $O(\log \log n + \log^{1.5} m)$ stages. The second policy that we consider is a Nash rerouting policy, which is a myopic strategic policy, in which {\em every} rerouting stage is a Nash equilibrium. For the Nash rerouting policy, we show similar termination bounds. Additionally we consider policies which are subgame perfect equilibrium in the discounted repeated game and derive there similar termination bounds. We study the structural properties of the Nash rerouting policies, and derive both existence result and an efficient algorithm for the case that the number of links is small. We also show that if the users have different weights then there exists a set of weights such that every Nash rerouting or subgame perfect equilibrium policy terminates in $\Omega(\sqrt{n})$ stages.