Fast Convergence of Routing Games with Splittable Flows

In this paper we investigate the splittable routing game in a seriesparallel network with two selfish players. Every player wishes to route optimally, i.e. at minimum cost, an individual flow demand from the source to the destination, giving rise to a non-cooperative game. We allow a player to split his flow along any number of paths. One of the fundamental questions in this model is the convergence of the best response dynamics to a Nash equilibrium, as well as the time of convergence. We prove that this game converges indeed to a Nash equilibrium in a logarithmic number of steps. Our results hold for increasing and convex player-specific latency functions. Finally, we prove that our analysis on the convergence time is tight for affine latency functions.