Symmetry in Network Congestion Games: Pure Equilibria and Anarchy Cost ⋆ Dimitris Fotakis 1 , Spyros Kontogiannis 2,3 , and Paul Spirakis 2 1 Dept. of Information and Communication Systems Engineering, University of the Aegean, 83200 Samos, Greece fotakis@aegean.gr 2 Research Academic Computer Technology Institute, 26500 Patras, Greece {kontog, spirakis}@cti.gr 3 Dept. of Computer Science, University of Ioannina, 45110 Ioannina, Greece Abstract. We study computational and coordination efficiency issues of Nash equilibria in symmetric network congestion games. We first propose a simple and natural greedy method that computes a pure Nash equi- librium with respect to traffic congestion in a network. In this algorithm each user plays only once and allocates her traffic to a path selected via a shortest path computation. We then show that this algorithm works for series-parallel networks when users are identical or when users are of varying demands but have the same best response strategy for any initial network traffic. We also give constructions where the algorithm fails if either the above condition is violated (even for series-parallel networks) or the network is not series-parallel (even for identical users). Thus, we essentially indicate the limits of the applicability of this greedy approach. We also study the price of anarchy for the objective of maximum latency. We prove that for any network of m uniformly related links and for identical users, the price of anarchy is Θ( log m log log m ). 1 Introduction Network congestion games provide a sound model for selfish routing of unsplit- table traffic and have recently been the subject of intensive research. The prevail- ing questions in recent work have to do with the performance degradation due to lack of users’ coordination (e.g., [23,12,10,1,3]) and the efficient computation of pure Nash equilibria (e.g., [8,11,10]). A natural greedy approach for computing a pure Nash equilibrium (PNE) is Greedy Best Response (GBR). Let us consider a dynamic setting with new users arriving in the network. The users play only once and irrevocably choose their strategy upon arrival. Each new user routes her traffic on the minimum delay path given the paths of the users currently in the network. Hopefully the existing ⋆ This work was partially supported by the EU within the Future and Emerging Technologies Programme under contract IST-2001-33135 (CRESCCO) and within the 6th Framework Programme under contract 001907 (DELIS). T. Erlebach and G. Persiano (Eds.): WAOA 2005, LNCS 3879, pp. 161–175, 2006. c  Springer-Verlag Berlin Heidelberg 2006