Automata, Languages and Programming
Volume 2719 of the series Lecture Notes in Computer Science pp 514526
Nashification and the Coordination Ratio for a Selfish Routing Game
 Rainer FeldmannAffiliated withDepartment of Computer Science, Electrical Engineering and Mathematics, University of Paderborn
 , Martin GairingAffiliated withDepartment of Computer Science, Electrical Engineering and Mathematics, University of Paderborn
 , Thomas LückingAffiliated withDepartment of Computer Science, Electrical Engineering and Mathematics, University of Paderborn
 , Burkhard MonienAffiliated withDepartment of Computer Science, Electrical Engineering and Mathematics, University of Paderborn
 , Manuel RodeAffiliated withDepartment of Computer Science, Electrical Engineering and Mathematics, University of Paderborn
Abstract
We study the problem of n users selfishly routing traffic through a network consisting of m parallel related links. Users route their traffic by choosing private probability distributions over the links with the aim of minimizing their private latency. In such an environment Nash equilibria represent stable states of the system: no user can improve its private latency by unilaterally changing its strategy.
Nashification is the problem of converting any given nonequilibrium routing into a Nash equilibrium without increasing the social cost. Our first result is an O(nm ^{2}) time algorithm for Nashification. This algorithm can be used in combination with any approximation algorithm for the routing problem to compute a Nash equilibrium of the same quality. In particular, this approach yields a PTAS for the computation of a best Nash equilibrium. Furthermore, we prove a lower bound of \( \Omega \left( {2^{\sqrt n } } \right) \) and an upper bound of O(2n) for the number of greedy selfish steps for identical link capacities in the worst case.
In the second part of the paper we introduce a new structural parameter which allows us to slightly improve the upper bound on the coordination ratio for pure Nash equilibria in [3]. The new bound holds for the individual coordination ratio and is asymptotically tight. Additionally, we prove that the known upper bound of \( \frac{{1 + \sqrt {4m  3} }} {2} \) on the coordination ratio for pure Nash equilibria also holds for the individual coordination ratio in case of mixed Nash equilibria, and we determine the range of m for which this bound is tight.
 Title
 Nashification and the Coordination Ratio for a Selfish Routing Game
 Book Title
 Automata, Languages and Programming
 Book Subtitle
 30th International Colloquium, ICALP 2003 Eindhoven, The Netherlands, June 30 – July 4, 2003 Proceedings
 Pages
 pp 514526
 Copyright
 2003
 DOI
 10.1007/3540450610_42
 Print ISBN
 9783540404934
 Online ISBN
 9783540450610
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 2719
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Jos C. M. Baeten ^{(1)}
 Jan Karel Lenstra ^{(2)}
 Joachim Parrow ^{(3)}
 Gerhard J. Woeginger ^{(4)}
 Editor Affiliations

 1. Dept. of Mathematics and Computer Science, Technische Universiteit Eindhoven
 2. School of Industrial and Systems Engineering, Georgia Institute of Technology
 3. Department of Information Technology, Uppsala University
 4. Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente
 Authors

 Rainer Feldmann ^{(5)}
 Martin Gairing ^{(5)}
 Thomas Lücking ^{(5)}
 Burkhard Monien ^{(5)}
 Manuel Rode ^{(5)}
 Author Affiliations

 5. Department of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, Fürstenallee 11, 33102, Paderborn, Germany
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