Nashification and the Coordination Ratio for a Selfish Routing Game
 Rainer Feldmann,
 Martin Gairing,
 Thomas Lücking,
 Burkhard Monien,
 Manuel Rode
 … show all 5 hide
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.
 P. Brucker, J. Hurink, and F. Werner. Improving local search heuristics for some scheduling problems. part ii. Discrete Applied Mathematics, 72:47–69, 1997. CrossRef
 Y. Cho and S. Sahni. Bounds for list schedules on uniform processors. SIAM Journal on Computing, 9(1):91–103, 1980. CrossRef
 A. Czumaj and B. Vöcking. Tight bounds for worstcase equilibria. In Proc. of SODA 2002, pp 413–420, 2002.
 J. Feigenbaum, C. Papdimitriou, and S. Shenker. Sharing the cost of multicast transmissions. In Proc. of STOC 2000, pp 218–227, 2000.
 G. Finn and E. Horowitz. A linear time approximation algorithm for multiprocessor scheduling. BIT, 19:312–320, 1979. CrossRef
 D. Fotakis, S. Kontogiannis, E. Koutsoupias, M. Mavronicolas, and P. Spirakis. The structure and complexity of nash equilibria for a selfish routing game. In Proc. of ICALP 2002, pp 123–134, 2002.
 M. Gairing, T. Lücking, M. Mavronicolas, B. Monien, and P. Spirakis. Extreme nash equilibria. Technical report, FLAGSTR0310, 2002.
 D.S. Hochbaum and D. Shmoys. A polynomial approximation scheme for scheduling on uniform processors: using the dual approximation approach. SIAM Journal on Computing, 17(3):539–551, 1988. CrossRef
 K. Jain and V. Vazirani. Applications of approximation algorithms to cooperative games. In Proc. of STOC 2001, pp 364–372, 2001.
 Y.A. Korilis, A.A. Lazar, and A. Orda. Architecting noncooperative networks. IEEE Journal on Selected Areas in Communications, 13(7):1241–1251, 1995. CrossRef
 E. Koutsoupias and C. Papadimitriou. Worstcase equilibria. In Proc. of STACS 1999, pp 404–413, 1999.
 M. Mavronicolas and P. Spirakis. The price of selfish routing. In Proc. of STOC 2001, pp 510–519, 2001.
 R.D. McKelvey and A. McLennan. Computation of equilibria in finite games. In H. Amman, D. Kendrick, and J. Rust, editors, Handbook of Computational Economics, 1996.
 J. Nash. Noncooperative games. Annals of Mathematics, 54(2):286–295, 1951. CrossRef
 N. Nisan. Algorithms for selfish agents. In Proc. of STACS 1999, pp 1–15, 1999.
 N. Nisan and A. Ronen. Algorithmic mechanism design. In Proc. of STOC 1999, pp 129–140, 1999.
 M.J. Osborne and A. Rubinstein. A Course in Game Theory. MIT Press, 1994.
 C.H. Papadimitriou. Algorithms, games, and the internet. In Proc. of STOC 2001, pp 749–753, 2001.
 T. Roughgarden and E. Tardos. How bad is selfish routing? In Proc. of FOCS 2000, pp 93–102, 2000.
 P. Schuurman and T. Vredeveld. Performance guarantees of load search for multiprocessor scheduling. In Proc. of IPCO 2001, pp 370–382, 2001.
 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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