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 non-equilibrium 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.
Partly supported by the DFG-SFB 376 and by the IST Program of the EU under contract numbers IST-1999-14186 (ALCOM-FT), and IST-2001-33116 (FLAGS).
International Graduate School of Dynamic Intelligent Systems
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
P. Brucker, J. Hurink, and F. Werner. Improving local search heuristics for some scheduling problems. part ii. Discrete Applied Mathematics, 72:47–69, 1997.
Y. Cho and S. Sahni. Bounds for list schedules on uniform processors. SIAM Journal on Computing, 9(1):91–103, 1980.
A. Czumaj and B. Vöcking. Tight bounds for worst-case 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.
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, FLAGS-TR-03-10, 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.
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.
E. Koutsoupias and C. Papadimitriou. Worst-case 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. Non-cooperative games. Annals of Mathematics, 54(2):286–295, 1951.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Feldmann, R., Gairing, M., Lücking, T., Monien, B., Rode, M. (2003). Nashification and the Coordination Ratio for a Selfish Routing Game. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds) Automata, Languages and Programming. ICALP 2003. Lecture Notes in Computer Science, vol 2719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45061-0_42
Download citation
DOI: https://doi.org/10.1007/3-540-45061-0_42
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40493-4
Online ISBN: 978-3-540-45061-0
eBook Packages: Springer Book Archive