Nashification and the Coordination Ratio for a Selfish Routing Game

  • Rainer Feldmann
  • Martin Gairing
  • Thomas Lücking
  • Burkhard Monien
  • Manuel Rode
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2719)

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(nm2) 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. Brucker, J. Hurink, and F. Werner. Improving local search heuristics for some scheduling problems. part ii. Discrete Applied Mathematics, 72:47–69, 1997.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Y. Cho and S. Sahni. Bounds for list schedules on uniform processors. SIAM Journal on Computing, 9(1):91–103, 1980.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Czumaj and B. Vöcking. Tight bounds for worst-case equilibria. In Proc. of SODA 2002, pp 413–420, 2002.Google Scholar
  4. 4.
    J. Feigenbaum, C. Papdimitriou, and S. Shenker. Sharing the cost of multicast transmissions. In Proc. of STOC 2000, pp 218–227, 2000.Google Scholar
  5. 5.
    G. Finn and E. Horowitz. A linear time approximation algorithm for multiprocessor scheduling. BIT, 19:312–320, 1979.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    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.Google Scholar
  7. 7.
    M. Gairing, T. Lücking, M. Mavronicolas, B. Monien, and P. Spirakis. Extreme nash equilibria. Technical report, FLAGS-TR-03-10, 2002.Google Scholar
  8. 8.
    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.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    K. Jain and V. Vazirani. Applications of approximation algorithms to cooperative games. In Proc. of STOC 2001, pp 364–372, 2001.Google Scholar
  10. 10.
    Y.A. Korilis, A.A. Lazar, and A. Orda. Architecting noncooperative networks. IEEE Journal on Selected Areas in Communications, 13(7):1241–1251, 1995.CrossRefGoogle Scholar
  11. 11.
    E. Koutsoupias and C. Papadimitriou. Worst-case equilibria. In Proc. of STACS 1999, pp 404–413, 1999.Google Scholar
  12. 12.
    M. Mavronicolas and P. Spirakis. The price of selfish routing. In Proc. of STOC 2001, pp 510–519, 2001.Google Scholar
  13. 13.
    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.Google Scholar
  14. 14.
    J. Nash. Non-cooperative games. Annals of Mathematics, 54(2):286–295, 1951.CrossRefMathSciNetGoogle Scholar
  15. 15.
    N. Nisan. Algorithms for selfish agents. In Proc. of STACS 1999, pp 1–15, 1999.Google Scholar
  16. 16.
    N. Nisan and A. Ronen. Algorithmic mechanism design. In Proc. of STOC 1999, pp 129–140, 1999.Google Scholar
  17. 17.
    M.J. Osborne and A. Rubinstein. A Course in Game Theory. MIT Press, 1994.Google Scholar
  18. 18.
    C.H. Papadimitriou. Algorithms, games, and the internet. In Proc. of STOC 2001, pp 749–753, 2001.Google Scholar
  19. 19.
    T. Roughgarden and E. Tardos. How bad is selfish routing? In Proc. of FOCS 2000, pp 93–102, 2000.Google Scholar
  20. 20.
    P. Schuurman and T. Vredeveld. Performance guarantees of load search for multiprocessor scheduling. In Proc. of IPCO 2001, pp 370–382, 2001.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Rainer Feldmann
    • 1
  • Martin Gairing
    • 1
  • Thomas Lücking
    • 1
  • Burkhard Monien
    • 1
  • Manuel Rode
    • 1
  1. 1.Department of Computer Science, Electrical Engineering and MathematicsUniversity of PaderbornPaderbornGermany

Personalised recommendations