Congestion Games with Linearly Independent Paths: Convergence Time and Price of Anarchy

  • Dimitris Fotakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4997)


We investigate the effect of linear independence in the strategies of congestion games on the convergence time of best response dynamics and on the pure Price of Anarchy. In particular, we consider symmetric congestion games on extension-parallel networks, an interesting class of networks with linearly independent paths, and establish two remarkable properties previously known only for parallel-link games. More precisely, we show that for arbitrary non-negative and non-decreasing latency functions, any best improvement sequence converges to a pure Nash equilibrium in at most n steps, and that for latency functions in class \(\mathcal{D}\), the pure Price of Anarchy is at most \(\rho(\mathcal{D})\).


Nash Equilibrium Latency Function Convergence Time Congestion Game Individual Cost 
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  1. 1.
    Ackerman, H., Röglin, H., Vöcking, B.: On the Impact of Combinatorial Structre on Congestion Games. In: FOCS 2006, pp. 613–622 (2006)Google Scholar
  2. 2.
    Ackerman, H., Röglin, H., Vöcking, B.: Pure Nash Equilibria in Player-Specific and Weighted Congestion Games. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 50–61. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)zbMATHGoogle Scholar
  4. 4.
    Aland, S., Dumrauf, D., Gairing, M., Monien, B., Schoppmann, F.: Exact Price of Anarchy for Polynomial Congestion Games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 218–229. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Awerbuch, B., Azar, Y., Epstein, A.: The Price of Routing Unsplittable Flow. In: STOC 2005, pp. 57–66 (2005)Google Scholar
  6. 6.
    Caragiannis, I., Flammini, M., Kaklamanis, C., Kanellopoulos, P., Moscardelli, L.: Tight Bounds for Selfish and Greedy Load Balancing. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 311–322. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Christodoulou, G., Koutsoupias, E.: The Price of Anarchy of Finite Congestion Games. In: STOC 2005, pp. 67–73 (2005)Google Scholar
  8. 8.
    Correa, J.R., Schulz, A.S., Stier Moses, N.E.: Selfish Routing in Capacitated Networks. Mathematics of Operations Research 29(4), 961–976 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Epstein, A., Feldman, M., Mansour, Y.: Efficient Graph Topologies in Network Routing Games. In: NetEcon+IBC 2007 (2007)Google Scholar
  10. 10.
    Epstein, A., Feldman, M., Mansour, Y.: Strong Equilibrium in Cost Sharing Connection Games. In: EC 2007, pp. 84–92 (2007)Google Scholar
  11. 11.
    Even-Dar, E., Kesselman, A., Mansour, Y.: Convergence Time to Nash Equilibria in Load Balancing. ACM Transactions on Algorithms 3(3) (2007)Google Scholar
  12. 12.
    Fabrikant, A., Papadimitriou, C., Talwar, K.: The Complexity of Pure Nash Equilibria. In: STOC 2004, pp. 604–612 (2004)Google Scholar
  13. 13.
    Fotakis, D.: Stackelberg Strategies for Atomic Congestion Games. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 299–310. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Rode, M.: Nash Equilibria in Discrete Routing Games with Convex Latency Functions. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 645–657. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Gairing, M., Lücking, T., Monien, B., Tiemann, K.: Nash Equilibria, the Price of Anarchy and the Fully Mixed Nash Equilibrium Conjecture. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 51–65. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Holzman, R., Law-Yone, N.: Strong Equilibrium in Congestion Games. Games and Economic Behaviour 21, 85–101 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Holzman, R., Law-Yone, N.: Network Structure and Strong Equilibrium in Route Selection Games. Mathematical Social Sciences 46, 193–205 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ieong, S., McGrew, R., Nudelman, E., Shoham, Y., Sun, Q.: Fast and compact: A simple class of congestion games. In: AAAI 2005, pp. 489–494 (2005)Google Scholar
  19. 19.
    Papadimitriou, C.H., Koutsoupias, E.: Worst-Case Equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  20. 20.
    Monien, B., Mavronicolas, M., Lücking, T., Rode, M.: A New Model for Selfish Routing. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 547–558. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  21. 21.
    Milchtaich, I.: Network Topology and the Efficiency of Equilibrium. Games and Economic Behaviour 57, 321–346 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Milchtaich, I.: The Equilibrium Existence Problem in Finite Network Congestion Games. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 87–98. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  23. 23.
    Rosenthal, R.W.: A Class of Games Possessing Pure-Strategy Nash Equilibria. International Journal of Game Theory 2, 65–67 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Roughgarden, T.: The Price of Anarchy is Independent of the Network Topology. Journal of Computer and System Sciences 67(2), 341–364 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Dimitris Fotakis
    • 1
  1. 1.Dept. of Information and Communication Systems EngineeringUniversity of the AegeanSamosGreece

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