Efficiency of Equilibria in Uniform Matroid Congestion Games

  • Jasper de Jong
  • Max Klimm
  • Marc Uetz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9928)


Network routing games, and more generally congestion games play a central role in algorithmic game theory, comparable to the role of the traveling salesman problem in combinatorial optimization. It is known that the price of anarchy is independent of the network topology for non-atomic congestion games. In other words, it is independent of the structure of the strategy spaces of the players, and for affine cost functions it equals 4/3. In this paper, we show that the situation is considerably more intricate for atomic congestion games. More specifically, we consider congestion games with affine cost functions where the players’ strategy spaces are symmetric and equal to the set of bases of a k-uniform matroid. In this setting, we show that the price of anarchy is strictly larger than the price of anarchy for singleton strategy spaces where it is 4/3. As our main result we show that the price of anarchy can be bounded from above by \(28/13 \approx 2.15\). This constitutes a substantial improvement over the price of anarchy bound 5/2, which is known to be tight for network routing games with affine cost functions.


Nash Equilibrium Strategy Space User Equilibrium Congestion Game Pure Nash Equilibrium 
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  1. 1.
    Abed, F., Correa, J.R., Huang, C.-C.: Optimal coordination mechanisms for multi-job scheduling games. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 13–24. Springer, Heidelberg (2014)Google Scholar
  2. 2.
    Ackermann, H., Röglin, H., Vöcking, B.: On the impact of combinatorial structure on congestion games. J. ACM 55(6), 1–22 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ackermann, H., Röglin, H., Vöcking, B.: Pure Nash equilibria in player-specific and weighted congestion games. Theoret. Comput. Sci. 410(17), 1552–1563 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Aland, S., Dumrauf, D., Gairing, M., Monien, B., Schoppmann, F.: Exact price of anarchy for polynomial congestion games. SIAM J. Comput. 40(5), 1211–1233 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Awerbuch, B., Azar, Y., Epstein, A.: The price of routing unsplittable flow. In: Proceedings of the 37th Annual ACM Symposium Theory Computing, pp. 57–66 (2005)Google Scholar
  6. 6.
    Beckmann, M., McGuire, C.B., Winsten, C.B.: Studies in the Economics and Transportation. Yale University Press, New Haven (1956)Google Scholar
  7. 7.
    Braess, D.: Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung 12, 258–0268 (1968). (German)MathSciNetzbMATHGoogle Scholar
  8. 8.
    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
  9. 9.
    Christodoulou, G., Koutsoupias, E.: The price of anarchy of finite congestion games. In: Proceedings of the 37th Annual ACM Symposium Theory Computing, pp. 67–73 (2005)Google Scholar
  10. 10.
    de Jong, J., Klimm, M., Uetz, M.: Efficiency of equilibria in uniform matroid congestion games. CTIT Technical report TR-CTIT-16-04, University of Twente (2016).
  11. 11.
    Dunkel, J., Schulz, A.S.: On the complexity of pure-strategy Nash equilibria in congestion and local-effect games. Math. Oper. Res. 33(4), 851–868 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Fotakis, D.: Stackelberg strategies for atomic congestion games. Theory Comput. Syst. 47, 218–249 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fujishige, S., Goemans, M.X., Harks, T., Peis, B., Zenklusen, R.: Matroids are immune to Braess paradox. arXiv:1504.07545 (2015)
  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., Schoppmann, F.: Total latency in singleton congestion games. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 381–387. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Goemans, M.X., Mirrokni, V.S., Vetta, A.: Sink equilibria and convergence. In: Proceedings of the 46th Annual IEEE Symposium Foundations of Computer Science, pp. 142–154 (2005)Google Scholar
  17. 17.
    Harks, T., Klimm, M., Peis, B.: Resource competition on integral polymatroids. In: Liu, T.-Y., Qi, Q., Ye, Y. (eds.) WINE 2014. LNCS, vol. 8877, pp. 189–202. Springer, Heidelberg (2014)Google Scholar
  18. 18.
    Harks, T., Peis, B.: Resource buying games. Algorithmica 70(3), 493–512 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  20. 20.
    Lücking, T., Mavronicolas, M., Monien, B., Rode, M.: A new model for selfish routing. Theoret. Comput. Sci. 406(3), 187–206 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Meyers, C., Problems, N.F., Games, C.: Complexity and approximation results. Ph.D. thesis, MIT, Operations Research Center (2006)Google Scholar
  22. 22.
    Milchtaich, I.: Congestion games with player-specific payoff functions. Games Econom. Behav. 13(1), 111–124 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Pigou, A.C.: The Economics of Welfare. Macmillan, London (1920)Google Scholar
  24. 24.
    Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Internat. J. Game Theory 2(1), 65–67 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Rosenthal, R.W.: The network equilibrium problem in integers. Networks 3, 53–59 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Roughgarden, T.: The price of anarchy is independent of the network topology. J. Comput. System Sci. 67, 341–364 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Roughgarden, T., Tardos, É.: How bad is selfish routing? J. ACM 49(2), 236–259 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Suri, S., Tóth, C.D., Zhou, Y.: Selfish load balancing and atomic congestion games. Algorithmica 47(1), 79–96 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Tran-Thanh, L., Polukarov, M., Chapman, A., Rogers, A., Jennings, N.R.: On the existence of pure strategy nash equilibria in integer–splittable weighted congestion games. In: Persiano, G. (ed.) SAGT 2011. LNCS, vol. 6982, pp. 236–253. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  30. 30.
    Wardrop, J.G.: Some theoretical aspects of road traffic research. Proc. Inst. Civ. Eng. 1(3), 325–362 (1952)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Universiteit TwenteEnschedeThe Netherlands
  2. 2.Technische Universität BerlinBerlinGermany

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