Restoring Pure Equilibria to Weighted Congestion Games

  • Konstantinos Kollias
  • Tim Roughgarden
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6756)


Congestion games model several interesting applications, including routing and network formation games, and also possess attractive theoretical properties, including the existence of and convergence of natural dynamics to a pure Nash equilibrium. Weighted variants of congestion games that rely on sharing costs proportional to players’ weights do not generally have pure-strategy Nash equilibria. We propose a new way of assigning costs to players with weights in congestion games that recovers the important properties of the unweighted model. This method is derived from the Shapley value, and it always induces a game with a (weighted) potential function. For the special cases of weighted network cost-sharing and atomic selfish routing games (with Shapley value-based cost shares), we prove tight bounds on the price of stability and price of anarchy, respectively.


congestion games network design Shapley value 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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
  2. 2.
    Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, É., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. SIAM Journal on Computing 38(4), 1602–1623 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Awerbuch, B., Azar, Y., Epstein, A.: The price of routing unsplittable flow. In: STOC, pp. 57–66 (2005)Google Scholar
  4. 4.
    Bhawalkar, K., Gairing, M., Roughgarden, T.: Weighted congestion games: Price of anarchy, universal worst-case examples, and tightness. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6347, pp. 17–28. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Chen, H., Roughgarden, T.: Network design with weighted players. Theory of Computing Systems 45(2), 302–324 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chen, H., Roughgarden, T., Valiant, G.: Designing network protocols for good equilibria. SIAM Journal on Computing 39(5), 1799–1832 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Fotakis, D., Kontogiannis, S.C., Spirakis, P.G.: Selfish unsplittable flows. Theoretical Computer Science 348(2-3), 226–239 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    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
  9. 9.
    Goemans, M.X., Mirrokni, V., Vetta, A.: Sink equilibria and convergence. In: FOCS, pp. 142–151 (2005)Google Scholar
  10. 10.
    Harks, T., Klimm, M.: On the existence of pure Nash equilibria in weighted congestion games. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 79–89. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Hart, S., Mas-Colell, A.: Potential, value, and consistency. Econometrica 57(3), 589–614 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Kalai, E., Samet, D.: On weighted Shapley values. International Journal of Game Theory 16(3), 205–222 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Milchtaich, I.: Congestion games with player-specific payoff functions. Games and Economic Behavior 13(1), 111–124 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Monderer, D., Shapley, L.S.: Fictitious play property for games with identical interests. Journal of Economic Theory 68, 258–265 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Monderer, D., Shapley, L.S.: Potential games. Games and Economic Behavior 14(1), 124–143 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Mosk-Aoyama, D., Roughgarden, T.: Worst-case efficiency analysis of queueing disciplines. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 546–557. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Moulin, H.: The price of anarchy of serial, average and incremental cost sharing. Economic Theory 36(3), 379–405 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Osborne, M.J., Rubinstein, A.: A Course in Game Theory. MIT Press, Cambridge (1994)zbMATHGoogle Scholar
  19. 19.
    Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory 2(1), 65–67 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Rosenthal, R.W.: The network equilibrium problem in integers. Networks 3(1), 53–59 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Roughgarden, T.: Intrinsic robustness of the price of anarchy. In: STOC, pp. 513–522 (2009)Google Scholar
  22. 22.
    Roughgarden, T., Tardos, É.: How bad is selfish routing? Journal of the ACM 49(2), 236–259 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Shapley, L.S.: Additive and Non-Additive Set Functions. PhD thesis, Department of Mathematics, Princeton University (1953)Google Scholar
  24. 24.
    Shenker, S.J.: Making greed work in networks: A game-theoretic analysis of switch service disciplines. IEEE/ACM Transactions on Networking 3(6), 819–831 (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Konstantinos Kollias
    • 1
  • Tim Roughgarden
    • 1
  1. 1.Stanford UniversityStanfordUSA

Personalised recommendations