Computing Approximate Pure Nash Equilibria in Shapley Value Weighted Congestion Games

  • Matthias Feldotto
  • Martin Gairing
  • Grammateia Kotsialou
  • Alexander Skopalik
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10660)


We study the computation of approximate pure Nash equilibria in Shapley value (SV) weighted congestion games, introduced in [19]. This class of games considers weighted congestion games in which Shapley values are used as an alternative (to proportional shares) for distributing the total cost of each resource among its users. We focus on the interesting subclass of such games with polynomial resource cost functions and present an algorithm that computes approximate pure Nash equilibria with a polynomial number of strategy updates. Since computing a single strategy update is hard, we apply sampling techniques which allow us to achieve polynomial running time. The algorithm builds on the algorithmic ideas of [7], however, to the best of our knowledge, this is the first algorithmic result on computation of approximate equilibria using other than proportional shares as player costs in this setting. We present a novel relation that approximates the Shapley value of a player by her proportional share and vice versa. As side results, we upper bound the approximate price of anarchy of such games and significantly improve the best known factor for computing approximate pure Nash equilibria in weighted congestion games of [7].


Approximate pure Nash equilibria Computation Shapley cost-sharing Weighted congestion games Approximate Price of Anarchy 


  1. 1.
    Ackermann, H., Röglin, H., Vöcking, B.: On the impact of combinatorial structure on congestion games. J. ACM 55(6), 25:1–25:22 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ackermann, H., Skopalik, A.: Complexity of pure Nash equilibria in player-specific network congestion games. Internet Math. 5(4), 323–342 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    Aziz, H., de Keijzer, B.: Shapley meets shapley. In: Mayr, E.W., Portier, N. (eds.) 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), STACS 2014, 5–8 March 2014, Lyon, France. LIPIcs, vol. 25, pp. 99–111. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2014)Google Scholar
  5. 5.
    Bachrach, Y., Markakis, E., Resnick, E., Procaccia, A.D., Rosenschein, J.S., Saberi, A.: Approximating power indices: theoretical and empirical analysis. Auton. Agent. Multi-Agent Syst. 20(2), 105–122 (2010)CrossRefGoogle Scholar
  6. 6.
    Caragiannis, I., Fanelli, A., Gravin, N., Skopalik, A.: Efficient computation of approximate pure Nash equilibria in congestion games. In: Ostrovsky, R. (ed.) IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, 22–25 October 2011, pp. 532–541. IEEE Computer Society (2011)Google Scholar
  7. 7.
    Caragiannis, I., Fanelli, A., Gravin, N., Skopalik, A.: Approximate pure Nash equilibria in weighted congestion games: existence, efficient computation, and structure. ACM Trans. Econ. Comput. 3(1), 2:1–2:32 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chien, S., Sinclair, A.: Convergence to approximate Nash equilibria in congestion games. Games Econ. Behav. 71(2), 315–327 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure Nash equilibria. In: Babai, L. (ed.) Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, 13–16 June 2004, pp. 604–612. ACM (2004)Google Scholar
  10. 10.
    Fotakis, D., Kontogiannis, S.C., Spirakis, P.G.: Selfish unsplittable flows. Theor. Comput. Sci. 348(2–3), 226–239 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gairing, M., Kollias, K., Kotsialou, G.: Tight bounds for cost-sharing in weighted congestion games. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 626–637. Springer, Heidelberg (2015). CrossRefGoogle Scholar
  12. 12.
    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
  13. 13.
    Gkatzelis, V., Kollias, K., Roughgarden, T.: Optimal cost-sharing in weighted congestion games. In: Liu, T.-Y., Qi, Q., Ye, Y. (eds.) WINE 2014. LNCS, vol. 8877, pp. 72–88. Springer, Cham (2014). Google Scholar
  14. 14.
    Gopalakrishnan, R., Marden, J.R., Wierman, A.: Potential games are necessary to ensure pure Nash equilibria in cost sharing games. Math. Oper. Res. 39(4), 1252–1296 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hansknecht, C., Klimm, M., Skopalik, A.: Approximate pure Nash equilibria in weighted congestion games. In: Jansen, K., Rolim, J.D.P., Devanur, N.R., Moore, C. (eds.) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, 4–6 September 2014, Barcelona, Spain. LIPIcs, vol. 28, pp. 242–257. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2014)Google Scholar
  16. 16.
    Harks, T., Klimm, M.: On the existence of pure Nash equilibria in weighted congestion games. Math. Oper. Res. 37(3), 419–436 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hart, S., Mas-Colell, A.: Potential, value, and consistency. Econometrica 57(3), 589–614 (1989)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Klimm, M., Schmand, D.: Sharing non-anonymous costs of multiple resources optimally. In: Paschos, V.T., Widmayer, P. (eds.) CIAC 2015. LNCS, vol. 9079, pp. 274–287. Springer, Cham (2015). CrossRefGoogle Scholar
  19. 19.
    Kollias, K., Roughgarden, T.: Restoring pure equilibria to weighted congestion games. ACM Trans. Econ. Comput. 3(4), 1–24 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    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
  21. 21.
    Liben-Nowell, D., Sharp, A., Wexler, T., Woods, K.: Computing shapley value in supermodular coalitional games. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 568–579. Springer, Heidelberg (2012). CrossRefGoogle Scholar
  22. 22.
    Maleki, S.: Addressing the computational issues of the Shapley value with applications in the smart grid. Ph.D. thesis, University of Southampton (2015)Google Scholar
  23. 23.
    Mann, I., Shapley, L.S.: Values of large games, 6: evaluating the electoral college exactly. Technical report, DTIC Document (1962)Google Scholar
  24. 24.
    Milchtaich, I.: Congestion games with player-specific payoff functions. Games Econ. Behav. 13(1), 111–124 (1996)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Monderer, D., Shapley, L.S.: Potential games. Games Econ. Behav. 14(1), 124–143 (1996)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2(1), 65–67 (1973)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Roughgarden, T., Schrijvers, O.: Network Cost-Sharing without Anonymity. ACM Trans. Econ. Comput. 4(2), 8:1–8:24 (2016)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Skopalik, A., Vöcking, B.: Inapproximability of pure Nash equilibria. In: Dwork, C. (ed.) Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, 17–20 May 2008, pp. 355–364. ACM (2008)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Matthias Feldotto
    • 1
  • Martin Gairing
    • 2
  • Grammateia Kotsialou
    • 3
  • Alexander Skopalik
    • 1
  1. 1.Paderborn UniversityPaderbornGermany
  2. 2.University of LiverpoolLiverpoolUK
  3. 3.King’s College LondonLondonUK

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