On the Complexity of Pure-Strategy Nash Equilibria in Congestion and Local-Effect Games

–Extended Abstract–
  • Juliane Dunkel
  • Andreas S. Schulz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4286)

Abstract

Congestion games are a fundamental class of noncooperative games possessing pure-strategy Nash equilibria. In the network version, each player wants to route one unit of flow on a path from her origin to her destination at minimum cost, and the cost of using an arc only depends on the total number of players using that arc. A natural extension is to allow for players sending different amounts of flow, which results in so-called weighted congestion games. While examples have been exhibited showing that pure-strategy Nash equilibria need not exist, we prove that it actually is strongly NP-hard to determine whether a given weighted network congestion game has a pure-strategy Nash equilibrium. This is true regardless of whether flow is unsplittable (has to be routed on a single path for each player) or not.

A related family of games are local-effect games, where the disutility of a player taking a particular action depends on the number of players taking the same action and on the number of players choosing related actions. We show that the problem of deciding whether a bidirectional local-effect game has a pure-strategy Nash equilibrium is NP-complete, and that the problem of finding a pure-strategy Nash equilibrium in a bidirectional local-effect game with linear local-effect functions (for which the existence of a pure-strategy Nash equilibrium is guaranteed) is PLS-complete. The latter proof uses a tight PLS-reduction, which implies the existence of instances and initial states for which any sequence of selfish improvement steps needs exponential time to reach a pure-strategy Nash equilibrium.

Keywords

Nash Equilibrium Truth Assignment Congestion Game Strategic Game Noncooperative Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ackermann, H., Röglin, H., Vöcking, B.: On the impact of combinatorial structure on congestion games. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, Berkeley, CA (to appear, 2006a)Google Scholar
  2. Ackermann, 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 (2006b)CrossRefGoogle Scholar
  3. Àlvarez, C., Gabarró, J., Serna, M.: Pure Nash equilibria in games with a large number of actions. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 95–106. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, É., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, Italy, pp. 295–304 (2004)Google Scholar
  5. Awerbuch, B., Azar, Y., Epstein, A.: The price of routing unsplittable flow. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, pp. 57–66 (2005)Google Scholar
  6. Brandt, F., Fischer, F., Holzer, M.: Symmetries and the complexity of pure Nash equilibrium. Electronic Colloquium on Computational Complexity, TR06-091 (2006)Google Scholar
  7. Chen, X., Deng, X.: 3-Nash is PPAD-complete. Electronic Colloquium on Computational Complexity, TR05-134 (2005)Google Scholar
  8. Chen, X., Deng, X.: Settling the complexity of 2-player Nash-equilibrium. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, Berkeley, CA (to appear, 2006)Google Scholar
  9. Daskalakis, C., Goldberg, P., Papadimitriou, C.: The complexity of computing a Nash equilibrium. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, pp. 71–78 (2006)Google Scholar
  10. Daskalakis, C., Papadimitriou, C.: Three-player games are hard. Electronic Colloquium on Computational Complexity, TR05-139 (2005)Google Scholar
  11. Dunkel, J.: The Complexity of Pure-Strategy Nash Equilibria in Non-Cooperative Games. Diplomarbeit, Institute of Mathematics, Technische Universität Berlin, Germany (July 2005)Google Scholar
  12. Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure Nash equilibria. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, pp. 604–612 (2004)Google Scholar
  13. Fischer, F., Holzer, M., Katzenbeisser, S.: The influence of neighbourhood and choice on the complexity of finding pure Nash equilibria. Information Processing Letters 99, 239–245 (2006)CrossRefMathSciNetMATHGoogle Scholar
  14. Fotakis, D., Kontogiannis, S., Spirakis, P.: Selfish unsplittable flows. Theoretical Computer Science 348, 226–239 (2005)MATHCrossRefMathSciNetGoogle Scholar
  15. Goemans, M., Mirrokni, V., Vetta, A.: Sink equilibria and convergence. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, Pittsburgh, PA, pp. 142–154 (2005)Google Scholar
  16. Goldberg, P., Papadimitriou, C.: Reducibility among equilibrium problems. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, pp. 61–70 (2006)Google Scholar
  17. Gottlob, G., Greco, G., Scarcello, F.: Pure Nash equilibria: Hard and easy games. Journal of Artificial Intelligence Research 24, 357–406 (2005)MATHMathSciNetGoogle Scholar
  18. Ieong, S., McGrew, R., Nudelman, E., Shoham, Y., Sun, Q.: Fast and compact: A simple class of congestion games. In: Proceedings of the 20th National Conference on Artificial Intelligence and the 17th Innovative Applications of Artificial Intelligence Conference, Pittsburgh, PA, pp. 489–494 (2005)Google Scholar
  19. Leyton-Brown, K., Tennenholtz, M.: Local-effect games. In: Proceedings of the 18th International Joint Conference on Artificial Intelligence, Acapulco, Mexico, pp. 772–780 (2003)Google Scholar
  20. Milchtaich, I.: Congestion games with player-specific payoff functions. Games and Economic Behavior 13, 111–124 (1996)MATHCrossRefMathSciNetGoogle Scholar
  21. Nash, J.: Non-cooperative games. Annals of Mathematics 54, 268–295 (1951)CrossRefMathSciNetGoogle Scholar
  22. Rosenthal, R.: A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory 2, 65–67 (1973)MATHCrossRefMathSciNetGoogle Scholar
  23. Schäffer, A., Yannakakis, M.: Simple local search problems that are hard to solve. SIAM Journal on Computing 20, 56–87 (1991)MATHCrossRefMathSciNetGoogle Scholar
  24. Schoenebeck, G., Vadhan, S.: The computational complexity of Nash equilibria in concisely represented games. In: Proceedings of the 7th ACM Conference on Electronic Commerce, Ann Arbor, MI, pp. 270–279 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Juliane Dunkel
    • 1
  • Andreas S. Schulz
    • 1
  1. 1.Massachusetts Institute of TechnologyOperations Research CenterCambridgeUSA

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