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On the Complexity of Pure Nash Equilibria in Player-Specific Network Congestion Games

  • Heiner Ackermann
  • Alexander Skopalik
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4858)

Abstract

Network congestion games with player-specific delay functions do not necessarily possess pure Nash equilibria. We therefore address the computational complexity of the corresponding decision problem, and show that it is NP-complete to decide whether such games possess pure Nash equilibria. This negative result still holds in the case of games with two players only. In contrast, we show that one can decide in polynomial time whether an equilibrium exists if the number of resources is constant.

In addition, we introduce a family of player-specific network congestion games which are guaranteed to possess equilibria. In these games players have identical delay functions, however, each player may only use a certain subset of the edges. For this class of games we prove that finding a pure Nash equilibrium is PLS-complete even in the case of three players. Again, in the case of a constant number of edges an equilibrium can be computed in polynomial time.

We conclude that the number of resources has a bigger impact on the computation complexity of certain problems related to network congestion games than the number of players.

Keywords

Nash Equilibrium Constant Number Delay Function Congestion Game Path Segment 
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. 1.
    Ackermann, H., Röglin, H., Vöcking, B.: On the impact of combinatorial structure on congestion games. In: Proc. of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 613–622 (2006)Google Scholar
  2. 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 (2006)CrossRefGoogle Scholar
  3. 3.
    Anshelevich, E., Dasgupta, A., Kleinberg, J.M., Tardos, É., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. In: Proc. of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 295–304 (2004)Google Scholar
  4. 4.
    Chakrabarty, D., Mehta, A., Nagarajan, V.: Fairness and optimality in congestion games. In: Proc. of the 6th ACM conference on Electronic Commerce (EC), pp. 52–57. ACM Press, New York (2005)CrossRefGoogle Scholar
  5. 5.
    Dunkel, J., Schulz, A.S.: On the complexity of pure-strategy Nash equilibria in congestion and local-effect games. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 62–73. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure Nash equilibria. In: Proc. of the 36th Annual ACM Symposium on Theory of Computing (STOC), pp. 604–612 (2004)Google Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)zbMATHGoogle Scholar
  8. 8.
    Johnson, D.S., Papadimtriou, C.H., Yannakakis, M.: How easy is local search? Journal on Computer and System Sciences 37(1), 79–100 (1988)zbMATHCrossRefGoogle Scholar
  9. 9.
    Meyers, C.: Network Flow Problems and Congestion Games: Complexitiy and Approximation. Massachusetts Institute of Technology (2006)Google Scholar
  10. 10.
    Milchtaich, I.: Congestion games with player-specific payoff functions. Games and Economic Behavior 13(1), 111–124 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    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
  12. 12.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.: Algorithmic Game Theory. Cambridge University Press, Cambridge (forthcoming)Google Scholar
  13. 13.
    Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. Journal of Game Theory 2, 65–67 (1973)zbMATHCrossRefGoogle Scholar
  14. 14.
    Schäffer, A.A., Yannakakis, M.: Simple local search problems that are hard to solve. SIAM Journal on Computing 20(1), 56–87 (1991)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Heiner Ackermann
    • 1
  • Alexander Skopalik
    • 1
  1. 1.Department of Computer Science, RWTH Aachen, D-52056 AachenGermany

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