Advertisement

Pure Nash Equilibria in Games with a Large Number of Actions

  • Carme Àlvarez
  • Joaquim Gabarró
  • Maria Serna
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)

Abstract

We study the computational complexity of deciding the existence of a Pure Nash Equilibrium in multi-player strategic games. We address two fundamental questions: how can we represent a game? and how can we represent a game with polynomial pay-off functions? Our results show that the computational complexity of deciding the existence of a pure Nash equilibrium in a strategic game depends on two parameters: the number of players and the size of the sets of strategies. In particular we show that deciding the existence of a Nash equilibrium in a strategic game is NP-complete when the number of players is large and the number of strategies for each player is constant, while the problem is Σ\(^{p}_{\rm 2}\)-complete when the number of players is a constant and the size of the sets of strategies is exponential (with respect to the length of the strategies).

Keywords

Strategic games Nash equilibria complexity classes 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Àlvarez, C., Gabarró, J., Serna, M.: Pure Nash equilibria in games with a large number of actions. Technical Report 31, Electronic Colloquium on Computational Complexity (2005)Google Scholar
  2. 2.
    Balcazar, J.L., Díaz, J., Gabarró, J.: Structural Complexity II. Springer, Heidelberg (1990)zbMATHGoogle Scholar
  3. 3.
    Balcazar, J.L., Díaz, J., Gabarró, J.: Structural Complexity I, 2nd edn. Springer, Heidelberg (1995)Google Scholar
  4. 4.
    Ben-Porath, E.: The complexity of computing a best response automaton in repeated games with mixed strategies. Games and Economic Behavior 2(1), 1–12 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chu, F., Halpern, J.: On the NP-completeness of finding an optimal strategy in games with commons pay-offs. International Journal of Game Theory (2001)Google Scholar
  6. 6.
    Conitzer, V., Sandholm, T.: Complexity results about Nash equilibra. In: IJCAI 2003, pp. 765–771 (2003)Google Scholar
  7. 7.
    Daskalakis, K., Papadimitriou, C.: The complexity of games on highly regular graphs. Technical report (2005), available at http://www.cs.berkeley.edu/christos/
  8. 8.
    Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure Nash equilibria. In: STOC 2004, pp. 604–612 (2004)Google Scholar
  9. 9.
    Fotakis, D.A., Kontogiannis, S.C., Koutsoupias, E., Mavronicolas, M., Spirakis, P.G.: The structure and complexity of nash equilibria for a selfish routing game. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 123–134. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Fotakis, D., Kontogiannis, S., Spirakis, P.: Selfish unsplittable flows. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 593–605. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    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
  12. 12.
    Gilboa, I., Zemel, E.: Nash and correlated equilibria. Games and Economic Behavior 1(1), 80–93 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gottlob, G., Greco, G., Scarcello, F.: Pure Nash equilibria: Hard and easy games. Theoretical Aspects of Rationality and Knowledge, 215–230 (2003)Google Scholar
  14. 14.
    Koller, D., Megiddo, M.: The complexity of two-person zero sum games in extensive form. Games and Economic Behavior 4(4), 528–552 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Nash, J.: Non-cooperative games. Annals of Mathematics, 286–295 (1951)Google Scholar
  16. 16.
    Osborne, M.J., Rubinstein, A.: A Course in Game Theory. MIT Press, Cambridge (1994)zbMATHGoogle Scholar
  17. 17.
    Papadimitriou, C.: On players with a bounded number of actions. Games and Economic Behavior 4(1), 122–131 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  19. 19.
    Papadimitriou, C.: Algorithms, games and the internet. In: STOC 2001, pp. 4–8 (2001)Google Scholar
  20. 20.
    Schoenebeck, G.R., Vadham, S.: The complexity of Nash equilibria in concisely represented games. Technical Report 52, Electronic Colloquium on Computational Complexity (2005)Google Scholar
  21. 21.
    Thierauf, T.: The computational complexity of equivalence and isomorphisms problems. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Carme Àlvarez
    • 1
  • Joaquim Gabarró
    • 1
  • Maria Serna
    • 1
  1. 1.ALBCOM Research Group.Universitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations