Polynomial Space Suffices for Deciding Nash Equilibria Properties for Extensive Games with Large Trees,

  • Carme Àlvarez
  • Joaquim Gabarró
  • Maria Serna
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3827)


We study the computational complexity of deciding the existence of a Pure Nash Equilibrium or a subgame perfect Nash equilibrium with a given payoff and other related problems in finite multi-player extensive games with perfect information. We propose three ways of representing a game with different degrees of succinctness for the components of the game. We show that when the number of moves of each player is large and the player function and the utilities are represented succinctly the considered problems are PSPACE-complete. In contraposition, when the game is described extensively by means of its associated tree all the problems are decidable in polynomial time.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Àlvarez, C., Gabarro, J., Serna, M.: Pure Nash equilibrium in strategic 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
  2. 2.
    Conitzer, V., Sandholm, T.: Complexity results about Nash equilibria. In: IJCAI 2003, pp. 765–771 (2003)Google Scholar
  3. 3.
    Daskalakis, K., Papadimitriou, C.: The complexity of games on highly regular graphs. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 71–82. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure Nash equilibria. In: STOC 2004, pp. 604–612 (2004)Google Scholar
  5. 5.
    Fotakis, D., Kontogiannis, S., Koutsoupias, E., Mavronicolas, M., Spirakis, P.: 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
  6. 6.
    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
  7. 7.
    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
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and intractability. A guide to the NP-completeness. Freeman, New York (1979)MATHGoogle Scholar
  9. 9.
    Gottlob, G., Greco, G., Scarcello, F.: Pure Nash equilibria: Hard and easy games. Theoretical Aspects of Rationality and Knowledge, 215–230 (2003)Google Scholar
  10. 10.
    Koller, D., Megiddo, N., Stengel, B.: Efficient computation of equilibria for extensive two-person games. Games and Economic Behavior 14, 247–259 (1996)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kuhn, H.W.: Extensive games and the problem of information. In: Kuhn, H.W., Tucker, A.W. (eds.) Contribution to the theory of games. Annals of Mathematics Studies, vol. II, 28, pp. 193–216. Princeton University press, Princeton (1953)Google Scholar
  12. 12.
    Osborne, M.J.: A Introductions to Game Theory. Oxford University Press, Oxford (2004)Google Scholar
  13. 13.
    Osborne, M.J., Rubinstein, A.: A Course in Game Theory. MIT Press, Cambridge (1994)MATHGoogle Scholar
  14. 14.
    Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  15. 15.
    Papadimitriou, C.: Algorithms, games and the Internet. In: STOC 2001, pp. 4–8 (2001)Google Scholar
  16. 16.
    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
  17. 17.
    Stengel, B.: Computational complexity of correlated equilibria for extensive games. Technical Report LSE-CDAM-2001-03, CDAM (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

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

Personalised recommendations