The Game World Is Flat: The Complexity of Nash Equilibria in Succinct Games

  • Constantinos Daskalakis
  • Alex Fabrikant
  • Christos H. Papadimitriou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4051)

Abstract

A recent sequence of results established that computing Nash equilibria in normal form games is a PPAD-complete problem even in the case of two players [11,6,4]. By extending these techniques we prove a general theorem, showing that, for a far more general class of families of succinctly representable multiplayer games, the Nash equilibrium problem can also be reduced to the two-player case. In view of empirically successful algorithms available for this problem, this is in essence a positive result — even though, due to the complexity of the reductions, it is of no immediate practical significance. We further extend this conclusion to extensive form games and network congestion games, two classes which do not fall into the same succinct representation framework, and for which no positive algorithmic result had been known.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aumann, R.J.: Subjectivity and Correlation in Randomized Strategies. Journal of Mathematical Economics 1, 67–95 (1974)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bürgisser, P.: On the structure of Valiant’s complexity classes. Discr. Math. Theoret. Comp. Sci. 3, 73–94 (1999)MATHGoogle Scholar
  3. 3.
    Chen, X., Deng, X.: 3-NASH is PPAD-Complete. ECCC, TR05-134 (2005)Google Scholar
  4. 4.
    Chen, X., Deng, X.: Settling the Complexity of 2-Player Nash-Equilibrium. ECCC, TR05-140 (2005)Google Scholar
  5. 5.
    Daskalakis, C., Papadimitriou, C.H.: Three-Player Games Are Hard. ECCC, TR05-139 (2005)Google Scholar
  6. 6.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The Complexity of Computing a Nash Equilibrium. In: Proceedings of 38th STOC (2006)Google Scholar
  7. 7.
    Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The Complexity of Pure Nash Equilibria. In: Proceedings of 36th STOC (2004)Google Scholar
  8. 8.
    Feigenbaum, J., Koller, D., Shor, P.: A game-theoretic classification of interactive complexity classes. In: IEEE Conference on Structure in Complexity Theory (1995)Google Scholar
  9. 9.
    Fortnow, L., Impagliazzo, R., Kabanets, V., Umans, C.: On the Complexity of Succinct Zero-Sum Games. In: IEEE Conference on Computational Complexity (2005)Google Scholar
  10. 10.
    Geanakoplos, J.: Nash and Walras Equilibrium via Brouwer. Economic Theory, 21 (2003)Google Scholar
  11. 11.
    Goldberg, P.W., Papadimitriou, C.H.: Reducibility Among Equilibrium Problems. In: Proceedings of 38th STOC (2006)Google Scholar
  12. 12.
    Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: How Easy is Local Search? J. Comput. Syst. Sci. 37(1), 79–100 (1988)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kearns, M., Littman, M., Singh, S.: Graphical Models for Game Theory. In: UAI (2001)Google Scholar
  14. 14.
    Lemke, C.E., Howson Jr., J.T.: Equilibrium points of bimatrix games. Journal of the Society for Industrial and Applied Mathematics 12, 413–423 (1964)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Nash, J.: Noncooperative Games. Annals of Mathematics 54, 289–295 (1951)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Osborne, M.J., Rubinstein, A.: A Course in Game Theory. MIT Press, Cambridge (1994)MATHGoogle Scholar
  17. 17.
    Papadimitriou, C.H.: Computing Correlated Equilibria in Multiplayer Games. In: STOC (2005)Google Scholar
  18. 18.
    Papadimitriou, C.H.: On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence. J. Comput. Syst. Sci. 3, 498–532 (1994)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Savani, R., von Stengel, B.: Exponentially many steps for finding a Nash equilibrium in a Bimatrix Game. In: Proceedings of 45th FOCS (2004)Google Scholar
  20. 20.
    von Stengel, B.: Computing equilibria for two-person games. In: Aumann, R.J., Hart, S. (eds.) Handbook of Game Theory, vol. 3, pp. 1723–1759. North-Holland, Amsterdam (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Constantinos Daskalakis
    • 1
  • Alex Fabrikant
    • 1
  • Christos H. Papadimitriou
    • 1
  1. 1.Computer Science DivisionUC Berkeley 

Personalised recommendations