Recent Developments in Equilibria Algorithms

  • Christos Papadimitriou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3828)


Nash proved in 1951 that every game has a mixed Nash equilibrium [6]; whether such an equilibrium can be found in polynomial time has been open since that time. We review here certain recent results which shed some light to this old problem.

Even though a mixed Nash equilibrium is a continuous object, the problem is essentially combinatorial, since it suffices to identify the support of a mixed strategy for each player; however, no algorithm better than exponential for doing so is known. For the case of two players we have a simplex-like pivoting algorithm due to Lemke and Howson that is guaranteed to converge to a Nash equilibrium; this algorithm has an exponential worst case [9]. But even such algorithms seem unlikely for three or more players: in his original paper Nash supplied an example of a 3-player game, an abstraction of poker, with only irrational Nash equilibria.


Nash Equilibrium Mixed Strategy Correlate Equilibrium Nash Equilibrium Problem Graphical Game 
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  1. 1.
    Beame, P., Cook, S., Edmonds, J., Impagliazzo, R., Pitassi, T.: The Relative Complexity of NP Search Problems. J. Comput. Syst. Sci. 57(1), 13–19 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The Complexity of Nash Equilibria. Electronic Colloquium in Computational Complexity (2005)Google Scholar
  3. 3.
    Goldberg, P.W., Papadimitriou, C.H.: Reducibility Among Equilibrium Problems. Electronic Colloquium in Computational Complexity TR-05-090 (2005)Google Scholar
  4. 4.
    Hirsch, M., Papadimitriou, C.H., Vavasis, S.: Exponential Lower Bounds for Finding Brouwer Fixpoints. J. Complexity 5, 379–416 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kearns, M., Littman, M., Singh, S.: Graphical Models for Game Theory. In: Proceedings of UAI (2001)Google Scholar
  6. 6.
    Nash, J.: Noncooperative Games. Annals of Mathematics 54, 289–295 (1951)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Papadimitriou, C.H.: Computing Correlated Equilibria in Multiplayer Games. In: Proceedings of STOC (2005)Google Scholar
  8. 8.
    Papadimitriou, C.H.: On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence. J. Comput. Syst. Sci. 48(3), 498–532 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Savani, R., von Stengel, B.: Exponentially many steps for finding a Nash equilibrium in a Bimatrix Game. In: Proceedings of 45th FOCS, pp. 258–267 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Christos Papadimitriou
    • 1
  1. 1.Department of EECSUC BerkeleyUSA

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