Efficient Computation of Nash Equilibria for Very Sparse Win-Lose Bimatrix Games

  • Bruno Codenotti
  • Mauro Leoncini
  • Giovanni Resta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)


It is known that finding a Nash equilibrium for win-lose bimatrix games, i.e., two-player games where the players’ payoffs are zero and one, is complete for the class PPAD.

We describe a linear time algorithm which computes a Nash equilibrium for win-lose bimatrix games where the number of winning positions per strategy of each of the players is at most two.

The algorithm acts on the directed graph that represents the zero-one pattern of the payoff matrices describing the game. It is based upon the efficient detection of certain subgraphs which enable us to determine the support of a Nash equilibrium.


Nash Equilibrium Mixed Strategy Pure Strategy Current Cycle Nash Equilibrium Strategy 
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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  1. 1.
    Abbott, T., Kane, D., Valiant, P.: On the Complexity of Two-Player Win-Lose Games. In: Proc. 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 113–122 (2005)Google Scholar
  2. 2.
    Chen, X., Deng, X.: 3-NASH is PPAD-Complete, ECCC TR05-134 (2005)Google Scholar
  3. 3.
    Chen, X., Deng, X.: Settling the Complexity of 2-Player Nash-Equilibrium, ECCC TR05-140 (2005)Google Scholar
  4. 4.
    Codenotti, B., Leoncini, M., Resta, G.: Efficient Computation of Nash Equilibria for Very Sparse Win-Lose Games, ECCC Technical Report TR06-012. Available at: http://eccc.hpi-web.de/eccc-reports/2006/TR06-012/index.html
  5. 5.
    Codenotti, B., Stefankovic, D.: On the computational complexity of Nash equilibria for (0,1)-bimatrix games. Information Processing Letters 94(3), 145–150 (2005)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Daskalakis, C., Goldberg, P., Papadimitriou, C.: The complexity of computing a Nash equilibrium, ECCC TR05-115 (2005)Google Scholar
  7. 7.
    Daskalakis, C., Papadimitriou, C.: Three-Player Games Are Hard, ECCC TR05-139 (2005)Google Scholar
  8. 8.
    Goldberg, P.W., Papadimitriou, C.: Reducibility Among Equilibrium Problems, ECCC TR05-090 (2005)Google Scholar
  9. 9.
    Lemke, C.E., Howson, J.T.: Equilibrium points in bimatrix games. Journal of the Society for Industrial and Applied Mathematics 12, 413–423 (1964)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Nash, J.: Non-Cooperative Games. Annals of Mathematics 54(2), 286–295 (1951)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Papadimitriou, C.: On the Complexity of the Parity Argument and other Inefficient Proofs of Existence. Journal of Computer and System Sciences 48, 498–532 (1994)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Savani, R., von Stengel, B.: Exponentially Many Steps for Finding a Nash Equilibrium in a Bimatrix Game. In: Proc. 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 258–267 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Bruno Codenotti
    • 1
  • Mauro Leoncini
    • 2
  • Giovanni Resta
    • 1
  1. 1.IIT-CNRPisaItaly
  2. 2.Dipartimento di Ingegneria dell’InformazioneUniversità di Modena e Reggio EmiliaModenaItaly

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