Random Bimatrix Games Are Asymptotically Easy to Solve (A Simple Proof)

  • Panagiota N. Panagopoulou
  • Paul G. Spirakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6982)


We focus on the problem of computing approximate Nash equilibria and well-supported approximate Nash equilibria in random bimatrix games, where each player’s payoffs are bounded and independent random variables, not necessarily identically distributed, but with common expectations. We show that the completely mixed uniform strategy profile, i.e. the combination of mixed strategies (one per player) where each player plays with equal probability each one of her available pure strategies, is with high probability a \(\sqrt{\frac{\ln n}{n}}\)-Nash equilibrium and a \(\sqrt{\frac{3\ln n}{n}}\)-well supported Nash equilibrium, where n is the number of pure strategies available to each player. This asserts that the completely mixed, uniform strategy profile is an almost Nash equilibrium for random bimatrix games, since it is, with high probability, an ε-well-supported Nash equilibrium where ε tends to zero as n tends to infinity.


Nash Equilibrium Independent Random Variable Mixed Strategy Pure Strategy Noncooperative Game 
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  1. 1.
    Althöfer, I.: On sparse approximations to randomized strategies and convex combinations. Linear Algebra and Applications 199, 339–355 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bárány, I., Vempala, S., Vetta, A.: Nash equilibria in random games. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), pp. 123–131 (2005)Google Scholar
  3. 3.
    Chen, X., Deng, X.: Settling the complexity of 2-player Nash-equilibrium. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006 (2005)Google Scholar
  4. 4.
    Chen, X., Deng, X., Teng, S.-H.: Computing Nash equilibria: Approximation and smoothed complexity. In: Electronic Colloquium on Computational Complexity, ECCC (2006)Google Scholar
  5. 5.
    Daskalakis, C., Goldberg, P., Papadimitriou, C.: The complexity of computing a Nash equilibrium. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC 2006), pp. 71–78 (2006)Google Scholar
  6. 6.
    Daskalakis, C., Papadimitriou, C.: Three-player games are hard. In: Electronic Colloquium on Computational Complexity, ECCC (2005)Google Scholar
  7. 7.
    Goldberg, P., Papadimitriou, C.: Reducibility among equilibrium problems. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC 2006), pp. 61–70 (2006)Google Scholar
  8. 8.
    Hoeffding, W.: Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58, 13–30 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lemke, C.E.: Bimatrix equilibrium points and mathematical programming. Management Science 11, 681–689 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lemke, C.E., Howson, J.T.: Equilibrium points of bimatrix games. J. Soc. Indust. Appl. Math. 12, 413–423 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lipton, R.J., Markakis, E., Mehta, A.: Playing large games using simple startegies. In: Proceedings of the 4th ACM Conference on Electronic Commerce (EC 2003), pp. 36–41 (2003)Google Scholar
  12. 12.
    Nash, J.: Noncooperative games. Annals of Mathematics 54, 289–295 (1951)CrossRefGoogle Scholar
  13. 13.
    Papadimitriou, C.H.: On inefficient proofs of existence and complexity classes. In: Proceedings of the 4th Czechoslovakian Symposium on Combinatorics (1991)Google Scholar
  14. 14.
    Savani, R., von Stengel, B.: Exponentially many steps for finding a nash equilibrium in a bimatrix game. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2004), pp. 258–267 (2004)Google Scholar
  15. 15.
    Tsaknakis, H., Spirakis, P.G.: An optimization approach for approximate Nash equilibria. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 42–56. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Panagiota N. Panagopoulou
    • 2
  • Paul G. Spirakis
    • 1
    • 2
  1. 1.Computer Engineering and Informatics DepartmentPatras UniversityGreece
  2. 2.Research Academic Computer Technology InstituteGreece

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