On Nash-Equilibria of Approximation-Stable Games

  • Pranjal Awasthi
  • Maria-Florina Balcan
  • Avrim Blum
  • Or Sheffet
  • Santosh Vempala
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6386)

Abstract

One reason for wanting to compute an (approximate) Nash equilibrium of a game is to predict how players will play. However, if the game has multiple equilibria that are far apart, or ε-equilibria that are far in variation distance from the true Nash equilibrium strategies, then this prediction may not be possible even in principle. Motivated by this consideration, in this paper we define the notion of games that are approximation stable, meaning that all ε-approximate equilibria are contained inside a small ball of radius Δ around a true equilibrium, and investigate a number of their properties. Many natural small games such as matching pennies and rock-paper-scissors are indeed approximation stable. We show furthermore there exist 2-player n-by-n approximation-stable games in which the Nash equilibrium and all approximate equilibria have support Ω(log n). On the other hand, we show all (ε,Δ) approximation-stable games must have an ε-equilibrium of support \(O(\frac{\Delta^{2-o(1)}}{\epsilon^{2}}{\rm log n})\), yielding an immediate \(n^{O(\frac{\Delta^{2-o(1)}}{\epsilon^2}log n)}\)-time algorithm, improving over the bound of [11] for games satisfying this condition. We in addition give a polynomial-time algorithm for the case that Δ and ε are sufficiently close together. We also consider an inverse property, namely that all non-approximate equilibria are far from some true equilibrium, and give an efficient algorithm for games satisfying that condition.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abbott, T.G., Kane, D.M., Valiant, P.: On the complexity of two-player win-lose games. In: Proc. 46th FOCS, pp. 113–122 (2005)Google Scholar
  2. 2.
    Balcan, M.F., Blum, A., Gupta, A.: Approximate clustering without the approximation. In: Proc. 20th SODA, pp. 1068–1077 (2009)Google Scholar
  3. 3.
    Balcan, M.F., Braverman, M.: Approximate Nash equilibria under stability conditions (2010) (manuscript)Google Scholar
  4. 4.
    Bárány, I., Vempala, S., Vetta, A.: Nash equilibria in random games. Random Struct. Algorithms 31(4), 391–405 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bosse, H., Byrka, J., Markakis, E.: New algorithms for approximate Nash equilibria in bimatrix games. Theor. Comput. Sci. 411(1), 164–173 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, X., Deng, X.: Settling the complexity of two-player Nash equilibrium. In: Proc. 47th FOCS, pp. 261–272 (2006)Google Scholar
  7. 7.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Daskalakis, C., Mehta, A., Papadimitriou, C.H.: Progress in approximate Nash equilibria. In: Proc. 8th ACM-EC (2007)Google Scholar
  9. 9.
    Etessami, K., Yannakakis, M.: On the complexity of Nash equilibria and other fixed points. SIAM J. Comput. 39(6), 2531–2597 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Feder, T., Nazerzadeh, H., Saberi, A.: Approximating nash equilibria using small-support strategies. In: Proc. 8th ACM-EC, pp. 352–354 (2007)Google Scholar
  11. 11.
    Lipton, R.J., Markakis, E., Mehta, A.: Playing large games using simple strategies. In: Proc. 4th ACM-EC, pp. 36–41 (2003)Google Scholar
  12. 12.
    Tsaknakis, H., Spirakis, P.: 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 2010

Authors and Affiliations

  • Pranjal Awasthi
    • 1
  • Maria-Florina Balcan
    • 2
  • Avrim Blum
    • 1
  • Or Sheffet
    • 1
  • Santosh Vempala
    • 2
  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.Georgia Institute of TechnologyAtlantaUSA

Personalised recommendations