On the Communication Complexity of Approximate Nash Equilibria

  • Paul W. Goldberg
  • Arnoud Pastink
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7615)

Abstract

We study the problem of computing approximate Nash equilibria, in a setting where players initially know their own payoffs but not the payoffs of other players. In order for a solution of reasonable quality to be found, some amount of communication needs to take place between the players. We are interested in algorithms where the communication is substantially less than the contents of a payoff matrix, for example logarithmic in the size of the matrix. At one extreme is the case where the players do not communicate at all; for this case (with 2 players having n×n matrices) ε-Nash equilibria can be computed for ε = 3/4, while there is a lower bound of slightly more than 1/2 on the lowest ε achievable. When the communication is polylogarithmic in n, we show how to obtain ε = 0.438. For one-way communication we show that ε = 1/2 is the exact answer.

Keywords

Nash Equilibrium Mixed Strategy Pure Strategy Communication Complexity 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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References

  1. 1.
    Bosse, H., Byrka, J., Markakis, E.: New Algorithms for Approximate Nash Equilibria in Bimatrix Games. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 17–29. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Chen, X., Deng, X.: Settling the complexity of two-player Nash equilibrium. In: Procs. of the 47th FOCS Symposium, pp. 261–272. IEEE (2006)Google Scholar
  3. 3.
    Chen, X., Deng, X., Teng, S.H.: Settling the complexity of computing two-player Nash equilibria. J. ACM 56, 14:1–14:57 (2009)Google Scholar
  4. 4.
    Conitzer, V., Sandholm, T.: Communication complexity as a lower bound for learning in games. In: Proceedings of the 21st ICML, pp. 24–32 (2004)Google Scholar
  5. 5.
    Dantzig, G.B.: Linear Programming and Extensions. Princeton Univ. Press (1963)Google Scholar
  6. 6.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Daskalakis, C., Mehta, A., Papadimitriou, C.: A Note on Approximate Nash Equilibria. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 297–306. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Goldberg, P.W.: Some discriminant-based PAC algorithms. Journal of Machine Learning Research 7, 283–306 (2006)MATHGoogle Scholar
  9. 9.
    Hart, S., Mansour, Y.: How long to equilibrium? the communication complexity of uncoupled equilibrium procedures. GEB 69(1), 107–126 (2010)MATHMathSciNetGoogle Scholar
  10. 10.
    Hoeffding, W.: Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58(301), 13–30 (1963)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. In: 16th STOC, pp. 302–311. ACM (1984)Google Scholar
  12. 12.
    Kushilevitz, E.: Communication complexity. Advances in Computers 44, 331–360 (1997)CrossRefGoogle Scholar
  13. 13.
    Lipton, R.J., Markakis, E., Mehta, A.: Playing large games using simple strategies. In: Procs. of the 4th ACM-EC, EC 2003, pp. 36–41 (2003)Google Scholar
  14. 14.
    Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    von Neumann, J.: Zur theorie der gesellschaftsspiele. Mathematische Annalen 100, 295–320 (1928)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Pastink, A.: Aspects of communication complexity for approximating Nash equilibria. MSc dissertation, Utrecht University (2012)Google Scholar
  17. 17.
    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
  18. 18.
    Yao, A.C.C.: Some complexity questions related to distributive computing (preliminary report). In: 11th STOC, pp. 209–213. ACM (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Paul W. Goldberg
    • 1
  • Arnoud Pastink
    • 2
  1. 1.Department of Computer ScienceUniversity of LiverpoolU.K.
  2. 2.Department of Information and Computing ScienceUtrecht UniversityUtrechtThe Netherlands

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