Logarithmic Query Complexity for Approximate Nash Computation in Large Games

  • Paul W. Goldberg
  • Francisco J. Marmolejo Cossío
  • Zhiwei Steven Wu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9928)


We investigate the problem of equilibrium computation for “large” n-player games where each player has two pure strategies. Large games have a Lipschitz-type property that no single player’s utility is greatly affected by any other individual player’s actions. In this paper, we assume that a player can change another player’s payoff by at most \(\frac{1}{n}\) by changing her strategy. We study algorithms having query access to the game’s payoff function, aiming to find \(\varepsilon \)-Nash equilibria. We seek algorithms that obtain \(\varepsilon \) as small as possible, in time polynomial in n.

Our main result is a randomised algorithm that achieves \(\varepsilon \) approaching \(\frac{1}{8}\) in a completely uncoupled setting, where each player observes her own payoff to a query, and adjusts her behaviour independently of other players’ payoffs/actions. \(O(\log n)\) rounds/queries are required. We also show how to obtain a slight improvement over \(\frac{1}{8}\), by introducing a small amount of communication between the players.


Nash Equilibrium Mixed Strategy Query Complexity Congestion Game Pure Nash Equilibrium 
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.


  1. 1.
    Azrieli, Y., Shmaya, E.: Lipschitz games. Math. Oper. Res. 38(2), 350–357 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Babichenko, Y.: Best-reply dynamics in large binary-choice anonymous games. Games Econ. Behav. 81, 130–144 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Babichenko, Y.: Query complexity of approximate Nash equilibria. In: Proceedings of 46th STOC, pp. 535–544 (2014)Google Scholar
  4. 4.
    Babichenko, Y., Barman, S.: Query complexity of correlated equilibrium. ACM. Trans. Econ. Comput. 4(3), 1–35 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, X., Cheng, Y., Tang, B.: Well-supported versus approximate Nash equilibria: Query complexity of large games (2015). ArXiv rept. 1511.00785Google Scholar
  6. 6.
    Fearnley, J., Savani, R.: Finding approximate Nash equilibria of bimatrix games via payoff queries. In: Proceedings of 15th ACM EC, pp. 657–674 (2014)Google Scholar
  7. 7.
    Fearnley, J., Gairing, M., Goldberg, P.W., Savani, R.: Learning equilibria of games via payoff queries. J. Mach. Learn. Res. 16, 1305–1344 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Foster, D.P., Young, H.P.: Regret testing: learning to play Nash equilibrium without knowing you have an opponent. Theor. Econ. 1(3), 341–367 (2006)Google Scholar
  9. 9.
    Germano, F., Lugosi, G.: Global Nash convergence of Foster and Young’s regret testing (2005).
  10. 10.
    Goldberg, P.W., Roth, A.: Bounds for the query complexity of approximate equilibria. In: Proceedings of the 15th ACM-EC Conference, pp. 639–656 (2014)Google Scholar
  11. 11.
    Goldberg, P.W., Turchetta, S.: Query complexity of approximate equilibria in anonymous games. In: Markakis, E., et al. (eds.) WINE 2015. LNCS, vol. 9470, pp. 357–369. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48995-6_26 CrossRefGoogle Scholar
  12. 12.
    Hart, S., Mansour, Y.: How long to equilibrium? the communication complexity of uncoupled equilibrium procedures. Games Econ. Behav. 69(1), 107–126 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hart, S., Mas-Colell, A.: A simple adaptive procedure leading to correlated equilibrium. Econometrica 68(5), 1127–1150 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hart, S., Mas-Colell, A.: Uncoupled dynamics do not lead to Nash equilibrium. Am. Econ. Rev. 93(5), 1830–1836 (2003)CrossRefGoogle Scholar
  15. 15.
    Hart, S., Nisan, N.: The query complexity of correlated equilibria (2013). ArXiv tech rept. 1305.4874Google Scholar
  16. 16.
    Kalai, E.: Large robust games. Econometrica 72(6), 1631–1665 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kearns, M., Pai, M.M., Roth, A., Ullman, J.: Mechanism design in large games: Incentives and privacy. Am. Econ. Rev. 104(5), 431–435 (2014). doi: 10.1257/aer.104.5.431 MathSciNetCrossRefGoogle Scholar
  18. 18.
    Young, H.P.: Learning by trial and error (2009).

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Paul W. Goldberg
    • 1
  • Francisco J. Marmolejo Cossío
    • 1
  • Zhiwei Steven Wu
    • 2
  1. 1.University of OxfordOxfordUK
  2. 2.University of PennsylvaniaPhiladelphiaUSA

Personalised recommendations