The Limits of Smoothness: A Primal-Dual Framework for Price of Anarchy Bounds

  • Uri Nadav
  • Tim Roughgarden
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6484)


We show a formal duality between certain equilibrium concepts, including the correlated and coarse correlated equilibrium, and analysis frameworks for proving bounds on the price of anarchy for such concepts. Our first application of this duality is a characterization of the set of distributions over game outcomes to which “smoothness bounds” always apply. This set is a natural and strict generalization of the coarse correlated equilibria of the game. Second, we derive a refined definition of smoothness that is specifically tailored for coarse correlated equilibria and can be used to give improved POA bounds for such equilibria.


Nash Equilibrium Equilibrium Concept Congestion Game Correlate Equilibrium Mixed 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aumann, R.J.: Subjectivity and correlation in randomized strategies. Journal of Mathematical Economics 1(1), 67–96 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blum, A., Even-Dar, E., Ligett, K.: Routing without regret: On convergence to Nash equilibria of regret-minimizing algorithms in routing games. In: Proceedings of the 25th Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 45–52 (2006)Google Scholar
  3. 3.
    Blum, A., Hajiaghayi, M., Ligett, K., Roth, A.: Regret minimization and the price of total anarchy. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC), pp. 373–382 (2008)Google Scholar
  4. 4.
    Blum, A., Mansour, Y.: Learning, regret minimization, and equilibria. In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V. (eds.) Algorithmic Game Theory, ch. 4, pp. 79–101. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  5. 5.
    Christodoulou, G., Koutsoupias, E.: On the price of anarchy and stability of correlated equilibria of linear congestion games. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 59–70. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Correa, J.R., Schulz, A.S., Stier Moses, N.E.: On the inefficiency of equilibria in congestion games. Games and Economic Behavior 64(2), 457–469 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure Nash equilibria. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pp. 604–612 (2004)Google Scholar
  8. 8.
    Hannan, J.: Approximation to Bayes risk in repeated plays. In: Dresher, M., Tucker, A., Wolfe, P. (eds.) Contributions to the Theory of Games, vol. 3, pp. 97–139. Princeton University Press, Princeton (1957)Google Scholar
  9. 9.
    Harks, T.: Stackelberg strategies and collusion in network games with splittable flow. In: Bampis, E., Skutella, M. (eds.) WAOA 2008. LNCS, vol. 5426, pp. 133–146. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Perakis, G.: The price of anarchy when costs are non-separable and asymmetric. Mathematics of Operations Research 32(3), 614–628 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Roughgarden, T.: Intrinsic robustness of the price of anarchy. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), pp. 513–522 (2009)Google Scholar
  12. 12.
    Vetta, A.: Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions. In: Proceedings of the 43rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 416–425 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Uri Nadav
    • 1
  • Tim Roughgarden
    • 2
  1. 1.Department of Computer ScienceStanford UniversityStanford
  2. 2.Department of Computer ScienceStanford UniversityStanford

Personalised recommendations