On the Inefficiency Ratio of Stable Equilibria in Congestion Games

  • Arash Asadpour
  • Amin Saberi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5929)


Price of anarchy and price of stability are the primary notions for measuring the efficiency (i.e. the social welfare) of the outcome of a game. Both of these notions focus on extreme cases: one is defined as the inefficiency ratio of the worst-case equilibrium and the other as the best one. Therefore, studying these notions often results in discovering equilibria that are not necessarily the most likely outcomes of the dynamics of selfish and non-coordinating agents.

The current paper studies the inefficiency of the equilibria that are most stable in the presence of noise. In particular, we study two variations of non-cooperative games: atomic congestion games and selfish load balancing. The noisy best-response dynamics in these games keeps the joint action profile around a particular set of equilibria that minimize the potential function. The inefficiency ratio in the neighborhood of these “stable” equilibria is much better than the price of anarchy. Furthermore, the dynamics reaches these equilibria in polynomial time.

Our observations show that in the game environments where a small noise is present, the system as a whole works better than what a pessimist may predict. They also suggest that in congestion games, introducing a small noise in the payoff of the agents may improve the social welfare.


Pure Strategy Evolutionary Game Theory Congestion Game Small Noise Noncooperative Game 
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.
    Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, É., Wexler, T., Roughgarden, T.: The Price of Stability for Network Design with Fair Cost Allocation. In: FOCS 2004, pp. 295–304 (2004)Google Scholar
  2. 2.
    Awerbuch, B., Azar, Y., Epstein, A.: The price of routing unsplittable flow. In: STOC 2005, pp. 57–66 (2005)Google Scholar
  3. 3.
    Awerbuch, B., Azar, Y., Epstein, A., Mirrokni, V.S., Skopalik, A.: Fast convergence to nearly optimal solutions in potential games. In: EC 2008, pp. 264–273 (2008)Google Scholar
  4. 4.
    Blum, A., Hajiaghayi, M., Ligett, K., Roth, A.: Regret minimization and the price of total anarchy. In: STOC 2008, pp. 373–338 (2008)Google Scholar
  5. 5.
    Blume, L.: The statistical mechanics of strategic interaction. Games and Economic Behavior 5, 387–426 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Caragiannis, I., Flammini, M., Kaklamanis, C., Kanellopoulos, P., Moscardelli, L.: Tight bounds for selfish and greedy load balancing. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 311–322. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Chien, S., Sinclair, A.: Convergence to approximate Nash equilibria in congestion games. In: SODA 2007, pp. 169–178 (2007)Google Scholar
  8. 8.
    Christodoulou, G., Koutsoupias, E.: The price of anarchy of finite congestion games. In: STOC 2005, pp. 67–73 (2005)Google Scholar
  9. 9.
    Chung, C., Ligett, K., Pruhs, K., Roth, A.: The Price of Stochastic Anarchy. In: Monien, B., Schroeder, U.-P. (eds.) SAGT 2008. LNCS, vol. 4997, pp. 303–314. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Ellison, G.: Learning, Local Interaction, and Coordination. Econometrica 61(5), 1047–1071 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Finn, G., Horowitz, E.: A linear time approximation algorithm for multiprocessor scheduling. BIT 19(3), 312–320 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fudenberg, D., Kreps, D.: Learning Mixed Equilibria. Games and Economic Behavior 5, 320–367 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fudenberg, D., Levine, D.K.: Theory of Learning in Games. MIT Press, Cambridge (1998)zbMATHGoogle Scholar
  14. 14.
    Kleinberg, R., Piliouras, G., Tardos, É.: Multiplicative updates outperform generic no-regret learning in congestion games. In: STOC 2009, pp. 533–542 (2009)Google Scholar
  15. 15.
    Montanari, A., Saberi, A.: Convergence to Equilibrium in Local Interaction Games and Ising Models. In: FOCS 2009 (to appear, 2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Arash Asadpour
    • 1
  • Amin Saberi
    • 1
  1. 1.Department of Management Science and EngineeringStanford UniversityStanford

Personalised recommendations