The Price of Stochastic Anarchy

  • Christine Chung
  • Katrina Ligett
  • Kirk Pruhs
  • Aaron Roth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4997)


We consider the solution concept of stochastic stability, and propose the price of stochastic anarchy as an alternative to the price of (Nash) anarchy for quantifying the cost of selfishness and lack of coordination in games. As a solution concept, the Nash equilibrium has disadvantages that the set of stochastically stable states of a game avoid: unlike Nash equilibria, stochastically stable states are the result of natural dynamics of computationally bounded and decentralized agents, and are resilient to small perturbations from ideal play. The price of stochastic anarchy can be viewed as a smoothed analysis of the price of anarchy, distinguishing equilibria that are resilient to noise from those that are not. To illustrate the utility of stochastic stability, we study the load balancing game on unrelated machines. This game has an unboundedly large price of Nash anarchy even when restricted to two players and two machines. We show that in the two player case, the price of stochastic anarchy is 2, and that even in the general case, the price of stochastic anarchy is bounded. We conjecture that the price of stochastic anarchy is O(m), matching the price of strong Nash anarchy without requiring player coordination. We expect that stochastic stability will be useful in understanding the relative stability of Nash equilibria in other games where the worst equilibria seem to be inherently brittle.


Nash Equilibrium Stochastic Stability Evolutionary Game Theory Pure Strategy Nash Equilibrium Evolutionarily Stable 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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  1. 1.
    Andelman, N., Feldman, M., Mansour, Y.: Strong price of anarchy. In: SODA 2007 (2007)Google Scholar
  2. 2.
    Awerbuch, B., Azar, Y., Richter, Y., Tsur, D.: Tradeoffs in worst-case equilibria. Theor. Comput. Sci. 361(2), 200–209 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blum, A., Even-Dar, E., Ligett, K.: Routing without regret: On convergence to Nash equilibria of regret-minimizing algorithms in routing games. In: PODC 2006 (2006)Google Scholar
  4. 4.
    Blum, A., Hajiaghayi, M., Ligett, K., Roth, A.: Regret minimization and the price of total anarchy. In: STOC 2008 (2008)Google Scholar
  5. 5.
    Blume, L.E.: The statistical mechanics of best-response strategy revision. Games and Economic Behavior 11(2), 111–145 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, X., Deng, X.: Settling the complexity of 2-player Nash-equilibrium. In: FOCS 2006 (2006)Google Scholar
  7. 7.
    Ellison, G.: Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution. Review of Economic Studies 67(1), 17–45 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Even-Dar, E., Kesselman, A., Mansour, Y.: Convergence time to Nash equilibria. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, Springer, Heidelberg (2003)Google Scholar
  9. 9.
    Fabrikant, A., Papadimitriou, C.: The complexity of game dynamics: Bgp oscillations, sink equilibria, and beyond. In: SODA 2008 (2008)Google Scholar
  10. 10.
    Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure nash equilibria. In: STOC 2004 (2004)Google Scholar
  11. 11.
    Fiat, A., Kaplan, H., Levy, M., Olonetsky, S.: Strong price of anarchy for machine load balancing. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, Springer, Heidelberg (2007)Google Scholar
  12. 12.
    Fischer, S., Räcke, H., Vöcking, B.: Fast convergence to wardrop equilibria by adaptive sampling methods. In: STOC 2006 (2006)Google Scholar
  13. 13.
    Fischer, S., Vöcking, B.: On the evolution of selfish routing. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, Springer, Heidelberg (2004)Google Scholar
  14. 14.
    Foster, D., Young, P.: Stochastic evolutionary game dynamics. Theoret. Population Biol. 38, 229–232 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Goemans, M., Mirrokni, V., Vetta, A.: Sink equilibria and convergence. In: FOCS 2005 (2005)Google Scholar
  16. 16.
    Josephson, J., Matros, A.: Stochastic imitation in finite games. Games and Economic Behavior 49(2), 244–259 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kandori, M., Mailath, G.J., Rob, R.: Learning, mutation, and long run equilibria in games. Econometrica 61(1), 29–56 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: 16th Annual Symposium on Theoretical Aspects of Computer Science, Trier, Germany, March 4–6, 1999, pp. 404–413 (1999)Google Scholar
  19. 19.
    Larry, S.: Stochastic stability in games with alternative best replies. Journal of Economic Theory 64(1), 35–65 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V. (eds.): Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)Google Scholar
  21. 21.
    Robson, A.J., Vega-Redondo, F.: Efficient equilibrium selection in evolutionary games with random matching. Journal of Economic Theory 70(1), 65–92 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Roughgarden, T., Tardos, É.: How bad is selfish routing. J. ACM 49(2), 236–259 (2002); In: FOCS 2000 (2000)Google Scholar
  23. 23.
    Suri, S.: Computational evolutionary game theory. In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V.V., Vazirani, V.V. (eds.) Algorithmic Game Theory, Cambridge University Press, Cambridge (2007)Google Scholar
  24. 24.
    Peyton Young, H.: The evolution of conventions. Econometrica 61(1), 57–84 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Peyton Young, H.: Individual Strategy and Social Structure. Princeton University Press, Princeton (1998)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Christine Chung
    • 1
  • Katrina Ligett
    • 2
  • Kirk Pruhs
    • 1
  • Aaron Roth
    • 2
  1. 1.Department of Computer ScienceUniversity of PittsburghUSA
  2. 2.Department of Computer ScienceCarnegie Mellon UniversityUSA

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