Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andelman, N., Feldman, M., Mansour, Y.: Strong price of anarchy. In: SODA 2007 (2007)
Awerbuch, B., Azar, Y., Richter, Y., Tsur, D.: Tradeoffs in worst-case equilibria. Theor. Comput. Sci. 361(2), 200–209 (2006)
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)
Blum, A., Hajiaghayi, M., Ligett, K., Roth, A.: Regret minimization and the price of total anarchy. In: STOC 2008 (2008)
Blume, L.E.: The statistical mechanics of best-response strategy revision. Games and Economic Behavior 11(2), 111–145 (1995)
Chen, X., Deng, X.: Settling the complexity of 2-player Nash-equilibrium. In: FOCS 2006 (2006)
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)
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)
Fabrikant, A., Papadimitriou, C.: The complexity of game dynamics: Bgp oscillations, sink equilibria, and beyond. In: SODA 2008 (2008)
Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure nash equilibria. In: STOC 2004 (2004)
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)
Fischer, S., Räcke, H., Vöcking, B.: Fast convergence to wardrop equilibria by adaptive sampling methods. In: STOC 2006 (2006)
Fischer, S., Vöcking, B.: On the evolution of selfish routing. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, Springer, Heidelberg (2004)
Foster, D., Young, P.: Stochastic evolutionary game dynamics. Theoret. Population Biol. 38, 229–232 (1990)
Goemans, M., Mirrokni, V., Vetta, A.: Sink equilibria and convergence. In: FOCS 2005 (2005)
Josephson, J., Matros, A.: Stochastic imitation in finite games. Games and Economic Behavior 49(2), 244–259 (2004)
Kandori, M., Mailath, G.J., Rob, R.: Learning, mutation, and long run equilibria in games. Econometrica 61(1), 29–56 (1993)
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)
Larry, S.: Stochastic stability in games with alternative best replies. Journal of Economic Theory 64(1), 35–65 (1994)
Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V. (eds.): Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)
Robson, A.J., Vega-Redondo, F.: Efficient equilibrium selection in evolutionary games with random matching. Journal of Economic Theory 70(1), 65–92 (1996)
Roughgarden, T., Tardos, É.: How bad is selfish routing. J. ACM 49(2), 236–259 (2002); In: FOCS 2000 (2000)
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)
Peyton Young, H.: The evolution of conventions. Econometrica 61(1), 57–84 (1993)
Peyton Young, H.: Individual Strategy and Social Structure. Princeton University Press, Princeton (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chung, C., Ligett, K., Pruhs, K., Roth, A. (2008). The Price of Stochastic Anarchy. In: Monien, B., Schroeder, UP. (eds) Algorithmic Game Theory. SAGT 2008. Lecture Notes in Computer Science, vol 4997. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79309-0_27
Download citation
DOI: https://doi.org/10.1007/978-3-540-79309-0_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79308-3
Online ISBN: 978-3-540-79309-0
eBook Packages: Computer ScienceComputer Science (R0)