Abstract
We study logit dynamics [Blume, Games and Economic Behavior, 1993] for strategic games. At every stage of the game a player is selected uniformly at random and she plays according to a noisy best-response dynamics where the noise level is tuned by a parameter β. Such a dynamics defines a family of ergodic Markov chains, indexed by β, over the set of strategy profiles. Our aim is twofold: On the one hand, we are interested in the expected social welfare when the strategy profiles are random according to the stationary distribution of the Markov chain, because we believe it gives a meaningful description of the long-term behavior of the system. On the other hand, we want to estimate how long it takes, for a system starting at an arbitrary profile and running the logit dynamics, to get close to the stationary distribution; i.e., the mixing time of the chain.
In this paper we study the stationary expected social welfare for the 3-player congestion game that exhibits the worst Price of Anarchy [Christodoulou and Koutsoupias, STOC’05], for 2-player coordination games (the same class of games studied by Blume), and for a simple n-player game. For all these games, we give almost-tight upper and lower bounds on the mixing time of logit dynamics.
Work partially supported by EU project FRONTS and by MIUR PRIN project COGENT.
This is a preview of subscription content, log in via an institution to check access.
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
Anshelevich, E., Dasgupta, A., Kleinberg, J.M., Tardos, É., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. SIAM Journal on Computing 38(4), 1602–1623 (2008)
Asadpour, A., Saberi, A.: On the inefficiency ratio of stable equilibria in congestion games. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 545–552. Springer, Heidelberg (2009)
Auletta, V., Ferraioli, D., Pasquale, F., Persiano, G.: Mixing time and stationary expected social welfare of logit dynamics. CoRR abs/1002.3474 (2010)
Blume, L.E.: The statistical mechanics of strategic interaction. Games and Economic Behavior 5, 387–424 (1993)
Chen, X., Deng, X., Teng, S.-H.: Settling the complexity of computing two-player Nash equilibria. Journal of the ACM 56(3) (2009)
Christodoulou, G., Koutsoupias, E.: The price of anarchy of finite congestion games. In: Proc. of the 37th Annual ACM Symposium on Theory of Computing (STOC 2005), pp. 67–73. ACM, New York (2005)
Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM Journal on Computing 39(1), 195–259 (2009)
Ellison, G.: Learning, local interaction, and coordination. Econometrica 61(5), 1047–1071 (1993)
Fabrikant, A., Papadimitriou, C.H.: The complexity of game dynamics: BGP oscillations, sink equilibria, and beyond. In: Proc. of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pp. 844–853. ACM, New York (2008)
Foster, D.P., Vohra, R.V.: Calibrated learning and correlated equilibrium. Games and Economic Behavior 21(1-2), 40–55 (1997)
Fudenberg, D., Levine, D.K.: The Theory of Learning in Games. MIT, Cambridge (1998)
Goemans, M., Mirrokni, V., Vetta, A.: Sink equilibria and convergence. In: Proc. of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), pp. 142–154. IEEE, Los Alamitos (2005)
Harsanyi, J.C., Selten, R.: A General Theory of Equilibrium Selection in Games. MIT Press, Cambridge (1988)
Hart, S., Mas-Colell, A.: A simple adaptive procedure leading to correlated equilibrium. Econometrica 68(5), 1127–1150 (2000)
Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. Computer Science Review 3(2), 65–69 (2009); Preliminary version in STACS 1999
Levin, D., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2008)
Montanari, A., Saberi, A.: Convergence to equilibrium in local interaction games. In: Proc. of the 50th Annual Symposium on Foundations of Computer Science (FOCS 2009). IEEE, Los Alamitos (2009)
Roth, A., Balcan, N., Kalai, A., Mansour, Y.: On the equilibria of alternating move games. In: Proc. of ACM-SIAM Symposium on Discrete Algorithms, SODA 2010 (2010)
Sandholm, W.H.: Population Games and Evolutionary Dynamics. MIT Press, Cambridge (2010)
Peyton Young, H.: Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton University Press, Princeton (1998)
Peyton Young, H.: The diffusion of innovations in social networks. Economics Working Paper Archive number 437, Johns Hopkins University, Department of Economics (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Auletta, V., Ferraioli, D., Pasquale, F., Persiano, G. (2010). Mixing Time and Stationary Expected Social Welfare of Logit Dynamics. In: Kontogiannis, S., Koutsoupias, E., Spirakis, P.G. (eds) Algorithmic Game Theory. SAGT 2010. Lecture Notes in Computer Science, vol 6386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16170-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-16170-4_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16169-8
Online ISBN: 978-3-642-16170-4
eBook Packages: Computer ScienceComputer Science (R0)