## Abstract

Logit dynamics (Blume in Games Econ Behav 5:387–424, 1993) are randomized best response dynamics for strategic games: at every time step a player is selected uniformly at random and she chooses a new strategy according to a probability distribution biased toward strategies promising higher payoffs. This process defines an ergodic Markov chain, over the set of strategy profiles of the game, whose unique stationary distribution is the long-term equilibrium concept for the game. However, when the mixing time of the chain is large (e.g., exponential in the number of players), the stationary distribution loses its appeal as equilibrium concept, and the transient phase of the Markov chain becomes important. It can happen that the chain is “metastable”, i.e., on a time-scale shorter than the mixing time, it stays close to some probability distribution over the state space, while in a time-scale multiple of the mixing time it jumps from one distribution to another. In this paper we give a quantitative definition of “metastable probability distributions” for a Markov chain and we study the metastability of the logit dynamics for some classes of coordination games. We first consider a pure *n*-player coordination game that highlights the distinctive features of our metastability notion based on distributions. Then, we study coordination games on the clique without a risk-dominant strategy (which are equivalent to the well-known Glauber dynamics for the Curie–Weiss model) and coordination games on a ring (both with and without risk-dominant strategy).

## Notes

### Acknowledgements

We wish to thank Paolo Penna for useful ideas, hints, and discussions. We also thank the anonymous referees for their comments and suggestions.

## References

- 1.Alós-Ferrer, C., Netzer, N.: The logit-response dynamics. Games Econ. Behav.
**68**(2), 413–427 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Alós-Ferrer, C., Netzer, N.: Robust stochastic stability. Econ. Theory
**58**(1), 31–57 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Asadpour, A., Saberi, A.: On the inefficiency ratio of stable equilibria in congestion games. In: Proceedings of the 5th International Workshop on Internet and Network Economics (WINE’09), Lecture Notes in Computer Science, vol. 5929, Springer, pp. 545–552 (2009)Google Scholar
- 4.Auletta, V., Ferraioli, D., Pasquale, F., Penna, P., Persiano, G.: Logit dynamics with concurrent updates for local interaction games. In: Proceedings of 21st Annual European Symposium on Algorithms–ESA 2013, Sophia Antipolis, France, September 2–4, pp. 73–84 (2013a)Google Scholar
- 5.Auletta, V., Ferraioli, D., Pasquale, F., Persiano, G.: Metastability of logit dynamics for coordination games. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17–19, pp. 1006–1024 (2012)Google Scholar
- 6.Auletta, V., Ferraioli, D., Pasquale, F., Persiano, G.: Mixing time and stationary expected social welfare of logit dynamics. Theory Comput. Syst.
**53**(1), 3–40 (2013b)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Auletta, V., Ferraioli, D., Pasquale, F., Penna, P., Persiano, G.: Convergence to equilibrium of logit dynamics for strategic games. Algorithmica
**76**(1), 110–142 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Berger, N., Kenyon, C., Mossel, E., Peres, Y.: Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Relat. Fields
**131**, 311–340 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Bianchi, A., Gaudilliere, A.: Metastable states, quasi-stationary and soft measures, mixing time asymprtotics via variational principles (2011). arXiv preprint arXiv:1103.1143
- 10.Blume, L.E.: The statistical mechanics of strategic interaction. Games Econ. Behav.
**5**, 387–424 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Borowski, H., Marden, J.R., Frew, E.W.: Fast convergence in semi-anonymous potential games. In: 2013 IEEE 52nd Annual Conference on Decision and Control (CDC), IEEE, pp. 2418–2423 (2013)Google Scholar
- 12.Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Relat. Fields
**119**, 99–161 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability and low lying spectra in reversible Markov chains. Commun. Math. Phys.
**228**, 219–255 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Camerer, C.: Behavioral Game Theory: Experiments in Strategic Interaction. Princeton University Press, Princeton (2003)zbMATHGoogle Scholar
- 15.Collet, P., Martínez, S., San Martín, J., Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems. Springer, Berlin (2012)Google Scholar
- 16.Ding, J., Lubetzky, E., Peres, Y.: The mixing time evolution of Glauber dynamics for the mean-field Ising model. Commun. Math. Phys.
**289**, 725–764 (2009a)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Ding, J., Lubetzky, E., Peres, Y.: Censored Glauber dynamics for the mean field Ising model. J. Stat. Phys.
**137**, 407–458 (2009b)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Ellison, G.: Learning, local interaction, and coordination. Econometrica
**61**(5), 1047–1071 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Ellison, G.: Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution. Rev. Econ. Stud.
**67**(1), 17–45 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure nash equilibria. In: STOC, pp. 604–612 (2004)Google Scholar
- 21.Ferraioli, D., Ventre, C.: Metastability of asymptotically well-behaved potential games. In: Italiano, G., Pighizzini, G., Sannella, D. (eds.) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science, vol. 9235, pp. 311–323. Springer, Berlin, Heidelberg (2015)Google Scholar
- 22.Freidlin, M., Koralov, L.: Metastable distributions of Markov chains with rare transitions. J. Stat. Phys.
**167**(6), 1355–1375 (2017)Google Scholar - 23.Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
- 24.Galam, S., Walliser, B.: Ising model versus normal form game. Phys. A Stat. Mech. Appl.
**389**(3), 481–489 (2010)MathSciNetCrossRefGoogle Scholar - 25.Harsanyi, J.C., Selten, R.: A General Theory of Equilibrium Selection in Games. MIT Press, Cambridge (1988)zbMATHGoogle Scholar
- 26.Keilson, J.: Markov Chain Modelsrarity and Exponentiality, vol. 28. Springer, Berlin (2012)Google Scholar
- 27.Kreindler, G.E., Young, H.P.: Rapid innovation diffusion in social networks. In: Proceedings of the National Academy of Sciences, vol. 111, no. Supplement 3, pp. 10881–10888 (2014)Google Scholar
- 28.Kreindler, G.E., Young, H.P.: Fast convergence in evolutionary equilibrium selection. Games Econ. Behav.
**80**, 39–67 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 29.Levin, D., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
- 30.Levin, D., Luczak, M., Peres, Y.: Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability. Probab. Theory Relat. Fields
**146**, 223–265 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 31.Marden, J.R., Shamma, J.S.: Revisiting log-linear learning: asynchrony, completeness and payoff-based implementation. In: 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), IEEE, pp. 1171–1172 (2010)Google Scholar
- 32.Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1997), Lecture Notes in Mathematics, vol. 1717, Springer, pp. 93–191 (1999)Google Scholar
- 33.Monderer, D., Shapley, L.S.: Potential games. Games Econ. Behav.
**14**, 124–143 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 34.Montanari, A., Saberi, A.: Convergence to equilibrium in local interaction games. In: Proceedings of the 50th Annual Symposium on Foundations of Computer Science (FOCS’09), IEEE (2009)Google Scholar
- 35.Sandholm, W.H.: Population Games and Evolutionary Dynamics. MIT Press, Cambridge (2010)zbMATHGoogle Scholar
- 36.Sandholm, W.H., Staudigl, M.: Large deviations and stochastic stability in the small noise double limit. Theor. Econ.
**11**(1), 279–355 (2016)MathSciNetCrossRefGoogle Scholar - 37.Shah, D., Shin, J.: Dynamics in congestion games. In: ACM SIGMETRICS Performance Evaluation Review, vol. 38, ACM, pp. 107–118 (2010)Google Scholar
- 38.Taylor, H.M., Karlin, S.: An Introduction to Stochastic Modeling. Academic, Cambridge (2014)zbMATHGoogle Scholar
- 39.Young, H.P.: The diffusion of innovations in social networks. Economics Working Paper Archive number 437, Johns Hopkins University, Department of Economics (2000)Google Scholar
- 40.Young, H.P.: Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton University Press, Princeton (1998)Google Scholar