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The logit-response dynamics. (English) Zbl 1207.91017
Summary: We develop a characterization of stochastically stable states for the logit-response learning dynamics in games, with arbitrary specification of revision opportunities. The result allows us to show convergence to the set of Nash equilibria in the class of best-response potential games and the failure of the dynamics to select potential maximizers beyond the class of exact potential games. We also study to which extent equilibrium selection is robust to the specification of revision opportunities. Our techniques can be extended and applied to a wide class of learning dynamics in games.

MSC:
91A26 Rationality and learning in game theory
91A05 2-person games
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[1] Alós-Ferrer, C.; Ania, A.B., The evolutionary stability of perfectly competitive behavior, Econ. theory, 26, 497-516, (2005) · Zbl 1106.91002
[2] Alós-Ferrer, C., Kirchsteiger, G., 2007. Learning and market clearing. Mimeo · Zbl 1367.91070
[3] Baron, R.; Durieu, J.; Haller, H.; Solal, P., A note on control costs and logit rules for strategic games, J. evolutionary econ., 12, 563-575, (2002)
[4] Baron, R.; Durieu, J.; Haller, H.; Solal, P., Control costs and potential functions for spatial games, Int. J. game theory, 31, 541-561, (2002) · Zbl 1052.91024
[5] Beggs, A., Waiting times and equilibrium selection, Econ. theory, 25, 599-628, (2005) · Zbl 1127.91009
[6] Bergin, J.; Lipman, B.L., Evolution with state-dependent mutations, Econometrica, 64, 943-956, (1996) · Zbl 0862.90142
[7] Blume, L., The statistical mechanics of strategic interaction, Games econ. behav., 5, 387-424, (1993) · Zbl 0797.90123
[8] Blume, L., Population games, (), 425-460
[9] Blume, L., How noise matters, Games econ. behav., 44, 251-271, (2003) · Zbl 1056.91011
[10] Dokumaci, E., Sandholm, W.H., 2007. Stochastic evolution with perturbed payoffs and rapid play. Mimeo, University of Wisconsin-Madison
[11] Ellison, G., Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution, Rev. econ. stud., 67, 17-45, (2000) · Zbl 0956.91027
[12] Freidlin, M.; Wentzell, A., Random perturbations of dynamical systems, (1988), Springer Verlag New York
[13] Harsanyi, J.; Selten, R., A general theory of equilibrium selection in games, (1988), The MIT Press Cambridge, MA · Zbl 0693.90098
[14] Hofbauer, J.; Sandholm, W., On the global convergence of stochastic fictitious play, Econometrica, 70, 2265-2294, (2002) · Zbl 1141.91336
[15] Hofbauer, J.; Sandholm, W., Evolution in games with randomly disturbed payoffs, J. econ. theory, 132, 47-69, (2007) · Zbl 1142.91343
[16] Hofbauer, J.; Sigmund, K., The theory of evolution and dynamical systems, (1988), Cambridge University Press Cambridge, UK
[17] Hofbauer, J.; Sorger, G., Perfect foresight and equilibrium selection in symmetric potential games, J. econ. theory, 85, 1-23, (1999) · Zbl 0922.90146
[18] Kandori, M.; Rob, R., Evolution of equilibria in the long run: A general theory and applications, J. econ. theory, 65, 383-414, (1995) · Zbl 0837.90139
[19] Kandori, M.; Mailath, G.J.; Rob, R., Learning, mutation, and long run equilibria in games, Econometrica, 61, 29-56, (1993) · Zbl 0776.90095
[20] Marden, J.R., Shamma, J.S., 2008. Revisiting log-linear learning: Asynchrony, completeness and payoff-based implementation. Mimeo, Caltech · Zbl 1239.91017
[21] Maruta, T., Binary games with state dependent stochastic choice, J. econ. theory, 103, 2, 351-376, (2002) · Zbl 1137.91324
[22] Mattsson, L.-G.; Weibull, J.W., Probabilistic choice and procedurally bounded rationality, Games econ. behav., 41, 1, 61-78, (2002) · Zbl 1024.91003
[23] McKelvey, R.D.; Palfrey, T.R., Quantal response equilibria for normal form games, Games econ. behav., 10, 1, 6-38, (1995) · Zbl 0832.90126
[24] Monderer, D.; Shapley, L., Potential games, Games econ. behav., 14, 124-143, (1996) · Zbl 0862.90137
[25] Morris, S.; Ui, T., Generalized potential and robust sets of equilibria, J. econ. theory, 124, 45-78, (2005) · Zbl 1100.91004
[26] Myatt, D.P.; Wallace, C., A multinomial probit model of stochastic evolution, J. econ. theory, 113, 286-301, (2003) · Zbl 1158.91307
[27] Okada, D., Tercieux, O., 2008. Log-linear dynamics and local potential. Mimeo · Zbl 1258.91021
[28] Sandholm, W.H., Simple and clever decision rules for a model of evolution, Econ. letters, 61, 165-170, (1998) · Zbl 0914.90280
[29] Sandholm, W.H., Pigouvian pricing and stochastic evolutionary implementation, J. econ. theory, 132, 367-382, (2007) · Zbl 1142.91344
[30] Thurstone, L., A law of comparative judgement, Psychological rev., 34, 273-286, (1927)
[31] van Damme, E.; Weibull, J.W., Evolution in games with endogenous mistake probabilities, J. econ. theory, 106, 296-315, (2002) · Zbl 1035.91006
[32] Vega-Redondo, F., The evolution of Walrasian behavior, Econometrica, 65, 375-384, (1997) · Zbl 0874.90049
[33] Voorneveld, M., Best-response potential games, Econ. letters, 66, 289-295, (2000) · Zbl 0951.91008
[34] Young, P., The evolution of conventions, Econometrica, 61, 57-84, (1993) · Zbl 0773.90101
[35] Young, P., Individual strategy and social structure, (1998), Princeton University Press Princeton, New Jersey
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