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Log-linear dynamics and local potential. (English) Zbl 1258.91021
Summary: We show that local potential maximizer [S. Morris and T. Ui, ibid. 124, No. 1, 45–78 (2005; Zbl 1100.91004)], a generalization of potential maximizer, is stochastically stable in the log-linear dynamic if the payoff functions are, or the associated local potential is, supermodular. Thus an equilibrium selection result similar to those on robustness to incomplete information [Zbl 1100.91004], and on perfect foresight dynamic [D. Oyama, S. Takahashi and J. Hofbauer, “Monotone methods for equilibrium selection under perfect foresight dynamics”, Theor. Econ. 3, No. 2, 155–192 (2008)] holds for the log-linear dynamic. An example shows that stochastic stability of an LP-max is not guaranteed for non-potential games without the supermodularity condition. We investigate sensitivity of the log-linear dynamic to cardinal payoffs and its consequence on the stability of weighted local potential maximizer. In particular, for \(2\times 2\) games, we examine a modified log-linear dynamic (relative log-linear dynamic) under which local potential maximizer with positive weights is stochastically stable. The proof of the main result relies on an elementary method for stochastic ordering of Markov chains.

MSC:
91A10 Noncooperative games
91A05 2-person games
91A22 Evolutionary games
91B55 Economic dynamics
60E15 Inequalities; stochastic orderings
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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[1] Alós-Ferrer, Carlos; Netzer, Nick, The logit-response dynamics, Games econ. behav., 68, 2, 413-427, (2010) · Zbl 1207.91017
[2] Bergin, James; Lipman, Barton L., Evolution with state-dependent mutations, Econometrica, 64, 4, 943-956, (1996) · Zbl 0862.90142
[3] Blume, Lawrence E., The statistical mechanics of strategic interaction, Games econ. behav., 5, 387-424, (1993) · Zbl 0797.90123
[4] Blume, Lawrence E., Population games, (), 425-460
[5] Frankel, David M.; Morris, Stephen; Pauzner, Ady, Equilibrium selection in global games with strategic complementarities, J. econ. theory, 108, 1-44, (2003) · Zbl 1044.91005
[6] Freidlin, Mark; Wentzell, Alexander, Random perturbations of dynamical systems, (1984), Springer-Verlag New York · Zbl 1267.60004
[7] Kajii, Atsushi; Morris, Stephen, The robustness of equilibria to incomplete information, Econometrica, 65, 6, 1283-1309, (1997) · Zbl 0887.90186
[8] Kamae, T.; Urengel, U.; OʼBrien, G.L., Stochastic inequalities on partitially ordered spaces, Ann. probab., 5, 6, 899-912, (1977) · Zbl 0371.60013
[9] Kandori, Michihiro; Mailath, George J.; Rob, Rafael, Learning, mutation, and long-run equilibria in games, Econometrica, 61, 29-56, (1993) · Zbl 0776.90095
[10] Jason R. Marden, S. Jeff Shamma, Revisiting log-linear learning: Asynchrony, completeness and payoff-based implementation, Games Econ. Behav., forthcoming. · Zbl 1239.91017
[11] Massey, William A., Stochastic orderings for Markov processes on partially ordered spaces, Math. oper. res., 12, 2, 350-367, (1987) · Zbl 0622.60098
[12] Monderer, Dov; Shapley, Lloyd S., Potential games, Games econ. behav., 14, 124-143, (1996) · Zbl 0862.90137
[13] Stephen Morris, Potential methods in interaction games, 1999.
[14] Morris, Stephen; Ui, Takashi, Generalized potential and robust sets of equilibria, J. econ. theory, 124, 45-78, (2005) · Zbl 1100.91004
[15] Müller, Alfred, Comparison methods for stochastic models and risks, (2002), John Wiley & Sons New York · Zbl 0999.60002
[16] Myerson, Roger B., Refinements of the Nash equilibrium concept, Int. J. game theory, 15, 133-154, (1978)
[17] Daijiro Okada, Olivier Tercieux, Log-linear dynamics and local potential, Economics Working Paper No. 85, Institute for Advanced Study, Princeton, 2008. · Zbl 1258.91021
[18] Oyama, Daisuke; Takahashi, Satoru; Hofbauer, Josef, Monotone methods for equilibrium selection under perfect foresight dynamics, Theoretical economics, 3, 2, 155-192, (2008)
[19] Oyama, Daisuke; Tercieux, Olivier, Iterated potential and the robustness of equilibria, J. econ. theory, 144, 4, 1726-1769, (2009) · Zbl 1229.91214
[20] Shostak, Robert, Deciding linear inequalities by computing loop residues, J. assoc. comput. Mach., 28, 769-779, (1981) · Zbl 0468.68073
[21] Smith, Hal L., Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Math. surv. monogr., vol. 41, (1995), American Mathematical Society Providence, RI · Zbl 0821.34003
[22] Stoyan, Dietrich, Comparison methods for queues and other stochastic models, (1984), John Wiley & Sons New York · Zbl 0556.62075
[23] Ui, Takashi, Robust equilibria of potential games, Econometrica, 69, 1373-1380, (2001) · Zbl 1041.91006
[24] Hiroshi Uno, Nested potentials and robust equilibria, Discussion Paper 2011/9, CORE, Université catholique de Louvain, 2011.
[25] Peyton Young, H., The evolution of conventions, Econometrica, 61, 1, 57-84, (1993) · Zbl 0773.90101
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