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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.

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)
Full Text: DOI
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