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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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