Evolutionary dynamics in finite populations with zealots.
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We investigate evolutionary dynamics of two-strategy matrix games with zealots in finite populations.Zealots are assumed to take either strategy regardless of the fitness.We illustrate our results with examples of various social dilemma games.
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Affiliation: Department of Mathematical Informatics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo , 113-8656, Japan.
ABSTRACT
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We investigate evolutionary dynamics of two-strategy matrix games with zealots in finite populations. Zealots are assumed to take either strategy regardless of the fitness. When the strategy selected by the zealots is the same, the fixation of the strategy selected by the zealots is a trivial outcome. We study fixation time in this scenario. We show that the fixation time is divided into three main regimes, in one of which the fixation time is short, and in the other two the fixation time is exponentially long in terms of the population size. Different from the case without zealots, there is a threshold selection intensity below which the fixation is fast for an arbitrary payoff matrix. We illustrate our results with examples of various social dilemma games. |
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Mentions: Equations (23) and (24) imply that in Eq. (26) is safely ignored near the singularity at because it would contribute at most to the fixation time. Therefore, we obtain28\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} t_0 \propto \sqrt{N}\exp (\gamma N), \end{aligned}$$\end{document}t0∝Nexp(γN),where is a constant that depends on , and . The dependence of on is shown in Fig. 2 for sample payoff matrices for the prisoner’s dilemma game (solid line) and snowdrift game (dotted line). For both games, monotonically decreases with , implying that the fixation time decreases with . In particular, is equal to zero, which corresponds to , when is larger than a threshold value.Fig. 2 |
View Article: PubMed Central - PubMed
Affiliation: Department of Mathematical Informatics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo , 113-8656, Japan.