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Evolutionary dynamics in finite populations with zealots.

Nakajima Y, Masuda N - J Math Biol (2014)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematical Informatics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo , 113-8656, Japan.

ABSTRACT
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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The exponent  for the mean fixation time [Eq. (28)] plotted against the density of zealots  for the prisoner’s dilemma game with , , , and  (solid line) and the snowdrift game with , , , , with  (dotted line). We calculated  on the basis of Eqs. (26), (27), and (55)
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Fig2: The exponent for the mean fixation time [Eq. (28)] plotted against the density of zealots for the prisoner’s dilemma game with , , , and (solid line) and the snowdrift game with , , , , with (dotted line). We calculated on the basis of Eqs. (26), (27), and (55)

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


Evolutionary dynamics in finite populations with zealots.

Nakajima Y, Masuda N - J Math Biol (2014)

The exponent  for the mean fixation time [Eq. (28)] plotted against the density of zealots  for the prisoner’s dilemma game with , , , and  (solid line) and the snowdrift game with , , , , with  (dotted line). We calculated  on the basis of Eqs. (26), (27), and (55)
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4289535&req=5

Fig2: The exponent for the mean fixation time [Eq. (28)] plotted against the density of zealots for the prisoner’s dilemma game with , , , and (solid line) and the snowdrift game with , , , , with (dotted line). We calculated on the basis of Eqs. (26), (27), and (55)
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

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematical Informatics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo , 113-8656, Japan.

ABSTRACT
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.

Show MeSH