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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 normalized mean fixation time for the prisoner’s dilemma game as a function of . We set , , , and . In a, we set  and . In b, we set  and . In c, we set  and . The dashed lines represent  divided by the  value for the neutral game
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Fig3: The normalized mean fixation time for the prisoner’s dilemma game as a function of . We set , , , and . In a, we set and . In b, we set and . In c, we set and . The dashed lines represent divided by the value for the neutral game

Mentions: The mean fixation time with and obtained by direct calculations of Eq. (18) is shown in Fig. 3a. In this and the following figures, the values are those normalized by that for the neutral game [Eq. (37)]. The behavior of is qualitatively different according to whether is larger or smaller than . If , the ratio of for the prisoner’s dilemma game to for the neutral game seems to approach a constant as . This is consistent with case (i). In contrast, if , grows rapidly, which is consistent with case (ii). To be more quantitative, divided by the value for the neutral game is shown by the dashed line in Fig. 3a. It should be noted that 400 is a constant for fitting and that value is theoretically determined as described in Sect. 3.3.2. The theory (dashed line) agrees well with the exact numerical results (thinnest solid line). We remark that the normalized behaves non-monotonically in ; it takes a minimum at an intermediate value of .Fig. 3


Evolutionary dynamics in finite populations with zealots.

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

The normalized mean fixation time for the prisoner’s dilemma game as a function of . We set , , , and . In a, we set  and . In b, we set  and . In c, we set  and . The dashed lines represent  divided by the  value for the neutral game
© Copyright Policy
Related In: Results  -  Collection

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

Fig3: The normalized mean fixation time for the prisoner’s dilemma game as a function of . We set , , , and . In a, we set and . In b, we set and . In c, we set and . The dashed lines represent divided by the value for the neutral game
Mentions: The mean fixation time with and obtained by direct calculations of Eq. (18) is shown in Fig. 3a. In this and the following figures, the values are those normalized by that for the neutral game [Eq. (37)]. The behavior of is qualitatively different according to whether is larger or smaller than . If , the ratio of for the prisoner’s dilemma game to for the neutral game seems to approach a constant as . This is consistent with case (i). In contrast, if , grows rapidly, which is consistent with case (ii). To be more quantitative, divided by the value for the neutral game is shown by the dashed line in Fig. 3a. It should be noted that 400 is a constant for fitting and that value is theoretically determined as described in Sect. 3.3.2. The theory (dashed line) agrees well with the exact numerical results (thinnest solid line). We remark that the normalized behaves non-monotonically in ; it takes a minimum at an intermediate value of .Fig. 3

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