Chromodynamics of Cooperation in Finite Populations

Arne Traulsen,Martin A Nowak,Stuart West

doi:10.1371/journal.pone.0000270

Arne Traulsen, Martin A Nowak + Show 1 more

Open Access

https://doi.org/10.1371/journal.pone.0000270

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Abstract

BackgroundThe basic idea of tag-based models for cooperation is that individuals recognize each other via arbitrary signals, so-called tags. If there are tags of different colors, then cooperators can always establish new signals of recognition. The resulting “chromodynamics” is a mechanism for the evolution of cooperation. Cooperators use a secret tag until they are discovered by defectors who then destroy cooperation based on this tag. Subsequently, a fraction of the population manages to establish cooperation based on a new tag.Methodology/Principal FindingsWe derive a mathematical description of stochastic evolutionary dynamics of tag-based cooperation in populations of finite size. Benefit and cost of cooperation are given by b and c. We find that cooperators are more abundant than defectors if b/c > 1+2u/v, where u is the mutation rate changing only the strategy and v is the mutation rate changing strategy and tag. We study specific assumptions for u and v in two genetic models and one cultural model.Conclusions/SignificanceIn a genetic model, tag-based cooperation only evolves if a gene encodes both strategy and tag. In a cultural model with equal mutation rates between all possible phenotypes (tags and behaviors), the crucial condition is b/c > (K+1)/(K−1), where K is the number of tags. A larger number of tags requires a smaller benefit-to-cost ratio. In the limit of many different tags, the condition for cooperators to have a higher average abundance than defectors becomes b > c.

Highlights

The green beard effect was introduced by William D
Hamilton as a thought experiment in sociobiology: a gene that leads both to a visible tag and the tendency to help others with the same tag allows evolution of cooperation [1,2,3,4,5,6,7,8,9,10,11]
In contrast to the model of Jansen and van Baalen where the coexistence of many tags is possible, the analytical description of our model considers only two tags at a time and tag diversity is only reflected by the mutation rates

Summary

Background

The basic idea of tag-based models for cooperation is that individuals recognize each other via arbitrary signals, so-called tags. If there are tags of different colors, cooperators can always establish new signals of recognition. A fraction of the population manages to establish cooperation based on a new tag. We derive a mathematical description of stochastic evolutionary dynamics of tag-based cooperation in populations of finite size. Benefit and cost of cooperation are given by b and c. We find that cooperators are more abundant than defectors if b/c . Tag-based cooperation only evolves if a gene encodes both strategy and tag. A larger number of tags requires a smaller benefit-to-cost ratio. In the limit of many different tags, the condition for cooperators to have a higher average abundance than defectors becomes b .

INTRODUCTION

RESULTS AND DISCUSSION

MATERIALS AND METHODS