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Eco-evolutionary Spatial Dynamics of Nonlinear Social Dilemmas

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Advances in Dynamic Games

Part of the book series: Annals of the International Society of Dynamic Games ((AISDG,volume 17))

Abstract

Spatial dynamics can promote the evolution of cooperation. While dispersal processes have been studied in simple evolutionary games, real-world social dilemmas are much more complicated. When the investment is low, for example, every additional unit of investment may substantially raise the public goods. However, the effect vanishes as the number of investments increases. Such nonlinear public goods are the norm in a variety of social as well as biological systems. Therefore, we investigate the effect of the nonlinearity on the evolution of cooperation. We show how the nonlinearity in payoffs, resulting in synergy or discounting of public goods, can alter the return on the cooperative investments compared to the linear game. The alteration affects the resulting spatial patterns, not just quantitatively, but in some cases, drastically changing the outcomes. Notably, in cases where a linear game would lead to extinction, synergy can support the coexistence of cooperators and defectors. The eco-evolutionary trajectory can thus be qualitatively different in cases on nonlinear social dilemmas.

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Acknowledgments

We thank Christoph Hauert for comments and suggestions in improving an early version of the manuscript. The authors thank the constructive comments of the reviewers. Both authors acknowledge generous support from the Max Planck Society.

Author information

Authors and Affiliations

  1. Research Group for Theoretical Models of Eco-evolutionary Dynamics, Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August Thienemann Str-2, 24306, Plön, Germany

    Chaitanya S. Gokhale

  2. Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August Thienemann Str-2, 24306, Plön, Germany

    Hye Jin Park

Authors
  1. Chaitanya S. Gokhale
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  2. Hye Jin Park
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Corresponding author

Correspondence to Chaitanya S. Gokhale .

Editor information

Editors and Affiliations

  1. Faculty of Computer Science and Management, Wrocław University of Science and Technology, Wrocław, Poland

    David M. Ramsey

  2. Toulouse School of Economics, University Toulouse Capitole and ANITI, Toulouse, France

    Jérôme Renault

Appendix

Appendix

1.1 Colour Coding

Similar to the colour coding used in [37] we use mint green (colour code: #A7FF70) and Fuchsia pink (colour code: #FF8AF3) colours for denoting the cooperator and defector densities, respectively, for each type. The colour spectrum and saturation is determined by the ratio of cooperators to defectors which results in the Maya blue colour for equal densities of cooperators and defectors. For convenience, we use HSB colour space which is a cylindrical coordinate system \((r, \theta , h)=\) (saturation, hue, brightness). The radius of circle r indicates saturation or the colour whereas \(\theta \) helps us transform the RGB space to HSB. The total density of the population \(\rho =u+v\) is represented by the brightness h of the colour. For better visualisation, we formulate the brightness h as

$$\begin{aligned} \frac{\log {a\rho +1}}{\log {a+1}}, \end{aligned}$$
(17)

where a control parameter a \((>-1\) and \(\ne 0)\) (see Fig. 6). The complete colour scheme so developed passes the standard tests for colour blindness.

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Gokhale, C.S., Park, H.J. (2020). Eco-evolutionary Spatial Dynamics of Nonlinear Social Dilemmas. In: Ramsey, D.M., Renault, J. (eds) Advances in Dynamic Games. Annals of the International Society of Dynamic Games, vol 17. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-56534-3_8

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