# Effects of Relatedness on the Evolution of Cooperation in Nonlinear Public Goods Games

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## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Demographical Dynamics

#### 2.2. Evolutionary Dynamics

## 3. Results

#### 3.1. Quadratic Benefit and Cost Functions

**Theorem**

**1.**

- Assume B is accelerating and that ${\mu}_{0}>{\mu}_{V}$, as is the case if C is decelerating. For $\mu <{\mu}_{V}$, we have ${r}^{*}\left(\mu \right)=0$ and selection is thus towards no cooperation. For ${\mu}_{V}<\mu <{\mu}_{0}$, ${r}^{*}\left(\mu \right)$ is decreasing and the evolutionary dynamics is bistable. For $\mu >{\mu}_{0}$, we have ${r}^{*}\left(\mu \right)=1$ and selection is thus towards full cooperation.
- Assume B is accelerating and that ${\mu}_{0}<{\mu}_{V}$. For $\mu <{\mu}_{0}$, we have ${r}^{*}\left(\mu \right)=0$ and selection is thus towards no cooperation. For ${\mu}_{0}<\mu <{\mu}_{V}$, we have ${r}^{*}\left(\mu \right)$ is increasing and convergence stable, i.e., an evolutionary attractor. For $\mu >{\mu}_{V}$, we have ${r}^{*}\left(\mu \right)=1$ and selection is thus towards full cooperation.
- Assume B is decelerating and ${\mu}_{0}>{\mu}_{V}$, as is the case if C is accelerating. For $\mu <{\mu}_{0}$, we have ${r}^{*}\left(\mu \right)=0$ and selection is thus towards no cooperation. For $\mu >{\mu}_{V}$, ${r}^{*}\left(\mu \right)$ is increasing and convergence stable, i.e., an evolutionary attractor.
- Assume B is decelerating and ${\mu}_{0}<{\mu}_{V}$. For ${\mu}_{0}<\mu <{\mu}_{V}$, ${r}^{*}\left(\mu \right)$ is increasing and convergence stable, i.e., an evolutionary attractor. For $\mu >{\mu}_{V}$, we have ${r}^{*}\left(\mu \right)=1$ and selection is thus towards full cooperation.

**Proof.**

#### 3.2. General Cost and Benefit Functions

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Analytical Investigations

#### Appendix A.1. Condition for Evolutionary Stability

#### Appendix A.2. Impossible Region in the μ − σ^{2}-Plane

#### Appendix A.3. Implicit Differentiation

## Appendix B. Individual-Based Simulations

## References

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**Figure 1.**Examples of cost and benefit functions. Parameters used are for decelerating benefit ${b}_{2}$ = −1, ${b}_{1}=20$; for accelerating benefit ${b}_{2}=0.5,\phantom{\rule{4pt}{0ex}}{b}_{1}=1$; for decelerating cost ${c}_{2}=-2.75,\phantom{\rule{4pt}{0ex}}{c}_{1}=6$; and for accelerating cost ${c}_{2}=2,\phantom{\rule{4pt}{0ex}}{c}_{1}=1$. In choosing the horizontal range for the plots of the benefit functions, we have assumed $N=10$.

**Figure 2.**Bifurcation diagrams showing how the evolutionary dynamics depend on average relatedness for all four possible combinations of the accelerating and decelerating benefit and cost functions in Figure 1. Green thin line represents convergence stability without evolutionary stability, blue thick line represents convergence stability with evolutionary stability, black dotted line represents neither convergence stability nor evolutionary stability, and black thick line represents evolutionary stability without convergence stability. We see that the four cases are qualitatively different. We also see that, when B and C are both decelerating, a certain range of relatedness leads to evolutionary branching. The parameters used were in all cases $N=10$, and ${b}_{1},{b}_{2},{c}_{1},{c}_{2}$ as in Figure 1.

**Figure 3.**Plots of variance in relatedness against average relatedness. The grey region represents combinations of mean and variance that are not logically possible (see Appendix A.2). We see that for decreasing cost and benefit, the region with evolutionary branching shrinks with increased variance in relatedness. We also see that. for increasing cost and benefit, the region with neither convergence stability nor evolutionary stability shrinks with increased variance in relatedness. In the latter case, with increased average relatedness or variance in relatedness, the singular strategy moves from evolutionary stability (ESS) without convergence stability to neither evolutionary stability nor convergence stability. The parameters used are in both cases $N=10$, and ${b}_{1},{b}_{2},{c}_{1},{c}_{2}$ as in Figure 1.

**Figure 4.**Simulations using group size $N=10$, and B and C both decelerating or both accelerating. As predicted by the analytical investigation, when B and C are decelerating, the population undergoes evolutionary branching when $\mu =0.4$, and it reaches evolutionary stability when $\mu =0.8$. When B and C are accelerating, bistability occurs when $\mu =0.5$. In all three cases, ${\sigma}^{2}=0$.

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**MDPI and ACS Style**

Coder Gylling, K.; Brännström, Å.
Effects of Relatedness on the Evolution of Cooperation in Nonlinear Public Goods Games. *Games* **2018**, *9*, 87.
https://doi.org/10.3390/g9040087

**AMA Style**

Coder Gylling K, Brännström Å.
Effects of Relatedness on the Evolution of Cooperation in Nonlinear Public Goods Games. *Games*. 2018; 9(4):87.
https://doi.org/10.3390/g9040087

**Chicago/Turabian Style**

Coder Gylling, Kira, and Åke Brännström.
2018. "Effects of Relatedness on the Evolution of Cooperation in Nonlinear Public Goods Games" *Games* 9, no. 4: 87.
https://doi.org/10.3390/g9040087