# Evolution of Cooperation in Social Dilemmas with Assortative Interactions

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## Abstract

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## 1. Introduction

## 2. Models

#### 2.1. Discrete Games

#### 2.1.1. Replicator Dynamics with Assortative Interactions

#### 2.1.2. Donation Game

#### 2.1.3. Snowdrift Game

#### 2.1.4. Sculling Game

#### 2.2. Continuous Games

#### 2.2.1. Adaptive Dynamics with Assortative Interactions

#### 2.2.2. Continuous Donation Game

#### Linear Cost and Benefit Functions

#### Convex Cost and Concave Benefit Functions

#### 2.2.3. Continuous Snowdrift Game

#### Concave Cost and Benefit Functions

#### 2.2.4. Continuous Tragedy of the Commons Game

#### Convex Cost and Sigmoidal Benefit Functions

#### 2.3. Individual-Based Model

#### 2.3.1. Discrete Games

#### 2.3.2. Continuous Games

## 3. Results from Individual-Based Simulations

#### 3.1. Discrete Games

#### 3.1.1. Donation Game

#### 3.1.2. Snowdrift Game

#### 3.1.3. Sculling Game

#### 3.2. Continuous Games

#### 3.2.1. Continuous Donation Game

#### 3.2.2. Continuous Snowdrift Game

#### 3.2.3. Continuous Tragedy of the Commons Game

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Phase line diagrams and the bifurcation diagram for the donation game with assortative interactions. (

**a**) In the phase line diagrams closed circles represent stable equilibrium points, open circles represent unstable equilibrium points, and the curved line connecting equilibrium points indicates the graph of the function on the right-hand side of the r-replicator equation. (

**b**) In the bifurcation diagram solid lines represent stable equilibrium points, dashed lines represent unstable equilibrium points, and arrows indicate the direction of evolutionary change.

**Figure 2.**Phase line diagrams and the bifurcation diagram for the snowdrift game with assortative interactions. (

**a**) In the phase line diagrams closed circles represent stable equilibrium points, open circles represent unstable equilibrium points, and the curved line connecting equilibrium points indicates the graph of the function on the right-hand side of the r-replicator equation. (

**b**) In the bifurcation diagram solid lines represent stable equilibrium points, dashed lines represent unstable equilibrium points, and arrows indicate the direction of evolutionary change.

**Figure 3.**Phase line diagrams and the bifurcation diagram for the sculling game with assortative interactions. (

**a**) In the phase line diagrams closed circles represent stable equilibrium points, open circles represent unstable equilibrium points, and the curved line connecting equilibrium points indicates the graph of the function on the right-hand side of the r-replicator equation. (

**b**) In the bifurcation diagram solid lines represent stable equilibrium points, dashed lines represent unstable equilibrium points, and arrows indicate the direction of evolutionary change.

**Figure 4.**Variation of the long-term frequency ${p}_{\infty}$ of cooperators with assortativity $r\in [0,1]$ and cost-to-benefit ratio $\rho \in (0,1)$ for the donation game. (

**a**) ${p}_{\infty}$ (analytically predicted) versus r and $\rho $. (

**b**) ${p}_{\infty}$ (simulated) versus r and $\rho $. (

**c**) ${p}_{\infty}$ versus r when $\rho =0.25$. (

**d**) ${p}_{\infty}$ versus $\rho $ when $r=0.25$. Parameters: $n=10,000$, ${p}_{0}=0.5$, and $\beta =1$.

**Figure 5.**Variation of the long-term frequency ${p}_{\infty}$ of cooperators with assortativity $r\in [0,1]$ and cost-to-benefit ratio $\rho \in (0,1)$ for the snowdrift game. (

**a**) ${p}_{\infty}$ (analytically predicted) versus r and $\rho $. (

**b**) ${p}_{\infty}$ (simulated) versus r and $\rho $. (

**c**) ${p}_{\infty}$ versus r when $\rho =0.75$. (

**d**) ${p}_{\infty}$ versus $\rho $ when $r=0.25$. Parameters: $n=10,000$, ${p}_{0}=0.5$, and $\beta =1$.

**Figure 6.**Variation of the long-term frequency ${p}_{\infty}$ of cooperators with assortativity $r\in [0,1]$ and cost-to-benefit ratio $\rho \in (\frac{1}{2},\frac{3}{2})$ for the sculling game. (

**a**) ${p}_{\infty}$ (analytically predicted) versus r and $\rho $. (

**b**) ${p}_{\infty}$ (simulated) versus r and $\rho $. (

**c**) ${p}_{\infty}$ versus r when $\rho =1.2$. (

**d**) ${p}_{\infty}$ versus $\rho $ when $r=0.25$. Parameters: $n=10,000$, ${p}_{0}=0.5$, and $\beta =1$.

**Figure 7.**Variation of the long-term mean strategy ${\overline{x}}_{\infty}$ with assortativity $r\in [0,1]$ and cost-to-benefit ratio $\rho \in (0,1)$ in the CD game with linear cost and benefit functions: $C\left(x\right)=cx$ and $B\left(x\right)=bx$ with $b>c$. (

**a**) ${\overline{x}}_{\infty}$ (analytically predicted) versus r and $\rho $. (

**b**) ${\overline{x}}_{\infty}$ (simulated) versus r and $\rho $. (

**c**) ${\overline{x}}_{\infty}$ versus r when $\rho =0.26$. (

**d**) ${\overline{x}}_{\infty}$ versus $\rho $ when $r=0.26$. Parameters: $n=10,000$, ${x}_{0}=0.1$, ${x}_{m}=1$, $\mu =0.01$, $\sigma =0.005$, and $\beta =1$.

**Figure 8.**Variation of the distribution of long-term strategy values ${x}_{\infty}$ with assortativity r in the CD game with quadratic cost and benefit functions: $C\left(x\right)={x}^{2}$ and $B\left(x\right)=-{x}^{2}+2x$. Parameters: ${x}_{0}=0.1$, ${x}_{m}=1$, $\mu =0.01$, $\sigma =0.005$, and $\beta =1$. Arrows indicate the direction of evolutionary change.

**Figure 9.**Variation of the distribution of asymptotic strategy values ${x}_{\infty}$ with assortativity r, in a CSD game with quadratic cost function and quadratic benefit function. (

**a**) $C\left(x\right)=-1.6{x}^{2}+4.8x$ and $B\left(x\right)=-{x}^{2}+5x$, and (

**b**) $C\left(x\right)=-1.5{x}^{2}+4x$ and $B\left(x\right)=-0.2{x}^{2}+3x$. Parameters: n = 10,000, ${x}_{0}=0.3$, ${x}_{m}=1$, $\mu =0.01$, $\sigma =0.005$, and $\beta =1$. Arrows indicate the direction of evolutionary change.

**Figure 10.**Variation of the distribution of asymptotic strategy values ${x}_{\infty}$ with assortativity r, in a continuous tragedy of the commons (CTOC) game with quadratic cost and cubic benefit functions: $C\left(x\right)={x}^{2}$ and $B\left(x\right)=-0.0834{x}^{3}+2{x}^{2}+x$. Parameters: $n=10,000$, ${x}_{0}=0.1$, ${x}_{m}=3$, $\mu =0.01$, $\sigma =0.005$, and $\beta =1$. Arrows indicate the direction of evolutionary change.

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Iyer, S.; Killingback, T. Evolution of Cooperation in Social Dilemmas with Assortative Interactions. *Games* **2020**, *11*, 41.
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**Chicago/Turabian Style**

Iyer, Swami, and Timothy Killingback. 2020. "Evolution of Cooperation in Social Dilemmas with Assortative Interactions" *Games* 11, no. 4: 41.
https://doi.org/10.3390/g11040041