# Fractional Punishment of Free Riders to Improve Cooperation in Optional Public Good Games

^{*}

## Abstract

**:**

## 1. Introduction

#### Sanctioning System

## 2. Materials and Methods

**Assumption**

**1.**

**Assumption**

**2.**

## 3. Results

#### 3.1. Boundary of the Simplex

#### 3.1.1. Border $xy$

**Lemma**

**1.**

#### 3.1.2. Border $zx$

#### 3.1.3. Border yz

#### 3.2. Interior of the Simplex

#### 3.2.1. Effect of d over the Value of z in the Equilibrium

**Lemma**

**2.**

**Lemma**

**3.**

#### 3.2.2. Effect of d over the Value of f in the Equilibrium

#### 3.2.3. Effect of d in the Interior Equilibrium Point

**Theorem**

**1.**

#### 3.3. The Equilibrium Point $(x,y,z)=(1,0,0)$

**Theorem**

**2.**

#### 3.4. Summary of the Fractional Punishment Effect in the System

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of the Lemma 1

**Proof.**

#### Appendix A.2. Proof of the Lemma 2

**Proof.**

#### Appendix A.3. Proof of the Lemma 3

**Proof.**

#### Appendix A.4. Proof of the Theorem 1

**Proof.**

#### Appendix A.5. Proof of the Theorem 2

**Proof.**

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**Figure 1.**Phase diagram of the system when d increases. Parameters $N=5$, $r=3$, and $\sigma =1$. For these values, ${d}_{1}=0.16667$ and ${d}_{2}=0.2857$. Notice that ${q}_{1}$, ${q}_{2}$, and ${q}_{3}$ are equilibrium points. (

**a**) $d=0$, (

**b**) $d=0.1$, (

**c**) $d={d}_{1}$, (

**d**) $d=0.25$, (

**e**) $d={d}_{2}$, and (

**f**) $d=0.99$.

**Figure 2.**Stability of equilibrium points with different values of d: a black dot represents a stable point, and a white dot represents an unstable point. (

**a**) With $d<{d}_{1}$, the $\widehat{x}=1$ is unstable and $\widehat{y}=1$ is asymptotically stable. (

**b**) If ${d}_{1}<d\le 1$, an unstable equilibrium $\widehat{x}=(n-r)/r(n-1)d$, denoted by $\tilde{x}$, appears, changing the stability of $\widehat{x}=1$, which becomes an stable equilibrium point.

**Figure 3.**(

**a**) $m\left(z\right)$ and $\tilde{g}(z,d)$ with $d=0.01$. In the whole interval $z\in (0,1)$, (

**b**) Zoom in the neighborhood of $m\left(z\right)=0$ and $\tilde{g}(z,d)=0$. It can be observed that d decreases the value of $\widehat{z}$ in the equilibria. It shows how d shifts the equilibrium point $\widehat{z}$ to the border of $z=0$. Parameters: $n=5$, $r=3$, and $\sigma =1$.

**Figure 4.**Graph of $\tilde{g}$ with several values of d. Observe that, if $z=0$, $\tilde{g}=0$ when $d={d}_{2}\left(0.2857\right)$. Parameters: $n=5$, $r=3$, and $\sigma =1$.

**Figure 5.**Interior equilibrium point $(\widehat{f}$ and $\widehat{z})$ for increasing values of d. Observe that, when d increases, the value of $\widehat{f}$ increases, and the value of $\widehat{z}$ decreases. Parameters: $n=5$, $r=3$, $\sigma =1$, and ${d}_{2}=(n-r)/(n\sigma +n-r)$.

**Figure 6.**(

**a**) Values of ${J}_{{R}_{12}}$ for $0\le {z}_{0}\le 1$ and $\widehat{{f}_{0}}=\sigma /(r-1)$; for ${\widehat{z}}_{0}$, the term is positive. (

**b**) Values of a for $0\le z<1$. (

**c**) Values of ${a}^{\prime}$ for $0\le z<1$. Parameters: $n=5$, $r=3$, and $\sigma =1$.

**Figure 7.**Effect of d on the interior equilibrium point. Parameters $n=5$, $r=3$, and $\sigma =1$, and values of d as (

**a**) $d=0.1$ and (

**b**) $d=0.2$. The white dot represents an unstable point.

**Figure 8.**Values of ${\dot{V}}_{\widehat{f}}$ for each possible state of the system. The orange area represent states ${\dot{V}}_{\widehat{f}}<0$; the purple area represent states ${\dot{V}}_{\widehat{f}}>0$. The red line between both regions represent the set of points where ${\dot{V}}_{\widehat{f}}$ changes sign. The black lines are trajectories from several initial values in the simplex. Parameters $n=5$, $r=3$, and $\sigma =1$, and increasing values of d: (

**a**) $d=0.25$ and (

**b**) $d=0.5$.

**Figure 9.**The behavior of x in the trajectories for each possible initial value. The green area represents initial values where the frequency of cooperators x increases along the orbit; the blue area represents initial values where at some point of the trajectories x decreases. The black lines are trajectories from several initial values in the simplex. $n=5$, $r=3$, and $\sigma =1$, and values of d are (

**a**) $d=0.22$ and (

**b**) $d=0.5$.

**Table 1.**Equilibrium points relationship between $(\widehat{x},\widehat{y},\widehat{z})$ and $(\widehat{f},\widehat{z})$. $\alpha =(r-1)+d(\sigma -(r-1\left)\right)$, $\beta =(N-r)/\left(r\right(N-1\left)d\right)$.

$(\widehat{\mathit{x}},\widehat{\mathit{y}},\widehat{\mathit{z}})$ | $(\widehat{\mathit{f}},\widehat{\mathit{z}})$ |
---|---|

$(\widehat{x},\widehat{y},\widehat{z})=(1,0,0)$ | $(\widehat{f},\widehat{z})=(1,0)$ |

$(\widehat{x},\widehat{y},\widehat{z})=(0,1,0)$ | $(\widehat{f},\widehat{z})=(0,0)$ |

$(\widehat{x},\widehat{y},\widehat{z})=(0,0,1)$ | $(\widehat{f},\widehat{z})=(0,1)$ |

$(\widehat{x},\widehat{y},\widehat{z})=(\frac{\sigma}{\alpha}(1-\widehat{z}),(1-\frac{\sigma}{\alpha})(1-\widehat{z}),\widehat{z})$ | $(\widehat{f},\widehat{z})=(\frac{\sigma}{\alpha},\widehat{z})$ |

$(\widehat{x}=f,\widehat{y}=1-f,\widehat{z}=0)$ | $(\widehat{f},\widehat{z})=(\beta ,0)$ |

**Table 2.**Effect of the parameter d on the entries of the Jacobian J evaluated at equilibrium point $({\widehat{f}}_{d},{\widehat{z}}_{d})$ and its corresponding eigenvalues. For $d=0$, $({\widehat{f}}_{d},{\widehat{z}}_{d})$ = $({\widehat{f}}_{0},{\widehat{z}}_{0})$. Parameters $N=5$, $r=3$, and $\sigma =1$; $\psi ={J}_{{R}_{22}}-{\epsilon}_{22}+{\epsilon}_{11}$ and $\Delta ={({J}_{{R}_{22}}-{\epsilon}_{22}+{\epsilon}_{11})}^{2}-4\phantom{\rule{0.166667em}{0ex}}({J}_{{R}_{21}}\pm {\epsilon}_{21})({J}_{{R}_{12}}+{\epsilon}_{12})$.

d | $\mathit{J}={\mathit{J}}_{\mathit{R}}+\mathit{d}\phantom{\rule{0.166667em}{0ex}}{\mathit{J}}_{\mathit{T}}$ | $({\widehat{\mathit{f}}}_{\mathit{d}},{\widehat{\mathit{z}}}_{\mathit{d}})$ | $\mathit{\lambda}=0.5\phantom{\rule{0.166667em}{0ex}}\left(\mathit{\psi}\right)\pm 0.5\phantom{\rule{0.166667em}{0ex}}\sqrt{\mathit{\Delta}}$ |
---|---|---|---|

$d=0.01$ | $\left(\right)open="["\; close="]">\begin{array}{cc}0& 0.24831\\ -0.47471& 0.00046\end{array}$ | $\begin{array}{c}(0.5025,0.4516)\end{array}$ | $\begin{array}{c}{\lambda}_{1}=0.00210-0.34182i\hfill \\ {\lambda}_{2}=0.00210+0.34182i\hfill \end{array}$ |

$d=0.001$ | $\left(\right)open="["\; close="]">\begin{array}{cc}0& 0.24688\\ -0.47454& 0.00005\end{array}$ | $\begin{array}{c}(0.5003,0.4604)\end{array}$ | $\begin{array}{c}{\lambda}_{1}=0.00021-0.34213i\hfill \\ {\lambda}_{2}=0.00021+0.34213i\hfill \end{array}$ |

$d=0$ | $\left(\right)open="["\; close="]">\begin{array}{cc}0& 0.24671\\ -0.47450& 0\end{array}$ | $\begin{array}{c}(0.5,0.4613)\end{array}$ | $\begin{array}{c}{\lambda}_{1}=-0.34215i\hfill \\ {\lambda}_{2}=+0.34215i\hfill \end{array}$ |

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**MDPI and ACS Style**

Botta, R.; Blanco, G.; Schaerer, C.E.
Fractional Punishment of Free Riders to Improve Cooperation in Optional Public Good Games. *Games* **2021**, *12*, 17.
https://doi.org/10.3390/g12010017

**AMA Style**

Botta R, Blanco G, Schaerer CE.
Fractional Punishment of Free Riders to Improve Cooperation in Optional Public Good Games. *Games*. 2021; 12(1):17.
https://doi.org/10.3390/g12010017

**Chicago/Turabian Style**

Botta, Rocio, Gerardo Blanco, and Christian E. Schaerer.
2021. "Fractional Punishment of Free Riders to Improve Cooperation in Optional Public Good Games" *Games* 12, no. 1: 17.
https://doi.org/10.3390/g12010017