# Cooperation and Coordination in Threshold Public Goods Games with Asymmetric Players

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## Abstract

**:**

## 1. Introduction

- What is the impact of various forms of inequality on cooperation?
- How do people coordinate when group members differ among multiple dimensions?

## 2. Model

## 3. Results

**Theorem**

**1.**

**Theorem**

**2.**

- (1)
- For relative contribution, $\left|{S}_{1}^{1}\right|<\dots <\left|{S}_{n}^{1}\right|$ if $0<\theta <{e}_{n}{p}_{n},$ and $\left|{S}_{1}^{1}\right|>\dots >\left|{S}_{n}^{1}\right|$ if ${\sum}_{i=1}^{n-1}{e}_{i}{p}_{i}<\theta <G$.
- (2)
- For absolute contribution, when there is endowment heterogeneity, i.e., ${e}_{1}>\dots >{e}_{n}$ and ${p}_{1}=\dots ={p}_{n}$, $\left|{S}_{1}^{2}\right|\ge \dots \ge \left|{S}_{n}^{2}\right|$ for all $0<\theta <G$. When there is productivity heterogeneity, i.e., ${e}_{1}=\dots ={e}_{n}$ and ${p}_{1}>\dots >{p}_{n}$, $\left|{S}_{1}^{2}\right|<\dots <\left|{S}_{n}^{2}\right|$ if $0<\theta <{e}_{n}{p}_{n},$ and $\left|{S}_{1}^{2}\right|>\dots >\left|{S}_{n}^{2}\right|$ if ${\sum}_{i=1}^{n-1}{e}_{i}{p}_{i}<\theta <G$.
- (3)
- For collective contribution, $\left|{S}_{1}^{3}\right|\ge \dots \ge \left|{S}_{n}^{3}\right|$ for all $0<\theta <G$. Furthermore, $\left|{S}_{1}^{3}\right|=\dots =\left|{S}_{n}^{3}\right|$ if $0<\theta <{e}_{n}{p}_{n}$.

## 4. Numerical Analysis

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

**Proof.**

**(i)****Existence conditions of a defective Nash equilibrium**

**(ii)****Cooperative Nash equilibria set and its existence condition**

## Appendix B. Proof of Theorem 2

**Proof Outline.**Theorem 2 includes three parts of results, namely, results for (1) the relative contribution, (2) the absolute contribution, and (3) the collective contribution. We begin by proving part (3) and then establish part (1) based on the approach developed in part (3). Finally, part (2) can be directly obtained from part (1) and part (3).

**Lemma**

**1.**

**Lemma**

**2.**

**Figure A1.**Comparison between $\left|{S}_{i}^{1}\right|$ and $\left|{S}_{i+1}^{1}\right|$ when $0<\theta <{e}_{n}{p}_{n}$.

**Proof of part (3).**The set ${S}_{i}^{3}$ is defined as

**Proof of part (1).**The set ${S}_{i}^{1}$ is defined as

**(i)**- When ${\sum}_{i=1}^{n-1}{e}_{i}{p}_{i}<\theta <G$, each ${S}_{i}^{1}$ is an ($n-1$)-dimensional convex polytope formed by $n$ vertices. Specifically, we define the following:$$\begin{array}{c}{\alpha}_{1}=\left(\frac{\theta -{\sum}_{k\ne 1}^{n}{e}_{k}{p}_{k}}{{e}_{1}{p}_{1}},1,1,\dots ,1\right);\\ {\alpha}_{2}=\left(1,\frac{\theta -{\sum}_{k\ne 2}^{n}{e}_{k}{p}_{k}}{{e}_{2}{p}_{2}},1,\dots ,1\right);\\ \vdots \\ {\alpha}_{n}=\left(1,1,1,\dots ,\frac{\theta -{\sum}_{k\ne n}^{n}{e}_{k}{p}_{k}}{{e}_{n}{p}_{n}}\right);\\ \mathrm{and}O=\left(\frac{\theta}{G},\frac{\theta}{G},\dots ,\frac{\theta}{G}\right).\end{array}$$

**(ii)**- When $0<\theta <{e}_{n}{p}_{n}$, we represent ${S}_{i}^{1}$ as${\bigcup}_{j=1}^{n}\left({S}_{i}^{1}\cap {S}_{j}^{3}\right)$ and focus on the set ${S}_{i}^{1}\cap {S}_{j}^{3}$. This set can be expressed as$${S}_{i}^{1}\cap {S}_{j}^{3}=\left\{\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)|\begin{array}{c}{\sum}_{k=1}^{n}{e}_{k}{p}_{k}{x}_{k}=\theta ,{x}_{k}\in \left[0,1\right],\forall k\in \left\{1,2,\dots ,n\right\},\\ \underset{k}{\mathrm{max}}\left\{{e}_{k}{p}_{k}{x}_{k}\right\}={e}_{j}{p}_{j}{x}_{j},\underset{k}{\mathrm{max}}\left\{{x}_{k}\right\}={x}_{i}\end{array}\right\}.$$

**Proof of Lemma**

**1.**

**Proof of Lemma**

**2.**

**Figure A2.**Comparison between $\left|{S}_{i}^{1}\cap {S}_{i}^{3}\right|$ and $\left|{S}_{i+1}^{1}\cap {S}_{i+1}^{3}\right|$ when $0<\theta <{e}_{n}{p}_{n}$.

## Notes

1 | Dragicevic [10] theoretically studied a TPGG in the context of the option fund market and found that payoff inequality between buyers and sellers can undermine coordination efforts. |

2 | Dong et al. [19] considered a climate game with two types of players, in which rich (or poor) players have higher (or lower) endowment and emission reduction cost (i.e., low productivity). Their theoretical analysis and behavioral experiment based on specific parameters showed that the effect of multiple inequalities on coordination is generally more complex. More general discussion on NE in a climate game with heterogeneous players can be found in [19]. |

3 | We note that at an NE, the absolute contribution of player $i$ cannot exceed ${r}_{i}$ even if ${r}_{i}<{e}_{i}$. Otherwise, this player can obtain a higher payoff by deviating to free-riding. Thus, this assumption does not affect the equilibrium structure of the game. |

4 | An alternative scenario is one in which players choose their strategies from a finite grid $\{0,\frac{1}{m},\dots ,1\}$ with sufficiently large $m$. In this case, the cooperative NE set consists of finite number of equilibria, and $\left|{S}_{i}^{k}\right|>\left|{S}_{j}^{k}\right|$ for all $j\ne i$ ($k=1,2,3$) implies that there are more equilibria in which player $i$ contributes the most. |

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**Figure 1.**Nash equilibria for different game scenarios and thresholds $\theta $. Utilizing parameters from Table 1, we set (

**a**) $\theta =10$ in the low $\theta $ panel, (

**b**) $\theta =30$ in the intermediate $\theta $ panel, and (

**c**) $\theta =50$ in the high $\theta $ panel for all five game scenarios. The colored segments denote the sets of cooperative NEs, and the colored filled points denote the defective NE. Squares, diamonds, and circles, respectively, denote the points of equal relative contributions, equal absolute contributions, and equal collective contributions in the cooperative NE set. Thus, these points are the segmentation points of sets ${S}_{1}^{k}$ (below the point) and ${S}_{2}^{k}$ (above the point) for $k=1,2,3$. When multiple points overlap, they are adjusted horizontally to be better differentiated. The grey area indicates that the threshold is reached.

**Table 1.**Nash equilibria and sizes of ${S}_{i}^{k}$ in two-player threshold public goods games. Column 1: Five game scenarios are considered, namely, full equality, endowment inequality, productivity inequality, aligned inequality, and misaligned inequality. Column 2: For each scenario, the threshold $\theta $ is categorized into three ranges according to Theorem 2. Column 3: The cooperative NE set is non-empty for all parameter combinations, and the defective NE exists only for high thresholds. Columns 4–6: Absolute sizes of the three sets ${S}_{i}^{1}$, ${S}_{i}^{2}$, and ${S}_{i}^{3}$. The size relationship between $\left|{S}_{1}^{k}\right|$ and $\left|{S}_{2}^{k}\right|$ is marked using different colors. Red: $\left|{S}_{1}^{k}\right|>\left|{S}_{2}^{k}\right|$. Black: $\left|{S}_{1}^{k}\right|=\left|{S}_{2}^{k}\right|$. Blue: $\left|{S}_{1}^{k}\right|<\left|{S}_{2}^{k}\right|$. Grey: the relationship between $\left|{S}_{1}^{k}\right|$ and $\left|{S}_{2}^{k}\right|$ depends on $\theta $. Three cells are highlighted. In these cells, $\left|{S}_{1}^{3}\right|>\left|{S}_{2}^{3}\right|$ for almost all $\theta $. The exceptions are $\theta =20$ in the first two cells and $\theta =12$ in the last cell.

$\mathbf{(}{\mathit{e}}_{1}\mathbf{,}{\mathit{e}}_{2}\mathbf{,}{\mathit{p}}_{1}\mathbf{,}{\mathit{p}}_{2}\mathbf{)}$ | Threshold | NE | $\mathbf{\left(}\mathbf{\right|}{\mathit{S}}_{1}^{1}\mathbf{|}\mathbf{,}\mathbf{|}{\mathit{S}}_{2}^{1}\mathbf{\left|}\mathbf{\right)}$ | $\mathbf{\left(}\mathbf{\right|}{\mathit{S}}_{1}^{2}\mathbf{|}\mathbf{,}\mathbf{|}{\mathit{S}}_{2}^{2}\mathbf{\left|}\mathbf{\right)}$ | $\mathbf{\left(}\mathbf{\right|}{\mathit{S}}_{1}^{3}\mathbf{|}\mathbf{,}\mathbf{|}{\mathit{S}}_{2}^{3}\mathbf{\left|}\mathbf{\right)}$ |
---|---|---|---|---|---|

Full equality $(30,30,1,1)$ | $0<\theta <30$ | ${x}_{1}+{x}_{2}=\frac{\theta}{30}$ | $\left(\frac{\sqrt{2}\theta}{60},\frac{\sqrt{2}\theta}{60}\right)$ | $\left(\frac{\sqrt{2}\theta}{60},\frac{\sqrt{2}\theta}{60}\right)$ | $\left(\frac{\sqrt{2}\theta}{60},\frac{\sqrt{2}\theta}{60}\right)$ |

$\theta =30$ | ${x}_{1}+{x}_{2}=1$ | $\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$ | $\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$ | $\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$ | |

$30<\theta <60$ | ${x}_{1}+{x}_{2}=\frac{\theta}{30};$ ${x}_{1}={x}_{2}=0$ | $\left(\sqrt{2}-\frac{\sqrt{2}\theta}{60},\sqrt{2}-\frac{\sqrt{2}\theta}{60}\right)$ | $\left(\sqrt{2}-\frac{\sqrt{2}\theta}{60},\sqrt{2}-\frac{\sqrt{2}\theta}{60}\right)$ | $\left(\sqrt{2}-\frac{\sqrt{2}\theta}{60},\sqrt{2}-\frac{\sqrt{2}\theta}{60}\right)$ | |

Endowment inequality $(40,20,1,1)$ | $0<\theta <20$ | $2{x}_{1}+{x}_{2}=\frac{\theta}{20}$ | ${\left(\frac{\sqrt{5}\theta}{120},\frac{\sqrt{5}\theta}{60}\right)}$ | $\left(\frac{\sqrt{5}\theta}{80},\frac{\sqrt{5}\theta}{80}\right)$ | $\left(\frac{\sqrt{5}\theta}{80},\frac{\sqrt{5}\theta}{80}\right)$ |

$20\le \theta \le 40$ | $2{x}_{1}+{x}_{2}=\frac{\theta}{20}$ | ${\left(\frac{\sqrt{5}\theta}{120},\frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{120}\right)}$ | ${\left(\frac{\sqrt{5}\theta}{80},\frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{80}\right)}$ | ${\left(\frac{\sqrt{5}\theta}{80},\frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{80}\right)}$ | |

$40<\theta <60$ | $2{x}_{1}+{x}_{2}=\frac{\theta}{20};$ ${x}_{1}={x}_{2}=0$ | ${\left(\sqrt{5}-\frac{\sqrt{5}\theta}{60},\frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{120}\right)}$ | ${\left(\frac{3\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{40},0\right)}$ | ${\left(\frac{3\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{40},0\right)}$ | |

Productivity inequality $(20,20,2,1)$ | $0<\theta <20$ | $2{x}_{1}+{x}_{2}=\frac{\theta}{20}$ | $\left({\frac{\sqrt{5}\theta}{120},\frac{\sqrt{5}\theta}{60}}\right)$ | ${\left(\frac{\sqrt{5}\theta}{120},\frac{\sqrt{5}\theta}{60}\right)}$ | $\left(\frac{\sqrt{5}\theta}{80},\frac{\sqrt{5}\theta}{80}\right)$ |

$20\le \theta \le 40$ | $2{x}_{1}+{x}_{2}=\frac{\theta}{20}$ | ${\left(\frac{\sqrt{5}\theta}{120},\frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{120}\right)}$ | ${\left(\frac{\sqrt{5}\theta}{120},\frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{120}\right)}$ | ${\left(\frac{\sqrt{5}\theta}{80},\frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{80}\right)}$ | |

$40<\theta <60$ | $2{x}_{1}+{x}_{2}=\frac{\theta}{20};$ ${x}_{1}={x}_{2}=0$ | ${\left(\sqrt{5}-\frac{\sqrt{5}\theta}{60},\frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{120}\right)}$ | ${\left(\sqrt{5}-\frac{\sqrt{5}\theta}{60},\frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{120}\right)}$ | ${\left(\frac{3\sqrt{5}}{2}-\frac{\sqrt{5}\theta}{40},0\right)}$ | |

Aligned inequality $(24,12,2,1)$ | $0<\theta <12$ | $4{x}_{1}+{x}_{2}=\frac{\theta}{12}$ | ${\left(\frac{\sqrt{17}\theta}{240},\frac{\sqrt{17}\theta}{60}\right)}$ | ${\left(\frac{\sqrt{17}\theta}{144},\frac{\sqrt{17}\theta}{72}\right)}$ | $\left(\frac{\sqrt{17}\theta}{96},\frac{\sqrt{17}\theta}{96}\right)$ |

$12\le \theta \le 48$ | $4{x}_{1}+{x}_{2}=\frac{\theta}{12}$ | ${\left(\frac{\sqrt{17}\theta}{240},\frac{\sqrt{17}}{4}-\frac{\sqrt{17}\theta}{240}\right)}$ | ${\left(\begin{array}{c}\mathrm{min}\left\{\frac{\sqrt{17}\theta}{144},\frac{\sqrt{17}}{4}\right\},\\ \mathrm{max}\left\{\frac{\sqrt{17}}{4}-\frac{\sqrt{17}\theta}{144},0\right\}\end{array}\right)}$ | ${\left(\begin{array}{c}\mathrm{min}\left\{\frac{\sqrt{17}\theta}{96},\frac{\sqrt{17}}{4}\right\},\\ \mathrm{max}\left\{\frac{\sqrt{17}}{4}-\frac{\sqrt{17}\theta}{96},0\right\}\end{array}\right)}$ | |

$48<\theta <60$ | $4{x}_{1}+{x}_{2}=\frac{\theta}{12};$ ${x}_{1}={x}_{2}=0$ | ${\left(\sqrt{17}-\frac{\sqrt{17}\theta}{60},\frac{\sqrt{17}}{4}-\frac{\sqrt{17}\theta}{240}\right)}$ | ${\left(\frac{5\sqrt{17}}{4}-\frac{\sqrt{17}\theta}{48},0\right)}$ | ${\left(\frac{5\sqrt{17}}{4}-\frac{\sqrt{17}\theta}{48},0\right)}$ | |

Misaligned inequality $(30,15,1,2)$ | $0<\theta <30$ | ${x}_{1}+{x}_{2}=\frac{\theta}{30}$ | $\left(\frac{\sqrt{2}\theta}{60},\frac{\sqrt{2}\theta}{60}\right)$ | ${\left(\frac{\sqrt{2}\theta}{45},\frac{\sqrt{2}\theta}{90}\right)}$ | $\left(\frac{\sqrt{2}\theta}{60},\frac{\sqrt{2}\theta}{60}\right)$ |

$\theta =30$ | ${x}_{1}+{x}_{2}=\frac{\theta}{30}$ | $\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$ | ${\left(\frac{2\sqrt{2}}{3},\frac{\sqrt{2}}{3}\right)}$ | $\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$ | |

$30<\theta <60$ | ${x}_{1}+{x}_{2}=\frac{\theta}{30};$ ${x}_{1}={x}_{2}=0$ | $\left(\sqrt{2}-\frac{\sqrt{2}\theta}{60},\sqrt{2}-\frac{\sqrt{2}\theta}{60}\right)$ | ${\left(\begin{array}{c}\mathrm{max}\left\{\sqrt{2}-\frac{\sqrt{2}\theta}{90},2\sqrt{2}-\frac{\sqrt{2}\theta}{30}\right\},\\ \mathrm{max}\left\{\sqrt{2}-\frac{\sqrt{2}\theta}{45},0\right\}\end{array}\right)}$ | $\left(\sqrt{2}-\frac{\sqrt{2}\theta}{60},\sqrt{2}-\frac{\sqrt{2}\theta}{60}\right)$ |

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**MDPI and ACS Style**

An, X.; Dong, Y.; Wang, X.; Zhang, B.
Cooperation and Coordination in Threshold Public Goods Games with Asymmetric Players. *Games* **2023**, *14*, 76.
https://doi.org/10.3390/g14060076

**AMA Style**

An X, Dong Y, Wang X, Zhang B.
Cooperation and Coordination in Threshold Public Goods Games with Asymmetric Players. *Games*. 2023; 14(6):76.
https://doi.org/10.3390/g14060076

**Chicago/Turabian Style**

An, Xinmiao, Yali Dong, Xiaomin Wang, and Boyu Zhang.
2023. "Cooperation and Coordination in Threshold Public Goods Games with Asymmetric Players" *Games* 14, no. 6: 76.
https://doi.org/10.3390/g14060076