# Watershed Ecological Compensation Mechanism for Mainstream and Branches Based on Stochastic Evolutionary Game: A Case of the Middle Yellow River

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Overview of Research Region

^{2}and it concentrated 63% of the province’s population and GDP. It is the political, economic and cultural center of Shaanxi Province [26]. Therefore, the Weihe River is considered to be Branch 2 and Shaanxi Provincial Government is considered to be Branch Government 2. The Fenhe River and the Weihe River merge into the mainstream of the Yellow River and then flows into Henan Province; thus, Henan Province is considered to be Mainstream Government. In recent years, the Fenhe River and the Weihe River have been seriously polluted due to the growth of the population, overexploitation and other reasons. Frequent occurrence of water pollution has seriously affected the ecological environment security and economic and social development of Shanxi, Shaanxi, Henan and even the entire basin [27].

#### 2.2. Method

#### 2.2.1. Model Assumptions

- Shanxi, Shaanxi and Henan provincial governments are all limited rational players, and it is difficult to determine individual optimal strategies in a single game. It takes multiple games to reach a consensus.
- With the goal of pollution control and emission reduction, Shanxi and Shaanxi governments have two strategies. The first is complete governance: that is, restricting the development of some local industries to achieve the goal of reducing pollutants. The second is incomplete governance: that is, not completely restricting the development of local industries. Thus, the goal of pollution control and emission reduction cannot be completely achieved, and the task of emission reduction will be transferred to the mainstream government and other branch governments.
- For the mainstream Henan provincial government, there are two strategies: one is compensation and reward. Since the Shanxi and Shaanxi provincial governments have given up on the development of local industries to improve the water environment, this has greatly reduced the pollution control cost of the Henan provincial government. Thus, the provincial government believes that compensation and reward should be given for the opportunities and benefits given up due to pollution control; the other is no compensation or reward. The Henan provincial government believes that it is the obligation of branch governments to eliminate pollutants without compensation or reward.

#### 2.2.2. Mechanism Design

#### 2.2.3. Model Construction

- (1)
- Definition of parameters

- (2)
- Certainty evolutionary system

#### 2.2.4. Model Solution

- (1)
- Stability judgment of stochastic evolution system

**Lemma 1.**

- (1)
- If there is a positive constant $\gamma $, such that $LV\left(t,x\right)\le \gamma V\left(t,x\right),t\ge 0$, the zero solution of Equation (19) p-order moment is exponentially stable and $E{\left|x\left(t,x\right)\right|}^{p}<\left({c}_{2}/{c}_{1}\right){\left|{x}_{0}\right|}^{p}{e}^{\gamma t},t\ge 0$ is achieved.
- (2)
- If there is a positive constant $\gamma $, such that $LV\left(t,x\right)\ge \gamma V\left(t,x\right),t\ge 0$, the zero solution of Equation (19) p-order moment is exponentially stable and $E{\left|x\left(t,x\right)\right|}^{p}\ge \left({c}_{2}/{c}_{1}\right){\left|{x}_{0}\right|}^{p}{e}^{\gamma t},t\ge 0$ is achieved.

- (2)
- System equilibrium

## 3. Results

#### 3.1. Influence Rule of the Initial Probability

#### 3.2. Influence Rule of ${\theta}_{1}$

- (1)
- Influence rule of changes of ${\theta}_{1}$ on the Branch Government 1

- (2)
- Influence rule of the change of ${\theta}_{1}$ on Branch Government 2

- (3)
- Influence rule of change of ${\theta}_{1}$ on the mainstream government

#### 3.3. Influence Rule of $J$

- (1)
- Influence rule of $J$ change on Branch Government 1

- (2)
- Influence rule of $J$ change on Branch Government 2

- (3)
- Influence rule of $J$ change on the mainstream government

#### 3.4. Influence Rule of Random Factors

## 4. Discussion

## 5. Conclusions and Suggestion

- (1)
- The initial probability affects the decisions of the three governments. With the increase of the initial probability, Branch Government 2 (the Shaanxi Provincial government) will first stabilize from the incomplete governance strategy to the complete governance strategy. In the case of high initial willingness, the three governments will stabilize to the optimal strategy (complete governance, complete governance, reward and compensation). As a result, strengthening the publicity of the ecological compensation policy and the guidance of the branch and mainstream governments plays an important role in the implementation of the ecological compensation mechanism.
- (2)
- With the increase of the transferred pollution amount of a branch, other branch governments are more unwilling to govern the pollution. When the proportion of transferred pollution amount exceeds a certain critical value, then other branch governments will choose the incomplete governance strategy. In addition, in the case of the high initial willingness of other branch governments to control pollution, the critical value of the proportion of transferred pollution will also increase. As a result, when carrying out the actual implementation of policies, it is essential to strengthen the supervision of a branch government on water pollution governance. When necessary, compensation mechanisms for watershed ecology can be combined with other policies, including China’s “The River Chief System” policy, to avoid a branch government’s “free-rider” phenomenon. This means that it is necessary to avoid the situation of a branch government transferring the pollution it shall treat to other branch governments.
- (3)
- The more the mainstream government (Henan provincial government) rewards the branch governments (Shanxi and Shaanxi provincial governments) for more pollution control, the more the branch governments will be encouraged to carry out pollution control. Thus, the mainstream government can increase the rewards to the branch governments within a certain range, to encourage them to increase pollution control efforts. To encourage the effective implementation of the mechanism, the central government can also offer some incentives to the river basin and branches where the ecological compensation mechanism is well implemented.
- (4)
- Branch governments (Shanxi and Shaanxi provincial governments) are greatly affected by random factors, and the greater random interference, the more unfavorable for the branch governments regarding pollution control. It is necessary to consider the influence of random factors when formulating the ecological compensation mechanism. According to the long time series data, the impact of random factors on branch governments should be quantified, and differentiated compensation standards should be set under different circumstances.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Influence rule of the initial probabilities on three governments: (

**a**) influence rule of the initial probabilities on three governments when x(0) = 0.2, y(0) = 0.2, z(0) = 0.2, (

**b**) influence rule of the initial probabilities on three governments when x(0) = 0.4, y(0) = 0.4, z(0) = 0.4, (

**c**) influence rule of the initial probabilities on three governments when x(0) = 0.6, y(0) = 0.6, z(0) = 0.6, and (

**d**) influence rule of the initial probabilities on three governments when x(0) = 0.8, y(0) = 0.8, z(0) = 0.8.

**Figure 3.**Influence rule of changes of $\theta 1$ on Branch Government 1: (

**a**) influence rule of changes of $\theta 1$ on Branch Government 1 when x(0) = 0.7, y(0) = 0.7, z(0) = 0.7, and (

**b**) influence rule of changes of $\theta 1$ on Branch Government 1 when x(0) = 0.2, y(0) = 0.2, z(0) = 0.2.

**Figure 4.**Influence rule of changes of $\theta 1$ on Branch Government 2: (

**a**) influence rule of changes of $\theta 1$ on Branch Government 2 when x(0) = 0.7, y(0) = 0.7, z(0) = 0.7, and (

**b**) influence rule of changes of $\theta 1$ on Branch Government 2 when x(0) = 0.2, y(0) = 0.2, z(0) = 0.2.

**Figure 5.**Influence rule of changes of $\theta 1$ on Mainstream Government: (

**a**) influence rule of changes of $\theta 1$ on Mainstream Government when x(0) = 0.7, y(0) = 0.7, z(0) = 0.7, and (

**b**) influence rule of changes of $\theta 1$ on Mainstream Government when x(0) = 0.5, y(0) = 0.5, z(0) = 0.5.

**Figure 6.**Influence rule of $J$ change on Branch Government 1: (

**a**) influence rule of $J$ change on Branch Government 1 when x(0) = 0.7, y(0) = 0.7, z(0) = 0.7, and (

**b**) influence rule of $J$ change on Branch Government 1 when x(0) = 0.2, y(0) = 0.2, z(0) = 0.2.

**Figure 7.**Influence rule of $J$ change on Branch Government 2: (

**a**) influence rule of $J$ change on Branch Government 2 when x(0) = 0.7, y(0) = 0.7, z(0) = 0.7, and (

**b**) influence rule of $J$ change on Branch Government 1 when x(0) = 0.2, y(0) = 0.2, z(0) = 0.2.

**Figure 8.**Influence rule of $J$ change on Mainstream Government: (

**a**) influence rule of $J$ change on Mainstream Government when x(0) = 0.7, y(0) = 0.7, z(0) = 0.7, and (

**b**) influence rule of $J$ change on Mainstream Government when x(0) = 0.4, y(0) = 0.4, z(0) = 0.4.

**Figure 9.**Influence rule of random factor $\sigma $ change on three governments: (

**a**) influence rule of random factor $\sigma $ change on Branch Government 1, (

**b**) influence rule of random factor $\sigma $ change on Branch Government 2, and (

**c**) influence rule of random factor $\sigma $ change on Mainstream Government.

Game Subject | Strategy | Selection Probability |
---|---|---|

Branch Government 1 (Shanxi Provincial Government) | Complete governance | $x$ |

Incomplete governance | $1-x$ | |

Branch Government 2 (Shaanxi Provincial Government) | Complete governance | $y$ |

Incomplete governance | $1-y$ | |

Mainstream Government (Henan Provincial Government) | Reward | $z$ |

No reward | $1-z$ |

**Table 2.**Game benefit matrix among Mainstream Government, Branch Government 1 and Branch Government 2.

Mainstream Government Executes Reward and Compensation $\left(\mathit{z}\right)$ | Mainstream Government Does Not Execute Reward and Compensation $(1-\mathit{z})$ | |||
---|---|---|---|---|

Branch Government 2 Achieves Complete Governance $\left(\mathit{y}\right)$ | Branch Government 2 Achieves Incomplete Governance $(1-\mathit{y})$ | Branch Government 2 Achieves Complete Governance $\left(\mathit{y}\right)$ | Branch Government 2 Achieves Incomplete Governance $(1-\mathit{y})$ | |

Branch Government 1 achieves complete governance $\left(x\right)$ | $-{C}_{1}{A}_{1}+{S}_{1}{A}_{1}+{B}_{1}$ | $\begin{array}{l}-{C}_{1}({A}_{1}+{\theta}_{2}{P}_{2})+{S}_{1}({A}_{1}+{\theta}_{2}{P}_{2})\\ +{B}_{1}+{\theta}_{2}J\end{array}$ | $-{C}_{1}{A}_{1}+{S}_{1}{A}_{1}$ | $\begin{array}{l}-{C}_{1}({A}_{1}+{\theta}_{2}{P}_{2})\\ +{S}_{1}({A}_{1}+{\theta}_{2}{P}_{2})\end{array}$ |

$-{C}_{2}{A}_{2}+{S}_{2}{A}_{2}+{B}_{2}$ | $\begin{array}{l}-{C}_{2}({A}_{2}-{P}_{2})+{S}_{2}({A}_{2}-{P}_{2})\\ -{F}_{2}-{W}_{2}\end{array}$ | $-{C}_{2}{A}_{2}+{S}_{2}{A}_{2}$ | $\begin{array}{l}-{C}_{2}({A}_{2}-{P}_{2})+{S}_{2}({A}_{2}\\ -{P}_{2})-{W}_{2}\end{array}$ | |

$L({A}_{1}+{A}_{2})-{B}_{1}-{B}_{2}$ | $\begin{array}{l}-{C}_{3}(1-{\theta}_{2}){P}_{2}+L({A}_{1}+{\theta}_{2}{P}_{2}\\ +{A}_{2}-{P}_{2})-{B}_{1}+{F}_{2}-{\theta}_{2}J\end{array}$ | $L({A}_{1}+{A}_{2})-{W}_{3}$ | $\begin{array}{l}-{C}_{3}(1-{\theta}_{2}){P}_{2}+L({A}_{1}+{\theta}_{2}{P}_{2}\\ +{A}_{2}-{P}_{2})-{W}_{3}\end{array}$ | |

Branch Government 1 achieves incomplete governance $(1-x)$ | $\begin{array}{l}-{C}_{1}({A}_{1}-{P}_{1})+{S}_{1}({A}_{1}-{P}_{1})\\ -{F}_{1}-{W}_{1}\end{array}$ | $\begin{array}{l}-{C}_{1}({A}_{1}-{P}_{1})+{S}_{1}({A}_{1}-{P}_{1})\\ -{F}_{1}-{W}_{1}\end{array}$ | $\begin{array}{l}-{C}_{1}({A}_{1}-{P}_{1})+{S}_{1}({A}_{1}\\ -{P}_{1})-{W}_{1}\end{array}$ | $-{C}_{1}({A}_{1}-{P}_{1})+{S}_{1}({A}_{1}-{P}_{1})$ |

$\begin{array}{l}-{C}_{2}({A}_{2}+{\theta}_{1}{P}_{1})+{S}_{2}({A}_{2}\\ +{\theta}_{1}{P}_{1})+{B}_{2}+{\theta}_{1}J\end{array}$ | $\begin{array}{l}-{C}_{2}({A}_{2}-{P}_{2})+{S}_{2}({A}_{2}-{P}_{2})\\ -{F}_{2}-{W}_{2}\end{array}$ | $\begin{array}{l}-{C}_{2}({A}_{2}+{\theta}_{1}{P}_{1})\\ +{S}_{2}({A}_{2}+{\theta}_{1}{P}_{1})\end{array}$ | $-{C}_{2}({A}_{2}-{P}_{2})+{S}_{2}({A}_{2}-{P}_{2})$ | |

$\begin{array}{l}-{C}_{3}(1-{\theta}_{1}){P}_{1}+L({A}_{1}\\ +{\theta}_{1}{P}_{1}+{A}_{2}-{P}_{1})-{B}_{2}\\ +{F}_{1}-{\theta}_{1}J\end{array}$ | $\begin{array}{l}-{C}_{3}({P}_{1}+{P}_{2})+L({A}_{1}+{A}_{2}\\ -{P}_{1}-{P}_{2})+{F}_{1}+{F}_{2}\end{array}$ | $\begin{array}{l}-{C}_{3}(1-{\theta}_{1}){P}_{1}+L({A}_{1}\\ +{\theta}_{1}{P}_{1}+{A}_{2}-{P}_{1})-{W}_{3}\end{array}$ | $\begin{array}{l}-{C}_{3}({P}_{1}+{P}_{2})+L({A}_{1}+{A}_{2}\\ -{P}_{1}-{P}_{2})\end{array}$ |

${P}_{1}$ | ${P}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ${S}_{1}$ | ${S}_{2}$ | ${S}_{3}$ | ${B}_{1}$ | ${B}_{2}$ |

15 | 10 | 7 | 9 | 6 | 8 | 8 | 3 | 4 |

${F}_{1}$ | ${F}_{2}$ | ${W}_{1}$ | ${W}_{2}$ | ${W}_{3}$ | ||||

3 | 2 | 5 | 5 | 3 |

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**MDPI and ACS Style**

Liu, Y.; Jiang, E.; Qu, B.; Zhu, Y.; Liu, C.
Watershed Ecological Compensation Mechanism for Mainstream and Branches Based on Stochastic Evolutionary Game: A Case of the Middle Yellow River. *Water* **2022**, *14*, 4038.
https://doi.org/10.3390/w14244038

**AMA Style**

Liu Y, Jiang E, Qu B, Zhu Y, Liu C.
Watershed Ecological Compensation Mechanism for Mainstream and Branches Based on Stochastic Evolutionary Game: A Case of the Middle Yellow River. *Water*. 2022; 14(24):4038.
https://doi.org/10.3390/w14244038

**Chicago/Turabian Style**

Liu, Ying, Enhui Jiang, Bo Qu, Yongwei Zhu, and Chang Liu.
2022. "Watershed Ecological Compensation Mechanism for Mainstream and Branches Based on Stochastic Evolutionary Game: A Case of the Middle Yellow River" *Water* 14, no. 24: 4038.
https://doi.org/10.3390/w14244038