Analyzing the Water Pollution Control Cost-Sharing Mechanism in the Yellow River and Its Two Tributaries in the Context of Regional Differences

Abstract

A river basin is a complex system of tributaries and a mainstream. It is vital to cooperatively manage the mainstream and the tributaries to alleviate water pollution and the ecological environment in the basin. On the other hand, existing research focuses primarily on upstream and downstream water pollution control mechanisms, ignoring coordinated control of the mainstream and tributaries, and does not consider the impacts of different environmental and economic conditions in each region on pollution control strategies. This study designed a differential game model for water pollution control in the Yellow River and two of its tributaries, taking regional differences into account and discussing the best pollution control strategies for each region under two scenarios: Nash noncooperative and cost-sharing mechanisms. Furthermore, the factors influencing regional differences in pollution control costs are analyzed, and their impacts on the cost-sharing mechanism of pollution control are discussed. The results show that: (1) The cost-sharing mechanism based on cooperative management can improve pollutant removal efficiency in the watershed and achieve Pareto improvement in environment and economy. (2) The greater the economic development pressure between the two tributaries, the fewer the effects of the cost-sharing mechanisms and the higher the proportion of pollution control costs paid by the mainstream government. (3) Industry water consumption, the proportion of the urban population to the total population at the end of the year, the value-added of secondary sectors as a percentage of regional GDP, the volume of industrial wastewater discharge, and granted patent applications all influence industrial wastewater pollution treatment investment. (4) The greater the coefficient of variation in pollution control costs between the two tributary areas, the less favorable the solution to water pollution management synergy. These findings can help governments in the basin regions negotiate cost-sharing arrangements.

1. Introduction

Water is indispensable to people. However, water pollution has worsened in many regions of the world, with high pollution threats identified across Europe, India, China, South America, and parts of Africa [1]. Due to the fluidity of water, pollutants generated in upstream areas can affect downstream areas, and pollutants generated in tributary areas can affect mainstream areas, leading to the deterioration of water quality and other transboundary water pollution problems [2,3]. Water pollution can pose serious threats to ecological environments, human health, and sustainable social development [4]. It is very necessary to control water pollution.

Government monitoring of polluters and collaboration between the parties involved in river basins to reduce pollution are the two primary categories of studies on transboundary water pollution control [5,6,7,8,9]. There are primarily two strategies for collaboration between the parties involved: one is an ecological compensation mechanism based on the internalization of the spatial overflow of public goods, and the other is a market-based cost-sharing mechanism. Capital supply, industrial transfer, water rights exchange, and other mechanisms are used to build a compensation relationship between the upstream and downstream portions of the river basin [10,11]. In many circumstances, however, under the influence of market effectiveness, central government coordination is required. The influence of reward, punishment, incentive coordination, and other mechanisms of the central government on the strategic choice of upstream and downstream governments should be considered [12,13,14]. Based on this, Yang et al. [15] developed an evolutionary game model of three-party transboundary water pollution control of watershed governments from the perspectives of left and right banks. They investigated the three-party strategy’s evolution pattern using a composite mechanism of reward, punishment, and compensation. Yang et al. [16] then built a tripartite evolutionary game model using the mainstream and two tributaries based on upstream and downstream transboundary water pollution control studies and taking into account mechanisms of compensation, repayment, and reward integration.

In a cost-sharing mechanism, the upstream and downstream regions of a basin collaborate to control pollution, with the downstream region benefiting from upstream pollution control and contributing a portion of the upstream region’s pollution control costs. The cost-sharing method is more dynamic and flexible than the fixed-amount ecological compensation approach. The compensator, for example, can determine the compensation share based on the compensator’s pollution control investment. Jiang et al. [17,18] analyzed the impacts of several factors on upstream and downstream pollution emissions in the Xin’an River basin as an example. If upstream and downstream governments invest in pollution control, welfare can be maximized by determining the optimal emission reduction investment level and share ratio [19]. Meanwhile, central government subsidies may impact the cost-sharing mechanism’s implementation [20].

Water quality in the YRB is in general worse than the national average, with the water quality of the tributaries worse than that of the mainstream. Several fundamental concerns impacting river water quality in specific tributary locations have remained unsolved [21]. In September 2019, China launched a national policy for ecological conservation and high-quality development of the YRB. Environmental protection should be strengthened to turn the Yellow River into a river of happiness for the benefit of the people.

To reduce pollution in the YRB, government regulation and cooperation between the stakeholders are also important. The core research on environmental compensation mechanisms, specifically at the macro and micro levels, focuses on collaboration among interested parties in the basin to control pollution.

At the macro level, research on ecological compensation mechanisms in the YRB focuses on priority regions, essential ideas, key directions, implementation paths, and the concept of measuring funds. For example, according to Han et al. [22], the types of and priority areas for ecological compensation should include the source of the Yellow River water conservation area, water and soil conservation, significant trans-province project construction, drinking water source areas, and downriver beaches. Dong et al. [23] examined the significant issues that have arisen during the current ecological compensation mechanism’s construction stage. They proposed a basic concept and framework for creating an ecological compensation mechanism in the YRB. Yang et al. [24] proposed ecological compensation for the YRB, including its principles, contents, and estimated cost.

The primary extant research focuses on compensation models and standards in a specialized field at the micro level. For example, in the Loess Plateau in northern Shaanxi, Zhuang et al. [25] enhanced ecological compensation models and methods. Zhao and Li [26] assessed compensation content and standards in the Yellow River downstream.

We can draw two conclusions from the above. First, most transboundary water contamination issues in the Yellow River basin are focused on upstream and downstream viewpoints, with less attention paid to the mainstream and tributaries. Differences in ecological and economic situations between regions have not been considered. Second, while ecological compensation for transboundary water pollution in the YRB has been extensively explored, cost sharing has received less attention. Regional disparities are also not taken into account.

On the aspect of regional disparities, we used the topic of air pollution control in surrounding regions as a starting point. Disparities are reflected in the damage caused by environmental pollution between developed and developing countries [27], particularly in the three aspects of production capacity, pollutant treatment costs, and environmental pollution damages in asymmetric regions [28]. In studies of differences between nearby regions, only one parameter is used to express the difference coefficient, but multiple factors contribute to regional differences. Scholars often use panel data methods to analyze the drivers of regional differences [29]. For example, Guo et al. [30] explored critical influences on the efficiency of investment in environmental management in China, and Li et al. [31] studied the impact of economic development on the efficiency of PM2.5 emission reduction in Chinese cities.

This paper, therefore, to systematically solve the transboundary water pollution problem in the YRB, constructed a cost-sharing mechanism in the mainstream and tributary regions in the context of regional differences.

The following is the order in which the paper is presented. The scientific foundation for this paper is a study on transboundary water pollution prevention, which is discussed in Section 1. Section 2 introduces the regional overview of the study area. The research technique is proposed in Section 3. Section 4 conducts simulation experiments and analyzes the results. Conclusions are presented in Section 5.

2. Study Area

In China, the YRB is a significant ecological barrier. It is also a vital hub for agricultural production, with abundant resources and energy. It plays an important role in China’s ecological security, as well as its economic and social development. The YRB spans 5464 km and covers 796,000 km², including Qinghai, Sichuan, Gansu, Ningxia, Inner Mongolia, Shaanxi, Shanxi, Henan, and Shandong provinces [32]. Figure 1 depicts the YRB’s geographic location. Thirteen principal tributaries flow into the YRB, of which ten tributaries are located in the middle and lower reaches. The climate across the YRB varies from arid and semi-arid in the northwest to semi-humid in the southeast. The mean annual temperature is 1 °C in the western regions and 16 °C in the eastern regions. The average annual precipitation ranges from 140 mm to 1200 mm, with the gradient decreasing from south to north [33].

3. Materials and Methods

The following are the primary contributions of this paper: (1) The upstream and downstream of the YRB are expanded into the mainstream and two tributaries in the analysis framework. The ecological vulnerability, cost of pollution management, and economic benefits of pollution control are all different between the two tributaries. (2) Factors driving the variations in pollution-control costs between regions are investigated using panel data from the YRB. The difference in pollution-control costs between tributaries is calculated as a coefficient. (3) Nash’s noncooperative and cost-sharing mechanisms are compared between the mainstream and the two tributaries using the differential game model. Figure 2 depicts the flowchart of the methodology of this paper.

3.1. Establishment and Solution of Differential Game Models

To eliminate pollutants and improve the water quality in the river basin, both the mainstream and the tributary governments need to carry out water pollution treatment at the same time. We examined the behavior of a tripartite government effort in transboundary water pollution control using Nash’s noncooperative mechanisms versus cost-sharing mechanisms. The following are some assumptions:

Assumption 1: The two tributaries in the YRB are denoted as tributary 1 (T1) and tributary 2 (T2), and the areas they flow in are the T1 area and the T2 area, respectively. Each regional government is responsible for water pollution control within its jurisdiction.

Assumption 2: Industrial production occurs in the two tributary areas, discharging sewage into the tributaries that carries many categories of pollutants such as organic. The total amounts of pollutants by industrial production are $q_1$ and $q_2$, where $q_1>0$ and $q_2>0$. The revenues obtained by industrial production are $J_1$ and $J_2$. The central government imposes an environmental protection tax $\omega$ on pollutants on the regional governments of the two tributaries.

Assumption 3: Pollution control means reducing the pollutants in sewage. The pollution-control efforts by T1, T2, and the mainstream regional government are expressed as $A$, $B$, and $Z$, respectively, where $0<A<1$, $0<B<1$, and $0<Z<1$. The pollution-control effort represents the governments’ policies, personnel, and financial investments. $\xi$, $\eta$, and $\theta$ represent the amounts of pollution eliminated per unit of pollution-control effort by T1, T2, and the mainstream regional government, respectively, where $\xi >0,\eta >0,\theta >0$.

Assumption 4: According to Li et al. [30], the cost of pollution control by the T1 regional government is a quadratic function of the form $\frac{1}{2}{A}^2$. Due to the different pollution-control costs between the two tributaries, the pollution-control cost for the T2 regional government is a quadratic function of the form $\frac{k}{2}{B}^2$ where $k$ represents the coefficient of difference in the costs of pollution control between the two tributaries. The effort cost of the mainstream regional government is $\frac{1}{2}c{Z}^2,c$represents the effort cost coefficient of the mainstream government.

Assumption 5: The environmental benefits of pollution control on the two tributaries and the mainstream are different due to the different ecological vulnerabilities of each region. The environmental benefit to a region from the elimination of unit pollutants by the governments of the two tributaries and the mainstream are $\xi A{r}_1$, $\eta B{r}_2$, and $\theta Z{r}_3$where $r_1$, $r_2$, and $r_3$ represent the differences in the environmental benefits per pollutant treated in the T1, T2, and mainstream regions, respectively, reflecting the ecological vulnerability of each area.

Assumption 6: The pollutant treatment in the basin results from tripartite efforts from the T1, T2, and mainstream regional governments. The total amounts of pollutants treatment can be expressed with the dynamic differential Equation (1):

$$\stackrel{·}{q}(t)=\xi A+\eta B+\theta Z-\delta q\left(t\right)$$

(1)

where $q(t)$ represents the total amounts of pollutants treated at time $t$ assuming the initial state of the system $q(0)\ge 0$. Then, $\delta$ indicates the attenuation factor of the reduction due to factors such as the aging of the sewage treatment equipment.

Assumption 7: The regional governments of both tributaries were unwilling to contribute to pollution-control policies or funds because of their economic development levels. According to the related studies [34,35,36,37], once the competitive coefficient $\alpha$ is set, the degree of economic development pressure on the two tributary governments is measured as $0<\alpha <1$.

Assumption 8: The level of effort by the regional government of T1 is a linear function of competition from the regional government of T2 and supervision from the regional government of the mainstream. $A$ is the effort level of the T1 regional government, $A=A_0-\alpha B+d_{1}Z$. Similarly, the level of effort of the T2 regional government is $B$, $B=B_0-\alpha A+d_{2}Z$. The initial pollution-control effort levels of the T1 and T2 regional governments are represented as $A_0$ and $B_0$when weighing their own benefits and costs without considering the competition factor or the supervision of the mainstream regional government: $A_0\ge 0$ and $B_0\ge 0$. The parameters $d_1$ and $d_2$represent the mainstream government’s influence on the efforts of T1 and T2, respectively.

Assumption 9: The social welfare effects of pollutants treated by the regional governments of the two tributaries and the mainstream are $S\left(t\right)=S_0+\lambda q\left(t\right)$, where $S_0>0$represents the initial basin welfare status and $\lambda >0$represents the coefficient of influence of pollution control on the social welfare effects in the basin.

Assumption 10: Due to the different economic development levels between the three regions, the effect coefficients are different for each basin’s social welfare benefit to its regional revenues. The effect coefficients for the T1, T2, and mainstream regional governments are ${\pi }_1$, ${\pi }_2,$ and ${\pi }_3,$, where ${\pi }_1>0$, ${\pi }_2>0$, and ${\pi }_3>0$. The three governments have the same discount rate $\rho$, with $\rho >0$.

3.1.1. The Nash Noncooperative Game Scenario

In the Nash noncooperative game scenario, the regional governments of both tributaries and the mainstream make their own decisions on pollution management initiatives. Based on the above assumptions, the T1, T2, and mainstream regional governments are all finite rational players. All players are trying to figure out the best way to maximize their profits. According to general assumptions, the revenues functions of the three parties are related to their efforts. Referring to the method of objective function setting in differential games [37], we get Equations (2)–(4).

The revenues function for the T1 government is

$${\displaystyle {\int }_0^t[\hspace{0.33em}{J}_1}-\frac{1}{2}{A}^2+{\xi Ar}_1+{\pi }_{1}S(t)-(q_1-\xi A)\omega \hspace{0.33em}+h_1\eta B]e^{-\rho t}dt$$

(2)

where $q_1-\xi A$ is the total amount of pollutants discharged from T1, and $h_1$ represents the revenues from the pollution-control efforts of the T2 government to relieve the pollution-control pressure on the T1 government.

The revenues function for the T2 government is

$${\displaystyle {\int }_0^t[\hspace{0.33em}{J}_2}+\eta Br2-\frac{k}{2}{B}^2+{\pi }_{2}S(t)-(q_2-\eta B)\omega \hspace{0.33em}+h_2\xi A]e^{-\rho t}dt$$

(3)

where $q_2-\eta B$ is the total amount of pollutants discharged from T2 and $h_2$ represents the revenues from the pollution control efforts of the T1 government to relieve the pollution-control pressure on the T2 government.

The revenues function for the mainstream government is

$${\displaystyle {\int }_0^t\left\{-\frac{c}{2}{Z}^2+{\theta Zr}_3+{\pi }_{3}S(t)+\left[q(t)-q_0\right]P\right\}}{e}^{-\rho t}dt$$

(4)

where $q_0$is the total amount of pollutants that need to be controlled by the three parties when the assessment section of the river reaches the standard. $P$ is the ecological compensation received (or paid) by the mainstream government for each pollutant unit that is better (worse) than that in the assessment section. The assessment section is set in the mainstream.

To achieve a unique continuous solution $q(t)$ to Equation (1), construct the Hamilton-Jacobi-Bellman-Fleming (HJB) equations for the T1 government, T2 government, and mainstream government, respectively [38], as Equations (5)–(7):

$${\rho V}_1(q)=\underset{A}{\max }\left\{\begin{array}{l}{J}_1-\frac{1}{2}{A}^2+{\xi Ar}_1+{\pi }_{1}S(t)-(q_1-\xi A)\omega \hspace{0.33em}+h_1\eta B\\ +V_1’(q)\left[\xi A+\eta B+\theta Z-\delta q\left(t\right)\right]\end{array}\right\}$$

(5)

$${\rho V}_2(q)=\underset{B}{\max }\left\{\begin{array}{l}{J}_2+\eta Br2-\frac{k}{2}{B}^2+{\pi }_{2}S(t)-(q_2-\eta B)\omega \hspace{0.33em}+h_2\xi A\\ +V_2’(q)\left[\xi A+\eta B+\theta Z-\delta q\left(t\right)\right]\end{array}\right\}$$

(6)

$${\rho V}_3(q)=\underset{Z}{\max }\left\{\begin{array}{l}-\frac{c}{2}{Z}^2+{\theta Zr}_3+{\pi }_{3}S(t)+\left[q(t)-q_0\right]P\hspace{0.33em}\\ +V_3’(q)\left[\xi A+\eta B+\theta Z-\delta q\left(t\right)\right]\end{array}\right\}$$

(7)

After solving the above models (see Appendix A), the game equilibrium solution can be solved as Equations (8)–(10):

$$A^{*}=\xi (\omega +r_1)+\frac{{\pi }_1\lambda }{\rho +\delta }\left(\xi -\eta \alpha \right)-h_1\eta \alpha$$

(8)

$$B^{*}=\frac{\eta (\omega +r_2)+\frac{{\pi }_2\lambda }{\rho +\delta }\left(\eta -\xi \alpha \right)-h_2\xi \alpha }{k}$$

(9)

$$Z^{*}=\frac{{\pi }_3\lambda +P}{c\left(\rho +\delta \right)}\times \left\{\left[\frac{\xi \left(d_1-\alpha d_2\right)}{1-{\alpha }^2}+\frac{\eta \left(d_2-\alpha d_1\right)}{1-{\alpha }^2}\right]+\theta \right\}+\frac{\theta r_3}{c}$$

(10)

The optimal revenues to the two tributary governments and the mainstream government in this equilibrium condition is calculated with Equations (11)–(13):

$$V_1^{*}=\frac{{\pi }_1\lambda }{\rho +\delta }{q}^{\ast }+\frac{1}{\rho }\left[\begin{array}{l}{J}_1+\left(-\frac{1}{2}{A}^{\ast }+\xi \omega +\xi r_1+\frac{{\pi }_1\lambda }{\rho +\delta }\xi \right)A^{\ast }+{\pi }_1{S}_0-q_1\omega \\ +\frac{{\pi }_1\lambda }{\rho +\delta }\left(\eta B^{\ast }+\theta Z^{\ast }\right)+h_1\eta B^{\ast }\end{array}\right]\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}$$

(11)

$$V_2^{\ast }=\frac{{\pi }_2\lambda }{\rho +\delta }{q}^{\ast }+\frac{1}{\rho }\left[\begin{array}{l}{J}_2+\left(-\frac{k}{2}{B}^{\ast }+\eta \omega +\eta r_2+\frac{{\pi }_2\lambda }{\rho +\delta }\eta \right)B^{\ast }+{\pi }_2{S}_0-q_2\omega \\ +\frac{{\pi }_2\lambda }{\rho +\delta }\left(\xi A^{\ast }+\theta Z^{\ast }\right)+h_2\xi A^{\ast }\end{array}\right]$$

(12)

$$V_3^{\ast }=\frac{{\pi }_3\lambda +P}{\rho +\delta }{q}^{\ast }+\frac{1}{\rho }\left[\begin{array}{l}\left(-\frac{c}{2}{Z}^{\ast }+\theta r_3+\frac{{\pi }_3\lambda +P}{\rho +\delta }\theta \right)Z^{\ast }+{\pi }_3{S}_0-P{q}_0\\ +\frac{{\pi }_3\lambda +P}{\rho +\delta }\left(\xi A^{\ast }+\eta B^{\ast }\right)\end{array}\right]$$

(13)

3.1.2. The Stackelberg Cost-Sharing Game Scenario

In this game, pollution control by the two tributaries will provide advantages and lower pollution-control costs for the mainstream government. The mainstream regional government will divide a percentage of pollution-control costs with the regional governments of the two tributaries. The mainstream regional government first decides on the pollution-control efforts and the cost-sharing ratios with the T1 and T2 regional governments and then determines their ideal solutions. The inverse induction approach is used to solve the optimal decisions of the T1 and T2 regional governments, and the objective function is satisfied the HJB Equations (14)–(16).

$${\rho V}_1(q)=\underset{A}{\max }\left\{\begin{array}{l}{J}_1-\left(1-m\right)\frac{1}{2}{A}^2+\xi A{r}_1+{\pi }_{1}S(t)-(q_1-\xi A)\omega \hspace{0.33em}+h_1\eta B\\ +V_1’(q)\left[\xi A+\eta B+\theta Z-\delta q\left(t\right)\right]\end{array}\right\}$$

(14)

$${\rho V}_2(q)=\underset{B}{\max }\left\{\begin{array}{l}{J}_2+\eta B{r}_2-\left(1-n\right)\frac{k}{2}{B}^2+{\pi }_{2}S(t)-(q_2-\eta B)\omega \hspace{0.33em}+h_2\xi A\\ +V_2’(q)\left[\xi A+\eta B+\theta Z-\delta q\left(t\right)\right]\end{array}\right\}$$

(15)

$${\rho V}_3(q)=\underset{Z}{\max }\left\{\begin{array}{l}-\frac{c}{2}{Z}^2+\theta Z{r}_3+{\pi }_{3}S(t)+\left[q(t)-q_0\right]P\hspace{0.33em}-\frac{1}{2}m{A}^2-\frac{k}{2}n{B}^2\\ +V_3’(q)\left[\xi A+\eta B+\theta Z-\delta q\left(t\right)\right]\end{array}\right\}$$

(16)

The Stackelberg equilibrium can be obtained using the backward induction proposed by Von Stakelberg [39] (Appendix B). The game equilibrium solution in the cost-sharing mechanism can be solved as Equations (17)–(21):

$$A^{**}=\frac{\xi (\omega +r_1)+\frac{{\pi }_1\lambda }{\rho +\delta }\left(\xi -\eta \alpha \right)-h_1\eta \alpha }{1-m^{*}}$$

(17)

$$B^{**}=\frac{\eta (\omega +r_2)+\frac{{\pi }_2\lambda }{\rho +\delta }\left(\eta -\xi \alpha \right)-h_2\xi \alpha }{k\left(1-n^{*}\right)}$$

(18)

$$Z^{**}=\frac{\left({\pi }_3\lambda +P\right)\theta +}{c\left(\rho +\delta \right)}+\frac{\theta r_3}{c}$$

(19)

$$m^{*}=\frac{2\frac{{\pi }_3\lambda +P}{\rho +\delta }\xi -\omega \xi -\xi r_1-\frac{{\pi }_1\lambda }{\rho +\delta }\left(\xi -\eta \alpha \right)+h_1\eta \alpha }{2\frac{{\pi }_3\lambda +P}{\rho +\delta }\xi +\omega \xi +\xi r_1+\frac{{\pi }_1\lambda }{\rho +\delta }\left(\xi -\eta \alpha \right)-h_1\eta \alpha }$$

(20)

$$n^{*}=\frac{2\frac{{\pi }_3\lambda +P}{\rho +\delta }\eta -\omega \eta -\eta r_2-\frac{{\pi }_2\lambda }{\rho +\delta }\left(\eta -\xi \alpha \right)+h_2\xi \alpha }{2\frac{{\pi }_3\lambda +P}{\rho +\delta }\eta +\omega \eta +\eta r_2+\frac{{\pi }_2\lambda }{\rho +\delta }\left(\eta -\xi \alpha \right)-h_2\xi \alpha }$$

(21)

The optimal revenues to the regional governments of the mainstream and the two tributaries are calculated with Equations (22)–(24):

$$V_1^{*}{}^{*}=\frac{{\pi }_1\lambda }{\rho +\delta }{q}^{**}+\frac{1}{\rho }\left\{\begin{array}{l}{J}_1+\left[-\frac{1}{2}\left(1-m^{*}\right)A^{**}+\xi (\omega +r_1)+\frac{{\pi }_1\lambda }{\rho +\delta }\xi \right]A^{**}\\ +{\pi }_1{S}_0-q_1\omega +\frac{{\pi }_1\lambda }{\rho +\delta }\left(\eta B^{**}+\theta Z^{**}\right)+h_1\eta B^{**}\end{array}\right\}$$

(22)

$$V_2^{**}=\frac{{\pi }_2\lambda }{\rho +\delta }{q}^{**}+\frac{1}{\rho }\left\{\begin{array}{l}{J}_2+\left[-\frac{k}{2}\left(1-n^{*}\right)B^{**}+\eta (\omega +r_2)+\frac{{\pi }_2\lambda }{\rho +\delta }\eta \right]B^{**}\\ +{\pi }_2{S}_0-q_2\omega +\frac{{\pi }_2\lambda }{\rho +\delta }\left(\xi A^{**}+\theta Z^{**}\right)+h_2\xi A^{**}\end{array}\right\}$$

(23)

$$V_3^{**}=\frac{{\pi }_3\lambda +P}{\rho +\delta }{q}^{**}+\frac{1}{\rho }\left[\begin{array}{l}\left(-\frac{c}{2}{Z}^{**}+\theta r_3+\frac{{\pi }_3\lambda +P}{\rho +\delta }\theta \right)Z^{**}+{\pi }_3{S}_0-P{q}_0\\ +\left(-\frac{1}{2}{m}^{*}{A}^{**}+\frac{{\pi }_3\lambda +P}{\rho +\delta }\xi \right)A^{**}\\ +\left(-\frac{k}{2}{n}^{*}{B}^{**}+\frac{{\pi }_3\lambda +P}{\rho +\delta }\eta \right)B^{**}\end{array}\right]$$

(24)

3.1.3. Numerical Specification

The Fenhe River is the largest river in Shanxi province and is a first-class tributary of the Yellow River. The Weihe River is the Yellow River’s largest tributary. It covers an area of 67,100 km² in Shaanxi province, concentrating 63% of the Shaanxi province’s population and GDP [40]. As a result, we selected the Fenhe River as T1 and the Weihe River as T2, designating Shanxi province as the T1 government and Shaanxi province as the T2 government. We chose Henan province as the mainstream government because the Fenhe and Weihe rivers flow through Henan province after converging into the Yellow River’s main channel.

After we collect data from the Statistical Yearbook of Shanxi Province and the Statistical Yearbook of Shaanxi Province in 2020, $q_1$ and $q_2$are set to 199,136 and 239,374.5 thousand tons, respectively, and $J_1$and $J_2$ are set to 656.951 and 2662.311 billion yuan, respectively. Similarly, data on the value-added of industries in Shanxi and Shaanxi provinces show that the higher the value of industrial production, the greater the advantage of reducing pollution-control pressure in the region. Therefore, $h_1$and $h_2$are assigned according to the proportions of the industrial added value of Shanxi province and Shaanxi province, respectively, in the sum of the industrial added value of the two provinces, so that $h_1$ = 0.2 and $h_2$ = 0.8, respectively. According to the Statistical Yearbook of Henan Province, the total investment for the treatment of industrial wastewater pollution ($c2$) is 308 million yuan. According to the Law of the People’s Republic of China on Environmental Protection Tax, $\omega$ is set to 1.4 yuan/m³. According to Yang et al. [34], take $d_1$ = 0.5 and $d_2$ = 0.5. According to Sheng and Webber [41], take $\delta =0.2$ and $\rho =0.2$.

$S_0$ is 9707.907 billion, which is calculated as the sum of the gross regional product of the three provinces. As a result of water pollution control, regional investment attraction will increase, leading to regional industrial development and increased social welfare in the basin. Social welfare in the three provinces is different as the economic development situations in the three provinces are different. Therefore, ${\pi }_1$, ${\pi }_2$, and ${\pi }_3$ are calculated as the GDP shares from Shanxi province, Shaanxi province, and Henan province in the sum of the regional GDP of each of the three provinces, ${\pi }_1=0$respectively. According to the ecological vulnerability calculation [42], $r_1$, $r_2$, and $r_3$ are set to 0.74, 0.23, and 0.62, respectively. According to the Horizontal Ecological Protection Compensation Agreement for the Yellow River Basin (Henan province–Shandong province Section), $P$ is set to 0.6.

3.2. Factors Influencing Inter-Regional Cost Differences and Coefficient of Cost Difference

3.2.1. Analysis of Factors Influencing Inter-Regional Cost Differences in Pollution Control

As shown in Figure 3, we used Stata 16 software to make a temporal trend graph of investments in industrial sewage treatment in nine provinces along the YRB. The provinces of Shaanxi, Gansu, Henan, Sichuan, and Shandong all fluctuate over time, while Shanxi is U-shaped, Inner Mongolia varies, and Ningxia and Qinghai remain stable. Industrial enterprise wastewater treatment investment appears to differ in the nine provinces.

Seven variables were selected from the natural environment and economic development for panel data analysis, based on relevant research [43,44] and the current development status of nine provinces along the YRB.

Table 1. Description and measurement of the variables.

Category		Variables	Description	Measurement	A Priori Sign
Explained variable		$Y$	Investment in industrial sewage treatment	10,000 yuan
Explanatory variable	Natural environment	$X_1$	Total amount of water resources	100 million cu. m	No prediction
	Economic development	$X_2$	Per capita GDP (PGDP)	yuan	Negative
		$X_3$	Proportion of urban population to total population at the end of the year	%	Positive
		$X_4$	Value-added of secondary industries as a proportion of regional GDP	%	Positive
		$X_5$	Water consumption of industry	100 million cu. m	Positive
		$X_6$	Volume of industrial wastewater discharge	10,000 tons	Positive
		$X_7$	Patent applications granted	piece	Negative

lists the factors in detail.

Variable data are taken from the China Statistical Yearbook on Environmental and the 2005–2019 Statistical Yearbook for each of the nine provinces along the Yellow River. Interpolation and extrapolation methods are used to fill in missing data.

Table 2. Descriptive statistics (period = 2005–2019, number of provinces = 9).

Variables		Mean	Standard Deviation	Minimum Value	Maximum Value	Observations
$Y$	overall	55,421.39	60,419.58	206	295,540.2	N = 135
	between		49,489.11	4367.747	176,512.6	n = 9
	within		38,173.77	−54,655.2	174,449	T = 15
$X_1$	overall	568.4675	723.2988	8.4	2953.79	N = 135
	between		752.0007	10.31033	2491.502	n = 9
	within		129.3782	−57.1945	1064.831	T = 15
$X_2$	overall	32,159.56	15,683.06	7332	70,129	N = 135
	between		7283.934	19,910.2	43,943.6	n = 9
	within		14,087.08	5523.956	62,375.09	T = 15
$X_3$	overall	0.4811088	0.0830462	0.3	0.6662526	N = 135
	between		0.0578442	0.3986667	0.5763305	n = 9
	within		0.0624523	0.3684421	0.5924421	T = 15
$X_4$	overall	0.4595013	0.0618296	0.33	0.62	N = 135
	between		0.0495491	0.3909102	0.5366667	n = 9
	within		0.0403025	0.3528346	0.5428346	T = 15
$X_5$	overall	22.59585	18.97095	2.4	88.56	N = 135
	between		19.46974	3.946667	56.83867	n = 9
	within		4.520003	3.697185	54.31719	T = 15
$X_6$	overall	57,135.34	54,754.15	7098	208,257	N = 135
	between		55,264.31	8163.844	172,783.7	n = 9
	within		16,216.13	−7046	100,457.3	T = 15
$X_7$	overall	20,005.76	29,063.94	79	146,481	N = 135
	between		23,018.8	887.4667	68,127	n = 9
	within		19,240.75	−37,378.2	98,359.76	T = 15

shows the descriptive statistics for the essential data attributes.

The sample observation is 135, as seen from the descriptive statistics in Table 2. Variables change with both persons and time demonstrated by the standard deviations of the variables. When comparing the within-group and between-group standard deviations for each variable, the within-group standard deviation for per capita GDP and the proportion of the urban population to the total population at the end of the year are more significant than the between-group standard deviation. We found that the two variables change dramatically over time. The remaining indicators are contrary. The original data for $Y$ and ($X_1\sim X_2,X_5\sim X_7$) take logarithms to mitigate heteroskedasticity, autocorrelation, and cross-sectional correlation, which leads to skewed estimates. After logarithmic processing, the data range is 1 to 10. To keep all data values in the same range, $X_3$and$X_4$are expanded by a factor of 10 to also range from 1 to 10.

The following section uses the Levin-Lin-Chu (LLC) test to perform a unit root test on the data, and the test results are shown in

Table 3. Unit root test results.

Variables		Statistics	P
$Y$	Adjust t *	−5.6423	0.0000
$X_1$	Adjust t *	−3.5988	0.0002
$X_2$	Adjust t *	−2.0491	0.0202
$X_3$	Adjust t *	−4.1176	0.0000
$X_4$	Adjust t *	−2.7843	0.0027
$X_5$	Adjust t *	−1.7622	0.0390
$X_6$	Adjust t *	−3.1458	0.0008
$X_7$	Adjust t *	−6.1683	0.0000

Adjust t* can ensure the results of statistics are unbiased.

. The test results show that the data form a smooth series.

The results of the F-test are shown in

Table 4. Model testing.

	Statistical Quantities	Critical Value Corresponding to 5% Significance Level	P
F(8, 126)	4.9619833	1.6445652	0.0000

.

The F-test rejects the mixed-effects model, indicating that a fixed-effects model should be adopted instead. The following is the model form:

$$\begin{array}{l}\ln Y{i}_t={\beta }_0+{\beta }_1\ln X_{1it}+{\beta }_2\ln X_{2it}+{\beta }_3{X}_{3it}+{\beta }_4{X}_{4it}+{\beta }_5\ln X_{5it}\\ +{\beta }_6\ln X_{6it}+{\beta }_7\ln X_{7it}+u_i+v_t+{\epsilon }_{it}\end{array}$$

(25)

where ${\beta }_1$ represents the coefficient of explanatory variable,$u_{\mathrm{i}}$represents the cross-sectional random error component, $v_{\mathrm{t}}$ represents the temporal random error component, and ${\epsilon }_{it}$ represents the random error component. The regression results are shown in

Table 5. Regression results.

Variable	Regression Coefficient	T	P
$\ln X_1$	0.2731058 (0.2778911)	0.98	0.328
$\ln X_2$	−0.6134181(1.166126)	−0.53	0.600
$X_3$	−1.840621 ** (0.7442428)	−2.52	0.018
$X_4$	0.6834359 ** (3.165576)	2.16	0.033
$\ln X_5$	−1.403508 *** (0.3040708)	−4.62	0.000
$\ln X_6$	0.6229956 * (0.3673876)	1.74	0.085
$\ln X_7$	−0.4754808 *(0.2891424)	−1.64	0.100

Standard errors are in parentheses. * p < 0.1; ** p < 0.05; *** p < 0.01.

.

The water consumption by industry, proportion of the urban population to the total population at the end of the year, value-added of secondary sectors as a proportion of regional GDP, volume of industrial wastewater discharge, and patent applications granted all influence investment in the treatment of industrial wastewater pollution at the 10% significance level, according to the regression results. PGDP and total water resources, on the other hand, are not statistically significant at the 10% level.

3.2.2. The Difference Coefficient for Pollution-Control Costs

The key factors impacting industrial water pollution-treatment costs varied between the two locations. We chose variables with 10% significance. Because each variable has a positive or negative impact on the price, the positive and negative differences are stated in Equation (26):

$$\mathrm{Positive} \mathrm{variables} \frac{\frac{1}{T}{\displaystyle \sum _{t=1}^T{X}_{i2}}}{\frac{1}{T}{\displaystyle \sum _{t=1}^T{X}_{i1}}}, \mathrm{Negative} \mathrm{variables} \frac{\frac{1}{T}{\displaystyle \sum _{t=1}^T{X}_{i1}}}{\frac{1}{T}{\displaystyle \sum _{t=1}^T{X}_{i2}}}$$

(26)

We use $k$ to represent the coefficient for the industrial water pollution treatment-cost difference between the two regions, where $k$ can be expressed as:

$$k={\displaystyle \sum _{i=3}^7\left|{\beta }_i\right|\frac{\frac{1}{T}{\displaystyle \sum _{t=1}^T{X}_{ij}}}{\frac{1}{T}{\displaystyle \sum _{t=1}^T{X}_{ij}}}},(j=1,2)$$

(27)

4. Results and Discussion

4.1. Comparison of Two Scenarios

According to Equations (8) and (17), we obtain

$$A^{\ast \ast }-A^{\ast }=\frac{m^{*}}{1-m^{*}}{A}^{\ast }>0$$

(28)

According to Equations (9) and (18), we obtain

$$B^{\ast \ast }-B^{\ast }=\frac{n^{*}}{1-n^{*}}{B}^{\ast }>0$$

(29)

According to Equations (10) and (19), we obtain

$$Z^{\ast \ast }-Z^{\ast }=-\frac{{\pi }_3\lambda +P}{c\left(\rho +\delta \right)}\times \left\{\left[\frac{\xi \left(d_1-\alpha d_2\right)}{1-{\alpha }^2}+\frac{\eta \left(d_2-\alpha d_1\right)}{1-{\alpha }^2}\right]\right\}<0$$

(30)

We can deduce from Equations (28)–(30) that in the Nash system, the regional governments of the two tributaries make lower pollution-control efforts than they do under the cost-sharing method. This finding suggests that the tributaries’ regional governments’ conscious environmental governance approaches are undesirable. In contrast, cost-sharing by the mainstream regional government with the tributaries’ regional governments can improve the two tributaries’ regional governments’ pollution-control efforts. The mainstream’s pollution-control effort in a cost-sharing system is less than that in the Nash noncooperative mechanism, suggesting that the cost-sharing model can help the government save money on pollution control. Due to the complexity of the equilibrium solution, the influence law of some parameters cannot be directly derived. Thus, we used data in Section 3.1.3 to analyze the influence law of some parameters. According to Section 3.2, the coefficient of difference in the cost of pollution control between two regions $k$ is calculated to be 5.08.

4.2. The Influence Law of Competition Coefficient $\alpha$

4.2.1. The Effect on Pollutants Treatment

Assume $\alpha$ = 0.3 and 0.8. The results are shown in Figure 4. The amounts of pollutants treated under the Nash and the cost-sharing mechanisms decrease as the competitiveness coefficient increases. Furthermore, the pollutant removal rate in the cost-sharing mechanism is lower than in the Nash mechanism.

4.2.2. The Effects on the Three Governments’ Pollution-Treatment Efforts

Figure 5 depicts the effects of the three governments’ pollution-control initiatives. As the competitiveness coefficient increases and drops more rapidly under the Nash and cost-sharing systems, the pollution remediation efforts of the Shanxi and Shaanxi regional governments decrease. This finding is because as the competitiveness coefficient increases, so does the pressure for economic development on the tributary governments. Tributaries’ provincial governments are less willing to invest in pollution-control strategies and capital. The Henan local government’s pollution treatment effort in the cost-sharing method is lower than in the Nash mechanism. This is because the Henan provincial government pays a share of the pollution treatment costs to the Shanxi and Shaanxi province governments under a cost-sharing mechanism to make pollution control easier for the two governments, and pollutants are eliminated in the tributary regions.

4.2.3. The Effects on the Three Governments’ Revenues

Figure 6 depicts the impacts of the competitiveness coefficient on the three governments’ revenues. As the competitiveness coefficient increases, tripartite government’s revenues will drop in both the Nash and cost-sharing systems, and the pressure on economic development grows for the two tributaries’ governments. As a result, the tributaries’ governments will cut their pollution-control activities, resulting in a reduction in pollution control. Even though pollution-control expenditures will fall, the environmental and economic benefits of pollution control will diminish, resulting in a decrease in revenues for the two tributaries’ governments due to the combined effect of the two factors. Shanxi province’s reduction is smaller than Shaanxi province’s because Shanxi province’s ecological sensitivity is greater than Shaanxi province’s. The amount of pollution treatment will be reduced to protect the local ecological environment, though the reduction will be minor in Shanxi province.

Henan province’s revenues decreases when the competition coefficient increases. On the one hand, Henan’s optimal cost-sharing ratio will increase. The mainstream’s social welfare advantages, on the other hand, will be decreased when pollution management in the two tributaries is reduced. The greater the development pressure on the two tributaries and the less effective implementation of the cost-sharing system, the higher the competitiveness coefficient.

4.2.4. The Effects on Optimal Cost-Sharing Ratio $m$ and $n$

From Figure 7, the optimal cost-sharing ratio $m$ and $n$ increases with $\alpha$. The massive pressure for economic development in Shanxi and Shaanxi provinces causes their governments to be less willing to spend policies, manpower, and cash on water pollution control as the competitiveness coefficient $\alpha$ rises. As a result, the Henan government will raise the cost-sharing ratio as an incentive for the two tributaries’ governments’ pollution-control measures.

The preceding analysis revealed the impacts of competitiveness coefficient $\alpha$ on the three governments, and the following section examines the effects on the three governments of the variables that produce cost disparities in industrial water pollution treatment.

4.3. The Influence Law of Urbanization Rate on Pollutants Treated and the Three Governments’ Revenues

4.3.1. The Effects on Pollutants Treated

Suppose $X_{31}$ = 0.32 and 0.72. Figure 8 depicts the amounts of pollutants treatment. As the urbanization rate in Shanxi province rises, the amounts of pollutants treated will drop under both the Nash noncooperative and cost-sharing processes. Because the urbanization rate in Shanxi province is increasing, the cost discrepancies between Shanxi and Shaanxi provinces are spreading even further. The expense of pollution control by the Shaanxi provincial government will be higher than that for Shanxi province. The provincial administration of Shaanxi will scale back its pollution-control measures.

Assume that $X_{32}$ = 0.38 and 0.78. Figure 9 depicts the results. As the urbanization rate in Shaanxi province rises, the amounts of pollutants treated will increase under both the Nash noncooperative and cost-sharing methods. The pollution-control cost for the Shaanxi provincial government decreases as the rate of urbanization increases. The local government in Shaanxi will step up its efforts to reduce pollution and improve the aquatic environment.

4.3.2. The Effects on the Three Governments’ Revenues

It is evident from Figure 10 that the urbanization rate in Shanxi province has a comparable impact on the three governments’ revenues. As the urbanization rate in Shanxi province increases, each government’s revenues will decrease under both the Nash noncooperative and cost-sharing mechanisms. As the urbanization rate in Shanxi province increases, the cost gap between the Shanxi and Shaanxi provincial governments widens even further. In comparison with Shanxi Province, the cost of pollution control in Shaanxi is higher. The provincial administration of Shaanxi will scale back its pollution-control measures. The Shanxi provincial government will need to devote more resources to environmental control to eradicate pollutants.

Pollution-control costs for the Henan provincial government will increase as the Shaanxi province discharges more pollutants. The basin’s overall environmental and social welfare consequently are harmed as the amounts of pollutants treated decreases. Under the cost-sharing mechanism, the three governments’ revenues decrease even more. The impacts on revenues for each government under the cost-sharing mechanism are more significant than those of the Nash noncooperative mechanism as the difference in the cost of pollution management between the two provinces widens.

As seen in Figure 11, the urbanization rate in Shaanxi province has a similar impact on all three governments’ revenues. As the urbanization rate in Shaanxi province increases, each government’s revenues will increase under both the Nash noncooperative and the cost-sharing mechanism. The cost of pollution control will decrease for the Shaanxi provincial government as the rate of urbanization increases. The provincial government of Shaanxi will enhance its pollution control efforts to reduce the amounts of pollutants and improve the basin’s overall socioeconomic welfare. All three governments will see an increase in revenues.

The Shanxi provincial government will be relieved of the pressure to manage pollution due to the Shaanxi provincial government’s intensified pollution-control measures. As a result, the Shanxi provincial government will be able to maintain its development while also increasing its earnings.

The enhanced efforts of the upstream Shaanxi provincial government to treat pollutants will minimize the cost of pollution control for the Henan provincial government. As a result, the provincial government of Henan will see an increase in revenues. The cost-sharing mechanism generates more revenues for the three governments when the urbanization rate in Shaanxi province increases than does the Nash noncooperative system.

4.4. The Influence Law of Industrialization on Pollutants Treated and Each Government’s Revenues

4.4.1. The Effects on Pollutants Treated

Assume that $X_{41}$ = 0.33 and 0.73. Figure 12 depicts the results. The amounts of pollutants treated will increase under the Nash noncooperative and cost-sharing mechanisms as industrialization in Shanxi province increases. The cost of pollution control for Shanxi’s provincial government will increase as industrialization increases. The cost disparity between the Shanxi and Shaanxi provincial governments, on the other hand, will decrease. As a result, when compared with Shanxi province, the cost of pollution control for the Shaanxi provincial government is decreasing. The provincial government of Shaanxi will step up its efforts to combat pollution. To summarize, both mechanisms will increase the amounts of pollutants treated.

Assume that $X_{42}$ = 0.29 and 0.69. Figure 13 depicts the results. As Shaanxi province becomes more industrialized, the amounts of pollutants treated under the Nash noncooperative and cost-sharing mechanisms will decrease. The cost of pollution control in Shaanxi province will increase as industrialization increases, so Shaanxi province will be reluctant to pay for more efforts to control water pollution. As a result, the amounts of pollutants treated decreases.

4.4.2. The Effects on the Three Governments’ Revenues

It is evident from Figure 14 that the Nash noncooperative mechanism will improve the revenues of each government, while the cost-sharing mechanism will increase industrialization in Shanxi province. The cost-sharing mechanism will increase revenues more than Nash’s noncooperative mechanism. As industrialization increases, the cost of pollution control for the Shanxi provincial government will increase, but the cost differential between the Shanxi and Shaanxi provincial governments will decrease. As a result, when compared with Shanxi province, the cost of pollution control for the Shaanxi provincial government decreases. The provincial government of Shaanxi will increase its efforts to reduce pollution, thereby improving the basin’s overall socioeconomic welfare. As a result, the revenues of the three governments will increase.

The pressure on the Shanxi provincial government to control pollution will be relieved due to the province’s strengthened pollution-control measures. Shanxi’s provincial government will be able to maintain its development while also increasing its revenues.

The cost of pollution control will be reduced for the Henan provincial government due to the upstream Shaanxi provincial government’s increased pollution-control efforts. The revenues is expected to increase. The revenues for each government increases more under the cost-sharing mechanism than under the Nash mechanism. Pollution elimination increases and offers more significant advantages to the basin because the Shanxi and Shaanxi provincial governments receive subsidies from Henan province for pollution treatment costs.

It is evident from Figure 15 that under the Nash noncooperative and cost-sharing mechanisms, when industrialization in Shaanxi province increases, the revenues of each government will decrease. The cost of pollution control for the Shaanxi province government will increase as its industrialization increases, so the Shaanxi provincial government will reduce its efforts to control pollution and transfer the pressure of water pollution control to the Shanxi provincial government. As a result, the provincial government of Shanxi will see a decrease in revenues. Under the cost-sharing mechanism, the Shanxi provincial government’s revenues decreases more slowly than under Nash’s noncooperative mechanism. Although the Shanxi provincial government is under pressure to treat pollution, it receive get subsidies from the Henan provincial government under the cost-sharing structure, allowing it to reduce its revenues gradually.

The revenues of the Henan provincial government will be reduced as a result of the Nash noncooperative and cost-sharing mechanisms. As the Shaanxi provincial government’s pollution-control costs increase, pollution-control efforts decrease, and pollutants treated increase, the Henan provincial government’s pollution-control costs increase.

The parameters $X_5$ and $X_7$, which have negative impacts on the cost of water pollution treatment, have a similar pattern of impact on pollutant-treatment efficacy and on each government’s revenues as $X_3$. The parameter $X_6$ has a similar pattern of effect on the effectiveness of pollutants treated and tripartite government’s revenues as $X_4$, which has a positive effect on the cost of water pollution treatment.

4.5. Discussion

We established differential game models for examining transboundary water pollution control and simulated the dynamic decision-making between a mainstream government and two tributaries’ governments on transboundary water pollution control under Nash noncooperative and cost-sharing scenarios using the case of China’s YRB.

The cost-sharing mechanism for transboundary water pollution control has been applied upstream and downstream with the goal of improving water quality and reducing pollution. Sheng and Webber [41] provided a differential game modeling approach to examine and compare the behavior of water pollution control in the middle route of China’s South-North Water Transfer Project. They verified the sewage stock and profit of upstream and downstream governments under the scenarios of a cost-sharing mechanism and a baseline mechanism. Our research extends upstream and downstream to mainstream and tributaries. Through comparison of governments’ revenues and sewage treatment efforts, we can see clearly that pollutants treated will continue to increase and eventually stabilize over time. The sewage treatment under cost sharing is significantly greater than that under Nash, the cost sharing can improve revenues for both the mainstream and tributary governments, and the increase in mainstream government revenues is most apparent. These regularities align with the conclusions of Sheng and Webber [41]. It can be seen that the method adopted in this article is correct and the results are reliable.

Gao [12,13] analyzed decision-making behavior related to water pollution control among upstream governments, downstream governments, and the central government on the eastern route of the South-to-North Water Transfer Project in China using evolutionary game theory. Based on the fluidity and long-term characteristics of water pollution in the river basin, the participants representing the interests of various regions needed to go through a dynamic process of interaction to achieve the Nash equilibrium. Compared with evolutionary games, differential game can describe the process of eliminating river basin water pollutant, express the dynamic process of each participant’s selecting strategy and achieve Nash equilibrium in a continuous time system. Therefore, this paper adopts the differential game method. Compared with Gao’s [12,13] results, we find that the ecological benefits of pollution control have impacts on government decision-making behavior.

Nevertheless, there are some limitations in this study. It is assumed that the degree of government effort in the two tributaries is a linear function of the competition coefficient for the convenience of the analysis, but further studies can consider the nonlinear relationships among them.

5. Conclusions

This paper extends upstream and downstream to the mainstream and tributaries for the cooperative control problem of transboundary water pollution. The choice of pollution control strategies for each region is examined under two scenarios: the Nash noncooperative mechanism and a cost-sharing mechanism, using a differential game model that considers the competitive connection between the two tributaries and regional variances. The following are the outcomes:

(1) A cost-sharing mechanism based on cooperative pollution control can result in Pareto improvements in the environment and in economies, with better pollutant treatment and benefits for the tributary and mainstream governments. So, a mechanism can be designed for cost-sharing water pollution control between a mainstream government and multiple tributary governments.

(2) As the competitiveness coefficient increases, the amounts of pollutants treated decreases, and the Nash noncooperative mechanism’s decrease rate is faster than the cost-sharing mechanism’s. At the same time, each government’s revenues decrease. The cost-sharing mechanism is less effective as the competitiveness coefficient increases. So, first, governments should strengthen the dual roles of pollution control and economic development and avoid one-sided emphasis on GDP-style economic development. Governments should take the harmonious unification of economic growth and green development as the starting point. Second, governments should encourage and attract private capital and foreign investment into the field of environmental protection to establish a diversified, multichannel, multilevel funding system. Lastly, the government should implement a reward and punishment mechanism to motivate enterprises to innovate on their own, so that technological advances can drive them to invest in sewage treatment.

(3) The volume of industrial wastewater discharge, the proportion of the urban population to the total population at the end of the year, the value-added of secondary industries as a proportion of regional GDP, and the number of patent applications granted are the main factors that influenced investment in treating industrial wastewater pollution.

(4) As the indicators of urbanization rate, industrial water use, and Shaanxi provincial government patent applications increase, the amounts of pollutants treated increase, and each government’s revenues increase. The coefficient of variation of pollution control costs increases when urbanization rate, industrial water use, and patent applications granted by the Shanxi provincial government increase, indicating a higher cost of pollution control in Shaanxi province. The amounts of pollutants treated and each government’s revenues decrease. The relationship between the amount of industrialization and the volume of industrial wastewater outflow is inverse. The major cost factors for pollution control should be controlled. Simultaneously, differentiated actions to minimize the cost difference should be considered.

The above findings explain the game situation using the YRB’s mainstream government’s and two tributaries’ governments’ efforts at water pollution control. It can be used as a guide for governments to determine their water pollution-control behavioral strategies and negotiate the establishment of cost-sharing mechanisms. Water is indispensable to people, so it is very necessary to control water pollution. Clean water can improve biodiversity and sustain ecological systems. This paper can be seen as a path for the delivery of the Green New Deal and has great significance for exploring the sustainability of natural resources and the environment.