# Cost-Sharing Mechanism of Water Pollution Control in Main and Subbasins Based on Stackelberg Game Model

Authors: Yaohong Yang; Ying Liu; Jing Dai; Yi Zeng
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Faced with goal to improve ecological environment, water pollution treatment in river basins has been one of research focuses in recent years. Based on the basic characteristics of the river basins including main stream and multiple or multilevel tributaries, considering competitive relationship and cost-sharing mechanism of water pollution control, a differential game model of pollution elimination including main stream government and two tributary governments was constructed, and the equilibrium strategies under two situations of noncooperation mechanism and cost-sharing mechanism were discussed. The results show that Stackelberg game model based on cost-sharing mechanism can improve the pollution elimination effect and achieve Pareto improvement of the environment and economy. Then, through numerical analysis, influence of factors such as competition coefficient on pollutant elimination effect, tripartite effort, and optimal cost-sharing ratio were discussed. The results show that, in order to effectively reduce pollutant and improve environmental benefits, the greater pressure on tributary governments to develop the economy and the smaller degree of pollution treatment efforts, the more cost-sharing mechanism needed, and the higher main stream government’s cost-sharing proportion to the tributary government. When the pollution eliminating-quantity on unit effort of one tributary government increases, the revenue of main stream and tributary governments will increase; simultaneously, when the effort degree of the tributary government increases, the other decreases. The optimal cost-sharing proportions of main stream government to two tributaries are different which is influenced by many factors, and the influence rule presents stable monotonic increase or decrease. The research conclusions can provide reference for the governments to negotiate and determine cost-sharing mechanism.


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## Body

## 1. Introduction

With the rapid progress of industrialization and urbanization, China has made great progress in economic and social development and other aspects, while facing increasingly serious water pollution problems. According to the “2019 China Ecological Environment Bulletin,” among the seven major basins in China, the Yellow River, the Songhua River, the Huaihe River, the Haihe River Basin, and the Liao River Basin are still lightly polluted in large areas and moderately polluted in some areas, which shows that the water pollution problem characterized by transboundary mobility is still serious [1]. The 14th Five-Year Plan, adopted on October 29, 2020, clearly proposes to achieve new progress in ecological civilization construction, continuous reduction of total emissions of major pollutants, and continuous improvement of ecological environment [2, 3]. There are widespread conflicts of transboundary pollution between upstream and downstream of the river basin or between main and tributary streams, and the “free rider” behavior in transboundary pollution control can easily curb the government’s motivation to control pollution [4]. So it is of great practical significance to study how to build a coordinated governance pattern of unified governance of upstream and downstream and joint governance of main and tributary streams to effectively reduce pollutant emissions [5].Based on the fluidity and long-term characteristics of water pollution in river basin, the participants representing the interests of various regions need to go through a dynamic process of interaction to achieve the Nash equilibrium. In terms of the research method, differential game can describe the elimination process of river basin water pollutants and express the dynamic process of each participant selecting strategy and achieving Nash equilibrium in a continuous time system [6, 7]. From the review of the existing literature, it can be seen that many scholars conduct dynamic analysis of river basin water pollution control based on the differential game theory. In terms of pollution control mechanisms, there are more results on ecological compensation mechanisms, and there is a relative lack of literature on cost-sharing mechanism [8, 9]. Meanwhile, in the analytical framework, mostly noncooperative and cooperative game problems between upstream and downstream are considered, and some studies consider the inclusion of central government and do analysis based on optimal control theory [10]. In fact, the river basin is a complex system consisting of a main stream and multiple, multilevel tributaries which interact with each other, and water pollution management in river basin is a system engineering that requires collaborative governance of upstream and downstream, main stream, and tributaries. The existing research results are mostly for upstream and downstream, and these results can be simply extended and applied between main stream and a tributary, but the synergistic governance of the main stream and multiple tributaries is yet to be studied [11–13].In this paper, we study the problem of synergistic governance of water pollution between two tributaries and main stream based on cost-sharing mechanism and differential game approach. Compared with the existing literature, the main contributions of this paper are as follows: (i) The research framework including not only the pollution governance competition between upstream and downstream (or called between the main stream and tributary) but also the pollution governance competition between two tributaries is considered, which is more complex and more in line with the actual situation of the basin; (ii) the cooperative mechanism and noncooperative mechanism among the three parties of the main stream and the two tributaries as well as the influence laws of various factors such as competition coefficients on the equilibrium results of the game are analyzed [14, 15].


## 2. Materials and Methods

### 2.1. Problem Description and Hypothesis

To eliminate pollutants and improve the water environment in river basin, both the main stream and the tributary governments need to carry out water pollution treatment at the same time [16]. To achieve the overall goal of pollutant elimination, the competition between main stream and tributary is formed, and the pollution treatment of the tributaries regions will also bring environmental benefits to the main stream region [17]. At the same time, due to the pressure of economic development, the competition between the two tributaries is also formed; if one tributary treats more pollution, the other tributary will treat less pollution for its own economic development. In this study, we focus on the impact of tributary pollution control on main stream and the competition between them, so we only consider the pollutants generated by the two tributaries and do not consider the impact of the pollutants generated by the main stream on itself [18].Assumption 1.
The main stream is fed by two tributaries, denoted as tributary 1 and tributary 2, respectively. Under the requirement of achieving high-quality and sustainable economic and social development, the water quality of the main stream is required to be stable to good, so both the two tributary governments and the main stream government will treat the water environment. Assuming that the degrees of tributary 1 and tributary 2 governments’ pollution treatment efforts areA and B, respectively, and 0<A<1<0<B<1, it represents the policies, personnel, and capital investment, etc. input by the two tributary governments for water pollution treatment.Assumption 2.
Based on the study of Li, we assume that the cost of pollution control by tributary 1 and tributary 2 governments is in the form of quadratic functionsc1/2A2 and c2/2B2, respectively, and we use c1 and c2 to denote the cost coefficients of pollution control.Assumption 3.
Since both tributaries are unwilling to input policies and funds in pollution treatment because of their own economic development, the relationship of water pollution treatment of the two tributaries is competitive for the requirements of pollutant elimination target. Referring to many scholars’ studies, the competitive coefficientα is set, where 0<α<1, to measure the economic development pressure of the two tributaries’ governments.Assumption 4.
The pollution control effort of the main stream government is expressed byZ, where 0<Z<1, which represents the water quality testing, policy making, personnel, and capital investment of the main stream pollution control. The effort cost of the main stream is c3/2Z2; the parameter c3>0 represents the effort cost coefficient of the main stream government. The central government levies environmental protection tax on water pollution emissions in the two tributaries, and a indicates the proportion of subsidies to the main stream government after the central government levies environmental protection tax, where 0≤a≤1.Assumption 5.
Pollutant emission reduction in main stream is the result of pollution treatment by the three governments in the main stream and the tributaries, and it is a dynamic change process with time. The pollutant elimination in the basin can be expressed by the following dynamic differential equation:(1)q˙t=ξA+ηB+θZ−δqt,where qt denotes the pollutant reduction in the two tributaries at time t. Assuming that the initial state of the system is q0≥0, the parameters ξ>0, η>0, and θ>0 denote the pollution elimination of tributary 1, tributary 2, and the main stream paying unit pollution control efforts, and the parameter δ denotes the attenuation coefficient of the reduction due to factors such as aging of wastewater treatment equipment.Assumption 6.
The pollution treatment effort of tributary 1 government is influenced by the competition coefficientα and also influenced by the supervision of the main stream government. Assume that the effort of tributary 1 government is a linear function of the competition of tributary 2 government and the supervision of the main stream government, i.e., A=A0−αB+d1Z; similarly, the effort of tributary 2 government is B=B0−αA+d2Z, A0≥0, B0≥0, representing tributary 1 and tributary 2 initial pollution control efforts when weighing their own benefits and costs, without considering the competition factor and the supervisory role of the main stream and where the parameters d1, d2 are parameters of main stream’s influence on the efforts of tributary 1 and tributary 2, respectively.Assumption 7.
The social welfare effects of water pollution treatment and reduction of pollutant discharges in the two tributaries and the main stream.(2)St=S0+λqt,where S0>0 denotes the initial welfare status of the basin, and λ>0 denotes the coefficient of the effect of basin emission reduction on the social welfare effect of the basin.Assumption 8.
The parameters of basin social welfare effect on tributary 1, tributary 2, and main stream government benefits areπ1, π2, and π3, respectively, π1>0, π2>0, and π3>0, and the three parties have the same discount ρ, with ρ>0.Assumption 9.
The marginal benefits of tributary pollution control for the main stream are greater than the subsidies allocated by the central government to the main stream government for pollution control.
In order to better study the cost-sharing mechanism of cooperative water pollution treatment in the main stream and tributary governments of the basin, the following is divided into two cases for comparison and analysis: Nash noncooperative decision when there is no cost-sharing, and Stackelberg master-slave decision when there is cost-sharing, hereinafter referred to as Nash and cost-sharing.


### 2.2. Model Building and Solving

#### 2.2.1. Nash Noncooperative Game Scenario

The game situation is that central government sets targets for water quality improvement or pollutant elimination in the mainstream, and both the main stream and tributary governments make their own decisions about pollution treatment efforts. The two tributary governments pay environmental protection tax based on the pollutants discharged, and the central government subsidizes the pollution treatment in the main stream [19, 20]. Based on the above assumptions, and considering that tributary 1, tributary 2, and main stream governments are all finite rational subjects, the goal of all three parties is to seek the optimal pollution control effort strategy that maximizes their own interests. According to the general assumption that the benefit functions achieved by the three parties through their respective efforts are related to the degree of effort, the revenue functions of the three parties can be obtained as follows:The revenue function of tributary 1 is(3)∫0tJ1+r1A−c12A2+π1St−q1−ξAω+h1ηBe−ρtdt.The total amount of discharge by tributary 1 isq1, q1>0. J1 is the benefit obtained from the discharge by tributary 1. To reduce pollutants, the tributary 1 government inputs corresponding policies, personnel, and capital, improves the local industrial structure, and introduces new green industries; r1 is the social benefit brought to the region by the pollution control efforts of the tributary 1 government. q1−ξA is the total pollutants discharged by tributary 1 to the main stream, ω is the environmental protection tax paid by the discharging units of taxable water pollutants of the two tributaries, and h1 represents the benefit due to pollution control of tributary 2 to relieve pollution control pressure of tributary 1, h1<c1.The revenue function for tributary 2 is(4)∫0tJ2+r2B−c22B2+π2St−q2−ηBω+h2ξAe−ρttdt.The total amount of discharge by tributary 2 isq2, q2>0, J2 is the benefit obtained from the discharge of tributary 2, r2 is the social benefit brought to the region by the pollution control efforts of the tributary 2 government, q2−ηB is the total amount of pollutants discharged by tributary 2 to the main stream, and h2 represents the benefit due to the treatment of tributary 1 to relieve the pollution control pressure of tributary 2, h2<c2.The revenue function of main stream is(5)∫0tr3Z−c32Z2+π3St+q1−ξAωa+q2−ηBωae−ρtdt,where r3 is social benefits for the region from pollution control efforts of the main stream government.To achieve a unique continuous solutionqt for equation (1), a set of bounded, continuous, and differentiable value functions V1q, V2q, and V3q need to be constructed, so that the Hamilton-Jacobi-Bellman-Fleming (HJB) for tributary 1 government, tributary 2 government, and main stream government is constructed as(6)ρV1q=maxAJ1+r1A−c12A2+π1St−q1−ξAω+h1ηB+V1′qξA+ηB−δqt,(7)ρV2q=maxBJ2+r2B−c22B2+π2St−q2−ηBω+h2ξA+V2′qξA+ηB−δqt,(8)ρV3q=maxZr3Z−c32Z2+π3St+q1−ξAωa+q2−ηBωa+V3′qξA+ηB−δqt.By the maximization first-order conditions of combining (6) and (7) and (8) for A, B, and Z, we get(9)A=r1+ξω+V1′qξ−ηα−h1ηαc1,(10)B=r2+ηω+V2′qη−ξα−h2ξαc2,(11)Z=1c3×r3+V3′q−ωaξd1−αd21−α2+ηd2−αd11−α2+V3′qθ.Substituting equations (9)–(11) into the HJB equation and assuming that the expression of the function Viq is in linear form, the game equilibrium solution can be obtained as follows:(12)A∗=r1+ξω+π1λ/ρ+δξ−ηα−h1ηαc1,(13)B∗=r2+ηω+π2λ/ρ+δη−ξα−h2ξαc2,(14)Z∗=1c3×r3+π3λρ+δ−ωaξd1−αd21−α2+ηd2−αd11−α2+π3λρ+δθ.And we get the optimal benefits of the tripartite governments under this equilibrium condition are(15)V1∗=π1λρ+δq∗+1ρJ1+r1−c12A∗+ξω+π1λρ+δξA∗+π1S0−q1ω+π1λρ+δηB∗+θZ∗+h1ηB∗,V2∗=π2λρ+δq∗+1ρJ2+r2−c22B∗+ηω+π2λρ+δηB∗+π2S0−q2ω+π2λρ+δξA∗+θZ∗+h2ξA∗,V3∗=π3λρ+δq∗+1ρr3−c32Z∗+π3λρ+δθZ∗+π3S0+q1−ξA∗ωa+q2−ηB∗ωa+π3λρ+δξA∗+ηB∗.


#### 2.2.2. The Stackelberg Game with Cost-Sharing in Main Stream and Tributaries

In this game situation, main stream government shares a certain proportion of the cost of pollution control with tributary governments, and we assume main stream will share cost-sharing ratiosm and n to tributary 1 government and tributary 2 government [21]. Decision sequence is that main stream government determines the pollution control efforts and cost-sharing ratio; then tributary governments determine their own optimal strategies. Using the inverse induction method, the optimal decision of tributary 1 and tributary 2 is first sought, and its objective function satisfies the HJB equation:(16)ρV1q=maxAJ1+r1A−1−mc12A2+π1St−q1−ξAω+h1ηB+V1′qξA+ηB−δqt,(17)ρV2q=maxBJ2+r2B−1−nc22B2+π2St−q2−ηBω+h2ξA+V2′qξA+ηB−δqt,where m and n are cost-sharing ratios of main stream government to tributary 1 and tributary 2 governments, respectively.By the maximization first-order condition forA and B in equations (18) and (19), we can get the results which are shown in equations (20) and (21).(18)A=r1+ξω+V1′qξ−ηα−h1ηαc11−m,(19)B=r2+ηω+V2′qη−ξα−h2ξαc21−n.Considering that the two tributaries can develop strategies based on (18) and (19), the HJB equation of the mainstream government is(20)ρV3q=maxZr3Z−c32Z2+π3St+q1−ξAωa+q2−ηBωa−c12mA2−c22nB2+V3′qξA+ηB−δqt.Substituting (18) and (19) into equation (22) and by the maximization first-order conditions for Z, m, and n , we get(21)Z=r3+V3′qθc3,m=−ωξ2a+1+2V3′qξ−r1−V1′qξ−ηα+h1ηα−ωξ2a−1+2V3′qξ+r1+V1′qξ−ηα−h1ηα,n=−ωη2a+1+2V3′qη−r2−V2′qη−ξα+h2ξα−ωη2a−1+2V3′qη+r2+V2′qη−ξα−h2ξα.Again, we assume that the expression of the functionViq is in linear form; the Stackelberg equilibrium solution for the tripartite governments under cost-sharing mechanism is obtained as(22)A∗∗=r1+ξω+π1λ/ρ+δξ−ηα−h1ηαc11−m∗,(23)B∗∗=r2+ηω+π2λ/ρ+δη−ξα−h2ξαc21−n∗,(24)Z∗∗=r3+π3λ/ρ+δθc3,(25)m=−ωξ2a+1+2π3λ/ρ+δξ−r1−π1λ/ρ+δξ−ηα+h1ηα−ωξ2a−1+2π3λ/ρ+δξ+r1+π1λ/ρ+δξ−ηα−h1ηα,(26)n=−ωη2a+1+2π3λ/ρ+δη−r2−π2λ/ρ+δη−ξα+h2ξα−ωη2a−1+2π3λ/ρ+δη+r2+π2λ/ρ+δη−ξα−h2ξα.The expression for the optimal revenue of the three parties in this game condition is(27)V1∗∗=π1λρ+δq∗∗+1ρJ1+r1−c121−m∗A∗∗+ξω+π1λρ+δξA∗∗+π1S0−q1ω+π1λρ+δηB∗∗+θZ∗∗+h1ηB∗∗,V2∗∗=π2λρ+δq∗∗+1ρJ2+r2−c221−n∗+ηω+π2λρ+δηB∗∗+π2S0−q2ω+π2λρ+δξA∗∗+θZ∗∗+h2ξA∗∗,V3∗∗=π3λρ+δq∗∗+1ρr3−c32Z∗∗+π3λρ+δθZ∗∗+π3S0+q1+q2ωa+−ξωa−c12m∗A∗∗+π3λρ+δξA∗∗+−ηωa−c22n∗B∗∗+π3λρ+δηB∗∗.


#### 2.2.3. Comparative Analysis of Equilibrium Results

According to the equilibrium solutions of the game for main stream and tributary governments in the two cases of Nash noncooperative game and Stackelberg game with cost-sharing, we can obtain the following.Corollary 1.
From equations (12) and (13) and (26) and (27), we can see that(28)A∗∗−A∗=m1−mA∗>0,B∗∗−B∗=n1−nB∗>0.In Nash mechanism, the pollution control efforts of two tributaries are lower than cost-sharing mechanism, indicating that the conscious environmental governance model of the tributary governments is not desirable. The cost-sharing of the mainstream government to the tributary governments can improve the pollution control efforts of the two tributary governments.Corollary 2.
From equations (14) and (28), based on hypothesis (9), it is known that(29)Z∗∗−Z∗=−1c3×π3λρ+δ−ωaξd1−αd21−α2+ηd2−αd11−α2<0.In cost-sharing mechanism, the pollution control effort of the main stream is lower than that of Nash noncooperative mechanism, indicating that the cost-sharing mechanism can reduce the pollution control cost of the mainstream.


## 2.1. Problem Description and Hypothesis

To eliminate pollutants and improve the water environment in river basin, both the main stream and the tributary governments need to carry out water pollution treatment at the same time [16]. To achieve the overall goal of pollutant elimination, the competition between main stream and tributary is formed, and the pollution treatment of the tributaries regions will also bring environmental benefits to the main stream region [17]. At the same time, due to the pressure of economic development, the competition between the two tributaries is also formed; if one tributary treats more pollution, the other tributary will treat less pollution for its own economic development. In this study, we focus on the impact of tributary pollution control on main stream and the competition between them, so we only consider the pollutants generated by the two tributaries and do not consider the impact of the pollutants generated by the main stream on itself [18].Assumption 1.
The main stream is fed by two tributaries, denoted as tributary 1 and tributary 2, respectively. Under the requirement of achieving high-quality and sustainable economic and social development, the water quality of the main stream is required to be stable to good, so both the two tributary governments and the main stream government will treat the water environment. Assuming that the degrees of tributary 1 and tributary 2 governments’ pollution treatment efforts areA and B, respectively, and 0<A<1<0<B<1, it represents the policies, personnel, and capital investment, etc. input by the two tributary governments for water pollution treatment.Assumption 2.
Based on the study of Li, we assume that the cost of pollution control by tributary 1 and tributary 2 governments is in the form of quadratic functionsc1/2A2 and c2/2B2, respectively, and we use c1 and c2 to denote the cost coefficients of pollution control.Assumption 3.
Since both tributaries are unwilling to input policies and funds in pollution treatment because of their own economic development, the relationship of water pollution treatment of the two tributaries is competitive for the requirements of pollutant elimination target. Referring to many scholars’ studies, the competitive coefficientα is set, where 0<α<1, to measure the economic development pressure of the two tributaries’ governments.Assumption 4.
The pollution control effort of the main stream government is expressed byZ, where 0<Z<1, which represents the water quality testing, policy making, personnel, and capital investment of the main stream pollution control. The effort cost of the main stream is c3/2Z2; the parameter c3>0 represents the effort cost coefficient of the main stream government. The central government levies environmental protection tax on water pollution emissions in the two tributaries, and a indicates the proportion of subsidies to the main stream government after the central government levies environmental protection tax, where 0≤a≤1.Assumption 5.
Pollutant emission reduction in main stream is the result of pollution treatment by the three governments in the main stream and the tributaries, and it is a dynamic change process with time. The pollutant elimination in the basin can be expressed by the following dynamic differential equation:(1)q˙t=ξA+ηB+θZ−δqt,where qt denotes the pollutant reduction in the two tributaries at time t. Assuming that the initial state of the system is q0≥0, the parameters ξ>0, η>0, and θ>0 denote the pollution elimination of tributary 1, tributary 2, and the main stream paying unit pollution control efforts, and the parameter δ denotes the attenuation coefficient of the reduction due to factors such as aging of wastewater treatment equipment.Assumption 6.
The pollution treatment effort of tributary 1 government is influenced by the competition coefficientα and also influenced by the supervision of the main stream government. Assume that the effort of tributary 1 government is a linear function of the competition of tributary 2 government and the supervision of the main stream government, i.e., A=A0−αB+d1Z; similarly, the effort of tributary 2 government is B=B0−αA+d2Z, A0≥0, B0≥0, representing tributary 1 and tributary 2 initial pollution control efforts when weighing their own benefits and costs, without considering the competition factor and the supervisory role of the main stream and where the parameters d1, d2 are parameters of main stream’s influence on the efforts of tributary 1 and tributary 2, respectively.Assumption 7.
The social welfare effects of water pollution treatment and reduction of pollutant discharges in the two tributaries and the main stream.(2)St=S0+λqt,where S0>0 denotes the initial welfare status of the basin, and λ>0 denotes the coefficient of the effect of basin emission reduction on the social welfare effect of the basin.Assumption 8.
The parameters of basin social welfare effect on tributary 1, tributary 2, and main stream government benefits areπ1, π2, and π3, respectively, π1>0, π2>0, and π3>0, and the three parties have the same discount ρ, with ρ>0.Assumption 9.
The marginal benefits of tributary pollution control for the main stream are greater than the subsidies allocated by the central government to the main stream government for pollution control.
In order to better study the cost-sharing mechanism of cooperative water pollution treatment in the main stream and tributary governments of the basin, the following is divided into two cases for comparison and analysis: Nash noncooperative decision when there is no cost-sharing, and Stackelberg master-slave decision when there is cost-sharing, hereinafter referred to as Nash and cost-sharing.


## 2.2. Model Building and Solving

### 2.2.1. Nash Noncooperative Game Scenario

The game situation is that central government sets targets for water quality improvement or pollutant elimination in the mainstream, and both the main stream and tributary governments make their own decisions about pollution treatment efforts. The two tributary governments pay environmental protection tax based on the pollutants discharged, and the central government subsidizes the pollution treatment in the main stream [19, 20]. Based on the above assumptions, and considering that tributary 1, tributary 2, and main stream governments are all finite rational subjects, the goal of all three parties is to seek the optimal pollution control effort strategy that maximizes their own interests. According to the general assumption that the benefit functions achieved by the three parties through their respective efforts are related to the degree of effort, the revenue functions of the three parties can be obtained as follows:The revenue function of tributary 1 is(3)∫0tJ1+r1A−c12A2+π1St−q1−ξAω+h1ηBe−ρtdt.The total amount of discharge by tributary 1 isq1, q1>0. J1 is the benefit obtained from the discharge by tributary 1. To reduce pollutants, the tributary 1 government inputs corresponding policies, personnel, and capital, improves the local industrial structure, and introduces new green industries; r1 is the social benefit brought to the region by the pollution control efforts of the tributary 1 government. q1−ξA is the total pollutants discharged by tributary 1 to the main stream, ω is the environmental protection tax paid by the discharging units of taxable water pollutants of the two tributaries, and h1 represents the benefit due to pollution control of tributary 2 to relieve pollution control pressure of tributary 1, h1<c1.The revenue function for tributary 2 is(4)∫0tJ2+r2B−c22B2+π2St−q2−ηBω+h2ξAe−ρttdt.The total amount of discharge by tributary 2 isq2, q2>0, J2 is the benefit obtained from the discharge of tributary 2, r2 is the social benefit brought to the region by the pollution control efforts of the tributary 2 government, q2−ηB is the total amount of pollutants discharged by tributary 2 to the main stream, and h2 represents the benefit due to the treatment of tributary 1 to relieve the pollution control pressure of tributary 2, h2<c2.The revenue function of main stream is(5)∫0tr3Z−c32Z2+π3St+q1−ξAωa+q2−ηBωae−ρtdt,where r3 is social benefits for the region from pollution control efforts of the main stream government.To achieve a unique continuous solutionqt for equation (1), a set of bounded, continuous, and differentiable value functions V1q, V2q, and V3q need to be constructed, so that the Hamilton-Jacobi-Bellman-Fleming (HJB) for tributary 1 government, tributary 2 government, and main stream government is constructed as(6)ρV1q=maxAJ1+r1A−c12A2+π1St−q1−ξAω+h1ηB+V1′qξA+ηB−δqt,(7)ρV2q=maxBJ2+r2B−c22B2+π2St−q2−ηBω+h2ξA+V2′qξA+ηB−δqt,(8)ρV3q=maxZr3Z−c32Z2+π3St+q1−ξAωa+q2−ηBωa+V3′qξA+ηB−δqt.By the maximization first-order conditions of combining (6) and (7) and (8) for A, B, and Z, we get(9)A=r1+ξω+V1′qξ−ηα−h1ηαc1,(10)B=r2+ηω+V2′qη−ξα−h2ξαc2,(11)Z=1c3×r3+V3′q−ωaξd1−αd21−α2+ηd2−αd11−α2+V3′qθ.Substituting equations (9)–(11) into the HJB equation and assuming that the expression of the function Viq is in linear form, the game equilibrium solution can be obtained as follows:(12)A∗=r1+ξω+π1λ/ρ+δξ−ηα−h1ηαc1,(13)B∗=r2+ηω+π2λ/ρ+δη−ξα−h2ξαc2,(14)Z∗=1c3×r3+π3λρ+δ−ωaξd1−αd21−α2+ηd2−αd11−α2+π3λρ+δθ.And we get the optimal benefits of the tripartite governments under this equilibrium condition are(15)V1∗=π1λρ+δq∗+1ρJ1+r1−c12A∗+ξω+π1λρ+δξA∗+π1S0−q1ω+π1λρ+δηB∗+θZ∗+h1ηB∗,V2∗=π2λρ+δq∗+1ρJ2+r2−c22B∗+ηω+π2λρ+δηB∗+π2S0−q2ω+π2λρ+δξA∗+θZ∗+h2ξA∗,V3∗=π3λρ+δq∗+1ρr3−c32Z∗+π3λρ+δθZ∗+π3S0+q1−ξA∗ωa+q2−ηB∗ωa+π3λρ+δξA∗+ηB∗.


### 2.2.2. The Stackelberg Game with Cost-Sharing in Main Stream and Tributaries

In this game situation, main stream government shares a certain proportion of the cost of pollution control with tributary governments, and we assume main stream will share cost-sharing ratiosm and n to tributary 1 government and tributary 2 government [21]. Decision sequence is that main stream government determines the pollution control efforts and cost-sharing ratio; then tributary governments determine their own optimal strategies. Using the inverse induction method, the optimal decision of tributary 1 and tributary 2 is first sought, and its objective function satisfies the HJB equation:(16)ρV1q=maxAJ1+r1A−1−mc12A2+π1St−q1−ξAω+h1ηB+V1′qξA+ηB−δqt,(17)ρV2q=maxBJ2+r2B−1−nc22B2+π2St−q2−ηBω+h2ξA+V2′qξA+ηB−δqt,where m and n are cost-sharing ratios of main stream government to tributary 1 and tributary 2 governments, respectively.By the maximization first-order condition forA and B in equations (18) and (19), we can get the results which are shown in equations (20) and (21).(18)A=r1+ξω+V1′qξ−ηα−h1ηαc11−m,(19)B=r2+ηω+V2′qη−ξα−h2ξαc21−n.Considering that the two tributaries can develop strategies based on (18) and (19), the HJB equation of the mainstream government is(20)ρV3q=maxZr3Z−c32Z2+π3St+q1−ξAωa+q2−ηBωa−c12mA2−c22nB2+V3′qξA+ηB−δqt.Substituting (18) and (19) into equation (22) and by the maximization first-order conditions for Z, m, and n , we get(21)Z=r3+V3′qθc3,m=−ωξ2a+1+2V3′qξ−r1−V1′qξ−ηα+h1ηα−ωξ2a−1+2V3′qξ+r1+V1′qξ−ηα−h1ηα,n=−ωη2a+1+2V3′qη−r2−V2′qη−ξα+h2ξα−ωη2a−1+2V3′qη+r2+V2′qη−ξα−h2ξα.Again, we assume that the expression of the functionViq is in linear form; the Stackelberg equilibrium solution for the tripartite governments under cost-sharing mechanism is obtained as(22)A∗∗=r1+ξω+π1λ/ρ+δξ−ηα−h1ηαc11−m∗,(23)B∗∗=r2+ηω+π2λ/ρ+δη−ξα−h2ξαc21−n∗,(24)Z∗∗=r3+π3λ/ρ+δθc3,(25)m=−ωξ2a+1+2π3λ/ρ+δξ−r1−π1λ/ρ+δξ−ηα+h1ηα−ωξ2a−1+2π3λ/ρ+δξ+r1+π1λ/ρ+δξ−ηα−h1ηα,(26)n=−ωη2a+1+2π3λ/ρ+δη−r2−π2λ/ρ+δη−ξα+h2ξα−ωη2a−1+2π3λ/ρ+δη+r2+π2λ/ρ+δη−ξα−h2ξα.The expression for the optimal revenue of the three parties in this game condition is(27)V1∗∗=π1λρ+δq∗∗+1ρJ1+r1−c121−m∗A∗∗+ξω+π1λρ+δξA∗∗+π1S0−q1ω+π1λρ+δηB∗∗+θZ∗∗+h1ηB∗∗,V2∗∗=π2λρ+δq∗∗+1ρJ2+r2−c221−n∗+ηω+π2λρ+δηB∗∗+π2S0−q2ω+π2λρ+δξA∗∗+θZ∗∗+h2ξA∗∗,V3∗∗=π3λρ+δq∗∗+1ρr3−c32Z∗∗+π3λρ+δθZ∗∗+π3S0+q1+q2ωa+−ξωa−c12m∗A∗∗+π3λρ+δξA∗∗+−ηωa−c22n∗B∗∗+π3λρ+δηB∗∗.


### 2.2.3. Comparative Analysis of Equilibrium Results

According to the equilibrium solutions of the game for main stream and tributary governments in the two cases of Nash noncooperative game and Stackelberg game with cost-sharing, we can obtain the following.Corollary 1.
From equations (12) and (13) and (26) and (27), we can see that(28)A∗∗−A∗=m1−mA∗>0,B∗∗−B∗=n1−nB∗>0.In Nash mechanism, the pollution control efforts of two tributaries are lower than cost-sharing mechanism, indicating that the conscious environmental governance model of the tributary governments is not desirable. The cost-sharing of the mainstream government to the tributary governments can improve the pollution control efforts of the two tributary governments.Corollary 2.
From equations (14) and (28), based on hypothesis (9), it is known that(29)Z∗∗−Z∗=−1c3×π3λρ+δ−ωaξd1−αd21−α2+ηd2−αd11−α2<0.In cost-sharing mechanism, the pollution control effort of the main stream is lower than that of Nash noncooperative mechanism, indicating that the cost-sharing mechanism can reduce the pollution control cost of the mainstream.


## 2.2.1. Nash Noncooperative Game Scenario

The game situation is that central government sets targets for water quality improvement or pollutant elimination in the mainstream, and both the main stream and tributary governments make their own decisions about pollution treatment efforts. The two tributary governments pay environmental protection tax based on the pollutants discharged, and the central government subsidizes the pollution treatment in the main stream [19, 20]. Based on the above assumptions, and considering that tributary 1, tributary 2, and main stream governments are all finite rational subjects, the goal of all three parties is to seek the optimal pollution control effort strategy that maximizes their own interests. According to the general assumption that the benefit functions achieved by the three parties through their respective efforts are related to the degree of effort, the revenue functions of the three parties can be obtained as follows:The revenue function of tributary 1 is(3)∫0tJ1+r1A−c12A2+π1St−q1−ξAω+h1ηBe−ρtdt.The total amount of discharge by tributary 1 isq1, q1>0. J1 is the benefit obtained from the discharge by tributary 1. To reduce pollutants, the tributary 1 government inputs corresponding policies, personnel, and capital, improves the local industrial structure, and introduces new green industries; r1 is the social benefit brought to the region by the pollution control efforts of the tributary 1 government. q1−ξA is the total pollutants discharged by tributary 1 to the main stream, ω is the environmental protection tax paid by the discharging units of taxable water pollutants of the two tributaries, and h1 represents the benefit due to pollution control of tributary 2 to relieve pollution control pressure of tributary 1, h1<c1.The revenue function for tributary 2 is(4)∫0tJ2+r2B−c22B2+π2St−q2−ηBω+h2ξAe−ρttdt.The total amount of discharge by tributary 2 isq2, q2>0, J2 is the benefit obtained from the discharge of tributary 2, r2 is the social benefit brought to the region by the pollution control efforts of the tributary 2 government, q2−ηB is the total amount of pollutants discharged by tributary 2 to the main stream, and h2 represents the benefit due to the treatment of tributary 1 to relieve the pollution control pressure of tributary 2, h2<c2.The revenue function of main stream is(5)∫0tr3Z−c32Z2+π3St+q1−ξAωa+q2−ηBωae−ρtdt,where r3 is social benefits for the region from pollution control efforts of the main stream government.To achieve a unique continuous solutionqt for equation (1), a set of bounded, continuous, and differentiable value functions V1q, V2q, and V3q need to be constructed, so that the Hamilton-Jacobi-Bellman-Fleming (HJB) for tributary 1 government, tributary 2 government, and main stream government is constructed as(6)ρV1q=maxAJ1+r1A−c12A2+π1St−q1−ξAω+h1ηB+V1′qξA+ηB−δqt,(7)ρV2q=maxBJ2+r2B−c22B2+π2St−q2−ηBω+h2ξA+V2′qξA+ηB−δqt,(8)ρV3q=maxZr3Z−c32Z2+π3St+q1−ξAωa+q2−ηBωa+V3′qξA+ηB−δqt.By the maximization first-order conditions of combining (6) and (7) and (8) for A, B, and Z, we get(9)A=r1+ξω+V1′qξ−ηα−h1ηαc1,(10)B=r2+ηω+V2′qη−ξα−h2ξαc2,(11)Z=1c3×r3+V3′q−ωaξd1−αd21−α2+ηd2−αd11−α2+V3′qθ.Substituting equations (9)–(11) into the HJB equation and assuming that the expression of the function Viq is in linear form, the game equilibrium solution can be obtained as follows:(12)A∗=r1+ξω+π1λ/ρ+δξ−ηα−h1ηαc1,(13)B∗=r2+ηω+π2λ/ρ+δη−ξα−h2ξαc2,(14)Z∗=1c3×r3+π3λρ+δ−ωaξd1−αd21−α2+ηd2−αd11−α2+π3λρ+δθ.And we get the optimal benefits of the tripartite governments under this equilibrium condition are(15)V1∗=π1λρ+δq∗+1ρJ1+r1−c12A∗+ξω+π1λρ+δξA∗+π1S0−q1ω+π1λρ+δηB∗+θZ∗+h1ηB∗,V2∗=π2λρ+δq∗+1ρJ2+r2−c22B∗+ηω+π2λρ+δηB∗+π2S0−q2ω+π2λρ+δξA∗+θZ∗+h2ξA∗,V3∗=π3λρ+δq∗+1ρr3−c32Z∗+π3λρ+δθZ∗+π3S0+q1−ξA∗ωa+q2−ηB∗ωa+π3λρ+δξA∗+ηB∗.


## 2.2.2. The Stackelberg Game with Cost-Sharing in Main Stream and Tributaries

In this game situation, main stream government shares a certain proportion of the cost of pollution control with tributary governments, and we assume main stream will share cost-sharing ratiosm and n to tributary 1 government and tributary 2 government [21]. Decision sequence is that main stream government determines the pollution control efforts and cost-sharing ratio; then tributary governments determine their own optimal strategies. Using the inverse induction method, the optimal decision of tributary 1 and tributary 2 is first sought, and its objective function satisfies the HJB equation:(16)ρV1q=maxAJ1+r1A−1−mc12A2+π1St−q1−ξAω+h1ηB+V1′qξA+ηB−δqt,(17)ρV2q=maxBJ2+r2B−1−nc22B2+π2St−q2−ηBω+h2ξA+V2′qξA+ηB−δqt,where m and n are cost-sharing ratios of main stream government to tributary 1 and tributary 2 governments, respectively.By the maximization first-order condition forA and B in equations (18) and (19), we can get the results which are shown in equations (20) and (21).(18)A=r1+ξω+V1′qξ−ηα−h1ηαc11−m,(19)B=r2+ηω+V2′qη−ξα−h2ξαc21−n.Considering that the two tributaries can develop strategies based on (18) and (19), the HJB equation of the mainstream government is(20)ρV3q=maxZr3Z−c32Z2+π3St+q1−ξAωa+q2−ηBωa−c12mA2−c22nB2+V3′qξA+ηB−δqt.Substituting (18) and (19) into equation (22) and by the maximization first-order conditions for Z, m, and n , we get(21)Z=r3+V3′qθc3,m=−ωξ2a+1+2V3′qξ−r1−V1′qξ−ηα+h1ηα−ωξ2a−1+2V3′qξ+r1+V1′qξ−ηα−h1ηα,n=−ωη2a+1+2V3′qη−r2−V2′qη−ξα+h2ξα−ωη2a−1+2V3′qη+r2+V2′qη−ξα−h2ξα.Again, we assume that the expression of the functionViq is in linear form; the Stackelberg equilibrium solution for the tripartite governments under cost-sharing mechanism is obtained as(22)A∗∗=r1+ξω+π1λ/ρ+δξ−ηα−h1ηαc11−m∗,(23)B∗∗=r2+ηω+π2λ/ρ+δη−ξα−h2ξαc21−n∗,(24)Z∗∗=r3+π3λ/ρ+δθc3,(25)m=−ωξ2a+1+2π3λ/ρ+δξ−r1−π1λ/ρ+δξ−ηα+h1ηα−ωξ2a−1+2π3λ/ρ+δξ+r1+π1λ/ρ+δξ−ηα−h1ηα,(26)n=−ωη2a+1+2π3λ/ρ+δη−r2−π2λ/ρ+δη−ξα+h2ξα−ωη2a−1+2π3λ/ρ+δη+r2+π2λ/ρ+δη−ξα−h2ξα.The expression for the optimal revenue of the three parties in this game condition is(27)V1∗∗=π1λρ+δq∗∗+1ρJ1+r1−c121−m∗A∗∗+ξω+π1λρ+δξA∗∗+π1S0−q1ω+π1λρ+δηB∗∗+θZ∗∗+h1ηB∗∗,V2∗∗=π2λρ+δq∗∗+1ρJ2+r2−c221−n∗+ηω+π2λρ+δηB∗∗+π2S0−q2ω+π2λρ+δξA∗∗+θZ∗∗+h2ξA∗∗,V3∗∗=π3λρ+δq∗∗+1ρr3−c32Z∗∗+π3λρ+δθZ∗∗+π3S0+q1+q2ωa+−ξωa−c12m∗A∗∗+π3λρ+δξA∗∗+−ηωa−c22n∗B∗∗+π3λρ+δηB∗∗.


## 2.2.3. Comparative Analysis of Equilibrium Results

According to the equilibrium solutions of the game for main stream and tributary governments in the two cases of Nash noncooperative game and Stackelberg game with cost-sharing, we can obtain the following.Corollary 1.
From equations (12) and (13) and (26) and (27), we can see that(28)A∗∗−A∗=m1−mA∗>0,B∗∗−B∗=n1−nB∗>0.In Nash mechanism, the pollution control efforts of two tributaries are lower than cost-sharing mechanism, indicating that the conscious environmental governance model of the tributary governments is not desirable. The cost-sharing of the mainstream government to the tributary governments can improve the pollution control efforts of the two tributary governments.Corollary 2.
From equations (14) and (28), based on hypothesis (9), it is known that(29)Z∗∗−Z∗=−1c3×π3λρ+δ−ωaξd1−αd21−α2+ηd2−αd11−α2<0.In cost-sharing mechanism, the pollution control effort of the main stream is lower than that of Nash noncooperative mechanism, indicating that the cost-sharing mechanism can reduce the pollution control cost of the mainstream.


## 3. Results

Based on the solution results mentioned above, numerical simulation calculations are carried out. Then, parameters are set as shown in Table1.Table 1
Parameter values.
r1r2r3ωξηπ1π2π3ρ0.40.40.50.50.70.60.250.250.50.1δc1c2c3λαS0q1q2d10.13360.80.44660.5d2J1J2A0B0aθh1h20.5650.20.20.40.20.30.3Through comparison of tripartite revenue and pollutant elimination, we can see clearly that pollutant elimination will continue to increase and eventually stabilize over time. The pollutant elimination in cost-sharing is significantly larger than that in Nash, the cost-sharing can improve revenue of main stream and tributaries governments, and the increase in main stream government revenue is most apparent. These regularities are consistent with literature conclusions on ecological compensation mechanisms and will not be detailed here.


### 3.1. Competitive Coefficient

#### 3.1.1. The Effect of Competitive Coefficient (α) on Pollutant Elimination

To compare the effect of competition coefficient in Nash and cost-sharing to eliminate river pollutants, we supposeα  = 0.3 and 0.7. The final results are shown in Figure 1.Figure 1
Effect competitive coefficient (α) to pollutant elimination.According to Figure1, we can see clearly that as competition coefficient increases, the pollutant elimination under Nash and cost-sharing decreases. Moreover, the pollutant elimination in cost-sharing decreases at a low rate; the pollutant elimination in Nash decreases but at a larger rate. The results show that the more pressure on tributary governments to develop economy, the more cost-sharing mechanism needed.


#### 3.1.2. The Effect of Competition Coefficient (α) on Tripartite Degree of Pollution Treatment Efforts

Takeα  = 0.3、0.4、0.5、0.6、0.7 separately. The final results are shown in Figure 2.Figure 2
Effect competitive coefficient (α) on pollution treatment efforts.According to Figure2, the tributaries’ pollution treatment effort decreases both in Nash and in cost-sharing as competition coefficient increases. The reason is that when the pressure for economic development in the two tributary regions increases, their governments are less willing to invest policies and capitals in pollution control. Therefore, the pollution treatment efforts degree of the two tributaries will decrease, and it decreases more rapidly in Nash.In Nash mechanism, the degree of pollution treatment effort on main stream decreases as competition coefficient increases. The reason is that pollution treatment in the two tributaries decreases; thus the ecological benefits for the main stream decrease. Main stream government can only compensate for the damage caused by tributary regions discharged through the central government’s allocation of sewage charges, so main stream government will reduce its effort to reach the optimal equilibrium solution.In cost-sharing situation, the pollution treatment effort of main stream government is lower than that in Nash. The reason is that main stream government gives tributary governments a share of the pollution treatment cost to facilitate the tributaries’ pollution treatment effort in cost-sharing situation, and pollutants are eliminated in tributary regions.


#### 3.1.3. The Effect of Competition Coefficient (α) on Tripartite Revenue

The influence of competition coefficient on the three parties’ revenue is shown in Figures3–5. With competitive coefficient increases, the revenue of both main stream and tributaries governments will decrease in Nash and cost-sharing mechanism. Moreover, the effect in Nash is harder than that in cost-sharing. Meanwhile, the larger competitive coefficient is, the more significant revenue improvement for both tributaries by carrying on cost-sharing mechanism is.Figure 3
Effect competitive coefficient (α) on tributary 1.Figure 4
Effect competitive coefficient (α) on tributary 2.Figure 5
Effect competitive coefficient (α) on main stream.


### 3.2. Pollution Elimination of Tributary 1 Paying Unit Pollution Control Efforts

#### 3.2.1. The Effect of Pollution Elimination of Tributary 1 Paying Unit Pollution Control Efforts (ξ) on Tripartite Degree of Pollution Treatment Efforts

Takingξ  = 0.3, 0.4, 0.5, 0.6, and 0.7 respectively, we can see from Figure 6 that, as ξ increases, the pollution treatment effort degree of tributary 1 increases linearly in both Nash and cost-sharing. The slope of the pollution control effort degree in cost-sharing is more significant than that in Nash. The reason is that as ξ increases, tributary 1 will increase its pollution treatment effort spontaneously. Meanwhile, the cost subsidizing from main stream will further promote tributary 1 to carry out pollution treatment. When pollution elimination of tributary 1 paying unit pollution control efforts increases, tributary 2 government is inclined to choosing “free rider” behavior, so the effort degree of tributary 2 will decrease, and the rate of decrease of tributary 2 in cost-sharing will be lower than that in Nash.Figure 6
Effectξ on effort degree.


#### 3.2.2. The Effect of Pollution Elimination of Tributary 1 Paying Unit Pollution Treatment Efforts (ξ) on Tripartite Revenue

ξ equals 0.4 and 0.7, respectively. The final results are shown in Figures 7–9. Tripartite revenues increase in both mechanisms with ξ increases, and the revenue of tributary 2 will increase more obviously. The reason is that, with increases, ecological benefits from the increased pollution treatment efforts of tributary 1 increase and are beneficial for all three parties. However, tributary 2 is inclined to choosing “free rider” behavior and reduces invest of pollution control at the same time, so the revenue gap is more significant for tributary 2.Figure 7
Effectξ on tributary 1 revenue.Figure 8
Effectξ on tributary 2 revenue.Figure 9
Effectξ on main stream revenue.


#### 3.2.3. The Effect of Pollution Elimination of Tributary 1 Paying Unit Pollution Treatment Efforts (ξ) on Basin Pollution Elimination

Takingξ  = 0.4 and 0.7, respectively, the final result is shown in Figure 10. Then, we can see that as ξ increases, pollution elimination increases in both Nash and cost-sharing, and pollution elimination in cost-sharing is greater than that in Nash. The reason is that as ξ increases, the revenue of tributary 1’s pollution treatment efforts will increase, and the cost subsidized from the mainstream will further promote tributary 1’s pollution treatment, and the pollutant elimination effect is better.Figure 10
Effortξ on basin pollutant elimination.


### 3.3. Environmental Protection Tax

#### 3.3.1. The Effect of the Environmental Protection Tax (ω) on Tripartite Degree of Pollution Treatment Efforts

Takingω  = 0.3, 0.4, 0.5, 0.6 and 0.7, respectively, the final result is shown in Figure 11. Then, we can see clearly that, as ω increases, pollution treatment effort degree of tributaries 1 and 2 in both Nash and cost-sharing increases linearly. On the contrary, pollution treatment effort degree of main stream government will decrease linearly. And the growth rate of two tributaries’ pollution control efforts in cost-sharing is lower than that in Nash. The reason is that, in Nash situation, the two tributary governments will increase their pollution control efforts to maximize their own revenue with ω increases. In cost-sharing situation, tributary governments’ effort degree shows slow growth with ω increases. At the same time, optimal incentive from main stream government to tributary government will decrease. The two reasons make effort degree of tributaries to treatment pollution slow growth.Figure 11
Effortω on degree of pollution treatment efforts.


#### 3.3.2. The Effect of Environmental Protection Tax (ω) on Tripartite Revenue

Takingω  = 0.4 and 0.7, respectively, the final results are shown in Figures 12–14 According to Figures 12 and 13, we can see that as ω increases, tripartite revenue in both Nash and cost-sharing will decrease significantly. The reason is that as ω increases, the more environmental protection taxes need to be paid. The tributaries will increase their efforts to reduce emissions, the cost of pollution treatment and the taxes paid will increase, and therefore, the revenues of tributaries will decrease significantly.Figure 12
Effort ofω on tributary 1 revenue.Figure 13
Effort ofω on tributary 2 revenue.Figure 14
Effort ofω on main stream revenue.We can see that main stream government revenue will increase from Figure14. As ω increases, tributaries will eliminate more pollutants, and the main stream will access higher environmental benefits.


#### 3.3.3. The Effect of Environmental Protection Tax (ω) on Pollution Elimination

Takingω  = 0.4 and 0.7 respectively, the final result is shown in Figure 15. We can see clearly that pollution elimination increases in both Nash and cost-sharing with increase of ω, but the improvement effect is not evident in the situation of cost-sharing. It can be seen that the pollution treatment efforts of the two tributaries in cost-sharing mechanism are not improved as ω increases, so the performance of improvement for pollution elimination is also not apparent.Figure 15
Effort ofω on pollutant elimination.


### 3.4. The Effect of Each Parameter on the Optimal Cost-Sharing Ratiosm and n

Through calculating and analyzing, the effect of individual factors on optimal cost-sharing ratio is shown in Table2.Table 2
The effect of each parameter on optimal cost-sharing ratios.
VariablemNξ ↑↑↑r1 ↑↓Not changeα ↑↑↑a ↑↓↓η ↑↑↑h1 ↑↑Not changeh2 ↑Not change↑r2 ↑Not change↓ω ↑↓↓π3 ↑↑↑ρ ↑↓↓δ ↑↓↓Notes:ξ represents pollution elimination of tributary 1 paying unit pollution control efforts; r1 represents the social benefit brought to the region by pollution treatment efforts of the tributary 1 government; α represents competitive coefficient; a represents the proportion of subsidies to the main stream after the central government levies environmental protection tax; η represents pollution elimination of tributary 2 paying unit pollution control efforts; h1 represents the benefit due to pollution treatment of tributary 2 to relieve pollution control pressure of tributary 1; h2 represents the benefit due to pollution treatment of tributary 1 to relieve pollution control pressure of tributary 2; r2 represents the social benefit brought to the region by pollution treatment efforts of the tributary 2 government; ω represents environmental protection tax; π3 represents coefficients of basin social welfare effect on main stream government benefits; ρ represents discount rate; δ represents the attenuation coefficient of the reduction due to factors such as aging of wastewater treatment equipment.Taking parameters forξ and η as example, influence laws are shown in Figures 16–17.Figure 16
Effortξ on optimal cost-sharing ratios.Figure 17
Effortη on optimal cost-sharing ratios.Takingξ  = 0.3, 0.4, 0.5, 0.6, and 0.7, respectively, we can see from Figures 16–17 that, as ξ increases, the optimal cost-sharing ratios of the main stream to tributary 1 government m will increase, but the growth rate gradually decreases; meanwhile, the optimal cost-sharing ratios of main stream to tributary 2 n increase, and the growth rate gradually increases. The reason is that as ξ increases, the tributary 1 government will tend to treatment water pollution without too much incentive from main stream government, and they will increase pollution treatment efforts spontaneously. Tributary 2 will choose a “free rider,” so the main stream needs to stimulate pollution treatment efforts of tributary 2 by increasing the pollution treatment subsidies for tributary 2. The optimal cost-sharing ratios of the main stream to tributary 2 government η follow the same coefficient as ξ.The influence law of competition coefficientα on optimal cost-sharing ratio is shown in Figure 18.Figure 18
Effect of competition coefficient (α) on optimal cost-sharing ratio.Takingα  = 0.3, 0.4, 0.5, 0.6, and 0.7, the final result is shown in Figure 18. We can see that optimal cost-sharing ratios m and n from main stream government are positively correlated with α. As competition coefficient α increases, the greater pressure of economic development in two tributaries is, the less willing to invest policies, personnel, and funds in water pollution control is. So, main stream government will increase cost-sharing ratio as an incentive to the pollution control efforts of two tributaries.


## 3.1. Competitive Coefficient

### 3.1.1. The Effect of Competitive Coefficient (α) on Pollutant Elimination

To compare the effect of competition coefficient in Nash and cost-sharing to eliminate river pollutants, we supposeα  = 0.3 and 0.7. The final results are shown in Figure 1.Figure 1
Effect competitive coefficient (α) to pollutant elimination.According to Figure1, we can see clearly that as competition coefficient increases, the pollutant elimination under Nash and cost-sharing decreases. Moreover, the pollutant elimination in cost-sharing decreases at a low rate; the pollutant elimination in Nash decreases but at a larger rate. The results show that the more pressure on tributary governments to develop economy, the more cost-sharing mechanism needed.


### 3.1.2. The Effect of Competition Coefficient (α) on Tripartite Degree of Pollution Treatment Efforts

Takeα  = 0.3、0.4、0.5、0.6、0.7 separately. The final results are shown in Figure 2.Figure 2
Effect competitive coefficient (α) on pollution treatment efforts.According to Figure2, the tributaries’ pollution treatment effort decreases both in Nash and in cost-sharing as competition coefficient increases. The reason is that when the pressure for economic development in the two tributary regions increases, their governments are less willing to invest policies and capitals in pollution control. Therefore, the pollution treatment efforts degree of the two tributaries will decrease, and it decreases more rapidly in Nash.In Nash mechanism, the degree of pollution treatment effort on main stream decreases as competition coefficient increases. The reason is that pollution treatment in the two tributaries decreases; thus the ecological benefits for the main stream decrease. Main stream government can only compensate for the damage caused by tributary regions discharged through the central government’s allocation of sewage charges, so main stream government will reduce its effort to reach the optimal equilibrium solution.In cost-sharing situation, the pollution treatment effort of main stream government is lower than that in Nash. The reason is that main stream government gives tributary governments a share of the pollution treatment cost to facilitate the tributaries’ pollution treatment effort in cost-sharing situation, and pollutants are eliminated in tributary regions.


### 3.1.3. The Effect of Competition Coefficient (α) on Tripartite Revenue

The influence of competition coefficient on the three parties’ revenue is shown in Figures3–5. With competitive coefficient increases, the revenue of both main stream and tributaries governments will decrease in Nash and cost-sharing mechanism. Moreover, the effect in Nash is harder than that in cost-sharing. Meanwhile, the larger competitive coefficient is, the more significant revenue improvement for both tributaries by carrying on cost-sharing mechanism is.Figure 3
Effect competitive coefficient (α) on tributary 1.Figure 4
Effect competitive coefficient (α) on tributary 2.Figure 5
Effect competitive coefficient (α) on main stream.


## 3.1.1. The Effect of Competitive Coefficient (α) on Pollutant Elimination

To compare the effect of competition coefficient in Nash and cost-sharing to eliminate river pollutants, we supposeα  = 0.3 and 0.7. The final results are shown in Figure 1.Figure 1
Effect competitive coefficient (α) to pollutant elimination.According to Figure1, we can see clearly that as competition coefficient increases, the pollutant elimination under Nash and cost-sharing decreases. Moreover, the pollutant elimination in cost-sharing decreases at a low rate; the pollutant elimination in Nash decreases but at a larger rate. The results show that the more pressure on tributary governments to develop economy, the more cost-sharing mechanism needed.


## 3.1.2. The Effect of Competition Coefficient (α) on Tripartite Degree of Pollution Treatment Efforts

Takeα  = 0.3、0.4、0.5、0.6、0.7 separately. The final results are shown in Figure 2.Figure 2
Effect competitive coefficient (α) on pollution treatment efforts.According to Figure2, the tributaries’ pollution treatment effort decreases both in Nash and in cost-sharing as competition coefficient increases. The reason is that when the pressure for economic development in the two tributary regions increases, their governments are less willing to invest policies and capitals in pollution control. Therefore, the pollution treatment efforts degree of the two tributaries will decrease, and it decreases more rapidly in Nash.In Nash mechanism, the degree of pollution treatment effort on main stream decreases as competition coefficient increases. The reason is that pollution treatment in the two tributaries decreases; thus the ecological benefits for the main stream decrease. Main stream government can only compensate for the damage caused by tributary regions discharged through the central government’s allocation of sewage charges, so main stream government will reduce its effort to reach the optimal equilibrium solution.In cost-sharing situation, the pollution treatment effort of main stream government is lower than that in Nash. The reason is that main stream government gives tributary governments a share of the pollution treatment cost to facilitate the tributaries’ pollution treatment effort in cost-sharing situation, and pollutants are eliminated in tributary regions.


## 3.1.3. The Effect of Competition Coefficient (α) on Tripartite Revenue

The influence of competition coefficient on the three parties’ revenue is shown in Figures3–5. With competitive coefficient increases, the revenue of both main stream and tributaries governments will decrease in Nash and cost-sharing mechanism. Moreover, the effect in Nash is harder than that in cost-sharing. Meanwhile, the larger competitive coefficient is, the more significant revenue improvement for both tributaries by carrying on cost-sharing mechanism is.Figure 3
Effect competitive coefficient (α) on tributary 1.Figure 4
Effect competitive coefficient (α) on tributary 2.Figure 5
Effect competitive coefficient (α) on main stream.


## 3.2. Pollution Elimination of Tributary 1 Paying Unit Pollution Control Efforts

### 3.2.1. The Effect of Pollution Elimination of Tributary 1 Paying Unit Pollution Control Efforts (ξ) on Tripartite Degree of Pollution Treatment Efforts

Takingξ  = 0.3, 0.4, 0.5, 0.6, and 0.7 respectively, we can see from Figure 6 that, as ξ increases, the pollution treatment effort degree of tributary 1 increases linearly in both Nash and cost-sharing. The slope of the pollution control effort degree in cost-sharing is more significant than that in Nash. The reason is that as ξ increases, tributary 1 will increase its pollution treatment effort spontaneously. Meanwhile, the cost subsidizing from main stream will further promote tributary 1 to carry out pollution treatment. When pollution elimination of tributary 1 paying unit pollution control efforts increases, tributary 2 government is inclined to choosing “free rider” behavior, so the effort degree of tributary 2 will decrease, and the rate of decrease of tributary 2 in cost-sharing will be lower than that in Nash.Figure 6
Effectξ on effort degree.


### 3.2.2. The Effect of Pollution Elimination of Tributary 1 Paying Unit Pollution Treatment Efforts (ξ) on Tripartite Revenue

ξ equals 0.4 and 0.7, respectively. The final results are shown in Figures 7–9. Tripartite revenues increase in both mechanisms with ξ increases, and the revenue of tributary 2 will increase more obviously. The reason is that, with increases, ecological benefits from the increased pollution treatment efforts of tributary 1 increase and are beneficial for all three parties. However, tributary 2 is inclined to choosing “free rider” behavior and reduces invest of pollution control at the same time, so the revenue gap is more significant for tributary 2.Figure 7
Effectξ on tributary 1 revenue.Figure 8
Effectξ on tributary 2 revenue.Figure 9
Effectξ on main stream revenue.


### 3.2.3. The Effect of Pollution Elimination of Tributary 1 Paying Unit Pollution Treatment Efforts (ξ) on Basin Pollution Elimination

Takingξ  = 0.4 and 0.7, respectively, the final result is shown in Figure 10. Then, we can see that as ξ increases, pollution elimination increases in both Nash and cost-sharing, and pollution elimination in cost-sharing is greater than that in Nash. The reason is that as ξ increases, the revenue of tributary 1’s pollution treatment efforts will increase, and the cost subsidized from the mainstream will further promote tributary 1’s pollution treatment, and the pollutant elimination effect is better.Figure 10
Effortξ on basin pollutant elimination.


## 3.2.1. The Effect of Pollution Elimination of Tributary 1 Paying Unit Pollution Control Efforts (ξ) on Tripartite Degree of Pollution Treatment Efforts

Takingξ  = 0.3, 0.4, 0.5, 0.6, and 0.7 respectively, we can see from Figure 6 that, as ξ increases, the pollution treatment effort degree of tributary 1 increases linearly in both Nash and cost-sharing. The slope of the pollution control effort degree in cost-sharing is more significant than that in Nash. The reason is that as ξ increases, tributary 1 will increase its pollution treatment effort spontaneously. Meanwhile, the cost subsidizing from main stream will further promote tributary 1 to carry out pollution treatment. When pollution elimination of tributary 1 paying unit pollution control efforts increases, tributary 2 government is inclined to choosing “free rider” behavior, so the effort degree of tributary 2 will decrease, and the rate of decrease of tributary 2 in cost-sharing will be lower than that in Nash.Figure 6
Effectξ on effort degree.


## 3.2.2. The Effect of Pollution Elimination of Tributary 1 Paying Unit Pollution Treatment Efforts (ξ) on Tripartite Revenue

ξ equals 0.4 and 0.7, respectively. The final results are shown in Figures 7–9. Tripartite revenues increase in both mechanisms with ξ increases, and the revenue of tributary 2 will increase more obviously. The reason is that, with increases, ecological benefits from the increased pollution treatment efforts of tributary 1 increase and are beneficial for all three parties. However, tributary 2 is inclined to choosing “free rider” behavior and reduces invest of pollution control at the same time, so the revenue gap is more significant for tributary 2.Figure 7
Effectξ on tributary 1 revenue.Figure 8
Effectξ on tributary 2 revenue.Figure 9
Effectξ on main stream revenue.


## 3.2.3. The Effect of Pollution Elimination of Tributary 1 Paying Unit Pollution Treatment Efforts (ξ) on Basin Pollution Elimination

Takingξ  = 0.4 and 0.7, respectively, the final result is shown in Figure 10. Then, we can see that as ξ increases, pollution elimination increases in both Nash and cost-sharing, and pollution elimination in cost-sharing is greater than that in Nash. The reason is that as ξ increases, the revenue of tributary 1’s pollution treatment efforts will increase, and the cost subsidized from the mainstream will further promote tributary 1’s pollution treatment, and the pollutant elimination effect is better.Figure 10
Effortξ on basin pollutant elimination.


## 3.3. Environmental Protection Tax

### 3.3.1. The Effect of the Environmental Protection Tax (ω) on Tripartite Degree of Pollution Treatment Efforts

Takingω  = 0.3, 0.4, 0.5, 0.6 and 0.7, respectively, the final result is shown in Figure 11. Then, we can see clearly that, as ω increases, pollution treatment effort degree of tributaries 1 and 2 in both Nash and cost-sharing increases linearly. On the contrary, pollution treatment effort degree of main stream government will decrease linearly. And the growth rate of two tributaries’ pollution control efforts in cost-sharing is lower than that in Nash. The reason is that, in Nash situation, the two tributary governments will increase their pollution control efforts to maximize their own revenue with ω increases. In cost-sharing situation, tributary governments’ effort degree shows slow growth with ω increases. At the same time, optimal incentive from main stream government to tributary government will decrease. The two reasons make effort degree of tributaries to treatment pollution slow growth.Figure 11
Effortω on degree of pollution treatment efforts.


### 3.3.2. The Effect of Environmental Protection Tax (ω) on Tripartite Revenue

Takingω  = 0.4 and 0.7, respectively, the final results are shown in Figures 12–14 According to Figures 12 and 13, we can see that as ω increases, tripartite revenue in both Nash and cost-sharing will decrease significantly. The reason is that as ω increases, the more environmental protection taxes need to be paid. The tributaries will increase their efforts to reduce emissions, the cost of pollution treatment and the taxes paid will increase, and therefore, the revenues of tributaries will decrease significantly.Figure 12
Effort ofω on tributary 1 revenue.Figure 13
Effort ofω on tributary 2 revenue.Figure 14
Effort ofω on main stream revenue.We can see that main stream government revenue will increase from Figure14. As ω increases, tributaries will eliminate more pollutants, and the main stream will access higher environmental benefits.


### 3.3.3. The Effect of Environmental Protection Tax (ω) on Pollution Elimination

Takingω  = 0.4 and 0.7 respectively, the final result is shown in Figure 15. We can see clearly that pollution elimination increases in both Nash and cost-sharing with increase of ω, but the improvement effect is not evident in the situation of cost-sharing. It can be seen that the pollution treatment efforts of the two tributaries in cost-sharing mechanism are not improved as ω increases, so the performance of improvement for pollution elimination is also not apparent.Figure 15
Effort ofω on pollutant elimination.


## 3.3.1. The Effect of the Environmental Protection Tax (ω) on Tripartite Degree of Pollution Treatment Efforts

Takingω  = 0.3, 0.4, 0.5, 0.6 and 0.7, respectively, the final result is shown in Figure 11. Then, we can see clearly that, as ω increases, pollution treatment effort degree of tributaries 1 and 2 in both Nash and cost-sharing increases linearly. On the contrary, pollution treatment effort degree of main stream government will decrease linearly. And the growth rate of two tributaries’ pollution control efforts in cost-sharing is lower than that in Nash. The reason is that, in Nash situation, the two tributary governments will increase their pollution control efforts to maximize their own revenue with ω increases. In cost-sharing situation, tributary governments’ effort degree shows slow growth with ω increases. At the same time, optimal incentive from main stream government to tributary government will decrease. The two reasons make effort degree of tributaries to treatment pollution slow growth.Figure 11
Effortω on degree of pollution treatment efforts.


## 3.3.2. The Effect of Environmental Protection Tax (ω) on Tripartite Revenue

Takingω  = 0.4 and 0.7, respectively, the final results are shown in Figures 12–14 According to Figures 12 and 13, we can see that as ω increases, tripartite revenue in both Nash and cost-sharing will decrease significantly. The reason is that as ω increases, the more environmental protection taxes need to be paid. The tributaries will increase their efforts to reduce emissions, the cost of pollution treatment and the taxes paid will increase, and therefore, the revenues of tributaries will decrease significantly.Figure 12
Effort ofω on tributary 1 revenue.Figure 13
Effort ofω on tributary 2 revenue.Figure 14
Effort ofω on main stream revenue.We can see that main stream government revenue will increase from Figure14. As ω increases, tributaries will eliminate more pollutants, and the main stream will access higher environmental benefits.


## 3.3.3. The Effect of Environmental Protection Tax (ω) on Pollution Elimination

Takingω  = 0.4 and 0.7 respectively, the final result is shown in Figure 15. We can see clearly that pollution elimination increases in both Nash and cost-sharing with increase of ω, but the improvement effect is not evident in the situation of cost-sharing. It can be seen that the pollution treatment efforts of the two tributaries in cost-sharing mechanism are not improved as ω increases, so the performance of improvement for pollution elimination is also not apparent.Figure 15
Effort ofω on pollutant elimination.


## 3.4. The Effect of Each Parameter on the Optimal Cost-Sharing Ratiosm and n

Through calculating and analyzing, the effect of individual factors on optimal cost-sharing ratio is shown in Table2.Table 2
The effect of each parameter on optimal cost-sharing ratios.
VariablemNξ ↑↑↑r1 ↑↓Not changeα ↑↑↑a ↑↓↓η ↑↑↑h1 ↑↑Not changeh2 ↑Not change↑r2 ↑Not change↓ω ↑↓↓π3 ↑↑↑ρ ↑↓↓δ ↑↓↓Notes:ξ represents pollution elimination of tributary 1 paying unit pollution control efforts; r1 represents the social benefit brought to the region by pollution treatment efforts of the tributary 1 government; α represents competitive coefficient; a represents the proportion of subsidies to the main stream after the central government levies environmental protection tax; η represents pollution elimination of tributary 2 paying unit pollution control efforts; h1 represents the benefit due to pollution treatment of tributary 2 to relieve pollution control pressure of tributary 1; h2 represents the benefit due to pollution treatment of tributary 1 to relieve pollution control pressure of tributary 2; r2 represents the social benefit brought to the region by pollution treatment efforts of the tributary 2 government; ω represents environmental protection tax; π3 represents coefficients of basin social welfare effect on main stream government benefits; ρ represents discount rate; δ represents the attenuation coefficient of the reduction due to factors such as aging of wastewater treatment equipment.Taking parameters forξ and η as example, influence laws are shown in Figures 16–17.Figure 16
Effortξ on optimal cost-sharing ratios.Figure 17
Effortη on optimal cost-sharing ratios.Takingξ  = 0.3, 0.4, 0.5, 0.6, and 0.7, respectively, we can see from Figures 16–17 that, as ξ increases, the optimal cost-sharing ratios of the main stream to tributary 1 government m will increase, but the growth rate gradually decreases; meanwhile, the optimal cost-sharing ratios of main stream to tributary 2 n increase, and the growth rate gradually increases. The reason is that as ξ increases, the tributary 1 government will tend to treatment water pollution without too much incentive from main stream government, and they will increase pollution treatment efforts spontaneously. Tributary 2 will choose a “free rider,” so the main stream needs to stimulate pollution treatment efforts of tributary 2 by increasing the pollution treatment subsidies for tributary 2. The optimal cost-sharing ratios of the main stream to tributary 2 government η follow the same coefficient as ξ.The influence law of competition coefficientα on optimal cost-sharing ratio is shown in Figure 18.Figure 18
Effect of competition coefficient (α) on optimal cost-sharing ratio.Takingα  = 0.3, 0.4, 0.5, 0.6, and 0.7, the final result is shown in Figure 18. We can see that optimal cost-sharing ratios m and n from main stream government are positively correlated with α. As competition coefficient α increases, the greater pressure of economic development in two tributaries is, the less willing to invest policies, personnel, and funds in water pollution control is. So, main stream government will increase cost-sharing ratio as an incentive to the pollution control efforts of two tributaries.


## 4. Conclusion

(1)
Moving from Nash noncooperative game mechanism to cost-sharing mechanism, it can lead to environmental and economic Pareto improvements, with improved pollutant elimination and improved revenue to both main stream and two tributary governments. As competition coefficient increases, tributary pollution treatment efforts decrease, pollution elimination in both Nash noncooperative mechanism and cost-sharing mechanism decreases, and pollution elimination in noncooperative mechanism decreases at a faster rate. At the same time, the larger the competition coefficient, the more significant the increase of revenue for both tributaries by using cost-sharing mechanism. The result shows that the greater pressure on tributary governments for economic development, the greater need to adopt cost-sharing mechanism.(2)
An increase of pollutants elimination paying unit pollution control efforts of one tributary can lead to the increase of revenue to all three parties of main stream and both tributaries. At the same time, the tributary’s pollution treatment effort degree increases, the other decreases, and the “free rider” phenomenon comes into being. As environmental protection taxes increase, pollution treatment efforts degree of both tributaries increases, and pollution treatment efforts degree of the main stream decreases. Meanwhile, pollution elimination increases. At the same time, revenue of both tributaries decreases and revenue of the main stream increases.(3)
The optimal cost-sharing ratio of the main stream to the two tributaries varies. This paper analyze many factors that influence the sharing balance, all of which show a steady monotonic increasing or decreasing relationship. For example, as competition coefficient increases, cost-sharing ratios of the main stream to the two tributaries increase. The greater the pollutants elimination paying per unit of effort of a tributary government is, the greater the cost-sharing ratio of the main stream to that tributary government is, but it increases at a low growth rate. In contrast, the cost-sharing ratio to another tributary government is smaller but increases at a high growth rate.(4)
It is suggested that the government adopts the stark Berg game scheme of cost-sharing between the main river and tributaries to manage the main river and tributaries.



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*Source: 6559840-2022-06-02.xml*