Does Fiscal Policy Promote Third-Party Environmental Pollution Control in China? An Evolutionary Game Theoretical Approach

: To promote third-party environmental pollution control in China, it is necessary to dissect the mechanism of ﬁscal policy in third-party environmental pollution control. This study ﬁrst discusses the acting paths of ﬁscal policies on third-party environmental pollution control in theory. A tripartite evolutionary game model involving the local government, the polluting enterprise, and the third-party enterprise is established. The replicator dynamics, evolutionary stability strategies, and numerical simulation of the behavior of the three participants are analyzed to further study the acting mechanism of ﬁscal policy in third-party environmental pollution control. In addition, the inﬂuences of other parameters on the implementation of third-party environmental pollution control are evaluated. The results show that the behaviors of the local government, the polluting enterprise, and the third-party enterprise inﬂuence each other. Furthermore, strengthening the relevant ﬁscal policy, reducing the risks of the polluting enterprise and third-party enterprise, and improving the beneﬁt to the local government are conducive to promoting third-party environmental pollution control in China. Based on these results, this study proposes the following policy implications: formulating ﬁscal policies for third-party environmental pollution control, implementing ﬁscal policies in a dynamic and progressive manner, improving the market mechanism of third-party environmental pollution control, and strengthening the environmental performance assessment of the local government.


Introduction
China has received high attention from countries worldwide due to its rapid economic growth, industrialization, and urbanization. However, the frequent occurrences of ecological damage and environmental pollution show that China is facing great resource and environment constraints [1][2][3]. The combination of environmental pollution control and economic development has become the "new normal" in China's modernization process [4][5][6].
As environmental issues have the characteristics of externality, polluters cannot control environmental pollution well; thus, the government needs to participate in environmental governance directly [7]. Some scholars believe that the fiscal competition among local governments [8] promotes environmental protection under certain conditions [9][10][11], while other scholars believe that such competition leads to a worse environmental quality [12,13]. To obtain short-term economic competitive advantages among regions [14,15], i.e., expanding the tax base and increasing tax revenue, local governments may ignore the environment and relax the supervision and governance of the environment [16][17][18][19]. In addition, the unclear division of environmental protection powers between (1) Could the implementation of fiscal policies promote third-party environmental pollution control? How do the actions of the government, the polluting enterprise and the third-party enterprise interact? (2) What other important factors influence the behavior of the government, the polluting enterprise and the third-party enterprise? How do these factors promote third-party environmental pollution control? (3) What fiscal policies and suggestions could be proposed to promote third-party environmental pollution control in China?
The structure of this paper is organized as follows. Section 2 introduces the theory and methodology. Section 3 describes the modeling according to the basic assumptions. Section 4 analyzes the evolutionary path and numerical simulation. Section 5 explores other influencing parameters. Finally, the conclusions and policy implications are summarized in Section 6.

Acting Paths of Fiscal Policies on Third-Party Environmental Pollution Control
(1) Acting paths of fiscal expenditure policies As shown in Figure 1, fiscal subsidies may be transferred as payments from the central government to the local government, from the central government to the polluting enterprise and the third-party enterprise, or from the local government to the polluting enterprise and the third-party enterprise. The central government and the local government implement fiscal subsidy policies for the third-party enterprise to reduce its operating costs and increase the supply of third-party pollution control, as shown in Figure 2a. The central government implements the transfer payment policy to the local government, which chooses the third-party control to reduce the associated costs and thereby increase the local government's demand for third-party control, as shown in Figure 2b. Similarly, the local government offers subsidies to the polluting enterprise, which chooses the third-party control to reduce costs and improve the enterprise's demand for third-party governance, as shown in Figure 2b.
(2) Acting paths of fiscal revenue policies As shown in Figure 1, preferential tax policies may be implemented by the central government and the local government to encourage the polluting enterprise and the third-party enterprise to engage in third-party pollution control. The central government and the local government implement preferential tax policies for the third-party enterprise to reduce operating costs and thereby improve the supply of third-party pollution control, as shown in Figure 2a. The central government and local Sustainability 2019, 11, 4434 4 of 18 government implement preferential tax policies for the polluting enterprise, which chooses third-party control to reduce costs and improve the enterprise's demand for third-party control, as shown in Figure 2b.
Sustainability 2019, 11, x FOR PEER REVIEW 4 of 18 government and local government implement preferential tax policies for the polluting enterprise, which chooses third-party control to reduce costs and improve the enterprise's demand for third-party control, as shown in Figure 2b.

Research Methodology
Evolutionary game theory combines game theory and a dynamic evolutionary analysis by considering that at a stable equilibrium in the economy, an economic subject with limited rationality engages in continuous imitation and learning of favorable strategies to finally reach a stable strategy, i.e., an evolutionary stable strategy (ESS) [54]. Replication dynamics refers to a dynamic differential equation that describes the proportion of a particular strategy adopted in a population as follows: where / is the rate of change in the proportion of game participants adopting strategy S over time. The value x is the proportion of game participants adopting strategy S. U is the expected return of a game participant adopting strategy S. U is the average expected return of the game  government and local government implement preferential tax policies for the polluting enterprise, which chooses third-party control to reduce costs and improve the enterprise's demand for third-party control, as shown in Figure 2b.

Research Methodology
Evolutionary game theory combines game theory and a dynamic evolutionary analysis by considering that at a stable equilibrium in the economy, an economic subject with limited rationality engages in continuous imitation and learning of favorable strategies to finally reach a stable strategy, i.e., an evolutionary stable strategy (ESS) [54]. Replication dynamics refers to a dynamic differential equation that describes the proportion of a particular strategy adopted in a population as follows: where / is the rate of change in the proportion of game participants adopting strategy S over time. The value x is the proportion of game participants adopting strategy S. U is the expected return of a game participant adopting strategy S. U is the average expected return of the game

Research Methodology
Evolutionary game theory combines game theory and a dynamic evolutionary analysis by considering that at a stable equilibrium in the economy, an economic subject with limited rationality engages in continuous imitation and learning of favorable strategies to finally reach a stable strategy, i.e., an evolutionary stable strategy (ESS) [54]. Replication dynamics refers to a dynamic differential equation that describes the proportion of a particular strategy adopted in a population as follows: where dx/dt is the rate of change in the proportion of game participants adopting strategy S over time. The value x is the proportion of game participants adopting strategy S. U S is the expected return of a game participant adopting strategy S. U is the average expected return of the game participant. The stability points of the ESS should have the following properties: if one player deviates from stable point x * due to an accidental error, the replication dynamic will still return x to x * . Mathematically, when the disturbance makes x lower than x * , F(x) = dx/dt must be greater than 0; thus, t ↑ , then, x ↑ to return x to x * ; when the disturbance makes x higher than x * , F(x) = dx/dt must be less than 0; thus, t ↑ , then x ↓ to return x to x * . According to Figure 3, when the evolution reaches a stable state, the derivative of the differential equation of the replicator dynamics should be less than 0, namely, F (x * ) < 0. participant. The stability points of the ESS should have the following properties: if one player deviates from stable point x * due to an accidental error, the replication dynamic will still return x to x * . Mathematically, when the disturbance makes x lower than x * , F(x) = dx/dt must be greater than 0; thus, t ↑, then, x ↑ to return x to x * ; when the disturbance makes x higher than x * , F(x) = dx/dt must be less than 0; thus, t ↑, then x ↓ to return x to x * . According to Figure 3, when the evolution reaches a stable state, the derivative of the differential equation of the replicator dynamics should be less than 0, namely, F′(x * ) < 0.

Basic Assumptions and Modeling
Third-party environmental pollution control includes the government's third-party control and enterprise's third-party control. The enterprise's third-party control in China has been implemented for a relatively short time, and the polluting enterprise and the third-party enterprise tend to be conservative in strategic choices. Therefore, it is necessary and urgent for the government to formulate and implement fiscal policies to promote the enterprise's third-party control. This study considers the case of the fiscal expenditure policy implemented by the local government for the enterprises' third-party control. As an indirect expenditure of the government, preferential tax policy is a type of fiscal expenditure. Thus, its function is shown in the analysis of the fiscal expenditure policy.
The game participants are as follows: the local government G, the polluting enterprise P, and the third-party enterprise T. The behavioral strategies are as follows: S = S , S , S = S , S , and S = S , S . S indicates that the local government implements a fiscal expenditure policy for third-party environmental pollution control to the polluting enterprise and the third-party enterprise. S indicates that the local government does not implement a fiscal policy. S indicates that the polluting enterprise accepts the fiscal policy and improves the demand for third-party control. S indicates that the polluting enterprise does not accept the fiscal policy. S indicates that the third-party enterprise accepts the fiscal policy and improves the supply of third-party control. S indicates that the third-party enterprise does not accept the fiscal policy.
The proportion of the adoption of behavioral strategies is as follows: during the initial stage of the game, the probability that the local government chooses S is x, the probability that the polluting enterprise chooses S is y, and the probability that the third-party enterprise chooses S is z.
There are eight game strategy profiles among the three participants as follows: The assumptions and implications of the return of the strategy portfolio are described as follows:

Basic Assumptions and Modeling
Third-party environmental pollution control includes the government's third-party control and enterprise's third-party control. The enterprise's third-party control in China has been implemented for a relatively short time, and the polluting enterprise and the third-party enterprise tend to be conservative in strategic choices. Therefore, it is necessary and urgent for the government to formulate and implement fiscal policies to promote the enterprise's third-party control. This study considers the case of the fiscal expenditure policy implemented by the local government for the enterprises' third-party control. As an indirect expenditure of the government, preferential tax policy is a type of fiscal expenditure. Thus, its function is shown in the analysis of the fiscal expenditure policy.
The game participants are as follows: the local government G, the polluting enterprise P, and the third-party enterprise T. The behavioral strategies are as follows: S G = {S 1 , S 2 }, S P = {S 3 , S 4 }, and S T = {S 5 , S 6 }. S 1 indicates that the local government implements a fiscal expenditure policy for third-party environmental pollution control to the polluting enterprise and the third-party enterprise. S 2 indicates that the local government does not implement a fiscal policy. S 3 indicates that the polluting enterprise accepts the fiscal policy and improves the demand for third-party control. S 4 indicates that the polluting enterprise does not accept the fiscal policy. S 5 indicates that the third-party enterprise accepts the fiscal policy and improves the supply of third-party control. S 6 indicates that the third-party enterprise does not accept the fiscal policy.
The proportion of the adoption of behavioral strategies is as follows: during the initial stage of the game, the probability that the local government chooses S 1 is x, the probability that the polluting enterprise chooses S 3 is y, and the probability that the third-party enterprise chooses S 5 is z.
The assumptions and implications of the return of the strategy portfolio are described as follows: (1) When the local government does not implement a fiscal expenditure policy and neither the polluting enterprise nor the third-party enterprise accepts the policy, the three subjects gain U, V, and W, respectively. (2) When the local government implements a fiscal expenditure policy for the polluting enterprise and the third-party enterprise with the intensity of a 1 and a 2 , respectively, and the polluting enterprise and the third-party enterprise accept the fiscal policy and improve the demand and supply for third-party control, the three parties will gain U − a 1 − a 2 + M, V + a 1 + N, and W + a 2 + R, respectively. M, N, and R represent the benefits that the local government, the polluting enterprise and the third-party enterprise obtained from the third-party control. For example, M represents a higher tax revenue, reduced regulatory costs and improved social welfare; N represents a reduction in the pollution control cost and improvement in enterprise competitiveness; and R represents a higher economic benefit or technological progress. (3) When the local government implements a fiscal expenditure policy, the polluting enterprise accepts the fiscal policy and improves the demand for third-party control, and the third-party enterprise accepts the fiscal subsidy without improving the supply for third-party control, the benefit to the local government is U − a 1 − a 2 , the benefit to the polluting enterprise is V + a 1 − b under the assumption that the search risk of the polluting enterprise increasing the demand for third-party control is b, and the benefit to the third-party enterprise increases to W + a 2 . (4) When the local government implements a fiscal expenditure policy for third-party environmental pollution control, the third-party enterprise accepts the fiscal policy and improves the supply for third-party control, and the polluting enterprise accepts the fiscal subsidy without improving the demand for third-party control, the benefit to the local government is U − a 1 − a 2 , the benefit to the polluting enterprise increases to V + a 1 , and the benefit to the third-party enterprise is W + a 2 − c under the assumption that the risk the third-party enterprise incurs when increasing the supply of third-party control is c. (5) When the local government implements a fiscal expenditure policy and neither the polluting enterprise nor the third-party enterprise accepts the policy, the three parties will gain U − a 1 − a 2 , V + a 1 , and W + a 2 , respectively. (6) When the local government does not implement a fiscal expenditure policy but the polluting enterprise and third-party enterprise believe that the government will implement the fiscal policy and accordingly improve the demand and supply for third-party control, the three parties will gain U + m, V + n, and W + r, respectively. Simultaneously, the trading volume of the third-party control market is lower than that of the fiscal policy implementation. The revenue increment of the three parties should be lower than the benefit when the fiscal policy is implemented, i.e., M > m, N > n, and R > r. (7) When the local government does not implement a fiscal expenditure policy and only the polluting enterprise believes that the government will implement a fiscal policy and chooses third-party control, the benefit to the local government is U, the benefit to the polluting enterprise is V − b, and the benefit to the third-party enterprise is W. (8) When the local government does not implement a fiscal expenditure policy and only the third-party enterprise believes that the government will implement a fiscal policy and accordingly improves the supply for third-party control, the benefit to the local government is U, the benefit to the polluting enterprise is V, and the benefit to the third-party enterprise is W − c.
Since fiscal expenditure plays an expansionary role in the GDP, increasing the fiscal expenditure can expand the aggregate demand with a multiplier greater than 1. In the long-term, third-party environmental pollution control will yield increasing taxes, reduce the regulatory costs and increase social welfare for the local government, i.e., M = (a 1 + a 2 ) k , (a 1 + a 2 ) > 1, k > 1.

Replicator Dynamics Analysis
(1) Analysis of the replicator dynamics of the local government's strategies When the local government implements and does not implement a fiscal policy, the expected benefits are described as follows: Thus, the average expected revenue of the local government is expressed as follows: The replicator dynamics equation when the local government implements a fiscal policy is described as follows: The stability of the evolution strategies of the local government is analyzed as follows: A. When z = ( ) , F(x) ≡ 0, suggesting that regardless of the proportion of the local government implementing the fiscal policy, the strategy will not change over time as shown in Figure 5a

Replicator Dynamics Analysis
(1) Analysis of the replicator dynamics of the local government's strategies When the local government implements and does not implement a fiscal policy, the expected benefits are described as follows: Thus, the average expected revenue of the local government is expressed as follows: The replicator dynamics equation when the local government implements a fiscal policy is described as follows: The stability of the evolution strategies of the local government is analyzed as follows: A. When z = a 1 +a 2 y(M−m) , F(x) ≡ 0, suggesting that regardless of the proportion of the local government implementing the fiscal policy, the strategy will not change over time as shown in Figure 5a Thus, the average expected revenue of the polluting enterprise is described as follows: The replicator dynamics equation of the polluting enterprise's willingness to accept the fiscal policy and improve the demand for third-party control is described as follows: The stability of the evolution strategies of the polluting enterprise is analyzed as follows: A. When x = ( ) − , F(y) ≡ 0, suggesting that regardless of the proportion of the polluting enterprise choosing to accept the fiscal policy, the strategy will not change over time as shown in Figure 6a.  (2) Analysis of the replicator dynamics of the polluting enterprise's strategies When the polluting enterprise chooses strategy S 3 and strategy S 4 , the expected benefits are expressed as follows: Thus, the average expected revenue of the polluting enterprise is described as follows: The replicator dynamics equation of the polluting enterprise's willingness to accept the fiscal policy and improve the demand for third-party control is described as follows: The stability of the evolution strategies of the polluting enterprise is analyzed as follows: A. When x = b z(N−n) − n+b N−n , F(y) ≡ 0, suggesting that regardless of the proportion of the polluting enterprise choosing to accept the fiscal policy, the strategy will not change over time as shown in Figure 6a.
Thus, the average expected revenue of the third-party enterprise is expressed as follows: The replicator dynamics equation of the third-party enterprise's willingness to accept the fiscal policy and improve the supply for third-party control is expressed as follows: The stability of the evolution strategies of the third-party enterprise is analyzed as follows: A. When x = ( ) − , F(z) ≡ 0 , suggesting that regardless of the proportion of the third-party enterprise choosing to accept the fiscal policy, the strategy will not change over time as shown in Figure 7a  (3) Analysis of the replicator dynamics of the third-party enterprise's strategies When the third-party enterprise chooses strategy S 5 and strategy S 6 , the expected benefits are described as follows: Thus, the average expected revenue of the third-party enterprise is expressed as follows: The replicator dynamics equation of the third-party enterprise's willingness to accept the fiscal policy and improve the supply for third-party control is expressed as follows: The stability of the evolution strategies of the third-party enterprise is analyzed as follows: A. When x = c y(R−r) − r+c R−r , F(z) ≡ 0, suggesting that regardless of the proportion of the third-party enterprise choosing to accept the fiscal policy, the strategy will not change over time as shown in Figure 7a

Stability of the Evolutionary Strategies Analysis
The stability analysis of the strategy combinations of the local government, the polluting enterprise and the third-party enterprise can be carried out through the Jacobian matrix [55]. The Jacobian matrix in this paper is shown as follows: According to Lyapunov's first method [56], regarding the ESS, the characteristic root of its corresponding Jacobian matrix must be less than 0. In Table 1, the symbols shown in brackets indicate the positive and negative values of the eigenvalues.  By analyzing the strategic stability of each subject, we obtain the stability strategies of the tripartite evolutionary game. Of the eight equilibrium points, two equilibrium points have local stability. The ESSs are (implementation, acceptance, acceptance) and (non-implementation, non-acceptance, non-acceptance).

Numerical Simulation
To more directly show the evolutionary path of the system, the evolutionary game model is analyzed through a numerical simulation. According to the above assumptions, i.e.,M > m, N > n, R > r, M = (a + a ) , (a + a ) > 1, and k > 1, this paper sets the initial values of the parameters as a = 1, a = 1, M = 14, N = 10, R = 11, m = 4, n = 2, r = 3, b = 3, and c = 5.
(1) For the fixed x initial value (e.g., x = 0.5) to achieve the ideal state of equilibrium, it needs to satisfy A = yz(M − m) − (a +a ) > 0. From the random initial value of y and z, the evolution

Stability of the Evolutionary Strategies Analysis
The stability analysis of the strategy combinations of the local government, the polluting enterprise and the third-party enterprise can be carried out through the Jacobian matrix [55]. The Jacobian matrix in this paper is shown as follows: According to Lyapunov's first method [56], regarding the ESS, the characteristic root of its corresponding Jacobian matrix must be less than 0. In Table 1, the symbols shown in brackets indicate the positive and negative values of the eigenvalues.  By analyzing the strategic stability of each subject, we obtain the stability strategies of the tripartite evolutionary game. Of the eight equilibrium points, two equilibrium points have local stability. The ESSs are (implementation, acceptance, acceptance) and (non-implementation, non-acceptance, non-acceptance).

Numerical Simulation
To more directly show the evolutionary path of the system, the evolutionary game model is analyzed through a numerical simulation. According to the above assumptions, i.e., M > m, N > n, R > r, M = (a 1 + a 2 ) k , (a 1 + a 2 ) > 1, and k > 1, this paper sets the initial values of the parameters as a 1 = 1, a 2 = 1, M = 14, N = 10, R = 11, m = 4, n = 2, r = 3, b = 3, and c = 5.
(1) For the fixed x initial value (e.g., x = 0.5) to achieve the ideal state of equilibrium, it needs to satisfy A = yz(M − m) − (a 1 + a 2 ) > 0. From the random initial value of y and z, the evolution curve of x (x = 1 e −At +1 ) is shown in Figure 8. As shown, the different initial values of y and z have an impact on the convergence rate of x, but x still monotonically increases and converges to 1, indicating that the proportion of the local governments choosing the "implementation" strategy continues to increase over time until all local governments eventually choose "implementation". (2) For the fixed y initial value (e.g., y = 0.5) to achieve the ideal state of equilibrium, it needs to satisfy B = xz(N − n) + z(n + b) − b > 0. From a random initial value of x and z, the evolution curve of y (y = 1 e −Bt +1 ) is shown in Figure 9. As shown, the different initial values of x and z have an impact on the convergence rate of y, but y still monotonically increases and converges to 1, indicating that the proportion of the polluting enterprises choosing the "acceptance" strategy continues to increase over time until all polluting enterprises eventually choose "acceptance".
(3) For the fixed z initial value (e.g., z = 0.5) to achieve the ideal state of equilibrium, it needs to satisfy C = xy(R − r) + y(r + c) − c > 0. From a random initial value of x and y, the evolution curve of z (z = 1 e −Ct +1 ) is shown in Figure 10. As shown, the different initial values of x and y have an impact on the convergence rate of z, but y still monotonically increases and converges to 1, indicating that the proportion of the third-party enterprises choosing the "acceptance" strategy continues to increase over time until all third-party enterprises eventually choose "acceptance". curve of x (x = ) is shown in Figure 8. As shown, the different initial values of y and z have an impact on the convergence rate of x, but x still monotonically increases and converges to 1, indicating that the proportion of the local governments choosing the "implementation" strategy continues to increase over time until all local governments eventually choose "implementation". (2) For the fixed y initial value (e.g., y = 0.5) to achieve the ideal state of equilibrium, it needs to satisfy B = xz(N − n) + z(n + b) − b > 0. From a random initial value of x and z, the evolution curve of y (y = ) is shown in Figure 9. As shown, the different initial values of x and z have an impact on the convergence rate of y, but y still monotonically increases and converges to 1, indicating that the proportion of the polluting enterprises choosing the "acceptance" strategy continues to increase over time until all polluting enterprises eventually choose "acceptance". (3) For the fixed z initial value (e.g., z = 0.5) to achieve the ideal state of equilibrium, it needs to satisfy C = xy(R − r) + y(r + c) − c > 0. From a random initial value of x and y, the evolution curve of z (z = ) is shown in Figure 10. As shown, the different initial values of x and y have an impact on the convergence rate of z, but y still monotonically increases and converges to 1, indicating that the proportion of the third-party enterprises choosing the "acceptance" strategy continues to increase over time until all third-party enterprises eventually choose "acceptance".   curve of x (x = ) is shown in Figure 8. As shown, the different initial values of y and z have an impact on the convergence rate of x, but x still monotonically increases and converges to 1, indicating that the proportion of the local governments choosing the "implementation" strategy continues to increase over time until all local governments eventually choose "implementation". (2) For the fixed y initial value (e.g., y = 0.5) to achieve the ideal state of equilibrium, it needs to satisfy B = xz(N − n) + z(n + b) − b > 0. From a random initial value of x and z, the evolution curve of y (y = ) is shown in Figure 9. As shown, the different initial values of x and z have an impact on the convergence rate of y, but y still monotonically increases and converges to 1, indicating that the proportion of the polluting enterprises choosing the "acceptance" strategy continues to increase over time until all polluting enterprises eventually choose "acceptance". (3) For the fixed z initial value (e.g., z = 0.5) to achieve the ideal state of equilibrium, it needs to satisfy C = xy(R − r) + y(r + c) − c > 0. From a random initial value of x and y, the evolution curve of z (z = ) is shown in Figure 10. As shown, the different initial values of x and y have an impact on the convergence rate of z, but y still monotonically increases and converges to 1, indicating that the proportion of the third-party enterprises choosing the "acceptance" strategy continues to increase over time until all third-party enterprises eventually choose "acceptance".

Parameter Analysis
In addition, the following equation set calculates a central point in the tripartite evolutionary game model of third-party environmental pollution control.
Assuming that M − m = ∆m, N − n = ∆n, R − r = ∆r, and a +a = A, we can obtain the central point solution.
The central point solution shows that x * , y * , z * are all related to the parameters a , a , b, c, M, N, R, m, n, and r. By solving a partial derivative of x * , y * , z * with respect to parameters N, R, m, n, and r, it can be observed that the outcome is uncertain and that these parameters are not related to the fiscal policy; therefore, this paper does not discuss the influence of parameters N, R, m, n, and r.
(1) Parameter A = a + a represents the fiscal expenditure policy formulated by the local government to encourage the polluting enterprise and third-party enterprise to choose third-party control. Solving a partial derivative of x * , y * , z * with respect to parameter A yields * > 0, * < 0, * < 0. Therefore, the change in parameter A will affect the decision-making behaviors of the three parties under the premise that the other parameters remain unchanged.
A. When z > ( ) , the local government will choose to implement the fiscal policy. Increasing A leads to decreasing y * and z * . Thus, z = ( ) moves inward, leading the

Parameter Analysis
In addition, the following equation set calculates a central point in the tripartite evolutionary game model of third-party environmental pollution control.
Assuming that M − m = ∆m, N − n = ∆n, R − r = ∆r, and a 1 + a 2 = A, we can obtain the central point solution.
The central point solution shows that x * , y * , z * are all related to the parameters a 1 , a 2 , b, c, M, N, R, m, n, and r. By solving a partial derivative of x * , y * , z * with respect to parameters N, R, m, n, and r, it can be observed that the outcome is uncertain and that these parameters are not related to the fiscal policy; therefore, this paper does not discuss the influence of parameters N, R, m, n, and r.
(1) Parameter A = a 1 + a 2 represents the fiscal expenditure policy formulated by the local government to encourage the polluting enterprise and third-party enterprise to choose third-party control.
Solving a partial derivative of x * , y * , z * with respect to parameter A yields ∂x * ∂A > 0, ∂y * ∂A < 0, ∂z * ∂A < 0. Therefore, the change in parameter A will affect the decision-making behaviors of the three parties under the premise that the other parameters remain unchanged.

A.
When z > a 1 +a 2 y(M−m) , the local government will choose to implement the fiscal policy. Increasing A leads to decreasing y * and z * . Thus, z = a 1 +a 2 y(M−m) moves inward, leading the probability of z > a 1 +a 2 y(M−m) to increase. Therefore, the possibility of the local government choosing to implement fiscal policy will increase.
(2) The strength of the fiscal policy, the risks to the polluting enterprise and the third-party enterprise, and the benefit to the local government from third-party control affect the speed of the convergence of the local government, the polluting enterprise and the third-party enterprise towards the ideal ESS (implementation, acceptance, acceptance). Improving the strength of the fiscal policy, reducing the risks to the polluting enterprise and third-party enterprise, and improving the benefit to the local government are conducive for promoting third-party environmental pollution control.

Policy Implications
According to the above mentioned conclusions, to promote third-party environmental pollution control in China, this paper offers fiscal policies and suggestions from the following perspectives.
(1) Formulate fiscal policies for third-party environmental pollution control The Chinese government could formulate further fiscal expenditure and revenue policies for third-party environmental pollution control.
Regarding fiscal expenditure policies, it is suggested to establish special financial funds and carry out the policy of "Awards replace subsidies". Firstly, international, national and provincial financial funds could be set up. Governments of different countries could cooperate to support third-party enterprises for participating in international third-party control projects. Chinese central government could provide special transfer funds to third-party control projects focusing on air, water and soil pollution control. Chinese local government could arrange provincial financial funds to support third-party control projects in key industries, key pollution sources, key regions, and river basins. Secondly, subsidies for third-party control projects should be replaced by awards according to the investment scale. The incentive funds should be used during the whole life cycle of the projects, including pre-project expense awards and operation awards.
Regarding fiscal revenue policies, it is suggested to implement preferential policies for environmental protection tax, value-added tax, real estate tax and land use tax. Firstly, the environmental protection tax incentives could be increased by 10% for polluting enterprises that entrust third-party enterprises to conduct pollution control, on the basis of the current environmental protection tax incentives. Secondly, the VAT rate could be calculated and levied in accordance with the 6% VAT rate of modern service industry by creating the catalogue of value-added tax preferences for third-party control products and services in the environmental service industry. Thirdly, the real estate tax and land tax involved in third-party environmental pollution control facilities could be exempted, with reference to the exemption of real estate tax and land use tax for the units appropriated by Chinese financial departments.
(2) Implement fiscal policies in a dynamic and gradual manner During the initial stage, the government should strengthen the implementation of fiscal policies to encourage third-party control. With the continuous improvement of the third-party control market, the implementation of fiscal policies can be gradually reduced. The fiscal policies should be adjusted and optimized according to the behaviors of the local government, the polluting enterprise, and the third-party enterprise. The optimal incentive policy system should be established to promote good interaction among the participants.
(3) Improve the market mechanism of third-party environmental pollution control There are risks in third-party environmental pollution control due to the imperfection of the market mechanisms. Such control requires the government, the polluting enterprise, the third-party enterprise and the public to take various measures. The government should provide normative guidance by clarifying the responsibilities and setting service standards. The participants should share information by establishing good faith archives and implementing credit ratings. Strict supervision could be conducted to solve the moral hazard problem through strict environmental law enforcement and public participation.
(4) Assess the environmental performance of local governments Along with social and economic indicators, environmental indicators should be included as an important part of the central government's appraisal of local governments' performance. A lifelong accountability system for ecological and environmental damage should be established, and leading officials who leave their posts should be audited. In addition, the indicators used in environmental performance assessment of local governments should be constantly updated according to the main function zoning.

Limitations
This study contends with several limitations. China has implemented third-party environmental pollution control for a short time and lacks the data of fiscal policies in third-party environmental pollution control. Therefore, the analysis in this paper is mainly at the theoretical level. Although evolutionary game method has achieved great success in analyzing the mechanism of policy, the model is established based on specific hypotheses and may ignore the possible changes of the local government, the polluting enterprise and the third-party enterprise by themselves. The conclusion of this paper is principally to provide reference for the government to formulate fiscal policies and other policies to promote the third-party environmental pollution control in China.