1. Introduction
The environmental problems arising from economic development are widely related to the government and the public, and it is difficult to balance the efficiency and effectiveness of traditional environmental pollution management approaches. For this reason, the state has tried a new governance idea in the environmental field, namely the third-party governance of environmental pollution, which means that independent third-party pollution control enterprises other than the sewage enterprises and government regulatory departments undertake the task of environmental pollution control [
1]. In 2013, the third-party management model of environmental pollution was first proposed in the Decision of the Central Committee of the Communist Party of China on Several Major Issues of Comprehensively Deepening Reform, and in 2015 the State Council issued the Opinions on Promoting Third-Party Management of Environmental Pollution, marking the full implementation of third-party management for environmental pollution from a pilot reform to the national level [
2]. The implementation of this system in China’s environmental pollution sector has achieved certain results, but the existing problems are also increasingly evident, including:
Government departments have “sometimes tight, sometimes loose” regulatory behavior, and the regulatory capacity needs to be improved [
3];
The reward and punishment system of the government regulatory department in the third-party environmental pollution management system is not yet perfect, and the existing reward and punishment system is too singular to mobilize the enthusiasm of the participants [
4];
Some pollution control enterprises and professional environmental testing institutions of the third-party environmental pollution management system take advantage of information asymmetry to adopt irregular governance and rent-seeking behaviors, which seriously undermine the effectiveness of the governance system [
1].
Therefore, considering the demands of governance practice, there is an urgent need to explore a set of effective regulatory strategies for the third-party governance of environmental pollution to stimulate the compliance behavior of pollution treatment participants and punish and restrain their violations, so as to provide support for the compliance governance of environmental pollution projects and the effectiveness of the governance system.
At present, the research on the regulation of the third-party governance of environmental pollution focuses on two aspects; one is to propose regulatory issues based on the institutional level, and the other is to analyze the behavior of governance subjects through quantitative models. Most scholars are committed to raising problems with regulation at the institutional level, such as Tang [
3] pointing out that the key to unblock the system of third-party governance is to regulate the behavior of governance subjects by using incentive regulation theory. The literature [
5,
6] indicates that the regulatory system of third-party governance for environmental pollution in China has not been perfected, and third-party governance needs to be incorporated into the legal track. Very few scholars have analyzed the behaviors of pollution treatment enterprises, pollution emission enterprises, the public, and the government by means of quantitative research; for example, Du [
7] constructed a two-party evolutionary game model of the government and third-party pollution treatment enterprises, indicating that the evolution of the government and pollution treatment enterprises at this stage depends on the relative payment of various strategies. Chu [
8] constructed a tripartite game model of the public, local government, and central government based on the prisoner game dilemma of public participation and local government, and the study showed that the public regulation of environmental impacts by local governments could be an alternative to central government regulation. In addition, scholars such as [
9,
10] also applied a quantitative analysis to study the strategy choices of governance subjects. The research results of regulatory strategies in other fields are quite fruitful, including the use of game theory to analyze the regulatory strategies of the different evolutionary paths of each subject and the benefits of public participation in regulation, which provide a lot of suggestions for high-quality regulation by government regulators. For example, Zhu and He [
11] studied the supervision strategy of regulating the quality of online goods. He [
12] studied the regulation of green product quality. Kong [
13] studied the regulation of product quality in industrial clusters. It can be seen that the above-mentioned literature on third-party governance focuses on analyzing the strategy choices of governance subjects, with less research on regulatory strategies and especially a lack of quantitative research on regulating the behavior of governance subjects through reward and punishment mechanisms.
On the other hand, there is less research on professional environmental testing institutions. Due to the information asymmetry between government regulators and pollution treatment enterprises, government regulators will commission professional environmental testing agencies to test the effectiveness of pollution treatment enterprises. However, it is difficult to curb the rent-seeking behavior of pollution treatment enterprises from professional environmental testing agencies. Pollution control enterprises that follow the rules of environmental pollution generally have high governance costs, technological innovation difficulties, and other problems, while false pollution control saves on the cost of governance and the benefits of space. Under the constraint of “sometimes tight, sometimes loose” government regulations, pollution control enterprises have a tendency to seek rent from professional environmental testing agencies to obtain governance approval [
1]. Driven by huge rent-seeking interests, professional environmental testing institutions are also bound to have the risk of rent-seeking intentions [
14]. It can also be seen through the literature [
1,
8,
14,
15,
16] that the subjects are mostly government regulators, sewage enterprises, pollution treatment enterprises, and the public when scholars research the third-party governance of environmental pollution.
In recent years, behavioral operations research has been widely used in various fields [
17,
18,
19], and game theory is a standard tool for analyzing strategy choices. If we disregard the assumption of “perfect rationality” in traditional game theory and consider the whole dynamic system of environmental governance, the strategy choices of government regulators, pollution control enterprises, and other subjects with limited rationality will no longer be a negligible factor [
20]. Evolutionary game theory takes into account not only the limited rationality and incomplete information state of the subject, but also the dependence and interaction of the subject’s strategy choice over time, which is actually more relevant than the traditional game theory. Therefore, in the state of limited rationality and incomplete information of the governance subjects, the evolutionary game theory is an effective method to study the regulatory strategy of environmental pollution third-party governance.
The research in this paper differs from the aforementioned scholars’ research in three aspects. First, it considers the problem of false pollution control by collusion between pollution control enterprises and professional environmental testing institutions. Second, under the regulatory incentive and constraint mechanism, it analyzes the performance behavior of each subject through evolution from the self-interest of decision makers, so that they spontaneously form the “compliance” behavior a long-term evolutionary stability strategy, rather than only restraining the behavior of subjects through laws and regulations. Third, the system optimization mechanism, based on the dynamic performance payment mechanism researched by scholars, proposes improvement measures, introduces a non-linear dynamic reward and punishment mechanism to control the stability of the evolutionary system, and provides suggestions and countermeasures to improve the governmental regulatory mechanism of the environmental pollution third-party treatment.
3. Model Analysis
3.1. Evolutionary Stability Analysis of the Strategy of Pollution Control Companies
Assuming that the expected return of a pollution treatment company to adopt a rule-based treatment strategy is
, the expected return to adopt a false treatment strategy is
, and the expected average return to adopt a mixed strategy is
, then
,
, and
are expressed as in Equations (1)–(3), respectively:
From Equations (1)–(3), the replication dynamics equation for the adoption of the rule-based pollution control strategy by the pollution treatment enterprises can be obtained as Equation (4):
The derivative of Function (5) is obtained by taking the derivative of Equation (4):
Let , whereby the two stable states of the replica dynamic equation, i.e., and , can be solved. Thus, the evolutionary stability of the replicated dynamic system can be discussed in the following three cases: when (0), ≡ 0, at which time the system reaches a stable state no matter what value is taken. When , , , at this time is the evolutionarily stable strategy and is the unstable strategy. When , , , at this time is the unstable strategy and is the evolutionarily stable strategy.
The evolutionary phase diagram of the strategy of pollution control companies is shown in
Figure 2.
Remember
from
Figure 1, where we can see that
, so the surface will be divided into two spaces
and
. When the initial region falls in
, the strategy of the pollution treatment companies will evolve into false pollution treatment, and when the initial region falls into
, the strategy of the pollution treatment companies will evolve into rule-based pollution treatment.
Proposition 1. The adoption of a compliance strategy by pollution treatment companies is correlated with the revenue obtained after environmental pollution treatment , the cost of speculation , the rent-seeking cost of pollution treatment companies , the incentive and punishment of government regulators , and the cost saved by the false pollution treatment relative to compliance , which is positively correlated with respect to , , , and negatively correlated with respect to .
Proof. Figure 2 shows that the probability of a pollution treatment company adopting a false treatment strategy is the volume
of
and the probability of adopting a compliance strategy is the volume
of
. Here, we calculate
as shown in Equation (6):
The partial derivative of yields , , , , . Thus, increasing , , , or decreasing can make increase, i.e., the probability of pollution treatment enterprises following the rules increases. □
From Proposition 1, it can be seen that guaranteeing the profits of the pollution treatment enterprises after the environmental treatment and appropriately increasing the strength of the rewards and punishments from the governmental regulatory departments can effectively motivate pollution treatment enterprises to adopt compliance strategies. At the same time, government regulators can also increase the speculative costs for pollution treatment enterprises by increasing the credibility of the media’s opinions and influence, and by expanding the competitiveness of enterprises to promote the adoption of compliance pollution treatment strategies.
3.2. Evolutionary Stability Analysis of the Strategies of Professional Environmental Testing Institutions
Assuming that the expected return of the professional environmental testing institutions adopting a rent-seeking rejection strategy is
, the expected return of adopting an intentional rent-seeking strategy is
, and the expected average return of adopting a mixed strategy is
, then
,
, and
are expressed as in Equations (7)–(9), respectively:
From Equations (7)–(9), the replication dynamic equation for a rent-seeking strategy adopted by a professional environmental testing organization can be obtained as:
The derivative function can be obtained by deriving (10) as shown in (11):
Let , which can be solved for the two stable states of the replica dynamic equations, namely and . Therefore, the evolutionary stability of the replica dynamic system can be discussed in the following three cases: when (0, ≡ 0, at this time the system reaches a stable state, regardless of the value of . When , , , at this time is an evolutionarily stable strategy and is an unstable strategy. When , , is the unstable strategy and is an evolutionarily stable strategy.
The evolutionary phase diagram of the professional environmental testing agency’s strategy is shown in
Figure 3.
From
Figure 3, it can be seen that the
surface divides
into two spaces,
and
. When the initial region falls in
, the strategy of the professional environmental testing institutions will evolve to rent-seeking rejection, and when the initial region falls in
, the strategy of the professional environmental testing organization will evolve to intentional rent-seeking.
Proposition 2. The adoption of a rent-seeking strategy by professional environmental testing institutions is correlated with the reward and punishment of professional environmental testing institutions by government regulators (), the speculative cost of professional environmental testing institutions , and the rent-seeking cost of pollution treatment enterprises . Specifically, it is positively correlated with respect to and and negatively correlated with respect to .
Proof. Figure 3 shows that the probability of a professional environmental testing institution adopting a rent-seeking strategy of rejection is the volume
of
, and the probability of it adopting an intentional rent-seeking strategy is the volume
of
. Here, we calculate
as in Equation (12).
The partial derivative of yields , , , . Thus, increasing or or decreasing or can make increase, i.e., the probability of rejecting rent-seeking by professional environmental testing institutions increases. □
From Proposition 2, it can be seen that government regulators appropriately increase the rewards and punishments for professional environmental testing institutions, as well as increasing their rent-seeking costs by exposing the rent-seeking behavior of professional environmental testing institutions through the media, which can reduce the speculative behavior of professional environmental testing institutions. In addition, active supervision by government regulators reduces the probability of rent-seeking behavior and facilitates fair testing by professional environmental testing institutions.
3.3. Analysis of the Evolutionary Stability of the Government Regulator’s Strategy
Assuming that the government regulator’s expected return from adopting a positive regulatory strategy is
, the expected return from adopting a negative regulatory strategy is
, and the expected average return from adopting a mixed strategy is
. Then,
,
, and
are expressed as in Equations (13)–(15), respectively:
From Equations (13)–(15), the replication dynamic equation for an aggressive regulatory strategy by government regulators can be obtained as:
The expression for the derivative function of (16) is shown in (17):
Let , whereby the two stable states of the replica dynamic equations, i.e., and , can be solved. Thus, the evolutionary stability of the replica dynamic system can be discussed in the following three cases: when (0), ≡ 0, at which time the system reaches a stable state, regardless of the value of . When , , , at this time is the evolutionarily stable strategy and is the unstable strategy. When , , , at this time is the unstable strategy and is the evolutionarily stable strategy.
The evolutionary phase diagram of the government regulator’s strategy is shown in
Figure 4.
From
Figure 4, it can be seen that the
surface divides
into two spaces,
and
. When the initial region falls in
, the government regulator’s strategy will evolve to positive regulation, and when the initial region falls in
, the government regulator’s strategy will evolve to negative regulation.
Proposition 3. The positive regulatory strategy adopted by government regulators is related to the rewards and punishments (,) of government regulators for pollution control enterprises and professional environmental testing institutions and the penalties imposed by higher government regulators on government regulators when they regulate negatively, and is positively correlated with respect to and negatively correlated with respect to , .
Proof. Figure 4 shows that the probability of a government regulator adopting a positive regulatory strategy is the volume
of
, and the probability of adopting a negative regulatory strategy is the volume
of
. The calculation of
is shown in Equation (18):
Taking the partial derivative of yields , , , , . Thus, increasing or decreasing , increases the volume of . □
From Proposition 3, it can be seen that the greater the punishment set by the government regulators for the pollution control enterprises and professional environmental testing institutions, the higher the rate will be of positive regulation by the government regulators and the more beneficial it will be to promote positive regulation by the government regulators, while higher amounts of rewards set will be detrimental to their own regulatory duties, leading to a decrease in the rate of positive regulation. Punishment via negative regulation by higher government regulators is an important factor for positive regulation by government regulators.
3.4. System Equilibrium Point Stability Analysis
Solving the system of equations consisting of Equations (4), (10) and (16), we can obtain nine equilibrium points in the game process of pollution control enterprises, professional environmental testing agencies, and government regulators, which are
(0,0,0),
(0,1,0),
(0,0,1),
(0,1,1),
(1,0,0),
(1,1,0),
(1,0,1),
(1,1,1),
(
,
,
), where
(
,
,
)
,
. Here,
,
,
. The stability of the evolving system can be judged using the Jacobi matrix, and according to the stability determination theorem of the Lyapunov ordinary differential equation, the system is asymptotically stable when the eigenvalues of the Jacobi matrix are all negative real parts; the system is unstable if the eigenvalues of the Jacobi matrix have at least one positive real part [
21,
22]. The expression of the Jacobi matrix
J is as follows:
The corresponding eigenvalues of each equilibrium point are solved by bringing each equilibrium point into the Jacobi matrix, and the stability analysis of the equilibrium point is shown in
Table 2 and
Table 3.
As can be seen from
Table 3, the system will evolve to
(0, 0, 1) and
(1, 1, 0) in different initial states when government regulators offer fewer rewards and penalties to pollution control companies and professional environmental testing agencies and when rent-seeking behavior by pollution control companies and third parties generates high interest. For the strategy combination of
(0, 0, 1), the pollution treatment enterprise falsely treats pollution, the professional environmental testing agency intends to seek rent, and the falsely treated environmental protection project passes inspections, causing social environmental pollution. In order to avoid this situation, government regulatory departments should reasonably develop reward and punishment programs to appropriately increase rewards and punishments for pollution treatment enterprises and professional environmental testing agencies to ensure that the project follows the rules of governance.
In summary, based on the model assumptions, two evolutionary stabilization strategies, (0, 0, 1) and (1, 1, 0), can be achieved. The strategy combination of false pollution control, intentional rent-seeking, and active regulation can be avoided under a reasonable reward and punishment system of government regulators. The strategy combination of compliance, rent-seeking, and negative regulation, however, means that the pollution control companies will spontaneously adopt the compliance strategy, which means that the pollution control companies can still evolve the compliance behavior even under the negative regulation of the government. In fact, from the perspective of the managers, the evolutionary stabilization strategy of the equilibrium point (1, 1, 1) is what we expect, in which the pollution control enterprises follow the rules and regulations, the third-party agencies refuse to seek rent, the government regulators actively supervise the process, and the three parties participate in the environmental pollution control projects. To achieve the equilibrium point (1, 1, 1), we need to reintroduce incentives and penalties for pollution control enterprises and professional environmental testing agencies, which will be discussed in depth in this paper.
4. Computational Experimental Simulation
Based on the above analysis, in order to more intuitively reflect the influencing factors of the evolutionary process of environmental project management, MATLAB 2016b is used for the numerical simulation. The simulation start time is set to 0, the end time is set to 30, the simulation unit is not specifically set, the threshold of the model in this paper is in the range of [0, 30], and the simulation end time is set to satisfy the iteration threshold of the evolutionary stabilization strategy of the model. Then, we are given array 1:
,
,
,
,
,
,
,
,
,
,
,
, where the unit of the parameters is one million; that is, the initial state of the point
(1,1,0) in
Table 3 is satisfied:
and
,
. In this case, the evolution of the pollution control enterprises, professional environmental testing agencies, and government regulators is as shown in
Figure 5, and the three-party evolutionary game system converges to
(1,1,0).
In order to clearly observe the evolutionary path of the three game subjects, we assume that the probability of implementing each strategy in the initial state is 0.5 for pollution control enterprises, professional environmental testing institutions, and government regulatory departments, and we conduct a simulation analysis for some factors.
(1) Incentives from government regulators for pollution treatment enterprises. Here, we decrease
to 3 and increase
to 12, respectively, comparing
, as shown in
Figure 6a. From
Figure 6a, it can be seen that as
increases, the probability of convergence of the pollution treatment enterprises to the compliance strategy increases, but the rate of convergence of government regulators to active regulation decreases significantly, then finally evolves to 0 and reaches a steady state. At the same time, there is a significant increase in the probability of professional environmental testing institutions rejecting rent-seeking in the
increase process. This indicates that under the condition of active supervision by governmental regulatory authorities, the reasonable formulation of incentive measures plays a positive role in the project’s rule-based pollution control and is conducive to the project’s environmental governance.
(2) The accountability of higher levels of government to government regulators. Here, we decrease
to 6 and increase
to 24, respectively, comparing
, as shown in
Figure 6b. From
Figure 6b, we can see that an increase in
leads to an increase in the probability of active regulation by government regulators. Through
Figure 5, we can also find an interesting phenomenon, where for any one curve the evolutionary curve of the government regulators first increases and then decreases, which indicates that the government regulators intend to adopt a positive regulation strategy in the early stage of evolution, but with the reward and punishment mechanism for pollution control enterprises and professional environmental testing agencies and under the passage of time, we find that positive regulation is not the dominant strategy, so government regulators will change the strategy and eventually choose negative regulation, but the supervision of government regulators by higher levels of government can reduce the rate of regulation by government regulatory departments who adopt negative regulatory strategies, which is conducive to the pollution treatment enterprises following the rules and regulations to control environmental pollution.
The MATLAB simulation shows that the conclusions of the theoretical analysis and simulation analysis are consistent, and the simulation results for the remaining parameters are consistent with the theoretical analysis; in fact, we can analyze the size and sensitivity analyses of the parameters in the arbitrary evolution process, but this is not repeated to save space.
6. Conclusions
6.1. Research Findings and Significance of the Study
Based on the third-party management of environmental pollution, this paper considered the rent-seeking behavior of pollution treatment enterprises and professional environmental testing institutions and constructed an evolutionary game model of pollution treatment enterprises, professional environmental testing institutions, and government regulatory departments. Firstly, the stability and influencing factors of the respective strategies of the three subjects were analyzed. Secondly, the evolutionary stability strategy of the whole dynamic system was analyzed and the evolutionary path of the three-party evolutionary game was revealed intuitively through computational experimental simulations to verify the correctness of the theoretical analysis. Finally, the system was optimized to achieve the desired evolutionary stability strategy. The main conclusions drawn from the theoretical and simulation analyses are as follows:
(1) Appropriate increases in rewards and penalties by government regulators can not only incentivize pollution control enterprises to follow the rules of pollution control, but can also regulate the behavior of professional environmental testing agencies, although excessive rewards will not be conducive to the performance of government regulators themselves;
(2) The existing static reward and punishment mechanism of the government regulatory department fails to make timely adjustments according to the strategic choices of each subject, and there is no legal system to regulate the behavior of governance subjects;
(3) The adoption of a linear dynamic punishment mechanism by government regulators plays a controlling role in the stability of the system, but it greatly increases the probability of rent-seeking behavior and poses a greater threat to environmental governance;
(4) The non-linear dynamic reward and punishment mechanism considers both dynamic incentives and dynamic constraints to make the system achieve the desired evolutionary stability strategy, i.e., the pollution control enterprises follow the rules and regulations, the professional environmental testing agencies refuse to seek rent, and the government actively supervises the system as the final evolutionary direction.
The main significance of the work done in this paper is that firstly we constructed an evolutionary model of environmental pollution with a third-party governance and regulatory strategy, analyzed the behavior of participants from the perspective of their own interests, and made their spontaneous “compliance” behavior a long-term evolutionary and stable strategy, which enriched the theoretical research of the environmental pollution third-party governance system. Second, based on the deficiencies of the dynamic performance payment mechanism, we innovatively introduced a non-linear dynamic reward and punishment mechanism to control the stability of the evolutionary system, which solved the single problem of rewards and punishments in the third-party environmental pollution governance system. Finally, this paper has a certain practical significance; that is, it can provide new ideas for environmental pollution governance in other countries or regions (partly).
6.2. Management Insights
The above findings give us management inspirations in the area of third-party environmental pollution management for the following areas:
(1) To improve the government service capacity, build a “sometimes tight, sometimes loose” type of government regulation and accountability mechanism, improve the institutional reform process for third-party environmental pollution management, and ensure that government regulations are effectively applied to the environmental pollution management project itself;
(2) Government regulatory departments should establish and improve their multi-directional supervision mechanism, give full play to the supervisory role of the public and social media, and set up a reporting reward mechanism to fundamentally improve the efficiency of the supervision process;
(3) A non-linear dynamic reward and punishment mechanism should be implemented to monitor the performance of the pollution control enterprises and professional environmental testing agencies in real time, to stimulate the compliance behaviors of enterprises and professional environmental testing agencies, and to severely punish false pollution control and rent-seeking behavior;
(4) The risk of technological innovation in pollution treatment enterprises should be reduced and the determination of pollution treatment enterprises to follow regulations should be strengthened. In terms of technological innovation, the government should provide technical support to encourage environmental pollution control technology innovation in enterprises so as to reduce the cost of technological innovation in pollution control enterprises.
6.3. Research Limitations of This Paper
One of the most common assumptions used in the environmental pollution third-party governance model constructed in this paper is that the participants are finitely rational subjects. However, in the real cases, there are inevitably non-finite rational participants due to the differences in political, economic, cultural, and environmental factors in some regions and countries. Considering the behavior of non-finite rational participants and narrowing the differences between the theoretical model and the real cases will be a future research topic for the authors.