The Evolutionary Game of Cooperative Air Pollution Management under Complex Networks

Abstract

In this paper, based on complex networks and evolutionary game theory, we use Text Mining and Analytics, MATLAB Simulation, and other technical means to study the decision-making process of each subject in the collaborative air pollution management network, taking the “limited rational” local government as the decision-making subject of the evolutionary game. The study finds that the cooperative network of small-scale air pollution management is a very important element in the evolutionary process. The small-scale air pollution collaborative governance network has the effect of significantly improving the evolution speed of local government collaborative governance decisions in the network. It can better mobilize local governments to participate in collaborative air pollution governance and realize the cooperative emergence with the ratio interval of income heterogeneity in [0.6, 1], preference heterogeneity in [1.2, 1.4], and allocation heterogeneity in [0.6, 1]. The results of the study are consistent with the actual situation, which verifies the validity and operability of the model. Finally, the article also proposes countermeasures to improve the dilemma of air pollution synergy caused by regional heterogeneity.

1. Introduction

In recent years, the prevention and control of air pollution in China have achieved significant results. However, in March 2014, Environmental Authorities assessed air quality in key cities and regions of China. The results showed that China’s air quality still needs continued improvement, especially cross-regional pollution in the Beijing-Tianjin-Hebei (Hereinafter referred to as “BTH”) region, which requires more sustained attention.

In 2010, the Ministry of Environmental Protection issued the “Guidance on Promoting Joint Prevention and Control of Air Pollution to Improve Regional Air Quality”. This is the first time to break through the limits of territorial governance and propose that air pollution requires the joint response of multiple local governments. In 2013, the State Council proposed a governance model combining territorial governance and collaborative governance of air pollution, and prepared the “Action Plan for the Prevention and Control of Air Pollution”. From 2018, the General Office of the State Council has issued relevant notices, requiring BTH, Yangtze River Delta, and other regions to set up special action groups for air pollution management, further standardizing and organizing the collaborative management of air pollution. The Development and Reform Commission, the Ministry of Ecology and Environment, and other ministries and commissions have issued several policy documents related to the collaborative treatment of air pollution in BTH in recent years, such as the “BTH Energy Cooperative Development Action Plan (2017–2020)”, “BTH and the surrounding areas in the autumn and winter of 2018–2019 comprehensive treatment of air pollution attack action plan”, “BTH and the surrounding areas 2019–2020 Autumn and Winter Air Pollution Comprehensive Management Tackling Action Program”, etc. These were proposed to jointly improve the level of BTH energy governance and management, strengthen the heavy pollution weather emergency linkage, and deepen regional cooperation and joint law enforcement. In the current process of air pollution control, due to the imbalance of development between regions, local governments do not have a sound mechanism of benefit distribution and coordination, as a limited rational “economic man” of some local governments have the idea of getting something for nothing, which is known as the “free-rider phenomenon” and “shirking phenomenon”. This leads to an unclear division of labor between departments, insufficient command and coordination, and poor overall linkage, which increases the cost of air pollution control and reduces the performance of air pollution control [1].

Therefore, it is an important problem to explore the dynamic factors that affect the difficult situation of coordinated control of air pollution in our region and solve the cross-regional and cross-sectoral problems of air pollution. Based on this, this paper analyzes the topology of the collaborative air pollution management network in China, establishing the evolution game model of air pollution co-management under the complex network; conducting the evolution game study, which is most in line with the actual situation between the subjects of air pollution co-governance and the whole network; laying a theoretical foundation for the practice of air pollution co-management in China; and further enriching the relevant research theories.

2. Literature Review

Decision-making is a complicated task [2]. Especially, it is hard to select the best choice when the environment is dynamically changing [3]. Deciding how to select the best solution and how to interpret solution results highly depends on the peculiarities of the methodology [4] and the personnel involved in the problem-solving process [5]. However, the selection of the problem solution approach involves local and international governance requirements [6] and features defining each particular problem [7].

With the advent of a risk society, air pollution occurs more and more frequently, and it is difficult for a single local government department to deal with complex air pollution alone, which requires cooperation among different local government subjects to jointly resolve the risks and hazards caused by air pollution [8,9]. In real situations, decision makers do not always have all decision information and do not always make correct judgments. Therefore, evolutionary game analysis, which considers limited rationality, has begun to be recognized and used [10,11]. Evolutionary game theory integrates the dynamic evolutionary process with game analysis, and the game subject no longer has perfect rationality, but achieves the equilibrium state of the system through continuous trial and error and continuous learning of higher-yielding strategies. Evolutionary game theory has been widely studied in the fields of social governance and public opinion communication [12,13,14]. For example, Suyong Zhang et al. studied an evolutionary game system to reduce manufacturers’ pollution emissions, in which government departments, as well as manufacturers as decision makers, analyzed the equilibrium strategy of the evolutionary game under a dynamic carbon trading price and the influence of government policies on manufacturers’ decisions through simulation [15]. To solve the problem of the difficulty of extracting features in crowd counting under low-quality conditions, Ruihan Hu et al. developed a model called the Audiovisual Multiscale Network (AVMSN) to achieve a lower mean absolute error [16].

In terms of collaborative air pollution governance, scholars have studied the collaborative governance mechanism of multiple subjects based on evolutionary game theory and explored the conditions that enable multiple subjects to reach long-term and stable cooperation. For example, Ma et al. used evolutionary game theory to analyze the behavior of participating subjects under the supervision/non-supervision of higher-level government and elaborated on the importance of higher-level government participation [17]. Gao et al. studied the factors affecting the formation of an air pollution collaborative governance coalition by building an evolutionary game model between central and local governments and found that the cost of cooperation and central constraints have important effects on the stability of collaborative governance [18]. Jidong Yang et al. found that individual local governments need behavioral consistency and will refer to and imitate the strategies of their game opponents to change their governance behavior and minimize the differences in the level of collaborative air pollution governance with other subjects [19]. Dass A et al. presented the two main classifications of air pollution, namely, outdoor and indoor pollution, discussing their effects, sources, and possible ways to reduce their growth levels. They focused on the analysis of pollution data from government sources and used artificial intelligence tools to predict likely future levels. Using vehicle emissions as the main source of outdoor pollution, atmospheric carbon monoxide (CO) levels are modeled and predicted. Fuzzy type-1 and type-2 systems have been used to model and predict CO levels in three metropolitan cities of India, namely, Bangalore, Delhi, and Mumbai. A comparative analysis of these two methods has also been carried out, and the simulation results demonstrate the superiority of the fuzzy type-2 model. The paper also shows the effect of different learning rates used for learning parameters [20].

In Axelrod’s study, it was first specified that the network relations between individual subjects on the network affect the evolution of the whole network [21]. The real study of evolutionary games on complex networks was carried out by scholars such as Martin A. Nowak. Their study, for the first time, combined networks and games, based on the prisoner’s dilemma model and considering only two simple participants: cooperative subjects and betrayed subjects, and studying the two-dimensional space in which these subjects have placed the results of the game in the network. In such networks, both cooperators and betrayers persist indefinitely (in the proportion of predictable long-term mean fluctuations), and the study has potential implications for the study of the dynamics of various spatially extended systems [22]). Subsequently, as the theory of complex networks has been further investigated, scholars have focused more attention on irregular networks, such as small-world networks, and Abramson et al., based on the prisoner’s dilemma model, studied the game scenarios on small-world networks. In their study, they clarified that each game subject adopts a strategy update rule that imitates the most optimal one in each game round, and they found that the evolution of this network is influenced by the network topology through a large number of simulations [23].

In recent years, the study of games on complex networks has been gradually integrated with real-world management problems. Xiangqian Li et al. summarized the development patterns of provincial innovation chains in China and analyzed the evolutionary paths in combination with time series. They found that there is a certain group proximity of provincial innovation chains in China and an ascending development trend. Among them, increasing investment and efficiency are the dominant drivers of development, and there is a gradual shift from high investment to high efficiency [24]. Based on the small-world network theory, Houming Fan et al. established a complex network model of the Arctic environmental emergency response and governance system. The strategies of each country in the complex network were verified through the evolutionary game simulation of the system model. The calculation results show that environmental emergency response and governance benefits affect the cooperation strategies of countries, and long-term benefits and payoff intensity are the key parameters of environmental emergency response and governance benefits [25]. In the study by Fazheng Chen et al., an ecological management system indicator system for rural agroecosystems was constructed based on complex network theory with big data as the research background. Fertilizer consumption, water pollution level, pest and disease level, carbon and nitrogen uptake, and agricultural economic benefits of rural agroecosystems in a region were used as the system indicators of the ecological management system. The relevant data in the network were collected and processed by using data mining technology in big data, and the agroecosystem was analyzed and understood through the complex network, and the data of each indicator was finally calculated and analyzed [26].

Based on the previous research literature, this paper addresses the dilemma of collaborative air pollution management in the region, from the perspective of dynamic analysis, under the guidance of complex networks and evolutionary game theory, with the “finite rational” local government as one of the decision makers in the evolutionary game. The internal factors include income heterogeneity, preference heterogeneity, and benefit distribution heterogeneity, while the external factors include, network size, strategy evolution rules, and network structure. The model can better reveal the dynamic evolutionary characteristics of cooperative behavior among subjects in the region, derive the dynamics of the behavioral choices of each subject participating in governance, and propose countermeasures for the coordination and cooperation optimization of collaborative air pollution management in China.

3. Materials and Methods

3.1. Network Topology Description and Structural Characterization

Considering the current situation, as well as the availability and operability of the sample data, this paper focuses on 28 urban areas and 26 government departments in each city in BTH and surrounding areas in the selection of the sample data. The informal agreements related to air pollution control on the official websites of each people’s government from 2013 to 2019 were crawled using Python. In addition, the data on the cooperation relationship between local government departments were obtained from the “Notice of the State Council on Issuing the Three-Year Action Plan to Win the Blue-Sky Defense War”. In addition, the advanced search function was used to conduct an advanced search in the newspaper database of the China Knowledge Network using “air pollution”, “air pollution”, and “air quality” as keywords, and then the data related to informal agreements were manually filtered. After manual screening and identification, invalid data were eliminated, and 279 data related to informal inter-city agreements were obtained.

The text analysis was used to quantitatively, objectively, and systematically describe the contents of the obtained informal agreements, identify the subjects and events contained in the text contents, and infer whether they have collaborative governance relationships. The contents of some of the coded informal agreements are shown in

Table 1. Part of the informal protocol coding content.

Informal Agreement Content	Subjects Involved	Cooperation Method	Data Source
Beijing officially signed a cooperation agreement with Langfang and Baoding on air pollution prevention and control, and Beijing arranged about 230 million CNY of special funds for air pollution to support the air pollution control work in Langfang and Baoding respectively.	Beijing, Langfang, Baoding	Financial Support	Beijing Daily
Tianjin arranged 200 million yuan of air pollution treatment funds to support Tangshan City and Cangzhou City to carry out the work related to the treatment of the atmosphere.	Tianjin, Tangshan, Cangzhou	Financial Support	Beijing Daily News
Zhuang Zhidong, deputy director of the Beijing Environmental Protection Bureau, visited Langfang for research on the implementation of the spirit of the fourth working meeting of the BTH and surrounding areas’ air pollution prevention and control collaboration group and deepening regional collaboration.	Beijing, Langfang	Research	Langfang Daily
Zhengzhou, Xinxiang, Jiaozuo, and other cities decided to establish an “air pollution control coordination community”, hand in hand to study the scheduling problem, and common fight, hand in hand cure.	Zhengzhou City, Xinxiang City, Jiaozuo City	Regional Planning	Zhengzhou Daily
For preventing and controlling air pollution, Cangzhou city, Tianjin Jinghai District and Langfang City signed a cooperation framework agreement, the specific implementation of an air pollution synergy program.	Cangzhou, Tianjin, Langfang	Cooperation Agreement	Cangzhou Local Government Network
Jinfu Wang and other comrades (Baoding City, Hebei Province, Environmental Protection Bureau) and his party of five people came to Dezhou City, Shandong Province to research and exchange on air pollution prevention and control work.	Dezhou, Baoding	Research	Dezhou Local Government Network
Zhengzhou City party and government delegation to Shijiazhuang City, Hebei Province for air pollution control work special study.	Zhengzhou, Shijiazhuang	Inspection	Zhengzhou Local Government Network

below.

3.1.1. Network Subject Relationship Analysis

• Inter-city network relationship analysis

After coding, the adjacency matrix is established to inscribe the connection relationship between city network nodes, according to the record of collaborative governance between city municipal governments. In the inter-city air pollution collaborative governance network relationship matrix of Beijing, Tianjin, Hebei, and surrounding areas, if the city municipal governments have a collaborative governance network connection relationship, fill in 1 in the table at the inter-city intersection, and vice versa, fill in 0 in the table at the inter-city intersection. Some of the inter-city air pollution collaborative governance relationships are shown in

Table 2. The adjacency matrix of the relationship between collaborative governance.

	BJ	TJ	SJZ	TS	LF	BD	CZ	HS	XT	HD	TY	YQ	CZ	JC	JN	ZB	JN	DZ
BJ	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
TJ	1	0	1	1	1	1	1	1	0	0	1	0	0	0	1	1	1	0
SJZ	1	1	0	1	1	1	1	1	1	1	1	0	0	0	1	0	0	0
TS	1	1	1	0	1	1	1	1	1	1	0	0	0	0	0	0	0	0
LF	1	1	1	1	0	1	1	1	1	1	1	0	0	0	0	0	0	0
BD	1	1	1	1	1	0	1	1	1	1	1	0	0	0	0	0	0	1
CZ	1	1	1	1	1	1	0	1	1	1	0	0	0	0	0	1	0	0
HS	1	1	1	1	1	1	1	0	1	1	1	0	0	0	0	0	0	1
XT	1	0	1	1	1	1	1	1	0	1	0	0	0	0	0	0	0	0
HD	1	0	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0	0
TY	1	1	1	1	1	1	0	1	0	0	0	1	1	1	1	0	0	0
YQ	1	0	0	0	0	0	0	0	0	0	1	0	1	1	0	0	0	0
CZ	1	0	0	0	0	0	0	0	0	0	1	1	0	1	0	1	0	0
JC	1	0	0	0	0	0	0	0	0	0	1	1	1	0	1	1	0	0
JN	1	1	1	0	0	0	0	0	0	0	1	0	0	1	0	1	1	1
ZB	1	1	0	0	0	0	1	0	0	0	0	0	1	1	1	0	1	1
JN	1	1	0	0	0	0	0	0	0	0	0	0	0	0	1	1	0	1
DZ	1	0	0	0	0	1	0	1	0	0	0	0	0	0	1	1	1	0

.

• Inter-governmental network relationship analysis

China has formed a preliminary emergency management organization system for collaborative air pollution management consisting of non-permanent emergency command institutions and permanent offices, relevant functional departments, etc. [27,28]. The system addresses the prevention and control of air pollution with 35 air pollution management tasks, and some of the management tasks and participating departments are shown in

Table 3. Tasks and participating departments of environmental emergencies prevention and control.

	Optimize Industrial Layout	Strictly Control the “Two High” Industry Capacity	Strengthen the Comprehensive Rectification of “Scattered and Disorganized” Enterprises	Deepening Industrial Pollution Control	Vigorously Cultivate Green Environmental Protection Industry	Carry Out Comprehensive Rectification of Coal-Fired Boilers	Improve Energy Efficiency	Optimize and Adjust the Structure of Cargo Transportation	Accelerate the Upgrading of the Structure of the Car and Boat	Accelerate the Quality Upgrade of Oil Products	Strengthen the Prevention of Mobile Source Pollution	Implement the Wind and Sand Stabilization Greening Project
Bureau of Ecology and Environment	P	S	P	P	S	P		S	S	S	P	S
Development and Reform Commission	S	P	S	S	P	S	P	P			S
Bureau of Industry and Information Technology	S	P	P	S	S	S	S		P		S
Bureau of Natural Resources and Planning	S		S									P
Bureau of Finance		S						S	S	P	S
Market Supervision Administration		S	S			P	P			S	S
Bureau of Science and Technology				S	S
Commerce Bureau				S					S	S
Energy Agency						S	S		S	P	S
Housing and Urban-Rural Development Bureau						S	P		S			P
Transport Bureau								P	P		P
Railroad Bureau								P	S
China Railway Corporation								P	S
Civil Aviation Authority									S		P
Public Security Bureau									S		S
Bureau of Agriculture											S	S
Forestry Bureau												S

Note: P stands for Lead Unit, and S stands for Support Coordination Unit.

below. The dichotomous diagram can accurately portray the dichotomous structure of “project one participant” that exists in a large number of social systems. One type of node is “project”, such as the projects of optimizing the industrial layout and deepening industrial pollution control in Table 3, while the other type of node is “participant”, such as the Ecological Environment Bureau and the Development and Reform Commission. If two nodes (i.e., government departments) are in the same project, they are considered to collaborate in the air pollution management task of the project, which is represented by the connected edges between nodes in the network.

Taking “deepening industrial pollution control” as an example, local government departments including the Ecological Environment Bureau, Development and Reform Commission, Industry and Information Technology Bureau, Science and Technology Bureau, and Commerce Bureau participate in the control, which is all represented by nodes in the network, and the connecting lines between nodes indicate their cooperation in air pollution cooperative control tasks. The government departments of the same project group are connected to form a complete map, and the complete maps of the 35 major working groups are connected to form the interdepartmental collaborative air pollution management network.

3.1.2. Network Topology and Characterization

In this paper, we use UCINET software to obtain a two-layer topology diagram of the collaborative air pollution management network, in which a node represents a city or government department involved in the management, and a connected edge indicates the existence of a collaborative relationship between two cities or government departments, which also reflects the interesting relationship among the subjects in the collaborative air pollution management, without considering the administrative responsibilities of each city and department and the size of the management capacity. The proposed collaborative air pollution management network is a non-directional and powerless network, as shown in Figure 1 below.

From Figure 1, it can be seen that in the collaborative air pollution management network, with the air pollution management event as the link, each governance subject influences and interacts with the others, constituting a huge, multi-factor-influenced, dynamic, and complex network. In the evolutionary game of air pollution cooperative governance under the complex network, there are intricate evolutionary game relations among the governance subjects, and the interaction relations among these subjects make the whole network adapt and self-organize from disorder to order and continuously promote air pollution cooperative governance.

The point degree centrality of atmospheric pollution is “the degree of contact between cities or government departments”. In this paper, the nrmdegree centrality formula is:

$$C_D\left(x_i\right)=D\left(x_i\right)/(N-1)$$

(1)

where $N$ is the size of the city/local government collaborative air pollution management network, and $D\left(x_i\right)$ is the number of subjects that have established collaborative management relationships with cities/local governments.

The intermediate centrality (betweenness centrality) measures the degree of resource control by the subject of collaborative air pollution management and is given by the formula:

$$C_B\left(x_i\right)=\left[{\displaystyle \sum _{j<k}\frac{G_{jk}\left(x_j\right)}{G_{jk}}}\right]\mid (N-1)(N-2)$$

(2)

where $G_{jk}(x_j)$ denotes the number of short-range lines between two cities or government departments containing $x_i$. $G_{jk}$ is the number of short-range lines that exist between $x_i$ and $x_k$.

Closeness centrality reflects the degree of betweenness between a subject and others in the collaborative air pollution management network. The reciprocal of the accumulation of the shortest path distance from one collaborative governance subject to all other collaborative governance subjects represents the betweenness centrality. That is, for a network node, the closer it is to other network nodes, the greater its betweenness centrality. The formula is:

$$C_c\left(x_i\right)=1/{\displaystyle \sum _{j=1}^n{D}_i}\left(x_i,x_j\right)$$

(3)

where $D_i\left(x_i,x_j\right)$ is the distance of the short-range line between $x_i$ and $x_j$.

With the above Equations (1)–(3), the nrmdegree centrality, closeness centrality, and betweenness centrality of the inter-city air pollution cooperative management network can be calculated using UCINET software (version 6.186) (Analytic Technologies, Lexington, KY, USA), as shown in Figure 2.

The nrmdegree centrality, closeness centrality, and betweenness centrality of the collaborative air pollution management network among government departments are similar to the inter-city situation.

To quantitatively study the overall topological characteristics of the collaborative air pollution management network, the average path length, clustering coefficient, and small-world quotient of the collaborative management network need to be analyzed.

(1)

Average path length

Depicting the distance between any two network nodes from the perspective of the whole collaborative air pollution management network, the average path length can be expressed as:

$$L=\frac{1}{\frac{1}{2}N(N-1)}{\displaystyle \sum _{i>j}{d}_{ij}}$$

(4)

where $N$ denotes the number of nodes in the network, $i,j$ denotes the network nodes, and $d_{i,j}$ denotes the shortest path length between network nodes $i,j$.

(2)

Clustering coefficient

In the collaborative air pollution management network, the clustering coefficient is used to indicate the clustering of the nodes in the network, i.e., how many of the neighbors of each of the small groups connected are common neighbors. The average value of the clustering coefficient of the network nodes can be expressed by the following formula:

$$G=\frac{1}{N}{\displaystyle \sum _{i=1}^N\frac{2E_i}{K_i\left(K_i-1\right)}}$$

(5)

where $G$ denotes the clustering coefficient of the whole collaborative air pollution control network, and $k_i$ denotes that node $i$ has $k_i$ edges connected to it. There are at most $k_i(k_i-1)/2$ possible edges between these $k_i$ nodes, and $G_i$ is the ratio of the actual number of $E_i$ edges to the possible number of $k_i(k_i-1)/2$ edges.

(3)

Small-world quotient

By comparing the air pollution collaborative management network with a random network of the same size, we can determine whether the air pollution collaborative management network has small-world characteristics. If the ratio of the clustering coefficients of the two networks is closer to 1, and the ratio of the characteristic path length is higher than 1, the more significant the small-worldness of the collaborative air pollution control network is. The following Equation (3) is used to express the obvious degree of small-worldness, where $Q$ is the small-world quotient:

$$Q=\frac{G/G_{\mathrm{Random}}}{L/L_{\mathrm{Random}}}$$

(6)

According to Equation (4), the average path length between government departments is 1.542 by substituting the relevant parameters, and the specific calculation results are shown in Figure 3.

The average distance between two government departments is 1.542, and the maximum distance is no more than 2. Only one other department is needed on average to obtain contact between two government departments. The relevant results calculated according to Equations (4)–(6) are shown in

Table 4. Parameters of collaborative governance network for emergent environmental events.

	Inter-Government Departments Network	Inter-City Network	Random Network
Average path length	1.542	1.664	1.6927
Clustering coefficient	0.845	0.762	0.5205
Average path length ratio	0.911	0.983
Clustering coefficient ratio	1.609	1.464
Small-world quotient	1.767	1.489

below.

From Table 4, we find that both inter-city and inter-governmental departments satisfy the condition of $\mathrm{L}<{\mathrm{L}}_{\mathrm{random}},\mathrm{G}>{\mathrm{G}}_{\mathrm{random}},\hspace{0.33em}\mathrm{Q}>1$. The network has a small average path length, a large clustering coefficient, and a high small-world quotient; thus, it can be judged to have the characteristics of a small-world network. The small average distance and large clustering coefficient presented by this network reflect the closeness of the relationship between subjects. In such a network structure, information, capital, and personnel can be better shared among the subjects.

3.2. Evolutionary Game Model Construction

3.2.1. Revenue Matrix

There is heterogeneity among the nodes of the collaborative air pollution governance network, which can be roughly divided into two groups. Hence, the relevant local governments are divided into core local government group 1 (which has a high public income level and a low economic preference) and the other local government group 2 (which has a low public income level, a high economic preference, and low environmental preference). The parameters of local governments in each governance group in the cooperative air pollution management strategy choice are set as shown in

Table 5. Parameter Setting of Each Main Body of Emergency Response to Sudden Environmental Events.

Coefficient	Meaning
$F$	Gain from the pursuit of economic growth
${\alpha }_i$	Economic preferences of local governments
$E_i$	Revenue status of local governments
$\mathrm{k}$	Reward/penalty factors for local government participation/non-participation in collaborative air pollution management
$R$	Total benefits from local government participation in collaborative air pollution management
$S$	Total loss due to non-participation of local governments in collaborative management of air pollution events
$\theta$	Heterogeneity value coefficient of environmental benefit distribution of atmospheric pollution events

.

Assuming that the parameters of each indicator of each group in each category will not change after each game, where ${\alpha }_2>{\alpha }_1$, $E_1>E_2$, the Pay-off matrix constructed from the elements in the above table is shown in

Table 6. Pay-off matrix of the evolutionary game for collaborative governance of emergent environmental events.

Core Local Government Groups 1	Other Local Government Groups 2
	Participation (Cooperation)	Non-Participation (Non-Cooperation)
Participation (Cooperation)	$\theta R$ $(1-\theta )R$	$\theta (R-{\alpha }_{2}S)+K{E}_2$ $(1-\theta )(R-{\alpha }_{2}S)-K{E}_2+{\alpha }_{2}F$
Non-participation (non-cooperation)	$\theta (R-{\alpha }_{1}S)-K{E}_1+{\alpha }_{1}F$ $(1-\theta )(R-{\alpha }_{1}S)+K{E}_1$	$\theta (R-{\alpha }_{1}S-{\alpha }_{2}S)+{\alpha }_{1}F$ $(1-\theta )(R-{\alpha }_{1}S-{\alpha }_{2}S)+{\alpha }_{2}F$

.

3.2.2. Evolutionary Game Analysis

In the collaborative air pollution management network, a pair of game subjects are randomly selected from two major subgroups for evolutionary game analysis, i.e., local government 1 and local government 2, which face two strategic choices in the collaborative air pollution management game, namely, participation and non-participation. Under the assumption of limited rationality, the probability of local government 1 choosing to participate is $x$, then the probability of choosing not to participate is $(1-x)$ and the probability of local government 2 choosing to participate is $y$, and vice versa, where $(1-y)$, $x$ and $y$ are functions of time t.

• Replication dynamic equation for local government 1

Expected benefits when local government 1 chooses to participate in collaborative air pollution management:

$$U_{11}=y(\theta R-C)+(1-y)[\theta (R-{\alpha }_{2}S)+K{E}_2]$$

(7)

Expected benefits when local government 2 chooses not to participate in collaborative air pollution management:

$$U_{12}=y[\theta (R-{\alpha }_{1}S)-K{E}_1+{\alpha }_{1}F]+(1-y)[\theta (R-{\alpha }_{1}S-{\alpha }_{2}S)+{\alpha }_1{F}_1]$$

(8)

The average return for Local Government 1:

$$\overline{U_1}=x{U}_{11}+\left(1-x\right)U_{12}$$

(9)

From Equations (7) and (8) above, the replication dynamic equation for the collaborative governance strategy of local government 1 can be obtained:

$$\begin{array}{l}F\left(x\right)=\frac{dx}{dt}=x(U_{11}-\overline{U_1})=x\left(1-x\right)(U_{11}-U_{12})\\ =x\left(1-x\right)[y(K{E}_1-K{E}_2)+\theta {\alpha }_{1}S+K{E}_2-{\alpha }_{1}F]\end{array}$$

(10)

Similarly, the replication dynamic equation for the choice of collaborative governance strategy by local government 2 is derived:

$$\begin{array}{l}F\left(y\right)=\frac{dy}{dt}=y(U_{21}-\overline{U_2})=y\left(1-y\right)(U_{21}-U_{22})\\ =y\left(1-y\right)[x(K{E}_2-K{E}_1) + (1-\theta ){\alpha }_{2}S+K{E}_1-{\alpha }_{2}F]\end{array}$$

(11)

From the above Equations (10) and (11), we can obtain the replicated dynamic equation system of the cooperative air pollution management game system:

$$\left\{\begin{array}{l}F(x)=\frac{dx}{dt}=x\left(1-x\right)[y(K{E}_1-K{E}_2)+\theta {\alpha }_{1}S+K{E}_2-{\alpha }_{1}F]\\ F(y)=\frac{dy}{dt}=y\left(1-y\right)[x(K{E}_2-K{E}_1) + (1-\theta ){\alpha }_{2}S+K{E}_1-{\alpha }_{2}F]\end{array}\right.$$

(12)

Let the replicated dynamic system of equations for this system result in 0, that is, for the above Equation (12), $\left\{\begin{array}{l}F(x)=0\\ F(y)=0\end{array}\right.$, five asymptotically stable equilibria of this system are obtained, which are ${\gamma }_1(0,0)$, ${\gamma }_2(0,1)$, ${\gamma }_3(1,0)$, ${\gamma }_4(1,1)$, ${\gamma }_5(\frac{(1-\theta ){\alpha }_{2}S+K{E}_1-{\alpha }_{2}F}{K(E_1-E_2)},\frac{\theta {\alpha }_{1}S+K{E}_2{}_1-{\alpha }_{1}F}{K(E_2-E_1)})$.

To further analyze the stability of the strategy, the Jacobi matrix of the system needs to be solved and analyzed. The Jacobi matrix of this system is:

$$\begin{gathered} \mathit{Det}(J)=\left[\begin{array}{l}\frac{\partial F(x)}{\partial F(x)}\frac{\partial F(x)}{\partial F(y)}\\ \frac{\partial F(y)}{\partial F(x)}\frac{\partial F(y)}{\partial F(y)}\end{array}\right]=\left[\begin{array}{l}\left(1-2x\right)[y(K{E}_1-K{E}_2)+\theta {\alpha }_{1}S+K{E}_2-{\alpha }_{1}F], x(1-x)(K{E}_1-K{E}_2)\\ y(1-y)(K{E}_1-K{E}_2), (1-2y)[x(K{E}_2-K{E}_1) + (1-\theta ){\alpha }_{2}S+K{E}_1-{\alpha }_{2}F]\end{array}\right] \\ \begin{array}{l}\mathit{Tr}(J)=\frac{\partial F(x)}{\partial x}+\frac{\partial F(y)}{\partial y}=\left(1-2x\right)[y(K{E}_1-K{E}_2)+\theta {\alpha }_{1}S+K{E}_2-{\alpha }_{1}F]+(1-2y)[x(K{E}_2-K{E}_1)\\ + (1-\theta ){\alpha }_{2}S+K{E}_1-{\alpha }_{2}F]\end{array} \end{gathered}$$

(13)

When Det(J) > 0 and Tr(J) < 0 hold at the same time, the equilibrium point will tend to a locally stable state, that is, the evolutionary game system has a stable evolutionarily strategy. The stability analysis of the above four local equilibria of the system is non-asymptotically stable; thus, its Det(J) and Tr(J) are not considered here, as shown in

Table 7. Values of the equilibrium points Det(J) and Tr(J).

Equilibrium Point	Det(J)	Tr(J)
$r_1$	$(\theta {\alpha }_{1}S+K{E}_2-{\alpha }_{1}F)[(1-\theta ){\alpha }_{2}S+K{E}_1-{\alpha }_{2}F]$	$(\theta {\alpha }_{1}S+K{E}_2-{\alpha }_{1}F)+[(1-\theta ){\alpha }_{2}S+K{E}_1-{\alpha }_{2}F]$
$r_2$	$(-\theta {\alpha }_{1}S-K{E}_2+{\alpha }_{1}F)[(1-\theta ){\alpha }_{2}S+K{E}_2-{\alpha }_{2}F]$	$-\theta {\alpha }_{1}S+{\alpha }_{1}F+(1-\theta ){\alpha }_{2}S-{\alpha }_{2}F$
$r_3$	$(K{E}_1+\theta {\alpha }_{1}S-{\alpha }_{1}F)[-(1-\theta ){\alpha }_{2}S-K{E}_1+{\alpha }_{2}F]$	$\theta {\alpha }_{1}S-{\alpha }_{1}F-(1-\theta ){\alpha }_{2}S+{\alpha }_{2}F$
$r_4$	$(K{E}_1+\theta {\alpha }_{1}S-{\alpha }_{1}F)[(1-\theta ){\alpha }_{2}S+K{E}_2-{\alpha }_{2}F]$	$-(K{E}_1+\theta {\alpha }_{1}S-{\alpha }_{1}F)-[(1-\theta ){\alpha }_{2}S+K{E}_2-{\alpha }_{2}F]$

.

The positive and negative Det(J) and Tr(J) of each equilibrium point are determined by several parameters together. Consequently, the judgment of the stability of this equilibrium point needs to be analyzed in conjunction with the actual situation, as shown in

Table 8. Stability analysis of equilibrium points.

Conditions	Local Equilibrium Point	Det(J)	Tr(J)	Stability
Scenario 1. $K{E}_1>{\alpha }_{2}F-(1-\theta ){\alpha }_{2}S,K{E}_2>{\alpha }_{1}F-\theta {\alpha }_{1}S$	${\gamma }_1(0,0)$	+	+	Unstable
	${\gamma }_2(0,1)$	−	Indefinite	Saddle Point
	${\gamma }_3(1,0)$	−	Indefinite	Saddle Point
	${\gamma }_4(1,1)$	+	-	ESS
Scenario 2. $K{E}_1<{\alpha }_{2}F-(1-\theta ){\alpha }_{2}S,K{E}_2<{\alpha }_{1}F-\theta {\alpha }_{1}S$	${\gamma }_1(0,0)$	+	-	ESS
	${\gamma }_2(0,1)$	−	Indefinite	Saddle Point
	${\gamma }_3(1,0)$	−	Indefinite	Saddle Point
	${\gamma }_4(1,1)$	+	+	Unstable
Scenario 3. $\begin{array}{l}{\alpha }_{1}F<K{E}_1+\theta {\alpha }_{1}S,{\alpha }_{2}F<(1-\theta ){\alpha }_{2}S+K{E}_2,\\ K{E}_1<{\alpha }_{2}F-(1-\theta ){\alpha }_{2}S\end{array}$	${\gamma }_1(0,0)$	−	+	Unstable
	${\gamma }_2(0,1)$	+	Indefinite	Saddle Point
	${\gamma }_3(1,0)$	+	Indefinite	Saddle Point
	${\gamma }_4(1,1)$	+	-	ESS
Scenario 4. $\begin{array}{l}{\alpha }_{1}F<K{E}_2+\theta {\alpha }_{1}S,K{E}_2<{\alpha }_{2}F-(1-\theta ){\alpha }_{2}S,\\ {\alpha }_{2}F<K{E}_1+(1-\theta ){\alpha }_{2}S\end{array}$	${\gamma }_1(0,0)$	+	+	Unstable
	${\gamma }_2(0,1)$	+	-	ESS
	${\gamma }_3(1,0)$	−	+	Unstable
	${\gamma }_4(1,1)$	−	Indefinite	Saddle Point
Scenario 5. $\begin{array}{l}{\alpha }_{1}F<K{E}_1+\theta {\alpha }_{1}S,{\alpha }_{2}F<(1-\theta ){\alpha }_{2}S+K{E}_2,\\ K{E}_1<{\alpha }_{2}F-(1-\theta ){\alpha }_{2}S,\\ K{E}_2<{\alpha }_{1}F-\theta {\alpha }_{1}S\end{array}$	${\gamma }_1(0,0)$	+	-	ESS
	${\gamma }_2(0,1)$	−	+	Unstable
	${\gamma }_3(1,0)$	−	+	Unstable
	${\gamma }_4(1,1)$	+	-	ESS

.

From Table 8, we can see that when the condition ${\alpha }_{1}F-\theta {\alpha }_{1}S<K{E}_1,{\alpha }_{2}F-(1-\theta ){\alpha }_{2}S<K{E}_2$ is satisfied, ${\gamma }_4(1,1)$ is a stable strategy point for the evolution of the air pollution game, which means that there is a situation in which both parties choose to participate in the collaborative air pollution management strategy. It can be found that the choice of local government strategy in collaborative air pollution management is influenced by the public income, economic preference, and benefit distribution of both parties, as well as the other party’s choice of strategy.

• Model Construction

The above analysis ignores factors such as the behavioral differences of government subjects and the total number of participants. Given this, this paper fully combines evolutionary game theory with complex network theory in order to construct an evolutionary game model of cooperative air pollution management under a complex network, as shown in Figure 4 below.

This paper gives several assumptions for constructing a complex network evolutionary game model for collaborative air pollution management:

• In this paper, we only consider the direct network effect in the external effect of the governance network and assume that the governance subjects will limit the scope of the game object to the “small world network” with which they have direct contact. Therefore, we define that each node $k_i$ of the collaborative air pollution management needs to participate in the game of $k_i+1$ subjects, and the game radius $r=1$, and its benefit is related to the strategy adopted by the neighboring subjects in the neighborhood.
• This paper assumes that the local government subject $i$ has only two strategies in the decision of collaborative governance: one is active participation and the other is non-participation. The combined strategy of the local government department is denoted by $S_i=\{\mathrm{C},D\}$. $C$ represents the local government subjects’ participation in this collaborative governance of air pollution, and $D$ represents the local government subjects’ non-participation in this collaborative governance of air pollution.
• In this paper, we assume that all local government subjects adopt the same strategy update rule, and each subject determines the behavioral strategy to be adopted in the next game based on the current game gain only at the time of strategy update.

• External network model

The complex network of collaborative air pollution management is a small-world network $G=(N,E,M)$, which is a powerless and undirected network, where $N$ denotes the number of points in the collaborative air pollution management network, that is, the number of local government subjects $N=\{1,2,3,4,5,\dots \dots n\}$. $E$ denotes the set of edges in the collaborative air pollution management network between local governments, that is, the set of evolutionary game relations between collaborative management subjects $E=\{(i,j)|e_{i,j}=1\}$, where $e_{i,j}=0$ means that there are no collaborative governance evolutionary game relations between two local governments, and $e_{i,j}=1$ means that there is a collaborative governance evolutionary game relation between them. There are three types of edges in this network, i.e., $CD$ (involved in cooperative governance—not involved in cooperative governance), $CC$ (involved in cooperative governance—involved in cooperative governance), and $DD$ (not involved in cooperative governance—not involved in cooperative governance). $M$ represents the set of neighboring nodes of a node in the air pollution collaborative management network, that is, the set of a local government and all neighboring local government subjects $M_i=\{M_1,M_2,M_3,\dots \dots , M_n\}$, where $M_i=\left\{j\right|{\mathrm{e}}_{i,j}=1\}$.

Strategy update: In this paper, we assume that each local government subject adopts the FEMI rule for strategy update, i.e., at each moment of the game on the collaborative governance network, the local government nodes will continuously learn, update, and adjust their collaborative governance strategies according to the real-time gain situation, and the specific rule is to randomly select a neighboring subject in the neighborhood and determine whether to participate in the collaborative governance game strategy with the probability of adjustment, i.e.,

$$W_{S_{i,t}\to S_{j,t}}=\frac{1}{1+\exp \left[\left(U_{i,t}-U_{j,t}\right)/\eta \right]}$$

(14)

where $S_{i,t},S_{j,t}$ are the strategies of local government node $i$ and local government node $j$ at moment $t$, $U_{i,t}$, $U_{j,t}$ are the total gains of the local government node $i$ and local government node $j$ at moment $t$, and $\eta$ is the environmental noise factor, which indicates the irrational choice of each subject in the evolutionary game of cooperative air pollution management. That is, when its gain is smaller than that of its neighboring local government, that local government still insists on not changing its cooperative management strategy, i.e., the probability that the local government will not change its cooperative management strategy when its benefit is smaller than that of its neighboring local government. When the value of $\eta$ is close to 0, each governing body has high rationality and updates its strategy strictly according to the size of its gain; when the value of $\eta$ tends to infinity, it means that each governing body is in a noisy environment and updates its strategy in the next round randomly.

Structural adjustment: The evolutionary game structure is adjusted by using the disconnected edge reconnection mechanism of the collaborative air pollution management network. Assuming that there are $N$ subjects, at each moment $t$, an edge is randomly selected with probability $p$ to disconnect the original connection and start reconnecting. In this paper, we use the disconnected edge reconnection mechanism with preference, and the probability of node $i$ connecting to node $j$ at time $t$ is:

$${\mathrm{p}}_{ij,t}={\displaystyle \sum _{i\in G}(U_{j,t}^{\alpha }/U_{i,t}^{\alpha })}$$

(15)

where $U_i$ is the payoff of node $i$; $\alpha$ is the preference propensity.

The initial degree distribution of the collaborative air pollution control network shall satisfy the following equation:

$$P(k)=\left\{\begin{array}{l}{\displaystyle \sum _{n=0}^{\min (k-K/2.K/2)}\left(\begin{array}{l}K/2\hfill \\ n\hfill \end{array}\right)}{(1-p)}^n{p}^{\left(\frac{K}{2} - n\right)}\frac{{(pK/2)}^{k-K/2-n}}{(k - K/2 - n)!}{e}^{-pK/2},k⩾K/2\hfill \\ 0,k<K/2\hfill \end{array}\right.$$

(16)

• Internal game model

In each evolutionary cycle $t$, each local government subject involved in air pollution will play with all local government subjects in its game radius, according to the four payoff scenarios in the game payoff matrix constructed in the previous section and obtain the corresponding payoffs, denoted by $\pi (S_i,S_j)$ as the game payoff of local government $i$, where C represents participation in the collaborative governance strategy and D represents non-participation in the collaborative governance strategy.

$$\begin{array}{c}\pi =\left(\begin{array}{c}{\pi }_i\left(C,C\right),{\pi }_i\left(C,D\right)\\ {\pi }_i\left(D,C\right),{\pi }_i\left(D,D\right)\end{array}\right)\hfill \\ Where, {\pi }_i\left(C,C\right)={\theta }_{i}R\hfill \\ {\pi }_i\left(C,D\right)={\theta }_i\left(R-{\alpha }_{j}S\right)+K{E}_j\hfill \\ {\pi }_i\left(D,C\right)={\theta }_i\left(R-{\alpha }_{i}S\right)-K{E}_i+{\alpha }_{i}F\hfill \\ {\pi }_i\left(D,D\right)={\theta }_i\left(R-{\alpha }_{i}S-{\alpha }_{j}S\right)+{\alpha }_{i}F \hfill \end{array}$$

Under the definition of the small-world network model, each local government subject plays a game with those other local government subjects with which it has a cooperative governance game relationship, and the benefit of each game subject is the cumulative sum of the benefits obtained from the evolutionary game with all its neighboring government subjects (${\mho }_i$). The benefit ${\displaystyle \sum _{j\subset {\mho }_i}{\pi }_{i,t}(S_{i,t},S_{j,t})}$ of its local government subject node $i$ is shown in Equation (17) below:

$$\begin{array}{l}{U}_{i,t}={\displaystyle \sum _{j\subset {\mho }_i}{\pi }_{i,t}(S_{i,t}, S_{j,t})}\\ U_{i,t}=\left\{\begin{array}{l}E(N_{CC}){\theta }_{i}R+E(N_{CD})\left[({\theta }_i(R-{\alpha }_{j}S)+K{E}_j\right], s_i=C\\ E(N_{DC})({\theta }_i(R-{\alpha }_{i}S)-K{E}_i+{\alpha }_{i}F)+E(N_{DD})\left[{\theta }_i(R-{\alpha }_{i}S-{\alpha }_{j}S)+{\alpha }_{i}F\right], s_i=D\end{array}\right.\end{array}$$

(17)

where $E(N_{CC})$ is the number of participating subjects in all neighbors when the government subject $i$ participates in collaborative governance, $E(N_{CD})$ is the number of non-participating subjects in all neighbors when the government subject $i$ participates in collaborative governance, $E(N_{DC})$ is the number of participating subjects in all neighbors when the government subject $i$ does not participate in collaborative governance, and $E(N_{DD})$ is the number of non-participating subjects in all neighbors when government subject $i$ does not participate in collaborative governance.

3.3. Simulation by Matlab

3.3.1. Simulation Framework and Parameter Settings

The model is programmed with MATLAB to perform game analysis and the overall network evolution measurement of local government evolution in a collaborative governance network. The simulation framework consists of five main components: (1) network generation; (2) game rules; (3) strategy update; (4) network structure adjustment; and (5) participation rate calculation.

In the simulation experiment, each node in the network represents a local government. In the process of air pollution management, the number of participating local government subjects is generally between 26 and 100. This paper takes an integer to set the network size as $30\le N\le 100$, and the parameters of the network structure of collaborative air pollution management are set as shown in

Table 9. Structural parameters of collaborative air pollution control network.

Size	Maximum Local Government Degree Value	Minimum Local Government Degree Value	Core Local Government Share (Degree 20)	Other Local Government Shares (Degree < 20)
$30\le N\le 100$	27	8	20%	80%

.

According to the financial budget reports of our local governments, the initial parameters are set after consulting experts in management science and complex network-related fields. Consequently, the relevant evolutionary game parameters are set as shown in

Table 10. Evolution Game Parameter Table of Collaborative Air Pollution Control Network (Unit: billion yuan).

Gain from the Pursuit of Economic Growth (F)	$\mathbf{Economic} \mathbf{Preferences} \left({\mathit{\alpha }}_{\mathit{i}}\right)$	$\mathbf{Public} \mathbf{Revenue} \mathbf{Status} \left({\mathit{E}}_{\mathit{i}}\right)$	Reward and Punishment Factor (k)	Losses Due to Non-Participation in Collaborative Governance (S)	$\mathbf{Earnings} \mathbf{Distribution} \mathbf{Heterogeneity} \mathbf{Value} \mathbf{Factor} \left({\mathit{\theta }}_{\mathit{i}}\right)$
4	1/1.2	600/400	0.005	12	0.6/0.4

.

In the simulation, the number of game parties choosing to participate in collaborative governance is set to 50%, and the number of non-participants is set to 50%, i.e., the initial cooperation ratio is set to 0.5, the noise factor h is taken as 0.1, and n is 100. For each group of parameters, 50 experiments are conducted, and the average of the participation rate is taken to observe the situation of comparing the degree of collaborative governance and the speed of evolution.

3.3.2. Simulation Experiments

• Simulation of external dynamic factors

Because the network structure and strategy update rules set in this paper are found to be the most consistent with the reality of collaborative air pollution management in China and are not easy to change, the analysis of external dynamics only analyzes the network size and investigates the game behavior of local government subjects in a 30-node, 50-node, and 100-node collaborative management network (small-world network). Each local government subject and the subjects in the domain play the game repeatedly. The evolution of the game at different network scales is discussed in comparison by changing the loss of local government choosing not to participate and pursuing economic growth to gain, and the initial parameters are set according to the condition ${\alpha }_{1}F<K{E}_1+\theta {\alpha }_{1}S,{\alpha }_{2}F<(1-\theta ){\alpha }_{2}S+K{E}_2$ that needs to be satisfied for the point ${\gamma }_4(1,1)$ to be the stable state of the game system, as shown in

Table 11. Relevant parameters of the evolutionary game of collaborative governance of air pollution under different collaborative governance networks.

Network Size			$\mathit{K}$	${\mathit{E}}_{\mathit{i}}$	${\mathit{\alpha }}_{\mathit{i}}$	$\mathit{F}$	S	$\mathit{\theta }$
30-Node	50-Node	100-Node
T1	T7	T13	0.005	600/400	1/1.2	5	12	0.6/0.4
T2	T8	T14	0.005	600/400	1/1.2	5	10	0.6/0.4
T3	T9	T15	0.005	600/400	1/1.2	13	12	0.6/0.4
T4	T10	T16	0.005	600/400	1/1.2	5	4	0.6/0.4
T5	T11	T17	0.005	600/400	1/1.2	5	5	0.6/0.4
T6	T12	T18	0.005	600/400	1/1.2	6	12	0.6/0.4

below.

• Simulation of internal dynamic factors

For the influence of internal dynamics factors on collaborative air pollution control, the network size is set to the middle value of 50 local government nodes for simulation, and the relevant parameters are set as shown in Table 9 and Table 10, where the ratios of income heterogeneity, preference heterogeneity, and benefit distribution heterogeneity are the ratios of other local governments to core local governments.

4. Results and Discussion

4.1. Simulation Results and Discussion of External Dynamic Factors

By conducting simulation experiments under a small-world network with 30 network nodes, 50 network nodes, and 100 network nodes, the results are shown in Figure 5a–c below.

In collaborative air pollution management, when the loss of local governments choosing not to participate in collaborative governance is greater than the gain from pursuing economic growth ($S>F$), as shown in T1, T2, and T6 in Figure 5a, T7, T8, and T12 in Figure 5b, and T13, T14, and T18 in Figure 5c, even with different sizes of collaborative governance networks, all local government subjects eventually choose to participate in collaborative governance, and the participation rate of collaborative governance is 100%. Taking T1, T7, and T13 as examples, we find that in the 30-node scale air pollution collaborative governance network, each local government reaches the 100% collaborative governance participation rate at step 26 (t = 26) of the game; similarly, in the 50-node scale, each local government reaches the 100% participation rate at step 31 (t = 31); and in the 100-node scale, it reaches 100% at step 41 (t = 41).

When $S<F$, as shown in T3, T4, T9, T10, T15, and T16, there are more and more local government subjects in different sizes of collaborative governance networks gradually choosing not to participate in collaborative governance. Taking T3, T9, and T15 as examples, we can find that the participation rate of collaborative governance reaches 0 in 3 different size networks of 30, 50, and 100 nodes, corresponding to the game at step 28 (t = 28), step 37 (t = 37) and step 50 (t = 50), respectively.

The last case is $S=F$, as shown in T5, T11, and T17, where the evolution of cooperation under different scales evolves in the direction of 0 in slow fluctuations, but does not reach a steady state at the 50th step of the game time. In addition, the evolution under 30 nodes is relatively faster than that under the 100-nodes scale.

By analyzing the results of the above simulation experiments, it can be found that the small-scale network is more sensitive to the relevant parameters in the evolution game of air pollution cooperative management and has the effect of significantly increasing the evolution speed of local government cooperative management decisions in the network, which can evolve into a fully cooperative network in a shorter period.

4.2. Simulation Results and Discussion of Internal Dynamic Factors

4.2.1. Impact of Income Heterogeneity on the Evolution of Collaborative Governance

To investigate the influence of revenue heterogeneity among local governments on the strategy choice of game subjects, the simulation was conducted by adjusting the public revenue ratio, and we found that the collaborative governance cooperation behavior of nodal local governments is influenced by the public revenue ratio (local government subjects with less public revenue/local government subjects with more public revenue), with other variables held constant. Under the FEMI strategy rule, the simulation results of the air pollution governance evolution game when the heterogeneity of benefit distribution takes different values are shown in Figure 6a–c, from which it can be found that: 0.6 is the critical value.

With other variables held constant, the cooperative governance behavior of local government nodes is influenced by income heterogeneity, and when the range of public income ratio is between [0.6, 1], it can motivate each local government node to participate in governance, and when the public income levels among local government nodes are comparable, the number of participants will gradually increase until all local governments in the network are willing to participate in air pollution coordination.

This is because the public income disparity between regions will affect the choice of local government behavior and eventually have an impact on collaborative air pollution management: when the public income disparity is large, the externality of air pollution management in economically developed regions is obviously smaller than that in economically backward regions, and they can get more benefits from air pollution management, thus local governments in this region will act as the driver of management behavior while economically relatively. On the contrary, when the public income gap is relatively small, local governments can imitate and learn from the strategy, and it is more likely to have a good situation of collaborative governance.

4.2.2. Impact of Preference Heterogeneity on the Evolution of Collaborative Governance

To investigate the influence of preference heterogeneity on the strategy choice of game subjects, the simulation was conducted by adjusting the economic preference ratio, and we found that the cooperative behavior of nodal local governments in collaborative governance is influenced by the economic preference ratio (subjects with higher economic preference/subjects with lower economic preference), with other variables held constant. Under the FEMI strategy rule, the simulation results are shown in Figure 6b, from which it can be found that: 1.4 is the critical value.

(1)

When the economic preference ratio (hereinafter referred to as EPR) is greater than 1.4, cooperative behavior cannot emerge. Take 1.5 as an example, the participation rate evolves to 0 in the fluctuation and evolves to 0.3 at the 50th step of the game, that is, only 30% of local government nodes participate in the collaborative air pollution control.

(2)

When EPR is equal to 1.4, cooperative behavior emerges, all local governments participate in governance at the 30th step of the game, and the collaborative governance network reaches a stable state.

(3)

When EPR is greater than 1.2 and less than 1.4, all government entities participate in governance and reach a steady state. Compared with the ratio of 1.4, when the economic preference ratio is equal to 1.2, each local government has a stronger willingness to participate, and the participation rate evolves to 1 in the game in step 11, that is, all local governments participate in governance.

(4)

When the EPR is less than 1.2, the number of government nodes involved in governance gradually increases during the evolutionary game with 1.1, but the collaborative governance network is not formed in the game time of 50 steps.

Holding other variables constant, the cooperative governance behavior of local government nodes is affected by preference heterogeneity, and the ratio of economic preference ratios in the range of [1, 1.4] all have a positive impact on collaborative governance, but the impact between [1.2, 1.4] can quickly reach a stable state of cooperative emergence, and the closer the economic preference ratio between local government nodes is to 1.2, the governance participation rate evolves faster toward 1. From the above analysis, it can be found that moderate preference heterogeneity can mobilize local governments to participate in collaborative air pollution management more quickly.

This is because to obtain the maximum benefit, the local government will adjust its strategy in real time according to the strategy of the game opponent. When the opponent with a higher economic preference chooses not to participate in collaborative governance, and the amount of compensation given by the opponent is large enough (larger than the benefit of not participating in collaborative governance to pursue economic preference), the local government will choose the strategy of participation, even if the opponent does not participate. At the same time, because the network is a small-world network, once the government with lower economic preference chooses to participate in governance and obtains higher benefits, the government with higher economic preference is likely to imitate this strategy, hence the moderate preference heterogeneity among local governments is positive for air pollution collaborative governance.

4.2.3. Impact of the Evolution of Synergistic Governance with Heterogeneity in the Distribution of Benefits

To investigate the influence of the heterogeneity of benefit allocation of collaborative governance on the strategy choice of game subjects, the simulation was conducted by adjusting the value of the benefit allocation ratio, and we found that the collaborative governance cooperation behavior of local governments is influenced by the benefit allocation ratio ${\theta }_2:{\theta }_1$ (smaller benefit subjects/greater benefit subjects), with other variables held constant. Under the FEMI strategy rule, the simulation results are shown in Figure 6c, from which it can be found that: 0.6 is the critical value.

(1)

When the benefit distribution ratio (hereinafter referred to as BDR) is less than 0.6, cooperative behavior cannot emerge. For example, when the BDR is 0.5, although the number of cooperative subjects increases at the beginning of the game, it gradually decreases after the 16th step of the game, and at the 50th step, the BDR is 0.32, about 68% of local government nodes do not participate, and the air pollution governance cannot realize the cooperative situation.

(2)

When BDR is equal to 0.6, cooperative behavior can emerge. The participation rate of cooperative air pollution management in this network is 1 at the 30th step of the game, which means that all local government subjects are involved in the management work, and the network reaches a stable state.

(3)

When BDR is greater than 0.6 and less than 1, for example, BDR is equal to 1, the participation rate of cooperative governance is 1 at step 17, and the stable state of system cooperative governance is reached.

The cooperative governance behavior of local government nodes is influenced by the heterogeneity of the benefit distribution when the range of the benefit distribution ratio is between [0.6, 1] to stimulate local government participation, and when the benefit distribution ratio is increased, the number of participants will gradually increase to all local governments in the network are willing to participate, and a benefit distribution ratio close to 1 will lead to long-lasting and deep cooperation, promoting the participation of all relevant local governments in collaborative air pollution management.

This is because the purpose of each local government to participate in collaborative governance is to obtain the maximum benefit for itself. If the benefits are not fairly distributed, it will inevitably discourage its participation, and there will inevitably be negative wait-and-see behavior, thus a fair distribution of benefits according to the costs paid is an important means to improve the participation rate of collaborative governance.

5. Conclusions

The conclusions of this paper are as follows:

(1)

The air pollution collaborative governance network has a small average path length, a large aggregation coefficient, and a high small-world quotient, so it can be judged that the air pollution collaborative governance network has the characteristics of a small-world network, and the information sharing, capital sharing, and personnel sharing related to air pollution collaborative governance can be better among various subjects.

(2)

In the evolutionary game of collaborative air pollution management, the small-scale collaborative air pollution management network has the effect of significantly increasing the evolutionary speed of local governments’ collaborative management decisions in the network, and the small-scale collaborative air pollution management is more sensitive to the relevant parameters in the collaborative management game. Therefore, in the process of collaborative air pollution management, we should seize the main areas and establish as small-scale, streamlined, and efficient a collaborative governance network as possible, which is more conducive to the circulation of relevant collaborative governance information and the development of cooperation.

(3)

In the evolutionary game of collaborative air pollution management, when the public income level among local government nodes is comparable, it will stimulate the initiative of local governments to participate in collaborative air pollution management.

(4)

In the air pollution collaborative governance game, when the preference heterogeneity is appropriate, i.e., the ratio of economic preferences (the ratio of core local governments to other local governments) is between [1, 1.4], it is favorable to promote local governments to engage in air pollution collaborative governance, but when the influence is between [1.2, 1.4], it can make the air pollution collaborative governance network reach a stable state of cooperative emergence quickly, so it should use appropriate mechanisms to moderately guide the behavior of local governments.

(5)

In the air pollution collaborative governance game, when the distribution of benefits is more equitable, i.e., when the ratio of benefit distribution (the ratio of core local governments to other local governments) is between [0.6, 1], it can motivate local governments to participate in air pollution collaborative governance, so a fair distribution of benefits according to the costs paid is an important means to increase the participation rate of collaborative governance.

(6)

Regions with lower heterogeneity among local governments are more likely to generate synergy, which makes local governments spontaneously participate in air pollution management, and this spontaneous collaborative governance is also more likely to evolve into a stable state. In collaborative air pollution management, the heterogeneity among individual local governments should be assessed first, and then corresponding policy mechanisms should be formulated according to the level of heterogeneity.

The research of this paper is based on the summary and analysis of the existing studies at home and abroad. However, since collaborative air pollution management in China is a very complex system with many influencing factors, it needs to consider many factors in all aspects. However, because collaborative air pollution management in China is a very complex system with many factors that need to be considered in all aspects, there are still many aspects to be explored in the future.

Firstly, the main body of the model established in this paper is limited to local governments. In the future, different governance bodies, such as the evolutionary game relationship of collaborative governance among the central government, local governments, enterprises, and the public, can be comprehensively considered to find all-around synergies affecting air pollution. The driving force of governance can better solve the dilemma of air pollution governance.

Secondly, with the continuous development of China’s social and economic level, the central government and local governments have paid more and more attention to the collaborative governance of air pollution, and the background of collaborative governance is also changing. In this way, the most scientific and rigorous conclusions can be drawn, which can provide a reference for solving the dilemma of air pollution collaborative governance.