The Evolutionary Game Analysis of Public Opinion on Pollution Control in the Citizen Journalism Environment

Abstract

In the context of the rapid development of new media such as network citizen journalism, it is of great theoretical and practical significance to use the online public opinion to supervise sewage discharge enterprises’ emission governance behaviors and improve the social opinion supervision mechanism. This paper considers the dynamic characteristics of the spread process of public opinion and the game process of social supervision on corporate pollution control; constructs a tripartite evolutionary game model of the local government, sewage discharge enterprises, and the public by coupling the susceptible–exposed–infected–removed (SEIR) model and the evolutionary game model; and discusses the influence laws of public opinion spread on the tripartite evolutionary game. The results show that (1) the public with higher influence or authority has a more significant restraint effect to restrain the pollution control behavior of the local government and pollutant companies by using online public opinion supervision. (2) Increasing the probability of transforming a latent person into a supervisor and the topic derivative rate or reducing the probability of a supervisor’s self-healing can increase the peak value of supervisors, expand the scope of social public opinion, and improve the effectiveness of public opinion supervision. (3) The relatively high authenticity of public opinion supervision makes public opinion supervision a substitute for local government supervision, but it has a relatively strong inhibitory effect on the over-standard pollutant discharge behavior of sewage discharge enterprises. These conclusions can provide a reference for improving the social supervision mechanism of pollution control in the era of network citizen journalism.

1. Introduction

With the rapid development of industrialization and urbanization, China has made great progress in economic and social development. However, it also faces increasingly serious environmental pollution problems, such as the Yangtze River annex gushing out of the ‘‘milk water’’ event, the Dayan River ‘‘bright red’’ wastewater discharge event, the Sanmenxia Reservoir discharge ‘‘soy sauce’’ event, etc. The occurrence of pollution incidents shows that production enterprises lack morality in the process of pollution discharge, and local governments are derelict in the process of fulfilling their regulatory duties. Many incidents are reflected by the public to the central supervision group before local governments carry out pollution control. How to form an effective pollution control mechanism has become one of the research focuses of environmental management and pollution control.

The government and sewage discharge enterprises are the main participants in emission control. To correctly guide an enterprise’s behavior, the government implements environmental regulation policies such as environmental taxes [1] and government subsidies [2]. Some scholars have studied the impact of environmental regulation policies on enterprise strategy. Compared with the positive effect of static government environmental regulation policy on corporate environmental behavior [3], the effect of dynamic combination measures is more obvious, such as the gradual increase in the dynamic environmental tax and the gradual decrease in the dynamic emission reduction subsidy optimization combination [4], dynamic punishment, and the dynamic subsidy bilateral strategy [5]. The interactions between government and government and between governments and enterprises in the process of pollution control have also received a large amount of attention. To strengthen supervision, inter-regional governments will promote green technology innovation, but too much supervision leads to local government supervision costs that are relatively high. The stability strategy of one government strictly supervising another government is not strict supervision evolution [6,7]. Chen et al. [8] pointed out that the central government’s regulatory oversight is necessary to restrict the deviation of policy implementation caused by local governments’ collusion behavior. Tao [9] and Pan et al. [10] found that the pollution control cost of sewage discharge enterprises is also an important factor affecting the strategy of local government environmental regulation.

However, to rely only on government regulation or the market means that controlling corporate pollution behavior is not enough; the active participation in environmental pollution control of the public, as a direct victim of environmental pollution, can effectively reduce local government regulatory failure and corporate pollution behavior [11]. Traditional public participation in the supervision of environmental behavior is mainly public complaints and proposals to the relevant competent departments, hearings, etc. More public reporting can not only promote the green technology innovation of enterprises but also reduce the cost of local government supervision [12]. The enthusiasm of the public is closely related to the cost of public participation and the psychological benefits [13,14]. Xu et al. studied the influence paths of different government behaviors and public participation on enterprise green technology innovation and found, on the basis of the exogenous guidance of government behavior, a focus on stimulating public participation and guiding enterprise green technology innovation are effective strategies for the endogenous evolution of pollution reduction [15]. In the process of the rise and rapid development of new media networks, self-media networks have become central to public opinion supervision. The public spread information through the media network to expand the social impact of pollution incidents, forcing the government to increase the supervision and punishment of enterprises over discharge and forcing enterprises to comply with pollutant discharge regulations. The public can access or share information more quickly and effectively through social media, making the public more actively involved in solving environmental-protection-related problems [16]. Ling et al. found that the new media help to build a fast information channel for the central government to monitor local governments and help the central government to effectively account for local governments’ information-hiding behavior in sudden environmental accidents [17]. Based on a provincial panel data analysis, Guo found that there are obvious regional differences in the impact of public opinion environmental supervision on SO₂ and CO₂ emissions in various provinces, but it has a certain inhibitory effect on emissions [18]. The public participation in supervision through self-media networks is an effective measure for environmental pollution control. Moreover, there are studies on the public supervision of new media in other areas, such as the impact of the sina media “opinion leader” model on consumers’ purchasing behavior [19], the impact of new media reports and their authenticity on the game strategies of food enterprises and government regulators [20,21,22], and the impact of new media on the choices of social organizations, government departments, and third-party evaluation institutions [23]. These studies have generalized the influence of network public opinion communication into a parameter of the influencing factor. In fact, network public opinion communication is a dynamic and complex process, including multiple subjects and various influencing factors, with its own regularity. It is necessary to consider the impact of multiple subjects and multiple factors on the subject’s behavior in the process of public opinion communication.

With the advent of the internet era, the public began to use the new media for public opinion supervision. However, when the existing public uses new media to participate in public opinion supervision, there are problems such as the low enthusiasm of public participation in supervision and the small influence of online public opinion supervision [24]. How to improve the enthusiasm of public network public opinion supervision and expand the influence of public network public opinion supervision is the main problem to be solved.

In 1964, Goffman et al. [25] first applied the SIR (susceptible–infected–recovered) model of infectious diseases to the field of information dissemination. Subsequently, scholars studied network public opinion dissemination based on the information dissemination models such as SIS (susceptible–infected–susceptible) [26], SEIR (susceptible–exposed–infected–removed) [27], and SCIR (susceptible–contacted–infected–removed) [28] and applied them to solve the different problems [29,30,31,32,33]. At present, the research on information dissemination in social networks mainly focuses on the research on the information dissemination process and information dissemination prediction. The key to the research on the information dissemination process is to establish an appropriate information dissemination model. The propagation models are mainly divided into the classical propagation model [34,35], the improved epidemic model based on the classical model [36,37], the variant epidemic model [38,39], the epidemic model with time delay [40,41], and other epidemic models [42,43,44]. These models comprehensively consider the psychological state of the communicator, the initial state of the communicator, the characteristics of the information source, and the environmental factors during the communication and are closer to the actual facts of network communication [45].

The control of online public opinion is mainly reflected in the monitoring, early warning, and emergency handling of information spread on social networks by the government and relevant functional departments. The theory of infectious disease dynamics starts from the prevention and control of epidemic disease transmission and can effectively describe the transmission mechanism of some infectious diseases [46]. Because the process of network information dissemination is similar to that of infectious diseases in the population, many scholars have studied the control of network public opinion using communication dynamics in order to solve the problem of information dissemination and control in the network [47,48,49,50,51]. Compared with the classic SIR model, the SEIR model [52] introduces the latent person, fully considers the psychological state of the communicator, and reflects the communication mechanism of the public network public opinion monitoring enterprise pollution behavior.

This paper aims to improve the public network public opinion supervision mechanism, improve the effect of public opinion supervision, and put forward suggestions for the formation of an effective system of enterprise pollution control mechanisms. Considering the dynamic nature of network public opinion dissemination through self-media networks and the dynamic nature of environmental pollution supervision in the self-media network environment, this paper analyzes the government’s regulatory behavior, the enterprises’ pollutant discharge behavior, and the public’s supervision behavior from the perspective of the influence scope of the network public opinion and the public opinion pressure on the government and pollution enterprises to analyze the mechanism of the dynamic nature of public opinion communication on the strategic choices of local governments and pollution enterprises in pollution treatment.

This article is framed as follows: In Section 2, we describe a coupled SEIR model and the evolutionary game model of local government, emission enterprises, and the public in the public opinion supervision of environmental pollution management. In Section 3, we analyze the impacts of different parameters, such as the public node degree, internode conversion probability, and the authenticity of public opinion supervision, on the evolutionary stable strategies of the three subjects and present an analysis of the role of public opinion dissemination on the strategy choices of local governments and sewage discharge enterprises in environmental pollution management. In Section 4, some suggestions for improving the effectiveness of new media (online) public opinion supervision are proposed.

2. SEIR Model and Evolutionary Game Model Building

2.1. SEIR Model Building

There are N individuals in online social network systems, which are divided into four categories [53]: (1) ignorant (S), a netizen who does not receive public opinion information; (2) lurker (E), a netizen who has access to information but is still hesitating; (3) supervisor (I), a netizen who publishes relevant public opinion information; and (4) immunized person (R), a netizen who receives information but loses interest without supervision. The transition process between the four types of node states is shown in Figure 1.

In Figure 1, ρ is the probability that an ignorant individual transforms into a lurker. Some latent persons are interested in public opinion information. The probability that they are transformed into supervisors is β, and the probability that they are directly transformed into immune persons is ε. Under external factors or their own interest, the probability of a supervisor transforming into an immune person is γ; when public opinion events generate new topics, the probability that an immune person chooses to monitor again is τ. The public opinion propagation model established in this paper is

$$\left\{\begin{array}{c}\frac{dS}{dt}=-[\rho k\theta (t)S(t)]/N\\ \frac{dE}{dt}=[\rho k\theta (t)S(t)]/N-[\beta k\theta (t)E(t)]/N-\epsilon E(t)\\ \frac{dI}{dt}=\beta \theta (t)E(t)-\gamma I(t)+\tau R(t)\\ \frac{dR}{dt}=\gamma I(t)+\epsilon E(t)-\tau R(t)\\ S(t)+E(t)+I(t)+R(t)=N\end{array}\right.$$

(1)

Online social networks have the characteristics of a small world, are scale-free, and are composed of heterogeneous individuals [54]. k is the node degree of the initial supervisor. θ(t) is the probability of any edge in the scale-free network connecting with the supervisor. The calculation formula of θ(t) is

$$\theta (t)=\frac{{\displaystyle \sum _{k}kp(k)I(t)}}{\langle k\rangle }$$

(2)

where p(k) is the degree distribution function and <k> is the network node centrality.

2.2. Evolutionary Game Model Building

Evolutionary game theory emphasizes the combination of game theory and the dynamic evolution process to study dynamic equilibrium in an interactive decision-making environment. The basic assumption of an evolutionary game is that the participants are not completely rational. The information possessed by participants is incomplete and asymmetric, and players often need to learn and imitate other decision makers to find better strategies. Zhen et al. [55]., Zhu et al. [56], and Xiahou et al. [57] used evolutionary game theory to analyze the pollution control behavior of enterprises under government supervision and public participation.

The local government is responsible for the natural environment within its jurisdiction in China. Pollutants discharged by sewage discharge enterprises are reasonably discharged according to the types, concentrations, emission methods, and emission directions of pollutants formulated by local governments. In the actual pollution discharge, the pollution discharge enterprises may have the behaviors of concealing the amount of pollution discharge, stealing, and leaking under the constraints of pollution discharge control technology and cost. Therefore, local governments should strive to carry out the work of protecting the watershed environment and strictly supervise the over-discharge and smuggled discharge of pollution discharge enterprises. However, local governments will choose to relax the regulation of corporate pollution, seek rapid regional economic growth, increase local fiscal revenue, and improve local government performance based on regulatory costs and regional economic development. In the process of the public network public opinion monitoring of enterprises’ pollution discharge behavior, the spread of public opinion is the diffusion of public opinion information as well as the process of the interaction and exchange of ideas, views, and values between communicators and receivers. When an individual receives a piece of public opinion information, the individual’s cognitive ability, the importance of the public opinion information, and the public’s actions jointly restrict the behavior of the individual. The behavior of the government, the pollutant discharge enterprises, and the public may change.

Based on the above analysis, the following basic assumptions are given:

Assumption 1: The model takes local governments, emission enterprises, and the public as the main players. Their emission control behaviors correspond to two strategies. The local government’s strategy is strict supervision and not strict supervision. The probabilities that the government chooses strict or not strict supervision are x and 1 − x, respectively. The pollution discharge enterprises’ strategies are standard emission and non-standard emission. The probabilities that the sewage discharge enterprises choose standard emission or non-standard emission are y and 1 − y, respectively. The public’s strategies are public opinion supervision and no public opinion supervision. The probability that the public chooses supervision is z; the probability of choosing no supervision is 1 − z.

Assumption 2: Local governments need to pay cost C₁ for strict supervision, including the strict investigation of corporate pollution behavior and the timely publication of relevant information about local governments’ inspection of corporate pollution behavior when public opinions arise. The cost of local government’s non-strict supervision is C₂. As long as the local government is strictly regulated, it must be able to find the phenomenon of excess discharge and the leakage of sewage discharge enterprises before the public. It will reduce the cost of enterprise development, accelerate the development of the local economy, and increase the economic benefits of local governments by K₁. At the same time, it may lead to pollution accidents where enterprises fail to discharge pollutants. The cost for local governments to control pollution is C₃. Whether it is local government supervision or public opinion supervision, the local government is fined L for as long as it is found that enterprises do not meet the discharge standards of pollutants. The central government verifies the public opinion information when the public opinion supervision exposes enterprises’ non-standard sewage discharge behavior. If enterprises have non-compliance behaviors and local governments do not strictly supervise them, the central government blames local governments and imposes a penalty of K₂. If enterprises do not have non-compliance behaviors, the central government punishes the public according to the influence of public opinion, such as deleting posts, labeling, fines, and investigating legal liability. The punishment of public untrue speech is based on the income obtained by the public. The greater the income, the greater the punishment, and the punishment coefficient is q. The central government’s clarification of the emission behavior of enterprises on the Internet restores the reputation of emission enterprises, reduces the opportunity cost, and even obtains the opportunity benefit without loss. The reputation recovery coefficient is d. Public opinion supervision finds that when the local government is strictly regulated the credibility of the local government increases by M. When the local government is not strictly regulated, the credibility of the local government decreases by N.

Assumption 3: If the sewage emission enterprises discharge pollutants according to the standard, the cost of reaching the standard is P₁, and the cost of not reaching the standard is P₂. If an enterprise fails to meet the discharge standards, it will cause losses to public health; the losses are A₂. Regardless of the local government supervision or public opinion supervision, the pollutant discharge enterprise pays compensation for public damage as long as it finds that an enterprise fails to meet the discharge standards; the compensation is B when public opinion supervision reveals that enterprises do not meet the emission standards. Under the strict supervision of local governments, enterprises lose reputation; the loss is F. When local governments do not strictly supervise, the reputation loss of enterprises is H.

Assumption 4: The cost of public opinion supervision is A₁, including the cost of obtaining information and the time cost. The public obtains revenue by exposing the non-compliance behavior of pollutant discharge enterprises on the Internet. In the case of local government regulation, the revenue is G; and in the case of a local government that is not strictly regulated, the revenue is Q. The information exposed by public opinion supervision is not necessarily true, and there is distortion. The probability of the authenticity of public opinion supervision is a.

Assumption 5: In the case of strict supervision by local governments, the numbers of ignorant individuals, latent individuals, supervisors, and immune individuals are S₁, E₁, I₁, and R₁, respectively. The probability of ignorant individuals transforming into latent individuals is ρ₁, the probability of latent individuals transforming into supervisors is β₁, the direct immune rate is ε₁, the probability of supervisor self-healing is γ₁, and the topic derivative rate is τ₁. When local governments do not strictly supervise, the numbers of ignorant individuals, latent individuals, supervisors, and immune individuals are, respectively, S₂, E₂, I₂, and R₂. The probability of ignorant individuals transforming into latent individuals is ρ₂, the probability of latent individuals transforming into supervisor is β₂, the direct immune rate is ε₂, the probability of supervisor self-healing is γ₂, and the topic derivative rate is τ₂. As local governments respond quickly and effectively to public opinion in strict supervision, public opinion has little communication power and in this case the spread of small. Therefore, ρ₁ < ρ₂, β₁ < β₂, ε₁ > ε₂, γ₁ > γ₂, τ₁ < τ₂. The network public opinion’s initial dissemination time is S₀ = S₁ = S₂, E₀ = E₁ = E₂, I₀ = I₁ = I₂, R₀ = R₁ = R₂.

Assumption 6: The influence of network public opinion dissemination on the incomes of the three parties in the evolutionary game lies in the scope of network public opinion dissemination, including the number of supervisors and immunized persons, and the influence of supervisors is higher than that of immunized persons. Therefore, the incomes of the three parties are determined by the maximum value of supervisors and immunized persons [58]. In the process of initial supervisors spreading public opinion information, each increase in a supervisor’s local government credibility unit change is m₁, the reputation loss of sewage discharge enterprises is f₁, and the public unit income is g₁. The unit of change in the credibility of the local government with each additional immunization is m₂, m₁ > m₂; the unit of reputation loss for the pollutant discharge enterprise is f₂, f₁ > f₂; and the public unit of income is g₂, g₁ > g₂.

Based on the above assumptions, the payoff matrix of the tripartite model was established, as shown in

Table 1. The payoff matrix.

Pollution Discharge Enterprises	Local Government Strict Supervision		Local Government Does Not Strictly Supervise
	Public Opinion Supervision	No Public Opinion Supervision	Public Opinion Supervision	No Public Opinion Supervision
Standard emission	$-C_1+(1-a)M$	$-C_1$	$-C_2-(1-a)N$	$-C_2$
	$-P_1-(1-a)(1-d)F$	$-P_1$	$-P_1-(1-a)(1-d)H$	$-P_1$
	$-A_1+(1-a)(1-q)G$	$0$	$-A_1+(1-a)(1-q)Q$	$0$
Non-standard emission	$-C_1-C_3+K_1+L+aM$	$-C_1-C_3+K_1+L$	$-C_2-C_3+K_1+L-K_2-aN$	$-C_2-C_3+K_1$
	$-P_2-L-B-aF$	$-P_2-L-B$	$-P_2-L-B-aH$	$-P_2$
	$-A_1-A_2+B+aG$	$-A_2+B$	$-A_1-A_2+B+aQ$	$-A_2$

Notes: $M=(\max I_1-I_0)m_1+(\max R_1-R_0)m_2$; $N=(\max I_2-I_0)m_1+(\max R_2-R_0)m_2$; $F=(\max I_1-I_0)f_1+(\max R_1-R_0)f_2$; $H=(\max I_2-I_0)f_1+(\max R_2-R_0)f_2$; $G=(\max I_1-I_0)g_1+(\max R_1-R_0)g_2$; $Q=(\max I_2-I_0)g_1+(\max R_2-R_0)g_2$.

.

2.3. Analysis of Evolutionary Stability Strategy

2.3.1. Replicator Dynamic Equation

The expected payoffs of the local government choosing strict supervision are

$$\begin{array}{l}{U}_{11}=yz[-C_1+(1-a)M]+(1-y)z(-C_1-C_3+K_1+L+aM)+\\ y(1-z)(-C_1)+(1-y)(1-z)(-C_1-C_3+K_1+L)\end{array}$$

(3)

The expected payoffs of the local government not choosing strict supervision are

$$\begin{array}{l}{U}_{12}=yz[-C_2-(1-a)N]+(1-y)z(-C_2-C_3+K_1+L-K_2-aN)+\\ y(1-z)(-C_2)+(1-y)(1-z)(-C_2-C_3+K_1)\end{array}$$

(4)

The average expected revenue of local governments is

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

(5)

The replicator dynamic equation of local governments choosing strict supervision is

$$\begin{array}{l}F(x)=\frac{dx}{dt}=x(1-x)\{-yL+z[K_2-L+a(M+N)]+\\ yz[L-K_2+(1-2a)(M+N)]-C_1+C_2+L\}\end{array}$$

(6)

Similarly, the expected benefits to pollutant discharge enterprises for choosing the emission compliance strategy and the emission non-compliance strategy are

$$\begin{array}{l}{U}_{21}=xz[-P_1-(1-a)(1-d)F]+x(1-z)(-P_1)\\ +(1-x)z[-P_1-(1-a)(1-d)H]+(1-x)(1-z)(-P_1)\end{array}$$

(7)

$$\begin{array}{l}{U}_{22}=xz(-P_2-L-B-aF)+x(1-z)(-P_2-L-B)+\\ (1-x)(1-z)(-P_2-L-B-aH)+(1-x)(1-z)(-P_2)\end{array}$$

(8)

The average expected income of pollutant discharge enterprises is

$$\overline{U_2}=y{U}_{21}+(1-y)U_{22}$$

(9)

Thus, the replicator dynamic equation of pollutant discharge enterprises’ choice of emission strategy is

$$\begin{array}{l}F(y)=\frac{dy}{dt}=y(1-y)\{x(L+B)+z[L+B+(2a-1+d-ad)H]\\ +xz[-L-B+(1-2a-d+ad)(H-F)]-P_1+P_2\}\end{array}$$

(10)

Similarly, the expected benefits of public opinion supervision are

$$\begin{array}{l}{U}_{31}=xy[-A_1+(1-a)(1-q)G]+x(1-y)(-A_1-A_2+B+aG)+\\ (1-x)y[-A_1+(1-a)(1-q)Q]+(1-x)(1-y)(-A_1-A_2+B+aQ)\end{array}$$

(11)

The expected benefits of non-public opinion supervision are

$$U_{32}=x(1-y)(-A_2+B)+(1-x)(1-y)(-A_2)$$

(12)

The average income of the public is

$$\overline{U_3}=z{U}_{31}+(1-z)U_{32}$$

(13)

The replication dynamic equation of public opinion supervision behavior is

$$\begin{array}{l}F(z)=\frac{dz}{dt}=z(1-z)\{x[a(G-Q)]+y[-B+(1-2a-q+aq)Q]\\ +xy(1-2a-q+aq)(G-Q)-A_1+B+aQ\}\end{array}$$

(14)

2.3.2. Analysis of Evolutionary Stable Points

Let F(x) = 0, F(y) = 0, and F(z) = 0. We obtain eight special equilibrium points and one mixed-strategy equilibrium solution. The special equilibrium points are, respectively, (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), and (1, 1, 1). The mixed-strategy equilibrium solution is x*, y*, and z*, as shown in Equation (15).

$$\left\{\begin{array}{c}{x}^{*}=\frac{-y[(1-2a-q+aq)Q-B]+A_1-aQ-B}{a(G-Q)+y(1-2a-q+aq)(G-Q)}\\ y^{*}=\frac{-z[a(M+N)-L+K_2]+C_1-C_2-L}{-L+z[(1-2a)(M+N)+L-K_2]}\\ z^{*}=\frac{-x(L+B)+P_1-P_2}{(2a-1+d-ad)H+L+B+x[(1-2a-d+ad)(H-F)-L-B]}\end{array}\right.$$

(15)

The evolutionary stable equilibrium of a multi population evolutionary game must be a strict Nash equilibrium. The mixed-strategy Nash equilibrium cannot resist the small interference accumulated many times, and finally evolves to a pure strategy. Therefore, the stability of the mixed-strategy equilibrium point is not discussed [59]. According to the method proposed by Friedman, the stability of the equilibrium point is judged by analyzing the local stability of the Jacobi matrix of the differential equation group [60]. For a linear stationary system, the equilibrium point is the evolutionary stable point when all eigenvalues in the Jacobi matrix are non-positive. The Jacobi matrix is

$$\begin{gathered} \mathbf{T}=\left(\begin{array}{ccc}(1-2x)\{-yL+z[K_2-L+ \\ a(M+N)]+yz[L-K_2+(1 \\ \hspace{1em}-2a\left)(M+N)\right]-C_1+C_2+L\}\hfill & x(1-x)\{-L+\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ z[L-K_2+(1-2a)(M+N)]\}\hfill & x(1-x)\{K_2-L+a(M+N)+ \\ y[L-K_2+(1-2a)(M+N)]\}\hspace{1em}\hfill \\ y(1-y)\{L+B+z[-L-B \\ +(1-2a-d+ad)(H-F)\left]\right\}\hfill & (1-2y)\left\{x\right(L+B)+z[L+B+(2a \\ -1+d-ad\left)H\right]+xz[-L-B+(1 \\ -2a-d+ad\left)(H-F)\right]-P_1-P_2\} \hfill & y(1-y)\{L+B+(2a-1+d \\ -ad)H+x[-L-B+(1-2a \\ -d+ad\left)\right(H-F\left)\right]\}\hspace{1em}\hspace{1em}\hspace{1em} \hfill \\ z(1-z)\left[a\right(G-Q)\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em} \\ +y(1-2a-q+aq)(G-Q)]\hfill & z(1-z)[-B+(1-2a-q+aq)Q \\ +x(1-2a-q+aq)(G-Q)]\hspace{1em}\hspace{1em}\hfill & (1-2z)\left\{x\right[a(G-Q)]+y[-B\hspace{1em}\hspace{1em} \\ +(1-2a-q+aq)Q]+xy(1-2a- \\ q+aq)(G-Q)-A_1+B+aQ\}\hspace{1em}\hfill \end{array}\right) \end{gathered}$$

(16)

Substituting the eight local equilibrium points into the Jacobi matrix, three eigenvalues corresponding to eight local equilibrium points are obtained, as shown in

Table 2. Eigenvalues of equilibrium points.

Equilibrium Points	${\lambda }_1$	${\lambda }_2$	${\lambda }_3$
(0, 0, 0)	$-C_1+C_2+L$	$-P_1+P_2$	$-A_1+B+aQ$
(0, 0, 1)	$-C_1+C_2+K_2+a(M+N)$	$-P_1+P_2+L+B+(2a-1+d-ad)H$	$-(-A_1+B+aQ)$
(0, 1, 0)	$-C_1+C_2$	$-(-P_1+P_2)$	$-A_1+(1-a-q+aq)Q$
(0, 1, 1)	$-C_1+C_2+(1-a)(M+N)$	$-[-P_1+P_2+L+B+(2a-1+d-ad)H]$	$-[-A_1+(1-a-q+aq)Q]$
(1, 0, 0)	$-(-C_1+C_2+L)$	$-P_1+P_2+L+B$	$-A_1+B+aG$
(1, 0, 1)	$-[-C_1+C_2+K_2+a(M+N)]$	$-P_1+P_2+L+B+(2a-1+d-ad)F$	$-(-A_1+B+aG)$
(1, 1, 0)	$-(-C_1+C_2)$	$-(-P_1+P_2+L+B)$	$-A_1+(1-a-q+aq)G$
(1, 1, 1)	$-[-C_1+C_2+(1-a)(M+N)]$	$-[-P_1+P_2+L+B+(2a-1+d-ad)F]$	$-[-A_1+(1-a-q+aq)G]$

.

In actual pollution control, whether it is local government supervision or public opinion supervision, the main purpose is to let the enterprises consciously deal with the pollutants to meet the discharge standard. Therefore, we only discuss the stability of the four equilibrium points where the probability that the enterprises choose the standard discharge strategy tends to 1. The cost of strict supervision by local governments is greater than that of non-strict supervision, i.e., C₁ > C₂. Therefore, (1, 1, 0) is an unstable point. Enterprises choose secret discharge, leakage, and other behaviors that can reduce the cost of pollutant treatment, which means that the cost of enterprises choosing to meet the discharge standard is greater than that of choosing to fail to meet the discharge standard, namely P₁ > P₂. That is, (0, 1, 0) is an unstable point.

When −C₁ + C₂ + (1 − a)(M + N) < 0, -P₁ + P₂ + L + B + (2a − 1 + d − ad)H > 0, and –A₁ + (1 − a − q + aq)Q > 0, (0, 1, 1) is the stable point. The income of local governments choosing a strict supervision strategy is less than that when choosing non-strict supervision. In the case of local governments choosing non-strict supervision, the income of enterprises choosing standard discharge is greater than that of enterprises choosing a non-standard discharge strategy, and the cost of public opinion supervision is less than the income.

When −C₁ + C₂ + (1 − a)(M + N) > 0, −P₁ + P₂ + L + B + (2a − 1 + d − ad)F > 0, and –A₁ + (1 − a − q + aq) G > 0, the evolutionary stable point is (1, 1, 1). This is because the income of local governments choosing strict supervision is greater than when choosing non-strict supervision. In the case of local governments choosing strict supervision, the income of enterprises choosing standard discharge is greater than that of enterprises choosing non-standard discharge, and the cost of public opinion supervision is less than the income.

Public network opinion supervision changes the credibility of local governments, the reputations of enterprises, and the public opinion supervision income through the dissemination of public opinion, which affects the change in the number of supervisors and immune persons and affects the stability strategy of local governments, enterprises, and the public. Therefore, the influence of public opinion propagation law on the tripartite evolutionary game cannot be obtained by simply analyzing the eigenvalues of the eight equilibrium points. Next, Python software is used for a numerical simulation to analyze the influence of public opinion propagation on the tripartite stability strategy.

3. Results and Discussion

In the process of network public opinion information dissemination, the time of public opinion information dissemination is relatively short. The inputs and outputs of network users are not considered in this paper. That is, the network structure and the total number of network users remain unchanged. Let N = 20,000, m = 2, and <k> = 10 [61]. According to the defined parameters, a BA scale-free network model is generated [62]. It is assumed that the public who chooses public opinion supervision is the initial supervisor of the network public opinion, and the nodes are ignorant, namely S₀ = 19,999, E₀ = 0, I₀ = 1, and R₀ = 0.

Let C₁ = 3, C₂ = 1, P₁ = 5, P₂ = 2, L = 2, K₂ = 2, A₁ = 0.3, B = 0.2, d = 0.2, q = 0.2, x₀ = 0.5, y₀ = 0.5, and z₀ = 0.5 [13,63].

3.1. Effect of Initial Supervisor Node Degree on the Evolution of Tripartite Strategy

Let ρ₁ = 0.1, ρ₂ = 0.2, β₁ = 0.1, β₂ = 0.2, ε₁ = 0.5, ε₂ = 0.4, γ₁ = 0.5, γ₂ = 0.4, τ₁ = 0.02, τ₂ = 0.05, m₁ = f₁ = g₁ = 0.0003, m₂ = f₂ = g₂ = 0.00008, and a = 0.7. The initial supervisor node degrees are 1, 5, 10, 30, and 50, respectively. The results are shown in Figure 2.

Figure 2a shows that local governments tend to choose strict supervision with the increase in the initial supervisor node degree. In addition, the greater the initial supervisor node degree, the faster the local governments tend to choose strict supervision. Figure 2b also shows that when the initial supervisor node degree is small, the probability of enterprises choosing standard discharge is stable at 0. In addition, when the initial supervisor node degree is large, the enterprises tend to choose the standard discharge strategy. Figure 2c shows that the public with a low node degree will choose not to supervise public opinion because of their low influence and authority and the low income obtained by public opinion supervision.

3.2. Effect of Transformation Probability on the Evolution of Tripartite Strategy

Let k = 20, m₁ = f₁ = g₁ = 0.0003, m₂ = f₂ = g₂ = 0.00008, and a = 0.7.

(1)

Let β₁ = 0.1, β₂ = 0.2, ε₁ = 0.5, ε₂ = 0.4, γ₁ = 0.5, γ₂ = 0.4, τ₁ = 0.02, and τ₂ = 0.05.

The ρ₁ values are, respectively, taken as 0.1, 0.3, 0.5, 0.7, and 0.9; the ρ₂ values are, respectively, taken as 0.2, 0.4, 0.6, 0.8, and 1. The results are shown in Figure 3.

It can be found from Figure 3 that changing the probability of transformation from an ignorant individual to a latent individual does not change the evolutionary game equilibrium strategy results or the final evolution trends of the local government, the enterprises, and the public. However, under different values of the probability of transformation from an ignorant individual to a latent individual, the change rate of the stable point (1, 0, 1) is different, and the rate where the probability of an enterprise choosing to meet the discharge standard tends to 0 is the most significant. With an increase in the probability of transformation from an ignorant individual to a latent individual, local governments tend to choose strict supervision faster, the enterprises tend to choose non-standard discharge slower, and the public tends to choose non-public opinion supervision faster.

(2)

Set ρ₁ = 0.1, ρ₂ = 0.2, ε₁ = 0.5, ε₂ = 0.4, γ₁ = 0.5, γ₂ = 0.4, τ₁ = 0.02, and τ₂ = 0.05.

The β₁ values are, respectively, taken as 0.1, 0.3, 0.5, 0.7, and 0.9. The β₂ values are, respectively, taken as 0.2, 0.4, 0.6, 0.8, and 1. The results are shown in Figure 4.

From Figure 4, it can be seen that changing the size of the probability of transformation from a latent individual to a supervisor does not change the results and trends of the evolutionary game equilibrium strategy of the local government and the public, but it affects the speed of the local government and the public to stabilize the point. With the increase in the probability of transformation from a latent individual to a supervisor, the local government tends to accelerate the choice of strict supervision, and the probability of public opinion supervision is stable at 1. However, Figure 4b shows that with the increase in the probability of conversion from a lurker to a supervisor, the enterprise changes from the original selection of non-standard discharge to the selection of standard discharge. The results show that increasing the probability of transition from a lurker to a supervisor can increase the peak value of supervisors, increase the influence of public opinion, and increase the reputation loss of enterprises for choosing non-standard discharge. Therefore, the enterprises finally choose to meet the standard.

(3)

Set ρ₁ = 0.1, ρ₂ = 0.2, β₁ = 0.1, β₂ = 0.2, γ₁ = 0.5, γ₂ = 0.4, τ₁ = 0.02, and τ₂ = 0.05.

The ε₁ values are 0.2, 0.4, 0.6, 0.8, and 1, respectively; the ε₂ values are 0.1, 0.3, 0.5, 0.7, and 0.9, respectively. The results are shown in Figure 5.

It can be seen from Figure 5 that the evolutionary strategies of local governments, enterprises, and the public are still stable at (1, 0, 1) with an increase in the direct immunization rate, but the probability of strict supervision by local governments to stabilize at 1 slows down, and the rate of enterprises tending to choose non-standard discharge becomes faster.

(4)

Set ρ₁ = 0.1, ρ₂ = 0.2, β₁ = 0.1, β₂ = 0.2, ε₁ = 0.5, ε₂ = 0.4, τ₁ = 0.02, and τ₂ = 0.05.

The γ₁ values are, respectively, taken as 0.2, 0.4, 0.6, 0.8, and 1; the γ₂ values are, respectively, taken as 0.1, 0.3, 0.5, 0.7, and 0.9. The results are shown in Figure 6.

Figure 6a shows that the probability of changing supervisor self-healing does not change the final outcome of local government evolution strategies, only slowing the rate at which local governments tend to choose strict regulation. From Figure 6b,c, it can be seen that the probability of the self-healing of supervisors affects the influence of public opinion. The larger the probability of the self-healing of supervisors, the smaller the influence, the smaller the reputation loss of the enterprises, and the smaller the public income. Therefore, the enterprises tend to choose non-standard discharge, and the public tends to choose no public opinion supervision.

(5)

Set ρ₁ = 0.1, ρ₂ = 0.2, β₁ = 0.1, β₂ = 0.2, ε₁ = 0.5, ε₂ = 0.4, γ₁ = 0.5, and γ₂ = 0.4.

The τ₁ values are, respectively, taken as 0.1, 0.2, 0.3, 0.5, and 0.7; the τ₂ values are, respectively, taken as 0.2, 0.4, 0.6, 0.8, and 1. The results are shown in Figure 7.

Figure 7 shows that with an increase in the topic derivative rate, the trend of local government to choose strict supervision is accelerated, the probability of enterprises choosing standard discharge behavior is stable at 1, and the public tends to choose public opinion supervision. The main reason is the increase in the topic derivative rate. It significantly increases the peak value of supervisors, expands the scope of network public opinion spread, the increases the impact on society. Due to the increase in the topic derivative rate, the local government chooses strict supervision because of the credibility of the increase, the enterprises choose standard discharge because of the increase in reputation loss, and the public chooses public opinion supervision because of the income increases.

It can be seen from Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 that the probability of transformation from an ignorant individual to a latent individual and the direct immunization rate have no effect on the final evolutionary strategy results of local governments, enterprises, and the public and only affect the evolutionary rates of the three parties. The probability of transformation from a latent person to a supervisor changes the evolution trend of the enterprises. When the probability of transformation from a latent person to a supervisor is relatively small, the probability of an enterprise choosing the standard discharge is stable at 0. In addition, with an increase in the probability of transformation from a latent person to a supervisor, the probability of a sewage discharge enterprise choosing the standard discharge is stable at 1. The probability of a supervisor self-healing has a great influence on the strategy evolution of the enterprises and the public. With a decrease in the probability of supervisor self-healing, the enterprises change from non-standard discharge behavior to standard discharge behavior, and the public changes from no public opinion supervision behavior to public opinion supervision behavior. The topic derivative rate also affects the results of the evolutionary game stability strategy of the enterprises. With an increase in the topic derivative rate, the probability of enterprises choosing standard discharge is finally stabilized at 1. The numerical results show that the parameters that have great influence on the evolution of the main tripartite strategy change the stability strategy of the tripartite model by changing the number of supervisors and the time to reach the peak value of supervisors. This is because the size of the peak value of supervisors directly determines the scope of the network public opinion as well as the impact of the public opinion on society, and a shorter time to reach the peak value of supervisors indicates a longer duration of public opinion. In order to strengthen the willingness of the public to supervise the discharge behavior of enterprises by network public opinion, improve the effectiveness of public network public opinion supervision, and enable enterprises to discharge according to the standards, the central government has taken some measures to increase the probability of transformation from a latent person to a supervisor and the topic derivative rate and reduce the probability of self-healing of supervisors.

3.3. Impact of the Authenticity of Public Opinion Supervision on Tripartite Strategy Choice

Set k = 30, ρ₁ = 0.1, ρ₂ = 0.2, β₁ = 0.1, β₂ = 0.2, ε₁ = 0.5, ε₂ = 0.4, γ₁ = 0.5, γ₂ = 0.4, τ₁ = 0.02, τ₂ = 0.05, m₁ = f₁ = g₁ = 0.0003, and m₂ = f₂ = g₂ = 0.00008. The authenticity of public opinion supervision (a) is, respectively, taken as 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. The results are shown in Figure 8.

From Figure 8a,b, when the authenticity of public opinion supervision is relatively low, with the increase in authenticity, local governments tend to choose strict supervision faster, and the public tends to choose public opinion supervision faster. However, when the authenticity of public opinion supervision increases to a certain value, the local government tends to slow the speed of strict supervision, and finally the evolution strategy of the local government is stable in not strict supervision. This shows that, when the authenticity of public opinion supervision is relatively low, the higher the authenticity, the more public supervision can significantly improve the supervision level of local governments. However, with an increase in the authenticity of public opinion supervision, public opinion supervision has a certain substitution effect on local government supervision.

From Figure 8b, the authenticity of public opinion supervision is positively correlated with the probability of enterprises choosing standard discharge. The higher the authenticity of public opinion supervision, the greater the reputation loss of the enterprises not choosing standard discharge. Therefore, the probability of the enterprises choosing standard discharge is stable at 1. The results show that although the high authenticity of public opinion supervision has a certain substitution effect on local government supervision, the binding force on the pollution control behavior of the enterprises has not weakened or it has even increased.

3.4. Impact of Unit Credibility Change and Unit Reputation Loss on Local Government and Enterprises

Let k = 30, ρ₁ = 0.1, ρ₂ = 0.2, β₁ = 0.1, β₂ = 0.2, ε₁ = 0.5, ε₂ = 0.4, γ₁ = 0.5, γ₂ = 0.4, τ₁ = 0.02, τ₂ = 0.05, g₁ = 0.0003, g₂ = 0.00008, and a = 0.7.

(1) Let f₁ = 0.0004 and f₂ = 0.00012. The m₁ values are, respectively, taken as 0.0002, 0.0003, 0.0004, 0.0005, and 0.0006; the m₂ values are, respectively, taken as 0.00005, 0.0008, 0.00012, 0.0002, and 0.0003. The results are shown in Figure 9.

From Figure 9, when the unit credibility of the local government increases, the probability of the local government choosing strict supervision changes from stable at 0 to stable at 1, and with an increase in unit credibility, the change rate of the local government tending to choose strict supervision becomes faster.

(2) Let m₁ = 0.0004 and m₂ = 0.00012. The f₁ values are, respectively, taken as 0.0002, 0.0003, 0.0004, 0.0005, and 0.0006; the f₂ values are, respectively, taken as 0.00005, 0.00008, 0.00012, 0.0002, and 0.0003. The results are shown in Figure 10.

Figure 10 shows that there is a positive correlation between the probability of the enterprises choosing standard discharge and the unit reputation loss. With an increase in the unit reputation loss of the enterprises, the cost of the enterprises choosing non-standard discharge increases, so the probability of the pollutant discharge enterprises choosing standard discharge is stable at 1.

The above analysis shows that the public’s participation in monitoring the pollution discharge behavior of enterprises through the self-media network can improve the regulatory efficiency of local governments and constrain the behavior of enterprises. It is an effective measure to control environmental pollution. This finding is consistent with previous studies [12,18] and shows that, in the process of public online opinion supervision, when considering the different interest needs of multiple participants, the participants’ strategy choices will fluctuate. This result reflects problems associated with the current environmental supervision mode of China. The central government can take measures to increase the local government’s unit credibility or increase the unit reputation loss of the enterprises and improve the effectiveness of public opinion supervision.

4. Conclusions and Suggestions

Aiming at the problem of environmental pollution control in the self-media network environment, the SEIR model and the evolutionary game model were coupled to describe the dynamics of public opinion dissemination and the dynamics of environmental pollution control. Based on the coupled model, the influence of public opinion dissemination law and public opinion authenticity on the selection of the stable evolutionary strategies of local governments, enterprises, and the public was analyzed. The conclusions and the corresponding policy suggestions are as follows:

(1) The public with high influence or authority gains higher returns than the general public through public opinion exposure of non-compliance by enterprises, and the supervision of the public with high influence or authority has obvious binding effects on local governments and polluting enterprises. Therefore, cultivating and supporting the public with high influence or authority is one of the important measures to improve the effectiveness of public opinion supervision and improve the environmental pollution control system. The central government should improve the public participation mechanism, strengthen the propaganda of environmental protection, and attract more public participation in environmental protection, especially the network news media with high attention and development potential, the websites of non-governmental environmental protection organizations, personal microblogs, WeChat public numbers, and so on. Through the establishment of special funds to support the public to participate in the supervision of environmental pollution control, and material and honorary awards for the public making outstanding contributions, the central government should enhance the participation of all individuals, especially the public with high influence or authority, in the supervision of pollution control and improve their initiative and enthusiasm.

(2) The probability of transformation between nodes is the key factor affecting the influence of network public opinion. The higher the influence of network public opinion, the higher the effectiveness of public opinion supervision of enterprise discharge behavior. The probability of transformation from a lurker to a supervisor, the probability of the self-healing of supervisors, and the topic derivative rate are the main factors affecting the influence of public opinion. Therefore, measures should be taken to increase the probability of transformation from a lurker to a supervisor and the topic derivative rate and decrease the probability of self-healing of supervisors. While strengthening environmental protection publicity, the central government can report and publish the online verification of public opinion supervision and typical pollution incidents, providing basic information for the dissemination of online public opinion. It can also cultivate online commentators; let them publish, reproduce, and comment on posts reflecting the truth on the Internet; and even cultivate ‘‘opinion leaders’’ to lead public opinion and increase public interest. In addition, news of other related real events can be released to improve the topic derivative rate and extend the time of public opinion dissemination when publishing or reprinting the facts of typical pollution events.

(3) The real network public opinion has a certain substitution effect on local government regulation, but it can more effectively restrain the non-compliance behavior of the enterprises. Government departments should strengthen the synchronous monitoring and timely verification of the non-compliance emission information on enterprises exposed by the network media, strengthen the standardization and management of the network media, improve relevant laws and regulations, and restrict or encourage self-media to improve the authenticity of exposure.

(4) The central government can not only effectively improve the effectiveness of the public opinion media network by expanding the influence range of public opinion, but it can also improve the pollution control system by reinforcing the credibility loss of local governments and the reputation loss of enterprises because of non-standard discharge. The central government’s performance evaluation system for local governments should include not only economic benefit indicators and environmental quality indicators but also the credibility of local governments. At the same time, measures such as publicly releasing information on non-standard discharge events, publishing blacklists of non-standard discharge, and reducing the environmental occupational health level of enterprises with non-standard discharge can increase the reputation loss of enterprises with non-standard discharge. Setting up emission barriers to discharge in government bidding projects can improve the pollution control consciousness of enterprises.

In this paper, we only consider the initial public’s benefits from network supervision. In further research, we can analyze the public’s supervision in different stages in the evolution of the public opinion network and consider the influence of time delay and the social strengthening mechanism on the behavior of public opinion network supervision.