Influence of the Government Department on the Production Capacity Reserve of Emergency Enterprises Based on Multi-Scenario Evolutionary Game

Abstract

With the frequent occurrence of world emergency events, the emergency management of government departments in various countries is facing great challenges. In the context of the COVID-19 pandemic, the shortage of various emergency resources is prominent, and the emergency supplies reserve combined by governments and enterprises is an important guarantee for dealing with emergency management problems. This paper mainly studies the impact of a government sustainability-oriented innovation policy on the production capacity reserve of security emergency enterprises (SEEs), and constructs an evolutionary game model between the local government and SEEs. Then, it focuses on the impact of a reputation mechanism on the government enterprise cooperative reserve. According to the condition of a country, the central government’s punishment mechanism is introduced, and the evolutionary and stable strategies of the behavior choices of the local government and SEEs are compared and analyzed. Combined with the evolution degree of emergency events, the numerical simulation analysis is used to deduce and simulate the multi scenario of the example model. The results show that the reputation mechanism can effectively promote the evolution of government and enterprises to the ideal state under various scenarios. When the punishment intensity of the central government is higher than the total benefit of the implementation of the mere formality policy by the local government, it can effectively prevent the omission of the local government. This study provides a new idea for the government to formulate reasonable policies to promote SEEs to reserve production capacity, which is more conducive to government enterprise cooperation to deal with emergency events.

1. Instruction

In general, emergency events are divided into natural disasters, accident disasters, public health events and social security events that occur suddenly and cause or may cause serious social harm, which require the need for emergency response measures. In the first three quarters of 2022, a total of 107 million people suffered from natural disasters in China, such as floods, wind and hail, droughts, typhoons, earthquakes and geological disasters. In total, 525 people died and disappeared, and 2.335 million people were urgently transferred for treatment. The direct economic loss was as high as CNY 209.59 billion (statistics are from the Ministry of Emergency Management of the People’s Republic of China). According to relevant research, China’s social security situation has entered a high-risk period [1,2]. Therefore, it is an important guarantee to have an efficient response to all kinds of emergencies in order to actively ensure successful management after the occurrence of emergency events, and to improve the current emergency supplies reserve capacity, support capacity and allocation capacity.

There is global demand for the security emergency industry. According to reported statistics, the market size of the global security emergency industry has increased from USD 531.11 billion in 2011 to USD 864.93 billion in 2017, and it is predicted that the market size of the global security emergency industry will reach USD 1425.65 billion in 2024. (Source: Report of Market Prospective and Investment Strategic Planning On China Emergency Industry (2022–2027)). Industrial development is an important basis for the reserve of production capacity. Security emergency enterprises (SEEs) is the general name for enterprises that can provide special products and emergency services, such as monitoring and early warning, disposal and rescue, prevention and protection for various emergency events.

Since the outbreak of the COVID-19 pneumonia, the demand for emergency supplies, such as medical protection and rescue disposal, has increased [3,4], and the humanitarian department has recognized the importance of having a strategic reserve of emergency supplies [5]. Joshi pointed out that resources can be mobilized from other channels to achieve the safe stock of materials, which is one of the ways to solve the problem of supply chain interruption [6]. Relying solely on the reserves of the government or humanitarian relief organizations leads to difficulty in meeting the huge material needs after emergency events [7]. Sun et al. proposed the application of new technology in the field of material reserves and improving the reserve efficiency by virtue of digital transformation [8]. Wang and Xie analyzed the demand, storage process and influencing factors of emergency materials and proposed the joint reserve of emergency materials [9]. Chen et al. [10] analyzed that the joint reserve of the government and enterprises before the accident is the optimal strategy through evolutionary game theory (EGT) and proposed the emergency supplies joint reserve mode (ESJRM) for government and enterprises. Men et al. [11] established a reserve model that combines physical reserves with contractual enterprise production capacity reserves, which can not only improve the efficiency of emergency reserves, but also reduce government costs while maintaining disaster relief capacity. Zhang et al. [12] built a tripartite evolutionary game model on the basis of ESJRM and made a preliminary beneficial exploration on the influencing factors of government enterprise cooperation. This paper focuses on the cooperation mechanism between the government and enterprises in emergency supplies reserves, builds an evolutionary game model of government enterprise cooperation, explores the process, conditions and results of the evolutionary strategy equilibrium and continues to find the main factors that affect the cooperation between the government and SEEs.

Different from the traditional classical evolutionary game theory, evolutionary game theory (EGT) combines the ideas of evolutionary biology and rational economics, regards game subjects as bounded rationality and believes that participants can achieve game equilibrium through constant trial and error and learning [13,14]. Fisher [15] conducted game analysis on the conflict and cooperation between animals and plants without relying on any rational assumptions, which is the earliest application of EGT. Smith and Price [16] put forward the concept of evolutionary stability strategy (ESS) for the first time, and later the concept of replicator dynamics was introduced by Taylor and Jonker [17]. The expansion and dynamics of these two concepts have become an important part of the development of evolutionary games. EGT has been widely used in the study of government and enterprises as participants, especially in the field of new energy vehicles [18], electric energy [19], public health [20], environmental governance [21], etc. In this paper, as the direct stakeholders of the emergency reserves problem, the government and SEEs will make decisions according to their own interests, and the optimal strategy choice can enable both parties to jointly obtain long-term interests.

This paper focuses on exploring the conditions for the local government and SEEs to achieve a stable equilibrium strategy through model construction and scenario simulation, and the research results provide management countermeasures for the government to cooperate with SEEs to conduct capacity reserves. The problems studied in this paper include the following three points:

(1)

What factors will affect the strategic choice of the local government and SEEs?

(2)

Under the reward and punishment mechanism of the central government, how is the optimal ESS of the material capacity reserve system of the local government and SEEs formed?

(3)

What are the differences in the factors that affect the strategic choices of the government and SEEs under different scenarios?

On the one hand, the innovation of this paper is reflected in the use of the EGT system to analyze the behavior evolution mode of the SEE’s emergency supplies production capacity reserve strategy selection under the influence of the central and local government policies, and multiple parameters are set from the perspective of the perceived reputation benefits and losses to expand the factors that affect the problem of government enterprise cooperation supplies reserves. On the other hand, it is reflected in simulating a variety of scenarios and displaying the influence of related factors on the evolution results in a more intuitive and scientific form with data simulation software, which has practical value for reference.

The rest of this paper is as follows: In Section 2, we summarize and review the existing relevant literature from two aspects: emergency supplies reserves and the application of evolutionary game theory. In Section 3, the research problem is described, and the meaning of variables and parameters involved in the evolutionary game model is explained in detail. An evolutionary game model including the central government, the local government and SEEs is constructed. Under possible assumptions, the stability of the model is qualitatively analyzed. In Section 4, the scenario deduction stability strategy analysis and numerical simulation experiment results analysis are carried out. In Section 5, summary and suggestions are made and future research directions are determined.

2. Literature Review

2.1. Research on Emergency Supplies Reserve

Most of the existing research on emergency material reserves are qualitative. Scholars mainly study the reserve mode, reserve cost and the response efficiency of reserve materials to emergencies. Whybark [22] pointed out that the capacity reserve is more flexible and can effectively reduce the static reserve of emergency supplies. On the basis of the current situation of emergency material reserves in China, Zhang pointed out that in addition to the physical reserve mode, the country should also reserve material production capacity [7]. Focusing on the Chinese emergency supplies reserve system, expanding the material reserve warehouse and flexibly managing the capacity reserve were proposed by Shen et al. [23] to deal with the problem of insufficient capacity reserves of medical supplies. From the perspective of minimizing the total cost to the government, Chen and Shi [24] put forward the optimal reserve scheme of government physical reserves and enterprise production capacity reserves. Some scholars actively explored the mode of government enterprise joint reserves from the perspective of option contract material supply procurement pricing [25,26,27]. Zhang [28] studied the reserve and allocation of emergency supplies between the government and the private sector from the perspective of reserve capacity and response efficiency, and proposed the optimal reserve allocation strategy for local government departments based on a variety of scenarios.

Quantitative research on emergency supplies mainly focuses on the location selection, inventory planning and prediction of physical storage. Cong and Yu [29] considered that the typhoon path was divided into typhoon scenarios and combined with multi-objective genetic algorithm NSGA-II to solve the multi-objective optimization model for the location of regional emergency supplies reserves. Ai et al. [30] proposed a discrete nonlinear integer programming model and combined it with a hybrid heuristic genetic algorithm to study the location allocation of emergency resources in the maritime emergency system. Coskun et al. [31] determined the optimal inventory quantity of emergency supplies of two humanitarian relief agencies by simulating the game of inventory transfer cooperation under an inventory shortage. Liu et al. [32] established a two-stage optimization framework before and after the accident to study the location of the material warehouse and the emergency supplies reserve plan under an uncertain environment, so as to provide effective emergency actions for the rapid response of emergency supplies. Olanrewaju et al. [33] brought the supplier’s decision into the disaster emergency preparedness stage, proposed a multi-stage stochastic programming model for disaster response material supplies, signed an agreement with the supplier and tried to reduce the total cost of material procurement from the supplier. Zhang et al. [34] suggested that decision-makers should use the combination of safety stock and production capacity to expand the reserve level with capital reserves when reserving emergency medical supplies. Sun et al. [35] used case-based reasoning (CBR) to forecast the demand for emergency supplies in order to reasonably configure and optimize railway emergency resources in case of emergency events, so as to effectively reserve emergency supplies.

At present, China’s security emergency supplies are mainly reserved by the government alone, which can provide material support in a timely manner. However, there are problems such as high procurement and management costs and material waste caused by poor management [36], and the number and types of supplies reserved are limited. The importance of emergency supplies reserves has been demonstrated in current literature, and it is worth applying the method of emergency supplies production capacity reserve. When the emergency supplies reserved by the government and the enterprises are not enough to deal with emergency events, the government can relieve the financial pressure and social losses caused by emergencies by building the enterprise production capacity reserve base in order to ensure the timely supply of emergency materials and avoid high costs caused by the temporary coordination of enterprise material procurement. Most of the above views are qualitative analyses. The quantitative research literature focuses on the location, inventory and demand forecast of emergency materials. Scholars have seldom used quantitative methods such as modeling analysis and numerical scenario simulation to find out the factors that affect the production capacity reserve of emergency supplies for government enterprise cooperation, and cannot provide targeted policies or recommendations for local government departments to solve the problem of emergency supplies reserves.

2.2. Application of Evolutionary Game Theory

At present, most scholars use evolutionary game models to study government decision-making behavior. The policies implemented by government departments and the related parameters, such as the possible losses and gains, were converted into specific values by scholars, and a mathematical model was established to calculate and deduce the stable relationship between the various subjects. Adida et al. [37] constructed a non-cooperative game model to study the problem of public emergency supplies reserves in hospitals under the condition of uncertain demand, finding that the model can reduce the inventory cost. Du and Qian [38] studied the impact of a government mobilization strategy on state-owned non-profit organizations and grass-roots non-profit organizations in disaster relief. With the help of the EGT, this paper makes a concrete analysis of ESSs from the perspectives of cooperation benefits, reward and punishment factors and coordination costs.

In the past, scholars have analyzed many factors that affect the cooperation between governments and enterprises. Heetun et al. [39] believed that effective cooperation between aid agencies is the key to post-disaster recovery and reconstruction. With the help of the evolutionary game model, it was found that cooperation between disaster relief agencies was affected by the reputation and cooperation potential of game players. Yang et al. [40] pointed out that one-time subsidies in the incentive policies of government departments play an important role in studying the development of national renewable energy projects. Qiu et al. [41] established an evolutionary game model and a system dynamics model to explore the optimal decision-making scheme for local emergency management departments and logistics enterprises to spontaneously dispatch emergency supplies across regions. At the same time, scholars pointed out that economic rewards and punishments have a decisive impact on cross-regional coordinated dispatching. Fan et al. [42] pointed out that the key for the government to deal with public health emergency events is to provide policy preferences, especially preferential tax policies. Liang and Liu [43] constructed a tripartite game model of the government, protective equipment enterprise and consumer to discuss the dynamic mechanism of medical protective equipment market supervision and introduced two incentive forms, finding that the nonlinear dynamic penalty subsidy mechanism had the best incentive effect. Qi and Yang [44] introduced the central government’s punishment mechanism into the governance of network public opinion in emergency crisis events and found that the punishment was higher than the supervision input cost of local governments, which would promote the active supervision of local governments. Zhang et al. [45] built an evolutionary game model to analyze the quality of government regulation in the dynamic interaction between the government and vaccine manufacturers under different regulatory models.

In past literature, scholars used scenarios to conduct scientific deduction and analysis when studying various security emergency management problems caused by emergencies. Based on the uncertainty characteristics of the evolution of social security emergencies, Chang et al. [46] integrated the scenario theory to structurally express social security incidents and analyzed the driving factors and methods of the evolution of social security incidents. Qi and Yang [44] used the EGT to build an evolutionary game model between online media and local governments, created a variety of scenarios for the governance of online public opinion in emergency crisis events and put forward governance suggestions for government departments in the face of online public opinion in emergency crisis events. You et al. [47] studied the interaction between stakeholders in the internal security inspection system of Chinese coal enterprises and the security inspection system of Chinese coal enterprises, and further analyzed the stability of stakeholder interaction under different scenarios. Li and Wang [48] extended the Dempster–Shafer theory and CBR to predict the demand for emergency supplies. A natural disaster scenario matching method was proposed to predict losses related to natural disasters in the absence of effective decision-making data.

It can be seen from the above literature that the evolutionary game method is also often used in the research of emergency management problems, involving governments, enterprises, social organizations and the public. Through the analysis of the evolution and stability strategies of the participants, it provides a scientific theoretical basis for government departments to solve emergency management problems. From the perspective of methodology, policy simulation in the evolutionary game framework is a new supplement to policy research tools. This paper studies how to promote the sustainability of production capacity reserve cooperation between government departments and SEEs, establishes an evolutionary game model between the government and SEEs, explores the factors that affect their ESS and provides a reference for government departments to formulate feasible supervision and management policies.

3. Model Construction and Analysis

3.1. Problem Descriptions

In order to implement the important directive spirit of national leaders to “improve the unified emergency supplies guarantee system and the national reserve system”, and in accordance with the requirements of the Hebei Emergency Industry Development Plan (2020–2025) and the Guidelines for the Establishment of Emergency Supplies Production Capacity Reserve Base in Hebei Province (Trial), the government will cooperate with the local SEEs to actively build a reserve base for the production of emergency supplies at the provincial level. During this process, the government needs to strengthen the quality assessment and real-time management of the reserve base’s development and take active encouragement policies to promote the reserve behavior of enterprises [41]. Enterprises are responsible for reserving a certain amount of the production capacity of emergency supplies, so that they can supply the emergency supplies in time after the occurrence of an emergency event. The research mechanism of this paper is shown in Figure 1.

3.2. Research Hypothesis

Many emergency management problems have been exposed due to the frequent occurrence of emergency events, among which the issue of emergency material reserve and supply has been widely concerned. Government departments and SEEs play a key role in providing an emergency supplies guarantee for emergency events. Once an emergency occurs, the corresponding behavior decisions are made by the government and the SEE under the limited rational conditions according to the needs of their own interests. After the outbreak of the emergency event, if the joint reserve capacity of government departments and SEEs is at a low level, it will cause a large-scale social crisis. As the main body to deal with emergency events, government departments not only need to be successful in emergency supplies reserves, but also ensure that they can play an effective incentive and regulatory role in the emergency supplies reserve work of SEEs. Therefore, in order to establish a joint reserve mode of emergency supplies that can deal with all kinds of emergency events, it is important for government departments and SEEs to choose their behavior strategies in the evolution process. Based on the above problem description, the following basic assumptions are proposed in this paper:

(1)

The strategy set adopted by government departments when facing the problem of the emergency supplies reserve of SEEs is {active encouragement policy, mere formality policy} [49], in which the probability of “active encouragement policy $G_1$” being selected by the local government is $x\left(0\le x\le 1\right)$, and the probability of “mere formality policy $G_2$” being selected by the local government is $1-x$. It is assumed that under the influence of government policies, the set of possible strategies taken by an enterprise is {reserve $E_1$, no reserve $E_2$}. The probability of choosing the strategy of “reserve $E_1$” is $y\left(0\le y\le 1\right)$ and the probability of choosing the strategy of “not reserve $E_2$” is $1-y$. It is assumed that the probability of an emergency event occurring in the enterprise’s reserve period is $a\left(0\le a\le 1\right)$, and the probability of no emergency events occurring is $1-a$.

(2)

When the local government chooses the strategy of “active encouragement policy $G_1$”, the local government will give preferential policies $P$ to SEEs. In addition, enterprises that implement the strategy of security emergency material production capacity reserve will be given a one-time subsidy $S$, reduced tax rates and tax reduction and exemption policies. The taxes to be paid by enterprises are recorded as $f_x\left(T{S}_x,r_x\right)$. When the enterprise chooses the “reserve $E_1$” strategy, the potential benefits obtained by the government department from the relatively relieved pressure to deal with emergencies is $R_1$. The local government department takes active encouragement policies and the enterprises actively reserve emergency supplies, so that the local government can obtain additional revenue, including recognition of the public and stability of the society, which is recorded as $\Delta R$. When the enterprise chooses the “no reserve” strategy under this policy, the additional revenue of local government departments will be reduced to $k_1\Delta R\left(0<k_1<1\right)$ in proportion. The regulatory cost and base construction cost paid by the government are $C_1$, and the enterprise’s income tax payable is $f_1\left(T{S}_1,r_1\right)$. It can be seen that $T{S}_1$ is the total sales revenue obtained after the enterprise chooses the reserve strategy, and $r_1$ is the reduced tax rate.

(3)

When the local government chooses the “mere formality policy” strategy, regardless of whether the enterprise chooses the “reserve” strategy or the “no reserve” strategy, the local government will not implement preferential policies $P$, one-time subsidy $S$, reduced tax rates, tax reduction or exemption policies measures for SEEs. At the same time, the local government will leave the supervision cost $C_1$ that should have been paid by the SEEs. As the local government adopts a policy of “mere formality” and the enterprise adopts a strategy of “no reserve of production capacity”, the government will not be able to properly respond to emergency events. The economic loss that the local government needs to bear is $W$, and the loss of public trust of the local government is $L_g$. Under this policy, if the enterprise chooses the “reserve” strategy, the loss of public trust in local government department will be reduced to $k_2{L}_g\left(0<k_2<1\right)$ in proportion.

(4)

When the security emergency enterprise chooses the “reserve $E_1$” strategy of emergency supplies, the basic income obtained by the enterprise choosing the reserve strategy is $R_2$. The enterprise’s income tax payable is $f_2\left(T{S}_1,r_2\right)$, of which $T{S}_1$ is the total sales revenue of the enterprise choosing the reserve strategy and $r_2$ is the standard tax rate of enterprise income tax levied by the government department. The cost required by the enterprise for the emergency supplies reserve is $C_2$. It includes the production capacity building, management cost and maintenance cost of material reserves. The reserve capacity for emergency supplies of the enterprise is recognized by the government and the public, and the social reputation benefit brought to the enterprise is $I$. Under the “mere formality policy” of the local government department, the security emergency materials are consciously stored by the enterprise. In the event of an emergency, the enterprise can actively assist the government department to provide sufficient emergency supplies, thus obtaining multiple public recognition, which is recorded as $nI\left(n\ge 1\right)$.

(5)

When the enterprise chooses the strategy of “no reserve $E_2$” of emergency supplies, the production capacity of some emergency materials will be surplus by the enterprise, and the income of the goods produced with this production capacity is $R_3$. The enterprise income tax payable is $f_3\left(T{S}_2,r_2\right)$, in which $T{S}_2$ is the total sales revenue of the enterprise choosing the “no reserve” strategy, and $r_2$ is the standard tax rate of the enterprise income tax levied by the government department. If the enterprise does not reserve materials, when an emergency event occurs, the enterprise cannot perform its social responsibilities well and is under the pressure of public opinion. This not only damages the image of the enterprise, but also affects the normal production and operation activities of the enterprise. The enterprise loss caused is recorded as $L_e$. Under the influence of the government’s “active encouragement policy”, the enterprise still chooses the strategy of no reserve of emergency supplies, which may cause double losses for the enterprise, and is recorded as $m{L}_e\left(m\ge 1\right)$. The specific information of relevant parameters is shown in

Table 1. Parameters and explanations.

Parameters	Explanations
$a\left(0\le a\le 1\right)$	Occurrence probability of emergency event.
$R_1$	The potential benefits obtained by the local government department due to the reserve strategy of security emergency enterprises.
$\Delta R$	Additional benefits obtained by the local government due to the positive encouragement policy to gain the recognition of the public and social stability.
$k_1\Delta R\left(0<k_1<1\right)$	When the enterprise chooses the no reserve strategy, the additional revenue of the local government department will be reduced proportionally.
$P$	The local government gives policy preference to security emergency enterprises.
$S$	One-time subsidy given by the local government to enterprises due to security emergency enterprises’ choice of production capacity reserve strategy.
$C_1$	Supervision cost of the local government department.
$W$	Economic losses to be borne by the local government department.
$L_g$	Loss of public trust to be borne by local governments.
$k_2{L}_g\left(0<k_2<1\right)$	As enterprises choose the reserve strategy, the loss of public trust of the local government department is reduced proportionally.
$R_2$	Basic income obtained from the enterprise’s choice of reserve strategy.
$C_2$	Cost incurred by enterprises for emergency supplies reserve.
$I$	The benefit from the social reputation of an enterprise.
$nI\left(n\ge 1\right)$	Under the government’s mere formality policy, enterprises have obtained multiple social reputation benefits due to their choice of reserve strategy.
$R_3$	The income obtained by the enterprise from the products produced by the remaining production capacity of the no reserve strategy.
$L_e$	Enterprise losses caused by the no reserve strategy.
$m{L}_e\left(m\ge 1\right)$	Under the active encouragement policy of the local government, enterprises have suffered multiple enterprise losses due to the no reserve strategy.
$f_1\left(T{S}_1,{\mathrm{r}}_1\right)$	Enterprise income tax payable levied by the local government department at tax rate ${\mathrm{r}}_1$; the total sales revenue of the reserve strategy selected by the enterprise is $T{S}_1$.
$f_2\left(T{S}_1,{\mathrm{r}}_2\right)$	Enterprise income tax payable levied by the local government department at tax rate ${\mathrm{r}}_2$; the total sales revenue of the reserve strategy selected by the enterprise is $T{S}_1$.
$f_3\left(T{S}_2,{\mathrm{r}}_2\right)$	Enterprise income tax payable levied by the local government department at tax rate ${\mathrm{r}}_2$; the total sales revenue of the no reserve strategy selected by the enterprise is $T{S}_2$.

. (Note: Among them, the income tax payable (refunded) in the current period or the income tax payable approved by the tax authority in the current period is $f_x\left(T{S}_x,r_x\right)$ = taxable income * tax rate—exemption of enterprise income tax for eligible enterprises—actual paid income tax, income tax payable = total income—non taxable income—non exempt income).

3.3. Construction of Evolutionary Game Model between the Local Government Department and SEEs

3.3.1. Evolutionary Game Model between the Local Government Departments and SEEs

According to the above hypothesis analysis, the evolutionary game payoff matrix between local government departments and SEEs is established, as shown in

Table 2. Payoff matrix of local government departments and SEEs.

		Security Emergency Enterprises
		Reserve E1(y)	No Reserve E2(1 − y)
Local government departments	active encouragement policy $G_1$($x$)	$U_{11}=a{R}_1+a\Delta R-P-aS+a{f}_1-C_1$ $V_{11}=a{R}_2+P+aS-a{f}_1+aI-C_2$	$U_{12}=a{k}_1\Delta R-P+f_3-C_1$ $V_{12}=R_3+P-f_3-am{L}_e$
	mere formality policy $G_2$($1-x$)	$U_{21}=a{R}_1+a{f}_2{+\mathrm{C}}_1-a{k}_2{L}_g$ $V_{21}=a{R}_2+anI-a{f}_2-C_2$	$U_{22}=C_1+f_3-aW-a{L}_g$ $V_{22}=R_3-f_3-a{L}_e$

.

In dealing with emergency events, local government departments and SEEs have limited rationality in their strategic choices. In order to describe the specific evolution process of both parties’ participants, we must find a stable strategy of group evolution by constructing a replicator dynamic equation of the behavior strategies of local government departments and SEEs. The detailed solution process is as follows:

Suppose that the expected utility of the local government department when it chooses the “active encouragement policy $G_1$” strategy is $U_1$, the expected utility of the “mere formality policy $G_2$” strategy is $U_2$, and the average expected utility of the local government department is $\overline{U}$, respectively.

$$\begin{array}{cc}{U}_1\hfill & =y{U}_{11}+\left(1-y\right)U_{12}\hfill \\ & =\left(a{R}_1+a\Delta R-a{k}_1\Delta R-aS+a{f}_1-f_3\right)y+a{k}_1\Delta R-P+f_3-C_1\hfill \end{array}$$

(1)

$$\begin{array}{cc}{U}_2\hfill & =y{U}_{21}+\left(1-y\right)U_{22}\hfill \\ & =\left(a{R}_1+a{f}_2+a{L}_g-a{k}_2{L}_g-f_3+aW\right)y+C_1+f_3-aW-a{L}_g\hfill \end{array}$$

(2)

The average expected utility of the local government department can be calculated by simultaneous Equations (1) and (2), then:

$$\begin{array}{cc}\overline{U}\hfill & =x{U}_1+\left(1-x\right)U_2\hfill \\ & =x\left[y{U}_{11}+\left(1-y\right)U_{12}\right]+\left(1-x\right)\left[y{U}_{21}+\left(1-y\right)U_{22}\right]\hfill \\ & =xy{U}_{11}+x\left(1-y\right)U_{12}+\left(1-x\right)y{U}_{21}+\left(1-x\right)\left(1-y\right)U_{22}\hfill \end{array}$$

(3)

According to the Malthusian equation, the growth of the local government department choosing the “active encouragement policy” strategy should be equal to the payoff $U_1$ minus the average payoff $\overline{U}$. The replicator dynamic equation of the local government department can be obtained by simultaneous Equations (1)–(3).

$$\begin{array}{cc}F\left(x\right)& =\frac{dx}{dt}=x\left(U_1-\overline{U}\right)==x\left(1-x\right)\left[y\left(U_{11}-U_{21}\right)+\left(1-y\right)\left(U_{12}-U_{22}\right)\right]\hfill \\ & =x\left(1-x\right)\left[\left(a\Delta R-a{k}_1\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-a{L}_g-aW\right)y+a{k}_1\Delta R-P-2C_1+aW+a{L}_g\right]\hfill \end{array}$$

(4)

Suppose that the expected utility of the SEE when it chooses the “reserve $E_1$” strategy is $V_1$, the expected utility of the “no reserve $E_2$” strategy is $V_2$, and the average expected utility of the SEE is $\overline{V}$, respectively.

$$\begin{array}{cc}{V}_1\hfill & =x{V}_{11}+\left(1-x\right)V_{21}\hfill \\ & =\left(P+aS+aI-anI-a{f}_1+a{f}_2\right)x+a{R}_2+anI-a{f}_2-C_2\hfill \end{array}$$

(5)

$$\begin{array}{cc}{V}_2\hfill & =x{V}_{12}+\left(1-x\right)V_{22}\hfill \\ & =\left(P+a{L}_e-am{L}_e\right)x+R_3-f_3-a{L}_e\hfill \end{array}$$

(6)

The average expected utility of the SEE can be calculated by simultaneous Equations (5) and (6), then:

$$\begin{array}{cc}\overline{V}\hfill & =y{V}_1+\left(1-y\right)V_2\hfill \\ & =y\left[x{V}_{11}+\left(1-x\right)V_{21}\right]+\left(1-y\right)\left[x{V}_{12}+\left(1-x\right)V_{22}\right]\hfill \\ & =xy{V}_{11}+\left(1-x\right)y{V}_{21}+x\left(1-y\right)V_{12}+\left(1-x\right)\left(1-y\right)V_{22}\hfill \end{array}$$

(7)

The replicator dynamic equation of SEEs can be obtained by simultaneous Equations (5)–(7).

$$\begin{array}{cc}F\left(y\right)\hfill & =\frac{dy}{dt}=y\left(V_1-\overline{V}\right)==y\left(1-y\right)\left[x\left(V_{11}-V_{12}\right)+\left(1-x\right)\left(V_{21}-V_{22}\right)\right]\hfill \\ & =\mathrm{y}\left(1-y\right)\left[\left(aS-a{f}_1+aI-anI+am{L}_e-a{L}_e+a{f}_2\right)x+a{R}_2+anI-a{f}_2-C_2-R_3+f_3+a{L}_e\right]\hfill \end{array}$$

(8)

The two-dimensional dynamical system (I) of local government departments and SEEs is established in simultaneous Equations (4) and (8), and then:

$$\{\begin{array}{l}F\left(x\right)=\frac{dx}{dt}==x\left(1-x\right)\left[\left(a\Delta R-a{k}_1\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-a{L}_g-aW\right)y+a{k}_1\Delta R-P-2C_1+aW+a{L}_g\right]\hfill \\ F\left(y\right)=\frac{dy}{dt}=\mathrm{y}\left(1-y\right)\left[\left(aS-a{f}_1+aI-anI+am{L}_e-a{L}_e+a{f}_2\right)x+a{R}_2+anI-a{f}_2-C_2-R_3+f_3+a{L}_e\right]\hfill \end{array}$$

(9)

Further, find the equilibrium point of system (I) as follows:

Let $F\left(x\right)=0$ to get $x_1=0$, $x_2=1$ and $y^{\ast }=\frac{-a{k}_1\Delta R+P+2C_1-aW-a{L}_g}{a\Delta R-a{k}_1\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-a{L}_g-aW}$

Let $F\left(y\right)=0$ to get $y_1=0$, $y_2=1$ and $x^{\ast }=\frac{-a{R}_2-anI+a{f}_2+C_2+R_3-f_3-a{L}_e}{aS-a{f}_1+aI-anI+am{L}_e-a{L}_e+a{f}_2}$

3.3.2. Stability Analysis of Evolutionary Game Model between Local Government Departments and SEEs

According to the above replicator dynamic system (I), the equilibrium points $\left(x,y\right)\in \left\{\left(x,y\right)|0\le x\le 1,0\le y\le 1\right\}$ obtained include: $\left(0,0\right)$, $\left(0,1\right)$, $\left(1,0\right)$ and $\left(1,1\right)$. When $0\le \frac{-a{k}_1\Delta R+P+2C_1-aW-a{L}_g}{a\Delta R-a{k}_1\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-a{L}_g-aW}\le 1$ and $0\le \frac{-a{R}_2-anI+a{f}_2+C_2+R_3-f_3-a{L}_e}{aS-a{f}_1+aI-anI+am{L}_e-a{L}_e+a{f}_2}\le 1$, there is a system equilibrium point $\left(x^{\ast },y^{\ast }\right)$.

According to the method proposed by Friedman, the partial derivative of system (I) is calculated to construct the Jacobian matrix $J_1\left(x,y\right)$, and the stability of each equilibrium point is analyzed with the help of the Jacobian matrix $J_1\left(x,y\right)$ obtained. The results are shown in Equations (10) and (11). According to EGT, if the ESS is to be obtained, its corresponding Jacobian matrix $J_1\left(x,y\right)$ needs to meet two conditions at the same time, namely, determinant $Det\left(J\right)>0$ and trace $Tr\left(J\right)<0$.

$$J_1\left(x,y\right)=\left[\begin{array}{cc}\frac{\partial F\left(x\right)}{\partial x}& \frac{\partial F\left(x\right)}{\partial y}\\ \frac{\partial F\left(y\right)}{\partial x}& \frac{\partial F\left(y\right)}{\partial y}\end{array}\right]$$

(10)

Then, each element in the Jacobian matrix $J_1\left(x,y\right)$ is specifically expressed as follows:

$$\begin{array}{c}\frac{\partial F\left(x\right)}{\partial x}=\left(1-2x\right)\left[\left(a\Delta R-a{k}_1\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-a{L}_g-aW\right)y+a{k}_1\Delta R-P-2C_1+aW+a{L}_g\right]\hfill \\ \frac{\partial F\left(x\right)}{\partial y}=x\left(1-x\right)\left(a\Delta R-a{k}_1\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-a{L}_g-aW\right)\hfill \\ \frac{\partial F\left(y\right)}{\partial x}=y\left(1-y\right)\left(aS-a{f}_1+a{f}_2+aI-anI+am{L}_e-a{L}_e\right)\hfill \\ \frac{\partial F\left(y\right)}{\partial y}=\left(1-2y\right)\left[\left(aS-a{f}_1+a{f}_2+aI-anI+am{L}_e-a{L}_e\right)x+a{R}_2+anI-C_2-R_3+f_3+a{L}_e\right]\hfill \end{array}$$

(11)

Next, the positive and negative of the determinant and trace conditions are further determined by calculating the eigenvalues of the Jacobian matrix. The calculation results of eigenvalues are shown in

Table 3. Eigenvalues of the Jacobian matrix.

Equilibrium Points ${\mathit{E}}_{\mathit{i}}\left(\mathit{x},\mathit{y}\right)$	Eigenvalues ${\mathit{\lambda }}_{\mathit{i}1}$	Eigenvalues ${\mathit{\lambda }}_{\mathit{i}2}$
$E_1\left(0,0\right)$	${\lambda }_{11}=a{k}_1\Delta R-P-2C_1+aW+a{L}_g$	${\lambda }_{12}=a{R}_2+anI-a{f}_2-C_2-R_3+f_3+a{L}_e$
$E_2\left(0,1\right)$	${\lambda }_{21}=a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1$	${\lambda }_{22}=-a{R}_2-anI+a{f}_2+C_2+R_3-f_3-a{L}_e$
$E_3\left(1,0\right)$	${\lambda }_{31}=-a{k}_1\Delta R+P+2C_1-aW-a{L}_g$	${\lambda }_{32}=aS-a{f}_1+aI+am{L}_e+a{R}_2-C_2-R_3+f_3$
$E_4\left(1,1\right)$	${\lambda }_{41}=-\left(a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1\right)$	${\lambda }_{42}=-\left(aS-a{f}_1+aI+am{L}_e+a{R}_2-C_2-R_3+f_3\right)$

$i=1,2,3,4$.

.

According to the eigenvalue of each equalization point calculated above, the determinant and trace of five equalization points are calculated, as shown in

Table 4. Determinants and traces of equilibrium points in evolutionary game systems.

Equilibrium Points ${\mathit{E}}_{\mathit{i}}\left(\mathit{x},\mathit{y}\right)$	$\mathit{D}\mathit{e}\mathit{t}\left(\mathit{J}\right)$	Sign	$\mathit{T}\mathit{r}\left(\mathit{J}\right)$	Sign
$E_1\left(0,0\right)$	${\lambda }_{11}\times {\lambda }_{12}$	$+/-$	${\lambda }_{11}+{\lambda }_{12}$	$+/-$
$E_2\left(0,1\right)$	${\lambda }_{21}\times {\lambda }_{22}$	$+/-$	${\lambda }_{21}+{\lambda }_{22}$	$+/-$
$E_3\left(1,0\right)$	${\lambda }_{31}\times {\lambda }_{32}$	$+/-$	${\lambda }_{31}+{\lambda }_{32}$	$+/-$
$E_4\left(1,1\right)$	${\lambda }_{41}\times {\lambda }_{42}$	$+/-$	${\lambda }_{41}+{\lambda }_{42}$	$+/-$
$E_5\left(x^{\ast },y^{\ast }\right)$	${\lambda }_{51}\times {\lambda }_{52}$	$+/-$	0	0

below.

After analyzing Table 4, it is found that the stability of equilibrium points $\left(0,0\right)$, $\left(0,1\right)$, $\left(1,0\right)$ and $\left(1,1\right)$ cannot be directly judged, and Taylor [17] pointed out that although point $\left(x^{\ast },y^{\ast }\right)$ is only a stable equilibrium point in the game system, it does not have asymptotic stability, and the system will not automatically stabilize to point $\left(x^{\ast },y^{\ast }\right)$. The above analysis shows that there is no ESS between local governments and SEEs, and the evolutionary trend will change with the change of some variables. Next, the initial variables are adjusted appropriately, the evolutionary trend of the game system is discussed and analyzed under four possible assumptions, and the ESS of the game system is determined:

(1)

When $a{k}_1\Delta R-P-2C_1+aW+a{L}_g<0$ (i.e., $a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1>0$), $0<x^{\ast }<1$ is divided into the following two cases for discussion and analysis:

Hypothesis 1. 

When $a{k}_1\Delta R-P-2C_1+aW+a{L}_g<0$ (i.e., $a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1>0$) and $aS+aI+am{L}_e+a{R}_2+f_3-a{f}_1-C_2-R_3>0$, the evolutionary stability strategies of the system are $\left(0,0\right)$ and $\left(1,1\right)$.

Hypothesis 2. 

When $a{k}_1\Delta R-P-2C_1+aW+a{L}_g<0$ (i.e., $a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1>0$) and $aS+aI+am{L}_e+a{R}_2+f_3-a{f}_1-C_2-R_3<0$, there is no evolutionary stability strategy in the system.

(2)

When $a{k}_1\Delta R-P-2C_1+aW+a{L}_g>0$ (i.e., $a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1<0$), $0<x^{\ast }<1$ is divided into the following two cases for discussion and analysis:

Hypothesis 3. 

When $a{k}_1\Delta R-P-2C_1+aW+a{L}_g>0$ (i.e., $a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1<0$) and $aS+aI+am{L}_e+a{R}_2+f_3-a{f}_1-C_2-R_3>0$, there is no evolutionary stability strategy in the system.

Hypothesis 4. 

When $a{k}_1\Delta R-P-2C_1+aW+a{L}_g>0$ (i.e., $a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1<0$) and $aS+aI+am{L}_e+a{R}_2+f_3-a{f}_1-C_2-R_3<0$, the evolutionary stability strategies of the system are $\left(0,1\right)$ and $\left(1,0\right)$.

In order to further verify the rationality of hypotheses 1–4, under the constraints of known assumptions, the stability analysis is conducted on the five evolutionary equilibrium points $\left(0,0\right)$, $\left(0,1\right)$, $\left(1,0\right)$, $\left(1,1\right)$ and $\left(x^{\ast },y^{\ast }\right)$. The specific results are shown in

Table 5. Stability analysis of the equilibrium point of the evolutionary game system.

Hypothesis 1				Hypothesis 2			Hypothesis 3			Hypothesis 4
Equilibrium Points	$\mathit{D}\mathit{e}\mathit{t}\left(\mathit{J}\right)$	$\mathit{T}\mathit{r}\left(\mathit{J}\right)$	Stability	$\mathit{D}\mathit{e}\mathit{t}\left(\mathit{J}\right)$	$\mathit{T}\mathit{r}\left(\mathit{J}\right)$	Stability	$\mathit{D}\mathit{e}\mathit{t}\left(\mathit{J}\right)$	$\mathit{T}\mathit{r}\left(\mathit{J}\right)$	Stability	$\mathit{D}\mathit{e}\mathit{t}\left(\mathit{J}\right)$	$\mathit{T}\mathit{r}\left(\mathit{J}\right)$	Stability
$E_1\left(0,0\right)$	$+$	$-$	ESS	$-$	$N$	Saddle point	$-$	$N$	Saddle point	$+$	$+$	Instability point
$E_2\left(0,1\right)$	$+$	$+$	Instability point	$+$	$+$	Instability point	$-$	$N$	Saddle point	$+$	$-$	ESS
$E_3\left(1,0\right)$	$+$	$+$	Instability point	$-$	$N$	Saddle point	$-$	$N$	Saddle point	$+$	$-$	ESS
$E_4\left(1,1\right)$	$+$	$-$	ESS	$-$	$N$	Saddle point	$-$	$N$	Saddle point	$+$	$+$	Instability point
$E_5\left(x^{\ast },y^{\ast }\right)$	$-$	0	Not equilibrium point	$+$	0	Not equilibrium point	$+$	0	Not equilibrium point	$-$	0	Not equilibrium point

“$N$” indicates that its sign cannot be determined.

.

It can be seen from Table 5 that there are stable points $\left(0,0\right)$, $\left(1,1\right)$ and $\left(0,1\right)$, $\left(1,0\right)$ in Hypothesis 1 and Hypothesis 4, respectively, and the rest are saddle points, instability points and non-equilibrium points. The results of the comprehensive analysis of Hypothesis 1 and Hypothesis 4 are as follows:

(1)

When the total payoff of the active encouragement policies adopted by the local government department are greater than the total payoff of the mere formality policies adopted by them, the government department will choose the active encouragement policy. Assuming that the active encouragement policy of the local government department can play a better role in promoting enterprises to choose emergency supplies reserves, the evolutionary track of local government departments and SEEs will tend to {active encouragement policy, reserve}. This situation shows that the local government departments have achieved the expected effect in terms of publicity and incentive policies on the issue of security emergency supplies reserves. Reasonable active encouragement policies not only play an effective incentive role for enterprises, but also make enterprises and the public aware of the importance of emergency supplies reserves. When an emergency occurs, this ideal strategy can enable the local government and SEEs to be recognized by the public and gain benefits from social stability at the same time, and also enable the good image of the government and enterprises to be established in the public mind.

(2)

When the active encouragement policies of local government departments are ineffective, the evolutionary track of local government departments and SEEs will tend to {active encouragement policy, no reserve}. On the one hand, it shows that the total payoff of the SEE when it chooses the reserve strategy is less than the total payoff of the no reserve strategy. At this time, the enterprise will suffer less losses and will also be supported by the policy of the government department. The enterprise will choose “free riding”, which leads to the failure to establish a joint emergency supplies reserve model between the government and the enterprise. On the other hand, when government departments need to deal with emergency events that cause large social losses, even if the problem of the emergency supplies reserve of government departments is regarded as a very essential thing, then enterprises should be encouraged to actively reserve through government incentive policies. However, if the government publicity is not enough, and the incentive threshold is set unreasonably, it is difficult to achieve coordinated reserves between the government and enterprises.

(3)

When the total payoff of the active encouragement policies adopted by the local government department are less than the total payoff of the mere formality policies adopted by them, assuming that the government’s policies do not play a role in enterprises, then the evolutionary track of local government departments and SEEs will tend to {mere formality policy, no reserve}. It shows that if the frequency of emergency events is low and cannot cause large social losses, the importance of the security emergency supplies reserve has not been fully recognized by the government and enterprises. Local government departments should have a sense of crisis, assess the probability of an emergency and the social losses it will cause, formulate reasonable incentive measures and make emergency plans. In the following, the central government punishment mechanism can be introduced to constrain the regulatory actions of local governments.

(4)

When the government policy can be effectively implemented, with the increasing policy support from the government, then the huge financial expenditure will bring heavy burden to the government departments. At this time, the evolutionary track of local government departments and SEEs will tend to {mere formality policy, reserve}. Therefore, when formulating incentive policies, government departments should not only promote the selection of emergency supplies reserve strategies by SEEs in the short term, but also consider the economic capacity of local governments themselves.

3.4. Construction and Analysis of Evolutionary Game Model between the Local Government and SEEs under Central Government Punishment Mechanism

3.4.1. Evolutionary Game Model between the Local Government and SEEs under Central Government Punishment Mechanism

In order to promote local governments and SEEs to adopt active encouragement strategies and reserve strategies when responding to emergency events and avoid large social losses, it is necessary to introduce the central government’s punishment mechanism to control the local government department. The local government who adopts a mere formality policy will be punished $F$ by the central government. At the same time, the supervision and control measures of the central government have enhanced the public’s concern about the emergency supplies reserve. With the help of the Internet and social platforms, the popularity of public opinion continues to increase, and enterprises are affected jointly and severally. With the expansion of public opinion pressure, the social reputation and social losses of enterprises have increased, which become $\overline{I}\left(\overline{I}>I\right)$ and ${\overline{L}}_e\left({\overline{L}}_e>L_e\right)$. The payoff matrix of both parties under the central government punishment mechanism is shown in

Table 6. Payoff matrix of the local government department and SEEs under central government punishment mechanism.

		Security Emergency Enterprises
		Reserve ${\mathit{E}}_1$(y)	No Reserve ${\mathit{E}}_2$$(1-\mathit{y})$
Local government departments	active encouragement policy $G_1$($x$)	$U_{11}=a{R}_1+a\Delta R-P-aS+a{f}_1-C_1$ $V_{11}=a{R}_2+P+aS-a{f}_1+a\overline{I}-C_2$	$U_{12}=a{k}_1\Delta R-P+f_3-C_1$ $V_{12}=R_3+P-f_3-am{\overline{L}}_e$
	mere formality policy $G_2$($1-x$)	$U_{21}=a{R}_1+a{f}_2{+\mathrm{C}}_1-a{k}_2{L}_g-F$ $V_{21}=a{R}_2+an\overline{I}-a{f}_2-C_2$	$U_{22}=C_1+f_3-aW-a{L}_g-F$ $V_{22}=R_3-f_3-a{\overline{L}}_e$

.

In order to make the introduced central government department punishment $F$ effective for the evolutionary game system composed of local governments and SEEs, and to achieve the goal of {active encouragement policy, reserve}, the following two conditions need to be met:

(1)

The total benefit of punishing local governments that adopt mere formality policies are less than the total benefit of the local governments that adopt active encouragement policies, which is expressed as $U_{11}+U_{12}>U_{21}+U_{22}$, that is, $F>\frac{-a\Delta R+2P+aS-a{f}_1+4C_1-a{k}_1\Delta R+a{f}_2-a{k}_2{L}_g+aW-a{L}_g}{2}$.

(2)

Under the punishment mechanism of the central government, the active encouragement policy of local governments needs to play their role. The total payoff of the SEEs that adopt the strategy of emergency supplies reserve under the influence of the local government’s incentive policies such as policy preference, one-time subsidy, reduced tax rates, tax reduction and exemption policies are greater than that of the SEEs that adopt the strategy of no reserve, which is expressed as $V_{11}>V_{12}$, that is, $aS+a\overline{I}+am{\overline{L}}_e+a{R}_2+f_3-a{f}_1-C_2-R_3>0$.

According to the solution process of Equations (4) and (8) above, we can get the replicator dynamic equation of the local government department under the central government punishment mechanism, as shown in Equation (12):

$$\begin{array}{cc}F\left(x\right)\hfill & =\frac{dx}{dt}=x\left(U_1-\overline{U}\right)==x\left(1-x\right)\left[y\left(U_{11}-U_{21}\right)+\left(1-y\right)\left(U_{12}-U_{22}\right)\right]\hfill \\ & =x\left(1-x\right)\left[\left(a\Delta R-a{k}_1\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-a{L}_g-aW\right)y+a{k}_1\Delta R-P-2C_1+aW+a{L}_g+F\right]\hfill \end{array}$$

(12)

The replicator dynamic equation of the security emergency enterprise under the central government punishment mechanism is shown in Equation (13):

$$\begin{array}{cc}F\left(y\right)\hfill & =\frac{dy}{dt}=y\left(V_1-\overline{V}\right)==y\left(1-y\right)\left[x\left(V_{11}-V_{12}\right)+\left(1-x\right)\left(V_{21}-V_{22}\right)\right]\hfill \\ & =\mathrm{y}\left(1-y\right)\left[\left(aS-a{f}_1+a\overline{I}-an\overline{I}+am{\overline{L}}_e-a{\overline{L}}_e+a{f}_2\right)x+a{R}_2+an\overline{I}-a{f}_2-C_2-R_3+f_3+a{\overline{L}}_e\right]\hfill \end{array}$$

(13)

Under the punishment mechanism of the central government, the two-dimensional dynamical system (II) of the local government department and SEEs is established through the simultaneous Equations (12) and (13), as follows:

$$\{\begin{array}{l}F\left(x\right)=\frac{dx}{dt}=x\left(1-x\right)\left[\left(a\Delta R-a{k}_1\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-a{L}_g-aW\right)y+a{k}_1\Delta R-P-2C_1+aW+a{L}_g+F\right]\hfill \\ F\left(y\right)=\frac{dy}{dt}=\mathrm{y}\left(1-y\right)\left[\left(aS-a{f}_1+a\overline{I}-an\overline{I}+am{\overline{L}}_e-a{\overline{L}}_e+a{f}_2\right)x+a{R}_2+an\overline{I}-a{f}_2-C_2-R_3+f_3+a{\overline{L}}_e\right]\hfill \end{array}$$

(14)

Further, solve the equilibrium point of system (II) as follows:

Let $F\left(x\right)=0$ to get $x_1=0$, $x_2=1$ and $y^{\ast }=\frac{-a{k}_1\Delta R+P+2C_1-aW-a{L}_g-F}{a\Delta R-a{k}_1\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-a{L}_g-aW}$

Let $F\left(y\right)=0$ to get $y_1=0$, $y_2=1$ and $x^{\ast }=\frac{-a{R}_2-an\overline{I}+C_2+R_3-a{\overline{L}}_e+a{f}_2-f_3}{aS-a{f}_1+a{f}_2+a\overline{I}-an\overline{I}+am{\overline{L}}_e-a{\overline{L}}_e}$

3.4.2. Stability Analysis of Evolutionary Game Model between the Local Government Department and SEEs under Central Government Punishment Mechanism

Similarly, the Jacobian matrix $J_2\left(x,y\right)$ of system (II) can be constructed. Then, each element in the Jacobian matrix is specifically expressed as follows:

$$\begin{array}{c}\frac{\partial F\left(x\right)}{\partial x}=\left(1-2x\right)\left[\left(a\Delta R-a{k}_1\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-a{L}_g-aW\right)y+a{k}_1\Delta R-P-2C_1+aW+a{L}_g+F\right]\hfill \\ \frac{\partial F\left(x\right)}{\partial y}=x\left(1-x\right)\left(a\Delta R-a{k}_1\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-a{L}_g-aW\right)\hfill \\ \frac{\partial F\left(y\right)}{\partial x}=y\left(1-y\right)\left(aS-a{f}_1+a{f}_2+a\overline{I}-an\overline{I}+am{\overline{L}}_e-a{\overline{L}}_e\right)\hfill \\ \frac{\partial F\left(y\right)}{\partial y}=\left(1-2y\right)\left[\left(aS-a{f}_1+a{f}_2+a\overline{I}-an\overline{I}+am{\overline{L}}_e-a{\overline{L}}_e\right)x+a{R}_2+an\overline{I}-C_2-R_3+f_3+a{\overline{L}}_e\right]\hfill \end{array}$$

(15)

The process of stability analysis is the same as above; firstly, calculate the eigenvalue of each equilibrium point according to the Jacobian matrix $J_2\left(x,y\right)$, and then determine the determinant and trace of the five equilibrium points. The specific process is shown in Appendix A. It can be seen from the analysis that the stability of the equilibrium point cannot be directly judged. Therefore, under the two initial conditions that the punishment mechanism of the central government department should use effectively, we discuss and analyze the evolutionary trend of the game system under two possible hypotheses, and finally determine the optimal ESS of the game system, as follows:

Hypothesis 5. 

When $a{k}_1\Delta R-P-2C_1+aW+a{L}_g+F<0$ (i.e., $a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1+F>0$) and $aS+a\overline{I}+am{\overline{L}}_e+a{R}_2+f_3-a{f}_1-C_2-R_3>0$, the evolutionary stability strategies of the system are $\left(0,0\right)$ and $\left(1,1\right)$.

Hypothesis 6. 

When $a{k}_1\Delta R-P-2C_1+aW+a{L}_g+F>0$ (i.e., $a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1+F<0$) and $aS+a\overline{I}+am{\overline{L}}_e+a{R}_2+f_3-a{f}_1-C_2-R_3>0$, there is no evolutionary stability strategy in the system.

In order to further verify the rationality of Hypotheses 5 and 6, under the constraints of known assumptions, stability analysis is conducted on the five evolutionary equilibrium points $\left(0,0\right)$, $\left(0,1\right)$, $\left(1,0\right)$, $\left(1,1\right)$ and s$\left(x^{\ast },y^{\ast }\right)$. The specific results are shown in

Table 7. Stability analysis of the equilibrium point of the evolutionary game under central government punishment mechanism.

	Hypothesis 5			Hypothesis 6
Equilibrium Points	Det(J)	$\mathit{T}\mathit{r}\left(\mathit{J}\right)$	Stability	$\mathit{D}\mathit{e}\mathit{t}\left(\mathit{J}\right)$	$\mathit{T}\mathit{r}\left(\mathit{J}\right)$	Stability
$E_1\left(0,0\right)$	$+$	−	ESS	$-$	$N$	Saddle point
$E_2\left(0,1\right)$	$+$	$+$	Instability point	$-$	$N$	Saddle point
$E_3\left(1,0\right)$	$+$	$+$	Instability point	$-$	$N$	Saddle point
$E_4\left(1,1\right)$	$+$	$-$	ESS	$-$	$N$	Saddle point
$E_5\left(x^{\ast },y^{\ast }\right)$	$-$	0	Not equilibrium point	$+$	0	Not equilibrium point

“$N$” indicates that its sign cannot be determined.

.

From the analysis results in Table 7, it can be concluded that the strategy choices of the local government and enterprises for emergency supplies reserves are affected by the punishment mechanism of the central government to some extent. By constantly raising the upper limit of penalties imposed by the central government, the local government will pay a higher price for choosing a mere formality policy, and then the probability of the local government taking an active encouragement policy will increase. At this time, the evolutionary track of the local government department and SEEs will tend to the ideal strategy {active encouragement policy, reserve}. Similarly, on the one hand, the punishment mechanism of the central government can restrict the inaction of local governments on the issue of emergency supplies reserves; on the other hand, the supervision behavior of the central government is widely spread on the Internet and on media platforms, which will attract the attention of the wider public. As stakeholders, enterprises tend to choose emergency material reserve strategies in order to avoid losses and gain a good corporate reputation.

4. Analysis of Stability Strategy and Numerical Simulation Results under Scenario Deduction

4.1. Scenario Hypothesis Analysis

Scenario deduction analysis is to discuss the impact of different government policies and enterprise behaviors on emergency reserves under the background of emergency events, and simulate the evolution process of participants under different scenarios. The scenario deduction analysis of this paper is based on the discussion of the impact of emergency events on local governments and SEEs. On this basis, it analyzes the evolution and stability strategies of local governments and SEEs for emergency supplies reserves and simulates the evolution process of governments and enterprises under different scenarios.

This paper takes H-Province as an example to analyze the impact of government policies of H-Province on the production capacity reserve of SEEs. In recent years, driven by the national security emergency industry policy environment, H-Province attaches great importance to the construction of the security emergency industry. It has not only issued a development plan for the security emergency industry from 2020 to 2025 to improve the overall level and core competitiveness of the security emergency industry, but has also built a provincial security emergency industry demonstration base and emergency supplies production capacity reserve base to expand the scale of the province’s emergency industry. In addition, the relevant management systems of the industrial demonstration base and emergency material production capacity reserve base in H-Province have been formulated to further improve the emergency material reserve system in H-Province. H-Province has unique advantages in its geographical location: it can use the advanced technology and scientific and technological resources of emergency industry development in Beijing and Tianjin to transform the scientific and technological achievements of the two cities into enterprises in H-Province, which not only improves the reserve capacity of emergency supplies in H-Province, but also surrounding provinces and cities can be provided with emergency supplies in a timely manner. It can be seen from Figure 2 that among the key enterprises in the security emergency industry of H-Province in 2020, 195 enterprises are suitable for public health events, accounting for 41.94%. There are 88 and 81 enterprises applicable to natural disaster events and accident disaster events, respectively, accounting for 18.92% and 17.42%. There are relatively few enterprises suitable to social security events and other events, 41 and 61 enterprises, respectively, accounting for 8.60% and 13.12% of the total number of key enterprises.

According to the number of employees at the end of the period, the key SEEs in H-Province are divided into micro-enterprises (less than 10 people), small enterprises (10–100 people), medium-sized enterprises (100–300 people) and large enterprises (more than 300 people). It can be seen from Figure 3 that the proportion of micro-enterprises, small enterprises and medium-sized enterprises is 80%, and most of the SEEs in H- Province are small and medium-sized enterprises.

Emergency events are characterized by suddenness, uncertainty, destructiveness and social nature. In order to avoid greater harm to society caused by emergencies, government departments need to establish a complete emergency supplies reserve system to provide the necessary emergency materials for a timely response to all kinds of emergency events. Based on the periodic characteristics of emergencies, there are four stages in the evolution of emergency events, including incubation period, outbreak period, impact period and end period. The incubation period of an emergency event is generally long, and when it accumulates to a certain extent, it will easily bring harm to the society. The outbreak and impact periods of emergency events basically overlap with each other, resulting in disasters that continue to exist, which still cause a great destructive force to the society. The end period begins after the hazards and impacts of the emergency event are controlled. This paper divides the scenarios based on the impact of emergency events on the government and SEEs in terms of income and loss. The specific analysis is as follows:

(1)

The occurrence of emergency events triggered an increase in public opinion, which is a double-edged sword for the government and enterprises. In the short term, the additional benefits brought by the formulation of feasible response policies by the local government department are far greater than other economic losses and losses of public trust, playing a positive role in promoting social development. Similarly, the impact of enterprise reputation gains and losses will also increase, which is consistent with scenario 1.

(2)

Once an emergency occurs, the government, as the supervisor and direct stakeholder of the society, will suffer a large social loss. The social risk is assessed by the government department and the government will pay attention to the reserve of the enterprise’s emergency supplies production capacity under the influence of perceived risk. As the public pays less attention to the enterprise, the enterprise is less affected by emergency events, and the value of enterprise reputation and loss is relatively small and at a low level, which is consistent with the deduction of Scenario 2.

(3)

The central government department will decide whether to reward or punish the local government according to the social harm degree of the emergency event and the control effect of the government department. When the local government departments become a mere formality and the enterprises do not reserve production capacity, the central government will give reasonable punishment to the local government for acts of inadequate supervision during the outbreak and impact of the emergency event, which is consistent with the deduction of Scenario 3.

In order to more clearly show the impact of parameter changes on system evolution and verify the feasibility of assumptions in a more scientific way, the following three scenarios are proposed during numerical simulation, as shown in

Table 8. Description and analysis of scenario types.

Scenario Classification	Specific Description	Initial Variable Adjustment Direction	Hypothesis
Scenario 1	The strategic evolution of governments and SEEs under the scenario that the issue of emergency supplies reserves has a relatively small impact on the loss of social public trust and economic losses of the local government department, and a relatively large impact on the reputation gains and losses of SEEs.	$L_g\downarrow$; $W\downarrow$; $\Delta R\uparrow$; $I\uparrow$; $L_e\uparrow$;	Hypothesis 1
Scenario 2	The strategic evolution of governments and SEEs under the scenario that the issue of emergency supplies reserves has a relatively large impact on the social public trust loss and economic loss of the local government department, and a relatively small impact on the reputation gains and losses of SEEs.	$L_g\uparrow$; $W\uparrow$; $\Delta R\downarrow$; $I\downarrow$; $L_e\downarrow$	Hypothesis 4
Scenario 3	In the outbreak and impact periods of emergency events, the local government department and SEEs behavior and strategy evolution on emergency supplies reserves under the scenario of introducing the central government’s punishment mechanism in the absence of the local government and SEEs.	$F\uparrow$	Hypothesis 5

The arrow only represents the adjustment of the initial variable value to make it meet the hypotheses.

:

4.2. Analysis on the Influencing Factors of Strategy Evolution of the Local Government Department and SEEs in Scenario 1

In this paper, according to the constraints in hypotheses 1–6 and the replication dynamic equation, the impact of parameter changes on the strategy evolution of the local government department and SEEs is clearly demonstrated through MATLAB simulation. The initial probability of the local government department choosing the active encouragement policy and the SEEs choosing the reserve strategy is set as [0.5, 0.5], and the evolution time is set to [0,1]. The simulation data set in this paper are mainly based on: (1) Refer to previous literature to determine the values of some parameters [50,51]. (2) The setting of some parameter values refers to the actual policies formulated by government departments and takes into account the reality of relevant emergency enterprises involved in this study. (3) The discussion and communication were conducted with experts in the emergency field.

According to the macro conditions of Scenario 1, and the three constraints of Hypothesis 1 need to be met at the same time:

(1)

$a{k}_1\Delta R-P-2C_1+aW+a{L}_g<0$

(2)

$a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1>0$

(3)

$aS+aI+am{L}_e+a{R}_2+f_3-a{f}_1-C_2-R_3>0$

The evolution simulation experiment of the local government and SEEs is carried out. The specific values of each parameter in the simulation experiment of the ESS are shown in

Table 9. Specific values of each parameter in the simulation experiment of ESS.

Parameters	Initial Values	Parameters Change Under Hypothesis 1
		$\mathit{a}$	$\mathbf{\Delta }\mathit{R}$	$\mathit{S}$	$\mathit{P}$	${\mathit{f}}_1$	$\mathit{W}$	${\mathit{L}}_{\mathit{g}}$	$\mathit{I}$	${\mathit{L}}_{\mathit{e}}$
$a$	0.1	0.11; 0.12; 0.13	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
$k_1$	0.5	0.5	0.5	0.5	0.5	0.5	0.5	0.5	0.5	0.5
$\Delta R$	3600	3600	3800; 4000; 4200	3600	3600	3600	3600	3600	3600	3600
$S$	50	50	50	200; 350; 500	50	50	50	50	50	50
$k_2$	0.5	0.5	0.5	0.5	0.5	0.5	0.5	0.5	0.5	0.5
$L_g$	300	300	300	300	300	300	300	500; 700; 900	300	300
$W$	100	100	100	100	100	100	300; 500; 700	100	100	100
$P$	100	100	100	100	80; 120; 140	100	100	100	100	100
$C_1$	100	100	100	100	100	100	100	100	100	100
$n$	1	1	1	1	1	1	1	1	1.5; 2; 3	1
$m$	1	1	1	1	1	1	1	1	1	1.5; 2; 3
$L_e$	1200	1200	1200	1200	1200	1200	1200	1200	1200	1800; 2400; 3600
$I$	1200	1200	1200	1200	1200	1200	1200	1200	1800; 2400; 3600	1200
$R_2$	1000	1000	1000	1000	1000	1000	1000	1000	1000	1000
$C_2$	200	200	200	200	200	200	200	200	200	200
$R_3$	500	500	500	500	500	500	500	500	500	500
$f_1$	350	350	350	350	350	250; 150; 50	350	350	350	350
$f_2$	500	500	500	500	500	500	500	500	500	500
$f_3$	400	400	400	400	400	400	400	400	400	400

below.

(1)

The impact of the change of emergency event probability on the dynamic evolution of local governments and SEEs.

Assume that the initial probability of an emergency is $a=0.1$. In recent years, natural disasters, public health events and other emergencies have emerged in an endless stream, causing tension at home and abroad. In order to meet the actual requirements, the probability of an emergency event may be continuously increased, and its impact on the evolutionary behavior of the government and SEEs will be observed. Keeping other parameters unchanged and under the constraint condition of hypothesis 1, the values of occurrence probability of emergency events are, respectively, set as: $a=0.1$; $a=0.11$; a = 0.12; $a=0.13$. It can be observed from Figure 4a that the evolution of government departments in the initial state is towards a mere formality policy. With the increase in the frequency of emergency events, the evolution of government departments is toward the direction of the active encouragement policy strategy. The higher the frequency of emergency events, the faster the government department will evolve to active encouragement policies. Under the same conditions, it can be observed in Figure 4b that the evolution of SEEs in the initial state is toward the direction of the no reserve strategy. In addition, under the condition that the frequency of emergency events increases, the SEE not only evolves towards emergency supplies reserves, but also shortens the time for the enterprise to evolve towards the emergency supplies reserve strategy. The evolution results show that the probability of emergency events affects the decision-making behavior of the local government department and SEEs, and the government department should actively assume social responsibility and regulatory functions to reduce the economic losses caused by the increase in the frequency of emergency events. For SEEs, if they want to obtain more economic benefits in response to emergency events, they can win more development opportunities and conditions for enterprises by entering into an agreement with the government on the reserve of production capacity of emergency supplies.

(2)

The impact of different probabilities of the government’s choice to active encouragement policies on the evolution of SEEs.

In Scenario 1, changing the value of $x$ is to explore the impact of the probability of the government’s choice of positive incentive policies on the evolution of corporate behavior strategies. When $x=0.3$, $x=0.5$, $x=0.7$ and $x=0.9$, the evolution results of enterprises under different reserve probabilities are shown in Figure 5. As shown in Figure 5a, when the probability of the government choosing an active encouragement policy is higher than a certain value, the security emergency response enterprise will evolve towards the direction of emergency material reserves. When the threshold value is not reached, it will evolve towards the direction of no reserve, and the lower the probability of the SEE choosing the reserve strategy, the faster it will evolve towards the direction of no reserve. By observing Figure 5b–d, the greater the probability of the government choosing an active encouragement policy, the lower the threshold for the enterprise to evolve toward the reserve strategy, which indicates that the government’s policy has an impact on the choice of the enterprise’s strategy.

(3)

The impact of the local government’s policy preference $P$, one-time subsidy $S$ and government subsidized tax rate $r_1$ on the evolution of government departments and SEEs.

It can be seen from Figure 6 that under the three constraints of hypothesis 1, the impact of the local government’s policy preferences, one-time subsidies and tax rates after government subsidies on the evolution of the government departments and SEEs is discussed, respectively. Assume that the initial value of policy preference is $P=100$, and the adjusted values are $P=80$, $P=120$ and $P=140$ within a certain range. As shown in Figure 6a,b, it can be found that the evolution of government departments towards active encouragement policies will not be affected by the appropriate granting of policy preferences to enterprises. However, with the increase of policy preferences, the government’s financial burden will increase, which will lead to the faster evolution of government departments towards mere formality policies. For enterprises, the evolution of enterprises towards positive reserves is promoted by policy preference within a certain range. However, with the increase of policy preference, enterprises perceive that no matter whether they choose the reserve strategy or not, they will receive the same policy preference. When enterprises have the speculative mentality of “free riding”, they will reduce the reserve cost expenditure and only enjoy the policy preference rather than cooperate with the government to reserve the production capacity of emergency supplies. In order to ensure that the policy preference granted by the government can play a better role, the amount of policy preference should be controlled within a reasonable range.

Assume that the initial value of the one-time subsidy is $S=50$, and the value of the increased one-time subsidy given by the government to enterprises is $S=200$, $S=350$ and $S=500$. Similarly, suppose that the initial value of the tax rate reduction policy given by the government department is ${\mathrm{r}}_1=17.5\%$, and the government will levy income tax on enterprises at a lower rate, which is ${\mathrm{r}}_1=12.5\%$, ${\mathrm{r}}_1=7.5\%$ and ${\mathrm{r}}_1=2.5\%$, respectively. It can be seen from Figure 6c–e that one-time subsidies and tax rate reduction and exemption policies have a reverse effect on the evolution of government departments. With the increase of one-time subsidies and tax rate reduction, government departments will accelerate their evolution to a mere formality policy. As for the effect of the above two policies on enterprises, it can be seen from Figure 6d–f that one-time subsidies and tax rate reduction policies can promote the evolution of enterprises towards reserve strategies in the short term, and the rate of evolution towards reserves is significantly accelerated with the increase of subsidies. As for the effect of the above two policies on enterprises, it can be seen from Figure 6d–f that one-time subsidies and tax rate reduction policies can promote the evolution of enterprises towards reserve strategies in the short term, and the rate of evolution towards reserves is significantly accelerated with the increase of subsidies. However, from a long-term point of view, enterprises will eventually evolve in the direction of the no reserve strategy. When appropriate supportive policies are adopted by local governments, enterprises will actively cooperate with government departments under various preferential policies in the short term. In the long term, with the increase of various government expenditures, on the one hand the financial burden of the government is increased; on the other hand, enterprises believe that their losses should be compensated by the government. There is a deviation between the actual situation and the expected government support, which ultimately makes the game players’ directions evolve towards {mere formality policy, no reserve} and reach stability. In the game process, the implementation degree of the incentive policy should be accurately grasped by the government to ensure that all measures are implemented within a reasonable threshold, so as to encourage enterprises to actively reserve production capacity.

(4)

The impact of the additional benefits $\Delta R$ on the evolution of the local government department.

It can be seen from Figure 7 that under the constraint conditions of hypothesis 1, when the strategy choice of local governments and SEEs is {active encouragement policy, reserve}, this decision-making behavior makes the local governments gain the recognition of the public, social stability and other additional benefits as $\Delta R$. In order to explore the impact of additional benefits on the decision-making behavior of government departments, $\Delta R=3600$, $\Delta R=3800$, $\Delta R=4000$ and $\Delta R=4200$ are taken, respectively, and other parameters remain unchanged. The study finds that the additional benefits obtained by the government department have a positive impact on their choice of positive incentive policy strategies. The greater the additional benefits obtained by the government department, the faster the government departments evolve towards active encouragement policies. The storage behavior of government joint enterprises provides an important guarantee for government departments to deal with various emergency events. The rescue and disposal effect of emergencies has been recognized by the public through social media, which has improved the social credibility of the government to a certain extent and maintained social stability.

(5)

The impact of the economic loss $W$ and loss of public trust $L_g$ on the evolution of the local government department.

It can be seen from Figure 8 that under the three constraints of hypothesis 1, the impact of the economic loss $W$ and loss of public trust $L_g$ on the evolution of the government department is discussed, respectively. The value of economic loss is $W=100$, $W=300$, $W=500$ and $W=700$. The loss of public trust is $L_g=300$, $L_g=500$, $L_g=700$ and $L_g=900$. It can be seen from Figure 8a that when the economic losses are at a low level, the government evolves towards a mere formality policy strategy and reaches a stable state, indicating that when the economic losses caused by emergency events are small, the losses are within the acceptable range of the government, so the government fails to pay attention to them. With economic losses increasing until reaching a certain limit, the government has improved its ability to respond to emergency events by adopting active encouragement policies to compensate for the losses caused by previous weak supervision. As shown in Figure 8b, when the loss of public trust is low at the beginning, the government suffers less interference from the loss of trust, and the final evolution result is stable in a mere formality policy. As the government perceives that the loss of public trust is increasing, the government’s ultimate stable strategy is to active encouragement policies and the rate of positive evolution is getting faster and faster as the loss increases.

(6)

The impact of enterprise social reputation benefits $I$ and enterprise losses $L_e$ on the evolution of SEEs.

It can be seen from Figure 9 that under the constraint of hypothesis 1, the impact of enterprise social reputation benefits and enterprise losses on the strategic choice behavior of SEEs is explored. The impact of social reputation on enterprise evolution can be divided into the following two situations: The first is the initial value of the overall change $I$, which is taken as $I=1200$, $I=1800$, $I=2400$ and $I=3600$, respectively. Other parameters shall remain unchanged, as shown in Figure 9a. It is very important for enterprises to choose the reserve strategy whether or not they can obtain good social reputation. The government should reasonably use the active encouragement policy to publicize the SEEs with strong social responsibility to the society, and give financial support and spiritual honor incentives to the enterprises that actively reserve to deal with emergency events, so as to facilitate the evolution of both sides toward {active encouragement policy, reserve}. When the initial value of perceived reputation remains unchanged, as shown in Figure 9b, the impact of the reputation benefits gap on the choice of the enterprise’s conscious reserve strategy is studied by changing $n$. It is found that when the social reputation benefits brought by the enterprise’s conscious reserve is large enough, the enterprise will still choose the reserve strategy without incentive policies. It can be seen from Figure 9c,d that enterprise losses and enterprise social reputation have roughly the same impact on the results of enterprise evolution. When the initial loss of the enterprise is small, the enterprise will first evolve to the direction of no reserve. When the perceived loss of the enterprise is increasing, the enterprise will convert the original no reserve strategy into a reserve strategy to avoid loss risk. The coefficient $m$ has a moderating effect. If the enterprise still does not choose the reserve strategy under the government’s active encouragement policy, compared with the no reserve strategy under the mere formality policy, the enterprise will bear double condemnation from the public after an emergency. With the increase of $m$, the enterprise will accelerate its evolution towards reserves.

4.3. Analysis on the Influencing Factors of Strategy Evolution of Local Government Departments and SEEs in Scenario 2

In Scenario 2, during the incubation period of an emergency event, the government bears the risk of potential large public trust losses and economic losses. At this time, the local government still chooses to invest in policy subsidies under the condition of less perceived benefits to prevent the huge losses caused to the government by the outbreak of emergency events, and also to avoid the social impact of emergency events, which increases the public panic and destroys social harmony and stability. We adjust the data on the basis of following principles: (1) the adjustment of data is within the reasonable assumption range; (2) all changes in values meet the constraint conditions of hypothesis 4. The three constraints of Hypothesis 4 need to be met at the same time:

(1)

$a{k}_1\Delta R-P-2C_1+aW+a{L}_g>0$

(2)

$a\Delta R-aS+a{f}_1-a{f}_2+a{k}_2{L}_g-P-2C_1<0$

(3)

$aS+aI+am{L}_e+a{R}_2+f_3-a{f}_1-C_2-R_3<0$

This simulation experiment aims to explore the evolution trend and change rule of the strategies of both governments and enterprises. Therefore, the above principles can be used to set values to achieve this goal. The specific parameter changes are shown in Appendix B 

Table A2. Specific values of some parameter changes in the simulation experiment of evolutionary stability strategy.

Parameters	Initial Values	Parameters Change under Hypothesis 4
		$\mathbf{\Delta }\mathit{R}$	$\mathit{P}$	$\mathit{W}$	${\mathit{L}}_{\mathit{g}}$	$\mathit{I}$	${\mathit{L}}_{\mathit{e}}$
$\Delta R$	200	400; 600; 800	200	200	200	200	200
$L_g$	2500	2500	2500	2500	3000; 3500; 4000	2500	2500
$W$	2500	2500	2500	3000; 3500; 4000	2500	2500	2500
$P$	280	280	290; 300; 310	280	280	280	280
$n$	1	1	1	1	1	2;3;4	1
$m$	1	1	1	1	1	1	1.5; 2; 3
$L_e$	600	600	600	600	600	600	1000; 1400; 1800
$I$	600	600	600	600	600	1000; 1400; 1800	600
$R_2$	800	800	800	800	800	800	800

Other unadjusted parameters are shown in Table 9.

.

(1)

The impact of the change of emergency event probability on the dynamic evolution of local government and SEEs.

Under the constraint conditions of hypothesis 4, the influence of $a$ on the evolution results of governments and enterprises is observed. Keep other parameters unchanged, and the values of $a$ are $a=0.10$, $a=0.11$, $a=0.12$ and $a=0.13$. It can be observed from Figure 10a that in a short period of time, the strategic choice of local governments has evolved into a mere formality policy. However, as the evolution time goes on, the government department is still stable in active encouragement policy strategies. Additionally, with the increase of the probability of emergency events, the smaller the degree of short-term inward reverse evolution of government departments, and the faster the forward evolution. It can be seen from Figure 10b that although the probability of emergencies increases, the enterprise is still stable in the strategy of no reserve.

(2)

The impact of local government’s policy preference $P$, one-time subsidy $S$ and government subsidized tax rate $r_1$ on the evolution of the government department and SEEs.

Under the three constraints of hypothesis 4, analyze the policy preference, one-time subsidy and tax rate reduction and exemption policies implemented by the local government under Scenario 2 when it suffers high economic losses and trust losses of the social public, and observe the stable state of the government and SEEs. At this time, set the initial value of the policy preference given by the government department as $P=280$, and continuously increase the value of $P$. It is found from Figure 11a that the increase of $P$ makes the government evolve into a mere formality policy in a short time. However, as time goes by, due to the threat of sustained economic losses and losses of public trust, government departments will still choose to implement active encouragement policies to solve the difficulties faced by government regulators. When $P$ reaches a certain amount, the government cannot afford the preferential policy expenditure; it will be stable in a mere formality policy. It can be seen from Figure 11c,e that with the increase of one-time subsidies and policy preferences, the evolution of government departments has turned into mere formality policies first and then stabilized in active encouragement policies. What is different from Scenario 1 is that various incentive policies issued by the government at this time make the government evolution stable in an ideal state. From Figure 11b,d,f, it can be seen that the enterprise doesn’t choose the reserve strategy under the government’s active encouragement policy, and the evolution of the government and the enterprise finally goes in the direction of {active encouragement policy, no reserve}. The enterprise has enjoyed the benefits brought by the government’s policy preference and other incentives, but has not taken action. At this time, the government’s policies don’t achieve the expected results, but stimulate the enterprise’s “free riding” behavior.

(3)

The impact of the additional benefits $\Delta R$ on the evolution of the local government department.

Under the constraint conditions of hypothesis 4, the government in Scenario 2 suffers huge economic losses and public trust losses. At this time, the value of additional benefits such as public recognition and social stability obtained by the local government is relatively small, which are recorded as $\Delta R=200$, $\Delta R=400$, $\Delta R=600$ and $\Delta R=800$, respectively. It can be seen from Figure 12 that the evolution of government departments under the influence of additional benefits is finally stabilized on the active encouragement policy. After a certain time point, the increase of benefits promotes the rate of positive evolution of government departments. Among them, since the overall value of $\Delta R$ is at a low level, the impact of the additional benefit at the initial stage of evolution is less than the government’s loss, and the role of the additional benefit is less perceived by the government, which will lead to evolving in the direction of mere formality policies in a short period of time.

(4)

The impact of the economic loss $W$ and loss of public trust $L_g$ on the evolution of the local government department.

Under the three constraints of hypothesis 4, this paper discusses the evolution results of the local government under the scenarios of large economic losses and losses of social public trust. Under the condition that other parameters remain unchanged, it can be seen from Figure 13a,b that if a good emergency reserve response mechanism is not established by the local government, huge economic losses will be caused during the outbreak and impact periods of emergency events. In the early days of the emergency event, it is very important to actively encourage enterprises to reserve production capacity, which is not fully recognized by the local government. With the increase of economic losses, the government is finally urged to take an active encouragement policy, and jointly build a reserve base for emergency supplies production capacity to avoid the risk losses. If the problem of emergency material guarantee and timely supply is not handled by the government department, the losses of public trust will continue to increase, which will pose a threat to the credibility of the government department. Ultimately, it will also enable the government to evolve towards the direction of active encouragement policies and reach a stable equilibrium.

(5)

The impact of enterprise social reputation benefits $I$ and enterprise losses $L_e$ on the evolution of SEEs.

Under the constraint conditions of hypothesis 4, analyze the evolution of enterprises in Scenario 2 where social reputation benefits and enterprise losses are at a low level. The initial values of $I$ and $L_e$ are $I=600$ and $L_e=600$, and other parameters are guaranteed to remain unchanged. It can be seen from Figure 14a,c that the numerical changes of enterprise social reputation benefits and risk losses within the scope of meeting the assumptions do not affect the evolution of the enterprise towards reserves, but the evolution of the enterprise is always stable in the strategy of not reserving production capacity. In order to reduce high losses, the government department can adopt active encouragement policies, which to some extent also increases the threshold of perceived reputation benefits and risks of enterprises. As shown in Figure 14b,d, compared with the numerical simulation of Scenario 1, the size of $n$ and $m$ changes under Scenario 2. It is found that only when the doubled reputation gain is large enough, the enterprise will evolve towards reserves, and the enterprise loss evolution under this scenario is always stable in the no reserve strategy. The government department should enhance the publicity of the good reputation of SEEs and maintain the image of enterprises with the help of news media and network platforms.

4.4. Analysis on the Influencing Factors of the Strategic Evolution of the Local Government Department and SEEs under the Central Government Punishment Mechanism in Scenario 3

(1)

The impact of introducing the central government punishment mechanism $F$ on the evolution of the local government department and SEEs.

The data simulation of Scenario 3 is conducted under the constraint conditions of hypothesis 5. When the central government punishment mechanism is not introduced, the initial state of the local government evolves towards a mere formality policy. It can be seen from Figure 15a that punishment $F$ is added to the game system and the initial value is set to $F=25$, and the punishment of the central government is constantly increased. It is found that the evolution of the government department is toward the direction of active encouragement policies. In order to promote the evolution of local governments and SEEs towards the optimal strategy {active encouragement policy, reserve}, it is necessary for the central government to punish local governments appropriately. The greater the punishment of the central government, the higher the cost for the local government to choose mere formality policies. Therefore, the probability for the local government to choose active encouragement policies increases. It can be seen from Figure 15b that the introduction of the central government’s punishment mechanism changes the evolution direction of the initial state of enterprises, and the increase of punishment promotes the evolution rate of enterprises, which further confirms that the central government’s punishment has a knock-on effect on enterprises.

(2)

The impact of the central government’s punishment mechanism $F$ on the dynamic evolution results of enterprise social reputation benefits and enterprise losses.

After the evolution of the initial social reputation of the enterprise in Scenario 1, the enterprise is affected by the punishment mechanism of the central government. It is found that the evolution of the enterprise has changed from no reserve to reserve, as shown in Figure 16a. Considering that the punishment of the central government will also have an impact on the initial state of the enterprise’s reputation, the enterprise social reputation is adjusted. It is found that the greater the impact on reputation under the same punishment, the faster the evolution towards the reserve direction. Similarly, it can be seen from Figure 16b that the central government’s punishment has the same effect on enterprise losses as enterprise social reputation, indicating that the macro control policies of the central government have an indirect impact on enterprise strategic choices.

5. Conclusions

In this paper, by building an evolutionary game model between the local government and SEEs, the evolution process of the behavior strategies of game players under different scenarios can be analyzed. This paper not only discusses the impact of government incentive policies on enterprise strategy choice, but also focuses on evaluating the impact of reputation mechanisms on the evolution of governments and SEEs. The study also finds that a reasonable central punishment mechanism is the key to the ESS of the local government and SEEs from $\left(0,0\right)$ to $\left(1,1\right)$. The game model is deduced from multiple scenarios based on the numerical example, and the following conclusions are drawn:

(1)

From the evolution results of Scenario 1, we can see that the evolution system of local governments and SEEs will have two possible stable results: {mere formality policy, no reserve} and {active encouragement policy, reserve}. The ideal state of evolution of the local government and SEEs can be achieved through appropriate policy preference. In a short period of time, the evolution of SEEs towards reserves can be promoted by one-off subsidies and tax rate reduction policies. For a long time, enterprises will have the speculative mentality of “free riding”, which is not conducive to the realization of the enterprise production capacity escrow model. Both the government and enterprises will pay attention to their social reputation gains, so publicizing and maintaining the good public image of the government and enterprises is conducive to the realization of the joint reserve model for both sides. Similarly, when the government and enterprises perceive large losses, they can also seek cooperation to avoid risks and reduce their own losses.

(2)

It can be seen from the evolution results of Scenario 2 that the evolution system of the local government and SEEs will have two possible stable results: {mere formality policy, reserve} and {active encouragement policy, no reserve}. As the local government is threatened by greater risk losses, the government department prefers to choose active encouragement policy to promote enterprise reserves, so as to relieve the pressure of government supervision. The implementation of incentive measures such as policy preference and one-time subsidy has been strengthened, but it has failed to change the direction of enterprises to reserve production capacity. On the contrary, it has increased the financial pressure of the government and stimulated the “free riding” behavior of enterprises. In this scenario, one of the most effective means is to achieve joint reserves by improving the social reputation benefits of the local government and enterprises [52].

(3)

From the evolution results of Scenario 3, it can be concluded that the evolution of the game system formed by the local government and SEEs towards the ideal state {active encouragement policy, reserve} can be effectively promoted by the central government’s punishment mechanism. The punishment of the central government has a positive knock-on effect on SEEs, and indirectly affects the strategic choice of enterprises.

In order to promote the government and SEEs to reach an agreement on the reserve of emergency supplies production capacity, the following suggestions are put forward:

(1)

The government’s innovation policy has a double effect, so we should pay attention to developing strengths and avoiding weaknesses. The government has introduced a variety of incentive policies. On the one hand, it is to improve the willingness of SEEs to reserve production capacity, which is conducive to the formation of a joint emergency reserve system between the government and enterprises. On the other hand, it is easy to expand the gap between social returns and corporate returns and stimulate the emergence of “free riding” behavior. Therefore, the SEEs escrow system needs to be established and improved by government departments, and the construction level of the emergency supplies production capacity base should be improved to create a scientific and efficient government enterprise cooperation mechanism.

(2)

The government department shall publicize the policies, so that enterprises can fully understand the government policies and the actual production capacity reserves of enterprises can be monitored. The government should pay attention to publicizing the good image of security emergency enterprises, and it is important to clarify the reputation mechanism for government enterprise cooperation. At the same time, the punishment implemented by the central government can effectively prevent the inaction of the local government and provide a second line of defense for security emergency reserves.

This study is based on the recognition of the importance of preparedness and pandemic response measures in the pandemic situation [53]. The research purpose of this paper is to promote the active development of the government and enterprise emergency material reserve, and to reduce social losses caused by various emergency events. The research model proposed in this paper makes a classified discussion on the possible situations, and also provides a reference for the research on government enterprise cooperation at the national and international levels. There are still some limitations in this paper: (1) The factors influencing the level of government enterprise cooperation still need to be expanded. The reputation benefit of stakeholders cannot be obtained without the support of information dissemination [54]. Future research will consider the impact of factors such as the design of a blockchain-enabled digital humanitarian network (BT-DHN) and the information resource orchestration on emergency supplies reserves [53,54]. The technical level is one of the key factors to reduce the cost of material reserves. In particular, if artificial intelligence (AI) is applied to the government enterprise cooperation model, the potential benefits can be analyzed through quantitative research in the future [55]. (2) Ignoring the impact of risk preference factors of decision makers on the results, future research can be further analyzed in combination with prospect theory.