Research on Psychological Crisis Intervention Strategies under Emergencies: An Analysis Based on the Four-Party Evolutionary Game

Abstract

The study of multi-subject psychological crisis intervention under emergencies is of great significance for maintaining the psychological states of public groups. A government’s strict regulation can stimulate social responsibility in medical institutions and communities, increase the probability that positive behavioral strategies might be chosen, and accelerate the implementation of psychological crisis intervention. In this paper, we constructed a four-party dynamic evolutionary game model containing the government, medical institutions, communities, and the public; analyzed the asymptotic stability conditions of the behavioral strategies of each player in the game; and explored the impact of the changes of the relevant key parameters in the model on the strategic choices of the players via use of Matlab 2020a simulation. The results of the study show that there are complex interactions and competitive relationships between the subjects of the game in emergencies, and that different intervention strategies can have different impacts on the behavior and outcomes of the subjects. The accountability of superiors increases the probability that there will be strict government regulation and enhances the robustness of medical institutions and communities to choose positive behaviors. A government’s decision to increase incentives and penalties may urge healthcare providers to provide active treatment and the community to provide safety and security, while also reducing the cost of public participation in supervision and reducing resource waste. By analyzing strategic choices made by subjects under a four-party game, a perfect countermeasure can be formulated to help the public form a positive psychological coping mechanism in the event of emergencies, and to provide support and help for their peers. Thus, the psychological health of the group can be better maintained, and extreme negative emotions and behaviors can be avoided. Finally, the simulation results demonstrate the rationality of the research conclusions and provide reference suggestions by which to improve the psychological crisis intervention system.

1. Introduction

With the continuous progress of modern society, there has been a great breakthrough in the construction of material civilization in China, and the construction of spiritual civilization has become an important part of “building a socialist harmonious society”. The state of the public’s mental health is the foundation of spiritual construction, and it is positive emotions that can lead the public to a happy future. In recent years, the occurrence of emergencies has had a serious impact on people’s lives, property, and psychology, and also exacerbates the social risk of information explosion and fermentation [1]. Examples include the SARS epidemic outbreak in Guangdong Province, China, in 2003; the 8.0 magnitude earthquake in Wenchuan County, Sichuan Province, China, in 2008; the H1N1 influenza outbreak in Mexico and the United States in 2009; the MERS epidemic outbreak in Saudi Arabia in 2012; the outbreak of Ebola virus disease (EVD) in West Africa in 2016; and the COVID-19 epidemic outbreak in Wuhan, Hebei Province, China, in 2019. This series of emergencies has shown that countries and regions across the globe are facing various crises and challenges that are accompanied by incidents of social instability. The public can experience social and psychological stress reactions when stimulated by emergencies [2]. Particularly following the occurrence of fatalities in emergencies, the public naturally lacks pre-existing knowledge, leading to a state of psychological distress marked by panic, anxiety, and other adverse emotions associated with imbalance and shock [3]. During emergencies, most public health organizations and the media focus primarily on the biological and physical outcomes of the event. As a result, less attention is paid to mental health issues [4].

In the developmental processes of human history, every major disaster and security issue has, to a certain extent, promoted the updating and iteration of normative standards and conceptual methods in social management and emergency response. This has, in turn, become one of the important reasons for promoting the healthy development of human society [5]. In the process of an emergency response, it is very important to provide the necessary psychological assistance and crisis intervention to specific groups. The psychological state of the public during emergencies frequently serves as a concentrated reflection of their collective maturity and social civility. A well-developed modern emergency management system should also pay attention to the psychosocial behavior of the public [6]. During the 19th National Congress of the Communist Party of China (CPC), General Secretary Xi Jinping placed special emphasis on the need to enhance the development of a social psychological service system and foster a societal mindset characterized by self-respect, self-confidence, rationality, calmness, and positivity [7]. As a result, the management of psychological crisis intervention in emergencies has become a major issue in terms of the capacity of governments to govern and build a shared future for humanity. This paper adopts an evolutionary game theory approach in order to investigate how various participants in psychological crisis intervention effectively respond to emergencies, alleviate psychological stress, and promote social harmony and stability, taking a multi-subject participation perspective.

This paper’s primary contributions are as follows.

(1) This paper constructs a game model encompassing government, medical institution, community, and the public to analyze the interactions and competitive relationships among different participants under emergencies. This model provides a novel theoretical framework for studying psychological crisis intervention under emergencies.

(2) This paper employs an evolutionary game approach, which, in comparison with traditional game models, is more suitable for analyzing psychological crisis intervention involving multiple parties under emergencies. The evolutionary game approach takes into account the temporal evolution and interactions of strategies among the subjects, making it more adept at capturing complex gaming relationships. This helps in the obtaining of a deeper understanding of the complex dynamic relationships behind multi-subject psychological crisis interventions.

(3) This paper underscores the close connection between psychological crisis intervention under emergencies and the core principles of sustainable development, including social responsibility, government accountability, community support, and resource management. It offers robust recommendations for safeguarding the mental well-being of the public under emergencies, enhancing social sustainability, and reducing resource wastage. A government’s strict regulation can stimulate a sense of social responsibility in medical institutions and communities, which is a core element of sustainable development. It can promote the provision of public services, reduce resource wastage, and ensure societal health and safety under emergencies. Government accountability can better drive sustainable policies and measures while emphasizing the importance of community support for social sustainability. This approach enhances societal resilience and stability.

The remainder of the paper is organized as follows. Section 2 summarizes the research content related to emergencies and psychological crisis intervention and proposes the research focus of this paper. Section 3 constructs a multi-subject game model for psychological crisis intervention under emergencies and conducts stability analysis. Section 4 assigns values to each key parameter in the model to perform the simulation results analysis. Section 5 discusses the simulation results. Section 6 focuses on introducing the research conclusions of this paper, including the main findings, policy recommendations, limitations, and future directions.

2. Literature Review

As living standards improve, the public’s concern for mental health has increased on annual basis. Consequently, the field of psychological crisis intervention within emergency management has begun to garner increasing attention from governments at all levels. Since 2006, China has promulgated the National Overall Emergency Response Plan for Public Emergencies [8] and the Law on Response to Emergencies [9], demonstrating the importance that China attaches to emergency management. A great deal of research has also been conducted by scholars to raise awareness of psychological crises and to help governments carry out psychological crisis intervention. According to a survey of elderly people in the Hong Kong community, there was a substantial increase in the suicide rate of elderly people during the SARS epidemic as compared with the pre-epidemic period. This was found to be related to the increase in social isolation, psychological stress, and anxiety during the epidemic [10]. A study conducted in Singapore has revealed the significant impact on the emotions and well-being of medical personnel. It emphasizes the importance for medical institutions to consider the mental health of medical personnel when treating patients during epidemics. Providing fundamental social and psychological support and intervention is deemed necessary for these personnel [11]. A study following the 2008 Wenchuan earthquake showed that survivors who lost loved ones experienced more severe symptoms of post-traumatic stress disorder and negative emotions than survivors who did not, and that more psychological crisis interventions should be given [12]. A survey conducted in Greece has indicated that, during the 2009 H1N1 influenza pandemic, over 50% of medical personnel in a tertiary hospital in Greece experienced moderate-to-high levels of anxiety and psychological distress [13]. Rabani et al. developed a multi-class machine learning classifier to identify the suicide risk levels indicated in social media posts. Their research indicates that the proposed suicide risk identification with enhanced feature engineering methods has shown significant improvements in precision, recall, and overall accuracy when compared with previous studies that used classical feature extraction mechanisms. These research findings offer valuable insights to scholars studying mental health issues and provide clues for the detecting of suicidal ideation in novel and effective ways [14]. Several studies in China have also found that H1N1 has a significant impact on the mental health of medical personnel and the public, and that the impact is related to the severity of the outbreak, with residents who have been severely affected experiencing more severe psychological crises. Numerous studies have shown varying degrees of physical and mental health problems among EVD survivors, EVD-affected persons, medical personnel, and epidemic prevention personnel. During EVD outbreaks, fear-related behaviors have a serious pathological impact on the public psyche at all stages of the post-epidemic, increasing distress and psychiatric symptoms [15]. The detrimental effects of COVID-19 on the global economy and on society, combined with the dissemination of false or negative information during the pandemic, have exacerbated public insecurity, panic, and anger [16]. A survey involving 2091 members of the general public revealed that, one month after the COVID-19 outbreak, the prevalence of post-traumatic stress disorder was 4.6% among mainland residents, and as high as 18.4% in a high-risk area (Wuhan). Subsequently, many scholars’ studies have also confirmed this viewpoint [17]. A research study conducted by Tian et al. in early 2020 on 1060 participants from across the country showed that, during the COVID-19 outbreak, all of the respondents had varying degrees of psychological symptoms, with more than 70% experiencing moderate and higher levels of psychological symptoms [18]. Chen et al. investigated the relationship between work stress, mental health, and employee job performance. Through a sample analysis of 196 small and medium-sized enterprises in China, it was found that, as a typical outcome of emergencies, work stress has a negative impact on employee performance, especially on their mental health [19]. Based on the literature analysis above, it is evident that emergencies can have far-reaching negative effects on an individual’s mental health. Issues such as anxiety, depression, and post-traumatic stress disorder may become more prevalent and pronounced under emergencies. However, it is crucial not to overlook the way in which psychological crisis intervention becomes paramount in the face of emergencies. Early identification and management of mental health issues, along with the provision of appropriate support and therapy, can assist individuals in returning to a normal psychological state.

In the context of emergency psychological crisis intervention systems, the varying roles, decisions, and actions adopted by different participating entities in emergency management [20] lead to a severe proliferation of redundant and contaminated information from various channels. The collaborative efforts among the involved stakeholders cannot be swiftly realized, leading to a collective action dilemma in the management of the emergency information required for psychological crisis intervention during emergencies [21]. In the actual decision-making process, due to the complexity of the decision-making environment and the ambiguity of human thinking, decision-making information often has a certain degree of uncertainty. It is very important to find a suitable information expression to describe the uncertain evaluation of decision-makers. Therefore, it is necessary to study the decision-making evolution process of each subject participating in an emergency psychological crisis intervention system as part of the event development process, based on a dynamic perspective. Game theory provides a methodology by which to guide emergency decision-making on the part of the various participating subjects involved in the outbreak of an emergency and to explore the interactive behavior between stakeholders [22]. In the process of psychological crisis intervention, the various participating subjects operate within a limited scope of rationality, and their choice of behavioral strategies is not static. Meanwhile, they dynamically adjust their strategy choices by observing and comparing the gains of other participants, thus steering an evolutionary steady state in the game of chance. The release and dissemination of public opinion information on emergencies affects the psychology of the public, and inaccurate public opinion information may lead to a psychological crisis. Based on the theory of participatory governance, Zheng studied the behavioral choice game strategy between government regulatory departments and new media organizations under limited rationality, encouraging the public to participate in participatory governance, raising awareness of social responsibility and psychological crisis prevention, and then effectively reducing the cost of psychological crisis intervention regulation via the dissemination of public opinion information [23]. Zhu et al. studied the cooperation between local governments and social organizations in the face of emergency management, and constructed an evolutionary game model to analyze the evolutionary characteristics of the game between the two parties. The results indicate that local governments can provide appropriate cooperative subsidies and training costs for psychological crisis intervention personnel in response to emergencies, reduce the economic cost of social organization cooperation in response to crisis, and provide a reasonable participation space for social organizations [24]. Zhang et al. analyzed the cooperative behavior between multiple subjects of emergency management under conditions of sudden disaster by constructing a centipede game model, proposing that the education of cooperative awareness should be strengthened, that the intervention of coercive force should be improved, that the game subjects should be guaranteed to suffer as little as possible from the loss of opportunity, and that the psychological hazards brought by disasters to the public should be reduced by high-efficiency decision-making [25]. Aiming at the problem of different demands for emergency resources in psychological crisis intervention after the outbreak of an emergency, Wang et al. have proposed a non-cooperative game model based on complete information. Their model effectively solves the problem of resource allocation in emergencies and provides effective decision-making for psychological crisis intervention in emergency management [26]. Luo et al. used an evolutionary game to study the choice of emergency material production capacity reserve strategies between the government and enterprises during psychological crisis intervention for emergencies, and put forward countermeasures and suggestions by which to achieve cooperation between the two parties [27]. Yang et al. constructed a dynamic and finite sequential game by analyzing the relationship between decision-makers and emergencies, thereby obtaining emergency information and proposing an optimal crisis rescue plan [28].

At present, there is little in the game analysis literature that applies evolutionary game theory to psychological crisis intervention in emergencies in order to study the behavioral choices of each participating subject. Even where there is an application of evolutionary games to study the participation of multi-party game subjects, it is limited only to two- or three-party games, which do not involve four-party game subjects. The existing research models or methods make it difficult to accurately describe the evolutionary laws of the game between multiple subjects in psychological crisis intervention under emergency [29]. Both emergency prevention and the control of emergencies involve multiple aspects, including health, the economy, people’s livelihoods, and public security. Therefore, each governance system is complex and covers government departments such as health, transportation, and public security, as well as enterprises, medical institutions, third-party organizations, communities, the public, and other pluralistic subjects [30]. In the process of psychological crisis intervention in emergencies, and in order to maximize the efficiency of the intervention, the participating subjects often need to carry out resource complementation and benefit maximization. Considering the limited resources during an emergency response, each participating subject in the psychological crisis intervention system has different benefit tendencies and pursues the maximization of their respective benefits (benefits are not only material, but also include the effectiveness of governance, social reputation, and so on), thus showing the characteristics of limited rationality in cooperation. Therefore, it is more realistic to use an evolutionary game to analyze the cooperation mechanism of each participating subject in the psychological crisis intervention process. Drawing on the Mitchell scoring method to define stakeholders in the emergency psychological crisis intervention system [31], the three dimensions of relevance, urgency, and impact were identified for each stakeholder. In this paper, we use evolutionary game theory to analyze the limited rationality of the four parties involved in psychological crisis interventions in emergencies, including the government, medical institutions, communities, and the public. An evolutionary game model is established to examine the dynamic adjustment process of the decision-making behaviors for each player in the game. Our study aims to investigate the influence of variations in the key parameters of the model on behavioral choices. In the end, it is hoped that the study will provide rationalized suggestions to the relevant departments for emergency prevention and control policies.

3. Construction and Analysis of Multi-Subject Game Models for Psychological Crisis Intervention under Emergencies

3.1. Theoretical Foundation

Game theory typically assumes that participants in the game are fully rational. This encompasses the requirement for analytical reasoning, strategic decision-making, memory, and the ability to take accurate actions, as well as the presence of rational awareness [32]. If any participant has deficiencies in these aspects, they are considered to possess bounded rationality. In bounded rationality games, equilibria that exhibit robust stability and accurate predictive power, remaining robust even in the face of minor perturbations, can be achieved by game agents through processes such as imitation, learning, and adjustment. To facilitate effective analysis and prediction, dynamic analysis methods must be employed, which are applicable to agents’ learning and strategy adjustments.

Evolutionary game theory is a theoretical framework that combines game theory with the analysis of dynamic evolutionary processes [33]. Its primary focus is on “populations”, emphasizing the analysis of changes in population structure rather than studying individual effects. Evolutionary game theory originated from the study of biological evolution and has been applied to the analysis of the formation and evolution of social customs, norms, institutions, or systems. The application of this theory has evolved into a new approach in economic research. The fundamental idea of the research is that, within a population, bounded rational individuals engage in repeated interactions and games, continuously imitating and improving their strategies in the quest for stable equilibrium points. When all participants opt for evolutionarily stable strategies, any individual attempting to deviate from this strategy will either gradually adjust their strategy during the evolutionary process or be eliminated.

“Replicator dynamics” and “evolutionary stable strategy (ESS)” are core concepts in evolutionary game theory [34]. ESS represents a stable state for a population, and a strategy is considered an ESS if and only if,

(1) s* forms a Nash equilibrium (That is, for any s, u(s*, s*) ≥ (s*, s));

(2) If s* ≠ s and u(s*, s*) = u(s*, s), then u(s*, s) > u(s, s).

As economic participants with bounded rationality, the various participants in psychological crisis interventions possess unique resource reserves and have a relatively high level of resource interdependence. This situation drives cooperation and competition among the participants, and through repeated coordination and adjustment, ultimately leads to a state of game equilibrium. In this equilibrium state, each participating subject achieves mutual benefit and resource sharing, enabling the entire psychological crisis intervention system to function effectively. This paper will employ an evolutionary game model to conduct a comprehensive analysis of the cooperative relationships between the government, medical institutions, communities, and the public. It will also analyze the process of competition and cooperation in game strategies. Through detailed research, we aim to clarify the critical links and constraining factors among the various participants in psychological crisis interventions, with the goal of further optimizing the psychological crisis intervention model.

3.2. Multi-Subject Analysis of Psychological Crisis Intervention

In the process of the emergency management of a psychological crisis intervention, efficient and effective management of emergency information, such as the progress of the incident, the situation at the scene, casualties, and rescue needs, is of great significance to the ability of the government, as well as of medical institutions, communities and the public, to make timely and correct emergency decisions.

3.2.1. Government

The government is the maker of laws regarding psychological crisis intervention, establishes psychological crisis intervention organizations, enforces propaganda during psychological crisis intervention, is the guarantor of funding for psychological crisis intervention, and is the cultivator of talents for psychological crisis intervention. The government should strengthen psychological consultation and psychological intervention, enhance public opinion guidance, and appropriately deal with various problems that may arise in the process of the prevention and control of emergencies, so as to ensure the stable development of society [35]. For the government, there is a great need for real-time and accurate emergency information on emergencies in order for it to make the right emergency decisions.

3.2.2. Medical Institutions

In the event of emergencies, medical institutions serve as centralized places for the treatment of patients and bear the burden of high pedestrian flow and tight supply of emergency supplies [36]. Medical institutions should improve their emergency prevention and control systems, and take multiple measures, such as personnel screening and hierarchical control, health information monitoring and sharing, patient management and crowd density control, and emergency supplies and deployment, to ensure that it is not only able to effectively treat patients in the event of emergencies, but also able to avoid the occurrence of larger gatherings and mass public health events. Eventually, the work, mental, and psychological states of medical personnel will be stable, medical institutions will operate in an orderly manner, public life will not be affected, and social order will remain stable.

3.2.3. Community

The community is the basic unit of defense against epidemics in the event of emergencies, as a community sustains the common life of its inhabitants. The core function of the community is to maintain relationships and link networks of resources, ensuring a wide range of social activities and social relationships for residents. Under emergencies, the community needs to actively guide its residents to unite in their efforts to rebuild an unimpeded social network, to actively build a community centered service system, and to provide a wide range of services, such as psychological consultations, to provide the public with support and a sense of security [37].

3.2.4. The Public

The public is the direct victim of emergencies and also one of the main subjects of the psychological crisis intervention system for emergencies. In this paper, the concept of “public” has a relatively narrow scope and primarily refers to people within the community. Under the stimulation of panic and anxiety in emergencies, the public urgently needs timely information and effective communication on the development direction of emergencies, in order to obtain more emotional, material, and informational support [38]. As vulnerable subjects, members of the public cannot change the occurrence of emergencies on their own but can reduce the impact of emergencies through active participation in prevention and control. The public pays more attention to their benefits in the face of emergencies, and their purpose in actively monitoring and cooperating with prevention and control is mainly to restore their normal working and living order as soon as possible and to reduce the impact on their lives. The public is at the most downstream position in the system, and reactions among members of the public influence each other and can affect governmental decisions. Supervision can include reporting improper conduct, participating in social activities, making suggestions, or demanding the establishment of transparency and accountability systems.

3.3. Multi-Subject Evolutionary Game Model Construction

3.3.1. Basic Assumptions

Assumption 1.

The government (g), medical institutions (m), communities (s), and the public (k) are the four key players in the four-party game of psychological crisis intervention in emergencies. In the game process, each participating subject follows the limited rationality assumption of “profit preference, loss avoidance”, is subject to rationality constraints, adheres to the value perception function, and pursues the maximization of their own benefits in the crisis intervention process [39]. The behavioral decisions of each game subject are determined by their value perceptions rather than their true utility.

Assumption 2.

In the psychological crisis intervention system, each participating subject will adjust their strategy selection based on the strategic benefits during the game process. Benefits are divided into general utility and psychological utility [40]. The general utility is the actual economic benefit, such as a medical institution gaining from treating the public in psychological crisis interventions, the government rewarding medical institution for active treatment, or the government penalizing the community if it finds that they do not provide safety and security in the course of psychological crisis interventions. The psychological utility is the psychological satisfaction utility of the game participating subject’s intrinsic knowledge of things, which also has an impact on the subject’s strategy choice.

Assumption 3.

The process of psychological crisis intervention has a degree of complexity and difficulty due to the multiple levels and sectors involved [41]. Therefore, the strategic decision-making of the government can be divided into strict regulation and loose regulation, and the strategy set is denoted as {$g_1,g_2$}. The strategic choices of a medical institution can be divided into active treatment and negative treatment, and the strategy set is denoted as {$m_1,m_2$}. The strategic choices of the community can be divided into providing security safeguards and not providing security safeguards, and the strategy set is denoted as {$s_1,s_2$}. The strategic choices of the public can be divided into supervision and non-supervision, and the strategy set is denoted as {$k_1,k_2$}.

Assumption 4.

In the four-party game model, assume that the probability that the government chooses to impose strict regulation is x, the probability that the medical institution chooses to active treatment is y, the probability that the community chooses to provide security safeguards is h, and the probability that the public chooses to supervise is t, and x, y, h, t ∈ (0,1).

Assumption 5.

The cost of government strict regulation is $C_1$. Because medical institutions and communities are the main participants in psychological crisis intervention, any behavioral deviation on the part of either party may lead to the spread of psychological crisis. When medical institutions provide active treatment and the community provides security safeguards, benefits accrue to the implementation of psychological crisis intervention work and this enhances the government’s social credibility. The social benefit to the government is $B_1$. The rewards (e.g., social benefits such as reputation, honor, recognition, etc.) given by the government to the two parties are $B_2$ and $B_3$, respectively. On the contrary, the loss of government credibility due to social disorder caused by the spread of public psychological crisis in emergencies is $D_1$. The corresponding punishment imposed by the government on a medical institution is $L_1$, and the punishment imposed on the community (such as social reputation) is $L_2$. When the government conducts loose regulation and the public chooses to supervise, the government will be punished by the superior department, with a penalty of $L_3$. When the government, influenced by the cost of regulation, adopts loose regulation, the probability of detecting negative treatment by medical institutions or of detecting the lack of security safeguards provided by the community becomes smaller due to the smaller cost of regulation. Therefore, the government does not make appropriate rewards and penalties.

Assumption 6.

The intervention cost invested by a medical institution for active treatment is $C_2$. The cost invested by the community in providing security safeguards and cooperating with medical institution for psychological crisis intervention is $C_3$. The short-term benefits (social benefits such as reputation and honor) obtained from negative treatment by a medical institution are $B_4$. The short-term benefits (savings in community resources and costs, reduction in community workload and responsibilities, etc.) obtained from not providing security safeguards by the community are $B_5$. The government and the public implement dynamic regulation/supervision of psychological crisis intervention. The rectification costs that need to be invested when a medical institution’s negative treatment or the community’s lack in providing security safeguards are discovered by the government or the public are $C_2^{\prime }$ and $C_3^{\prime }$, respectively. From this, certain penalties of $R_1$ and $R_2$ will be paid to the government separately. Government penalties imposed upon a medical institution or a community are obtained with a coefficient of $\theta$. These are obtained in full by the government when there is no public supervision.

Assumption 7.

A medical institution providing active treatment and a community providing security safeguards are conducive to the implementation of psychological crisis intervention, bringing benefits to the public as $B_6$. Conversely, the public’s losses are $D_2$. The cost of supervision by the public is $C_4$. When the public chooses to supervise and a medical institution’s negative treatment or a community’s refusal to provide security safeguards are punished by the government, the penalties paid by both parties will be transferred to the public as a reward, with a coefficient of $1-\theta$. However, at this time, when the government’s loose regulation is held accountable by the higher-level government, it is fully obtained by the public. If the public chooses not to supervise, they will not be rewarded.

3.3.2. Benefit Matrix Construction and Parameter Description

According to the assumption, if the government, medical institution, community, and the public respectively choose the first strategy, then this constitutes a situation as ($g_1, m_1, s_1, k_1$). In this situation, the values of the expected benefits (payments) to the government, medical institution, community, and the public are denoted as $G_{ij}, M_{ij}, S_{ij}$ and $K_{ij}$, respectively. The expected benefits to the government, medical institution, community, and the public under the strategic situation ($g_1, m_1, s_1, k_1$), namely $i=1$ row and $j=1$ column (as shown in

Table 1. A normative representation of the four-party game between government, medical institution, community, and the public.

Strategy				Community
				${\mathit{s}}_1(\mathit{h})$		${\mathit{s}}_2(1-\mathit{h})$
				The Public		The Public
				${\mathit{k}}_1(\mathit{t})$	${\mathit{k}}_2(1-\mathit{t})$	${\mathit{k}}_1(\mathit{t})$	${\mathit{k}}_2(1-\mathit{t})$
Government	$g_1(x)$	Medical institution	$m_1(y)$	$(G_{11},M_{11},S_{11},K_{11})$	$(G_{12},M_{12},S_{12},K_{12})$	$(G_{13},M_{13},S_{13},K_{13})$	$(G_{14},M_{14},S_{14},K_{14})$
			$m_2(1-y)$	$(G_{21},M_{21},S_{21},K_{21})$	$(G_{22},M_{22},S_{22},K_{22})$	$(G_{23},M_{23},S_{23},K_{23})$	$(G_{24},M_{24},S_{24},K_{24})$
	$g_2(1-x)$	Medical institution	$m_1(y)$	$(G_{31},M_{31},S_{31},K_{31})$	$(G_{32},M_{32},S_{32},K_{32})$	$(G_{33},M_{33},S_{33},K_{33})$	$(G_{34},M_{34},S_{34},K_{34})$
			$m_2(1-y)$	$(G_{41},M_{41},S_{41},K_{41})$	$(G_{42},M_{42},S_{42},K_{42})$	$(G_{43},M_{43},S_{43},K_{43})$	$(G_{44},M_{44},S_{44},K_{44})$

), are denoted as $G_{11}, M_{11}, S_{11}$ and $K_{11}$, respectively. Among these, $i=1,2,3,4$ represents the row number of the matrix, $j=1,2,3,4$ represents the column number of the matrix, and other explanations are similar.

As can be seen from Table 1, a total of 16 strategy combinations were formed under different strategy choices by the four-party game players, namely, the government, the medical institution, the community, and the public. Therefore, the expected benefit (payment) values for the government, medical institution, community, and the public under various strategies are shown in

Table 2. The four-party game payoff matrix for the government, medical institution, community, and the public.

Strategy				Community
				${\mathit{s}}_1(\mathit{h})$		${\mathit{s}}_2(1-\mathit{h})$
				The Public		The Public
				${\mathit{k}}_1(\mathit{t})$	${\mathit{k}}_2(1-\mathit{t})$	${\mathit{k}}_1(\mathit{t})$	${\mathit{k}}_2(1-\mathit{t})$
Government	$g_1(x)$	Medical institution	$m_1(y)$	$B_1-C_1-B_2-B_3$ $B_2-C_2$ $B_3-C_3$ $B_6-C_4$	$B_1-C_1-B_2-B_3$ $B_2-C_2$ $B_3-C_3$ $B_6$	$L_1+L_2+\theta R_2-C_1-D_1$ $-C_2-L_1$ $B_5-C_3^{\prime }-L_2-R_2$ $(1-\theta ) R_2-C_4-D_2$	$L_1+L_2+R_2-C_1-D_1$ $-C_2-L_1$ $B_5-C_3^{\prime }-L_2-R_2$ $-D_2$
			$m_2(1-y)$	$L_1+L_2+\theta R_1-C_1-D_1$ $B_4-C_2^{\prime }-L_1-R_1$ $-C_3-L_2$ $(1-\theta ) R_1-C_4-D_2$	$L_1+L_2+R_1-C_1-D_1$ $B_4-C_2^{\prime }-L_1-R_1$ $-C_3-L_2$ $-D_2$	$L_1+L_2+\theta (R_1+R_2)-C_1-D_1$ $B_4-C_2^{\prime }-L_1-R_1$ $B_5-C_3^{\prime }-L_2-R_2$ $(1-\theta ) (R_1+R_2)-C_4-D_2$	$L_1+L_2+R_1+R_2-C_1-D_1$ $B_4-C_2^{\prime }-L_1-R_1$ $B_5-C_3^{\prime }-L_2-R_2$ $-D_2$
	$g_2(1-x)$	Medical institution	$m_1(y)$	$B_1-L_3$ $-C_2$ $-C_3$ $B_6-C_4$	$B_1$ $-C_2$ $-C_3$ $B_6$	$-D_1-L_3$ $-C_2$ $B_5-C_3^{\prime }-R_2$ $R_2-C_4-D_2$	$-D_1$ $-C_2$ $B_5$ $-D_2$
			$m_2(1-y)$	$-D_1-L_3$ $B_4-C_2^{\prime }-R_1$ $-C_3$ $R_1-C_4-D_2$	$-D_1$ $B_4$ $-C_3$ $-D_2$	$-D_1-L_3$ $B_4-C_2^{\prime }-R_1$ $B_5-C_3^{\prime }-R_2$ $R_1+R_2-C_4-D_2$	$-D_1$ $B_4$ $B_5$ $-D_2$

.

3.4. Analysis of the Stability of Each Game Subject’s Strategy

3.4.1. Analysis of the Stability of Government Strategies

The expected benefits of “strict regulation” and “loose regulation” behavior by the government are $E_1$ and $E_2$, respectively.

$$\begin{array}{l}{E}_1=yht{G}_{11}+yh(1-t)G_{12}+y(1-h)t{G}_{13}+y(1-h)(1-t)G_{14}+\\ \hspace{1em}\hspace{1em}(1-y)ht{G}_{21}+(1-y)h(1-t)G_{22}+(1-y)(1-h)t{G}_{23}+(1-y)(1-h)(1-t)G_{24}\end{array}$$

(1)

$$\begin{array}{l}{E}_2=yht{G}_{31}+yh(1-t)G_{32}+y(1-h)t{G}_{33}+y(1-h)(1-t)G_{34}+\\ \hspace{1em}\hspace{1em}(1-y)ht{G}_{41}+(1-y)h(1-t)G_{42}+(1-y)(1-h)t{G}_{43}+(1-y)(1-h)(1-t)G_{44}\end{array}$$

(2)

The average return to government participation in the game is:

$${\overline{E}}_g=x{E}_1+(1-x)E_2$$

(3)

The replication dynamic equation for the government participation game is:

$$F_{(g)}=x(E_1-{\overline{E}}_g)=x(1-x)(E_1-E_2)=x(1-x)\phi (y,h,t)$$

(4)

In the equation:

$$\begin{array}{l}\phi (y,h,t)=L_1+L_2-C_1+R_1+R_2-y{R}_1-h{R}_2+t{L}_3-t{R}_1-t{R}_2+yt{R}_1+\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} ht{R}_2-yh{B}_2-yh{B}_3-yh{L}_1-yh{L}_2+t\theta R_1+t\theta R_2-yt\theta R_1-ht\theta R_2\end{array}$$

(5)

its first-order derivative is:

$$F_{(g)}^{\prime }=(1-2x)\varphi (y,h,t)$$

(6)

According to the stability theorem of differential equations, when the government chooses strict regulation, it must meet the following conditions to be in a stable state: $F_{(g)}=0$ and $F_{(g)}^{\prime }<0$.

3.4.2. Analysis of the Stability of Medical Institutions Strategies

The expected benefits of “active treatment” and “negative treatment” behavior by a medical institution are $E_3$ and $E_4$, respectively.

$$\begin{array}{l}{E}_3=xht{M}_{11}+xh(1-t)M_{12}+x(1-h)t{M}_{13}+x(1-h)(1-t)M_{14}+\\ \hspace{1em}\hspace{1em}(1-x)ht{M}_{31}+(1-x)h(1-t)M_{32}+(1-x)(1-h)t{M}_{33}+(1-x)(1-h)(1-t)M_{34}\end{array}$$

(7)

$$\begin{array}{l}{E}_4=xht{M}_{21}+xh(1-t)M_{22}+x(1-h)t{M}_{23}+x(1-h)(1-t)M_{24}+\\ \hspace{1em}\hspace{1em}(1-x)ht{M}_{41}+(1-x)h(1-t)M_{42}+(1-x)(1-h)t{M}_{43}+(1-x)(1-h)(1-t)M_{44}\end{array}$$

(8)

The average return to medical institution participation in the game is:

$${\overline{E}}_m=y{E}_3+(1-y)E_4$$

(9)

The replication dynamic equation for medical institution participation game is:

$$F_{(m)}=y(E_3-{\overline{E}}_m)=y(1-y)(E_3-E_4)=y(1-y)\Psi (x,h,t)$$

(10)

In the equation:

$$\Psi (x,h,t)=xh(B_2+L_1)+x(C_2^{\prime }+R_1)+t(1-x)(C_2^{\prime }+R_1)-B_4-C_2$$

(11)

its first-order derivative is:

$$F_{(m)}^{\prime }=(1-2y)\Psi (x,h,t)$$

(12)

According to the stability theorem of differential equations, when the medical institution chooses active treatment, it must meet the following conditions to be in a stable state: $F_{(m)}=0$ and $F_{(m)}^{\prime }<0$.

3.4.3. Analysis of the Stability of Community Strategies

The expected benefits of “provide security safeguards” and “not provide security safeguards” behavior by community are $E_5$ and $E_6$, respectively.

$$\begin{array}{l}{E}_5=xyt{S}_{11}+xy(1-t)S_{12}+x(1-y)t{S}_{21}+x(1-y)(1-t)S_{22}+\\ \hspace{1em}\hspace{1em}(1-x)yt{S}_{31}+(1-x)y(1-t)S_{32}+(1-x)(1-y)t{S}_{41}+(1-x)(1-y)(1-t)S_{42}\end{array}$$

(13)

$$\begin{array}{l}{E}_6=xyt{S}_{13}+xy(1-t)S_{14}+x(1-y)t{S}_{23}+x(1-y)(1-t)S_{24}+\\ \hspace{1em}\hspace{1em}(1-x)yt{S}_{33}+(1-x)y(1-t)S_{34}+(1-x)(1-y)t{S}_{43}+(1-x)(1-y)(1-t)S_{44}\end{array}$$

(14)

The average return to community participation in the game is:

$${\overline{E}}_s=h{E}_5+(1-h)E_6$$

(15)

The replication dynamic equation for community participation game is:

$$F_{(s)}=h(E_5-{\overline{E}}_s)=h(1-h)(E_5-E_6)=h(1-h)\varphi (x,y,t)$$

(16)

In the equation:

$$\varphi (x,y,t)=xy(B_3+L_2)+x(C_3^{\prime }+R_2)+t(1-x)(C_3^{\prime }+R_2)-B_5-C_3$$

(17)

its first-order derivative is:

$$F_{(s)}^{\prime }=(1-2h)\phi (x,y,t)$$

(18)

According to the stability theorem of differential equations, when a community chooses to provide security safeguards, it must meet the following conditions to be in a stable state: $F_{(s)}=0$ and $F_{(s)}^{\prime }<0$.

3.4.4. Analysis of the Stability of the Public Strategies

The expected benefits of “supervision” and “non-supervision” behavior by community are $E_7$ and $E_8$, respectively.

$$\begin{array}{l}{E}_7=xyh{K}_{11}+xy(1-h)K_{13}+x(1-y)h{K}_{21}+x(1-y)(1-h)K_{23}+\\ \hspace{1em}\hspace{1em}(1-x)yh{K}_{31}+(1-x)y(1-h)K_{33}+(1-x)(1-y)h{K}_{41}+(1-x)(1-y)(1-h)K_{43}\end{array}$$

(19)

$$\begin{array}{l}{E}_8=xyh{K}_{12}+xy(1-h)K_{14}+x(1-y)h{K}_{22}+x(1-y)(1-h)K_{24}+\\ \hspace{1em}\hspace{1em}(1-x)yh{K}_{32}+(1-x)y(1-h)K_{34}+(1-x)(1-y)h{K}_{42}+(1-x)(1-y)(1-h)K_{44}\end{array}$$

(20)

The average return to the public participation in the game is:

$${\overline{E}}_k=t{E}_7+(1-t)E_8$$

(21)

The replication dynamic equation for the public participation game is:

$$F_{(k)}=t(E_7-{\overline{E}}_k)=t(1-t)(E_7-E_8)=t(1-t)\delta (x,y,h)$$

(22)

In the equation:

$$\delta (x,y,h)=(1-x\theta )[(1-y)R_1+(1-h)R_2]-C_4$$

(23)

its first-order derivative is:

$$F_{(k)}^{\prime }=(1-2t)\delta (x,y,h)$$

(24)

According to the stability theorem of differential equations, when the public chooses supervision, it must meet the following conditions to be in a stable state: $F_{(k)}=0$ and $F_{(k)}^{\prime }<0$.

3.5. Stability Analysis of Strategy Combination Evolution

In the four-party game replication dynamic system of psychological crisis intervention, the stability of the strategy combination of the government, medical institution, community, and the public can be judged based on Lyapunov’s first law. The equilibrium point is locally asymptotically stable when all eigenvalues of the Jacobian matrix corresponding to that point have negative real parts. If all of the eigenvalues of the associated Jacobian matrix have at least one positive real part, the point represents an unstable fixed point.

According to the analysis of the stability of the strategies of each game subject in Section 3.4, we can obtain the system of replicated dynamic equations of the four-party game of the government, medical institution, community, and the public as follows:

$$\left\{\begin{array}{c}{F}_{(g)}=x(1-x)\phi (y,h,t)\\ F_{(m)}=y(1-y)\Psi (x,h,t)\\ F_{(s)}=h(1-h)\varphi (x,y,t)\\ F_{(k)}=t(1-t)\delta (x,y,h)\end{array}\right.$$

(25)

According to equilibrium theory, let

$$\left\{\begin{array}{c}{F}_{(g)}=0\\ F_{(m)}=0\\ F_{(s)}=0\\ F_{(k)}=0\end{array}\right.$$

(26)

Solving the non-linear equation system (25) can obtain the equilibrium point of the evolutionary game among the government, medical institution, community, and the public. This includes 16 pure strategy equilibrium points E₁(0,0,0,0), E₂(0,0,0,1), E₃(0,0,1,0), E₄(0,0,1,1), E₅(0,1,0,0), E₆(0,1,0,1), E₇(0,1,1,0), E₈(0,1,1,1), E₉(1,0,0,0), E₁₀(1,0,0,1), E₁₁(1,0,1,0), E₁₂(1,0,1,1), E₁₃(1,1,0,0), E₁₄(1,1,0,1), E₁₅(1,1,1,0), E₁₆(1,1,1,1), and mixed strategy equilibrium point E* (x*, y*, h*, t*).

Ritzberger et al. [42] and Selten [43] have pointed out that the stable solution in multi-group evolutionary games is a strict Nash equilibrium. Furthermore, the strict Nash equilibrium must indeed be a pure strategy. Therefore, this study will analyze the stability of 16 pure strategy equilibrium points in the four-party evolutionary game.

By abstractly describing practical problems, a replication dynamic equation system involving government, medical institution, community, and the public was constructed. However, in the abstract process of practical problems, small changes in the initial state and parameters of the behavior strategy selection of the government, medical institution, community, and the public can cause changes in the equilibrium point of the differential equation system. Therefore, after finding the equilibrium points of the differential equation system involving the government, medical institution, community, and the public, it is necessary to study the stability of each equilibrium point. This is because only stable equilibrium points are evolutionary stable strategies.

According to the replication dynamic equation of each game subject of the psychological crisis intervention, the Jacobian matrix of the four-party evolutionary game replication dynamic system is obtained as follows:

$$J=\left[\begin{array}{cccc}\partial F_{(g)}/\partial x& \partial F_{(g)}/\partial y& \partial F_{(g)}/\partial h& \partial F_{(g)}/\partial t\\ \partial F_{(m)}/\partial x& \partial F_{(m)}/\partial y& \partial F_{(m)}/\partial h& \partial F_{(m)}/\partial t\\ \partial F_{(s)}/\partial x& \partial F_{(s)}/\partial y& \partial F_{(s)}/\partial h& \partial F_{(s)}/\partial t\\ \partial F_{(k)}/\partial x& \partial F_{(k)}/\partial y& \partial F_{(k)}/\partial h& \partial F_{(k)}/\partial t\end{array}\right]$$

(27)

3.5.1. Analysis of the Stability of the Strategy Combinations under Government

Strict Regulation

In the process of a psychological crisis intervention under emergency, when the government’s stability strategy is strict regulation, the asymptotic stability analysis of the equilibrium point in the replicated dynamic system of this evolutionary game is shown in

Table 3. Asymptotic stability analysis of equilibrium points of evolutionary game replication dynamical systems under government strict regulation.

Equilibrium Point	Eigenvalue λ1, λ2, λ3, λ4	Positivity and Negativity	Stability
(1,0,0,0)	$C_2^{\prime }$ − C2 − B4$+R_1, C_3^{\prime }$ − C3 − B5 + R2, C1 − L1 − L2 − R1 − R2, R1 + C4 + R2 + θR1 + θR2	(U, U, U, +)	instability
(1,0,0,1)	$C_2^{\prime }$ − C2 − B4$+R_1, C_3^{\prime }$ − C3 − B5 + R2, C4 + R1 + R2 + θR1 + θR2, C1 − L1 − L2 − L3 − θR1 − θR2	(U, U, +, U)	instability
(1,0,1,0)	R1 + C4 + θR1, B5 + C3 − $C_3^{\prime }$ − R2, C1 − L1 − L2 − R1, B2 − B4 − C2 $+C_2^{\prime }$ + L1 + R1	(+, U, U, U)	instability
(1,0,1,1)	C4 + R1 + θR1, B5 + C3 − $C_3^{\prime }$ − R2, C1 − L1 − L2 − L3 − θR1, B2 − B4 − C2 $+C_2^{\prime }$ + L1 + R1	(+, U, U, U)	instability
(1,1,0,0)	R2 + C4 + θR2, B4 + C2 − $C_2^{\prime }$ − R1, C1 − L1 − L2 − R2, B3 − B5 − C3 $+C_3^{\prime }$ + L2 + R2	(+, U, U, U)	instability
(1,1,0,1)	C4 + R2 + θR2, B4 + C2 − $C_2^{\prime }$ − R1, C1 − L1 − L2 − L3 − θR2, B3 − B5 − C3 $+C_3^{\prime }$ + L2 + R2	(+, U, U, U)	instability
(1,1,1,0)	−C4$, B_2+B_3-C_1, B_4-B_2+C_2-C_2^{\prime }$ − L1 − R1, B5 − B3 + C3 − $C_3^{\prime }$ − L2 − R2	(−, U, U, U)	ESS (condition 1)
(1,1,1,1)	C4, B2 + B3 + C1−L3, B4−B2 + C2 − $C_2^{\prime }$ − L1 − R1, B5 − B3$+C_3-C_3^{\prime }$ − L2 − R2	(+, U, U, U)	instability

Notes: U indicates that the symbol is uncertain. ESS denotes evolutionary stabilization strategy. Condition 1: B₂ + B₃ − C₁ < 0, B₄ − B₂ + C₂ − ${C^{\prime }}_2$ − L₁ − R₁ < 0, B₅ − B₃ + C₃ − ${C^{\prime }}_3$ − L₂ − R₂ < 0.

.

From Table 3, it can be seen that, under government strict regulation, there may be a stable pure strategy equilibrium point E₁₅(1,1,1,0). When condition 1 is established, this indicates that the regulatory cost paid by the government is greater than the rewards given to the medical institution and community, and that the cost paid by medical institution and community for taking positive actions is greater than the penalty imposed by the government for taking negative actions. At this point, the evolutionary stability strategy of the four-party evolutionary game is as follows: the government’s stability strategy is strict regulation, the medical institution’s stability strategy is active treatment, and the community provides security safeguards. As a result, the psychological crisis intervention process can be maximized and runs smoothly without public supervision, i.e., E₁₅(1,1,1,0).

From this, it can be concluded that, to avoid the spread of psychological crises in emergencies and provide effective treatment for those with psychological crises, the government needs to regulate and pay corresponding regulatory costs. The government’s strict regulation of the medical institution can effectively prevent negative treatment from becoming an asymptotic stabilization strategy. Through the adoption of effective regulation and penalty mechanisms, the replication of the dynamic system of psychological crisis intervention moves away from the spread of psychological crises due to passive treatment by medical institution and the lack of security safeguards provided by the community, and towards positive intervention behaviors that promote social stability.

3.5.2. Analysis of the Stability of Strategy Combinations under Government

Loose Regulation

In the process of psychological crisis intervention under emergencies, when the government’s stability strategy is one of loose regulation, the asymptotic stability analysis of the equilibrium point in the replicated dynamic system of this evolutionary game is shown in

Table 4. Asymptotic stability analysis of equilibrium points of evolutionary game replication dynamical systems under government loose regulation.

Equilibrium Point	Eigenvalue λ1, λ2, λ3, λ4	Positivity and Negativity	Stability
(0,0,0,0)	−B4 − C2, −B5 − C3, R1 − C4 + R2, L1 + C1 + L2 + R1 + R2	(−, −, U, +)	instability
(0,0,0,1)	C4 − R1 − R2$, C_2^{\prime }$ − C2 − B4 + $R_1, C_3^{\prime }$ − C3 − B5 + R2, L1 + C1 + L2 + L3 + θR1 + θR2	(U, U, U, +)	instability
(0,0,1,0)	R1 − C4, −B4 − C2, B5 + C3, L1 − C1 + L2 + R1	(U, −, +, U)	instability
(0,0,1,1)	C4 + $R_1, C_2^{\prime }$ − C2 − B4 + R1, B5 + C3 − $C_3^{\prime }$ − R2, L1 − C1 + L2 + L3 + θR1	(+, U, U, U)	instability
(0,1,0,0)	R2 − C4, −B5 − C3, B4 + C2, L1 − C1 + L2 + R2	(U, −, +, U)	instability
(0,1,0,1)	C4 + R2, B4 + C2 − $C_2^{\prime }$ − R1$, C_3^{\prime }$ − C3 − B5 + R2, L1 − C1 + L2 + L3 + θR2	(+, U, U, U)	instability
(0,1,1,0)	−C4, B4 − C2, B5 − C3, −B2 − B3 − C1	(−, U, U, −)	ESS (condition 2)
(0,1,1,1)	C4, B4 + C2 − $C_2^{\prime }$ − R1, B5 + C3 − $C_3^{\prime }$ − R2, L3 − B3 − C1 − B2	(+, U, U, U)	instability

Notes: U indicates that the symbol is uncertain. ESS denotes evolutionary stabilization strategy. Condition 2: B₄ − C₂ < 0, B₅ − C₃ < 0.

.

From Table 4, it can be seen that, under government loose regulation, there may be a stable pure strategy equilibrium point E₇(0,1,1,0). When condition 2 is established, this indicates that the cost incurred by the medical institution and community in adopting positive behavior is greater than the short-term benefits obtained by adopting negative behavior. However, due to the government’s loose regulatory strategy, in the long-term evolution process, the medical institution and community will also choose negative behavioral strategies driven by benefit maximization.

From this, it can be seen that loose government regulation is not conducive to the implementation of psychological crisis intervention when emergencies occur. To avoid negative psychological crisis intervention behaviors from becoming a stabilizing strategy, and to encourage active treatment and cooperation from medical institutions and communities, the government can adopt rewards and penalties mechanisms by which to make psychological crisis intervention in positive behavior a stable strategy. Thus, in the event of emergencies, the problem of a psychological crisis will be solved at the lowest cost and the fastest speed, in order to better protect the mental health of the public and maintain the stability and development of society.

4. Analysis of Simulation Results

Matlab simulation technology is a potent computer simulation approach for investigating the feedback dynamics of intricate systems [44,45,46]. In a multi-party game, individuals adjust their strategic choices by observing and comparing the benefits of others, continuously imitating and learning and thus forming feedback behavior in the group. To verify the correctness of the conclusion of evolutionary stability analysis and demonstrate the impact of key factors in replicating dynamic systems on the behavior evolution of various subjects in psychological crisis intervention, based on the actual significance of the cost of benefits between the government [47,48], the medical institution, the community, and the public in the four-party game, numerical simulations were conducted using Matlab 2020a on the evolution trajectories of each subject. This helps to provide a decision-making basis from which the government and other relevant departments can respond to psychological crisis intervention for emergencies, as well as to formulate reasonable and effective control measures.

To verify the rationality of the model constructed in this article and the accuracy of the analysis results, this paper explores the impact of changes in various parameters on the evolution path and evolution equilibrium strategy from the initial distribution of the four-party behavior subjects and their respective behavior strategy choices.

4.1. Numerical Simulation Analysis of Stable Equilibrium Point

To visually represent the system’s evolution path more clearly, the equilibrium points of the evolutionary game obtained in Section 3.5 are analyzed and verified through numerical simulation.

For the system to eventually evolve to the ideal state point E₇(0,1,1,0), the initial values of each parameter must meet the following conditions: B₄ − C₂ < 0, B₅ − C₃ < 0. The parameters are set as L₁ = 4, L₂ = 3, C₁ = 20, R₁ = 8, R₂ = 6, L₃ = 6, B₂ = 7, B₃ = 5, θ = 0.55, $C_2^{\prime }$=17, B₄ = 4, C₂ = 10, $C_3^{\prime }$ = 15, B₅ = 3, C₃ = 9, C₄ = 3. Under the above conditions, the initial value of one party’s probability is fixed, and the initial values of the probabilities for the other three parties are selected randomly to assess the impact of these initial values on the variation of the fixed party’s probability over time. Using x = 0.5 as an example, Figure 1 illustrates the evolution curve of x. From Figure 1, it can be seen that different initial values of y, h, and t affect the rate of convergence of monotonically decreasing x. This indicates that the probability of the government choosing a strict regulation strategy will consistently decline over time, and that, ultimately, and regardless of the initial values of y, h, and t, the government will choose a loose regulation strategy. Taking y = 0.5 and h = 0.5 as examples, the evolution curves of y and h are shown in Figure 2 and Figure 3. From Figure 2 and Figure 3, it can be seen that, when the probability of government regulation is greater than 0.5, the medical institution and community will adopt active treatment and provide security safeguards. Otherwise, they adopt negative treatment and opt to not provide security safeguards, respectively, which is not conducive to the implementation of psychological crisis intervention and treatment work. Using t = 0.5 as an example, Figure 4 illustrates the evolution curve of t. As shown in Figure 4, when the probability of government regulation is greater than 0.5, the public gradually adopts a non-supervision strategy. Otherwise, the public adopts a supervision strategy to ensure that their psychological problems can be treated as soon as possible. This finding contradicts the stability of the equilibrium point E₇(0,1,1,0) of the evolutionary system, therefore E₇(0,1,1,0) is not an ideal stable point for psychological crisis intervention systems.

For the system to eventually evolve to the ideal state point E₁₅(1,1,1,0), the initial values of each parameter must meet the following conditions: B₂ + B₃ − C₁ < 0, B₄ − B₂ + C₂ − $C_2^{\prime }$ − L₁ − R₁ < 0, B₅ − B₃ + C₃ − $C_3^{\prime }$ − L₂ − R₂ < 0. The parameters are set as L₁ = 4, L₂ = 3, C₁ = 20, R₁ = 12, R₂ = 8, L₃ = 25, B₂ = 7, B₃ = 5, θ = 0.55, $C_2^{\prime }$ = 17, B₄ = 4, C₂ = 10, $C_3^{\prime }$ = 15, B₅ = 3, C₃ = 9, and C₄ = 3. Similarly, under the conditions outlined above, the evolution curves of x, y, h, and t over time are displayed in Figure 5, Figure 6, Figure 7 and Figure 8, respectively. Figure 5 reveals that varying the initial values of y, h, and t impacts the convergence rate of monotonically increasing x. This indicates that the likelihood of the government selecting a strict regulation strategy will continue to increase over time, ultimately adopting a strict regulation strategy to accelerate the process of psychological crisis intervention and ensure the mental health of the public. Through continuous improvement of the relevant regulations and policies and a strict system of rewards and penalties, the public will be provided with better mental health services, so as to avoid the occurrence of serious psychological crises in the event of emergencies. From Figure 6 and Figure 7, it can be seen that the strategic choices of the medical institution and community tend to favor negative behaviors when the likelihood of the government opting for strict regulation is exceedingly low. As the probability of government strict regulation increases, to reduce the penalties for negative behavior by the government, the medical institution and community will ultimately choose positive behavior to accelerate the intervention of psychological crisis in emergencies. From Figure 8, it can be seen that the different initial values of x, y, and h affect the convergence rate of the monotonically decreasing t-evolution curve. This indicates that, as the probability that the government, the medical institution, and the community choosing positive behavior (i.e., strict regulation, active treatment, and provide security safeguards) gradually leans towards 1, then, over time, the public’s strategy will ultimately be to choose non-supervision. At this time, the psychological crisis intervention work is effectively carried out with the active behavior of the government, medical institution, and community. The public only needs to do a good job of cooperating with the work so that the psychological problems can be solved and normal life can be resumed. This is in accordance with the stability of the equilibrium point E₁₅(1,1,1,0) of the evolutionary system, therefore E₁₅(1,1,1,0) is an ideal stable point for psychological crisis intervention systems.

As can be seen from Figure 5, Figure 6, Figure 7 and Figure 8, in the psychological crisis intervention system, when the initial probability of one of the game parties is determined, the change of the initial probability of the remaining three-party game subjects does not affect their final strategy choice.

For a more precise depiction of the interplay between the four parties in the games, the initial probability of the government is set to x = 0.7, the initial probability of medical institution is set to y = 0.6, the initial probability of the community is set to h = 0.4, and the initial probability of the public is set to t = 0.2. The evolution curves of the strategy choices of each game party are shown in Figure 9, Figure 10, Figure 11 and Figure 12, respectively.

From Figure 9, Figure 10, Figure 11 and Figure 12, it can be seen that, in the process of the four-party game, when the government adopts a strict regulation strategy, the medical institution and community choose to engage in positive behavior to ensure that their profits outweigh penalties and costs, that they provide high-quality psychological crisis intervention and treatment, and that they increase public awareness of the psychological problems being faced. At the same time, information sharing should be strengthened to provide better mental health monitoring and management services for the public. At this point, the government gains good credibility and reputation and establishes prestige in the minds of the public. The public trusts the government to efficiently solve the psychological crisis outbreaks under emergency conditions. Thus, regardless of the initial probabilities of x, y, h, and t, the parties to the game choose the optimal strategy for the psychological crisis intervention game. The higher initial probability results in a shorter duration in which to attain the optimal strategy selection, while, in the converse situation there is a longer time to reach the optimal strategy selection.

4.2. Analysis of the Impact of Government Strict Regulation Costs

Assuming C₁ = {14, 20, 24}, the outcomes of the strategy evolution process for the four-party game subjects are depicted in Figure 13, Figure 14 and Figure 15, respectively.

From Figure 13, Figure 14 and Figure 15, it can be seen that the high or low cost of government strict regulation not only affects the evolution trend of government strategy selection, but also affects the evolution trend of other third-party game strategy selection. With an increasing cost of government regulation, the probability of the government adopting strict regulation rises at a slower rate, and at this time, the medical institution and community tend to adopt negative behaviors. When the government adopts strict regulation to reach its maximum, the medical institution and community gradually adopt active behavior to avoid penalties. Therefore, by reducing the cost of strict government regulation, the medical institution and community can be forced to take active measures by which to address psychological crisis issues as soon as possible.

4.3. Analysis of the Impact of Being Penalized by Superiors Due to Unfavorable

Government Regulation

Assuming L₃ = {15, 25, 35}, the simulation results of the dynamic equation system replicated by the four-party game subject evolving 50 times over time are shown in Figure 16.

From Figure 16, it can be seen that, as the L₃ value of the government being penalized by superiors for inadequate regulation increases, the probability of strict regulation by the government will also increase. The higher penalty value leads to a shorter time for the government to opt for strict regulatory strategies, and the speed of the choice of a stable strategy continues to accelerate. Therefore, when the government loosens regulation, the psychological crisis of the public cannot be solved when emergencies occur. There will be chaos in society as a result, and the government will be penalized by its superiors. The government loses credibility and opts for a strict regulation strategy to save its lost image.

4.4. Analysis of the Impact of the Costs of Adopting Positive Behaviors in the Medical Institution and Community, and of Government Rewards and Penalties

To verify the impact of the cost of positive behavior in the medical institution and community, and of government rewards and penalties, on the strategic choices of various game subjects, let C₂ = {14, 12, 10}, C₃ = {13, 11, 9}, B₂ = {3, 5, 7}, B₃ = {1, 3, 5}, L₁ = {1, 2, 4}, L₂ = {1, 2, 3}, R₁ = {8, 10, 12}, and R₂ = {4, 6, 8}. The results of the evolution process of the subject strategy selection in the four-party game are shown in Figure 17.

From Figure 17, it can be seen that, as the cost of adopting positive behavior strategies by the medical institution and community decreases and the rewards and penalties chosen by the government for their behavior strategies increase, that is, when the difference between the government rewards obtained by the medical institution and community under active behavior and their costs is greater than the difference between the short-term benefits obtained under negative behavior and the rectification costs and fines paid, the probability of positive behavior by the medical institution and community gradually evolves from the initial probability and stabilizes at 1. In the process of evolution, the probability of the public adopting supervision first increases and then decreases. This is because, at the beginning of the emergency, the government’s loose regulation leads to the negative behavior of the medical institution and the community, which damages the public’s benefits. The public promotes the smooth implementation of psychological crisis intervention work through supervision. After the government implements strict regulation, the public gradually adopts a non-supervision strategy to save their supervision costs.

4.5. Stability Point Simulation Analysis of Combined Strategies

Based on the estimation of the actual situation, the parameters assigned in the evolutionary game are as follows: L₁ = 4, L₂ = 3, C₁ = 20, R₁ = 12, R₂ = 8, L₃ = 25, B₂ = 7, B₃ = 5, θ = 0.55, $C_2^{\prime }$ = 17, B₄ = 4, C₂ = 10, $C_3^{\prime }$ = 15, B₅ = 3, C₃ = 9, and C₄ = 3. It follows that the above parameter assignment satisfies condition 1. Therefore, when the public chooses a non-supervision behavior strategy, i.e., t = 0, the evolutionary game process of the government, medical institution, and community is shown in Figure 18, and the system ultimately stabilizes at the stable point of the E₁₅(1,1,1,0) combination strategy.

As can be seen in Figure 18, the system stability point indicates that the government chooses to enforce strict regulation, the medical institution chooses active treatment, the community chooses to provide security safeguards, and the public chooses non-supervision. The government establishes a rewards and penalties mechanism for the medical institution and community, maintaining a certain level of strict regulation probability. Therefore, the medical institution adopts active treatment strategies, while the community provides security safeguards to ensure the smooth implementation of psychological crisis intervention processes, to treat the public’s psychological problems, and to restore social security and stability.

5. Discussion

The question of the four-party evolutionary game of psychological crisis intervention is an urgent problem that needs to be solved. To effectively promote the development of this field, it is necessary to combine game theory to conduct an in-depth exploration of psychological crisis intervention, providing psychologists and sociologists with a new research framework, so that they can better understand conflict resolution methods in human social life.

The four-party evolutionary game refers to the situation wherein individuals and groups interact to form a balance between four parties in the process of conflict resolution. In this process, the four parties continuously engage in games and competitions, ultimately achieving a relatively stable balance.

When emergencies occur, the public suffering from disasters will experience varying degrees of psychological crisis [49]. During this process, people are prone to feeling helpless or unsafe, and in severe cases, they may experience psychological problems such as anxiety, panic, and insomnia [50]. Therefore, all levels of society must pay close attention to the provision of effective psychological crisis intervention to the public, as this has a positive effect on the protection of their physical and mental health [51].

Firstly, to ensure the mental health of the public, we need to establish a comprehensive crisis intervention system. The system should cover multiple fields, from psychological counseling to psychotherapy, in order to meet the needs of different types of psychological problems. In addition, the psychological crisis intervention system should also work closely with the relevant medical institution and community to ensure that effective measures can be taken quickly in emergencies [52].

Secondly, psychological crisis intervention should not only focus on individual psychological problems, but also the mental health of the entire social group [53]. There is a need to enhance public education on mental health and raise awareness of psychological problems so that the public can recognize and seek professional help promptly when emergencies occur. In addition, regularly conducting mental health education activities, such as psychological lectures and mental health promotion, can also help the public establish correct mental health concepts and reduce the incidence of psychological problems.

In addition, the government and all sectors of society should actively invest resources to support the implementation of psychological crisis intervention work [54]. The government can establish special funds to support the development of mental health education, psychological counseling services, crisis intervention, and other aspects. At the same time, all sectors of society should take corresponding responsibilities and promote the smooth implementation of psychological crisis intervention work by increasing the rewards and penalties for medical institutions and communities. Medical institutions and communities should fully leverage their respective advantages and roles, form a collaborative and complementary relationship in the process of crisis intervention, and jointly promote psychological crisis intervention work to achieve the best effect of psychological crisis intervention. Medical institutions mainly provide psychological crisis intervention services for the public, while communities mainly coordinate resources, provide social support, such as the provision of security safeguards, and assist in cooperating with medical institutions for psychological crisis intervention. At the same time, the two should establish an information-sharing mechanism, share relevant information on crisis intervention in a timely fashion and avoid information asymmetry and misunderstandings. Medical institutions can provide professional guidance and training to communities, improving the psychological intervention ability and level of community workers.

In short, psychological crisis intervention is a complex and important field that requires the joint efforts of the government, society, and individuals to establish a sound crisis intervention system and achieve social harmony, stability, and development. Only by fully paying attention to and solving psychological crisis problems can we better protect the mental health of the public and maintain social harmony and stability in the face of emergencies.

6. Conclusions

6.1. Main Findings

Psychological crisis intervention is an area that needs to be focused on, along with the current high frequency of emergencies. This is not only conducive to helping the public to resolve psychological problems, but also helps to promote social stability and development. Psychological crisis intervention research has become an important piece of post-disaster treatment when emergencies occur. Because of the specificity and complexity of the participating subjects, this paper constructs an evolutionary game model between four benefit subjects, namely, the government, medical institution, community, and the public, based on evolutionary game theory. By analyzing the payment matrix of the four-party game, the evolutionary stability strategies for the behavior choices of each game subject were analyzed. Based on the actual situation, the key parameters were assigned and simulated to explore the impact of each participant on the evolution of psychological crisis intervention. Combined with the results of the simulation analysis, countermeasures to promote the acceleration of the psychological crisis intervention process are proposed.

(1) The strategy choices of each game subject are influenced by multiple external variables. When the relevant parameters of each game subject meet certain conditions, they may ultimately reach the corresponding stable equilibrium point. By calculating the payment matrix of the game model and analyzing the simulation model, it can be concluded that there are 16 evolutionary stable states in the psychological crisis intervention four-party game model. Only when the equilibrium point E₁₅(1,1,1,0) reaches the evolutionary stable equilibrium ESS, that is, when the government adopts strict regulation strategies, the medical institution adopts active treatment strategies, the community provides security safeguards, and the public adopts non-supervision strategies, do psychological interventions reach a stable state most rapidly. As all parties involved in the process of psychological crisis intervention are driven by the maximization of benefits and may choose negative behavior, there is a resulting inability to achieve the best state of psychological crisis intervention, which can easily lead to social unrest and serious psychological crisis problems among the public. In the process of psychological crisis intervention, the occurrence of the remaining 15 evolutionary stable equilibrium states should be avoided.

(2) Through simulation analysis of the four-party evolutionary game model, it can be concluded that, in the psychological crisis intervention system, the government bears the main leading responsibility and has a certain guiding effect on the strategic choices of other game subjects. The strategic choices of the four-party game subjects influence each other. The government’s reward and penalty measures for the medical institution and the community are conducive to the promotion of the development of psychological crisis intervention systems towards a stable state and have a significant impact on their positive behavior choices. However, the size of the reward intensity will affect the government’s choice of regulatory strategies.

(3) Based on the research in this paper, corresponding countermeasures are proposed to facilitate psychological crisis intervention from the perspective of the game quadrangle.

6.2. Policy Recommendations

By analyzing the results of the four-party evolutionary game of the psychological crisis intervention system, the following countermeasures can be proposed:

(1) Fully leverage the leading role of the government [55]. The government should implement strict reward and penalty mechanisms for medical institutions and communities. It is necessary to increase the penalties for medical institutions and communities that choose negative behaviors, encourage them to choose active behaviors, and ensure the smooth implementation of psychological crisis intervention work. At the same time, appropriate rewards should also be given to them. In addition, the government’s reward intensity should be reasonably set. If the government provides too much reward to medical institutions and communities, high regulatory costs will be incurred for the government, which is not conducive to the stable development of the psychological crisis intervention system.

(2) Medical institutions play a crucial role in intervening in psychological crises. However, to maximize their benefits, they often engage in behaviors that violate laws and regulations in the process of psychological crisis intervention, which puts the psychological crisis intervention work into an unprecedented crisis [56]. Therefore, it is very importance that the regulation of psychological crisis interventions in medical institutions is strengthened, so that they can carry out their work under the framework of laws and regulations and safeguards both the benefits of patients and social stability. The government should strengthen the regulation of medical institutions, establish a sound self-discipline mechanism for them and improve laws and regulations to crack down on violations, ensuring the effective protection of patients’ physical and mental health. Furthermore, it is essential to enhance the training and management of the psychological crisis intervention team in a medical institution and improve their professional quality and service level. Psychological crisis intervention is a highly professional task that requires professional personnel and technical support. Therefore, strengthening the training and management of psychological crisis intervention teams in medical institutions and improving their professional quality and service level are important measures by which to promote the development of psychological crisis intervention.

(3) The key task of the community is to provide relevant security safeguards to assist medical institutions in psychological crisis intervention, and the challenge facing the community is to ensure that the implementation of this work complies with ethical standards and laws and regulations. This is especially in terms of personal privacy and information protection, where it is necessary to strictly follow relevant regulations for operation [57]. There is a need to establish a sound information protection mechanism to ensure that all personal privacy information related to psychological crisis intervention is strictly protected. Additionally, it is necessary to collaborate with medical institutions to develop a process for the collection and use of information that meets the needs of psychological crisis intervention, in order to ensure that all intervention activities are carried out within the scope allowed by law. In addition, there is a need to work closely with professionals in the community to jointly study how to better carry out psychological crisis intervention work. This includes conducting research on common psychological problems, understanding current treatment methods and effects, and developing more effective intervention measures. There is also a need to establish a comprehensive regulatory mechanism to supervise and evaluate the implementation of relevant work, to ensure that all work can achieve the expected results. Only in this way can the key work of the community be carried out more smoothly.

6.3. Limitation and Future Direction

There are some shortcomings in this study. There may be a collusion between medical institutions and the communities in the process of psychological crisis intervention. Some medical institutions and communities may have vested interests. For instance, certain medical institutions may provide services to a community, and in return, the community may bring more patients to this institution, leading to potential instances of mutual collusion.

In the process of psychological crisis intervention, the behavioral decisions of the public, as the intervened object, also have a significant impact on the game results. In this study, the progress of the implementation of psychological crisis intervention work considers the public’s behavioral decision-making as supervision and non-supervision. The research results indicate that, under the strict regulation strategy of the government, the four-party game system can reach a stable state when the public adopts non-supervision measures to reduce supervision costs. In fact, the public’s response to psychological crisis intervention under emergencies is often influenced by a multitude of factors. By framing the public’s strategy choice as ‘supervision’ or ‘non-supervision,’ this model overlooks the intricacies involved in potential public responses of cooperation and non-cooperation, thereby simplifying the complex spectrum of public reactions.

Additionally, drawing from the research findings in this paper, potential directions for future research can be proposed.

(1) Deepen research on multi-subject cooperation strategies. Future research will be considered to analyze the behavioral strategies of the four-party game subjects in the psychological crisis intervention system in the case of possible collusion between medical institutions and communities. Additionally, research can change the public’s behavioral decision-making into cooperation and non-cooperation and study the issue of psychological crisis intervention under emergencies from the perspective of whether it can be smoothly carried out.

(2) Comparative research. Future research could focus on comparative studies across different regions, countries, or types of crises. For instance, analyzing how the dynamics of the four-way game vary in response to different types of crises (natural disasters, economic crises, health emergencies) could provide valuable insights into the adaptability of the model.

(3) Research on the application of technology and communication. With the increasing role of technology in crisis response, future studies can explore how technological advancements and communication platforms impact the effectiveness of crisis intervention. This may involve investigating the roles of social media, artificial intelligence, and data analysis in real-time decision-making for multi-stakeholder interventions.