Evolution Game and Simulation Analysis of Disturbance Emergency Disposal of In-Flight Cabin: China Civil Aviation Security Strategy Study

Abstract

The disturbance of disruptive passengers in the cabin seriously affects aviation security in China. Reducing the occurrence of disturbances in the cabin is an inevitable choice to strengthen the security of civil aviation in China and improve the emergency disposal of airlines. Therefore, this paper investigates the strength of government support, the benefits of cooperation and the punishment of disposers, and uses the event tree method to analyze the relevant subjects affecting the consequences of cabin disruptions. To this end, a cabin disturbance emergency management game model comprised of regulatory authority, airline security and disruptive passengers was then established. The results of the game playing and simulation highlight the inherent relationships in emergency management in China’s civil aviation. First, the choice of behavior of the regulatory authority and disruptive passengers are mainly influenced by the strategy of airline security. Second, the strength of support from the regulatory authority determines whether the airline security can dispose of the passengers successfully. Third, the disruptive inclination of disruptive passengers is directly influenced by the intensity of punishment by airport police. Finally, an empirical analysis and simulation is conducted with the example of using an open flame or smoking in the cabin. This study provides new ideas for enhancing aviation security and improving emergency management.

1. Introduction

The improvement of the national security system and the safe functioning of the civil aviation system both depend on the effectiveness of cabin security, which is a key component of civil aviation security. Cabin disruptions caused by disruptive passengers have always been a difficult point for in-flight emergency disposal. According to the Global Safety-Related Incident Reporting System that the International Air Transport Association (IATA) set up, there is one disruption in the cabin for every 885 flights in 2021. Cabin disruptions are being closely watched around the world. The Federal Aviation Administration (FAA) collected 14,418 unruly passenger reports in 2021–2022. The European Union Aviation Safety Agency (EASA) counted that over 1000 flights in Europe are threatened by unruly or disruptive passengers each year, with one incident occurring every three hours. Similarly, aviation security in China is threatened by cabin disruptions. According to the China Civil Aviation Safety and Security Incident Information Statistical Report, 4029 incidents of passenger disturbance occurred from 2017–2021, such as the use of an open flame or smoking in the cabin, fighting and provocation, as well as the violation of the use of electronic devices and disobedience. Of the three types of incidents, there were 1312, 375 and 277, respectively, and the number of incidents of an open flame being used or smoking in the cabin especially increased continuously for five years. On 19 May 2016, an EgyptAir plane taking off from Paris-Charles de Gaulle Airport in France bound for Cairo crashed due to a fire caused by the pilot’s illegal smoking on board, killing all 66 people on board. On 10 July 2018, an oxygen mask dislodgement occurred on Air China flight CA106 from Hong Kong to Dalian in Guangzhou airspace. The oxygen masks fell off due to the co-pilot smoking an e-cigarette in the cockpit and incorrectly operating it, resulting in a lack of oxygen in the cabin. The Civil Aviation Administration decided to cut 10% of the total flight capacity of the Air China Headquarters 737 and revoked the pilot’s license. In addition to endangering the lives of the passengers and crew, damaging aircraft and causing damage to passenger property, cabin disruptions also impair other flights’ ability to operate normally, create panic and anxiety among other passengers, and harm the reputation and trust of the airline industry. Therefore, it has become necessary to discuss whether security expenditure and penalty should be increased, as well as how to make effective emergency decisions and dispose of passenger disturbance in the cabin.

Although the cabin disturbance emergency disposal has obvious national judicial characteristics, the effectiveness of domestic and international countermeasures to passenger disturbance has not been significant. In December 2022, the Version 2 of “Even safer and more enjoyable air travel for all: A strategy for reducing unruly and disruptive passenger incidents” was released by IATA. In 2019, the Manual on the Legal Aspects of Unruly and Disruptive Passengers (Doc10117) was published by the International Civil Aviation Organization (ICAO). “Public Security Administration Punishment Law of the People’s Republic of China” (Order of the President of the People’s Republic of China (No.38)), “Rules for In-Flight Safety and Security in Public Air Passenger Transport” (CCAR-332-R1) and other relevant regulations impose restrictions on in-flight disturbances. However, until November 2022, approximately 60% of disruptive passenger cases were not prosecuted based on IATA statistics for unruly passengers [1]. When a disruption happens in the cabin in China, airline security must first evaluate the situation before determining whether to dispose of it. The disruptive passengers must be handed over to the airport police after the plane has arrived so that they can impose the proper penalty in accordance with the law. Thus, the successful disposal of disruptions on board depends on the ability of airline security to dispose of the situation and the support of the regulatory authority. In addition, inadequate security capabilities, a lack of in-depth research and scientific evaluation of decision-making effectiveness, and difficulties disposing of complex and changing emergency scenarios are other problems that currently plague the emergency disposal of cabin disturbances.

In view of this, an event tree is used to analyze the relevant subjects and disposal processes affecting cabin disruptions. Based on the assumption of a limited rationality of subjects, this paper builds an asymmetric dynamic evolutionary game model of the pertinent subjects, studies the relevant subject’s interaction mechanism and applies evolutionary game theory and a system dynamics simulation to analyze the evolutionary stabilization strategy of emergency disturbance disposal in the cabin and the stabilization conditions for each subject’s strategy to reach the ideal state. By examining the process of the emergency disposal of cabin disruptions, evolutionary process and disruptions reports in China from 2017–2021, and analyzing the evolutionary stabilization strategy of emergency disposals, the current research aims to examine the dynamic evolutionary law of behavioral choices of the regulatory authority, airline security, airport police and disruptive passengers in the emergency disposal of cabin disruptions from the perspective of evolutionary game theory.

2. Literature Review

At present, domestic and international research on the emergency disposal of disturbances is currently focused on the delineation of legal boundaries, civil aviation operational security and emergency decision-making analysis. Firstly, the study of regulatory boundaries focuses on the causes of disturbance [2], the configuration of law enforcement powers [3] and legal responsibilities [4,5], and analyzes the ambiguity of law enforcement boundaries that exist in the decision-making of disturbances and emergency disposal subjects. Secondly, civil aviation operation studies have focused on the lack of aviation security capacity, such as Paul McFarlane [6] who highlighted the link between aviation security failures and human-mediated error through a literature review. Bell et al. [1], based on a data assessment, noted a lack of airline support for frontline staff in follow-up actions against disruptive passengers. Macdonald Russell D. et al. [7] proposed a simple pre-transport mental health risk assessment tool using 498 cases. Rhoden Steven et al. [8], looking at the safety of disruptive airline passengers, pointed out that longer and frequent training sessions for crews would enable the crew to better dispose of Disruptive Airline Passenger Behavior (DAPB). Daniel Kwasi Adjekum et al. [9] assessed the relationship between collegiate aviation programs and safety culture and made recommendations. Gharehbaghi Koorosh et al. [10] studied the environmental impact of civil transportation infrastructure. Wenwen Xu et al. [11] investigated the current situation of in-flight infection prevention and control (IPC) training for cabin crew in China, pointed out the shortcomings of in-flight IPC training for cabin crew in China and suggested measures. Coral J. Dando et al. [12] experimentally demonstrated that Controlled Cognitive Engagement (CCE) is more effective in detecting deception in airline passengers. Yanqing Wang et al. [13] studied the relationship between aviation security personnel and passengers’ abnormal behavior identification, and pointed out that there was a significant positive correlation between the gaze duration and sweep magnitude of aviation security personnel and their ability to identify abnormal behavior. Thirdly, in terms of emergency decision-making research, Wengang Feng [14] proposed a new method for identifying the actors of civil aviation security events and conducted a study on aviation security from a game perspective [15,16,17], which provides a reference for the analysis of the behavioral decision-making of disturbance emergency disposal subjects. Dennis Keiser et al. [18] proposed a dynamic method to evaluate the impact of different aircraft–cabin configurations, which in turn helped airlines achieve sustainability goals by optimizing cabin layouts. Jiayang Yu et al. [19] applied the Bow-tie-DT-FTA model to quantitatively analyze the decision problem of optimal preventive measures for aviation accidents and verified the validity of the model using an aircraft tail-icing accident as an example. Xia Feng et al. [20] constructed an integrated prediction model of the aviation security event risk level (ECSDNN) to provide a decision basis for the proactive risk management of aviation safety systems. Since the emergency disposal decision-making of cabin disturbances involves multiple groups, their decision-making behavior is influenced by complex factors in different scenarios. Currently, there is a lot of literature on the strategy. Tanveer Muhammad et al. [21] used multiple methods to quantitatively analyze the context, major trends and themes, salient patterns and future research directions of waste management (WTM) and technological innovation (TI) in the circular economy (CRE) context over the past two decades to provide assistance and support to professionals and policy makers. Qingchao Li et al. [22] used the ABAQUS platform to analyze in depth the influencing factors affecting the hydrate reservoir characteristics during hydrate development operations. Fuling Wang et al. investigated a preparation strategy for the synergistic effect between low-loading the Pt precious metal and non-precious metals in an electrocatalytic system [23]. Lean Yu et al. [24] studied the supply chain reconfiguration strategy under the risk of COVID-19. Junsheng Huang [25] et al. studied carbon reduction strategies in China’s transportation sector under a carbon neutral strategy. These apers provide research ideas for the strategy research of this paper. Existing methods for studying decision-making behavior include Large Group Decision-Making (LGDM) [26], dual attitude theory [27], semi-structured interviews [28] and evolutionary game theory [29]. Evolutionary game theory can explain the dynamic process and state of group evolution in an event that changes over time. In terms of theoretical studies, Scaramangas Alan et al. used evolutionary game theory to study signals in prey–predator systems and explored the co-evolution of symbiotic traits in the context of evolutionary stability by building mathematical models [30]. Venkateswaran Vandana Revathi et al. [31] shifted the theory of evolutionary games from single-field games to multi-field games and from two-player games to multi-player games, proposing evolutionary dynamics of complex multiple games. Su Qi et al. [32] built an evolutionary dynamics model with a transition game considering the impact of a changing environment. In terms of emergency decision making, Xiaoling Xiong et al. [33] applied the evolutionary game theory to the management of heavy metal pollution in arable land, and found an effective solution to the conflicting interests of the participants by building a three-party evolutionary game model. Yuwei Zhang et al. [15] and Wengang Feng et al. [17] applied evolutionary games to the field of aviation security to study the game between two or three parties involved in security. Huiyan Zhao et al. [34] built a platform ecological system game model consisting of industrial Internet platform enterprises, industry chain complementors and innovation chain complementors, which provides new ideas for the governance of the industrial Internet platform-based ecosystem. Using a quadratic evolutionary game theory method, Yuechao Chao et al. [35] evaluated the evolutionary stabilization strategy of a public–private partnership for the coordinated development of solar and coal-fired power generation, providing relatively appropriate reference values for governments and carbon exchange enterprises.

Nevertheless, there is a lack of quantitative research on the decision-making of emergency responses to cabin disruptions in the context of the actual process of emergency disposal in civil aviation operations. On the one hand, it fails to include the subjects involved in the emergency disposal of cabin disturbances (i.e., regulatory authority, airline security, airport police and disruptive passengers) into a unified research scope from the perspective of the whole process of emergency disposal, such as integrating the regulatory authority and airport police into one side, or discussing the behavioral strategies of aviation security subjects only from the perspective of the game between two or three parties, which is inconsistent with the actual disposal process of cabin disturbances in China. On the other hand, the limitations of cabin security disposal is not fully considered and the behavioral strategies of the participating subjects are analyzed from the perspective of the benefits of the parties involved in the disturbance, e.g., the benefit measurement of aviation security mainly considers the explicit value (e.g., fines for disturbance), while one of the important factors for the regulatory authority to support the airlines to strengthen their emergency disposal efforts is the invisible value (e.g., government image, corporate trust, etc.).

3. Event Tree Analysis of Cabin Disruptions

3.1. Cabin Disruption ETA Model Construction

Event Tree Analysis (ETA) can predict the event process and unsafe factors, quantitatively calculate the probability of each stage of the event and the final event, visually reflect the dynamic change process of the whole cabin disturbance event, estimate the possible consequences of the event and conduct quantitative analysis. The event tree consists of initial disruptive events, path events and an event tree branch logical structure. According to the Appendix Civil Aviation Safety and Security Incident Information Comprehensive Statistical Tables (Categories) in the 2018–2021 report, the event tree for cabin disruption emergency disposal is constructed as follows. (1) The initial disruptive event is the first element of building an event tree, which is a disturbance that may lead to an aircraft accident when the accident has not occurred. For example, smoking, using fire, fighting and brawling in the cabin. Initial disruptive events are determined according to “Rules for In-Flight Safety and Security in Public Air Passenger Transport” (CCAR-332-R1). (2) The path event is the event that follows the initial disruptive event and is a relevant factor that affects the occurrence of an aviation accident or limits the escalation of the disruptive event. According to the 2017–2021 report to determine the path events, the events are as follows: airline security in-flight study of incidents, airline security in-flight dispose of the incidents, disruptive passengers stop the disturbance, then they are transferred under the plane to the airport police and then there is an airport police penalty. (3) The event tree branch logical structure generally follows a left-to-right order, starting from the initial disruptive event and unfolding layer by layer according to the event development process, with branches representing the event development path. The upper branch is generally used to indicate system success and the lower branch to indicate failure. Different branches of the event tree correspond to different consequences of aviation accidents.

The resulting event tree model for disruptions in the cabin is shown in Figure 1. The probability of failure of the use of an open flame or smoking path event in the cabin is shown in

Table 1. Use of open flame or smoking path event failure probability in the cabin.

Serial Number	Use of Open Flame or Smoking Path Event in the Cabin	Formula	Data	Failure Probability
1	Use of open flame or smoking in the cabin	Formula (1)	$\frac{902}{44,144}$	$2.0430\times {10}^{-2}$
2	Airline security in-flight study of incidents	Formula (2)	$\frac{502+61}{902}$	$6.2417\times {10}^{-1}$
3	Airline security in-flight dispose incidents	Formula (3)	$\frac{502+61-21-1}{902}$	$5.9978\times {10}^{-1}$
4	Use of open flame or smoking passengers to stop the behavior	Formula (4)	$\frac{858}{902}$	$9.5122\times {10}^{-1}$
5	Transferred under the plane to the airport police	Formula (5)	$\frac{29+862+0}{902}$	$9.8780\times {10}^{-1}$
6	Airport police impose penalty	Formula (6)	$\frac{862+0}{902}$	$9.5565\times {10}^{-1}$

. A disruption involving the use of an open flame or smoking in the cabin is shown in Figure 1.

$$\frac{Number of smoking incidents}{Security incidents}$$

(1)

$$\frac{Number of incidents in which no measures were taken after study+Number of incidents in which emergency}{Number of smoking incidents}$$

(2)

$$\frac{Number of incidents studied-Number of incidents that affected flight operations}{Number of smoking incidents}$$

(3)

$$\frac{Number of incidents in which flights were operating normally}{Number of smoking incidents}$$

(4)

$$\frac{Number of incidents disposed+Number of incidents penalised}{Number of smoking incidents}$$

(5)

$$\frac{Number of incidents penalised}{Number of smoking incidents}$$

(6)

The event tree model and calculation results for the use of an open flame or smoking in the cabin are shown in Figure 1.

3.2. Analysis of Event Tree Results for Use of Open Flame or Smoking in the Cabin

When a disruption occurs in the cabin, the probability of the event is represented by the logical product of the failure probabilities of the safety measures at each stage of the emergency disposal process for the cabin, and the corresponding measures are taken at any stage of the event to adopt any safety measures.

Based on Figure 1, there are 14 possible combinations of events that affect cabin safety when using an open flame or smoking in the cabin, including the impact of the airline security assessment, the success of event disposal, whether the passenger has stopped using the open flame or smoking, whether the case has been transferred to the airport police and whether the airport police have imposed penalties. Among them, the probability of the (0,*,1,*,*) event occurring is higher than the other 13 events, indicating that the emergency disposal capability of airline security has a decisive role in disposing disruptions in the cabin. This is followed by the (1,1,1,1,1) and (1,0,1,1,1) events, indicating that the current regulatory authority support for airline security, airline security disposal processes and airport police penalty methods and deterrence have been relatively improved in the event of using an open flame or smoking in the cabin. In addition, other related subjects will also have an impact on the consequences of the event. Overall, whether a disruption occurs in the cabin and ultimately affects the normal operation of the flight is closely related to the strategy choices of the regulatory authority, airline security, airport police and disruptive passengers.

4. Model Assumptions and Construction

4.1. Question Description

Based on the two-party model of civil aviation passenger security [15], the three-party model of civil aviation security [17] and other related evolutionary game models [29,31,32], the evolutionary game model of cabin disturbance emergency disposal is proposed, with the participation of airport public security added, and the regulatory authority, airline security, airport police and disruptive passengers are placed under a unified framework to establish a four-party model of evolutionary game theory, with the model participants defined as follows.

According to relevant regulations, the emergency disposal for passenger disturbance in the cabin refers to the security work of airline security in disposing passenger disturbance. In this paper, the regulatory authority refers to the Civil Aviation Administration of China and its regional administrations, which guide, supervise, and inspect the security work during the flight. Airline security refers to crew members who perform tasks on civil aircraft during the flight, including pilots, flight attendants, aviation safety personnel and other cabin crew members, who are responsible for security work during the flight. Airport police refer to the direct units of the provincial public security department, responsible for public security management within their jurisdiction, with the power of administrative penalties and criminal case investigations. Disruptive passengers refer to passengers who may disobey regulations or fail to follow crew member instructions and disrupt the good order in the cabin.

Based on the event tree of cabin disruptions and the actual security operation situation in China, an emergency disposal flowchart for cabin disruptions was constructed, consisting of the regulatory authority, airline security, airport police and disruptive passengers (see Figure 2). The four main subjects have the following relationships.

Under the bounded rationality of game players, the optimal strategies of the regulatory authority, airline security, airport police and disruptive passengers cannot be determined before making decisions. When a security incident occurs in the cabin, disruptive passengers may choose to obey the law and stop their disturbance or continue to disobey. In response, airline security may choose to take action to protect passenger safety or choose not to due to unclear legal boundaries or inadequate security capabilities. After airline security disposes disruptive passengers who have broken the law, airport police may choose to impose strong or weak penalties to prevent similar incidents. The regulatory authority may choose to support security measures to ensure aviation security or choose not to due to cost issues. Therefore, there are a total of 16 possible combination strategies in the evolutionary game for the regulatory authority, airline security, airport police and disruptive passengers.

The strategy space of the regulatory authority is $S_1$ = ($P_1$,$P_2$) = (support, nonsupport). Here, “support” means that the regulatory authority clarifies the responsibilities of airlines in security and gives them more law enforcement powers through legislation; “nonsupport” means that the regulatory authority does not take additional measures to address the issues of unclear boundaries and inadequate security capabilities of airlines’ security.

The strategy space of airline security is $S_2$ = ($Q_1$,$Q_2$) = (action, inaction). Here, “action” means that when disposing of a disturbance in the cabin, airline security should initiate emergency plans and procedures if the disruptive passenger continues to cause disturbances after being warned and educated by the crew, which may lead to more serious consequences, and hand over the passenger to airport police upon landing; “inaction” means that when disposing of the disturbance in the cabin, airline security judges that no measures need to be taken if the disruptive passenger stops causing disturbances after being warned and educated by the crew, and hand over the passenger to airport police upon landing.

The strategy space of airport police is $S_3$ = ($M_1$,$M_2$) = (strong penalty, weak penalty). Here, “strong penalty” means that airport police initiate criminal cases against disruptive passengers who violate the law and adopt penalty methods such as fines and detention; “weak penalty” means that airport police only impose administrative penalties on disruptive passengers, or the amount for the fines is relatively small.

The strategy space of disruptive passengers is $S_4$ = ($N_1$,$N_2$) = (keep the law, break the law). Here, “keep the law” means that when security incidents occur, disruptive passengers comply with civil aviation laws and regulations, and do not continue to cause disruption; “break the law” means that disruptive passengers continue to violate laws and regulations by occupying seats, fighting, smoking e-cigarettes, etc.

4.2. Basic Assumptions

To construct a game model for the emergency disposal system of disturbance in the cabin involving subjects such as the regulatory authority, airline security, airport police and disruptive passengers, the following assumptions are made.

Hypothesis 1.

The probability of strategy selection for the four subjects in the evolutionary game are as follows: the regulatory authority chooses to support with probability  $x$ and nonsupport with probability $(1-x$); the airline security chooses action with probability $y$ and inaction with probability $\left(1-y\right)$; the airport police choose to impose a strong penalty with probability  $z$ and a weak penalty with probability $\left(1-z\right)$; and the disruptive passenger chooses to keep the law with probability  $w$ and break the law with probability $\left(1-w\right)$, $0\le x, y, z,w\le 1$.

Hypothesis 2.

The cost of the regulatory authority support and the benefit of regulatory authority support are represented as $C_1$ and $G_1$, respectively. If regulatory authority support is not provided, there may be a negative impact on airline security disposal, resulting in reduced revenue, represented as $G_2$ ($G_1>G_2$). In addition, a supportive strategy can enhance the regulator’s credibility, represented as $L_g$. If regulatory authority support is not provided and airline security enforcement becomes difficult and disrupts passenger interests, the regulator’s credibility will be damaged, represented as $D_g$.

Hypothesis 3.

The benefit of airline security action is represented as $G_3$. If airline security does not take action, the disturbance caused by disruptive passengers may result in more serious consequences, leading to reduced revenue for the airline, represented as  $G_4$ ($G_4>0$). In addition, with regulatory authority support, airline security action will have more responsibilities and rights, represented as $L_a$. Conversely, without regulatory authority support, airline security action may encounter enforcement difficulties in disposing of the disturbance, represented as $D_a$. Strong penalty from airport police can help airline security action reduce their operational costs, represented as $G_6$.

Hypothesis 4.

The revenue of airport police mainly comes from a strong penalty and weak penalty for disruptive passengers, represented as $R_2$ and $R_3$ ($R_2>R_3$), respectively. The cost of a strong penalty and weak penalty is represented as $C_2$ and $C_3$ ($C_2>C_3$), respectively. If airport police strictly penalize disruptive passengers, they will gain positive deterrence, represented as $L_s$. Conversely, weak penalty for a series of disturbances by disruptive passengers will damage their image and deterrence, represented as $D_s$.

Hypothesis 5.

It costs a certain amount $C_4$ for disruptive passengers to choose to break the law. If disruptive passengers succeed in breaking the law, they will receive additional revenue $G_5$. If airline security disposes disruptive passengers, the loss to disruptive passengers is represented as $R_1$. If airport police strongly penalize the disturbance by disruptive passengers, the loss to disruptive passengers is represented as  $R_2$. Conversely, if airport public security weakly penalizes disturbance by disruptive passengers, the loss to disruptive passengers is represented as $R_3$. The social cost of a disturbance is borne jointly by the regulatory authority, airline security and airport police, represented as  $S_1$. The probability of a disturbance causing social costs is $a$ ($0<a<1$). If there is regulatory authority support, if airline security takes action, and if airport police strongly penalize the disturbance, the probability of the disturbance causing social costs is 0.

The meanings of the variable used in the text are shown in

Table 2. Parameter setting of quadratic evolutionary game model for emergency disposal of cabin disturbances and its explanations.

Subjects	Variable	Explanations	Notes
Regulatory authority	$x$	The probability of regulatory authority choosing support.	$0<x<1$
	$C_1$	The costs of regulatory authority support.	—
	$G_1$	The benefits of regulatory authority support.	$G_1>G_2$
	$G_2$	The benefits of regulatory authority nonsupport.	$G_2>0$
	$L_g$	The enhancement of regulatory authority credibility when it chooses support.	—
	$D_g$	The loss of regulatory authority credibility when it chooses nonsupport.	—
Airline security	$y$	The probability of airline security choosing action.	$0<y<1$
	$G_3$	The benefits of airline security action.	$G_3>G_4$
	$G_4$	The benefits of airline security inaction.	$G_4>0$
	$L_a$	The acquisition of airline security duties and rights when it chooses action.	—
	$D_a$	The increase in enforcement difficulties when regulatory authority chooses nonsupport and airline security chooses action.	—
	$G_6$	The reduction in disposal costs when airport police choose strong penalty.	—
Airport police	$z$	The probability of airport police choosing strong penalty.	$0<z<1$
	$C_2$	The costs of airport police choosing strong penalty.	${\mathrm{C}}_2>{\mathrm{C}}_3$
	$C_3$	The costs of airport police choosing weak penalty.	—
	$L_s$	The enhancement of deterrence capability when disruptive passengers choose to break the law and airport police choose strong penalty.	—
	$D_s$	The loss of deterrence capability when disruptive passengers choose to break the law and airport police choose weak penalty.	—
Disruptive passengers	$w$	The probability of disruptive passengers choosing to keep the law.	$0<w<1$
	$C_4$	The costs of disruptive passengers choosing to break the law.	—
	$G_5$	The additional benefits of disruptive passengers choosing to break the law.	—
	$R_1$	The penalty of disruptive passengers choosing to break the law when airline security chooses action.	—
	$R_2$	The penalty of disruptive passengers choosing to break the law when airport police choose strong penalty.	$R_2>R_3$
	$R_3$	The penalty of disruptive passengers choosing to break the law when airport police choose weak penalty.	—
	$a$	The probability of social loss when disruptive passengers choose to break the law.	$0<a<1$
	$S_1$	The social loss of disruptive passengers choosing to break the law.	—

.

4.3. Model Building

Based on the assumptions of the model, the behavior strategy combinations and their payoff matrix for the game among the regulatory authority, airline security, airport police and disruptive passengers under different strategy choices can be obtained, as shown in

Table 3. Regulatory authority, airline Security, airport police and disruptive passenger behavior strategy gain matrix.

Strategy		Strong Penalty		Weak Penalty
		Keep the Law	Break the Law	Keep the Law	Break the Law
Support	Action	$-{\mathrm{C}}_1+{\mathrm{G}}_1+{\mathrm{L}}_{\mathrm{g}}$ ${\mathrm{G}}_3+{\mathrm{L}}_{\mathrm{a}}+{\mathrm{G}}_6$ $-{\mathrm{C}}_2+{\mathrm{L}}_{\mathrm{s}}$ $0$	$-{\mathrm{C}}_1+{\mathrm{G}}_1$ ${\mathrm{G}}_3+{\mathrm{L}}_{\mathrm{a}}+{\mathrm{G}}_6+{\mathrm{R}}_1$ $-{\mathrm{C}}_2+{\mathrm{L}}_{\mathrm{s}}+{\mathrm{R}}_2$ $-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$	$-{\mathrm{C}}_1+{\mathrm{G}}_1+{\mathrm{L}}_{\mathrm{g}}$ ${\mathrm{G}}_3+{\mathrm{L}}_{\mathrm{a}}$ $-{\mathrm{C}}_3-{\mathrm{D}}_{\mathrm{s}}$ $0$	$-{\mathrm{C}}_1+{\mathrm{G}}_1-\frac{1}{3}{\mathrm{aS}}_1$ ${\mathrm{G}}_3+{\mathrm{L}}_{\mathrm{a}}-\frac{1}{3}{\mathrm{aS}}_1+{\mathrm{R}}_1$ $-{\mathrm{C}}_3-{\mathrm{D}}_{\mathrm{s}}-\frac{1}{3}{\mathrm{aS}}_1+{\mathrm{R}}_3$ $-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_3$
	Inaction	$-{\mathrm{C}}_1+{\mathrm{G}}_1$ ${\mathrm{G}}_4$ $-{\mathrm{C}}_2+{\mathrm{L}}_{\mathrm{s}}$ $0$	$-{\mathrm{C}}_1+{\mathrm{G}}_1-\frac{1}{3}{\mathrm{aS}}_1$ ${\mathrm{G}}_4-\frac{1}{3}{\mathrm{aS}}_1$ $-{\mathrm{C}}_2+{\mathrm{L}}_{\mathrm{s}}-\frac{1}{3}{\mathrm{aS}}_1+{\mathrm{R}}_2$ $-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$	$-{\mathrm{C}}_1+{\mathrm{G}}_1$ ${\mathrm{G}}_4$ $-{\mathrm{C}}_3-{\mathrm{D}}_{\mathrm{s}}$ $0$	$-{\mathrm{C}}_1+{\mathrm{G}}_1-\frac{1}{3}{\mathrm{aS}}_1$ ${\mathrm{G}}_4-\frac{1}{3}{\mathrm{aS}}_1$ $-{\mathrm{C}}_3-{\mathrm{D}}_{\mathrm{s}}-\frac{1}{3}{\mathrm{aS}}_1+{\mathrm{R}}_3$ $-{\mathrm{C}}_4+{\mathrm{G}}_5-{\mathrm{R}}_3$
Nonsupport	Action	${\mathrm{G}}_2-{\mathrm{D}}_{\mathrm{g}}$ ${\mathrm{G}}_3-{\mathrm{D}}_{\mathrm{a}}+{\mathrm{G}}_6$ $-{\mathrm{C}}_2+{\mathrm{L}}_{\mathrm{s}}$ $0$	${\mathrm{G}}_2$ ${\mathrm{G}}_3-{\mathrm{D}}_{\mathrm{a}}+{\mathrm{G}}_6$ $-{\mathrm{C}}_2+{\mathrm{L}}_{\mathrm{s}}+{\mathrm{R}}_2$ $-{\mathrm{C}}_4-{\mathrm{R}}_2$	${\mathrm{G}}_2-{\mathrm{D}}_{\mathrm{g}}$ ${\mathrm{G}}_3-{\mathrm{D}}_{\mathrm{a}}$ $-{\mathrm{C}}_3-{\mathrm{D}}_{\mathrm{s}}$ $0$	${\mathrm{G}}_2-\frac{1}{3}{\mathrm{aS}}_1$ ${\mathrm{G}}_3-{\mathrm{D}}_{\mathrm{a}}-\frac{1}{3}{\mathrm{aS}}_1$ $-{\mathrm{C}}_3-{\mathrm{D}}_{\mathrm{s}}-\frac{1}{3}{\mathrm{aS}}_1+{\mathrm{R}}_3$ $-{\mathrm{C}}_4-{\mathrm{R}}_3$
	Inaction	${\mathrm{G}}_2$ ${\mathrm{G}}_4$ $-{\mathrm{C}}_2+{\mathrm{L}}_{\mathrm{s}}$ $0$	${\mathrm{G}}_2-\frac{1}{3}{\mathrm{aS}}_1$ ${\mathrm{G}}_4-\frac{1}{3}{\mathrm{aS}}_1$ $-{\mathrm{C}}_2+{\mathrm{L}}_{\mathrm{s}}-\frac{1}{3}{\mathrm{aS}}_1+{\mathrm{R}}_2$ $-{\mathrm{C}}_4-{\mathrm{R}}_2$	${\mathrm{G}}_2$ ${\mathrm{G}}_4$ $-{\mathrm{C}}_3$ $0$	${\mathrm{G}}_2-\frac{1}{3}{\mathrm{S}}_1$ ${\mathrm{G}}_4-\frac{1}{3}{\mathrm{S}}_1$ $-{\mathrm{C}}_3-\frac{1}{3}{\mathrm{S}}_1+{\mathrm{R}}_3$ $-{\mathrm{C}}_4+{\mathrm{G}}_5-{\mathrm{R}}_3$

.

4.4. Replicator Dynamic Equation and Evolutionary Stable Strategy

According to the above research, the expected benefit for the regulatory authority choosing the “support” strategy is denoted as ${\mathrm{U}}_{11}$, and the expected benefit for choosing the “not support” strategy is denoted as ${\mathrm{U}}_{12}$. The average expected benefit is denoted as ${\mathrm{U}}_1$.

$${\mathrm{U}}_{11}={\mathrm{G}}_1-{\mathrm{C}}_1+\left(\mathrm{y}-\mathrm{yz}\left(1-\mathrm{w}\right)\right){\mathrm{L}}_{\mathrm{g}}-\frac{1}{3}\left(1-\mathrm{w}\right)\left(1-\mathrm{yz}\right){\mathrm{aS}}_1$$

$${\mathrm{U}}_{12}=\left(2\mathrm{yzw}-\mathrm{yw}-\mathrm{zw}-2\mathrm{yz}+\mathrm{y}+\mathrm{z}\right)\left({\mathrm{G}}_2-\frac{1}{3}{\mathrm{aS}}_1\right)+\mathrm{yw}\left({\mathrm{G}}_2-{\mathrm{D}}_{\mathrm{g}}\right)+\left(1-\mathrm{y}\right)\left(1-\mathrm{z}\right)\left(1-\mathrm{w}\right)\left({\mathrm{G}}_2-\frac{1}{3}{\mathrm{S}}_1\right)+\left(1-\mathrm{y}\right){\mathrm{wG}}_2$$

$${\mathrm{U}}_1={\mathrm{xU}}_{11}+\left(1-\mathrm{x}\right){\mathrm{U}}_{12}$$

The replicator dynamic equation for the regulatory authority is:

$$\begin{array}{cc}\hfill \mathrm{F}\left(\mathrm{x}\right)=\mathrm{x}\left(1-\mathrm{x}\right)& \left({\mathrm{U}}_{11}-{\mathrm{U}}_{12}\right)\hfill \\ & =\mathrm{x}\left(1-\mathrm{x}\right)({\mathrm{G}}_1-{\mathrm{C}}_1-{\mathrm{G}}_2+\frac{1}{3}(1-\mathrm{y}-\mathrm{z}-\mathrm{w}+\mathrm{yw}+\mathrm{zw}+\mathrm{yz}\hfill \\ & -\mathrm{yzw}\left)\left(1-\mathrm{a}\right){\mathrm{S}}_1\right)+\left(\mathrm{y}-\mathrm{yz}+\mathrm{yzw}\right){\mathrm{L}}_{\mathrm{g}}+{\mathrm{ywD}}_{\mathrm{g}})\hfill \end{array}$$

(7)

Similarly, the expected benefits for airline security to choose “action” and “inaction” strategies are denoted as ${\mathrm{U}}_{21}$ and ${\mathrm{U}}_{22}$, respectively, and the average expected benefit is ${\mathrm{U}}_2$.

$${\mathrm{U}}_{21}={\mathrm{G}}_3+{\mathrm{zG}}_6+{\mathrm{xL}}_{\mathrm{a}}-\left(1-\mathrm{x}\right){\mathrm{D}}_{\mathrm{a}}+\mathrm{x}\left(1-\mathrm{w}\right){\mathrm{R}}_1-\frac{1}{3}\left(1-\mathrm{w}-\mathrm{z}+\mathrm{zw}\right){\mathrm{aS}}_1$$

$${\mathrm{U}}_{22}={\mathrm{wG}}_4+\left(\mathrm{xzw}-\mathrm{xw}-\mathrm{zw}-\mathrm{xz}+\mathrm{x}+\mathrm{z}\right)\left({\mathrm{G}}_4-\frac{1}{3}{\mathrm{aS}}_1\right)+\left(1-\mathrm{x}\right)\left(1-\mathrm{z}\right)\left(1-\mathrm{w}\right)\left({\mathrm{G}}_4-\frac{1}{3}{\mathrm{S}}_1\right)$$

$${\mathrm{U}}_2={\mathrm{yU}}_{21}+\left(1-\mathrm{y}\right){\mathrm{U}}_{22}$$

The replicator dynamic equation for airline security is:

$$\begin{array}{cc}\hfill \mathrm{F}\left(\mathrm{y}\right)=\mathrm{y}\left(1-\mathrm{y}\right)& \left({\mathrm{U}}_{21}-{\mathrm{U}}_{22}\right)\hfill \\ & =\mathrm{y}\left(1-\mathrm{y}\right)({\mathrm{G}}_3-{\mathrm{G}}_4+{\mathrm{zG}}_6+{\mathrm{xL}}_{\mathrm{a}}-\left(1-\mathrm{x}\right){\mathrm{D}}_{\mathrm{a}}+\mathrm{x}\left(1-\mathrm{w}\right){\mathrm{R}}_1\hfill \\ & +\frac{1}{3}\mathrm{z}\left(1-\mathrm{w}\right){\mathrm{aS}}_1+\frac{1}{3}\left(1-\mathrm{x}-\mathrm{z}-\mathrm{w}+\mathrm{xw}+\mathrm{zw}+\mathrm{xz}-\mathrm{xzw}\right)(1\hfill \\ & -\mathrm{a}\left){\mathrm{S}}_1\right)\hfill \end{array}$$

(8)

The expected benefits for airport police to choose “strong penalty” and “weak penalty” strategies are denoted as ${\mathrm{U}}_{31}$ and ${\mathrm{U}}_{32}$, respectively, and the average expected benefit is ${\mathrm{U}}_3$.

$${\mathrm{U}}_{31}={\mathrm{L}}_{\mathrm{s}}-{\mathrm{C}}_2+\left(1-\mathrm{w}\right){\mathrm{R}}_2-\frac{1}{3}\left(1-\mathrm{y}-\mathrm{w}+\mathrm{yw}\right){\mathrm{aS}}_1$$

$$\begin{array}{cc}\hfill {\mathrm{U}}_{32}=-\left(1-\mathrm{x}\right)& \left(1-\mathrm{y}\right){\mathrm{wC}}_3-\left(1-\mathrm{x}\right)\left(1-\mathrm{y}\right)\left(1-\mathrm{w}\right)\left({\mathrm{C}}_3-{\mathrm{R}}_3+\frac{1}{3}{\mathrm{S}}_1\right)+(\mathrm{xyw}\hfill \\ & -\mathrm{yw}-\mathrm{xw})\left({\mathrm{C}}_3+{\mathrm{D}}_{\mathrm{s}}\right)+\left(\mathrm{xy}+\mathrm{xw}+\mathrm{yw}-\mathrm{xyw}-\mathrm{x}-\mathrm{y}\right)({\mathrm{C}}_3\hfill \\ & +{\mathrm{D}}_{\mathrm{s}}-{\mathrm{R}}_3+\frac{1}{3}{\mathrm{aS}}_1)\hfill \end{array}$$

$${\mathrm{U}}_3={\mathrm{zU}}_{31}+\left(1-\mathrm{z}\right){\mathrm{U}}_{32}$$

The replicator dynamic equation for airport police is:

$$\begin{array}{cc}\hfill \mathrm{F}\left(\mathrm{z}\right)=\mathrm{z}\left(1-\mathrm{z}\right)& \left({\mathrm{U}}_{31}-{\mathrm{U}}_{32}\right)\hfill \\ & =\mathrm{z}\left(1-\mathrm{z}\right)({\mathrm{C}}_3-{\mathrm{C}}_2+{\mathrm{L}}_{\mathrm{s}}+\left(1-\mathrm{w}\right)\left({\mathrm{R}}_2-{\mathrm{R}}_3\right)+(\mathrm{x}+\mathrm{y}\hfill \\ & -\mathrm{xy}){\mathrm{D}}_{\mathrm{s}}+\frac{1}{3}\left(1-\mathrm{x}-\mathrm{y}-\mathrm{w}+\mathrm{xw}+\mathrm{xy}+\mathrm{yw}-\mathrm{xyw}\right)\left(1-\mathrm{a}\right){\mathrm{S}}_1\hfill \\ & +\frac{1}{3}\mathrm{y}\left(1-\mathrm{w}\right){\mathrm{aS}}_1)\hfill \end{array}$$

(9)

The expected benefits for disruptive passengers to choose “keep the law” and “break the law” strategies are denoted as ${\mathrm{U}}_{41}$ and ${\mathrm{U}}_{42}$, respectively, and the average expected benefit is ${\mathrm{U}}_4$.

$${\mathrm{U}}_{41}=0$$

$${\mathrm{U}}_{42}=-{\mathrm{C}}_4-{\mathrm{xyR}}_1-{\mathrm{zR}}_2-\left(1-\mathrm{z}\right){\mathrm{R}}_3+\left(1-\mathrm{y}-\mathrm{z}+\mathrm{yz}\right){\mathrm{G}}_5$$

$${\mathrm{U}}_4={\mathrm{wU}}_{41}+\left(1-\mathrm{w}\right){\mathrm{U}}_{42}$$

The replicator dynamic equation for disruptive passengers is:

$$\begin{array}{cc}\hfill \mathrm{F}\left(\mathrm{w}\right)=\mathrm{w}\left(1-\mathrm{w}\right)& \left({\mathrm{U}}_{41}-{\mathrm{U}}_{42}\right)\hfill \\ & =-\mathrm{w}\left(1-\mathrm{w}\right)(-{\mathrm{C}}_4-{\mathrm{xyR}}_1-{\mathrm{zR}}_2-\left(1-\mathrm{z}\right){\mathrm{R}}_3+(1-\mathrm{y}-\mathrm{z}\hfill \\ & +\mathrm{yz}\left){\mathrm{G}}_5\right)\hfill \end{array}$$

(10)

4.5. Evolutionary Stability Analysis of the Strategy of Four Subjects

Let the replicator dynamic equations (Equations (7)–(10)) be equal to 0. That is, the 16 equilibrium points of the system can be obtained when the rate of change of the system strategy choice is 0. The 16 equilibrium points are: ${\mathrm{D}}_1\left(0, 0, 0, 0\right)$, ${\mathrm{D}}_2\left(1, 0, 0, 0\right)$, ${\mathrm{D}}_3\left(0, 1, 0, 0\right)$, ${\mathrm{D}}_4\left(0, 0, 1, 0\right)$, ${\mathrm{D}}_5\left(0, 0, 0, 1\right)$, ${\mathrm{D}}_6\left(1, 1, 0, 0\right)$, ${\mathrm{D}}_7\left(1, 0, 1, 0\right)$, ${\mathrm{D}}_8\left(0, 1, 1, 0\right)$, ${\mathrm{D}}_9\left(1, 0, 0, 1\right)$, ${\mathrm{D}}_{10}\left(0, 1, 0, 1\right)$, ${\mathrm{D}}_{11}\left(0, 0, 1, 1\right)$, ${\mathrm{D}}_{12}\left(1, 1, 1, 0\right)$, ${\mathrm{D}}_{13}\left(1, 1, 0, 1\right)$, ${\mathrm{D}}_{14}\left(1, 0, 1, 1\right)$, ${\mathrm{D}}_{15}\left(0, 1, 1, 1\right)$, ${\mathrm{D}}_{16}\left(1, 1, 1, 1\right)$. In an asymmetric game, mixed-strategy equilibrium points are not evolutionarily stable, so only pure-strategy equilibrium points are discussed for stability [36]. Following the method proposed by FRIEDMAN (1991) [37], the system’s Evolutionarily Stable Strategy (ESS) is determined by analyzing the local stability of the Jacobian matrix:

$$J=\left[\begin{array}{cccc}\frac{\partial F\left(x\right)}{\partial x}& \frac{\partial F\left(x\right)}{\partial y}& \frac{\partial F\left(x\right)}{\partial z}& \frac{\partial F\left(x\right)}{\partial w}\\ \frac{\partial F\left(y\right)}{\partial x}& \frac{\partial F\left(y\right)}{\partial y}& \frac{\partial F\left(y\right)}{\partial z}& \frac{\partial F\left(y\right)}{\partial w}\\ \frac{\partial F\left(z\right)}{\partial x}& \frac{\partial F\left(z\right)}{\partial y}& \frac{\partial F\left(z\right)}{\partial z}& \frac{\partial F\left(z\right)}{\partial w}\\ \frac{\partial F\left(w\right)}{\partial x}& \frac{\partial F\left(w\right)}{\partial y}& \frac{\partial F\left(w\right)}{\partial z}& \frac{\partial F\left(w\right)}{\partial w}\end{array}\right]=\left[\begin{array}{cc}\begin{array}{cc}\left(\begin{array}{c}\begin{array}{c}\left((1-x\right)-x)(C_4+\\ z{R}_2-(1-y-z+\end{array}\\ \begin{array}{c}yz)G_5+\left(1-z\right)R_3+\\ xy{R}_1)+x\left(1-x\right)y{R}_1\end{array}\end{array}\right)& \left(\begin{array}{c}x\left(1-x\right)\\ \left(x{R}_1+\left(1-z\right)G_5\right)\end{array}\right)\\ y^2\left(1-y\right)R_1& \left(\begin{array}{c}\left(\left(1-y\right)-y\right)X+\\ y\left(1-y\right)\\ \left(x{R}_1+\left(1-y\right)G_5\right)\end{array}\right)\end{array}& \begin{array}{cc}\left(\begin{array}{c}x\left(1-x\right)(-R_3+\\ R_2+\left(1-y\right)G_5\end{array}\right)& 0\\ y\left(1-y\right)Y& 0\end{array}\\ \begin{array}{cc}yz\left(1-z\right)R_1& \left(\begin{array}{c}z\left(1-z\right)\\ \left(x{R}_1+\left(1-z\right)G_5\right)\end{array}\right)\\ yw\left(1-w\right)R_1& \left(\begin{array}{c}w\left(1-w\right)\\ \left(x{R}_1+\left(1-z\right)G_5\right)\end{array}\right)\end{array}& \begin{array}{cc}\left(\begin{array}{c}\left(\left(1-z\right)-z\right)X+\\ z\left(1-z\right)Y\end{array}\right)& 0\\ w\left(1-w\right)Y& (\left(1-w\right)-w)X\end{array}\end{array}\right]$$

Note:

$$X=C_4+z{R}_2-\left(1-y-z+yz\right)G_5+\left(1-z\right)R_3+xy{R}_1;Y=-R_3+R_2+\left(1-y\right)G_5$$

The equilibrium points of the four-player evolutionary game system and their eigenvalues are shown in

Table 4. Eigenvalue of the equilibrium point.

Equilibrium Points	Eigenvalue λ1	Eigenvalue λ2	Eigenvalue λ3	Eigenvalue λ4
$\left(0,0,0,0\right)$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$
$\left(1,0,0,0\right)$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5-{\mathrm{R}}_3$
$\left(0,1,0,0\right)$	$-{\mathrm{C}}_4-{\mathrm{R}}_3$	${\mathrm{C}}_4+{\mathrm{R}}_3$	${\mathrm{C}}_4+{\mathrm{R}}_3$	${\mathrm{C}}_4+{\mathrm{R}}_3$
$\left(0,0,1,0\right)$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	${\mathrm{C}}_4+{\mathrm{R}}_2$	${\mathrm{C}}_4+{\mathrm{R}}_2$	${\mathrm{C}}_4+{\mathrm{R}}_2$
$\left(0,0,0,1\right)$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5-{\mathrm{R}}_3$
$\left(1,1,0,0\right)$	${\mathrm{C}}_4+{\mathrm{R}}_1+{\mathrm{R}}_3$	${\mathrm{C}}_4+{\mathrm{R}}_1+{\mathrm{R}}_3$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_3$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_3$
$\left(1,0,1,0\right)$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	${\mathrm{C}}_4+{\mathrm{R}}_2$	${\mathrm{C}}_4+{\mathrm{R}}_2$
$\left(0,1,1,0\right)$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	${\mathrm{C}}_4+{\mathrm{R}}_2$	${\mathrm{C}}_4+{\mathrm{R}}_2$
$\left(1,0,0,1\right)$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5-{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5-{\mathrm{R}}_3$
$\left(0,1,0,1\right)$	$-{\mathrm{C}}_4-{\mathrm{R}}_3$	$-{\mathrm{C}}_4-{\mathrm{R}}_3$	${\mathrm{C}}_4+{\mathrm{R}}_3$	${\mathrm{C}}_4+{\mathrm{R}}_3$
$\left(0,0,1,1\right)$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	${\mathrm{C}}_4+{\mathrm{R}}_2$	${\mathrm{C}}_4+{\mathrm{R}}_2$
$\left(1,1,1,0\right)$	${\mathrm{C}}_4+{\mathrm{R}}_1+{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$
$\left(1,1,0,1\right)$	${\mathrm{C}}_4+{\mathrm{R}}_1+{\mathrm{R}}_3$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_3$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_3$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_3$
$\left(1,0,1,1\right)$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	${\mathrm{C}}_4+{\mathrm{R}}_2$
$\left(0,1,1,1\right)$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_2$	${\mathrm{C}}_4+{\mathrm{R}}_2$
$\left(1,1,1,1\right)$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$

.

Since it is impossible to determine the signs of the eigenvalues, the equilibrium points that satisfy the stability conditions are shown in

Table 5. The equilibrium point where the conditions are met.

Equilibrium Point	$\mathbf{Eigenvalue} {\mathit{\lambda }}_1$	$\mathbf{Eigenvalue} {\mathit{\lambda }}_2$	$\mathbf{Eigenvalue} {\mathit{\lambda }}_3$	$\mathbf{Eigenvalue} {\mathit{\lambda }}_4$
$\left(0,0,0,0\right)$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$	${\mathrm{C}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3$
$\left(1,1,1,1\right)$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$	$-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2$

. Based on the payoffs of the participating subjects, an analysis is conducted.

When the condition of ${\mathrm{G}}_4-{\mathrm{G}}_5+{\mathrm{R}}_3<0$ is satisfied, that is, the additional benefits of disruptive passengers breaking the law are higher than the costs of disruptive passengers breaking the law and the penalty under weak penalty strategy, the equilibrium point $\left(0, 0, 0, 0\right)$ is the evolutionarily stable point, and the strategy choices of each player are (nonsupport, inaction, weak penalty, break the law).

When the condition of $-{\mathrm{C}}_4-{\mathrm{R}}_1-{\mathrm{R}}_2<0$ is satisfied, that is, the costs of disruptive passengers breaking the law, the penalty of disruptive passengers breaking the law under strong penalty strategy and the penalty of disruptive passengers breaking the law under weak penalty strategy are higher than 0, the equilibrium point $\left(1, 1, 1, 1\right)$ is the evolutionarily stable point, and the strategy choices of each player are (support, action, strong penalty, keep the law).

System dynamics provides an effective tool for analyzing the evolution process of evolutionary game models under conditions of bounded rationality and incomplete information [15]. Federico Barnabè et al. [38] used system dynamics to explain the behavior and decisions of the participants in the game. Therefore, a system dynamics model for emergency disposal of cabin disturbances was constructed, and different parameter initial values were analyzed using system dynamics simulation tools to investigate their effects on the evolution process of the game.

5. System Dynamics Analysis and Simulation

Based on the replicator dynamic equation calculated in the previous section, a system dynamics model was constructed using the Vensim PLE 9.3.4 software to simulate the evolving game among the regulatory authority, airline security, airport police and disruptive passengers. The model consists of eight level variables: the probability of the regulatory authority choosing support $x$, the probability of the regulatory authority choosing nonsupport $\left(1-x\right)$, the probability of airline security choosing action $y$, the probability of airline security choosing inaction $\left(1-y\right)$, the probability of airport police choosing strong penalty $z$, the probability of airport police choosing weak penalty $\left(1-z\right)$, the probability of disruptive passengers choosing to keep the law $w$ and the probability of disruptive passengers choosing to break the law $\left(1-w\right)$. The model also includes four rate variables: the rate of strategy change for the regulatory authority, airline security, airport police and disruptive passengers, as well as 29 auxiliary variables. The model diagram is shown in Figure 3.

Assume the simulation model starts at INITIAL TIME = 0, ends at FINAL TIME = 60, with the time step set as TIME STEP = 1, in units of months. Based on the “Regulations of the People’s Republic of China on Public Security Administration Punishments,” the variables are classified into three categories: explicit costs, implicit costs and probabilities. Explicit costs refer to costs that can be directly measured by prices, such as fines for disruptive passengers’ disturbance, measured in hundreds of yuan (CNY). Implicit costs refer to costs that cannot be directly measured by prices, such as the enhancement of the regulatory authority credibility when it chooses support, and the human and time resources consumed by the regulatory authority for supporting strategies, which are converted into prices for simulation purposes, measured in hundreds of yuan (CNY). The initial simulation parameters are set as shown in

Table 6. Model variables and defaults.

Variable	Explanation	Default
$C_1$	The costs of regulatory authority support.	1
$G_1$	The benefits of regulatory authority support.	2
$G_2$	The benefits of regulatory authority nonsupport.	1
$L_g$	The enhancement of regulatory authority credibility when it chooses support.	0.2
$D_g$	The loss of regulatory authority credibility when it chooses nonsupport.	0.2
$G_3$	The benefits of airline security action.	1
$G_4$	The benefits of airline security inaction.	0.5
$L_a$	The acquisition of airline security duties and rights when it chooses action.	0.1
$D_a$	The increase in enforcement difficulties when regulatory authority chooses nonsupport and airline security chooses action.	0.1
$G_6$	The reduction in disposal costs when airport police choose strong penalty.	0.2
$C_2$	The costs of airport police choosing strong penalty.	1
$C_3$	The costs of airport police choosing weak penalty.	0.5
$L_s$	The enhancement of deterrence capability when disruptive passengers choose to break the law and airport police choose strong penalty.	0.1
$D_s$	The loss of deterrence capability when disruptive passengers choose to break the law and airport police choose weak penalty.	0.1
$C_4$	The costs of disruptive passengers choosing to break the law.	0.1
$G_5$	The additional benefits of disruptive passengers choosing to break the law.	0.5
$R_1$	The penalty of disruptive passengers choosing to break the law when airline security chooses action.	1
$R_2$	The penalty of disruptive passengers choosing to break the law when airport police choose strong penalty.	2
$R_3$	The penalty of disruptive passengers choosing to break the law when airport police choose weak penalty.	1
$a$	The probability of social loss when disruptive passengers choose to break the law.	0.5
$S_1$	The social loss of disruptive passengers choosing to break the law.	5

.

5.1. System Equilibrium Point Simulation Analysis

According to the hypothesis, in the game process of disposing of passenger disturbance in the cabin, the two initial strategies are (0, 0, 0, 0) and (1, 1, 1, 1), respectively. The evolutionary game process is shown in Figure 4a,b. (0, 0, 0, 0) indicates that the optimal strategy is that the regulatory authority chooses nonsupport, airline security chooses inaction, airport police choose a weak penalty and disruptive passengers choose to break the law. (1, 1, 1, 1) indicates that the optimal strategy is that the regulatory authority chooses support, airline security chooses action, airport police choose a strong penalty and disruptive passengers choose to keep the law. Pure strategy selection shows that each subject is unwilling to change the equilibrium state in the system, but if one subject adjusts, the other subjects will also adjust according to their interests. The strategy (0, 0, 0, 0) is adjusted for (0.9, 0.1, 0.1, 0.1) and the simulation results are shown in Figure 4c. The simulation results show that when the regulatory authority chooses support, even if the probability of airline security action is only 0.1, the system reaches an equilibrium state tending to 1 under the influence of the regulatory authority’s support. The probability of a strong penalty by airport police increases under its influence and when airline security reaches a stable state, the probability of a strong penalty by airport police ultimately tends to 0. The probability of disruptive passengers choosing to keep the law oscillates around 1 and gradually decreases with time until it reaches a steady oscillation. The support of the regulatory authority has a positive effect on the airline security’s decision to dispose of the situation, and the airline security and airport police have the same deterrent effect on disruptive passengers, which helps disruptive passengers choose to keep the law.

The strategy was adjusted to (0.1, 0.1, 0.1, 0.1) and the simulation results are shown in Figure 4d. Even if the initial probabilities chosen by the four subjects are the same and small, the time for the game participants to reach equilibrium is not the same. As the emergency disposal party for disturbances, airline security tends to reach equilibrium faster than the regulatory authority and airport police. This is because the benefits of airline security for disposing disruptive disturbances are higher and reach equilibrium faster over time.

5.2. Evolutionary Game Simulation under Dynamic Strategy

5.2.1. Evolutionary Game Simulation under Dynamic Support Strategy

To verify the impact of support intensity $L_a$ on the behavior selection of both the regulatory authority and airline security in the emergency disposal of passenger disturbances, different values were assigned to the support intensity of the regulatory authority towards the airline security’s disposing disruptive passengers $L_a$. The support intensity $L_a$ was set to 0, 1 and 2, respectively. $L_a=0$ means that the regulatory authority does not take any improvement measures for the current problems of airline security; $L_a=1$ means that the regulatory authority provides financial and resource support to airlines to support airline security for improving security capability; $L_a=$ 2 means that the regulatory authority not only provides financial and other support, but also gives airline security more enforcement rights by making regulations and other means. The simulation results are shown in Figure 5a,b. The simulation results show that under different support intensities, the probability of the regulatory authority choosing support tends to be stable, and finally reaches an equilibrium state that tends to 1. With the $L_a$ increase, the rate of the regulatory authority choosing support slows down, indicating that the regulatory authority should consider the costs and benefits of support. The probability of airline security choosing action tends to 1, and when it increases, the rate of airline security tending to 1 becomes faster. The equilibrium state tending to 1 fluctuates with the increase, indicating that when the regulatory authority gives too much support and too many rights to airline security, airline security may choose inaction because of laxity.

5.2.2. Evolutionary Game Simulation under Dynamic Penalty Strategy

To verify the impact of the penalty severity ${\mathrm{R}}_2$ on the behavior of airport police and disruptive passengers, different values were assigned and the simulation results were observed. The penalty severity of airport police towards disruptive passengers ${\mathrm{R}}_2$ was set to 0, 1 and 2. ${\mathrm{R}}_2=0$ means that airport police do not take any penalty for disruptive passengers who break the law; ${\mathrm{R}}_2=1$ means the airport police take only administrative penalty for disruptive passengers who break the law; ${\mathrm{R}}_2=2$ means that airport police take administrative and criminal penalties for disruptive passengers who break the law. The simulation results in Figure 6a,b show that the probability of airport police choosing to impose a strong penalty on disruptive passengers tends to stabilize and eventually reaches an equilibrium state tending towards 0. As the severity of the penalty increases, the probability of airport police imposing a strong penalty also increases. The probability of disruptive passengers choosing to keep the law oscillates around 1, and as the severity of the penalty increases, the amplitude of oscillation increases but the rate of convergence towards 1 also increases. This indicates that as the penalty severity of airport police increases, disruptive passengers are more likely to choose to keep the law, but excessive penalty severity may lead to breaking the law among disruptive passengers.

5.3. Evolutionary Game Simulation Trend Analysis

Through the simulation of the model for 60 months, we can observe and analyze the behavior choices of the game players. In order to predict the long-term behavior choices of the game players, the model is simulated for 100 months. Assume that the starting TIME of the model simulation is INITIAL TIME = 0, the FINAL TIME = 100, the unit is month, and the STEP is TIME STEP = 1. The initial strategies are (0, 0, 0, 0) and (1, 1, 1, 1). The simulation results are the same as those of 60 months. The strategy (0, 0, 0, 0) is adjusted to (0.9, 0.1, 0.1, 0.1). The simulation results are shown in Figure 7a. By comparison, the regulatory authority, airline security and airport police still maintain an equilibrium state of 1 in the last 40 months, and disruptive passengers after 40 months show constant amplitude oscillation. This indicates that the behavior choice of the current game players is ideal over time.

Setting the simulated values of strategy (0, 0, 0, 0) to (0.1, 0.1, 0.1, 0.1), the simulation result is shown in Figure 7b. By comparison, it takes longer for the regulatory authority to reach the equilibrium state of tending towards 1, indicating that longer-term support requires the regulatory authority to spend more time.

5.4. Empirical Analysis and Simulation

5.4.1. Data and Parameters

The use of an open flame or smoking by disruptive passengers in the cabin is a typical disruption, posing a public safety threat and fire hazard to the enclosed high-pressure oxygen cabin. According to ICAO statistics, 80% of aircraft cabin fires are caused by passengers using an open flame to ignite combustibles. Based on reports, the number of security incidents involving the use of an open flame or smoking in the cabin in China has increased each year from 2017 to 2021, with 164, 228, 247, 238 and 435 incidents, respectively. Recently, with the adjustment of China’s epidemic prevention and control policies and the increase in passenger transportation volume, incidents of using an open flame and smoking in the cabin have occurred frequently, such as on flights MU5553, KY8225 and HU7237 on 30 May, 10 July and 26 December 2022, respectively. In such cases, airline security should immediately dissuade the disruptive passengers from smoking and upon landing, transfer them to airport police for penalty. Here, taking the disposal of such incidents as an example, based on the previous assumptions, simulations are conducted to analyze the decision-making behavior and strategy evolution of each entity. On 3 April 2023, a man smoked an e-cigarette in the cabin of a flight from Changsha, Hunan, China to Beijing. After the cabin crew reminded the man that e-cigarettes were not allowed on the plane, the man refused to listen to the advice and continued to smoke, after which the crew immediately stopped and called the police. After the plane landed, the man was handed over to the Capital Airport, where the police gave the man seven days of administrative detention. The simulation parameters were set on this basis. The initial simulation parameters are shown in

Table 7. The use of an open flame or smoking in the cabin variables and defaults.

Variable	Explanation	Default
$C_1$	The costs of regulatory authority support.	1
$G_1$	The benefits of regulatory authority support.	1.5
$G_2$	The benefits of regulatory authority nonsupport.	0.6
$L_g$	The enhancement of regulatory authority credibility when it chooses support.	0.1
$D_g$	The loss of regulatory authority credibility when it chooses nonsupport.	0.1
$G_3$	The benefits of airline security action.	2
$G_4$	The benefits of airline security inaction.	1
$L_a$	The acquisition of airline security duties and rights when it chooses action.	0.2
$D_a$	The increase in enforcement difficulties when regulatory authority chooses nonsupport and airline security chooses action.	0.2
$G_6$	The reduction in disposal costs when airport police choose strong penalty.	0.2
$C_2$	The costs of airport police choosing strong penalty.	1
$C_3$	The costs of airport police choosing weak penalty.	0.5
$L_s$	The enhancement of deterrence capability when disruptive passengers choose to break the law and airport police choose strong penalty.	0.1
$D_s$	The loss of deterrence capability when disruptive passengers choose to break the law and airport police choose weak penalty.	0.1
$C_4$	The costs of disruptive passengers choosing to break the law.	0.01
$G_5$	The additional benefits of disruptive passengers choosing to break the law.	0.1
$R_1$	The penalty of disruptive passengers choosing to break the law when airline security chooses action.	1
$R_2$	The penalty of disruptive passengers choosing to break the law when airport police choose strong penalty.	2
$R_3$	The penalty of disruptive passengers choosing to break the law when airport police choose weak penalty.	1
$a$	The probability of social loss when disruptive passengers choose to break the law.	0.7
$S_1$	The social loss of disruptive passengers choosing to break the law.	1

.

5.4.2. Simulation Results

According to the assumptions and set parameters, in the game process of using an open flame or smoking in the cabin, with two initial strategies of (0, 0, 0, 0) and (1, 1, 1, 1), the evolutionary game process is the same as before. The simulation value of the strategy (0, 0, 0, 0) was adjusted to (0.9, 0.1, 0.1, 0.1) and the simulation results in Figure 8a showed that under the influence of regulatory authority support, the equilibrium state where airline security and disruptive passengers choose to comply and not smoke can still reach a probability of 1. After the equilibrium state where airline security chooses to dispose and disruptive passengers choose not to smoke is reached, the regulatory authority eventually reaches an equilibrium state with a probability of 0 under its influence. This indicates that airline security choosing action and disruptive passengers choosing to keep the law can reduce regulatory authority costs. Despite the high probability and small losses associated with disruptive passengers using an open flame or smoking, the probability of disruptive passengers choosing to keep the law eventually reaches an equilibrium state approaching 1. This suggests that when the social losses caused by disruptive passengers using an open flame or smoking are relatively small, the regulatory authority can choose short-term support to achieve the goal of strengthening the security equilibrium at a lower cost.

The strategy is adjusted to (0.1, 0.1, 0.1, 0.1). The simulation results are shown in Figure 8b. The simulation results show that the equilibrium rate of disruptive passengers as the implementer of using an open flame or smoking is faster than that of the regulatory authority, airline security and airport police. This is because the benefits brought by disruptive passengers through non-use of an open flame or non-smoking behaviors reach the equilibrium faster over time.

6. Conclusions

6.1. Results

(1) To clarify the influencing factors and interaction mechanisms of emergency disposal strategies for cabin disturbance, an event tree model for cabin disruption and a system dynamics model for emergency disposal of the cabin disturbance were constructed. The evolution path of the four-party behavior was simulated and analyzed using system dynamics tools.

(2) The emergency disposal capability of airline security plays a decisive role in disposing of disruptions in the cabin. Currently, the support of the regulatory authority for airline security, the process of airline security disposal measures and the airport police penalty methods and deterrence are relatively sound for the use of an open flame or smoking events in the cabin, while the impact of other relevant subjects on the consequences of the event will also affect the disposal.

(3) The main influencing factor for airline security to take action against passengers who disrupt cabin order is the probability of the regulatory authority choosing support. When the regulatory authority provides support, the benefits of airline security choosing to take action outweigh the costs, and all parties’ interests will be maximized. This is an ideal state, indicating that the cost of airline security’s actions will be reduced when the regulatory authority provides support, thus increasing their initiative and willingness to take action. The regulatory authority’s decision to support or not depends mainly on the probability of airline security taking action. When airline security takes action, regulatory authorities’ expenditures can be reduced.

(4) The airport police’s decision to impose a strong or weak penalty mainly depends on the probability that the disruptive passengers choose to break the law. Additionally, airline security’s action to disruptive passengers’ disturbance will also affect the airport police’s strategy. When airline security’s action on disruptive passengers is severe enough to deter them from breaking the law, the airport police will choose to impose a weak penalty. Whether the disruptive passengers break the law or not mainly depends on the probability of airline security choosing action. When airline security takes action, disruptive passengers choose to keep the law to avoid penalty.

(5) Based on the simulation analysis results, taking the use of an open flame or smoking on aircraft as an example, the regulatory authority should increase their investment in airline security, improve the emergency disposal capabilities of cabin security personnel through legislation, security testing, and other means, and set up a reasonable reward and punishment mechanism to improve the initiative and willingness of airlines and airport police. Airlines, as the first line of disposing a cabin disturbance, should strengthen the training and inspection of cabin security personnel, improve their ability to coordinate emergency disposal and dispose of disruptive passengers’ disturbance, and increase their efforts to dispose of disruptive passengers who break the law to reduce the occurrence of cabin disturbances. Airport police should enhance their ability to coordinate and cooperate with cabin security personnel, improve the deterrence of onboard and ground actions against disruptive behavior, and promote the development of civil aviation security in China and the improvement of emergency disposal by airlines.

6.2. Recommendations

In view of the importance of the emergency disposal of cabin disruptions, various measures can be taken to improve the emergency disposal capability and overall safety level of cabin disruptions in order to promote the improvement of China’s civil aviation security and airlines’ emergency disposal capability, and provide effective guarantee for ensuring aviation security.

First, the regulatory authority needs to increase the investment in airline security and establish a reasonable reward and punishment mechanism. This will motivate airlines to strengthen security measures, invest more resources to train and equip highly qualified security personnel, and provide advanced security equipment and technology. The regulatory authority should also conduct regular security audits and assessments to ensure that airlines comply with relevant security standards and regulations.

Secondly, airlines should enhance the comprehensive capabilities of cabin security personnel, including skills training, such as physical protection, emergency handling and crisis management, to ensure that security personnel have the necessary skills and knowledge to dispose of disruptive incidents. In addition, emphasis should be placed on psychological training so that security personnel can respond calmly in stressful situations. Airlines should also organize regular drills and simulation training to improve security personnel’s emergency disposal and disposal capabilities.

At the same time, airport police need to strengthen the coordination and cooperation ability with cabin security personnel. They should establish a close cooperation mechanism to ensure smooth information flow and a rapid response so that timely action can be taken in case of disruptions or unexpected situations. Airport police should also provide the necessary support and resources, such as equipment and manpower, to help cabin security personnel better perform their duties.

In summary, by increasing investments and establishing reward and punishment mechanisms by the regulatory authority, improving the comprehensive capabilities of security personnel by airlines, and strengthening coordination and cooperation capabilities by airport police, the emergency disposal capability and overall safety level of cabin disruptions can be improved, which will help safeguard the security of China’s civil aviation and provide safer and more reliable aviation services to passengers.

6.3. Limitations and Future Research

Several limitations are worth noting, which may provide ideas for the future research direction. First, one of the most common assumptions of the fourth-party evolutionary game model of cabin disturbance emergency disposal in this paper is that the participants are finite rational subjects. However, in real cases, there are inevitably non-finite rational participants due to the differences in emergency disposal. Future research can further refine the model’s parameters to consider the behavior of non-finite rational participants and narrow the discrepancy between the theoretical model and the actual case. Second, the data in this paper are from China only, and data from other countries will be studied subsequently. This paper does not consider the security configurations of other countries. Therefore, our subsequent research direction will be constructing evolutionary game models of other security-related subjects and making innovative suggestions to explore globally relevant security strategies.