An evolutionary game study of traffic safety risk regulation considering public participation behavior

Abstract

With the development of transportation technology, the improvement of transportation efficiency also increases the complexity and diversity of traffic safety risks; the public’s concern for traffic safety risks is increasing, and the traditional regulatory model is difficult to meet the improvement of the effectiveness of the safety regulation of the huge transportation network and the public’s changing safety needs. In order to enhance the effectiveness of safety regulation, a game model of safety risk regulation among the public, transport enterprises, and regulators is constructed to simulate the effects of different traffic risks, penalty levels, and exposure rates on the behavior of transport enterprises and regulators. The results of the study show that: traffic risk has a positive facilitating effect on the participation of transport enterprises and the public in the regulation of safety risks, which will accelerate the process of stabilization of their strategies; transport enterprises have a high sensitivity to changes in the level of punishment, and only when the level of punishment exceeds a certain threshold will they be able to promote the choice of complying with the transport; and a higher exposure will lead to a greater tendency for transport enterprises to choose complying with the transport.

Introduction

With sustained economic growth and rising transport demand, the road transport industry plays a crucial role in promoting economic development and meeting people’s travel needs. But at the same time, according to data published by the World Health Organization, road traffic accidents are now the seventh leading cause of death globally for all age groups. Therefore, while transport systems are constantly being upgraded and optimized, safety regulation therein remains crucial, especially as overloading and speeding in the transport of goods and long-distance passenger transport have become one of the most prominent issues in road traffic risk. In 2021, the World Health Organization (WHO) issued the “Decade of Action for Road Safety Global Plan,” advocating for countries to take proactive measures and utilize an integrated “safety system” that regards road safety as a key driver for sustainable development. This system encompasses roads, vehicles, and road traffic safety behaviors to collaboratively achieve the goal of reducing global road traffic fatalities and injuries by at least 50% by 2030¹. According to relevant statistics, China’s traffic management authorities have investigated more than a million cases of overloaded transportation between 2019 and 2021, jeopardizing road safety and the long-term development of the transportation industry, with overloaded vehicles nearly three times more likely to be involved in accidents than normally loaded vehicles. When trucks are overloaded, the ability to merge into lanes, avoid obstacles, and change lanes will be drastically reduced, and the braking efficiency will accelerate the decline, which is easy to cause traffic accidents². The collision of overloaded trucks with other vehicles of small mass increases the probability of casualties. Overloading in cargo transportation not only shortens the service life of roads and increases maintenance costs, but may also lead to a decrease in vehicle braking performance and increase the probability of accidents. In addition, damage to road surfaces caused by overloaded vehicles in the course of traveling³ creates safety hazards and threatens the safety of other road users. Subjective transportation violations by transporters lead to frequent problems of road safety risks and increased pressure for green and low-carbon transformation of transportation⁴. Overloading of long-distance passenger transportation is equally dangerous, as it not only increases the instability of the vehicle, but may also lead to difficulties in evacuating passengers in case of emergency and increase the casualty rate of accidents. Speeding further amplifies these risks, as increased speed reduces driver reaction time and increases the likelihood and severity of accidents. With this in mind, this paper will focus on the risk of overloading and speeding in cargo and long-distance passenger transportation, and explore how public regulation can improve transportation safety. Against this background, the public are no longer just the beneficiaries of the transport system, but they also bear the responsibility of monitoring and reporting overloading and speeding as other non-operational categories of participants in road traffic, such as drivers of private cars, holders of small car licenses and passengers. Their involvement is critical to the timely detection and cessation of these violations. Transport enterprises, including all types of large-scale freight and passenger transport carriers, as direct participants in road transport, their actions have a direct impact on road safety. Transportation enterprises need to take responsibility for compliant transportation, ensure that vehicles are not overloaded and speeding, and safeguard the safety of passengers and goods. Regulatory authorities, which mainly include official supervisory and law enforcement agencies such as the Road Enforcement Police (REP) and the Traffic Control Department (TCD), who are responsible for enacting and enforcing traffic laws and regulations, and imposing penalties for non-compliance with these laws and regulations. However, limited regulatory resources make it difficult for the regulator to fully cover all road transport activities. The public is often overlooked for its important regulatory role in the supervision of traffic safety risks. The public can help the regulators to discover potential safety risks in time through reporting and complaints, and also supervise the behavior of the regulators, enhance the sense of responsibility and pressure on the regulators and transport enterprises, promote the improvement of traffic safety and regulatory work, and better protect the safety of all kinds of participants in the transportation system, so the public’s participation is of great significance in bridging the regulatory gap and improving regulatory efficiency.

Based on this, this paper analyzes the interests, decision-making basis and synergies among the three major stakeholders, namely, the public, transport enterprises and regulators, in the process of traffic safety risk supervision, and studies the role of public participation in the supervision of traffic safety risk, the transport enterprises consciously fulfilling their responsibility of complying with the transport, and enhancing their enthusiasm in practicing complying transport, and explores the stability of the three-party synergism to enhance the safety of transportation. It is of great theoretical and practical significance to realize traffic safety and promote the sustainable development of transportation.

Literature review

Road transportation is a large part of the smart transportation system and the focus of the practice of safety risk regulation in smart transportation, in which safety risk regulation is the key to protect people and goods from injuries and reduce supply disruptions during transportation on the road⁵. In the study of road risk, Ye et al.(2023)⁶ investigated the intrinsic formation mechanism of dangerous riding behaviors of takeaway electric riders at urban intersections, and showed that the adoption of mandatory restraints is extremely important in reducing and preventing riders’ dangerous behaviors. Petrov et al.(2023)⁷ measured the probability of traffic accidents in three types of urban road safety systems in Russia in terms of relative entropy, and the coefficient of road users’ sense of danger varied in the opposite direction to the level of road safety. Jafarzadeh et al.(2023)⁸ used a combination of SWARA and MARCOS in a spherical fuzzy environment to assess road risk and found that anthropogenic sources of risk are important factors contributing to road accidents. Ahamd et al.(2024)⁹ found that speeding, drowsiness, and frontal collisions due to driving in the opposite direction are the main risk factors contributing to the increased probability of serious injuries in traffic accidents. Yang et al.(2022)¹⁰ studied the mechanism of highway collision occurrence and collision precursor identification based on traffic flow. A series of measures such as highway digitization and intelligent transportation systems have been adopted in road design and safety supervision¹¹. The introduction of intelligent transportation systems helps to improve transportation efficiency, traffic safety, and the use of transportation resources, integrates transportation participants and related facilities into an interconnected network, improves the effectiveness of safety supervision, and reduces the dangers of intelligent transportation¹². Chen et al.(2021)¹³ examined the association of pseudo-tense time delivery schedules of U.S. truck drivers with their unsafe driving behaviors and perceptions of safety needs, and the construction of more stops would be effective in improving safety. Michelaraki et al.(2023)¹⁴, Singh et al.(2021)¹⁵ showed that driver distraction or inattention largely reduces their level of perception as well as their decision-making and vehicle control. Boua et al.(2022)¹⁶, Domenichini et al.(2019)¹⁷ investigated the relationship between control beliefs, risk perception, and road safety behaviors, and that drivers with a high level of control awareness of road risks tended to perceive these risks less and adopt fewer safety behaviors. Mohammadfam et al.(2020)¹⁸ used Bayesian network and fuzzy inference system to assess the quantitative risk in road transportation of hazardous materials and the combination of Bayesian network and fuzzy inference system can be used as an accurate tool to assess the quantitative risk of hazardous chemicals transportatio. Cafiso et al.(2013)¹⁹ assessed the awareness and perceptions of bus managers on safety issues and the introduction of new technologies is considered as an important factor in improving the safety of public transportation. Alemdar et al.(2023)²⁰ discussed the negative scenarios due to drivers waiting for their cell phones at traffic lights and calculated the delays at selected intersections due to cell phone use. In the study of road safety transport, Parviziomran et al.(2023)²¹ developed a time-varying model to simulate the cost of decarbonization of the heavy road transport sector, and the study concluded that the charging strategy road transport sustainability transition consists of an important contribution. Nævestad et al.(2018)²² developed an environmental small road transport company to develop an OSM strategy from traditional factors (vehicle, driver, road, etc.) with organizational and managerial factors to improve the potential of transport safety. Barreno et al.(2022)²³ developed a fuzzy expert system to assess the state of a vehicle on a two-lane road, and a fuzzy risk index was proposed to assist in the adjustment of road speed limits. Thompson et al.(2020)²⁴ investigated the relationship between city type and road traffic injury rates, and the transportation networks of different types of cities differed in several dimensions, and the design of urban transportation networks was closely related to road transport injuries. Ahmad et al.(2023)²⁵ compared game models of evolutionary game theory in traffic management, route operation and transportation safety, and evolutionary games have good performance in studying the behavior and decision making of transportation systems. Adler et al.(2021)²⁶ discussed the methods and techniques of modeling based on the transportation market, and constructed a more general and effective transportation game model. Besselink et al.(2016)²⁷ proposed a cyber-physical approach for controlling and coordinating large fleets of heavy vehicles to optimize transportation planning and vehicle routing, effectively reduce freight fuel consumption, and improve transportation efficiency and supply chain low-carbon levels²⁸ and sustainability. Ogryzek et al.(2021)²⁹ considered different stages of public participation in decision-making and proposed a model solution based on an integrated approach to public participation in road infrastructure planning for smart cities within sustainable transportation systems, where public participation has a positive impact on road planning and investment formulation. Zhao et al.(2022)³⁰ emphasized the value of public participation in transportation infrastructure projects, shifted from unidimensional assessment to multidimensional assessment, constructed a holistic assessment framework based on VFM, and calibrated the assessment results of transportation infrastructure projects.

Through a review of existing research, studies on road transport risk and safety in transportation have focused on risk factor identification³¹, driver behavioral characteristics³², and the application of emerging technologies in transportation network design and route optimization to achieve the improvement of transportation safety or the purpose of sustainable transformation. Most of the studies analyze the interests of transportation participants in terms of traditional factors, and there is not enough research on safety risk regulation and public participation in safety risk regulation. The innovations of this paper are (1) Consider the public’s involvement behavior in traffic safety risks, and examine the impact of public exposure to transport companies’ non-compliant transport behavior on the subject’s strategy choices. (2) In the area of public participation in transportation regulation, different combinations of strategies for intelligent transportation safety risk regulation among the public, transportation companies and regulators were studied. (3) Focusing on the effects of traffic risk, public exposure, punishment, and additional benefits on system evolution, it provides a reference for the optimization of traffic safety risk regulation in reality.

Problem description and hypothesis

In road transport, transport companies play an increasingly important role, however, with the increase in the scale of traffic and the complexity of road conditions, the frequency of traffic accidents, the safety of transport further attracts attention, and to strengthen the order and compliance of the road transport network is a viable option to effectively reduce the risk of road transport. In reality, on the one hand, transport enterprises may adopt some non-compliant transport behaviors in order to pursue more economic benefits. For example, reducing regular vehicle maintenance can save money, and overloading transportation can improve the efficiency of a single transportation, but serious overloading may trigger serious accidents such as tire blowouts, sudden veering, brake failures, and rollovers, which undermine road safety. On the other hand, regulators are mainly official agencies responsible for enacting, enforcing and monitoring traffic regulations, such as the traffic police and road transport authorities. These include, but are not limited to, enacting traffic regulations, enforcing traffic inspections, penalizing violations, conducting road safety education and campaigns, and supervising the compliance of transportation companies in order to ensure a safe and orderly transportation system. However, the regulatory authority is faced with the problems of limited regulatory resources, lagging regulatory tools and lax regulation. In addition, there may be financial interests or human resources influences between the regulator and the transportation companies, resulting in the regulation not being strictly enforced as required. Therefore, relying only on the regulatory authorities to regulate the non-compliance behavior in transport will inevitably lead to ineffective or negligent regulation. Meanwhile, according to the relevant theory of public choice³³, the behavior of law enforcement agencies is influenced by public interest and pressure. The public refers to all non-operating class participants involved in road traffic, including private car drivers, passengers, and pedestrians. The public can improve traffic safety by reporting violations, participating in road safety education, and monitoring the compliance of regulatory authorities and transportation enterprises. Transportation enterprises are commercial entities engaged in the transportation of goods or passengers, including large trucks and long-distance buses, etc. Transportation enterprises should ensure compliance of their operations, including vehicle maintenance, driver training, and compliance with traffic regulations. Non-compliance behavior mainly refers to overloading (exceeding the load limit specified for the vehicle) and speeding (exceeding the maximum speed limit specified for the road). Strict regulation means that the regulator conducts regular traffic inspections, strictly enforces traffic laws and regulations, imposes timely penalties for non-compliance, and has transparent regulatory records and feedback mechanisms. Non-strict regulation refers to infrequent inspections by the regulator, untimely or lax penalties for violations, non-transparent regulatory records, and lack of effective feedback and improvement mechanisms. Active public participation in monitoring can increase the probability of detection and exposure of non-compliant behaviors, thereby forcing government regulators to adopt more stringent regulatory measures in response to public expectation and pressure. Public scrutiny is a good facilitator of CSR fulfilment³⁴, and on another level it motivates transport companies to maintain compliant transport. Therefore, this study is a two-way regulation of regulators and transportation companies by introducing public participation in the regulation of traffic safety risks, and the logical relationship between the three-way evolutionary game subjects of traffic safety risk regulation constructed in this paper is shown in Fig. 1.

Fig. 1

Logical relationship between the subjects of the three-party evolutionary game.

Due to the existence of more uncertainties in the supervision of traffic safety risks, in order to further clarify the relevant issues, this paper puts forward the following hypotheses in conjunction with the real situation:

Hypothesis 1

The subjects of the evolutionary system of traffic safety risk supervision considering public participation behavior include transportation enterprises, regulatory authorities and the public. The participating subjects are all limited rational individuals with different interests and different strategy choices over time, and the final strategy choices gradually evolve over time and stabilize at the optimal strategy.

Hypothesis 2

Evolutionary game theory suggests that under high regulatory intensity, the cost of non-compliance increases, and in the long run, non-compliant transport firms will be penalized more and gradually eliminated from the market or forced to shift to compliant operations. Stringent regulation will therefore push transport companies to opt more for compliant operations to avoid the high cost of non-compliance. Public participation can mitigate information asymmetry and increase information transparency between government regulators and transport companies to improve road safety, and public participation behavior influences and is influenced by the behavior of the other two parties. Based on this, it is assumed that the public will expose the transport companies’ transport violations for the sake of their own travel safety, and the exposure rate is $\mu (0 \leqslant \mu \leqslant 1)$, which will affect the public’s participation behaviors. The probability that public $P$ chooses to participate in transportation regulation is $\alpha$$(0 \leqslant \alpha \leqslant 1)$, and the probability that it does not is $(1 - \alpha )$. The benefit of public participation in road transport regulation is ${R_{p1}}$, the benefit of non-participation in regulation is ${R_{p2}}$, if the transport enterprises are allowed to overloading and other non-compliant transport behavior, not to be strictly regulated, it will accelerate the quality of traffic roads, bridges and other transport infrastructure to produce loss, but also due to the excess transport to produce more pollution emissions, affecting the quality of highway infrastructure, affecting the normal order of the traffic, and increase the risk of traffic. When transport companies are not penalized for non-compliant transport, the exposure of the public to traffic risks and threats to physical and mental health is ${C_p}$. ${C_p}$ is the loss caused by public inaction. The greater the loss value, the higher the participation. The direct benefits of participating in regulation ${R_{p1}}$ include increased travel safety and reduced potential harm from non-compliant behavior. If one does not participate, the gain is ${R_{p2}}$, which avoids the time and effort costs of participation, but incurs the loss ${C_p}$ that might exist if one does not participate. Public choice theory suggests that an individual’s behavior in public affairs is similarly governed by the principle of the rational economic man, pursuing the maximization of personal interests, and the theory of loss aversion in behavioral economics states that people are usually more sensitive to losses than they are to gains. In this scenario, the motivation for public participation is to reduce traffic risks and health threats to individuals from non-compliant behaviors of transport companies. Thus, when ${C_p}$ is higher, the public may be more inclined to actively engage in regulation to reduce their own potential losses.

Hypothesis 3

The probability of compliant transport is $\beta (0 \le \beta \le 1)$, and the probability of non-compliant transport is $(1 - \beta )$. When non-compliant transport is carried out, the transport enterprise will carry out regulatory capture behavior to the regulatory authorities. The revenue generated by compliance transportation is ${R_{t1}}$, and the input cost is ${C_{t1}}$. The additional revenue generated by non-compliant transport is ${R_{t2}}$, the cost during non-compliant transport is ${C_{t2}}$(${C_{t2}}<{C_{t1}}$), the traffic risk increased by non-compliant transport is ${V_{t2}}$, the cost of colluding with the regulatory authorities is ${C_c}$, the fine imposed by the regulatory authorities is $\theta {V_{t2}}$ ($\theta$ is the penalty imposed by the regulatory authorities on the transport enterprises), and the reputation loss exposed by the transport enterprises due to non-compliant transport is ${B_t}$.

Hypothesis 4

The probability of strict supervision by the regulatory department is $\gamma$$(0 \leqslant \gamma \leqslant 1)$, and the probability of lax supervision is $1 - \gamma$. Under strict supervision, the regulatory department will punish the transport enterprises that violate the rules of transportation, and accept the regulatory capture of the transport enterprises when implementing lax supervision. The benefit of strict supervision by the regulatory department is ${R_s}$, the cost of supervision is ${C_s}$, and the environmental benefit obtained by strict supervision is $W$; When the supervision is relaxed, the transportation enterprises carry out regulatory capture to the regulatory authorities, and the additional revenue obtained by the regulatory authorities is ${R_{bs}}$, which is exposed by the public that the regulatory authorities accept the regulatory capture of the transportation enterprises and are punished by the higher authorities as ${B_s}$.

Model construction and analysis

Based on the problem description and assumptions in the preceding chapters, the parameters and variables involved in the model of this paper are determined, and the constraints between the three subjects are sorted out. In this chapter, the evolutionary game model of this paper is gradually constructed from the interest payment matrix of the public, transportation enterprises, and regulators to the replication of the dynamic system of equations and stability analysis.

Payment matrix

According to the above problem description and parameter setting, the interest demands and restriction relationship between the subjects in the evolutionary system are further clarified. The public puts forward safety expectations to the regulatory authorities and transportation enterprises from the safety needs as the starting point and supervises the behavior of the subjects in this process. Meanwhile, the behavior of the public is also affected by the strategic choice of the transportation enterprises and the regulatory authorities. Therefore, in order to further analyze the evolutionary game process among the three, a tripartite game income matrix of transport safety co-governance by the public, transport enterprises, and regulatory authorities is constructed, as shown in Table 1.

Table 1 Payment matrix.

The expected utility functions of the three agents in the evolutionary system are as follows:

(1) The expected revenue of the public when it chooses to participate in the regulation is:

$${\pi _{P1}}={R_{p1}} - (1 - \beta )(1 - \gamma )(1 - \mu ){C_p}$$

(1)

The expected benefits when the public opts out of regulation are:

$${\pi _{P2}}={R_{P2}} - (1 - \beta )(1 - \gamma ){C_P}$$

(2)

The average expected benefit of the public is:

$${\bar {\pi }_p}=\alpha {\pi _{p1}}+(1 - \alpha ){\pi _{p2}}=\alpha {R_{p1}}+(1 - \alpha ){R_{p2}} - (1 - \mu )(1 - \beta )(1 - \alpha \mu ){C_p}$$

(3)

The public replication dynamic equation is:

$$\begin{gathered} {G_p}(\alpha ,\beta ,\gamma )=\frac{{d\alpha }}{{dt}}=\alpha ({\pi _{p1}} - {{\bar {\pi }}_p}) \hfill \\ =\alpha ({\pi _{p1}} - (\alpha {\pi _{p1}}+(1 - \alpha ){\pi _{p2}}) \hfill \\ =\alpha (1 - \alpha )({\pi _{p1}} - {\pi _{p2}}) \hfill \\ =\alpha (1 - \alpha )({R_{P1}} - {R_{P2}}+(1 - \beta )(1 - \gamma )\mu {C_P}) \hfill \\ \end{gathered}$$

(4)

(2) The expected benefits of compliant transportation for transportation enterprises are:

$$\begin{gathered} {\pi _{t1}}=\alpha \gamma ({R_{t1}} - {C_{t1}})+\gamma (1 - \alpha )({R_{t1}} - {C_{t1}})+(1 - \gamma )\alpha ({R_{t1}} - {C_{t1}})+(1 - \gamma )(1 - \alpha )({R_{t1}} - {C_{t1}}) \hfill \\ ={R_{t1}} - {C_{t1}} \hfill \\ \end{gathered}$$

(5)

The expected revenue of non-compliant transportation of transportation enterprises is:

$${\pi _{t2}}={R_{t1}}+{R_{t2}} - {C_{t2}} - (1 - \gamma ){C_c} - \gamma \theta {V_{t2}} - \alpha \mu {B_t}$$

(6)

The average expected revenue of transportation enterprises is:

$${\bar {\pi }_t}=\beta {\pi _{t1}}+(1 - \beta ){\pi _{t2}}={R_{t1}}+(1 - \beta )({R_{t2}} - {C_{t2}} - (1 - \gamma ){C_c} - \alpha \mu {B_t} - \gamma \theta {V_{t2}})+\beta {C_{t1}}$$

(7)

The replication dynamic equation of transport enterprise is as follows:

$$\begin{gathered} {G_t}(\alpha ,\beta ,\gamma )=\frac{{d\beta }}{{dt}}=\beta ({\pi _{t1}} - {{\bar {\pi }}_t}) \hfill \\ =\beta (1 - \beta )({\pi _{t1}} - {\pi _{t2}})=\beta (1 - \beta )( - {C_{t1}} - {R_{t2}}+{C_{t2}}+(1 - \gamma ){C_c}+\alpha \mu {B_t}+\gamma \theta {V_{t2}}) \hfill \\ \end{gathered}$$

(8)

The expected benefits of strict supervision by the regulatory authorities are:

$${\pi _{s1}}=W+{R_s} - {C_s}+(1 - \beta )\theta {V_{t2}}$$

(9)

The expected benefits of the regulatory authorities’ lax supervision are as follows:

$${\pi _{s2}}={R_S}+(1 - \beta ){R_{bs}} - {C_s} - (1 - \beta )\alpha \mu {B_s}$$

(10)

The average expected revenue of the regulatory authorities is:

$${\bar {\pi }_s}=\gamma {\pi _{s1}}+(1 - \gamma ){\pi _{s2}}=\gamma W+\gamma (1 - \beta )\theta {V_{t2}}+(1 - \beta )(1 - \gamma ){R_{bs}} - (1 - \beta )(1 - \gamma )\alpha \mu {B_s}+{R_s} - {C_s}$$

(11)

The regulatory authority’s replication dynamic equation is:

$$\begin{gathered} {G_s}(\alpha ,\beta ,\gamma )=\frac{{d\gamma }}{{dt}}=\gamma ({\pi _{s1}} - {{\bar {\pi }}_s})=\gamma (1 - \gamma )({\pi _{s1}} - {\pi _{s2}}) \hfill \\ =\gamma (1 - \gamma )(W+(1 - \beta )\theta {V_{t2}}+(1 - \beta )\alpha \mu {B_s} - (1 - \beta ){R_{bs}}) \hfill \\ \end{gathered}$$

(12)

The replicated dynamic equations of the evolutionary game are differential equations describing the change of the proportion of adopting a particular strategy among the subjects over time, and are composed by Eqs. (4–4) (4–8) (4–12) to form a set of replicated dynamic equations for the analysis of the three-party evolutionary game in the regulation of traffic safety risks considering the behavior of public participation:

$$\left\{ {\begin{array}{*{20}{c}} {{G_p}(\alpha ,\beta ,\gamma )=\frac{{d\alpha }}{{dt}}=\alpha (1 - \alpha )({R_{P1}} - {R_{P2}}+(1 - \beta )(1 - \gamma )\mu {C_P})} \\ {{G_t}(\alpha ,\beta ,\gamma )=\frac{{d\beta }}{{dt}}=\beta (1 - \beta )( - {C_{t1}} - {R_{t2}}+{C_{t2}}+(1 - \gamma ){C_c}+\alpha \mu {B_t}+\gamma \theta {V_{t2}})} \\ {{G_s}(\alpha ,\beta ,\gamma )=\frac{{d\gamma }}{{dt}}=\gamma (1 - \gamma )(W+(1 - \beta )\theta {V_{t2}}+(1 - \beta )\alpha \mu {B_s} - (1 - \beta ){R_{bs}})} \end{array}} \right.$$

(13)

Stability analysis of game players

The first derivation of the replication dynamic equation of the public, transport enterprises and regulatory authorities is obtained:

$$\frac{{d{G_p}(\alpha ,\beta ,\gamma )}}{{d\alpha }}=(1 - 2\alpha )({R_{P1}} - {R_{P2}}+(1 - \beta )(1 - \gamma )\mu {C_P})$$

(14)

$$\frac{{d{G_t}(\alpha ,\beta ,\gamma )}}{{d\beta }}=(1 - 2\beta )( - {C_{t1}} - {R_{t1}}+{C_{t2}}+(1 - \gamma ){C_c}+\alpha \mu {B_t}+\gamma \theta {V_{t2}})$$

(15)

$$\frac{{d{G_s}(\alpha ,\beta ,\gamma )}}{{d\gamma }}=(1 - 2\gamma )(W+(1 - \beta )\theta {V_{t2}}+(1 - \beta )\alpha \mu {B_s} - (1 - \beta ){R_{bs}})$$

(16)

Set:

$$F(\gamma )=({R_{P1}} - {R_{P2}}+(1 - \beta )(1 - \gamma )\mu {C_P})$$

(17)

$$F(\alpha )=( - {C_{t1}} - {R_{t1}}+{C_{t2}}+(1 - \gamma ){C_c}+\alpha \mu {B_t}+\gamma \theta {V_{t2}})$$

(18)

$$F(\beta )=(W+(1 - \beta )\theta {V_{t2}}+(1 - \beta )\alpha \mu {B_s} - (1 - \beta ){R_{bs}})$$

(19)

To achieve a steady state of public policy choice, ${G_p}(\alpha ,\beta ,\gamma )=0$ and $\frac{{d{G_p}(\alpha ,\beta ,\gamma )}}{{d\alpha }}<0$ are required. Since $F(\gamma )$ is an increasing function, when $\gamma =1+\frac{{{R_{p1}} - {R_{p2}}}}{{(1 - \beta )\mu {C_p}}}$, there is $F(\gamma )=0$ and $\frac{{d{G_p}(\alpha ,\beta ,\gamma )}}{{d\alpha }} \equiv 0$, the public cannot be sure of its stability strategy at this time. When $\frac{{d{G_p}(\alpha ,\beta ,\gamma )}}{{d\alpha }}\left| {_{{\alpha =0}}} \right.<0$, $F(\gamma )<0$, that is, $\gamma >1+\frac{{{R_{p1}} - {R_{p2}}}}{{(1 - \beta )\mu {C_p}}}$, $\alpha =0$ is the evolutionary stable point, and the public’s choice not to participate in supervision is its stable strategy. When $\frac{{d{G_p}(\alpha ,\beta ,\gamma )}}{{d\alpha }}\left| {_{{\alpha =1}}} \right.<0$, $F(\gamma )>0$, that is, $\gamma <1+\frac{{{R_{p1}} - {R_{p2}}}}{{(1 - \beta )\mu {C_p}}}$, $\alpha =1$ is the evolutionary stable point, and the public chooses to participate in supervision as its stability strategy. The phase diagram of public policy evolution is shown in Fig. 2 (a).

The $\alpha$ corresponding to all points of the section in the phase diagram (Fig. 2 (a)) is the stability strategy of the public, and the $\alpha$ on the upper side of the section tends to be $\alpha =1$, and the stability strategy of the public tends to participate in supervision. When the government has relaxed supervision, the transport enterprises have not complied with the transportation regulations, and the transport enterprises have not been punished for violating the transportation regulations, the loss of traffic risks and physical and mental health threats to the public is reduced, The stability strategy of the public will change to unsupervised, that is, the lower part of the cross-section, and the evolutionary stability point tends to $\alpha =0$.

To achieve the steady state of the transport enterprise’s policy choice, ${G_t}(\alpha ,\beta ,\gamma )=0$ and $\frac{{d{G_t}(\alpha ,\beta ,\gamma )}}{{d\beta }}<0$ are required. Since $F(\alpha )$ is a decreasing function, when $\alpha =1+\frac{{{C_{t1}}+{R_{t2}} - {C_{t2}} - (1 - \gamma ){C_c} - \gamma \theta {V_{t2}}}}{{\mu {B_t}}}$, $F(\alpha )=0$, $\frac{{d{G_t}(\alpha ,\beta ,\gamma )}}{{d\beta }} \equiv 0$, then the transportation enterprise can not determine its stability strategy. When $\frac{{d{G_t}(\alpha ,\beta ,\gamma )}}{{d\beta }}\left| {_{{\beta =1}}} \right.<0$, there is $F(\alpha )>0$, that is $\alpha >1+\frac{{{C_{t1}}+{R_{t2}} - {C_{t2}} - (1 - \gamma ){C_c} - \gamma \theta {V_{t2}}}}{{\mu {B_t}}}$, $\beta =1$ is the point of evolutionary equilibrium. And compliance transportation is the stable strategy of transportation enterprises. When $\frac{{d{G_t}(\alpha ,\beta ,\gamma )}}{{d\beta }}\left| {_{{\beta =0}}} \right.<0$, $F(\alpha )<0$, that is, $\alpha <1+\frac{{{C_{t1}}+{R_{t2}} - {C_{t2}} - (1 - \gamma ){C_c} - \gamma \theta {V_{t2}}}}{{\mu {B_t}}}$, $\beta =0$ is the evolutionary stability point, and the transportation enterprise chooses non-compliant transportation as its stability strategy. The strategy evolution phase diagram of the constructed transportation enterprise is shown in Fig. 2 (b).

In the phase diagram (Fig. 2 (b)), the $\beta$ corresponding to all points of the section is the stability strategy of the transport enterprise, and the $\beta$ at the lower side of the section tends to be $\beta =0$, and the stability strategy of the transport enterprise tends to choose non-compliant transportation. If the additional revenue of non-compliant transportation of transport enterprises decreases, the government strictly supervises, the punishment is intensified, the public participates in supervision, and the probability of exposing non-compliant transportation is increased, the strategic choice of transport enterprises will turn to compliant transportation, that is, the upper part of the cross-section, and the evolutionary stability point tends to be $\beta =1$.

To achieve the stable state of the regulator’s policy selection, ${G_s}(\alpha ,\beta ,\gamma )=0$ and $\frac{{d{G_s}(\alpha ,\beta ,\gamma )}}{{d\gamma }}<0$ are required. Since $F(\beta )$ is a decreasing function, when $\beta =1+\frac{W}{{\theta {V_{t2}}+\alpha \mu {B_s} - {R_{ts}}}}$, $F(\beta )=0$, $\frac{{d{G_s}(\alpha ,\beta ,\gamma )}}{{d\gamma }} \equiv 0$, then the regulator can not determine its stability strategy. When $\frac{{d{G_s}(\alpha ,\beta ,\gamma )}}{{d\gamma }}\left| {_{{\gamma =1}}} \right.<0$, $F(\alpha )>0$, that is, $\beta <1+\frac{W}{{\theta {V_{t2}}+\alpha \mu {B_s} - {R_{ts}}}}$, $\gamma =1$ is the evolutionary stability point, and the regulatory authorities choose strict supervision as the evolutionary stability strategy. When $\frac{{d{G_s}(\alpha ,\beta ,\gamma )}}{{d\gamma }}\left| {_{{\gamma =0}}} \right.<0$, $F(\gamma )<0$, that is, $\beta >1+\frac{W}{{\theta {V_{t2}}+\alpha \mu {B_s} - {R_{ts}}}}$, $\gamma =0$ is the evolutionary stability point, and the regulatory department chooses loose supervision as its stability strategy. The phase diagram of the strategy evolution of the constructed regulatory department is shown in Fig. 2 (c).

In the phase diagram (Fig. 2 (c)), the $\gamma$ corresponding to all points on the cross section is the stability strategy of the transportation enterprise, and the $\gamma$ on the upper side of the cross section tends to be $\gamma =0$, and the stability strategy of the regulatory department tends to choose loose supervision. If the regulatory capture of transport enterprises brings less additional revenue to the regulatory authorities, the public exposes that the regulatory authorities accept the regulatory capture of transport enterprises and thus receive more penalties from the higher authorities, and the government imposes more penalties on non-compliant enterprises under strict supervision, the regulatory authorities will shift their strategic choice to strict supervision, that is, the lower part of the cross-section, and the evolutionary stability point tends to $\gamma =1$.

Fig. 2

Phase diagram of tripartite agent strategy evolution.

Stability analysis of equilibrium point of tripartite evolutionary game system

Based on the principle of maximizing each player’s own interests, the players of evolutionary game adjust their own strategies according to the strategies of other players, and finally make the strategies of participating players tend to be stable, which is called the stable strategy of evolutionary game (ESS). ESS is the subset of equilibrium points of the corresponding replicating dynamic equation set. In order to obtain the equilibrium point when the public, transportation enterprises and regulatory authorities all reach the stable state of the evolutionary game, so that ${G_p}(\alpha ,\beta ,\gamma )=0$, ${G_t}(\alpha ,\beta ,\gamma )=0$, ${G_s}(\alpha ,\beta ,\gamma )=0$, to obtain the eight equilibrium points of the replicated dynamic equations, ${E_1}(0,0,0)$, ${E_2}(1,0,0)$, ${E_3}(0,1,0)$, ${E_4}(0,0,1)$,${E_5}(1,1,0)$, ${E_6}(1,0,1)$, ${E_7}(0,1,1)$, ${E_8}(1,1,1)$.

The Jacobi matrix corresponding to the tripartite evolutionary game system is:

$\begin{gathered} J=\left[ {\begin{array}{*{20}{c}} {\frac{{\partial {G_p}(\alpha ,\beta ,\gamma )}}{{\partial \alpha }}}&{\frac{{\partial {G_p}(\alpha ,\beta ,\gamma )}}{{\partial \beta }}}&{\frac{{\partial {G_p}(\alpha ,\beta ,\gamma )}}{{\partial \gamma }}} \\ {\frac{{\partial {G_t}(\alpha ,\beta ,\gamma )}}{{\partial \alpha }}}&{\frac{{\partial {G_t}(\alpha ,\beta ,\gamma )}}{{\partial \beta }}}&{\frac{{\partial {G_t}(\alpha ,\beta ,\gamma )}}{{\partial \gamma }}} \\ {\frac{{\partial {G_s}(\alpha ,\beta ,\gamma )}}{{\partial \alpha }}}&{\frac{{\partial {G_s}(\alpha ,\beta ,\gamma )}}{{\partial \gamma \beta }}}&{\frac{{\partial {G_s}(\alpha ,\beta ,\gamma )}}{{\partial \gamma }}} \end{array}} \right] \hfill \\ = \hfill \\ \left[ {\begin{array}{*{20}{c}} {(1 - 2\alpha )(({R_{P1}} - {R_{P2}}+(1 - \beta )(1 - \gamma )\mu {C_P}))}&{ - \alpha (\alpha - 1)(\gamma - 1)\mu {C_P}}&{ - \alpha (\alpha - 1)(\beta - 1)\mu {C_P}} \\ {\mu \beta (1 - \beta ){B_t}}&{(1 - 2\beta )({C_{t2}} - {C_{t1}} - {R_{t2}}+(1 - \gamma ){C_c}+\alpha \mu {B_t}+\gamma \theta {V_{t2}})}&{\beta (1 - \beta )(\theta {V_{t2}} - {C_c})} \\ {\alpha (1 - \alpha )(1 - \beta )\mu {B_s}}&{\gamma (\gamma - 1)(\theta {V_{t2}}+\alpha \mu {B_t} - {R_{bs}})}&{(1 - 2\gamma )(W+(1 - \beta )(\theta {V_{t2}}+\alpha \mu {B_s} - {R_{bs}})} \end{array}} \right] \hfill \\ \end{gathered}$

According to Lyapunov’s first method, if all the eigenvalues of the Jacobi matrix have non-positive real parts, then the equilibrium point is asymptotically stable. Therefore, the asymptotic stability of the equilibrium point can be judged by the eigenvalues of the Jacobi matrix. If all the eigenvalues have negative real parts, the equilibrium point is the evolutionary equilibrium point of the evolutionary game. The stability of each equilibrium point is analyzed as shown in Table 2.

Table 2 Stability analysis of equilibrium points.

Proposition 1

The system has and has only one evolutionary stable point ${E_6}(1,0,1)$ when ${C_{t2}} - {C_{t1}} - {R_{t2}}+\theta {V_{t2}}+\mu {B_t}<0$ and ${R_{p2}} - {R_{p1}}<0$ .

Proof

the condition of the system’s evolutionary stable point is to satisfy all the eigenvalues are negative; as shown in Table 2, the eigenvalues respectively satisfy the given conditions; the system has only one evolutionary stable point; the system has an evolutionary stable strategy; it is easy to know that Proposition 1 is valid; the proof is complete.

Proposition 2

When ${C_{t1}} - {C_{t2}}+{R_{t2}} - \theta {V_{t2}} - \mu {B_t}<0$ and ${R_{p2}} - {R_{p1}}<0$, the system has and has only one evolutionary stable point ${E_8}(1,1,1)$.

Proof

the condition of the system evolutionary stable point is to satisfy all the eigenvalues are negative, as shown in Table 2 eigenvalues respectively satisfy the given conditions, the system has and only one evolutionary stable point, the system has an evolutionary stable strategy, easy to know that the proposition 2 is valid, the proof is complete.

Numerical simulation

In order to verify the validity of the stability analysis and the influence of important parameters on the evolution of the system’s main body behavior, a simulation analysis of the system’s evolutionary stability equilibrium point ${E_8}(1,1,1)$ is carried out in conjunction with the parameter settings and ranges in the problem assumptions. According to the real traffic violation penalty data, trucks exceed more than 30% of the approved mass or violation of the provisions of the passenger, a fine of 500 yuan or more than 2,000 yuan, and the load exceeds the maximum permissible total mass of more than 50% of the departure of 2,000 yuan. The starting price of freight for lorries is between RMB 400–800, and when the starting mileage is exceeded, the freight rate is RMB 2.5/km for small lorries (blue light trucks), RMB 4/km for medium lorries (yellow light trucks), and RMB 6/km for heavy lorries (yellow heavy trucks), which varies from one region to another. As you can see, the corresponding penalties vary in different regions and for different offenses, and the benefits also vary. Parameter values are set with reference to reality and relevant government and industry reports, and at the same time, since the units of the parameters, such as the benefits of each subject and the traffic risk, are inconsistent in the regulation of the traffic safety risk, a uniform scale processing is carried out so that the unit of each parameter is 1, and the numerical simulation is carried out.

The return ${{\text{R}}_{p1}}$ of public participation in security risk supervision is 20, the return ${{\text{R}}_{p2}}$ of non-participation in supervision is 10, the probability $\mu$ of exposing noncompliant behavior is 0.7, and the loss ${{\text{C}}_p}$ caused by inaction is 5. The benefit ${{\text{R}}_{{\text{t1}}}}$ and cost ${{\text{C}}_{{\text{t1}}}}$ of compliance transportation are 30 and 20 respectively. The additional revenue ${{\text{R}}_{{\text{t2}}}}$ obtained by non-compliant transportation is 10, the cost ${{\text{C}}_{{\text{t2}}}}$ is 10, the penalty for non-compliance $\theta$ is 0.2, the traffic risk is 10, the loss ${{\text{B}}_t}$ due to exposure is 60, and the cost ${{\text{C}}_{\text{c}}}$ of capturing the regulatory authorities is 5. The income ${{\text{R}}_{\text{s}}}$ of strict supervision by the supervisory department is 20, the environmental benefit ${\text{W}}$ is 15, the regulatory cost ${{\text{C}}_{\text{s}}}$ is 5, the additional benefit ${{\text{R}}_{bs}}$ obtained by the regulatory capture is 10, and the penalty ${{\text{B}}_s}$ by the superior department if it is found to accept the regulatory capture is 15.

According to the replicated dynamic equations and parameter assignments, as shown in Fig. 3, if the replicated dynamic equations evolve 100 times over time, no matter what the initial strategy probability of the game player is, the system will eventually converge to ${E_8}(1,1,1)$, which is consistent with the evolutionary stability analysis results and verifies the effectiveness of the model.

Fig. 3

Stability test of equilibrium point ${E_8}(1,1,1)$.

Fig. 4

Evolution trend of the system when traffic risk changes.

As shown in Fig. 4, when other values remain unchanged, changing the value of traffic risk ${V_{t2}}$ will not change the stability of system evolution, and the evolutionary stability point is still ${E_8}(1,1,1)$. The increase of traffic risk will increase the evolution speed of stable and compliant transportation of transportation enterprises. With the increase of ${V_{t2}}$, the probability of public participation in safety risk supervision will increase. Therefore, when traffic risk rises, transportation companies and the public will enhance the fulfillment of their responsibilities and actively participate in the regulation of traffic safety risk in order to reduce traffic safety risk.

Fig. 5

Evolutionary trend of the system when the penalty strength $\theta$ changes.

As in Fig. 5, when other parameters are kept constant, changing the value of the penalty strength $\theta$ does not change the stability of the system’s evolution, and the evolutionary stability point is still ${E_8}(1,1,1)$. An increase in the penalty strength increases the evolutionary speed of the transport firms to stabilize the compliant transport, and therefore the transport firms are forced to tend to the compliant transport by the high penalties when the penalties are rising.

Fig. 6

Influence of punishment intensity on the evolution strategy of transportation enterprises.

As shown in Fig. 6, when the punishment intensity $\theta$ gradually increased, the strategic choice of the transportation enterprise is gradually stable to the compliant transportation; when the punishment intensity is between 20% and 50%, the probability of the enterprise choosing the compliant transportation is less than 40%; when the punishment intensity is greater than 70%, the evolution of the enterprise’s strategic choice is stable to the acceleration of the compliant transportation rate. And eventually stabilize to a compliant transportation strategy.

Fig. 7

Influence of penalty intensity on regulatory authorities’ evolutionary strategies.

As shown in Fig. 7, with the increase in the severity of punishment, the strategy of regulatory authorities to choose strict regulation will accelerate with the increase in the severity of punishment.

Fig. 8

Evolution trend of the system when exposure rate changes.

As shown in Fig. 8, the influence of changes in the exposure rate of public participation in safety risk supervision on the behavior evolution and system stability of the system is studied. By changing the exposure rate, it can be seen that when the exposure rate is small, the equilibrium point of the system will gradually evolve to ${E_6}(1,0,1)$, and when the exposure rate gradually increases, the stability point of the system evolution will gradually evolve to ${E_8}(1,1,1)$. It shows that when the public participates in supervision and the exposure rate increases, the probability of reputation loss of transportation enterprises due to exposure of non-compliant transportation gradually increases, and the reputation loss caused by exposure exceeds the additional benefits of non-compliant transportation. Transport enterprises will choose compliant transport, and the strategic choice of the public, transport enterprises and regulatory authorities is stable in the combination {participation, compliant transport, strict supervision}. In the face of public scrutiny and a high level of exposure, the system of transportation companies and regulators will choose to fulfill their responsibilities, road transport compliance transportation, strict supervision, to enhance the effectiveness of intelligent transportation risk safety monitoring and road safety.

Fig. 9

Influence of exposure rate on the evolution strategy of transportation enterprises.

As shown in Fig. 9, the public’s exposure to non-compliant transportation has an effect on the strategic choice of transportation enterprises. As the exposure increases, the convergence rate of transportation enterprises’ choice of compliant transportation gradually increases. When the exposure rate is below 0.6, transportation enterprises have a fluke mentality, and the probability that transportation enterprises tend to choose non-compliant transportation increases, and finally stabilize and choose non-compliant transportation strategies. When the exposure probability is greater than 0.6, transportation enterprises tend to choose compliant transportation strategies.

Fig. 10

Impact of exposure on regulators’ evolutionary strategies.

As shown in Fig. 10, when the exposure rate increases, the regulator’s strategy choice stabilizes at strict regulation, and as the exposure rate gradually increases, the convergence rate of the regulator’s choice of strict regulation gradually accelerates, and the regulator’s choice of strategy will not be affected by the changes in the exposure rate, but only by the convergence rate of its strategy choice.

Fig. 11

Impact of public inaction losses on public evolutionary strategies.

As can be observed from Fig. 11, as the loss of the public in the face of the transportation enterprises in violation of transportation to take the choice of inaction increases, the public’s strategy choice tends to choose to participate in the regulation of transportation, and tends to choose to participate in the regulation of the strategy of the rate of convergence is increasingly fast. It shows that when the public recognizes that the greater the loss of not participating in the regulation of transportation, as one of the participants in transportation, in order to better protect their own safety, will be willing to choose to participate in the regulatory system.

Fig. 12

Impact of changes in the additional revenue of transport firms on the evolution of the system for different exposure rates.

When varying the additional benefits gained from non-compliant transport by transport companies, the system evolution diagram is shown in Fig. 12. It can be seen that the stability point of the system tends to ${E_6}(1,0,1)$ when the additional benefit from non-compliant transport by transport companies increases with an exposure rate of 0.6, and the stability point of the system remains at ${E_8}(1,1,1)$ when the exposure rate is 0.7, even though the benefit from non-compliant transport is increasing. It suggests that when public scrutiny of transport companies intensifies and exposure rises, transport companies will choose to transport in compliance even if the benefit gained by transport companies from non-compliance increases.

Conclusion

In this paper, based on the consideration of public participation behavior, through the construction of the three-party evolutionary game model, combined with the finite rationality subject, we explored the systematic evolution process of the public, transportation enterprises, and the regulatory authorities and analyzed the influence of different factors on the behavioral tendency, strategy selection, and system stability of the evolutionary subjects in the process of implementation of the regulatory process through numerical simulation. The main research conclusions drawn are: (1) Traffic risk has a positive contribution to the participation of transport companies and the public in the co-regulation of road transport regulation, and the rate of stabilization of their strategies accelerates as traffic risk increases. Secondly, for the public, the greater the loss of their inaction, the more they are motivated to participate in regulation. (2) Compared with the regulatory authorities, transport enterprises are more sensitive to changes in the intensity of punishment, and only when the intensity of punishment exceeds a certain threshold will transport enterprises be encouraged to choose compliant transport. (3) The exposure rate has an important impact on the stability of the system evolution, and a higher exposure rate promotes compliant transport by transport companies and strict supervision by the regulator. Meanwhile, the exposure rate affects the strategic choices of transport companies, and when the exposure rate exceeds a threshold value, transport companies will not choose non-compliant transport even if the additional revenue from non-compliant transport increases, indicating that the exposure rate has a good effect on suppressing the non-compliant transport behaviors of transport companies. Although the exposure rate does not affect the evolutionary stability strategy of the regulator, its rate of convergence to strict regulation accelerates with exposure.

According to the above research results, we put forward the corresponding management implications:

• (1) Strengthening the mechanism of traffic safety risk regulation and exposure: Traffic risks have a facilitating effect on the participation of transport enterprises and the public in traffic safety risk regulation; therefore, violations and corresponding risks are publicized so that the participating entities have a concrete perception of the risks. Second, financial rewards or other forms of incentives should be given to members of the public who report non-compliant transportation behaviors in order to increase the motivation of public participation, and a convenient online reporting and feedback platform should be established so as to improve the channels for public participation in transportation supervision, thereby achieving the construction of safer transportation.

• (2) Increase the severity of punishment and set a threshold: transportation enterprises are highly sensitive to the change in punishment intensity, and only when it exceeds a certain threshold will they be prompted to choose compliant transportation. Therefore, a modest increase in the severity of penalties could be considered, while setting clear thresholds to ensure that adequate punitive measures are taken for violations. This can guide transportation companies to choose compliant modes of transportation and increase the safety and reliability of the entire transportation system.

• (3) Enhance the two-way monitoring role of the public for transport enterprises and regulators: the public has the dual role of participants and monitors, through the use of self-media and other means of exposure of violations to make up for the lack of resources or rent-seeking behaviors that may be faced by the official regulators, but also conducive to the standardization of their own transport behaviors by the transport enterprises. Dynamically adjust the exposure rate and regulatory efforts to enhance the governance effectiveness of the transportation system, achieve collaborative governance among the public, transportation enterprises, and regulatory authorities, and safeguard traffic safety and social stability.

Although this study examines the role of public participation in the regulation of traffic safety risks, it analyzes that the collaborative governance among the public, transport companies and regulators as well as the two-way supervisory role of the public are conducive to the development of traffic safety in a better direction. However, regarding the insufficient consideration of the dynamic change scenarios of regulation and the limitation of data research and acquisition, this study also has certain limitations, and in the future, it can also further continue to deepen the study to compensate for the limitations. (1) There are some limitations in terms of research subjects and modeling scenarios. There will be more traffic participant roles in further digitally-intelligent traffic scenarios, such as intelligent regulators, industry associations, media, etc., and the relationships between subjects will be more complex, failing to fully capture the diversity and complexity that may exist in the actual decision-making process. (2) In further future research, the model can be extended with dynamic regulatory scenarios to incorporate more relevant stakeholders and complex interactions, capture the diversity and complexity of the actual decision-making process, and focus on more specific research scenarios, such as the transportation of hazardous materials, and the incorporation of digital smart technology scenarios, in order to further refine the research.