Emergency Capacity Pool to Respond to Unconventional Emergencies Based on Principal–Agent Theory

Abstract

To address the conflict of interest between the government and enterprises regarding urban emergency transportation resources in unconventional emergencies and to enhance resource allocation and response efficiency. This paper proposes a collaborative government–enterprise model for emergency transport capacity reserves and develops an incentive model based on principal–agent theory. First, by comprehensively considering enterprise characteristics, high-quality enterprises are selected to collaborate with the government in building an emergency capacity pool of social vehicles. Second, to address potential conflicts of interest between the government and enterprises within the emergency capacity pool, this paper uses principal–agent theory to analyze the interest game process under information asymmetry, constructs a corresponding incentive model, and determines the government’s optimal incentive coefficient, the enterprise’s optimal actual capacity supply ratio, and the benefit distribution between both parties. Finally, numerical simulations and sensitivity analyses were used to verify the model’s applicability. The findings reveal that transport effort cost, economic requisition compensation, and government supervision cost influence the optimal decisions and outcomes in government–enterprise interactions. This study provides theoretical guidance and managerial insights for coordinating emergency transport scheduling between the government and enterprises during unconventional emergencies.

1. Introduction

Unconventional emergencies refer to sudden events with insufficient precursors, significant complexity, potential secondary disasters, and challenges that cannot be effectively addressed through standard management methods [1]. With the rapid development of global economic, social, and environmental changes, the frequency of unconventional emergencies has increased. Examples include the “9/11 terrorist attack” in New York (2001), the nuclear radiation crisis following Japan’s 2008 earthquake, the Tianjin “8/12” explosion in 2015, and the global COVID-19 pandemic in 2020. How to effectively warn about, respond to, and manage unconventional emergencies has become a pressing scientific challenge for countries worldwide, including China. In emergency rescue operations, efficiently completing emergency transportation tasks is critical to improving the response’s effectiveness. However, limited transport capacity prevents the government from meeting the surge in demand, leading to supply chain disruptions and transportation delays [2]. Against this backdrop, the complementary capabilities of the government and transport enterprises in risk management, along with their collaboration in joint planning, knowledge management, and resource sharing, can help to prevent the escalation of unconventional emergencies, easing the burden on both parties [3]. Therefore, establishing a collaborative model between the government and transport enterprises is essential.

Strategic interactions between the two parties are crucial in a government–enterprise cooperation model. As the principal, the government oversees coordination and decision-making, while transport enterprises, as agents, execute transportation tasks. Due to information asymmetry, the government has limited access to information, whereas transport enterprises possess extensive knowledge and resources. This disparity can result in misaligned objectives. Specifically, the government aims to maximize social benefits and enhance emergency response capabilities, while transport enterprises focus on economic interests, potentially leading to profit conflicts. Therefore, aligning the interests of both parties for mutual benefit is essential to achieving successful cooperation. Incentive mechanisms are an effective approach to addressing these challenges [4]. As part of the cooperation agreement, the government should provide appropriate subsidies to enhance the commercial viability of collaborative capacity reserves.

Research on government–enterprise cooperation in emergency transport capacity has examined various aspects, including cooperation models, capacity coordination, and the influence of policies [5,6,7]. The public–private partnership (PPP) model, which enhances the provision of public goods and services, has been widely applied to collaborative emergency resource reserves. In terms of capacity coordination, scholars have explored flexible resource allocation during emergencies through optimization models to enhance response efficiency [8,9,10]. Furthermore, researchers have analyzed the influence of policies on corporate participation in emergency responses, such as the effects of government subsidies, incentive mechanisms, and industry standards [11,12]. Despite the growing body of research, systematic discussions on collaborative reserves for emergency capacity, particularly regarding resource preparation and reserve strategies, still need to be improved. This research gap results in a need for more adequate theoretical support for resource reserves in actual emergency management, negatively affecting the efficiency of resource allocation and the timeliness of emergency responses. Therefore, deeper theoretical and practical research is essential to optimize emergency management.

To address the identified research gaps, this paper proposes a transportation reserve model from the perspective of emergency transportation resource management. The model primarily relies on contracted transportation capacity, supplemented by dispersed social transportation resources, termed the “Emergency Capacity Pool of Social Vehicles”. Its objective is to optimize collaboration between the government and transport enterprises through strategic design and management, ensuring efficient allocation of emergency transportation resources. This, in turn, enhances emergency management and maximizes benefits. To achieve these objectives, the following research questions are explored:

(1)

What are the selection criteria for transport enterprises?

(2)

How can the government’s optimal incentive coefficient, the enterprise’s optimal actual capacity supply ratio, and their respective benefits be effectively determined?

(3)

How do changes in transport effort cost, government supervision cost, and economic requisition compensation influence the decisions and benefits of both the government and enterprises?

This paper systematically selects potential transport enterprises based on four dimensions: grading, categorization, distribution, and integrity, and establishes the reserve structure of the emergency capacity pool of social vehicles. The principal–agent theory is employed to explore the strategic relationship between the government and transport enterprises within the pool. An incentive model is developed to determine the optimal decisions and utilities for both parties. Furthermore, through dynamic perspective and numerical simulation, this paper analyzes the evolution of decision-making paths for the government and transport enterprises, exploring key factors that motivate transport enterprises to participate actively in emergency responses. In conclusion, this study constructs a more efficient emergency transport capacity reserve structure, providing theoretical support for government–enterprise cooperation in emergency management.

The remainder of this paper is organized as follows: Section 2 reviews the relevant literature on collaborative emergency transport capacity reserve. Section 3 explains the selection criteria for transport enterprises and the framework for emergency transport capacity. Section 4 constructs an incentive model for the emergency capacity pool of social vehicles based on principal–agent theory to analyze the optimal strategies and expected utilities of the government and transport enterprises. Section 5 validates the applicability of the model through case analysis. Section 6 analyzes and compares the results of the study. Section 7 summarizes the conclusions, discusses limitations, and proposes future research directions.

2. Literature Review

This section reviews research from four perspectives: government–enterprise emergency cooperation model, emergency transportation resource planning, the application of principal–agent theory, and government subsidies for incentivizing enterprise participation.

2.1. Government–Enterprise Emergency Cooperation Model

Collaborative models between industries and governments have become essential strategies for effectively managing unconventional emergencies in emergency management. According to Jiaqi’s research [13], collaboration between the government and medical enterprises during epidemics has significantly enhanced the efficiency of public health responses. For example, the UK government collaborated with medical enterprises to create a new supply chain framework, enabling the rapid procurement and distribution of personal protective equipment. However, challenges like non-transparent public funding and preferential access substantially reduced the effectiveness of these efforts [14]. Similarly, partnerships between the transportation industry and governments have played a crucial role in emergency responses. Wang et al. observed that during the COVID-19 outbreak, the Chinese government rapidly partnered with logistics companies to carry out emergency transportation, ensuring that essential goods were delivered at minimal cost and maximum speed despite constrained transportation resources [15]. However, this model encountered obstacles like low coordination efficiency and limited corporate participation, which could have improved its effectiveness.

In summary, while partnerships between industries and governments have delivered many positive outcomes in emergency management, they also encounter notable limitations. This paper proposes a government–enterprise partnership model for emergency transportation capacity, aiming to explore the alignment of interests between governments and transport enterprises, strengthen collaboration, and enhance responses to unconventional emergencies.

2.2. Emergency Transportation Resource Planning

Emergency transportation resource planning is critical in emergency management, as it directly impacts rescue efficiency and post-disaster recovery, especially during sudden, unforeseen events [16]. Adequate planning includes the types and quantities of vehicles and the optimization of routes, both of which are essential for enhancing emergency responses. Recent research in this area has primarily focused on route optimization. Liu et al. combined ant colony optimization with dynamic programming to develop a dynamic path model that flexibly adjusts emergency supply routes based on real-time traffic [17]. Wang et al. applied vehicle-sharing and time-window strategies to optimize resource allocation, enhancing emergency logistics network efficiency [18]. Hao et al. analyzed route selection and priority control strategies to improve the allocation of emergency transportation resources [19]. Despite progress in route optimization, research on vehicle types and quantities still needs to be improved, potentially hindering efficient resource use in emergencies and impacting overall response effectiveness. This paper proposes a multi-party collaborative approach for capacity reserve planning through an emergency capacity pool of social vehicles. This model integrates resources from government and transport enterprises, creating a shared platform for dynamically adjusting emergency capacity, enabling efficient allocation and use of vehicles.

2.3. Application of Principal–Agent Theory

During emergency management, stakeholders often face unequal access to information. This information asymmetry results in resource allocation and decision-making inefficiencies [20]. Principal–agent theory, the standard framework for addressing information asymmetry, has recently gained attention as scholars have explored its application in emergency cooperation. Its core is to enhance collaboration between parties by designing incentives, establishing constraints, and optimizing information transmission and collection. First, the principal can encourage the agent to share information actively by implementing goal-driven incentives. Xiao et al. examined the interactions between nonprofit organizations and capacity-reserving enterprises under information asymmetry and developed an incentive model for reserving emergency medical supplies. Their findings indicated that linking incentive policies directly to enterprise performance increased the likelihood of businesses, providing relevant reserve information [21]. Second, in the principal–agent relationship, the principal can implement supervisory mechanisms to regulate the agent’s information-sharing behavior. These mechanisms reduce the chances of information concealment and strengthen the agent’s accountability. For example, Ma et al. studied the principal–agent relationship between higher- and lower-level governments in early warning systems for heavy rain, showing that supervisory mechanisms enhance the effectiveness of information flow and feedback [22]. Finally, using information-sharing platforms and data systems can improve information transmission and collection, ensuring timely and accurate data acquisition while facilitating multidimensional integration and analysis [23].

In summary, principal–agent theory offers systematic solutions for addressing information asymmetry in emergency cooperation by designing incentives, establishing constraints, and optimizing information flow. This paper proposes an incentive model grounded in principal–agent theory to mitigate information asymmetry between the government and transport enterprises, intending to enhance emergency capacity cooperation mechanisms.

2.4. Government Subsidies for Incentivizing Enterprise Participation

In government–enterprise emergency cooperation, companies often encounter complex environments and significant risks from sudden events, which can hinder profitability or even result in losses. Thus, the government must provide subsidies to encourage active corporate participation in emergency cooperation. Guan et al. examined how government subsidies affect the private sector’s risk attitudes under disaster uncertainty and influence decision-making during emergencies. Their study found that moderate subsidies reduce enterprises’ risk aversion, enhancing their willingness to participate in emergency cooperation [24]. Yang et al. examined how different government subsidies and coordination strategies influence pricing decisions, pandemic response efforts, and cost-sharing ratios within emergency supply chains. Their findings indicate that cost-sharing subsidies result in the highest levels of pandemic response effectiveness and social welfare [25]. Zhang et al. investigated how subsidy policy design and duration influence emergency supply levels in private manufacturing. Their findings emphasized that appropriate subsidy durations significantly enhance enterprises’ supply capabilities [26].

In summary, government subsidies in emergency cooperation significantly impact corporate decision-making and coordination mechanisms, providing essential support to strengthen risk resilience. This paper aims to analyze government subsidy measures for transport enterprises, focusing on designing incentive mechanisms that effectively motivate participation in emergency responses while safeguarding the enterprises’ reasonable interests.

2.5. Research Gaps

In joint government–enterprise emergency transportation reserves, many studies have addressed emergency resource planning, applications of principal–agent theory, and the incentive effects of subsidies on corporate motivation. However, most research has focused on optimizing transport routes. There is limited research on the types and quantities of vehicles, which restricts existing collaboration mechanisms and resource management strategies from adapting to changing demands and resource constraints. A comprehensive theoretical framework is urgently needed to guide collaborative reserve and dispatch strategies for transportation resources under different scenarios, thereby enhancing emergency response effectiveness. Furthermore, research on government–enterprise roles and interaction mechanisms in emergency transportation reserves remains limited, particularly concerning information sharing, responsibility allocation, and incentive design. This gap limits theoretical development and challenges practical resource allocation efficiency. Future research should prioritize systematic models for collaborative emergency transportation reserves, exploring multi-stakeholder collaboration to advance scientific and practical innovation in emergency management.

To explore collaborative emergency transportation mechanisms and analyze multi-stakeholder models, this paper proposes an emergency capacity pool of social vehicles co-built by government and transport enterprises. An incentive model based on principal–agent theory is developed to examine optimal decisions and benefits for government and transport companies in emergencies. Additionally, a sensitivity analysis of government compensation is conducted to examine its impact on government–enterprise decisions and benefits, revealing how subsidies influence transport enterprise participation in emergency responses.

3. Building the Emergency Capacity Pool of Social Vehicles

3.1. Principles for Selecting Transport Enterprises

By establishing rigorous selection criteria, suitable transport enterprises are evaluated and selected to form a collaborative emergency capacity pool of social vehicles. This study, grounded in China’s national context, evaluates transport enterprises comprehensively across four dimensions: ownership structure, core business type, regional distribution, and historical reputation.

3.1.1. Grading Principle

The stability and capacity of the emergency capacity pool of social vehicles directly affect the government’s emergency response efficiency during unconventional emergencies. Given state-owned enterprises’ reliability and national security alignment, they should be prioritized as the primary builders. However, since state-owned enterprises alone may not meet rising demand, the government should also include high-quality private enterprises to leverage their flexibility and resources, thereby enhancing emergency capacity reserves. Combining state-owned and private enterprises creates a flexible and scalable emergency transportation network, facilitating a comprehensive and resilient emergency response system capable of addressing unconventional emergencies.

3.1.2. Categorization Principle

The core function of the emergency capacity pool of social vehicles lies in the efficient transportation of various emergency supplies and rescue personnel. To achieve this, it is essential to have diversified emergency transportation capacity to address challenges in different scenarios. Therefore, the government should systematically assess enterprises’ transport capacity, equipment configurations, and experience in handling specific materials and emergency logistics. By establishing an emergency capacity pool of social vehicles with diversified capacity, the government can meet the demands of various emergencies, ensuring smooth operations and effective outcomes.

3.1.3. Distribution Principle

The distribution of transport enterprises is crucial for enhancing a city’s emergency response capabilities during unconventional emergencies. The layout should follow the principles of scientific allocation and dynamic adjustment, considering factors such as disaster distribution, potential risks, population density, and transportation to continuously optimize the regional distribution of reserve capacity. Through careful consideration and dynamic optimization, the government can achieve full coverage of transportation capacity while focusing on critical areas for monitoring.

3.1.4. Integrity Principle

Integrity is critical to sustainable enterprise development and maintaining trust between the government and enterprises. The government typically evaluates the reputation of transport enterprises based on historical interactions. Cooperating with enterprises with a good track record and honoring contracts help to reduce transaction costs, minimize risks of information concealment, and enable the rapid mobilization and efficient allocation of resources. Moreover, reliable partners can enhance the resilience of the overall emergency system, ensuring necessary support and services during unconventional emergencies.

3.2. Structure of the Emergency Capacity Pool of Social Vehicles

The emergency capacity pool of social vehicles integrates state-owned and private transport enterprises with excellent historical performance and reasonable regional distribution. It systematically categorizes capacity based on different emergency transport demands and sets corresponding scales for each category to ensure that general, specialized, and critical transport needs are met. This layered and structured strategy can improve the efficiency of resource allocation, ensuring that the emergency capacity pool maintains high performance and rapid responses during unconventional emergencies. The structure is shown in Figure 1. On the left, guided by the grading principle, it shows that the emergency capacity pool of social vehicles comprises state-owned and high-quality private transport enterprises in urban areas, with state-owned enterprises as the core. On the right, following the categorization principle, the pool is divided into different capacity types based on varied transportation requirements.

3.3. Integration of Social Vehicle Transportation Capacity

The municipal government should establish a central coordination platform within the emergency capacity pool of social vehicles. Based on a real-time information-sharing system, this platform allows various levels of government and contracted transport enterprises to register available resources (such as semi-trailers, vans, and refrigerated trucks), enabling systematic integration of transportation resources. The platform should dynamically update operational status, location, and capacity information to facilitate rapid deployment in response to evolving emergencies.

4. Development of an Incentive Model for the Emergency Capacity Pool of Social Vehicles Based on Principal–Agent Theory

4.1. Problem Description

This study analyzes the principal–agent relationship between a government and a transport enterprise, focusing on their interaction mechanism while constructing an emergency capacity pool of social vehicles. Figure 2 illustrates the research process. The government enters a principal–agent agreement with the selected transport enterprise, specifying the quantity of emergency capacity available for deployment during unconventional emergencies. During the capacity reserve period, the government subsidizes the transport enterprise and monitors its operations. During the emergency response period, the government coordinates transportation based on the capacity supplied by the enterprise. After the emergency, the government compensates the transport enterprise based on actual capacity utilization.

4.2. Model Assumptions and Symbol Definitions

This section systematically explains the model’s assumptions and associated variables.

4.2.1. Model Assumptions

The model mainly involves two participants: the government as the principal and the transport enterprise as the agent. The government selects state-owned and private enterprises to jointly construct the emergency capacity pool of social vehicles.

(1)

In constructing an emergency capacity pool of social vehicles, the government faces information asymmetry with transport enterprises, making it difficult to monitor their actions and decision-making processes directly. This study assumes that the ratio of actual capacity β_iq_i supplied during emergencies to the promised capacity α_iq_i, expressed as β_i/α_i, can serve as an indirect measure of the effort exerted by enterprises in supplying transportation capacities, where α_i (0 < α_i < 1) represents the proportion of capacity pledged to the emergency capacity pool of social vehicles; β_i (0 < β_i < 1) represents the proportion of capacity actually supplied, and q_i represents the available quantity of various transportation capacities.

(2)

During emergency responses to unconventional emergencies, the government derives social benefits mainly from reducing accident-related losses. These benefits depend on the effort levels of the government and its collaborators and are influenced by external environmental factors [27]. Thereby, this paper sets the output function of effort as $\pi \left({\beta }_1/{\alpha }_1,{\beta }_2/{\alpha }_2,\cdots ,{\beta }_n/{\alpha }_n\right)={\displaystyle {\sum }_{i=1}^n{r}_i\left({\beta }_i/{\alpha }_i\right)}+{\theta }_i$, where r_i represents the marginal social benefit of each capacity type and θ_i ~ N (0, σ_i²) represents random factors unrelated to the conversion of social benefits, reflecting external environmental uncertainty. The larger the value of σ_i², the stronger the uncertainty.

(3)

When a transport enterprise increases its effort to supply one type of emergency transportation capacity, its effort in other tasks decreases proportionally. However, this does not imply that the unit cost of effort for different types of capacity will change. Therefore, it is assumed that the cost of effort functions independently for each capacity type [28]. Referring to [29], the effort cost function of the transport enterprise is assumed to be $C\left({\beta }_1/{\alpha }_1,{\beta }_2/{\alpha }_2,\cdots ,{\beta }_n/{\alpha }_n\right)=0.5{{\displaystyle {\sum }_{i=1}^n{n}_i\left({\beta }_i/{\alpha }_i\right)}}^2$, where n_i represents the marginal cost of supplying each capacity type. This quadratic cost function meets the condition of increasing marginal production costs [30], i.e., ${c^{″}}_i\left({\beta }_i\right)>0$.

(4)

According to Holmstrom and Milgrom’s principal–agent model [31], the government provides a linear incentive contract to the transport enterprise, represented as $S={\omega }^{\prime }+{\displaystyle {\sum }_{i=1}^n{\omega }_i{\pi }_i}$, where ω’ represents a one-time subsidy from the government and ω_i represents the incentive coefficient for each type of emergency capacity provided, with 0 < ω_i < 1 (i = 1, 2, …, n).

(5)

Unconventional emergencies involve high uncertainty and complexity, posing significant challenges for transport enterprises, which tend to exhibit risk aversion in such complex environments. This paper assumes that the government is risk-neutral. At the same time, enterprises are risk-averse, following a constant absolute risk aversion function: $\mu \left(f_1\right)=-e^{-\rho f_1}$, where ρ represents the risk aversion coefficient of the transport enterprise and f₁ represents the enterprise’s utility function.

4.2.2. Variable Definitions

Table 1. Relevant variables and definitions.

Variable	Definition
αi	Percentage of each type of capacity committed by the transport enterprise to the emergency capacity pool of social vehicles.
qi	Existing amount of each type of capacity in the transport enterprise.
βi	Proportion of capacity actually supplied.
βi*	Optimal proportion of capacity actually supplied.
ri	Social benefits are generated by each type of transportation capacity.
θi	Random factors unrelated to the conversion of social benefits.
ni	Marginal cost of each type of capacity supplied by the transport enterprise.
ω’	One-time subsidy provided by the government to the transport enterprise.
ωi	Incentive coefficient given by the government for each type of emergency transportation capacity.
ωi*	Optimal incentive coefficient is given by the government for each type of emergency transportation capacity.
ρ	Risk aversion coefficient of the transport enterprise.
hi	Economic compensation is given by the government for each type of emergency transportation capacity.
bi	Supervision costs incurred by the government for each type of emergency transportation capacity.
f’	Maximum opportunity return from the actual capacity provided by the transport enterprise.
f1	Payoff function of the transport enterprise.
f2	Payoff function of government.
Π1	Expected utility value for transport enterprises.
Π2	Expected utility value for government.

provides a detailed description of the relevant variables, clarifying the model construction process and each function’s purpose.

4.3. Building an Incentive Model for the Emergency Capacity Pool of Social Vehicles

This section derives the utility functions by analyzing the costs and benefits of establishing an emergency capacity pool of social vehicles for the government and transport enterprises.

4.3.1. Utility Function Analysis for Transport Enterprises

(1)

Emergency Transportation Capacity Input Costs

The input costs for transport enterprises are determined by their effort costs. According to Assumption 3, the calculation formula for the input costs is:

$$C\left({\beta }_1/{\alpha }_1,{\beta }_2/{\alpha }_2,\cdots ,{\beta }_n/{\alpha }_n\right)=0.5{{\displaystyle {\sum }_{i=1}^n{n}_i\left({\beta }_i/{\alpha }_i\right)}}^2$$

(1)

where n_i represents the marginal cost of each type of capacity supplied by the transport enterprise.

(2)

Emergency Transportation Capacity Incentives

To encourage enterprises to provide as much emergency capacity as possible, the government offers a portion of societal benefits as incentives to transport enterprises. According to Assumption 4, the formula for calculating incentives for enterprises’ efforts in providing emergency transportation capacity is:

$$S={\omega }^{\prime }+{\displaystyle {\sum }_{i=1}^n{\omega }_i{r}_i\left({\beta }_i/{\alpha }_i+{\theta }_i\right)}$$

(2)

(3)

Emergency Transportation Capacity Compensation

For actual emergency transportation capacity used, the government should implement an economic compensation mechanism based on the types of transportation capacity β_iq_i provided by the enterprise. Fixed compensation alone cannot entirely reflect the actual contribution of the enterprise during emergencies. Therefore, adjustments based on the enterprise’s performance should reflect government rewards or penalties. The formula for calculating the actual capacity compensation received by the transport enterprise is:

$$Z={\displaystyle \sum _{i=1}^n{h}_i{\beta }_i{q}_i}$$

(3)

where h_i represents the economic compensation given by the government for each type of emergency transportation capacity.

(4)

Utility of the Transport Enterprise

Therefore, the payoff function of the transport enterprise is expressed as:

$$f_1=S+Z-C={\omega }^{\prime }+{\displaystyle \sum _{i=1}^n\left[\frac{{\omega }_i{r}_i{\beta }_i}{{\alpha }_i}+{\omega }_i{\theta }_i+h_i{\beta }_i{q}_i-\frac{n_i{\beta }_i^2}{2{\alpha }_i^2}\right]}$$

(4)

Given that θ_i ~ N (0, σ_i²), we obtain:

$$f_1\sim N\left(\omega \text{'}+{\displaystyle \sum _{i=1}^n\left[\frac{{\omega }_i{r}_i{\beta }_i}{{\alpha }_i}+h_i{\beta }_i{q}_i-\frac{n_i{\beta }_i^2}{2{\alpha }_i^2}\right]}, {\displaystyle \sum _{i=1}^n{\left({\omega }_i{\sigma }_i\right)}^2}\right)$$

According to Assumption 5, the enterprise is risk-averse, and its utility function is $u\left(f_1\right)=-e^{-\rho f_1}$; thus, the expected value of its utility function is:

$$E\left[u\left(f_1\right)\right]=E\left(-e^{-\rho f_1}\right)={\displaystyle {\int }_{-\infty }^{+\infty }-e^{-\rho f_1}\frac{1}{\sqrt{2\pi \mathrm{var}\left(f_1\right)}}d{f}_1=}-e^{-\rho \left[E\left(f_1\right)-{\displaystyle \frac{1}{2}}\rho \mathrm{var}\left(f_1\right)\right]}$$

For ease of solving the model, the expected value of the enterprise’s utility function is expressed using certainty-equivalent income (CE). According to the definition of CE: $E\left[u\left(f_1\right)\right]=u\left(CE\right)$ [32], we obtain:

$$-e^{-\rho \left[E\left(f_1\right)-{\displaystyle \frac{1}{2}}\rho \mathrm{var}\left(f_1\right)\right]}=-e^{-\rho \left(CE\right)}$$

Therefore, $CE=E(f1)-0.5\rho \mathrm{var}(f1)$, and the certainty-equivalent income (CE) of the enterprise’s utility function is:

$${\prod }_1=\omega \text{'}+{\displaystyle \sum _{i=1}^n\left[\frac{{\omega }_i{r}_i{\beta }_i}{{\alpha }_i}+h_i{\beta }_i{q}_i-\frac{n_i{\beta }_i^2}{2{\alpha }_i^2}-\frac{\rho {\left({\omega }_i{\sigma }_i\right)}^2}{2}\right]}$$

(5)

4.3.2. Utility Function Analysis for Government Departments

(1)

Government Supervision Costs

The government incurs supervision costs M when monitoring transport enterprises’ emergency capacity reserves. Supervision costs are lower when transport enterprises actively meet their emergency reserve tasks; otherwise, they increase. The formula for government supervision costs is assumed as:

$$M={\displaystyle \sum _{i=1}^n{b}_i\left(1-\frac{{\beta }_i}{{\alpha }_i}\right)}$$

(6)

where b_i represents the supervision costs incurred by the government for each type of emergency transportation capacity.

(2)

Incentive and Compensation Costs

As per the contract, the government provides incentives and compensation for emergency capacity, calculated as:

$$S+Z=\omega \text{'}+{\displaystyle \sum _{i=1}^n\left[\frac{{\omega }_i{r}_i{\beta }_i}{{\alpha }_i}+{\omega }_i{\theta }_i+h_i{\beta }_i{q}_i\right]}$$

(7)

(3)

Social Benefits

Social benefits refer to the outcome of transport enterprises maintaining sufficient emergency capacity, effectively reducing accident losses during unconventional emergencies. Social benefits are determined by the effort level of transport enterprises β_i/α_i and the uncertainty of the external environment θ_i, reflecting the type and severity of unconventional emergencies. According to Assumption 2, the formula for the social benefits π is:

$$\pi ={\displaystyle \sum _{i=1}^n{r}_i\left({\beta }_i/{\alpha }_i\right)+{\theta }_i}$$

(8)

(4)

Government Utility

Therefore, the government’s payoff function is:

$$f_2=\pi -S-Z-M={\displaystyle \sum _{i=1}^n\left[\frac{\left[\left(1-{\omega }_i\right)r_i+b_i\right]{\beta }_i}{{\alpha }_i}-h_i{\beta }_i{q}_i+\left(1-{\omega }_i\right){\theta }_i-b_i\right]}-\omega \text{'}$$

(9)

With the government being risk-neutral, its utility function is:

$${\prod }_2={\displaystyle \sum _{i=1}^n\left[\frac{\left[\left(1-{\omega }_i\right)r_i+b_i\right]{\beta }_i}{{\alpha }_i}-h_i{\beta }_i{q}_i-b_i\right]}-\omega \text{'}$$

(10)

4.4. Incentive Model

In the emergency capacity pool of social vehicles, the government seeks to select the incentive coefficient ω_i to maximize its expected returns. The government faces two constraints from transport enterprises: participation constraints (their expected return from joining the pool must be no less than their highest opportunity return) and incentive constraints (enterprises, as rational actors, will always choose effort levels that maximize their expected returns).

4.4.1. Objective Function

As the principal in the emergency capacity pool of social vehicles, the government’s main task is to set incentive coefficients to increase the percentage of actual emergency transportation capacity β_i provided by the enterprises and maximize social satisfaction. Therefore, the objective function of this paper’s incentive model is:

$$\underset{{\omega }_i}{\max }{\prod }_2={\displaystyle \sum _{i=1}^n\left[\frac{\left[\left(1-{\omega }_i\right)r_i+b_i\right]{\beta }_i}{{\alpha }_i}-h_i{\beta }_i{q}_i-b_i\right]}-\omega \text{'}$$

(11)

4.4.2. Participation Constraint

Since transport enterprises aim to maximize profits, they will compare returns from the emergency capacity pool (both committed and actual provided) with returns from alternative projects. The return from these alternative projects is called the opportunity return, with the highest opportunity return denoted as f’. If the expected utility value Π₁ from participating in the emergency capacity pool is less than the highest opportunity return f’, the transport enterprises will not contract with the government to join the pool. Therefore, the government must ensure that the enterprises’ returns are not lower than their highest opportunity return, as expressed in the participation constraint Formula (12):

$$\omega \text{'}+{\displaystyle \sum _{i=1}^n\left[\frac{{\omega }_i{r}_i{\beta }_i}{{\alpha }_i}+h_i{\beta }_i{q}_i-\frac{n_i{\beta }_i^2}{2{\alpha }_i^2}-\frac{\rho {\left({\omega }_i{\sigma }_i\right)}^2}{2}\right]}\ge f\text{'}$$

(12)

4.4.3. Incentive Constraint

As rational actors, transport enterprises will always choose actions that maximize returns. Therefore, the enterprises’ incentive constraint is:

$$\underset{{\beta }_i}{\max }{\prod }_1=\omega \text{'}+{\displaystyle \sum _{i=1}^n\left[\frac{{\omega }_i{r}_i{\beta }_i}{{\alpha }_i}+h_i{\beta }_i{q}_i-\frac{n_i{\beta }_i^2}{2{\alpha }_i^2}-\frac{\rho {\left({\omega }_i{\sigma }_i\right)}^2}{2}\right]}$$

(13)

In summary, the incentive model for the emergency capacity pool of social vehicles is constructed as follows:

$$\underset{{\omega }_i}{\max }{\prod }_2={\displaystyle \sum _{i=1}^n\left[\frac{\left[\left(1-{\omega }_i\right)r_i+b_i\right]{\beta }_i}{{\alpha }_i}-h_i{\beta }_i{q}_i-b_i\right]}-\omega \text{'}$$

(14)

$$s.t.\left\{\begin{array}{l}\omega \text{'}+{\displaystyle \sum _{i=1}^n\left[\frac{{\omega }_i{r}_i{\beta }_i}{{\alpha }_i}+h_i{\beta }_i{q}_i-\frac{n_i{\beta }_i^2}{2{\alpha }_i^2}-\frac{\rho {\left({\omega }_i{\sigma }_i\right)}^2}{2}\right]}\ge f\text{'}\\ \underset{{\beta }_i}{\max }{\prod }_1=\omega \text{'}+{\displaystyle \sum _{i=1}^n\left[\frac{{\omega }_i{r}_i{\beta }_i}{{\alpha }_i}+h_i{\beta }_i{q}_i-\frac{n_i{\beta }_i^2}{2{\alpha }_i^2}-\frac{\rho {\left({\omega }_i{\sigma }_i\right)}^2}{2}\right]}\\ 0\le {\omega }_i\le 1, 0\le {\beta }_i\le 1\end{array}\right.$$

(15)

4.5. Model Solution

This paper applies backward induction to solve the model. After analyzing both constraints, they are substituted into the objective function to solve for the incentive coefficient ω_i.

4.5.1. Analysis of Incentive Constraints

This paper assumes that the government cannot directly observe the effort levels of transport enterprises in providing various types of emergency transportation capacity β_i/α_i. In such cases, enterprises will choose moderate effort to maximize their expected utility Π₁. Since a fixed transportation capacity ratio α_i is already agreed upon, only the variation in the actual capacity supply ratio β_i needs consideration. By differentiating the certainty-equivalent revenue Π₁, the first-order and second-order derivatives concerning the actual capacity supply ratio β_i are obtained, as shown in Formula (16):

$$\begin{array}{l}{\displaystyle \frac{\partial {\prod }_1}{\partial {\beta }_i}}=h_i{q}_i-{\displaystyle \frac{{\omega }_i{r}_i}{{\alpha }_i}}-{\displaystyle \frac{n_i{\beta }_i}{{\alpha }_i^2}}\\ {\displaystyle \frac{{\partial }^2{\prod }_1}{\partial {\beta }_i^2}}=-{\displaystyle \frac{n_i}{{\alpha }_i^2}}<0\end{array}$$

(16)

From Formula (16), the second-order derivative of the expected utility value Π₁ concerning the actual capacity ratio β_i is negative, indicating that the expected utility is a convex function of the enterprises’ effort level. Therefore, there exists an optimal actual capacity supply ratio β_i* that maximizes the enterprises’ expected utility, with the formula for the optimal ratio provided in Formula (17):

$${\beta }_i^{*}={\displaystyle \frac{\left({\omega }_i{r}_i+h_i{\alpha }_i{q}_i\right){\alpha }_i}{n_i}}$$

(17)

4.5.2. Analysis of Participation Constraints

The incentive constraints for transport enterprises are expressed through Formula (16). For the participation constraints, the Lagrangian function is constructed as follows:

$$\begin{array}{c}F\left({\omega }_i,{\gamma }_i,{\lambda }_1,{\lambda }_2\right)={\displaystyle \sum _{i=1}^n\left[\frac{\left[\left(1-{\omega }_i\right)r_i+b_i\right]{\beta }_i}{{\alpha }_i}-h_i{\beta }_i{q}_i-b_i\right]}-\omega \text{'}+{\lambda }_1\left[{\displaystyle \frac{\left({\omega }_i{r}_i+h_i{\alpha }_i{q}_i\right){\alpha }_i}{n_i}}-{\beta }_i^{\ast }\right]\\ +{\lambda }_2\left\{\omega \text{'}+{\displaystyle \sum _{i=1}^n\left[\frac{{\omega }_i{r}_i{\beta }_i}{{\alpha }_i}+h_i{\beta }_i{q}_i-\frac{n_i{\beta }_i^2}{2{\alpha }_i^2}-\frac{\rho {\left({\omega }_i{\sigma }_i\right)}^2}{2}\right]}-f\text{'}\right\}\end{array}$$

According to the KKT (Karush–Kuhn–Tucker) conditions, $\partial F/\partial {\omega }^{\prime }={\lambda }_2-1=0$ implies that λ₂ = 1. Additionally, since ${\lambda }_2\left\{\omega \text{'}+{\displaystyle \sum _{i=1}^n\left[\frac{{\omega }_i{r}_i{\beta }_i}{{\alpha }_i}+h_i{\beta }_i{q}_i-\frac{n_i{\beta }_i^2}{2{\alpha }_i^2}-\frac{\rho {\left({\omega }_i{\sigma }_i\right)}^2}{2}\right]}-f\text{'}\right\}=0$, the participation constraint for transport enterprises can be expressed as:

$$\omega \text{'}+{\displaystyle \sum _{i=1}^n\left[\frac{{\omega }_i{r}_i{\beta }_i}{{\alpha }_i}+h_i{\beta }_i{q}_i-\frac{n_i{\beta }_i^2}{2{\alpha }_i^2}-\frac{\rho {\left({\omega }_i{\sigma }_i\right)}^2}{2}\right]}=f\text{'}$$

(18)

4.5.3. Variable Solution

Substituting Formulas (17) and (18) into the objective function (14) yields:

$$\underset{{\omega }_i,{\gamma }_i}{\max }{\prod }_2={\displaystyle \sum _{i=1}^n\left\{\left(\frac{r_i+b_i-h_i{\alpha }_i^5{q}_i}{n_i}\right)r_i{\omega }_i-\left(\frac{r_i^2{\alpha }_i^4+n_i\rho {\sigma }_i^2}{2n_i}\right){\omega }_i^2+\frac{h_i{\alpha }_i{q}_i\left(2r_i+2b_i-h_i{\alpha }_i^5{q}_i\right)}{2n_i}-b\right\}}-f\text{'}$$

(19)

By solving the first-order and second-order partial derivatives of Π₂ concerning the government’s incentive coefficient ω_i, we obtain:

$$\begin{array}{l}{\displaystyle \frac{\partial {\prod }_2}{\partial {\omega }_i}}={\displaystyle \frac{\left(r_i+b_i-h_i{\alpha }_i{q}_i\right)r_i}{n_i}}-{\displaystyle \frac{r_i^2{\alpha }_i^4+n_i\rho {\sigma }_i^2}{n_i}}{\omega }_i\\ {\displaystyle \frac{{\partial }^2{\prod }_2}{\partial {\omega }_i^2}}=-{\displaystyle \frac{r_i^2{\alpha }_i^4+n_i\rho {\sigma }_i^2}{n_i}}<0\end{array}$$

(20)

Based on the property that the second-order derivative is less than 0, it is determined that an optimal incentive coefficient ω_i* exists:

$${\omega }_i^{\ast }={\displaystyle \frac{\left(r_i+b_i-h_i{\alpha }_i^5{q}_i\right)r_i}{r_i^2{\alpha }_i^4+n_i\rho {\sigma }_i^2}}$$

(21)

4.6. Model Analysis

This section evaluates how variations in factors like the committed supply ratio and economic compensation coefficient affect the incentive model outcomes.

4.6.1. Analysis of Optimal Actual Supply Ratio for Transport Enterprises

Theorem 1. 

The optimal actual capacity supply ratio β_i* increases with the committed supply ratio α_i, economic compensation coefficient h_i, and marginal social benefit r_i, and decreases with the marginal cost of supply n_i.

Proof of Theorem 1. 

The first-order derivatives of Equation (17) with respect to α_i, h_i, r_i, and n_i are as follows:

$$\begin{array}{l}{\displaystyle \frac{\partial {\beta }_i}{\partial {\alpha }_i}}={\displaystyle \frac{2hq\alpha +r\omega }{n}}>0\\ {\displaystyle \frac{\partial {\beta }_i}{\partial h_i}}={\displaystyle \frac{q{\alpha }^2}{n}}>0\\ {\displaystyle \frac{\partial {\beta }_i}{\partial r_i}}={\displaystyle \frac{\alpha \omega }{n}}>0\\ {\displaystyle \frac{\partial {\beta }_i}{\partial n_i}}=-{\displaystyle \frac{\alpha \left(hq\alpha +r\omega \right)}{n^2}}<0\end{array}$$

(22)

Equation (22) shows that the optimal actual capacity supply ratio β_i* is positively correlated with the committed supply ratio α_i, the economic compensation coefficient h_i, and the marginal social benefit r_i, while it is negatively correlated with the marginal cost of supply n_i.

The effort levels of various emergency supply ratios β_i/α_i represent the relationship between the actual capacity supply ratio β_i and the committed supply ratio α_i. Since the incentives are linearly related to effort levels, enterprises must fully consider this dynamic mechanism when developing supply strategies. When the committed supply ratio α_i increases, transport enterprises will realize that more effort will yield more significant incentives, prompting them to adjust their actual capacity supply ratio β_i to maximize returns. Conversely, if α_i decreases, enterprises may reduce their actual capacity supply ratio β_i to minimize costs and risks. This adjustment reflects enterprises’ sensitivity to the incentive mechanism and quick response to market changes.

The economic compensation for transport enterprises is closely related to the economic compensation coefficient h_i and the actual quantity of utilized capacity β_iq_i. When the economic compensation coefficient h_i increases, compensation per unit of utilized capacity rises, driving transport enterprises to increase the supply of various emergency capacities. This compensation mechanism directly influences the enterprises’ supply decisions. As the economic compensation coefficient h_i increases, enterprises are increasingly motivated to enhance their actual capacity supply ratio β_i, maximizing external incentives to achieve higher returns and social benefits. This positive feedback relationship further strengthens enterprises’ motivation to provide emergency transportation capacity, enabling them to meet social demands effectively.

Furthermore, when the marginal social benefit r_i increases, the efforts of transport enterprises are more easily converted into social benefits, indicating that the same effort generates more social benefits, offering stronger incentives for enterprises. When enterprises realize that their efforts can translate into significant social benefits, their motivation to increase effort rises, directly boosting the actual capacity supply ratio β_i. This positive relationship highlights the interaction between social benefits and enterprise efforts. It emphasizes the critical role of marginal social benefits in the incentive mechanism, further promoting the adequate supply of emergency transportation capacities.

Finally, when the marginal cost of supply n_i increases, transport enterprises face greater cost pressures in providing capacity services, necessitating more effort to maintain supply levels. In this case, economic returns are squeezed, potentially leading to a decline in effort levels. In weighing the benefits and costs, enterprises find that the returns from their efforts are insufficient to compensate for the increased costs, affecting their actual capacity supply ratio β_i for emergency transportation capacities. This cost–benefit shift makes it difficult for enterprises to maintain or increase the actual supply levels of various emergency transportation capacities, which may result in insufficient fulfillment of social demand. □

4.6.2. Analysis of the Government’s Optimal Incentive Coefficient

Theorem 2. 

The optimal incentive coefficient ω_i* decreases with the economic compensation coefficient h_i, effort cost of transportation capacity n_i, risk aversion coefficient ρ, and social benefit–risk level σ_i², while it increases with the government’s supervision cost b_i.

Proof of Theorem 2. 

The government will sign contracts with transport enterprises only if the social benefits r_i from the provided emergency capacities exceed the government’s compensation h_iα_iq_i; thus, $r_i>h_i{\alpha }_i{q}_i>h_i{\alpha }_i^5{q}_i$. The first-order derivatives of Equation (21) concerning h_i, n_i, ρ, σ_i², and b_i are as follows:

$$\begin{array}{l}{\displaystyle \frac{\partial {\omega }_i}{\partial h_i}}=-{\displaystyle \frac{q_i{r}_i{\alpha }_i^5}{r_i^2{\alpha }_i^4+n_i\rho {\sigma }_i^2}}<0\\ {\displaystyle \frac{\partial {\omega }_i}{\partial n_i}}=-{\displaystyle \frac{\left(b_i+r_i-h_i{q}_i{\alpha }_i^5\right)r_i\rho {\sigma }_i^2}{{\left(r_i^2{\alpha }_i^4+n_i\rho {\sigma }_i^2\right)}^2}}<0\\ {\displaystyle \frac{\partial {\omega }_i}{\partial \rho }}=-{\displaystyle \frac{\left(b_i+r_i-h_i{q}_i{\alpha }_i^5\right)r_i{n}_i{\sigma }_i^2}{{\left(r_i^2{\alpha }_i^4+n_i\rho {\sigma }_i^2\right)}^2}}<0\\ {\displaystyle \frac{\partial {\omega }_i}{\partial {\sigma }_i^2}}=-{\displaystyle \frac{\left(b_i+r_i-h_i{q}_i{\alpha }_i^5\right)r_i{n}_i\rho }{{\left(r_i^2{\alpha }_i^4+n_i\rho {\sigma }_i^2\right)}^2}}<0\\ {\displaystyle \frac{\partial {\omega }_i}{\partial b_i}}={\displaystyle \frac{r_i}{r_i^2{\alpha }_i^4+n_i\rho {\sigma }_i^2}}>0\end{array}$$

(23)

From Equation (23), the government’s optimal incentive coefficient ω_i* is negatively correlated with the economic compensation coefficient h_i, transport effort cost n_i, risk aversion coefficient ρ, and social benefit–risk level σ_i², while it is positively correlated with government supervision cost b_i.

The government’s incentive coefficient ω_i is negatively correlated with the economic compensation coefficient h_i. This phenomenon stems from their functional similarity, as both serve as government incentives to enhance enterprises’ ability to supply emergency capacities. Therefore, social benefit incentives and economic compensation for capacity requisition are substitutable tools. When designing the incentive framework, the government needs to know how these mechanisms interact to ensure enterprises are effectively incentivized to provide emergency capacities, maximizing social benefits.

When the effort cost n_i increases, transport enterprises typically decrease their actual capacity supply ratio β_i to reduce overall costs, based on a cost–risk trade-off. This may cause capacity shortages during emergencies, negatively impacting the timeliness and effectiveness of responses. In this scenario, the government should increase the incentive coefficient ω_i to boost enterprises’ expected returns and reduce cost pressures, encouraging them to maintain or expand capacity supply during emergencies.

Enterprises with high risk aversion ρ generally resist income uncertainty. This leads enterprises to adopt conservative strategies during emergencies, maintaining low actual capacity supply ratios β_i to minimize revenue fluctuations. Therefore, the government may reduce the social benefit incentive coefficient ω_i when designing incentive mechanisms and instead offer more stable and predictable incentives. This approach effectively lowers enterprises’ perceived risks and encourages them to provide as much emergency capacity as possible during emergencies.

An increase in the social benefit-risk level σ_i² indicates rising uncertainty in the external environment. Under such circumstances, the correlation between the social benefits r_i from transportation capacities and enterprise effort levels β_i/α_i weakens. This is because changes in the external environment often dominate fluctuations in social benefits, making it harder for enterprise efforts to translate into social returns. Therefore, increasing the incentive coefficient ω_i has a limited impact on the actual capacity supply ratio β_i, making it hard to fully realize the incentive mechanism’s positive effects.

As the government’s supervision costs b_i increase, the pressure and risks faced by transport enterprises also increase, prompting more cautious decision-making and avoiding speculative behavior. This trend encourages transport enterprises to continuously improve their effort levels β_i/α_i to meet regulatory requirements. Higher effort levels enhance enterprises’ emergency response capabilities, improving efficiency and generating more social benefits. □

5. Case Study Analysis

5.1. Numerical Simulation of the Incentive Model

This study uses numerical simulation to verify the optimal solution of the incentive model and explore the complex relationships among its variables.

Table 2. Transport enterprise-related parameters.

Transportation Capacity Type	αi (%)	qi (t)	Var (θi)	ni (Yuan)	ρ (%)
Vans	60	2500	0.5	150,000	80
Refrigerated Trucks	80	800	0.6	250,000
Semi-Trailers	50	3200	0.2	1,500,000

shows the situation of transport enterprises.

The government-related parameters are shown in

Table 3. Government-related parameters (unit: yuan).

Transportation Capacity Type	ri	hi	bi	ω’	f’
Vans	20,000	100	1000	500,000	700,000
Refrigerated Trucks	70,000	200	5000
Semi-Trailers	120,000	1200	3000

.

By substituting the above parameters into Formulas (17) and (18), we can calculate the optimal actual transportation capacity supply ratio for transport enterprises and the government’s optimal incentive coefficient. At the same time, using Formulas (5) and (10), we can also determine the expected utility values for both enterprises and the government. The specific results are shown in

Table 4. Numerical results of the incentive model.

	Vans	Refrigerated Trucks	Semi-Trailers
Government’s Optimal Incentive Coefficient ωi*	0.6012	0.7872	0.3999
Optimal Actual Capacity Supply Ratio for Transport Enterprises βi*	0.6481	0.5859	0.6560
Expected Utility Value for Transport Enterprises Π1	3,259,582.21
Expected Utility Value for Government Π2	−3,161,109.15

.

5.2. Sensitivity Analysis

To further explore the influence of various factors on the optimal decision strategies ω_i* and β_i*, as well as the expected utility values Π₁ and Π₂ for both the government and transport enterprises, this study uses simulation methods to analyze dynamic changes under variations in transport effort cost n_i, economic requisition compensation h_i, and government supervision cost b_i.

5.2.1. Transport Effort Cost

In the revenue structure of transport enterprises, transport effort cost n_i is the only factor fully controlled by the enterprises. To gain greater initiative in the emergency capacity pool of social vehicles, enterprises must thoroughly understand the interactions between effort cost n_i and the optimal decision strategies ω_i*, β_i*, and utility Π₁ for both the government and the enterprises. To explore these relationships, this study sets the variation range for n_i as: n_i_start = [100,000, 200,000, 1,000,000], n_i_end = [200,000, 400,000, 2,000,000]. Given the large differences in the values of n_i, the trends of ω_i* and β_i* for the three types of transportation capacities are compared using n_i/n_i_start as the scale on the graph’s horizontal axis. The relevant results are shown in Figure 3.

Figure 3a shows that the correlation between transport effort cost n_i and the government’s incentive coefficient ω_i is relatively low. This is because transport enterprises, when making decisions on effort input, focus more on their operational efficiency and market demands, rather than relying solely on the government’s social benefit incentives. This difference in decision-making orientation creates a disconnect between enterprises’ effort costs and the government’s social benefit incentives, limiting the impact of the former on the latter. In contrast, a significant negative correlation exists between the actual capacity supply ratio β_i and transport effort cost n_i. According to Theorem 1 and Figure 3b, the optimal actual capacity supply ratio β_i for enterprises is a decreasing function of transport effort cost n_i. This indicates that, when enterprises increase investment in effort cost, their actual capacity supply ratio β_i will correspondingly decrease, reflecting the need to balance costs and benefits in resource allocation.

Next, transport effort costs are plotted as coordinate points (n₁, n₂, n₃), with the size and color of the spheres representing the values of the utility function Π₁ for transport enterprises. The red sphere denotes the enterprises’ maximum utility value, and the changes in the utility function are shown in Figure 4.

As shown in Figure 4, a decrease in transport effort cost n_i reduces the overall costs of providing emergency transport capacity, thus strengthening enterprises’ willingness to raise their actual capacity supply ratio β_i. As emergency transportation resources increase, emergency response and service quality improve social benefits. This prompts the government to provide more economic incentives, increasing the compensation received by enterprises. Therefore, increasing supply enhances social benefits and generates higher profits for transport enterprises, encouraging their continued participation in the emergency capacity pool of social vehicles.

5.2.2. Government Supervision Cost

In the revenue structure of the government and transport enterprises, many factors are not fully controlled by the government, with government supervision cost b_i being one adjustable parameter. Therefore, it is crucial to investigate the interaction between government supervision cost b_i and the optimal decision strategies ω_i* and β_i* for both the government and enterprises. For this, this study sets the variation range for government supervision cost b_i as b_i_start = [2000, 3000, 2000], b_i_end = [4000, 6000, 4000]. Given the large differences in b_i values, b_i/b_i_start is used as the scale on the graph’s horizontal axis, and the relevant results are shown in Figure 5.

According to Theorem 2 and Figure 5a,b, there is a positive correlation between government supervision cost b_i, the government’s incentive coefficient ω_i, and the transport enterprise’s optimal actual capacity supply ratio β_i*. This implies that, when the government increases its supervision costs, the oversight risks for transport enterprises increase, prompting them to adopt more cautious operational decisions and reduce speculative behavior. Consequently, this regulatory environment compels enterprises to continually improve their effort levels β_i/α_i in capacity supply to gain the government’s trust. However, increasing effort levels β_i/α_i inevitably leads to higher operational costs, including labor, materials, and technology investments. To offset these costs and maintain profitability, transport enterprises will demand higher incentive coefficients as compensation, ensuring that they can achieve sustainable development and profitability while improving transportation capacity supply.

Based on the horizontal axis in Figure 5a,b, the reasonable range of b_i is [2000, 3000, 2000] to [3600, 6000, 4000]. Using the government supervision cost as coordinate points (b₁, b₂, and b₃), the size and color of the spheres represent the values of the government’s utility function Π₂. A blue sphere indicates the maximum utility, and the changes in the government’s utility function are shown in Figure 6.

From the analysis of Figure 5a,b and Figure 6, it is evident that there is a significant positive correlation between government supervision cost b_i and the government’s incentive coefficient ω_i. In contrast, the correlation with the actual capacity supply ratio β_i of transport enterprises is relatively small. This indicates that, when government supervision cost b_i decreases, the actual capacity supply ratio β_i of transport enterprises only slightly decreases, while the government’s incentive coefficient ω_i decreases significantly. This suggests that, when the social benefits generated by transport enterprises fluctuate minimally, the government can retain more social benefits, thus increasing its expected utility.

5.2.3. Economic Requisition Compensation

To explore the relationship between the economic compensation coefficients h₁, h₂, and h₃ provided by the government to transport enterprises and the optimal decision strategies ω_i*, β_i*, along with the expected utility values Π₁, Π₂, a variation range of h_i was set. Specifically, h_i_start = [200, 400, 1000], and h_i_end = [400, 800, 2000]. Given the large variations in the economic compensation coefficients h_i, this study uses h_i/h_i_start as the scale on the horizontal axis to compare the trends of ω_i* and β_i* across three types of transportation capacity. The relevant trend results are shown in Figure 7.

Figure 7a and Theorem 2 show that ω_i and h_i are negatively correlated, indicating that ω_i shows a decreasing trend as h_i increases. Additionally, Theorem 1 indicates that the actual capacity supply ratio β_i of transport enterprises is positively correlated with both h_i and ω_i. Therefore, when h_i increases and ω_i decreases, the change in β_i depends on the correlation strength between these two factors. Figure 7b shows that as h_i increases, β_i decreases, indicating that the social benefit incentive coefficient ω_i has a greater impact on β_i than the economic compensation coefficient h_i. Moreover, based on the horizontal axis scale in Figure 7a,b, the reasonable range of h_i is determined to be [280, 720, 1500] to [320, 760, 1600].

Next, using the economic compensation coefficients (h₁, h₂, and h₃) as coordinate points, the size and color intensity of the spheres represent the utility functions values Π₁ and Π₂ for the transport enterprises and the government. The maximum utility values for enterprises and the government are marked in red and blue, respectively, as shown in Figure 8.

Figure 8a shows that, when the economic compensation coefficient h_i decreases, the expected utility of transport enterprises increases. This can be attributed to the negative correlation between h_i and the incentive coefficient ω_i. The revenue of transport enterprises depends more on social benefit incentives than on economic compensation alone. Thus, as h_i increases, the incentive coefficient ω_i decreases, ultimately decreasing the revenue of transport enterprises. In contrast, Figure 8b shows that the expected utility value Π₂ for the government follows the opposite trend to that of enterprises. For the government, the challenge lies in how to appropriately balance incentive and compensation mechanisms to achieve coordinated development among economic growth, social benefits, and resource allocation.

6. Discussion

This study employs numerical simulations and sensitivity analysis to clarify how key parameters influence interactions between the government and transport enterprises. It identifies a weak positive correlation between transport effort cost n_i and the incentive coefficient ω_i, underscoring the limitations of the incentive mechanism on enterprise cost structures. As the incentive mechanism partly relies on social benefits, it does not directly affect enterprises’ operational costs. Under financial strain, transport enterprises often reduce their actual capacity supply ratio β_i. This contrasts with the report by Guan et al., who argue that cost-sharing subsidies strengthen the link between external incentives and internal costs, improving the correlation between effort costs and the incentive coefficient. This comparison provides new insights into the effectiveness of various incentive strategies [24].

Furthermore, government supervision costs b_i and the incentive coefficient ω_i are significantly positively correlated. With stricter regulations, enterprises in high-pressure environments are more likely to meet contractual obligations. However, such environments can cause enterprise dissatisfaction, requiring higher-incentive coefficients to mitigate discontent. Excessive incentives combined with supervision costs may strain government finances. Thus, institutional design should balance supervision costs and incentives to enhance supply and regulatory efficiency. This aligns with the report by Zhang et al., who highlighted the crucial role of regulation and incentives in corporate risk-taking [33].

A reduction in the economic compensation coefficient h_i increases transport enterprises’ utility Π₁, reflecting greater reliance on government-provided social benefit incentives. This finding contrasts sharply with those of Liu et al., who stress the significance of economic compensation [11]. In summary, the government should flexibly adjust the balance between benefit incentives and economic compensation mechanisms when designing subsidy policies to optimize policy outcomes and enterprise behavior. This approach promotes a positive government–enterprise cooperation dynamic and ensures improved emergency response capabilities.

Human factors are critical in the decision-making process for emergency transportation supply, significantly shaping corporate behavior and decisions. An enterprise’s decision-making culture and its management’s risk preferences directly affect supply strategy choices. When management exhibits strong innovation and a willingness to take risks, they are more likely to maintain or increase the actual capacity supply ratio β_i amid market uncertainty and high-pressure costs. This perspective aligns with that of Todaro et al., who emphasized the critical role of corporate culture and management’s risk attitudes in decision-making and their profound impact on strategic choices [34]. Additionally, trust between the government and enterprises is another essential human factor. Enterprises that perceive government regulatory policies as fair and transparent are more likely to cooperate and comply with regulations, enhancing emergency capacity supply. Conversely, skepticism about government regulatory measures may lead enterprises to adopt conservative strategies, lowering their actual capacity supply ratio β_i. This view echoes that of Mizrahi et al., who found that enterprise cooperation in policy implementation strongly correlates with their trust in government regulation [35].

External environmental factors play a significant role in shaping the decision-making processes of governments and transport enterprises during responses to unconventional emergencies. Intensified external environmental disturbances, reflected by a rising societal benefit–risk level σ_i², lead to greater fluctuations in societal benefits. These changes diminish the effectiveness of societal benefits derived from the emergency transportation services offered by enterprises. As societal benefits form a key part of government incentives for enterprises, their decline directly reduces the actual capacity supply ratio β_i provided by enterprises. In this context, a clearly defined compensation mechanism is more effective in motivating transportation enterprises than frequently altering incentive mechanisms. This perspective aligns with that of Liu et al., who found that, under heightened external environmental uncertainty, fixed compensation incentives outperform reward and reputation incentives in effectively and rapidly enhancing the motivation of social enterprises [11].

Based on the above analysis, the government should implement targeted incentive measures that align with enterprises’ cost structures and transportation supply capacities. A cost-sharing mechanism can effectively boost enterprise motivation and increase actual capacity supply within the incentive framework. Additionally, the government should consider corporate culture and management’s risk preferences, employing more adaptable regulatory and incentive strategies to motivate enterprises to sustain or enhance their actual capacity supply ratio in the face of market uncertainties. Strengthening the trust between the government and enterprises and ensuring fair and transparent policies can significantly increase enterprises’ willingness to cooperate, thereby increasing emergency capacity supply.

7. Conclusions

This paper discusses the joint emergency transportation capacity reserve between the government and transport enterprises. It considers the actual distribution of urban transportation resources and the selection principles of transport enterprises. An incentive model for an emergency capacity pool of social vehicles is developed based on principal–agent theory. By analyzing the revenue structures of government agencies and enterprises, this paper explores the government’s optimal incentive coefficient and the optimal actual transportation capacity supply ratio of enterprises. Finally, numerical simulations and sensitivity analysis are employed to examine how different model elements influence the optimal strategies and utilities of government–enterprise cooperation, offering new insights into emergency management from theoretical and practical perspectives.

• The systematic selection of transport enterprises is based on four dimensions: grading, categorization, distribution, and integrity. This approach provides practical strategies to meet diverse emergency transportation needs. First, a reserve structure is proposed in which state-owned transport enterprises form the backbone, supplemented by high-quality private enterprises, ensuring stable and adequate capacity provision during emergencies. Second, functional classification increases the specificity and flexibility of emergency responses. Geographically, a well-planned distribution of transportation resources ensures that high-frequency event areas are prioritized while enabling rapid capacity provision in other regions. Finally, transport enterprises with strong integrity records help to reduce transaction costs and mitigate risks associated with incomplete or concealed information. This framework offers essential theoretical guidance for the government’s selection of transport enterprises;
• There is a positive correlation between the government’s incentive coefficient and the actual capacity supply ratio of transport enterprises. This suggests that a higher incentive coefficient effectively motivates transport enterprises, increasing their willingness and ability to provide emergency capacity. This finding suggests that, if government compensation policies are effectively aligned with enterprises’ actual capacity supply ratios, it will significantly improve their service capacity during crises, thereby enhancing the efficiency of the entire emergency management system;
• A substitution relationship exists between the government’s social benefit incentives and the economic compensation for requisition. This result indicates that a reasonable regulatory mechanism should be established during policy formulation to balance social benefit incentives with economic compensation, preventing over-reliance on either. Achieving this balance enables the government to enhance enterprises’ sense of responsibility and participation while more effectively achieving its policy goals. This important contribution offers valuable guidance for policymakers in designing and implementing support policies for transport enterprises and advances the development of emergency transportation management research.

While this study proposes selection criteria for transport enterprises, these standards currently function as a theoretical framework. Future research could develop specific evaluation metrics and quantitative standards for each principle to improve their practical applicability. This paper also applies principal–agent theory to examine the strategic interactions between the government and transport enterprises in constructing an emergency capacity pool of social vehicles. Due to the limitations of the researchers, the model considers only a single-period, one-to-one scenario between the government and enterprises, with assumptions that may not fully apply to real-world situations. While the numerical analysis of the model offers theoretical insights, its effectiveness depends on specific scenarios and parameter settings. Future research should explore multi-period and one-to-many principal–agent models between the government and enterprises to examine dynamic interactions across diverse scenarios and timeframes. Incorporating real-world data into refined analyses is crucial to improving research findings’ practical relevance and validity. This study does not address constraints on the economic compensation governments provide to transport enterprises. However, given the practical significance of governmental budget limitations, future research should incorporate such constraints into the model for analysis. Lastly, differences in risk preferences between governments and transport enterprises influence their optimal decisions and utilities. Future studies should explore how these differences impact decision-making, offering more profound insights into the role of risk management in emergency capacity planning.