Incentive Mechanism of Utility Tunnel PPP Projects with User Involvement

Abstract

Users of utility tunnel public–private partnership (PPP) projects are productive corporate entities with a specific scope, a limited quantity, and strong bargaining power. They possess the ability and motivation to participate in PPP projects through equity investments. Tunnel PPP projects involving user participation are implemented jointly by consortia comprising construction contractors and users. Given that the government, construction contractors, and users have different interests, significant conflicts of interest arise among them. Furthermore, there exists dual information asymmetry between the government and construction contractors, as well as between the government and users. The government lacks a direct observation of the true ability endowments of construction contractors and users prior to signing PPP contracts, and it cannot directly observe the effort levels exerted by the construction contractors and users after contract signing. The combination of dual information asymmetry and conflicting interests may result in adverse selection and moral hazard on the part of construction contractors and users. Drawing on principal–agent theory, this study constructed incentive models with and without user involvement, examined the impact of user involvement on the effort level of construction contractors and the performance of PPP projects, and designed a reasonable incentive mechanism. The objective is to ensure that construction contractors and users truthfully report their ability endowments during the bidding stage and exert optimal efforts during the construction and operation stages. This research provides new insights for promoting the sustainable development of tunnel PPP projects. A case analysis of a PT tunnel PPP project and an SC tunnel PPP project demonstrates the reasonableness and feasibility of the research findings.

1. Introduction

The public–private partnership (PPP), derived from the public finance initiative (PFI) in the United Kingdom, represents an innovative approach for governments to provide public goods [1]. PPP transforms the delivery of public goods from solely a government responsibility to a collaborative effort between the government and private partners. This approach allows for the exploration and utilization of the private sector’s comparative advantages, including technical expertise, innovation capabilities, and information integration, thereby enhancing the efficiency of public goods provision. Over the past decade, PPP arrangements have been increasingly adopted by numerous Chinese cities, with construction contractors emerging as the predominant private entities in the Chinese PPP market [2].

In 2020, construction contractors successfully secured a total of 567 PPP projects, amounting to RMB 1.369 trillion, representing 47.67% of all projects and averaging RMB 2.414 billion per project. In comparison, operators obtained a total of 617 PPP projects, valued at RMB 475.4 billion, accounting for 16.53% of projects, with an average project value of RMB 0.769 billion. However, the dominance of construction contractors in tunnel PPP projects may pose significant challenges to their sustainable development. Contractors often prioritize their construction-related benefits through engineering procurement and construction while disregarding project operations. Consequently, PPP projects led by contractors may fall into the trap of prioritizing construction over operation, potentially compromising the quality of infrastructure and public services, thus failing to meet the original expectations.

The “Guiding Opinions of the General Office of the State Council on Promoting the Construction of Urban Underground Utility Tunnels,” issued by the General Office of the State Council of China in August 2015, mandates local governments to prioritize the establishment or collaboration between hosted pipeline units and other private organizations to form Special Purpose Vehicles (SPVs) [2]. These SPVs serve as vehicles for user involvement in tunnel PPP projects. For instance, in a tunnel PPP project in Changsha city, users such as Changsha Water Group and Changsha Gas Group hold a 14% equity stake in the SPV, granting them veto power over crucial matters related to public welfare, including price adjustments [2]. The involvement of users through equity investment in tunnel PPP projects has a significant impact on the behavior of other participants and infuses new vitality and momentum into the sustainable development of the projects, as illustrated in Figure 1. Firstly, user involvement can enhance the availability and quality of project outputs. Users clarify their demands, leading to improved availability, and exercise effective supervision, contributing to enhanced output quality. Secondly, users and other investors possess distinct comparative advantages, allowing for the allocation of responsibilities among stakeholders. By leveraging the capabilities of users, responsibilities can be shared with other investors. Lastly, user involvement fosters increased enthusiasm and control over quality and cost, thereby exerting a demonstration effect that attracts potential users. These outcomes promote market development, reduce project market risks, mitigate financing risks, and ultimately positively impact the entire life cycle performance of PPP projects.

The government, users, and contractors have divergent interests, resulting in significant conflicts among them. In the absence of user involvement, a dual information asymmetry exists between the government and the contractors. The government is unable to ascertain the contractors’ capacity endowments during the bidding stage and cannot directly observe their effort levels during the construction and operation stages. In tunnel PPP projects with user involvement, consortiums consisting of contractors and users are responsible for implementation. Apart from the contractors, there is also a dual information asymmetry between the government and the users. The government cannot observe the users’ true capacity endowment prior to PPP contract signing, nor can it directly observe their effort levels afterward. These information asymmetries, combined with interest conflicts, can lead to adverse selection and moral hazard among the contractors and users. Drawing upon principal–agent theory, the incentive model without user involvement aims to incentivize contractors to truthfully report their capacity endowments through well-designed incentive mechanisms. This approach facilitates the assessment of contractors’ capabilities and encourages optimal efforts in both the construction and operation stages. The incentive model with user involvement explores the influence of user participation on contractors’ effort levels and project performance. It seeks to ensure the truthful reporting of capacity endowments by contractors and users during the bidding stage through reasonable incentive mechanisms while also motivating them to exert optimal efforts throughout the construction and operation stages. This research direction offers a fresh perspective on promoting the sustainable development of tunnel PPP projects.

The theoretical contribution of this study is the introduction of a novel, innovative approach called “user involvement” in tunnel PPP projects with corporate users. It examines the incentive issues faced by private partners in PPP projects with user involvement, thereby addressing a gap in the existing literature. The practical significance of this study lies in its potential to enhance the efficiency of fiscal expenditure and promote the sustainable development of tunnel PPP projects. By establishing a reasonable revenue-sharing mechanism with user involvement, the performance of PPP projects can be improved. This, in turn, reduces the challenges and costs associated with selecting private partners and lowers the financing expenses of PPP projects. Consequently, it lessens the government’s investment and subsidies for PPP projects, enhances the efficiency of fiscal expenditure, and facilitates the realization of project sustainability.

2. Literature Review

2.1. Users and Sustainability of PPP Projects

Users of PPP projects can be categorized as “people” [3,4,5] or rational consumers with individual needs, or they may represent the general public. In the context of Chinese PPP-related regulations and practices, users refer to individuals or entities directly utilizing or benefiting from project outputs. Users play a pivotal role as core stakeholders in PPP projects [6], and effective communication with users is essential for accurately understanding their needs [5], thereby ensuring the success of PPP projects [7]. Conversely, neglecting or disregarding users can impede timely and appropriate responses to their concerns by both the government and private entities [8], potentially leading to user opposition [9]. User opposition refers to users safeguarding their own interests by refusing to pay for or utilize project outputs, and it is a significant factor contributing to the failure of sustainable development in PPP projects [10,11].

User opposition often arises in projects that employ a “user-pays” approach in the public sector’s adoption of PPP [12]. This can be attributed to the conflicting revenue and profitability objectives of the project consortium and the economic interests of users [13]. To mitigate user opposition in PPP projects, it is necessary to foster cross-organizational cooperation for managing user needs [10,11] and establish more flexible and sustainable management strategies among key stakeholders [11,14,15]. Researchers have proposed incorporating users or the public as the core stakeholders in PPP arrangements, establishing the P4 model to enhance project sustainability. Ahmed and Ali [16] advocated for shifting users’ roles from passive service recipients to active service partners in PPP projects for solid waste management. Building upon this idea, Majamaa et al. [9] proposed the implementation of user-oriented public services in real estate PPP projects, thereby developing the P4 model and outlining its fundamental principles. Ng et al. [17] suggested a P4 process framework based on Hong Kong PPP practices, which incorporates bottom-up participative strategies to make user engagement more transparent in infrastructure planning, policy making, and decision making. Such strategies empower citizens by diverting decision-making authority from policy makers and enabling proactive engagement. These approaches can facilitate the establishment of a decision-making mechanism with clearly defined responsibilities and rights, promote public participation, and mitigate the risk of public opposition. Kumaraswamy et al. [4] proposed incorporating users into PPP projects through the P4 scheme to strengthen formal–informal relationships and enhance the projects’ “social infrastructures”, ultimately leading to more resilient partnerships and infrastructure that offers higher value for money. Drawing on a case study in Finland, Torvinen and Ulkuniemi [18] introduced the P4 model, which transforms users into value co-creators in PPP projects.

2.2. Information Asymmetry of PPP Projects

According to Wang et al. [19], dual information asymmetry is commonly observed in PPP projects. This means that private entities possess two types of private information. The first type pertains to the real enterprise information, including their capacity endowments. Before the signing of concession contracts, the government is aware that the private entities fall into one of several possible types, but they cannot determine the specific type, resulting in the challenge of adverse selection [19]. Adverse selection typically occurs during the procurement stages of principal–agent relationships, wherein inefficient private entities may win franchise rights for PPP projects due to prior information-hiding behaviors [20,21]. The second type of private information concerns the behavior of private entities. After the signing of concession contracts, the government lacks the ability to observe the effort exerted by the private entities [19], which leads to moral hazard stemming from speculation and opportunism on the part of the private entities [22,23]. This situation gives rise to unobservable defaults, such as substandard engineering quality and inefficient public services [19].

Wang et al. [19] developed an incentive mechanism model that addresses the issues of adverse selection and moral hazard in PPP projects. The model utilizes project output performance as an incentive approach to encourage private entities to reveal their true capability endowments and exert optimal efforts, thereby enhancing project outputs. Through the design of reasonable contracts, the incentive mechanism model ensures that private entities are motivated to perform effectively. In a study conducted by Liu et al. [23] on the opportunistic behaviors of private entities in the early termination of PPP projects, it was found that government supervision plays a crucial role. The research suggests that the government can effectively control the opportunistic behavior of private entities by increasing penalties in cases of early termination or non-default termination due to a government default. However, in cases where the termination is caused by the SPV, it is recommended that the government adopt more frequent supervision, as the default behaviors of private entities cannot be effectively controlled through punishment.

2.3. Research Gap

Although previous studies have highlighted the significance of user involvement in the sustainable development of PPP projects and explored the establishment and enhancement of public participation mechanisms or the public–private–people partnership (P4) model [4,17,24], as well as examined the incentive issues of PPP projects under information asymmetry [21] and investigated the incentive problems faced by private entities under dual information asymmetry [19], there is still a research gap concerning the incentive issue of productive corporate users participating as private partners in PPP projects under dual information asymmetry to improve project performance and ensure project sustainability. User involvement refers to corporate users becoming shareholders of the SPVs through equity investment, engaging in PPP projects throughout their lifecycle, and influencing project performance by clarifying user needs, establishing communication and coordination with other private partners, establishing channels for information sharing, and sharing benefits and risks. For PPP projects that involve productive corporate users with a specific scope, a limited number, and strong bargaining power, such as tunnel PPP projects, this study proposes a fundamental incentive model and an incentive model with user involvement based on principal–agent theory. The aim is to screen capability endowments and motivate the best efforts of both users and contractors under dual information asymmetry, thus advancing the research on incentive mechanisms and sustainability in PPP projects.

Although existing studies have emphasized the importance of user involvement in the sustainability development of PPP projects and explored the establishment and improvement of public participation mechanisms or the public–private–people partnership (P4) mode [4,17,24] and also discussed the incentive issues of PPP projects under information asymmetry [21], as well as investigated the incentive problems of the private entities under dual information asymmetry [19], there is still a research gap on the incentive issue of some productive corporate users involved as the private partners of PPP projects under dual information asymmetry to enhance project performance and fulfill project sustainability. Involvement means that users become shareholders of SPVs through equity investment, participate in PPP projects throughout the whole life cycle, and affect the performance of PPP projects by clarifying user needs, establishing communication and coordination with other private partners, building information-sharing channels, and sharing benefits and risks. For PPP projects having productive corporate users with a specific scope, a limited number, and strong bargaining power, such as tunnel PPP projects, this study thus built a basic incentive model and an incentive model with user involvement based on principal–agent theory to achieve the purposes of capability endowment screening and motivating the best efforts of users and contractors under dual information asymmetry, so as to enhance the relevant research on the incentive mechanism and sustainability of PPP projects.

3. Incentive Model of Tunnel PPP Projects with User Involvement under Dual Information Asymmetry

3.1. Model Hypotheses

3.1.1. Game Subjects

According to our previous study [2], construction contractors were found to be the primary private partners in tunnel PPP projects. Therefore, in the absence of user involvement, it is presumed that the private partners of tunnel PPP projects consist solely of construction contractors. However, when users are involved, the private partners are formed by consortia comprising both construction contractors and users.

In the absence of user involvement, the government and construction contractors have distinct interest demands, leading to a game-like relationship between them. However, when users are involved in tunnel PPP projects, the dynamics change, and principal–agent relationships emerge between the government and the consortium. The government and the consortium become key stakeholders in the tunnel PPP projects, engaging in game-like interactions due to their differing interest demands.

Users of utility tunnels in tunnel PPP projects typically include municipal pipeline units, such as water, hot water and steam, natural gas, sewage, and rainwater systems [25]. These users possess distinct professional backgrounds, operate in different business fields, and possess technical expertise that differs from construction contractors. Given these differences, construction contractors do not possess comparative advantages in the operation and maintenance of tunnels. When users become involved in tunnel PPP projects, they can have various impacts on the projects through the following channels. First, construction contractors remain responsible for the actual construction of tunnel PPP projects since users are not specialized in project construction. However, users influence project outputs by sharing their demand quantities, demand qualities, and demand forms with construction contractors [26]. Second, during the operation stages, users and contractors jointly manage project operations. Users are responsible for the maintenance of their respective pipelines, while contractors oversee the maintenance of the public area of the utility tunnels. For example, specific arrangements are explicitly stated in the franchise agreement of the Changsha tunnel PPP project, including the implementation of the Engineering, Procurement, and Construction (EPC) mode for newly built tunnels and the joint responsibility of users and contractors for the design and operation of the tunnel PPP project.

3.1.2. Game Incomes

(1)

Without user involvement

The life cycle of a tunnel PPP project can be divided into a construction phase and operation phase. In the situation without user involvement, the construction output of the tunnel PPP project is determined by the contractor’s construction effort $e_{i10}$ ($i=h,l$, where $h$ represents high-capability contractors, and $l$ represents low-capability contractors) and the government’s construction coordination effort $a_{10}$, i.e., $E[{\vartheta }_{i0}|e_{i10},a_{10}]=e_{i10}+a_{10}$ (${\vartheta }_{i0}$ represents the construction quality of the tunnel PPP project when users have not been involved). The government’s construction coordination effort cost is $a_{10}^2$. Referring to the study by Bhaskaran and Krishnan [27], the contractors’ construction effort cost is $I{e}_{i10}^2+c_i{t}_1{e}_{i10}$. Here, $I{e}_{i10}^2$ is the fixed construction cost; $I$ is contractors’ fixed cost coefficient; $c_i{t}_1{e}_{i10}$ is the variable construction cost; $t_1$ is the construction duration of the tunnel PPP project; and the coefficient $c_i$ is contractors’ variable cost coefficient, which is used to measure contractors’ capabilities of construction and operation. Since the capability endowments are the private information of the contractors, referring to the research by Wang et al. [19], it is assumed that the government infers the capability endowments of the contractors from Equation (1).

$$c_i=\left\{\begin{array}{l}{c}_h, \mathrm{With} \mathrm{the} \mathrm{probability} \mathrm{of} \rho \\ \\ c_l, \mathrm{With} \mathrm{the} \mathrm{probability} \mathrm{of} 1-\rho \end{array}\right.$$

(1)

Referring to the study by Xiao and Xu [28], the operation income of the tunnel PPP project depends on the following: (1) the construction quality of the tunnel PPP project ${\vartheta }_{i0}$; (2) contractors’ operation efforts $e_{i20}$; (3) the operation coordination effort of the government $a_{20}$; (4) the influence of bias, $\epsilon \sim N(0,{\delta }^2)$. Therefore, the operation income is ${\chi }_{i0}={\vartheta }_{i0}+v{e}_{i20}+\mu a_{20}+\epsilon$. Here, the coefficient $v$ ($0<v<1$) is contractors’ operation contribution coefficient, indicating the importance of the contractors in the tunnel PPP project’s operation stage; $\mu$ ($0<\mu <1$) is the government’s operation contribution coefficient, representing the contribution degree of the government to the operation of the tunnel PPP project; and $v+\mu =1$ when the users are not involved. According to the research by Liu et al. [29], the operation effort cost of the contractors is $I{e}_{i20}^2+c_i{t}_2{e}_{i20}$, in which the fixed operation cost is $I{e}_{i20}^2$, the variable operation cost is $c_i{t}_2{e}_{i20}$, and $t_2$ is the entire operation duration.

(2)

User involvement

When users are involved in the tunnel PPP project, the construction output is related to the contractors’ construction effort $e_{i11}$, users’ information-sharing effort $m_{k11}$ ($k=g,b$, where $g$ represents good-capability users, and $b$ represents poor-capability users), and the construction coordination effort of the government, i.e., $E[{\vartheta }_{ik}|e_{i11},m_{k11},a_{11}]=e_{i11}+m_{k11}+a_{11}$ (${\vartheta }_{ik}$ represents the construction quality of the tunnel PPP project with user involvement). According to the research by Liu et al. [29], users’ information-sharing effort cost is $I{m}_{k11}^2+c_k{t}_1{m}_{k11}$, and $c_k$ is users’ variable cost coefficient. For simplicity, it is assumed that dual information asymmetry exists merely between the government and the consortiums, and the government infers the ability of the consortiums from Equation (2). When the consortium is a strong-capability consortium, $c_i=c_h$ and $c_k=c_g$; when the consortium is a weak-capability consortium, $c_i=c_l$ and $c_k=c_b$. It is clear that $c_b>c_g$, $c_h>c_l$.

$$\left\{\begin{array}{l}{c}_i=c_h \mathrm{and} c_k=c_g, \mathrm{With} \mathrm{the} \mathrm{probability} \mathrm{of} \rho \\ \\ c_i=c_l \mathrm{and} c_k=c_b, \mathrm{with} \mathrm{the} \mathrm{probability} \mathrm{of} 1-\rho \end{array}\right.$$

(2)

When users are involved, the operation income of the tunnel PPP project is ${\chi }_{ik1}={\vartheta }_{ik}+v{e}_{i21}+\eta m_{k21}+\mu a_{21}+\epsilon$. In the formula, $e_{i21}$ stands for contractors’ operation effort, $m_{k21}$ stands for users’ operation effort, $a_{21}$ is the government’s operation coordination effort, and $\eta$ ($0<\eta <1$ and $v+\eta +\mu =1$) is users’ operation contribution coefficient, indicating the importance of the users in the PPP project operation stage. According to the research by Liu et al. [29], the users’ operation effort cost is $I{m}_{k21}^2+c_k{t}_2{m}_{k21}$, of which the fixed operation cost is $I{m}_{k21}^2$, and the variable operation cost is $c_k{t}_2{m}_{k21}$.

(3)

Income structure

Principal–agent theory shows that a linear income distribution structure possesses optimal stability. Many scholars in the PPP domain, therefore, have adopted a linear structure to measure the income of private partners [19,30]. Referring to the research by Zhang et al. [30], private partners’ income is divided into fixed income and shared income. Specifically, contractors’ income is ${\omega }_{i0}+{\alpha }_{i0}{\chi }_{i0}$ in the basic incentive model (i.e., incentive model without user involvement), among which ${\omega }_{i0}$ is fixed income, referring to contractors’ contract income as professional companies to undertake the PPP project [30], and the contract income is assumed to be paid after the completion of the project and independent of contractors’ effort; ${\alpha }_{i0}$ represents contractors’ operation-revenue-sharing coefficient, which is related to contractors’ effort [30]. In the incentive model with user involvement, contractors’ income is ${\omega }_{i1}+{\alpha }_{i1}{\chi }_{ik1}$; similar to the basic incentive model, ${\omega }_{i1}$ is fixed income, which is unrelated to contractors’ effort, and ${\alpha }_{i1}$ is contractors’ operation-revenue-sharing coefficient, which is related to contractors’ effort. Users’ income is ${\omega }_{k1}+{\alpha }_{k1}{\chi }_{ik1}$, which also includes fixed income and shared income; users do not participate in project construction, and users are equivalent to financial investors during the construction stage [31]. Fixed income ${\omega }_{k1}$ refers to respecting return on investment, which is irrelevant to users’ effort; shared income ${\alpha }_{k1}{\chi }_{ik1}$ means that the users, as the investors of the tunnel PPP project, share the project’s operation income in a proportion of ${\alpha }_{k1}$ during the operation phase, which is related to users’ effort.

3.1.3. Game Sequences

(1)

Without user involvement

The game sequence between the government and contractors in the incentive model without user involvement is shown in Figure 2. First, the government invites bids for the tunnel PPP project and proposes an initial contract $\left\{{\alpha }_{i0},{\omega }_{i0}\right\}$ at time $T_0$. Second, contractors weigh their own capabilities and interests to determine which contract to choose at time $T_1$; the contractors may truthfully report their capability endowments and choose contract $\left\{{\alpha }_{i0},{\omega }_{i0}\right\}$ that matches their capabilities; the contractors may also falsely report their capability endowments and choose contract $\left\{{\alpha }_{j0},{\omega }_{j0}\right\}$($j=h,l$ and $j\ne i$) that does not match their capabilities. Third, the contractors independently determine the construction effort $e_{i10}$ from time $T_1$ to $T_3$, and the government simultaneously determines the construction coordination effort $a_{10}$; the PPP project is successfully constructed at time $T_3$, the contractors obtain the fixed income (${\omega }_{i0}$ or ${\omega }_{j0}$) corresponding to the prior selected contract type, and the project enters the operation phase. At last, time $T_3$ to $T_4$ is the operation phase of the PPP project; the contractors independently decide the operation effort $e_{i20}$, the government independently decides the operation coordination effort $a_{20}$, and the contractors share the operation income according to the sharing coefficient (${\alpha }_{i0}$ or ${\alpha }_{j0}$) specified in the selected contract.

(2)

With user involvement

The game sequence between the government and the consortiums in the incentive mechanism with user involvement is shown in Figure 3. First, the government proposes an initial contract $\left\{{\alpha }_{i1},{\alpha }_{k1},{\omega }_{i1},{\omega }_{k1}\right\}$ at time $T_0$, where $i=h$, $k=g$ or $i=l$, $k=b$. Second, the consortiums comprehensively consider their own capabilities and interests to decide the type of contract to choose at time $T_1$. Type $ik$ consortiums may choose the contract $\left\{{\alpha }_{i1},{\alpha }_{k1},{\omega }_{i1},{\omega }_{k1}\right\}$ that matches their ability endowments, or they may choose a contract $\left\{{\alpha }_{j1},{\alpha }_{n1},{\omega }_{j1},{\omega }_{n1}\right\}$($n=g,b$ and $n\ne k$) that falsely matches their ability endowments. Third, from time $T_1$ to $T_3$, the contractors in the consortiums independently determine the construction effort, the users independently determine the information-sharing effort, and the government independently determines the construction coordination effort. The PPP project will be successfully constructed at time $T_3$, the contractors and the users receive the corresponding fixed income ($\left({\omega }_{i1},{\omega }_{k1}\right)$ or $\left({\omega }_{j1},{\omega }_{n1}\right)$) according to their selected contract, and the project enters the operation phase. Fourth, from time $T_3$ to $T_4$, the contractors and the users independently decide their own operation effort, the government simultaneously independently decides its operation coordination effort, and the contractors and users each share the operation income according to the specified sharing coefficient ($\left({\alpha }_{i1},{\alpha }_{k1}\right)$ or $\left({\alpha }_{j1},{\alpha }_{n1}\right)$).

3.2. Construction, Solution, and Discussion of Basic Incentive Mechanism

3.2.1. Construction of Basic Incentive Mechanism

Based on principal–agent theory and the reverse induction method, a basic incentive model without user involvement was constructed.

(1)

Contractors truthfully report their capability endowments

When contractors truthfully report their capacity endowments, after the successful construction of the PPP project, the type $i$ contractor will receive fixed income ${\omega }_{i0}$ and share the operation income according to the coefficient ${\alpha }_{i0}$ during the operation phase. Comprehensively analyzing the model hypotheses, the expected benefit to contractors ${\pi }_{s0}\left(i,i\right)$ and the expected benefit to the government $G_0\left(i,i\right)$ at the initial stage of the PPP project can be expressed as follows when contractors truthfully report their capacity endowments.

$${\pi }_{s0}^{}\left(i,i\right)=\underset{e_{i10}\ge 0,e_{i20}\ge 0}{\max }{\alpha }_{i0}\left(e_{i10}+a_{10}+v{e}_{i20}+\mu a_{20}\right)-I\left(e_{i10}^2+e_{i20}^2\right)-c_i\left(t_1{e}_{i10}+t_2{e}_{i20}\right)+{\omega }_{i0}$$

(3)

$$G_0\left(i,i\right)=\underset{a_{10}\ge 0,a_{20}\ge 0}{\max }\left(1-{\alpha }_{i0}\right)\left(e_{i10}+a_{10}+v{e}_{i20}+\mu a_{20}\right)-a_{20}^2-a_{10}^2-{\omega }_{i0}$$

(4)

Calculating Equations (3) and (4), when contractors truthfully report their capacity endowments, the expected benefit to the contractors and the government at the initial stage of the PPP project can be expressed as follows:

$${\pi }_{s0}^{}\left(i,i\right)={\alpha }_{i0}\left[\frac{\left(1+{\mu }^2\right)\left(1-{\alpha }_{i0}\right)}{2}+\frac{{\alpha }_{i0}\left(1+v^2\right)-2c_i\left(t_1+v{t}_2\right)}{4I}\right]+\frac{c_i^2\left(t_1^2+t_2^2\right)}{4I}+{\omega }_{i0}$$

(5)

$$G_0\left(i,i\right)=\left(1-{\alpha }_{i0}\right)\left[\frac{\left(1+{\mu }^2\right)\left(1-{\alpha }_{i0}\right)}{4}+\frac{{\alpha }_{i0}\left(1+v^2\right)-c_i\left(t_1+v{t}_2\right)}{2I}\right]-{\omega }_{i0}$$

(6)

(2)

Contractors falsely report their capability endowments

When contractors falsely report their capacity endowments, after the successful construction of the PPP project, the type $i$ contractor will receive fixed income ${\omega }_{j0}$ and share the operation income according to the coefficient ${\alpha }_{j0}$ during the operation phase. Therefore, when contractors falsely report capacity endowments, the expected benefit to the contractors ${\pi }_{s0}\left(i,j\right)$ and the expected benefit to the government $G_0\left(i,j\right)$ at the initial stage of the PPP project can be expressed as follows:

$${\pi }_{s0}^{}\left(i,j\right)=\underset{e_{i10}\ge 0,e_{i20}\ge 0}{\max }{\alpha }_{j0}\left(e_{i10}+a_{10}+v{e}_{i20}+\mu a_{20}\right)-I\left(e_{i10}^2+e_{i20}^2\right)-c_i\left(t_1{e}_{i10}+t_2{e}_{i20}\right)+{\omega }_{j0}$$

(7)

$$G_0\left(i,j\right)=\underset{a_{10}\ge 0,a_{20}\ge 0}{\max }\left(1-{\alpha }_{j0}\right)\left(e_{i10}+a_{10}+v{e}_{i20}+\mu a_{20}\right)-a_{20}^2-a_{10}^2-{\omega }_{j0}$$

(8)

Solving Equations (7) and (8), when contractors falsely report their capacity endowments, the expected benefit to the contractors and the government at the initial stage of the PPP project can be respectively expressed as follows:

$${\pi }_{s0}^{}\left(i,j\right)={\alpha }_{j0}\left[\frac{\left(1+{\mu }^2\right)\left(1-{\alpha }_{j0}\right)}{2}+\frac{{\alpha }_{j0}\left(1+v^2\right)-2c_i\left(t_1+v{t}_2\right)}{4I}\right]+\frac{c_i^2\left(t_1^2+t_2^2\right)}{4I}+{\omega }_{j0}$$

(9)

$$G_0\left(i,j\right)=\left(1-{\alpha }_{j0}\right)\left[\frac{\left(1+{\mu }^2\right)\left(1-{\alpha }_{j0}\right)}{4}+\frac{{\alpha }_{j0}\left(1+v^2\right)-c_i\left(t_1+v{t}_2\right)}{2I}\right]-{\omega }_{j0}$$

(10)

(3)

Inventive mechanism without user involvement

It can be concluded from Equation (6) that, when cooperating with high-capability contractors and the contractor truthfully reports its capability endowment, the government’s optimal expected revenue $G_0\left(h,h\right)$ at the initial stage of the PPP project can be expressed as Equation (11).

$$G_0\left(h,h\right)=\left(1-{\alpha }_{h0}\right)\left[\frac{\left(1+{\mu }^2\right)\left(1-{\alpha }_{h0}\right)}{4}+\frac{{\alpha }_{h0}\left(1+v^2\right)-c_h\left(t_1+v{t}_2\right)}{2I}\right]-{\omega }_{h0}$$

(11)

Similarly, when cooperating with low-capability contractors and the contractor truthfully reports its capability endowment, the government’s expected revenue $G_0\left(l,l\right)$ at the initial stage of the PPP project can be expressed as Equation (12).

$$G_0\left(l,l\right)=\left(1-{\alpha }_{l0}\right)\left[\frac{\left(1+{\mu }^2\right)\left(1-{\alpha }_{l0}\right)}{4}+\frac{{\alpha }_{l0}\left(1+v^2\right)-c_l\left(t_1+v{t}_2\right)}{2I}\right]-{\omega }_{l0}$$

(12)

To encourage contractors to participate in the project and truthfully report its capacity endowment, the government’s incentive mechanism design at the initial stage of the PPP project should satisfy the following trade-offs.

$$\begin{gathered} \underset{{\alpha }_{h0}\ge 0,{\alpha }_{l0}\ge 0,{\omega }_{h0},{\omega }_{l0}}{\max }\rho G_0\left(h,h\right)+\left(1-\rho \right)G_0\left(l,l\right) \\ \mathrm{s}.\mathrm{t}. (I{C}_{h0}){\pi }_{s0}\left(h,h\right)\ge {\pi }_{s0}\left(h,l\right) \\ (I{C}_{l0}){\pi }_{s0}\left(l,l\right)\ge {\pi }_{s0}\left(l,h\right) \\ (I{R}_{h0}){\pi }_{s0}\left(h,h\right)\ge 0 \\ (I{R}_{l0}){\pi }_{s0}\left(l,l\right)\ge 0 \end{gathered}$$

(13)

Incentive compatibility constraints $I{C}_{h0}$ and $I{C}_{l0}$ are satisfied, which means that the expected benefits to contractors (including high-capability and low-capability contractors) from truthfully reporting their capability endowments are no less than those from falsely reporting their capability endowments and ensures that the high-capacity contractor chooses contract $\left\{{\alpha }_{h0},{\omega }_{h0}\right\}$, and the low-capacity contractor chooses contract $\left\{{\alpha }_{l0},{\omega }_{l0}\right\}$. The establishment of participation constraints $I{R}_{h0}$ and $I{R}_{l0}$ means that the benefits to contractors participating in the PPP project are not less than their retained income, which ensures that the contractors are willing to accept the contract.

3.2.2. Solution of Basic Incentive Model

By solving the basic incentive model in Equation (13), the optimal solution in

Table 1. Optimal solution of the basic incentive model.

Items	Expressions
${\alpha }_{h0}^{*}$	$\frac{1+v^2}{1+I+v^2+I{\mu }^2}$
${\alpha }_{l0}^{*}$	$\frac{\left(1+v^2\right)\left(1-\rho \right)+\rho \left(c_h-c_l\right)\left(t_1+v{t}_2\right)}{\left(1-\rho \right)\left(1+I+v^2+I{\mu }^2\right)}$
$e_{i10}^{*}$	$\frac{{\alpha }_{i0}^{*}-c_i{t}_1}{2I}$
$e_{i20}^{*}$	$\frac{v{\alpha }_{i0}^{*}-c_i{t}_2}{2I}$

can be obtained.

3.2.3. Discussion of Basic Incentive Model

Proposition 1.

The presence of an upper threshold is observed for the optimal revenue-sharing coefficient for contractors. Specifically, the optimal revenue-sharing coefficient for high-capability contractors remains independent of the government’s belief and remains fixed at the upper threshold. In contrast, the optimal revenue-sharing coefficient for low-capability contractors demonstrates a negative relationship with the government’s belief and is either equal to or less than the upper threshold. Moreover, when the market share of high-capability contractors surpasses a certain threshold, it results in the exclusion of low-capability contractors from the PPP market.

Proof. 

The relationship between the contractors’ optimal revenue-sharing coefficient and the contractors’ market distribution is shown in Figure 4. The following can be summarized from Figure 3. (1) There is an upper threshold for the optimal revenue-sharing coefficient of the contractors in the PPP market, which is the optimal revenue-sharing coefficient of high-capability contractors. (2) Let $\rho =0$. It can be calculated that ${\alpha }_{h0}^{*}={\alpha }_{l0}^{*}$, which shows that, when the government believes that there are only low-capability contractors in the market, low-capability contractors can also obtain the same benefit as high-capability contractors. (3) Let ${\alpha }_{l0}^{*}=0$ to solve for $\rho$. When the market distribution of high-capability contractors reaches $\tilde{\rho }=\frac{1+v^2}{1+v^2+\left(c_l-c_h\right)\left(t_1+v{t}_2\right)}$, low-capability contractors will be forced to withdraw from the PPP market. □

Proposition 1 emphasizes the crucial role of addressing dual information asymmetry when formulating incentive contracts. It underscores the need for the government to establish distinct and optimal revenue-sharing coefficients for contractors with high and low capabilities. When determining the optimal profit-sharing coefficient for high-capability contractors, several factors should be taken into account, including the project’s fixed cost, the government’s operating contribution coefficient, and the contractors’ operating contribution coefficient. On the other hand, the optimal revenue-sharing coefficient for low-capability contractors exhibits a negative correlation with the market distribution of high-capability contractors. In instances where past experiences reveal a higher prevalence of high-capability contractors in the PPP market, indicating a stronger likelihood of government collaboration with competent private entities, the optimal revenue-sharing coefficient for low-capability contractors will be comparatively lower.

Proposition 2.

When the capability endowment gap between high- and low-capability contractors is not greater than  ${\tau }_{10}$, the incentive mechanism can identify the capabilities of the contractors. When the capability endowment gap between the high- and low-capability contractors is greater than ${\tau }_{10}$, the incentive mechanism will drive low-capability contractors out of the PPP market.

Proof. 

Referring to Table 1, let ${\alpha }_{l0}^{*}=0$ to solve for $c_l$. It can be determined that ${\tau }_{10}=\frac{\left(1-\rho \right)\left(1+v^2\right)}{\rho \left(t_1+v{t}_2\right)}$ and $\frac{\partial {\tau }_{10}}{\partial \rho }=-\frac{1+v^2}{{\rho }^2\left(t_1+v{t}_2\right)}<0$, which shows that ${\tau }_{10}$ is negatively related to the government’s belief. When the government considers that the contractors have a high probability of being high-capability contractors (that is, it is inferred from past experience that high-capability contractors occupy a larger market share), low-capability contractors will be driven out of the PPP market by the incentive mechanism, even if there is only a small capability endowment gap between low-capability and high-capability contractors. In contrast, according to past experiences, the smaller the proportion of high-capability contractors in the PPP market, the lower the eviction threshold: even if the capability difference between low-capability and high-capability contractors is relatively large, the incentive mechanism will allow them to exist, and the eviction threshold will be significantly lowered. □

According to Proposition 2, when formulating incentive contracts, it is essential for the government to comprehensively evaluate the operational contribution coefficients of both the government and the private partner, as well as the fixed costs involved. These factors will help determine the upper threshold of the optimal benefit-sharing coefficient, which is also applicable to high-capability contractors. Additionally, the government should consider the market distribution of high-capability contractors and the disparity in capabilities among contractors of different levels when establishing the optimal revenue-sharing coefficient for low-capability contractors. Drawing on past experience, if the proportion of high-capability contractors in the PPP market is substantial, the government can set higher entry barriers for low-capability contractors, resulting in their expulsion from the market, even if there is a relatively small gap in capability endowments between low-capability and high-capability contractors. Conversely, if the proportion of high-capability contractors in the PPP market is relatively small, the government should lower the entry threshold and allow the existence of low-capability contractors, even if their abilities are significantly inferior to those of high-capability contractors.

Proposition 3.

The optimal incentive mechanism with dual information asymmetry can encourage high-capability contractors to exert more construction and operation efforts than low-capability contractors, and it can fully tap the potential and value of high-capability contractors.

Proof. 

${\alpha }_{h0}^{*}-{\alpha }_{l0}^{*}=\frac{\rho \left(c_l-c_h\right)\left(t_1+v{t}_2\right)}{\left(1-\rho \right)\left(1+I+v^2+I{\mu }^2\right)}>0$, which indicates that the optimal incentive mechanism without user involvement will give high-capability contractors higher shared benefits. $e_{h10}^{*}-e_{l10}^{*}=\frac{{\alpha }_{h0}^{*}-{\alpha }_{l0}^{*}}{2I}+\frac{\left(c_l-c_h\right)t_1}{2I}>0$, which indicates that the optimal incentive mechanism will prompt higher construction effort from high-capability contractors. $e_{h20}^{*}-e_{l20}^{*}=\frac{v\left({\alpha }_{h0}^{*}-{\alpha }_{l0}^{*}\right)}{2I}+\frac{\left(c_l-c_h\right)t_2}{2I}>0$, which indicates that the optimal incentive mechanism can ensure that high-capability contractors exert more operation effort than low-capability contractors. □

3.3. Construction, Solution, and Discussion of Incentive Model with User Involvement

3.3.1. Construction of Incentive Model with User Involvement

(1)

Consortiums truthfully report their capability endowments

When consortiums truthfully report their capacity endowments, after the successful construction of the PPP project, type $i$ contractors will receive fixed income ${\omega }_{i1}$, and type $k$ users will receive fixed income ${\omega }_{k1}$; in the operation phase, type $i$ contractors will share the income according to sharing coefficient ${\alpha }_{i1}$, and type $k$ users will share the income according to sharing coefficient ${\alpha }_{k1}$. The operation income of the tunnel PPP projects with user involvement is ${\chi }_{i1}=e_{i11}+m_{k11}+a_{11}+v{e}_{i21}+\eta m_{k21}+\mu a_{21}$. Therefore, the expected benefits of contractors, users, and the government at the initial stage of the PPP project can be expressed as Equations (14) to (16) when the consortium truthfully reports their capacity endowments.

$${\pi }_{s1}^{}\left(i,i,k,k\right)=\underset{e_{i11}\ge 0,e_{i21}\ge 0}{\max }{\alpha }_{i1}\left(e_{i11}+m_{k11}+a_{11}+v{e}_{i21}+\eta m_{k21}+\mu a_{21}\right)-I\left(e_{i11}^2+e_{i21}^2\right)-c_i\left(t_1{e}_{i11}+t_2{e}_{i21}\right)+{\omega }_{i1}$$

(14)

$${\pi }_{u1}^{}\left(i,i,k,k\right)=\underset{m_{11}\ge 0,m_{21}\ge 0}{\max }{\alpha }_{k1}\left(e_{i11}+m_{k11}+a_{11}+v{e}_{i21}+\eta m_{k21}+\mu a_{21}\right)-I\left(m_{k11}^2+m_{k21}^2\right)-c_k\left(t_1{m}_{k11}+t_2{m}_{k21}\right)+{\omega }_{k1}$$

(15)

$$G_1^{}\left(i,i,k,k\right)=\underset{a_{11}\ge 0,a_{21}\ge 0}{\max }\left(1-{\alpha }_{i1}-{\alpha }_{k1}\right)\left(e_{i11}+m_{k11}+a_{11}+v{e}_{i21}+\eta m_{k21}+\mu a_{21}\right)-a_{11}^2-a_{21}^2-{\omega }_{i1}-{\omega }_{k1}$$

(16)

Solving Equations (14)–(16), when consortiums truthfully report capacity endowments, the optimal expected benefits of contractors, users, and the government at the initial stage of the PPP project are as follows:

$${\pi }_{s1}\left(i,i,k,k\right)={\alpha }_{i1}\left[\frac{\left(1-{\alpha }_{i1}-{\alpha }_{k1}\right)\left(1+{\mu }^2\right)}{2}+\frac{{\alpha }_{k1}\left(1+{\eta }^2\right)-c_k\left(t_1+\eta t_2\right)}{2I}\right]+{\alpha }_{i1}\left[\frac{{\alpha }_{i1}\left(1+v^2\right)-2c_i\left(t_1+v{t}_2\right)}{4I}\right]+\frac{c_i\left(t_1^2+t_2^2\right)}{4I}+{\omega }_{i1}$$

(17)

$${\pi }_{u1}\left(i,i,k,k\right)={\alpha }_{k1}\left[\frac{\left(1-{\alpha }_{i1}-{\alpha }_{k1}\right)\left(1+{\mu }^2\right)}{2}+\frac{{\alpha }_{i1}\left(1+v^2\right)-c_i\left(t_1+v{t}_2\right)}{2I}\right]+{\alpha }_{k1}\left[\frac{{\alpha }_{k1}\left(1+{\eta }^2\right)-2c_k\left(t_1+\eta t_2\right)}{4I}\right]+\frac{c_k\left(t_1^2+t_2^2\right)}{4I}+{\omega }_{k1}$$

(18)

$$G_1\left(i,i,k,k\right)=\left(1-{\alpha }_{i1}-{\alpha }_{k1}\right)\left[\frac{\left(1-{\alpha }_{i1}-{\alpha }_{k1}\right)\left(1+{\mu }^2\right)}{4}+\frac{{\alpha }_{i1}\left(1+v^2\right)-c_i\left(t_1+v{t}_2\right)}{2I}\right]+\left(1-{\alpha }_{i1}-{\alpha }_{k1}\right)\left[\frac{{\alpha }_{k1}\left(1+{\eta }^2\right)-c_k\left(t_1+\eta t_2\right)}{2I}\right]-{\omega }_{i1}-{\omega }_{k1}$$

(19)

(2)

Consortiums falsely report their capability endowments

When consortiums falsely report capacity endowments, after the successful construction of the PPP project, type $i$ contractors will receive fixed income ${\omega }_{j1}$, and type $k$ users will receive fixed income ${\omega }_{n1}$; in the operation phase, type $i$ contractors will share operation income according to sharing coefficient ${\alpha }_{j1}$, and type $k$ users will share operation income according to sharing coefficient ${\alpha }_{n1}$. The expected benefits of contractors, users, and the government at the initial stage of the PPP project can be expressed as follows when consortiums falsely report capability endowments:

$${\pi }_{s1}^{}\left(i,j,k,n\right)=\underset{e_{11}\ge 0,e_{21}\ge 0}{\max }{\alpha }_{j1}\left(e_{i11}+m_{k11}+a_{11}+v{e}_{i21}+\eta m_{k21}+\mu a_{21}\right)-I\left(e_{i11}^2+e_{i21}^2\right)-c_i\left(t_1{e}_{i11}+t_2{e}_{i21}\right)+{\omega }_{j1}$$

(20)

$${\pi }_{u1}^{}\left(i,j,k,n\right)=\underset{m_{11}\ge 0,m_{21}\ge 0}{\max }{\alpha }_{n1}\left(e_{i11}+m_{k11}+a_{11}+v{e}_{i21}+\eta m_{k21}+\mu a_{21}\right)-I\left(m_{k11}^2+m_{k21}^2\right)-c_k\left(t_1{m}_{k11}+t_2{m}_{k21}^{}\right)+{\omega }_{n1}$$

(21)

$$G_1^{}\left(i,j,k,n\right)=\underset{a_{11}\ge 0,a_{21}\ge 0}{\max }\left(1-{\alpha }_{j1}-{\alpha }_{n1}\right)\left(e_{i11}+m_{k11}+a_{11}+v{e}_{i21}+\eta m_{k21}+\mu a_{21}\right)-a_{11}^2-a_{21}^2-{\omega }_{j1}-{\omega }_{n1}$$

(22)

Calculating Equations (20)–(22), when consortiums falsely report their capability endowments, the optimal expected benefits to contractors, users, and the government at the initial stage of the PPP project can be expressed as follows:

$${\pi }_{s1}\left(i,j,k,n\right)={\alpha }_{j1}\left[\frac{\left(1-{\alpha }_{j1}-{\alpha }_{n1}\right)\left(1+{\mu }^2\right)}{2}+\frac{{\alpha }_{n1}\left(1+{\eta }^2\right)-c_k\left(t_1+\eta t_2\right)}{2I}\right]+{\alpha }_{j1}\left[\frac{{\alpha }_{j1}\left(1+v^2\right)-2c_i\left(t_1+v{t}_2\right)}{4I}\right]+\frac{c_i\left(t_1^2+t_2^2\right)}{4I}+{\omega }_{j1}$$

(23)

$${\pi }_{u1}\left(i,j,k,n\right)={\alpha }_{n1}\left[\frac{\left(1-{\alpha }_{j1}-{\alpha }_{n1}\right)\left(1+{\mu }^2\right)}{2}+\frac{{\alpha }_{j1}\left(1+v^2\right)-c_i\left(t_1+v{t}_2\right)}{2I}\right]+{\alpha }_{n1}\left[\frac{{\alpha }_{n1}\left(1+{\eta }^2\right)-2c_k\left(t_1+\eta t_2\right)}{4I}\right]+\frac{c_k\left(t_1^2+t_2^2\right)}{4I}+{\omega }_{n1}$$

(24)

$$G_1\left(i,j,k,n\right)=\left(1-{\alpha }_{j1}-{\alpha }_{n1}\right)\left[\frac{\left(1-{\alpha }_{j1}-{\alpha }_{n1}\right)\left(1+{\mu }^2\right)}{4}+\frac{{\alpha }_{j1}\left(1+v^2\right)-c_i\left(t_1+v{t}_2\right)}{2I}\right]+\left(1-{\alpha }_{j1}-{\alpha }_{n1}\right)\left[\frac{{\alpha }_{n1}\left(1+{\eta }^2\right)-c_k\left(t_1+\eta t_2\right)}{2I}\right]-{\omega }_{j1}-{\omega }_{n1}$$

(25)

(3)

Incentive mechanism with user involvement

It can be seen from Equation (19) that when cooperating with a strong-capability consortium and the consortium truthfully reports its capability endowment, the government’s expected revenue $G_1\left(h,h,g,g\right)$ at the initial stage of the PPP project can be expressed as follows:

$$G_1\left(h,h,g,g\right)=\left(1-{\alpha }_{h1}-{\alpha }_{g1}\right)\left[\frac{\left(1-{\alpha }_{h1}-{\alpha }_{g1}\right)\left(1+{\mu }^2\right)}{4}+\frac{{\alpha }_{h1}\left(1+v^2\right)-c_h\left(t_1+v{t}_2\right)}{2I}\right]+\left(1-{\alpha }_{h1}-{\alpha }_{g1}\right)\left[\frac{{\alpha }_{g1}\left(1+{\eta }^2\right)-c_g\left(t_1+\eta t_2\right)}{2I}\right]-{\omega }_{h1}-{\omega }_{g1}$$

(26)

It can be seen from Equation (19) that when cooperating with a weak-ability consortium and the consortium truthfully reports its capability endowment, the government’s expected revenue $G_1\left(l,l,b,b\right)$ at the initial stage of the PPP project can be expressed as follows:

$$G_1\left(l,l,b,b\right)=\left(1-{\alpha }_{l1}-{\alpha }_{b1}\right)\left[\frac{\left(1-{\alpha }_{l1}-{\alpha }_{b1}\right)\left(1+{\mu }^2\right)}{4}+\frac{{\alpha }_{l1}\left(1+v^2\right)-c_l\left(t_1+v{t}_2\right)}{2I}\right]+\left(1-{\alpha }_{l1}-{\alpha }_{b1}\right)\left[\frac{{\alpha }_{b1}\left(1+{\eta }^2\right)-c_b\left(t_1+\eta t_2\right)}{2I}\right]-{\omega }_{l1}-{\omega }_{b1}$$

(27)

To motivate consortiums to actively participate in the PPP project and truthfully report their ability endowments, the government’s incentive mechanism design with user involvement at the initial stage of the project should satisfy the trade-offs shown in Equation (28) (the constraints in Equation (28) have been simplified, and unnecessary constraints have been removed).

$$\begin{gathered} \underset{{\alpha }_{h1},{\alpha }_{l1},{\alpha }_{g1},{\alpha }_{b1}}{\max }\rho G_1\left(h,h,g,g\right)+\left(1-\rho \right)G_1\left(l,l,b,b\right) \\ \mathrm{s}.\mathrm{t}.(I{C}_{h1}){\pi }_{s1}^{}\left(h,h,g,g\right)\ge {\pi }_{s1}^{}\left(h,l,g,b\right)\hspace{1em} (I{C}_{g1}){\pi }_{u1}^{}\left(h,h,g,g\right)\ge {\pi }_{u1}^{}\left(h,l,g,b\right) \\ (I{C}_{l1}){\pi }_{s1}^{}\left(l,l,b,b\right)\ge {\pi }_{s1}^{}\left(l,h,b,g\right)\hspace{1em} (I{C}_{b1}){\pi }_{u1}^{}\left(l,l,b,b\right)\ge {\pi }_{u1}^{}\left(l,h,b,g\right) \\ (I{R}_{l1}){\pi }_{s1}^{}\left(l,l,b,b\right)\ge 0\hspace{1em} (I{R}_{b1}){\pi }_{u1}^{}\left(l,l,b,b\right)\ge 0 \end{gathered}$$

(28)

Incentive compatibility constraints $I{C}_{h1}$, $I{C}_{l1}$, $I{C}_{g1}$, and $I{C}_{b1}$ are satisfied, which means that the incomes of high-capability private entities truthfully reporting their capability endowments are not less than those of such entities falsely reporting their capability endowments and ensures that high-capability private entities choose contract $\left\{{\alpha }_{h1},{\alpha }_{g1},{\omega }_{h1},{\omega }_{g1}\right\}$ and low-capability private entities choose contract $\left\{{\alpha }_{l1},{\alpha }_{b1},{\omega }_{l1},{\omega }_{b1}\right\}$; incentive participation constraints $I{R}_{l1}$ and $I{R}_{b1}$ are satisfied, which means that the incomes obtained by contractors and users from participating in the PPP project are not less than their retained income and ensures that consortiums are willing to accept the contract.

3.3.2. Solution of Incentive Model with User Involvement

By solving the incentive model in Equation (28), the optimal solution in

Table 2. Optimal solution of the incentive model with user involvement.

Items	Expressions
${\alpha }_{h1}^{*}$	$\frac{I\left(1+{\mu }^2\right)\left(v^2-{\eta }^2\right)+\left(1+{\eta }^2\right)\left(1+v^2\right)}{I\left(1+{\mu }^2\right)\left(2+v^2+{\eta }^2\right)+\left(1+{\eta }^2\right)\left(1+v^2\right)}$
${\alpha }_{g1}^{*}$	$\frac{I\left(1+{\mu }^2\right)\left({\eta }^2-v^2\right)+\left(1+{\eta }^2\right)\left(1+v^2\right)}{I\left(1+{\mu }^2\right)\left(2+v^2+{\eta }^2\right)+\left(1+{\eta }^2\right)\left(1+v^2\right)}$
${\alpha }_{l1}^{*}$	$\frac{\rho \left(1+{\eta }^2\right)F_1\left(c_h,c_l,c_g,c_b\right)/\left(\rho -1\right)+F_2\left(c_h,c_l,c_g,c_b\right)}{F_3\left(c_h,c_l,c_g,c_b\right)}$
${\alpha }_{b1}^{*}$	$\frac{\rho \left(1+v^2\right)F_1\left(c_h,c_l,c_g,c_b\right)/\left(\rho -1\right)+F_4\left(c_h,c_l,c_g,c_b\right)}{F_3\left(c_h,c_l,c_g,c_b\right)}$
$e_{i11}^{*}$	$\frac{{\alpha }_{i1}^{*}-c_i{t}_1}{2I}$
$e_{i21}^{*}$	$\frac{v{\alpha }_{i1}^{*}-c_i{t}_2}{2I}$
$m_{k11}^{*}$	$\frac{{\alpha }_{k1}^{*}-c_k{t}_1}{2I}$
$m_{k21}^{*}$	$\frac{\eta {\alpha }_{k1}^{*}-c_k{t}_2}{2I}$

Note: $F_1\left(c_h,c_l,c_g,c_b\right)=\left(t_1+v{t}_2\right)\left(c_l-c_h\right)+\left(t_1+{\eta }_2\right)\left(c_b-c_g\right)$; $F_2\left(c_h,c_l,c_g,c_b\right)=\left(1+v^2\right)\left(1+{\eta }^2\right)+I\left(1+{\mu }^2\right)\left(v^2-{\eta }^2\right)$;.$F_3\left(c_h,c_l,c_g,c_b\right)=\left(1+v^2\right)\left(1+{\eta }^2\right)+I\left(1+{\mu }^2\right)\left(2+{\eta }^2+v^2\right)$; $F_4\left(c_h,c_l,c_g,c_b\right)=\left(1+v^2\right)\left(1+{\eta }^2\right)+I\left(1+{\mu }^2\right)\left({\eta }^2-v^2\right)$.

can be obtained.

3.3.3. Discussion of Incentive Model with User Involvement

Proposition 4.

The optimal incentive mechanism with user involvement, through a revenue-sharing coordination mechanism between contractors and users, establishes an internal competition mechanism in the private consortiums, restrains the speculative behavior of the private entities, and prevents them from receiving excess returns.

Proof. 

The solutions of the incentive model with user involvement show that ${\alpha }_{h1}^{*}=\frac{1+v^2-I{\alpha }_{g1}^{*}\left(1+{\mu }^2\right)}{1+I+v^2+I{\mu }^2}$ and ${\alpha }_{g1}^{*}=\frac{1+{\eta }^2-I{\alpha }_{h1}^{*}\left(1+{\mu }^2\right)}{1+I+{\eta }^2+I{\mu }^2}$, indicating that the optimal revenue-sharing coefficients of high-capability contractors and good-capability users are negatively correlated: that is, the shared revenue of these two is “one increasing and the other decreasing”. ${\alpha }_{h1}^{*}-{\alpha }_{g1}^{*}=\frac{2I\left(1+{\mu }^2\right)\left(v^2-{\eta }^2\right)}{I\left(1+{\mu }^2\right)\left(2+v^2+{\eta }^2\right)+\left(1+{\eta }^2\right)\left(1+v^2\right)}$, where the private entities with higher operation contributions will obtain more shared benefits, indicating that there is a competition mechanism between contractors and users, and contractors or users can “encroach” on the other party’s shared benefits by increasing their operation contribution, which will, to a certain extent, force the other party to make improvements. □

Proposition 4 demonstrates a discernible competitive relationship arising between contractors and users upon user involvement. In light of this, it is essential for the government to establish a sound performance evaluation mechanism that adequately assesses the operational contributions of both contractors and users. Consequently, private partners exhibiting higher operational contributions should be entitled to increased shared benefits as a means of fostering fair competition and incentivizing enhanced performance.

Proposition 5.

With a greater a priori probability that contractors or users are high-capability entities, the optimal incentive mechanism with user involvement, by squeezing the expected shared benefits of low-capability entities and by restricting or even excluding the activity space of low-capability entities, will maximize the likelihood of the government to cooperate with high-capability entities.

Proof. 

The partial derivative of the difference in the optimal revenue-sharing coefficient between high-capability contractors and low-capability contractors $\Delta {\alpha }_{s1}$ to $\rho$ is $\frac{\partial \Delta {\alpha }_{s1}}{\partial \rho }=\frac{\left(1+{\eta }^2\right)\left(\left(c_b-c_g\right)\left(t_1+\eta t_2\right)+\left(c_l-c_h\right)\left(t_1+v{t}_2\right)\right)}{{\left(1-\rho \right)}^2\left(\left(1+{\eta }^2\right)\left(1+I+v^2+I{\mu }^2\right)+I\left(1+{\mu }^2\right)\left(1+v^2\right)\right)}>0$. This indicates that, in the incentive mechanism with user involvement, if past experience shows that the higher the proportion of high-capability contractors in the PPP market, the greater the gap in the upper thresholds of the sharing coefficient between high-capability contractors and users, and the lower the possibility of their participation in the project.

Similarly, $\frac{\partial \Delta {\alpha }_{u1}}{\partial \rho }=\frac{\left(1+v^2\right)\left(\left(c_b-c_g\right)\left(t_1+\eta t_2\right)+\left(c_l-c_h\right)\left(t_1+v{t}_2\right)\right)}{{\left(1-\rho \right)}^2\left(\left(1+{\eta }^2\right)\left(1+I+v^2+I{\mu }^2\right)+I\left(1+{\mu }^2\right)\left(1+v^2\right)\right)}>0$, which indicates that the higher the proportion of high-capability contractors, the less the shared benefits to low-capability contractors in the optimal inventive mechanism with user involvement, and the lower the possibility and enthusiasm of poor-capability users to participate in the PPP project. □

Proposition 6.

The optimal incentive mechanism with user involvement can motivate high-capability contractors and good-capability users to put in more effort than low-capability entities and fully tap the potential and value of high-capability entities.

Proof. 

$e_{h11}^{*}-e_{l11}^{*}=\frac{{\alpha }_{h1}^{*}-{\alpha }_{l1}^{*}}{2I}+\frac{\left(c_l-c_h\right)t_1}{2I}>0$, it shows that the optimal incentive mechanism with user involvement will motivate high-capability contractors to exert greater construction effort than low-capability contractors. $e_{h21}^{*}-e_{l21}^{*}=\frac{v\left({\alpha }_{h1}^{*}-{\alpha }_{l1}^{*}\right)}{2I}+\frac{\left(c_l-c_h\right)t_2}{2I}>0$, which indicates that the optimal incentive mechanism with user involvement will motivate high-capability contractors to expend more operation effort than low-capability contractors.

$m_{g11}^{*}-m_{b11}^{*}=\frac{{\alpha }_{g1}^{*}-{\alpha }_{b1}^{*}}{2I}+\frac{\left(c_b-c_g\right)t_1}{2I}>0$, which shows that the optimal incentive mechanism with user involvement will motivate good-capability users to expend more information-sharing effort than poor-capability users. $m_{g21}^{*}-m_{b21}^{*}=\frac{\eta \left({\alpha }_{g1}^{*}-{\alpha }_{b1}^{*}\right)}{2I}+\frac{\left(c_b-c_g\right)t_2}{2I}>0$, it indicates that the optimal incentive mechanism with user involvement will motivate good-capability users to pay more operation efforts than poor-capability users. □

4. Comparative Analyses of Models

4.1. Theoretical Contrastive Analysis

Proposition 7.

The optimal incentive mechanism with user involvement generates a “catfish” effect by introducing users, which helps to encourage high-capability contractors to continuously improve their own PPP project operation contributions and to further tap the potential and value of high-capability contractors while ensuring that high-capability contractors do not obtain excess returns.

Proof. 

In the basic incentive model and the incentive model with user involvement, relationships between the optimal revenue-sharing coefficients of high-capability entities are shown in Figure 5. Figure 5 shows that the upper threshold of the optimal revenue-sharing coefficient of high-capability contractors with user involvement is ${\alpha }_{h0}^{*}$, indicating that the optimal incentive mechanism with user involvement can constrain the shared revenue of high-capability contractors within a reasonable range to protect the public interest by preventing high-capability contractors from gaining excess returns. Figure 5 additionally illustrates that, in the incentive mechanism with user involvement, private entities possessing higher operation contributions have stronger voices and can obtain more shared benefits. Therefore, when the operation contribution of high-capability contractors is low (as shown in curve ${\alpha }_{h1}^{*}\left(v<\eta \right)$), by furtherly leveraging their own comparative advantages in technology and management to improve their operation contributions to the PPP project, high-capability contractors can make the revenue curve move toward ${\alpha }_{h1}^{*}\left(v=\eta \right)$ or even ${\alpha }_{h1}^{*}\left(v>\eta \right)$, which will fully explore and utilize the potentiality and value of high-capability contractors. □

Proposition 8.

Incentive mechanisms with user involvement will give consortiums a “synergy premium” to achieve sufficient incentives for high-capability consortiums.

Proof. 

Combining Table 1 and Table 2, ${\alpha }_{h1}^{*}+{\alpha }_{g1}^{*}-{\alpha }_{h0}^{*}=\frac{1+v^2-I{\alpha }_{g1}^{*}\left(1+{\mu }^2\right)}{1+I+{\eta }^2+I{\mu }^2}+{\alpha }_{g1}^{*}-\frac{1+v^2}{1+I+{\eta }^2+I{\mu }^2}$, a simplification can be obtained, i.e., ${\alpha }_{h1}^{*}+{\alpha }_{g1}^{*}-{\alpha }_{h0}^{*}=\frac{-I{\alpha }_{g1}^{*}\left(1+{\mu }^2\right)}{1+I+{\eta }^2+I{\mu }^2}+{\alpha }_{g1}^{*}=\frac{{\alpha }_{g1}^{*}\left(1+v^2\right)}{1+I+{\eta }^2+I{\mu }^2}={\alpha }_{g1}^{*}\times {\alpha }_{h0}^{*}>0$, which shows that in the optimal incentive mechanism with user involvement, the shared benefits given by the government to high-capability consortiums are higher than those given to high-capability contractors before the users are involved.

Similarly, ${\alpha }_{l1}^{*}+{\alpha }_{b1}^{*}-{\alpha }_{l0}^{*}={\alpha }_{b1}^{*}\times {\alpha }_{l0}^{*}+\frac{\rho \left(c_b-c_g\right)\left(t_1+\eta t_2\right)}{\left(1-\rho \right)\left(1+I+{\eta }^2+I{\mu }^2\right)}>0$, which indicates that the shared benefits of low-capability consortiums are higher than those of low-capability contractors in the basic incentive model, which means that the incentive mechanism with user involvement also gives low-capability consortiums a certain “synergy premium”. □

Proposition 9.

Interest conflicts exist within consortiums after users are involved, which causes contractors and users to adopt speculative behaviors. Whether the optimal incentive mechanism with user involvement can restrain the speculative behaviors of contractors and users during the construction process depends on whether the “synergy premium” can cover the increased variable costs of users due to information sharing.

Proof. 

It is clear that $e_{h11}^{*}+m_{g11}^{*}-e_{h10}^{*}=\frac{{\alpha }_{h1}^{*}+{\alpha }_{g1}^{*}-{\alpha }_{h0}^{*}-c_g{t}_1}{2I}=\frac{{\alpha }_{g1}^{*}\times {\alpha }_{h0}^{*}-c_g{t}_1}{2I}$. When ${\alpha }_{g1}^{*}\times {\alpha }_{h0}^{*}>c_g{t}_1$, $e_{h11}^{*}+m_{g11}^{*}-e_{h10}^{*}>0$, which indicates that the “synergy premium” is sufficient to cover the variable costs of good-capability users due to information sharing; in the optimal incentive mechanism with user involvement, the optimal effort of high-capability consortiums in the construction stage is higher than that of high-capability contractors that solely implement the PPP project.

Conversely, when ${\alpha }_{g1}^{*}\times {\alpha }_{h0}^{*}<c_g{t}_1$, $e_{h11}^{*}+m_{g11}^{*}-e_{h10}^{*}<0$. This indicates that when the “synergy premium” is insufficient to cover the variable costs of good-capability users in the construction stage, the optimal effort of high-capability contractors when solely implementing PPP projects is higher than that of high-capability consortiums.

For low-capability private entities, $e_{l11}^{*}+m_{b11}^{*}-e_{l10}^{*}=\frac{{\alpha }_{l1}^{*}+{\alpha }_{b1}^{*}-{\alpha }_{l0}^{*}-c_l{t}_1}{2I}$. When ${\alpha }_{l1}^{*}+{\alpha }_{b1}^{*}-{\alpha }_{l0}^{*}>c_l{t}_1$, i.e., ${\alpha }_{b1}^{*}\times {\alpha }_{l0}^{*}+\frac{\rho \left(c_b-c_g\right)\left(t_1+\eta t_2\right)}{\left(1-\rho \right)\left(1+I+{\eta }^2+I{\mu }^2\right)}>c_l{t}_1$, $e_{l11}^{*}+m_{b11}^{*}>e_{l10}^{*}$, which indicates that when the “synergy premium” of low-capability consortiums can completely cover their variable costs in the construction stage, the optimal incentive mechanism with user involvement can inhibit low-capability consortiums’ speculative behaviors and encourages low-capability consortiums to expend higher optimal efforts. When ${\alpha }_{l1}^{*}+{\alpha }_{b1}^{*}-{\alpha }_{l0}^{*}<c_l{t}_1$, ${\alpha }_{b1}^{*}\times {\alpha }_{l0}^{*}+\frac{\rho \left(c_b-c_g\right)\left(t_1+\eta t_2\right)}{\left(1-\rho \right)\left(1+I+{\eta }^2+I{\mu }^2\right)}<c_l{t}_1$, $e_{l11}^{*}+m_{b11}^{*}<e_{l10}^{*}$, which indicates that speculative behaviors will inhibit the effort of low-capability consortiums when the “synergy premium” cannot cover the variable information-sharing costs of low-capability consortiums. □

Proposition 10.

Although the optimal incentive mechanism with user involvement will give the consortium the “synergy premium”, it will also limit the bargaining power of a single type of private entity and prevent the single type of private entity from obtaining excess returns.

Proof. 

For high-capability contractors, ${\alpha }_{h1}^{*}={\alpha }_{h0}^{*}-\frac{I{\alpha }_{g1}^{*}\left(1+{\mu }^2\right)}{1+I+v^2+I{\mu }^2}\le {\alpha }_{h0}^{*}$, which means that the revenue-sharing coefficient of high-capability contractors is not higher than that of high-capability contractors who independently implement PPP projects in the optimal incentive mechanism with user involvement. For low-capability contractors, it is ${\alpha }_{l1}^{*}={\alpha }_{l0}^{*}-\frac{\rho \left(c_b-c_g\right)\left(t_1+\eta t_2\right)+I{\alpha }_{b1}^{*}\left(1-\rho \right)\left(1+{\mu }^2\right)}{\left(1-\rho \right)\left(1+I+v^2+I{\mu }^2\right)}\le {\alpha }_{l0}^{*}$, which indicates that, in the optimal incentive mechanism with user involvement, the optimal revenue-sharing coefficient of low-capability contractors also has an upper threshold equal to the optimal revenue-sharing coefficient in the basic incentive model.

In addition, when considering high-capability contractors individually, because of ${\alpha }_{h1}^{*}=\frac{1+v^2-I{\alpha }_{g1}^{*}\left(1+{\mu }^2\right)}{1+I+v^2+I{\mu }^2}$ and ${\alpha }_{h1}^{*}\ge 0$, the reasonable interval of the optimal revenue-sharing coefficient for good-capability users is $\left[0,\frac{1+v^2}{I\left(1+{\mu }^2\right)}\right]$. However, after comprehensively considering high-capability contractors and good-capability users, the reasonable interval of the optimal revenue-sharing coefficient for good-capability users is $\left[0,\frac{1+{\eta }^2}{1+I+{\eta }^2+I{\mu }^2}\right]$. Due to $\frac{1+v^2}{I\left(1+{\mu }^2\right)}>\frac{1+{\eta }^2}{1+I+{\eta }^2+I{\mu }^2}$, the internal competition of the private entities generated by user involvement limits the bargaining power of high-capability contractors. Similarly, the optimal incentive mechanism with user involvement also limits the bargaining power of high-capability contractors and good-capability users, which can prevent private entities from obtaining excess returns. □

4.2. Numerical Comparative Analysis

4.2.1. Parameter Assignment

In this section, the aforementioned theoretical analysis is empirically validated through numerical simulation while also conducting a comparative analysis between the basic incentive model and the model incorporating user involvement. Drawing upon the existing literature, the model’s constraints are comprehensively evaluated, and the following parameter assignments are employed:

• Variable cost coefficient for high-capability contractors: $c_h=0.10$;
• Variable cost coefficient for low-capability contractors: $c_l=0.20$;
• Variable cost coefficient for high-capability users: $c_g=0.10$;
• Variable cost coefficient for low-capability users: $c_b=0.125$;
• Fixed cost coefficient: $I=0.60$;
• Operation contribution coefficient for contractors: $v=0.50$;
• Operation contribution coefficient for users: $\eta =0.40$;
• Operation contribution coefficient for users: $\mu =0.10$;
• Government belief: $\rho =0.70$;
• Construction duration of the tunnel PPP project: $t_1=0.4$;
• Operation duration of the tunnel PPP project: $t_2=0.6$.

These parameter assignments are determined based on established references and align with the research’s objective of comprehensive measurement and analysis.

4.2.2. Private Partners’ Optimal Expected Revenues

In this study, the variable cost coefficient for high-capability contractors, represented as $c_h$, is selected as a specific example. The aim is to conduct a thorough comparative analysis and evaluation of the basic incentive model and the model incorporating user involvement. The focus is on investigating the performance and outcomes of these models across various variable cost coefficients.

Under different variable cost coefficients of the high-capability contractor, the optimal expected revenues of the private partners are illustrated in Figure 6. The findings from Figure 6 can be summarized as follows:

(1)

Increasing the variable cost coefficients of the high-capability contractor leads to lower optimal expected revenues for both the high-capability contractor and the high-capability consortium in both the basic incentive model and the incentive model with user involvement.

(2)

Irrespective of the variable cost coefficient of the high-capability contractor, the incentive model with user involvement, when combined with high-capability users in implementing the tunnel PPP project, ensures higher returns for the high-capability contractor compared to the independent implementation of the PPP project.

(3)

The optimal expected revenues of high-capability users are negatively correlated with the variable cost coefficients of high-capability contractors, regardless of whether the basic incentive model or the incentive model with user involvement is considered.

(4)

In both the basic incentive model and the incentive model with user involvement, private partners with strong capabilities achieve greater optimal expected returns compared to partners with low capabilities, indicating the presence of a “capability premium” for highly capable partners.

(5)

The incentive mechanism provides a larger “capability premium” to high-capability private partners as their capabilities strengthen, and conversely, the “capability premium” decreases for partners with lower capabilities.

4.2.3. Government’s Optimal Expected Revenue

Under varying variable cost coefficients of the high-capability contractor, the optimal expected revenues of the government are presented in Figure 7. An analysis of Figure 7 reveals the following observations:

(1)

Irrespective of whether the basic incentive mechanism or the incentive mechanism with user involvement is employed, the government’s optimal expected revenues from collaborating with high-capability private partners exhibit a negative correlation with the variable cost coefficients of the high-capability contractor. Conversely, the government’s optimal expected revenues from collaborating with low-capability private partners show a positive correlation with the variable cost coefficients of the high-capability contractor.

(2)

When the high-capability contractor possesses a significantly strong ability, denoted as $c_h<0.1347$ under different incentive mechanisms, the relationship among the government’s optimal expected revenues from cooperating with private partners of varying capabilities satisfies $G_1^{*}\left(h,h,g,g\right)>G_0^{*}\left(h,h\right)>G_1^{*}\left(l,l,b,b\right)>G_0^{*}\left(l,l\right)$.

(3)

Under different incentive mechanisms, when the high-capability contractor demonstrates relatively strong ability, represented as $0.1347<c_h<0.2$, the relationship among the government’s optimal expected revenues from cooperating with private partners with different capabilities complies with $G_1^{*}\left(h,h,g,g\right)>G_1^{*}\left(l,l,b,b\right)>G_0^{*}\left(h,h\right)>G_0^{*}\left(l,l\right)$.

(4)

Due to the competition mechanism established by the incentive mechanism with user involvement between the contractor and user, there is an improvement in the effort levels of private partners. Consequently, when the high-capability contractor’s ability in the PPP market is exceptionally strong ($c_h<0.1347$ ), low-capability private partners significantly enhance their effort levels under the guidance of the benchmark. As a result, even if the government collaborates with a low-capability consortium, it can obtain greater benefits compared to collaborating solely with a high-capability contractor.

4.2.4. Total Optimal Expected Revenue

Under varying variable cost coefficients of the high-capability contractor, the total optimal expected revenues of the game subjects (government and private partners) are depicted in Figure 8. An analysis of Figure 8 reveals the following findings:

(1)

Under both the basic incentive mechanism and the incentive mechanism with user involvement, when the government collaborates with high-capability private partners, the total optimal expected revenues of the game subjects display a negative correlation with the variable cost coefficient of the high-capability contractor.

(2)

Similarly, under the basic incentive mechanism and the incentive mechanism with user involvement, when the government partners with low-capability private partners, the total optimal expected revenues of the game subjects exhibit a positive correlation with the variable cost coefficients of high-capability contractors.

(3)

In the case where the ability of the high-capability contractor is exceptionally strong (denoted as $c_h<0.17$), the total optimal expected revenues of the game subjects satisfy ${\prod }_1^{*}\left(h,h,g,g\right)>{\prod }_0^{*}\left(h,h\right)>{\prod }_1^{*}\left(l,l,b,b\right)>{\prod }_0^{*}\left(l,l\right)$ across various scenarios.

(4)

Furthermore, when the ability of the high-capability contractor is relatively strong (represented as $0.17<c_h<0.2$), under different incentive mechanisms, the total optimal expected revenue of the game subjects satisfies ${\prod }_1^{*}\left(h,h,g,g\right)>{\prod }_1^{*}\left(l,l,b,b\right)>{\prod }_0^{*}\left(h,h\right)>{\prod }_0^{*}\left(l,l\right)$.

By comparing Figure 6, Figure 7 and Figure 8, it is evident that the incentive mechanism with user involvement represents a Pareto improvement over the basic incentive mechanism. This improvement is characterized by the ability of both the government and the contractor to achieve higher expected benefits after user involvement, regardless of their collaboration with high-capability or low-capability private partners. Consequently, when the high-capability contractor demonstrates a strong ability, the government’s selection order for private partners should prioritize the high-capability consortium, followed by the high-capability contractor, the low-capability consortium, and finally, the low-capability contractor. Conversely, when the ability of the high-capability contractor is relatively weak, the government’s selection order for private partners should prioritize the high-capability consortium, then the low-capability consortium, followed by the high-capability contractor, and lastly, the low-capability contractor.

By comparing Figure 6, Figure 7 and Figure 8, it can be seen that the incentive mechanism with user involvement is a Pareto improvement of the basic incentive mechanism, which is manifested in that both the government and the contractor can obtain higher expected benefits after user involvement, regardless of whether the government cooperates with high-capability or low-capability private partners. Therefore, when the ability of the high-capability contractor is strong, the order of the government’s selection of the private partners should be a high-capability consortium, high-capability contractor, low-capability consortium, and high-capability contractor. When the ability of the high-capability contractor is relatively weak, the order of the government’s selection of private partners should be a high-capability consortium, low-capability consortium, high-capability contractor, and high-capability contractor.

5. Case Study

5.1. PT Utility Tunnel PPP Project

Throughout the bidding process of the PT tunnel PPP project, the government facilitated competitive negotiations by offering a contract menu for private entities to choose from in the competitive negotiation document. The maximum bid price was limited, allowing private entities to indirectly communicate their own capacity endowment through different bid quotations. The project implementation plan served as the basis for determining the maximum bid limit price, stating that the total project investment was RMB 3.818 billion and the post-tax internal rate of return (IRR) of capital was 6%. The government anticipated an annual feasibility gap subsidy (excluding turnover tax) of approximately RMB 241.97 million (control price). This control mechanism ensured that the government maintained reasonable control over the maximum potential return rate for private entities during the project bidding phase. Importantly, this return rate was independent of the government’s ability to assess private entities and estimate the market distribution, thereby validating the rationale of Proposition 1.

The competitive negotiation document of the PT tunnel PPP project outlined the qualifications required for private entities, such as meeting the minimum investment and financing capacity estimated for the project, possessing net assets of RMB 3 billion or more, and demonstrating a track record of at least 1 km of utility tunnel, 3 km of subway, 500 m of tunnel, or 500 m of bridge construction experience (limited to domestic performance). These criteria established the entry threshold for private entities participating in the bidding process. In situations where both high-capability contractors and low-capability contractors competed for PPP projects, the high-capability contractors exhibited superior asset status, existing performance, operational ability, and staffing compared to the low-capability contractors. Consequently, the government’s entry threshold effectively served as the minimum requirement for low-capability contractors. When determining the entry thresholds for private entities, the government took into account the project’s specific requirements and their understanding of the private entities’ past performance. Additionally, as the proportion of high-capability contractors in the PPP market increases, the government tends to set higher entry thresholds, aligning with the findings of Proposition 2.

5.2. SC utility Tunnel PPP Project

The SC tunnel PPP project, with a total investment of RMB 3.995 billion and a length of 42.69 km, operates under the “TOT + BOT” model. The collaboration period spans 28 years. This project is a user-involved PPP initiative, with users holding a 14% share and other investors holding 66%. The SPV was established in July 2016 to facilitate project implementation. The project was successfully completed by the end of 2017, and operations commenced in January 2018, adhering to the predefined schedule.

The upper threshold for the internal rate of return (IRR) was set at 6.45% for the total investment of the SC tunnel PPP project, while the upper threshold for capital IRR was set at 8%. In comparison to the PT tunnel PPP project, the government’s IRR threshold for the SC tunnel PPP project is higher. This higher threshold demonstrates that the government provided a “synergy premium” to the consortium, aiming to incentivize and promote the consortium’s full engagement. This finding supports the validity of Proposition 8, which highlights the rationality behind the government’s decision-making process.

Since 2018, the government has evaluated the operation and management performance of the SC tunnel PPP project across five key areas: tunnel facility operation and maintenance, safety production management, ownership unit supervision, emergency response measures, and stakeholder satisfaction. The assessment scores for the project were as follows: 97.55 in 2018, 93.05 in 2019, and 95.45 in 2020. In comparison to the PT tunnel PPP project, it is evident that the performance assessment requirements for the SC tunnel PPP project are relatively less stringent. However, the operation and maintenance assessment results for the SC tunnel PPP project outperformed those of the PT tunnel PPP project. This outcome indicates that user involvement can effectively incentivize contractors to exert greater effort, thereby validating the rationale behind Proposition 7.

6. Conclusions and Recommendations

Users, particularly productive corporate users, exhibit a higher level of concern for the life cycle performance of tunnel PPP projects compared to other participants. When productive corporate users are involved, they have the capability to actively engage in tunnel PPP projects. By clarifying user needs, enhancing output availability, mitigating market and financing risks, and fostering internal supervision, the involvement of productive corporate users through equity investment in SPVs can significantly enhance project performance. However, a dual information asymmetry exists between the users and the government. The government lacks the ability to observe the capability endowments of users before signing contracts, and it also cannot directly observe users’ effort levels after contract signing. This information gap increases the possibility of users engaging in speculative behaviors. To address these challenges, this study applied principal–agent theory and developed both a basic incentive model and an incentive model with user involvement. These models aim to identify the capability endowments of contractors and users, as well as motivate them to exert optimal efforts. By leveraging these models, the study aims to mitigate the adverse effects of information asymmetry, improve project performance, and foster the sustainable development of tunnel PPP projects.

The main findings of the basic incentive model are as follows. (1) The existence of a threshold for the capability endowment gap between high-capability and low-capability contractors. When the capability gap is below the threshold, the incentive mechanism can effectively differentiate between contractors with different capability levels. However, when the capability gap exceeds the threshold, the incentive mechanism will drive low-capability contractors out of the PPP market. (2) The proportion of high-capability contractors in the PPP market influences the incentive mechanism’s setting of entry and eviction thresholds for low-capability contractors. A higher proportion of high-capability contractors leads to a higher entry threshold for low-capability contractors and a lower eviction threshold. (3) Considering only the construction and operation durations of the PPP project, there are upper and lower thresholds for the optimal revenue-sharing coefficient of low-capability contractors. As the construction duration increases, the optimal revenue-sharing coefficient gradually decreases from the upper threshold to the lower threshold. Conversely, with an increase in the operation duration, the optimal revenue-sharing coefficient gradually increases from the lower threshold to the upper threshold. These conclusions provide valuable insights into the design of incentive mechanisms for PPP projects and contribute to the understanding of how to effectively allocate resources, enhance project performance, and promote sustainable development in the context of tunnel PPP projects.

The main conclusions of the incentive mechanism with user involvement in this study are as follows. (1) The optimal incentive mechanism employs revenue-sharing reconciliation rules characterized by a “one increasing and the other decreasing” relationship between contractors and users. This approach establishes an internal competition mechanism among private entities, curbing speculative behavior and preventing excessive returns for these entities. (2) The optimal incentive mechanism maximizes the cooperation potential between the government and high-capability contractors by considering the a priori probability of contractors or users being high-capability private entities. It achieves this by reducing the expected shared benefits for low-capability contractors and imposing restrictions or even exclusions on their activity space.

The comparative analysis of the basic incentive model and the incentive model with user involvement reveals the following findings. (1) The optimal incentive mechanism with user involvement creates a “catfish” effect by engaging users to further explore the potential and value of high-capability contractors. (2) The optimal incentive mechanism with user involvement provides consortiums with a synergy premium, which ensures sufficient incentives for high-capability consortiums. (3) The effectiveness of the incentive mechanism with user involvement in curbing speculative behaviors of contractors and users during the construction process depends on whether the “synergy premium” adequately covers the variable costs incurred by users due to information sharing. (4) While the incentive mechanism with user involvement grants consortiums a synergy premium, it also limits the bargaining power of individual private entities.

Based on the main research conclusions of this study, the following recommendations are proposed:

(1)

When selecting private partners, the government should prioritize users who possess significant advantages in operation, market control, and management. It is essential to design reasonable incentives, such as policy support and price concessions, to encourage user participation in PPP projects. This will enhance output availability, reduce market and financing risks, and ultimately improve project performance.

(2)

The government should give priority to selecting private entities that have greater advantages in operation, market control, and management. It is important to establish reasonable incentives, such as policy support and price concessions, to encourage user participation in PPP projects. This will increase the availability of outputs, mitigate market and financing risks, and ultimately enhance project performance.

(3)

After involving users in PPP projects, they assume dual roles as customers and shareholders of the SPV (Special Purpose Vehicle). Conflicts of interest between users and contractors, as well as issues related to the equivalence of strengths, may give rise to new governance challenges. Therefore, it is crucial to establish an effective communication and coordination mechanism between the government, users, and contractors. This will allow for the full utilization of the advantageous resources brought about by user involvement while addressing any adverse reactions that may arise.

(4)

A practical and comprehensive performance appraisal plan should be developed to objectively and fairly evaluate the performance of private partners. This assessment should consider multiple factors, such as investment, effort, and contract performance. By combining the results of the government’s performance appraisal and taking a multi-dimensional approach, an internal revenue-sharing scheme for private partners can be formulated.

7. Limitations and Future Research

The limitations of the study and future research directions are outlined below.

(1)

Information Asymmetry: The theoretical model in this study primarily focuses on the information asymmetry between the government and contractors, as well as between the government and the consortium comprising contractors and users. However, it does not consider the information asymmetry between contractors and users. Future research can relax the model assumptions to incorporate the information asymmetry between contractors and users, thereby expanding the existing model.

(2)

Output and Cost Measurement: The measurement of output and cost for both contractors and users in this study is largely symmetrical. Moreover, the study only considers the impact of synergies on costs. In future studies, it is recommended to separately measure the output and cost of contractors and users and further explore the synergistic benefits between them. This will allow for a more comprehensive analysis and expansion of the existing models.