1. Introduction
Engineering-procurement-construction (EPC) refers to the mode in which the construction enterprise implements partial or whole process contracting for the design, procurement, construction, and trial operation of the engineering project according to the contract signed with the owner. PPP refers to the establishment of partnerships between the public and private sectors in order to help the government provide public goods and services and reduce pressure on the government. The EPC+PPP mode is a process of integrating the EPC and PPP modes and their complementary advantages, as well as a process of win-win cooperation, risk sharing, and benefit sharing between the government and social capital. Although the EPC+PPP mode integrates the advantages of EPC and PPP, its financing problem is still prominent. Financing will lead to many problems, among which risk sharing is a more complicated problem in the PPP model. Meanwhile, in the context of the EPC+PPP framework, prominent engineering projects exhibit significant characteristics such as extended timeliness and the involvement of multiple stakeholders. Throughout the extended duration of these projects, risks stem from diverse sources, while the associated knowledge systems exhibit substantial variation. Consequently, stakeholders may find it challenging to independently mitigate all the risks they encounter [
1]. Therefore, to effectively manage the risks arising from these projects, it becomes imperative to distribute the anticipated losses arising from these risks, considering the distinct knowledge attributes of each participant and their respective risk management capabilities. If the risk cannot be properly shared, it will lead to the failure of the project. The failure of a project can cause a chain reaction that prevents the smooth implementation of other projects [
2].
Nevertheless, the process of risk sharing introduces secondary risks. To illustrate, we consider a certain highway project operating within the EPC+PPP framework. The government assesses the highway operational risk as substantial and plans to shift a portion of this risk to social capital through consolidation. However, the social capital entity might lack the necessary operational experience and technical competence to effectively oversee and manage the highway. Consequently, this gives rise to secondary risks such as inadequate road maintenance and increased traffic safety hazards. These secondary risks could potentially result in insufficient road upkeep and an increase in traffic-related safety issues. An adverse impact on project stability and the interests of all involved parties. While both the government and social capital share the initial intent of better managing operational risks, the emergence of secondary risks may deviate the outcome from the intended goal. Consequently, overlooking secondary risks when determining risk-sharing arrangements might lead to outcomes contrary to the envisioned effects, thus impending project success. As such, delving into risk-sharing methodologies that account for secondary risks holds substantial theoretical and practical significance.
To address the aforementioned challenges, this paper presents a risk-sharing approach for major engineering projects operating within the EPC+PPP framework, employing an evolutionary game-based methodology. Initially, a comprehensive cost and benefit model encompassing secondary risk considerations is established to guide the risk-sharing process. Subsequently, we create an evolutionary game model that incorporates factors such as secondary risk, discrepancies in risk control capabilities, and other pertinent variables. The resulting evolutionary stable strategy for risk sharing is examined across diverse situations, and the model is deduced to derive important insights. The structure of the article is as follows:
Section 2 reviews the literature related to PPP project risk and secondary risk management.
Section 3 discusses the risk control return model. In
Section 4, the risk-sharing model is delivered. A case study is presented in
Section 5. The conclusion is given in
Section 6.
4. Risk Sharing Model
Risks should be identified and evaluated before risk sharing in EPC+PPP projects. Effective risk identification entails a comprehensive and accurate exploration of potential risks, encompassing the identification of secondary risks and their sources. This process extends to risk assessment, which involves both risk estimation and risk evaluation. Risk estimation investigates elements such as the likelihood of risk occurrence and the extent of potential harm. Complementary to this, the risk evaluation aims to determine the project’s overall risk level, risk grades, and interconnections, enabling a comprehensive appraisal of project risk. Only with a comprehensive grasp of these insights can a rational and equitable approach to risk sharing be embarked upon.
4.1. Benefit Distribution Model
Individual income dictates the choice of risk-sharing schemes by public and private entities, necessitating the prior establishment of an income distribution model. Assuming an expected income
for a PPP project, the income distribution ratio of the public and private parties is determined by their resource inputs (such as human, material, and financial resources) denoted as
and the degree of risk sharing (
, representing the sharing ratios of different risks for both parties). Here
represents the private and public sectors, respectively. The income distribution ratio for the private sector is denoted as
. As the interest distribution follows a pattern where greater risk corresponds to higher income, the principle holds that
,
. The project benefits for both the public and private sectors are calculated as
. The following relationships emerge from the aforementioned formula:
Formula (6) holds valid based on Formula (10):
When examining the risk sharing of the study assumes that the sharing ratios for other risks remain constant, and that the returns of both parties solely depend on their respective risk-sharing ratios. Simultaneously, the interference factor , reflecting the impact of other risk losses on the revenues of both public and private entities, is introduced. Consequently, the benefits for both parties can be simplified as. It is also noted that holds true.
4.2. Basic Assumption of Risk Sharing
- (1)
The game involves the participation of two main entities: the public sector and the private sector. The public sector operates with a public-oriented approach, aiming to maximize societal benefits, while the private sector operates in a profit-driven manner, striving to optimize its returns;
- (2)
Game strategy: both entities engage in the game with the option of sharing or not sharing risk i. The resulting strategy sets for both parties encompass: {share, share the}, {partake, do not share the}, {do not share, share the}, and {do not share, do not share the}, represented as ;
- (3)
In cases where the private sector opts to share the risk, the public sector introduces incentive-based risk compensation. Additionally, the public sector, leveraging its authority, can transfer certain risks onto the private sector while pursuing its risk-sharing efforts [
28];
- (4)
Risk control ability refers to the likelihood of successfully managing risk, with varying levels of proficiency among the involved parties. Successful risk control leads to the attainment of risk-related benefits, whereas failure results in assuming the costs of risk loss and risk control;
- (5)
Instances of unattended risks causing losses translate to a reduction in societal welfare, which the government bears. The resulting loss in societal welfare significantly outweighs any potential benefits.
4.3. Construction of Evolutionary Game Model
The symbolic representation of the game model entails the following: the private sector assumes a risk with a proportion denoted as , while the public sector’s share of the same risk is represented by . The public sector has the potential to transfer risk to the private sector, as indicated by . When the private sector bears the risk, the public sector receives compensation denoted as. The private sector’s risk control capability for risk is defined as , and the public sector’s risk control ability for the same risk is . The likelihood of risk occurring is represented by , while the loss benefit linked to societal welfare is . Here,, reflecting that when one party does not take on the risk, or . Similarly, , signifying that neither party can fully control the risk. Furthermore, , indicating that the proportion of risk transferred by the public sector is less than the proportion it bears itself.
The strategic choices of the public and private entities result in the following benefits:
- (1)
Private sector risk sharing:
- (1)
Risk sharing by the public sector:
- (2)
The public sector does not share risks:
- (2)
The private sector does not share risks:
- (1)
Risk sharing by the public sector:
- (2)
The public sector does not share risks:
The results of the above public-private sector risk sharing evolutionary game are shown in
Table 1.
Let us consider the probabilities of the public entity (
sharing a certain risk
i as
, and not sharing it as
. Similarly, for the private entity (
the probability of sharing risk
is
and not sharing
. The fitness of
sharing strategy is then calculated as follows:
And for the non-sharing strategy:
The average fitness for
G is obtained
Similarly, for the private entity
P, the average fitness is derived by:
Using the Malthusian equation, a two-dimensional dynamical system incorporating both selective strategies for
and
can be derived as follows:
The Jacobian determinant for this system is determined by:
The equilibrium solutions for this evolutionary game system are (0,0), (0,1), (1,0), (1,1), and the solution (
). The determinants and traces at each equilibrium point are illustrated in
Table 2.
Where , , , .
4.4. Result Discussion
Table 3 displays the essential stability conditions for each point. Building upon these findings, the stability strategy of the risk-sharing evolutionary game is discussed.
- (1)
Regarding the point (0,0), due to , it can be interfered that (0,0) does not constitute an evolutionarily stable strategy (ESS);
- (2)
For the point (0,1), to qualify as an ESS, the following conditions need to hold: . If , then it follows that
. If , it implies that
;
- (3)
To establish the point (1,0), to be an ESS, the following conditions should be satisfied: . If , it implies that
;
- (4)
For point (1,1), to be an ESS, the following conditions need to hold:
. If , then
. If , then
.
Conclusion 1. There must be takers of risk.
Proof. The point (0,0) is non-ESS; moreover from a realistic point of view, the public sector must ensure project continuity for the greater public good. □
Conclusion 2. In two situations, both the public and private entities are inclined to share risks, with the evolutionarily stable strategy being ().
- (1)
When the risk control ability ;
- (2)
When risk consumption (encompassing risk loss and costs) is less than the disparity between income from risk-sharing and non-risk-sharing choices, specifically , where .
Proof. - (1)
Because , when , are all simultaneously greater than 0. Moreover, significantly surpasses and . This confirms the conclusion.
- (2)
If , then As a result, . This confirms the conclusion. □
Conclusion 3. When
, the private sector must refrain from taking risks. Conversely, when , the private sector is bound to choose to take risks.
Proof. Convert the expression from the point (1,0) to . By introducing the variable , the derivative is calculated. Let, then . Consequently, decreases, indicating its maximum value is achieved when . As , decreases, it reaches its minimum value at ; thus, . With the binding points (0,1) and (1,0), it is deduced that when , the private sector opts not to share risks. However, for Points (0,1), and (1,1), when , the private sector evolves strategies to partake in risk sharing. This conclusion is validated. □
Corollary 1. When, the private sector proactively engages in risk sharing. In such cases, there is no necessity for the public sector to compensate the private sector for assuming risks.
Proof. When are simultaneously greater than 0. This implies that the private sector actively shares risks irrespective of any compensation from the public sector. This conclusion is supported through inference. □
Corollary 2. Effective risk-sharing incentives for the private sector can be achieved through risk compensation strategies in the public sector.
Proof. When , are simultaneously greater than 0. The public sector utilizes risk compensation to ensure thereby inducing risk-sharing tendencies in the private sector. Additionally, it is essential to note that only when the risk compensation reaches a specific threshold ( or ), does the private sector become motivated to partake in risk sharing. Increasing the risk compensation after the private sector engages in risk sharing does not result in additional incentives. □
Through the analysis of the essential conditions for risk sharing between public and private parties, we can derive the risk-sharing limits depicted in
Figure 3. Where
.
The discussion of the results highlights that multiple parameters exert an influence on risk-sharing outcomes between the public and private sectors. The impact of parameter variations on outcomes is presented in
Table 4. As shown in the table (The direction of the arrow indicates the direction of the value change, ↑ indicates that the value becomes larger, ↓ indicates that the value becomes smaller, and ↕ indicates that the direction of change is not necessarily), there is an increase in
incentives for both public and private entities to proactively assume risks, and parties with robust risk control capabilities should bear the available risks. Decreasing
and
encourages active risk-taking by both the public and private sectors, emphasizing the need to minimize losses and prudently manage secondary risks to reduce control costs. The increase in
prompts the private sector to engage in risk-taking, aligning with the findings of Inference 2. The influence of other parameters on risk-sharing outcomes necessitates the equivalent values of
, and
followed by further analysis of the results.
5. Case Study
The region is investing in an EPC+PPP project to comprehensively utilize bridges. Both parties need to establish a construction risk-sharing mode for the project.
Table 5 presents the parameters for both parties. The unit of gain and loss is USD 1000.
To control the risk of cost overruns in project construction, the adoption of new technologies may be pursued to save costs. However, this approach introduces a potential risk of technical failure, and addressing such technical risks might come at the expense of the natural environment, leading to environmental pollution and health risks for residents. Concurrently, the adoption of new materials could enhance construction quality and manage construction-related risks. Nonetheless, the risk of material incompatibility arises, potentially necessitating adjustments to the construction plan. The impact of other secondary risks on losses is considered minimal.
Figure 4 depicts the secondary risk loss associated with construction risk, while
Table 6 itemizes the parameters for secondary risk loss and control costs.
When accounting for secondary risks, the cost of mitigating the primary risk amounts to 1600 units (500 + 500 + 600). Conversely, addressing secondary risk entails a net cost reduction of −200 units (100 + 100 − 350 + 100 + 100 + 100 + 200 − 450). Further accounting for secondary-secondary risk results in reduced costs of −20 units (80 − 90 + 60 − 70, where , and thus remains uncontrolled). Consequently, the comprehensive risk control cost factoring in secondary risks totals 1380 units.
- (1)
Do not consider secondary risks
Not considering secondary risks implies leaving them uncontrolled despite their presence. For the private sector, the dynamic replicator equation is , and for any value of β, when the
, . Consequently, for evolutionary stability, signifying that the private sector opts against risk-sharing. Similarly, for the public sector, the dynamic replicator equation is , and for any value of when , leading to an evolutionarily stable point at , indicating risk-sharing by the public sector. Hence, in this scenario, the evolutionary stability strategy manifests as (risk-sharing, not sharing).
- (2)
Consider secondary risks
Considering secondary risks involves managing these risks. For the private sector, the dynamic replicator equation is . When , ; for , , resulting in as the evolutionarily stable point, indicating the private sector’s inclination to embrace risks. Conversely, when , , and becomes the evolutionarily stable point, implying the private sector’s choice to avoid risk-sharing. For the public sector, the dynamic replicator equation is . Regardless of value, , when a = 1, making the evolutionary stable point, reflecting the public sector’s predisposition to share risks.
Figure 5,
Figure 6 and
Figure 7 depict the evolutionary trends of strategies, contrasting scenarios with and without consideration of secondary risks.
Figure 5a illustrates the strategic evolution of both parties while accounting for secondary risks, while
Figure 5b represents their strategic evolution without such consideration. A comparison between
Figure 5a,b reveals that the inclusion of secondary risk has a discernible impact on the evolutionary stability strategies of both parties, transitioning from the initial state of (share, not share) to (share, share).
Figure 6a (with the horizontal axis representing evolution time) showcases the progression of public sector strategies, considering secondary risks, whereas
Figure 6b (also with the horizontal axis representing evolution time) portrays the evolution of public sector strategies without considering secondary risks. By analyzing the contrast between
Figure 6a and
Figure 6b it becomes evident that accounting for secondary risks can expedite the decision-making process of the public sector regarding risk-sharing strategies. Similarly,
Figure 7a (with the horizontal axis representing evolution time) illustrates the strategy evolution of the private sector when secondary risks are considered, and
Figure 7b (also with the horizontal axis representing evolution time) showcases the strategy evolution of the private sector without accounting for secondary risks. A comparison between
Figure 7a,b, underscores the significant impact of secondary risks on the private sector’s risk-sharing strategy, transitioning from the initial non-sharing stance to the predominant strategy in most cases. This underscores that by attentively addressing and effectively controlling secondary risks, both public and private entities can be incentivized to actively participate in risk-sharing.
6. Conclusions
This paper addresses the issue of secondary risk, which emerges as a consequence of implementing certain risk response measures and significantly influences risk outcomes. Despite its impact, existing research has often overlooked secondary risks. The study analyzes secondary risk within a project context and derives a risk control profit model that accounts for secondary risk. Subsequently, an evolutionary game model is formulated based on factors such as the income distribution and risk control capabilities of both parties, enabling an examination of how various factors shape the risk-sharing strategies of both entities. The findings are as follows:
- (1)
Secondary risks exert a notable influence on the risk-sharing strategies of both the public and private sectors; effectively managing secondary risks can motivate these sectors to engage in risk-sharing;
- (2)
The risk control abilities of the parties play an important role in determining their inclination toward risk sharing. Greater risk control capabilities correspond to a greater propensity for embracing risks;
- (3)
Risk compensation offered by the public sector to the private sector serves as an incentive for risk sharing, and there exists a lower limit of risk compensation that encourages the private sector to participate in risk sharing;
- (4)
When the risk consumption is lower than the difference in returns between risk- and non-risk sharing, both the public and private sectors actively partake in risk sharing.
The study also proposes risk-sharing suggestions for EPC+PPP projects as follows:
- (1)
The public and private parties should actively participate in the risk sharing of PPP projects and realize the Pareto optimization of both through reasonable negotiation. If one party chooses not to bear the risk, it will cause huge losses for the project, which will affect its success;
- (2)
In the context of risk sharing, a thorough consideration of the impact of secondary risks on risk control costs and benefits is essential, and a comprehensive understanding and management of risk losses is important;
- (3)
The public and private parties should accurately assess the hazards of secondary risks to assess whether risk sharing can be carried out or its proportion. At the same time, both companies can balance the relationship between the effect of the risk coping strategy and the secondary risk hazards so as to choose the appropriate risk coping strategy;
- (4)
Before risk sharing, a comprehensive evaluation of one’s risk control capabilities is recommended, followed by the selection of appropriate risk-sharing plans based on risk control costs and benefits. Risk sharing should ideally be undertaken by the party with robust risk control capabilities;
- (5)
The public sector should provide reasonable compensation for private sector risks. However, in cases of high risk-sharing ratios or inadequate risk control capabilities, higher levels of risk compensation might be necessary to safeguard one’s interests.
Due to the large number of subjects in EPC+PPP projects, such as government departments, general contractors, and financial institutions, the risk sharing study only from the perspective of public and private parties ignores the roles and functions of some subjects in risk management. Therefore, in future studies, risk sharing among government departments, general contractors, and financial institutions can be considered so as to analyze the impression of different roles on the result of risk sharing and better guide practice.