Evolutionary Game-Based Research on Risk Sharing in Major Projects under the EPC+PPP Mode Considering Secondary Risks

Abstract

Existing research on risk sharing management often ignores the adverse consequences of secondary risks. This study addresses secondary risks that emerge from the implementation of specific risk mitigation measures. Addressing the limitations of existing research that overlooks secondary risks and exploring the impact of secondary risks on the outcome of risk sharing, this paper integrates secondary risks into a game model that examines risk sharing between the public and private sectors in EPC+PPP projects. Utilizing a risk-control benefit model, an evolutionary game model is established to determine the evolutionary stability strategy under various conditions. This encompasses factors such as project income distribution, risk control capability, imbalanced status, and risk compensation. The findings indicate that secondary risks impact the risk-sharing strategies of both parties. Furthermore, a stronger risk-control ability correlates with a greater inclination toward risk-taking. The public sector can motivate risk sharing for the private sector through risk compensation, with a discernible lower limit. Ultimately, risk sharing becomes an active choice for both public and private entities when the risk cost is below the difference in returns between risks.

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.

2. Literature Review

2.1. Research on PPP or EPC Proejct Risk Identification and Evaluation

Iyer and Sagheer [3] employed the interpretive structure model to establish a hierarchical framework of PPP risks, illustrating risk interdependencies and pinpointing critical factors such as financial settlement delays, cost overruns, and time overruns. In a different approach, Xu et al. [4] employed the fuzzy comprehensive evaluation method to formulate a risk assessment model for PPP projects. This uncovered 17 key risk factors encompassing macroeconomic, construction, operational, and market-related risks. Their analysis indicated that PPP highway projects exhibited a risk level ranging from “medium” to “high”. Meanwhile, Ke et al. [5], utilizing two rounds of the Delphi method, appraised the likelihood and potential consequences of significant risks, identifying key factors such as government intervention, change in market demand, and financial risk. Based on the Delphi survey outcomes, Carbonara et al. [6] established guidelines for both public and private entities to outline a list of substantial risks inherent in PPP highway projects. Furthermore, they proposed effective allocation strategies and suitable mitigation approaches. Adopting a social network perspective, Wang et al. [7] established a risk network for PPP projects, revealing varying degrees of risk influence and key intermediary risks. Lastly, Ahmadabadi et al. [8] adopted a structural equation model to devise a risk assessment framework for extensive PPP projects, encompassing risk interactions and stakeholders’ expectations. They effectively identified pivotal risk pathways. Li et al. [9] established a risk assessment index system for hydropower engineering EPC projects and utilized an improved fuzzy evidence reasoning model to evaluate risks. Song and Hao [10] used the ISM model to analyze the correlation between political risk, economic risk, contract risk, and other risks of EPC projects and put forward corresponding risk coping strategies.

2.2. Research on PPP Project Risk Sharing

Wang et al. [11] examined the government’s minimum return guarantee using the traditional principal-agent theory to integrate the reciprocity preference theory to analyze the optimal risk-sharing ratio for the government in PPP projects. To address the uncertainty inherent in PPP projects, Carbonara et al. [12] employed the Monte Carlo simulation method to propose an optimal risk-sharing framework and benefit distribution mechanism for both public and private stakeholders. Shrestha et al. [13] introduced a risk allocation framework for PPP projects that identifies stakeholder-generated risks and assigns them to the party with the most effective risk control. Analyzing sponge city PPP projects, Zhao et al. [14] initially identified risks, assigned risk weights and responsibilities to each entity, and determined the shared risk proportion based on utility theory. Nguyen [15] highlighted that distinct pre- and post-event risk-sharing strategies predominantly influence risk control costs, with post-event strategies taking precedence. For effective risk allocation, Li et al. [16] suggested that macro and micro risks be borne or jointly shared by the public and private sectors, while assigning medium risks to the private sector. Applying game theory and system dynamics, Gao et al. [17] studied the tripartite game evolution and strategic behavior of the government, investors, and the public. Sensitivity analysis revealed the impact of key factor changes on behavioral strategies. Employing a bargaining game model, Li et al. [18] studied the risk allocation balance between public and private entities, revealing a connection between the risk allocation ratio and alternate offers, discount factors, and asymmetry. Wang et al. [19] devised an incentive mechanism and employed game theory to examine the relationship between investor incentive policies and their collaborative stance toward the government. The study highlights that the indiscriminate increase in rewards and penalties does not effectively guide investors’ conduct. Upholding the principle of balancing risk and return, Xu and Li [20] employed the Shapley value method to normalize returns, ascertain risk-bearing proportions, and allocate risk amounts for each stakeholder in the proposed public-private partnership (PPP) project. In the same vein, Jin and Zhang [21] constructed, trained, validated, and tested artificial neural network (ANN) models grounded in transaction cost economics to stimulate risk allocation decisions in PPP projects. Findings indicated participants’ risk management mechanism maturity and cooperation history impacted optimal risk allocation strategy formulation. Building upon this, Xu et al. [22] developed a fuzzy comprehensive assessment model to ascertain the equitable risk distribution between the government and private sector. Martiniello et al. [23] studied the benefit distribution of EPC+PPP projects, considering risk factors in order to maintain the balance between public and private interests in EPC-PPP contracts.

2.3. Secondary Risk Management Research

Bai et al. [24] introduced a method for assessing secondary risks and constructed a framework for mitigating such risks. Addressing the influence of secondary risks originating from risk responses, Zuo et al. [25] aimed to minimize the expected loss by determining optimal response plans for primary and secondary risks within resource limitations. Parsaei and Bamdad [26], considering environmental factors and secondary risks, developed a decision model for risk response. Their objective was to minimize unfavorable deviations from anticipated shifts in project cost, time, and quality, ultimately selecting optimal strategies for both primary and secondary risks.

2.4. Literature Summary

At present, numerous studies focus on identifying, evaluating, and sharing risks in PPP projects, offering valuable guidance for project risk management. Nevertheless, existing research in PPP risk management tends to overlook the significance of secondary risks as crucial influencing elements. The inability to adequately handle or control secondary risk losses might lead to potential failure in PPP projects. Consequently, it becomes imperative to integrate secondary risk factors into risk-sharing frameworks and propose efficient methods for managing these secondary risks.

3. Risk Control Return Model

3.1. Secondary Risk Analysis

We first define the risk arising from the response to the original risk as secondary risk, the risk arising from the response to the secondary risk as secondary-secondary risk, and the risk arising from the response to the secondary-secondary risk as the third secondary risk, and the following risk names can be followed by analogies. Before delving into secondary risk analysis, the meanings of symbols are explained. In this context, $C$ denotes risk control full cost (including risk control coat and expected lost of secondary risk); $L$ stands for risk expected loss; $P$ represents the probability of risk occurrence; $S$ signifies risk potential loss; $T$ quantifies the cost of risk control; $Y$ indicates whether risk control is undertaken, where $Y$ equals 1 for controlled risks and 0 otherwise. E reflects the loss mitigated due to risk control. With $Y=0$, $T<E$. Given the initial risk known within a collection of $n_1$ PPP projects, the anticipated loss of the initial risk $i_1$, denoted as $L_1^{i_1}=f(P_1^{i_1},S_1^{i_1})=P_1^{i_1}\times S_1^{i_1}$ [27]. In scenarios where the control of the initial risk $i_1$ yield $n_2$ secondary risks, each secondary risk $i_2$ carries an expected loss of $L_2^{i_1{i}_2}=f\left(P_2^{i_1{i}_2},S_2^{i_1{i}_2}\right)=P_2^{i_1{i}_2}\times S_2^{i_1{i}_2}$ [19]. Consequently, the cost of controlling the initial risk $i_1$ is expressed as:

$$C_1^{i_1}=f\left(P_2^{i_1{i}_2},S_2^{i_1{i}_2},T_1^{i_1},Y_1^{i_1}\right)={\sum }_{i_2=1}^{n_2}(P_2^{i_1{i}_2}\times S_2^{i_1{i}_2})Y_1^{i_1}+T_1^{i_1}\times Y_1^{i_1}$$

(1)

When all secondary risks $i_2$ are brought under control, leading to the emergence of $n_3$ additional secondary risks, the ensuing expected loss of secondary risks $i_3$ can be expressed as follows: $L_3^{i_1{i}_2{i}_3}=f\left(P_3^{i_1{i}_2{i}_3},S_3^{i_1{i}_2{i}_3}\right)=P_3^{i_1{i}_2{i}_3}\times S_3^{i_1{i}_2{i}_3}$. Subsequently, the cost linked to controlling these secondary risks is outlined as:

$$C_2^{i_2}=f\left(P_3^{i_1{i}_2{i}_3},S_3^{i_1{i}_2{i}_3},T_2^{i_1{i}_2},{E_2^{i_1{i}_2},Y}_2^{i_2}\right)={\sum }_{i_2=1}^{n_2}{\sum }_{i_3=1}^{n_3}{(P}_3^{i_1{i}_2{i}_3}\times S_3^{i_1{i}_2{i}_3})Y_2^{i_2}+{\sum }_{i_2=1}^{n_2}(T_2^{i_1{i}_2}-E_2^{i_1{i}_2})Y_2^{i_2}$$

(2)

When considering the interdependence of risks $i_1,i_2,i_3$, where $i_1$ leads to secondary risk $i_2$ and $i_2$ leads to secondary-secondary risk $i_3$, $P_3^{i_1{i}_2{i}_3},{\mathrm{and} S}_3^{i_1{i}_2{i}_3}$ values are non-zero, and conversely. As illustrated in Figure 1, only $P_3^{111},P_3^{112},P_3^{123},P_3^{124}$ and $S_3^{111},S_3^{112},S_3^{123},S_3^{124}$ exhibit non-zero values. In this model, the relationship among $P,S,T,\mathrm{a}\mathrm{n}\mathrm{d} E$ must be maintained, as depicted in Figure 1. This study solely focuses on secondary-secondary risk, thus disregarding the expected loss of third-secondary risk.

By analogy, the control cost of secondary risk $i_3$ is determined as follows:

$$C_3^{i_3}=f\left(P_3^{i_3},S_3^{i_3}\right)+{(T}_2^{i_2}-E_2^{i_2})$$

(3)

3.2. Risk Control Return

During the risk-sharing process, the recovered loss resulting from mitigated risks is perceived as a hidden benefit, referred to as the risk control benefit. $g_{i_1}$ denotes the probability of successfully managing the risk. The decision-maker faces a likelihood of failure $(1-g_{i_1})$ in controlling risk $i_1$, and only by successfully managing the risk can the decision-maker obtain the benefit of risk control. In instances where risk control falls short, the decision-maker must bear the burden of both risk loss and risk control costs. The quantification of the risk control benefits to decision-makers emerges from the formula:

$$B_{i_1}=P_1^{i_1}\left[g_{i_1}\right(L_1^{i_1}-C_{i_1})-\left(1-g_{i_1}\right)\left(L_1^{i_1}+C_{i_1}\right)]$$

(4)

When considering secondary risks without implementing control measures, the adjusted risk control benefit becomes:

$$B_{i_1}^{\prime }=P_1^{i_1}\left[g_{i_1}\right(L_1^{i_1}-C_1^{i_1})-\left(1-g_{i_1}\right)\left(L_1^{i_1}+C_1^{i_1}\right)]$$

(5)

When a risk decision maker anticipates a risk control benefit denoted as B, they face a critical determination of whether to partake in risk sharing. This decision hinges on a careful evaluation of both the associated risk control costs and the potential benefits. The decision-maker’s choices can manifest in three distinct schemes: 1. if $B_{i_1}^{\prime }>B$, the decision leans toward risk sharing. Moreover, in cases where the management of secondary risks promises loss reduction, the decision is to control those secondary risks, and vice versa. 2. In scenarios where $B_{i_1}^{\prime }<B$, a potential strategy is to reduce losses through the control of secondary risks, aligning outcomes with expected returns, and thereby opting for risk sharing. 3. In situations where $B_{i_1}^{\prime }<B$ simultaneous secondary risk management cannot ensure the desired returns, the choice is to avoid risk sharing. The alterations in risk control returns across these three schemes and their respective risk-sharing strategies are shown in Figure 2.

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 $V$ 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 $I_j$ and the degree of risk sharing ($k_1^j,k_2^j,\cdots ,k_{n_1}^j$, representing the sharing ratios of different risks for both parties). Here $j=\mathrm{1,2}$ represents the private and public sectors, respectively. The income distribution ratio for the private sector is denoted as ${\alpha }_j(I_j,k_1^j,k_2^j,\cdots ,k_{n_1}^j)$. As the interest distribution follows a pattern where greater risk corresponds to higher income, the principle holds that $\frac{\partial {\alpha }_j}{\partial I_j}>0,\frac{{\partial }^2{\alpha }_j}{\partial {I_j}^2}<0$, $\frac{\partial {\alpha }_j}{\partial k_i^j}>0,\frac{{\partial }^2{\alpha }_j}{\partial {k_i^j}^2}<0, where i=\mathrm{1,2},\cdots ,i_1,\cdots n_1$. The project benefits for both the public and private sectors are calculated as $V_j=V{\alpha }_j\left(I_j,k_1^j,k_2^j,\cdots ,k_{n_1}^j\right)$. The following relationships emerge from the aforementioned formula:

$${\sum }_{j=1}^2{k}_i^j(i=\mathrm{1,2},\cdots ,i_1,\cdots n_1)={\sum }_{j=1}^2{\alpha }_j(I_j,k_1^j,k_2^j,\cdots ,k_{n_1}^j)={\sum }_{j=1}^2{I}_j=1$$

(6)

Formula (6) holds valid based on Formula (10):

$${\alpha }_2\left(I_2,k_1^2,k_2^2,\cdots ,k_{n_1}^2\right)={\alpha }_2\left({1-I}_1,{1-k}_1^1,{1-k}_2^1,\cdots ,1-k_{n_1}^1\right)={1-\alpha }_1\left(I_1,k_1^1,k_2^1,\cdots ,k_{n_1}^1\right)$$

(7)

When examining the risk sharing of $i_1,$ 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 ${\epsilon }_j(where {\epsilon }_j>0)$, 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$V_j\left(k_{i_1}^j\right)-{\epsilon }_j$. It is also noted that$V_j^{\prime }\left(k_{i_1}^j\right)>0, {\mathrm{and} V}_j^{″}\left(k_{i_1}^j\right)<0, \mathrm{while}\mathrm{the}\mathrm{relationship}{V}_1\left(k_{i_1}^1\right)+V_2\left({1-k}_{i_1}^1\right)=V$ holds true.

4.2. Basic Assumption of Risk Sharing

(1)

The game involves the participation of two main entities: the public sector $\left(G\right)$ and the private sector$\left(P\right)$. 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 $\left(G_1,P_1\right),\left(G_1,P_2\right),\left(G_2,P_1\right),\mathrm{a}\mathrm{n}\mathrm{d} \left(G_2,P_2\right)$;

(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 $i_1$ with a proportion denoted as $k_{i_1}^1$, while the public sector’s share of the same risk is represented by $k_{i_1}^2$. The public sector has the potential to transfer risk $i_1$ to the private sector, as indicated by $r_{i_1}$. When the private sector bears the risk, the public sector receives compensation denoted as$T_{i_1}$. The private sector’s risk control capability for risk $i_1$ is defined as $g_{i_1}^1$, and the public sector’s risk control ability for the same risk is $g_{i_1}^2$. The likelihood of risk $i_1$ occurring is represented by $P_1^{i_1}$, while the loss benefit linked to societal welfare is $U_{i_1}$. Here,$k_{i_1}^j\in [0, 1]$, reflecting that when one party does not take on the risk, $k_{i_1}^j=0$ or $1$. Similarly, $g_{i_1}^j\in (0, 1)$, signifying that neither party can fully control the risk. Furthermore, $r_{i_1}<k_{i_1}^2$, 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:

For the private sector:

$$X_{\mathrm{1,1}}=\left[V_1\left(k_{i_1}^1\right)-{\epsilon }_1+T_{i_1}\right]+{P_1^{i_1}\left(k_{i_1}^1+r_{i_1}\right)[g_{i_1}^1\gamma }_1\left(L_1^{i_1}-C_{i_1}\right)-(1-g_{i_1}^1){\delta }_1(L_1^{i_1}+C_{i_1})]$$

(8)

For the public sector:

$$Y_{\mathrm{1,1}}=[V_2\left(k_{i_1}^2\right)-{\epsilon }_2-T_{i_1}]+{P_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)[g_{i_1}^2\gamma }_2(L_1^{i_1}-C_{i_1})-(1-g_{i_1}^2){\delta }_2(L_1^{i_1}+C_{i_1})]$$

(9)

(2)

The public sector does not share risks:

For the private sector:

$$X_{\mathrm{0,1}}={[V}_1\left(1\right)-{\epsilon }_1+T_{i_1}]+{P_1^{i_1}[g_{i_1}^1\gamma }_1(L_1^{i_1}-C_{i_1})-(1-g_{i_1}^1\left){\delta }_1\right(L_1^{i_1}+C_{i_1}\left)\right]$$

(10)

For the public sector:

$$Y_{\mathrm{0,1}}=V_2\left(0\right)-{\epsilon }_2-T_{i_1}$$

(11)

(2)

The private sector does not share risks:

(1)

Risk sharing by the public sector:

For the private sector:

$$X_{\mathrm{1,0}}={[V}_1\left(0\right)-{\epsilon }_1]+{P_1^{i_1}{r}_{i_1}[g_{i_1}^1\gamma }_1\left(L_1^{i_1}-C_{i_1}\right)-(1-g_{i_1}^1\left){\delta }_1\right(L_1^{i_1}+C_{i_1}\left)\right]$$

(12)

For the public sector:

$$Y_{\mathrm{1,0}}={[V}_2\left(1\right)-{\epsilon }_2]+{P_1^{i_1}\left(1-r_{i_1}\right)[g_{i_1}^2\gamma }_2(L_1^{i_1}-C_{i_1})-(1-g_{i_1}^2\left){\delta }_2\right(L_1^{i_1}+C_{i_1}\left)\right]$$

(13)

(2)

The public sector does not share risks:

For the private sector:

$$X_{\mathrm{0,0}}=V_1\left(0\right)-{\epsilon }_1$$

(14)

For the public sector:

$$Y_{\mathrm{0,0}}=\{V-V_1\left(0\right)-{\epsilon }_2\}-P_1^{i_1}{U}_{i_1}$$

(15)

The results of the above public-private sector risk sharing evolutionary game are shown in

Table 1. The game matrix depicting the interactions between the public and private entities in an evolutionary game.

Both Sides of the Game		Private Sector
		${\mathit{P}}_1$	${\mathit{P}}_2$
Public sector	$G_1$	$Y_{\mathrm{1,1}}$	$Y_{\mathrm{1,0}}$
		$X_{\mathrm{1,1}}$	$X_{\mathrm{1,0}}$
	$G_2$	$Y_{\mathrm{0,1}}$	$Y_{\mathrm{0,0}}$
		$X_{\mathrm{0,1}}$	$X_{\mathrm{0,0}}$

.

Let us consider the probabilities of the public entity ($G)$ sharing a certain risk i as $\alpha$, and not sharing it as $1-\alpha$. Similarly, for the private entity ($P),$ the probability of sharing risk $i$ is $\beta ,$ and not sharing $\mathrm{it}\mathrm{is} 1-\beta$. The fitness of $G\prime s$ sharing strategy is then calculated as follows:

$$h\left(G_1\right)=\beta Y_{\mathrm{1,1}}+(1-\beta )Y_{\mathrm{1,0}}$$

(16)

And for the non-sharing strategy:

$$h\left(G_2\right)=\beta Y_{\mathrm{0,1}}+(1-\beta )Y_{\mathrm{0,0}}$$

(17)

The average fitness for G is obtained

$$\overline{h}\left(G\right)=\alpha [\beta Y_{\mathrm{1,1}}+(1-\beta \left)Y_{\mathrm{1,0}}\right]+\left(1-\alpha \right)[\beta Y_{\mathrm{0,1}}+(1-\beta \left)Y_{\mathrm{0,0}}\right]$$

(18)

Similarly, for the private entity P, the average fitness is derived by:

$$\overline{h}\left(P\right)=\beta [\alpha X_{\mathrm{1,1}}+(1-\alpha \left)X_{\mathrm{0,1}}\right]+\left(1-\beta \right)[\alpha X_{\mathrm{1,0}}+(1-\alpha \left)X_{\mathrm{0,0}}\right]$$

(19)

Using the Malthusian equation, a two-dimensional dynamical system incorporating both selective strategies for $G$ and $P$ can be derived as follows:

$$\left\{\begin{array}{c}\dot{\alpha }=\alpha \left(1-\alpha \right)[\beta \left(Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}\right)+(1-\beta \left)\right(Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}}\left)\right]\\ \dot{\beta }=\beta \left(1-\beta \right)[\alpha \left(X_{\mathrm{1,1}}-X_{\mathrm{1,0}}\right)+(1-\alpha \left)\right(X_{\mathrm{0,1}}-X_{\mathrm{0,0}}\left)\right]\end{array}\right.$$

(20)

The Jacobian determinant for this system is determined by:

$$\left[\begin{array}{cc}\left(1-2\alpha \right)\left[\beta \left(Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}\right)+\left(1-\beta \right)\left(Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}}\right)\right]& \alpha \left(1-\alpha \right)[\left(Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}\right)-(Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}}\left)\right]\\ \beta \left(1-\beta \right)\left[\left(X_{\mathrm{1,1}}-X_{\mathrm{1,0}}\right)-\left(X_{\mathrm{0,1}}-X_{\mathrm{0,0}}\right)\right]& (1-2\beta )[\alpha \left(X_{\mathrm{1,1}}-X_{\mathrm{1,0}}\right)+(1-\alpha \left)\right(X_{\mathrm{0,1}}-X_{\mathrm{0,0}}\left)\right]\end{array}\right]$$

(21)

The equilibrium solutions for this evolutionary game system are (0,0), (0,1), (1,0), (1,1), and the solution (${\alpha }^{\mathrm{*}},{\beta }^{\mathrm{*}}$). The determinants and traces at each equilibrium point are illustrated in

Table 2. The determinant and trace values at each point.

Equalization Point	Strategy Combination	$\mathbf{det}\mathit{J}$	$\mathbf{tr}\mathit{J}$
(0,0)	($G_2,P_2$)	$(Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}})(X_{\mathrm{0,1}}-X_{\mathrm{0,0}})$	$(Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}})+(X_{\mathrm{0,1}}-X_{\mathrm{0,0}})$
(0,1)	($G_2,P_1$)	$-\left(Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}\right)(X_{\mathrm{0,1}}-X_{\mathrm{0,0}})$	$\left(Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}\right)-(X_{\mathrm{0,1}}-X_{\mathrm{0,0}})$
(1,0)	($G_1,P_2$)	$-(Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}}) \left(X_{\mathrm{1,1}}-X_{\mathrm{1,0}}\right)$	$-\left(Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}}\right)+(X_{\mathrm{1,1}}-X_{\mathrm{1,0}})$
(1,1)	($G_1,P_1$)	$\left(Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}\right)\left(X_{\mathrm{1,1}}-X_{\mathrm{1,0}}\right)$	$-\left(Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}\right)-\left(X_{\mathrm{1,1}}-X_{\mathrm{1,0}}\right)$
(${\alpha }^{*},{\beta }^{*}$)		$\frac{\left(Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}\right)(Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}})\left(X_{\mathrm{1,1}}-X_{\mathrm{1,0}}\right)(X_{\mathrm{0,1}}-X_{\mathrm{0,0}})}{[\left(Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}\right)-\left(Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}}\right)]\left[\left(X_{\mathrm{1,1}}-X_{\mathrm{1,0}}\right)-\left(X_{\mathrm{0,1}}-X_{\mathrm{0,0}}\right)\right]}$	0

.

Where $Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}}={P_1^{i_1}\left(1-r_{i_1}\right)[g_{i_1}^2\gamma }_2(L_1^{i_1}-C_{i_1})-(1-g_{i_1}^2){\delta }_2(L_1^{i_1}+C_{i_1})]+P_1^{i_1}{U}_{i_1}>0$, $Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}=[{V_2\left(k_{i_1}^2\right)-V_2\left(0\right)]+P}_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)\left[g_{i_1}^2(L_1^{i_1}-C_{i_1})-\left(1-g_{i_1}^2\right)(L_1^{i_1}+C_{i_1})\right]$, $X_{\mathrm{0,1}}-X_{\mathrm{0,0}}=\left[V_1\left(1\right)-V_1\left(0\right)\right]+T_{i_1}+P_1^{i_1}\left[g_{i_1}^1\right(L_1^{i_1}-C_{i_1})-\left(1-g_{i_1}^1\right)(L_1^{i_1}+C_{i_1}\left)\right]$, $X_{\mathrm{1,1}}-X_{\mathrm{1,0}}=[{V_1\left(k_{i_1}^1\right)-V_1\left(0\right)]+T_{i_1}+P}_1^{i_1}{k}_{i_1}^1\left[g_{i_1}^1(L_1^{i_1}-C_{i_1})-\left(1-g_{i_1}^1\right)(L_1^{i_1}+C_{i_1})\right]$.

4.4. Result Discussion

Table 3. Analysis of the requirements for stabilizing points.

Equilibrium Solution	Conditions for a Stable Point
(0,0)	$Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}}<0,X_{\mathrm{0,1}}-X_{\mathrm{0,0}}<0$
(0,1)	$Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}<0,X_{\mathrm{0,1}}-X_{\mathrm{0,0}}>0$
(1,0)	$Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}}>0, X_{\mathrm{1,1}}-X_{\mathrm{1,0}}<0$
(1,1)	$Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}>0,X_{\mathrm{1,1}}-X_{\mathrm{1,0}}>0$

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 $Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}}>0$, 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: $Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}<0,\mathrm{and}{X}_{\mathrm{0,1}}-X_{\mathrm{0,0}}>0$. If $Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}<0$, then it follows that

$g_{i_1}^2<\frac{P_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)\left(L_1^{i_1}+C_{i_1}\right)-\left(V_2\left(k_{i_1}^2\right)-V_2\left(0\right)\right)}{2P_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)L_1^{i_1}}$. If $X_{\mathrm{0,1}}-X_{\mathrm{0,0}}>0$, it implies that

$g_{i_1}^1>\frac{P_1^{i_1}\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{L}_1^{i_1}}$;

(3)

To establish the point (1,0), to be an ESS, the following conditions should be satisfied: $Y_{\mathrm{1,0}}-Y_{\mathrm{0,0}}>0,X_{\mathrm{1,1}}-X_{\mathrm{1,0}}<0$. If $X_{\mathrm{1,1}}-X_{\mathrm{1,0}}<0$, it implies that

$g_{i_1}^1<\frac{P_1^{i_1}{k}_{i_1}^1\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{k_{i_1}^{1}L}_1^{i_1}}$;

(4)

For point (1,1), to be an ESS, the following conditions need to hold:

$Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}>0,\mathrm{and}{X}_{\mathrm{1,1}}-X_{\mathrm{1,0}}>0$. If $Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}}>0$, then

$g_{i_1}^2>\frac{P_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)\left(L_1^{i_1}+C_{i_1}\right)-\left(V_2\left(k_{i_1}^2\right)-V_2\left(0\right)\right)}{2P_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)L_1^{i_1}}$. If $X_{\mathrm{1,1}}-X_{\mathrm{1,0}}>0$, then

$g_{i_1}^1>\frac{P_1^{i_1}{k}_{i_1}^1\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{k_{i_1}^{1}L}_1^{i_1}}$.

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 ($G_1,P_1$).

(1) 

When the risk control ability $g_{i_1}^j\ge \frac{L_1^{i_1}+C_{i_1}}{2L_1^{i_1}}$;

(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 $P_1^{i_1}{k}_{i_1}^j\left(L_1^{i_1}+C_{i_1}\right)<\left[\left(V_j\left(k_{i_1}^j\right)-V_j\left(0\right)\right)+x_{j}T\right]$, where $k_{i_1}^j\in \left[0, 1\right],x_1=1,x_2=0$.

Proof. 

(1)

Because $\frac{\partial {\alpha }_j}{\partial k_i^j}>0$, when $g_{i_1}^j>\frac{L_1^{i_1}+C_{i_1}}{2L_1^{i_1}}$, $Y_{\mathrm{1,1}}-Y_{\mathrm{0,1}},X_{\mathrm{0,1}}-X_{\mathrm{0,0}},\mathrm{and}{X}_{\mathrm{1,1}}-X_{\mathrm{1,0}}$ are all simultaneously greater than 0. Moreover, $\frac{L_1^{i_1}+C_{i_1}}{2L_1^{i_1}}$ significantly surpasses $\frac{P_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)\left(L_1^{i_1}+C_{i_1}\right)-\left(V_2\left(k_{i_1}^2\right)-V_2\left(0\right)\right)}{2P_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)L_1^{i_1}}$ and $\frac{P_1^{i_1}{k}_{i_1}^1\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{k_{i_1}^{1}L}_1^{i_1}}$. This confirms the conclusion.

(2)

If $P_1^{i_1}{k}_{i_1}^2\left(L_1^{i_1}+C_{i_1}\right)<\left(V_2\left(k_{i_1}^2\right)-V_2\left(0\right)\right)$, then $\left(k_{i_1}^2-r_{i_1}\right)\left(L_1^{i_1}+C_{i_1}\right)<\left(V_2\left(k_{i_1}^2\right)-V_2\left(0\right)\right).$ As a result, $g_{i_1}^2>\frac{P_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)\left(L_1^{i_1}+C_{i_1}\right)-\left(V_2\left(k_{i_1}^2\right)-V_2\left(0\right)\right)}{2P_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)L_1^{i_1}},{ \mathrm{and}g}_{i_1}^1>\frac{P_1^{i_1}{k}_{i_1}^1\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{k_{i_1}^{1}L}_1^{i_1}}$. This confirms the conclusion. □

Conclusion 3.

When  $g_{i_1}^1<\frac{P_1^{i_1}{k}_{i_1}^1\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{k_{i_1}^{1}L}_1^{i_1}}$, the private sector must refrain from taking risks. Conversely, when $g_{i_1}^1>\frac{P_1^{i_1}\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{L}_1^{i_1}}$, the private sector is bound to choose to take risks.

Proof. 

Convert the expression $\frac{P_1^{i_1}{k}_{i_1}^1\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{k_{i_1}^{1}L}_1^{i_1}}$ from the point (1,0) to $\frac{P_1^{i_1}\left(L_1^{i_1}+C_{i_1}\right)-\frac{\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)+T}{k_{i_1}^1}}{2P_1^{i_1}{L}_1^{i_1}}$. By introducing the variable $H=\frac{\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)+T}{k_{i_1}^1}$, the derivative $H^{\prime }=\frac{V_1^{\prime }\left(k_{i_1}^1\right)k_{i_1}^1-V_1\left(k_{i_1}^1\right)+V_1\left(0\right)-T}{{k_{i_1}^1}^2}$ is calculated. Let$Y=V_1^{\prime }\left(k_{i_1}^1\right)k_{i_1}^1-V_1\left(k_{i_1}^1\right)$, then $Y^{\prime }=V_1^{″}\left(k_{i_1}^1\right)k_{i_1}^1<0$. Consequently, $Y$ decreases, indicating its maximum value is achieved when $k_{i_1}^1\to 0$. As $H^{\prime }<0$, $H$ decreases, it reaches its minimum value at $k_{i_1}^1=1$; thus, $\frac{P_1^{i_1}\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{L}_1^{i_1}}>\frac{P_1^{i_1}{k}_{i_1}^1\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{k_{i_1}^{1}L}_1^{i_1}}$. With the binding points (0,1) and (1,0), it is deduced that when $g_{i_1}^1<\frac{P_1^{i_1}{k}_{i_1}^1\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{k_{i_1}^{1}L}_1^{i_1}}$, the private sector opts not to share risks. However, for Points (0,1), and (1,1), when $g_{i_1}^1>\frac{P_1^{i_1}\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{L}_1^{i_1}}$, the private sector evolves strategies to partake in risk sharing. This conclusion is validated. □

Corollary 1.

When$g_{i_1}^1>\frac{P_1^{i_1}\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(1\right)-V_1\left(0\right)\right)}{2P_1^{i_1}{L}_1^{i_1}}$, 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 $g_{i_1}^1>\frac{P_1^{i_1}\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(1\right)-V_1\left(0\right)\right)}{2P_1^{i_1}{L}_1^{i_1}},{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h} X}_{\mathrm{0,1}}-X_{\mathrm{0,0}},{\mathrm{and} X}_{\mathrm{1,1}}-X_{\mathrm{1,0}}$ 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 $g_{i_1}^1<\frac{P_1^{i_1}\left(L_1^{i_1}+C_{i_1}\right)-\frac{\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)}{k_{i_1}^1}}{2P_1^{i_1}{L}_1^{i_1}}$, $\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h} X_{\mathrm{0,1}}-X_{\mathrm{0,0}},\mathrm{and}{X}_{\mathrm{1,1}}-X_{\mathrm{1,0}}$ are simultaneously greater than 0. The public sector utilizes risk compensation to ensure $X_{\mathrm{0,1}}-X_{\mathrm{0,0}}>0\mathrm{and}{X}_{\mathrm{1,1}}-X_{\mathrm{1,0}}>0$ 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 ($X_{\mathrm{0,1}}-X_{\mathrm{0,0}}>0$ or $X_{\mathrm{1,1}}-X_{\mathrm{1,0}}>0$), 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 $A=\frac{P_1^{i_1}{k}_{i_1}^1\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(k_{i_1}^1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{k_{i_1}^{1}L}_1^{i_1}},B=\frac{P_1^{i_1}\left(L_1^{i_1}+C_{i_1}\right)-\left(V_1\left(1\right)-V_1\left(0\right)\right)-T}{2P_1^{i_1}{L}_1^{i_1}},C=\frac{P_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)\left(L_1^{i_1}+C_{i_1}\right)-\left(V_2\left(k_{i_1}^2\right)-V_2\left(0\right)\right)}{2P_1^{i_1}\left(k_{i_1}^2-r_{i_1}\right)L_1^{i_1}}$.

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. Impact of parameter changes on the load sharing result.

Argument	${\mathit{Y}}_{\mathrm{1,0}}-{\mathit{Y}}_{\mathrm{0,0}}$	${\mathit{Y}}_{\mathrm{1,1}}-{\mathit{Y}}_{\mathrm{0,1}}$	${\mathit{X}}_{\mathrm{0,1}}-{\mathit{X}}_{\mathrm{0,0}}$	${\mathit{X}}_{\mathrm{1,1}}-{\mathit{X}}_{\mathrm{1,0}}$
$k_{i_1}^1$↑	$-$	$-$	$-$	↕
$k_{i_1}^2$↑	$-$	↕	$-$	$-$
$P_1^{i_1}$↑	↕	↕	↕	↕
$r_{i_1}$↑	↕	↕	$-$	$-$
$g_{i_1}^1$↑	$-$	$-$	↑	↑
$g_{i_1}^2$↑	↑	↑	$-$	$-$
$L_1^{i_1}$↓	↑	↑	↑	↑
$C_{i_1}$↓	↑	↑	↑	↑
$T_{i_1}$↑	$-$	$-$	↑	↑
$U_{i_1}$↑	↑	$-$	$-$	$-$

. 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 $g_{i_1}^j$ incentives for both public and private entities to proactively assume risks, and parties with robust risk control capabilities should bear the available risks. Decreasing $L_1^{i_1}$ and $C_{i_1}$ 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 $T_{i_1}$ 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 $g_{i_1}^j,L_1^{i_1},C_{i_1}$, and $C_{i_1}$ 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. Parameter values.

Parameter	Value	Parameter	Value	Parameter	Value	Parameter	Value
$L_1^{i_1}$	3000	$k_{i_1}^1$	0.4	$g_{i_1}^1$	0.5	$P_1^{i_1}$	0.6
$r_i$	0.1	$k_{i_1}^2$	$0.6$	$g_{i_1}^2$	0.8	$U_{i_1}$	20,000
$V$	5000	$V_1\left(k_{i_1}^1\right)$	1800	$V_1\left(0\right)$	1600	$V_1\left(1\right)$	2200
$V_2\left(k_{i_1}^2\right)$	3200	$V_2\left(0\right)$	2800	$V_2\left(1\right)$	3400

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. Secondary risk parameters.

$\mathit{P}$		$\mathit{S}$		$\mathit{T}$		$\mathit{E}$
$P_2^{11}$	0.8	$S_2^{11}$	625	$P_2^{11}$	0.8	$E_2^{11}$	350
$P_2^{12}$	0.6	$S_2^{12}$	1000	$P_2^{12}$	0.6	$E_2^{12}$	450
$P_3^{111}$	0.5	$S_3^{111}$	200	$P_3^{111}$	0.5	$E_3^{111}$	90
$P_3^{122}$	0.5	$S_3^{122}$	200	$P_3^{122}$	0.5	$E_3^{122}$	70
$P_3^{123}$	0.8	$S_3^{123}$	250	$P_3^{123}$	0.8	$E_3^{123}$	90

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 $T_3^{123}>E_3^{123}$, 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 $P(\alpha ,\beta )=\beta \left(1-\beta \right)(176\alpha -210)$, and for any value of β, when the $\beta =1$, $P\prime \left(\alpha ,1\right)<0$. Consequently, for evolutionary stability, $\beta =0,$ signifying that the private sector opts against risk-sharing. Similarly, for the public sector, the dynamic replicator equation is $G(\alpha ,\beta )=\alpha \left(1-\alpha \right)(12108-11648\beta )$, and for any value of $\beta$ when $\alpha =1$ $G^{\prime }\left(1,\beta \right)<0$, leading to an evolutionarily stable point at $\alpha =1$, 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 $P(\alpha ,\beta )=\beta \left(1-\beta \right)(96.8\alpha -78)$. When $\alpha =0.81$, $P\left(0.81,\beta \right)=0$; for $\alpha >0.81$, $P\prime \left(\alpha ,1\right)<0$, resulting in $\beta =1$ as the evolutionarily stable point, indicating the private sector’s inclination to embrace risks. Conversely, when $\alpha <0.81$, $P\prime \left(\alpha ,0\right)<0$, and $\beta =0$ becomes the evolutionarily stable point, implying the private sector’s choice to avoid risk-sharing. For the public sector, the dynamic replicator equation is $G(\alpha ,\beta )=\alpha \left(1-\alpha \right)(12226.8-11700.8\beta )$. Regardless of $\beta \prime s$ value, $G^{\prime }\left(1,\beta \right)<0$, when a = 1, making $\alpha =1$ 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.