Sustainable Quality Control Decisions in Water Conservancy Supply Chains: A Principal-Agent Approach Considering Early Completion Benefits

Abstract

Water conservancy infrastructure is essential for sustainable socio-economic development, but quality failures caused by information asymmetry among stakeholders may undermine long-term project performance. Existing studies rarely integrate early completion benefits into quality control decisions or investigate governance strategies under different information conditions. This study develops a three-level principal-agent model involving the owner, supervisor, and contractor by incorporating early completion incentives into the quality supervision framework. The optimal quality monitoring level (Pa), quality guarantee deposit (S), and penalty mechanism (F) are derived under symmetric, asymmetric, and incomplete information scenarios, where probability density functions are introduced to characterize behavioral uncertainty under incomplete information. The results show that early completion incentives require a balance between schedule acceleration and quality risk control. Under sufficient contractor quality probability-based behavior, increased time effort reduces the owner’s supervision demand, whereas schedule compression combined with weak quality control increases the need for financial constraints. Under asymmetric and incomplete information, higher penalties and adaptive deposit strategies are required to mitigate moral hazard. Moreover, contractor probability-based behavior and supervisor performance exhibit substitution effects on owner supervision, indicating that strong partner probability-based behavior can reduce monitoring costs. These findings provide practical guidance for designing adaptive contracts, including differentiated supervision strategies and probability-based behavior-based deposit mechanisms, thereby enhancing the resilience and sustainability of water infrastructure supply chains.

1. Introduction

Water conservancy projects constitute critical infrastructure underpinning sustainable development, delivering essential services such as flood control, irrigation, water supply, and ecological regulation that directly contribute to the United Nations Sustainable Development Goals (SDGs) [1]. As sustainable infrastructure development increasingly demands the integration of environmental, social, and economic objectives throughout the project lifecycle [2], effective project governance must address these multidimensional requirements. As a project-oriented system, the engineering supply chain is characterized by dynamic resource flows that emphasize flexibility and collaboration. Within this context, the economic value of time is significant; early completion not only reduces the occupation of resources but also generates social value through the timely realization of functions like flood control [3,4,5]. However, the construction process involves information asymmetry between owners and contractors. Driven by profit or time incentives, contractors may reduce quality investments, thereby triggering moral hazard risks [6]. Moreover, empirical research has confirmed that contract types and incentive structures significantly influence project performance, underscoring the need for carefully designed governance mechanisms in construction projects [7]. Therefore, designing effective mechanisms to balance the benefits of early completion with quality supervision has become a key issue in project management.

In terms of time value, Irfan et al. [8] developed a model for highway projects considering supply chain factors, highlighting the impact of contracts and costs on project duration. Wang et al. [9] proposed a machine-learning-based contingency approach for the time–cost trade-off in construction projects, demonstrating how data-driven models can support dynamic scheduling decisions under uncertainty. Ballesteros-Pérez et al. [10] developed non-linear time–cost trade-off models of activity crashing and applied them to construction scheduling and project compression with fast-tracking, highlighting the complex relationship between schedule acceleration and cost escalation. Zong et al. [11] integrated commuting time value into a generalized travel cost function based on cumulative prospect theory to construct a transportation mode selection model. Research on construction megaprojects has further highlighted the evolving complexity of managing large-scale infrastructure projects and the need for adaptive management approaches [12]. In addition, supply chain maturity models have demonstrated that integrated governance frameworks significantly improve construction project delivery outcomes [13]. Zhang et al. [14] applied the maximum principle to derive optimal quality prevention solutions for suppliers when seller evaluation information is hidden.

Regarding quality supervision and coordination under information asymmetry, Zhang et al. [15] applied the maximum principle to derive optimal quality prevention solutions for suppliers when seller evaluation information is hidden. Lang et al. [16] proposed a collaborative mechanism combining order quantity, penalties, and bonuses to coordinate manufacturers and suppliers under asymmetric information. Cachon et al. [17] proposed incentive-compatible contracts to coordinate the interests of the supply chain under private information, providing a theoretical basis for the coordination of implicit information. In the context of infrastructure projects, research on public–private partnerships has identified governance structures and risk allocation mechanisms as critical success factors for effective project delivery [18]. Moreover, quality supervision game models that incorporate the bounded rationality of project owners have provided more realistic insights into construction quality management under asymmetric information [19]. Mokhtari and Hosseinian [20] employed principal-agent theory to compare optimal outcome-sharing mechanisms among clients, builders, and designers in collaborative construction contracts (DBB vs. DB), highlighting how risk-sharing and information asymmetry affect incentive design. Lei et al. [21] studied the coordination of supply chains under asymmetric production cost information and inventory inaccuracy. Gao et al. [22] studied the game process between suppliers with private cost information and retailers facing inventory inaccuracy under stochastic demand. Yang et al. [23] analyzed coordination parameters and benefits in a three-level supply chain considering retailer financing. Shen et al. [24] constructed a tripartite evolutionary game model to analyze stakeholder behavioral strategies in infrastructure projects. Furthermore, trust-based interface management has been demonstrated to mitigate coordination challenges in international engineering–procurement–construction projects, and recent research on megaproject complexity has revealed the persistent tension between rational planning approaches and emergent complexity in large-scale infrastructure delivery [25]. Finally, Yang [26] studied quality control strategies from the owner’s perspective, considering deposit retention and penalties for supervisor inaction.

Existing literature extensively analyzes time value [8,9,10,11,12,13,14] and quality supervision under information asymmetry [15,16,17,18,19,20,21,22,23,24,25,26]. However, critical gaps remain. First, regarding the time dimension, most studies treat time merely as a delay constraint or a penalty factor, failing to integrate “early completion benefits” as a proactive incentive variable within the quality supervision framework. This limitation overlooks the potential of time incentives to drive supply chain value creation. Second, concerning supply chain structure, existing models predominantly focus on two-level “owner–contractor” relationships. Although some studies introduce a supervisor, they often confine the analysis to static information scenarios. Consequently, there is a lack of dynamic analysis on how owners can adjust strategies when facing varying degrees of information opacity in a three-level setting.

To address these deficiencies, this paper constructs a comparative analytical framework covering symmetric, asymmetric, and incomplete information scenarios. Our primary contribution lies in employing probability density functions to characterize the uncertainty of the contractor’s and supervisor’s behavior under incomplete information, enabling the robust optimization of quality monitoring (Pa), deposit retention (S), and penalties (F). Following this logical thread, the paper first establishes a principal-agent game model involving the owner, contractor, and supervisor to define the fundamental decision environment. Subsequently, it derives and contrasts the optimal strategies under three distinct information states to reveal the impact of information transparency on incentive mechanisms. Finally, numerical simulations are conducted to validate the theoretical findings and provide managerial insights, offering practical guidance for owners to mitigate risks arising from information asymmetry and to enhance the long-term sustainability of water conservancy projects.

2. Problem Statement

In the three-level supply chain of hydraulic engineering, the owner, supervisor, and contractor constitute the core governance entities whose interactions collectively determine whether a project delivers its intended sustainability outcomes. Unlike previous studies focusing on the “owner-general contractor-subcontractor” structure, this study incorporates the Supervisor as an independent third party to explore the role of early completion in engineering quality supervision. Based on this tripartite collaborative framework, we analyze the optimal quality supervision strategy of the owner under different information conditions. Specifically, the contractor determines the quality control level (P_b) and early completion effort level (P_e), while the supervisor determines the supervision level (P_s). The owner determines the direct supervision level (P_a) and contract incentive mechanisms, including the quality guarantee deposit (S) and penalty mechanism (F), to mitigate moral hazard.

To ensure consistency, these behavioral variables are represented as probability-based decision variables. Specifically, (P_b) denotes the probability of quality compliance achieved through contractor effort; (P_s) and (P_a) represent the probabilities of defect detection by the supervisor and owner, respectively; and (P_e) represents the normalized early completion effort level. All variables are normalized within the interval [0, 1] and are consistently applied throughout the model and simulation analysis.

3. Model Description and Construction

3.1. Model Description

The quality control process of water conservancy projects involves three main participants: the owner, contractor, and supervisor. These participants interact through quality control decisions and contractual incentive mechanisms. The contractor is responsible for project execution and determines the quality control level and early completion effort. The supervisor acts as the owner’s delegated agent and determines the supervision level. The owner designs the direct supervision strategy and contractual incentive mechanisms to coordinate participant behaviors and mitigate potential moral hazard.

The decision-making process is described as follows.

Stage 1: Contractor decision.

The contractor determines the quality control level (P_b) and early completion effort level (P_e) according to project benefits, quality-related costs, and contractual incentives. The contractor seeks to maximize expected utility while balancing quality investment and early completion benefits.

Stage 2: Supervisor decision.

The supervisor determines the supervision level (P_s) based on supervision revenue, supervision costs, and potential penalty risks. A higher supervision level increases the probability of detecting quality defects but also leads to additional supervision costs.

Stage 3: Project outcome realization.

The final project quality outcome is jointly determined by the owner’s supervision probability (P_a), contractor’s quality compliance probability (P_b), and supervisor’s defect detection probability (P_s). Different combinations of these behavioral variables determine the realization of quality states, corresponding penalties, and potential losses. The probability relationships among these variables are incorporated into the expected utility functions developed in the following section.

Based on information availability among participants, three information structures are considered: symmetric information, asymmetric information, and incomplete information. Under symmetric information, all behavioral decisions are observable to the owner. Under asymmetric information, the contractor’s and supervisor’s behavioral decisions are private information. Under incomplete information, the owner cannot observe the exact behavioral levels and must evaluate them through probability distributions.

3.2. Model Construction

3.2.1. Definition of Model Parameters and Variables

Based on the decision variables defined in Section 2, the remaining parameters required for constructing the expected utility functions are summarized in

Table 1. Symbol definitions and descriptions.

Symbol	Meaning	Unit
U1	Benefit to the owner when the project is free of defects	10 k CNY
U2	Benefit to the owner when the project has defects	10 k CNY
G	Extra revenue from early completion brought by the contractor at maximum effort	10 k CNY
α	Owner’s incentive intensity to the contractor for early completion (percentage of extra earnings)	-
Pa	Probability of defect detection resulting from the owner’s supervision effort	-
Pb	Probability of quality compliance resulting from the contractor’s effort level	-
Ps	Probability of defect detection resulting from the supervisor’s effort	-
Pe	Normalized early completion effort level of the contractor (intensity of schedule compression)	-
Ca	Cost function of owner’s quality control	10 k CNY
Cb	Cost function of contractor’s quality control	10 k CNY
Cc	Cost function of supervisor’s quality control	10 k CNY
Ce	Contractor’s early completion effort cost	10 k CNY
ka	Owner’s quality supervision cost coefficient	10 k CNY
kb	Contractor’s quality control cost coefficient	10 k CNY
ks	Supervisor’s quality control cost coefficient	10 k CNY
k	Contractor’s time effort cost coefficient	10 k CNY
S	Amount of the quality guarantee deposit retained by the owner	10 k CNY
SL, SH	Lower and upper limits of the quality guarantee deposit amount	10 k CNY
F	Penalty imposed by the owner on the supervisor for dereliction of duty	10 k CNY
Q	Rework cost for quality defects identified and corrected in a timely manner	10 k CNY
V1	Contract amount agreed between the owner and the contractor	10 k CNY
V2	Contract amount agreed between the owner and the supervisor	10 k CNY
M	Opportunity cost (reservation utility) of the contractor	10 k CNY
N	Opportunity cost (reservation utility) of the supervisor	10 k CNY

. These parameters describe the economic benefits, contractual incentives, cost structures, and participation constraints of the three participants.

The economic benefit parameters include the project contract revenues (V₁) and (V₂), the additional benefit from early completion (G), and the corresponding incentive coefficient (α). The cost-related parameters include the quality control and supervision costs ((C_a), (C_b), (C_s), and (C_e)) and their associated cost coefficients ((k_a), (k_b), (k_s), and (k_e)).

The contractual parameters include the quality guarantee deposit (S), penalty mechanism (F), and the feasible range of deposit levels ((S_L) and (S_H)) under incomplete information. In addition, (M) and (N) represent the reservation utilities of the contractor and supervisor, respectively. These parameters are consistently applied in the subsequent utility formulation and optimization analysis.

3.2.2. Quality Control Process and Probability Relationship

To establish the expected utility functions, the interactions among the owner, supervisor, and contractor are modeled as a sequential quality control process, as illustrated in Figure 1. The project outcome is determined by the joint realization of the owner’s supervision probability (P_a), supervisor’s defect detection probability (P_s), and contractor’s quality compliance probability (P_b).

According to the decision tree, the probability of each terminal outcome is obtained by multiplying the probabilities along the corresponding decision path.

The probability that project quality is successfully guaranteed or effectively controlled is expressed as:

$${\mathrm{\Omega }}_1=P_a+(1-P_a)[P_b+P_s(1-P_b)]$$

(1)

The probability that quality defects remain undetected is:

$${\mathrm{\Omega }}_2=(1-P_a)(1-P_s)(1-P_b)$$

(2)

The probability that supervisor failure leads to penalty is:

$${\mathrm{\Omega }}_3=(1-P_s)(1-P_b)$$

(3)

The probability that quality defects are detected by either the owner or supervisor is:

$${\mathrm{\Omega }}_4=[P_a+(1-P_a)P_s](1-P_b)$$

(4)

where Ω₁, Ω_2, Ω₃, and Ω₄ represent the probabilities of different project outcomes. These probability terms are used to simplify the subsequent expected utility formulations.

Other outcome probabilities can be derived similarly based on different combinations of P_a, P_s, and P_b. These probabilities determine the corresponding benefit, penalty, and cost outcomes, which are incorporated into the expected utility functions.

Furthermore, the early completion incentive mechanism is introduced through the contractor’s schedule compression effort (P_e). The additional completion benefit is allocated between the owner and contractor according to the incentive coefficient α:

$$A_a=(1-\alpha )G{P}_e$$

(5)

$$A_b=\alpha G{P}_e$$

(6)

where A_a and A_b denote the additional benefits obtained by the owner and contractor, respectively.

3.2.3. Expected Utility Formulation

Based on the quality outcome probabilities established in Section 3.2.2, the expected utilities of the owner, contractor, and supervisor are formulated by considering project benefits, contractual incentives, penalty mechanisms, and effort-related costs.

The quality control and early completion costs are assumed to increase with the corresponding behavioral efforts. Following the quadratic cost assumption, the cost functions of the three participants are defined as:

$$C_a={\displaystyle \frac{1}{2}}{k}_a{P}_a^2$$

(7)

$$C_b={\displaystyle \frac{1}{2}}{k}_b{P}_b^2$$

(8)

$$C_c={\displaystyle \frac{1}{2}}{k}_c{P}_c^2$$

(9)

$$C_e={\displaystyle \frac{1}{2}}k{P}_e^2$$

(10)

where C_a, C_b, C_c, and C_e represent the owner’s supervision cost, contractor’s quality control cost, supervisor’s supervision cost, and contractor’s early completion effort cost, respectively. The parameters k_a, k_b, k_s, and k denote the corresponding cost coefficients.

The owner’s expected utility is formulated as:

$$E_1=(U_1+A_a){\mathrm{\Omega }}_1+(U_2+S){\mathrm{\Omega }}_2+F{\mathrm{\Omega }}_3-C_a-V_1-V_2$$

(11)

where U₁ represents the owner’s benefit when the project is free of defects; U₂ represents the owner’s benefit when defects occur; S represents the retained quality guarantee deposit; F represents the penalty imposed on the supervisor due to supervision failure.

The contractor’s expected utility is:

$$E_2=V_1+A_b{\mathrm{\Omega }}_1-Q{\mathrm{\Omega }}_4-S{\mathrm{\Omega }}_2-C_b-C_e$$

(12)

where V₁ represents the contract payment received from the owner; A_b represents the contractor’s share of early completion benefits; Q represents the rework cost caused by detected quality defects; C_b and C_e represent quality control and early completion effort costs, respectively.

The supervisor’s expected utility is:

$$E_3=V_2-F{\mathrm{\Omega }}_3-C_c$$

(13)

where V₂ represents the supervision contract payment; FΩ₃ represents the expected penalty caused by supervision failure; C_c represents the supervisor’s supervision cost.

3.3. Quality Control Strategy for Symmetrical Information from the Perspective of the Owner

3.3.1. Optimization Problem Formulation

Under symmetric information conditions, all participants’ behavioral decisions and related information are observable to the owner. Therefore, the owner can design the quality control strategy and contractual incentive mechanisms based on complete information.

Based on the expected utility functions established in Section 3.2.3, the owner acts as the principal and determines the supervision strategy and contractual parameters to maximize the expected utility. Specifically, the owner’s decision variables include the direct supervision probability $P_a$, the quality guarantee deposit $S$, and the penalty mechanism $F$. The variables $P_b$, $P_s$, and $P_e$ are treated as behavioral responses of the contractor and supervisor rather than direct decision variables of the owner. The incentive coefficient $\alpha$, representing the proportion of additional early-completion benefits allocated to the contractor, is treated as an exogenous contractual parameter.

Accordingly, the owner’s optimization problem under symmetric information can be formulated as:

$$\underset{P_a,S,F}{max}{E}_1$$

(14)

Subject to the participation constraints of the contractor and supervisor:

$$E_2\ge M$$

(15)

$$E_3\ge N$$

(16)

where M and N denote the reservation utilities of the contractor and supervisor, respectively. These constraints ensure that both participants obtain sufficient expected benefits to participate in the quality control process.

Since the behavioral variables represent probability-based decision variables, their feasible ranges are defined as:

$$0\le P_a,P_b,P_s,P_e\le 1$$

(17)

where $P_a$, $P_b$, $P_s$, and $P_e$ represent the owner’s supervision probability, contractor’s quality compliance probability, supervisor’s defect detection probability, and contractor’s early completion effort level, respectively.

The quality guarantee deposit $S$ is introduced as a contractual incentive mechanism to regulate contractor behavior and is restricted within the feasible contractual range:

$$S_L\le S\le S_H$$

(18)

where $S_L$ and $S_H$ represent the lower and upper bounds of the quality guarantee deposit.

The penalty mechanism $F$ is adopted by the owner to motivate the supervisor to fulfill quality supervision responsibilities. Considering that penalties cannot take negative values, the following constraint is imposed:

$$F\ge 0$$

(19)

Therefore, the complete optimization problem can be summarized as:

$$\begin{array}{c}\begin{array}{cc}\underset{P_a,S,F}{\max }\hspace{1em}& E_1\\ s.t.\hspace{1em}& E_2\ge M\\ & E_3\ge N\\ & 0\le P_a,P_b,P_s,P_e\le 1\\ & S_L\le S\le S_H\\ & F\ge 0\end{array}\end{array}$$

(20)

This optimization framework provides the basis for deriving the optimal supervision decision and contractual incentive mechanisms under symmetric information.

3.3.2. Derivation of the Reduced Owner Utility Function

According to the optimization problem formulated in Section 3.3.1, the owner determines the contractual incentive mechanisms while ensuring that both the contractor and the supervisor are willing to participate in the project. Therefore, the participation constraints of the contractor and supervisor are assumed to be binding at the optimum.

For the contractor, the participation constraint is expressed as:

$$E_2=M$$

(21)

where M denotes the contractor’s reservation utility. Solving Equation (21) for the quality guarantee deposit yields.

$$S={\displaystyle \frac{V_1+\alpha G{P}_e{\mathrm{\Omega }}_1-Q{\mathrm{\Omega }}_2-C_b-C_e-M}{{\mathrm{\Omega }}_2}}$$

(22)

Since the quality guarantee deposit is restricted by the contractual agreement, its feasible value satisfies.

$$S=\left\{\begin{array}{l}{S}^H\hspace{1em}\hspace{1em}{S}^H\le S\\ S\hspace{1em}\hspace{1em}{S}^L<S<S^H\\ S^L\hspace{1em}\hspace{1em}S\le S^L\end{array}\right.$$

(23)

Similarly, the participation constraint of the supervisor is:

$$E_3=N$$

(24)

where N denotes the supervisor’s reservation utility. Solving Equation (24) with respect to the contractual penalty gives.

$$F={\displaystyle \frac{N+C_s-V_2}{(1-P_s)(1-P_b)}}$$

(25)

Substituting Equations (22)–(25) into the owner’s expected utility function in Equation (11) and Equations (7)–(10) eliminates the contractual variables S and F. Consequently, the owner’s expected utility can be rewritten as:

$$\begin{array}{l}{E}_1=(U_1+(1-\alpha )G{P}_e){\mathrm{\Omega }}_1+(U_2-Q){\mathrm{\Omega }}_2\\ -{\displaystyle \frac{1}{2}}{k}_a{P}_a^2-{\displaystyle \frac{1}{2}}{k}_b{P}_b^2-{\displaystyle \frac{1}{2}}{k}_s{P}_s^2-{\displaystyle \frac{1}{2}}k{P}_e^2\\ -V_1-V_2-M-N\end{array}$$

(26)

Equation (26) represents the reduced expected utility of the owner after incorporating the participation constraints of the contractor and supervisor. Compared with the original formulation, the contractual variables S and F have been eliminated, allowing the owner’s optimal supervision decision to be derived directly in the following section.

3.3.3. Optimal Owner Supervision Decision

After eliminating the contractual variables S and F, the owner’s reduced expected utility function obtained in Section 3.3.2 only contains the behavioral decision variables. Under symmetric information, the owner can directly observe the contractor’s and supervisor’s behaviors and determine the optimal direct supervision probability P_a.

To derive the optimal supervision strategy, the first-order condition of the reduced expected utility with respect to P_a is obtained as:

$${\displaystyle \frac{\partial E_1}{\partial P_a}}=0$$

(27)

where E₁ represents the reduced expected utility of the owner defined in Equation (26).

Solving the first-order condition yields the optimal direct supervision probability:

$$P_a^{\ast }={\displaystyle \frac{\left[U_1+(1-\alpha )G{P}_e-U_2+Q\right](1-P_s)(1-P_b)}{k_a}}$$

(28)

To verify the optimality of the solution, the second-order derivative of the owner’s expected utility with respect to P_a is calculated as:

$${\displaystyle \frac{{\partial }^2{E}_1}{\partial P_a^2}}=-k_a<0$$

(29)

Since the second-order derivative is negative, the owner’s expected utility function is concave with respect to P_a. Therefore, Equation (28) represents the unique optimal supervision strategy under symmetric information.

3.3.4. Comparative Statics Analysis of Owner Supervision Decision

Based on the optimal supervision strategy in Equation (28), the effects of contractor quality compliance probability P_b and supervisor defect detection probability P_s on the owner’s optimal supervision probability are analyzed.

Taking the partial derivative of $P_a^{\ast }$ with respect to $P_b$ gives:

$${\displaystyle \frac{\partial P_a^{\ast }}{\partial P_b}}=-{\displaystyle \frac{\left[U_1+(1-\alpha )G{P}_e-U_2+Q\right](1-P_s)}{k_a}}<0$$

(30)

Similarly, the effect of $P_s$ on $P_a^{\ast }$ is obtained as:

$${\displaystyle \frac{\partial P_a^{\ast }}{\partial P_s}}=-{\displaystyle \frac{\left[U_1+(1-\alpha )G{P}_e-U_2+Q\right](1-P_b)}{k_a}}<0$$

(31)

Equations (30) and (31) indicate that the owner’s optimal supervision probability decreases with increases in contractor quality compliance probability and supervisor defect detection probability. This implies that effective quality control behaviors of the contractor and supervisor can substitute for the owner’s direct supervision effort.

3.4. Quality Control Strategy Under Asymmetric Information

Under asymmetric information conditions, the behavioral decisions of different participants cannot be completely observed by other parties. Specifically, the contractor’s quality compliance probability P_b and early completion effort level P_e, as well as the supervisor’s defect detection probability P_s, are private information. Meanwhile, the owner’s supervision strategy P_a and contractual mechanisms S and F cannot be fully observed by other participants.

Therefore, each participant determines its own quality control behavior based on its expected utility, while the owner designs contractual mechanisms to coordinate participant behaviors and reduce moral hazard.

3.4.1. Behavioral Decision of Contractor and Supervisor

Under asymmetric information, the contractor determines its quality compliance probability P_b and early completion effort level P_e by maximizing its expected utility. The first-order optimality condition is obtained as:

$${\displaystyle \frac{\partial E_2}{\partial P_b}}=0$$

(32)

Solving the above equation gives the contractor’s optimal quality compliance probability:

$$P_b^{\ast }=1-\sqrt{1-{\displaystyle \frac{2(\alpha G{P}_e-\frac{1}{2}k{P}_e^2+V_1-M)}{k_b}}}$$

(33)

Similarly, the supervisor determines the optimal defect detection probability P_s by maximizing its expected utility:

$${\displaystyle \frac{\partial E_3}{\partial P_s}}=0$$

(34)

Therefore, the optimal supervision probability of the supervisor is:

$$P_s^{\ast }=1-\sqrt{1-{\displaystyle \frac{2(V_2-N)}{k_s}}}$$

(35)

Equations (33) and (35) describe the equilibrium behavioral decisions of the contractor and supervisor when their actions are not directly observed by other participants.

3.4.2. Optimal Contract Design Under Asymmetric Information

Under asymmetric information conditions, the contractor’s and supervisor’s behavioral decisions are determined by their own expected utility maximization. Therefore, the owner designs contractual mechanisms based on their equilibrium responses $P_b^{\ast }$ and $P_s^{\ast }$.

Substituting the optimal behavioral decisions obtained in Section 3.4.1 into the owner’s supervision strategy gives:

$$P_a^{\ast }={\displaystyle \frac{\left[U_1+(1-\alpha )G{P}_e-U_2+Q\right](1-P_s^{\ast })(1-P_b^{\ast })}{k_a}}$$

(36)

Compared with the symmetric information case, the owner’s supervision decision is affected by the estimated behavioral responses of the contractor and supervisor rather than their directly observed effort levels.

The quality guarantee deposit is determined according to the contractor’s participation constraint:

$$S^{\ast }={\displaystyle \frac{V_1+\alpha G{P}_e{\mathrm{\Omega }}_1^{\ast }-Q{\mathrm{\Omega }}_2^{\ast }-\frac{1}{2}k{P}_e^2-\frac{1}{2}{k}_b{(P_b^{\ast })}^2-M}{(1-P_a^{\ast })(1-P_s^{\ast })(1-P_b^{\ast })}}$$

(37)

where ${\mathrm{\Omega }}_1^{\ast }=P_a^{\ast }+(1-P_a^{\ast })(P_b^{\ast }+P_s^{\ast }(1-P_b^{\ast }))$, ${\mathrm{\Omega }}_2^{\ast }=(1-P_a^{\ast })(1-P_s^{\ast })(1-P_b^{\ast })$.

Similarly, the penalty mechanism for the supervisor is obtained as:

$$F^{\ast }={\displaystyle \frac{V_2-\frac{1}{2}{k}_s{(P_s^{\ast })}^2-N}{(1-P_s^{\ast })(1-P_b^{\ast })}}$$

(38)

The above results show that under asymmetric information, the owner adjusts supervision intensity, quality guarantee deposit, and penalty mechanism according to the equilibrium behavioral responses of other participants.

3.5. Quality Control Strategy Under Incomplete Information

Under incomplete information conditions, the owner cannot obtain the exact behavioral decisions of the contractor and supervisor. However, the owner can estimate their behavioral characteristics through probability distributions. Therefore, $P_b$, $P_s$, and the uncertain quality-related parameter $P_e$ are characterized by probability density functions.

The probability density function of contractor quality compliance and supervisor defect detection is expressed as:

$$f(P_b,P_s,P_e)=f(P_b)f(P_s)f(P_e)$$

(39)

where $f\left(P_b\right)$, $f\left(P_s\right)$, and $f\left(P_e\right)$ represent the probability density functions of contractor quality compliance, supervisor inspection, and owner’s related quality decision behavior, respectively.

Based on the expected utility formulation in Section 3.2.3, the owner’s expected utility under incomplete information is given by:

$${\overline{E}}_1={\displaystyle {\int }_0^1{\displaystyle {\int }_0^1{\displaystyle {\int }_0^1{E}_1(P_b,P_s,P_e)f(P_b)f(P_s)f(P_e)d{P}_{b}d{P}_{s}d{P}_e}}}$$

(40)

where $f\left(P_b\right)$, $f\left(P_s\right)$ and $f\left(P_e\right)$ denote the probability density functions of the uncertain behavioral variables.

The owner determines the optimal supervision strategy and contractual mechanisms by maximizing expected utility:

$$\underset{P_a,S,F}{\max }{\overline{E}}_1$$

(41)

Given the participation constraints, the sand fare is endogenously determined by $P_a$. Therefore, the optimization problem reduces to determining $P_a$, and accordingly, the optimal supervision probability is obtained as:

$$P_a^{\ast \ast \ast }=\arg \underset{P_a}{\max }{\overline{E}}_1$$

(42)

After determining $P_a^{\ast \ast \ast }$, the quality guarantee deposit and penalty mechanism are obtained according to the participation constraints of the contractor and supervisor:

$$S^{\ast \ast \ast }=S(P_a^{\ast \ast \ast },f(P_b),f(P_e))$$

(43)

$$F^{\ast \ast \ast }=F(f(P_b),f(P_s))$$

(44)

Finally, the expected utility of the owner under incomplete information is calculated as:

$${\overline{E}}_1^{\ast }={\overline{E}}_1(P_a^{\ast \ast \ast },S^{\ast \ast \ast },F^{\ast \ast \ast })$$

(45)

Compared with symmetric and asymmetric information conditions, incomplete information introduces uncertainty into behavioral estimation. Therefore, the owner needs to design quality control strategies based on expected behavioral outcomes rather than deterministic decisions.

4. Analysis of Numerical Examples

4.1. Parameter Setting and Data Sources

To further investigate the impact of factors such as the contractor’s quality compliance probability, early completion effort level, and the supervisor’s defect detection probability on the owner’s strategy selection, this study verifies the proposed model through numerical analysis. To ensure the validity and representativeness of the simulation, the parameter settings are grounded in a combination of empirical data from a typical water conservancy project in China and relevant regulatory and research documents. The detailed classification of parameters, assigned values, and their specific sources are presented in

Table 2. Classification and justification of numerical parameters.

Parameter Category	Assigned Values/Ranges	Source & Justification
Regulatory & Contractual Parameters	V1 = 2500, V2 = 8000; M = 2000, N = 7000; SL = 500, SH = 1500	[8,27,28]
Cost Coefficients	ka = 1000, ks = 3000, kb = 1500, k = 1000	[15,21,29]
Project Benefit & Quality Parameters	U1 = 22,000, U2 = 18,000, Q = 2000, α = 0.25, G = 8000	[27,30]
Decision Variable Ranges	Pₐ, Pᵦ, Ps, Pe ∈ [0.3, 0.8]	[16,22]

.

The contractual variables $S$ and $F$ are not predetermined parameters in the simulation because they are endogenously determined by the optimization model. Therefore, only their feasible constraints and related parameters are specified.

4.2. Symmetric Information Simulation

Under symmetrical information conditions, the decision-making behaviors of the contractor and the supervisor directly affect the quality control decision-making behaviors of the owner. The following presents an analysis of the owner’s quality control decision-making behaviors. The effects of contractor quality compliance probability $P_b$, contractor time effort level $P_e$, and supervisor defect detection probability $P_s$ on the optimal owner supervision probability $P_a$ are investigated, as shown in Figure 2, Figure 3 and Figure 4.

Figure 2 illustrates the interaction mechanism between the owner’s supervision level (P_a) and the supervisor’s supervision level (P_s) under high-risk conditions, specifically when the contractor’s quality control is low (P_b = 0.3) and schedule compression is high (P_e = 0.8. Sensitivity analysis reveals a clear substitution effect: the surface declines as P_s increases. Analytically, this inverse relationship stems from the principal-agent structure where the supervisor acts as the owner’s delegate. A higher P_s implies a stronger probability-based behavior to detect quality defects, thereby reducing the marginal benefit of the owner’s direct supervision. Consequently, to minimize total supervision costs while maintaining quality standards, the optimal P_a decreases. Managerially, this suggests that the owner should adopt a dynamic resource allocation strategy. When the supervisor demonstrates high professional probability-based behavior (high P_s), the owner can appropriately reduce direct intervention efforts to save supervision costs. Conversely, if the supervisor’s effort is low, the owner must significantly intensify direct supervision to mitigate moral hazard risks. This finding underscores the importance of evaluating supervisor performance as a prerequisite for optimizing the owner’s own management input.

Figure 3 illustrates the sensitivity of the owner’s supervision level (P_a) to the contractor’s early completion effort level (P_e), revealing a conditional relationship dependent on the contractor’s quality compliance probability (P_b) and the supervisor’s supervision level (P_s). When Pe is directed at schedule compression (e.g., P_b = 0.3, P_s = 0.5), higher P_e increases quality risk (due to resource strain and procedural shortcuts), necessitating a non-linear increase in Pa to mitigate moral hazard (positive relationship). Conversely, when Pe is allocated to process optimization (e.g., P_b = 0.8, P_s = 0.7), higher P_e reduces quality risk (by improving workflow efficiency), allowing the owner to lower Pa (negative relationship). Managerially, owners should differentiate supervision strategies—intensifying P_a for schedule compression and relaxing it for process optimization. Notably, even with high supervisor effort (P_s), low contractor quality control (P_b) and high P_e require high P_a, indicating third-party supervision cannot fully substitute for direct owner oversight under severe schedule risks. Thus, implementing schedule incentives (e.g., early completion bonuses) requires parallel investment in quality supervision, especially for contractors with weak quality capabilities.

Figure 4 illustrates the sensitivity of the owner’s supervision level (P_a) to the contractor’s quality compliance probability (P_b). Sensitivity analysis reveals a significant substitution effect: the surfaces shift downward as P_b increases, indicating an inverse relationship between the contractor’s internal quality probability-based behavior and the owner’s external supervision requirement. Analytically, this demonstrates a “risk buffering” mechanism. A high P_b signifies that the contractor possesses a robust internal quality assurance system, which serves as the primary barrier against defects. This effectively mitigates the information asymmetry regarding the contractor’s actual behavior, thereby reducing the marginal benefit of the owner’s direct supervision. Consequently, even under conditions of high schedule compression (P_e = 0.8) and moderate supervisor effort (P_s), a high P_b allows the owner to maintain P_a at a medium-low level. Managerially, this implies that owners should adopt a differentiated supervision strategy based on contractor probability-based behavior. Selecting contractors with strong quality reputations (high P_b) is a cost-effective risk management approach, as it significantly lowers the necessary investment in direct supervision resources. Conversely, for contractors with weak quality controls (low P_b), intensive owner oversight is indispensable to prevent quality failures.

The effects of contractor quality compliance probability $P_b$, contractor time effort level $P_e$, and supervisor defect detection probability $P_s$ on the optimal quality guarantee deposit $\mathrm{S}$ are further analyzed under different parameter settings, as shown in Figure 5, Figure 6 and Figure 7.

Figure 5 illustrates the determinants of the quality guarantee deposit (S) and the regulatory role of the contractor’s early completion effort level (P_e). Sensitivity analysis reveals a significant synergistic effect between the contractor’s quality control (P_b) and the supervisor’s effort (P_s). When P_e is low, high levels of both P_b and P_s lead to a substantial reduction in S, indicating that effective internal control and external supervision can act as substitutes for financial collateral, thereby lowering the contractor’s capital burden. However, as P_e increases to 0.7, the sensitivity of S to P_b intensifies sharply; a low P_b triggers a surge in S even if P_s is high. Analytically, this suggests that schedule compression exacerbates the marginal risk of weak quality control, rendering external supervision insufficient as a standalone safeguard. Consequently, the owner must increase the financial constraint (S) to mitigate the heightened risk of default. Managerially, this implies that the deposit retention strategy should be dynamic rather than fixed. Owners should implement a “risk-pricing” mechanism: reducing the deposit burden for contractors with strong quality capabilities to incentivize performance, while imposing stricter financial constraints when strict deadlines coincide with weak quality control to ensure project integrity.

Figure 6 illustrates the sensitivity of the quality guarantee deposit (S) to schedule compression (P_e) and supervision intensity (P_s) under varying contractor capabilities. Sensitivity analysis reveals a critical asymmetry: when contractor quality control is weak (P_b = 0.3), S exhibits high sensitivity to P_e but rigidity (inelasticity) to P_s. Analytically, this indicates that under high schedule pressure, the marginal risk mitigation provided by external supervision diminishes significantly. The combination of aggressive scheduling (high P_e) and inherent probability-based behavior deficits (low P_b) creates a structural risk that monitoring alone cannot resolve; consequently, the owner is compelled to demand a high financial guarantee (S) to cover potential default risks. Conversely, when P_b is high, S remains stable at a low level even if P_e rises and P_s is low. This highlights that the contractor’s self-management (inherent prevention) plays a dominant role in risk control, whereas supervision acts merely as an auxiliary constraint. Managerially, this underscores the importance of contractor selection for fast-track projects. Owners must recognize the limitations of supervision; when facing strict deadlines, prioritizing contractors with proven quality capabilities is the most effective risk management strategy, as it fundamentally prevents the escalation of guarantee costs.

Figure 7 elucidates the asymmetric regulatory effect of the supervisor’s defect detection probability (P_s) on the quality guarantee deposit (S). Sensitivity analysis demonstrates a “stabilizing effect”: a higher P_s significantly reduces the volatility of S in response to variations in contractor probability-based behavior (P_b) and schedule pressure (P_e). Analytically, when P_s is low, the owner faces high uncertainty regarding project quality, forcing a defensive strategy where S acts as a highly sensitive financial buffer against external risks. Conversely, a high P_s provides a reliable external monitoring signal, allowing the owner to moderate the deposit requirement. However, a critical observation arises: even under high P_s, the combination of low P_b and high P_e forces S to remain near its upper limit. This confirms the principle of “non-substitutability of subject responsibility.” External monitoring, no matter how rigorous, cannot fully internalize the contractor’s quality behavior or eliminate the structural risks posed by probability-based behavior deficits and aggressive scheduling. Managerially, this implies that while enhancing supervision can stabilize risk expectations, it cannot replace financial guarantees when the contractor’s internal risk factors are excessive. Owners should establish a multi-layered risk defense system where supervision and deposits serve distinct functions—supervision for detection and control, and deposits for financial assurance—rather than treating them as fully interchangeable risk mitigation tools.

The change in the retention amount S of the quality guarantee deposit results from the joint influence of the contractor’s quality control P_b, early completion effort level P_e, and supervision P_s. When the contractor significantly increases the level of time effort P_e in pursuit of schedule rewards, if the quality control P_b is not strengthened synchronously or there is a lack of effective supervision P_s by the Supervisor, the potential risk of quality defects during the construction process will significantly accumulate, forcing the owner to increase the retention ratio of the security deposit. On the contrary, if the contractor actively optimizes the quality control system while compressing the construction period, or if the supervisor intervenes through high-frequency and high-precision supervision, it can effectively suppress the irrational growth of S.

The effects of $P_b$, $P_e$, and $P_s$ on the owner expected utility ${\mathrm{E}}_1$ under different parameter settings are shown in Figure 8, Figure 9 and Figure 10.

Figure 8 illustrates the sensitivity of the owner’s expected utility (E₁) to the interplay between contractor probability-based behavior (P_b) and schedule pressure (P_e). Sensitivity analysis reveals a significant “synergistic value premium”: under low schedule pressure (P_e = 0.3), the combination of high P_b and high Ps maximizes E₁, demonstrating that internal quality assurance and external monitoring jointly minimize defect-related losses. However, this synergy is fragile; as P_e rises, the utility surface collapses if P_b remains low. Analytically, this “value erosion” under high P_e suggests that schedule compression imposes an “efficiency tax” on quality. When the contractor operates at the margin of their capacity (low P_b), aggressive acceleration exponentially increases the probability of failure, a risk that external supervision (P_s) cannot fully mitigate. Managerially, this implies a “Probability-based behavior-First” scheduling strategy: high-intensity schedules (fast-tracking) should only be applied to contractors with proven high-quality capabilities (high P_b). Attempting to compensate for a low-probability-based behavior contractor via supervision during a compressed schedule results in diminishing returns and significant utility loss for the owner.

Figure 9 demonstrates the dominance of the contractor’s quality compliance probability (P_b) in determining the owner’s expected utility (E₁) under schedule pressure. Sensitivity analysis reveals that when P_b is low (0.3), E₁ exhibits a sharp non-linear decline as P_e increases, showing significant rigidity to improvements in supervision (P_s). Analytically, this indicates that quality is primarily an “endogenous attribute” determined by production probability-based behavior, rather than an “exogenous outcome” of inspection. Under low probability-based behavior, the “production constraint” becomes the bottleneck; increasing supervision intensity (P_s) merely identifies defects but cannot prevent the utility loss caused by the contractor’s inability to execute under tight deadlines. Conversely, a high P_b serves as a robust safeguard, maintaining high utility even when P_e is high and P_s is low. Managerially, this underscores a shift from “outcome inspection” to “source control.” For projects with aggressive schedules, the owner’s priority should be selecting high-probability-based behavior contractors rather than relying on intensive supervision. Investing in the screening of contractor capabilities yields higher marginal returns than increasing supervision efforts when facing inherent probability-based behavior deficits.

Figure 10 elucidates the stabilizing role of the supervisor’s quality supervision level (P_s) on the owner’s expected utility (E₁). Sensitivity analysis reveals a distinct “variance reduction” effect: when Ps is high (0.7), E1 exhibits strong stability against fluctuations in P_b and P_e, indicating that effective supervision acts as a buffer against operational risks. Analytically, this suggests that under high P_s, the supervisor absorbs a significant portion of the information asymmetry risk, shielding the owner from the direct volatility of the contractor’s performance. However, the analysis also identifies a “structural ceiling”: even with high P_s, the combination of low P_b and high P_e results in a persistently low E₁. This confirms that supervision is an ex post detection mechanism with diminishing returns, unable to fully compensate for the ex ante lack of production probability-based behavior. Managerially, this implies that while strong supervision can stabilize returns under normal variance, it cannot rectify systemic failures caused by the combination of weak contractors and aggressive scheduling. Owners must view supervision as a tool for risk mitigation, not as a cure for fundamental probability-based behavior deficits.

Overall, reasonable investment in P_e can promote the progress of the project and improve E₁, but excessive rush may cause quality problems and damage E₁. P_e will change the degree of influence of P_b and P_s on E₁; for example, at higher P_e, the gain effect of P_b enhancement on E₁ is more significant; P_b also affects the relationship between P_s, P_e, and E₁. When the quality control level of the contractor is high, the marginal effect of supervision and time effort on E₁ may change; And P_a not only affects E₁ on its own, but also interacts with P_b and P_e. Only by matching appropriate owner supervision with contractor quality control and time investment can E₁ be maximized.

4.3. Asymmetric Information Simulation

The effects of $M$, $V_1$, $N$, and $V_2$ on the optimal supervision probability $P_a^{\ast }$ under different parameter settings are shown in Figure 11 and Figure 12.

Figure 11 reveals the complex sensitivity of the owner’s supervision level (P_a) to the contractor’s expected return (M) and the owner’s potential loss (V₁). Sensitivity analysis indicates that P_a exhibits a non-linear, fluctuating response to M, while showing a consistent positive correlation with V₁. Analytically, the fluctuation of P_a with respect to M reflects a trade-off between the “commitment effect” and the “moral hazard effect.” While high returns may incentivize the contractor to invest in quality (commitment), they also increase the opportunity cost of compliance, potentially motivating profit-seeking shortcuts that necessitate higher P_a (moral hazard). This ambiguity forces the owner to adjust supervision dynamically. In contrast, the positive relationship between V₁ and P_a is driven by risk aversion; as potential losses escalate, the marginal cost of supervision becomes acceptable compared to the risk of project failure. Managerially, this implies that revenue incentives (M) alone are insufficient to ensure quality; owners must design comprehensive contracts that align incentives with oversight. Furthermore, for projects with high potential loss (V₁), a proactive “high-supervision” strategy is imperative regardless of the contractor’s revenue expectation.

Figure 12 elucidates the sensitivity of the owner’s supervision level (P_a) to the supervisor’s expected return (N) and contract value (V₂). Sensitivity analysis indicates that P_a exhibits a non-monotonic, fluctuating response to N, while maintaining a consistent positive correlation with V₂. Analytically, the fluctuation of P_a regarding N reflects a trade-off between the “incentive effect” and the “collusion risk.” While high returns may motivate the supervisor to perform diligently (reducing the need for owner oversight), they simultaneously increase the potential surplus for collusion with the contractor, necessitating heightened vigilance from the owner. This ambiguity leads to the non-linear trend. In contrast, the positive correlation with V₂ is driven by asset specificity and risk aversion; as the owner’s financial commitment increases, the marginal cost of supervision decreases relative to the potential loss of project failure, prompting higher P_a. Managerially, this implies that simply increasing supervisor remuneration is insufficient to guarantee quality assurance. Owners must design contracts that separate performance incentives from potential corruption rents. Furthermore, for high-value contracts (V₂), establishing a robust “supervision of the supervisor” mechanism is essential to ensure the return on investment in supervisory services.

The effects of $M$, $V_1$, $N$, and $V_2$ on the optimal quality guarantee deposit $\mathrm{S}$ and penalty $\mathrm{F}$ under different parameter settings are shown in Figure 13 and Figure 14.

Figure 13 illustrates the sensitivity of the quality guarantee deposit (S) to financial incentives (M), potential loss (V₁), and time effort (P_e). Sensitivity analysis reveals a consistent positive correlation between S and both M and V₁, while showing a negative correlation with P_e. Analytically, the increase in S alongside M reflects the principle of “incentive compatibility.” As the contractor’s expected return (M) rises, the temptation to prioritize speed over quality increases; therefore, the owner must proportionally increase the financial constraint (S) to maintain an effective deterrence mechanism, ensuring the cost of default outweighs the benefits of opportunism. Similarly, the positive response to V₁ represents a “risk hedging strategy,” where higher deposits are required to cover potential losses in high-stakes projects. Conversely, the lower S at high P_e levels indicates a “signaling effect.” A high early completion effort level (P_e) signals the contractor’s commitment and resource input, reducing information asymmetry and allowing the owner to relax financial constraints. Managerially, this suggests a “Dynamic Deposit Mechanism”: owners should scale the deposit in proportion to project value and profit margins, but offer “performance relaxations”—reducing deposit requirements for contractors who demonstrate high effort levels (P_e)—to optimize capital efficiency while maintaining risk control.

Figure 14 elucidates the determinants of the penalty for supervision dereliction (F), demonstrating its positive sensitivity to both the supervisor’s expected return (N) and the contract value (V₂). Sensitivity analysis reveals that F scales proportionally with the economic significance of the supervision contract. Analytically, the positive relationship between N and F reflects the “responsibility-alignment” principle. Higher expected returns (N) indicate a greater scope of authority and potential benefits; to prevent moral hazard, the penalty (F) must be sufficiently high to outweigh the benefits of shirking or opportunistic behavior, ensuring effective deterrence. Similarly, the increase in F with V₂ represents a “proportional accountability” mechanism. As the contract value (V₂) rises, the negative externality of supervision failure magnifies, necessitating a higher financial commitment to cover potential losses. Managerially, this implies that penalty clauses should not be static but rather “indexed” to contract value and expected returns. Owners should design dynamic penalty agreements where the severity of punishment escalates with the supervisor’s income and contract scale, thereby maintaining a credible threat of punishment that aligns the supervisor’s incentives with the owner’s quality objectives.

The effects of $M$ and $V_1$ on the owner expected utility $E_1$ under different parameter settings are shown in Figure 15.

Figure 15 illustrates the sensitivity of the owner’s expected utility (E₁) to the contractor’s expected return (M), project scale (V₁), and time effort (P_e). Sensitivity analysis reveals a distinct “Inverted U-shaped” relationship between M and E1, while E1 exhibits a consistent positive correlation with V₁ and P_e. Analytically, the non-linear trend of M reflects a trade-off between the “incentive effect” and the “opportunism effect.” Initially, increasing M aligns the contractor’s interests with the owner’s, motivating resource investment and quality improvement. However, beyond an optimal threshold, excessive returns induce moral hazard, where the contractor prioritizes short-term profit maximization over quality, leading to diminishing marginal utility for the owner. The positive impact of P_e confirms that time effort acts as a reliable signal of commitment, significantly elevating the utility surface. Managerially, this implies that owner utility is maximized within a “moderate profit interval.” Owners should design contracts that avoid both under-incentivizing (leading to low effort) and over-incentivizing (inducing profiteering). Furthermore, promoting high early completion effort levels (P_e) is a robust strategy to sustain high utility regardless of profit fluctuations.

4.4. Incomplete Information Simulation

Under the condition of incomplete information, the quality control decision-making behavior of the owner is closely related to the behaviors of the contractor and the supervisor. The following analysis focuses on the owner’s quality control decision-making behavior under asymmetric information. The effects of $P_b^H$, $P_s^L$, and $P_e^L$ on the optimal supervision probability $P_a^{\ast }$ under different fixed parameter combinations are analyzed, as shown in Figure 16, Figure 17 and Figure 18.

Figure 16 elucidates the sensitivity of the owner’s quality supervision level (P_a) to the lower limit of supervisor probability-based behavior (P_s^L) and the upper limit of contractor probability-based behavior (P_b^H). Sensitivity analysis reveals a distinct “substitution effect,” where P_a exhibits a negative correlation with both P_s^L and P_b^H. Analytically, the increase in P_s^L establishes a reliable “baseline defense” against quality risks. As the supervisor’s minimum competence rises, the uncertainty in the agency relationship diminishes, allowing the owner to substitute costly direct monitoring (P_a) with the supervisor’s delegated oversight. Similarly, a high P_b^H signifies strong “endogenous reliability” of the contractor, reducing the probability of moral hazard and thereby lowering the necessity for external intervention. Managerially, this suggests a “threshold-based resource optimization” strategy. Owners should prioritize ex ante screening mechanisms—setting high entry thresholds for both contractor and supervisor capabilities—over ex post intensive monitoring. Investing in probability-based behavior screening yields long-term cost savings by significantly reducing the marginal need for owner supervision.

Figure 17 illustrates the sensitivity of the owner’s supervision level (P_a) to the supervisor’s lower probability-based behavior bound (P_s^L) and the contractor’s upper probability-based behavior bound (P_b^H), with the contractor’s upper time effort bound (P_e^H) as a key moderating factor. Sensitivity analysis reveals that P_a is highly elastic to the improvement of ex ante probability-based behavior thresholds, but the direction of P_b^H’s impact on Pa depends on P_b^H. When P_b^H is high (e.g., aggressive schedule compression) and P_b^H is low (e.g., weak quality control), P_a increases to counter heightened quality risk (positive P_b^H − P_a relationship). Conversely, when PeH is high but P_b^H is high (e.g., strong quality control), P_a decreases as internal quality assurance buffers the risk (negative P_b^H − Pa relationship). This confirms that P_e’s impact on P_a is not universal—it is mediated by the contractor’s quality probability-based behavior and the nature of time effort. Analytically, raising P_s^L establishes a robust “risk control baseline.” When the minimum supervision level is guaranteed, the probability of undetected defects decreases, allowing the owner to reduce costly direct monitoring (P_a) without compromising quality assurance. Similarly, a higher P_b^H signifies the contractor’s technical potential and reliability, effectively reducing information asymmetry and the necessity for intensive oversight. Managerially, this suggests a strategic shift from “ex post monitoring” to “ex ante screening.” Owners should prioritize setting stringent qualification thresholds (high P_s^L and P_b^H) during the bidding phase, as investment in probability-based behavior screening yields higher marginal returns than subsequent intensive supervision. Furthermore, dynamic incentive mechanisms should be designed to encourage contractors to operate near their upper probability-based behavior bound (P_b^H), thereby optimizing the overall governance cost structure.

Figure 18 elucidates the sensitivity of the owner’s supervision level (P_a) to the supervisor’s lower probability-based behavior bound (P_s^L) and the contractor’s lower effort bound (P_e^L). Sensitivity analysis reveals a consistent negative correlation, indicating that P_a is highly elastic to improvements in the minimum performance thresholds. Analytically, raising P_s^L enhances the “risk absorption capacity” of the supervision system. A higher minimum supervision level ensures a baseline quality of defect detection, allowing the owner to reduce direct monitoring intensity without increasing exposure to residual risk. Similarly, a higher P_e^L establishes a “floor” for project execution, reducing the volatility of the contractor’s performance. This constraint on the worst-case scenario minimizes the probability of severe quality failures, thereby decreasing the necessity for intensive owner oversight. Managerially, this suggests a “threshold-based governance” strategy. Owners should prioritize enforcing strict minimum entry and performance criteria (setting high P_s^L and P_e^L) over continuous high-intensity monitoring. Establishing rigorous “bottom lines” in contracts acts as an effective substitute for ex post supervision, optimizing the allocation of management resources.

Overall, the Supervisor’s lower limit of supervision level P_s^L has a significant impact on the owner’s decision-making. When P_s^L is improved, the Supervisor can more effectively identify and handle problems during construction, reducing the owner’s concerns about project quality and thus lowering the quality supervision level P_a. At the same time, although the upper limit P_s^H of the supervision level of the Supervisor is not intuitively reflected on the coordinate axis, different colored surfaces represent different combinations of values related to the contractor’s variables, which will adjust the relationship between P_s^L and P_a, reminding the owner to consider multiple factors comprehensively when evaluating the supervisory role. From the perspective of the contractor’s factors, whether it is the increase in the upper limit P_b^H or lower limit P_e^L of the contractor’s quality control level, it signals to the owner that the engineering quality is more guaranteed. As the contractor’s quality compliance behavior is enhanced, the possibility of quality problems occurring is reduced, and the owner can correspondingly reduce the investment in quality supervision.

The effects of $P_s^L$, $P_e^H$, and $P_b^H$ on the optimal quality guarantee deposit $\mathrm{S}$ under different fixed parameter combinations are analyzed, as shown in Figure 19 and Figure 20.

Figure 19 elucidates the sensitivity of the quality guarantee deposit (S) to the supervisor’s lower probability-based behavior bound (P_s^L), the contractor’s upper probability-based behavior bound (P_b^H), and the upper time effort bound (P_e^H). Sensitivity analysis reveals divergent trends: S exhibits a negative correlation with P_s^L and PbH, but a positive correlation with P_e^H. Analytically, the negative response to P_s^L and PbH reflects a “probability-based behavior substitution effect.” Higher technical probability-based behavior (P_b^H) and guaranteed supervision baselines (P_s^L) serve as internal risk buffers, reducing the owner’s reliance on external financial constraints like S. Conversely, the positive correlation with P_e^H indicates a “speed–quality trade-off.” A high upper bound of time effort (P_e^H) signals an aggressive pursuit of schedule compression, which statistically correlates with increased quality risks. Consequently, the owner must increase S as a “risk premium” to hedge against the potential quality failures induced by rushing. Managerially, this suggests implementing a “differentiated deposit strategy.” Owners should reduce deposits for high-probability-based behavior contractors to optimize their capital flow, while imposing “acceleration premiums” on deposits for projects with tight schedules to enforce quality accountability.

Figure 20 illustrates the multidimensional sensitivity of the quality guarantee deposit (S) to the supervisor’s lower probability-based behavior bound (P_s^L) and the contractor’s upper probability-based behavior bound (P_b^H), moderated by the supervisor’s upper bound (P_s^H) and the contractor’s lower bound (P_b^L). Sensitivity analysis confirms a consistent negative correlation between probability-based behavior bounds (P_s^L, P_b^H) and S, reflecting a “probability-based behavior-collateral substitution effect.” As the technical reliability of both parties improves, the owner can substitute costly financial retention with trust in technical competence, thereby optimizing the contractor’s capital liquidity. Furthermore, the divergent surfaces representing different (P_s^H, P_b^L) combinations demonstrate significant interaction effects. A higher floor for contractor quality (P_b^L) or a higher ceiling for supervision (P_s^H) further reduces the required deposit level, indicating that system-wide reliability is determined by the interplay of all four parameters. Managerially, this suggests a “Parametric Deposit Strategy”: rather than applying a fixed rate, owners should adopt a dynamic deduction mechanism where the deposit ratio is calculated based on a comprehensive probability-based behavior matrix. Contractors with higher proven performance bounds should be rewarded with reduced financial burdens to incentivize continued excellence.

The effects of $P_s^L$ and $P_b^H$ on the optimal penalty $\mathrm{F}$ under different fixed parameter combinations are analyzed, as shown in Figure 21.

Figure 21 elucidates the sensitivity of the penalty for supervision dereliction (F) to the supervisor’s lower probability-based behavior bound (P_s^L) and the contractor’s upper probability-based behavior bound (P_b^H), while highlighting the moderating effects of P_s^H and P_b^L. Sensitivity analysis confirms a negative correlation between probability-based behavior parameters (P_s^L, P_b^H) and F. Analytically, this trend reflects a “necessity attenuation mechanism” for penalties. A higher P_s^L guarantees a baseline of supervision quality, reducing the probability of severe negligence and thereby diminishing the need for high punitive deterrence. Similarly, a higher P_b^H creates a “performance buffer,” where the contractor’s superior probability-based behavior mitigates the quality risks associated with potential supervision lapses, effectively lowering the penalty burden on the supervisor. Furthermore, the interaction of the moderating variables (P_s^H, P_b^L) demonstrates that system reliability is multiplicative; a robust supervisory ceiling (P_s^H) and a high contractor floor (P_b^L) further compress the required penalty level. Managerially, this suggests implementing a “probability-based behavior-based penalty exemption” strategy. Owners should calibrate penalty clauses dynamically, reducing penalty thresholds for high-probability-based behavior supervisors or when engaging high-performance contractors, thereby aligning punitive risks with actual exposure to quality failures.

The effects of $P_s^L$, $P_b^H$, and $P_e^L$ on the owner expected utility $E_1$ under different fixed parameter combinations are analyzed, as shown in Figure 22 and Figure 23.

Figure 22 elucidates the sensitivity of the owner’s expected utility (E₁) to the supervisor’s lower probability-based behavior bound (P_s^L) and the contractor’s upper probability-based behavior bound (P_b^H), with significant moderating effects from P_s^H and P_b^L. Sensitivity analysis confirms that E₁ is positively correlated with both P_s^L and P_b^H. Analytically, P_s^L acts as a “risk control baseline,” where raising the supervision floor minimizes the probability of quality failures and stabilizes returns. Conversely, P_b^H represents the “value creation ceiling,” where a higher technical limit enables the contractor to deliver superior quality and efficiency, directly expanding the owner’s profit margin. Furthermore, the steeper utility surface observed at high P_s^H and P_b^L values indicates a “synergistic enhancement effect.” This suggests that in a high-probability-based behavior ecosystem, improvements in boundary parameters yield disproportionately higher marginal returns. Managerially, this implies a “dual-threshold-driven strategy.” Owners should simultaneously mandate strict entry thresholds for supervision (P_s^L) to secure the baseline, while incentivizing contractors to unlock their maximum technical potential (P_b^H) to maximize upside gains.

Figure 23 illustrates the sensitivity of the owner’s expected utility (E₁) to the supervisor’s lower probability-based behavior bound (P_s^L) and the contractor’s lower effort bound (P_e^L), moderated by their respective upper bounds (P_s^H, P_e^H). Sensitivity analysis confirms a positive correlation between the lower bounds (P_s^L, P_e^L) and E₁. Analytically, raising these lower bounds establishes a robust “risk floor,” minimizing the probability of worst-case performance scenarios and ensuring project stability. Crucially, the interaction effects reveal a “complementary synergy.” The marginal utility gain from increasing lower bounds is significantly amplified when the upper bounds (P_s^H, P_e^H) are also high. This indicates that a high-potential environment (high upper bounds) facilitates the translation of baseline improvements into superior outcomes, whereas a low-ceiling environment would constrain such benefits. Managerially, this suggests a “dual-track driving strategy.” Owners should not only enforce strict minimum performance standards (P_s^L, P_e^L) to secure a baseline but also invest in enhancing the system’s maximum potential (P_s^H, P_e^H). The simultaneous elevation of both “floor” and “ceiling” generates a multiplicative effect on expected returns, outperforming isolated improvements.

Overall, the expected return E₁ of the owner reaches its highest value when the supervisor and contractor’s supervision and quality control levels are high, and drops to its lowest value when both are low. This change trend reflects the synergistic effect between the Supervisor and the contractor in the supply chain. It also indicates that the owner needs to guide the Supervisor and the contractor to maintain a high level of performance during the construction process through reasonable contract design and incentive mechanisms. The owner can improve their expected returns by allocating resources reasonably, increasing the lower limit of supervision level P_s^L of the supervisor and the lower limit of quality control level P_e^L of the contractor, such as providing more professional training for the supervisor and requiring the contractor to improve their quality management system.

The research results indicate that under incomplete information conditions, the owner needs to guide the supervisor and contractor to maintain a high level of performance during the construction process through reasonable contract design and incentive mechanisms to maximize their expected benefits. This discovery provides a theoretical basis and practical guidance for water conservancy engineering supply chain management. In practical engineering, the owner can dynamically adjust the incentive mechanism to ensure that all parties in the supply chain maintain high-quality standards and time management efficiency during the construction process, thereby achieving overall project optimization.

5. Conclusions

This paper considers the early completion benefits in water conservancy supply chains and constructs a three-level principal-agent model involving the owner, supervisor, and contractor to derive optimal quality supervision strategies under symmetric, asymmetric, and incomplete information. The main conclusions are as follows:

(1)

Theoretical Findings: Under symmetric information, early completion incentives compel owners to increase the quality guarantee deposit (S) to mitigate risks, while the supervision level (Pa) decreases as the contractor’s time effort (Pe) increases—provided the contractor’s quality compliance probability is high. Under asymmetric and incomplete information, owners must impose higher penalties (F) on supervisors and adjust deposits to prevent moral hazard.

(2)

Numerical Trends: There is a substitution effect between Pa and the contractor’s quality compliance probability (high quality control reduces Pa). The relationship between Pa and Pe is conditional: Pa rises with Pe when efforts focus on schedule compression (increasing risk) but falls when efforts focus on process optimization (reducing risk). The owner’s expected utility is maximized when both supervisor and contractor performance are high.

(3)

Practical Implications: Owners should differentiate supervision by the nature of schedule efforts—intensifying oversight when compression raises risk, and relaxing it when process optimization reduces it. Deposit mechanisms should scale with contractor probability-based behavior, reducing burdens for high performers while tightening guarantees for weaker ones. Ex ante screening of capable partners proves more cost-effective than ex post monitoring. Collectively, these strategies constitute a governance framework that curbs supervision costs while preserving the long-term functionality of water infrastructure—ensuring flood control, water supply, and ecological services across its designed lifespan.

Model Limitations: This study assumes fully rational participants and utilizes simplified cost functions; additionally, the model lacks validation with actual project data. Future research could extend the framework to multi-contractor or stochastic project contexts to enhance generalizability.