In economic activities, when multiple actors join forces through alliances or partnerships, they have the potential to achieve greater gains or minimize losses compared to working individually, thereby creating a win-win or multi-win situation. The theory of Shapley Value focuses on the distribution of benefits or sharing of risks among the members of an alliance. It takes into account the level of contribution made by each member towards the overall goals of the alliance and reflects the interactive process of cooperation among alliance members. The theory of Shapley Value is primarily applied to address issues related to the distribution of cooperation costs and benefits among alliance members (Yu et al., 2014; Xu et al., 2016).

In recent years, the application of the Shapley Value Theory has extended to various areas such as investment analysis, compensation allocation, risk allocation, and burden apportionment. In the water sector, the Shapley Value has emerged as a prominent method for solving cooperative game problems. It provides a quantitative allocation ratio that is practical and straightforward to apply (Yang, 2016; Wang et al., 2018).

The calculation of the Shapley value is made in view of the costs and capabilities of both parties involved in managing a particular risk. Its results can reflect the disparity in risk management abilities between the parties. However, some factors such as the ratio of proportion of capital contributions and bargaining positions of the parties are not taken into account. Given the theoretical limitations of the Shapley Value and the unique characteristics of PPP projects, it is necessary to revise the index system of the Shapley Value method. This revision allows for a more comprehensive analysis, considering factors specific to PPP projects (Fan, 2014; Sheng and An, 2019).

Pinpointing the optimal allocation scheme or “solution” is indeed the most challenging aspect of the Cooperative Game Theory. The Shapley Value method, as a type of cooperative game, presents a straightforward calculation process and allocates benefits based on the partners’ marginal benefit to the alliance, which aligns with Adams’ equity theory. The latter theory suggests that distributing benefits based on the contribution rate can enhance team cooperation and output, in line with the principles of benefit sharing and risk sharing in PPP projects. It can also help emphasize each partner’s importance within the alliance.

Let N be the set of alliances consisting of n cooperating parties, and N = 1,2 , . . . , n . The cooperating parties within the alliance are non-antagonistic and the overall cooperation gains increase in step with participating parties. Let S be a subset of N , representing a cooperative alliance formed among the parties, such that S ⊆ N . v S denotes the profit of the cooperative alliance S , and N , V represents an n -person cooperative game with the characteristic function v . Specifically, v φ = 0 , v S 1 ∪ S 2 ≥ v S 1 + v S 2 , where S 1 ∩ S 2 = φ and φ represents the empty set. By applying the Shapley Value method to evaluate the marginal contributions of each party to the overall cooperation, a fair and reasonable allocation of interests for each party can be obtained.

The total benefits obtained by the cooperative alliance surpass the sum of the individual benefits obtained by each party working alone, i.e., v S 1 > ∑ i ∈ S v i (1)

Here, v i represents the profits obtained by participant i when working alone.

Let X i represent the distribution of benefits obtained by member i from the cooperative alliance. X = X 1 , X 2 , . . . , X n indicates that the cooperative participants’ benefits in the alliance are no less than the benefits obtained by independently completing the project, where i = 1,2 , . . . , n . X i ≥ v i (2) ∑ i ∈ S X i ≥ v s (3)

Assuming that the members of the cooperative alliance join the alliance N in a random order, there are a total of n ! possible orderings. Each ordering has an equal probability of occurrence, which is 1 n ! . When party i forms an alliance S with the preceding ( s − 1 ) individuals, the marginal contribution of party i to the alliance S ∈ N / i is v S − v S \ i . Since there are S − 1 ! permutations of the preceding ( s − 1 ) parties and n − s ! permutations of the remaining n − s parties, the total number of permutations for the alliance S formed is n − 1 ! S − 1 ! . The probability of forming the alliance S is n − S ! S − 1 ! n ! . Therefore, the expected value of the marginal contribution of participant i in alliance S is precisely the Shapley value.

The calculation formula for the Shapley value φ v = ( φ 1 v , φ 2 v , . . . , φ n v ) is as follows: φ i v = ∑ S ∈ S i W S v S − v S \ i (4)

Substituting W S = n − S ! S − 1 ! n ! , we have: φ i v = n − S ! S − 1 ! n ! v S − v S \ i (5)

Here, i = 1,2 , . . . , n , S represents the number of cooperating parties in the alliance; S \ i represents all subsets of set N that contain i ; W S is the weighting factor; v S \ i represents the overall benefit obtained by the alliance S when party i is excluded; and v S − v S \ i represents the marginal contribution of party i to the alliance S .

The research in this paper focuses on the risk sharing of quasi-public-welfare water conservancy PPP projects, which involves the allocation of costs within a cooperative alliance. To derive cost sharing, the term “revenue” in the previous formulas needs to be replaced with “cost”. The specific calculation formulas are as follows: φ i c = ∑ S ∈ S i W S c S − c S \ i (6) φ i c = n − S ! S − 1 ! n ! c S − c S \ i (7)

But before that we need to explore the four axioms of the Shapley Value to determine its suitability for risk-sharing studies in PPP projects. The study of risk sharing in PPP projects is grounded on the principle that benefit sharing is premised on risk sharing, and both parties voluntarily cooperate for mutual benefit. Figure 5 demonstrates the analysis of risk sharing in PPP projects based on the four axioms of the Shapley Value:

First, all partners share all the risks of the project, without any risk being left unshared, which aligns with the effectiveness axiom of the Shapley Value. Secondly, although the government department is the initiator of a PPP project and the social capital party plays a dominant role in the PPP project company, the risk-sharing results only take into account factors such as risk-control ability, risk-bearing ability, and risk willingness of each party and the legal equality of each partner is not considered during risk sharing. This satisfies the axiom of symmetry. Thirdly, sum of the risks respectively borne by each party is amount to the total risks of the PPP project, in line with the axiom of additivity. Finally, there is a clear division of work and responsibilities among the participants with each being required to share both project benefits and risks. As a result, no partner solely bears risks without enjoying benefits or solely enjoys benefits without bearing risks, aligning with the axiom of virtuality.

In conclusion, PPP projects, including quasi-public-welfare water PPP projects, align with the four axioms of the Shapley Value method. Consequently, the Shapley Value Theory can be effectively applied to analyze and study the risk-sharing dynamics in quasi-public-welfare water PPP projects. This approach provides a solid foundation for understanding the fair distribution of risks and benefits among the project stakeholders, promoting the overall effectiveness and success of such projects.

The Utility Theory, originally proposed by mathematician D. Bernoulli (1954), is a decision-making theory widely used in economics. It provides a framework for decision-makers to make choices by considering their individual utility or satisfaction levels. Since different decision-makers may have different utility preferences, even when presented with the same information, they still have diverse perceptions and evaluations.

In the context of risk-sharing decisions, the Utility Theory employs utility values as quantitative indicators. Higher utility values represent higher satisfaction with the expected returns, while lower utility values indicate lower satisfaction. The Utility Theory takes into account various factors such as the proportion of capital contributions of the parties involved and their bargaining positions. It enables the measurement of evaluations, preferences, and attitudes of different individuals toward risks and other factors.

Through the risk-sharing game that considers expected benefits and costs, the utility theory model enables the determination of a reasonable risk-sharing ratio., which provides valuable insights and references for decision-makers involved in the project. By utilizing this model, decision-makers can make more well-grounded and strategic choices, leading to optimized risk management and maximized project revenue.

In quasi-public-welfare water conservancy PPP projects, the utility functions U of the government department and the social capital party can be represented as functions of project benefits V and the cost of project risk control C , i.e., U = U V , C . Considering that PPP projects usually involve risk sharing and co-operation between multiple participants, each participant has a different sensitivity and preference for risk. With a fixed total investment, each party’s perception of risk taking has a non-linear characteristic, i.e., a higher risk control cost leads to a greater loss of utility, while a lower risk control cost has a lesser impact on the increase of utility, and as a quasi-public welfare projects the contribution of economic returns to utility is linear. Therefore, this paper adopts a quadratic utility function for quasi-public welfare PPP projects can comparatively describe this nonlinear relationship, the formula is: U = − 1 a C 2 + V + 1 (8)

The parameter a > 0 is estimated or adjusted according to individual preferences and risk attitudes. Larger values of a imply that individuals are more resistant to the cost of taking risks, and government departments are more sensitive to costs than social capitalists in Quasi-Public-Welfare Water Conservancy PPP Projects.

The optimization model for minimizing project risk-control costs is represented as follows: Max U 1 V 1 , C 1 + U 2 V 2 , C 2 (9) M i n C r C 1 , C 2 (10)

Here, U 1 , V 1 , and C 1 represent the utility function, benefits, and risk-bearing cost of the government department, respectively. U 2 , V 2 , and C 2 represent the utility function, benefits, and risk-bearing cost of the social capital party.

Assuming the expected total risk cost is C e , where K e represents the proportion of risk shared by the government department (with 0 ≤ K ≤ 1 ) and the proportion of risk shared by the social capital party is 1 − K . The actual cost required for risk control is represented as C r . C e = C e g + C e s (11) C r = C r g + C r s (12) C e g = K C e , C e s = 1 − K C e (13) C r g = K C r , C r s = 1 − K C r (14)

Here, C e g and C r g represent the expected risk cost borne by the government department and the actual risk cost, respectively. C e s and C r s represent the expected risk cost borne by the social capital party and the actual risk cost, respectively.

The actual cost of risk is a function of the effectiveness of risk control and management, as well as the shared estimated risk cost. The actual risk cost is determined by the effectiveness function of risk control and the function of shared expected risk costs, represented as: C r g = F K , C e g (15) C r s = G 1 − K , C e s (16)

Here, F and G are the effectiveness functions of risk control for both the public and private sectors. The spillover benefits respectively obtained by the public and private sectors from effective risk control are as follows: V 1 = C e g C r g = K C e C r (17) V 2 = C e s C r s = 1 − K C e C r (18)

Furthermore, the maximum utility function can be derived as: max U 1 K C e C r , K C r + U 2 1 − K C e C r , 1 − K C r (19) min C r (20)

To maximize the overall efficiency of the quasi-public-welfare water conservancy PPP project, a comprehensive utility objective function is constructed for the government and the social capital: f U 1 , U 2 = a 1 U 1 ∙ U 1 0 + a 2 U 2 ∙ U 2 0 (21)

Here, a 1 and a 2 are the weighting coefficients of the negotiation between the government and the social capital, and a 1 + a 2 =1. Generally, a is inversely proportional to the risk-sharing ratio. U 1 0 and U 2 0 are the initial utility values for the government and the social capital without risk impact, and U 1 ∙ U 1 0 and U 2 ∙ U 2 0 represent the spillover utilities obtained by the government and the social capital due to risk assumption. Since the overall objective is to maximize the comprehensive benefits, the spillover utilities are positive values, i.e., U 1 ∙ U 1 0 > 0 and U 2 ∙ U 2 0 > 0 .

Due to the long cooperation period and complex risks involved in the quasi-public-welfare water conservancy PPP project, it is challenging to determine the total risk cost. The minimum risk cost represents the joint expected cost of risk for both the government and the social capital, which can be expressed as: C r = E C ¯ (22)

The optimization model for risk-sharing is as follows: m a x f U 1 , U 2 = max ⁡ { a 1 U 1 K C e − C r , K ∙ C r − U 1 0 + a 2 U 2 1 − K ∙ C e − C r , 1 − K ∙ C r − U 2 0 (23) s . t .   C r = E C ¯

In this model, the objective is to maximize the function f U 1 , U 2 , which represents the comprehensive utility of the government and the social capital. The constraint ensures that the actual risk cost C r is equal to the expected cost E C ¯ , which represents the desired level of risk control.

To solve for the optimal risk cost-sharing ratio for the government, we can take the partial derivative of f U 1 , U 2 with respect to K . Given that C r is a function of K and C e is constant, we obtain: ∂ f ∂ K = a 1 C e − C r ∂ U 1 ∂ V 1 − K ∂ U 1 ∂ V 1 ∂ C r ∂ K + C r ∂ U 1 ∂ C 1 + K ∂ U 1 ∂ C 1 ∂ C r ∂ K + a 2 ∂ C e − C r ∂ U 2 ∂ V 2 − K ∂ U 2 ∂ V 2 ∂ C r ∂ K + C r ∂ U 2 ∂ C 2 + K ∂ U 2 ∂ C 2 ∂ C r ∂ K (24)

By applying the Lagrange multiplier method to find the optimal solution, we let a 1 = a and denote the optimal risk cost-sharing ratio as K 0 = u ∙ a . Thus, we have: L = f U 1 , U 2 + λ E C ¯ − C r = a U 1 K 0 C e − C r , K 0 ∙ C r − U 1 0 + 1 − a U 2 1 − K 0 C e − C r , 1 − K 0 ∙ C r − U 2 0 + λ E C ¯ − C r (25)

Taking ∂ L ∂ a = 0 and ∂ L ∂ λ = 0 , we obtain: ∂ L ∂ a = a C e − C r ∂ U 1 ∂ V 1 u ′ a + C r ∂ U 1 ∂ C 1 u ′ a + U 1 K 0 C e − C r , K 0 ∙ C r − U 1 0 − 1 − a C e − C r ∂ U 2 ∂ V 2 u ′ a + C r ∂ U 2 ∂ C 2 u ′ a − U 2 1 − K 0 ∙ C e − C r , 1 − K 0 ∙ C r − U 2 0 = 0 (26) ∂ L ∂ λ = E C ¯ − C r = 0 (27)

By Eq. 27, can be determined. Substituting C_r into Eq. 25 yields the value of a , which allows the determination of K . Consequently, the risk allocation between the parties involved will be established. The determined risk allocation not only effectively motivates the cooperation of all participants, enabling risk sharing, but also reduces the overall project risk cost.

The application of the Utility Theory for quasi-public-welfare water PPP projects assumes that the following conditions are all satisfied.

(1) Cost minimization and benefit maximization as key performance indicators for both partners;

Though taking into account factors such as the proportion of capital contributions and bargaining position of both parties, the Utility Theory neglects the disparity in the parties’ risk-management abilities. On the other hand, the Shapley values calculation is based on the cost and ability of the parties to handle a specific risk, providing insights into their differing risk-management capabilities. To address these considerations comprehensively, a combined approach of the Shapley Value and the Utility Theory is employed. The Shapley value is calculated for different sharing agents, reflecting their respective abilities in risk handling. Meanwhile, the Utility Theory is used to calculate the risk-sharing ratio based on the preferences and evaluations of the sharing parties. The Shapley value is then adopted to modify the risk-sharing ratio. By integrating these two approaches, a risk-sharing model based on the Shapley Value and the Utility Theory is established. Firstly, the relationship between the risk loss of heterogeneous subjects and the corresponding value scale and utility attribute is calibrated by comprehensively analyzing the factors such as the proportion of capital contribution, negotiation status and risk disposal ability of heterogeneous subjects, and the transformation characteristics and relationship of risk preference of heterogeneous subjects on time and space scale are summarized. Secondly, the utility value of the heterogeneous subject to the evaluation utility is used as the quantitative index to construct the comprehensive utility objective function of the heterogeneous subject. Finally, The Shapley Value is then applied to modify the risk-sharing ratio based on the Utility Theory. The risk-sharing process based on the two theories is depicted in Figure 6, illustrating the comprehensive and balanced consideration of these factors.

The steps are as follows.

The first step: Determining risk-sharing parties.

During the risk management process, it is essential to establish the allocation of risks between government departments and the social capital party provided that those mutually shared risks have also been identified. Since the objective of risk-sharing is to address the risks that both parties are responsible for, before initiating the sharing process, it is crucial to establish a clear understanding of which risks fall under the shared responsibility of both parties.

The second step: Calculating the share of risk based on Utility Theory.

Provided that the risk-sharing objectives are determined, based on the risk preference, control ability, and proportion of capital contribution of the government and social capital towards the risk, the expected loss brought by risk disposal and negotiation weighting coefficients are determined. The utility functions of the government and social capital in quasi-public-welfare water conservancy PPP projects can be expressed as a function of project returns V and the cost of project risk control C , that is, U = U V , C .

Based on Eqs 8–20, we construct the comprehensive utility objective function for the government and social capital as follows: f U 1 , U 2 = a 1 U 1 U 1 0 + a 2 U 2 U 2 0 (28) where a 1 and a 2 represent the weighting coefficients of the negotiation between the government and social capital, respectively, and a 1 + a 2 =1. In general, a is inversely proportional to the risk-sharing ratio. U 1 0 and U 2 0 denote the initial utility values of the government and social capital respectively, in the absence of risk influence.

The optimization model for risk-sharing can be formulated as follows: m a x f U 1 , U 2 = max ⁡ { a 1 U 1 K C e C r , K ∙ C r U 1 0 + a 2 U 2 1 − K ∙ C e C r , 1 − K ∙ C r U 2 0 (29) s . t .   C r = E C ¯

Taking partial derivatives of K with respect to f U 1 , U 2 , where C r is a function of K and C e is a constant, we can derive the optimal risk-cost sharing ratio K 1 for the government sector and the optimal risk-cost sharing ratio K 2 = 1 − K 1 for the private capital sector.

The third step: Calculate the Shapley values of risk sharing parties.

In order to quantify the cost of risk disposal in consideration of the differences in risk-management capabilities between government departments and social capital partners, the cost of individual risk disposal and the cost of collaborative risk disposal are evaluated. The Shapley values calculation formula, as shown in Eq. 30, is used to calculate the Shapley values for both government departments and social capital partners. φ i c = n − S ! S − 1 ! n ! c S − c S \ i (30)

Where: i represents the subject sharing the risk and can take values from 1 to n . n is the total number of risk-sharing subjects, with n = 2 . c S denotes the cost saved when both parties collaborate in risk management. c S \ i represents the cost saved by the subject sharing the risk. φ i c indicates the Shapley values of the risk-sharing subject.

The last step: By modifying the sharing ratio calculated in the Utility Theory with the Shapley values, the desired sharing ratio and sharing amounts of risks for both parties are obtained.

Based on the calculation results of Steps (2) and (3), the revised risk-sharing amounts and sharing ratios are obtained with Eqs 31, 32: φ i c = φ i c + K i − 1 / n × c S \ i (31) K i = φ i c / ∑ i = 1 n φ i c (32)

In the above equations, i represents the risk-sharing subject and can take values of i = 1,2 , . . . , n ; n represents the number of risk-sharing subjects; K i is the risk-sharing ratio of the risk-sharing party i calculated by the Utility Theory; K i is the risk-sharing ratio of the risk-sharing party i modified with the Shapley Value; φ i c is the Shapley value of the risk-sharing party i ; φ i c is the revised risk-sharing amount of the risk-sharing party i ; and c S \ i is the contribution of risk sharing in the alliance S after excluding i .

The Chongqing Fengsheng Reservoir project construction-and-operation company has a registered capital of 10 million yuan. Social investors who have won the bidding hold a 75% stake, while the PPP project implementation agencies hold the rest 25% stake. The construction of the reservoir constitutes a significant portion of the overall investment, with a preliminary estimated budget of 574.74 million yuan. The project construction will be carried out based on the total amount of investment determined through the bidding process.

To support the project, the successful social investors will contribute 114.94 million yuan, which accounts for 20% of the total project investment. This amount will be injected as a one-time cash investment into the project construction-and-operation company, specifically for expenses related to land acquisition, resettlement, and construction costs. Additionally, the central and municipal governments will provide investment subsidies amounting to 459.8 million yuan.

The Chongqing Fengsheng Reservoir project involves the participation of two entities: the government departments and the social capital party. A preliminary risk-sharing scheme for the project is outlined in Table 1. Among the various risks associated with the project, those that can be controlled by both the public and private parties will be managed through individual risk-control methods, with each party independently bearing the losses incurred by such risks.

For risks that require shared responsibility, it is essential to establish a scientifically and reasonably determined risk-sharing ratio. For the present analysis, a typical example of a risk shared by both parties within the sharing scheme is examined and the risk-sharing ratio between the two entities is analyzed.

A scenario is now assumed where a risk occurs, resulting in a loss cost of 12 million yuan for the government department and 20 million yuan for the social capital party, with the total loss cost of the project amounting to 32 million yuan. If the government department handles the risk independently, due to its special position and power in risk management, it would need to spend 20 million yuan while saving 12 million yuan.

On the other hand, if the social capital party handles the risk independently, leveraging its extensive experience and high management efficiency, it would need to spend 12 million yuan while saving 20 million yuan.

Alternatively, if both parties collaborate and jointly address the risk, they would need to spend 10 million yuan while saving 22 million yuan. These figures illustrate the potential costs and savings respectively associated with each party’s independent risk-management efforts as well as their collaborative approach.

(1) Calculating the sharing ratio obtained from the Utility Theory

During the risk-control negotiation process, due to the differences in contribution between the two parties, the weighting coefficients for the government department, representing the advantaged party, and the social capital side, representing the disadvantaged party, are 0.75 and 0.25 respectively. Assuming utility function of the risk for the government department is: U 1 = − 1 580 C 1 2 + V 1 + 1 (33)

And assuming utility function of the risk for the social capital side is: U 2 = − 1 400 C 2 2 + V 2 + 1 (34)

Here, U 1 , V 1 , and C 1 represent the utility function, benefits, and risk-bearing cost of the government department, respectively. U 2 , V 2 , and C 2 represent the utility function, benefits, and risk-bearing cost of the social capital party. The values of a are assumed to be 580 and 400 in U 1 and U 2 , respectively.

The actual cost of this risk is C million yuan C ≥ 0 , and the initial utility values for the government department and social capital side, excluding risk impact, are U 1 0 = 1 and U 2 0 = 1 respectively.

Based on Eq. 22 in the risk-sharing model, the average expected cost for both parties is 16 million RMB. Assuming the risk-sharing ratio for the government department is denoted as K , by substituting a 1 = 0.75 and a 2 = 0.25 into Eq. 23 and performing rearrangements. We obtain the following expression: m a x f U 1 , U 2 = max ⁡ { 0.75 U 1 K ∙ 1200 − 1600 , K ∙ 1600 − 1 + 0.25 U 2 1 − K ∙ 2000 − 1600 , 1 − K ∙ 1600 − 1 (35)

By substituting the risk utility function and by setting ∂ F ∂ K = 0 , we can solve for K as follows: ∂ F ∂ K = 186200 29 − 284800 K 29 = 0 (36)

Solving this equation yields K = 0.3462 . Consequently, the risk-sharing ratio for the social capital side is given by 1 − K = 1 − 0.3462 = 0.6538 . Hence, the project’s risk will be shared with the government department assuming 34.62% of the risk, while the social capital side will bear 65.38% of the risk.

(2) Calculating the Shapley values

In the calculation of risk sharing in PPP projects, controlling risks in a reasonable manner can lead to a reduction in project losses, which turns into the increase of project returns. The initial Shapley values calculation formula for the Fengsheng Reservoir PPP project in Chongqing is as follows: φ i c = ∑ s ⊆ N n − S ! S − 1 ! n ! (37)

Where: i = 1 represents the government side; i = 2 represents the social capital side; n = 2 ; c S = 22 million RMB; c S \ 1 = 12 million RMB; v S \ 2 = 20 million RMB. we can solve for Shapley value as follows: φ 1 = 2 − 2 ! 1 − 1 ! 2 ! 2000 − 0 + 2 − 2 ! 2 − 1 ! 2 ! 2200 − 1000 = 16   m i l l i o n (38) φ 2 = 2 − 1 ! 1 − 1 ! 2 ! 1200 − 0 + 2 − 2 ! 2 − 1 ! 2 ! 2200 − 2000 = 7   m i l l i o n (39)

Therefore, the initial Shapley value for the government side of the Fengsheng Reservoir PPP project in Chongqing is 16 million RMB, and the initial Shapley value for the social capital side is 7 million RMB.

(3) Modify the Utility-Theory-based sharing ratio with the Shapley Value

Integration of the Shapley Value and the Utility Theory can provide a more comprehensive approach to determining the risk-sharing ratio. While the Utility Theory considers factors like proportion of capital contribution and bargaining positions, it may not account for the varying abilities of the parties to handle risks. On the other hand, the Shapley values calculation precisely considers the costs and abilities of each party in managing specific risks, thereby reflecting their differential risk-management capabilities.

By combining the two theories, the risk-sharing ratio can be adjusted. The Shapley Value can help modify the Utility-Theory-based ratio by considering the parties’ abilities to handle risks. This approach provides a more nuanced perspective that incorporates both the overall contribution and bargaining positions of the involved parties, as well as their risk-management capabilities.

According to the above Utility-Theory formula (23), it is calculated that the government department of the Chongqing Fengsheng Reservoir PPP project bears the proportion K 1 = 0.3462 and the social capital party bears the proportion K 2 = 1 − K 1 = 1 − 0.3462 = 0.6538 .

From the Shapley value Eq. 30, we calculate that Initial Shapley value φ 1 = 16 million for the government side and φ 2 = 7 million for the social capital side.

Calculations are performed according to Eqs 31, 32 to derive the final risk-sharing amount and sharing ratio of each party K ′ . φ 1 c = 1600 + 0.3462 − 0.5 × 2200 = 12.6164   m i l l i o n (40) φ 2 c = 700 + 0.6538 − 0.5 × 2200 = 9.3836   m i l l i o n (41) K 1 ′ = φ 1 c ′ / φ 1 c ′ + φ 2 c ′ = 0.5735 (42) K 2 ′ = φ 2 c ′ / φ 1 c ′ + φ 2 c ′ = 0.4265 (43)

The calculation results are shown in Table 3, the government department bears the amount φ 1 c = 12.6164 million RMB, the social capital party bears the amount φ 2 c = 9.3836   m i l l i o n RMB, the government department bears the risk ratio of 0.5735, and the social capital party bears the ratio of 0.4265.

As the expected loss of the government sector to the risk is smaller than the social capital party, the government’s contribution ratio in the project is larger than that of the social capital party, the government sector as a management organization often occupies an advantageous position in the negotiation on risk sharing, while the utility theory mainly takes into account the two sides of the capital contribution ratio, negotiation position and other factors, so when a single use of the utility theory of the Chongqing Fengsheng Reservoir PPP project, both sides of the risk-sharing ratio When calculating, the results are often favorable to the government sector. As can be seen in Table 4.3, the proportion of the risk loss borne by the government sector calculated using the utility theory is 34.62%, and the proportion borne by the social capital party is 65.38%, and the proportion of the risk borne by the government sector is much smaller than that borne by the social capital party. Consistent with the previously expected results, this result is unfair to the social capital side, and the use of a single utility theory to calculate the risk-sharing ratio will hinder the enthusiasm of social capital to participate in the quasi-public welfare water conservancy PPP project.

On the contrary, when government departments deal with risks, their processing capacity is often weaker than that of social capital, and for various reasons, their cost is larger than that of social capital, while Shapley value theory considers more factors such as the cost and ability of both parties to deal with a certain risk. Therefore, when Shapley value theory is used alone to calculate the risk-sharing ratio of both parties in Chongqing Fengsheng Reservoir PPP project, the results are often beneficial to social capital. It can be seen from Figure 4 that the proportion of government departments to bear the risk loss calculated by Shapley value theory is 69.57%, and the proportion of social capital is 30.43%. The proportion of government departments to bear the risk is much higher than that of social capital. Consistent with the previous expected results, this result is unfair to government departments. Using a single Shapley value theory to calculate the risk-sharing ratio will lead to excessive burden on government departments and cannot give full play to the advantages of the PPP model.

However, the introduction of the PPP model for quasi-public welfare water projects can bring the advantages of financial support, professional management, risk sharing, technological innovation and other aspects, which can help promote the smooth implementation and long-term development of the project. Facilitating quasi-public welfare water PPP projects requires full consideration of the responsibilities and interests of the partners to ensure that the cooperation is fair and equitable in order to achieve a win-win situation. A single risk-sharing calculation method using utility theory or Shapley value theory cannot ensure fairness for all parties.

Therefore, in order to improve the enthusiasm of the partners of Chongqing Fengsheng Reservoir PPP project and reduce the overall risk of the project, this study is based on a multi-party perspective, coupling the Shapley value with the utility theory, amending the risk sharing ratio of the utility theory with the Shapley value, and taking into account the differences in the proportion of the parties’ contributions, negotiation status, risk disposal ability, etc. The Shapley value is used to modify the risk sharing ratio of utility theory, and the risk sharing calculation is carried out by taking into account the contribution ratio, negotiation status, risk disposal ability difference and other factors of each party. The results of the study show that: when the risk sharing model based on Shapley value and utility theory is used, the proportion of risk borne by the government is 57.35%, which is slightly higher than that of the social capital party (42.65%), and the calculation result is in between the results of the single utility theory or the Shapley value theory. The risk sharing model demonstrates significant advantages over the single method in considering various factors among all parties, which supports the previously proposed hypothesis. The derived risk sharing ratio is shown to be fairer and more reasonable, effectively mitigating the issue of unfairness arising from the use of a single method to calculate results. The model thoroughly accounts for the distinct characteristics and strengths of government departments and social capital parties, leading to a more rational and widely accepted risk-sharing approach. Consequently, it successfully mobilizes the enthusiasm and benefits of all involved parties, thereby reducing overall project risk.

The risk sharing model brings various benefits to quasi-public water resources PPP projects. Firstly, a fair and reasonable risk sharing ratio facilitates the formation of cooperation consensus among PPP project stakeholders. During the initial stages of project collaboration, mutual recognition of the fairness in risk sharing serves as the foundation for establishing cooperation consensus. Consequently, acceptance of the equitable risk allocation reduces cooperation concerns and promotes successful project implementation.

Secondly, it encourages active participation of both social capital parties and government departments in quasi-public water resources PPP projects. The risk sharing model takes into account the characteristics and advantages of both parties, balancing their interests to derive a more equitable and acceptable risk sharing proportion.

Lastly, the model enhances the success rate of quasi-public water resources PPP projects. The fair and reasonable risk sharing mechanism reduces disputes and dissatisfaction over risk allocation, consequently improving the project’s success rate. When all parties are satisfied with and perceive the risk sharing ratio as reasonable, they are more likely to fully commit to project implementation, thus reducing potential risks and uncertainties and increasing the likelihood of project success.

The successful application of this model provides robust support for the sustainable development of quasi-public water resources PPP projects and offers new research ideas and approaches for risk sharing in similar collaborative projects. By promoting active participation of all stakeholders through equitable risk sharing, the model achieves the dual objectives of risk reduction and project success. Hence, this model holds significant applicability value for future projects with similar characteristics.