Behavioral Decision-Making of Key Stakeholders in Public-Private Partnerships: A Hybrid Method and Benefit Distribution Study

1  Introduction

Although concessions have been used by governments to develop public works since the 18th century, the Public Finance Initiative (PFI), which was introduced in the United Kingdom in 1992, is the modern form of public-private partnership (PPP) model that has gained global recognition [1]. Since then, PPP has been regarded as a way to innovatively produce and manage public services worldwide. Besides, PPP is widely viewed as a means to improve the overall quality and efficiency of public services [2]. That said, many countries around the world regard PPPs as a successful mechanism for solving infrastructural bottlenecks which alleviates the government from the financial pressures of a project and public debt [3,4]. However, even though the PPP model has developed rapidly and achieved remarkable results in China, project delivery costs, such as higher transaction and borrowing costs for private parties, as well as inflated operating costs of PPP projects hinder the attainment of mission-critical goals. Consequently, despite tremendous support being given by the government to facilitate social capital development, PPP has not yielded desired results—especially when value-for-money assessment requirements are incorporated [5]. Moreover, the practical application of PPP models entails that the utilization of social capital is still insufficient, which drives the factors influencing the decision-making of social capital investment in PPP projects to be reconsidered. Likewise, as the initiator of most projects, the government tries to cooperate with social capital by broadening financing channels, resolving local debt risks, improving their efficiency levels and quality of public services, and then attempts to protect public rights simultaneously. Expectedly, the government’s goal in a PPP project is to maximize public welfare. However, the motive of social capital is the potential profitability of the project, which may harm public welfare [6]. Notably, to attract substantial social capital investment, the government should endeavor to satisfy its precondition of providing both social and economic benefits to its citizens and private parties using taxes, subsidies, and incentives as instruments of optimum allocation.

As China is in the rapid development period of infrastructure upgrading, the infrastructure has become more abundant in various facets of the economy, and the PPP model is being encouraged to enter more fields. However, when compared with traditional PPP projects, the market risk faced by social capital participation in new PPP project models is higher than before, as a result of the uncertainty of the future market in the new field in recent years. For instance, the old residential reconstruction project has emerged recently in China and the government encourages social capital participation [7], notwithstanding that the implementation mode is still being explored. Thus, there exist uncertainties in the future market which portends significant risks that might affect the long-term operation of PPP projects. Although the Chinese government fully supports innovation in the traditional PPP model, uncertainties about whether social capital can get the corresponding return when implementing innovation in PPP projects imply that the attitude of social capital towards innovation might be quite different [8]. Therefore, social capital responds not only to the uncertainties arising from policy changes by local governments but also to the unpredictability of market demand, technological construction, and management changes. Furthermore, current realities indicate that there exists great ambiguity about the degree to which the infrastructure and public services provided by social capital can be accepted by the market. In addition, the prospect of most PPP projects is tentative for both the government and social capital, and both parties will jointly face the uncertainty of the project benefit. The traditional stochastic models cannot well explain the complex behavior of PPP projects. In some practical cases, due to the limitation of the system and time, it is difficult to collect data [9]. To effectively manage PPP projects, the fuzzy theory is an important theoretical tool to solve fuzzy phenomena in management decision-making and game problems when compared with probability theory. Besides, a thick description of these uncertainties plays an important role in solving fuzzy uncertainties.

In the same vein, since profit and loss are an important basis for stakeholders to make informed decisions as to whether to participate in PPP projects or not, a reasonable benefit distribution mechanism is the basic qualification for the success of an excellent PPP project. For instance, the fairness of benefit distribution systems provides incentives for social capital to participate in decision-making processes that promote its efforts and positively influence the performance of PPP projects [10,11]. There are some stakeholders involved in PPP projects, who have different and overlapping interests, and each stakeholder has its characteristics in the attitude towards project risks. Therefore, an in-depth analysis of the relationship between the behavioral decision-making of various stakeholders and project performance is of great significance for the successful implementation of PPP projects. However, the behavioral decision-making of stakeholders in PPP projects has not been paid enough attention to, and the existing research on the relationship between decision-making and performance in PPP projects is also very few. In this respect, this study attempts to explore the reasonable distribution of benefits in public-private partnerships and find the influence mechanism of benefit distribution of PPP projects in different circumstances, thus affecting the behavioral decision-making of stakeholders and then improving the performance of PPP projects. Stakeholders of a PPP project include government departments, project companies, social capital, clients, and so on [12,13]. But the government and social capital are selected as research objects in this paper, which have an interest in the project and have the ability to influence the project.

Apart from the above motivation, another decisive task is to adopt a best-fitted methodology to supply the gaps. As there are many uncertain conditions in PPP projects, these uncertain conditions cannot be ignored in this study, so we should consider how stakeholders of a PPP project make their own behavioral decisions in the fuzzy decision-making environment. Unlike the traditional approach, where the possible uncertainties in a PPP project have been solved based on probability theory. But probability distributions are usually predicted based on historical data. When statistical data are unreliable, the models based on determining these probability distributions may not be the best choice [14]. The application of fuzzy theory has been considered in many fields and achieved satisfactory results. Thus, we apply fuzzy theory and game theory to explore the best performance of a PPP project and the behavioral decision-making of stakeholders with different risk preferences. To the best of our knowledge, no significant study has been found that explores the issue using fuzzy theory, particularly since the research idea is entirely missing in PPP literature. The main contributions and innovations of this paper are as follows. Firstly, the expectation model and the opportunity-constrained programming model are introduced into the game model of public-private behavioral decisions in a PPP project, enriching the theoretical study in this field. Secondly, it is taken into account that the government and social capital are decision-making subjects with different risk preferences and the utility differences are caused by the cognition of actors, then four game models are put forward. Thirdly, the coordination mechanism of benefit distribution of stakeholders under decentralized decision-making and centralized decision-making is studied, and sensitivity analysis is made on the influencing factors. Through the research, the government is provided with a theoretical basis and reference path to improve the performance of a PPP project more scientifically and effectively.

The rest of this paper is structured as follows. Section 2 introduces the existing research results. Section 3 lists some preliminaries, including the definition of the expected value of fuzzy variables, and the concept of triangular fuzzy numbers. Section 4 describes the problem and proposes assumptions. Section 5 is the process of model optimization and solution. Section 6 designs the benefit distribution between public entities and private parties under centralized decision-making. Section 7 demonstrates the four models under decentralized decision-making and centralized decision-making through a specific example, and analyzes the sensitivity of the factors that affect the behavioral decision-making of public entities and private parties. Then, by comparing with other similar studies, the outstanding advantages of the four models proposed in this paper are shown. Section 8 puts forward conclusions and suggestions, and finally summarizes the limitations and future study of this paper in Section 9.

2  Literature Review

The present study examines the link between the performance of a project and behavioral decision-making of stakeholders in PPP models by considering the role of benefit distribution, risk preference, and effort level of public entities and private parties in the case of China.

Economic benefits are the main driving force for the private party involved in the PPPs, hence, there should be a reasonable and efficient distribution of benefit system. In the existing studies, Love et al. [15,16] pointed out that fair distribution between public entities and their private partners was crucial for the delivery performance of infrastructure projects. Similarly, Dyer [17,18] were of the view that it was only when the benefit was fairly distributed in PPP projects, that efforts could be rewarded, which motivated various parties to develop enthusiasm to participate in PPP projects. Interestingly, Wang et al. [19] put forward that a fairer and more reasonable benefit distribution system should include capital investments, asset monetization, and private contributions, as well as participation in PPP projects. That is why Fainstein [20] introduced the benefit distribution of PPPs in Singapore. However, since the public sector controls the benefit distribution of PPPs in Singapore, which is dominated by the state, there is a tendency that private parties might not be satisfied with the benefit distribution system. From the above-mentioned literature review, it is obvious that the distribution of the benefit of PPP projects is a contentious matter, notwithstanding its importance. Although there exist some limitations in parameter selection and application environment in previous studies, they still provide development ideas for further research exploring the relationship between benefit distribution and the project performance of PPPs. Besides, it had been proven beyond reasonable doubt that reciprocal preference in the exchange of potential information among team members, and the development of a good cooperative relationship had positive effects on project performance [21]. This is the reason why the Chinese government could make use of reciprocal preferences to encourage social capital to make concerted efforts in the process of cooperation, as well as avoid any moral hazard and opportunistic [22]. In this paper, social capital is one of the stakeholders, which can be regarded as a private party. Therefore, in PPP projects, how to encourage social capital to invest more resources and efforts in the project, and how to rationally distribute the total output and income of the project play an important role in studying the behavior of participants’ decision-making and the PPP project performance.

Consistent with prior studies, the researcher observes that the impact of benefit distribution has been investigated using empirical techniques such as the Shapley value method, Nash negotiation model-based distribution method, the game quadratic programming (GQP) method, as well as the simplified minimum cost-remaining savings (MCRS) method. In addition, some researchers have also used game theory to study the distribution of benefits between public entities and private parties in PPP projects. According to Ding et al. [23], a quantitative method based on simulating an unfair aversion of alliance members to analyze the risk-sharing arrangement in PPP projects was found to be effective when studying the impact of benefit distribution in PPPs. Cao et al. [24] established a tripartite rent-seeking game model to explore the game dilemma among government regulators, private enterprises and government agents involved in the bidding operation of PPP projects. Feng et al. [25] regarded the decision-making activities of the concession period as a negotiation game process between public entities and private parties, thereafter, solved the optimization problem using a backward induction technique. Besides, under the current Chinese PPP model, the status of the government and social capital is different, since it depicts a principal-agency relationship. Given that the decision-making process of both parties involves the government deciding on an appropriate investment plan, while social capital follows the same strategy, it is completely suitable to be analyzed by the Stackelberg game model [26]. In the studies of PPP projects using the Stackelberg model to analyze the behavior decisions of both public and private parties, Wang et al. [27] constructed a Stackelberg game model to find the best subsidy plan given different levels of the government financial capacity (GFC). Liu et al. [28] established a bi-level programming model with the Stackelberg game to solve the multi-objective programming problem of public-private asymmetry, and verified the effectiveness of the model and algorithm through an example. Shang et al. [29] proposed a model based on the Stackelberg game to help public institutions design the payment mechanism for PPP transportation projects. Based on the transaction cost theory, Zhang et al. [30] constructed the incentive mechanism and value-added Stackelberg game model of PPP projects under the early intervention scenario of financial institutions.

Fascinatingly, in contemporary PPP literature, the fuzzy theory is widely applied to risk analysis, scheme selection, influencing factor analysis, and so on. For instance, Ameyaw et al. [19,31] studied the allocation of risks in PPP water-infrastructure and waste recycling projects based on a fuzzy approach. Correspondingly, Mazher et al. [32] proposed a non-additive fuzzy integral based multiple attribute risk allocation decision approach to effectively aggregate each stakeholder’s risk management capability assessment on the accepted risk allocation principles in PPPs. And Liu et al. [33] proposed an entropy-based gained and lost dominance score (GLDS) method for the social capital selection of PPP projects using their fuzzy information. Okudan et al. [34] established an alternative dispute resolution selection framework to settle disputes involving PPP projects. Given the anomalies that might result from PPP evaluation, Wang et al. [35] assessed the impact of the critical success factors of PPP projects using fuzzy theory. Due to the large investment scale of PPP projects and their complicated financing structure [36–38], managing PPP projects is a difficult task [39,40]. Moreover, the environmental emergencies related to the complexity and environmental uncertainty of PPP projects pose serious challenges [41]. Studies on the existing PPP projects emphasized the importance of risk assessment [42–48], rational design of concession period [49–53], and the government subsidy incentive mechanism to avoid contract renegotiation and ensure the successful operation of the project [54–56]. For the uncertain factors in a PPP project, Monte Carlo simulation of random variables was mostly used [57–62]. The results of the Monte Carlo simulation were represented by the probability distribution function (PDF) and cumulative distribution function (CDF) [63]. Traditional Stackelberg game models also deal with uncertain variables based on probability theory.

From the literature review, it can be concluded that the fair and reasonable distribution of benefits in a PPP project can attract the investment of social capital and put more effort into it. However, there is little research on the influence of benefit distribution combined with other factors on the behavioral decision-making of stakeholders, which is still a research gap. For instance, these studies rarely involve the comparison of public-private benefit distribution schemes under decentralized decision-making and centralized decision-making. Some studies only consider the risk preference of social capital, ignoring the risk preference of the government in practice. Some studies have studied how to improve the participation enthusiasm of social capital, but these references focus on the research of the government subsidy mechanism, ignoring the objective effort and endogenous motivation of social capital. In addition, some studies have not considered the uncertainty in a PPP project, and given certain values to some uncertain variables such as demand and market size, which to some extent makes the research results unreasonable. Besides, although the fuzzy theory has been applied in the field of PPP, most existing PPP studies use the probability distribution of known parameters to reflect the uncertainty in a PPP project, which makes the characterization of uncertain variables influenced by the subjectivity of researchers, and the parameter data used to input probability distributions into the simulation model is easy to change and difficult to predict.

Therefore, in this study, we examine the link between the performance of the project and the behavioral decision-making of key stakeholders in a PPP project based on fuzzy expected value and game theory. Moreover, the benefit distribution, risk preference and effort level are considered.

3  Preliminaries

Let ξ be a fuzzy variable in the possibility space (θ,P(θ),Pos) (The concept of the possibility space was explained in prior literature [64]). θ is a non-empty set, P(θ) is a power set of θ. Pos indicates the possibility of P(θ). The above-mentioned variables were assigned according to the definitions specified below. Some lemmas were also drawn from examples in this process.

Definition 1. [65] Assume that ξ is a fuzzy variable, and it is not negative, namely Pos{ξ≤0}=0.

Definition 2. [65] Assume that ξ is a fuzzy variable, and α∈(0,1]. ξαL=inf{r|Pos{ξ≤r}≥α} and ξαU=sup{r|Pos{ξ≥r}≥α} are respectively the α-pessimistic value and α-optimistic value of fuzzy variable ξ. The α-pessimistic value and α-optimistic value of ξ are used to express the fuzzy variable values of decision makers when they are pessimistic risk preferences and optimistic risk preferences, respectively.

Definition 3. [66] A triangular fuzzy number A~=(a1,a2,a3) is a fuzzy set defined on the set R of real numbers, where a1<a2<a3 is called triangular fuzzy number, and its membership function is as follows:

µA~(x)={L(x)=x−a1a2−a1,x∈[a1,a2]R(x)=a3−xa3−a2,x∈(a2,a3]0,x∉(a1,a3)(1)

where a1, a2 and a3 are called the lower bound, the median and the upper bound of the triangular fuzzy number A~, respectively. If a1>0, then the triangular fuzzy number A~=(a1,a2,a3) is called a positive triangular fuzzy number.

Example 1: Assume that ξ=(a,b,c) is a triangular fuzzy number, then its pessimistic value and optimistic value of α are expressed as:

ξαL=bα+a(1−α),ξαU=bα+c(1−α)

Lemma 1 [67]: Assume that fuzzy variables ξ and η are independent of each other, then the following formula holds for any α∈(0,1].

1)   (ξ+η)αL=ξαL+ηαL,(ξ+η)αU=ξαU+ηαU

2)   If λ>0, then (λξ)αL=λξαL,(λξ)αU=λξαU.

3)   (ηξ)αL=ηαLξαL,(ηξ)αU=ηαUξαU

4)   (ξ−η)αL=ξαL−ηαU,(ξ−η)αU=ξαU−ηαL

Lemma 2 [67]: If ξ is a fuzzy variable, its limited expected value can be expressed as:

E[ξ]=12∫01(ξαL+ξαU)dα(2)

Lemma 3 [67]: Assume that fuzzy variables ξ and η are independent of each other, for any real number a and b, then the following formula holds:

E[aξ+bη]=aE[ξ]+bE[η](3)

Example 2: Assume that ξ=(a,b,c) is a triangular fuzzy number, then its expected value is as follows:

E[ξ]=12∫01(bα+a(1−α)+ba+c(1−α))dα=a+2b+c4(4)

Example 3: Assume that ξ=(a,b,c) and η=(l,m,n) are triangular fuzzy numbers, then their expected values are as follows:

E[ξη]=12∫01((ξη)αL+(ξη)αU)dα=12∫01(ξαLηαL+ξαUηαU)dα=2al+am+3bl+3bn+cm+2cn12(5)

Definition 4 [67]: Assume that fuzzy variables ξ and η are independent of each other, for any α∈(0,1], if ξ>η, then there is ξαL>ηαL and ξαU>ηαU.

4  Description and Hypothesis of Problem

The motivation for this study is that the rational distribution of benefits between the government and social capital is analyzed, which affects the behavioral decision-making of both parties, and then improves the performance of a PPP project. The existing studies used the probability distribution of given parameters to reflect the uncertainty of benefits and market demand in a PPP project. However, the risk of infrastructure construction at the present stage is obviously higher than in the past, and there is great uncertainty about the extent to which the infrastructure projects and public services provided by social capital can be accepted by the market. Moreover, the complicated internal and external environment of a PPP project adds more technical and management uncertainties. If the unknown parameters are described by some random variables, it will cause a big deviation, so it is more difficult to estimate the distribution of random variables accurately. Therefore, the corresponding parameters in the demand function predicted by the experience of industry experts are more valuable for reference. Market scale, the effort cost coefficient of the government and social capital are fuzzy and uncertain. Market demand sensitivity coefficient to price and effort level of both public entities and private parties are related to the performance of both parties during the process of project operation. Fuzzy theory is an effective tool to solve fuzzy phenomena in management decision-making and game problems, and it plays an important role in solving such fuzzy and uncertain problems. The parameters and variables symbol used in this paper are shown in Table 1.

To simplify the research environment, this paper puts forward the following hypotheses.

Hypotheses 1: The key stakeholders of this project are the government and social capital. The government is the initiator of the project, and also provides various policy support for the project. The overall investment in the PPP construction period is I, the proportion of the government investment is λ, and the proportion of social capital investment is 1−λ.

Hypotheses 2: After the completion of a PPP project, the demand function is related to the price of public products, as well as the effort level of both public entities and private parties. With reference to the functional expression of logistics service demand [68], the fuzzy demand function of the market can be expressed as: Q(P,ei)=B~−θ~P+β~e1+γ~e2(B~>0,α~>0,β~>0).

Hypotheses 3: The government’s effort is manifested in the subsidies and taxation of the project, supervision responsibility of the quality and the price of public goods, as well as the management of project construction risks. Therefore, the government’s effort cost can be expressed as: C1(e1)=12µ~1e12. µ~1 denotes the effort cost coefficient of the government, reflecting the government’s ability to formulate project subsidies, taxation and other systems, as well as its ability to supervise social capital. The greater the value, and the higher the cost at the same effort. e1 denotes the effort level of the government to participate in the project. The greater the value, the higher the effort level. The effort cost of social capital can be expressed as: C2(e2)=12µ~2e22. µ~2 denotes the effort cost coefficient of social capital, reflecting the profitability of social capital, and the investment intensity of the advanced technology and equipment. e2 represents the effort level of social capital to participate in the project, including technical capital and equipment investment in the whole process of the project. In a PPP project, social capital has financial support and large capital investment, so µ~1<µ~2 is set up. The expressions of mutual effort cost show that marginal cost increases with each party’s efforts to performance [22,69].

Hypotheses 4: The government always takes the maximization of public welfare as its goal, while social capital pursues economic benefit. During the concession period of the project, it is agreed that the proportion of the government in the overall benefit of a PPP project (the benefit obtained by the government from the project is public welfare) is φ(0<φ<1), and the proportion of benefit distribution of social capital can be expressed as: 1−φ.

Hypotheses 5: Traditionally, the government’s motivation for adopting PPP projects is the technical experience and the financial advantage provided by social capital [70]. Generally, the government mainly bears the macro-political risk that has little impact on the construction of PPP projects, while the social capital mainly bears the risk at the meso level [71]. In addition, given that the government performs the function of supervising and subsidizing social capital, the government might be risk-averse. Therefore, the risk preference of the government is set to pessimistic in this paper. At the same time, because of the long operation cycle of PPP projects, and many uncertain factors, the decision-making of social capital in the process of project implementation is not always rational, so the risk preference of social capital is set to neutral, pessimistic and optimistic. Neutral decision makers are more rational, and the benefit is the expected value of the decision-makers’ benefit. Pessimistic decision makers are cautious about risks, and will not take radical actions, however, the benefit is the pessimistic value of decision makers(α). Optimistic decision-makers are willing to accept the idea of high risk and high benefit and make project decisions with a high-risk attitude, so the benefit is the optimistic value of decision makers(α).

Based on the assumptions above, the following models were developed as follows:

1. The utility function of government is:

π~1=φP(B~−θ~P+β~e1+γ~e2)−λI−12µ~1e12(6)

2. The utility function of social capital is:

π~2=(1−φ)P(B~−θ~P+β~e1+γ~e2)−(1−λ)I−12µ~2e22(7)

3. The overall utility function of the project is:

π~T=P(B~−θ~P+β~e1+γ~e2)−I−12µ~1e12−12µ~2e22(8)

5  Optimization and Solution of Model

In this section, four game models are presented in a fuzzy environment and the effort and performance of both parties are studied. It is that social capital with neutral, pessimistic and optimistic risk preference is the follower when the pessimistic government with risk aversion is the leading role, and the fourth model is the centralized decision-making with complete coordination of benefits and complete symmetry of information between both parties.

5.1 Decentralized Decision-Making

Under the decentralized decision-making model, both the government and social capital aim at maximizing their own benefits. According to the benefit distribution scheme provided by the government and the effort level of the government, the social capital will make investment decisions on the premise of maximizing expected utility, but the government does not have private information for social capital’s decision-making. The problem faced by the government in the decision-making of benefit distribution of a PPP project is how to design a suitable distribution scheme to induce social capital to invest in construction at the optimal effort level. The specific game process can be divided into two stages: the first stage is to determine the proportion of benefit distribution and the effort level of the government. The second stage is to determine the effort level of social capital. While φ and ei are decision variables, and superscript D indicates decentralized decision.

5.1.1 Decision-Making Model of Government Pessimism and Social Capital Neutrality

A pessimistic government with risk aversion plays a leading role in the Stackelberg game model. Firstly, we determine the proportion of benefit distribution (φ) and the level of effort of the government (e1). Subsequently, social capital with neutral risk preference is used to ascertain the effort level e2. Furthermore, based on the idea of fuzzy opportunity-constrained programming, the decision-making model of the pessimistic value of a pessimistic government (α) and social capital with a neutral risk preference is determined as follows:

{maxe1minπ~1Dπ~1D1s.t.Pos{φP(B~−θ~P+β~e1+γ~e2)−λI−12µ~1e12≤π~1D1}≥αPos{B~−θ~P+β~e1+γ~e2≤0}=0e2∗is the solution to the following problem{maxe2E[π~2D1(e2)]=E[(1−φ)P(B~−θ~P+β~e1+γ~e2)−(1−λ)I−12µ~2e22)]Pos{B~−θ~P+β~e1+γ~e2≤0}=0(9)

where α is the confidence level given in advance, min π~1D1 denotes the benefit of the pessimistic value of government (α). While the expected value of E[π~2D1(e2)] is used to indicate the benefit of social capital with a neutral risk preference. Besides, the above opportunity-constrained programming model is equated to the following model:

{maxe1[π~1D1(e1)]αL=[φP(B~−θ~P+β~e1+γ~e2)−λI−12µ~1e12]αLPos{B~−θ~P+β~e1+γ~e2≤0}=0e2∗is the solution to the following problem{maxe2E[π~2D1(e2)]=E[(1−φ)P(B~−θ~P+β~e1+γ~e2)−(1−λ)I−12µ~2e22)]Pos{B~−θ~P+β~e1+γ~e2≤0}=0(10)

Thus, in a fuzzy environment, which is dominated by a pessimistic government with risk aversion and followed by social capital with a neutral risk preference, the following conclusions can be drawn from the decision-making model of PPP projects.

Theorem 1: E[π~2D1(e2)] is the expected benefit of social capital with a neutral risk preference. When the effort level (e1) of the government is given, if Pos{B~−θ~P+β~e1+γ~e2≤0}=0 is established. Hence, the response function of the optimal level (e2) of social capital effort is e2∗=(1−φ)E[γ~]PE[µ~2].

Proof: The benefit of social capital with neutral risk preference is expressed as follows:

E[π~2D1(e2)]=E[(1−φ)P(B~−θ~P+β~e1+γ~e2)−(1−λ)I−12µ~2e22)](11)

then

E[π~2D1(e2)]=(1−φ)P(E[B~]−E[θ~]P+E[β~]e1+E[γ~]e2)−(1−λ)I−12E[µ~2]e22(12)

The first derivative and second derivatives of E[π~2D1(e2)] can be obtained as follows in Eqs. (13) and (14).

dπ~2D1(e2)de2D1=(1−φ)PE[γ~]−e2E[µ~2](13)

d2π~2D1(e2)d(e2D1)2=−E[µ~2](14)

Since d2π~2D1(e2)de22=−E[µ~2]<0, E[π~2D1(e2)] is a concave function of e2, the optimal effort level of social capital with a neutral risk preference can be achieved by making the first derivative of E[π~2D(e2)] equal to zero, that is:

dπ~2D1(e2)de2D1=(1−φ)PE[γ~]−e2E[µ~2]=0(15)

The optimal effort level of social capital obtained from Eq. (15) is as follows:

e2D1∗=(1−φ)E[γ~]PE[µ~2](16)

Theorem 1 is proved.

Theorem 2: [π~1D1(e1)]αL is the benefit of the accrues to a pessimistic government with risk aversion. If Pos{B~−θ~P+β~e1+γ~e2≤0}=0 and 2E[γ~]µ~1αUE[µ~2]>(β~αL)2, there is a unique strategy, that is φD1∗=µ~1αU[E[µ~2](B~αL−θ~αUP)+γ~αLE[γ~]P]2γ~αLE[γ~]Pµ~1αU−P(β~αL)2E[µ~2], e1D1∗=φD1∗Pβ~αLµ~1αU=β~αLPµ~1αU[E[µ~2](B~αL−θ~αUP)+γ~αLE[γ~]P]µ~1αU(2γ~αLE[γ~]Pµ~1αU−P(β~αL)2E[µ~2]), and e2D1∗=(1−φD1∗)E[γ~]PE[µ~2]=E[γ~][γ~αLE[γ~]Pµ~1αU−P(β~αL)2E[µ~2]−µ~1αU(B~αL−θ~αUP)E[µ~2]](2γ~αLE[γ~]µ~1αU−(β~αL)2E[µ~2])E[µ~2]. Consequently, the above solution is the Stackelberg-Nash equilibrium solution of Eq. (10).

Proof: The second stage of the game is considered, and backward induction is used to analyze the strategic choices of both parties, which is the maximization of the utility of the government and social capital.

Consequently, the benefit of a pessimistic government with risk aversion is as follows:

[π~1D1(e1)]αL=[φP(B~−θ~P+β~e1+γ~e2D1∗)−λI−12µ~1e12]αL(17)

If e2D1∗ of Eq. (16) is put into Eq. (17), we derive the following equation:

[π~1D1(e1)]αL=φP(B~αL−θ~αUP+β~αLe1+γ~αL(1−φ)E[γ~]PE[µ~2])−λI−12µ~1αUe12(18)

The first derivative of [π~1D1(e1,φ)]αL relative to (e1,φ) can be obtained in Eq. (18), therefore, its Hessian matrix and its corresponding determinant are H=[−µ~1αUPβ~αLPβ~αL−2P2E[γ~]E[µ~2]] and |H|=[−µ~1αUPβ~αLPβ~αL−2P2E[γ~]E[µ~2]]=2E[γ~]µ~1αUE[µ~2]−(β~αL)2, respectively. When 2E[γ~]µ~1αUE[µ~2]>(β~αL)2, the Hessian matrix is negative, thus this function has a unique equilibrium solution.

We derive the first partial derivative of [π~1D1(e1,φ)]αL to (e1,φ), respectively, and then set it to zero, as follows:

d[π~1D1(e1)]αLde1D1=φPβ~αL−µ~1αUe1=0(19)

d[π~1D1(e1)]αLdφD1=P(B~αL−θ~αUP+β~αLe1)+γ~αL(1−2φ)E[γ~]P2E[µ~2](20)

The following equation can be obtained by combining Eqs. (19) and (20).

φD1∗=µ~1αU[E[µ~2](B~αL−θ~αUP)+γ~αLE[γ~]P]2γ~αLE[γ~]Pµ~1αU−P(β~αL)2E[µ~2](21)

e1D1∗=β~αLPµ~1αU[E[µ~2](B~αL−θ~αUP)+γ~αLE[γ~]P]µ~1αU(2γ~αLE[γ~]Pµ~1αU−P(β~αL)2E[µ~2])(22)

If Eq. (21) is combined with Eq. (16), it is rewritten as follows:

e2D1∗=E[γ~][γ~αLE[γ~]Pµ~1αU−P(β~αL)2E[µ~2]−µ~1αU(B~αL−θ~αUP)E[µ~2]](2γ~αLE[γ~]µ~1αU−(β~αL)2E[µ~2])E[µ~2](23)

Therefore, the Stackelberg-Nash equilibrium solution of Eq. (10) is (e2D1∗,e1D1∗,φD1∗), then the maximum benefits accrued to both the government and social capital is stated as follows:

[π~1D1∗(e1)]αL=φD1∗P(B~αL−θ~αUP+β~αLe1D1∗+γ~αLe2D1∗)−λI−12µ~1αU(e1D1∗)2(24)

E[π~2D1∗(e2)]=(1−φD1∗)P(E[B~]−E[θ~]P+E[β~]e1D1∗+E[γ~]e2D1∗)−(1−λ)I−12E[µ~2](e2D1∗)2(25)

The overall benefit of the project is π~TD1∗=[π~1D1(e1)]αL+E[π~2D1(e2)].

Theorem 2 is proved.

5.1.2 Decision-Making Model of Government Pessimism and Social Capital Pessimism

As mentioned above, the proportion of benefit distribution (φ) and the effort level (e1) are first determined by the government. Subsequently, social capital with a pessimistic risk preference determines the degree of effort of e2 as a follower. Using the idea of fuzzy opportunity-constrained programming, the decision-making model of the pessimistic value of a pessimistic government (α) and the pessimistic value of pessimistic social capital (α) is as follows:

{maxe1minπ~1Dπ~1Ds.t.Pos{φP(B~−θ~P+β~e1+γ~e2)−λI−12µ~1e12≤π~1D}≥αPos{B~−θ~P+β~e1+γ~e2≤0}=0e2∗is the solution to the following problem{maxe2minπ~2Ds.t.Pos{(1−φ)P(B~−θ~P+β~e1+γ~e2)−(1−λ)I−12µ~2e22≤π~2D]≥αPos{B~−θ~P+β~e1+γ~e2≤0}=0(26)

The fuzzy opportunity-programming model can solve the problem of maximizing the pessimistic benefit of the participants’ expectation objective function, based on the premise that constraint conditions are established at a certain confidence level, where α is the confidence level given in advance. Besides, min π~1D and minπ~2D denote the benefit of the pessimistic value of both the government and social capital respectively, which implies that it can be achieved at least with confidence level α. Thus, the above opportunity-constrained programming model is equivalent to the following model:

{maxe1[π~1D(e1)]αL=[φP(B~−θ~P+β~e1+γ~e2)−λI−12µ~1e12]αLPos{B~−θ~P+β~e1+γ~e2≤0}=0e2∗is the solution to the following problem{maxe2[π~2D(e2)]αL=[(1−φ)P(B~−θ~P+β~e1+γ~e2)−(1−λ)I−12µ~2e22)]αLPos{B~−θ~P+β~e1+γ~e2≤0}=0(27)

In the fuzzy environment, the following conclusions can be drawn from the decision-making model of a PPP project, which is dominated by the pessimistic government with risk aversion and followed by the pessimistic social capital with risk aversion.

Theorem 3: [π~2D2(e2)]αL is the benefit of social capital with a pessimistic risk preference. When the effort level of the government (e1) is given, if Pos{B~−θ~P+β~e1+γ~e2≤0}=0 is established, the response function of the optimal effort level of social capital (e2) is e2D2∗=(1−φ)Pγ~αLµ~2αU.

Proof: The benefit of a pessimistic government with risk aversion is expressed as follows:

[π~1D2(e2)]αL=[(1−φ)P(B~−θ~P+β~e1+γ~e2∗)−(1−λ)I−12µ~2e22]αL(28)

Then

[π~1D2(e2)]αL=(1−φ)P(B~αL−θ~αUP+β~αLe1+γ~αLe2)−(1−λ)I−12µ~2αUe22(29)

The first derivative and second derivatives of [π~1D2(e2)]αL can be obtained from Eq. (29).

d[π~2D2(e2)]αLde2D2=(1−φ)Pγ~αL−µ~2αUe2(30)

d[π~2D2(e2)]αLd(e2D2)2=−µ~2αU(31)

Since d[π~2D2(e2)]αLde2D2=−µ~2αU<0, [π~1D2(e2)]αL is a concave function of e2, the optimal effort level of social capital with a pessimistic risk preference can be achieved by setting the first derivative of [π~1D2(e2)]αL equal to zero, that is:

d[π~2D2(e2)]αLde2D2=(1−φ)Pγ~αL−µ~2αUe2=0(32)