Cost Sharing in Insurance Communities: A Hybrid Approach Based on Multiple-Choice Objective Programming and Cooperative Games

Abstract

At present, utilizing the insurance community is a common method to deal with investment risks along the Belt and Road; however, there is no clear method or mechanism to deal with the decision-making optimization and cost allocation of the insurance community participants. We propose a hybrid approach to solve this problem. First, we construct an underwriting decision optimization model for the insurance community using the multi-choice goal programming method, which generates the cost characteristic function based on a cooperative alliance. Second, we use the cooperative game method combined with the modified Shapley value method to take risk factors into consideration, which allows us to optimize the cost allocation among members of the insurance community. Finally, our simulation analysis results show that the multi-choice goal programming method can optimize the insurance community’s underwriting decisions. Specifically, the total underwriting cost is lower than the sum of the underwriting costs under the insurance company’s single-action strategy, and the total underwriting scale is as large as possible. Compared with the classical Shapley value method, the modified Shapley value method can better reflect differences in the underwriting risks of different regions, encouraging governments to take measures to reduce underwriting risks. To conclude, we propose some suggestions based on our research findings. The possible contributions of this paper are as follows: our research provides a hybrid optimization method based on multiple-choice objective programming and cooperative games to solve the cost allocation problem facing the insurance community, and it has some reference value for improving the cost-sharing system of the insurance community.

1. Introduction

The Belt and Road Initiative has policy communication, facility connection, smooth trade, financial integration, and people-to-people contact as its core. It creates a community of interests, destiny, and responsibility with mutual political trust, economic integration, and cultural inclusion. The Belt and Road Initiative has become a new platform for mutually beneficial and win-win cooperation on a global scale. As a result, China has made great progress in its integration into the world economy and has become a major trading country with global influence.

The Belt and Road Initiative has made continuous progress through cooperation since it was put forward. It is now a platform for international cooperation with broad participation that vividly demonstrates and promotes a shared-future community for mankind. However, we should be aware that most of the countries along the Belt and Road are developing or emerging markets with large geographical differences and diverse risks. These include political [1], social [2], economic [3], default [4], legal [5], and natural disaster risks [6], among many others. The insurance mechanism is a common risk-prevention tool in international economic affairs and has several characteristics that can be used to determine if the insurance service has natural advantages in the construction of the Belt and Road. Therefore, the insurance mechanism should play a greater role in reducing the worries of enterprises participating in the construction of the Belt and Road [7]. Regarding the risks of investment projects along the Belt and Road, the biggest challenge is to effectively solve the problems of risk differentiation in different regions, the unbalanced development of insurance markets in various countries, and the insufficient underwriting capacity. The investment scales of the countries and regions along the Belt and Road have considerably increased since its launch. According to the investment aggregation effect, huge losses will transpire if risks occur in a gathering place [8]. Therefore, for large-scale investment projects, insurance businesses are urgently needed to protect the corresponding businesses from catastrophe. However, imbalances between supply and demand in the catastrophe insurance market are a global problem. From the supply perspective, private insurance institutions are reluctant to operate catastrophe insurance mainly because it requires those institutions to hold many current assets; however, insurance companies refuse to accept clients because of factors such as tax and solvency [9]. The catastrophe insurance market has a high marginal capital requirement, and when insurance companies are faced with problems, such as insufficient solvency, high capital costs, and uncertain loss, insurance institutions often choose not to enter this market [10]. In addition, a lack of financial cooperative insurance mechanisms leads to a lack of mutual trust among insurance companies, which makes it difficult to form stable cooperation income [11]. From the perspective of demand, catastrophe insurance is different from ordinary insurance because the correlation of catastrophe insurance claims necessitates payment to more customers at one time, resulting in liquidity pressure. This correlation causes positive and inclined demand and lower equilibrium prices, thus leading to the potential failure of the catastrophe insurance market [12].

Insurance companies must form an insurance community to address the large risk of damages, catastrophe insurance market supply and demand imbalances, etc. [13]. The insurance community refers to a community of long-term interests formed by several insurance companies in accordance with the agreed articles of association, which jointly provide insurance protection and risk management for specific risks [14]. When the investment projects of some countries or enterprises are faced with risks, the insurance companies of various countries can gather to form an insurance community that jointly undertakes insurance, which increases the underwriting capacity of the insurance industry and disperses the underwriting pressure of each insurance company [15]. The international community also uses the insurance community to provide insurance for special catastrophe risks, such as nuclear [16], earthquake [17], natural disaster [18], and systematic financial [19] risks. Well-known insurance community organizations include the Turkey Catastrophe Insurance Community (TCIP), the Caribbean Catastrophe Risk Insurance Fund (CCRIF), and the Pacific Catastrophe Risk Assessment and Financing Initiative Fund (PCRAFIF).

When different insurance companies jointly form an insurance community, it is necessary to formulate differentiated underwriting strategies according to the different scales of each subject and risk factor. This strategy is constrained by minimum coverage ratios and total costs; specifically, the total cost of underwriting cannot be higher than the sum of the costs of the internal members alone or the insurance company will withdraw from the insurance community. In addition, the strategy also needs to meet the expectations of the internal members for cost optimization, which is a complex multi-comprehensive programming problem [20,21]. Multi-objective programming problems are sometimes called multi-objective optimization problems, and they mainly focus on the simultaneous optimization of multiple numerical objectives under various constraints. Multi-objective programming has been applied in many important fields, such as portfolio [22], energy [23], engineering design [24], and information resource planning [25]. Our research aims to solve the multi-objective planning problems that are characteristic of insurance community underwriting, mainly from the perspective of the alliance’s overall operational efficiency. We also perform a comprehensive evaluation between the maximization of the underwriting scale and the minimization of the underwriting cost. Our main research methods include data envelopment analysis [26,27], the TOPSIS method [28,29], quality function deployment [30,31], the MCGP (multi-choice goal programming) method [32,33], etc. Most existing studies regarding the underwriting decisions of each member of the cooperative alliance use non-cooperative game theory, and multi-objective programming models are rarely used to solve the problem of underwriting decision optimization [34].

Although the insurance community model has a considerable effect on catastrophe risk mitigation [35], the insurance community does not have a clear plan for cost-sharing among its members at present. Under normal circumstances, the cooperative alliance uses the core [36], Nash negotiation [37], and Shapley value methods [38] for cost allocation. The core method uses a calculation process that is extremely complex, and the Nash negotiation method may lack an objective conclusion. In contrast, the Shapley value method can better balance fairness and efficiency. The Shapley value method also has considerable mathematical properties and practical operability [39,40]; therefore, it can avoid the disadvantages of egalitarianism [41]. Overall, the Shapley value method is recognized as one of the most effective methods for cost allocation. In the insurance research field, scholars have applied the Shapley value method to the pricing of insurance [42,43], the negotiation of insurance contracts [44], the control of mutual insurance processes [45], liability insurance claims [46], etc. However, the Shapley value method ignores all contributions except for marginal contributions and thus cannot fully reflect a fair distribution [47,48]. Therefore, when using the Shapley value method, scholars introduce different correction factors to modify the traditional method [49,50]. Risk factors cannot be ignored when an insurance company conducts business activities [51]. However, few existing studies incorporate risk factors into the cost allocation problem facing the insurance community, and research on the cost allocation problem in the insurance community is inadequate. Therefore, we added risk factors to the traditional Shapley value method and propose using this modified method as a cost allocation strategy that considers risk factors.

To summarize, when forming an insurance community for underwriting, it is necessary to consider the constraint of total cost and the expectation of cost optimization among internal members, which is a complex multi-objective programming problem [52]. In addition, a differentiated underwriting strategy requires that insurance community operators consider how to reasonably share the total investment and operating costs among its members. This sharing must reflect the effective incentives for different insurance companies to join the insurance community when striving for underwriting, which is a typical cooperative game problem [53]. The above two problems affect and restrict each other; therefore, they require integrated thinking and the systematic design of insurance community operators.

In conclusion, at present, there are many studies on the use of multi-objective programming or cooperative games for cost allocation. However, there is a lack of holistic research regarding the comprehensive problem of simultaneously using multi-objective programming and cooperative games to carry out cost allocation. This paucity is especially relevant to research exploring the management-decision problem facing insurance community alliances. In terms of theoretical methods, the multi-choice goal programming method finds potential solutions by setting multiple expectation levels for each optimization goal. A cooperative game in the form of a characteristic function can provide more allocation schemes than a traditional contract. Therefore, by combining the two, we use the multi-choice goal programming method to construct an underwriting decision optimization model that considers the maximum underwriting ratio, the minimum underwriting cost, and the cost characteristic function of a cooperative alliance. Then, we propose a modified Shapley value method for incorporating investment risk, which is used to allocate the total underwriting cost among the insurance community members. Finally, we use simulation analysis to verify the rationality and effectiveness of our proposed method. Our research provides a hybrid optimization method to solve the cost allocation problem facing the insurance community, which has reference value for improving the cost-sharing system of the insurance community.

2. Preliminaries

Many insurance companies in the countries along the Belt and Road have the willingness and strength to participate in an insurance community. To simplify our study, we select an insurance company from each country as a representative and mark the set of all insurance companies included in the study as $N=\{1,2,\dots ,n\}$ and $\left|n\right|>2$. All the insurance companies involved in our study use one of two underwriting strategies: (1) the go-it-alone strategy, in which an insurance company chooses to cover the Belt and Road projects in their own country by themselves or (2) the cooperation strategy, in which the insurance company chooses to build an insurance community with insurers from other countries to share risks and benefits. Considering these two strategies, we establish a cooperative game model of insurance community cost allocation based on the multi-choice goal programming method. The game process is divided into three interdependent and progressive phases. The logical relationship diagram is shown in the Figure 1 below:

Phase one concerns the first stage of the go-it-alone strategy. In phase one, the insurance company underwriting costs are carried out in accordance with a quantitative calculation of the insurance rate. Phase two concerns the second-stage cooperative strategy. In phase two, we use the sum of the insurance companies’ individual costs as the cost constraint objective of the insurance community decision-making optimization model under the second-stage cooperative strategy; that is. Specifically, after the insurance community is formed by different insurance companies, the cost of the insurance community should not be higher than the sum of the costs of the enterprise alone. In addition, in phase two, the total underwriting demand of the insurance community is the sum of the underwriting demands of all countries under the go-it-alone strategy. Therefore, phase one provides the necessary input variables and constraints for the decision optimization model in phase two.

Under the cooperation strategy of phase two, considering factors such as investment risk difference and government subsidy policy, we assume that the insurance community can set different underwriting ratios for countries with different investment risks and that the optimization goal is to improve the overall underwriting scale and to reduce the underwriting cost. Therefore, we used a multi-choice goal programming method to construct an insurance community decision optimization model. The second-stage model optimizes the total underwriting cost after calculating all possible insurance company alliances in the insurance community. Therefore, we use the optimized cost to construct the characteristic function of the cooperative game model in phase three. The characteristic function of the alliance refers to the maximum benefit or the minimum cost saved by the alliance when the players participate in alliance cooperation, which reflects the negotiation ability of the alliance to participate in the game.

In phase three, we design the alliance characteristic function according to the underwriting cost that was optimized in phase two. Then, we establish the cooperative game model for the cost allocation problem among the members of the insurance community. A cooperative game model usually contains two elements, N and v, where $N=\{1,2,\dots ,n\}$ is the set of players and v is a mapping of $2^N\to R$, called the characteristic function, which specifies $v(\varnothing )=0$. Each nonempty subset $S\subseteq N$ is called a coalition. N is defined as the grand coalition and V represents the utility generated by the cooperation of the players in coalition S. The model can be solved at this stage after determining a set of underwriting cost allocation schemes based on characteristic functions that consider the investment risk differentiation of the different countries. These schemes must ensure the insurance community’s stability and motivate the insurance companies to actively provide risk protection for the Belt and Road projects.

For the above analysis, model symbols and assumptions are explained as follows (

Table 1. Model notations and hypothesis specification.

Parameter Symbol	Parameter Meaning
$Q_i$	The total demand for insurance in the country where insurance company $i$ is located.
$S_i$	The investment risk in the country where the insurer i is located is used to measure the likelihood of a disaster in the country where insurance company i is located.
${\tilde{Q}}_i$	Insurance company $i$’s coverage size.
${\tau }_i$	The coverage ratio in the country where insurance company $i$ is located, ${\tau }_i=\frac{{\tilde{Q}}_i}{Q_i}$.
$C_i({\tau }_i)$	The cost of underwriting for insurance company $i$.
${\tau }_{\min }$	Market access mechanism and minimum underwriting ratio.
$B$	The scale parameter of the insurance company.
$\gamma$	The coefficient of the government subsidy.
$Q_S$	Underwriting requirements under the insurance community, $Q_S={\displaystyle {\sum }_{i\in S}{Q}_i}$.
${\tilde{Q}}_s$	The coverage size of the insurance community, ${\tilde{Q}}_S={\displaystyle {\sum }_{i\in S}{Q}_i}{\tau }_i$.
${\tilde{\tau }}_S$	The coverage ratio of the insurance community, ${\tilde{\tau }}_S=\frac{{\tilde{Q}}_S}{Q_S}$.

):

(1) Assume that the insurance company $i$ covers only one type of risk and that the total demand for coverage of the risk in its country is $Q_i(Q_i>0)$, where $Q_i$ can be obtained from the Belt and Road project evaluation report and other data. The investment risk of different countries is defined as $S_i(S_i>0)$, where $S_i$ depends on many factors, such as political risk and meteorological risk in different countries, which reflects the insurability of investment projects. Both $Q_i$ and $S_i$ are exogenous variables.

(2) Assume that the insurance company’s coverage ratio is ${\tau }_i$, which is used to measure the willingness of an insurance company to cover, and that $0<{\tau }_i<1$, where ${\tau }_i$ are the decision variables. Therefore, the insurance company determines the coverage size ${\tilde{Q}}_i=Q_i{\tau }_i$ by determining ${\tau }_i$. Consequently, under the go-it-alone strategy, the insurance company $i$’s underwriting cost is denoted as $C_i({\tau }_i)=Q_i{\tau }_i{S}_i+B{\tau }_i^2$, where $B{\tau }_i^2$ is the fixed investment spent by the insurance company and $B>0$ is the scale parameter according to Savaskan et al. [54], which makes the results more intuitive and easy to observe. To ensure our study’s rationality, $B$ is a sufficiently large constant and $B{\tau }_i^2>0$, which indicates that with an increase in the underwriting ratio, the underwriting cost increases at a faster rate. Because increasing the underwriting scale is equivalent to increasing the underwriting cost, insurance companies will choose the minimum underwriting scale when using the go-it-alone strategy; therefore, the condition $C_i({\tau }_{\min })=Q_i{\tau }_{\min }{S}_i+B{\tau }_{\min }^2$ is satisfied. The total underwriting demand under the insurance community model is denoted as $Q_S(Q_S>0)$, which is the sum of the total underwriting demand of the insurance companies participating in the construction of the insurance community; therefore, the condition $Q_S={\displaystyle {\sum }_{i\in S}{Q}_i}$ is satisfied. The insurance community is formed by the cooperation of insurance companies providing insurance services for the Belt and Road. Therefore, the underwriting scale of an insurance community is the sum of the underwriting scale of all insurance companies, which is denoted as ${\tilde{Q}}_S={\displaystyle {\sum }_{i\in S}{Q}_i}{\tau }_i$.

(3) To ensure the study’s rationality, we followed Atasu and Souza [55] and assumed that there is an access mechanism in the market. Under this assumption, the insurance companies and the insurance community have the same minimum underwriting ratio ${\tau }_{\min }$ after reaching the minimum coverage ratio, and the insurance community can decide the coverage scale in each country. We assume that each government, as the regulator of the degree of the insurance community system, provides financial subsidies to insurance companies participating in the insurance community. Then, we quantitatively calculate the number of subsidies according to the ratio exceeding the minimum insurance coverage in the market. Therefore, when ${\tau }_i>{\tau }_{\min }$, the government gives subsidies according to the insurance scale exceeding the minimum value. The subsidy function is $R_{{\tau }_i}=\gamma Q_i({\tau }_i-{\tau }_{\min })$, and $\gamma$ is the coefficient of the government subsidy, which is a fixed constant when ${\tau }_i\le {\tau }_{\min }$ and $\gamma =0$.

(4) To maintain generality, let the size parameter of the insurance community also be B. According to Zu-Jun M. et al. [56], the total cost of underwriting is $C_S^U({\tau }_i)={\displaystyle {\sum }_{i\in S}{Q}_i}{\tau }_i{S}_i+B{\displaystyle {\sum }_{i\in S}{\tau }_i^2}$ after the insurance company forms an insurance community. Under the government subsidy mechanism, the total cost of the insurance community is $C_S^{UT}({\tau }_i)={\displaystyle {\sum }_{i\in S}{Q}_i}{\tau }_i{S}_i+B{\displaystyle {\sum }_{i\in S}{\tau }_i^2}-{\displaystyle {\sum }_{i\in S}\max \{\gamma Q_i({\tau }_i-{\tau }_{\min }),0\}}$. According to Winternitz et al. [57], when establishing an insurance community, the original intention of different insurance companies is to pursue a scale effect so the total cost of the insurance community is not higher than the sum of costs under the single-operator strategy. Therefore, $C_S^U({\tau }_i)\le C_L^S({\tau }_{\min })={\displaystyle {\sum }_{i\in S}{Q}_i}{\tau }_{\min }{S}_i+B{\displaystyle {\sum }_{i\in S}{\tau }_{\min }^2}$.

3. Insurance Community Decision Optimization Model Based on the Multi-Choice Goal Programming Method

In this section, an insurance community decision optimization model based on the multi-choice goal programming method is proposed and the cost space of the insurance community is calculated. The results of these steps are used to form the coalition characteristic function of the insurance community cost allocation cooperative game model in the next phase.

3.1. Establishment of a Decision Optimization Model for the Insurance Community

When establishing an insurance community, an insurance company’s main goals are to provide adequate risk protection for the Belt and Road, to reduce costs and underwriting pressure, and to avoid the inability to pay huge losses and face business risks when extreme events occur. We consider the restrictive relationship between the insurance community’s total underwriting scale and the underwriting cost objective function when determining the insurance community’s decision-making optimization objective. Under the constraints of the insurance community’s underwriting costs and market access mechanisms, how should the insurance community set the underwriting ratio ${\tau }_i$ for different individuals so that the total underwriting scale and the total cost can be optimized between them? Based on the above description, we construct the insurance community’s multi-objective optimization model $M_1$ as follows:

Model $M_1$:

$$Max{\tilde{Q}}_S={\displaystyle {\sum }_{i\in S}{Q}_i}{\tau }_i,$$

(1)

$$Min{C}_S^{UT}({\tau }_i)={\displaystyle {\sum }_{i\in S}{Q}_i}{\tau }_i{s}_i+B{\displaystyle {\sum }_{i\in S}{\tau }_i^2}-{\displaystyle {\sum }_{i\in S}\max \{\gamma Q_i({\tau }_i-{\tau }_{\min }),0\}},$$

(2)

$$s.t. 0\le {\tau }_i\le 1,$$

(3)

$${\tau }_{\min }\le \frac{{\tilde{Q}}_S}{Q_S}\le 1,$$

(4)

$$C_S^U({\tau }_I)\le C_S^L({\tau }_{\min }).$$

(5)

In model $M_1$, Equations (1) and (2) are the underwriting scale and cost objectives of the insurance community, respectively, and these two goals conflict with each other. Equation (3) indicates that the value range of the decision variable ${\tau }_i$ is constrained between 0 and 1 to ensure the research’s rationality. Equation (4) indicates that the insurance community’s underwriting ratio should meet the market access mechanism. Equation (5) indicates that an insurance community’s underwriting cost must be lower than the sum of the underwriting costs when its members work alone; otherwise, an insurance community’s members will choose to leave the alliance and adopt the going-it-alone strategy.

3.2. Solution of the Decision Optimization Model for the Insurance Community

We use the MCGP method proposed by Chang [58] to solve the above multi-objective decision-making optimization model. MCGP is a goal-planning method that helps decision-makers find better solutions to decision problems by setting multiple expectation levels for the optimization objectives.

In classical goal programming, $d^{+}$ and $d^{-}$ represent the positive and negative deviations of the objective function $f_i(X)$ with respect to the expected value $g_i$, respectively, and ${\omega }_i$ is the weight of the positive and negative bias variables $d^{+}$ and $d^{-}$. In contrast to classical goal programming, the MCGP model uses $d^{+}$ and $d^{-}$ to represent the positive and negative deviations of the objective function $f_i(X)$ with respect to the expected value $y_i$, respectively. The MCGP model also uses the following values: $g_{i,\max }$ and $g_{i,\min }$ are the maximum and minimum expected values of the decision maker for the $i$th objective, respectively; $e^{+}$ and $e^{-}$ are the positive and negative deviations of the expected value variable $y_i$ from $g_{i,\max }$ and $g_{i,\min }$, respectively; ${\omega }_i$ is the weight of the positive and negative bias variables $d^{+}$ and $d^{-}$, respectively; and ${\mu }_i$ is the weight of the bias variables $e^{+}$ and $e^{-}$. The decision objective of both models is to minimize the total deviation from the target value under the condition of limited resources. Compared with the classical goal programming model, the MCGP model allows the decision maker to have a higher expectation level beyond the expected goal and to make an optimal decision based on it.

For the cost optimization objective in model $M_1$, the decision maker of the insurance community hopes that the total contracting cost will not be higher than the sum of costs compared with members going it alone and that the underwriting cost will be as low as possible. This is a classic MCGP decision problem. Based on the MCGP principle, we refer to Chang’s [59] linear utility function design method and propose the cost-utility function of the insurance community, which we define as follows:

Definition 1.

The cost-utility function of the insurance community is:

$$U_c=\left\{\begin{array}{cc}1,& y_c=0,\hfill \\ \frac{C_S^L({\tau }_{\min })-y_c}{C_S^L({\tau }_{\min })},& 0<y_c<C_S^L({\tau }_{\min }),\hfill \\ 0,& y_c\ge C_S^L({\tau }_{\min }).\hfill \end{array}\right.$$

(6)

In Equation (6), $y_c$ is the actual cost of the insurance community and $U_c$ is the satisfaction degree of all members of the insurance community with cost control. The above equation also reflects the collective rationality of an insurance community: the underwriting cost after members cooperate to form an insurance community cannot exceed the sum of the costs when all members work alone. Otherwise, some members will always leave the insurance community and choose the self-underwriting strategy.

According to the definition of $U_c$, the cost objectives in model $M_1$ can be transformed into cost-utility objectives, and the MCGP model can be modified accordingly.

Model $M_2$:

$$Min\{{\omega }_1(d_1^{-}+d_1^{+})+{\mu }_1{e}_1^{-}+{\omega }_2(d_2^{+}+d_2^{-})+{\mu }_2{e}_2^{-}\},$$

(7)

$$s.t. {\displaystyle \sum _{i\in S}{Q}_i}{\tau }_i-d_1^{+}+d_1^{-}=y_1,S\subseteq N,$$

(8)

$$y_1+e_1^{-}=Q_S,S\subseteq N,$$

(9)

$$C_S^U({\tau }_i)-d_2^{+}+d_2^{-}=y_c,S\subseteq N,$$

(10)

$${\lambda }_c=\frac{C_S^L({\tau }_{\min })-y_c}{C_S^L({\tau }_{\min })},$$

(11)

$${\lambda }_c+e_2^{-}=1,$$

(12)

$$d_1^{+},d_1^{-},d_2^{+},d_2^{-},e_1^{-},e_2^{-}\ge 0,$$

(13)

$$C_S^{UT}({\tau }_i)={\displaystyle \sum _{i\in S}{Q}_i{\tau }_i{s}_i}+B{\displaystyle \sum _{i\in S}{\tau }_i^2}-{\displaystyle \sum _{i\in S}\max \{\gamma Q_i({\tau }_i-{\tau }_{\min }),0\}},$$

(14)

$$0\le {\tau }_i\le 1,$$

(15)

$${\tau }_{\min }\le \frac{{\tilde{Q}}_S}{Q_S}\le 1.$$

(16)

In model $M_2$, ${\lambda }_c$ is the cost-utility variable because the value of the weight of the bias variable does not affect the result. To maintain generality, all deviant variables are set to have an equal weight equal to 1, and the other symbols are set to be the same as in model $M_1$. To verify that the value of the weights has no influence on the results, we conducted a sensitivity analysis on the value of the weights in The Simulation Analysis section.

The following proposition proves that model $M_2$ has the same optimization effect as model $M_1$.

Proposition. 

In model $M_2$, ${\lambda }_c$ approaches 1 under the action of minimizing $e_2^{-}$, which makes $y_c$ go to 0, minimizes $C_S^U({\tau }_i)$, and makes it go to 0; therefore, optimizing ${\lambda }_c$ is essentially the same as optimizing $C_S^{UT}({\tau }_i)$, and the optimization effect of models $M_1$ and $M_2$ are the same.

Proof. 

$$\begin{array}{ll}{C}_S^{UT}({\tau }_i)& ={\displaystyle \sum _{i\in S}{Q}_i{\tau }_i{S}_i}+B{\displaystyle \sum _{i\in S}{\tau }_i^2}-{\displaystyle \sum _{i\in S}\max \{\gamma Q_i({\tau }_i-{\tau }_{\min }),0\}}\\ & ={\displaystyle \sum _{i\in S,{\tau }_i\le {\tau }_{\min }}{Q}_i{\tau }_i{S}_i}+B{\displaystyle \sum _{i\in S,{\tau }_i\le {\tau }_{\min }}{\tau }_i^2+{\displaystyle \sum _{i\in S,{\tau }_i>{\tau }_{\min }}{Q}_i{\tau }_i{S}_i+B{\displaystyle \sum _{i\in S,{\tau }_i>{\tau }_{\min }}{\tau }_i^2}}}\\ & ={\displaystyle \sum _{i\in S,{\tau }_i\le {\tau }_{\min }}{Q}_i{\tau }_i{S}_i}+B{\displaystyle \sum _{i\in S,{\tau }_i\le {\tau }_{\min }}{\tau }_i^2+}{\displaystyle \sum _{i\in S,{\tau }_i>{\tau }_{\min }}{Q}_i{\tau }_i(S_i-\gamma )}+B{\displaystyle \sum _{i\in S,{\tau }_i>{\tau }_{\min }}{\tau }_i^2}+{\displaystyle \sum _{i\in S,{\tau }_i>{\tau }_{\min }}\gamma Q_i{\tau }_{\min }}.\end{array}$$

$$C_S^U({\tau }_i)={\displaystyle \sum _{i\in S,{\tau }_i\le {\tau }_{\min }}{Q}_i{\tau }_i{S}_i}+B{\displaystyle \sum _{i\in S,{\tau }_i\le {\tau }_{\min }}{\tau }_i^2}+{\displaystyle \sum _{i\in S,{\tau }_i>{\tau }_{\min }}{Q}_i{\tau }_i{S}_i}+B{\displaystyle \sum _{i\in S,{\tau }_i>{\tau }_{\min }}{\tau }_i^2}.$$

The parameters $Q_i,S_i,B,\gamma$, and ${\tau }_{\min }$ are constants. Therefore, optimizing $C_S^U({\tau }_i)$ is equivalent to optimizing $C_S^{UT}({\tau }_i)$. □

In addition, model $M_2$ is greater than 0 due to the weight vector of the single objective. Therefore, all components of the weight vector are greater than 0 if ${\tau }_i^{*}$ is the optimal solution of the single-objective model $M_2$ according to the relationship between the effective solution of the multi-objective model and the optimal solution of the single-objective model. The optimal solution of the single-objective model $M_2$ is also the effective solution of the multi-objective model $M_1$. Therefore, the optimal values of the decision variables of these two models are consistent.

4. Cooperative Game Model of Cost Allocation in the Insurance Community

4.1. Cooperative Game and Shapley Value

A cooperative game model usually contains two elements, N and v, where $N=\{1,2,\dots ,n\}$ is the set of players and v is a mapping of $2^N\to R$ called the characteristic function, which specifies $v(\varnothing )=0$. Each nonempty subset $S\subseteq N$ is called a coalition. N is defined as the grand coalition and v represents the utility generated by the cooperation of the players in coalition S.

The Shapley value was introduced by S. Shapley in 1953 [38], and it is a method to solve the cooperative game of n players, namely the Shapley value method. The cost allocation of alliance members based on the Shapley value reflects the marginal contribution of each member to the alliance. Egalitarianism in the distribution is avoided, which makes the Shapley value more reasonable and fairer. It also reflects the cooperative game process of each member of the alliance. In the Shapley value method, let us write $N=\{1,2,\dots ,n\}$ as the combination of n people working together, $\phi (v)$ is the total revenue of all parties under the cooperation, and $\phi (v)=({\phi }_1(v),{\phi }_2(v),\dots ,{\phi }_n(v))$.${\phi }_i(v)$ represents the cost-sharing value of the $i$th member under the cooperation N.

$$v(N)\le {\displaystyle \sum v(i)},i=1,2,\dots ,n.$$

The above equation states that, in the cooperative game, the total cost for all members after cooperation is not greater than the sum of the costs of individual members.

$${\phi }_i(v)\le v(i),$$

$$v(N)={\displaystyle \sum v(i)},i=1,2,\dots ,n.$$

The above equation states that, in the cooperative game, the cost borne by all members after cooperation is not greater than the cost under the strategy of going it alone, and the sum of costs borne by each member after cooperation is equal to the total cost of the insurance community.

4.2. Construction of the Cooperative Game Model of the Insurance Community

The insurance community’s cost-sharing cooperative game is represented by the binary group $(N,v)$, where $v$ is a real function defined on all subsets $S\subseteq N$ of $N$ called the coalition characteristic function, which is represented as $v(S)$, and $v(\varnothing )=0$. Leng and Parlar [60] and Zheng et al. [61] defined the coalition characteristic function of the cooperative game model of cost allocation among members of the insurance community as follows:

Definition 2.

The alliance characteristic function $v(S)(S\ne \varnothing )$is the optimal total cost that can be achieved by the insurance community formed by the insurance company alliance $S$:

$$v(S)=C_S^U({\tau }_i)={\displaystyle \sum _{i\in S}{Q}_i{\tau }_i{S}_i}+B{\displaystyle \sum _{i\in S}{\tau }_i^2}-{\displaystyle \sum _{i\in S}\max \{\gamma Q_i({\tau }_i-{\tau }_{\min }),0\}}.$$

(17)

The main cooperative game that we perform in this study is the cost allocation problem in the insurance community formed by the big alliance $v(N)$. The corresponding allocation method is called a solution, where the single-valued solution is also called a value. The cooperative game based on the characteristic function of the alliance can provide more allocation rules for the decision maker to help them choose an appropriate allocation scheme according to realistic problems. In the cooperative game, the Shapley value is the most classical single-valued solution, denoted $f_i^{SH}$, which is defined as:

$$f_i^{SH}(N,v)={\displaystyle \sum _{S\subseteq N\backslash i}\frac{S!(n-S-1)!}{n!}}(v(S\cup i)-v(S)),i\in N.$$

(18)

Although the Shapley value determines the final allocation scheme according to the marginal contribution of players, it provides a better solution for cost allocation. However, the Shapley value method ignores other contributions except for marginal contributions and so cannot fully reflect a fair distribution when stimulating the measures that different countries to take to reduce underwriting risks and thus reduce costs to the insurance community. Therefore, we take the risk factors of each country into account and use the marginal underwriting costs of each country’s insurers to reflect risk (the marginal underwriting costs are defined as $S_i$). In this analysis, the higher the marginal underwriting cost, the higher the risk.

Therefore, we propose a modified Shapley value that considers this factor, which is denoted as $f_i^{SHE}$ and is defined as follows:

Let ${\alpha }_i=\frac{S_i}{{\displaystyle {\sum }_{i\in S}{S}_i}}$ and ${\displaystyle {\sum }_{i\in S}{\alpha }_i}=1$ be the underwriting factor, which measures the relative insurability of risks across countries, and let $\beta (0<\beta <1)$ be the coefficient of coordination for a given ${\alpha }_i$ and $\beta$, which define the value of $f_i^{SHE}$ to be:

$$f_i^{SHE}(N,v)=\beta f_i^{SH}(N,v)+(1-\beta )v(N){\alpha }_i,i\in N.$$

(19)

In the above equation, $f_i^{SHE}$ is the modified Shapley value of the insurance company, $f_i^{SH}$ is a set of Shapley values determined by the insurance company $i$ based on the characteristic function, and the underwriting factor ${\alpha }_i$ is used to describe the insurability of risks in different countries. Larger ${\alpha }_i$ values indicate higher insurability risk in the country where the insurance company $i$ is located; therefore, the larger $v(N){\alpha }_i$ is, the more alliance costs are allocated to the insurance company $i$. $\beta$ is the coordination coefficient, and $\beta$ and $1-\beta$ denote the weights assigned to the different components of $f_i^{SHE}$. As the value of $\beta$ increases, more weight is given to $f_i^{SH}$. Thus, when the value of $f_i^{SHE}$ is larger, more attention is paid to the influence of the coalition characteristic function on the distribution result. Furthermore, as the value of $1-\beta$ increases, more weight is given to $v(N){\alpha }_i$, which indicates that $f_i^{SHE}$ pays more attention to the impact of risk factors of various countries on the cost allocation results. This encourages the government of the country where the insurance company $i$ is located to take measures to improve the insurability of risks.

5. The Simulation Analysis

To verify and analyze our proposed model, we designed a simulation analysis that is solved using Lingo and the MATLAB toolbox. We used the simulation analysis to study the influence of different parameter changes on the underwriting rate decisions and objective functions. We used four insurance companies from different countries in the simulation analysis, which are denoted as $N=\{1,2,3,4\}$, and the underwriting needs and investment risks of each country are shown in the table below (

Table 2. An insurance company’s underwriting needs and investment risks.

Test Group	Underwriting Demand and Investment Risk
	${\mathit{Q}}_1,{\mathit{S}}_1$	${\mathit{Q}}_2,{\mathit{S}}_2$	${\mathit{Q}}_3,{\mathit{S}}_3$	${\mathit{Q}}_4,{\mathit{S}}_4$
Test group 1	(1000, 0.1)	(2000, 0.2)	(3000, 0.3)	(4000, 0.4)
Test group 2	(2500, 0.1)	(2500, 0.2)	(2500, 0.3)	(2500, 0.4)

). We set other parameters to $B=5000$ and $\beta =0.8$.

5.1. Sensitivity Analysis of the Multi-Objective Optimization Effect

Using the data of test group 1, this section analyzes the impact of changes in the market access mechanism ${\tau }_{\min }$ and the size parameter $B$ on the underwriting ratio decision ${\tau }_i^{*}$ and the total underwriting ratio ${\tau }_S^{*}$ in an insurance community formed by the grand coalition $N$, as shown in the figure below.

Figure 2a shows that as ${\tau }_{\min }$ increases, the total coverage ratio ${\tau }_S^{*}$ also increases. Furthermore, when $0<{\tau }_{\min }<1$, ${\tau }_S^{*}$ is considerably higher than ${\tau }_{\min }$, under the same underwriting cost, the proposed decision optimization model can meet more underwriting needs and achieve a Pareto improvement over the sum of underwriting scales when the insurance company goes it alone. These results prove the effectiveness and superiority of our proposed model. Figure 2b shows that with the increase of ${\tau }_{\min }$, our model guarantees that the underwriting ratio of all countries increases. However, by comparing the insurance ratios of different countries, we found that countries with higher investment risks (e.g., 3 and 4) have higher insurance ratios, whereas countries with lower investment risks (e.g., 1 and 2) have lower insurance ratios. This counterintuitive result is caused by the law of diminishing marginal benefits in the underwriting ratio where an increase in the underwriting ratio leads to a faster increase in underwriting costs. When the size parameter $B$ is large, the effect of the diminishing margin of the underwriting ratio on the total underwriting cost exceeds the effect of the investment risk on the underwriting cost. Therefore, the model optimization direction makes the insurance community tend to underwrite projects in countries with higher investment risks to obtain economies of scale.

To further verify the impact of the size parameter $B$ on the underwriting ratio decision of the insurance community, we use the data of test group 1 and set the minimum underwriting ratio to ${\tau }_{\min }=0.5$ under the market access mechanism. Figure 3a,b show that when the scale parameter $B$ is small, the underwriting ratio ${\tau }_S^{*}$ of an insurance community decreases with the increase in the scale parameter, and the underwriting ratio of countries with lower investment risk is higher. This indicates that when $B$ is small, $S_i$ has a greater impact on the total cost, and the decision results tend to cover those countries with lower $S_i$. On the contrary, when $B$ is large (as shown in Figure 3c,d), the marginal benefit of the underwriting ratio decreases, and the cost difference caused by different $S_i$ is small. Therefore, the decision results tend to cover projects in countries with large underwriting needs. In addition, Figure 3a,c show that no matter how $B$ changes, ${\tau }_S^{*}$ will not be lower than ${\tau }_{\min }$ under the same underwriting cost, which again confirms the optimization effect of our model.

5.2. Sensitivity Analysis for the Weight of Bias Variables

The insurance community’s decision-making optimization objective is to solve the following: under the constraints of the insurance community’s underwriting costs and market access mechanisms, how should the insurance community set the underwriting ratio ${\tau }_i$ for different individuals so that the total underwriting scale and the total cost can be optimized between them? When we established the insurance community’s multi-choice objective programming model, we set all deviant variables to have equal weight (equal to 1) to maintain generality. Next, we set different bias variability weights to calculate whether the underwriting ratio ${\tau }_i$ of different individuals will change, thereby testing the robustness of the optimization decision. We use the data of test group 1 as an example, with ${\tau }_{\min }=0.5$ and $B=5000$. We randomly set different values of ${\omega }_1$, ${\mu }_1$, ${\omega }_2^{}$, and ${\mu }_2$ in Equation (7) and calculated the corresponding values of ${\tau }_1$, ${\tau }_2$, ${\tau }_3$, and ${\tau }_4$. The results are listed in

Table 3. Comparison of the optimal underwriting ratios under different bias variable weights.

$({\omega }_1,{\mu }_1,{\omega }_2,{\mu }_2)$	${\tau }_1$	${\tau }_2$	${\tau }_3$	${\tau }_4$
(0.2, 0.3, 0.1, 0.5)	0.2009818	0.3819637	0.5429455	0.6839273
(0.3,0.4, 0.2, 0.6)	0.2009818	0.3819637	0.5429455	0.6839273
(0.4, 0.5, 0.3, 0.7)	0.2009818	0.3819637	0.5429455	0.6839273
(0.5, 0.6, 0.7, 0.8)	0.2009818	0.3819637	0.5429455	0.6839273
(0.6, 0.7, 0.8, 0.9)	0.2009818	0.3819637	0.5429455	0.6839273
(0.5, 0.1, 0.3, 0.2)	0.2009818	0.3819637	0.5429455	0.6839273
(0.6, 0.2, 0.4, 0.3)	0.2009818	0.3819637	0.5429455	0.6839273
(0.7, 0.3, 0.5, 0.4)	0.2009818	0.3819637	0.5429455	0.6839273
(0.8, 0.4, 0.6, 0.5)	0.2009818	0.3819637	0.5429455	0.6839273
(0.9, 0.5, 0.7, 0.6)	0.2009818	0.3819637	0.5429455	0.6839273

.

Table 3 shows that when the weights of the bias variables ${\omega }_1$, ${\mu }_1$,${\omega }_2^{}$, and ${\mu }_2$ are randomly assigned different values, the values of ${\tau }_1$, ${\tau }_2$, ${\tau }_3$, and ${\tau }_4$ remain unchanged. This shows that the weight of the bias variable does not affect the result of the decision optimization, which proves the robustness of the decision optimization result. Therefore, to maintain generality, we set all deviant variables to have equal weight (equal to 1).

5.3. Sensitivity Analysis of the Cost Allocation Results

Using the data of test group 1, in this section, we first investigate the impact of changes in the minimum underwriting ratio ${\tau }_{\min }$, scale parameter $B$, and coordination coefficient $\beta$ on the cost allocation results of an insurance community formed by the grand alliance $N$.

By comparing the characteristic function with the Shapley value data in

Table 4. Comparison of $v(i)$, $f_i^{SH}$, and $f_i^{SHE}$ under different minimum coverage ratios.

${\tau }_{\min }$	$v(1)$	$v(2)$	$v(3)$	$v(4)$	$f_1^{SH}$	$f_2^{SH}$	$f_3^{SH}$	$f_4^{SH}$	$f_1^{SHE}$	$f_2^{SHE}$	$f_3^{SHE}$	$f_4^{SHE}$
0.1	60	90	140	210	24.43	79.66	128.68	197.69	28.16	80.95	128.77	192.59
0.2	220	280	380	520	123.48	221.55	353.67	447.94	121.71	223.11	351.73	450.08
0.3	480	570	720	930	311.77	473.06	664.04	798.83	294.37	468.36	666.10	818.88
0.4	840	960	1160	1440	603.26	818.90	1079.20	1241.62	557.47	804.84	1087.94	1292.73
0.5	1300	1450	1700	2050	993.71	1264.58	1594.15	1783.02	907.67	1237.08	1613.45	1877.25
0.6	1860	2040	2340	2760	1483.57	1810.17	2208.99	2423.62	1345.38	1765.19	2242.78	2573.01
0.7	2520	2730	3080	3570	2075.82	2458.46	2915.52	3166.48	1872.98	2391.42	2969.39	3382.49
0.8	3280	3520	3920	4480	2914.21	3275.76	3652.20	4064.26	2609.50	3176.87	3756.15	4363.93
0.9	4140	4410	4860	5490	3952.44	4246.34	4646.34	5209.68	3523.05	4119.27	4800.36	5612.12
1	5100	5400	5900	6600	5100.00	5400.00	5900.00	6600.00	4540.00	5240.00	6100.00	7120.00

, we show that no matter how ${\tau }_{\min }$ changes, $f_i^{SH}$ can reflect the changing law of the characteristic function of the coalition, where $f_i^{SH}\le (V_i),\forall i\in N$. Therefore, the Shapley value in our model is in line with the individual rationality of the insurance companies. Furthermore, the insurance companies are willing to join the big alliance to form an insurance community, which can reduce individual insurance costs and increase the total insurance scale. Table 4 shows a comparison between the alliance characteristic function and the modified Shapley value data. We observed that under different ${\tau }_{\min }$, the apportioned costs of insurance companies 1 and 2 are still less than their individual costs, that is, $f_i^{SHE}\le (V_i),\forall i\in \{1,2\}$. However, the Shapley values of insurance companies 3 and 4, do not satisfy individual rationality upon the occurrence of an extreme case (${\tau }_{\min }=0.9,1$), which is due to the differences caused by the extreme case and different investment risks. By comparing the Shapley value and the modified Shapley value, we found that $f_i^{SHE}<f_i^{SH},\forall i\in \{1,2\}$ and $f_i^{SHE}>f_i^{SH},\forall i\in \{3,4\}$ are valid when ${\tau }_{\min }>0.2$, which reflects the distribution difference of $f_i^{SHE}$ due to the different investment risks in different countries. Therefore, insurance companies in countries with small $S_i$ should share less cost, whereas insurance companies in countries with large $S_i$ should share more cost. The insurability of a country’s investment risk is indirectly reflected by $S_i$. Thus, when $S_i$ is larger, a government’s efforts to reduce investment risk are reduced. Therefore, the distribution of the $f_i^{SHE}$ reaction mechanism in extreme cases does not fulfill individual rationality. However, this mechanism can help motivate government departments to take measures to reduce the project’s domestic investment risk, thereby promoting the long-term sustainable development of the insurance community.

According to the comparison of the alliance characteristic functions ($f_i^{SH}$ and $f_i^{SHE}$ value data in

Table 5. Comparison of $v(i)$, $f_i^{SH}$, and $f_i^{SHE}$ under different scale parameters $B$.

$B$	$v(1)$	$v(2)$	$v(3)$	$v(4)$	$f_1^{SH}$	$f_2^{SH}$	$f_3^{SH}$	$f_4^{SH}$	$f_1^{SHE}$	$f_2^{SHE}$	$f_3^{SHE}$	$f_4^{SHE}$
100	75	225	475	825	3.84	15.05	56.00	406.00	12.69	31.28	73.65	363.27
200	100	250	500	850	3.09	8.07	10.25	349.98	9.90	21.31	30.48	309.69
300	125	275	525	875	85.67	56.73	185.35	380.59	82.70	73.71	190.78	361.14
400	150	300	550	900	125.92	130.17	341.28	567.26	124.03	150.72	342.90	546.98
500	175	325	575	925	121.15	227.99	422.40	711.15	126.58	241.70	426.88	687.53
600	200	350	600	950	121.38	298.47	481.02	822.23	131.57	307.70	488.21	795.64
700	225	375	625	975	124.84	342.39	546.80	905.88	138.27	350.71	552.63	878.30
800	250	400	650	1000	128.11	379.10	593.57	961.49	143.74	385.77	598.59	934.17
900	275	425	675	1025	122.76	390.22	617.00	975.83	140.32	396.41	619.95	949.13
1000	300	450	700	1050	122.16	398.30	643.39	988.47	140.78	404.73	643.85	962.97
2000	550	700	950	1300	295.57	567.01	871.04	1132.98	293.78	568.27	868.83	1135.71
3000	800	950	1200	1550	519.62	788.44	1106.74	1331.38	490.62	780.60	1110.16	1364.80
4000	1050	1200	1450	1800	754.08	1023.63	1349.00	1552.02	696.84	1006.05	1359.92	1615.92
5000	1300	1450	1700	2050	993.71	1264.58	1594.15	1783.02	907.67	1237.08	1613.45	1877.25
6000	1550	1700	1950	2300	1236.31	1508.47	1840.83	2019.69	1121.15	1470.99	1868.98	2144.17
7000	1800	1950	2200	2550	1480.77	1754.08	2088.40	2259.81	1336.27	1706.58	2125.71	2414.49
8000	2050	2200	2450	2800	1726.47	2000.76	2336.56	2502.20	1552.49	1943.25	2383.21	2687.04
9000	2300	2450	2700	3050	1973.03	2248.17	2585.10	2746.15	1769.48	2180.63	2641.23	2961.12
10,000	2550	2700	2950	3300	2220.23	2496.09	2833.93	2991.23	1987.01	2418.53	2899.63	3236.31

), similar to the ${\tau }_{\min }$ law change, no matter how the scale parameter $B$ changes, the $f_i^{SH}$ and $f_i^{SHE}$ value data are in line with individual rationality. Therefore, $f_i^{SH}\le v(i),\forall i\in N$ and $f_i^{SHE}<v(i),\forall i\in N$ are satisfied. By comparing the Shapley value and the modified Shapley value, we found that when the scale parameter $B$ is large, there are $f_i^{SHE}<f_i^{SH},\forall i\in \{1,2\}$ and $f_i^{SHE}>f_i^{SH},\forall i\in \{3,4\}$, which is in line with the established goal of the modified Shapley value. Therefore, under the condition that all members meet individual rationality, insurance companies in countries with high investment risks bear more costs and insurance companies in countries with low investment risks bear lower costs. When the scale parameter $B$ is small, $f_i^{SH}>f_i^{SHE},\forall i\in \{1,2,3\}$ and $f_i^{SH}<f_i^{SHE},\forall i\in \left\{4\right\}$ because the marginal diminishing effect of the underwriting ratio is not considerable when the scale parameter is small. Due to the existence of a government subsidy mechanism, the insurance community can underwrite investment projects in countries with low investment risks at very low costs. In view of the modified Shapley value calculation model, the optimized effect of the modified Shapley value makes the cost allocation effect show counterintuitive results. $S_i$ indirectly reflects a country’s investment risk in insurable property. Governments put less effort towards reducing the investment risk when $S_i$ is large. This mechanism is helpful for motivating government departments to take measures to reduce the project’s domestic investment risks, thus promoting a sustainable insurance community.

We separately tested the effect of the coordination coefficient $\beta$ on the resulting $f_i^{SHE}$ using the data of test group 2. For this analysis, we set the underwriting demand of each country to an equal constant $Q_i=2500$ and then investigated the impact of $\beta$ on $f_i^{SHE}$ in the case of ${\tau }_{\min }=0.5$.

Table 6. Comparison of $v(i)$, $f_i^{SH}$, and $f_i^{SHE}$ under different coordination coefficients.

$\beta$	$v(1)$	$v(2)$	$v(3)$	$v(4)$	$f_1^{SHE}$	$f_2^{SHE}$	$f_3^{SHE}$	$f_4^{SHE}$
0	1375	1500	1625	1750	612.15	1224.31	1836.46	2448.61
0.1	1375	1500	1625	1750	683.96	1249.89	1813.40	2374.28
0.2	1375	1500	1625	1750	755.77	1275.47	1790.34	2299.95
0.3	1375	1500	1625	1750	827.58	1301.05	1767.28	2225.61
0.4	1375	1500	1625	1750	899.39	1326.64	1744.22	2151.28
0.5	1375	1500	1625	1750	971.20	1352.22	1721.16	2076.95
0.6	1375	1500	1625	1750	1043.01	1377.80	1698.10	2002.62
0.7	1375	1500	1625	1750	1114.82	1403.39	1675.04	1928.28
0.8	1375	1500	1625	1750	1186.63	1428.97	1651.98	1853.95
0.9	1375	1500	1625	1750	1258.44	1454.55	1628.92	1779.62
1	1375	1500	1625	1750	1330.24	1480.13	1605.86	1705.29
					$f_1^{SH}$	$f_2^{SH}$	$f_3^{SH}$	$f_4^{SH}$
					1330.24	1480.13	1605.86	1705.29

shows that with the decrease in the coordination coefficient $\beta$, $f_1^{SHE}$ and $f_2^{SHE}$ gradually decrease, whereas $f_3^{SHE}$ and $f_4^{SHE}$ gradually increase. Furthermore, the decrease of $f_1^{SHE}$ is greater than that of $f_2^{SHE}$, and the increase of $f_4^{SHE}$ is greater than that of $f_3^{SHE}$. This shows that $\beta$ plays a considerable coordinating role and effectively reflects the influence of $S_i$ on the final cost allocation result. It shows that $f_i^{SHE}$ can better fit the insurance community system degree than $f_i^{SH}$. Overall, these results ensure that the insurance community system plays the role of co-insuring and sharing among its members and encourages relevant countries to improve their domestic investment environment and fundamentally reduce the risks of investment projects along the Belt and Road.

6. Conclusions and Recommendations

At present, utilizing the insurance community is a common method to deal with investment risks along the Belt and Road. However, there is no clear method or mechanism to deal with the decision-making optimization and cost allocation of the insurance community participants. Therefore, we propose a hybrid optimization method based on multiple-choice objective programming and cooperative games to solve this problem. Overall, our method showed reference value for improving the cost-sharing system of the insurance community. In our study, we considered the ways to reasonably distribute the underwriting costs among different members when many insurance companies form an insurance community. We established a hybrid model in three phases. In phase one, we used the MCGP model to optimize the underwriting decisions of insurance community members and generated the cost characteristic function based on the cooperative alliance. In phase two, we used the cooperative game and modified Shapley value methods to allocate the cost. In phase three, we verified the effectiveness and superiority of the proposed model using numerical simulation. We draw three valuable conclusions from our study results. First, insurance companies choose cooperation strategies to construct an insurance community that underwrites project risks in countries along the Belt and Road. This can reduce costs and further expand the underwriting scale for insurance companies. Under the insurance community model, the total coverage ratio increases as the minimum coverage ratio increases, and when the scale parameter is large, the effect of the diminishing margin of the underwriting ratio on the insurance community is enhanced. Second, the modified Shapley value method, which takes risk factors into account, is a more realistic cost allocation method than the original Shapley value method. The modified Shapley value method satisfies the individual rationality of each member and allocates more costs to individuals with higher investment risks and fewer costs to individuals with lower investment risks. Third, by reducing the coordination coefficient using the modified Shapley value method, the cost coordination effect becomes more obvious.

In conclusion, the game model proposed in our paper solves the decision equilibrium problem in a multi-objective conflict environment. Compared with the traditional optimization method, the game process naturally guides the coordination between conflicting objectives through the rational law of each component mitigation and ultimately achieves the optimal equilibrium. Therefore, stakeholders, such as the crew members in the insurance community, are closely related to the equilibrium mechanisms of natural coordination. As such, we conclude with the following management observations:

(1) Innovating the underwriting mechanism. From the above analysis, we discovered that the cooperation of insurance companies from different countries to form an insurance community is more advantageous compared to the situation where insurance companies from different countries undertake insurance alone. Through cooperation, insurance companies of various countries can reasonably formulate their own coverage ratio, which can reduce the underwriting cost of insurance companies in various countries and maximize the total underwriting scale. Therefore, innovative underwriting mechanisms to differentiate underwriting strategies should be adopted according to the specific conditions of the insurance companies in countries along the Belt and Road. This practice is conducive to the development of the insurance industry along the Belt and Road.

(2) Optimizing management to reduce the impact of an increasing underwriting scale on costs. According to our analysis, the insurance community tends to underwrite countries with less risk when the cost-scale parameter is small. Therefore, optimizing management and reducing the size parameter (that is, the impact of size on cost) can reduce the underwriting risk of the insurance community and promote the benign development of the insurance community.

(3) Optimizing the cost allocation method. After different insurance companies cooperate to build an insurance community, costs can be apportioned according to the marginal contribution of each country’s insurance companies to the insurance community (that is, through the Shapley method). This kind of allocation avoids egalitarianism; however, it does not consider the different risks of different countries. Therefore, we proposed a modified Shapley value method to share the cost, satisfying the individual rationality of each member and ensuring the countries with higher risk bear more cost compared to countries with lower risk. This allocation is more realistic and can also provide incentives for different countries to take measures to reduce underwriting risks, which is conducive to the long-term development of the insurance community.

(4) Government departments should establish underwriting compensation mechanisms. Because of the large risks involved in the Belt and Road, the underwriting capacities of ordinary insurance companies are insufficient. Therefore, in the early stages of insurance community development, the government can develop a corresponding underwriting compensation mechanism to bear part of the cost endured by the insurance companies and encourage additional insurance companies to participate. As a result, after the insurance community has been able to steadily develop, the government can turn the compensation mechanism into a supervision mechanism and, through supervision, the insurance community will gradually return to market development.

(5) Government departments should reasonably control regulation intensity when making policies. When the coverage ratio has diminishing marginal effects, mandating coverage ratios can lead to extremely high underwriting costs, affecting the enthusiasm of insurance companies to underwrite. At the same time, investment risks in different regions should be considered, and differentiated cost-sharing strategies should be adopted to stimulate the enthusiasm of insurance companies in various countries to underwrite, thereby assisting the construction of the Belt and Road.

(6) Use blockchain technology to reshape the new insurance industry. The importance of the application of blockchain technology in the insurance field stems from the natural combination of its internal mechanisms such as less centralized, openness, high autonomy, data immutability, and high anonymity with the functional characteristics of insurance itself. These mechanisms can break through the technical barriers inherent in the traditional insurance field and carry out the technological innovation and business transformation of insurance products. The insurance community, in which insurers in different countries along the Belt and Road participate and share investment risk, potentially can be developed into a decentralized ecosystem. Therefore, blockchain technology in distributed ledger mode provides a guarantee for data transmission and sharing. It greatly reduces the cost of information acquisition, reduces the phenomenon of “data silos”, and realizes data interconnection and a real-time update. This will significantly increase the operational efficiency of the insurance community and reduce underwriting risks, which will ultimately reduce regulatory difficulties and underwriting costs.