A study on two-sided matching game of virtual business partners based on heterogeneous multi-attribute preferences and subject's psychological behavior

Partner selection and matching is a key part of virtual enterprise formation, which directly affects the growth and development of virtual enterprises. The selection and matching of virtual enterprise partners can be regarded as multi-attribute two-sided matching problem. This paper considers the current situation of attribute preferences in the mutual evaluation process of virtual partner selection, introduces a heterogeneous multi-attribute preference evaluation language system, analyzes the psychological behavior characteristics of both subjects in the decision-making process of selection and matching, gives the strategies and conditions of bilateral matching of virtual enterprise partners, and realizes an effective matching with balanced satisfaction of both virtual enterprise partners.


Introduction
In the Internet era, virtual enterprises are created on an increasing scale, and their requirements in terms of dynamic resource reorganization and allocation are increasing in order to cope with the increasing market competition (Martinez et al. 2001); therefore, the selection and matching of virtual enterprise partners becomes the key to their growth and their competitiveness (Ip et al. 2003;Petersen 2007;Vaez-Alaei, et al. 2021).The problem of virtual enterprise partner selection and matching is a current frontier topic of academic research.
In two-sided matching (TSM) of virtual business partners, subjects (participants) can be divided into two disjoint sets, and each subject seeks to match one (or more) objects with the other for the purpose of aggregating innovation resources and improving innovation performance (Lin, 2019;Yildiz, 2022).In reality, cluster subjects often need to examine multiple attributes (indicators) to arrive at an accurate and comprehensive evaluation, and multi-attribute two-sided matching (MATSM) is a common type of decision problem (Zhang et al. 2020).Due to the complexity of the matching environment, the uncertainty of the matching information and the ambiguity of human thinking, in the actual matching decision, the matching subject will use the most familiar natural language to express the decision preference, i.e., the two-sided subject will give the preference information in the form of natural language (Zhang et al. 2019;Yu et al. 2020;Haas 2021), for example, enterprise A attaches great importance to the service quality when selecting a partner, and will mostly use the language in the form of ''very good quality'', ''average quality'', ''bad quality'', etc., while enterprise B attaches great importance to the degree of technical matching between the two parties when selecting a partner, and the subject will choose the language in the form of ''perfect'', ''suitable'', ''unsuitable'', etc.The evaluation language preferences of different subjects are different, which the multi-attribute preference information mining and has important value and significance for solving the two-sided matching problem of virtual enterprises (Wu Y et al. 2020;Zhang et al. 2021).In view of this, considering the linguistic preferences of evaluation subjects and the competing relationship, there is an iterative game relationship in the matching process between the two parties, and the psychological behavior based on self-interest maximization of the game subject will choose the matching strategy according to the benefit and cost payment changes, this paper integrates three evaluation preferences of satisfaction value, preference order and preference relationship, introduces game idea into the virtual partner bilateral matching process, and designs virtual enterprise partner matching decision method to provide.This paper introduces the game idea into the virtual partner bilateral matching process, designs the virtual enterprise partner matching decision method, and provides new ideas and methods for the selection of virtual enterprise partners.
In general, the main new contributions of this original work are depicted as follows: (a) The linguistic preferences of evaluation subjects are considered in the matching process, and the mining of multi-attribute preference information is more in line with the real situation.(b) Considering the matching process as a dynamic learning process of both parties, considering the psychological traits of both parties based on maximizing their own interests, introducing game theory, portraying the dynamic process of strategy adaptation of both matching subjects, and comparing the matching results of static and dynamic game models to obtain a more friendly matching result.(c) This paper breaks through the traditional ally perspective and takes collective satisfaction as the objective function, which takes into account the interests of the subjects on both sides of the match and obtains a more stable matching scheme.
The paper is divided into five parts: Set. 1 is the introduction, which mainly discusses the background, research value and significance of this paper; Set. 2 is a literature review, which summarizes the frontiers of research related to virtual enterprise partner selection methods at home and abroad and discusses the innovation points of this paper; Set. 3 the core concepts and model building, mainly the definition of two-sided matching, heterogeneous multi-attribute preferences, two-sided matching game model construction and solution methods; Set. 4 is the case analysis and static game model comparison, which consists of three components: case overview and solution analysis, and further discussion, which is mainly to verify the reliability and advancedness of the model method.Set. 5, which summarizes the problems in the virtual partner two-sided matching game model and its solution as well as its application value.

Literature overview
Evaluation and selection of partners is the starting point of virtual enterprise creation.Scholars at home and abroad pay much attention to the MATSM problem faced by virtual enterprise partner matching and put forward a variety of matching models, which can be classified according to the different perspectives of matching subjects: partner selection model based on ally perspective, partner matching model based on two-sided satisfaction/maximum benefit, and partner selection model based on two-sided satisfaction equilibrium.

Partner selection model based on ally perspective
The partner selection model based on the alliance owner's perspective starts from the core enterprise or there are enterprises in the supply chain, so as to select the matching partner according to their own needs, mainly by constructing the evaluation index system and evaluation model, so as to realize the partner selection, the representative studies are : Talluri, et al. (1996) proposed a twostage partner selection framework, from the perspective of designing business alliances, based on the initial partner selection using data envelopment analysis, and the final partner selection using overall goal planning;  Zhang et al. (2021) proposed a new ordinal value basis based on the identification of the dominance structure of the matching subject's characteristic attributes, and then constructed a two-sided matching model method based on the subject's objective evaluation of the dominance attribute scale, in response to the variability of the actor subject; Liang et al. (2022) proposed a probabilistic hesitant fuzzy two-sided matching decision method based on regret theory by considering the psychological behavioral characteristics of two-sided subjects' regret avoidance.The above model methods fully consider the satisfaction of both parties to form a matching scheme, which can provide some guidance for virtual enterprise decision making in reality, but most of the above models are a static decision making model without considering the psychological process of the actor subject, and it is difficult for the subject to maintain complete rationality and thus make a decision according to the maximum collective interest, for which the matching method needs to be further improved.

A partner matching game model approach based on two-sided satisfactory equilibrium
In the partner selection process, virtual enterprises as finite rational human, their decision-making process is still based on the maximization of their own utilization, therefore, virtual enterprises in the cooperation process is always based on their own interests to make decisions, while constantly according to each other's strategies to optimize the strategy, in general, the two sides in the selection of the best mutual choice implied game behavior, and finally achieve a mutually satisfactory and balanced cooperative alliance, so as to complete the two-sided Matching process (Korkmaz et al. 2008;Jia, et al. 2022).For this reason, many scholars introduce game theory to study the twosided matching game model of virtual enterprise partners, for example, Ye et al. (2003) 2021) to address the conflict of interests in the low-carbon production and green investment process between small and medium-sized manufacturers and suppliers, a bi-objective nonlinear programming model is constructed based on game theory to ensure that the matching cooperation can make the maximum green benefits of the whole production process.Huang et al. (2022) used fuzzy numbers to inscribe the decision payment functions of both parties for different attribute preferences of game subjects.Li, et al. (2021) established a two-sided matching game model with equilibrium of individual and group interests based on the expected matching ordinal number (EMO) according to the multi-attribute characteristics of two-sided subjects.Qin, et al. (2022) for the two-sided matching problem with uncertain preference order, based on the maximum satisfaction criterion of the game subject and the matrix game thinking under individual rationality, proposed the benefit (satisfaction) matrix under uncertain order and constructed a game matching optimization model that takes into account the overall and individual benefits.
From the above-mentioned literature, most of the studies still take the perspective of the alliance owner and take the maximum collective satisfaction as the objective function, while ignoring the interests of the matching parties.At the same time, the existing two-sided matching of virtual enterprise partners pays less attention to the influence of heterogeneous subject preferences on the two-sided matching results, while in reality the evaluation subjects always prefer to use their own preferred language for evaluation, so it is important to make full use of and mine these multi-attribute preference information to solve the virtual enterprise.Therefore, it is of great value and significance to make full use of these multi-attribute preference information to solve the two-sided matching problem of virtual enterprises.
3 Core concepts and model building

Definition of two-sided matching
Given the definition of two-sided matching problem of virtual enterprises.Suppose that in order to complete a key technology collaborative innovation project, virtual enterprises need to form an innovation alliance so as to ensure the smooth implementation of the key technology collaborative innovation project.Assume that there are innovation groups A ¼ A 1 ; A 2 ; . ..;A m f gand innovation groups B ¼ B 1 ; B 2 ; . ..; B n f gin the virtual enterprise, and for the sake of not losing generality, let m n; innovation groups A and B are heterogeneous groups, and the first i subject A i in group A and the first j subject B j in group B have good complementarity in innovation resources, and two-sided cooperation can greatly improve innovation performance (Yazici, 2017).Considering the cost of cooperation and the distribution of future benefits, it is agreed that each subject is matched with only one individual of the other group, i.e., there is only ''one-to-one'' cooperation, not ''one-to-many'' cooperation, thus ensuring that the interests of the innovation alliance subjects are maximized (Chakraborty, et al. 2010).
Definition (Chakraborty, et al. 2010;Lin, et al. 2017): Name the mapping p: is a matching pair, and the virtual business subjects form a matching pair and further form a matching scheme.

Heterogeneous multi-attribute preferences
Let the set of attributes of the A i subject be C ¼ c 1 ; c 2 ; . ..; c h f g , and the weight of the first / attribute c / be w / ; the set of attributes of the subject B j be , and the weight of the first u attribute d u be w 0 u .The constraint condition satisfied by w / and w0 u are shown in Eq. (1).
The matching subject will use the most familiar natural language to express the decision preference, i.e., the twosided subject will give the preference information in the form of natural language, and when A i B j À Á assigned the value of d u c / À Á , the attribute of B j A i ð Þ, they can be divided into numerical, ordinal and relational types.Different methods to cope with them are as follows.
(1) Numerical: and the larger the value, the higher the satisfaction of the subject.Numerical values do not need to be transformed and can be used directly for modeling.
(2) Ordinal: m, and in this paper, we uses a 1 to 9 scale formulation to indicate the ranking order of the subject.Smaller ordinal values indicate higher subject satisfaction, and reference (Yue, et al. 2012) converts ordinal values into numerical values, as shown in Eq. (2).
As a result, different types of information are unified into numerical preferences, i.e.
Then, the satisfaction (or benefit) matrices of the virtual enterprise matching parties are obtained by further fusing the attributes and corresponding weight information with the help of the classical OWA operator.Denote U nÂm as the satisfaction matrices of group A and group B, respectively.

Construction of two-sided matching game
model based on subject's psychological behavior

Basic assumptions and payment functions
According to the components and basic theories of game theory, it is assumed that the two-sided matching game system of virtual business partners is composed of three elements: insider, game strategy and payment function, so a basic two-sided matching game model can be represented by Eq. ( 5).
where N refers to the set of firms involved in the game, including virtual business group A and virtual business group B; S i denotes the set of possible strategies of the enterprise i and the virtual enterprise strategy choice is simplified into two strategies of ''cooperation'' and ''noncooperation'', which are denoted as S A ¼ S B ¼ s ac ; s re f g; P denotes the satisfaction (or payment) function assumed by virtual firm A, and let p i i ¼ 1; 2 ð Þ be the satisfaction function of A i ; H denotes the satisfaction (or payment) function assumed by virtual firm B, h j j ¼ 1; 2 ð Þthe satisfaction (or payment) function of B j .
Hypothesis 1 Matching in the enterprise decision process can only choose a pure strategy, there is no mixed strategy.For this purpose, let P A ¼ p ac ; p re f g be the probability distribution of A i strategy selection and Q A ¼ q ac ; q re f gbe the probability distribution of B j strategy selection, according to the basic theorem of probability distribution and two-sided matching strategy selection, the following constraints can be obtained in Eq. ( 6).
Hypothesis 2 In a two-sided game system, the objects that any subject can match are not unique, and the subjects can obtain cooperative benefits.When group A and group B in the virtual cluster partner two-sided matching game system start matching decision, it is assumed that the known sum p i;j and v i;j is the respective gain obtained when the subject A i and the subject B j match; pi;Àj is the potential gain obtained when the subject A i does not match with the subject B j and matches with the rest of the objects in group B; similarly, the potential gain obtained if the subject B j does not match with the subject A i and chooses to match with other individuals in group A is hÀi;j .
According to the above research hypothesis, the payment matrix of the virtual enterprise two-sided matching game system can be obtained as represented in Table 1.
As can be seen from Table 2, the general expression for the subject A i expected return in the two-sided matching game system is shown in Eq. ( 7).
where p i;j is the matching probability of subject A i and subject B j , q i;j is the probability that the subject B j B j B j B j matches the subject A i , b j is a constraint parameter indicating that the number of subjects matching the subject A i is not more than 1.Similarly the expected return h j of the subject B j can be found.

Modeling and solution
Both subjects of the two-sided matching game system naturally tend to adopt a strategy that maximizes their own interests.Let s Ã i be the optimal strategy choice of A i and s Ã j be the optimal strategy choice of B j , we can obtain the mathematical expression in Eq. ( 8).
Taking Eq. ( 9) as the game subject matching objective, Eqs. ( 7) and ( 8) are combined to construct the optimal model of the benefits of both subjects of the two-sided matching game system.s:t: The multi-objective optimization model shown in Eq. ( 10) is transformed into a single-objective linear model as shown in Eq. ( 11).
The P Ã ¼ p Ã i;j is noted as the acquired optimal strategy of subject A; Q Ã ¼ q Ã i;j is the optimal strategy of subject B. If P Ã ¼ Q Ã , then the optimal solution is also the Nash equilibrium solution of the game model.The proof procedure is as follows.
When P Ã ¼ Q Ã , to prove the existence of a unique solution to Eq. ( 10), we introduce the decision variable K from 0 to 1.
As a constraint, the following model is then developed.
Solve the model to obtain the optimal solution The following inference can be obtained from Eq. ( 12).
Corollary 1: The optimal solution of Eq. ( 12) is the Nash equilibrium solution.
Proof Eq. ( 12) is a 0-1 programming model with m and n variables, then there are at most 2 m?n valid solutions; note that the strategy combination of the optimal solution is p Ã 1j 1 ; p Ã 2;j 2 ; . ..; p Ã m;j m , at this point P Ã ¼ Q Ã ; assume that there exists a subject A k , whose strategy choice is p Ã k;j k .If the outcome is not the optimal response to the strategy choice of other subjects, then u p then the subject A k , based on the psychology of maximizing their own interests, must adjust the strategy so that the total expected return increases to u p k;j k À Á , which is in contradiction with p Ã k;j k is the optimal solution.Therefore, the optimal solution of the model is the Nash equilibrium solution.
When P Ã 6 ¼ Q Ã , at this time, both parties have not reached the most satisfactory state, but virtual enterprises, in order for the innovation cooperation to be carried out, as finite rational subjects, usually retreat to the second best and choose the strategy that both parties can be satisfied with, thus forming a virtual innovation alliance.At this time, the available matching solutions are not unique, i.e., there are multiple stable solutions.

Background
Industrial software design and development usually requires large-scale collaborative innovation, requiring the core enterprise to lead the top-level design, and then seek the participation of different software design and development companies, individuals and unofficial organizations, etc.After the core enterprise releases its R&D tasks, it needs to identify and match partners through the network platform, and conversely, the partners are also seeking projects and platforms for their own value realization, for which the two sides are a two-sided matching problem.An industrial software design company, which has been awarded a major national R&D project, needs to seek virtual partners to form an innovation consortium so as to pool the innovative resources and endowments of all parties and complete the project R&D tasks as soon as possible.The enterprise is jointly responsible for the software core module project by its five affiliated enterprises, namely: Enterprise A1, Enterprise A2, Enterprise A3, Enterprise A4 and Enterprise A5.These five enterprises need to cooperate with a database research and development enterprise to form a project team so as to achieve complementary advantages.At the same time, database R&D enterprises are more willing to participate in major national projects in order to gain revenue and improve their market position.After the core enterprises publish the project tasks in the platform, there are a large number of database R&D enterprises applying for it, and after the expert evaluation, six virtual enterprises with high R&D capability, R&D qualification and credit evaluation are selected (noted as B1, B2, …, B6).In order to select the most matching partner, both parties need to further evaluate each other and select each other.

Numerical analysis
Party A enterprises select their partners by evaluating three indicators: R&D quality (d1), time management (d2), and service quality (d3), with indicator weights of (0.4, 0.3, 0.3); Party B enterprises select their partners by evaluating three indicators: project profitability (c1), enterprise size (c2), and corporate reputation (c3), with weights of (0.35, 0.35, 0.3) for each indicator.The A-side evaluation of natural language information is shown in Table 2. Natural language information for the evaluation of party B is specified in Tables 3 and 4. According to Table 2, it can be seen that d1 and d3 in the natural language information evaluated by party A are numerical and d2 is ordinal, which needs to be transformed using Eq.(2) and converted to numerical before calculation.
According to Table 3, it can be seen that the natural language information evaluated by B-party has c1 as numerical type, c2 as ordinal value type, and c3 as relational type.c2 and c3 need to be transformed using Eqs.( 2) and (3), respectively, and converted to numerical type before calculation.
According to the heterogeneous multi-attribute preference information shown in Tables 2, 3, 4, the solution steps are as follows.
Step 1: Identify the types of evaluation information given by both sides, and convert all types to numerical types.
Step 2: Based on the attribute and weight information, the two-sided match satisfaction matrix is calculated, and the calculation results are shown in Tables 5 and 6.

Static game model matching comparison
Based on the static game, the optimal solution agreed by both partners of the virtual enterprise is not the optimal solution of the problem.The matching solutions obtained from different parties' gain perspective may be different.If the proposed model is applied to solve Case 1, the optimal strategic solution and the Nash equilibrium solution for both parties are obtained as shown in Eq. ( 14).
The corresponding matching scheme is shown in Table 7: As can be seen from Table 7, compared to the two-sided equilibrium solution, the A-party preferred solution is better, and no subject is matched with the least satisfied object.The equilibrium solution has two subjects A5 and B5 in the worst situation, which is similar to the ''prisoner's dilemma'' in the static game, but not the concept of ''prisoner's dilemma'' in the classical game theory, and only forms a least desirable matching solution.That is, the Nash equilibrium solution is not an optimal solution, but only a possible stable solution.Many scholars have added new constraints to find the Nash equilibrium solution, for example, Cao et al. (2004) as a stable constraint, but it still cannot solve the problem of conflicting interests between subjects, and the added constraints cannot ensure the stability of the matching result when the subject's psychological behavior maximizes its own interests.In conclusion, starting from the subject's psychological behavior and taking the maximization of self-interest as the condition and basis of its decision is more in line with the realistic phenomenon of repeated games between two parties in the matching process.

Conclusions
Virtual enterprise partner selection is related to the growth of its virtual enterprise and its innovation performance, and virtual enterprise partner selection is blind due to information asymmetry.The virtual enterprise partner matching problem can be regarded as MATSM problem.The current models for solving MATSM problems are mostly based on maximizing collective interests, while ignoring the psychological behavior of matching subjects, i.e., maximizing their own interests in the decision-making process.Therefore, it is difficult to make decisions based on maximizing collective interests in the virtual enterprise partner matching process, and the insiders are learning and adjusting strategies in the process of continuous game play, so the matching process is a dynamic learning and competition process rather than a static matching result, which leads to deviations in many methods in solving practical problems and limits the application value of the model.To address the shortcomings of existing research, this paper draws on the idea of classical matrix game and integrates heterogeneous multi-attribute preferences to propose a two-sided matching game model of virtual corporate partners based on heterogeneous multi-attribute preferences and subject's psychological behavior.The model integrates the features of evaluating subjects' linguistic choice preferences in real decision-making; at the same time, based on the rational psychological perspective of subjects, it takes the maximization of their own interests as the basis for decision-making.More importantly, the model can be solved to obtain a variety of stable matching solutions, and the conclusions obtained are more suitable for the matching decision process of both parties in the real process, which makes the model can be applied in many different scenarios, such as the selection of battery suppliers for new energy vehicles and the selection of screen suppliers for smartphones, and thus has more pragmatic value and significance.
It should be noted that there are some shortcomings in the setting conditions of this model method, mainly because some individuals in the group have cooperated and allied, thus forming a community of interests, and the decision of individuals in the group will be constrained by more conditions in the game process, which makes the results of this model deviate.The game matching under fuzzy information will be further studied in the future, and the reference basis of subject's psychological decision making will be considered, and the influence of subject's psychological change can be considered to be studied in depth in combination with prospect theory. ð13Þ d u 2 E P , is denoted as d P u;i c P

Table 1
Two-sided matching game system payment matrix

Table 3
Evaluation information of Party B for c1 and c2

Table 5
Satisfaction matrix of party A regarding party B

Table 7
Matching results