# A Competitive Multiattribute Group Decision-Making Approach for the Game between Manufacturers

Authors: Quan Zhang; KeXin Jiang; ManTing Yan; JiYun Ma
Journal: Computational Intelligence and Neuroscience
 (2019)

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## Abstract

Under the competitive market environment, the game between manufacturers comes down to the competitive multiattribute group decision-making problem. In this study, the evaluation information of experts is given in the form of 2-dimension 2-tuple linguistic variables, and an approach is proposed for the competitive multiattribute group decision-making problem based on game theory and evidence theory. Firstly, based on the evidence theory, the attribute values of each situation are obtained by aggregating the 2-dimension 2-tuple linguistic evaluation information given by experts. Secondly, according to the attribute values of every situation, the evidence theory is applied for the second aggregation to obtain the overall values of every situation, and then the game matrix of competitive multiattribute group decision problem is formed. Then, according to the bivariate game matrix, the Nash equilibrium point of competitive multiattribute group decision-making problem is determined based on game theory. Finally, a practical case about the alternative selections for a duopoly problem is used to illustrate the effectiveness and applicability of the proposed approach for the competitive multiattribute group decision-making problem.


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## Body

## 1. Introduction

With the economic globalization and market competition intensification, the game between manufacturers has become very common, which can be viewed as a multiple attribute group decision-making problem (MAGDM). MAGDM is the process to select the most desirable alternative with respect to multiple conflicting attributes based on the evaluation information given by the invited experts [1]. In a competitive multiattribute group decision-making problem, facing a decision matrix, two decision makers usually have to compete with each other when selecting their most desirable alternative [2]. In [2], a competitive multiattribute decision-making framework was proposed. The alternative selected by one decision maker affects the benefits of the other one. The competing decision makers must search a stable situation to ensure their benefits to be maximum or their loss to be minimum. In the course of competitive multiattribute group decision making, the stable situation is searched with respect to the alternatives by the two competing decision makers, which is called the Nash equilibrium. The evaluation information of every alternative is given by experts to determine the Nash equilibrium.Because of the complexity of multiple attribute decision-making problems, the evaluation information of every alternative given by experts is always uncertain. In competitive multiattribute group decision-making problems, the uncertain evaluation information from the experts can be expressed in various forms, such as fuzzy linguistic information [3–7] and linguistic term sets [8–12]. Compared with linguistic term sets, the fuzzy linguistic information lacks precision in the final results. But, the original linguistic term sets cannot describe the situation where the evaluation information proposed by experts is between two standard linguistic terms. In order to solve this problem, Herrera and Martínez proposed the concept of 2-tuple linguistics and added the unitary information to the original linguistic variables in order to describe the degree to which the evaluation information deviated from the standard linguistic terms [13]. However, 2-tuple linguistic information does not take into account the degree of uncertainty of evaluation information. For this reason, Zhu et al. [14] proposed the 2-dimension 2-tuple linguistic concept by adding one-dimensional information on the basis of 2-tuple linguistic information to describe the degree of uncertainty of evaluation information. The 2-dimension 2-tuple linguistic information can describe the evaluation information of experts more accurately than original linguistic information.In order to improve the accuracy of decision-making results as compared with the study in [2], this paper proposes an approach for the competitive multiattribute group decision-making problem of the game between manufacturers, where the experts’ evaluation information is in 2-dimension 2-tuple linguistics. To solve the competitive multiattribute group decision-making problem with 2-dimension 2-tuple linguistic evaluation information, it is necessary to aggregate experts’ evaluation information. The weighted sum is a common approach to aggregate 2-dimension 2-tuple linguistic evaluation information. However, the evidence theory possesses the ability to describe uncertainty and ignorance. Thus, an approach for aggregating 2-dimension 2-tuple linguistic evaluation information is proposed in this paper based on the evidence theory (e.g., Dempster–Shafer theory). In addition, based on the game matrix of competitive multiattribute group decision problem, the Nash equilibrium point of the problem is determined based on game theory.The organization of this paper is as follows. Section2 reviews the current research on the multiple attribute group decision-making problems and the methods for aggregating 2-dimension 2-tuple linguistic information as well as the applications of evidence theory and game theory in multiple attribute decision-making problems. Section 3 gives preliminary knowledge about 2-tuple linguistic information, 2-dimension 2-tuple linguistic information, and evidence theory. The competitive multiattribute decision-making problem with two decision makers is described in Section 4. Section 5 proposes an approach for the competitive multiattribute group decision-making problem with 2-dimension 2-tuple linguistic evaluation information. In Section 6, the application of the proposed approach is illustrated by an example. Section 7 summarizes the results of this study.


## 2. Literature Review

In order to determine the best solution, in [15], two intuitionistic fuzzy aggregation operators were proposed to aggregate the experts’ information and attribute information. Garg and Nancy [16] proposed some new linguistic priority aggregation operators, based on which a multiattribute group decision-making method was proposed too. By considering the priority of attributes and experts at the same time, Verma studied the multiattribute group decision-making problems under the trapezoidal fuzzy linguistic information and proposed a new weighted average priority operator [17].Xu and Cai [18] established a nonlinear programming model to determine the experts’ weights based on the principle that the experts’ weights should produce the highest consistency of the experts. On this basis, the authors of [19] pointed out the shortcomings of applying genetic algorithm to solve the nonlinear programming model and proposed a new method to determine the experts’ weights by linearizing the nonlinear programming model. The authors of [20] proposed the concept of probabilistic uncertain linguistic term sets and the method for probabilistic uncertain linguistic multiattribute group decision-making problems.In recent years, many scholars studied the 2-dimension 2-tuple linguistic information. In [21], some power aggregation operators were proposed, including 2-dimension uncertain linguistic power generalized aggregation operator (2DULPGA) and 2-dimension uncertain linguistic power generalized weighted aggregation operator (2DULPGWA), and some properties were discussed, as well as the special cases of them.The authors of [22] proposed a 2-dimension linguistic lattice implication algebra (2DL-LIA) as a linguistic evaluation set with lattice structure. Motivated by the ideas of dependent aggregation operator, in [23], some 2-dimension linguistic dependent aggregation operators were developed. Yu et al. [24] developed a 2-dimension linguistic weighted averaging (2DLWA) operator and a 2-dimension linguistic ordered weighted averaging (2DLOWA) operator. In [25], some density aggregation operators were proposed based on 2DULVs. Liu and Wang [26] proposed three operators: 2-dimension uncertain linguistic weighted averaging (2DULWA) operator, 2-dimension uncertain linguistic weighted geometric (2DULWG) operator, and 2-dimension uncertain linguistic generalized weighted average (2DULGWA) operator. Wu et al. [27] put forward 2-dimension interval type-2 trapezoidal fuzzy ordered weighted average (2DIT2TFOWA) operator and quasi-2-dimension interval type-2 trapezoidal fuzzy ordered weighted average (quasi-2DIT2TFOWA) operator. However, most of the research on the aggregation methods for 2-dimension linguistic variables as discussed above relied on the weighted average operator. The weighted average operator mainly combines the 2-dimension linguistic information by taking the smallest method. Then, the fusion of the highest-level 2-dimension linguistic information with the lowest-level 2-dimension linguistic information is the lowest level of fusion information, which is very unreasonable. Therefore, the approach proposed in this paper is to use the 2-dimension linguistic information as the reliability of the experts, and the above unreasonable situation will not occur.Because of the advantages of 2-dimension 2-tuple linguistic information, it is very meaningful to study the competitive multiattribute group decision-making problems with 2-dimension 2-tuple linguistic evaluation information.Evidence theory was widely used in multiattribute group decision-making problems. The authors of [28] proposed a DS-AHP approach for the multiattribute decision-making problem with incomplete information. The authors of [29] proposed an evidence reasoning approach for the multiattribute decision-making problem with incomplete decision matrix. The authors of [30] applied the Dempster–Shafer theory of evidence to develop a new 2-tuple linguistic representation model for decision making and then put forward some linguistic aggregation operators.Game theory is widely used in modern management science and produces fruitful research results. Chen and Larbani [31] applied game theory in fuzzy multiattribute decision-making problems, by assuming that fuzzy multiattribute decision making is a game between decision makers and nature. Zhou et al. [32] applied game theory to solve multiattribute decision-making problems and thought that the multiattribute decision-making problem was a game between the attributes. And sometimes, manufacturers are often in a fiercely competitive environment when making decisions. Therefore, this paper proposes a multiattribute group decision-making approach for manufacturers to conduct alternative selections in a competitive environment.


## 3. Preliminaries

Some background knowledge about 2-tuple linguistic information and 2-dimension 2-tuple linguistic information is introduced in this section, as well as evidence theory [13, 14, 33].


### 3.1. 2-Tuple Linguistic Variables

Definition 1.
Letβ be the result of an aggregation of the indexes of a set of labels assessed in a linguistic term set TSTS=S0,S1,…,Sg, i. e., the result of a symbolic aggregation operation. β∈0,g, with g+1 being the cardinality of TS. Let i=roundβ and α=β−i be two values such that i∈0,g and α∈−0.5,0.5; then, α is called symbolic translation [13].
The symbolic translation of a linguistic termSi, is a numerical value assessed in −0.5,0.5 that supports the “difference of information” between counting of information β∈0,g obtained after a symbolic aggregation operation and the closest value in 0,…,g that indicates the index of the closest linguistic term in TS i=roundβ [13].
A 2-tupleSi,α is usually used to represent the linguistic information, Si∈TS, and α∈−0.5,0.5, as follows:Si represents the linguistic label center of the informationαi is a numerical value expressing the value of the translation from the original result β to the closest index label, i, in the linguistic term set TSS1,…,Sg, i.e., the symbolic translation [13]Definition 2.
LetTS=S0,S1,…,Sg be a linguistic term set and β∈0,g be a value representing the result of a symbolic aggregation operation; then, the 2-tuple that expresses the equivalent information to β is obtained with the following function [13]:(1)Δ:0,g⟶TS×−0.5,0.5,Δβ=Si,α,withSi,i=roundβ,αi=β−i,αi∈−0.5,0.5,where round⋅ is the usual round operation, Si has the closest index label to “β,” and “α” is the value of the symbolic translation.Definition 3.
LetTS=S0,S1,…,Sg be a linguistic term set and Si,αi be a 2-tuple. There is always a Δ−1 function such that from a 2-tuple it returns its equivalent numerical value [13]; β∈0,g⊂R:(2)Δ−1:TS×−0.5,0.5⟶0,g.(3)Δ−1Si,αi=i+αi=β.Definition 4.
LetS1,α1 and S2,α2 be two linguistic term sets; then, their distance is obtained with the following function [33]:(4)dS1,α1,S2,α2=Δ−1S1,α1−Δ−1S2,α2.


### 3.2. 2-Dimension 2-Tuple Linguistic Variables

The definition of 2-dimension 2-tuple linguistic variables is given as follows.Definition 5.
LetTS=S0,S1,…,Sg1 be the set of linguistic information evaluation in the first dimension. Let TS∗=S0∗,S1∗,…,Sg2∗ be the set of linguistic information evaluation in the second dimension. Then, TWS=Si,αi,Sj∗,αj∗i=0,1,…,g1,j=0,1,…,g2 is defined as a 2-dimension 2-tuple linguistic term [14].


### 3.3. Evidence Theory

Definition 6.
LetΩ be a set of mutually exclusive and collectively exhaustive events indicated by(5)Ω=E1,E2,…,Ei,…,En,where set Ω is called a frame of discernment. The power set of Ω is indicated by 2Ω:(6)2Ω=ϕ,E1,…,En,E1,E2,…,E1,E2,…,Ei,…,Ω.
Dempster’s rule of combination, denoted bym=m1⊕m2, is defined as follows [2]:(7)m1⊕2⊕⋯⊕pA=11−k∑∩Ai=A∏i=1pmiAi,A≠ϕ;0,A=ϕ;with(8)k=∑∩Ai=ϕ∏i=1pmiAi.


## 3.1. 2-Tuple Linguistic Variables

Definition 1.
Letβ be the result of an aggregation of the indexes of a set of labels assessed in a linguistic term set TSTS=S0,S1,…,Sg, i. e., the result of a symbolic aggregation operation. β∈0,g, with g+1 being the cardinality of TS. Let i=roundβ and α=β−i be two values such that i∈0,g and α∈−0.5,0.5; then, α is called symbolic translation [13].
The symbolic translation of a linguistic termSi, is a numerical value assessed in −0.5,0.5 that supports the “difference of information” between counting of information β∈0,g obtained after a symbolic aggregation operation and the closest value in 0,…,g that indicates the index of the closest linguistic term in TS i=roundβ [13].
A 2-tupleSi,α is usually used to represent the linguistic information, Si∈TS, and α∈−0.5,0.5, as follows:Si represents the linguistic label center of the informationαi is a numerical value expressing the value of the translation from the original result β to the closest index label, i, in the linguistic term set TSS1,…,Sg, i.e., the symbolic translation [13]Definition 2.
LetTS=S0,S1,…,Sg be a linguistic term set and β∈0,g be a value representing the result of a symbolic aggregation operation; then, the 2-tuple that expresses the equivalent information to β is obtained with the following function [13]:(1)Δ:0,g⟶TS×−0.5,0.5,Δβ=Si,α,withSi,i=roundβ,αi=β−i,αi∈−0.5,0.5,where round⋅ is the usual round operation, Si has the closest index label to “β,” and “α” is the value of the symbolic translation.Definition 3.
LetTS=S0,S1,…,Sg be a linguistic term set and Si,αi be a 2-tuple. There is always a Δ−1 function such that from a 2-tuple it returns its equivalent numerical value [13]; β∈0,g⊂R:(2)Δ−1:TS×−0.5,0.5⟶0,g.(3)Δ−1Si,αi=i+αi=β.Definition 4.
LetS1,α1 and S2,α2 be two linguistic term sets; then, their distance is obtained with the following function [33]:(4)dS1,α1,S2,α2=Δ−1S1,α1−Δ−1S2,α2.


## 3.2. 2-Dimension 2-Tuple Linguistic Variables

The definition of 2-dimension 2-tuple linguistic variables is given as follows.Definition 5.
LetTS=S0,S1,…,Sg1 be the set of linguistic information evaluation in the first dimension. Let TS∗=S0∗,S1∗,…,Sg2∗ be the set of linguistic information evaluation in the second dimension. Then, TWS=Si,αi,Sj∗,αj∗i=0,1,…,g1,j=0,1,…,g2 is defined as a 2-dimension 2-tuple linguistic term [14].


## 3.3. Evidence Theory

Definition 6.
LetΩ be a set of mutually exclusive and collectively exhaustive events indicated by(5)Ω=E1,E2,…,Ei,…,En,where set Ω is called a frame of discernment. The power set of Ω is indicated by 2Ω:(6)2Ω=ϕ,E1,…,En,E1,E2,…,E1,E2,…,Ei,…,Ω.
Dempster’s rule of combination, denoted bym=m1⊕m2, is defined as follows [2]:(7)m1⊕2⊕⋯⊕pA=11−k∑∩Ai=A∏i=1pmiAi,A≠ϕ;0,A=ϕ;with(8)k=∑∩Ai=ϕ∏i=1pmiAi.


## 4. Problem Descriptions

LetA and B be two manufacturers involved in the competitive multiattribute group decision problems; then, their collections of alternatives, attributes, and experts are stated as follows.The set of alternatives for manufacturersA and B is stated as AS=AS1,AS2,…,ASy and BS=BS1,BS2,…,BSt, respectively.The set of attributes for manufacturersA and B is C11,C21,…,Cn1 and C12,C22,…,Ce2, respectively.The set of experts isE11,E21,…,Eh1 and E12,E22,…,Es2 for manufacturers A and B, respectively. In this study, the experts express their evaluation information on the alternatives against every attributes in the form of 2-dimension 2-tuple linguistic variables, according to the different strategies of opponent. The standard linguistic term set of the first dimension is TS=S0,S1,…,Sg1. The standard linguistic term set of the second dimension is TS∗=S0∗,S1∗,…,Sg2∗. The utility value of the first dimension linguistic term set is uTS=uS0,uS1,…,uSg1. Thus, the competitive multiattribute group decision matrix is obtained as follows:(9)D1=aijlBSy×n,D2=bijrASt×e,where aijl=TLij1l,TLij1l∗=Sij1pl,αij1l,Sij1ql∗,αij1l∗,i=1,2,…,y,l=1,2,…,h,j=1,2,…,n,p=0,1,…,g1,q=0,1,…,g2 is the evaluation value of the alternative ASi under the attribute Cj1 given by expert El1 for different strategies chosen by manufacturer B and bijr=TLij2r,TLij2r∗=Sij2xr,αij2r,Sij2vr∗,αij2r∗,i=1,2,…,t,r=1,2,…,s,j=1,2,…,e,x=0,1,…g1,v=0,1,…,g2 is the evaluation value of the alternative BSi under the attribute Cj2 given by expert Erz2 for different strategies chosen by manufacturer A.As shown in Figure1, the problem focused in this study is about the game between manufacturer A and manufacturer B, where experts of both sides give their evaluation information on the alternatives against the corresponding attributes in the form of 2-dimension 2-tuple linguistic variables. The purpose is to determine the Nash equilibrium for manufacturer A and manufacturer B.Figure 1
The competitive multiattribute decision framework.


## 5. An Approach to the Competitive Multiattribute Group Decision-Making Problems with 2-Dimension 2-Tuple Linguistic Variables

In order to solve the competitive multiattribute group decision-making problems with 2-dimension 2-tuple linguistic variables, three stages are employed to obtain the game matrix and determine the Nash equilibrium. In stage one, the evidence theory of Dempster and Shafer is applied and modified to aggregate the experts’ evaluation information of 2-dimension 2-tuple linguistic variables. In stage two, the attribute values are integrated to obtain the game matrix. In stage three, according to the obtained game matrix, the Nash equilibrium point for the competitive multiattribute group decision-making problem is determined based on game theory.


### 5.1. Aggregate Experts’ Evaluation Information of 2-Dimension 2-Tuple Linguistic Variables Based on Evidence Theory

Based on evidence theory, the following steps are employed to aggregate 2-dimension 2-tuple linguistic information given by experts:Step 1. Calculate the basic probability distribution functionSupposeTLij1l=Sij1pl,αij1l,TLij1l∗=Sij1ql∗,αij1l∗, then the basic probability distribution function is as follows:(1)
When0<αij1l<0.5, mlSij1pl=1−αij1l, mlSij1p+1l=αij1l(2)
Whenαij1l=0,mlSij1pl=1(3)
When−0.5<αij1l<0,mlSij1p−1l=αij1l,mlSij1pl=1−αij1lStep 2. Calculate experts’ reliabilityDefinition 7.
Based on the degree of certainty of 2-dimension 2-tuple linguistic information given by experts and their experience, an expert’s reliability is defined as follows:(10)λl=λfldSij1ql∗,αij1l∗,S0∗,0∑k=1hλfkdSij1qk∗,αij1k∗,S0∗,0,l=1,…,h,whereλfl is the coefficient that reflects experts’ experience and preference, 0.9<λfl<1; λl is called the expert reliability; dSij1ql∗,αl∗,S0∗,0 is the distance between the second dimension linguistic information given by experts and S0∗,0 as defined in equation (6); and Δ−1 is the equivalent transformation function between 2-tuple linguistic information and the numerical value as defined in equations (2) and (3).With respect to the different values ofSql∗, some special cases of λl are shown as follows:(1)
WhenSij1ql∗,αij1l∗=S0∗,0,λl=0(2)
WhenSij1ql∗,αij1l∗≠S0∗,0 and ∀w≠l,1≤w≤hSij1qw∗,αij1w∗=S0∗,0,λl=1Step 3. Aggregate experts’ information
This paper applies evidence theory to aggregate experts’ information and proposes a new evidence combination rule as follows:(11)mij1Sij1k=∑Sij1p1∩Sij1p2∩⋯∩Sij1ph=Sij1km1Sij1p1m2Sij1p2⋯mhSij1ph+K∑l=1hλlmlSij1kl,where K=∑Sij1p1∩Sij1p2∩⋯∩Sij1ph=ϕm1Sij1p1m2Sij1p2⋯mhSij1ph and mij1Sij1k represents the basic credibility that the evaluation value of alternative ASi1 is Sij1k against attribute Cj1.Theorem 1.
LetTLij1l=Sij1pl,αij1l,i=1,2,…,y,l=1,2,…,y,j=1,2,…,n,p=0,1,…,g1 be 2-tuple linguistic information given by experts. When h>2 and ∀l∈1,2,…,hΔ−1Sij1pl,αij1l∈p,p+1 or h=2 and Δ−1Sij1p1,0=Δ−1Sij1p2,0, then the experts’ evaluation information aggregated by evidence theory can be expressed as(12)mij1Sij1k=∑Sij1p1∩Sij1p2∩⋯∩Sij1ph=Sij1km1Sij1p1m2Sij1p2⋯mhSij1ph,where Δ−1 is the equivalent transformation function between 2-tuple linguistic information and the numerical value defined in equations (2) and (3).Proof:.
(1)
Forh>2, since ∀l∈1,2,…,hΔ−1Sij1pl,αij1l∈p,p+1then∀l∈1,2,…,h∩Sij1pl≠ϕthenK = 0thus equation (12) holds(2)
Forh=2, since Δ−1Sij1p1,0=Δ−1Sij1p2,0thenSij1p1∩Sij1p2≠ϕthenK = 0thus equation (12) holds
When there are more than two experts and their evaluation information are aggregated between the same two standard linguistic term sets or there are two experts and their evaluation information are identical, there are no conflicts between the experts. In this case, equation (12) is equivalent to equation (7).Theorem 2.
LetTLij1l=Sij1pl,αij1l,l=1,2,…,h,i=1,2,…,y,j=1,2,…,n,p=0,1,…,g1 be 2-tuple linguistic information given by experts. When ∀l∈1,2,…,hαij1l=0 and Sij1p1≠Sij1p2≠⋯≠Sij1ph, then the experts’ evaluation information aggregated by evidence theory can be expressed as(13)mijSij1k=∑l=1hλlmlSij1kl.Proof:.
When∀l∈1,2,…,hαij1l=0 and Sij1p1≠Sij1p2≠⋯≠Sij1ph, since ∀l∈1,2,…,h∩Sij1pl=ϕthenK = 1 and ∑Sij1p1∩Sij1p2∩⋯∩Sij1ph=Sij1km1Sij1p1m2Sij1p2⋯mhSij1ph=0thus equation (13) holds.WhenK = 1, equations (7) and (8) cannot be used to aggregate the experts’ evaluation information.


### 5.2. Integrate Attribute Values of Alternatives Based on Evidence Theory

On the basis of Section5.1, the integration steps of attribute values are given as follows:Step 1. Calculate attribute reliabilityThe reliability of attributes should be related to the consistency of experts. The higher the consistency of experts, the higher the reliability of attributes. Firstly, the expert consistency coefficient is calculated, and then the expert consistency coefficient is used to measure the reliability of attributes. The distance matrix between the evaluation information of experts and the standard linguistic sets of the first dimension is established as follows:(14)Dj=d11d12⋯d1g1+1d21d22⋯d2g1+1⋯⋯⋯⋯dh1dh2⋯dhg1+1,wheredlθ=∑i=1ydSij1pl,αij1l,Sθ−1,0Sij1pl,Sθ−1∈TS represents the distance between the first dimension linguistic evaluation information given by experts and the standard linguistic information Sθ−1Sθ−1∈TS.Definition 8.
Letγj=1/minθ∑i=1hdlθ,j=1,2,…,n be expert consistency coefficient; then, the attribute reliability is defined as follows:(15)ωj=γj∑ε=1nγε,j=1,2,...,n.Obviously,0<ωj<1.Step 2. Aggregate attribute values
According to formula (11), attribute values are integrated as(16)mi1Si1G=∑Si11k∩Si21k∩⋯∩Sin1k=Si1Gmi11Si11kmi21Si21k⋯min1Sin1k+K∑j=1nωjmij1Sij1G,where mi1Si1G represents the basic credibility when the evaluation value of alternative ASi1 is SG and K=∑Si11k∩Si21k∩⋯∩Sin1k=ϕmi11Si11kmi21Si21k⋯min1Sin1k.


### 5.3. Determine Nash Equilibrium Point

According to the discussions in Sections5.1 and 5.2, the basic probability distribution in each alternative can be obtained. Thus, the overall values of every alternative are obtained as follows:(17)zi1=∑G=0g1mi1Si1GuSi1G,i=1,2,…,y.Therefore, the overall value of manufacturerA choosing different strategies under the condition that manufacturer B chooses different strategies is obtained. Under the condition that manufacturer A chooses different strategies, manufacturer B chooses the same method to determine the overall values of different strategies.According to the overall values determined by formula (17), the game matrix is constructed as follows:(18)Z=z111;z112z121;z122⋯z1t1;z1t2z211;z212z221;z222⋯z2t1;z2t2⋯⋯⋯⋯zy11;zy12zy21;zy22⋯zyt1;zyt2,where zijk represents the overall value of decision maker k choosing strategy ASi1 under the condition that competitor chooses strategy ASj2.According to the game matrix, the Nash equilibrium point is determined by game theory, i.e.,(19a)maxz1p,q∗=∑i=1ypi∑j=1tqj∗zij1,maxz2p∗,q=∑j=1tqj∑i=1ypi∗zij2,(19b)s.t.∑i=1ypi=1∑j=1tqj=1pi,qj≥0.By solving the objective programming models (19a) and (19b), the Nash equilibrium point p∗,q∗ of competitive multiattribute group decision-making problem is obtained under the mixed strategy. The Nash equilibrium of mixed strategy refers to the situation that manufacturers are trying to guess each other first and trying not to let the other guess his or her strategy (e.g., mora, poker game, and game). This paper takes the case of literature [2] as the background and uses the data in the literature of [1, 21, 22, 26] to give the following illustrations.


## 5.1. Aggregate Experts’ Evaluation Information of 2-Dimension 2-Tuple Linguistic Variables Based on Evidence Theory

Based on evidence theory, the following steps are employed to aggregate 2-dimension 2-tuple linguistic information given by experts:Step 1. Calculate the basic probability distribution functionSupposeTLij1l=Sij1pl,αij1l,TLij1l∗=Sij1ql∗,αij1l∗, then the basic probability distribution function is as follows:(1)
When0<αij1l<0.5, mlSij1pl=1−αij1l, mlSij1p+1l=αij1l(2)
Whenαij1l=0,mlSij1pl=1(3)
When−0.5<αij1l<0,mlSij1p−1l=αij1l,mlSij1pl=1−αij1lStep 2. Calculate experts’ reliabilityDefinition 7.
Based on the degree of certainty of 2-dimension 2-tuple linguistic information given by experts and their experience, an expert’s reliability is defined as follows:(10)λl=λfldSij1ql∗,αij1l∗,S0∗,0∑k=1hλfkdSij1qk∗,αij1k∗,S0∗,0,l=1,…,h,whereλfl is the coefficient that reflects experts’ experience and preference, 0.9<λfl<1; λl is called the expert reliability; dSij1ql∗,αl∗,S0∗,0 is the distance between the second dimension linguistic information given by experts and S0∗,0 as defined in equation (6); and Δ−1 is the equivalent transformation function between 2-tuple linguistic information and the numerical value as defined in equations (2) and (3).With respect to the different values ofSql∗, some special cases of λl are shown as follows:(1)
WhenSij1ql∗,αij1l∗=S0∗,0,λl=0(2)
WhenSij1ql∗,αij1l∗≠S0∗,0 and ∀w≠l,1≤w≤hSij1qw∗,αij1w∗=S0∗,0,λl=1Step 3. Aggregate experts’ information
This paper applies evidence theory to aggregate experts’ information and proposes a new evidence combination rule as follows:(11)mij1Sij1k=∑Sij1p1∩Sij1p2∩⋯∩Sij1ph=Sij1km1Sij1p1m2Sij1p2⋯mhSij1ph+K∑l=1hλlmlSij1kl,where K=∑Sij1p1∩Sij1p2∩⋯∩Sij1ph=ϕm1Sij1p1m2Sij1p2⋯mhSij1ph and mij1Sij1k represents the basic credibility that the evaluation value of alternative ASi1 is Sij1k against attribute Cj1.Theorem 1.
LetTLij1l=Sij1pl,αij1l,i=1,2,…,y,l=1,2,…,y,j=1,2,…,n,p=0,1,…,g1 be 2-tuple linguistic information given by experts. When h>2 and ∀l∈1,2,…,hΔ−1Sij1pl,αij1l∈p,p+1 or h=2 and Δ−1Sij1p1,0=Δ−1Sij1p2,0, then the experts’ evaluation information aggregated by evidence theory can be expressed as(12)mij1Sij1k=∑Sij1p1∩Sij1p2∩⋯∩Sij1ph=Sij1km1Sij1p1m2Sij1p2⋯mhSij1ph,where Δ−1 is the equivalent transformation function between 2-tuple linguistic information and the numerical value defined in equations (2) and (3).Proof:.
(1)
Forh>2, since ∀l∈1,2,…,hΔ−1Sij1pl,αij1l∈p,p+1then∀l∈1,2,…,h∩Sij1pl≠ϕthenK = 0thus equation (12) holds(2)
Forh=2, since Δ−1Sij1p1,0=Δ−1Sij1p2,0thenSij1p1∩Sij1p2≠ϕthenK = 0thus equation (12) holds
When there are more than two experts and their evaluation information are aggregated between the same two standard linguistic term sets or there are two experts and their evaluation information are identical, there are no conflicts between the experts. In this case, equation (12) is equivalent to equation (7).Theorem 2.
LetTLij1l=Sij1pl,αij1l,l=1,2,…,h,i=1,2,…,y,j=1,2,…,n,p=0,1,…,g1 be 2-tuple linguistic information given by experts. When ∀l∈1,2,…,hαij1l=0 and Sij1p1≠Sij1p2≠⋯≠Sij1ph, then the experts’ evaluation information aggregated by evidence theory can be expressed as(13)mijSij1k=∑l=1hλlmlSij1kl.Proof:.
When∀l∈1,2,…,hαij1l=0 and Sij1p1≠Sij1p2≠⋯≠Sij1ph, since ∀l∈1,2,…,h∩Sij1pl=ϕthenK = 1 and ∑Sij1p1∩Sij1p2∩⋯∩Sij1ph=Sij1km1Sij1p1m2Sij1p2⋯mhSij1ph=0thus equation (13) holds.WhenK = 1, equations (7) and (8) cannot be used to aggregate the experts’ evaluation information.


## 5.2. Integrate Attribute Values of Alternatives Based on Evidence Theory

On the basis of Section5.1, the integration steps of attribute values are given as follows:Step 1. Calculate attribute reliabilityThe reliability of attributes should be related to the consistency of experts. The higher the consistency of experts, the higher the reliability of attributes. Firstly, the expert consistency coefficient is calculated, and then the expert consistency coefficient is used to measure the reliability of attributes. The distance matrix between the evaluation information of experts and the standard linguistic sets of the first dimension is established as follows:(14)Dj=d11d12⋯d1g1+1d21d22⋯d2g1+1⋯⋯⋯⋯dh1dh2⋯dhg1+1,wheredlθ=∑i=1ydSij1pl,αij1l,Sθ−1,0Sij1pl,Sθ−1∈TS represents the distance between the first dimension linguistic evaluation information given by experts and the standard linguistic information Sθ−1Sθ−1∈TS.Definition 8.
Letγj=1/minθ∑i=1hdlθ,j=1,2,…,n be expert consistency coefficient; then, the attribute reliability is defined as follows:(15)ωj=γj∑ε=1nγε,j=1,2,...,n.Obviously,0<ωj<1.Step 2. Aggregate attribute values
According to formula (11), attribute values are integrated as(16)mi1Si1G=∑Si11k∩Si21k∩⋯∩Sin1k=Si1Gmi11Si11kmi21Si21k⋯min1Sin1k+K∑j=1nωjmij1Sij1G,where mi1Si1G represents the basic credibility when the evaluation value of alternative ASi1 is SG and K=∑Si11k∩Si21k∩⋯∩Sin1k=ϕmi11Si11kmi21Si21k⋯min1Sin1k.


## 5.3. Determine Nash Equilibrium Point

According to the discussions in Sections5.1 and 5.2, the basic probability distribution in each alternative can be obtained. Thus, the overall values of every alternative are obtained as follows:(17)zi1=∑G=0g1mi1Si1GuSi1G,i=1,2,…,y.Therefore, the overall value of manufacturerA choosing different strategies under the condition that manufacturer B chooses different strategies is obtained. Under the condition that manufacturer A chooses different strategies, manufacturer B chooses the same method to determine the overall values of different strategies.According to the overall values determined by formula (17), the game matrix is constructed as follows:(18)Z=z111;z112z121;z122⋯z1t1;z1t2z211;z212z221;z222⋯z2t1;z2t2⋯⋯⋯⋯zy11;zy12zy21;zy22⋯zyt1;zyt2,where zijk represents the overall value of decision maker k choosing strategy ASi1 under the condition that competitor chooses strategy ASj2.According to the game matrix, the Nash equilibrium point is determined by game theory, i.e.,(19a)maxz1p,q∗=∑i=1ypi∑j=1tqj∗zij1,maxz2p∗,q=∑j=1tqj∑i=1ypi∗zij2,(19b)s.t.∑i=1ypi=1∑j=1tqj=1pi,qj≥0.By solving the objective programming models (19a) and (19b), the Nash equilibrium point p∗,q∗ of competitive multiattribute group decision-making problem is obtained under the mixed strategy. The Nash equilibrium of mixed strategy refers to the situation that manufacturers are trying to guess each other first and trying not to let the other guess his or her strategy (e.g., mora, poker game, and game). This paper takes the case of literature [2] as the background and uses the data in the literature of [1, 21, 22, 26] to give the following illustrations.


## 6. Illustrations

Consider a duopoly market where two oligarchs produce and sell two identical products, and each oligarch’s market demand curve is open to the other. Their combined output determines the market price of the product, and the oligarchs influence each other. Denote the two oligarchs as manufacturerA and manufacturer B. It is noticed that the decisions made by manufacturer A affect the decisions made by manufacturer B. In addition, the decisions made by manufacturer A are influenced by decisions made by manufacturer B.Now, the research and development department of manufacturerA has discovered a new technology that can greatly improve the production efficiency of products. Faced with this situation, manufacturers A and B must reevaluate and adjust their respective market strategies. Suppose that manufacturer A could employ three strategies: to keep the original market strategy unchanged (AS1), reduce the cost with new technology and take the initiative to reduce the price (AS2), and reduce the unit cost with new technology but keep the original output and price unchanged (AS3). Manufacturer B can also employ three strategies as to increase R&D investment to achieve technological breakthrough as soon as possible (BS1), reduce the market size to maintain the market price (BS2), and maintain the current production scale and reduce the price (BS3).The attributes for evaluations include sustainability, future development, and benefits. Each side invited three experts for evaluation and supposed thatλ1=0.92,0.93,0.95,λ2=0.91,0.94,0.95. The standard linguistic term set of the first dimension is(20)TS=S0=verypoor,S1=poor,S2=littlepoor,S3=general,S4=littlegood,S5=good,S6=verygood.The standard linguistic term set of the second dimension is(21)TS∗=S0∗=verypoor,S1∗=poor,S2∗=littlepoor,S3∗=general,S4∗=littlegood,S5∗=good,S6∗=verygood,uTSis0,0.1,0.3,0.6,0.7,0.9,1.Under the condition that manufacturerB chooses each strategy, for the strategies chosen by manufacturer A, experts give their evaluation information, as shown in Tables 1–3, respectively.Table 1
Experts’ evaluation information of every strategy for manufacturerA under the condition that manufacturer B chooses strategy BS1.
Expert 1
Expert 2
Expert 3
C
1
C
2
C
3
C
1
C
2
C
3
C
1
C
2
C
3
AS
1
S
2
,
−
0.2
,
S
3
∗
,
0.3
S
1
,
−
0.05
,
S
3
∗
,
0.2
S
2
,
−
0.4
,
S
3
∗
,
0
S
1
,
−
0.1
,
S
3
∗
,
−
0.15
S
2
,
−
0.2
,
S
3
∗
,
0.35
S
1
,
0.35
,
S
2
∗
,
0.4
S
2
,
−
0.15
,
S
3
∗
,
−
0.3
S
1
,
0.1
,
S
3
∗
,
−
0.5
S
2
,
0
,
S
3
∗
,
−
0.35
AS
2
S
2
,
0
,
S
3
∗
,
−
0.45
S
1
,
0
,
S
3
∗
,
0.35
S
2
,
0
,
S
3
∗
,
0.05
S
3
,
−
0.4
,
S
2
∗
,
0.398
S
2
,
0.24
,
S
2
∗
,
0.484
S
3
,
−
0.4
,
S
2
∗
,
0.484
S
2
,
0.12
,
S
4
∗
,
0
S
2
,
0.24
,
S
4
∗
,
0
S
3
,
−
0.24
,
S
4
∗
,
0
AS
3
S
2
,
0.36
,
S
4
∗
,
0
S
3
,
−
0.16
,
S
4
∗
,
0
S
3
,
−
0.16
,
S
4
∗
,
0
S
3
,
0.6
,
S
4
∗
,
0
S
3
,
0.32
,
S
3
∗
,
−
0.214
S
3
,
0.24
,
S
3
∗
,
−
0.214
S
3
,
0.08
,
S
2
∗
,
−
0.111
S
4
,
−
0.4
,
S
2
∗
,
−
0.111
S
4
,
−
0.32
,
S
2
∗
,
−
0.111Table 2
Experts’ evaluation information of every strategy for manufacturerA under the condition that manufacturer B chooses strategy BS2.
Expert 1
Expert 2
Expert 3
C
1
C
2
C
3
C
1
C
2
C
3
C
1
C
2
C
3
AS
1
S
5
,
−
0.32
,
S
2
∗
,
0.398
S
5
,
−
0.32
,
S
4
∗
,
0
S
4
,
−
0.4
,
S
2
∗
,
0.398
S
4
,
0.32
,
S
4
∗
,
0
S
4
,
0.16
,
S
2
∗
,
0.398
S
5
,
−
0.16
,
S
4
∗
,
0
S
4
,
−
0.24
,
S
2
∗
,
0.484
S
4
,
0.36
,
S
3
∗
,
−
0.214
S
4
,
−
0.12
,
S
2
∗
,
−
0.484
AS
2
S
4
,
0.28
,
S
3
∗
,
−
0.214
S
4
,
−
0.32
,
S
4
∗
,
0
S
4
,
0.4
,
S
4
∗
,
0
S
4
,
−
0.08
,
S
2
∗
,
−
0.111
S
5
,
−
0.08
,
S
4
∗
,
0
S
3
,
0.4
,
S
4
∗
,
0
S
4
,
0.4
,
S
4
∗
,
0
S
4
,
0.28
,
S
4
∗
,
0
S
4
,
−
0.21
,
S
3
∗
,
0
AS
3
S
4
,
0.21
,
S
3
∗
,
0
S
5
,
−
0.19
,
S
3
∗
,
0
S
5
,
−
0.19
,
S
3
∗
,
0
S
4
,
−
0.4
,
S
3
∗
,
0
S
4
,
0.01
,
S
3
∗
,
0
S
5
,
−
0.19
,
S
3
∗
,
0
S
5
,
−
0.19
,
S
3
∗
,
0
S
4
,
0.4
,
S
3
∗
,
0
S
5
,
−
0.2
,
S
3
∗
,
0Table 3
Experts’ evaluation information of every strategy for manufacturerA under the condition that manufacturer B chooses strategy BS3.
Expert 1
Expert 2
Expert 3
C
1
C
2
C
3
C
1
C
2
C
3
C
1
C
2
C
3
AS
1
S
3
,
0.21
,
S
2
∗
,
0
S
4
,
0
,
S
3
∗
,
0
S
4
,
0
,
S
2
∗
,
0
S
4
,
0.4
,
S
2
∗
,
0
S
4
,
−
0.19
,
S
2
∗
,
0
S
4
,
0.2
,
S
3
∗
,
0
S
4
,
0.19
,
S
2
∗
,
0
S
5
,
−
0.2
,
S
3
∗
,
0
S
5
,
−
0.2
,
S
3
∗
,
0
AS
2
S
3
,
−
0.2
,
S
2
∗
,
0
S
3
,
0.2
,
S
2
∗
,
0
S
3
,
−
0.01
,
S
2
∗
,
0
S
3
,
0.19
,
S
3
∗
,
0
S
4
,
0.4
,
S
2
∗
,
0
S
5
,
0.2
,
S
3
∗
,
0
S
4
,
0
,
S
3
∗
,
0
S
4
,
0.4
,
S
3
∗
,
0
S
5
,
−
0.2
,
S
2
∗
,
0
AS
3
S
5
,
0.01
,
S
3
∗
,
0
S
5
,
−
0.19
,
S
3
∗
,
0
S
5
,
0.2
,
S
3
∗
,
0
S
3
,
−
0.19
,
S
3
∗
,
0
S
3
,
0.4
,
S
2
∗
,
0
S
3
,
0.01
,
S
2
∗
,
0
S
4
,
−
0.19
,
S
3
∗
,
0
S
4
,
0.39
,
S
3
∗
,
0
S
5
,
−
0.02
,
S
4
∗
,
0Under the condition that manufacturerA chooses each strategy, for the strategies chosen by manufacturer B, experts give their evaluation information, as shown in Tables 4–6, respectively.Table 4
Experts’ evaluation information of every strategy for manufacturerB under the condition that manufacturer A chooses strategy AS1.
Expert 1
Expert 2
Expert 3
C
1
C
2
C
3
C
1
C
2
C
3
C
1
C
2
C
3
BS
1
S
3
,
−
0.4
,
S
2
∗
,
0
S
3
,
0.4
,
S
3
∗
,
0
S
3
,
−
0.19
,
S
2
∗
,
0
S
4
,
−
0.39
,
S
3
∗
,
0
S
4
,
0.3182
,
S
3
∗
,
0
S
5
,
−
0.4142
,
S
2
∗
,
0
S
5
,
−
0.1892
,
S
2
∗
,
0
S
5
,
0
,
S
2
∗
,
0
S
4
,
−
0.4495
,
S
3
∗
,
0
BS
2
S
4
,
−
0.2795
,
S
4
∗
,
0
S
4
,
−
0.0598
,
S
1
∗
,
0
S
4
,
0
,
S
2
∗
,
0
S
4
,
0.1388
,
S
2
∗
,
0
S
4
,
0.3182
,
S
1
∗
,
0
S
4
,
−
0.2795
,
S
4
∗
,
0
S
4
,
0.1388
,
S
3
∗
,
0
S
4
,
−
0.3784
,
S
4
∗
,
0
S
4
,
−
0.2134
,
S
4
∗
,
0
BS
3
S
4
,
0
,
S
2
∗
,
0
S
4
,
0.3182
,
S
3
∗
,
0
S
4
,
0.4349
,
S
3
∗
,
0
S
5
,
−
0.4142
,
S
2
∗
,
0
S
3
,
−
0.3437
,
S
4
∗
,
0
S
3
,
−
0.1623
,
S
3
∗
,
0
S
3
,
0.0093
,
S
3
∗
,
0
S
3
,
0.1716
,
S
3
∗
,
0
S
3
,
0.0093
,
S
3
∗
,
0Table 5
Experts’ evaluation information of every strategy for manufacturerB under the condition that manufacturer A chooses strategy AS2.
Expert 1
Expert 2
Expert 3
C
1
C
2
C
3
C
1
C
2
C
3
C
1
C
2
C
3
BS
1
S
3
,
0.1716
,
S
4
∗
,
0
S
3
,
0.41
,
S
3
∗
,
0
S
4
,
−
0.4495
,
S
4
∗
,
0
S
4
,
0.2679
,
S
3
∗
,
0
S
4
,
0.4349
,
S
3
∗
,
0
S
4
,
0.4349
,
S
3
∗
,
0
S
4
,
0.4142
,
S
3
∗
,
0
S
3
,
0.087
,
S
4
∗
,
0
S
3
,
0.2892
,
S
1
∗
,
0
BS
2
S
4
,
−
0.4495
,
S
5
∗
,
0
S
4
,
−
0.3784
,
S
4
∗
,
0
S
4
,
−
0.2795
,
S
1
∗
,
0
S
4
,
−
0.2134
,
S
1
∗
,
0
S
4
,
0
,
S
4
∗
,
0
S
4
,
0.4349
,
S
1
∗
,
0
S
4
,
0.3182
,
S
3
∗
,
0
S
4
,
0.1388
,
S
1
∗
,
0
S
3
,
−
0.0801
,
S
4
∗
,
0
BS
3
S
3
,
0.087
,
S
4
∗
,
0
S
3
,
0.2892
,
S
2
∗
,
0
S
3
,
0.4851
,
S
3
∗
,
0
S
4
,
−
0.3784
,
S
3
∗
,
0
S
4
,
−
0.2134
,
S
2
∗
,
0
S
2
,
0.064
,
S
4
∗
,
0
S
2
,
0.2776
,
S
3
∗
,
0
S
3
,
0.2892
,
S
1
∗
,
0
S
4
,
−
0.2795
,
S
2
∗
,
0Table 6
Experts’ evaluation information of every strategy for manufacturerB under the condition that manufacturer A chooses strategy AS3.
Expert 1
Expert 2
Expert 3
C
1
C
2
C
3
C
1
C
2
C
3
C
1
C
2
C
3
BS
1
S
4
,
0.1388
,
S
3
∗
,
0
S
4
,
0.1388
,
S
4
∗
,
0
S
4
,
0.2217
,
S
3
∗
,
0
S
4
,
0.4349
,
S
3
∗
,
0
S
5
,
−
0.4953
,
S
3
∗
,
0
S
5
,
−
0.4142
,
S
3
∗
,
0
S
5
,
−
0.1892
,
S
3
∗
,
0
S
3
,
0.3679
,
S
2
∗
,
0
S
3
,
0.4851
,
S
2
∗
,
0
BS
2
S
3
,
0.087
,
S
2
∗
,
0
S
3
,
0.2168
,
S
2
∗
,
0
S
3
,
−
0.2237
,
S
2
∗
,
0
S
3
,
−
0.0801
,
S
2
∗
,
0
S
2
,
0.4324
,
S
4
∗
,
0
S
3
,
−
0.4087
,
S
2
∗
,
0
S
3
,
−
0.4087
,
S
2
∗
,
0
S
3
,
−
0.2237
,
S
3
∗
,
0
S
3
,
0
,
S
4
∗
,
0
BS
3
S
4
,
0.2217
,
S
3
∗
,
0
S
4
,
0.3182
,
S
2
∗
,
0
S
4
,
0.2217
,
S
3
∗
,
0
S
3
,
0.3182
,
S
3
∗
,
0
S
4
,
0.4349
,
S
2
∗
,
0
S
5
,
−
0.4953
,
S
3
∗
,
0
S
5
,
−
0.4142
,
S
3
∗
,
0
S
5
,
−
0.3161
,
S
2
∗
,
0
S
5
,
−
0.3161
,
S
2
∗
,
0The results obtained by aggregating the information of experts are shown in Tables7–12. The results obtained by integrating the attribute values are shown in Tables 13–18.Table 7
The aggregation results of experts’ evaluation information for manufacturerA under the condition that manufacturer B chooses strategy BS1.
C
1
C
2
C
3
AS
1
m
S
0
=
0.0312
,
m
S
1
=
0.4253
,
m
S
2
=
0.5435
m
S
1
=
0.7146
,
m
S
2
=
0.2854
m
S
1
=
0.2688
,
m
S
2
=
0.7312
AS
2
m
S
2
=
0.8612
,
m
S
3
=
0.1388
m
S
1
=
0.2614
,
m
S
2
=
0.5614
m
S
3
=
0.1772
m
S
2
=
0.5667
,
m
S
3
=
0.4333
AS
3
m
S
3
=
0.682
,
m
S
4
=
0.318
m
S
2
=
0.0563
,
m
S
3
=
0.7614
m
S
4
=
0.1823
m
S
3
=
0.6986
,
m
S
4
=
0.3014Table 8
The aggregation results of experts’ evaluation information for manufacturerA under the condition that manufacturer B chooses strategy BS2.
C
1
C
2
C
3
AS
1
m
S
3
=
0.057
,
m
S
4
=
0.6719
,
m
S
5
=
0.272
m
S
4
=
0.6094
,
m
S
5
=
0.3906
m
S
3
=
0.1288
,
m
S
4
=
0.5259
m
S
5
=
0.3453
AS
2
m
S
3
=
0.0104
,
m
S
4
=
0.8235
m
S
5
=
0.1661
m
S
3
=
0.0955
,
m
S
4
=
0.5092
m
S
5
=
0.3953
m
S
3
=
0.2238
,
m
S
4
=
0.6598
m
S
5
=
0.1164
AS
3
m
S
3
=
0.1209
,
m
S
4
=
0.5663
,
m
S
5
=
0.3128
m
S
4
=
0.6386
,
m
S
5
=
0.3614
m
S
4
=
0.0977
,
m
S
5
=
0.9023Table 9
The aggregation results of experts’ evaluation information for manufacturerA under the condition that manufacturer B chooses strategy BS3.
C
1
C
2
C
3
AS
1
m
S
3
=
0.2331
,
m
S
4
=
0.5897
,
m
S
5
=
0.1772
m
S
3
=
0.0396
,
m
S
4
=
0.7046
m
S
5
=
0.2558
m
S
4
=
0.6813
,
m
S
5
=
0.3187
AS
2
m
S
2
=
0.0492
,
m
S
3
=
0.4989
,
m
S
4
=
0.4519
m
S
3
=
0.2086
,
m
S
4
=
0.5245
m
S
5
=
0.2669
m
S
2
=
0.0028
,
m
S
3
=
0.279
m
S
4
=
0.5163
,
m
S
5
=
0.1164
m
S
6
=
0.0855
AS
3
m
S
3
=
0.2742
,
m
S
4
=
0.309
,
m
S
5
=
0.3658
m
S
2
=
0.0473
,
m
S
6
=
0.0037
m
S
4
=
0.4302
,
m
S
5
=
0.4273
m
S
3
=
0.1425
m
S
4
=
0.0112
,
m
S
5
=
0.7045
m
S
3
=
0.2187
,
m
S
6
=
0.0656Table 10
The aggregation results of experts’ evaluation information for manufacturerB under the condition that manufacturer A chooses strategy AS1.
C
1
C
2
C
3
BS
1
m
S
3
=
0.3351
,
m
S
4
=
0.318
,
m
S
5
=
0.2356
m
S
2
=
0.1113
m
S
3
=
0.2199
,
m
S
4
=
0.4046
m
S
5
=
0.3755
m
S
4
=
0.3584
,
m
S
5
=
0.1681
m
S
2
=
0.0528
,
m
S
3
=
0.4207
BS
2
m
S
5
=
0.0365
,
m
S
3
=
0.0566
,
m
S
4
=
0.9069
m
S
3
=
0.1589
,
m
S
4
=
0.8093
m
S
5
=
0.0318
m
S
4
=
0.914
,
m
S
3
=
0.086
BS
3
m
S
3
=
0.4294
,
m
S
4
=
0.4031
,
m
S
5
=
0.1675
m
S
4
=
0.2286
,
m
S
5
=
0.0845
m
S
3
=
0.5297
,
m
S
2
=
0.1572
m
S
4
=
0.1868
,
m
S
5
=
0.1413
m
S
3
=
0.6174
,
m
S
2
=
0.0545Table 11
The aggregation results of experts’ evaluation information for manufacturerB under the condition that manufacturer A chooses strategy AS2.
C
1
C
2
C
3
BS
1
m
S
3
=
0.3
,
m
S
4
=
0.5074
,
m
S
5
=
0.1926
m
S
3
=
0.5324
,
m
S
4
=
0.3391
m
S
5
=
0.1285
m
S
4
=
0.5655
,
m
S
5
=
0.1506
m
S
3
=
0.2839
BS
2
m
S
5
=
0.0913
,
m
S
3
=
0.1677
,
m
S
4
=
0.7409
m
S
3
=
0.0767
,
m
S
4
=
0.916
m
S
5
=
0.0073
m
S
4
=
0.2101
,
m
S
3
=
0.6637
m
S
2
=
0.0538
,
m
S
5
=
0.0724
BS
3
m
S
3
=
0.5986
,
m
S
4
=
0.2011
,
m
S
5
=
0.2003
m
S
4
=
0.471
,
m
S
3
=
0.529
m
S
4
=
0.2663
,
m
S
5
=
0.3181
m
S
3
=
0.4156Table 12
The aggregation results of experts’ evaluation information for manufacturerB under the condition that manufacturer A chooses strategy AS3.
C
1
C
2
C
3
BS
1
m
S
2
=
0.0642
,
m
S
3
=
0.2751
,
m
S
4
=
0.4696
m
S
5
=
0.1911
m
S
3
=
0.1211
,
m
S
4
=
0.6844
m
S
5
=
0.1945
m
S
4
=
0.6336
,
m
S
5
=
0.2556
m
S
3
=
0.1108
BS
2
m
S
5
=
0.0142
,
m
S
3
=
0.0833
,
m
S
4
=
0.9025
m
S
3
=
0.7231
,
m
S
4
=
0.0345
m
S
2
=
0.2424
m
S
2
=
0.0848
,
m
S
3
=
0.9152
BS
3
m
S
3
=
0.2054
,
m
S
4
=
0.5516
,
m
S
5
=
0.243
m
S
4
=
0.5281
,
m
S
3
=
0.4719
m
S
4
=
0.5654
,
m
S
5
=
0.4346Table 13
The integration results of the attribute values for manufacturerA under the condition that manufacturer B chooses strategy BS1.
Strategies
The integration results of the attribute values
AS
1
m
S
0
=
0.0098
,
m
S
1
=
0.4475
,
m
S
2
=
0.5427
AS
2
m
S
1
=
0.0522
,
m
S
2
=
0.7606
,
m
S
3
=
0.1872
AS
3
m
S
2
=
0.0097
,
m
S
3
=
0.8026
,
m
S
4
=
0.1877Table 14
The integration results of the attribute values for manufacturerA under the condition that manufacturer B chooses strategy BS2.
Strategies
The integration results of the attribute values
AS
1
m
S
3
=
0.0426
,
m
S
4
=
0.6728
,
m
S
5
=
0.2846
AS
2
m
S
3
=
0.0693
,
m
S
4
=
0.758
,
m
S
5
=
0.1727
AS
3
m
S
3
=
0.0408
,
m
S
4
=
0.4366
,
m
S
5
=
0.5226Table 15
The integration results of the attribute values for manufacturerA under the condition that manufacturer B chooses strategy BS3.
Strategies
The integration results of the attribute values
AS
1
m
S
3
=
0.0628
,
m
S
4
=
0.7483
,
m
S
5
=
0.1889
AS
2
m
S
2
=
0.0155
,
m
S
3
=
0.3343
,
m
S
4
=
0.3719
,
m
S
5
=
0.2559
,
m
S
6
=
0.0224
AS
3
m
S
2
=
0.013
,
m
S
3
=
0.1886
,
m
S
4
=
0.2457
,
m
S
5
=
0.5359
,
m
S
6
=
0.0168Table 16
The integration results of the attribute values for manufacturerB under the condition that manufacturer A chooses strategy AS1.
Strategies
The integration results of the attribute values
BS
1
m
S
2
=
0.0512
,
m
S
3
=
0.3258
,
m
S
4
=
0.3722
,
m
S
5
=
0.2508
BS
2
m
S
3
=
0.0306
,
m
S
4
=
0.9461
,
m
S
5
=
0.0233
BS
3
m
S
2
=
0.0576
,
m
S
3
=
0.579
,
m
S
4
=
0.2504
,
m
S
5
=
0.113Table 17
The integration results of the attribute values for manufacturerB under the condition that manufacturer A chooses strategy AS2.
Strategies
The integration results of the attribute values
BS
1
m
S
3
=
0.3799
,
m
S
4
=
0.4846
,
m
S
5
=
0.1355
BS
2
m
S
2
=
0.0128
,
m
S
3
=
0.2355
,
m
S
4
=
0.7082
,
m
S
5
=
0.0435
BS
3
m
S
3
=
0.5692
,
m
S
4
=
0.3044
,
m
S
5
=
0.1264Table 18
The integration results of the attribute values for manufacturerB under the condition that manufacturer A chooses strategy AS3.
Strategies
The integration results of the attribute values
BS
1
m
S
2
=
0.0169
,
m
S
3
=
0.1364
,
m
S
4
=
0.6699
,
m
S
5
=
0.1768
BS
2
m
S
2
=
0.2223
,
m
S
3
=
0.5953
,
m
S
4
=
0.1779
,
m
S
5
=
0.0045
BS
3
m
S
3
=
0.1887
,
m
S
4
=
0.6227
,
m
S
5
=
0.1886The overall value of each alternative is calculated and the results are as follows.Under the condition that manufacturerB chooses strategy BS1:The overall value ofAS1:0.0098∗0+0.4475∗0.1+0.5427∗0.3=0.2076The overall value ofAS2:0.0522∗0.1+0.7606∗0.3+0.1872∗0.6=0.3457The overall value ofAS3:0.0097∗0.3+0.8026∗0.6+0.1877∗0.7=0.6159Under the condition that manufacturerB chooses strategy BS2:The overall value ofAS1:0.0426∗0.6+0.6728∗0.7+0.2846∗0.9=0.7527The overall value ofAS2:0.0693∗0.6+0.758∗0.7+0.1727∗0.9=0.7276The overall value ofAS3:0.0408∗0.6+0.4366∗0.7+0.5226∗0.9=0.8004Under the condition that manufacturer B chooses strategyBS3:The overall value ofAS1:0.0628∗0.6+0.7483∗0.7+0.1889∗0.9=0.7315The overall value ofAS2:0.0155∗0.3+0.3343∗0.6+0.3719∗0.7+0.2559∗0.9+0.0224∗1=0.7183The overall value ofAS3:0.013∗0.3+0.1886∗0.6+0.2457∗0.7+0.5359∗0.9+0.0168∗1=0.7882Under the condition that manufacturerA chooses strategy AS1:The overall value ofBS1:0.0512∗0.3+0.3258∗0.6+0.3722∗0.7+0.2508∗0.9=0.6971The overall value ofBS2:0.0306∗0.6+0.9461∗0.7+0.0233∗0.9=0.7016The overall value ofBS3:0.0576∗0.3+0.579∗0.6+0.2504∗0.7+0.113∗0.9=0.6417Under the condition that manufacturerA chooses strategy AS2:The overall value ofBS1:0.3799∗0.6+0.4846∗0.7+0.1355∗0.9=0.6891The overall value ofBS2:0.0128∗0.3+0.2355∗0.6+0.7082∗0.7+0.0435∗0.9=0.68The overall value ofBS3:0.5692∗0.6+0.3044∗0.7+0.1264∗0.9=0.6684Under the condition that manufacturerA chooses strategy AS3:The overall value ofBS1:0.0169∗0.3+0.1364∗0.6+0.6699∗0.7+0.1768∗0.9=0.715The overall value ofBS2:0.3417∗0.3+0.5953∗0.6+0.2974∗0.7+0.0045∗0.9=0.5525The overall value ofBS3:0.1887∗0.6+0.6227∗0.7+0.1886∗0.9=0.7188According to the above results, the bivariate decision matrix is constructed as follows:(22)Z=0.2076;0.69710.7527;0.68910.7315;0.7150.3457;0.70160.7276;0.680.7182;0.55250.6159;0.64170.8004;0.66840.7882;0.7188.Obviously, there is a pure strategy Nash equilibrium in this caseAS3,BS3. The corresponding revenue is 0.7882,0.7188.


## 7. Conclusions

In this paper, based on game theory and evidence theory, an approach is proposed for the competitive multiattribute group decision-making problem of the game between manufacturers, with the attribute values being the 2-dimension 2-tuple linguistic evaluation information. Firstly, in order to aggregate experts’ evaluation information, a method is proposed to determine the 2-dimension 2-tuple linguistic basic probability distribution function, and the experts’ reliability is defined. After that, the evidence theory is applied again to conduct secondary aggregation with attribute values as the evidence source, and the consistency degree of experts is taken as the criterion to measure the reliability of attribute evidence source. Finally, the Nash equilibrium point of competitive multiattribute group decision-making problem is determined by game theory.Compared with the traditional competitive multiattribute group decision-making problem, the 2-dimension 2-tuple linguistic competitive multiattribute group decision-making problem has higher certainty and accuracy, which makes the decision results more reliable and have very important significance.In addition, in the course of aggregating experts’ evaluation information, the proposed approach could solve the conflict of evidence, that is, from the subjective dimension and the objective dimension, to determine the reliability of experts’ evidence. The subjective dimension uses background and authority of experts [30], while the objective dimension uses the second semantic information of the degree of certainty, which overcomes the problem of too much subjectivity when the original evidence theory solves conflicts.



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*Source: 8389035-2019-11-11.xml*