# Nucleolus of Vague Payoff Cooperative Game

Authors: Huimin Xiao; Shiwei Li; Youlei Xu; Chuangchuang Qiu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper focuses on the problem of cooperative game with payoff of vague value and its nucleolus. Firstly, the paper defines the score function and accuracy function of vague sets and the method for ranking of vague sets and proposes the concept of core and nucleolus of vague payoff cooperative game. Based on this, the model of vague payoff cooperative game is built. Then, the relationship between the core and the nucleolus of vague payoff cooperative game is further discussed, and the existence and unique characteristics of the nucleolus are proved. We use the ranking method defined in the paper to transform the problem of finding the nucleolus solution into a nonlinear programming problem. Finally, the paper verifies the feasibility and effectiveness of the method for finding the nucleolus with an experimental analysis.


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## Body

## 1. Introduction

In the cooperative game, the most concerned problem is how to achieve a fair and reasonable distribution of the benefits of the coalition, that is, to find the solution of the cooperative game. In the process of solving this problem, scholars have given various forms of solutions; the set form of solutions mainly includes the core [1], nucleolus [2, 3], kernel [4], stable set [5–7], and so on; the single-valued form of solutions mainly includes Shapley value [8–10], Owen value [10, 11], etc. Each solution satisfies certain rationality principles and rational behavior. In 1969, Schmeidler [3] proposed the nucleolus solution, and the greatest advantage of nucleolus solution is existence and uniqueness. That is, the nucleolus is composed of a unique distribution and it always exists. There are various uncertainties in real life. When the players form a coalition, the players in game not only have two options of participating in the coalition and not but also participate in the coalition with a certain degree of membership between 0 and 1. The income of the coalition is no longer a specific real number, sometimes an interval value. Therefore, fuzzy cooperative game is proposed, and compared with classical cooperative games, fuzzy cooperative game can solve various practical problems better.In 1981, Aubin [12] proposed the fuzzy cooperative game, which has been widely studied and applied in various fields of modern society. At present, the research on fuzzy cooperative games mainly focuses on the fuzzy information integration theory of Xu et al. [13] and intuitionistic fuzzy cooperative games of Guo and Gao and Priyadharsini and Balasubramaniam [14, 15]. In 1993, Gau and Buehrer [16] proposed the concept of vague set after analyzing the characteristics of fuzzy set, emphasizing that the characteristic of vague set is that it could provide the degree of supporting evidence and opposing evidence at the same time, and it can express the degree of hesitation, that is, a state between supporting and opposing. The characteristics of vague set make it have unique advantages in expressing the uncertainty of real information. Vague set is more suitable to express the fuzzy characteristics of research objects than other fuzzy sets, and it is suitable as a tool to solve uncertain problems. At present, the research on vague set mainly focuses on the application of vague set theory to multiattribute decision-making problems [17–19] and the application of it to the field of multicriteria decision making [20–22] studied by Zeng and Ye et al. The main content of this paper is the cooperative game problem with payoff value of vague value and its nucleolus solution.


## 2. Preliminaries

### 2.1. Vague Set Theory

Definition 1 (see [16]).
LetU be the universe of discourse, with a generic element of U denoted by x. A vague set A in the universe of discourse U is characterized by a truth-membership function tA and a false-membership function fA:(1)tA:U⟶0,1,fA:U⟶0,1,where tAx is a lower bound on the grade of membership of x derived from the evidence for x and fAx is a lower bound on the negation of x derived from the evidence against x. Both tAx and fAx associate a real number in the interval [0, 1] with each point in U, where the functions tAx and fAx are constrained by the condition tAx+fAx≤1. The grade of membership of x in the vague set A is bounded to a subinterval tAx,1−fAx of 0,1. In other words, the vague set tAx,1−fAx indicates that the exact grade of membership μAx of x may be unknown, but it is bounded by tAx≤μAx≤1−fAx. The unknown degree or hesitancy degree of the vague value x is denoted by πAx and is defined by πAx=1−tAx−fAx, 0≤πAx≤1. If πAx is small, our knowledge about x is relatively precise; if πAx is large, it shows that we have a lot of unknown information about x. If πAx=0 (i.e., 1−fAx=tAx), our knowledge about x is very exact, and the theory reverts back to that of fuzzy sets.
In general, a vague valueα can be written as α=x,t,1−f or be written as t,1−f. A vague set A can be written as A=x,tAx,1−fAx|x∈U. In this paper, all the real-valued vague values in the universe of discourse U are denoted as RVVVU; they are usually abbreviated as RVVV. All the real-valued vague sets in the universe of discourse U are denoted as RVVSU; they are usually abbreviated as RVVS.Definition 2 (see [16]).
LetA and B be two vague sets of the universe U, where A=x,tAx,1−fAx|x∈U and B=x,tBx,1−fBx|x∈U. There are several definitions of operations between Vague sets, including containment, equal, intersection, and union. These definitions are as follows:(1)
Vague setA is contained in the other vague set B, A⊆B, if and only if tAx≤tBx, 1−fAx≤1−tBx.(2)
Two vague setsA and B are equal, written as A=B, if and only if A⊆B and A⊇B, that is, tAx=tBx and 1−fAx=1−tBx.(3)
The intersection of two vague setsA and B is a vague set C, written as C=A∩B, whose truth-membership and false-membership functions are related to those of A and B by(2)tC=mintA,tB,1−fC=min1−fA,1−fB=1−maxfA,fB.(4)
The union of two vague setsA and B is a vague set C, written as C=A∪B, whose truth-membership and false-membership functions are related to those of A and B by(3)tC=maxtA,tB,1−fC=max1−fA,1−fB=1−minfA,fB.


### 2.2. The Ranking Method of Vague Set

There are several representative score functions and accuracy functions of vague sets in the available literature as follows:(1)
Chen and Tan’s method [23]: they proposed score function of vague sets Sα; the score function expression is defined as Sα=tα−fα, Sα∈−1,1, where α=tα,1−fα is a vague value. The larger the value of Sα, the higher the ranking position of α in the vague set. However, it failed to use the score function to compare vague values when the score function values of two or more vague values are equal. For example, assume two vague values α1=0.6,0.9 and α2=0.7,0.8; then, according to equation Sα=tα−fα, we obtain Sα1=0.6−0.1=0.5, Sα2=0.7−0.2=0.5. In this case, the score function fails to compare two vague values.(2)
Hong and Choi’s method [24]: they presented an accuracy function of vague sets Hα to replace the score function Sα of Chen and Tan’s method. The accuracy function Hα is defined as Hα=tα+fα, where Hα∈0,1. The larger the value of Hα, the higher the degree of accuracy in the grades of membership of the vague value α in the vague sets. However, it is not hard to find that the accuracy function Hα sometimes does not work and the result of ranking is inaccurate. For instance, assume two vague values α3=0.6,1 and α4=0.8,0.8; then, according to equation Hα=tα+fα, we obtain Hα3=0.6, Hα4=1, and hence α3<α4. However, in fact, the nonmembership degree of α3 is 0, and people tend to place α3 in front of α4, that is, α3>α4. In this case, the result of ranking is not correct.(3)
Liu et al.’s [25] method: after analyzing the voting model, Liu et al. divided the abstention into three parts: support, opposition, and continued abstention. Then, they mainly proposed an improved score function SLα, which is defined as SLα=tα+1−tα−fαtα=1+παtα. However, if the membership degree of two vague values is both 0, it is impossible to compare the two vague values. For example, assume two vague values α5=0,0.6 and α6=0,0.8; then, according to equation SLα=1+παtα, we obtain SLα5=0, SLα6=0. In this case, the score function fails to compare two vague values.(4)
Ye Jun’s [26] method: Ye Jun proposed score function of vague sets SYα; the score function expression is defined as SYα=tα−fα+μπα, where 0≤μ≤1, and the value of parameter μ depends on the player’s attitude towards unknown information πα. There is no objective evidence for the value of parameter μ, which may lead to unreasonable or inaccurate results.Obviously, the score functionsSα and accuracy function Hα only consider the true membership and the nonmembership. However, they all ignore to consider the influence of the degree of hesitancy. Although Liu and Ye Jun et al. improved the method proposed by Chen and Tan, there are still some problems. In order to overcome the above problems, we propose an improved score function and accuracy function of vague sets as follows.Definition 3.
LetU be the universe of discourse and α=x,t,1−f be a vague value; the score function and the accuracy function are defined as(4)Sα=xt−f+t−fπx,(5)Hα=xt+f+t+fπx.
Based on the above score function and accuracy function, a ranking method of vague sets is proposed. Letα1, α2 be two vague values; the ranking rules are as follows:(1)
IfSα1>Sα2, then α1>α2(2)
IfSα1=Sα2 and Hα1>Hα2, then α1>α2IfHα1<Hα2, then α1<α2IfHα1=Hα2, then α1=α2
In the above ranking method of vague sets, the score function and accuracy function not only consider the degree of true membership and the nonmembership but also consider the influence of degree of the hesitancy. The larger the value of the score function, the higher the ranking position ofα in the vague sets. We need to use accuracy function to obtain the ranking of vague sets when the score function values of two or more vague values are equal. The larger the value of the accuracy function, the higher the ranking position of α in the vague sets.


## 2.1. Vague Set Theory

Definition 1 (see [16]).
LetU be the universe of discourse, with a generic element of U denoted by x. A vague set A in the universe of discourse U is characterized by a truth-membership function tA and a false-membership function fA:(1)tA:U⟶0,1,fA:U⟶0,1,where tAx is a lower bound on the grade of membership of x derived from the evidence for x and fAx is a lower bound on the negation of x derived from the evidence against x. Both tAx and fAx associate a real number in the interval [0, 1] with each point in U, where the functions tAx and fAx are constrained by the condition tAx+fAx≤1. The grade of membership of x in the vague set A is bounded to a subinterval tAx,1−fAx of 0,1. In other words, the vague set tAx,1−fAx indicates that the exact grade of membership μAx of x may be unknown, but it is bounded by tAx≤μAx≤1−fAx. The unknown degree or hesitancy degree of the vague value x is denoted by πAx and is defined by πAx=1−tAx−fAx, 0≤πAx≤1. If πAx is small, our knowledge about x is relatively precise; if πAx is large, it shows that we have a lot of unknown information about x. If πAx=0 (i.e., 1−fAx=tAx), our knowledge about x is very exact, and the theory reverts back to that of fuzzy sets.
In general, a vague valueα can be written as α=x,t,1−f or be written as t,1−f. A vague set A can be written as A=x,tAx,1−fAx|x∈U. In this paper, all the real-valued vague values in the universe of discourse U are denoted as RVVVU; they are usually abbreviated as RVVV. All the real-valued vague sets in the universe of discourse U are denoted as RVVSU; they are usually abbreviated as RVVS.Definition 2 (see [16]).
LetA and B be two vague sets of the universe U, where A=x,tAx,1−fAx|x∈U and B=x,tBx,1−fBx|x∈U. There are several definitions of operations between Vague sets, including containment, equal, intersection, and union. These definitions are as follows:(1)
Vague setA is contained in the other vague set B, A⊆B, if and only if tAx≤tBx, 1−fAx≤1−tBx.(2)
Two vague setsA and B are equal, written as A=B, if and only if A⊆B and A⊇B, that is, tAx=tBx and 1−fAx=1−tBx.(3)
The intersection of two vague setsA and B is a vague set C, written as C=A∩B, whose truth-membership and false-membership functions are related to those of A and B by(2)tC=mintA,tB,1−fC=min1−fA,1−fB=1−maxfA,fB.(4)
The union of two vague setsA and B is a vague set C, written as C=A∪B, whose truth-membership and false-membership functions are related to those of A and B by(3)tC=maxtA,tB,1−fC=max1−fA,1−fB=1−minfA,fB.


## 2.2. The Ranking Method of Vague Set

There are several representative score functions and accuracy functions of vague sets in the available literature as follows:(1)
Chen and Tan’s method [23]: they proposed score function of vague sets Sα; the score function expression is defined as Sα=tα−fα, Sα∈−1,1, where α=tα,1−fα is a vague value. The larger the value of Sα, the higher the ranking position of α in the vague set. However, it failed to use the score function to compare vague values when the score function values of two or more vague values are equal. For example, assume two vague values α1=0.6,0.9 and α2=0.7,0.8; then, according to equation Sα=tα−fα, we obtain Sα1=0.6−0.1=0.5, Sα2=0.7−0.2=0.5. In this case, the score function fails to compare two vague values.(2)
Hong and Choi’s method [24]: they presented an accuracy function of vague sets Hα to replace the score function Sα of Chen and Tan’s method. The accuracy function Hα is defined as Hα=tα+fα, where Hα∈0,1. The larger the value of Hα, the higher the degree of accuracy in the grades of membership of the vague value α in the vague sets. However, it is not hard to find that the accuracy function Hα sometimes does not work and the result of ranking is inaccurate. For instance, assume two vague values α3=0.6,1 and α4=0.8,0.8; then, according to equation Hα=tα+fα, we obtain Hα3=0.6, Hα4=1, and hence α3<α4. However, in fact, the nonmembership degree of α3 is 0, and people tend to place α3 in front of α4, that is, α3>α4. In this case, the result of ranking is not correct.(3)
Liu et al.’s [25] method: after analyzing the voting model, Liu et al. divided the abstention into three parts: support, opposition, and continued abstention. Then, they mainly proposed an improved score function SLα, which is defined as SLα=tα+1−tα−fαtα=1+παtα. However, if the membership degree of two vague values is both 0, it is impossible to compare the two vague values. For example, assume two vague values α5=0,0.6 and α6=0,0.8; then, according to equation SLα=1+παtα, we obtain SLα5=0, SLα6=0. In this case, the score function fails to compare two vague values.(4)
Ye Jun’s [26] method: Ye Jun proposed score function of vague sets SYα; the score function expression is defined as SYα=tα−fα+μπα, where 0≤μ≤1, and the value of parameter μ depends on the player’s attitude towards unknown information πα. There is no objective evidence for the value of parameter μ, which may lead to unreasonable or inaccurate results.Obviously, the score functionsSα and accuracy function Hα only consider the true membership and the nonmembership. However, they all ignore to consider the influence of the degree of hesitancy. Although Liu and Ye Jun et al. improved the method proposed by Chen and Tan, there are still some problems. In order to overcome the above problems, we propose an improved score function and accuracy function of vague sets as follows.Definition 3.
LetU be the universe of discourse and α=x,t,1−f be a vague value; the score function and the accuracy function are defined as(4)Sα=xt−f+t−fπx,(5)Hα=xt+f+t+fπx.
Based on the above score function and accuracy function, a ranking method of vague sets is proposed. Letα1, α2 be two vague values; the ranking rules are as follows:(1)
IfSα1>Sα2, then α1>α2(2)
IfSα1=Sα2 and Hα1>Hα2, then α1>α2IfHα1<Hα2, then α1<α2IfHα1=Hα2, then α1=α2
In the above ranking method of vague sets, the score function and accuracy function not only consider the degree of true membership and the nonmembership but also consider the influence of degree of the hesitancy. The larger the value of the score function, the higher the ranking position ofα in the vague sets. We need to use accuracy function to obtain the ranking of vague sets when the score function values of two or more vague values are equal. The larger the value of the accuracy function, the higher the ranking position of α in the vague sets.


## 3. Model of Vague Payoff Cooperative Game

LetN=1,2,…,n be the set of n players, and the set of all subsets of N is written as PN; then, any element of PN is called a coalition. A two-tuple N,v˜ is called a cooperative game with vague payoffs on a player set N, where v˜ is the vague payoff characteristic function defined on PN, that is, v˜:PN⟶RVVV and v˜Φ=0. We define a cooperative game with vague payoff characteristic function as a vague payoff cooperative game, written as RVVΓ=N,v˜.Vague payoff characteristic function expression is defined asv˜S=vS,tS,1−fS, where vS is the expected payoff value of a coalition S∈PN. The truth-membership function tS indicates the degree of support of the coalition S for the payoff value S for the payoff value vS; fS indicates the degree of opposition of the coalition S for the payoff value vS. In addition, we define a hesitancy function πS and the hesitancy function expression is defined as πS=1−fS−tS, which indicates the degree of hesitancy of the coalition S for the coalition’s payoff value vS.Definition 4.
LetRVVΓ=N,v˜ be a vague payoff cooperative game. There is an n-dimensional vector x˜=x˜1,x˜2,…,x˜n, where x˜i is the vague value, and x˜i satisfies(1)
x˜i≥v˜i, ∀i∈N(2)
∑i∈Nx˜i=v˜N,where x˜ is called the vague payoff imputation of RVVΓ=N,v˜ and player i obtains payoff value x˜i. We denote all vague payoff imputations for cooperative game RVVΓ=N,v˜ written as RVVEv˜.Definition 5.
LetRVVΓ=N,v˜ be a vague payoff cooperative game. The imputation set of RVVΓ=N,v˜ is denoted by X˜N,v˜ and is defined by(6)X˜N,v˜=x˜∈RVVS|∑i∈Nx˜i=v˜N;x˜i≥v˜i;i∈N.Definition 6.
LetRVVΓ=N,v˜ be a vague payoff cooperative game.x˜,y˜∈RVVEv˜, where x˜=x˜1,x˜2,⋯,x˜n and y˜=y˜1,y˜2,…,y˜n are two vague payoffs of cooperative game ∀S∈PN, and S≠Φ. If x˜i and y˜i satisfy the following conditions:(1)
x˜i≥y˜i, ∀i∈S(2)
v˜S≥∑i∈Sx˜i,then the relationship between x˜ and y˜ is called the vague payoff imputation x˜ on S vague dominate y˜, written as x˜≻Sy˜. Vague dominate condition (1) shows that all the players in coalition S agree that x˜ is better than y˜. That is, all the players in coalition S will choose x˜ and reject y˜. Vague dominate condition (2) shows that the players can obtain the imputation that coalition S promised them, and it is not bounced check.Definition 7.
LetRVVΓ=N,v˜ be a vague payoff cooperative game. The set of all imputation on RVVEv˜ that are not vague dominate by other vague payoff imputation is called the core of vague payoff cooperative game and is defined by(7)RVVCv˜=x˜∈RVVEv˜|x˜≻Sy˜,y˜∈RVVEv˜.
Therefore, every imputation of the core of vague payoff cooperative game can be accepted by any coalition. In other words, no coalition can propose a better imputation for himself. In a word,RVVCv˜ as a solution of vague payoff cooperative game is reasonable.Definition 8.
LetRVVΓ=N,v˜ be a vague payoff cooperative game. Simply take any coalition S∈PN, where S≠Φ and x˜∈RVVEv˜. eS,x˜=v˜S−x˜S is the vague excess of coalition S on the imputation x˜. The excess value reflects the attitude of coalition S towards the imputation x˜: the larger eS,x˜ is, the less popular imputation x˜ is with coalition. A fixed imputation x˜∈RVVEv˜; then, it is possible to list 2N vague excesses of coalition N on imputation x˜. The 2N excesses are arranged from the largest to the smallest to obtain an n-dimensional vector θx˜=θ1x˜,θ2x˜,…,θ2Nx˜, where θix˜=eSi,x˜ and i=1,2,…,2N. S1,S2,…,S2N is a permutation of all subsets of coalition N and the permutation is related to x˜. That is, eS1,x˜≥eS2,x˜≥⋯≥eS2N,x˜.
Letλ=λ1,λ2,λ3,…,λn and μ=μ1,μ2,μ3,…,μn be two vectors. If there is 1≤k≤n such that λi=μi and λk<μk, where i=1,2,…,k−1, then the relation between λ and μ is called vector λ is less than vector μ in lexicographic order [27] and is denoted by λ<Lμ.Definition 9.
LetRVVΓ=N,v˜ be a vague payoff cooperative game. Nucleolus of vague payoff cooperative game is the set that contains all imputations that minimize θx˜ in lexicographic order. The nucleolus is written as RVVNuv˜ and defined by(8)RVVNuv˜=x˜∈RVVEv˜|θx˜≤Lθy˜,∀y∈RVVEv˜.Theorem 1.
If the imputation is vague set, thenRVVEv˜ is a bounded closed convex set.Proof.
From Definition4, it is obvious that RVVEv˜ is a bounded closed set. The following proves that RVVEv˜ is a convex set. Simply take any ∀x˜,y˜∈RVVEv˜, where x˜=x˜1,x˜2,…,x˜n,x˜i=xi,txi,1−fxi, y˜=y˜1,y˜2,…,y˜n, y˜i=yi,tyi,1−fyi. Let z˜=z˜1,z˜2,…,z˜n; then, z˜=λx˜+1−λy˜, where λ∈0,1 and z˜i=λx˜i+1−λy˜i.(1)
From the definition of vague payoff imputation,x˜i≥v˜i, y˜i≥v˜i, i∈N. Thus, z˜i=λx˜i+1−λy˜i≥λv˜i+1−λv˜i=v˜i, that is, z˜i≥v˜i.(2)
From the definition of the score function and accuracy function of the vague set, we can obtain(9)∑i=1ntxi−fxi+txi−fxiπxixi=tx−fx+tx−fxπxvN,∑i=1ntxi+fxi+txi+fxiπxixi=tx+fx+tx+fxπxvN,∑i=1ntyi−fyi+tyi−fyiπyiyi=ty−fy+ty−fyπyvN,∑i=1ntyi+fyi+tyi+fyiπyiyi=ty+fy+ty+fyπyvN.
Thus,(10)∑i=1ntzi−fzi+tzi−fziπzizi=∑i=1ntzizi−∑i=1nfzizi+∑i=1ntziziπzi−∑i=1nfziziπzi=∑i=1nλtxixi+1−λtyiyi−∑i=1nλfxixi+1−λfyiyi+∑i=1nλtxixiπxi+1−λtyiyiπyi−∑i=1nλfxixiπxi+1−λfyiyiπyi=λ∑i=1ntxi−fxi+txi−fxiπxixi+1−λ∑i=1ntyi−fyi+tyi−fyiπyiyi=λtx−fx+tx−fxπxvN+1−λty−fy+ty−fyπyvN=λtx+1−λtyvN−λfx+1−λfyvN+λtxπx+1−λtyπyvN−λfxπx+1−λfyπyvN=tzvN−fzvN+tzπzvN−fzπzvN=tz−fz+tz−fzπzvN.
Evidenced by the same token,∑i=1ntzi+fzi+tzi+fziπzizi=tz+fz+tz+fzπzvN; then, z˜∈RVVEv˜. Therefore, RVVEv˜ is a convex set.Theorem 2.
The nucleolus of vague payoff cooperative game is written asRVVNuv˜. The nucleolus RVVNuv˜ has two properties, existence and uniqueness.Proof.
The existence ofRVVNuv˜. Let E0=RVVEv˜; then,(11)E1=x˜∈E0|θ1x˜=miny˜∈E0θ1y˜E2=x˜∈E1|θ2x˜=miny˜∈E1θ2y˜⋮Ek=x˜∈Ek−1|θkx˜=miny˜∈Ek−1θky˜,k=1,2,…,2N.
Thus, we recursively define the sequence of a setE0,E1,…,E2N. Obviously, we can obtain that E0⊇E1⊇…⊇E2N. Because E0 is the nonempty closed bounded subset and θ1x˜ is continuous function, E1 is a nonempty closed bounded subset. Because θ2x˜ is continuous function, then E2 is a nonempty closed bounded subset, and so on; every Ei is a nonempty closed bounded subset, i=1,2,…,2N. So far, we obtain RVVNuv˜=E2N.
In fact, letx˜∈E2N and simply take any y˜∈RVVEv˜. If θx˜≠θy˜, then assume θkx˜ is the first component different from θky˜. That is, where i<k, then θix˜=θiy˜, and θkx˜≠θky˜. Thus, y˜∈Ek−1; according to the expression of Ek, we know θkx˜<θky˜, and we can obtain θx˜<Lθy˜. Thus, if y˜∈RVVEv˜, then θx˜≤Lθy˜. That is, E2N⊆RVVNuv˜. In addition, from Definition 8, we know that if x˜∈RVVNuv˜, then x˜∈RVVEv˜. If x˜∉E2N, let k be the smallest positive integer satisfying x˜∉Ek; then, x˜∈Ei, where i<k. In particular, x˜∈Ek−1; according to the expression of Ek and x˜∉Ek, there must exist y˜∈Ek−1 satisfying θky˜<θkx˜. According to the expression of E0⊇E1⊇⋯⊇E2N, we know θix˜=θiy˜, where i<k. That is, θy˜<Lθx˜, and the result is in conflict with x˜∈RVVNuv˜. Therefore, RVVNuv˜⊆E2N. It is concluded that the nucleolus of vague payoff cooperative game must exist.
The uniqueness is proved by the reduction to absurdity.
Letx˜,y˜∈RVVNuv˜=E2N and x˜≠y˜; then, θx˜=θy˜. Because RVVEv˜ is the convex set, then z˜=1/2x˜+1/2y˜∈RVVEv˜. The proof of θz˜≤Lθx˜ is as follows.Letθx˜=eS1,x˜,eS2,x˜,…,eS2N,x˜,andeSk,x˜=eSk,y˜⟹eSk,y˜≥eSk+1,y˜.IfeSk,x˜=eSk,y˜, k=1,2,…,2N,thenei,x˜=ei,y˜, ∀i∈N.Thus,x˜i=y˜i, i=1,2,…,N, and it is in conflict with x˜=y˜.Therefore, there existl=mink|eSk,x˜≠eSk,y˜,1≤k≤2N, written as q=eSl,x˜.Obviously,eS,z˜=1/2eS,x˜+1/2eS,y˜, ∀S⊆N.Then,eSk,x˜=eSk,y˜, ∀k<l,Similar as above,eSk,z˜=eSk,y˜, ∀k<l.At this point, we have provedeSk,x˜≤q, eSk,y˜≤q, ∀k≥l.
IfeSk,y˜>q, then the number of coalitions that satisfies eSl,y˜>q is at least one more than the number of coalitions that satisfies eSl,x˜>q. It is in conflict with θx˜=θy˜. In particular, eSl,y˜<eSl,x˜=q. There is a k>l such that eSk,y˜=q. Then, according to eSk,x˜=eSk,y˜⇒eSk,y˜≥eSk+1,y˜, we can obtain eSk,x˜<q. Thus, eSk,z˜=1/2eSk,x˜+1/2eSk,y˜<q, ∀k≥l.
Finally,θkz˜=θkx˜, θlz˜=θlx˜, ∀k<l, that is, θz˜<Lθx˜.
θz˜<Lθx˜ is in conflict with x˜∈RVVNuv˜. Therefore, the uniqueness of RVVNuv˜ is proved.Theorem 3.
Assuming that the nucleolus of vague payoff cooperative game is not an empty set, that is,RVVCv˜≠Φ, then RVVNuv˜⊆RVVCv˜.Proof.
Due to the existence and uniqueness of the nucleolus of vague payoff cooperative game,RVVNuv˜ is a single point set. Let RVVNuv˜=x˜¯; then, x˜¯∈RVVEv˜. It can be obtained according to Definitions 7 and 8 that RVVCv˜=x˜∈RVVEv˜|eS,x˜≤0,∀S⊆N. If y˜∈RVVCv˜, then eS,y˜≤0,∀S⊆N. According to x˜¯∈RVVNuv˜, we can obtain θx˜¯≤Lθy˜. Thus, ∀S⊆N; then, eS,x˜¯≤θ1x˜¯≤θ1y˜=maxS⊆NeS,y˜≤0.
Therefore,x˜¯∈RVVCv˜, that is, RVVNuv˜⊆RVVCv˜.


### 3.1. The Solution of the Model

The nucleolus solution of vague payoff cooperative game is an imputation of the imputation set ofRVVΓ=N,v˜, that is, x˜=x,t,1−f∈RVVEv˜. Finding the nucleolus solution of the vague payoff cooperative game is equivalent to solving a linear programming problem, as described in the following procedure.(12)minα˜|∑i∈Sx˜i+α˜≥v˜S,S∈N,x˜∈RVVEv˜.According to the ranking method of vague set, the nucleolus solution of vague payoff cooperative game is transformed into a nonlinear programming model as follows:(13)mina,bs.t.S∑i∈Sx˜i+a≥Sv˜S,H∑i∈Sx˜i+b≥Hv˜S,x˜N=v˜N.According to the definition of the score function and the accuracy function, the transformation result of the above model is as follows:(14)mina,bs.t.∑i∈Stxi−fxi+txi−fxiπxix˜i+a≥tS−fS+tS−fSπSv˜S,∑i∈Stxi+fxi+txi+fxiπxix˜i+b≥tS+fS+tS−fSπSv˜S,∑i=1ntxi−fxi+txi−fxiπxix˜i=tN−fN+tN−fNπNv˜N,∑i=1ntxi+fxi+txi+fxiπxix˜i=tN+fN+tN+fNπNv˜N,∑i=1nx˜i=v˜N.This is a multiobjective programming problem. We can use the linear weighted sum method to transform the multiobjective programming problem into solving a single-objective linear programming problem.(15)mina+b2s.t.∑i∈Stxi−fxi+txi−fxiπxix˜i+a≥tS−fS+tS−fSπSv˜S,∑i∈Stxi+fxi+txi+fxiπxix˜i+b≥tS+fS+tS−fSπSv˜S,∑i=1ntxi−fxi+txi−fxiπxix˜i=tN−fN+tN−fNπNv˜N,∑i=1ntxi+fxi+txi+fxiπxix˜i=tN+fN+tN+fNπNv˜N,∑i=1nx˜i=v˜N.The result of solving the above model isx˜i,t,1−f, where x˜i=xi,ti,1−fi is the imputation that each player can obtain from the income of the coalition. In addition, t is the degree of support of the coalition N about the imputation of all players, and 1−f is the degree of opposition of the coalition N about the imputation of all players.


## 3.1. The Solution of the Model

The nucleolus solution of vague payoff cooperative game is an imputation of the imputation set ofRVVΓ=N,v˜, that is, x˜=x,t,1−f∈RVVEv˜. Finding the nucleolus solution of the vague payoff cooperative game is equivalent to solving a linear programming problem, as described in the following procedure.(12)minα˜|∑i∈Sx˜i+α˜≥v˜S,S∈N,x˜∈RVVEv˜.According to the ranking method of vague set, the nucleolus solution of vague payoff cooperative game is transformed into a nonlinear programming model as follows:(13)mina,bs.t.S∑i∈Sx˜i+a≥Sv˜S,H∑i∈Sx˜i+b≥Hv˜S,x˜N=v˜N.According to the definition of the score function and the accuracy function, the transformation result of the above model is as follows:(14)mina,bs.t.∑i∈Stxi−fxi+txi−fxiπxix˜i+a≥tS−fS+tS−fSπSv˜S,∑i∈Stxi+fxi+txi+fxiπxix˜i+b≥tS+fS+tS−fSπSv˜S,∑i=1ntxi−fxi+txi−fxiπxix˜i=tN−fN+tN−fNπNv˜N,∑i=1ntxi+fxi+txi+fxiπxix˜i=tN+fN+tN+fNπNv˜N,∑i=1nx˜i=v˜N.This is a multiobjective programming problem. We can use the linear weighted sum method to transform the multiobjective programming problem into solving a single-objective linear programming problem.(15)mina+b2s.t.∑i∈Stxi−fxi+txi−fxiπxix˜i+a≥tS−fS+tS−fSπSv˜S,∑i∈Stxi+fxi+txi+fxiπxix˜i+b≥tS+fS+tS−fSπSv˜S,∑i=1ntxi−fxi+txi−fxiπxix˜i=tN−fN+tN−fNπNv˜N,∑i=1ntxi+fxi+txi+fxiπxix˜i=tN+fN+tN+fNπNv˜N,∑i=1nx˜i=v˜N.The result of solving the above model isx˜i,t,1−f, where x˜i=xi,ti,1−fi is the imputation that each player can obtain from the income of the coalition. In addition, t is the degree of support of the coalition N about the imputation of all players, and 1−f is the degree of opposition of the coalition N about the imputation of all players.


## 4. Experimental Analysis

We find the solution of a vague payoff n-person cooperative game. Let the fuzzy vector be0.08,0.14,0.06,0.16. The characteristic function values of each player's payoff are as follows:(16)N=1,2,3,4,v˜N=100,tN,1−fNv˜1=v˜2=v˜3=v˜4=0,1,1,v˜1,2=50,0.8,0.8,v˜1,3=50,0.8,0.9,v˜1,4=50,0.8,0.9,v˜2,3=50,0.6,0.7,v˜2,4=50,0.6,0.7,v˜3,4=50,0.7,0.8,v˜1,2,3=95,0.6,0.7,v˜1,2,4=85,0.7,0.8,v˜1,3,4=80,0.7,0.9,v˜2,3,4=55,0.8,0.8.where a vague payoff value v˜1,2,3=95,0.6,0.7 means that the coalition S consists of players 1, 2, and 3, the income value of coalition S is 95, 0.6 represents the degree of support of coalition S for the income, and 1–0.7 = 0.3 indicates the degree of opposition of the coalition S. Other vague values can give the explanation in a similar way. A linear programming model is built by combining the nucleolus solution of the vague payoff cooperative game with the characteristic function values of the example, and the model is solved by software LINGO. The results are as follows: a∗=0, b∗=10.03, and(17)X∗=34.87,0.92,1;25.18,0.71,0.85;24.77,0.92,0.98;15.17,0.53,0.69,0.80,0.91.Similarly, there is model 2, which uses the score function of Chen and Tan and the accuracy function of Hong and Choi. Model 2 only considers the true membership and the nonmembership of payoff. The results of model 2 are as follows:(18)m∗=0,n∗=9.24,Y∗=34.24,0.80,0.88;26.02,0.56,0.70;23.81,0.75,0.81;15.93,0.83,0.99,0.73,0.83.The above two results indicate that the nucleolus solution of the vague payoff cooperative game is essentially consistent with the solution of model 2, with a slight difference in imputations of players. Except for player 4, other players in the game have slightly increased their support for their imputation. This is owing to player 4 having the lowest contribution to the coalition and obtaining the lowest imputation. Therefore, after the model of vague payoff cooperative game considers the hesitancy degree, then the imputation of player 4 could be dropped slightly and the degree of support for his imputation tends to decline. This is consistent with the facts. In addition, the nucleolus solution includes the degree of support and opposition of the coalitionN about the imputation of all players. The support degree of the coalition N of vague payoff cooperative game is 0.8, and the support degree of the coalition N of model 2 is 0.73. The coalition N prefers to accept the nucleolus solution of the model of vague payoff cooperative game. The result of experimental analysis indicates that the nucleolus solution of the model of vague payoff cooperative game is feasible and effective.


## 5. Conclusions

In this work, vague set theory and cooperative game are combined and applied to fuzzy cooperative game. After considering the support, opposition, and hesitation of player in the coalition, the concepts of score function and accuracy function of vague set are proposed. Then, we define the ranking method of vague set and build a model of vague payoff cooperative game. Based on the classical cooperative games, the paper defines and discusses the relationship between the core and nucleolus of vague payoff cooperative game. We propose a method of finding the nucleolus solution of vague payoff cooperative game. Finally, an example is used to verify the feasibility and effectiveness of this method. Vague payoff cooperative game is more relevant to reality than classical cooperative game, which has greater practical application value and wider application scope in practice. At present, there are not many research studies on the vague payoff cooperative game, and we will continue to work on finding other effective solutions of the vague payoff cooperative game in the future.



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*Source: 5519468-2021-03-11.xml*