Received 20 February 2020, Accepted 28 October 2020, Available Online 17 November 2020.
1. INTRODUCTION
Cooperative games mainly study how to distribute the payoff value of a grand coalition fairly and reasonably, which has become one of the important scientific methods for analyzing people's decision-making behavior. Moreover, cooperative games have been widely used in economic management, humanities and engineering issues [1–3]. The allocation scheme of the cooperative game is called the solution of cooperative game. Since each solution of cooperative game has certain deficiencies, researchers have perfected the problem by convexly combining the existing solutions of cooperative games.
In recent years, there exist some investigations on the combinatorial solutions of cooperative games. Brink et al. [4] studied two classes of equal surplus shared solutions. The first class consists of all convex combinations of the equal division (ED) solution and the center-of-gravity of the imputation-set (CIS) value. The second class is the dual class consisting of all convex combinations the (ED) solution and the egalitarian non-separable contribution (ENSC) value, respectively. The two combinatorial solutions make the payoffs of all players of equal surplus shared solutions more fair and reasonable than the payoff obtained only in the CIS value or the ENSC value. But the two solutions of CIS value and ENSC value do not consider the case of sub-coalition. Wang et al. [5], Brink et al. [6], Casajus and Huettner [7] studied that the Shapley value and ED are convexly combined by the social selfish coefficient α∈[0,1], denoted as α average Shapley value, and some properties of the α average Shapley value are studied. Xu et al. [8] studied the α-CIS value that is convex combinations of the ED value and the CIS value, and they proposed some properties of the α-CIS value.
In addition, Ju et al. [9–11] obtained a consensus value through negotiating recursive method, which is average of the Shapley value with the CIS value, and α-consensus value is also proposed. However, the formation mechanism of the α-consensus value is not given. Moreover, individual rationality is one of the important properties of cooperative game, individual rationality is defined as: for a cooperative game v∈GN, let x=x1,x2,⋯,xn be an n-dimensional vector, satisfy xi⩾v{i}, i=1,2,⋯,n. That is to say, each player will get more payoff value in the grand coalition than that of the player alone. The ranges of α, which makes the combinatorial solutions of cooperative games satisfy the individual rationality, have not been considered in the existing researches. Therefore, in this paper, according to Wang et al. [5] studied the formation mechanisms of α average Shapley value, we give the formation mechanism of the α-consensus value of cooperative game. Furthermore, we study the ranges of α, which make the α-consensus value satisfy the individual rationality. Thus, α-consensus value can make up for the shortcoming that Shapley value not satisfy the individual rationality by adjusting the value of α. So α-consensus value is more reasonable than Shapley value.
Furthermore, the existing researches about the combinatorial solutions are only for crisp cooperative games. As we all known, due to the influence of economic, political and other social factors, in some scenarios, there are some uncertainties in the game process. So fuzzy cooperative games have been extensively studied [11–14]. Fuzzy sets can represent the uncertain information of cooperative games, but they cannot reflect the hesitation degrees of the players. Thus, using the intuitionistic fuzzy sets (IFS) given Atanassov [15,16] to represent the uncertain information of cooperative games, intuitionistic fuzzy cooperative game has been studied. Nan et al. [17] studied the Shapley function of intuitionistic fuzzy cooperative games. Based on the extended Hukuhara difference of intuitionistic fuzzy numbers, the specific expression of the Shapley value of intuitionistic fuzzy cooperative games with multilinear extension form is obtained, and its existence and uniqueness are discussed. Liu and Zhao [18] studied the least squares pre-nucleolus of cooperative game with characteristic function of trapezoidal intuitionistic fuzzy numbers. As far as we know, most of the studies on fuzzy cooperative games and intuitionistic fuzzy cooperative games are focused on the single value solutions. However, there exists no investigation on the combination solutions of fuzzy cooperative games and intuitionistic fuzzy cooperative games. As the intuitionistic fuzzy cooperative games are expansions of fuzzy cooperative games, in this paper, the α-consensus value of intuitionistic fuzzy cooperative game is studied, and its some properties are proved.
The rest part of this paper is organized as follows: in Section 2, some basic concepts of intuitionistic fuzzy sets are given and Hukuhara difference of intuitionistic fuzzy numbers is introduced as well as the ranking order relation of intuitionistic fuzzy numbers. In Section 3, some definitions of the α-consensus value of cooperative game are reviewed. Then, the formation mechanism and some important conclusions of the α-consensus value of cooperative game are given. Section 4 studies the definition and property of the α-consensus value of the intuitionistic fuzzy cooperative game. In Section 5, a numerical example is offered to illustrate the validity and applicability of the α-consensus proposed in this paper. Section 6 is the summary of this paper.
2. PRELIMINARIES
2.1. Intuitionistic Fuzzy Set
The concept of an IFS was firstly introduced by Atanassov [15].
Definition 1.
[15,16] Let X={x1,x2,⋯,xn} be a finite universe set. An IFS à in X may be mathematically expressed as Ã={〈xl,μÃ(xl),υÃ(xl)〉|xl∈X}, where μÃ: X↦[0,1] and υÃ: X↦[0,1] are the membership degree and the non-membership degree of an element xl∈X to the set Ã⊆X, respectively, such that they satisfy the following condition: 0⩽μÃ(xl)+υÃ(xl)⩽1 for all xl∈X.
Let πÃ(xl)=1−μÃ(xl)−υÃ(xl), which is called the intuitionistic index (or hesitancy degree), which is the degree of indeterminacy membership of the element xl to the set Ã. Obviously, 0⩽π(xl)⩽1.
2.2. Triangle Intuitionistic Fuzzy Numbers and Related Operations
Definition 2.
[19,20] Suppose that ã=〈(a1−,a,a1+);(a2−,a,a2+)〉, is the triangular intuitionistic fuzzy number (TIFN) on the real number set R, then the membership function and the non-membership function of ã are defined:
μã(x)=0,(x<a1−,x>a1+),(x−a1−)∕(a−a1−),(a1−⩽x<a),1,(x=a),(a1+−x)∕(a1+−a),(a⩽x<a1+),
υã(x)=1,(x<a2−,x>a2+),(a−x)∕(a−a2−),(a2−⩽x<a),0,(x=a),(x−a)∕(a2+−a),(a⩽x<a2+).
The set of all TIFNs is denoted by ℜ̃.
Definition 3.
[18,19] Let ã=〈(a1−,a,a1+);(a2−,a,a2+)〉 and b̃=〈(b1−,b,b1+); (b2−,b,b2+)〉 be two TIFNs on the set ℜ̃ and r∈R is any real number. TIFNs arithmetic operations are given as follows:
(1)a~+b~=⟨(a1−+b1−,a+b,a1++b1+);(a2−+b2−,a+b,a2++b2+)⟩,
(2)rã=〈(ra1−,ra,ra1+);(ra2−,ra,ra2+)〉,r⩾0,〈(ra1+,ra,ra1−);(ra2+,ra,ra2−)〉,r<0.
According to extended Hukuhara difference of interval numbers given Meng et al. [21], the extended Hukuhara difference of TIFNs are defined as follows.
Definition 4.
[17] Let ã=〈(a1−,a,a1+);(a2−,a,a2+)〉, b̃=〈(b1−,b,b1+);(b2−,b,b2+)〉 and c̃=〈(c1−,c,c1+);(c2−,c,c2+)〉 be TIFNs. If bi−−ai−>bi+−ai+, c̃=ã−Hb̃ is said to the “imaginary” Hukuhara difference. The extended Hukuhara difference of TIFNs is defined ã−Hb̃=〈(a1−−b1−,a−b,a1+−b1+);(a2−−b2−,a−b,a2+−b2+)〉.
The ranking order relation of intuitionistic fuzzy number is an important problem for intuitionistic fuzzy cooperative game. Based on the λ weighted mean areas of intuitionistic fuzzy numbers, the ranking order relation of TIFNs is defined as follows.
Definition 5.
Let ã=〈(a1−,a,a1+);(a2−,a,a2+)〉 and b̃=〈(b1−,b,b1+);(b2−,b,b2+)〉 be TIFNs, Sλ(ã) and Sλ(b̃) are the index values of λ weighted mean areas of ã and b̃ respectively, λ∈[0,1],
where
Sλ(ã)=λ(a1−+2a+a1+)∕4+(1−λ)(a2−+2a+a2+)∕4, Sλ(b̃)=λ(b1−+2b+b1+)∕4+(1−λ)(b2−+2b+b2+)∕4,
then
if Sλ(ã)>Sλ(b̃), then ã>b̃;
if Sλ(ã)<Sλ(b̃), then ã<b̃;
if Sλ(ã)=Sλ(b̃), then ã=b̃.
3. THE α-CONSENSUS VALUE OF COOPERATIVE GAME
3.1. The Solutions of Cooperative Game
An n-person cooperative game is a pair (N,v) where N={1,2,⋯,n} is a finite set of players with N⩾2 and v:2N→R is a characteristic function on N such as v(ϕ)=0. For each coalition S⊆N, v(S) is called the payoff of coalition S. which is the value of the members of coalition S can obtain by agreeing to cooperate. The set of all n-person cooperative games are denoted by GN.
Shapley value and CIS value are two important solutions of cooperative game. For a cooperative game v∈GN, let Shapley value of a cooperative game be expressed as
Shi(v)=ρ[v(S)−v(S\{i})],(1)
where
ρ=∑S⊆N(n−|S|)!(|S|−1)!n!,S⊆N,v∈GN.
For a cooperative game v∈GN, let CIS value of a cooperative game be expressed as
CISi(v)=v{i}+1nv(N)−∑j∈Nv{j}.(2)
Ju et al. [9,10] studied the α–consensus value, which is the combinatorial solution of Shapley value and CIS value of cooperative games.
Definition 6.
For a cooperative game v∈GN, let α–consensus value of the cooperative game be expressed as
γ(v)=αSh(v)+(1−α)CIS(v),
where
Sh(v)=ρ[v(S)−v(S\{i})], and
CIS(v)=v{i}+1nv(N)−∑j∈Nv{j},
α∈[0,1].
3.2. Formation Mechanism of the α–Consensus Value
In this subsection, the formation mechanism of the α-consensus value of the cooperative game is studied. For a cooperative game (N,v)∈GN, the payoff of player i participating in the grand coalition is xi consisting of two parts xi=αxi1+1−αxi2. The first part of the value xi1 is obtained when player joins a grand coalition, and the second part xi2 is obtained when player leaves the grand coalition. The parameter α∈[0,1] represents a proportion of the profit when player i joins the grand coalition, 1−α is a proportion of the payoff when player i leaves the grand coalition. Let π(N) be set for all possible permutations on N. For player i∈N and any permutation π∈π(N), Siπ(k)={π(k)∈N|k⩽π−1(i)} represents that player i joins into the coalition of his predecessors to form the new coalition, where π−1(i) stands for player i in the order π. Piπ(k)={π(k)∈N|k⩾π−1(i)}, represents that player i departs from the coalition of his predecessors to form the new coalition.
The formation process of α–consensus value of the cooperative game is given as follows.
Step 1: The players arrive in or depart from a random order π, and all orders in π(N) have the same probability.
Step 2: Player i∈N joins a coalition and forms a new coalition Siπ(k). Before leaving, the coalition is Piπ(k). The first player who joins or leaves always obtains a payoff value of αv{i}.
Step 3: The joining player i∈N obtains his payoff value αv(Siπ(k)), and his marginal contribution surplus v(Siπ(k))−v(Siπ(k)\{i})−αv(Siπ(k)) is evenly distributed to the subsequent players. Moreover, the leaving player i∈N obtains a payoff value αv{i} and his marginal contribution surplus is equally distributed to the remaining player in the coalition.
Step 4: The last player π(n), joining into grand coalition N can only get his marginal contribution v(N)−v(N∖π(n)), while the last player who leaves the coalition obtains payoff value v(π(n)). Note that the whole quantity v(N), for every permutation π, is distributed among all players.
Step 5: The payoff value γiπ(N,v) of each player i is composed of two parts in the grand coalition according to the order π, one is from participating in the coalition and the other is from departure the coalition. Assuming that α is the allocation ratio of player i joining in coalition payoff value Shiπ in order π, and 1−α is the allocation ratio of player i leaving the coalition payoff value CISiπ(v), we have γiπ(v)=αShiπ(v)+(1−α)CISiπ(v).
Step 6: The final payoff value γi(N,v) of the player i in grand coalition is the expected value of the payoff value γiπ(N,v) for all orders. Then, for crisp cooperative game (N,v)∈GN, and all order π∈π(N), the payoff value γi(N,v) of player i∈N is determined from Steps 1-6 as follows:
γiπ(v)=αv{i},π−1(i)=1,αv(Siπk)+λiπ+(1−α)v{i}+μiπ,1<π−1(i)<n,αv(N)−ṽ(N\{i})+λiπ+(1−α)v{i}+μiπ,π−1(i)=n,
where
λiπ=∑k=1π−1(i)−1vSiπ(k)−vSiπ(k)\π(k)−αvSiπ(k)n−k and
μiπ=∑k=1π−1(i)−1vPiπ(k)−vPiπ(k)\π(k)−αv(π(k))n−k.
Thus, we can get
γi(v)=1n!∑Π(N)γiπ(v),i∈N,α∈[0,1].
Furthermore,
γi(v)=1n!∑Π(N)γiπ(v) =1n!∑Π(N)αShiπ(v)+1−αCISiπ(v) =α1n!∑Π(N)Shiπ(v)+1−β1n!∑Π(N)CISiπ(v) =αShi(v)+1−αCISi(v).
So, for v∈GN, the α–consensus value of cooperative game is to be expressed as
γi(v)=αShi(v)+(1−α)CISi(v).(3)
for all
i∈N, where
α∈[0,1].
3.3. The Range of α for the α–Consensus Value of the Cooperative Games
The individual rationality is an important condition for evaluating solutions of cooperative games. The coefficient α∈[0,1] of existing of convex combinatorial solutions, such as α–consensus value, α average Shapley value and α–CIS value, is given prior, that is not to make these combinatorial solutions satisfy the individual rationality. Thus, it is crucial to find the range of the coefficient α such that the α–consensus value satisfies the individual rationality. For cooperative game v∈GN, we have
γi(v)=αShi(v)+(1−α)CISi(v) =αShi(v)+CISi(v)−αCISi(v) =α(Shi(v)−CISi(v))+CISi(v) ⩾v{i}.
By simple calculating, we get
α⩾v{i}−CISi(v)(Shi(v)−CISi(v)).
If Shi(v)>CISi(v), then we can get
v{i}−CISi(v)Shi(v)−CISi(v)⩽0.
Hence, for any α∈[0,1], the α–consensus value of cooperative game satisfies the individual rationality.
If Shi(v)<CISi(v), then we have
0⩽α⩽v{i}−CISi(v)Shi(v)−CISi(v).
If Shi(v)<v{i}, we obtain
0⩽α⩽v{i}−CISi(v)Shi(v)−CISi(v)<1.
Thus, for α∈0,v{i}−CISi(v)Shi(v)−CISi(v), the α–consensus value of cooperative game satisfies the individual rationality.
Obviously, when Shi(v)⩾v{i}, for any α∈[0,1], α–consensus value satisfies individual rationality.
3.4. Some Conclusions of Procedural Values
Malawski [22] gives the concept of procedural values “a procedural value is determined by an underlying procedure of sharing marginal contributions to coalitions formed by players joining in random order.” Obviously, the α–consensus value is the procedural value, so α–consensus value has the following conclusions.
Lemma 1.
Every linear efficient value γi(v) having the equal treatment property is of the form:
γi(v)=∑S⊆N,i∈Sptv(S)t−∑S⊂N,i∉Sptv(S)n−t,
for every
v and
i, where
p1,p2,⋯,pn∈R,
pn=1.
Thus, the value of each player is a weighted sum of all coalitions with weights depending only on cardinalities of coalitions. We call p1,p2,⋯,pn coefficients of the value γi(v).
Lemma 2.
For the value γi(v)∈GN determined by a procedure q=(q1,q2,⋯,qn), the coefficients p1,p2,⋯,pn are expressed as follow:
pn=1,pt=qt+1nt,fort<n.
Corollary 3.
Every linear efficient value with equal treatment property on v∈GN with coefficients p1,p2,⋯,pn satisfying
pn=1,0⩽pt⩽1ntfort=1,2,⋯,n−1.
is procedural, and the coefficients of its procedure are
q1=1,qk=pk−1⋅nk−1fork=2,3,⋯,n.
Lemma 1 indicates that when the procedure value satisfies the validity and linearity, then it can be expressed as a linear combination of characteristic functions of coalitions. Lemma 2 indicates by the definition of procedure, a procedural value of a player is always a linear combination of characteristic functions of coalitions in the game, and we only need to find the coefficients of this combination. Corollary 3 is the generalization of Lemma 2. The proof of Lemma 2 is similar to Lemma 2 of [21], we omit it for unnecessary repetition. Since consensus value is the procedure value, it satisfies Lemmas 1 and 2 and Corollary 3.
4. THE α–CONSENSUS VALUE OF INTUITIONISTIC FUZZY COOPERATIVE GAMES AND PROPERTIES
4.1. The α–Consensus Value of Intuitionistic Fuzzy Cooperative Game
Let (N,ṽ) be n-person intuitionistic fuzzy cooperative game with characteristic functions of TIFNs, where N={1,2,⋯,n} is a finite set of players with N⩾2, and ṽ:2N→ℜ̃ is a characteristic function on N such as ṽ(ϕ)=〈0,0,0〉. For each coalition S⊆N, TIFN ṽ(S)=〈(v1−(S),v(S),v1+(S));v2−(S),v(S),v2+(S)〉 is called the payoff of coalition S. This is what the members of coalition S can obtain by agreeing to cooperate. The set of all n-person intuitionistic fuzzy cooperative games (N,ṽ) is denoted by G̃N.
Nan et al. [17] studied the Shapley function of intuitionistic fuzzy cooperative games. Based on the extended Hukuhara difference of intuitionistic fuzzy numbers, the specific expression of the Shapley value of intuitionistic fuzzy cooperative games is defined as follows:
Definition 7.
[17] For an intuitionistic fuzzy cooperative game ṽ∈G̃N, let Shapley value be expressed as
Shi(ṽ)=ρ[ṽ(S)−Hṽ(S\{i})].(4)
where
ρ=∑S⊆N(n−|S|)!(|S|−1)!n!,
S⊆N,
ṽ∈G̃N.
Based on the extended Hukuhara difference of intuitionistic fuzzy numbers, the CIS value of intuitionistic fuzzy cooperative games (N,ṽ) is defined as follows:
Definition 8.
For an intuitionistic fuzzy cooperative game ṽ∈G̃N, let CIS value of the cooperative game be expressed as
CISi(ṽ)=ṽ{i}+1nṽ(N)−H∑j∈Nṽ{j}.(5)
Thus, extending the consensus value of cooperative game, the α–consensus value of intuitionistic fuzzy cooperative game is obtained as follows:
γi(ṽ)=αShi(ṽ)+(1−α)CISi(ṽ).(6)
4.2. Properties of the α–Consensus Values of Intuitionistic Fuzzy Cooperative Game
Form Eq. (6), it is easily seen that the consensus value of intuitionistic fuzzy cooperative game is calculated using the Hukuhara difference of TIFNs, and satisfies some properties similar to those of cooperative game.
Let γ:G̃N→R̃n, we consider the following properties.
Efficiency: For any intuitionistic fuzzy cooperative game ṽ∈G̃N, ∑i∈Nγi(ṽ)=ṽ(N).
Symmetry: For any intuitionistic fuzzy cooperative game ṽ∈G̃N, two players i,j∈N are symmetric if for every coalition S⊆N∖{i,j}, ṽ(S⋃{i})=ṽ(S⋃{j}), then γi(ṽ)=γj(ṽ).
Linearity: For any intuitionistic fuzzy cooperative game ṽ,w̃∈G̃N and b,c∈R, γi(bṽ+cw̃)=bγi(ṽ)+cγi(w̃), where bṽ+cw̃ is given by (bṽ+cw̃)(S)=bṽ(S)+cw̃(S), for all S⊆N.
Variability: For any intuitionistic fuzzy cooperative game ṽ,w̃∈G̃N, a>0 and d∈Rn, where w̃ is given by w̃(S)=aṽ(S)+∑i∈Sdi for all S⊆N, then γi(w̃)=aγi(ṽ)+d.
Dummy: For any intuitionistic fuzzy cooperative game ṽ∈G̃N, if player i∈N is a dummy player, for each coalition S⊆N\i, ṽ(S⋃{i})−Hṽ(S)=ṽ{i}, then γi(ṽ)=ṽ{i}.
Theorem 4.
For any intuitionistic fuzzy cooperative game ṽ∈G̃N, the α–consensus value satisfies the following properties: Efficiency, Linearity, Variability and Dummy.
Proof.
Efficiency: Based on the effectiveness of Shapley values, CIS values and combine Eq. (6), one obtains that
∑i=1nγi(ṽ)=∑i=1nαShi(ṽ)+(1−α)CISi(ṽ).
By Eqs. (1) and (2) of Definition 3, we can get
∑i=1nγi(ṽ)=∑i=1n(αρ[ṽ(S)−Hṽ(S\i)])+∑i=1n((1−α)[ṽ{i} +1nṽ(N)−H∑j∈Nṽ{j} =α∑i=1n(ρ[ṽ(S)−Hṽ(S\i)])+(1−α)∑i=1n[ṽ{i} +1nṽ(N)−H∑j∈Nṽ{j} =αṽ(N)+(1−α)ṽ(N) =ṽ(N).
Symmetry:
Similarly, it follows from Eq. (6)
γi(ṽ)=(αρ[ṽ(S⋃{i})−Hṽ(S)])+(1−α)[ṽ{i} +1nṽ(N)−H∑j∈Nṽ{j}.
and
γj(ṽ)=αρṽ(S⋃{j})−Hṽ(S)+(1−α)[ṽ{j} +1nṽ(N)−H∑i∈Nṽ{i}.
According to the characteristic function ṽ(S⋃{i})=ṽ(S⋃{j}), it can be obtained that
γi(ṽ)=γj(ṽ).
Linearity:
By Eqs. (1) and (2) of Definition 3 and Eq. (4), we have
γi(bṽ+cw̃) =(αρ[b(ṽ(S)−Hṽ(S\{i}))+c(w̃(S) −Hw̃(S\{i}))])+(1−α)∑J∈Nb(ṽ{i}+cw̃{i}) +1nbṽ(N)−H∑j∈Nṽ{j}+cw̃(N)−H∑j∈Nw̃{j}.
It follows that
γi(bṽ+cw~)=αρb(ṽ(S)−Hṽ(S\{i}))+αρc(w~(S)−Hw~(S\{i}))+(1−α)bṽ{i}+1nṽ(N)−H∑j∈Nṽ{j}+(1−α)cw~{i}+1nw~(N)−H∑j∈Nw~{j}=(∑J∈Nαρb(ṽ(S)−Hṽ(S\{i}))+(1−α)b∑J∈N∑J∈Nṽ{i}+1n(ṽ(N)−H∑j∈Nṽ{j}+∑j∈Nαρc(w~(S)−Hw~(S\{i}))+(1−α)cw~{i}+1nw~(N)−H∑j∈Nw~{j}=bγi(ṽ)+cγi(w~).
Variability:
Similarly, it follows from Eq. (6)
γi(w̃)=αShi(w̃)+(1−α)CISi(w̃) =αρ[w̃(S)−Hw̃(S\{i})]+(1−α)(w̃{i} +1nw̃(N)−H∑j∈Nw̃{j}.
According to w̃(S)=aṽ(S)+∑i∈Sdi, γi(w̃) can be calculated as follows:
γi(w̃)=αρaṽ(S)+∑i∈Sdi−aṽS\{i}+∑S\{i}d +(1−α)(aṽ{i}+di)+1naṽ(N)+∑i∈Ndi −H∑j∈Naṽ{j}+dj =αρ(aṽ(S)−Haṽ(S\{i}))+∑i∈Sdi−∑S\{i}d +(1−α)((aṽ{i}+di)+1naṽ(N)−H∑j∈Naṽ{j} +∑i∈Ndi−∑j∈Ndj.
Thus, one has
γi(w̃)=aαρ[(ṽ(S)−Hṽ(S\{i})]+αρdi +a(1−α)(ṽ{i})+1nṽ(N)−H∑j∈Nṽ{j}+(1−α)di =a[αρ[(ṽ(S)−Hṽ(S\{i})]+(1−α)(ṽ{i}+ 1nṽ(N)−H∑j∈Nṽ{j}+(αρdi+(1−α)di) =aγi(ṽ)+d.
Dummy:
From Eq. (6), we have
γi(ṽ)=αShi(ṽ)+(1−α)CISi(ṽ) =αρ[ṽ(S)−Hṽ(S\{i})]+(1−α)(∑J∈Nṽ{i} +1nṽ(N)−H∑j∈Nṽ{j}.
According to the characteristic function
ṽ(S⋃{i})−Hṽ(S)=ṽ({i}), it follows that
γi(ṽ)=αṽ{i}+(1−α)ṽ{i} =ṽ{i}.
4.3. The Range of α for the α–Consensus Value of the Intuitionistic Cooperative Games
In this subsection, similarly to the α–consensus value of cooperative games, the range of α, which makes the α–consensus value of intuitionistic cooperative games satisfy the individual rationality, is obtained.
For the intuitionistic fuzzy cooperative game ṽ∈G̃N, we have
γi(ṽ)=αShi(ṽ)+(1−α)CISi(ṽ) =αShi(ṽ)+CISi(ṽ)−αCISi(ṽ) =α(Shi(ṽ)−CISi(ṽ))+CISi(ṽ) ⩾ṽ{i}.
By simple calculation, we get
α⩾ṽ{i}−CISi(ṽ)Shi(ṽ)−CISi(ṽ).(7)
It is easily seen that Eq. (7) involves the operations of subtraction and division of TIFNs, which are difficult. In order to avoid these operations of TIFNs, the λ weighted mean areas of TIFNs are used to transform the TIFNs into real numbers. Thus, according to Definition 5, Eq. (7) is transformed to
α⩾Sλ(ṽ{i})−Sλ(CISi(ṽ))Sλ(Shi(ṽ))−Sλ(CISi(ṽ)).
Then, according to the range of α of α–consensus value of cooperative games, given in Subsection 3.3, we have the following conclusions.
If Sλ(Shi(ṽ))>Sλ(CISi(ṽ)), then we can get
Sλ(ṽ{i})−Sλ(CISi(ṽ))Sλ(Shi(ṽ))−Sλ(CISi(ṽ))<0.
So, for arbitrary α∈[0,1], the α–consensus value of the intuitionistic fuzzy cooperative game satisfies the individual rationality.
If Sλ(Shi(ṽ))<Sλ(CISi(ṽ)), then we have
0⩽α⩽Sλ(ṽ{i})−Sλ(CISi(ṽ))Sλ(Shi(ṽ))−Sλ(CISi(ṽ)).
And if Sλ(Shi(ṽ))<Sλ(ṽ{i}), we obtain
0⩽α⩽Sλ(ṽ{i})−Sλ(CISi(ṽ))Sλ(Shi(ṽ))−Sλ(CISi(ṽ))<1.
Thus, for α∈0,Sλ(ṽ{i})−Sλ(CISi(ṽ))Sλ(Shi(ṽ))−Sλ(CISi(ṽ)), the α–consensus value of intuitionistic fuzzy cooperative game satisfies the individual rationality.
While Sλ(Shi(ṽ))>Sλ(ṽ{i}), for any α∈[0,1], the α–consensus value of the intuitionistic fuzzy cooperative game satisfies individual rationality.
5. NUMERICAL EXAMPLE
Suppose that there are three factories (i.e., players)1, 2 and 3, who have the ability to produce separately. Denoted the set of players by N={1,2,3}. Given the coalition S⊆N, let characteristic function ṽ(S)=〈(v1−(S),v(S),v1+(S));v2−(S),v(S),v2+(S)〉 be TIFN, all the characteristic function values of coalitions are obtained in Table 1.
| Clear Coalition S |
Intuitionistic Fuzzy Characteristic Functions |
| {1} |
〈(8, 11, 6), (6, 11, 21)〉 |
| {2} |
〈(11, 16, 21), (9, 16, 23)〉 |
| {3} |
〈(21, 26, 31), (18, 26, 36)〉 |
| {1, 2} |
〈(12, 18, 24), (10, 18, 30)〉 |
| {1, 3} |
〈(24, 30, 36), (20, 30, 44)〉 |
| {2, 3} |
〈(25, 32, 38), (22, 32, 46)〉 |
| {1, 2, 3} |
〈(41, 54, 69), (36, 54, 81)〉 |
Table 1The intuitionistic fuzzy characteristic functions.
|
Shapley Value and CIS Value of the Three-person Intuitionistic Cooperative Game
|
|
Shi |
CISi |
| 1 |
〈(8.7, 12, 17), (7.2, 12, 21.2)〉 |
〈(8.3, 11.3, 16.3), (6.3, 11.3, 21.3)〉 |
| 2 |
〈(10.7, 15.5, 20.5), (9.7, 15.5, 23.2)〉 |
〈(11.3, 16.3, 21.3), (9.3, 16.3, 23.3)〉 |
| 3 |
〈(21.7, 26.5, 31.5), (19.2, 26.5, 36.7)〉 |
〈(21.3, 26.3, 31.3), (18.3, 26.3, 36.3)〉 |
Table 2The intuitionistic fuzzy Shapley value and CIS value of the three factories.
According to Eqs. (4) and (5), we get Table 2.
According to Tables 1 and 2 and Definition 5, one can obtain Sh2(ṽ)<ṽ(2). Hence, Sh2(ṽ) does not satisfy the individual rationality.
According to condition (ii) of 4.3 and let λ=12(λ∈[0,1]), we can get
0⩽α⩽Sλ(ṽ{i})−Sλ(CISi(ṽ))Sλ(Shi(ṽ))−Sλ(CISi(ṽ))=0.558<1.
Thus, for α∈[0,0.558], the α–consensus value of the intuitionistic fuzzy cooperative game satisfies the individual rationality. Specially, let α=0.4, the consensus values of the three factories are obtained as follows, respectively.
γ1(ṽ)=〈(8.46,11.58,16.58),(6.66,11.58,21.26)〉, γ2(ṽ)=〈(11.06,15.98,20.98),(9.46,15.98,23.26)〉, γ3(ṽ)=〈(21.46,26.38,31.38),(18.66,26.38,36.48)〉.
6. CONCLUSION
In this paper, the α–consensus value of a cooperative game is improved and a formation mechanism of the α–consensus value of the cooperative game is given. Furthermore, we study the ranges of α, which make the α–consensus value of the cooperative game satisfy the individual rationality. In addition, the α–consensus value of an intuitionistic fuzzy cooperative game is proposed, and its some properties are formulated and proved. By adjusting the parameter α, the α–consensus value can make up for the disadvantage that Shapley value does not satisfy the individual rationality under the same cooperative game. This paper studies the α–consensus value of the intuitionistic fuzzy cooperative game, other combination solutions of an intuitionistic fuzzy cooperative game can be researched in near future.
ACKNOWLEDGMENTS
The authors would like to thank the associate editor and also appreciate the constructive suggestions from the anonymous referees. This research was supported by the Natural Science Foundation of China (Nos. 72061007 and 71961004).
REFERENCES
3.Z.X. Zou, Q. Zhang, and M.L. Li, The weighted Shapley value for games with restricted cooperation, Syst. Eng. Theory Pract., Vol. 38, 2018, pp. 145-163. 15.K.T. Atanassov, Intuitionistic Fuzzy Sets, 1983. VII ITKR Session, Sofia, (Deposed in Centr. Sci.-Techn. Library of the Bulg. Acad. of Sci., 1697/84) 17.J.X. Nan, H. Bo, and C.L. Wei, Cooperative games with the intuitionistic fuzzy coalitions and intuitionistic fuzzy characteristic functions, D.F. Li, X.G. Yang, M. Uetz, and G.J. Xu (editors), Game Theory and Applications, Springer, Singapore, Vol. 758, 2017, pp. 337-352. 20.G.S. Mahapatra and T.K. Roy, Intuitionistic fuzzy number and its arithmetic operation with application a system failure, J. Uncertain Syst., Vol. 7, 2013, pp. 92-107. 
