# Solidarity Value and Solidarity Share Functions for TU Fuzzy Games

Authors: Rajib Biswakarma; Surajit Borkotokey; Radko Mesiar
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

TU games under both crisp and fuzzy environments describe situations where players make full (crisp) or partial (fuzzy) binding agreements and generate worth in return. The challenge is then to decide how to distribute the profit among them in a rational manner: we call this a solution. In this paper, we introduce the notion of solidarity value and the solidarity share function as a suitable solution to TU fuzzy games. Two special classes of TU fuzzy games, namely, TU fuzzy games in Choquet integral form and in multilinear extension form, are studied and the corresponding solidarity value and the solidarity share functions are characterized.


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## Body

## 1. Introduction

A cooperative game with transferable utility, or simply a TU game, is a pair(N,v), where N is a set of n players, called the grand coalition, and v is the characteristic function defined on 2N that assigns to every subset (coalition) a real number called its worth which gives zero worth to the empty coalition. Let G0 denote the class of TU games. A solution for any TU game is a function on G0 which assigns to the TU game a distribution of payoffs for its players. If there is no ambiguity on the player set N, we denote by G0(N) the class of all TU games with the fixed N.Among the various one-point solutions for TU games, the Shapley value [1] and the solidarity value [2] are perhaps the most popular ones. The Shapley value builds on the axioms of efficiency, linearity, anonymity, and the null player. The solidarity value on the other hand is characterized by efficiency, linearity, anonymity, and the axiom of ν-null player. The null player axiom of the Shapley value rewards nothing to the nonperforming players. However in recent years solidarity has been considered as an important human attribute influencing both rationality (limited rationality) and social preference for fairness [3–5]. Therefore the role of solidarity in TU games is essentially discussed in the literature and the notion of the solidarity value was proposed as an alternative to the Shapley value. It follows that, unlike the Shapley value, the solidarity value expresses solidarity to both the nonperforming and performing players; see [2].As an alternative to the values, the share functions are proposed in [6] as useful solution concepts for TU games that assign to every game a vector whose components add up to one. A share function determines how much share a player can get from the worth of the grand coalition and therefore is devoid of the efficiency requirement as opposed to the other standard value functions. Therefore a share function simplifies the model formulation to a great extent. In [7], it is shown that, on a ratio scale, meaningful statements can be made for a certain class of share functions, whereas all statements with respect to the value functions are meaningless. The share function corresponding to the solidarity value is called the solidarity share function. It is obtained by dividing the solidarity value of each player by the sum of the solidarity values of all the players. In [8] the solidarity share function for TU games is studied in detail.Cooperative games with fuzzy coalitions or simply TU fuzzy games are a generalization of the ordinary TU games in the sense that participation of the players in a fuzzy coalition belongs to the interval[0,1]; see [9]. A fuzzy coalition is a fuzzy subset of the player set N which assigns a membership grade to its members. This membership of a player in a coalition represents her rate of participation in it. When distinction between the two classes of games is needed, we call the standard TU game the crisp TU game or simply the TU game. TU fuzzy games derived from their crisp counterparts are found in the literature; see, for example, [9–15]. The Shapley share function for TU fuzzy games is studied in [16]. The relevance of the solidarity value and the corresponding share function for TU fuzzy games can be realized in situations where players with partial participations are marginally unproductive but being the part of the cooperative endeavor may be rewarded with some nonzero payoffs. In this paper, we introduce the notion of solidarity value and solidarity share functions for TU fuzzy games. A set of axioms to characterize these functions is proposed. We define two classes of TU fuzzy games, namely, the TU fuzzy games in Choquet integral form due to [15] and TU fuzzy games in multilinear extension form due to [17]. These two classes are continuous with respect to the standard metric and also monotonic when the associated crisp game is monotonic. Moreover they build on the idea of nonadditive interactions among the players; for more details we refer to [15–17].The rest of the paper proceeds as follows. In Section2, we compile the related definitions and results from the existing literature. Section 3 discusses the solidarity share functions for TU fuzzy games. In Section 4 we discuss the solidarity share functions for TU fuzzy games in Choquet integral form followed by some illustrative example. Section 5 concludes the paper.


## 2. Preliminaries

In this section we compile the definitions and results necessary for the development of the present study from [2, 6, 8, 9, 11, 12, 15, 16, 18]. We start with the notion of solidarity values and share functions in crisp games.


### 2.1. The Solidarity Value and the Share Function for TU Games

Let the player setN be fixed so that the class of TU games can be taken as G0(N). We define the following.Definition 1.
Let∅≠T∈2N and v∈G0(N); the quantity νv(T)=1/T∑k∈T[v(T)-v(T\k)] is called the average marginal contribution of a player of the coalition T.Definition 2.
Given a gamev∈G0(N), player i∈N is called a ν-null player if νv(T)=0, for every coalition T⊆N containing i.Consider a functionΦ:G0(N)→R+n2N that assigns to any game v∈G0(N) a 2N→R+n mapping. For any fixed v∈G0(N) and set W⊆2N, we denote the corresponding n-ary vector in R+n as (Φ1(W,v),…,Φn(W,v)). We define the solidarity value as follows.Definition 3.
A functionΦ:G0N→R+n2N is said to be the solidarity value on G0(N) if it satisfies the following four axioms.
Axiom C 1 (Efficiency). If v∈G0N and W∈2N, then (1)∑i∈WΦiW,v=vW,ΦiW,v=0∀i∉W.
Axiom C 2 ( ν -Null Player). If v∈G0(N) and i∈W∈2N are a ν-null player, then (2)ΦiW,v=0∀i∈T⊂W.
Axiom C 3 (Symmetry). If v∈G0(N), W∈2N, and i,j∈W are symmetric, that is, v(S∪i)=vS∪j holds for any S∈2W\i,j, then (3)ΦiW,v=ΦjW,v.
Axiom C 4 (Additivity). For v1,v2∈G0(N), define v1+v2∈G0(N) by (v1+v2)(S)=v1(S)+v2(S) for each S∈2N. If v1,v2∈G0(N) and W∈2N, then (4)ΦiW,v1+v2=ΦiW,v1+ΦiW,v2.Theorem 4.
Define a functionΦ:G0(N)→R+n2N by (5)ΦiW,v=∑S∈PiWδS,WνvS,ifi∈W0,otherwise, where Pi(W)=S∈2W∣S∋i and δ(|S|,|W|)=S-1!W-S!/W!. Then the function Φ is the unique solidarity value on G0(N).Proof.
We refer to [8] for a detailed proof of Theorem 4.From now onward we denote the solidarity value byΦsol(W,v) where W⊆N and v∈G0(N).Definition 5.
LetC⊆G0(N) be a set of TU games, and let μ:C→R be a given function. A μ-share function on a set of games C⊆G0(N) is a function Ψμ:C→R+n2N that satisfies the following AxiomsCS1,CS2, andCS3 and either AxiomCS4 orCS5:
Axiom CS 1 (μ-Efficiency). If v∈C and K∈2N, then (6)∑i∈KΨiμK,v=1,ΨiμK,v=0,i∉K.
Axiom CS 2 ( μ -Symmetry). If v∈C and K∈2N, i,j∈K, and v(S∪i)= v(S∪j) hold for any S⊆K\i,j, then Ψiμ(K,v)=Ψjμ(K,v).Axiom CS 3 ( μ ν -Null Player). If v∈C and i∈K∈2N is an ν-null player, that is, νv(T)=0, then (7)ΨiμK,v=0∀i∈T⊂K.Axiom CS 4 ( μ -Additivity). For any pair v1,v2∈C such that v1+v2∈C, it holds that μ(K,v1+v2)Ψiμ(K,v1+v2) = μ(K,v1)Ψiμ(K,v1) + μ(K,v2)Ψiμ(K,v2).Axiom CS 5 ( μ -Linearity). For any pair v1, v2 of games in C and for any pair of real numbers a and b such that av1+bv2∈C, it holds that(8)μK,av1+bv2ΨiμK,av1+bv2=aμK,v1ΨiμK,v1+bμK,v2ΨiμK,v2.Theorem 6.
Letμ:C→R be a positive function on C. Then on the subclass C there exists a unique solidarity μ-share function Ψμ:C→R+n2N satisfying the axiomsCS1–CS4 if and only if μ is additive on C.


### 2.2. TU Games with Fuzzy Coalitions

Now we make a brief discourse of TU games with fuzzy coalitions or simply TU fuzzy games with the player setN. A fuzzy coalition is a fuzzy subset of N, which is identified with a characteristic function from N to [0,1]. Let L(N) be the set of all fuzzy coalitions in N. For a fuzzy coalition S∈L(N) and player i∈N, S(i) represents the membership grade of i in S. The empty fuzzy coalition denoted by ∅ is one where all the players provide zero membership. If no ambiguity arises we use the same notations to represent crisp and fuzzy coalitions as crisp coalitions are special fuzzy coalitions with memberships 0 or 1.The support of a fuzzy coalitionS is denoted by suppS={i∈N∣S(i)>0}. We use the notation S⊆T if and only if S(i)≤T(i) for all i∈N. Let ∨ and ∧, respectively, represent the maximum and the minimum operators. The union and intersections of fuzzy coalitions S and T given by S∪T and S∩T are defined as (S∪T)(i)=S(i)∨T(i) and (S∩T)(i)=S(i)∧T(i), respectively, for each i∈N. Following are some special fuzzy coalitions.Fori∈N and S∈L(N), the fuzzy coalitions Si and S-i are given by the following: (9)Sij=Siifj=i0otherwise,S-ij=0ifj=i.Sjotherwise.Definition 7.
A TU game with fuzzy coalitions or simply a TU fuzzy game is a pair(U,v) where U∈L(N) and v:L(N)→R are a set function, satisfying v(∅)=0.LetFG(N) denote the class of TU fuzzy games with player set N. Now we define the two classes of TU fuzzy games, namely, the TU fuzzy games in Choquet integral form due to [15] and TU fuzzy games in multilinear extension form due to [17]. As an a priori requirement, the following definition is given.Definition 8.
LetU∈LN and i, j∈N. For any S∈LU, define βij[S] by (10)βijSk=Sj,ifk=iSi,ifk=jSk,otherwise.Definition 9.
GivenS∈LN, let QS=Si∣S(i)>0,i∈N and let qS be the cardinality of QS. Write the elements of QS in the increasing order as h1<⋯<hqs. Then a game v∈FGN is said to be a TU fuzzy game in Choquet integral form if and only if (11)vS=∑l=1qSvShlhl-hl-1foranyS∈LN,whereh0=0. The set of all TU fuzzy games in Choquet integral form is denoted by FGCN.Definition 10 (see [12]).
For any givenU∈L(N) and v′∈G0(N), a TU fuzzy game v∈FG(N) generated by v′ and given by (12)vU=min⁡∑T0⊆suppU∏i∈T0Uiv′T0,v′suppU is said to be a TU fuzzy game “in multilinear extension form.” The set of all TU fuzzy games with form is denoted by multilinear extension form FGMN.Following section includes the main contribution of the present study.


## 2.1. The Solidarity Value and the Share Function for TU Games

Let the player setN be fixed so that the class of TU games can be taken as G0(N). We define the following.Definition 1.
Let∅≠T∈2N and v∈G0(N); the quantity νv(T)=1/T∑k∈T[v(T)-v(T\k)] is called the average marginal contribution of a player of the coalition T.Definition 2.
Given a gamev∈G0(N), player i∈N is called a ν-null player if νv(T)=0, for every coalition T⊆N containing i.Consider a functionΦ:G0(N)→R+n2N that assigns to any game v∈G0(N) a 2N→R+n mapping. For any fixed v∈G0(N) and set W⊆2N, we denote the corresponding n-ary vector in R+n as (Φ1(W,v),…,Φn(W,v)). We define the solidarity value as follows.Definition 3.
A functionΦ:G0N→R+n2N is said to be the solidarity value on G0(N) if it satisfies the following four axioms.
Axiom C 1 (Efficiency). If v∈G0N and W∈2N, then (1)∑i∈WΦiW,v=vW,ΦiW,v=0∀i∉W.
Axiom C 2 ( ν -Null Player). If v∈G0(N) and i∈W∈2N are a ν-null player, then (2)ΦiW,v=0∀i∈T⊂W.
Axiom C 3 (Symmetry). If v∈G0(N), W∈2N, and i,j∈W are symmetric, that is, v(S∪i)=vS∪j holds for any S∈2W\i,j, then (3)ΦiW,v=ΦjW,v.
Axiom C 4 (Additivity). For v1,v2∈G0(N), define v1+v2∈G0(N) by (v1+v2)(S)=v1(S)+v2(S) for each S∈2N. If v1,v2∈G0(N) and W∈2N, then (4)ΦiW,v1+v2=ΦiW,v1+ΦiW,v2.Theorem 4.
Define a functionΦ:G0(N)→R+n2N by (5)ΦiW,v=∑S∈PiWδS,WνvS,ifi∈W0,otherwise, where Pi(W)=S∈2W∣S∋i and δ(|S|,|W|)=S-1!W-S!/W!. Then the function Φ is the unique solidarity value on G0(N).Proof.
We refer to [8] for a detailed proof of Theorem 4.From now onward we denote the solidarity value byΦsol(W,v) where W⊆N and v∈G0(N).Definition 5.
LetC⊆G0(N) be a set of TU games, and let μ:C→R be a given function. A μ-share function on a set of games C⊆G0(N) is a function Ψμ:C→R+n2N that satisfies the following AxiomsCS1,CS2, andCS3 and either AxiomCS4 orCS5:
Axiom CS 1 (μ-Efficiency). If v∈C and K∈2N, then (6)∑i∈KΨiμK,v=1,ΨiμK,v=0,i∉K.
Axiom CS 2 ( μ -Symmetry). If v∈C and K∈2N, i,j∈K, and v(S∪i)= v(S∪j) hold for any S⊆K\i,j, then Ψiμ(K,v)=Ψjμ(K,v).Axiom CS 3 ( μ ν -Null Player). If v∈C and i∈K∈2N is an ν-null player, that is, νv(T)=0, then (7)ΨiμK,v=0∀i∈T⊂K.Axiom CS 4 ( μ -Additivity). For any pair v1,v2∈C such that v1+v2∈C, it holds that μ(K,v1+v2)Ψiμ(K,v1+v2) = μ(K,v1)Ψiμ(K,v1) + μ(K,v2)Ψiμ(K,v2).Axiom CS 5 ( μ -Linearity). For any pair v1, v2 of games in C and for any pair of real numbers a and b such that av1+bv2∈C, it holds that(8)μK,av1+bv2ΨiμK,av1+bv2=aμK,v1ΨiμK,v1+bμK,v2ΨiμK,v2.Theorem 6.
Letμ:C→R be a positive function on C. Then on the subclass C there exists a unique solidarity μ-share function Ψμ:C→R+n2N satisfying the axiomsCS1–CS4 if and only if μ is additive on C.


## 2.2. TU Games with Fuzzy Coalitions

Now we make a brief discourse of TU games with fuzzy coalitions or simply TU fuzzy games with the player setN. A fuzzy coalition is a fuzzy subset of N, which is identified with a characteristic function from N to [0,1]. Let L(N) be the set of all fuzzy coalitions in N. For a fuzzy coalition S∈L(N) and player i∈N, S(i) represents the membership grade of i in S. The empty fuzzy coalition denoted by ∅ is one where all the players provide zero membership. If no ambiguity arises we use the same notations to represent crisp and fuzzy coalitions as crisp coalitions are special fuzzy coalitions with memberships 0 or 1.The support of a fuzzy coalitionS is denoted by suppS={i∈N∣S(i)>0}. We use the notation S⊆T if and only if S(i)≤T(i) for all i∈N. Let ∨ and ∧, respectively, represent the maximum and the minimum operators. The union and intersections of fuzzy coalitions S and T given by S∪T and S∩T are defined as (S∪T)(i)=S(i)∨T(i) and (S∩T)(i)=S(i)∧T(i), respectively, for each i∈N. Following are some special fuzzy coalitions.Fori∈N and S∈L(N), the fuzzy coalitions Si and S-i are given by the following: (9)Sij=Siifj=i0otherwise,S-ij=0ifj=i.Sjotherwise.Definition 7.
A TU game with fuzzy coalitions or simply a TU fuzzy game is a pair(U,v) where U∈L(N) and v:L(N)→R are a set function, satisfying v(∅)=0.LetFG(N) denote the class of TU fuzzy games with player set N. Now we define the two classes of TU fuzzy games, namely, the TU fuzzy games in Choquet integral form due to [15] and TU fuzzy games in multilinear extension form due to [17]. As an a priori requirement, the following definition is given.Definition 8.
LetU∈LN and i, j∈N. For any S∈LU, define βij[S] by (10)βijSk=Sj,ifk=iSi,ifk=jSk,otherwise.Definition 9.
GivenS∈LN, let QS=Si∣S(i)>0,i∈N and let qS be the cardinality of QS. Write the elements of QS in the increasing order as h1<⋯<hqs. Then a game v∈FGN is said to be a TU fuzzy game in Choquet integral form if and only if (11)vS=∑l=1qSvShlhl-hl-1foranyS∈LN,whereh0=0. The set of all TU fuzzy games in Choquet integral form is denoted by FGCN.Definition 10 (see [12]).
For any givenU∈L(N) and v′∈G0(N), a TU fuzzy game v∈FG(N) generated by v′ and given by (12)vU=min⁡∑T0⊆suppU∏i∈T0Uiv′T0,v′suppU is said to be a TU fuzzy game “in multilinear extension form.” The set of all TU fuzzy games with form is denoted by multilinear extension form FGMN.Following section includes the main contribution of the present study.


## 3. Solidarity Value for TU Fuzzy Games

We now discuss the notion of a solidarity value to the classFG(N) of TU fuzzy games with player set N along the line of [2]. Begin with the following definition.Definition 11.
Let∅≠T⊆U∈LN and v∈FG(N), and the quantity (13)fνvT=1suppT∑k∈suppTvT-vT\k is called the average marginal contribution of a player of the fuzzy coalition T.Definition 12.
Given a gamev∈FG(N), player i∈N is called an fν-null player if fνv(T)=0, for every coalition T∈LU with T(i)∈0,1.Note that Definitions11 and 12 are the fuzzy extensions of their counterparts in crisp setting in the sense that if for all T∈FG(N), T(i)=1 and all j∈N, T(j)∈{0,1}, then fνv(T)=νv(T).Definition 13.
A solidarity value function onFG(N) is a function Ωsol:FG(N)→R+nL(N) that satisfies the following four axioms, that is, AxiomsF1–F4.
Axiom F 1 (Efficiency). If v∈FG(N) and U∈L(N), then (14)∑i∈NΩisolU,v=vU,ΩisolU,v=0∀i∉suppU.
Axiom F 2 (Symmetry). If v∈FG(N), U∈L(N) and v(S)=v(βij[S]). For any given S∈L(U) and i,j∈suppU, then Ωisol(U,v)=Ωjsol(U,v).
Axiom F 3 (Additivity). For any u,v∈FGN, Ωsol(U,u+v)=Ωsol(U,u)+Ωsol(U,v).
Axiom F 4 ( f ν -Null Player). If v∈FG(N) and i∈N are a fν-null player, that is, fνv(T)=0 for every fuzzy coalition T∈L(U) with T(i)∈(0,1], then Ωisol(U,v)=0.Note that AxiomsF1–F4 are standard axioms derived from their crisp counterparts and therefore can be applied to any class of fuzzy games. Moreover, when we revert back to the class of crisp games these axioms become the standard AxiomsCS1–CS4.


### 3.1. Solidarity Value for the ClassFGC(N)

We now find the solidarity value for the classFGC(N) by use of the following theorem.Theorem 14.
Letv∈FGCN and U∈LN. A function Ωsol:FGC(N)→R+nL(N), defined by (15)ΩisolU,v=∑l=1qUΦsolUhl,v·hl-hl-1, is a solidarity value function in U for v∈FGC(N), where (16)ΦisolUhl,v=∑i∈T⊆UhlT-1!Uhl-T!Uhl!νTv,ifi∈Uhl0,elsewhere.Proof.
Recall from Theorem4 that there exists a unique function Φsol satisfyingAxioms CS1–CS4. We use this to prove that the function Ωsol satisfiesAxioms F1–F4.
Axiom F 1 (Efficiency). Let v∈FGC(N) and U∈L(N). Since ∑i∈suppUΦisol(Uhl,v)=v(Uhl) holds for any l∈1,…,q(U), we obtain (17)∑i∈NΩisolU,v=∑l=1qU∑i∈NΦsolUhl,v·hl-hl-1=∑l=1qUvUhlhl-hl-1=vU. Since i∉suppU implies i∉Uhl, we must have Φisol(Uhl,v)=0.
It follows thatΩisol(U,v)=∑l=1qUΦsolUhl,v·(hl-hl-1)=0.
Axiom F 2 (Symmetry).Let v∈FGC(N) and U∈L(N). We have the following: v(S)-v(βij[S])=0,∀S∈L(U)⇔v(S)-v(βij[S])=0,∀S∈L(U), such that S(j)=0, S(k)∈{S(i),0}∀k∈suppU, ⇔v(S)-v(βij[S])=0,∀S∈L(U), such that S(i)=h, S(j)=0, and S(k)∈{h,0}∀k∈suppU,∀h∈(0,U(i)], ⇔vS′h∪i-vS′h∪j·h=0,∀S′∈L(U), such that S′(i)=S′(j)=0, and S′(k)∈{h,0}∀k∈suppU,∀h∈(0,U(i)], ⇔{v(S′h∪{i})-v(S′h∪{j})}=0,∀S′∈L(U), such that S′(i)=S′(j)=0, and S′(k)∈{h,0}∀k∈suppU,∀h∈(0,U(i)], ⇔{v(T∪{i})-v(T∪{j})}=0, ∀T∈P(Uh\{i,j}),∀h∈(0,U(i)]. Consequently, if v(S)=v(βij[S]) for any S∈L(U), then v(T∪{i})=v(T∪{j}) for any T∈P(Uh\{i,j}) and h∈(0,U(i)]. Hence we have Φisol(Uh,v)=Φjsol(Uh,v) for any h∈(0,U(i)] and Φisol(Uh,v)=Φjsol(Uh,v)=0 for any h∈(U(i),1]. Therefore, Φisol(Uh,v)=Φjsol(Uh,v) for any h∈(0,1]. It follows that Ωisol(U,v)=Ωjsol(U,v).
Axiom F 3 (Additivity). Since Φsol is additive so for any u, v∈FGCN and by the definition of Ωisol we can easily prove that Ωisol(U,u+v)=Ωisol(U,u)+Ωisol(U,v).
Axiom F 4 ( f ν -Null Player). Let v∈FGC(N) and i∈N is a fν-null player; that is, (18)fνvT=0,for  every  fuzzy  coalitionT∈LUwithTi∈0,1⇓1suppT∑k∈suppTvT-vT\k=0⇓1Thl∑k∈ThlvThl-vThl\k+1Thl∑k∉ThlvThl-vThl\k=0⇓1Thl∑k∈Thl[vThl-vThl\k=0⇓νvThl=0⇓ΦsolUhl,v=0⇓∑l=1qUΦsolUhl,v·hl-hl-1=0⇓ΩisolU,v=0.This completes the proof.


### 3.2. Solidarity Value for the ClassFGM(N)

Theorem 15.
For∅≠T⊆U∈L(N), the game u^T, that is,(19)u^TS=suppSsuppT-1ifS⊃T0otherwise,has the following properties:(i)
u^T(T)=1;(ii)
ifsuppS=suppT∪E with ∅≠E⊂suppU\suppT∈L(N), then (20)u^TS=1suppS∑i∈suppSu^TS-i and every player i∈suppU\suppT is fν-null in the game u^T.Theorem 16.
Letv∈FGMN and U∈LN. A function Ωsol:FGC(N)→R+nL(N) defined by(21)ΩisolU,v=∑T⊆UsuppT-1!suppU-suppT!suppU!AvT,∀i∈suppU, where (22)AvT=1suppT∑k∈suppT∑S0⊆suppT∏j∈S0Ujv′S0-∑S0⊆suppT-k∏j∈S0Ujv′S0,is the unique solidarity value for v∈FGM(N) in U.Proof.
Let us construct any value functionη on FGM(N) satisfying efficiency, symmetry, additivity, and fν-null player axioms by(23)ηiU,κu^T=κsuppUsuppT-1suppT,ifi∈suppT0,otherwise. Now, we know that v∈FGM(N) can be expressed by(24)vS=∑ϕ≠T⊆UcTvu^TS, where (25)cTv=∑S⊆T-1suppT-suppS∑H0⊆suppS∏i∈H0Uiv′H0;clearly Ωsol given by (21) and (22) satisfies symmetry and fν-null player axioms. Moreover, Ωsol is a linear mapping. Hence additivity is satisfied.
Using now linearity ofΩsol, we get (26)∑i∈suppUΩisolU,v=∑∅≠T⊆UcT∑i∈suppUΩisolU,u^T=∑cTu^TU=vU which proves that Ωsol is efficient. It is obvious that η(U,u^T)=Ωisol(U,u^T) for each game u^T. Thus η(U,v)=Ωisol(U,v) for every v∈FGM(N).


## 3.1. Solidarity Value for the ClassFGC(N)

We now find the solidarity value for the classFGC(N) by use of the following theorem.Theorem 14.
Letv∈FGCN and U∈LN. A function Ωsol:FGC(N)→R+nL(N), defined by (15)ΩisolU,v=∑l=1qUΦsolUhl,v·hl-hl-1, is a solidarity value function in U for v∈FGC(N), where (16)ΦisolUhl,v=∑i∈T⊆UhlT-1!Uhl-T!Uhl!νTv,ifi∈Uhl0,elsewhere.Proof.
Recall from Theorem4 that there exists a unique function Φsol satisfyingAxioms CS1–CS4. We use this to prove that the function Ωsol satisfiesAxioms F1–F4.
Axiom F 1 (Efficiency). Let v∈FGC(N) and U∈L(N). Since ∑i∈suppUΦisol(Uhl,v)=v(Uhl) holds for any l∈1,…,q(U), we obtain (17)∑i∈NΩisolU,v=∑l=1qU∑i∈NΦsolUhl,v·hl-hl-1=∑l=1qUvUhlhl-hl-1=vU. Since i∉suppU implies i∉Uhl, we must have Φisol(Uhl,v)=0.
It follows thatΩisol(U,v)=∑l=1qUΦsolUhl,v·(hl-hl-1)=0.
Axiom F 2 (Symmetry).Let v∈FGC(N) and U∈L(N). We have the following: v(S)-v(βij[S])=0,∀S∈L(U)⇔v(S)-v(βij[S])=0,∀S∈L(U), such that S(j)=0, S(k)∈{S(i),0}∀k∈suppU, ⇔v(S)-v(βij[S])=0,∀S∈L(U), such that S(i)=h, S(j)=0, and S(k)∈{h,0}∀k∈suppU,∀h∈(0,U(i)], ⇔vS′h∪i-vS′h∪j·h=0,∀S′∈L(U), such that S′(i)=S′(j)=0, and S′(k)∈{h,0}∀k∈suppU,∀h∈(0,U(i)], ⇔{v(S′h∪{i})-v(S′h∪{j})}=0,∀S′∈L(U), such that S′(i)=S′(j)=0, and S′(k)∈{h,0}∀k∈suppU,∀h∈(0,U(i)], ⇔{v(T∪{i})-v(T∪{j})}=0, ∀T∈P(Uh\{i,j}),∀h∈(0,U(i)]. Consequently, if v(S)=v(βij[S]) for any S∈L(U), then v(T∪{i})=v(T∪{j}) for any T∈P(Uh\{i,j}) and h∈(0,U(i)]. Hence we have Φisol(Uh,v)=Φjsol(Uh,v) for any h∈(0,U(i)] and Φisol(Uh,v)=Φjsol(Uh,v)=0 for any h∈(U(i),1]. Therefore, Φisol(Uh,v)=Φjsol(Uh,v) for any h∈(0,1]. It follows that Ωisol(U,v)=Ωjsol(U,v).
Axiom F 3 (Additivity). Since Φsol is additive so for any u, v∈FGCN and by the definition of Ωisol we can easily prove that Ωisol(U,u+v)=Ωisol(U,u)+Ωisol(U,v).
Axiom F 4 ( f ν -Null Player). Let v∈FGC(N) and i∈N is a fν-null player; that is, (18)fνvT=0,for  every  fuzzy  coalitionT∈LUwithTi∈0,1⇓1suppT∑k∈suppTvT-vT\k=0⇓1Thl∑k∈ThlvThl-vThl\k+1Thl∑k∉ThlvThl-vThl\k=0⇓1Thl∑k∈Thl[vThl-vThl\k=0⇓νvThl=0⇓ΦsolUhl,v=0⇓∑l=1qUΦsolUhl,v·hl-hl-1=0⇓ΩisolU,v=0.This completes the proof.


## 3.2. Solidarity Value for the ClassFGM(N)

Theorem 15.
For∅≠T⊆U∈L(N), the game u^T, that is,(19)u^TS=suppSsuppT-1ifS⊃T0otherwise,has the following properties:(i)
u^T(T)=1;(ii)
ifsuppS=suppT∪E with ∅≠E⊂suppU\suppT∈L(N), then (20)u^TS=1suppS∑i∈suppSu^TS-i and every player i∈suppU\suppT is fν-null in the game u^T.Theorem 16.
Letv∈FGMN and U∈LN. A function Ωsol:FGC(N)→R+nL(N) defined by(21)ΩisolU,v=∑T⊆UsuppT-1!suppU-suppT!suppU!AvT,∀i∈suppU, where (22)AvT=1suppT∑k∈suppT∑S0⊆suppT∏j∈S0Ujv′S0-∑S0⊆suppT-k∏j∈S0Ujv′S0,is the unique solidarity value for v∈FGM(N) in U.Proof.
Let us construct any value functionη on FGM(N) satisfying efficiency, symmetry, additivity, and fν-null player axioms by(23)ηiU,κu^T=κsuppUsuppT-1suppT,ifi∈suppT0,otherwise. Now, we know that v∈FGM(N) can be expressed by(24)vS=∑ϕ≠T⊆UcTvu^TS, where (25)cTv=∑S⊆T-1suppT-suppS∑H0⊆suppS∏i∈H0Uiv′H0;clearly Ωsol given by (21) and (22) satisfies symmetry and fν-null player axioms. Moreover, Ωsol is a linear mapping. Hence additivity is satisfied.
Using now linearity ofΩsol, we get (26)∑i∈suppUΩisolU,v=∑∅≠T⊆UcT∑i∈suppUΩisolU,u^T=∑cTu^TU=vU which proves that Ωsol is efficient. It is obvious that η(U,u^T)=Ωisol(U,u^T) for each game u^T. Thus η(U,v)=Ωisol(U,v) for every v∈FGM(N).


## 4. Solidarity Share Functions for TU Fuzzy Games

We now extend the notion of a share function to the classFG(N) of TU fuzzy games with player set N. In the line of its crisp counterpart we assume here also that the share function assigns to each player her share in the payoff v(U) of the fuzzy coalition U∈L(N). Therefore we provide the following definitions as an extension to their crisp versions.Definition 17.
A real valued functionμ:L(N)×FG(N)→R is called f-additive if, for U∈L(N) and any pair v1,v2∈FG(N) such that v1+v2∈FG(N), it holds that (27)μU,v1+v2=μU,v1+μU,v2.Definition 18.
A real valued functionμ:L(N)×FG(N)→R is called f-linear on the class FG(N) of games if it is f-additive and if for any v on FG(N) and U∈L(N) it holds that μ(U,αv)=αμ(U,v) for any real number α such that αv∈FG(N).Definition 19.
A real valued functionμ:L(N)×FG(N)→R is called positive if μ(U,v)≥0  ∀v∈FG(N),U∈L(N).Definition 20.
Given a functionμ:L(N)×FG(N)→R, a solidarity μ-share function on FG(N) is a function Ψμ:FG(N)→R+nL(N) that satisfies the following axioms, that is, AxiomsFS1–FS3 along with AxiomFS4 or AxiomFS5.
Axiom FS 1 (f-Efficiency). For U∈L(N) we have ∑i∈NΨiμ(U,v)=1 and Ψiμ(U,v)=0, for each i∉suppU.
Axiom FS 2 ( f ν -Null Player). If v∈FG(N) and i∈N are a fν-null player, that is, fνv(T)=0 for every fuzzy coalition T∈L(U) with T(i)∈(0,1], then Ψiμ(U,v)=0.
Axiom FS 3 (f-Symmetry). If v∈FG(N), U∈L(N), and v(S)=v(βij[S]) for any given S∈L(U) and i, j∈suppU, then Ψiμ(U,v)=Ψjμ(U,v).
Axiom FS 4 ( f μ -Additivity). For any pair v1,v2∈FG(N) such that v1+v2∈FG(N), it holds that (28)μU,v1+v2ΨiμU,v1+v2=μU,v1ΨiμU,v1+μU,v2ΨiμU,v2,∀i∈N.
Axiom FS 5 ( f μ -Linearity). For any pair v1,v2∈FG(N) such that v1+v2∈FG(N), it holds that μ(U,av1+bv2)Ψi(U,av1+bv2) = aμ(U,v1)Ψiμ(U,v1) + bμ(U,v2)Ψiμ(U,v2), for any pair of real numbers a and b such that av1+bv2∈FG(N) for all i∈N.Note thatAxioms FS1–FS5 are intuitive of their crisp counterparts in the sense that reverting back to the crisp formulation we get the standard axioms of share functions. It follows that for any v∈FG(N) a solidarity μ-share function Ψμ gives a payoff ΨiU,v·v(U) to player i when she is involved in the fuzzy coalition U and satisfies the above-mentioned axioms.


### 4.1. Solidarity Share Functions forFGC(N)

In this section we prove the existence and uniqueness of the solidarityμ-share function for the class FGC(N) of fuzzy games in Choquet integral form. To discuss the existence and uniqueness of the solidarity μ-share function for TU fuzzy game in FGC(N) we have to use some classical results from [1, 2]. Recall that, given a coalition T⊂K∈P(N), the game wT is defined as follows: (29)wTS=ST-1,ifS⊃T0,otherwise.Due to Theorem 6, for any T∈2K, each v∈G0(N) can be expressed as v=∑T∈2KcTwT where cT(v)=∑R⊆T-1T-Rv(R). Denote C+=T:cT(v)≥0 and C-=T:cT(v)<0. Then(30)v=∑T∈C+cTvwT-∑T∈C-cTvwT.Following similar procedure as in Lemma 3.2. of [10], we can have, for v∈FGC(N), (31)v=∑T∈2NcTvzT,where (32)cTv=∑R⊆T-1T-RvR,zTU=∑l=1qUwTUhl·hl-hl-1∀U∈LN.It follows from the above discussion that v(U) can be rewritten as (33)vU=∑T∈2NcTv≥0cTvzTU-∑T∈2NcTv<0-cTvzTU.Theorem 21.
Letμ:FGC(N)→R be a real valued function. There exists a unique solidarity μ-share function Ψμ:FGC(N)→R+nL(N) that satisfies the axioms of f-efficiency(FS1), fν- null player(FS2), f-symmetry (FS3), and fμ- additivity (FS4) if and only if μ is f-additive on FGC(N).Proof.
The proof proceeds in the line of [6]. First we suppose that Ψμ satisfies f-efficiency and fμ- additivity. It follows that Ψμ is μ-additive on G0(N). Thus we have (34)μU,v1+v2∑i∈NΨiμU,v1+v2=μU,v1+v2Ψ1μU,v1+v2+⋯+ΨsuppUμU,v1+v2=μU,v1∑i∈NΨiU,v1+μU,v2∑i∈NΨiU,v2 for any v1,v2∈FGC(N) such that v1+v2∈FGC(N). f-efficiency then implies that μ(U,v1+v2) = μ(U,v1) + μ(U,v2). Hence μ is f-additive.
Secondly we will show that we can have at most one solidarity share functionΨμ:FGC(N)→R+nL(N) satisfying the four axioms. Let Ψμ:FGC(N)→R+nL(N) be a function satisfying the four axioms. For a positively scaled unanimity game αzT∈FGC(N), α>0 and consequently for αzT∈FGC(N), we obtain (i)
Ψiμ(U,αzT)=1/suppU, when i∈suppU.(ii)
Ψiμ(U,αzT)=0, when i∉suppU.
Again forαzT,α>0 from (i) and (ii) clearly Ψμ satisfies all the four axioms. Thus it follows that for any αzT,α>0 the function Ψμ given by (i) and (ii) is the solidarity μ-share function satisfying the axioms of f-efficiency, fν-null player, and f-symmetry if and only if μ is f-additive.
The uniqueness ofΨμ(U,v) follows immediately. We next show that Ψμ(U,v) satisfies the four axioms for an arbitrary v. The assumption of f-additivity of μ ensures f-efficiency as in the case of crisp games. Consequently the fν-null player axiom also follows. Third, for any U∈L(N) and S∈L(U) with i,j∈suppU, v(S)=v(βij[S]) implies v(T∪{i})=v(T∪{j}),∀T∈P([U]h\{i,j})],∀h∈(0,U(i)], then ci(v) = cj(v), whereas for any other U∈L(N) with nonzero weight cT(v), i and j either both have nonzero memberships in U or both have zero memberships in U. Hence it follows that Ψiμ(U,v)=Ψjμ(U,v)=0 when, j∉suppU.
Next fori, j∈suppU, (35)ΨiμU,v=1μU,v∑T∈C+μU,cTvzT1suppU-∑T∈C-μU,-cTvzT1suppU,ΨjμU,v=1μU,v∑T∈C+μU,cTvzT1suppU-∑T∈C-μU,-cTvzT1suppU.So Ψμ satisfies the symmetry (FS3) axiom. Finally for any two games v1,v2∈FGC(N) we have that (v1+v2)(U)=(∑T∈2N(cT(v1)+cT(v2))zT)(U). Following f-additivity of μ this implies μ(U,v1+v2)Ψμ(U,v1+v2) = μ(U,v1)Ψμ(U,v1) + μ(U,v2)Ψ(U,v2) and hence Ψμ is fμ-additive.Theorem 22.
For given positive numbersωk with k=1,2,…,n, let the function μω be defined by (36)μωU,v=∑i∈N∑l=1qU∑i∈T⊆UhlωkνvThl-hl-1. Then the solidarity μ-share function Ψμω defined by (37)ΨiμωU,v=∑l=1qU∑i∈T⊆UhlωkνvThl-hl-1μωU,v is the unique solidarity μ-share function satisfying the axioms of f-efficiency, fν-null player, f-symmetry, and fμω- additivity on FGC(N) wherever μω is positive.Proof.
By definition,μω is f-additive. Hence the existence and uniqueness of the solidarity μ-share function follows from Theorem 21. We show that Ψμω satisfies the four axioms with respect to μω on the class FGC(N) of μω-positive games. Next we show that Ψμω satisfies the above four axioms. The f-efficiency and fν-null player axioms are direct consequences of their crisp counterparts. Now for any U∈L(N) and S∈L(U) with i, j∈suppU, v(S)=v(βij[S]) implies v(T∪{i})=v(T∪{j}),∀T∈P([U]h\{i,j})],∀h∈(0,U(i)], then we have that fνvS = fνvβij[S] implies νvT∪i = νvT∪j  ∀T∈P([U]h\{i,j}). Following the fact that ωk depends only on the size of T, the symmetry axiom holds. Finally we have μωU,vΨiμωU,v=∑l=1q(U)∑i∈T⊆Uhlωkνv(T)(hl-hl-1). For all T containing i, it holds that νau+bvT=aνvT+bνvT; it follows that Ψμω is μω-additive.In the following theorem, we take a particular form of the functionμ and obtain the corresponding solidarity share function for the class FGC(N). This exemplifies the existence of a wide range of such share functions generated by the various choices of the function μ.Theorem 23.
Let the functionμsol be defined by μsol(U,v)=v(U). Then the solidarity μ-share function Ψμsol is the unique solidarity μ-share function satisfying the axioms of f-efficiency, fν-null player, f-symmetry, and fμsol-linearity on FGC(N).Proof.
ForT⊆Uhl with T=k, take ωk=k-1!|Uhl|-k!/Uhl!. Then, we have that μωsol as defined in Theorem 15 given by(38)μωU,v=∑i∈N∑l=1qU∑i∈T⊆UhlωkνvThl-hl-1=vU=μsolU,v. Further, the share function Ψμsol as defined in Theorem 15 is given by (39)ΨiμsolU,v=∑l=1qU∑i∈T⊆UhlωkνvThl-hl-1μsolU,v=∑l=1qU∑i∈T⊆Uhlk-1!Uhl-k!/Uhl!νvThl-hl-1μsolU,v=ΦisolU,vvU=ΨiμsolU,v,i∈suppU. This completes the proof.


### 4.2. Solidarity Share Functions forFGM(N)

Here we discuss the existence and uniqueness of the solidarityμ-share function for TU fuzzy games in FGM(N) following the definition of the game u^T.Theorem 24.
Letμ:GM(N)→R be a real valued function on the class GM(N) of games. Then on GM(N) there exists a unique μ-share function Ψμ:GM(N)→(R)L(N) that satisfies the axioms of f-efficiency, fν-null player property, f-symmetry, and fμ-additivity if and only if μ is f-additive on GM(N).Proof.
The proof goes exactly in the same line of Theorem16 and hence is omitted.Theorem 25.
For given positive numbersωs with s=1,2,…,n, let the function μω be defined by (40)μωU,v=∑i∈suppU∑T⊆UωsAvT, where(41)AvT=1suppT∑k∈suppT∑S0⊆suppT∏j∈S0Ujv′S0-∑S0⊆suppT-k∏j∈S0Ujv′S0.Then the solidarity μ-share function Ψμω defined by (42)ΨiμωU,v=∑T⊆UωsAvTμωU,v is the unique solidarity μ-share function satisfying the axioms of f-efficiency (FS1), fν-null player (FS2), f-symmetry (FS3), and fμω-additivity (FS4) on FGM(N) wherever μω is positive.Proof.
Along the line of Theorem21, we can easily get the result.Next we obtain a particularμ to exemplify the wide variety of the class of μ-share functions.Theorem 26.
Let the functionμsol be defined by μsol(U,v)=v(U)=W(v). Then the solidarity μ-share function Ψμsol is the unique solidarity μ-share function corresponding to the solidarity value function satisfying the axioms of f-efficiency, fν-null player, f-symmetry, and fμsol-linearity on FGM(N).Proof.
The proof proceeds exactly in the same line of Theorem22 so it is omitted.


## 4.1. Solidarity Share Functions forFGC(N)

In this section we prove the existence and uniqueness of the solidarityμ-share function for the class FGC(N) of fuzzy games in Choquet integral form. To discuss the existence and uniqueness of the solidarity μ-share function for TU fuzzy game in FGC(N) we have to use some classical results from [1, 2]. Recall that, given a coalition T⊂K∈P(N), the game wT is defined as follows: (29)wTS=ST-1,ifS⊃T0,otherwise.Due to Theorem 6, for any T∈2K, each v∈G0(N) can be expressed as v=∑T∈2KcTwT where cT(v)=∑R⊆T-1T-Rv(R). Denote C+=T:cT(v)≥0 and C-=T:cT(v)<0. Then(30)v=∑T∈C+cTvwT-∑T∈C-cTvwT.Following similar procedure as in Lemma 3.2. of [10], we can have, for v∈FGC(N), (31)v=∑T∈2NcTvzT,where (32)cTv=∑R⊆T-1T-RvR,zTU=∑l=1qUwTUhl·hl-hl-1∀U∈LN.It follows from the above discussion that v(U) can be rewritten as (33)vU=∑T∈2NcTv≥0cTvzTU-∑T∈2NcTv<0-cTvzTU.Theorem 21.
Letμ:FGC(N)→R be a real valued function. There exists a unique solidarity μ-share function Ψμ:FGC(N)→R+nL(N) that satisfies the axioms of f-efficiency(FS1), fν- null player(FS2), f-symmetry (FS3), and fμ- additivity (FS4) if and only if μ is f-additive on FGC(N).Proof.
The proof proceeds in the line of [6]. First we suppose that Ψμ satisfies f-efficiency and fμ- additivity. It follows that Ψμ is μ-additive on G0(N). Thus we have (34)μU,v1+v2∑i∈NΨiμU,v1+v2=μU,v1+v2Ψ1μU,v1+v2+⋯+ΨsuppUμU,v1+v2=μU,v1∑i∈NΨiU,v1+μU,v2∑i∈NΨiU,v2 for any v1,v2∈FGC(N) such that v1+v2∈FGC(N). f-efficiency then implies that μ(U,v1+v2) = μ(U,v1) + μ(U,v2). Hence μ is f-additive.
Secondly we will show that we can have at most one solidarity share functionΨμ:FGC(N)→R+nL(N) satisfying the four axioms. Let Ψμ:FGC(N)→R+nL(N) be a function satisfying the four axioms. For a positively scaled unanimity game αzT∈FGC(N), α>0 and consequently for αzT∈FGC(N), we obtain (i)
Ψiμ(U,αzT)=1/suppU, when i∈suppU.(ii)
Ψiμ(U,αzT)=0, when i∉suppU.
Again forαzT,α>0 from (i) and (ii) clearly Ψμ satisfies all the four axioms. Thus it follows that for any αzT,α>0 the function Ψμ given by (i) and (ii) is the solidarity μ-share function satisfying the axioms of f-efficiency, fν-null player, and f-symmetry if and only if μ is f-additive.
The uniqueness ofΨμ(U,v) follows immediately. We next show that Ψμ(U,v) satisfies the four axioms for an arbitrary v. The assumption of f-additivity of μ ensures f-efficiency as in the case of crisp games. Consequently the fν-null player axiom also follows. Third, for any U∈L(N) and S∈L(U) with i,j∈suppU, v(S)=v(βij[S]) implies v(T∪{i})=v(T∪{j}),∀T∈P([U]h\{i,j})],∀h∈(0,U(i)], then ci(v) = cj(v), whereas for any other U∈L(N) with nonzero weight cT(v), i and j either both have nonzero memberships in U or both have zero memberships in U. Hence it follows that Ψiμ(U,v)=Ψjμ(U,v)=0 when, j∉suppU.
Next fori, j∈suppU, (35)ΨiμU,v=1μU,v∑T∈C+μU,cTvzT1suppU-∑T∈C-μU,-cTvzT1suppU,ΨjμU,v=1μU,v∑T∈C+μU,cTvzT1suppU-∑T∈C-μU,-cTvzT1suppU.So Ψμ satisfies the symmetry (FS3) axiom. Finally for any two games v1,v2∈FGC(N) we have that (v1+v2)(U)=(∑T∈2N(cT(v1)+cT(v2))zT)(U). Following f-additivity of μ this implies μ(U,v1+v2)Ψμ(U,v1+v2) = μ(U,v1)Ψμ(U,v1) + μ(U,v2)Ψ(U,v2) and hence Ψμ is fμ-additive.Theorem 22.
For given positive numbersωk with k=1,2,…,n, let the function μω be defined by (36)μωU,v=∑i∈N∑l=1qU∑i∈T⊆UhlωkνvThl-hl-1. Then the solidarity μ-share function Ψμω defined by (37)ΨiμωU,v=∑l=1qU∑i∈T⊆UhlωkνvThl-hl-1μωU,v is the unique solidarity μ-share function satisfying the axioms of f-efficiency, fν-null player, f-symmetry, and fμω- additivity on FGC(N) wherever μω is positive.Proof.
By definition,μω is f-additive. Hence the existence and uniqueness of the solidarity μ-share function follows from Theorem 21. We show that Ψμω satisfies the four axioms with respect to μω on the class FGC(N) of μω-positive games. Next we show that Ψμω satisfies the above four axioms. The f-efficiency and fν-null player axioms are direct consequences of their crisp counterparts. Now for any U∈L(N) and S∈L(U) with i, j∈suppU, v(S)=v(βij[S]) implies v(T∪{i})=v(T∪{j}),∀T∈P([U]h\{i,j})],∀h∈(0,U(i)], then we have that fνvS = fνvβij[S] implies νvT∪i = νvT∪j  ∀T∈P([U]h\{i,j}). Following the fact that ωk depends only on the size of T, the symmetry axiom holds. Finally we have μωU,vΨiμωU,v=∑l=1q(U)∑i∈T⊆Uhlωkνv(T)(hl-hl-1). For all T containing i, it holds that νau+bvT=aνvT+bνvT; it follows that Ψμω is μω-additive.In the following theorem, we take a particular form of the functionμ and obtain the corresponding solidarity share function for the class FGC(N). This exemplifies the existence of a wide range of such share functions generated by the various choices of the function μ.Theorem 23.
Let the functionμsol be defined by μsol(U,v)=v(U). Then the solidarity μ-share function Ψμsol is the unique solidarity μ-share function satisfying the axioms of f-efficiency, fν-null player, f-symmetry, and fμsol-linearity on FGC(N).Proof.
ForT⊆Uhl with T=k, take ωk=k-1!|Uhl|-k!/Uhl!. Then, we have that μωsol as defined in Theorem 15 given by(38)μωU,v=∑i∈N∑l=1qU∑i∈T⊆UhlωkνvThl-hl-1=vU=μsolU,v. Further, the share function Ψμsol as defined in Theorem 15 is given by (39)ΨiμsolU,v=∑l=1qU∑i∈T⊆UhlωkνvThl-hl-1μsolU,v=∑l=1qU∑i∈T⊆Uhlk-1!Uhl-k!/Uhl!νvThl-hl-1μsolU,v=ΦisolU,vvU=ΨiμsolU,v,i∈suppU. This completes the proof.


## 4.2. Solidarity Share Functions forFGM(N)

Here we discuss the existence and uniqueness of the solidarityμ-share function for TU fuzzy games in FGM(N) following the definition of the game u^T.Theorem 24.
Letμ:GM(N)→R be a real valued function on the class GM(N) of games. Then on GM(N) there exists a unique μ-share function Ψμ:GM(N)→(R)L(N) that satisfies the axioms of f-efficiency, fν-null player property, f-symmetry, and fμ-additivity if and only if μ is f-additive on GM(N).Proof.
The proof goes exactly in the same line of Theorem16 and hence is omitted.Theorem 25.
For given positive numbersωs with s=1,2,…,n, let the function μω be defined by (40)μωU,v=∑i∈suppU∑T⊆UωsAvT, where(41)AvT=1suppT∑k∈suppT∑S0⊆suppT∏j∈S0Ujv′S0-∑S0⊆suppT-k∏j∈S0Ujv′S0.Then the solidarity μ-share function Ψμω defined by (42)ΨiμωU,v=∑T⊆UωsAvTμωU,v is the unique solidarity μ-share function satisfying the axioms of f-efficiency (FS1), fν-null player (FS2), f-symmetry (FS3), and fμω-additivity (FS4) on FGM(N) wherever μω is positive.Proof.
Along the line of Theorem21, we can easily get the result.Next we obtain a particularμ to exemplify the wide variety of the class of μ-share functions.Theorem 26.
Let the functionμsol be defined by μsol(U,v)=v(U)=W(v). Then the solidarity μ-share function Ψμsol is the unique solidarity μ-share function corresponding to the solidarity value function satisfying the axioms of f-efficiency, fν-null player, f-symmetry, and fμsol-linearity on FGM(N).Proof.
The proof proceeds exactly in the same line of Theorem22 so it is omitted.


## 5. Conclusion

We have discussed the notion of solidarity value and solidarity share function on a class of TU fuzzy games. The solidarity share function on the two classesFGCN and FGM(N) is illustrated. Few consequent properties and relationships have been investigated. Other solution concepts of TU fuzzy games can also be studied in a similar way which is kept for our future work.



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*Source: 3502949-2018-04-15.xml*