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Solidarity Value and Solidarity Share Functions for TU Fuzzy Games.

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 [2.sup.N] that assigns to every subset (coalition) a real number called its worth which gives zero worth to the empty coalition. Let [G.sub.0] denote the class of TU games. A solution for any TU game is a function on [G.sub.0] 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 [G.sub.0](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 v-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 Section 2, 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 set N be fixed so that the class of TU games can be taken as [G.sub.0](N). We define the following.

Definition 1. Let 0 [not equal to] T [member of] [2.sup.N] and v [member of] [G.sub.0](N); the quantity [v.sup.v](T) = (1/[absolute value of T]) [[summation].sub.k[member of]T][v(T) - v(T\k)] is called the average marginal contribution of a player of the coalition T.

Definition 2. Given a game v [member of] [G.sub.0](N), player i [member of] N is called a v-null player if [v.sup.v](T) = 0, for every coalition T [subset or equal to] N containing i.

Consider a function [mathematical expression not reproducible] that assigns to any game v [member of] [G.sub.0](N) a [2.sup.N] [right arrow] [R.sup.n.sub.+] mapping. For any fixed v [member of] [G.sub.0](N) and set W [subset or equal to] [2.sup.N], we denote the corresponding n-ary vector in [R.sup.n.sub.+] as ([[PHI].sub.1](W, v), ..., [[PHI].sub.n] (W, v)). We define the solidarity value as follows.

Definition 3. A function [mathematical expression not reproducible] is said to be the solidarity value on [G.sub.0](N) if it satisfies the following four axioms.

Axiom [C.sub.1] (Efficiency). If v [member of] [G.sub.0](N) and W [member of] [2.sup.N], then

[summation over i [member of] W] [[PHI].sub.i] (W, v) = v (W),

[[PHI].sub.i] (W, v) = 0 [for all]i [not member of] W. (1)

Axiom [C.sub.2] (v-Null Player). If v [member of] [G.sub.0](N) and i [member of] W [member of] [2.sup.N] are a v-null player, then

[[PHI].sub.i] (W, v) = 0 [for all]i [member of] T [subset] W. (2)

Axiom [C.sub.3] (Symmetry). If v [member of] [G.sub.0](N), W [member of] [2.sup.N], and i, j [member of] W are symmetric, that is, v(S [union] i) = v(S [union] j) holds for any S [member of] [2.sup.W\{I,j}], then

[[PHI].sub.i] (W, v) = [[PHI].sub.j] (W, v). (3)

Axiom [C.sub.4] (Additivity). For [v.sub.1], [v.sub.2] [member of] [G.sub.0](N), define [v.sub.1] + [v.sub.2] [member of] [G.sub.0](N) by ([v.sub.1] + [v.sub.2])(S) = [v.sub.1](S) + [v.sub.2](S) for each S [member of] [2.sup.N]. If [v.sub.1], [v.sub.2] [member of] [G.sub.0](N) and W [member of] [2.sup.N], then

[[PHI].sub.i] (W, [v.sub.1] + [v.sub.2]) = [[PHI].sub.i] (W, [v.sub.1]) + [[PHI].sub.i] (W, [v.sub.2]). (4)

Theorem 4. Define a function [mathematical expression not reproducible] by

[mathematical expression not reproducible], (5)

where [P.sub.i](W) = {S [member of] [2.sup.W] | S [not contains as member] i} and [delta]([absolute value of S], [absolute value of W]) = ([absolute value of S] - 1)!([absolute value of W] - [absolute value of S])!/[absolute value of W]!. Then the function [PHI] is the unique solidarity value on [G.sub.0](N).

Proof. We refer to [8] for a detailed proof of Theorem 4.

From now onward we denote the solidarity value by [[PHI].sup.sol](W, v) where W [subset or equal to] N and v [member of] [G.sub.0](N).

Definition 5. Let C [subset or equal to] [G.sub.0](N) be a set of TU games, and let [mu] : C [right arrow] R be a given function. A [mu]-share function on a set of games C [subset or equal to] [G.sub.0](N) is a function [mathematical expression not reproducible] that satisfies the following Axioms C[S.sub.1], C[S.sub.2], and C[S.sub.3] and either Axiom C[S.sub.4] or C[S.sub.5]:

Axiom C[S.sub.1] ([mu]-Efficiency). If v [member of] C and K [member of] [2.sup.N], then

[summation over (i [member of] K)] [[PSI].sup.[mu].sub.i] (K, v) = 1,

[[PSI].sup.[mu].sub.i] (K, v) = 0,

i [not contains as member] K. (6)

Axiom C[S.sub.2] ([mu]-Symmetry). If v [member of] C and K [member of] [2.sup.N], i, j [member of] K, and v(S [union] {i}) = v(S [union] {j}) hold for any S [subset or equal to] K\{i, j}, then [[PSI].sup.[mu].sub.i] (K, v) = [[PSI].sup.[mu].sub.j](K, v).

Axiom C[S.sub.3] ([mu]v-Null Player). If v [member of] C and i [member of] K [member of] [2.sup.N] is an v-null player, that is, [v.sup.v](T) = 0, then

[[PSI].sup.[mu].sub.j] (K, v) = 0 [for all]i [member of] T [subset] K. (7)

Axiom C[S.sub.4] ([mu]-Additivity). For any pair [v.sub.1], [v.sub.2] [member of] C such that [v.sub.1] + [v.sub.2] [member of] C, it holds that [mu](K, [v.sub.1] + [v.sub.2])[[PSI].sup.mu].sub.i](K, [v.sub.1] + [v.sub.2]) = [mu](K, [v.sub.1]) [[PSI].sup.[mu].sub.i] (K, [v.sub.1]) + [mu](K, [v.sub.2]) [[PSI].sup.[mu].sub.i](K, [v.sub.2]).

Axiom C[S.sub.5] ([mu]-Linearity). For any pair [v.sub.1], [v.sub.2] of games in C and for any pair of real numbers a and b such that a[v.sub.1] + b[v.sub.2] [member of] C, it holds that

[mu] (K, a[v.sub.1] + b[v.sub.2]) [[PSI].sup.[mu].sub.i] (K, a[v.sub.1] + b[v.sub.2])

= a[mu] (K, [v.sub.1]) [[PSI].sup.[mu].sub.i] (K, [v.sub.1]) + b[mu] (K, [v.sub.2]) [[PSI].sup.[mu].sub.i] (K, [v.sub.2]). (8)

Theorem 6. Let [mu] : C [right arrow] R be a positive function on C. Then on the subclass C there exists a unique solidarity [mu]-share function [mathematical expression not reproducible] satisfying the axioms C[S.sub.1]-C[S.sub.4] if and only if [mu] 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 set N. 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 [member of] L(N) and player i [member of] N, S(i) represents the membership grade of i in S. The empty fuzzy coalition denoted by 0 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 coalition S is denoted by supp S = {i [member of] N | S(i) > 0}. We use the notation S [subset or equal to] T if and only if S(i) [less than or equal to] T(i) for all i [member of] N. Let [disjunction] and [conjunction], respectively, represent the maximum and the minimum operators. The union and intersections of fuzzy coalitions S and T given by S [union] T and S [intersection] T are defined as (S [union] T)(i) = S(i) [disjunction] T(i) and (S [intersection] T)(i) = S(i) [conjunction] T(i), respectively, for each i [member of] N. Following are some special fuzzy coalitions.

For i [member of] N and S [member of] L(N), the fuzzy coalitions [S.sub.i] and [S.sub.-i] are given by the following:

[mathematical expression not reproducible]. (9)

Definition 7. A TU game with fuzzy coalitions or simply a TU fuzzy game is a pair (U, v) where U [member of] L(N) and v : L(N) [right arrow] R are a set function, satisfying v(0) = 0.

Let FG(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. Let U [member of] L(N) and i, j [member of] N. For any S [member of] L(U), define [[beta].sub.ij] [S] by

[mathematical expression not reproducible]. (10)

Definition 9. Given S [member of] L(N), let Q(S) = {S(i) | S(i) > 0, i [member of] N} and let q(S) be the cardinality of Q(S). Write the elements of Q(S) in the increasing order as [h.sub.1] < ... < [h.sub.q(s)]. Then a game v [member of] FG(N) is said to be a TU fuzzy game in Choquet integral form if and only if

[mathematical expression not reproducible] (11)

for any S [member of] L (N), where [h.sub.0] = 0.

The set of all TU fuzzy games in Choquet integral form is denoted by [FG.sub.C](N).

Definition 10 (see [12]). For any given U [member of] L(N) and v [member of] [G.sub.0](N), a TU fuzzy game v [member of] FG(N) generated by v' and given by

[mathematical expression not reproducible] (12)

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 [FG.sub.M](N).

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 class FG(N) of TU fuzzy games with player set N along the line of [2]. Begin with the following definition.

Definition 11. Let 0 [not member of] T [subset or equal to] U [member of] L(N) and v [member of] FG(N), and the quantity

f[v.sup.v] (T) = 1/[absolute value of supp T] [summation over (k [member of] supp T)] [v (T) - v (T\k)] (13)

is called the average marginal contribution of a player of the fuzzy coalition T.

Definition 12. Given a game v [member of] FG(N), player i [member of] N is called an fv-null player if f[v.sup.v](T) = 0, for every coalition T [member of] L(U) with T(i) [member of] (0,1].

Note that Definitions 11 and 12 are the fuzzy extensions of their counterparts in crisp setting in the sense that if for all T [member of] FG(N), T(i) = 1 and all j [member of] N, T(j) [member of] {0,1}, then f[v.sup.v](T) = [v.sup.v](T).

Definition 13. A solidarity value function on FG(N) is a function [[OMEGA].sup.sol] : FG(N) [right arrow] [([R.sup.n.sub.+]).sup.L(N)] that satisfies the following four axioms, that is, Axioms [F.sub.1]-[F.sub.4].

Axiom [F.sub.1] (Efficiency). If v [member of] FG(N) and U [member of] L(N), then

[summation over (i [member of] N)] [[OMEGA].sup.sol.sub.i] (U, v) = v(U),

[[OMEGA].sup.sol.sub.i] (U, v) = 0 [for all]i [not member of] supp U. (14)

Axiom [F.sub.2] (Symmetry). If v [member of] FG(N), U [member of] L(N) and v(S) = V([[beta].sub.ij][S]). For any given S [member of] L(U) and i, j [member of] supp U, then [[OMEGA].sup.sol.sub.i](U, v) = [[OMEGA].sup.sol.sub.j](U, v).

Axiom [F.sub.3] (Additivity). For any u, v [[OMEGA].sup.sol.sub.i] FG(N), [[OMEGA].sup.sol](U, u + v) = [[OMEGA].sup.sol](U, u) + [[OMEGA].sup.sol](U, v).

Axiom [F.sub.4] (fv-Null Player). If v [member of] FG(N) and i [member of] N are a fv-null player, that is, f[v.sup.v](T) = 0 for every fuzzy coalition T [member of] L(U) with T(i) [member of] (0,1], then [[OMEGA].sup.sol.sub.i](U, v) = 0.

Note that Axioms [F.sub.1]-[F.sub.4] 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 Axioms C[S.sub.1]-C[S.sub.4].

3.1. Solidarity Value for the Class [FG.sub.C](N). We now find the solidarity value for the class [FG.sub.C](N) by use of the following theorem.

Theorem 14. Let v [member of] [FG.sub.C](N) and U [member of] L(N). A function [[OMEGA].sup.sol] : [FG.sub.C](N) [right arrow] [([R.sup.n.sub.+]).sup.L(N)], defined by

[mathematical expression not reproducible], (15)

is a solidarity value function in U for v [member of] [FG.sub.C](N), where

[mathematical expression not reproducible]. (16)

Proof. Recall from Theorem 4 that there exists a unique function [[PHI].sup.sol] satisfying Axioms C[S.sub.1]-C[S.sub.4]. We use this to prove that the function [[OMEGA].sup.sol] satisfies Axioms [F.sub.1]-[F.sub.4].

Axiom [F.sub.1] (Efficiency). Let v [member of] [FG.sub.C](N) and U [member of] L(N). Since [mathematical expression not reproducible] holds for any l [member of] 1, ..., q(U), we obtain

[mathematical expression not reproducible]. (17)

Since i [not member of] supp U implies [mathematical expression not reproducible], we must have [mathematical expression not reproducible].

It follows that [mathematical expression not reproducible].

Axiom [F.sub.2] (Symmetry). Let v [member of] [FG.sub.C](N) and U [member of] L(N). We have the following: v(S) - v([[beta].sub.ij][S]) = 0, [for all]S [member of] L(U) [??] v(S) - v([[beta].sub.ij][S]) = 0, [for all]S [member of] L(U), such that S(j) = 0, S(k) [member of] {S(i),0} [for all]k [member of] supp U, [??] v(S) - v([[beta].sub.ij][S]) = 0, [for all]S [member of] L(U), such that S(i) = h, S(j) = 0, and S(k) [member of] {h, 0} [for all]k [member of] suppU, [for all]h [member of] (0,U(i)], [mathematical expression not reproducible], such that S'(i) = S'(j) = 0, and S'(k) [member of] {h, 0} [for all]k [member of] suppU, [for all]h [member of] (0, U(i)], [??] {v([[S'].sub.h] [union] {i}) - v([[S'].sub.h] [union] {j})} = 0, [for all]S' [member of] L(U), such that S'(i) = S'(j) = 0, and S'(k) [member of] {h,0} [for all]k [member of] suppU, [for all]h [member of] (0,U(i)], [??] {v(T [union] {i}) - v(T [union] {j})} = 0, [for all]T [member of] P([[U].sub.h]\{i, j}), [for all]h [member of] (0, U(i)]. Consequently, if v(S) = v([[beta].sub.ij][S]) for any S [member of] L(U), then v(T [union] {i}) = v(T [union] {j}) for any T [member of] P([[U].sub.h]\{i, j}) and h [member of] (0, U'(i)]. Hence we have [[OMEGA].sup.sol.sub.i]([[U].sub.h], v) = [[OMEGA].sup.sol.sub.j]([[U].sub.h], v) for any h [member of] (0, U(i)] and [[OMEGA].sup.sol.sub.i]([[U].sub.h], v) = [[OMEGA].sup.sol.sub.j]([[U].sub.h], v) = 0 for any h [member of] (U(i), 1]. Therefore, [[OMEGA].sup.sol.sub.i]([[U].sub.h], v) = [[OMEGA].sup.sol.sub.j]([[U].sub.h], v) for any h [member of] (0,1]. It follows that [[OMEGA].sup.sol.sub.i](U,) v) = [[OMEGA].sup.sol.sub.j](U, v).

Axiom [F.sub.3] (Additivity). Since [[OMEGA].sup.sol] is additive so for any u, v [member of] [FG.sub.C](N) and by the definition of [[OMEGA].sup.sol.sub.i] we can easily prove that [[OMEGA].sup.sol.sub.i](G, u + v) = [[OMEGA].sup.sol.sub.i] (U, u) + [[OMEGA].sup.sol.sub.i] (U, v).

Axiom [F.sub.4] (fv-Null Player). Let v [member of] FGC(N) and i [member of] N is a fv-null player; that is,

f[v.sup.v] (T) = 0,

for every fuzzy coalition T [member of] L(U) with T (i) [member of] (0,1]

[mathematical expression not reproducible]. (18)

This completes the proof.

3.2. Solidarity Value for the Class [FG.sub.M](N)

Theorem 15. For 0 [not equal to] T [subset or equal to] U [member of] L(N), the game [[??].sub.T], that is,

[mathematical expression not reproducible], (19)

has the following properties:

(i) [[??].sub.T](T) = 1;

(ii) if supp S = supp T [union] E with 0 [not equal to] E [subset] supp U\supp T [member of] L(N), then

[[??].sub.T] (S) = 1/[absolute value of supp S] [summation over (i [member of] supp S] [[??].sub.T] ([S.sub.-i]) (20)

and every player i [member of] supp U\supp T is fv-null in the game [[??].sub.T].

Theorem 16. Let v [member of] [FG.sub.M](N) and U [member of] L(N). A function [[OMEGA].sup.sol] : [FG.sub.C](N) [right arrow] [([R.sup.n.sub.+]).sup.L(N)] defined by

[mathematical expression not reproducible], (21)

where

[mathematical expression not reproducible], (22)

is the unique solidarity value for v [member of] [FG.sub.M](N) in U.

Proof. Let us construct any value function [eta] on [FG.sub.M](N) satisfying efficiency, symmetry, additivity, and f v-null player axioms by

[mathematical expression not reproducible]. (23)

Now, we know that v [member of] [FG.sub.M](N) can be expressed by

v(S) = ([summation over ([phi] [not equal to] T [subset or equal to] U)] [c.sub.T](v) [[??].sub.T] (S)), (24)

where

[mathematical expression not reproducible]; (25)

clearly [[OMEGA].sup.sol] given by (21) and (22) satisfies symmetry and fv-null player axioms. Moreover, [[OMEGA].sup.sol] is a linear mapping. Hence additivity is satisfied.

Using now linearity of [[OMEGA].sup.sol], we get

[summation over (i [member of] supp U)] [[OMEGA].sup.sol.sub.i] (U, v) = [summation over (0 [not equal to] T [subset or equal to] U)] [c.sub.T] [summation over (I [member of] supp U)] [[OMEGA].sup.sol.sub.i] (U, [[??].sub.T])

= [summation] [c.sub.T][[??].sub.T] (U) = v (U) (26)

which proves that [[OMEGA].sup.sol] is efficient. It is obvious that [eta](U, [[??].sub.T]) = [[OMEGA].sup.sol.sub.i] (U, [[??].sub.T]) for each game [[??].sub.T]. Thus [eta](U, v) = [[OMEGA].sup.sol.sub.i] (U, v) for every v [member of] [FG.sub.M](N).

4. Solidarity Share Functions for TU Fuzzy Games

We now extend the notion of a share function to the class FG(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 [member of] L(N). Therefore we provide the following definitions as an extension to their crisp versions.

Definition 17. A real valued function [mu] : L(N) x FG(N) [right arrow] R is called f-additive if, for U [member of] L(N) and any pair [v.sub.1], [v.sub.2] [member of] FG(N) such that [v.sub.1] + [v.sub.2] [member of] FG(N), it holds that

[mu](U, [v.sub.1] + [v.sub.2]) = [mu] (U, [v.sub.1]) + [mu] (U, [v.sub.2]). (27)

Definition 18. A real valued function [mu] : L(N) x FG(N) [right arrow] 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 [member of] L(N) it holds that [mu](U, [alpha]v) = [alpha][mu](U, v) for any real number [alpha] such that [alpha]v [member of] FG(N).

Definition 19. A real valued function [mu] : L(N) x FG(N) [right arrow] R is called positive if [mu](U, v) [greater than or equal to] 0 [for all]v [member of] FG(N), U [member of] L(N).

Definition 20. Given a function [mu] : L(N) x FG(N) [right arrow] R, a solidarity [mu]-share function on FG(N) is a function [[PSI].sup.[mu]] : FG(N) [right arrow] [([R.sup.n.sub.+]).sup.L(N)] that satisfies the following axioms, that is, Axioms [FS.sub.1]-[FS.sub.3] along with Axiom [FS.sub.4] or Axiom [FS.sub.5].

Axiom [FS.sub.1] (f-Efficiency). For U [member of] L(N) we have [[summation].sub.i [member of] N] [[PSI].sup.[mu].sub.i] (U, v) = 1 and [[PSI].sup.[mu].sub.i] (U, v) = 0, for each i [not member of] supp U.

Axiom [FS.sub.2] (fv-Null Player). If v [member of] FG(N) and i [member of] N are a fv-null player, that is, f[v.sup.v](T) = 0 for every fuzzy coalition T [member of] L(U) with T(i) [member of] (0,1], then [[PSI].sup.[mu].sub.i] (U, v) = 0.

Axiom [FS.sub.3] (f-Symmetry). If v [member of] FG(N), U [member of] L(N), and v(S) = v([[beta].sub.ij][S]) for any given S [member of] L(U) and i, j [member of] supp U, then [[PSI].sup.[mu].sub.i](U, v) = [[PSI].sup.[mu].sub.j] (U, v).

Axiom [FS.sub.4] (f[mu]-Additivity). For any pair [v.sub.1], [v.sub.2] [member of] FG(N) such that [v.sub.1] + [v.sub.2] [member of] FG(N), it holds that

[mathematical expression not reproducible]. (28)

Axiom [FS.sub.5] (f[mu]-Linearity). For any pair [v.sub.]1, [v.sub.2] [member of] FG(N) such that [v.sub.1] + [v.sub.2] [member of] FG(N), it holds that [mu](U, a[v.sub.1] + b[v.sub.2]) [[PSI].sub.i](U, a[v.sub.1] + b[v.sub.2]) = a[mu](U, [v.sub.1]) [[PSI].sup.[mu].sub.i](U, [v.sub.1]) + b[mu](U, [v.sub.2])[[PSI].sup.[mu].sub.i](U, [v.sub.2]), for any pair of real numbers a and b such that a[v.sub.1] + b[v.sub.2] [member of] FG(N) for all i [member of] N.

Note that Axioms [FS.sub.1]-[FS.sub.5] 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 [member of] FG(N) a solidarity [mu]-share function [[PSI].sup.[mu]] gives a payoff [[PSI].sub.i](U, v)xv(U) to player i when she is involved in the fuzzy coalition U and satisfies the above-mentioned axioms.

4.1. Solidarity Share Functions for [FG.sub.C](N). In this section we prove the existence and uniqueness of the solidarity [mu]-share function for the class [FG.sub.C](N) of fuzzy games in Choquet integral form. To discuss the existence and uniqueness of the solidarity [mu]-share function for TU fuzzy game in [FG.sub.C](N) we have to use some classical results from [1,2]. Recall that, given a coalition T [subset] K [member of] P(N), the game [w.sub.T] is defined as follows:

[mathematical expression not reproducible]. (29)

Due to Theorem 6, for any T [member of] [2.sup.K], each v [member of] [G.sub.0](N) can be expressed as [mathematical expression not reproducible] where [c.sub.T](v) = [[summation].sub.R [subset or equal to] T] [(-1).sup.[absolute value of T]-[absolute value of R]]v(R). Denote [C.sup.+] = {T : [c.sub.T](v) [greater than or equal to] 0} and [C.sup.-] = {T : [c.sub.T](v) < 0}. Then

v = ([summation over (T [member of] [C.sup.+]] [c.sub.T](v)[w.sub.T]) - ([summation over (T [member of] [C.sup.-]] [c.sub.T](v)[w.sub.T]). (30)

Following similar procedure as in Lemma 3.2. of [10], we can have, for v [member of] [FG.sub.C](N),

v = (([summation over (T [member of] [2.sup.N]] [c.sub.T](v)[z.sub.T])), (31)

where

[mathematical expression not reproducible]. (32)

It follows from the above discussion that v(U) can be rewritten as

[mathematical expression not reproducible]. (33)

Theorem 21. Let [mu] : [FG.sub.C](N) [right arrow] R be a real valued function. There exists a unique solidarity [mu]-share function [[PSI].sup.[mu]] : [FG.sub.C](N) [right arrow] [([R.sup.n.sub.+]).sup.L(N)] that satisfies the axioms of f-efficiency ([FS.sub.1]), fv- null player ([FS.sub.2]), f-symmetry ([FS.sub.3]), and f[mu]-additivity ([F.sub.S4]) if and only if f [mu] is f-additive on [FG.sub.C](N).

Proof. The proof proceeds in the line of [6]. First we suppose that [[PSI].sup.[mu]] satisfies f-efficiency and f[mu]-additivity. It follows that [[PSI].sup.[mu]] is [mu]-additive on [G.sub.0](N). Thus we have

[mathematical expression not reproducible] (34)

for any [v.sub.1], [v.sub.2] [member of] [FG.sub.C](N) such that [v.sub.1] + [v.sub.2] [member of] [FG.sub.C](N). f- efficiency then implies that [mu](U, [v.sub.1] + [v.sub.2]) = [mu](U, [v.sub.1]) + [mu](U, [v.sub.2]). Hence [mu] is f-additive.

Secondly we will show that we can have at most one solidarity share function [[PSI].sup.[mu]] : [FG.sub.C](N) [right arrow] [([R.sup.n.sub.+]).sup.L(N)] satisfying the four axioms. Let [[PSI].sup.[mu]] : [FG.sub.C](N) [right arrow] [([R.sup.n.sub.+]).sup.L(N)] be a function satisfying the four axioms. For a positively scaled unanimity game [alpha][z.sub.T] [member of] [FG.sub.C](N), [alpha] > 0 and consequently for [alpha][z.sub.T] [member of] [FG.sub.C](N), we obtain

(i) [[PSI].sup.[mu].sub.i](U, [alpha][z.sub.T]) = 1/[absolute value of supp U], when i [member of] supp U.

(ii) [[PSI].sup.[mu].sub.i](U, [alpha][z.sub.T]) = 0, when i [not member of] supp U.

Again for [alpha][z.sub.T], [alpha] > 0 from (i) and (ii) clearly [[PSI].sup.[mu]] satisfies all the four axioms. Thus it follows that for any [alpha][z.sub.T], [alpha] > 0 the function [[PSI].sup.[mu]] given by (i) and (ii) is the solidarity [mu]-share function satisfying the axioms of f-efficiency, fv-null player, and f-symmetry if and only if [mu] is f-additive.

The uniqueness of [[PSI].sup.[mu]](U, v) follows immediately. We next show that [[PSI].sup.[mu]] (U, v) satisfies the four axioms for an arbitrary v. The assumption of f-additivity of [mu] ensures f-efficiency as in the case of crisp games. Consequently the fv-null player axiom also follows. Third, for any U [member of] L(N) and S [member of] L(U) with i, j [member of] supp U, v(S) = v([[beta].sub.ij][S]) implies v(T [union] {i}) = v(T [union] {j}), [for all]T [member of] P([[U].sub.h]\{i, j})], [for all]h [member of] (0, U(i)],then [c.sub.i](v) = [c.sub.j](v), whereas for any other U [member of] L(N) with nonzero weight [c.sub.T](v), i and j either both have nonzero memberships in U or both have zero memberships in U. Hence it follows that [[PSI].sup.[mu].sub.i] (U, v) = [[PSI].sup.[mu].sub.j] (U, v) = 0 when, j [not member of] supp U.

Next for i, j [member of] supp U,

[mathematical expression not reproducible]. (35)

So [[PSI].sup.[mu]] satisfies the symmetry ([FS.sub.3]) axiom. Finally for any two games [v.sub.1], [v.sub.2] [member of] [FG.sub.C](N) we have that [mathematical expression not reproducible]. Following f-additivity of [mu] this implies [mu](U, [v.sub.1] + [v.sub.2]) [[PSI].sup.[mu]] (U, [v.sub.1] + [v.sub.2]) = [mu](U, [v.sub.1]) [[PSI].sup.[mu]] (U, [v.sub.1]) + [mu](U, [v.sub.2]) [PSI] (U, [v.sub.2]) and hence [[PSI].sup.[mu]] is f[mu]-additive.

Theorem 22. For given positive numbers [[omega].sub.k] with k = 1, 2, ..., n, let the function [[mu].sup.[omega]] be defined by

[mathematical expression not reproducible]. (36)

Then the solidarity [mu]-share function [mathematical expression not reproducible] defined by

[mathematical expression not reproducible] (37)

is the unique solidarity [mu]-share function satisfying the axioms of f-efficiency, fv-null player, f-symmetry, and f[[mu].sub.[omega]]-additivity on [FG.sub.C](N) wherever [[mu].sub.[omega]] is positive.

Proof. By definition, [[mu].sub.[omega]] is f-additive. Hence the existence and uniqueness of the solidarity [mu]-share function follows from Theorem 21. We show that [mathematical expression not reproducible] satisfies the four axioms with respect to [[mu].sub.[omega]] on the class [FG.sub.C](N) of [[mu].sub.[omega]]-positive games. Next we show that [mathematical expression not reproducible] satisfies the above four axioms. The f-efficiency and fv-null player axioms are direct consequences of their crisp counterparts. Now for any U [member of] L(N) and S [member] L(U) with i, j [member of] supp U, v(S) = v([[beta].sub.ij][S]) implies v(T [union] {i}) = v(T [union] {j}), [for all]T [member of] P([[U].sub.h]\{i, j})], [for all]h [member of] (0, U(i)], then we have that f[v.sup.v](S) = f[v.sup.v]([[beta].sub.ij][S]) implies [v.sup.v](T [union] {i}) = [v.sup.v](T [union] {j}) [for all]T [member of] P([[U].sub.h]\{i, j}). Following the fact that [[omega].sub.k] depends only on the size of T, the symmetry axiom holds. Finally we have [mathematical expression not reproducible]. For all T containing i, it holds that [v.sup.(au+fcv)](T) = a[v.sup.v](T) + b[v.sup.v](T); it follows that [mathematical expression not reproducible] is [[mu].sub.[omega]]-additive.

In the following theorem, we take a particular form of the function [mu] and obtain the corresponding solidarity share function for the class [FG.sub.C](N). This exemplifies the existence of a wide range of such share functions generated by the various choices of the function [mu].

Theorem 23. Let the function [[mu].sub.sol] be defined by [[mu].sub.sol](U, v) = v(U). Then the solidarity [mu]-share function [mathematical expression not reproducible] is the unique solidarity [mu]-share function satisfying the axioms of f-efficiency, fv-null player, f-symmetry, and f[[mu].sub.sol]-linearity on [FG.sub.C](N).

Proof. For [mathematical expression not reproducible] with [absolute value of T] =k, take [mathematical expression not reproducible]. Then, we have that [mathematical expression not reproducible] as defined in Theorem 15 given by

[mathematical expression not reproducible]. (38)

Further, the share function [mathematical expression not reproducible] as defined in Theorem 15 is given by

[mathematical expression not reproducible]. (39)

This completes the proof.

4.2. Solidarity Share Functions for [FG.sub.M](N). Here we discuss the existence and uniqueness of the solidarity [mu]-share function for TU fuzzy games in [FG.sub.M](N) following the definition of the game [[??].sub.T].

Theorem 24. Let [mu] : [G.sub.M](N) [right arrow] R be a real valued function on the class [G.sub.M](N) of games. Then on [G.sub.M](N) there exists a unique [mu]-share function [[PSI].sup.[mu]] : [G.sub.M](N) [right arrow] [(R).sup.L(N)] that satisfies the axioms of f-efficiency, fv-null player property, f-symmetry, and f[mu]-additivity if and only if [mu] is f-additive on [G.sub.M](N).

Proof. The proof goes exactly in the same line of Theorem 16 and hence is omitted.

Theorem 25. For given positive numbers [[omega].sub.s] with s = 1, 2, ..., n, let the function [[mu].sup.[omega]] be defined by

[[mu].sup.[omega]] (U, v) = [summation over (i [member of] supp U)] [summation over (T [subset or equal to] U)] [[omega].sub.s][A.sup.v] (T), (40)

where

[mathematical expression not reproducible]. (41)

Then the solidarity [mu]-share function [mathematical expression not reproducible] defined by

[mathematical expression not reproducible] (42)

is the unique solidarity [mu]-share function satisfying the axioms off-efficiency ([FS.sub.1]), fv-null player ([FS.sub.2]), f-symmetry ([FS.sub.3]), and f[[mu].sub.[omega]]-additivity ([FS.sub.4]) on [FG.sub.M](N) wherever [[mu].sub.[omega]] is positive.

Proof. Along the line of Theorem 21, we can easily get the result.

Next we obtain a particular [mu] to exemplify the wide variety of the class of [mu]-share functions.

Theorem 26. Let the function [[mu].sub.sol be defined by [[mu].sub.sol](U, v) = v(U) = W(v). Then the solidarity [mu]-share function [mathematical expression not reproducible] is the unique solidarity y-share function corresponding to the solidarity value function satisfying the axioms of f-efficiency, fv-null player, f-symmetry, and f[[mu].sub.sol]-linearity on [FG.sub.M](N).

Proof. The proof proceeds exactly in the same line of Theorem 22 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 classes [FG.sub.C](N) and [FG.sub.M](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.

https://doi.org/10.1155/2018/3502949

Conflicts of Interest

The authors declare that funding listed in "Acknowledgments" did not lead to any conflicts of interest regarding the publication of this manuscript. There are also no other conflicts of interest in the manuscript.

Acknowledgments

This work is partially funded by the UKIERI Grants nos. 18415/2017(IC) and VEGA 1/0420/15.

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Rajib Biswakarma, (1) Surajit Borkotokey (iD), (1) and Radko Mesiar (2,3)

(1) Department of Mathematics, Dibrugarh University, Dibrugarh, India

(2) Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia

(3) Faculty of Science, Palacky University Olomouc, No. 12, 77146 Olomouc, Czech Republic

Correspondence should be addressed to Surajit Borkotokey; surajitbor@yahoo.com

Received 29 November 2017; Accepted 8 March 2018; Published 15 April 2018

Academic Editor: Amit Kumar
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Title Annotation:Research Article
Author:Biswakarma, Rajib; Borkotokey, Surajit; Mesiar, Radko
Publication:Advances in Fuzzy Systems
Date:Jan 1, 2018
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