Advances in Fuzzy Systems
Volume 2018 (2018), Article ID 3502949, 9 pages
https://doi.org/10.1155/2018/3502949

Research Article

Solidarity Value and Solidarity Share Functions for TU Fuzzy Games

Rajib Biswakarma,¹ Surajit Borkotokey,¹ and Radko Mesiar2,3

¹Department of Mathematics, Dibrugarh University, Dibrugarh, India
²Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia
³Faculty of Science, Palacký 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

Copyright © 2018 Rajib Biswakarma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

A cooperative game with transferable utility, or simply a TU game, is a pair , where is a set of players, called the grand coalition, and is the characteristic function defined on that assigns to every subset (coalition) a real number called its worth which gives zero worth to the empty coalition. Let denote the class of TU games. A solution for any TU game is a function on which assigns to the TU game a distribution of payoffs for its players. If there is no ambiguity on the player set , we denote by the class of all TU games with the fixed .

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 -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 ; see [9]. A fuzzy coalition is a fuzzy subset of the player set 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 be fixed so that the class of TU games can be taken as . We define the following.

Definition 1. Let and ; the quantity is called the average marginal contribution of a player of the coalition .

Definition 2. Given a game , player is called a -null player if , for every coalition containing .

Consider a function that assigns to any game a mapping. For any fixed and set , we denote the corresponding -ary vector in as . We define the solidarity value as follows.

Definition 3. A function is said to be the solidarity value on if it satisfies the following four axioms.
Axiom C₁ (Efficiency). If and , then

Axiom C₂ (-Null Player). If and are a -null player, then

Axiom C₃ (Symmetry). If , , and are symmetric, that is, holds for any , then

Axiom (Additivity). For , define by for each . If and , then

Theorem 4. Define a function by

where and . Then the function is the unique solidarity value on .

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

From now onward we denote the solidarity value by where and .

Definition 5. Let be a set of TU games, and let be a given function. A -share function on a set of games is a function that satisfies the following Axioms CS₁, CS₂, and CS₃ and either Axiom CS₄ or CS₅:
Axiom CS₁ (μ-Efficiency). If and , then

Axiom CS₂ (-Symmetry). If and , , and = hold for any , then .

Axiom CS₃ (-Null Player). If and is an -null player, that is, , then

Axiom CS₄ (-Additivity). For any pair such that , it holds that = + .

Axiom CS₅ (-Linearity). For any pair , of games in and for any pair of real numbers and such that , it holds that

Theorem 6. Let be a positive function on . Then on the subclass there exists a unique solidarity -share function satisfying the axioms CS₁–CS₄ if and only if is additive on .

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 . A fuzzy coalition is a fuzzy subset of , which is identified with a characteristic function from to . Let be the set of all fuzzy coalitions in . For a fuzzy coalition and player , represents the membership grade of in . 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 coalition is denoted by . We use the notation if and only if for all . Let and , respectively, represent the maximum and the minimum operators. The union and intersections of fuzzy coalitions and given by and are defined as and , respectively, for each . Following are some special fuzzy coalitions.

For and , the fuzzy coalitions and are given by the following:

Definition 7. A TU game with fuzzy coalitions or simply a TU fuzzy game is a pair where and are a set function, satisfying

Let denote the class of TU fuzzy games with player set . 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 and , . For any , define by

Definition 9. Given , let and let be the cardinality of . Write the elements of in the increasing order as . Then a game is said to be a TU fuzzy game in Choquet integral form if and only if

The set of all TU fuzzy games in Choquet integral form is denoted by .

Definition 10 (see [12]). For any given and , a TU fuzzy game generated by and given by

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 .

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 of TU fuzzy games with player set along the line of [2]. Begin with the following definition.

Definition 11. Let and , and the quantity

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

Definition 12. Given a game , player is called an -null player if , for every coalition with .

Note that Definitions 11 and 12 are the fuzzy extensions of their counterparts in crisp setting in the sense that if for all , and all , , then .

Definition 13. A solidarity value function on is a function that satisfies the following four axioms, that is, Axioms F₁–F₄.
Axiom F₁ (Efficiency). If and , then

Axiom F₂ (Symmetry). If , and . For any given and , then .
Axiom F₃ (Additivity). For any , .
Axiom F₄ (-Null Player). If and are a -null player, that is, for every fuzzy coalition with , then .

Note that Axioms F₁–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 Axioms CS₁–CS₄.

3.1. Solidarity Value for the Class

We now find the solidarity value for the class by use of the following theorem.

Theorem 14. Let and . A function , defined by

is a solidarity value function in for , where

Proof. Recall from Theorem 4 that there exists a unique function satisfying Axioms CS₁–CS₄. We use this to prove that the function satisfies Axioms F₁–F₄.
Axiom F₁ (Efficiency). Let and . Since holds for any , we obtain

Since implies , we must have .
It follows that
Axiom F₂ (Symmetry). Let and . We have the following: , such that , , , such that , , and , , such that , and , , such that , and , , . Consequently, if for any , then for any and . Hence we have for any and for any Therefore, for any . It follows that .
Axiom F₃ (Additivity). Since is additive so for any , and by the definition of we can easily prove that
Axiom F₄ (-Null Player). Let and is a -null player; that is,

This completes the proof.

3.2. Solidarity Value for the Class

Theorem 15. For , the game , that is,

has the following properties:

(i);(ii)if with , then

and every player is -null in the game .

Theorem 16. Let and . A function defined by

where

is the unique solidarity value for in .

Proof. Let us construct any value function on satisfying efficiency, symmetry, additivity, and -null player axioms by

Now, we know that can be expressed by

where

clearly given by (21) and (22) satisfies symmetry and -null player axioms. Moreover, is a linear mapping. Hence additivity is satisfied.
Using now linearity of , we get

which proves that is efficient. It is obvious that for each game . Thus for every

4. Solidarity Share Functions for TU Fuzzy Games

We now extend the notion of a share function to the class of TU fuzzy games with player set . In the line of its crisp counterpart we assume here also that the share function assigns to each player her share in the payoff of the fuzzy coalition . Therefore we provide the following definitions as an extension to their crisp versions.

Definition 17. A real valued function is called -additive if, for and any pair such that , it holds that

Definition 18. A real valued function is called -linear on the class of games if it is -additive and if for any on and it holds that for any real number such that .

Definition 19. A real valued function is called positive if   .

Definition 20. Given a function , a solidarity -share function on is a function that satisfies the following axioms, that is, Axioms FS₁–FS₃ along with Axiom FS₄ or Axiom FS₅.
Axiom FS₁ (f-Efficiency). For we have and , for each .
Axiom FS₂ (-Null Player). If and are a -null player, that is, for every fuzzy coalition with , then .
Axiom FS₃ (f-Symmetry). If , , and for any given and , , then .
Axiom FS₄ (-Additivity). For any pair such that , it holds that

Axiom FS₅ (-Linearity). For any pair such that , it holds that = + , for any pair of real numbers and such that for all .

Note that Axioms 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 a solidarity -share function gives a payoff to player when she is involved in the fuzzy coalition and satisfies the above-mentioned axioms.

4.1. Solidarity Share Functions for

In this section we prove the existence and uniqueness of the solidarity -share function for the class of fuzzy games in Choquet integral form. To discuss the existence and uniqueness of the solidarity -share function for TU fuzzy game in we have to use some classical results from [1, 2]. Recall that, given a coalition , the game is defined as follows:

Due to Theorem 6, for any , each can be expressed as where . Denote and . Then

Following similar procedure as in Lemma 3.2. of [10], we can have, for ,

where

It follows from the above discussion that can be rewritten as

Theorem 21. Let be a real valued function. There exists a unique solidarity -share function that satisfies the axioms of -efficiency (FS₁), - null player (FS₂), f-symmetry (FS₃), and - additivity (FS₄) if and only if is -additive on .

Proof. The proof proceeds in the line of [6]. First we suppose that satisfies -efficiency and - additivity. It follows that is -additive on . Thus we have

for any such that . -efficiency then implies that = + . Hence is -additive.
Secondly we will show that we can have at most one solidarity share function satisfying the four axioms. Let be a function satisfying the four axioms. For a positively scaled unanimity game , and consequently for , we obtain

(i), when .(ii), when .

Again for from (i) and (ii) clearly satisfies all the four axioms. Thus it follows that for any the function given by (i) and (ii) is the solidarity -share function satisfying the axioms of -efficiency, -null player, and -symmetry if and only if is -additive.
The uniqueness of follows immediately. We next show that satisfies the four axioms for an arbitrary . The assumption of -additivity of ensures -efficiency as in the case of crisp games. Consequently the -null player axiom also follows. Third, for any and with , implies , then = , whereas for any other with nonzero weight , and either both have nonzero memberships in or both have zero memberships in . Hence it follows that when, .
Next for , ,

So satisfies the symmetry () axiom. Finally for any two games , we have that . Following -additivity of this implies = + and hence is -additive.

Theorem 22. For given positive numbers with , let the function be defined by

Then the solidarity -share function defined by

is the unique solidarity -share function satisfying the axioms of -efficiency, -null player, -symmetry, and - additivity on wherever is positive.

Proof. By definition, is -additive. Hence the existence and uniqueness of the solidarity -share function follows from Theorem 21. We show that satisfies the four axioms with respect to on the class of -positive games. Next we show that satisfies the above four axioms. The -efficiency and -null player axioms are direct consequences of their crisp counterparts. Now for any and with , , implies , then we have that = implies =   . Following the fact that depends only on the size of , the symmetry axiom holds. Finally we have . For all containing , it holds that ; it follows that is -additive.

In the following theorem, we take a particular form of the function and obtain the corresponding solidarity share function for the class . This exemplifies the existence of a wide range of such share functions generated by the various choices of the function .

Theorem 23. Let the function be defined by . Then the solidarity -share function is the unique solidarity -share function satisfying the axioms of -efficiency, -null player, -symmetry, and -linearity on .

Proof. For with , take . Then, we have that as defined in Theorem 15 given by

Further, the share function as defined in Theorem 15 is given by

This completes the proof.

4.2. Solidarity Share Functions for

Here we discuss the existence and uniqueness of the solidarity -share function for TU fuzzy games in following the definition of the game .

Theorem 24. Let be a real valued function on the class of games. Then on there exists a unique -share function that satisfies the axioms of -efficiency, -null player property, -symmetry, and -additivity if and only if is f-additive on .

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

Theorem 25. For given positive numbers with , let the function be defined by

where

Then the solidarity -share function defined by

is the unique solidarity -share function satisfying the axioms of f-efficiency (FS₁), -null player (FS₂), f-symmetry (FS₃), and -additivity (FS₄) on wherever is positive.

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

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

Theorem 26. Let the function be defined by . Then the solidarity -share function is the unique solidarity -share function corresponding to the solidarity value function satisfying the axioms of -efficiency, -null player, -symmetry, and -linearity on .

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 and 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.

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. 184-15/2017(IC) and VEGA 1/0420/15.

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