Generalized Fuzzy Shapley Function for Fuzzy Games

Abstract

A generalized fuzzy Shapley function for fuzzy games is proposed. First, a game with fuzzy characteristic function is introduced. Based on Hukuhara difference, the fuzzy Hukuhara-Shapley function is proposed as a solution concept to this class of fuzzy games. Some of its properties are shown. An equivalent axiomatic characterization of the fuzzy Hukuhara-Shapley function is given. Furthermore, a generalized fuzzy Shapley function for games with fuzzy coalition and fuzzy characteristic function is developed. It is shown that the simplified expression of the generalized fuzzy Shapley function can be regarded as the generalization of the fuzzy Shapley function defined for some particular games with fuzzy coalition and fuzzy characteristic function.

1. Introduction

The Shapley value [1] is a well-known solution concept in cooperative game theory, which has been investigated by a number of researchers. Most of them treat games with crisp coalitions. However, there are some situations where some agents do not fully participate in a coalition, but to a certain extent. In a class of production games, partial participation in a coalition means offering a part of resources, while full participation means offering all of resources. A coalition including some players who participate partially can be treated as a so-called fuzzy coalition, introduced by Aubin [2, 3]. It shows to what extent a player transfers his/her representability [4] and is also called a rate of participation. Thus it can describe different participation levels of different players in different game situations, varying from noncooperation to full cooperation.

After the pioneering work by Aubin [3], the Shapley function for games with fuzzy coalition has received more and more attentions. Butnariu [5] gave the expression of the Shapley function on a limited class of fuzzy games with proportional values form. However, most games with proportional values are neither monotone nondecreasing nor continuous with regard to rates of players’ participation. In order to overcome this limitation, Tsurumi et al. [6] defined a new Shapley value on a new class of fuzzy games with Choquet integral form. This class of fuzzy games is both monotone nondecreasing and continuous with regard to rates of players’ participation. Branzei et al. [7] also introduced a concept of Shapley value for games with fuzzy coalition, which was defined by the associated crisp game corresponding to fuzzy game. Butnariu and Kroupa [8] extend this kind of fuzzy games with proportional values to fuzzy games with weighted function, and the corresponding Shapley function was given. Li and Zhang [9] proposed a simplified expression of the Shapley function for games with fuzzy coalition, which can be regarded as the generalization of Shapley functions defined in some particular games with fuzzy coalition.

On the other hand, by using cooperative game theory, Owen considered linear production programming problems in which multiple decision-makers pool resources to produce some goods [10]. An objective function of the linear production programming problem was represented as total revenue from selling some kinds of goods, and the problem was formulated as a linear programming problem in which, subject to the resource constraints, the revenue is maximized. In many decision-making situations, imprecision and uncertainty are often present due to (a) incomplete information, (b) conflicting evidence, (c) ambiguous information, and (d) subjective information. It can be seen that the possible values of parameters of this kind of production games model are often only imprecisely or ambiguously known to decision-makers. With this observation in mind, it would be certainly more appropriate to interpret the decision-makers’ understanding of the parameters as fuzzy numerical data which can be represented by means of fuzzy sets of the real line known as fuzzy numbers. It reflects the decision-makers’ ambiguous or fuzzy understanding of the nature of the parameters in the problem-formulation process [11]. The resulting production games problem involving fuzzy parameters would be viewed as a more realistic version than the conventional one [12, 13].

From fuzzy mathematical programming perspective, Nishizaki and Sakawa [11] investigated cooperative game problems with fuzzy characteristic functions. Mares [14, 15] and Mares and Vlach [16] were also concerned with the uncertainty in the value of the characteristic function associated with a game, where the characteristic function was expressed by fuzzy number. At the same time, they discussed the fuzzy Shapley values of this kind of fuzzy game. Borkotokey [17] considered a cooperative game with fuzzy coalitions and fuzzy characteristic function simultaneously. A Shapley function in the fuzzy sense was proposed as a solution concept to this class of fuzzy games. Yu and Zhang [18] studied a class of particular games with fuzzy coalitions and a fuzzy characteristic function with Choquet integral form and gave the explicit form of the Shapley value for this class of fuzzy games. However, as Li and Zhang [9] pointed out, there were other many approaches to extend cooperative games to fuzzy games besides Choquet integral method [6] and proportional values method [5]. Which one is more natural? The specific game situation is needed to be considered, because each fuzzy game may only be suitable for a certain case. Therefore, as a kind of special fuzzy game with Choquet integral form, the fuzzy Shapley function of the fuzzy game was only suitable for a certain case. In order to improve this limit, in this paper, we pay more attentions to generalized fuzzy game with fuzzy coalitions and fuzzy characteristic function, and a generalized expression of the fuzzy Shapley function is proposed for the generalized fuzzy game.

In order to do this, this paper is organized as follows. In Section 2, we briefly review some concepts of interval numbers and fuzzy numbers and introduce the Hukuhara difference on interval numbers and fuzzy numbers. In Section 3, a game with fuzzy characteristic functions is introduced, and its fuzzy Hukuhara-Shapley function is proposed, and some of its properties are investigated. Furthermore, an applicable example is given. In Section 4, we investigate a new class of games with fuzzy coalitions and fuzzy characteristic function and give a simplified expression of the generalized fuzzy Shapley function for the new fuzzy games. It is shown that it can be regarded as a generalization of the fuzzy Shapley function defined in the proposed fuzzy games with some particular games with fuzzy coalition and fuzzy characteristic function. Finally, in Section 5, we summarize the main conclusions of the paper.

2. Preliminaries

In this section, we start by providing a summary of some concepts of interval numbers and fuzzy numbers, which will be used throughout the paper.

2.1. Interval Numbers

Let ℝ be (−∞,∞), that is, the set of all real numbers. Given 𝑥⁻, 𝑥⁺∈ℝ and 𝑥⁻≤𝑥⁺, the closed interval [𝑥⁻,𝑥⁺] defines an interval number 𝑥=[𝑥⁻,𝑥⁺]={𝑥∈ℝ∣𝑥⁻≤𝑥≤𝑥⁺}. Obviously, when 𝑥⁻=𝑥⁺, the interval number 𝑥 reduces to a real number 𝑥⁻ or 𝑥⁺. We say that a real number 𝑥 is a member of an interval number 𝑥, written as 𝑥∈𝑥, if 𝑥⁻≤𝑥≤𝑥⁺. Let us denote by 𝕀ℝ the class of all closed and bounded intervals in ℝ. Throughout this paper, when we say that 𝑥 is a closed interval, we implicitly mean that 𝑥 is also bounded in ℝ.

In the following, we will briefly review the order relation and basic operation of interval numbers [19].

Definition 1. For any interval number 𝑥 and 𝑦, 𝑥=[𝑥⁻,𝑥⁺], 𝑦=[𝑦⁻,𝑦⁺]∈𝕀ℝ.
Consider the following:(1)𝑥⊆𝑦 iff 𝑥⁻≥𝑦⁻, 𝑥⁺≤𝑦⁺,(2)𝑥≤𝑦 iff 𝑥⁻≤𝑦⁻, 𝑥⁺≤𝑦⁺,(3)𝑥=𝑦 iff 𝑥⁻=𝑦⁻, 𝑥⁺=𝑦⁺.

Definition 2. For any interval numbers 𝑥=[𝑥⁻,𝑥⁺], 𝑦=[𝑦⁻,𝑦⁺]∈𝕀ℝ, the arithmetic operations on interval numbers are defined as follows:(1)𝑥+𝑦=[𝑥⁻+𝑦⁻,𝑥⁺+𝑦⁺],(2)𝑥−𝑦=[𝑥⁻−𝑦⁺,𝑥⁺−𝑦⁻],(3)𝑥×𝑦=[min(𝑥⁻𝑦⁻,𝑥⁻𝑦⁺,𝑥⁺𝑦⁻,𝑥⁺𝑦⁺),max(𝑥⁻𝑦⁻, 𝑥⁻𝑦⁺,𝑥⁺𝑦⁻,𝑥⁺𝑦⁺)],(4)𝛼(𝑥+𝑦)=𝛼𝑥+𝛼𝑦.
It is known that scalar multiplication is a special case of interval numbers multiplication operation. Scalar multiplication can be expressed by 𝛼𝑥=[𝛼𝑥⁻,𝛼𝑥⁺], if 𝛼≥0; 𝛼𝑥=[𝛼𝑥⁺,𝛼𝑥⁻], if 𝛼<0.
In general, according to Definition 2, for 𝑥,𝑦,𝑧∈𝕀ℝ, 𝑥+𝑦=𝑧 does not mean that 𝑥=𝑧−𝑦. For example, [1,3]−[1,3]=[−2,2]≠0 (unless 𝑥={𝑥} is a singleton). To overcome this situation, next we introduce H-difference of interval numbers, called Hukuhara difference [20–22], which is denoted by −_H in this paper. Let 𝑥=[𝑥⁻,𝑥⁺] and 𝑦=[𝑦⁻,𝑦⁺] be two closed intervals in ℝ. For a closed interval 𝑧=[𝑧⁻,𝑧⁺] such that 𝑥=𝑦+𝑧, 𝑧 is called the Hukuhara difference. Since 𝑥=𝑦+𝑧, it is easy to see that 𝑥⁻=𝑦⁻+𝑧⁻ and 𝑥⁺=𝑦⁺+𝑧⁺; that is, 𝑧⁻=𝑥⁻−𝑦⁻ and 𝑧⁺=𝑥⁺−𝑦⁺. Therefore, the closed interval 𝑧 exists if 𝑥⁻−𝑦⁻≤𝑥⁺−𝑦⁺. In this case, 𝑧=[𝑥⁻−𝑦⁻,𝑥⁺−𝑦⁺] and we also denote 𝑧=𝑥−_H𝑦. When we say that the Hukuhara difference 𝑧=𝑥−_H𝑦 exists, we implicitly mean that 𝑥⁻−𝑦⁻≤𝑥⁺−𝑦⁺. An important property of −_H is that 𝑥−_H𝑥=0, and if 𝑥−_H𝑦 exists, H-difference is unique [22]. For any positive number 𝛼>0, 𝛼(𝑥−_H𝑦)=𝛼𝑥−_H𝛼𝑦. In general, 𝑥−𝑦≠𝑥−_H𝑦.

Definition 3. For any 𝑖=1,2,…, let 𝑎_𝑖=[𝑎⁻_𝑖,𝑎⁺_𝑖]∈𝕀ℝ and define _𝑖^⊕𝑎_𝑖=_𝑎𝑖≥0𝑎_𝑖−_{H𝑎𝑖≱0}−𝑎_𝑖. (1)
It is easy to prove the following conclusions.

Proposition 4. (i) Let 𝑁={1,2,…,𝑛}, 𝑎_𝑖=[𝑎⁻_𝑖,𝑎⁺_𝑖], 𝑁₁∩𝑁₂=Ø, and 𝑁₁∪𝑁₂=𝑁; then ^⊕_𝑖∈𝑁𝑎_𝑖=_𝑖∈𝑁1^⊕𝑎_𝑖+_𝑖∈𝑁2^⊕𝑎_𝑖. (2)
(ii) Let 𝑎=[𝑎⁻,𝑎⁺]≥0, for any 𝜆_𝑖∈ℝ; then _𝑖^⊕𝜆_𝑖𝑎=_𝑖𝜆_𝑖𝑎. (3)

2.2. Fuzzy Numbers

A general fuzzy set over a given set (or space) 𝑋 of elements (the universe) is usually defined by its membership function 𝑚∶𝑋→𝑇⊆[0,1] and a fuzzy (sub)set 𝑢 of 𝑋 is uniquely characterized by the pairs (𝑥,𝑚_𝑢(𝑥)) for each 𝑥∈𝑋; the value 𝑚_𝑢(𝑥)∈[0,1] is the membership grade of 𝑥 to the fuzzy set 𝑢 and 𝑚_𝑢 is the membership function of a fuzzy set 𝑢 over 𝑋 [23, 24] for the origins of fuzzy set theory. The support of 𝑢 is the (crisp) subset of points of 𝑋 at which the membership grade 𝑚_𝑢(𝑥) is positive: supp(𝑢)={𝑥∣𝑥∈𝑋,𝑚_𝑢(𝑥)>0}. For 𝜆∈[0,1], the 𝜆-level-cut of 𝑢 (or simply the 𝜆-cut) is defined by [𝑢]_𝜆={𝑥∣𝑥∈𝑋,𝑚_𝑢(𝑥)≥𝜆} and for 𝜆=0 (or 𝜆→+0) by the closure of the support[𝑢]₀=cl{𝑥∣𝑥∈𝑋,𝑚_𝑢(𝑥)>0}, where cl denotes the closure of sets.

The following properties characterize the normal, convex, and upper semicontinuous fuzzy sets (in terms of the level-cuts):(B1)[𝑢]_𝜆 is the spaces of nonempty compact and compact convex sets of ℝ^𝑛 for all 𝜆∈[0,1];(B2)[𝑢]_𝜆⊆[𝑢]_𝛽 for 𝜆≥𝛽;(B3)[𝑢]_𝜆=⋂^∞_𝑘=1[𝑢]_𝜆𝑘 for all increasing sequences 𝜆_𝑘↑𝜆 converging to 𝜆.

Furthermore, any family {𝑈_𝜆∣𝜆∈[0,1]} satisfying conditions (B1)–(B3) represents the level-cuts of a fuzzy set 𝑢 having [𝑢]_𝜆=𝑈_𝜆.

To quantify fuzzy concepts, we use the following fuzzy numbers [25].

Definition 5. A fuzzy number, denoted by 𝑎, is a fuzzy subset of ℝ with membership function 𝑚_𝑎∶ℝ→[0,1] satisfying the following conditions.(i)There exists at least one number 𝑎₀∈ℝ such that 𝑚_𝑎(𝑎₀)=1;(ii)𝑚_𝑎(𝑥) is nondecreasing on (−∞,𝑎₀] and nonincreasing on [𝑎₀,+∞);(iii)𝑚_𝑎(𝑎₀) is upper semicontinuous; that is, lim_{𝑥→𝑥0+}𝑚_𝑎(𝑥)=𝑚_𝑎(𝑥₀) if 𝑥₀<𝑎₀ and lim_{𝑥→𝑥0−}𝑚_𝑎(𝑥)=𝑚_𝑎𝑥₀if𝑥₀>𝑎₀; (4)(iv)Supp(𝑎) is compact, where Supp(𝑎)={𝑥∈ℝ∣𝑚_𝑎(𝑥)>0}.
An important type of fuzzy numbers in common use is the trapezoidal fuzzy number [21] whose membership function has the form 𝑚_𝑎(𝑥)=1,if𝑥∈𝑎_𝑏,𝑎_𝑐,𝑥−𝑎_𝑙𝑎_𝑏−𝑎_𝑙,if𝑥∈𝑎_𝑙,𝑎_𝑏,𝑥−𝑎_𝑟𝑎_𝑐−𝑎_𝑟,if𝑥∈𝑎_𝑐,𝑎_𝑟,0,otherwise, (5) where 𝑎_𝑙,𝑎_𝑏,𝑎_𝑐,𝑎_𝑟∈ℝ with 𝑎_𝑙≤𝑎_𝑏≤𝑎_𝑐≤𝑎_𝑟 and the trapezoidal fuzzy number 𝑎 is simply denoted by (𝑎_𝑙,𝑎_𝑏,𝑎_𝑐,𝑎_𝑟). It is called nonnegative if 𝑎_𝑙≥0. The trapezoidal fuzzy number degenerates to be a triangular fuzzy number when 𝑎_𝑏=𝑎_𝑐, while it becomes an interval number (i.e., a rectangular fuzzy) when 𝑎_𝑙=𝑎_𝑏 and 𝑎_𝑐=𝑎_𝑟. Any crisp real number 𝑎 can be regarded as a special trapezoidal fuzzy number with 𝑎_𝑙=𝑎_𝑏=𝑎_𝑐=𝑎_𝑟=𝑎. In this paper, the set of all fuzzy numbers on ℝ is denoted by 𝔽ℝ.
Let any fuzzy number 𝑎∈𝔽ℝ have membership function 𝑚_𝑎, and the level set (or 𝜆-cut) is defined as 𝑎_𝜆={𝑥∈ℝ∣𝑚_𝑎≥𝜆}, 𝜆∈[0,1]. It follows from the properties of the membership function of a fuzzy number 𝑎 that each of its 𝜆-cuts 𝑎_𝜆 is an interval number, denoted by 𝑎_𝜆=[𝑎⁻_𝜆,𝑎⁺_𝜆].

Definition 6 (see [25]). Let 𝑎 be a fuzzy number; 𝑎=⋃_𝜆∈[0,1]𝜆𝑎_𝜆 (decomposition theory), where 𝑎_𝜆={𝑥∈ℝ∣𝑚_𝑎≥𝜆}=[𝑎⁻_𝜆,𝑎⁺_𝜆] is the 𝜆-cut set of 𝑎 for any 𝜆∈[0,1]. For all 𝑏̃∈𝔽ℝ,(i)𝑎⊂𝑏̃ iff 𝑎_𝜆⊂𝑏̃_𝜆;(ii)𝑎≤𝑏̃ iff 𝑎_𝜆≤𝑏̃_𝜆;(iii)𝑎=𝑏̃ iff 𝑎_𝜆=𝑏̃_𝜆;(iv)if 𝜆₁≤𝜆₂, then 𝑎_𝜆1⊇𝑎_𝜆2.
Let 𝑎,𝑏̃∈𝔽ℝ, and let ∗ be a binary operation on ℝ. 𝜆-cuts of the fuzzy number 𝑎∗𝑏̃ can be calculated because of the following: 𝑎+𝑏̃_𝜆=𝑎_𝜆+𝑏̃_𝜆=𝑎⁻_𝜆+𝑏⁻_𝜆,𝑎⁺_𝜆+𝑏⁺_𝜆,𝑎−𝑏̃_𝜆=𝑎_𝜆−𝑏̃_𝜆=𝑎⁻_𝜆−𝑏⁺_𝜆,𝑎⁺_𝜆−𝑏⁻_𝜆,(𝑚𝑎)_𝜆=𝑚𝑎_𝜆=𝑚𝑎⁻_𝜆,𝑚𝑎⁺_𝜆,∀𝑚∈ℝ,𝑚>0. (6)
Employing the 𝜆-cut representation, arithmetic operations on fuzzy numbers are defined in terms of the well-established arithmetic operations on interval numbers [25].

Definition 7. Given any pair of fuzzy numbers, 𝑎,𝑏̃∈𝔽ℝ, the basic operations on the 𝜆-cuts of 𝑎 and 𝑏̃ are defined for all 𝜆∈[0,1] by the general formula 𝑎∗𝑏̃=_[]𝜆∈0,1𝜆𝑎_𝜆∗𝑏̃_𝜆. (7)
However, similar to the interval number operations, in the fuzzy contexts, equation 𝑎=𝑏̃+𝑐̃ is not equivalent to 𝑐̃=𝑎−𝑏̃=𝑎+(−1)𝑏̃ or 𝑏̃=𝑎−𝑐̃=𝑎+(−)𝑐̃. This has motivated the introduction of the following Hukuhara difference [22].

Definition 8. Given 𝑎,𝑏̃∈𝔽ℝ, if there exists  𝑐̃∈𝔽ℝ such that 𝑎=𝑏̃+𝑐̃∈𝔽ℝ, then 𝑐̃ is called the Hukuhara difference (H-difference), denoted by 𝑎−_H𝑏̃=𝑐̃.
Clearly, 𝑎−_H𝑎=0; if 𝑎−_H𝑏̃ exists, it is unique. And the 𝜆-cuts of H-difference are (𝑎−_H𝑏̃)_𝜆=[𝑎⁻_𝜆−𝑏⁻_𝜆,𝑎⁺_𝜆−𝑏⁺_𝜆]; and 𝛼(𝑎−_H𝑏̃)=𝛼𝑎−_H𝛼𝑏̃, for 𝛼∈ℝ, 𝛼>0. The H-difference inverts the addition of fuzzy numbers. But the Hukuhara difference between two fuzzy numbers does not always exist. Regarding the existence of the Hukuhara difference, there is an extensive literature described in the study by Dubois et al. in [21].

Proposition 9. Let 𝑎,𝑏̃∈𝔽ℝ. The Hukuhara difference 𝑎−_H𝑏̃ exists if and only if 𝑎⁻_𝜆−𝑏⁻_𝜆≤𝑎⁻_𝛽−𝑏⁻_𝛽≤𝑎⁺_𝛽−𝑏⁺_𝛽≤𝑎⁺_𝜆−𝑏⁺_𝜆,∀𝜆,𝛽∈(0,1],𝛽>𝜆. (8)

Definition 10. Let 𝑋 be a nonempty set; 𝑃(𝑋) denotes its power set. For mapping 𝐻𝐻∶[0,1]⟶𝑃(𝑋),𝛼⟼𝐻(𝛼). (9)(10)
If, for any 𝛼₁,𝛼₂∈[0,1], we have 𝛼₁≤𝛼₂⟹𝐻𝛼₁⊇𝐻𝛼₂, (11) then 𝐻 is called a nested set of 𝑋.

Proposition 11. Given 𝑎,𝑏̃∈𝔽ℝ, for any 𝜆∈[0,1], let 𝐻(𝜆)=𝑎_𝜆−_H𝑏̃_𝜆=[𝑎⁻_𝜆−𝑏⁻_𝜆,𝑎⁺_𝜆−𝑏⁺_𝜆]; then 𝐻(𝜆) is a nested set.

Proof. For all 𝑎∈[0,1], we have 𝐻(𝑎)=𝑎⁻_𝛼−𝑏⁻_𝛼,𝑎⁺_𝛼−𝑏⁺_𝛼=𝑎−_H𝑏̃_𝑎. (12)
If 𝜆≤𝛼, according to Definition 7, we have 𝐴−_H𝐵_𝜆⊇𝐴−_H𝐵_𝛼,thatis,𝐻(𝜆)⊇𝐻(𝑎). (13)
Thus for any 𝜆∈[0,1], 𝐻(𝜆) is a nested set.
This completes the proof of Proposition 11.

For 𝑖=1,2,…,𝑎_𝑖∈𝔽ℝ, we define _𝑖^⊕𝑎_𝑖=_𝑎𝑖≥0𝑎_𝑖−_{H𝑎𝑖≱0}−𝑎_𝑖. (14)

According to representation theorem [25] and Proposition 4, it is easy to obtain the following conclusions.

Proposition 12. (i) Let 𝑁={1,2,…,𝑛}, 𝑎_𝑖∈𝔽ℝ, 𝑁₁∩𝑁₂=Ø, and 𝑁₁∪𝑁₂=𝑁; then ^⊕_𝑖∈𝑁𝑎_𝑖=_𝑖∈𝑁1^⊕𝑎_𝑖+_𝑖∈𝑁2^⊕𝑎_𝑖. (15)
(ii) Given fuzzy number 𝑎≥0, for 𝜆_𝑖∈ℝ, then _𝑖^⊕𝜆_𝑖𝑎=_𝑖𝜆_𝑖𝑎. (16)

3. Fuzzy Hukuhara-Shapley Function for Games with Fuzzy Characteristic Function

3.1. Crisp Cooperative Games and Shapley Value

A finite transferable utility cooperative game (from now on, simply a game or cooperative game) is a pair (𝑁,𝑣), where 𝑁={1,2,…,𝑛} is a finite set of players and 𝑣∶𝑃(𝑁)→ℝ⁺ is called characteristic function satisfying 𝑣(Ø)=0. 𝑃(𝑁) is the family of crisp subsets of 𝑁; that is, 𝑃(𝑁) is equivalent to 2^|𝑁|. We will refer to a subset 𝑆 of 𝑁 as a coalition or crisp coalition and to 𝑣(𝑆) as the worth of 𝑆, which can be seen as the amount of utility the coalition obtains when the players in 𝑆 work together. The class of all crisp games with player set 𝑁 is denoted by 𝐺(𝑁). |𝑆| denotes the cardinality of 𝑆. Given a game (𝑁,𝑣) and a coalition 𝑆, we write (𝑁,𝑣) for the subgame obtained by restricting 𝑣 to subsets of 𝑆 only (i.e., to 2^|𝑆|). If there is no fear of confusion, a (cooperative) game (𝑁,𝑣) is replaced by 𝑣. For any 𝐴,𝐵⊆𝑁, 𝐴∩𝐵=Ø, if 𝑣(𝐴∪𝐵)≥𝑣(𝐴)+𝑣(𝐵), then 𝑣 is called superadditive game. The set of all superadditive games is denoted by 𝐺₀(𝑁). For the sake of simplicity, the bracket is often omitted when a set is written in this paper. For example, we write 𝑖𝑗𝑘 instead of {𝑖,𝑗,𝑘}.

As an important solution concept for crisp cooperative games, the Shapley value is defined as follows.

Definition 13. The Shapley value 𝜑_𝑖(𝑣) of player 𝑖 with respect to a game 𝑣∈𝐺₀(𝑁) is a weighted average value of the marginal contribution 𝑣(𝑇∪𝑖)−𝑣(𝑇) of player 𝑖 alone in all combinations, which is defined by 𝜑_𝑖(𝑣)=_{𝑇⊆𝑁⧵𝑖}(|𝑁|−|𝑇|−1)!|𝑇|!|𝑁|![𝑣(𝑇∪𝑖)−𝑣(𝑇)]. (17)
Equation (17) is a unique expression which satisfies three axiomatic characterizations of Shapley value (see the study by Shapley [1]).

Proposition 14 (see [1]). For any 𝑆,𝑆⊆𝑁 and 𝑆≠Ø, if the 𝑆-unanimity game 𝜇_𝑠 is denoted by 𝑢_𝑆(𝐴)=1,𝐴⊇𝑆,0,otherwise, (18) then, {𝑢_𝑆}_𝑆∈2^𝑁_⧵Ø is a basis in the linear space 𝐺₀(𝑁) of all games, and, for 𝑣, it can be uniquely written as 𝑣=_{𝑆∈2𝑁,𝑆≠Ø}𝑐_𝑆(𝑣)𝑢_𝑆, (19) where 𝑐_𝑆(𝑣)=_{𝐵∈2𝑁,𝐵⊆𝑆}(−1)^{|𝑆|−|𝐵|}𝑣(𝐵)=_{𝐵∈2𝑁,𝐵⊆𝑆}(−1)^𝑠−𝑏𝑣(𝐵), (20) where 𝑠,𝑏 denote the cardinality of crisp coalitions  𝑆 and 𝐵, respectively.

3.2. Cooperative Games with Fuzzy Characteristic Function

In a crisp cooperative game, characteristic function describes a cooperative game and associates a crisp coalition 𝑆 with the worth 𝑣(𝑆), which is interpreted as the payoff that the coalition 𝑆 can acquire only through the action of 𝑆. The cooperative crisp game is based on the assumption that all players and coalitions know the payoff value 𝑣 before the cooperation begins.

The traditional cooperative game assumes that all data of a game are known exactly by players. However, in real game situations often the players are not able to evaluate exactly some data of the game due to a lack of information or/and imprecision of the available information on the environment or on the behavior of the other players. Taking imprecision of information in decision-making problems into account, this assumption is not realistic because there are many uncertain factors during negotiation and coalition formation. In many situations, the players can have only vague ideas about the real payoff value. Therefore, it is more suitable to incorporate fuzzy characteristic function, represented by fuzzy numbers, into cooperative games. In this section, the fuzzy Hukuhara-Shapley function for games with a fuzzy characteristic function is proposed.

Definition 15. A cooperative game with fuzzy characteristic function form is an ordered pair (𝑁,𝑣̃) where 𝑣̃∶𝑃(𝑁)→𝔽ℝ⁺={𝑎∈𝔽ℝ∣𝑎≥0} with 𝑣̃(Ø)=0.

Fuzzy characteristic function 𝑣̃(𝑆) can be interpreted as the maximal fuzzy worth or cost savings that the members of 𝑆 can obtain when they cooperate. Often we identify the game (𝑁,𝑣̃) with its fuzzy characteristic function 𝑣̃. The class of games with fuzzy characteristic function is denoted by 𝐹𝐺(𝑁). In this paper, we mainly discuss the superadditive games with fuzzy characteristic function; that is, for any two crisp coalitions 𝑆,𝑇∈𝑃(𝑁) such that 𝑆∩𝑇=Ø, for any 𝜆∈(0,1], 𝑣̃_𝜆(𝑆∪𝑇)≥𝑣̃_𝜆(𝑆)+𝑣̃_𝜆(𝑇). Then, for any crisp coalitions 𝑆,𝑇∈𝑃(𝑁), 𝑇⊆𝑆 and any 𝜆,𝛽∈(0,1] such that 𝛽>𝜆, the superadditive game (𝑁,𝑣̃) also satisfies 𝑣⁻_𝜆(𝑆)−𝑣⁻_𝜆(𝑇)≤𝑣⁻_𝛽(𝑆)−𝑣⁻_𝛽(𝑇)≤𝑣⁺_𝛽(𝑆)−𝑣⁺_𝛽(𝑇)≤𝑣⁺_𝜆(𝑆)−𝑣⁺_𝜆(𝑇). (21)

According to Proposition 9 it is easy to see that the Hukuhara difference 𝑣̃(𝑆)−_H𝑣̃(𝑇) exists for the superadditive games with fuzzy characteristic function. The set of all superadditive games with fuzzy characteristic functionis denoted by 𝐹𝐺₀(𝑁).

Definition 16. Given 𝑣̃∈𝐹𝐺₀(𝑁), a carrier of 𝑣̃ is any set 𝑇⊆𝑁 with 𝑣̃(𝑆)=𝑣̃(𝑆∩𝑇), ∀𝑆⊆𝑁.
Obviously, players outside any carrier have no influence on the play since they contribute nothing to any coalition. Such a player that does not contribute anything to any coalition is called a null player; that is, for any 𝑇⊆𝑁⧵𝑖, 𝑣̃(𝑇∪𝑖)=𝑣̃(𝑇).
Let (𝑁,𝑣̃),(𝑁,𝑤)∈𝐹𝐺₀(𝑁); for ∀𝑆⊆𝑁 define 𝑣̃+𝑤∶(𝑣̃+𝑤)(𝑆)=𝑣̃(𝑆)+𝑤(𝑆). It is easily seen that (𝑁,𝑣̃+𝑤)∈𝐹𝐺₀(𝑁).

Definition 17. Let 𝑁={1,2,…,𝑛}; for any 𝑣̃∈𝐹𝐺₀(𝑁), a function 𝜑∶𝐹𝐺₀(𝑁)→(𝔽ℝ⁺)^𝑛, where 𝜑=(𝜑₁,…,𝜑_𝑛), is said to be a fuzzy Hukuhara-Shapley function on 𝐹𝐺₀(𝑁) if it satisfies the following three axioms.
Axiom A₁. If 𝑣̃∈𝐹𝐺₀(𝑁)and𝑇⊆𝑁 is any carrier of 𝑣̃, then _𝑖∈𝑇𝜑_𝑖(𝑣̃)=𝑣̃(𝑇). (22)
Axiom A₂. If 𝑣̃∈𝐹𝐺₀(𝑁)and𝜋 is a permutation of the player set, that is, 𝜋∶𝑁→𝑁, then, for any 𝑖∈𝑁, 𝜑_𝜋𝑖(𝜋𝑣̃)=𝜑_𝑖(𝑣̃). (23)
Axiom A₃. If 𝑣̃,𝑤∈𝐹𝐺₀(𝑁), then 𝜑_𝑖(𝑣̃+𝑤)=𝜑_𝑖(𝑣̃)+𝜑_𝑖(𝑤)∀𝑖∈𝑁. (24)
It is easy to obtain the following conclusion.

Proposition 18. If 𝑣̃∈𝐹𝐺₀(𝑁), for any 𝑎∈𝔽ℝ⁺, then (𝑁,𝑎𝑣̃)∈𝐹𝐺₀(𝑁).

Lemma 19. For any 𝑎∈𝔽ℝ⁺, 𝑎≥0, 𝑆⊆𝑁, 𝑆≠Ø, fuzzy Hukuhara-Shapley value of 𝑎𝑢_𝑆 is 𝜑_𝑖𝑎𝑢_𝑆=𝑎0,𝑖∉𝑆|𝑆|,𝑖∈𝑆. (25)

Proof. It is obvious that 𝑆 is a carrier of 𝑎𝑢_𝑆. For 𝑖∈𝑁⧵𝑆, since 𝑆 and 𝑆∪𝑖 are also carriers of 𝑎𝑢_𝑆, according to Axiom A₁, we have _𝑗∈𝑆𝜑_𝑗𝑎𝑢_𝑆=𝑎𝑢_𝑆(𝑆)=𝑎𝑢_𝑆()=𝑆∪𝑖_{𝑗∈𝑆∪𝑖}𝜑_𝑗𝑎𝑢_𝑆=_𝑗∈𝑆𝜑_𝑗𝑎𝑢_𝑆+𝜑_𝑖𝑎𝑢_𝑆. (26)
So 𝜑_𝑖𝑎𝑢_𝑆=_𝑗∈𝑆𝜑_𝑗𝑎𝑢_𝑆−_H𝑗∈𝑆𝜑_𝑗𝑎𝑢_𝑆=0. (27)
For 𝑖,𝑗∈𝑆 and 𝑖≠𝑗, let 𝜋 be a permutation of 𝑁 such that 𝜋(𝑖)=𝑗, 𝜋(𝑗)=𝑖, 𝜋(𝑘)=𝑘, 𝑘=𝑁⧵{𝑖,𝑗}. First, we prove that, for any 𝑇⊆𝑁, 𝑎𝑢_𝑆(𝜋𝑇)=𝑎𝑢_𝑆(𝑇). (28)
For 𝑇⊇𝑆, 𝜋𝑇⊇𝜋𝑆=𝑆. We have 𝑎𝑢_𝑆(𝜋𝑇)=𝑎=𝑎𝑢_𝑆(𝑇). (29)
For 𝑇⊉𝑆, if 𝑖∉𝑇, then 𝑗∉𝜋𝑇. If 𝑗∉𝑇, then 𝑖∉𝜋𝑇. If 𝑘∉𝑇 and 𝑘∈𝑆⧵{𝑖,𝑗}, then 𝑘∉𝜋𝑇. So 𝜋𝑇⊂𝑆.
Therefore, 𝑎𝑢_𝑆(𝜋𝑇)=0=𝑎𝑢_𝑆(𝑇). (30)
According to Axiom A₂, we have 𝜑_𝑗𝑎𝑢_𝑆=𝜑_𝜋(𝑖)𝜋𝑎𝑢_𝑆=𝜑_𝑖𝑎𝑢_𝑆. (31)
According to the above analysis and Axiom A₁, we obtain that _𝑖∈𝑆𝜑_𝑖𝑎𝑢_𝑆=|𝑆|𝜑_𝑖𝑎𝑢_𝑆=𝑎𝑢_𝑆(𝑆)=𝑎,∀𝑖∈𝑆. (32)
That is, 𝜑_𝑖𝑎𝑢_𝑆=𝑎|𝑆|,∀𝑖∈𝑆. (33)
This completes the proof of Lemma 19.

Lemma 20. If 𝑣̃∈𝐹𝐺(𝑁), for any 𝑆,Ø≠𝑆⊆𝑁, then 𝑣̃ is a linear combination of 𝑢_𝑆: 𝑣̃(𝑇)=^⊕_𝑆⊆𝑄𝑐_𝑆̃(𝑣̃)𝑢_𝑆(𝑇),∀𝑇⊆𝑁, (34) where 𝑄 is a carrier of 𝑣̃. The coefficients 𝑐_𝑆̃ are independent of 𝑄 and are given by 𝑐_𝑆̃(𝑣̃)=^⊕_𝑅⊆𝑆(−1)^{|𝑆|−|𝑅|}𝑣̃(𝑅)=^⊕_𝑅⊆𝑆(−1)^𝑠−𝑟𝑣̃(𝑅), (35) where 𝑠, 𝑟 denote the cardinality of coalitions  𝑆 and 𝑅, respectively.

Proof. If 𝑇⊆𝑄, then 𝑐_𝑆̃^⊕_𝑆⊆𝑄(𝑣̃)𝑢_𝑆(𝑇)=^⊕_𝑆⊆𝑄^⊕_𝑅⊆𝑆(−1)^𝑠−𝑟𝑣̃(𝑅)𝑢_𝑆(𝑇)=^⊕_{𝑆⊆𝑄𝑇⊇𝑆}^⊕_𝑅⊆𝑆(−1)^𝑠−𝑟𝑣̃(𝑅)𝑢_𝑆(𝑇)+^⊕_{𝑆⊆𝑄𝑇⊂𝑆}^⊕_𝑅⊆𝑆(−1)^𝑠−𝑟𝑣̃(𝑅)𝑢_𝑆(𝑇)=^⊕_{𝑆⊆𝑄𝑇⊇𝑆}^⊕_𝑅⊆𝑆(−1)^𝑠−𝑟𝑣̃(𝑅)=^⊕_𝑆⊆𝑇^⊕_𝑅⊆𝑆(−1)^𝑠−𝑟𝑣̃(𝑅)(since𝑇⊆𝑄)=^⊕_𝑅⊆𝑇^⊕_{𝑅⊆𝑆𝑆⊆𝑇}(−1)^𝑠−𝑟𝑣̃(𝑅)=^⊕_𝑅⊆𝑇^𝑡_𝑠=𝑟(−1)^𝑠−𝑟𝑣̃(𝑅)=𝑡−𝑟𝑠−𝑟)(let𝑘=𝑠−𝑟_{⊕𝑅⊆𝑇𝑡−𝑟𝑘=0}(−1)^𝑘𝑘𝑣̃(𝑅)𝑡−𝑟=^⊕_{𝑅⊆𝑇𝑅=𝑇}^𝑡−𝑟_𝑘=0(−1)^𝑘𝑘𝑣̃(𝑅)𝑡−𝑟+^⊕_{𝑅⊆𝑇𝑅≠𝑇}^𝑡−𝑟_𝑘=0(−1)^𝑘𝑘𝑣̃(𝑅)𝑡−𝑟=^⊕_{𝑅⊆𝑇𝑅=𝑇}𝑣̃(𝑅)+^⊕_{𝑅⊆𝑇𝑅≠𝑇}(1−1)^𝑡−𝑟𝑣̃(𝑅)=𝑣̃(𝑇). (36)
In general, according to the definition of carrier, we have 𝑣̃(𝑇)=𝑣̃(𝑇∩𝑄)=_{𝑆⊆𝑄𝑆≠Ø}^⊕𝑐_𝑆̃(𝑣̃)𝑢_𝑆(𝑇∩𝑄)=_{𝑆⊆𝑄𝑆≠Ø}^⊕𝑐_𝑆̃(𝑣̃)𝑢_𝑆(𝑇). (37)
This completes the proof of Lemma 20.

Remark 21. For 𝑣̃∈𝐹𝐺₀(𝑁) and any 𝜆∈(0,1], it is obvious that crisp games 𝑣⁻_𝜆,𝑣⁺_𝜆∈𝐺₀(𝑁). Assume 𝑐_𝑇𝑣⁺_𝜆=_{𝐵∈𝑃(𝑁),𝐵⊆𝑇}(−1)^{|𝑇|−|𝐵|}⋅𝑣⁺_𝜆(𝐵),𝑐_𝑇𝑣⁻_𝜆=_{𝐵∈𝑃(𝑁),𝐵⊆𝑇}(−1)^{|𝑇|−|𝐵|}⋅𝑣⁻_𝜆(𝐵). (38)
According to Proposition 14, it is seen that 𝑣⁻_𝜆 and 𝑣⁺_𝜆 can be uniquely written as 𝑣⁺_𝜆(𝐵)=_{𝑇∈𝑃(𝑁),𝑇≠Ø}𝑐_𝑇𝑣⁺_𝜆⋅𝑢_𝑇(𝐵),𝑣⁻_𝜆(𝐵)=_{𝑇∈𝑃(𝑁),𝑇≠Ø}𝑐_𝑇𝑣⁻_𝜆⋅𝑢_𝑇(𝐵). (39)
Obviously, ∑_{𝑇∈𝑃(𝑁),𝑇≠Ø}(𝑐_𝑇(𝑣⁺_𝜆)−𝑐_𝑇(𝑣⁻_𝜆))⋅𝑢_𝑇(𝐵)=𝑣⁺_𝜆(𝐵)−𝑣⁻_𝜆(𝐵)≥0. According to representation theorem of fuzzy set [25], it is easy to obtain the same conclusion as Lemma 20.

Proposition 22. If 𝑣̃,𝑤 and (𝑣̃−_H𝑤)∈𝐹𝐺₀(𝑁), then 𝜑_𝑖𝑣̃−_H𝑤=𝜑_𝑖(𝑣̃)−_H𝜑_𝑖(𝑤). (40)

Proof. Let 𝑧=𝑣̃−_H𝑤; then 𝑣̃=𝑤+𝑧. According to Axiom A₃, we have 𝜑_𝑖(𝑣̃)=𝜑_𝑖(𝑤)+𝜑_𝑖(𝑧). (41) So 𝜑_𝑖(𝑧)=𝜑_𝑖(𝑣̃)−_H𝜑_𝑖(𝑤); that is, 𝜑_𝑖(𝑣̃−_H𝑤)=𝜑_𝑖(𝑣̃)−_H𝜑_𝑖(𝑤).
This completes the proof of Proposition 22.

From Proposition 18, if 𝑐_𝑆̃(𝑣̃)≥0, then 𝑐_𝑆̃(𝑣̃)𝑢_𝑆∈𝐹𝐺₀(𝑁); otherwise, (−𝑐_𝑆(𝑣̃))𝑢_𝑆∈𝐹𝐺₀(𝑁).

Lemma 23. If 𝑣̃∈𝐹𝐺₀(𝑁), then 𝜑_𝑖(𝑣̃)=^⊕_{𝑖∈𝑆⊆𝑁}𝑐_𝑆̃(𝑣̃)|𝑆|. (42)

Proof. According to Lemmas 19 and 20 and Axiom A₃, we have 𝜑_𝑖(𝑣̃)=𝜑_𝑖^⊕_𝑆⊆𝑁𝑐_𝑆̃(𝑣̃)𝑢_𝑆=𝜑_{𝑖̃𝑐𝑆⊆𝑁𝑆(̃𝑣)≥0}𝑐_𝑆̃(𝑣̃)𝑢_𝑆−_{H̃𝑐𝑆⊆𝑁𝑆(̃𝑣)≱0}−𝑐_𝑆̃(𝑣̃)𝑢_𝑆=𝜑_{𝑖̃𝑐𝑆⊆𝑁𝑆(̃𝑣)≥0}𝑐_𝑆̃(𝑣̃)𝑢_𝑆−_H𝜑_{𝑖̃𝑐𝑆⊆𝑁𝑆(̃𝑣)≱0}−𝑐_𝑆̃(𝑣̃)𝑢_𝑆=_{̃𝑐𝑆⊆𝑁𝑆(̃𝑣)≥0}𝜑_𝑖𝑐_𝑆̃(𝑣̃)𝑢_𝑆−_{H̃𝑐𝑆⊆𝑁𝑆(̃𝑣)≱0}𝜑_𝑖−𝑐_𝑆̃(𝑣̃)𝑢_𝑆=_{̃𝑐𝑆⊆𝑁,𝑖∈𝑆𝑆(̃𝑣)≥0}𝜑_𝑖𝑐_𝑆̃(𝑣̃)𝑢_𝑆+_{̃𝑐𝑆⊆𝑁,𝑖∉𝑆𝑆(̃𝑣)≥0}𝜑_𝑖𝑐_𝑆̃(𝑣̃)𝑢_𝑆−_H×_{̃𝑐𝑆⊆𝑁,𝑖∈𝑆𝑆(̃𝑣)≱0}𝜑_𝑖−𝑐_𝑆̃(𝑣̃)𝑢_𝑆−_{H̃𝑐𝑆⊆𝑁,𝑖∉𝑆𝑆(̃𝑣)≱0}𝜑_𝑖−𝑐_𝑆̃(𝑣̃)𝑢_𝑆=_{̃𝑐𝑆⊆𝑁,𝑖∈𝑆𝑆(̃𝑣)≥0}𝜑_𝑖𝑐_𝑆̃(𝑣̃)𝑢_𝑆−_{H𝑐𝑆⊆𝑁,𝑖∈𝑆𝑆(̃𝑣)≱0}𝜑_𝑖−𝑐_𝑆̃(𝑣̃)𝑢_𝑆=_{̃𝑐𝑆⊆𝑁,𝑖∈𝑆𝑆(̃𝑣)≥0}𝑐_𝑆̃(𝑣̃)|𝑆|−_{H̃𝑐𝑆⊆𝑁,𝑖∈𝑆𝑆(̃𝑣)≱0}−𝑐_𝑆̃(𝑣̃)|𝑆|=^⊕_{𝑆⊆𝑁𝑖∈𝑆}𝑐_𝑆̃(𝑣̃)|𝑆|. (43)
This completes the proof of Lemma 23.

Theorem 24. Given 𝑣̃∈𝐹𝐺₀(𝑁), there exists a unique fuzzy Hukuhara-Shapley function 𝜑(𝑣̃), satisfying Axioms A₁–A₃, with element 𝜑_𝑖(𝑣̃)=_{𝑅⊆𝑁⧵𝑖}𝑟!(𝑛−𝑟−1)!𝑣̃()−𝑛!𝑅∪𝑖_H𝑣̃(𝑅),𝑖=1,2,…,𝑛, (44) where 𝑟, 𝑛 denote the cardinality of crisp coalitions  𝑅 and 𝑁, respectively.

Proof. According to Lemmas 23 and 19, we have 𝜑_𝑖(𝑣̃)=^⊕_{𝑖∈𝑆⊆𝑁}𝑐_𝑆̃(𝑣̃)|𝑆|=^⊕_{𝑖∈𝑆⊆𝑁}^⊕_𝑅⊆𝑆(−1)^𝑠−𝑟𝑣̃(𝑅)𝑠=^⊕_𝑅⊆𝑁^⊕_{𝑅⊆𝑆⊆𝑁𝑅∪𝑖⊆𝑆}(−1)^𝑠−𝑟𝑠𝑣̃(𝑅)=^⊕_{𝑅⊆𝑁𝑖∈𝑅}^⊕_{𝑅⊆𝑆⊆𝑁𝑅∪𝑖⊆𝑆}(−1)^𝑠−𝑟𝑠𝑣̃(𝑅)+^⊕_{𝑅⊆𝑁𝑖∉𝑅}^⊕_{𝑅⊆𝑆⊆𝑁𝑅∪𝑖⊆𝑆}(−1)^𝑠−𝑟𝑠𝑣̃(𝑅)=^⊕_{𝑅⊆𝑁𝑖∉𝑅}^⊕_{𝑅⊆𝑆⊆𝑁𝑅∪𝑖⊆𝑆}(−1)^{𝑠−(𝑟+1)}𝑠𝑣̃(𝑅∪𝑖)+^⊕_{𝑅⊆𝑁𝑖∉𝑅}^⊕_{𝑅⊆𝑆⊆𝑁𝑅∪𝑖⊆𝑆}(−1)^𝑠−𝑟𝑠𝑣̃(𝑅)=^⊕_{𝑅⊆𝑁𝑖∉𝑅}^𝑛_𝑠=𝑟+1(−1)^{𝑠−(𝑟+1)𝑛−𝑟−1}_{𝑠−𝑟−1}𝑠𝑣̃(𝑅∪𝑖)+^⊕_{𝑅⊆𝑁𝑖∉𝑅}^𝑛_𝑠=𝑟+1(−1)^{𝑠−𝑟𝑛−𝑟−1}_{𝑠−𝑟−1}𝑠𝑣̃(𝑅). (45)
Since ^𝑛_𝑠=𝑟+1(−1)^{𝑠−𝑟−1}𝑠=𝑛−𝑟−1()!𝑠−𝑟−1𝑟!𝑛−𝑟−1,𝑛!_{𝑛𝑠=𝑟+1}(−1)^{𝑠−𝑟𝑛−𝑟−1}_{𝑠−𝑟−1}𝑠()!=−𝑟!𝑛−𝑟−1.𝑛! (46)
Therefore, 𝜑_𝑖(𝑣̃)=^⊕_{𝑅⊆𝑁𝑖∉𝑅}𝑟!(𝑛−𝑟−1)!𝑣̃()𝑛!𝑅∪𝑖+^⊕_{𝑅⊆𝑁𝑖∉𝑅}−𝑟!(𝑛−𝑟−1)!𝑣̃(𝑅)𝑛!=_{𝑅⊆𝑁𝑖∉𝑅}𝑟!(𝑛−𝑟−1)!𝑣̃()−𝑛!𝑅∪𝑖_H×_{𝑅⊆𝑁𝑖∉𝑅}𝑟!(𝑛−𝑟−1)!𝑣̃(𝑅)()!𝑛!since𝑟!𝑛−𝑟−1𝑛!>0=_{𝑅⊆𝑁⧵𝑖}𝑟!(𝑛−𝑟−1)!𝑣̃()−𝑛!𝑅∪𝑖_H𝑣̃(𝑅). (47)
From Lemma 20, it is known that 𝑐_𝑆̃(𝑣̃) is well defined by 𝑆 and 𝑣̃. Thus according to Lemma 23, 𝜑_𝑖(𝑣̃) is well defined by 𝑣̃, 𝑁, and 𝑖; that is, 𝜑_𝑖(𝑣̃) is well defined by 𝑣̃ and 𝑁.
In the following, we prove that the function defined by (44) satisfies Axioms A₁–A₃ in Definition 17.
According to Lemmas 19 and 20, it is easy to verify that 𝜑_𝑖(𝑣̃) satisfies Axioms A₁ and A₂.
Let 𝑣,𝑤∈𝐹𝐺₀(𝑁) and 𝑣+𝑤∈𝐹𝐺₀(𝑁); then 𝜑_𝑖(𝑣̃+𝑤)=_{𝑅⊆𝑁⧵𝑖}𝑟!(𝑛−𝑟−1)!̃𝑤)−𝑛!(𝑣+)(𝑅∪𝑖_H(𝑣̃+𝑤)(𝑅)=_{𝑅⊆𝑁⧵𝑖}𝑟!(𝑛−𝑟−1)!𝑣̃()+𝑤(−𝑛!(𝑅∪𝑖𝑅∪𝑖))_H×(𝑣̃(𝑅)+𝑤(𝑅))=_{𝑅⊆𝑁⧵𝑖}𝑟!(𝑛−𝑟−1)!𝑣̃()−𝑛!𝑅∪𝑖_H𝑣̃(𝑅)+𝑤(𝑅∪𝑖)−_H𝐻𝑤(𝑅)=_{𝑅⊆𝑁⧵𝑖}𝑟!(𝑛−𝑟−1)!𝑣̃()−𝑛!𝑅∪𝑖_H𝑣̃(𝑅)+_{𝑅⊆𝑁⧵𝑖}𝑟!(𝑛−𝑟−1)!𝑤()−𝑛!𝑅∪𝑖_H𝑤(𝑅)=𝜑_𝑖(𝑣̃)+𝜑_𝑖(𝑤). (48)
These complete the proof of Proposition 14.

Remark 25. Given 𝑣̃∈𝐹𝐺₀(𝑁), from Theorem 24, for any 𝜆∈(0,1] and 𝑣̃_𝜆, there exists a unique interval Hukuhara-Shapley function 𝜑(𝑣̃_𝜆) with element 𝜑_𝑖𝑣̃_𝜆=_{𝑅⊆𝑁⧵𝑖}𝑟!(𝑛−𝑟−1)!𝑣̃𝑛!^𝜆(𝑅∪𝑖)−_H𝑣̃_𝜆(𝑅),𝑖=1,2,…,𝑛. (49)
For the interval Hukuhara-Shapley function 𝜑(𝑣̃_𝜆), 𝜆∈(0,1], and it has the following properties.

Proposition 26. (i) For any pair (𝜆₁,𝜆₂) satisfying 0≤𝜆₁<𝜆₂≤1, 𝜑_𝑖(𝑣̃_𝜆1)⊇𝜑_𝑖(𝑣̃_𝜆2).
(ii) For any 𝜆∈(0,1], ⋂_𝜆<𝜆^∗𝜑_𝑖(𝑣̃_𝜆)=𝜑_𝑖(𝑣̃_𝜆^∗).

Proof. (i) According to (49), it is easy to prove the conclusion.
(ii) Let 𝛾_𝑅=𝑟!(𝑛−𝑟−1)!/𝑛!; from (49), for any 𝜆^∗∈(0,1], we have _𝜆<𝜆∗𝜑_𝑖𝑣̃_𝜆=_{𝜆<𝜆∗𝑅⊆𝑁⧵𝑖}𝛾_𝑅𝑣̃_𝜆(𝑅∪𝑖)−_H𝑣̃_𝜆(𝑅=_{𝜆<𝜆∗𝑅⊆𝑁⧵𝑖})𝛾_𝑅𝑣⁻_𝜆(𝑅∪𝑖)−𝑣⁻_𝜆(𝑅)_{𝑅⊆𝑁⧵𝑖},𝛾_𝑅𝑣⁺_𝜆(𝑅∪𝑖)−𝑣⁺_𝜆(𝑅=)lim_{𝜆→𝜆∗𝑅⊆𝑁⧵𝑖}𝛾_𝑅𝑣⁻_𝜆(𝑅∪𝑖)−𝑣⁻_𝜆(𝑅),lim_{𝜆→𝜆∗𝑅⊆𝑁⧵𝑖}𝛾_𝑅𝑣⁺_𝜆(𝑅∪𝑖)−𝑣⁺_𝜆(𝑅=_{𝑅⊆𝑁⧵𝑖})𝛾_𝑅𝑣⁻_𝜆^∗(𝑅∪𝑖)−𝑣⁻_𝜆∗(𝑅)_{𝑅⊆𝑁⧵𝑖},𝛾_𝑅𝑣⁺_𝜆^∗(𝑅∪𝑖)−𝑣⁺_𝜆^∗(𝑅)=_{𝑅⊆𝑁⧵𝑖}𝛾_𝑅𝑣̃_𝜆^∗(𝑅∪𝑖)−_H𝑣̃_𝜆∗(𝑅=)𝜑_𝑖𝑣̃_𝜆^∗. (50)
These complete the proof of Proposition 26.

Call players 𝑖 and 𝑗 symmetric in the game (𝑁,𝑣̃) if 𝑣̃(𝑆∪𝑖)=𝑣̃(𝑆∪𝑗) for every coalition 𝑆⊆𝑁⧵{𝑖,𝑗}. Call a player 𝑖 in a game (𝑁,𝑣̃) a dummy (player) if 𝑣̃(𝑆∪𝑖)−_H𝑣̃(𝑆)=𝑣̃(𝑖) for all 𝑆⊆𝑁⧵{𝑖}.

According to Theorem 24, it is easy to obtain the following conclusion.

Corollary 27. Given 𝑣̃∈𝐹𝐺₀(𝑁), if 𝑖 and 𝑗 are symmetric in game (𝑁,𝑣̃), then 𝜑_𝑖(𝑣̃)=𝜑_𝑗(𝑣̃). If 𝑖 is a dummy in game (𝑁,𝑣̃), then 𝜑_𝑖(𝑣̃)=𝑣̃(𝑖). If 𝑖 is a null player in game (𝑁,𝑣̃), then 𝜑_𝑖(𝑣̃)=0.

Symmetric players have the same contribution to any coalition, and therefore it seems reasonable that they should obtain the same payoff according to the value. A dummy player only contributes his/her own worth to every coalition, and that is what he/she should be paid. A null player does not contribute anything to any coalition; in particular also 𝑣̃(𝑖)=0. So it seems reasonable that such a player obtains zero according to the value.

Corollary 28. Given 𝑣̃∈𝐹𝐺₀(𝑁), for all 𝑖∈𝑁, 𝜑_𝑖(𝑣̃)≥𝑣̃(𝑖). (51)

Proof. Let 𝑄 be a carrier of 𝑣. If 𝑖∈𝑁 and 𝑖∉𝑄, then 𝑣̃(𝑅∪𝑖)=𝑣̃(𝑅).
From 𝑣̃(𝑅∪𝑖)≥𝑣̃(𝑅)+𝑣̃(𝑖), we have 𝑣̃(𝑖)=0.
From Corollary 27, we have 𝜑_𝑖(𝑣̃)=0, so 𝜑_𝑖(𝑣̃)=𝑣̃(𝑖).
If 𝑖∈𝑁 and 𝑖∈𝑄, since 𝑣̃(𝑅∪𝑖)≥𝑣̃(𝑅)+𝑣̃(𝑖), we have 𝑣̃(𝑅∪𝑖)−_H𝑣̃(𝑅)≥𝑣̃(𝑖).
Hence, according to (44), we easily have 𝜑_𝑖(𝑣̃)≥𝑣̃(𝑖).
The proof is completed.

According to Theorem 24, it is easy to obtain the following conclusions, too.

Corollary 29. Define a function 𝜑∶𝐹𝐺₀(𝑁)→(𝔽ℝ⁺)^𝑛 by 𝜑_𝑖(𝑣̃)(𝑊)=_{𝑅⊆𝑊⧵𝑖}𝑟!(𝑤−𝑟−1)!𝑣̃()−𝑤!𝑅∪𝑖_H𝑣̃(𝑅,𝜑)𝑖𝑓𝑖∈𝑊,_𝑖(𝑣̃)(𝑊)=0,𝑖𝑓𝑖∉𝑊. (52)
Then the function 𝜑 is the unique fuzzy Shapley function on 𝐹𝐺₀(𝑊).

Corollary 30. For any two 𝑣̃,𝑤∈𝐹𝐺₀(𝑁), if 𝑣̃(𝑆∪𝑖)−_H𝑣̃(𝑆)≤𝑤(𝑆∪𝑖)−_H𝑤(𝑆) for any 𝑆∈𝑃(𝑁), then, for any 𝑖∈𝑁, 𝜑_𝑖(𝑣̃)≤𝜑_𝑖(𝑤). (53)

According to the above analysis, we can use another equivalent axiomatic system to define the fuzzy Hukuhara-Shapley function on 𝐹𝐺₀(𝑁).

Theorem 31. Given 𝑣̃∈𝐹𝐺₀(𝑁), let 𝜓∶𝐹𝐺₀(𝑁)→(𝔽ℝ⁺)^𝑛 be a value. Then 𝜓 satisfies the following three axioms, if and only if 𝜓(𝑣̃) is the fuzzy Hukuhara-Shapley function 𝜑(𝑣̃).
Axiom F₁. If 𝑣̃∈𝐹𝐺₀(𝑁), then _𝑖∈𝑁𝜓_𝑖(𝑣̃)=𝑣̃(𝑁). (54)
Axiom F₂. If players 𝑖 and 𝑗 are symmetricin the game (𝑁,𝑣̃), then 𝜓_𝑖(𝑣̃)=𝜓_𝑗(𝑣̃). (55)
Axiom F₃. For any two 𝑣,𝑤∈𝐹𝐺₀(𝑁), if 𝑣̃(𝑆∪𝑖)−_H𝑣̃(𝑆)≤𝑤(𝑆∪𝑖)−_H𝑤(𝑆) for any 𝑆∈𝑃(𝑁), then, for ∀𝑖∈𝑁, 𝜓_𝑖(𝑣̃)≤𝜓_𝑖(𝑤). (56)

Proof. Obviously, 𝜑(𝑣̃) satisfies the three axioms. Conversely, suppose 𝜓(𝑣̃) satisfies the three axioms.(i)Let 𝑧∈𝐹𝐺₀(𝑁) that is identically zero. In this fuzzy game, all players are symmetric, so Axiom F₂ and Axiom F₁ together imply 𝜓(𝑧)=0.(ii)Let 𝑖 be a null player in 𝑣̃∈𝐹𝐺₀(𝑁). Then the condition in Axiom F₃ applies to 𝑧 and 𝑣̃ with all inequalities being equalities. So Axiom F₃ yields 𝜓_𝑖(𝑣̃)≥𝜓_𝑖(𝑧) and 𝜓_𝑖(𝑧)≥𝜓_𝑖(𝑣̃). Hence, by (i), 𝜓_𝑖(𝑣̃)=0.(iii)Let 𝑆⊆𝑁⧵Ø and 𝑐_𝑆̃∈𝔽ℝ. According to Lemma 20, we have 𝜓_𝑖𝑐_𝑆̃𝑢_𝑆𝜓=0forevery𝑖∈𝑁⧵𝑆,_𝑖𝑐_𝑆̃𝑢_𝑆=𝑐_𝑆̃|𝑆|forevery𝑖∈𝑆. (57) Hence, 𝜓(𝑐_𝑆̃𝑢_𝑆)=𝑐_𝑆̃|𝑆|⁻¹𝑒^𝑆, where 𝑒^𝑆=(𝑠₁,𝑠₂,…,𝑠_𝑛) satisfying 𝑠_𝑖=1 when 𝑖∈𝑆 and 𝑠_𝑖=0 when 𝑖∈𝑁⧵𝑆.(iv)According to Lemma 23 we have 𝑣̃=_{𝑆∈𝑃(𝑁)⧵Ø}𝑐_𝑆̃𝑢_𝑆. (58)The proof of 𝜓(𝑣̃)=𝜑(𝑣̃) is completed by induction on the number 𝛼(𝑣̃) of terms in 𝑣̃=∑_{𝑆∈𝑃(𝑁)⧵Ø}𝑐_𝑆̃𝑢_𝑆 with 𝑐_𝑆̃≠0.
From (i), 𝜓(𝑣̃)=𝜑(𝑣̃)=0 if 𝛼(𝑣̃)=0 and, from (iii), 𝜓(𝑣̃)=𝜑(𝑣̃) if 𝛼(𝑣̃)=1 because of 𝜙(𝑐_𝑆̃𝑢_𝑆)=𝑐_𝑆̃|𝑆|⁻¹𝑒^𝑇. Suppose 𝜓(𝑤)=𝜑(𝑤) for all 𝑤∈𝐹𝐺₀(𝑁) with 𝛼(𝑤)<𝑘, where 𝑘≥2. Let 𝑣̃∈𝐹𝐺₀(𝑁) with 𝛼(𝑣̃)=𝑘. Then there exist coalitions 𝑆₁,𝑆₂,…,𝑆_𝑘 and fuzzy numbers 𝑐₁̃,𝑐₂̃,…,𝑐_𝑘̃ unequal to zero, such that 𝑣̃=∑^𝑘_𝑟=1𝑐_𝑟̃𝑢_𝑆𝑟. Let 𝐷=⋂^𝑘_𝑟=1𝑆_𝑟.
For 𝑖∈𝑁⧵𝐷, define 𝑤^𝑖=∑_{𝑟∶𝑖∈𝑆𝑟}𝑐_𝑟̃𝜇_𝑆𝑟. Because 𝛼(𝑤^𝑖)<𝑘, the induction hypothesis implies 𝜓_𝑖𝑤^𝑖=𝜑_𝑖𝑤^𝑖. (59) Further, for every 𝑇∈𝑃(𝑁), 𝑣̃(𝑇∪𝑖)−_H𝑣̃(𝑇)=^𝑘_𝑟=1𝑐_𝑟̃𝑢_𝑆𝑟(𝑇∪𝑖)−_H^𝑘_𝑟=1𝑐_𝑟̃𝑢_𝑆𝑟(𝑇)=_{𝑟∶𝑖∈𝑆𝑟}𝑐_𝑟̃𝑢_𝑆𝑟(𝑇∪𝑖)−_{H𝑟∶𝑖∈𝑆𝑟}𝑐_𝑟̃𝑢_𝑆𝑟(𝑇)=𝑤^𝑖(𝑇∪𝑖)−_H𝑤^𝑖(𝑇). (60) So, by Axiom F₂, it follows that 𝜓_𝑖(𝑣̃)=𝜓_𝑖(𝑤^𝑖)=𝜙_𝑖(𝑤^𝑖)=𝜙_𝑖(𝑣̃). So, for all 𝑖∈𝑁⧵𝐷, we have 𝜓_𝑖(𝑣̃)=𝜑_𝑖(𝑣̃). (61) Further, according to Axiom F₁, combining with (61), we have _𝑖∈𝐷𝜓_𝑖(𝑣̃)=_𝑖∈𝐷𝜑_𝑖(𝑣̃). (62) Let 𝑖,𝑗∈𝐷; then, for every 𝑇⊆𝑁⧵𝑖,𝑗, 𝑣̃(𝑇∪𝑖)=^𝑘_𝑟=1𝑐_𝑟̃𝑢_𝑆𝑟()=𝑇∪𝑖_𝑘𝑟=1𝑐_𝑟̃𝑢_𝑆𝑟𝑇∪𝑗=𝑣̃𝑇∪𝑗, (63) so 𝑖 and 𝑗 are symmetric. Hence, by Axiom F₂, we have 𝜓_𝑖(𝑣̃)=𝜓_𝑗(𝑣̃),𝜑_𝑖(𝑣̃)=𝜑_𝑗(𝑣̃). (64) From (61), (62), and (64), we have 𝜓(𝑣̃)=𝜑(𝑣̃).
The proof is completed.

Example 32. Consider three economic companies, named 1, 2, and 3 (𝑖=1,2,3). They possess different resources. Now they want to do a joint project by means of pooling their resources. It is natural for these three decision-makers to try to evaluate the revenue of the joint project in the early period of the project in order to decide whether the project can be realized or not. However, the average profit of the joint project is dependent on a number of companies. And the average profit of the joint project is an approximate evaluation, which is represented by fuzzy numbers as follows: 𝑣̃(𝑖𝑥)=[][]𝑣̃(𝑥)=𝑥)(𝑥−4,𝑥∈4,56−𝑥,𝑥∈5,60,otherwise,𝑖=1,2,3.1,2)(5−6,𝑥∈[30,35]383−𝑥3[]𝑣̃(𝑥)=𝑥,𝑥∈35,380,otherwise,1,3)(2−232,𝑥∈[23,25]272−𝑥2[]𝑣̃(𝑥)=𝑥,𝑥∈25,270,otherwise,2,3)(5−3,𝑥∈[15,20]233−𝑥3[]𝑣̃(𝑥)=𝑥,𝑥∈20,230,otherwise,1,2,3)(5[]𝑥−9,𝑥∈45,5011−5[]𝑣̃(Ø),𝑥∈50,550,otherwise,=0. (65)
According to (44), we can calculate fuzzy Hukuhara-Shapley function of each decision-maker as follows: 𝜑₁(𝑣̃𝑥)=6)(7𝑥−1137,𝑥∈1136,20433−23𝑥,𝑥∈20,432𝜑0,otherwise,₂(𝑣̃𝑥)=3)(8𝑥−8916,𝑥∈896,352394−12𝑥,𝑥∈352,392𝜑0,otherwise,₃(𝑣̃𝑥)=6)(7𝑥−687,𝑥∈343,252283−23𝑥,𝑥∈252,2420,otherwise. (66)
By judging the allocations of fuzzy Hukuhara-Shapley function of each decision-maker, decision-makers can conclude whether the joint project can be realized or not. According to fuzzy Hukuhara-Shapley function, it is very convenient for us to obtain their interval Hukuhara-Shapley values of different 𝜆-level as shown in Table 1.

Table 1 

Interval Hukuhara-Shapley values for different 𝜆-level of decision-makers.

4. Generalized Fuzzy Shapley Function for Generalized Fuzzy Games

4.1. Cooperative Games with Fuzzy Coalition

For a finite set of players 𝑁={1,2,…,𝑛}, we call 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛), 𝑥_𝑖∈[0,1], the fuzzy coalition variable and call 𝑥^#={𝑦∣𝑦_𝑖=𝑥_𝑖or𝑦_𝑖=0foreach𝑖∈𝑁} the set of fuzzy coalition variables created by 𝑥. The level variable set of 𝑥 is denoted by 𝐿𝑥={𝑥₁,𝑥₂,…,𝑥_𝑛}. For any 𝐴⊆𝑁, we denote 𝐿𝐴={𝑥_𝑖∣𝑖∈𝐴}. 𝑠=(𝑠₁,𝑠₂,…,𝑠_𝑛) is called a fuzzy coalition; here, 𝑠_𝑖 is a constant which denotes the participation level of player 𝑖. Similar to 𝑥^#, we denote by  𝑠^# the class of fuzzy coalitions created by 𝑠, which is defined by 𝑠^#={𝑡∣𝑡_𝑖=𝑠_𝑖or𝑡_𝑖=0foreach𝑖∈𝑁}. The set of fuzzy coalitions is denoted by 𝐿(𝑁); the empty coalition in a fuzzy setting is denoted by 𝑒^Ø=(0,0,…,0). 𝑒^𝑆=(𝑠₁,𝑠₂,…,𝑠_𝑛) satisfying 𝑠_𝑖=1 when 𝑖∈𝑆 and 𝑠_𝑖=0 when 𝑖∈𝑁⧵𝑆, with 𝑆∈𝑃(𝑁), denotes a crisp coalition. It corresponds to the situation where the players within 𝑆 fully cooperate; that is, they have participation level 1, and the players outside 𝑆 are not involved in at all; that is, they have participation level 0. 𝑒^𝑁=(1,1,…,1) is called the grand coalition. We write 𝑒^𝑖 instead of 𝑒^{𝑖}. Let 𝑇⊆𝑁; the fuzzy coalition variable 𝑥_𝑇 corresponding to 𝑇 is denoted by 𝑥_𝑇=∑_𝑘∈𝑇𝑥_𝑘𝑒^𝑘. When 𝑥=𝑠, the fuzzy coalition 𝑠_𝑇 corresponding to 𝑇 is denoted by 𝑠_𝑇=∑_𝑘∈𝑇𝑥_𝑘𝑒^𝑘.

Cooperative games with fuzzy coalitions are represented by a pair (𝑁,𝑣), where function 𝑣∶𝐿(𝑁)→ℝ⁺ such that 𝑣(Ø)=0. If there is no fear of confusion, it is denoted by 𝑣 as well. The class of cooperative games with fuzzy coalitions and player set 𝑁 is denoted by 𝐿𝐺(𝑁).

In what follows, 𝑠∨𝑡 and 𝑠∧𝑡 are those elements of 𝐿(𝑁) with the 𝑖th coordinate equal to max(𝑠_𝑖,𝑡_𝑖) and min(𝑠_𝑖,𝑡_𝑖), respectively. The set operations ∨ and ∧ play the same role for the fuzzy coalitions as the union and intersection for crisp coalitions. For any 𝑠,𝑡∈𝐿(𝑁), 𝑠∧𝑡=Ø, if 𝑣(𝑠∨𝑡)≥𝑣(𝑠)+𝑣(𝑡), then 𝑣 is called superadditive game.

Definition 33. Given 𝑣∈𝐿𝐺(𝑁), 𝑠∈𝐿(𝑁). Based on 𝑠, the participation level 𝑠_𝑖 of player 𝑖 is called a dummy participation level if, for any 𝑡∈(𝑠−∑_{𝑘∈𝑁⧵𝑖}𝑠_𝑘𝑒^𝑘)^#, 𝑣(𝑠_𝑖𝑒^𝑖∨𝑡)=𝑣(𝑠_𝑖𝑒^𝑖)+𝑣(𝑡). And the participation level 𝑠_𝑖 of player 𝑖 is called a null participation level if, for any 𝑡∈(𝑠−∑_{𝑘∈𝑁⧵𝑖}𝑠_𝑘𝑒^𝑘)^#, 𝑣(𝑠_𝑖𝑒^𝑖∨𝑡)=𝑣(𝑡).

Definition 34. Given 𝑣∈𝐿𝐺₀(𝑁), 𝑠∈𝐿(𝑁); based on 𝑠, a fuzzy coalition 𝑠∈𝑠^# is called a carrier of 𝑠 in 𝑣 if, for any 𝑡∈𝑠^#, 𝑣(𝑡)=𝑣(𝑡∧𝑠).
The set of carriers of 𝑠 in 𝑣 is denoted by 𝑆𝐶(𝑣,𝑠).
Let 𝜋 be a permutation of 𝑁. For any 𝑠∈𝐿(𝑁) and any 𝑣∈𝐿𝐺₀(𝑁), the permutation of fuzzy coalition 𝑠 is defined by 𝜋𝑠=𝑠⋅𝜋⁻¹ and 𝜋𝑣(𝑠)=𝑣(𝜋⁻¹𝑠). Then, 𝜋𝑣∶𝑠↦𝜋𝑣(𝑠) from 𝐿(𝑁) to ℝ⁺ is still a fuzzy game in 𝐿𝐺₀(𝑁).
A crisp coalition 𝑆∈2^𝑁 corresponds in a canonical way with fuzzy coalition 𝑒_𝑆, where 𝑒^𝑆∈[0,1]^𝑁 is the vector 𝑒^𝑆={𝑒^𝑆(1),𝑒^𝑆(2),…,𝑒^𝑆(𝑛)}, where the 𝑖th coordinate 𝑒^𝑆(𝑖)=1 if and only if 𝑖∈𝑆; otherwise, 𝑒^𝑆(𝑖)=0. From the viewpoint of geometry meaning, the set 2^𝑁 of all admissible coalitions in 𝑁 is the set of all vertexes of [0,1]^𝑁. Therefore, the set 2^𝑁 of all admissible coalitions can be equivalently denoted by {0,1}^𝑁∶𝑆∈2^𝑁→𝑒_𝑆∈{0,1}^𝑁; that is, the correspondence between 2^𝑁 and {0,1}^𝑁 is one to one. Through this identification of coalitions with their characteristic vectors, a cooperative game is a function 𝑣∶{0,1}^𝑁→ℝ⁺ with 𝑣(𝑂)=0, where 𝑂=(0,0,…,0) corresponding to the empty coalition.
There exist different methods to extend 𝑛-person cooperative games to 𝑛-person cooperative games with fuzzy coalitions, such as proportional values method [5] and Choquet integral method [6, 26]. However, Li and Zhang [9] pointed out that which one is more natural may be insignificant if the specific game situation is not considered. Each fuzzy game may only be suitable for a certain case. When a fuzzy game established in one case is applied to another situation, the invalidity of it may be inevitable. Inspired by their work, in the following, we propose a generalized fuzzy game, game with fuzzy coalitions and fuzzy characteristic function, and a simplified expression of the fuzzy Shapley function is given for this generalized fuzzy game. It is also proved that the simplified expression is regarded as the generalization of fuzzy Shapley function defined in some particular fuzzy games, such as fuzzy games with proportional value and Choquet integral form.

4.2. Generalized Fuzzy Shapley Function for Generalized Fuzzy Games

In this section, we incorporate fuzzy coalition and fuzzy characteristic function into the game, which is called a generalized fuzzy game or fuzzy game.

A generalized fuzzy game is represented by a pair (𝑁,𝑤), where the fuzzy characteristic function 𝑤∶𝐿(𝑁)→𝔽ℝ⁺ with 𝑤(Ø)=0.

Obviously, a fuzzy game degenerates to be a game with fuzzy coalitions when the range of the mapping 𝑤 is ℝ⁺ and to be a game with a fuzzy characteristic function when the definitional domain of the mapping 𝑤 is 𝑃(𝑁). It degenerates to be a crisp game when the definitional domain and range of the mapping 𝑤 are 𝑃(𝑁) and ℝ⁺, respectively. Thus, our discussion and game models can be applicable to games involving both crisp and fuzzy games. The set of games with fuzzy coalition and fuzzy characteristic function is denoted by 𝐿𝐹𝐺(𝑁).

Definition 35. Given 𝑤∈𝐿𝐹𝐺(𝑁), 𝑠∈𝐿(𝑁). A function 𝜑(𝑤^𝑠)∶𝐿𝐹𝐺(𝑁)→(𝔽ℝ⁺)^𝑛 is called generalized fuzzy Shapley function based on 𝑠 if it satisfies the following three axioms.
Axiom S₁. If 𝑤∈𝐿𝐹𝐺(𝑁), 𝑠∈𝑆𝐶(𝑤,𝑠), then _𝑖∈𝑁𝜑^𝑠_𝑖(𝑤)=𝑤(𝑠). (67)
Axiom S₂. If 𝑤∈𝐿𝐹𝐺(𝑁) and 𝜋 is a permutation of 𝑠, then, for any 𝑖∈𝑁, 𝜑^𝑠_𝜋𝑖(𝜋𝑤)=𝜑^𝑠_𝑖(𝑤). (68)
Axiom S₃. For any 𝑤₁, 𝑤₂∈𝐿𝐹𝐺(𝑁), if a game is defined by (𝑤₁+𝑤₂)(𝑡)=𝑤₁(𝑡)+𝑤₂(𝑡) for any 𝑡∈𝑠^#, then 𝜑^𝑠_𝑖𝑤₁+𝑤₂=𝜑^𝑠_𝑖𝑤₁+𝜑^𝑠_𝑖𝑤₂∀𝑖∈𝑁. (69)

Theorem 36. Given 𝑤∈𝐿𝐹𝐺(𝑁), 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) is fuzzy coalition variable, 𝑠=(𝑠₁,𝑠₂,…,𝑠_𝑛)∈𝐿(𝑁). Then 𝜑_𝑥𝑖(𝑤)_𝑥=𝑠=_{𝑅⊆𝑁⧵𝑖}𝑟!(𝑛−𝑟−1)!𝑤𝑛!_{𝑗∈𝑅∪𝑖}𝑠_𝑗𝑒^𝑗−_H𝑤_𝑗∈𝑅𝑠_𝑗𝑒^𝑗,𝑖=1,2,…,𝑛 (70) is a generalized fuzzy Shapley function of player 𝑖 with participation level 𝑠_𝑖 based on 𝑠.

Proof. It is similar to the proof of fuzzy Hukuhara-Shapley function in Theorem 24; therefore, here it is omitted.

By means of Choquet integral, Yu and Zhang [18] proposed a class of fuzzy games with indeterminate integral form and the fuzzy Shapley function was developed.

Definition 37. Given 𝑠∈𝐿(𝑁), let 𝑄(𝑠)={𝑠_𝑖∣𝑠_𝑖>0,𝑖∈𝑁} and let 𝑞(𝑠) be the cardinality of 𝑄(𝑠); that is, 𝑞(𝑠)=|𝑄(𝑠)|. The elements of 𝑄(𝑠) are written in the increasing order as ℎ₁<⋯<ℎ_𝑞(𝑠). Then a game 𝑤∈𝐿𝐹𝐺(𝑁) is said to be a fuzzy game with “indeterminate integral form” if and only if the following holds: 𝑤(𝑠)=^𝑞(𝑠)_𝑙=1𝑤[𝑠]_ℎ𝑙ℎ_𝑙−ℎ_𝑙−1, (71) for any 𝑠∈𝐿(𝑁), where ℎ₀=0.
The set of games with indeterminate integral form in 𝐿𝐹𝐺(𝑁) is denoted by 𝐿𝐹𝐺_𝐶(𝑁).
For 𝑤∈𝐿𝐹𝐺_𝐶(𝑁), Yu and Zhang [18] obtained the following conclusion.

Theorem 38. The function 𝑓∶𝐿𝐹𝐺_𝐶(𝑁)→(𝔽ℝ^𝑛₊)^𝐿(𝑁) defined by 𝑓_𝑖(𝑤)(𝑠)=^𝑞(𝑠)_𝑙=1𝜑_𝑖(𝑤)[𝑠]_ℎ𝑙ℎ_𝑙−ℎ_𝑙−1 (72) is a fuzzy Shapley function on 𝐿𝐹𝐺_𝐶(𝑁), where 𝜑 is the function given by (44).
Besides Choquet integral method, we can apply the proportional values method proposed by Butnariu [5] to obtain a special game with fuzzy coalition and fuzzy characteristic function.
Let 𝑠 be a fuzzy coalition and 𝑟∈[0,1]. 𝑟-section of 𝑠 is denoted by 𝑠^𝑟={𝑖∣𝑖∈𝑁,𝑠_𝑖=𝑟}. The 𝑟-section of 𝑠 is the set of players, in which each player’s participation level is 𝑟. Based on the definition of 𝑟-section, we define a fuzzy game with proportional values.

Definition 39. The game 𝑤∈𝐿𝐹𝐺(𝑁) is said to be a fuzzy game with proportional values if and only if for any 𝑠 in 𝐿(𝑁) the following holds: 𝑤(𝑠)=_𝑟∈[0,1]𝑤𝑠^𝑟⋅𝑟. (73)
The set of the fuzzy games with proportional values in 𝐿𝐹𝐺(𝑁) will be denoted by 𝐿𝐹𝐺_𝑃(𝑁). For the sake of simplicity, a fuzzy game with proportional values form is called a fuzzy game if there is no fear of confusion. There is a one-to-one correspondence between a game with a fuzzy characteristic function and a fuzzy game. We call the game with a fuzzy characteristic function corresponding to a fuzzy game the associated game with a fuzzy characteristic function, and we call the fuzzy game corresponding to a game with a fuzzy characteristic function the associated fuzzy game.

Theorem 40. A function 𝑓∶𝐿𝐹𝐺_𝑃(𝑁)→(𝔽ℝ^𝑛₊)^𝐿(𝑁) defined by 𝑔_𝑖(𝑤)(𝑠)=𝜑_𝑖(𝑤)𝑠^𝑟⋅𝑠_𝑖 (74) is a fuzzy Shapley function of player 𝑖 with participation level 𝑠_𝑖 on 𝐿𝐹𝐺_𝐶(𝑁), where 𝜑 is the function given by (44).

Proof. We only need to prove that the function defined by (74) satisfies Axioms S₁–S₃ in Definition 35. Similar to the proof of Theorem 24, it is easy to obtain the result. Therefore, here it is omitted.

In the following, we make a comparison among the above fuzzy Shapley functions.

For any 𝑆⊆𝑁, we denote by  𝑁(𝑆) the set of 𝑛-players with the players of 𝑁⧵𝑆 being null players of 𝑣. Then, for any 𝑇⊆𝑁(𝑆) and 𝑣̃∈𝐹𝐺(𝑁), we have 𝑣̃(𝑇)=𝑣̃(𝑇∩𝑆).

Lemma 41. Let 𝑣̃∈𝐹𝐺(𝑁); 𝜑_𝑖(𝑣̃) is a fuzzy Hukuhara-Shapley function in Theorem 24. Then, for any 𝑆∈𝑃(𝑁), 𝜑_𝑖(𝑣̃)(𝑆)=𝜑_𝑖(𝑣̃)(𝑁(𝑆)). (75)

Proof. According to the definition of 𝜑_𝑖(𝑣̃), the proof of Lemma 41 is similar to that of Lemma 1 in Li and Zhang [9] and therefore here it is omitted.

Theorem 42. Let 𝑤∈𝐿𝐹𝐺_𝐶(𝑁), 𝑠=(𝑠₁,𝑠₂,…,𝑠_𝑛)∈𝐿(𝑁). Then, for any 𝑖∈𝑁, 𝜑_𝑥𝑖(𝑤)_𝑥=𝑠=𝑓_𝑖(𝑤)(𝑠). (76)

Proof. Let 𝑄(𝑠)={𝑠_𝑖∣𝑠_𝑖>0,𝑖∈𝑁}, 𝑞=|𝑄(𝑠)|. The elements of 𝑄(𝑠) are written in the increasing order as ℎ₁<⋯<ℎ_𝑞. Suppose 𝑠_𝑖=ℎ_𝑘; then 𝑓_𝑖(𝑤)(𝑠)=^𝑞(𝑠)_𝑙=1𝜑_𝑖(𝑤)[𝑠]_ℎ1ℎ_𝑙−ℎ_𝑙−1=𝜑_𝑖(𝑤)[𝑠]_ℎ1ℎ₁+𝜑_𝑖(𝑤)[𝑠]_ℎ2ℎ₂−ℎ₁+⋯+𝜑_𝑖(𝑤)[𝑠]_ℎ𝑞ℎ_𝑞−ℎ_𝑞−1. (77) By Corollary 29, it is obvious that 𝜑_𝑖(𝑤)([𝑠]_ℎ𝑗)=0 for any ℎ_𝑗>ℎ_𝑘. Then 𝑓_𝑖(𝑤)(𝑠)=𝜑_𝑖(𝑤)[𝑠]_ℎ1ℎ₁+𝜑_𝑖(𝑤)[𝑠]_ℎ2×ℎ₂−ℎ₁+⋯+𝜑_𝑖(𝑤)[𝑠]_ℎ𝑘×ℎ_𝑘−ℎ_𝑘−1. (78) According to Lemma 41, the remaining part of the proof of Theorem 42 is similar to that of Theorem 4.3 in Li and Zhang [9]; therefore, here it is omitted.

Theorem 43. Let 𝑤∈𝐿𝐹𝐺_𝑃(𝑁), 𝑠=(𝑠₁,𝑠₂,…,𝑠₃)∈𝐿(𝑁). Then, for any 𝑖∈𝑁, 𝜑_𝑥𝑖(𝑤)_𝑥=𝑠=𝑔_𝑖(𝑤)(𝑠). (79)

Proof. From Lemma 41, the proof of Theorem 43 is similar to that of Theorem 4.1 in Li and Zhang [9]; therefore, here it is omitted.

Remark 44. From the above analysis, itis easily seen that the simplified expression of the generalized fuzzy Shapley function given by (70) can be regarded as the generalization of fuzzy Shapley functions defined in some particular fuzzy games with fuzzy coalition. The simplified expression of the fuzzy Shapley function is equivalent to Yu and Zhang’s definition [18] when fuzzy characteristic function is a fuzzy game with Choquet integral forms and is equivalent to that when fuzzy characteristic function is a fuzzy game with proportional values.

Remark 45. The generalized fuzzy Shapley function given by (70) can be applied to crisp games, games with fuzzy coalitions, and games with fuzzy characteristic functions by restricting the domain. Equation (70) coincides with (44) or (52) when restricted to a crisp coalition; (70) is equal to (17) when restricted to a real-valued characteristic function and a crisp coalition. Briefly, the generalized fuzzy Shapley function defined by (70) is a general Shapley function, which serves as the connection between games with fuzzy coalitions and games with fuzzy characteristic functions. The most important aspect is that we do not need to transform the Shapley function when dealing with different kinds of cooperative games.

5. Conclusion

Shapley value is a well-known solution concept in cooperative game theory. In crisp game, it has been applied widely in many cases. In fuzzy game, Shapley functions of game with fuzzy coalition or with fuzzy characteristic function, as an important solution concept, have been paid more attention. Games, subject to fuzzy coalitions as well as those pertaining to fuzzy characteristic function, are separately investigated in the literature. In this paper, based on the Hukuhara difference, a fuzzy Hukuhara-Shapley function on the class of games with fuzzy characteristic function is investigated. And some interesting properties are shown. Further, a new fuzzy game model that admits both fuzzy coalitions and fuzzy characteristic functions is proposed. A simplified expression of the generalized fuzzy Shapley function for the generalized fuzzy games is given. It is shown that the simplified expression of the generalized fuzzy Shapley function is equivalent to that of fuzzy games with indeterminate integral form in the study by Yu and Zhang [18]. When the domain of the characteristic function or coalition is restricted, the generalized fuzzy Shapley function can be applicable to crisp games, games with fuzzy coalitions, and games with fuzzy characteristic functions. Thus, the simplified expression of the generalized fuzzy Shapley function for this kind of fuzzy games can be widely applied in many cases to address more realistic situations.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partly supported by the Funds for Creative Research Groups of China (no. 71221061), the Projects of Major International Cooperation NSFC (no. 71210003), the National Natural Science Foundation of China (nos. 71271217 and 71201089), and the Program for New Century Excellent Talents in University of China (no. NCET-12-0541).