Generalized Shapley function for cooperative games with fuzzy payoffs

Abstract

Based on the extended generalized Hukuhara difference, we introduce and study a new Shapley type of value for cooperative games with fuzzy payoffs. We first propose and characterize a new interval Shapley value for interval-valued cooperative games. Those results are then extended to cooperative games with fuzzy payoffs, and the generalized Shapley function is introduced. We characterize the generalized Shapley function using the properties of generalized efficiency, generalized dummy player, generalized symmetry, and generalized additivity. At the same time, the necessary and sufficient condition for the existence of the generalized Shapley function is given. This study also shows that the generalized Shapley function is a generalization of the Hukuhara-Shapley function defined by Yu and Zhang [22]. Meanwhile, an arbitrary cooperative game with payoffs of center triangular fuzzy numbers has a unique generalized Shapley function.

Keywords

Cooperative game Shapley value generalized Hukuhara difference fuzzy payoffs

1 Introduction

The Shapley value [18] is one of the well-known solution concepts in cooperative game theory, and it has been investigated by many scholars. The majority of such scholars treat games with real-valued characteristic function. However, in some situations, the values of coalitions cannot be precisely known. The theory of games with fuzzy characteristic function commenced with the investigations of Mareš [12, 13]and Mareš and Vlach [14], where the opinions of cooperative games with fuzzy payoffs and the fuzzy Shapley value were introduced. Branzei et al. [8] established another special class of games with fuzzy characteristic function, interval-valued cooperative games, and considered two Shapley-like values for these games.

The research on the Shapley value for games with fuzzy characteristic function is often based on the Hukuhara difference [5, 9]. On the one hand, based on the Hukuhara difference of two interval numbers, Alparslan Gök [2 –4] characterized the interval Shapley value for interval-valued cooperative games using the properties of additivity, efficiency, symmetry, and dummy player. Subsequently, many scholars [1 , 11] devoted efforts to the axiomatization approach of the interval Shapley value. On the other hand, based on the Hukuhara difference of two fuzzy numbers, Yu and Zhang [22] defined the Hukuhara-Shapley function for the games given in [12, 13]. In [22], the relationship that each α-cut game of cooperative games with fuzzy payoffs is an interval-valued cooperative game plays a key role in researching the Hukuhara-Shapley function. Following the work of Yu and Zhang [22], Tan and Chen [21] presented a characterization of the generalized fuzzy Shapley function for games with a fuzzy coalition and fuzzy characteristic function. Recently, based on the extended Hukuhara difference, which is an extension of the Hukuhara difference, Meng et al. [15 –17] investigated interval-valued cooperative games and fuzzy multichoice games with fuzzy payoffs and discussed the Shapley function for these game models. The extended Hukuhara difference defined by Meng et al. [15 –17] can be used in more situations than the Hukuhara difference; hence, these models and methods [15 –17] extended the research scope of cooperative games with fuzzy payoffs.

Note that there is another extension of the Hukuhara difference called the generalized Hukuhara difference, which is a recent and very promising concept proposed by Stefanini [19, 20] and further studied by Bede and Stefanini [6]. However, there are some shortcomings of the generalized Hukuhara difference when dealing with fuzzy equations; therefore, a novel extension of the generalized Hukuhara difference, called the extended generalized Hukuhara difference, is introduced in this paper.

The purpose of this paper is to use the extended generalized Hukuhara difference to define and characterize a Shapley type of value for cooperative games with fuzzy payoffs. For this propose, we first propose and characterize a new interval Shapley value for interval-valued cooperative games; then, those results are extended to cooperative games with fuzzy payoffs, and the generalized Shapley function is introduced. Two main results are obtained. The first main result is the characterization of generalized interval Shapley value as the only rule that satisfies efficiency, symmetry, dummy player and additivity. The second main result is the characterization of the generalized Shapley function as the only rule that satisfies generalized efficiency, generalized symmetry, generalized dummy player and generalized additivity. Besides, some new properties are provided, which are very different from previous results. For instance, the generalized interval Shapley value for an arbitrary interval-valued cooperative game always exists; an arbitrary cooperative game with payoffs of center triangular fuzzy numbers has a unique generalized Shapley function.

This paper is organized as follows. In Section 2, we recall basic definitions and notations, including interval number, fuzzy number, generalized Hukuhara difference, and cooperative games with fuzzy payoffs. In Section 3, some limitations of the generalized Hukuhara difference are shown, and the so-called extended generalized Hukuhara difference is introduced. In Section 4, we discuss the generalized interval Shapley value for interval-valued cooperative games. In Section 5, we thoroughly investigate the generalized Shapley function, including the definition, axiomatization, sufficient and necessary condition for the existence of this value, and some properties. In Section 6, we provide an illustrative example. The conclusion is presented in Section 7.

2 Preliminaries

2.1 A review of interval numbers and fuzzy numbers

We begin with a brief overview of some concepts of interval numbers and fuzzy numbers. Let $ℝ$ be (- ∞ , + ∞), i.e., the set of all real numbers.

Let $a^{-}, a^{+} \in ℝ$ with a^- ≤ a⁺. The closed interval [a^-, a⁺] defines an interval number $\bar{a} = [a^{-}, a^{+}] = {x \in ℝ ∣ a^{-} \leq x \leq a^{+}}$ . When a^- = a⁺, the interval number $\bar{a}$ reduces to a real number. In addition, an interval number $\bar{a}$ can be written as $〈 m (\bar{a}); w (\bar{a}) 〉$ , where $m (\bar{a}) = \frac{1}{2} (a^{-} + a^{+})$ represents the interval midpoint and $w (\bar{a}) = \frac{1}{2} (a^{+} - a^{-})$ represents the interval radius. The set of all interval numbers in $ℝ$ is denoted by $𝕀ℝ$ .

We denote by $ℱ (ℝ)$ the family of fuzzy numbers, i.e., normal, fuzzy convex, upper semi-continuous, and compactly supported fuzzy set defined over $ℝ$ . For any fuzzy number $\tilde{A} \in ℱ (ℝ)$ , the membership of $\tilde{A}$ is a function $μ_{\tilde{A}}$ : $ℝ \to [0, 1]$ . An important type of fuzzy number in common use is the triangular fuzzy number (TrFN for short), whose membership function has the form $\begin{matrix} μ_{\tilde{A}} (x) = {\begin{matrix} \frac{1}{m - l} (x - l) & if l \leq x \leq m, \\ \frac{1}{r - m} (r - x) & if m \leq x \leq r, \\ 0 & otherwise . \end{matrix} \end{matrix}$ where $l, m, r \in ℝ$ with l ≤ m ≤ r. This triangular fuzzy number $\tilde{A}$ is denoted as (l, m, r). In particular, $\tilde{A}$ is called a center triangular fuzzy number (simply CTrFN) if m - l = r - m.

Definition 2.1. (See [6, 22]) Let $\tilde{A} \in ℱ (ℝ)$ be a fuzzy number. The α-cut of $\tilde{A}$ is defined by $A_{α} = {x \in ℝ ∣ μ_{\tilde{A}} (x) \geq α}$ for α ∈ (0, 1] and by the closure of the support $A_{0} = cl {x \in ℝ ∣ μ_{\tilde{A}} (x) > 0}$ for α = 0.

For each α ∈ [0, 1], $A_{α} = [A_{α}^{-}, A_{α}^{+}]$ is an interval number, where $A_{α}^{-} = inf (A_{α})$ and $A_{α}^{+} = sup (A_{α})$ .

Definition 2.2. (See [3, 4]) For interval numbers $\bar{a} = [a^{-}, a^{+}]$ and $\bar{b} = [b^{-}, b^{+}]$ , we derive

$\bar{a} \geq \bar{b}$ iff a^- ≥ b^- and a⁺ ≥ b⁺,

$\bar{a} = \bar{b}$ iff a^- = b^- and a⁺ = b⁺,

$\bar{a} \subseteq \bar{b}$ iff a^- ≥ b^- and a⁺ ≤ b⁺.

Definition 2.3. (See [21, 22]) Let $\tilde{A}, \tilde{B} \in ℱ (ℝ)$ be any two fuzzy numbers. Then, $\tilde{A} = \tilde{B}$ if and only if $A_{α}^{-} = B_{α}^{-}$ and $A_{α}^{+} = B_{α}^{+}$ for all α ∈ [0, 1].

Definition 2.4. (See [23]) Let $\tilde{A}, \tilde{B} \in ℱ (ℝ)$ and $m \in ℝ$ . The addition $\tilde{A} + \tilde{B}$ and scalar multiplication $m \tilde{A}$ are defined as having the level cuts

${(\tilde{A} + \tilde{B})}_{α} = A_{α} + B_{α} = [A_{α}^{-} + B_{α}^{-}, A_{α}^{+} + B_{α}^{+}]$ ,

${(m \tilde{A})}_{α} = m A_{α} = [m A_{α}^{-}, m A_{α}^{+}]$ , if m ≥ 0; ${(m \tilde{A})}_{α} = [m A_{α}^{+}, m A_{α}^{-}]$ , if m < 0.

Proposition 2.1.(See [6]) Let $A^{-} : α \to A_{α}^{-}$ and $A^{+} : α \to A_{α}^{+}$ be the lower and upper branches of $\tilde{A}$ , respectively. Then, the fuzzy number $\tilde{A}$ is completely determined by a pair $\tilde{A} = [A^{-}, A^{+}]$ .

Proposition 2.2.(See [6]) $\tilde{A}$ is a fuzzy number if and only if (1) each α-cut of $\tilde{A}$ is a nonempty compact interval of the form $A_{α} = [A_{α}^{-}, A_{α}^{+}]$ and (2) for any $α, β \in ℝ$ such that 0 ≤ β < α ≤ 1 yields A_α ⊆ A_β. Moreover, $\tilde{A} = ⋃_{α \in [0, 1]} α \cdot A_{α}$ .

This “nested” property plays a key role in verifying a fuzzy number.

2.2 Generalized Hukuhara difference

In this subsection, we will briefly review the generalized Hukuhara difference.

From the extension principle on fuzzy sets [23], we generally cannot obtain $\tilde{A} + \tilde{B} - \tilde{B} = \tilde{A}$ for all $\tilde{A}, \tilde{B} \in ℱ (ℝ)$ . The Hukuhara difference [5, 9] can manage this issue well.

Definition 2.5. (See [5, 9]) Let $\tilde{A}, \tilde{B} \in ℱ (ℝ)$ . If there is a fuzzy number $\tilde{C} \in ℱ (ℝ)$ such that $\tilde{A} = \tilde{B} + \tilde{C}$ , then $\tilde{C}$ is called the Hukuhara difference of $\tilde{A}$ and $\tilde{B}$ , denoted by $\tilde{C} = \tilde{A} ⊖_{H} \tilde{B}$ .

If $\bar{a}$ and $\bar{b}$ are two interval numbers with $w (\bar{a}) \geq w (\bar{b})$ , the Hukuhara difference of $\bar{a}$ and $\bar{b}$ is given by $\bar{a} ⊖_{H} \bar{b} = [a^{-} - b^{-}, a^{+} - b^{+}]$ .

Note that the existence of the Hukuhara difference is under very restrictive conditions. Subsequently, Stefanini [19, 20] proposed an extension of the Hukuhara difference called the generalized Hukuhara difference (gH-difference for short), which can be used in more situations than the Hukuhara difference.

Definition 2.6. (See [19, 20]) The gH-difference of two interval numbers $\bar{a} = [a^{-}, a^{+}]$ and $\bar{b} = [b^{-}, b^{+}]$ is given by

$\bar{a} ⊖_{gH} \bar{b} = \bar{c} \Leftrightarrow {\begin{matrix} (i) & \bar{a} = \bar{b} + \bar{c}, \\ or (ii) & \bar{b} = \bar{a} + (- 1) \bar{c} . \end{matrix}$ (1)

Clearly, Eq. (1) is equivalent to $\bar{a} ⊖_{gH} \bar{b} = [min {a^{-} - b^{-}, a^{+} - b^{+}}, max {a^{-} - b^{-}, a^{+} - b^{+}}]$ . Case (i) of Eq. (1) is suitable for $w (\bar{a}) \geq w (\bar{b})$ , and case (ii) of Eq. (1) is suitable for $w (\bar{a}) \leq w (\bar{b})$ .

Definition 2.7. (See [19, 20]) Let $\tilde{A}, \tilde{B} \in ℱ (ℝ)$ . If there is a fuzzy number $\tilde{C} \in ℱ (ℝ)$ that satisfies

$\tilde{A} ⊖_{gH} \tilde{B} = \tilde{C} \Leftrightarrow {\begin{matrix} (i) & \tilde{A} = \tilde{B} + \tilde{C}, \\ or (ii) & \tilde{B} = \tilde{A} + (- 1) \tilde{C}, \end{matrix}$ (2) then $\tilde{C}$ is called the gH-difference of $\tilde{A}$ and $\tilde{B}$ .

Note that the gH-difference of two arbitrary interval numbers always exists; however, the gH-difference of two arbitrary fuzzy numbers does not always exist. If $\tilde{A} ⊖_{gH} \tilde{B}$ exists, then ${(\tilde{A} ⊖_{gH} \tilde{B})}_{α} = {(\tilde{A})}_{α} ⊖_{gH} {(\tilde{B})}_{α} = A_{α} ⊖_{gH} B_{α}$ for all α ∈ [0, 1].

2.3 Cooperative games with fuzzy payoffs

A situation in which a finite set of players N = {1, 2, …, n} can generate fuzzy payoffs by cooperation can be described by a cooperative game with fuzzy payoffs, being a pair $(N, \tilde{v})$ , where $\tilde{v} : 2^{N} \to ℱ (ℝ)$ is a characteristic function and satisfies $\tilde{v} (\emptyset) = 0$ . For any S ⊆ N, the fuzzy number $\tilde{v} (S)$ is interpreted as the fuzzy value that S can obtain on its own. In particular, if all the values of coalitions reduce to interval numbers, i.e., $\bar{v} : 2^{N} \to 𝕀ℝ$ , we call $(N, \bar{v})$ an interval-valued cooperative game. The set of all cooperative games with fuzzy payoffs on player set N is denoted by $FG (N)$ , and the set of all interval-valued cooperative games on player set N is denoted by $IG (N)$ . If there is no confusion, we will identify a game with its characteristic function $\tilde{v}$ (or $\bar{v}$ ). For convenience, the cardinality of any S ⊆ N is denoted by the corresponding lowercase s.

For any $\tilde{v}, \tilde{w} \in FG (N)$ , the sum game $\tilde{v} + \tilde{w} \in FG (N)$ is defined by $(\tilde{v} + \tilde{w}) (S) = \tilde{v} (S) + \tilde{w} (S)$ for every S ⊆ N.

Let $\bar{v} \in IG (N)$ . A game v^- is called the left-point game of $\bar{v}$ if $v^{-} (S) = {(\bar{v} (S))}^{-}$ for every S ⊆ N. A game v⁺ is called the right-point game of $\bar{v}$ if $v^{-} (S) = {(\bar{v} (S))}^{-}$ for every S ⊆ N.

3 A further discussion on gH-difference

3.1 Several limitations on gH-difference of two interval numbers

In many applications, the gH-difference appears to have several limitations and to be very restrictive.

First, the gH-difference cannot be directly used to solve the following fuzzy equations.

(1) The equation $\bar{b} + \bar{x} = \bar{a}$ for $\bar{x}$ when $0 < w (\bar{a}) < w (\bar{b})$ .

(2) The equation $\bar{a} + (- 1) \bar{x} = \bar{b}$ for $\bar{x}$ when $w (\bar{a}) > w (\bar{b}) > 0$ .

Moreover, the following fuzzy equations have two different solutions, which may lead to some ambiguous assertions in interval-valued cooperative games.

(3) The equation $\bar{a} ⊖_{gH} \bar{x} = \bar{b}$ for $\bar{x}$ when $w (\bar{a}) \geq w (\bar{b}) > 0$ .

(4) The equation $\bar{x} ⊖_{gH} \bar{a} = \bar{b}$ for $\bar{x}$ when $w (\bar{a}) \geq w (\bar{b}) > 0$ .

Example 3.1. Consider a game $\bar{v}$ , and its characteristic function is given by $\bar{v} (\emptyset) = [0, 0]$ , $\bar{v} ({1}) = \bar{v} ({2}) = [1, 2]$ , $\bar{v} ({3}) = [1, 7]$ , $\bar{v} ({1, 3}) = [2, 12]$ , $\bar{v} ({2, 3}) = [6, 8]$ , $\bar{v} ({1, 2}) = [3, 8]$ , and $\bar{v} ({1, 2, 3}) = [5, 14]$ .

For any S ∈ {∅ , {3}}, $\begin{matrix} \bar{v} (S \cup 1) ⊖_{gH} \bar{v} (S) = \bar{v} (S \cup 2) ⊖_{gH} \bar{v} (S) . \end{matrix}$ This implies that players 1 and 2 are symmetric in $\bar{v}$ under the operation of the gH-difference. However, there is a significant difference between player 1 and player 2 because $\bar{v} ({1, 3}) \neq \bar{v} ({2, 3})$ .

Furthermore, when an expression has admixture operations of "+" and "⊖_gH", the different calculation order of "+" and "⊖_gH" typically leads to different solutions.

Example 3.2. Let $\bar{a}, \bar{b}, \bar{c} \in 𝕀ℝ$ and $\bar{b} = \bar{a} + (- 1) \bar{x}$ . It follows that $\bar{a} ⊖_{gH} \bar{b} = \bar{x} = 〈 m (\bar{a}) - m (\bar{b}); w (\bar{b}) - w (\bar{a}) 〉$ . We obtain

$\bar{c} + \bar{a} ⊖_{gH} \bar{b} = 〈 m (\bar{c}) + m (\bar{a}) - m (\bar{b}); | w (\bar{c}) + w (\bar{a}) - w (\bar{b}) | 〉$ , and

$\bar{c} + \bar{x} = 〈 m (\bar{c}) + m (\bar{a}) - m (\bar{b}); w (\bar{c}) + w (\bar{b}) - w (\bar{a}) 〉$ .

Clearly, $\bar{c} + \bar{a} ⊖_{gH} \bar{b} = \bar{c} + \bar{x}$ if and only if $w (\bar{c}) = 0$ or $w (\bar{a}) = w (\bar{b})$ .

It is well known that the calculation of the Shapley value involves mixed operations of addition and subtraction. One can conclude from Example 3.2 that some properties, which are described by the addition operation, of the Shapley value calculated using the gH-difference may not hold.

3.2 Extended generalized Hukuhara difference

The reason for the limitations discussed above is that the gH-difference of two interval numbers is introduced based on two different additions. It is more plausible to distinguish the interval number between that from case (i) of Eq. (1) and that from case (ii) of Eq. (1). Now, we provide the following definition.

Definition 3.1. Let $\bar{a} = [a^{-}, a^{+}]$ and $\bar{b} = [b^{-}, b^{+}] \in 𝕀ℝ$ . The extended generalized Hukuhara difference (egH-difference for short) of $\bar{a}$ and $\bar{b}$ is defined as

$\bar{a} ⊖_{egH} \bar{b} = {\begin{matrix} \bar{c} & (i) \bar{a} = \bar{b} + \bar{c}, \\ {\bar{c}}^{*} & (ii) \bar{b} = \bar{a} + (- 1) {\bar{c}}^{*} . \end{matrix}$ (3) where $\bar{c} = [a^{-} - b^{-}, a^{+} - b^{+}]$ and ${\bar{c}}^{*} = [a^{+} - b^{+},$ a^- - b^-] ^*.

Eq. (3) is equivalent to $\bar{a} ⊖_{egH} \bar{b} = {\begin{matrix} [a^{-} - b^{-}, a^{+} - b^{+}] & if w (\bar{a}) \geq w (\bar{b}), \\ {[a^{+} - b^{+}, a^{-} - b^{-}]}^{*} & if w (\bar{a}) \leq w (\bar{b}) . \end{matrix}$ (4)

We claim that $\bar{c}$ and ${\bar{c}}^{*}$ are different types of interval numbers. In this setting, the order relations in Definition 2.2 are based on interval numbers of the same type, which is similar to the same fuzzy numbers in Definition 2.3.

For example, let $\bar{a} = [2, 12]$ , $\bar{b} = [6, 8]$ and $\bar{c} = [1, 7]$ ; thus, we have $\bar{a} ⊖_{egH} \bar{c} = [1, 5]$ , $\bar{b} ⊖_{egH} \bar{c} = {[1, 5]}^{*}$ , and $\bar{a} ⊖_{egH} \bar{c} \neq \bar{b} ⊖_{egH} \bar{c}$ .

Definition 3.2. (See [15]) For any $\bar{a} = [a^{-}, a^{+}]$ , if a^- > a⁺, $\bar{a}$ is said to be an "imaginary" interval number.

Such an “imaginary” interval number $\bar{a} = [a^{-}, a^{+}]$ can be described as $〈 m (\bar{a}); w (\bar{a}) 〉$ , where $m (\bar{a}) = \frac{1}{2} (a^{-} + a^{+})$ and $w (\bar{a}) = \frac{1}{2} (a^{+} - a^{-})$ . In this case, $w (\bar{a}) < 0$ . Clearly, $\bar{a} = [m (\bar{a}) - w (\bar{a}), m (\bar{a}) + w (\bar{a})]$ , which is the same as the expression of an interval number.

Each interval number ${\bar{c}}^{*} = {[c^{+}, c^{-}]}^{*}$ corresponds to an "imaginary" interval number $\bar{c} = [c^{-}, c^{+}]$ . When $\bar{c}$ is the "imaginary" form corresponding to ${\bar{c}}^{*}$ , we obtain that $\bar{b} = \bar{a} + (- 1) {\bar{c}}^{*}$ is equivalent to $\bar{a} = \bar{b} + \bar{c}$ . To ensure the consistency of addition, we claim that ${\bar{c}}^{*}$ is replaced by its "imaginary" form in the calculation. For example, [2, 5] ^* + [3, 4] = [5, 2] + [3, 4] = [8, 6] = [6, 8] ^*.

Therefore, Eq. (4) can be rewritten as

$\bar{a} ⊖_{egH} \bar{b} = [a^{-} - b^{-}, a^{+} - b^{+}] .$ (5)

Note that Eq. (5) is coincident with the concept of the extended Hukuhara difference introduced by Meng et al. [15 –17]. But the meanings of these differences are essentially different.

It is obvious from Eq. (5) that $\begin{matrix} \bar{a} ⊖_{egH} \bar{b} = \bar{c} \Leftrightarrow \bar{b} + \bar{c} = \bar{a} . \end{matrix}$

Example 3.3. Consider Example 3.2 using the egH-difference. We obtain that ${\bar{x}}^{*} = \bar{a} ⊖_{egH} \bar{b} = [a^{+} - b^{+}, a^{-} - b^{-}]^{*}$ , and its "imaginary" form is $[a^{-} - b^{-}, a^{+} - b^{+}] = 〈 m (\bar{a}) - m (\bar{b}); w (\bar{a}) - w (\bar{b}) 〉$ , where $w (\bar{a}) - w (\bar{b}) \leq 0$ . Hence, $\bar{c} + {\bar{x}}^{*} = 〈 m (\bar{c}) + m (\bar{a}) - m (\bar{b}); w (\bar{c}) + w (\bar{a}) - w (\bar{b}) 〉$ .

If $w (\bar{c}) + w (\bar{a}) - w (\bar{b}) \geq 0$ , we have $\bar{c} + \bar{a} ⊖_{egH} \bar{b} = \bar{c} + {\bar{x}}^{*} = [c^{-} + a^{-} - b^{-}, c^{+} + a^{+} - b^{+}]$ .

If $w (\bar{c}) + w (\bar{a}) - w (\bar{b}) < 0$ , we have $\bar{c} + \bar{a} ⊖_{egH} \bar{b} = \bar{c} + {\bar{x}}^{*} = {[c^{+} + a^{+} - b^{+}, c^{-} + a^{-} - b^{-}]}^{*}$ .

Therefore, $\bar{c} + \bar{a} \underset{egH}{⊖} \bar{b} = \bar{c} + {\bar{x}}^{*}$ always holds.

Note that $\underset{egH}{⊖}$ can be regarded as the opposite operation for an interval number. Thus, for any $k \in ℝ$ and any $\bar{a} \in 𝕀ℝ$ , we define $(\underset{egH}{⊖} k) \cdot (\underset{egH}{⊖} k) = {(\underset{egH}{⊖} k)}^{2} = k^{2}$ and $\underset{egH}{⊖} (k \bar{a}) = (\underset{egH}{⊖} k) \bar{a}$ . Then, ${(\underset{egH}{⊖} 1)}^{t} \cdot \bar{a} = {\begin{matrix} \bar{a} & if t \in {2 k ∣ n \in ℤ}, \\ \underset{egH}{⊖} \bar{a} & if t \in {2 k + 1 ∣ n \in ℤ} . \end{matrix}$ (6)

Proposition 3.1.For any $\bar{a}, \bar{b}, \bar{c} \in 𝕀ℝ$ ,

The equation $\bar{a} + \bar{x} = \bar{b}$ for $\bar{x}$ has a unique solution by using the egH-difference.

The equation $\bar{a} \underset{egH}{⊖} \bar{b} = \bar{x}$ for $\bar{x}$ has a unique solution, and this solution satisfies $\bar{x} + \bar{b} = \bar{a}$ .

The equation $\bar{a} \underset{egH}{⊖} \bar{x} = \bar{c}$ for $\bar{x}$ has a unique solution, and this solution satisfies $\bar{c} + \bar{x} = \bar{a}$ .

The equation $\bar{x} \underset{egH}{⊖} \bar{b} = \bar{c}$ for $\bar{x}$ has a unique solution, and this solution satisfies $\bar{c} + \bar{b} = \bar{x}$ .

Proof. These assertions are easily obtained from Eq.(5). □

Without proof, we provide the following straightforward properties on the egH-difference.

Proposition 3.2.Let ${\bar{a}}_{i} \in 𝕀ℝ$ , i = 1, 2, …, n. The egH-difference ⊖_egHhas the following properties:

${\bar{a}}_{1} \underset{egH}{⊖} {\bar{a}}_{1} = 0$ .

${\bar{a}}_{1} = [a_{1}^{-}, a_{1}^{+}]$ is an interval number; then, $\underset{egH}{⊖} {\bar{a}}_{1} = [- a_{1}^{-}, - a_{1}^{+}]$ is an "imaginary" interval number.

${\bar{a}}_{1} + (\underset{egH}{⊖} {\bar{a}}_{2}) = {\bar{a}}_{1} \underset{egH}{⊖} {\bar{a}}_{2}$ .

$({\bar{a}}_{1} + {\bar{a}}_{2}) \underset{egH}{⊖} ({\bar{a}}_{3} + {\bar{a}}_{4}) = ({\bar{a}}_{1} \underset{egH}{⊖} {\bar{a}}_{3}) + ({\bar{a}}_{2} \underset{egH}{⊖} {\bar{a}}_{4})$ .

${\bar{a}}_{1} \underset{egH}{⊖} {\bar{a}}_{2} = \underset{egH}{⊖} {\bar{a}}_{2} + {\bar{a}}_{1} = \underset{egH}{⊖} ({\bar{a}}_{2} \underset{egH}{⊖} {\bar{a}}_{1})$ .

$\underset{egH}{⊖} {\bar{a}}_{1} \underset{egH}{⊖} {\bar{a}}_{2} = \underset{egH}{⊖} ({\bar{a}}_{1} + {\bar{a}}_{2})$ .

$\underset{egH}{⊖} \sum_{i \in N} {\bar{a}}_{i} = [- \sum_{i \in N} a_{i}^{-}, - \sum_{i \in N} a_{i}^{+}]$ .

Corollary 3.1.For any ${\bar{a}}_{i} \in 𝕀ℝ$ and $s_{i} \in ℕ$ (i ∈ N),

$\begin{matrix} \sum_{i \in N} {(⊖_{egH} 1)}^{s_{i}} {\bar{a}}_{i} = \\ \sum_{i \in {i ∣ s_{i} = 2 k, k \in ℕ}} {\bar{a}}_{i} ⊖_{egH} \sum_{i \in {i ∣ s_{i} = 2 k + 1, k \in ℕ}} {\bar{a}}_{i} . \end{matrix}$ (7)

Proof.Eq.(7) can be directly deduced from Eq.(6) and items (3)–(7) of Proposition 3.2.

The egH-difference of interval numbers is the basis for the egH-difference of fuzzy numbers.

Definition 3.3. For $\tilde{A}, \tilde{B} \in ℱ (ℝ)$ , the egH-difference $\tilde{A} \underset{egH}{⊖} \tilde{B}$ is defined levelwise by

${[\tilde{A} ⊖_{egH} \tilde{B}]}_{α} = {\begin{matrix} C_{α} & (i) w (A_{α}) \geq w (B_{α}), \\ C_{α}^{*} & (ii) w (A_{α}) \leq w (B_{α}) . \end{matrix}$ (8) for all α ∈ [0, 1] if $\tilde{A} \underset{egH}{⊖} \tilde{B}$ exists. Here, $C_{α} = [A_{α}^{-} - B_{α}^{-}, A_{α}^{+} - B_{α}^{+}]$ , and $C_{α}^{*} = [A_{α}^{+} - B_{α}^{+},$ ${A_{α}^{-} - B_{α}^{-}]}^{*}$ .

Proposition 3.3.Given two fuzzy numbers $\tilde{A}, \tilde{B} \in ℱ (ℝ)$ , let $A_{α} = [A_{α}^{-}, A_{α}^{+}]$ and $B_{α} = [B_{α}^{-}, B_{α}^{+}]$ be the α-cuts of $\tilde{A}$ and $\tilde{B}$ , respectively. Then, $\tilde{A} ⊖_{egH} \tilde{B}$ exists if and only if

$C_{α}^{-} = A_{α}^{-} - B_{α}^{-}$ and $C_{α}^{+} = A_{α}^{+} - B_{α}^{+}$ , and it holds that $C_{α}^{-}$ is increasing of α, $C_{α}^{+}$ is decreasing of α, and w (A_α) ≥ w (B_α) for all α ∈ [0, 1]. Alternatively,

$C_{α}^{-} = A_{α}^{+} - B_{α}^{+}$ and $C_{α}^{+} = A_{α}^{-} - B_{α}^{-}$ , and it holds that $C_{α}^{-}$ is decreasing of α, $C_{α}^{+}$ is increasing of α, and w (A_α) ≤ w (B_α) for all α ∈ [0, 1].

Proof. One can easily check this assertion from Definition 3.3 and Proposition 2.2. □

4 An axiomatization for generalized interval Shapley value

In this section, we consider a new Shapley type of value for interval-valued cooperative games based on the egH-difference.

Definition 4.1. Let $\bar{v} \in IG (N)$ . Player i ∈ N is called a dummy player in $\bar{v}$ if $\bar{v} (S \cup i) ⊖_{egH} \bar{v} (S) = \bar{v} (i)$ for all S ⊆ N ∖ i.

Definition 4.2. Let $\bar{v} \in IG (N)$ . For i, j ∈ N, players i and j are said to be symmetric in $\bar{v}$ if $\bar{v} (S \cup i) \underset{egH}{⊖} \bar{v} (S) = \bar{v} (S \cup j) \underset{egH}{⊖} \bar{v} (S)$ for all S ⊆ N ∖ {i, j}.

If players i and j are symmetric in $\bar{v}$ , from item (4) of Proposition 3.1, we obtain $\bar{v} (S \cup i) = \bar{v} (S \cup j)$ for all S ⊆ N ∖ {i, j}.

A value on $IG (N)$ is a map that assigns a payoff vector $φ (\bar{v}) \in {𝕀ℝ}^{N}$ to every interval-valued cooperative game $\bar{v} \in IG (N)$ .

Definition 4.3. The generalized interval Shapley value $φ (\bar{v})$ satisfies the following axioms:

Efficiency (EFF). A value $φ (\bar{v})$ satisfies EFF if $\sum_{i \in N} {\bar{φ}}_{i} (\bar{v}) = \bar{v} (N)$ for each game $\bar{v} \in IG (N)$ .

Dummy player (DUM). Let i be a dummy player in $\bar{v} \in IG (N)$ . A value $φ (\bar{v})$ satisfies DUM if ${\bar{φ}}_{i} (\bar{v}) = \bar{v} (i)$ .

Symmetry (SYM). Let i and j be symmetric in $\bar{v} \in IG (N)$ . A value $φ (\bar{v})$ satisfies SYM if ${\bar{φ}}_{i} (\bar{v}) = {\bar{φ}}_{j} (\bar{v})$ .

Additivity (ADD). Let $\bar{v}, \bar{w} \in IG (N)$ . A value $φ (\bar{v})$ satisfies ADD if ${\bar{φ}}_{i} (\bar{v} + \bar{w}) = {\bar{φ}}_{i} (\bar{v}) + {\bar{φ}}_{i} (\bar{w})$ for all i ∈ N.

Definition 4.4. (See [7]) For any nonempty set S ⊆ N, the unanimity gameu_S is defined as $\begin{matrix} u_{S} (T) = {\begin{matrix} 1 & if T \supseteq S, \\ 0 & otherwise . \end{matrix} \end{matrix}$

The family of unanimity games is a basis for the class of cooperative games. That is, any cooperative game v can be uniquely represented by unanimity games, i.e., v = ∑_{S⊆N:S≠∅}c_S · u_S, where c_S = ∑_T⊆S (-1) ^s-tv (T).

Let $\bar{a} \in 𝕀ℝ$ be an interval number. A game $\bar{a} u_{S}$ is an interval-valued unanimity game if $(\bar{a} u_{S}) (T) = \bar{a} u_{S} (T)$ for all T ⊆ N.

Lemma 4.1.Let any $\bar{a} \in 𝕀ℝ$ and anyS ⊆ N, S≠ ∅. The generalized interval Shapley value for $\bar{a} u_{S}$ is given by

${\bar{φ}}_{i} (\bar{a} u_{S}) = {\begin{matrix} \frac{\bar{a}}{s} & if i \in S, \\ 0 & if i \notin S . \end{matrix}$ (9)

Proof. It is easy to check that each i ∈ N ∖ S is a dummy player in $\bar{a} u_{S}$ . Then by DUM, we have $\begin{matrix} {\bar{φ}}_{i} (\bar{a} u_{S}) = (\bar{a} u_{S}) (i) = 0, \forall i \in N ∖ S . \end{matrix}$

For any i, j ∈ S, we obtain $(\bar{a} u_{S}) (T \cup i) = (\bar{a} u_{S}) (T \cup j)$ for all T ⊆ N ∖ {i, j}. This implies that i and j is a pair of symmetric players. By SYM, ${\bar{φ}}_{i} (\bar{a} u_{S}) = {\bar{φ}}_{j} (\bar{a} u_{S})$ . Then, $\begin{matrix} \sum_{i \in N} {\bar{φ}}_{i} (\bar{a} u_{S}) & = & \sum_{i \in S} {\bar{φ}}_{i} (\bar{a} u_{S}) + \sum_{j \in N ∖ S} {\bar{φ}}_{j} (\bar{a} u_{S}) \\ = & \sum_{i \in S} {\bar{φ}}_{i} (\bar{a} u_{S}) = s \cdot {\bar{φ}}_{i} (\bar{a} u_{S}) . \end{matrix}$ Moreover, from EFF we have $\begin{matrix} \sum_{i \in N} {\bar{φ}}_{i} (\bar{a} u_{S}) = (\bar{a} u_{S}) (N) = \bar{a} . \end{matrix}$ Hence, for any i ∈ S, we obtain ${\bar{φ}}_{i} (\bar{a} u_{S}) = \frac{\bar{a}}{s}$ . □

Lemma 4.2. Every game $\bar{v} \in IG (N)$ has the form

$\bar{v} = \sum_{S \subseteq N : S \neq \emptyset} {\bar{c}}_{S} \cdot u_{S},$ (10) where ${\bar{a}}_{i} \in 𝕀ℝ$ is the interval-valued Harsanyi dividend of coalition S in $\bar{v}$ defined as ${\bar{c}}_{S} = \sum_{T \subseteq S} {(\underset{egH}{⊖} 1)}^{s - t} \bar{v} (T)$ (11) $= [\sum_{T \subseteq S} {(- 1)}^{s - t} v^{-} (T), \sum_{T \subseteq S} {(- 1)}^{s - t} v^{+} (T)] .$

Proof. Let v^- and v⁺ be the left-point game and the right-point game of $\bar{v}$ , respectively. Since v^- is a classical cooperative game, we have $v^{-} = \sum_{S \subseteq N : S \neq \emptyset} c_{S}^{-} \cdot u_{S}$ , where $c_{S}^{-} = \sum_{T \subseteq S} {(- 1)}^{s - t} v^{-} (T)$ . Similarly, $v^{+} = \sum_{S \subseteq N : S \neq \emptyset} c_{S}^{+} \cdot u_{S}$ , where $c_{S}^{+} = \sum_{T \subseteq S} {(- 1)}^{s - t} v^{+} (T)$ .

From the above, it follows that $\begin{matrix} \bar{v} & = & [v^{-}, v^{+}] \\ = & [\sum_{S \subseteq N : S \neq \emptyset} c_{S}^{-} \cdot u_{S}, \sum_{S \subseteq N : S \neq \emptyset} c_{S}^{+} \cdot u_{S}] \\ = & \sum_{S \subseteq N : S \neq \emptyset} [c_{S}^{-} \cdot u_{S}, c_{S}^{+} \cdot u_{S}] \\ = & \sum_{S \subseteq N : S \neq \emptyset} [c_{S}^{-}, c_{S}^{+}] \cdot u_{S} . \end{matrix}$ Therefore, we obtain $\begin{matrix} {\bar{c}}_{S} & = & [c_{S}^{-}, c_{S}^{+}] \\ = & [\sum_{T \subseteq S} {(- 1)}^{s - t} v^{-} (T), \sum_{T \subseteq S} {(- 1)}^{s - t} v^{+} (T)] \\ = & \sum_{T \subseteq S} [{(- 1)}^{s - t} v^{-} (T), {(- 1)}^{s - t} v^{+} (T)] \\ = & \sum_{(s - t) \in {2 k ∣ k \in ℕ} T \subseteq S :} [{(- 1)}^{s - t} v^{-} (T), {(- 1)}^{s - t} v^{+} (T)] + \\ \sum_{(s - t) \in {2 k + 1 ∣ k \in ℕ} T \subseteq S :} [{(- 1)}^{s - t} v^{-} (T), {(- 1)}^{s - t} v^{+} (T)] \\ = & \sum_{T \subseteq S : (s - t) \in {2 k ∣ k \in ℕ}} [v^{-} (T), v^{+} (T)] + \\ \sum_{T \subseteq S : (s - t) \in {2 k + 1 ∣ k \in ℕ}} [- v^{-} (T), - v^{+} (T)] \\ = & \sum_{T \subseteq S : (s - t) \in {2 k ∣ k \in ℕ}} [v^{-} (T), v^{+} (T)] ⊖_{egH} \\ \sum_{T \subseteq S : (s - t) \in {2 k + 1 ∣ k \in ℕ}} [v^{-} (T), v^{+} (T)] \\ = & \sum_{T \subseteq S} {(\underset{egH}{⊖} 1)}^{s - t} \cdot [v^{-} (T), v^{+} (T)] \\ = & \sum_{T \subseteq S} {(\underset{egH}{⊖} 1)}^{s - t} \cdot \bar{v} (T), \end{matrix}$ where the seventh equation holds from Eq.(7).

This shows the assertion. □

Remark 4.1. When all the values of coalitions reduce to real numbers, all the interval-valued Harsanyi dividends are real numbers. This means that Eq.(10) is a natural extension of the formula v = ∑_{S⊆N:S≠∅}c_S · u_S.

Remark 4.2. Lemma 4.2 holds for arbitrary interval-valued cooperative games. Because every interval-valued cooperative game can be expressed by its left-point game and its right-point game, there are no restrictions on its left-point game and its right-point game under the operation of the egH-difference.

Lemma 4.3.Let $\bar{v} \in IG (N)$ . The generalized interval Shapley value is given by the expression

${\bar{φ}}_{i} (\bar{v}) = \sum_{S \subseteq N : i \in S} \frac{{\bar{c}}_{S}}{s}, \forall i \in N,$ (12)where ${\bar{c}}_{S} = \sum_{T \subseteq S} {(⊖_{egH} 1)}^{s - t} \bar{v} (T)$ .

Proof. According to ADD, Lemmas 4.1 and 4.2, it obtains that $\begin{matrix} {\bar{φ}}_{i} (\bar{v}) & = & {\bar{φ}}_{i} (\sum_{S \subseteq N : S \neq \emptyset} {\bar{c}}_{S} \cdot u_{S}) = \sum_{S \subseteq N : S \neq \emptyset} {\bar{φ}}_{i} ({\bar{c}}_{S} \cdot u_{S}) \\ = & \sum_{S \subseteq N : i \in S} \frac{{\bar{c}}_{S}}{s} . \end{matrix}$ □

Remark 4.3. ${\bar{c}}_{S}$ and ${\bar{φ}}_{i} (\bar{v})$ can be "imaginary" interval numbers.

Theorem 4.1.The generalized interval Shapley value $φ (\bar{v})$ on $IG (N)$ is the unique value that satisfies EFF, SYM, DUM, and ADD. Moreover, for everyi ∈ N, the explicit form is

${\bar{φ}}_{i} (\bar{v}) = \sum_{S \subseteq N ∖ i} γ_{S} (\bar{v} (S \cup i) \underset{egH}{⊖} \bar{v} (S)),$ (13)where $γ_{S} = \frac{s! (n - s - 1)!}{n!}$ .

Proof. Existence. We have to prove that Eq.(13) satisfies EFF, SYM, DUM, and ADD.

(1) EFF. Summing over all i ∈ N on Eq.(13),

$\sum_{i \in N} {\bar{φ}}_{i} (\bar{v}) = \sum_{i \in N} \sum_{S \subseteq N ∖ i} γ_{S} (\bar{v} (S \cup i) ⊖_{egH} \bar{v} (S)) .$ (14)

The coefficient of minuend item $\bar{v} (S \cup i)$ in Eq.(14) is $\begin{matrix} γ_{S} \cdot (s + 1) = \frac{(s + 1)! (n - s - 1)!}{n!}, \end{matrix}$ for any S ∪ i ⊆ N. It follows that this coefficient is equal to $\frac{t! (n - t)!}{n!}$ when S ∪ i = T. In particular, the coefficient is equal to 1 when S ∪ i = N.

The coefficient of subtrahend item $\bar{v} (S)$ in Eq. (14) is $\begin{matrix} \underset{egH}{⊖} γ_{S} \cdot (n - s) = ⊖_{egH} \frac{s! (n - s)!}{n!}, \end{matrix}$ for any S ⊆ N, S ≠ N. It follows that this coefficient is equal to $\underset{egH}{⊖} \frac{t! (n - t)!}{n!}$ when S = T.

Thus, we obtain $\begin{matrix} \sum_{i \in N} {\bar{φ}}_{i} (\bar{v}) \\ = \sum_{T \subset N} (\frac{t! (n - t)!}{n!} \underset{egH}{⊖} \frac{t! (n - t)!}{n!}) \bar{v} (T) + v (N) \\ = v (N) . \end{matrix}$

(2) SYM. Let i and j be a pair of symmetric players in $\bar{v}$ . Namely, for any S ⊆ N ∖ {i, j}, $\bar{v} (S \cup i) = \bar{v} (S \cup j)$ . Then, $\begin{matrix} {\bar{φ}}_{i} (\bar{v}) & = & \sum_{S \subseteq N ∖ i} γ_{S} (\bar{v} (S \cup i) ⊖_{egH} \bar{v} (S)) \\ = & \sum_{S \subseteq N ∖ {i, j}} γ_{S} (\bar{v} (S \cup i) ⊖_{egH} \bar{v} (S)) \\ + \sum_{S \subseteq N ∖ {i, j}} γ_{S \cup j} (\bar{v} (S \cup {i, j}) ⊖_{egH} \bar{v} (S \cup j)) \\ = & \sum_{S \subseteq N ∖ {i, j}} γ_{S} (\bar{v} (S \cup j) ⊖_{egH} \bar{v} (S)) \\ + \sum_{S \subseteq N ∖ {i, j}} γ_{S \cup i} (\bar{v} (S \cup {i, j}) ⊖_{egH} \bar{v} (S \cup i)) \\ = & \sum_{S \subseteq N ∖ j} γ_{S} (\bar{v} (S \cup i) ⊖_{egH} \bar{v} (S)) \\ = & {\bar{φ}}_{j} (\bar{v}) . \end{matrix}$

(3) DUM. Let i be a dummy player in $\bar{v}$ . Namely, for any S ⊆ N ∖ i, $\bar{v} (S \cup i) ⊖_{egH} \bar{v} (S) = \bar{v} (i)$ . Then, $\begin{matrix} {\bar{φ}}_{i} (\bar{v}) & = & \sum_{S \subseteq N ∖ i} \frac{s! (n - s - 1)!}{n!} (\bar{v} (S \cup i) ⊖_{egH} \bar{v} (S)) \\ = & \sum_{S \subseteq N ∖ i} \frac{s! (n - s - 1)!}{n!} \bar{v} (i) \\ = & (\sum_{s = 0}^{n - 1} \frac{s! (n - s - 1)!}{n!} (\begin{array}{l} n - 1 \\ s \end{array})) \cdot \bar{v} (i) \\ = & (\sum_{s = 0}^{n - 1} \frac{1}{n}) \cdot \bar{v} (i) \\ = & \bar{v} (i) . \end{matrix}$

(4) ADD. Let $\bar{v}, \bar{w} \in IG (N)$ be any two games. Then for $\bar{v} + \bar{w} \in IG (N)$ , we have that $(\bar{v} + \bar{w}) (S) = \bar{v} (S) + \bar{w} (S)$ for all S ⊆ N. Then, $\begin{matrix} {\bar{φ}}_{i} (\bar{v} + \bar{w}) \\ = & \sum_{S \subseteq N ∖ i} γ_{S} ((\bar{v} + \bar{w}) (S \cup i) ⊖_{egH} (\bar{v} + \bar{w}) (S)) \\ = & \sum_{S \subseteq N ∖ i} γ_{S} ((\bar{v} (S \cup i) + \bar{w} (S \cup i)) ⊖_{egH} \\ (\bar{v} (S) + \bar{w} (S))) \end{matrix}$ $\begin{matrix} = & \sum_{S \subseteq N ∖ i} γ_{S} ((\bar{v} (S \cup i) ⊖_{egH} \bar{v} (S)) \\ + (\bar{w} (S \cup i) ⊖_{egH} \bar{w} (S))) \\ = & \sum_{S \subseteq N ∖ i} γ_{S} (\bar{v} (S \cup i) ⊖_{egH} (\bar{v} (S)) \\ + \sum_{S \subseteq N ∖ i} γ_{S} (\bar{w} (S \cup i) ⊖_{egH} \bar{w} (S)) \\ = & {\bar{φ}}_{i} (\bar{v}) + {\bar{φ}}_{i} (\bar{w}), \end{matrix}$ where in the third equation we use the item (5) of Proposition 3.2.

Uniqueness. From Lemma 4.3, we have to show for any i ∈ N, $\begin{matrix} {\bar{φ}}_{i} (\bar{v}) & = & \sum_{S \subseteq N : i \in S} \frac{{\bar{c}}_{S}}{s} \\ = & \sum_{T \subseteq N ∖ i} \frac{t! (n - t - 1)!}{n!} (\bar{v} (T \cup i) ⊖_{egH} \bar{v} (T)) . \end{matrix}$

To show this, since $\begin{matrix} \sum_{S \subseteq N : i \in S} \frac{{\bar{c}}_{S}}{s} \\ = \sum_{S \subseteq N : i \in S} \sum_{T \subseteq S} \frac{{(⊖_{egH} 1)}^{s - t} \bar{v} (T)}{s} \\ = \sum_{T \subseteq N} \sum_{S \supseteq T : i \in S \subseteq N} \frac{{(⊖_{egH} 1)}^{s - t}}{s} \bar{v} (T) \\ = \sum_{T \subseteq N : i \in T} (\sum_{S \supseteq T : i \in S \subseteq N} \frac{{(⊖_{egH} 1)}^{s - t}}{s}) \bar{v} (T) \\ + \sum_{T \subseteq N : i \notin T} (\sum_{S \supseteq T : i \in S \subseteq N} \frac{{(⊖_{egH} 1)}^{s - t}}{s}) \bar{v} (T) \\ = \sum_{T \subseteq N : i \notin T} (\sum_{S \supseteq T : i \in S \subseteq N} \frac{{(⊖_{egH} 1)}^{s - (t + 1)}}{s}) \bar{v} (T \cup i) \\ + \sum_{T \subseteq N : i \notin T} (\sum_{S \supseteq T : i \in S \subseteq N} \frac{{(⊖_{egH} 1)}^{s - t}}{s}) \bar{v} (T) \\ = \sum_{T \subseteq N : i \notin T} (\sum_{s = t + 1}^{n} \frac{{(⊖_{egH} 1)}^{s - t - 1}}{s} (\begin{array}{l} n - t - 1 \\ s - t - 1 \end{array})) \bar{v} (T \cup i) \\ + \sum_{T \subseteq N : i \notin T} (\sum_{s = t + 1}^{n} \frac{{(⊖_{egH} 1)}^{s - t}}{s} (\begin{array}{l} n - t - 1 \\ s - t - 1 \end{array})) \bar{v} (T) . \end{matrix}$ Moreover, $\begin{matrix} \sum_{s = t + 1}^{n} \frac{{(⊖_{egH} 1)}^{s - t - 1}}{s} (\begin{array}{l} n - t - 1 \\ s - t - 1 \end{array}) = \frac{t! (n - t - 1)!}{n!}, \end{matrix}$ and $\begin{matrix} \sum_{s = t + 1}^{n} \frac{{(⊖_{egH} 1)}^{s - t}}{s} (\begin{array}{l} n - t - 1 \\ s - t - 1 \end{array}) = \underset{egH}{⊖} \frac{t! (n - t - 1)!}{n!} . \end{matrix}$

Therefore, we obtain $\begin{matrix} {\bar{φ}}_{i} (\bar{v}) \\ = \sum_{T \subseteq N : i \notin T} γ_{T} \bar{v} (T \cup i) + \sum_{T \subseteq N : i \notin T} (\underset{egH}{⊖} γ_{T}) \bar{v} (T) \\ = \sum_{T \subseteq N : i \notin T} γ_{T} \bar{v} (T \cup i) + \sum_{T \subseteq N : i \notin T} γ_{T} (\underset{egH}{⊖} \bar{v} (T)) \\ = \sum_{T \subseteq N : i \notin T} γ_{T} (\bar{v} (T \cup i) \underset{egH}{⊖} \bar{v} (T)), \end{matrix}$ where $γ_{T} = \frac{t! (n - t - 1)!}{n!}$ .

The proof is completed.

Theorem 4.1 shows that the generalized interval Shapley value ${\bar{φ}}_{i} (\bar{v})$ is the expected marginal contribution $\bar{v} (S \cup i) ⊖_{egH} \bar{v} (S)$ of a player i to all coalitions S ⊆ N ∖ i.

The following property is clear based on Lemmas 4.2 and 4.3 and Theorem 4.1.

Corollary 4.1. The generalized interval Shapley value for an arbitrary interval-valued cooperative game always exists.

According toEq. (5), one can conclude thatEq. (13)can be rewritten as $\begin{matrix} {\bar{φ}}_{i} (\bar{v}) = [φ_{i} (v^{-}), φ_{i} (v^{+})], \end{matrix}$ where $φ_{i} (v^{-}), φ_{i} (v^{+}) \in ℝ$ are given by $\begin{matrix} φ_{i} (v^{-}) = \sum_{S \subseteq N ∖ i} \frac{s! (n - s - 1)!}{n!} (v^{-} (S \cup i) - v^{-} (S)), \end{matrix}$

$\begin{matrix} φ_{i} (v^{+}) = \sum_{S \subseteq N ∖ i} \frac{s! (n - s - 1)!}{n!} (v^{+} (S \cup i) - v^{+} (S)) . \end{matrix}$

Example 4.1. Consider a game $(N, \bar{v})$ with N = {1, 2, 3}, and the characteristic function is $\bar{v} (\emptyset) = [0, 0]$ , $\bar{v} ({1}) = [20, 30]$ , $\bar{v} ({2}) = [20, 40]$ , $\bar{v} ({3}) = [10, 35]$ , $\bar{v} ({1, 3}) = [50, 80]$ , $\bar{v} ({1, 2})$ = [60, 65], $\bar{v} ({2, 3}) = [60, 90]$ , $\bar{v} ({1, 2, 3}) = [120, 140]$ .

From Eq.(13), we have $\begin{matrix} {\bar{φ}}_{1} (\bar{v}) = \frac{2}{6} [20, 30] + \frac{1}{6} {[25, 40]}^{*} + \frac{1}{6} [40, 45] \\ + \frac{2}{6} {[50, 60]}^{*} \\ = [\frac{115}{3}, 40]^{*}, \end{matrix}$ $\begin{matrix} {\bar{φ}}_{2} (\bar{v}) = \frac{2}{6} [20, 40] + \frac{1}{6} {[35, 40]}^{*} + \frac{1}{6} [50, 55] \\ + \frac{2}{6} {[60, 70]}^{*} \\ = [45, \frac{145}{3}], \end{matrix}$ $\begin{matrix} {\bar{φ}}_{3} (\bar{v}) = \frac{2}{6} [10, 35] + \frac{1}{6} [30, 50] + \frac{1}{6} [40, 50] \\ + \frac{2}{6} [60, 75] \\ = [35, \frac{160}{3}] . \end{matrix}$

5 Generalized Shapley function and its axiomatization

In this section, we perform research on cooperative games with fuzzy payoffs based on the previous discussion of interval-valued cooperative games.

Definition 5.1. (See [22]) Let $\tilde{v} \in FG (N)$ and α ∈ [0, 1]. The game v_α is called the α-cut game of $\tilde{v}$ if $v_{α} (S) = {(\tilde{v} (S))}_{α} = [(\tilde{v} (S))_{α}^{-}, (\tilde{v} (S))_{α}^{+}]$ for all S ⊆ N.

Each α-cut game v_α is an interval-valued cooperative game; thus, v_α can be written as $[v_{α}^{-}, v_{α}^{+}]$ . Hence, from Proposition 2.2, it holds that $\tilde{v}$ can be described as $\tilde{v} = ⋃_{α \in [0, 1]} α \cdot v_{α}$ . For each S ⊆ N, let v^- (S): $α \to (\tilde{v} (S))_{α}^{-}$ and v⁺ (S): $α \to (\tilde{v} (S))_{α}^{+}$ . From Proposition 2.1, we obtain that $\tilde{v} (S)$ is completely determined by a pair of functions $\tilde{v} (S) = [v^{-} (S), v^{+} (S)]$ . Then, every game $\tilde{v}$ can be described in LU representation form $\tilde{v} = [v^{-}, v^{+}]$ , where $\tilde{v} (S) = [v^{-}, v^{+}] (S) = [v^{-} (S), v^{+} (S)]$ for all S ⊆ N.

Definition 5.2. For any $\tilde{v} \in FG (N)$ , the generalized Shapley function ${\tilde{φ}}_{i} (\tilde{v})$ is the expected marginal contribution of a player i to all coalitions S ⊆ N ∖ i. Moreover, for every i ∈ N, the explicit form is ${\tilde{φ}}_{i} (\tilde{v}) = \sum_{S \subseteq N ∖ i} \frac{s! (n - s - 1)!}{n!} (\tilde{v} (S \cup i) ⊖_{egH} \tilde{v} (S)) .$ (15)

Let $\begin{matrix} φ_{i}^{-} (\tilde{v}) = \sum_{S \subseteq N ∖ i} \frac{s! (n - s - 1)!}{n!} (v^{-} (S \cup i) - v^{-} (S)), \end{matrix}$ and $\begin{matrix} φ_{i}^{+} (\tilde{v}) = \sum_{S \subseteq N ∖ i} \frac{s! (n - s - 1)!}{n!} (v^{+} (S \cup i) - v^{+} (S)) . \end{matrix}$ Clearly, $φ_{i}^{-} (\tilde{v})$ and $φ_{i}^{+} (\tilde{v})$ are functions of α. Note that $φ_{i}^{-} (\tilde{v})$ and $φ_{i}^{+} (\tilde{v})$ always exist, but ${\tilde{φ}}_{i} (\tilde{v})$ does not always exist. If ${\tilde{φ}}_{i} (\tilde{v})$ exists, then Eq. (15) can be written as $\begin{matrix} {\tilde{φ}}_{i} (\tilde{v}) = [φ_{i}^{-} (\tilde{v}), φ_{i}^{+} (\tilde{v})] or {\tilde{φ}}_{i} (\tilde{v}) = {[φ_{i}^{+} (\tilde{v}), φ_{i}^{-} (\tilde{v})]}^{*} . \end{matrix}$

Lemma 5.1.Let $\tilde{v} \in FG (N)$ . The generalized Shapley function $φ (\tilde{v})$ exists if and only if

For any S ⊆ T, α, β ∈ [0, 1] and β > α,

$[\tilde{v} (T)]_{α}^{-} - [\tilde{v} (S)]_{α}^{-} \leq [\tilde{v} (T)]_{β}^{-} - [\tilde{v} (S)]_{β}^{-} \leq [\tilde{v} (T)]_{β}^{+} - [\tilde{v} (S)]_{β}^{+} \leq [\tilde{v} (T)]_{α}^{+} - [\tilde{v} (S)]_{α}^{+}$ , or

$[\tilde{v} (T)]_{α}^{+} - [\tilde{v} (S)]_{α}^{+} \leq [\tilde{v} (T)]_{β}^{+} - [\tilde{v} (S)]_{β}^{+} \leq [\tilde{v} (T)]_{β}^{-} - [\tilde{v} (S)]_{β}^{-} \leq [\tilde{v} (T)]_{α}^{-} - [\tilde{v} (S)]_{α}^{-}$ .

For any i ∈ N, α, β ∈ [0, 1] and β > α,

$[{\tilde{φ}}_{i} (\tilde{v})]_{α}^{-} \leq [{\tilde{φ}}_{i} (\tilde{v})]_{β}^{-} \leq [{\tilde{φ}}_{i} (\tilde{v})]_{β}^{+} \leq [{\tilde{φ}}_{i} (\tilde{v})]_{α}^{+}$ , or

$[{\tilde{φ}}_{i} (\tilde{v})]_{α}^{+} \leq [{\tilde{φ}}_{i} (\tilde{v})]_{β}^{+} \leq [{\tilde{φ}}_{i} (\tilde{v})]_{β}^{-} \leq [{\tilde{φ}}_{i} (\tilde{v})]_{α}^{-}$ .

Proof. For case (1). Let i ∈ N, α, β ∈ [0, 1] andβ > α.

If S, S ∪ i satisfy (i), namely, $\begin{matrix} [\tilde{v} (S \cup i)]_{α}^{-} - [\tilde{v} (S)]_{α}^{-} \leq [\tilde{v} (S \cup i)]_{β}^{-} - [\tilde{v} (S)]_{β}^{-} \leq \\ [\tilde{v} (S \cup i)]_{β}^{+} - [\tilde{v} (S)]_{β}^{+} \leq [\tilde{v} (S \cup i)]_{α}^{+} - [\tilde{v} (S)]_{α}^{+} . \end{matrix}$ Then, $\begin{matrix} A_{α} = {[\tilde{v} (S \cup i)]}_{α} ⊖_{egH} {[\tilde{v} (S)]}_{α} \\ = [(\tilde{v} (S \cup i))_{α}^{-} - (\tilde{v} (S))_{α}^{-}, (\tilde{v} (S \cup i))_{α}^{+} - (\tilde{v} (S))_{α}^{+}], \end{matrix}$ and $\begin{matrix} A_{β} = {[\tilde{v} (S \cup i)]}_{β} ⊖_{egH} {[\tilde{v} (S)]}_{β} \\ = [(\tilde{v} (S \cup i))_{β}^{-} - (\tilde{v} (S))_{β}^{-}, (\tilde{v} (S \cup i))_{β}^{+} - (\tilde{v} (S))_{β}^{+}] . \end{matrix}$ It follows that A_β ⊆ A_α holds for any α, β ∈ [0, 1] with β > α. This implies that the interval numbers in α-cuts set {A_α ∣ α ∈ [0, 1]} form a "nested" relation. Thus, $\tilde{A} = ⋃_{α \in [0, 1]} α \cdot A_{α}$ is a fuzzy number such that ${(\tilde{A})}_{α} = A_{α}$ for all α ∈ [0, 1]. Therefore, we obtain that $\tilde{v} (S \cup i) ⊖_{egH} \tilde{v} (S)$ exists, and $\tilde{v} (S \cup i) ⊖_{egH} \tilde{v} (S) = \tilde{A}$ , which satisfies case (i) of Eq.(8).

If S, S ∪ i satisfy (ii), namely, $\begin{matrix} [\tilde{v} (S \cup i)]_{α}^{+} - [\tilde{v} (S)]_{α}^{+} \leq [\tilde{v} (S \cup i)]_{β}^{+} - [\tilde{v} (S)]_{β}^{+} \leq \\ [\tilde{v} (S \cup i)]_{β}^{-} - [\tilde{v} (S)]_{β}^{-} \leq [\tilde{v} (S \cup i)]_{α}^{-} - [\tilde{v} (S)]_{α}^{-} . \end{matrix}$ Then, $\begin{matrix} B_{α} = {[\tilde{v} (S \cup i)]}_{α} ⊖_{egH} {[\tilde{v} (S)]}_{α} \\ = [(\tilde{v} (S \cup i))_{α}^{+} - {(\tilde{v} (S))_{α}^{+}, (\tilde{v} (S \cup i))_{α}^{-} - (\tilde{v} (S))_{α}^{-}]}^{*}, \end{matrix}$ and $\begin{matrix} B_{β} = {[\tilde{v} (S \cup i)]}_{β} ⊖_{egH} {[\tilde{v} (S)]}_{β} \\ = [(\tilde{v} (S \cup i))_{β}^{+} - {(\tilde{v} (S))_{β}^{+}, (\tilde{v} (S \cup i))_{β}^{-} - (\tilde{v} (S))_{β}^{-}]}^{*} . \end{matrix}$ It follows that B_β ⊆ B_α holds for any α, β ∈ [0, 1] with β > α. This implies that the interval numbers in α-cuts set {B_α ∣ α ∈ [0, 1]} form a "nested" relation. Thus, $\tilde{B} = ⋃_{α \in [0, 1]} α \cdot B_{α}$ is a fuzzy number such that ${(\tilde{B})}_{α} = B_{α}$ for all α ∈ [0, 1]. Therefore, $\tilde{v} (S \cup i) ⊖_{egH} \tilde{v} (S)$ exists, and $\tilde{v} (S \cup i) ⊖_{egH} \tilde{v} (S) = \tilde{B}$ , which satisfies case (ii) of Eq.(8).

From the above we conclude that if $\tilde{v}$ satisfies case (1), then $\tilde{v} (S \cup i) ⊖_{egH} \tilde{v} (S)$ exists for any i ∈ N and any S ⊆ N ∖ i.

For case (2). Let i ∈ N, α, β ∈ [0, 1] and β > α.

If ${\tilde{φ}}_{i} (\tilde{v})$ satisfies (i), then $\begin{matrix} [({\tilde{φ}}_{i} (\tilde{v}))_{β}^{-}, ({\tilde{φ}}_{i} (\tilde{v}))_{β}^{+}] \subseteq [({\tilde{φ}}_{i} (\tilde{v}))_{α}^{-}, ({\tilde{φ}}_{i} (\tilde{v}))_{α}^{+}], \end{matrix}$ i.e., ${({\tilde{φ}}_{i} (\tilde{v}))}_{β} \subseteq {({\tilde{φ}}_{i} (\tilde{v}))}_{α}$ . Denote ${\tilde{φ}}_{i} (\tilde{v}) = ⋃_{α \in [0, 1]} α \cdot {({\tilde{φ}}_{i} (\tilde{v}))}_{α}$ . We obtain that ${\tilde{φ}}_{i} (\tilde{v}) \in ℱ (ℝ)$ and ${\tilde{φ}}_{i} (\tilde{v}) = [φ_{i}^{-} (\tilde{v}), φ_{i}^{+} (\tilde{v})]$ .

If ${\tilde{φ}}_{i} (\tilde{v})$ satisfies (ii), then $\begin{matrix} {[({\tilde{φ}}_{i} (\tilde{v}))_{β}^{+}, ({\tilde{φ}}_{i} (\tilde{v}))_{β}^{-}]}^{*} \subseteq {[({\tilde{φ}}_{i} (\tilde{v}))_{α}^{+}, ({\tilde{φ}}_{i} (\tilde{v}))_{α}^{-}]}^{*}, \end{matrix}$ i.e., ${({\tilde{φ}}_{i} (\tilde{v}))}_{β} \subseteq {({\tilde{φ}}_{i} (\tilde{v}))}_{α}$ . Denote ${\tilde{φ}}_{i} (\tilde{v}) = ⋃_{α \in [0, 1]} α \cdot {({\tilde{φ}}_{i} (\tilde{v}))}_{α}$ . We obtain that ${\tilde{φ}}_{i} (\tilde{v}) \in ℱ (ℝ)$ and ${\tilde{φ}}_{i} (\tilde{v}) = {[φ_{i}^{+} (\tilde{v}), φ_{i}^{-} (\tilde{v})]}^{*}$ .

This completes the proof of Lemma 5.1. □

Remark 5.1. Case (1) ensures the feasibility of calculating the marginal contribution. Case (2) ensures that the α-cuts of the generalized Shapley function form a "nested" relation.

Next, we provide an axiomatization of the generalized Shapley function based on the axiomatization of the generalized interval Shapley value.

Definition 5.3. Player i is called a dummy player in $\tilde{v}$ if $\tilde{v} (S \cup i) ⊖_{egH} \tilde{v} (S) = \tilde{v} (i)$ for all S ⊆ N ∖ i.

Definition 5.4. Players i and j are called symmetric players in $\tilde{v}$ if $\tilde{v} (S \cup i) ⊖_{egH} \tilde{v} (S) = \tilde{v} (S \cup j) ⊖_{egH} \tilde{v} (S)$ for all S ⊆ N ∖ {i, j}.

Definition 5.5. Let $\tilde{v} \in FG (N)$ . The generalized Shapley function $φ (\tilde{v}) \in {[ℱ (ℝ)]}^{N}$ satisfies the following four axioms:

Generalized efficiency (GE). A value $φ (\tilde{v})$ satisfies GE if $\sum_{i \in N} {\tilde{φ}}_{i} (\tilde{v}) = \tilde{v} (N)$ .

Generalized dummy player (GD). Let i be a dummy player in $\tilde{v}$ . A value $φ (\tilde{v})$ satisfies GD if ${\tilde{φ}}_{i} (\tilde{v}) = \tilde{v} (i)$ .

Generalized symmetry (GS). Let i and j be a pair of symmetric players in $\tilde{v}$ . A value $φ (\tilde{v})$ satisfies GS if ${\tilde{φ}}_{i} (\tilde{v}) = {\tilde{φ}}_{j} (\tilde{v})$ .

Generalized additivity (GA). Let $\tilde{v}$ and $\tilde{w}$ be two games. A value $φ (\tilde{v})$ satisfies GA if for every i ∈ N, ${\tilde{φ}}_{i} (\tilde{v} + \tilde{w}) = {\tilde{φ}}_{i} (\tilde{v}) + {\tilde{φ}}_{i} (\tilde{w})$ .

Theorem 5.1.If the generalized Shapley function $φ (\tilde{v})$ exists, then $φ (\tilde{v})$ is the unique value on $FG (N)$ that satisfies GE, GD, GS, and GA.

Proof. Let $\tilde{v} \in FG (N)$ . Then $\tilde{v}$ can be written as $\tilde{v} = ⋃_{α \in [0, 1]} α \cdot v_{α}$ , where v_α is an α-cut game of $\tilde{v}$ . If $φ (\tilde{v}) = {({\tilde{φ}}_{i} (\tilde{v}))}_{i \in N}$ exists, then it follows that $\begin{matrix} {\tilde{φ}}_{i} (\tilde{v}) = ⋃_{α \in [0, 1]} α \cdot {\bar{φ}}_{i} (v_{α}), \end{matrix}$ where $φ (v_{α}) = {({\bar{φ}}_{i} (v_{α}))}_{i \in N}$ is the generalized interval Shapley value for v_α.

Since each v_α is an interval-valued cooperative game, then from EFF we have $\sum_{i \in N} {\bar{φ}}_{i} (v_{α}) = v_{α} (N)$ . Thus, $\begin{matrix} \sum_{i \in N} {\tilde{φ}}_{i} (\tilde{v}) & = & \sum_{i \in N} (⋃_{α \in [0, 1]} α \cdot {\bar{φ}}_{i} (v_{α})) \\ = & ⋃_{α \in [0, 1]} (α \cdot \sum_{i \in N} {\bar{φ}}_{i} (v_{α})) \end{matrix}$ $\begin{matrix} = ⋃_{α \in [0, 1]} (α \cdot v_{α} (N)) \\ = \tilde{v} (N), \end{matrix}$ showing that $φ (\tilde{v})$ satisfies GE.

Let i be a dummy player in $\tilde{v}$ . For each α ∈ [0, 1], one can easily check that i is a dummy player in v_α. Then by DUM, ${\bar{φ}}_{i} (v_{α}) = v_{α} (i)$ . We immediately conclude that $\begin{matrix} {\tilde{φ}}_{i} (\tilde{v}) & = & ⋃_{α \in [0, 1]} (α \cdot {\bar{φ}}_{i} (v_{α})) \\ = & ⋃_{α \in [0, 1]} (α \cdot v_{α} (i)) \\ = & \tilde{v} (i), \end{matrix}$ showing that $φ (\tilde{v})$ satisfies GD.

Let i and j be a pair of symmetric players in $\tilde{v}$ . For each α ∈ [0, 1], one can easily check that i and j is a pair of symmetric players in v_α. By SYM, we obtain ${\bar{φ}}_{i} (v_{α}) = {\bar{φ}}_{j} (v_{α})$ . Thus $\begin{matrix} {\tilde{φ}}_{i} (\tilde{v}) & = & ⋃_{α \in [0, 1]} (α \cdot {\bar{φ}}_{i} (v_{α})) \\ = & ⋃_{α \in [0, 1]} (α \cdot {\bar{φ}}_{j} (v_{α})) \\ = & {\tilde{φ}}_{j} (\tilde{v}), \end{matrix}$ showing that $φ (\tilde{v})$ satisfies GS.

Let $\tilde{v}, \tilde{w} \in FG (N)$ . Clearly, ${(\tilde{v} + \tilde{w})}_{α} = v_{α} + w_{α}$ for all α ∈ [0, 1]. Then from ADD, we obtain ${\bar{φ}}_{i} ({(\tilde{v} + \tilde{w})}_{α}) = {\bar{φ}}_{i} (v_{α} + w_{α}) = {\bar{φ}}_{i} (v_{α}) + {\bar{φ}}_{i} (w_{α})$ for every i ∈ N. Thus, $\begin{matrix} {\tilde{φ}}_{i} (\tilde{v} + \tilde{w}) & = & ⋃_{α \in [0, 1]} (α \cdot {[{\tilde{φ}}_{i} (\tilde{v} + \tilde{w})]}_{α}) \\ = & ⋃_{α \in [0, 1]} (α \cdot {\bar{φ}}_{i} ({(\tilde{v} + \tilde{w})}_{α})) \\ = & ⋃_{α \in [0, 1]} (α \cdot [{\bar{φ}}_{i} (v_{α}) + {\bar{φ}}_{i} (w_{α})]) \\ = & ⋃_{α \in [0, 1]} (α \cdot {\bar{φ}}_{i} (v_{α})) \\ + ⋃_{α \in [0, 1]} (α \cdot {\bar{φ}}_{i} (w_{α})) \\ = & {\tilde{φ}}_{i} (\tilde{v}) + {\tilde{φ}}_{j} (\tilde{v}), \end{matrix}$ showing that $φ (\tilde{v})$ satisfies GA.

Furthermore, from Theorem 4.1, we obtain that $\bar{φ} (v_{α}) = ({\bar{φ}}_{i} (v_{α}))_{i \in N}$ is the unique value that satisfies EFF, DUM, SYM, and ADD. Therefore, ${\tilde{φ}}_{i} (\tilde{v})$ is uniquely determined by ${\tilde{φ}}_{i} (\tilde{v}) = ⋃_{α \in [0, 1]} α \cdot {\bar{φ}}_{i} (v_{α})$ .

This proves the assertion. □

Remark 5.2. The approach to prove Theorem 5.1 is based on the existence of the generalized Shapley function.

The following example shows an important result that we cannot directly use ${\tilde{φ}}_{i} (\tilde{v}) =$ $⋃_{α \in [0, 1]} α \cdot {\bar{φ}}_{i} (v_{α})$ to structure the generalized Shapley function or the Hukuhara-Shapley function defined by Yu and Zhang [22].

Example 5.1. Consider a game $(N, \tilde{v})$ with N = {1, 2, 3}, and the characteristic function is

$\tilde{v} (\emptyset) = 0$ , $\tilde{v} ({1}) = 1$ , $\tilde{v} ({2}) = (5, 10, 12)$ ,

$\tilde{v} ({3}) = (4, 7, 11)$ , $\tilde{v} ({1, 2}) = (10, 14, 18)$ ,

$\tilde{v} ({1, 3}) = (9, 13, 17)$ , $\tilde{v} ({2, 3}) = (10, 18, 26)$ ,

$\tilde{v} ({1, 2, 3}) = (15, 25, 35)$ .

Then, the characteristic function of the α-cut game v_α is

v_α (∅) =0, v_α ({1}) =1,

v_α ({2}) = [5 + 5α, 12 - 2α],

v_α ({3}) = [4 + 3α, 11 - 4α],

v_α ({1, 2}) = [10 + 4α, 18 - 4α],

v_α ({1, 3}) = [9 + 4α, 17 - 4α],

v_α ({2, 3}) = [10 + 8α, 26 - 8α],

v_α ({1, 2, 3}) = [15 + 10α, 35 - 10α].

Consider player 1. From Eq. (13), we have $\begin{matrix} {\bar{φ}}_{1} (v_{α}) = \frac{2}{6} + \frac{1}{6} \cdot [5 - α, 6 - 2 α] + \frac{1}{6} \cdot [5 + α, 6] \\ + \frac{1}{6} \cdot [5 + 2 α, 9 - 2 α] \\ = \frac{1}{6} [17 + 2 α, 23 - 4 α] . \end{matrix}$ It is easy to verify that ${\bar{φ}}_{1} (v_{α})$ can form a "nested" relation. Then, $\begin{matrix} \tilde{A} = ⋃_{α \in [0, 1]} α \cdot {\bar{φ}}_{1} (v_{α}) = (\frac{17}{6}, \frac{19}{6}, \frac{23}{6}) \end{matrix}$ such that $A_{α} = {\bar{φ}}_{1} (v_{α})$ for all α ∈ [0, 1].

However, from Proposition 3.3, one can easily know that $\tilde{v} ({1, 2}) ⊖_{egH} \tilde{v} ({2})$ does not exist. This implies that the generalized Shapley function ${\tilde{φ}}_{1} (\tilde{v})$ does not exist. Therefore, $\begin{matrix} {\tilde{φ}}_{1} (\tilde{v}) \neq ⋃_{α \in [0, 1]} α \cdot {\bar{φ}}_{1} (v_{α}) . \end{matrix}$

Corollary 5.1.Let $\tilde{v} \in FG (N)$ , and letv_αbe an α-cut game of $\tilde{v}$ . If $φ (\tilde{v})$ exists, then for everyi ∈ N,

${\tilde{φ}}_{i} (\tilde{v}) = ⋃_{α \in [0, 1]} α \cdot {\bar{φ}}_{i} (v_{α}) .$ (16)

Lemma 5.2.Let $\tilde{v} \in FG (N)$ . If for anyS ⊆ T, and any α, β ∈ [0, 1] and β > α such that $\begin{matrix} [\tilde{v} (T)]_{α}^{-} - [\tilde{v} (S)]_{α}^{-} \leq [\tilde{v} (T)]_{β}^{-} - [\tilde{v} (S)]_{β}^{-} \leq \\ [\tilde{v} (T)]_{β}^{+} - [\tilde{v} (S)]_{β}^{+} \leq [\tilde{v} (T)]_{α}^{+} - [\tilde{v} (S)]_{α}^{+}, \end{matrix}$ then the generalized Shapley function $φ (\tilde{v})$ exists.

Proof. Let i ∈ N and S ⊆ N ∖ i. From Lemma 5.1, it is obtained that $\tilde{v} (S \cup i) ⊖_{egH} \tilde{v} (S)$ exists. Denote $\tilde{v} (S \cup i) ⊖_{egH} \tilde{v} (S) = {\tilde{m}}_{i} (S) = [m_{i}^{-} (S), m_{i}^{+} (S)]$ .

With this condition, we obtain that $m_{i}^{-} (S)$ is an increasing function of level sets α, $m_{i}^{+} (S)$ is a decreasing function of level sets α, and ${[m_{i}^{-} (S)]}_{α = 1} \leq {[m_{i}^{+} (S)]}_{α = 1}$ .

Moreover,

$φ_{i}^{-} (\tilde{v}) = \sum_{S \subseteq N ∖ i} γ_{S} \cdot m_{i}^{-} (S)$ , and

$φ_{i}^{+} (\tilde{v}) = \sum_{S \subseteq N ∖ i} γ_{S} \cdot m_{i}^{+} (S)$ , where $γ_{S} = \frac{s! (n - s - 1)!}{n!}$ .

With the above, it follows that $φ_{i}^{-} (\tilde{v})$ is an increasing function of level sets α, $φ_{i}^{+} (\tilde{v})$ is a decreasing function of level sets α, and ${[φ_{i}^{-} (\tilde{v})]}_{α = 1} \leq {[φ_{i}^{+} (\tilde{v})]}_{α = 1}$ . This implies that ${\tilde{φ}}_{i} (\tilde{v}) = [φ_{i}^{-} (\tilde{v}), φ_{i}^{+} (\tilde{v})]$ is a fuzzy number, which provides the result. □

In the above condition, the generalized Shapley function coincides with the Hukuhara-Shapley function defined by Yu and Zhang [22].

Lemma 5.3.An arbitrary cooperative game with payoffs of CTrFNs has a unique generalized Shapley function, which is also a CTrFN for everyi ∈ N.

Proof. Let $\tilde{A} = (a - m_{1}, a, a + m_{1})$ and $\tilde{B} = (b - m_{2}, b, b + m_{2})$ be arbitrary two CTrFNs (include such number satisfying case (ii) of Eq.(8)).

First, we have to show that egH-difference of any two CTrFNs is also a CTrFN.

For any α ∈ [0, 1], we have

A_α = [a - m₁ (1 - α) , a + m₁ (1 - α)], and

B_α = [b - m₂ (1 - α) , b + m₂ (1 - α)]. Then,

$A_{α} ⊖_{egH} B_{α} = [C_{α}^{-}, C_{α}^{+}] = [(a - m_{1} (1 - α)) - (b - m_{2} (1 - α)), (a + m_{1} (1 - α)) - (b + m_{2} (1 - α))]$ ,

If m₁ ≥ m₂. One easily checks that $C_{α}^{-}$ is an increasing function of α, $C_{α}^{+}$ is a decreasing function of α, and $C_{α}^{-} \leq C_{α}^{+}$ for all α ∈ [0, 1]. Then, it follows from Proposition 3.3 that $\tilde{A} ⊖_{egH} \tilde{B}$ exists, and $\tilde{A} ⊖_{egH} \tilde{B} = (a - b - m_{1} + m_{2}, a - b, a - b + m_{1} - m_{2})$ is a CTrFN.

If m₁ ≤ m₂. One easily checks that $C_{α}^{-}$ is a decreasing function of α, $C_{α}^{+}$ is an increasing function of α, and $C_{α}^{-} \geq C_{α}^{+}$ for all α ∈ [0, 1]. Then, it follows from Proposition 3.3 that $\tilde{A} ⊖_{egH} \tilde{B}$ exists, and $\tilde{A} ⊖_{egH} \tilde{B} = (a - b + m_{1} - m_{2}, a - b, a - b - m_{1} + m_{2})^{*}$ is a CTrFN.

Therefore, the marginal contribution of any player can be calculatd using egH-difference.

Next, we have to show that the generalized Shapley function is a CTrFN. To show this, we only need to show that the sum of any two CTrFNs is a CTrFN.

Indeed, $\tilde{A} + \tilde{B} = (a + b - m_{1} - m_{2}, a + b, a + b + m_{1} + m_{2})$ if m₁ + m₂ ≥ 0; $\tilde{A} + \tilde{B} = {(a + b + m_{1} + m_{2}, a + b, a + b - m_{1} - m_{2})}^{*}$ if m₁ + m₂ ≤ 0. Namely, $\tilde{A} + \tilde{B}$ is a CTrFN.

This completes the proof the assertion stated in Lemma 5.3. □

6 An illustrative example

Three companies decide to cooperate to complete a project. Let N = {1, 2, 3} denote the set of these companies. The profit of the joint project is an approximate evaluation, which is represented by center triangular fuzzy numbers as follows: $\begin{matrix} \tilde{v} (\emptyset) = (0, 0, 0), \tilde{v} ({1}) = (20, 25, 30), \\ \tilde{v} ({2}) = (20, 30, 40), \tilde{v} ({3}) = (10, 22.5, 35), \\ \tilde{v} ({1, 3}) = (50, 65, 80), \tilde{v} ({1, 2}) = (60, 62.5, 65), \\ \tilde{v} ({2, 3}) = (60, 75, 90), \tilde{v} ({1, 2, 3}) = (120, 130, 140) . \end{matrix}$

One can easily check that the Hukuhara-Shapley function defined by Yu and Zhang [22] does not exist. However, from Lemma 5.3, we know that this game has a unique generalized Shapley function. Now we calculate the generalized Shapley function.

The characteristic function of the α-cut game of $\tilde{v}$ is

v_α (∅) = [0, 0],

v_α ({1}) = [20 + 5α, 30 - 5α],

v_α ({2}) = [20 + 10α, 40 - 10α],

v_α ({3}) = [10 + 12.5α, 35 - 12.5α],

v_α ({1, 3}) = [50 + 15α, 80 - 15α],

v_α ({1, 2}) = [60 + 2.5α, 65 - 2.5α],

v_α ({2, 3}) = [60 + 15α, 90 - 15α],

v_α ({1, 2, 3}) = [120 + 10α, 140 - 10α].

Then from Eq.(13), the generalized interval Shapley value for the α-cut game is $\begin{matrix} {\bar{φ}}_{1} (v_{α}) = [\frac{115}{3} + \frac{5}{6} α, 40 - \frac{5}{6} α]^{*}, \\ {\bar{φ}}_{2} (v_{α}) = [45 + \frac{5}{3} α, \frac{145}{3} - \frac{5}{3} α], \\ {\bar{φ}}_{3} (v_{α}) = [35 + \frac{55}{6} α, \frac{160}{3} - \frac{55}{6} α] . \end{matrix}$

According to Eq. (16), we obtain the generalized Shapley function of each player (i.e. company) as follows: $\begin{matrix} {\tilde{φ}}_{1} (\tilde{v}) = (\frac{115}{3}, \frac{235}{6}, 40)^{*}, \\ {\tilde{φ}}_{2} (\tilde{v}) = (45, \frac{140}{3}, \frac{145}{3}), \\ {\tilde{φ}}_{3} (\tilde{v}) = (35, \frac{265}{6}, \frac{160}{3}) . \end{matrix}$

These values can also be obtained using Eq. (15).

This example cannot be perfectly solved using previous methods, but we now provide a rational scheme for distributing the total fuzzy profit $\tilde{v} ({1, 2, 3})$ among the members of {1, 2, 3}. This allocation scheme is reasonable and useful since this value satisfies several axioms, especially the generalized efficiency axiom, namely, ${\tilde{φ}}_{1} (\tilde{v}) + {\tilde{φ}}_{2} (\tilde{v}) + {\tilde{φ}}_{3} (\tilde{v}) = (120, 130, 140)$ .

7 Conclusion

Based on the extended generalized Hukuhara difference of two fuzzy numbers, we investigate the generalized Shapley function for cooperative games with fuzzy payoffs. To consider this class of games, it is necessary to discuss the interval-valued cooperative games on the basis of the extended generalized Hukuhara difference of two interval numbers; because each α-cut game of cooperative games with fuzzy payoffs is an interval-valued cooperative game. Thus, the definition and an axiomatization of the generalized interval Shapley value are provided. Then, considering the relationship between cooperative games with fuzzy payoffs and interval-valued cooperative games, the generalized Shapley function for cooperative games with fuzzy payoffs is studied.

To show that the generalized Shapley function is a fair and reasonable allocation, we provide an axiomatic characterization for it using the properties of generalized efficiency, generalized dummy player, generalized symmetry, and generalized additivity. Meanwhile, the necessary and sufficient condition for the existence of the generalized Shapley function is given. In a special condition, the generalized Shapley function coincides with the Hukuhara-Shapley function defined by Yu and Zhang [22]. Furthermore, a discussion shows that the generalized Shapley function cannot be directly structured by the generalized interval Shapley value of each α-cut game, even in the situation that these generalized interval Shapley values can form a “nested” relation. The study also shows a novel result that an arbitrary cooperative game with payoffs of CTrFNs has a unique generalized Shapley function. Besides, some results on the generalized interval Shapley value are obtained. For instance, the generalized interval Shapley value for an arbitrary interval-valued cooperative game always exists, which is quite different from the previous results [1–4 , 8].

Footnotes

Acknowledgements

The authors would like to thank three anonymous reviewers and editors for their extremely valuable comments. This work was supported by the National Natural Science Foundation of China (Nos.71771025, 71371030, 71571192, 71561022 and 71401003), the Natural Science Foundation of Beijing (No.9152002) and the Youth Fund Project for Humanities and Social Sciences Research of MOE of China (No.17YJC630203).

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