Cooperative game with fuzzy coalition and payoff value in the generalized integral form

Abstract

In this paper, a new class of games with fuzzy coalitions and fuzzy payoff value is proposed. This class of fuzzy games is based on the generalized integral form, which contains several kinds of other fuzzy games, such as fuzzy cooperative game with fuzzy payoff value, the multilinear extension game introduced by Owen, the game with proportional value proposed by Butnariu and the game with Choquet integral form given by Tsurumi et al. Also, the proposed fuzzy game is also further extension of the fuzzy game in Choquet integral form proposed by Yu et al., which is also is a kind of fuzzy games in the condition that coalition and fuzzy payoff value are both fuzzy information. The fuzzy Shapley value for this kind of fuzzy games is represented by Shapley value of corresponding fuzzy cooperative game with fuzzy payoff value. Based on Hukuhara-difference, we give the explicit Shapley value for the proposed fuzzy game. It has been seen that most of properties hold well in the proposed fuzzy game, which are processed by cooperation game with fuzzy payoff value, and fuzzy coalition game, respectively. Finally, a practical application of the proposed model is also provided.

Keywords

Cooperative game fuzzy coalition payoff value shapley value

1 Introduction

The fuzzy coalition is introduced by Aubin [1, 2], in which some players participate partially. Butnariu [3 –5] defined the games with proportional value and showed the explicit form of the Shapley function on that class of fuzzy games. Tsurumi et al. [6] defined a new class of fuzzy games with Choquet integral form and another explicit form of Shapley function. Li et al. [7] showed that the explicit form of Shapley value introduced by Butnariu and Tsurumi et al. can be written in the same expression. Yu et al. [8] defined the fuzzy core for any kind of games with fuzzy coalition and gave the explicit form of fuzzy cores in crisp cores for the multilinear extension introduced by Owen [9], the games with proportional value proposed by Butnariu and the games with Choquet integral form given by Tsurumi et al.

On the other hand, Mares [10, 11] and Vlach [12] were concerned about the uncertainty in the value of the characteristic function associated with a cooperative game. In their game models, the domain of the characteristic function of a game remains to be the class of crisp coalition but the coalition payoff values allocated to players are fuzzy numbers. By introducing fuzzy information into cooperative games, we know that the implicit assumption that all players and coalitions know the expected payoffs even before the cooperation negotiation is unrealistic. There are many uncertain factors during the process of negotiation and coalition forming, so in most situations players can only know the imprecise information of the real outcome of cooperation. Yu et al. [13] take research on fuzzy games of which the coalition and the payoff value are both fuzzy information. The proposed fuzzy game model is a connection between the fuzzy coalition game and the fuzzy game with payoff value, which have been separately studied before. However, the fuzzy game in Ref. [13] may only be seen an extension of fuzzy coalition in Ref. [6]. Considering there are three main kinds fuzzy coalition, such as the multilinear extension introduced by Owen, the game with proportional value proposed by Butnariu and the game with Choquet integral form given by Tsurumi et al., it is necessary to propose a new class of fuzzy games with fuzzy coalition and payoff value as an generalized extension of fuzzy coalition game.

The purpose of this paper is to study fuzzy Shapley value for a new class of games with fuzzy coalition and fuzzy payoff value. Note that the proposed class fuzzy cooperative game may contain several kinds of fuzzy games, such as the fuzzy characteristic game (i.e. the game with fuzzy payoff value), the multilinear extension extension game introduced by Owen, the game with proportional value proposed by Butnariu and the game with Choquet integral form given by Tsurumi et al. Hence, the new class of fuzzy games introduced in this paper may be seen as a further generalized form of fuzzy games. The fuzzy Shapley value of this class of fuzzy game is expressed by fuzzy Shapley value of fuzzy characteristic game, while the proposed Shapley value may also be applied to the fuzzy coalition game, the fuzzy characteristic game, and crisp game. The expression of fuzzy Shapley value is the generalized conclusion in Ref. [13], which can only be seen as the solution for the extension of the game with Choquet integral form given by Tsurumi et al.

The paper will be organized as follows. In Section 2, we introduce the concepts of fuzzy numbers, Hukuhara-difference on fuzzy numbers, and some basic definitions on crisp cooperative game firstly. Also, we introduce some existing concepts about fuzzy coalition game, fuzzy characteristic game, and the game with both fuzzy coalition and payoff value. In Section 3, a generalized kind of fuzzy games with fuzzy coalition and payoff value is given, and its properties is shown. In Section 4, Shapley value of the proposed fuzzy game is also shown based on the Hukuhara-difference. Finally, some conclusions will be discussed in Section 5.

2 Preliminaries

2.1 Preparation on fuzzy numbers

Let us start with reminding the most general definition of a fuzzy number. Let $ℝ$ be (- ∞ , ∞), i.e., the set of all real numbers.

Definition 2.1. A fuzzy number, denoted by $\tilde{a}$ , is a fuzzy subset of $ℝ$ with membership function $u_{\tilde{a}} : ℝ \to [0, 1]$ satisfying the following conditions:

There exists at least one number $a_{0} \in ℝ$ such that $u_{\tilde{a}} (a_{0}) = 1$ ;

$u_{\tilde{a}} (x)$ is nondecreasing on (- ∞ , a₀) and nonincreasing on (a₀, ∞);

$u_{\tilde{a}} (x)$ is upper semi-continuous, i.e., $lim_{x \to x_{0}^{+}} u_{\tilde{a}} (x) = u_{\tilde{a}} (x_{o})$ if x_o < a_o and $lim_{x \to x_{0}^{-}} u_{\tilde{a}} (x) = u_{\tilde{a}} (x_{o})$ if x_o > a_o;

$\int_{- \infty}^{\infty} u_{\tilde{a}} (x) d x < \infty .$

The set of all fuzzy numbers is denoted by $R$ . An important type of regular fuzzy numbers in common use is the triangular fuzzy number [14] whose membership function has a form $u_{\tilde{a}} (x) = {\begin{matrix} 1 & if x = a_{b}, \\ \frac{x - a_{l}}{a_{b} - a_{l}} & if x \in [a_{l}, a_{b}], \\ \frac{x - a_{r}}{a_{b} - a_{r}} & if x \in [a_{b}, a_{r}] \\ 0 & otherwise \end{matrix}$ where $a_{l}, a_{b}, a_{r} \in ℝ$ with a_l ≤ a_b ≤ a_r, and this triangular fuzzy number $\tilde{a}$ is denoted by (a_l, a_b, a_r).

For a fuzzy number $\tilde{a}$ , the level set is defined as ${\tilde{a}}_{λ} = {x \in ℝ | u_{\tilde{a}} (x) \geq λ}$ , λ ∈ [0, 1].

It follows from the properties of the membership function of a fuzzy number $\tilde{a}$ that each its λ-cut ${\tilde{a}}_{λ}$ is a closed interval, denoted by ${\tilde{a}}_{λ} = [{\tilde{a}}_{λ}^{L}, {\tilde{a}}_{λ}^{R}]$ , λ ∈ (0, 1], where ${\tilde{a}}_{λ}^{L}$ and ${\tilde{a}}_{λ}^{R}$ means the lower and upper bounds of ${\tilde{a}}_{λ}$ . For any two fuzzy numbers $\tilde{a}, \tilde{b} \in R$ , we have the following order relation

$\tilde{a} \geq \tilde{b}$ if and only if ${\tilde{a}}_{λ}^{L} \geq {\tilde{b}}_{λ}^{L}$ , ${\tilde{a}}_{λ}^{R} \geq {\tilde{b}}_{λ}^{R}$ , ∀λ ∈ (0, 1];

$\tilde{a} = \tilde{b}$ if and only if $\tilde{a} \geq \tilde{b}$ and $\tilde{b} \geq \tilde{a}$ ;

$\tilde{a} \subseteq \tilde{b}$ if and only if ${\tilde{a}}_{λ}^{L} \geq {\tilde{b}}_{λ}^{L}$ and ${\tilde{a}}_{λ}^{R} \leq {\tilde{b}}_{λ}^{R}$ , ∀λ ∈ (0, 1].

Let * ∈ {+ , ×} denotes two operations. Now the operations on fuzzy numbers can be defined asfollows.

Definition 2.2. For any $\tilde{a}, \tilde{b} \in R$ , $\tilde{a} * \tilde{b}$ is a fuzzy number with the membership function: $u_{\tilde{a} * \tilde{b}} (z) = sup_{x * y = z} min {u_{\tilde{a}} (x), u_{\tilde{b}} (y)}, z \in ℝ$ (1)

It is not easy to apply Equation (1) in calculation directly. However, calculatingλ-cuts of the fuzzy number $\tilde{a} * \tilde{b}$ is an easy task in each case because $(\tilde{a} + \tilde{b})_{λ} = {\tilde{a}}_{λ} + {\tilde{b}}_{λ} = [{\tilde{a}}_{λ}^{L} + {\tilde{b}}_{λ}^{L}, {\tilde{a}}_{λ}^{R} + {\tilde{b}}_{λ}^{R}],$ (2) $(m \tilde{a})_{λ} = m {\tilde{a}}_{λ} = [m {\tilde{a}}_{λ}^{L}, m {\tilde{a}}_{λ}^{R}], \forall m > 0$ (3)

Let $\bar{ℝ}$ be the class of all closed intervals in $ℝ$ . For any interval number, denoted by $\bar{a}$ , we also adopt thenotation $\bar{a} = [a^{L}, a^{R}]$ , where a^L and a^R means the lower and upper bounds of $\bar{a}$ . For any $\bar{a}, \bar{b} \in \bar{ℝ}$ , the following operations on interval numbers will be carried out in this paper.

$\bar{a} + \bar{b} = [a^{L} + b^{L}, a^{R} + b^{R}];$

$m \bar{a} = [{ma}^{L}, {ma}^{R}], \forall m \in R and m > 0 .$

As to the difference between fuzzy numbers, we extend the Hukuhara difference between interval numbers [15] as follows.

Definition 2.3. Let $\tilde{a}, \tilde{b} \in R$ . If there exists $\tilde{c} \in R$ such that $\tilde{a} = \tilde{b} + \tilde{c}$ , then $\tilde{c}$ is called the Hukuhara difference, denoted by $\tilde{c} = \tilde{a} -_{H} \tilde{b}$ .

Proposition 2.1.[13] Let $\tilde{a}, \tilde{b} \in R$ . If $\tilde{a} -_{H} \tilde{b}$ exists, then for anyλ ∈ (0, 1], ${(\tilde{a} -_{H} \tilde{b})}_{λ} = {\tilde{a}}_{λ} -_{H} {\tilde{b}}_{λ} = [{\tilde{a}}_{λ}^{L} - {\tilde{b}}_{λ}^{L}, {\tilde{a}}_{λ}^{R} - {\tilde{b}}_{λ}^{R}] .$

Theorem 2.1. [13] Let $\tilde{a}, \tilde{b} \in R$ . The Hukuhara difference $\tilde{a} -_{H} \tilde{b}$ exists if and only if $\begin{matrix} a_{λ}^{L} - b_{λ}^{L} \leq a_{β}^{L} - b_{β}^{L} \leq a_{β}^{R} - b_{β}^{R} \leq a_{λ}^{R} - b_{λ}^{R}, \\ \forall λ, β \in (0, 1], β > λ . \end{matrix}$

2.2 Crisp cooperative game and its Shapley value

A crisp cooperative game is defined by (N, v), in which N is the set of players. The class of all fuzzy subsets of U is denoted by F(U), and the class of all crisp subsets of a crisp set N by P(N). Then characteristic function (i.e. fuzzy payoff value) $v : P (N) \to ℝ_{+} = {r \in ℝ | r \geq 0}$ satisfies that v (∅) =0.

If S and T are disjoint crisp coalitions, it is clear that they can accomplish at least as much by joining forces as by remaining separate. Hence, we mainly discuss the supperaddtive crisp cooperative games in this paper, i.e. $\begin{matrix} v (S \cup T) \geq v (S) + v (T), \forall S, T \in P (N), \\ s . t . S \cap T = \emptyset, \end{matrix}$

and we denote by G₀(N) all the supperadditve crisp cooperative games.

Also, the convexity and the imputation of crisp game are defined as follows.

Definition 2.4. An imputation for a crisp cooperative game v ∈ G₀ (N) is a vector x = (x₁, …, x_n) satisfying

∑_i∈Nx_i = v (N) ,

x_i ≥ v ({i}) , ∀ i ∈ N .

We shall use the notation E (v) for the set of all imputations of the crisp game v ∈ G₀ (N).

The Shapley value is an important solution concepts for crisp cooperative game. The Shapley value f_i (v) of player i with respect to a game v ∈ G₀ (N) is a weighted average value of the marginal contribution v (S) - v (S ∖ {i}) of player i alone in all combinations, which is defined by $f_{i} (v) = \sum_{i \in S \in P (N)} \frac{(n - s)! (s - 1)!}{n!} [v (S) - v (S ∖ {i}]$ (4)

where n, s are the cardinality of N, S, i.e., n = |N|, s = |S|.

Equation (4) is the unique expression that satisfies three axiomatic characterization of Shapley value.

2.3 Fuzzy extension of cooperative game and fuzzy Shapley value

Let N = 1,...,n be the set of players. A fuzzy coalition K is a fuzzy subset of N, which is a vector K = K(1),...,K(n) with coordinates K(i) contained in the interval [0, 1]. The number K(i) describes the membership grade of i in K. For two (crisp or fuzzy) coalitions K and U, K ⊆ U means that K (i) ≤ U (i) , ∀ i ∈ N. For a fuzzy set K, the level set is defined as [K] _λ = {i ∈ N|K (i) ≥ λ} for any λ ∈ [0, 1], and support set is denoted by Supp (K) = {i ∈ N|K (i) >0}.

A cooperative game with fuzzy coalition is a pair $(N, \tilde{v})$ in which the characteristic function $\tilde{v} : F (N) \to ℝ_{+}$ is such that $\tilde{v} (\emptyset) = 0$ .

Taking imprecision of information in decision making problems into account, fuzzy characteristic function can be incorporated into cooperative game, which is represented by fuzzy number w. Therefore, the characteristic function of such game associates a coalition crisp S with a fuzzy number w (S). Assessing such fuzzy numbers for any fuzzy coalition K ⊆ N, we can define fuzzy characteristic game by $(N, \tilde{w})$ , where the characteristic function $\tilde{w} : F (N) \to R_{+} = {\tilde{r} \in R | \tilde{r} \geq 0}$ is such that $\tilde{w} (\emptyset) = 0$ .

Obviously, a fuzzy game degenerates to be a game with fuzzy coalition when the range of the mapping $\tilde{w}$ is $ℝ_{+}$ and to be a fuzzy characteristic game when definitional domain of the mapping $\tilde{w}$ is P (N), while to be a crisp game when definitional domain and range of the mapping $\tilde{w}$ are P (N) and $ℝ_{+}$ , respectively. Thus, our discussion and game models will be applicable to games involving both crisp and fuzzy games.

In this paper, we adopt the usual definition of union and intersection of fuzzy subset given by the maximum and minimum operators, i.e., $\begin{matrix} (K \cup U) (i) = max {K (i), U (i)} \forall i \in N, \\ (K \cap U) (i) = min {K (i), U (i)} \forall i \in N . \end{matrix}$

Definition 2.5. [13] A fuzzy game $(N, \tilde{w})$ is said to be supperadditive if, for any λ ∈ [0, 1] and any two coalitions K, U ∈ F (N) such that K∩ U = ∅, ${\tilde{w}}_{λ}^{L} (K \cup U) \geq {\tilde{w}}_{λ}^{L} (K) + {\tilde{w}}_{λ}^{L} (U),$ ${\tilde{w}}_{λ}^{R} (K \cup U) \geq {\tilde{w}}_{λ}^{R} (K) + {\tilde{w}}_{λ}^{R} (U) .$

We denote the set of all the supperadditive fuzzy games $(N, \tilde{w})$ as G (N).

Let S_U ∈ F (U) be defined by $S_{U} (i) = {\begin{matrix} U (i), & if i \in S, \\ 0, & otherwise, \end{matrix}$

for any U ∈ F (N) and S ⊆ N.

Definition 2.6. A fuzzy-valued function $x : F (N) \to R_{+}^{n}$ is said to be an imputation of a fuzzy game $(N, \tilde{w})$ if it satisfies

x_i (U) =0, ∀ i ∉ Supp (U) ;

$\sum_{i \in Supp (U)} x_{i} (U) = \tilde{w} (U);$

$x_{i} (U) \geq \tilde{w} ({i}_{U}), \forall i \in Supp (U),$

for ∀U ∈ F (N), where x (U) = (x₁ (U) , x₂ (U) , …, x_n (U)).

Form the above two definitions, it is not hard to see that supperadditivity and imputation in a fuzzy game can be applicable to both crisp game and game with fuzzy coalition by restricting the domain.

Prior to the definitions of carrier and fuzzy Shapley value, we give some notations introduced by Tsurumi et al. For any U ∈ F (N), K ∈ F (U) and i ∈ N, $K_{i}^{U} (j) = {\begin{matrix} U (j) & if j = i, \\ K (j) & otherwise, \end{matrix}$ $K_{ij}^{U} (k) = {\begin{matrix} min {K (i), U (j)} & if k = i, \\ min {K (j), U (i)} & if k = j, \\ K (j) & otherwise, \end{matrix}$ $p_{ij} [K] (k) = {\begin{matrix} K (j) & if k = i, \\ K (i) & if k = j, \\ K (k) & otherwise . \end{matrix}$ Clearly, $K_{i}^{U}$ , $K_{ij}^{U}$ and p_ij [K] are fuzzy subsets of U.

Definition 2.7. [13] Let γ satisfy 0 ≤ γ < U (i), U ∈ F (N). Player i ∈ Supp (U) is said to be a γ-null player in U for $(N, \tilde{w})$ if for any λ ∈ [0, 1] and any K ∈ F (U) satisfying K (i) > γ, ${\tilde{w}}_{λ}^{L} (K) = {\tilde{w}}_{λ}^{L} (K_{i}^{U}), {\tilde{w}}_{λ}^{R} (K) = {\tilde{w}}_{λ}^{R} (K_{i}^{U}) .$

Definition 2.8. [13] Let U ∈ F (N), K ∈ F (U). K is called an f-carrier in U for fuzzy game $(N, \tilde{w})$ if, for any λ ∈ [0, 1] and any M ∈ F (U), coalition K satisfies ${\tilde{w}}_{λ}^{L} (K \cap M) = {\tilde{w}}_{λ}^{L} (M),$ ${\tilde{w}}_{λ}^{R} (K \cap M) = {\tilde{w}}_{λ}^{R} (M) .$ The set of all f-carrier in U for $(N, \tilde{w})$ is denoted by $C (U | \tilde{w})$ .

Definition 2.9. [13] A function $f : G (N) \to (R_{+}^{n})^{F (N)}$ is said to be a fuzzy Shapley function on G(N) if it satisfies the following four axioms.

Axiom 1. If $\tilde{w} \in G (N)$ and U ∈ F (N) then ${\begin{matrix} \sum_{i \in I} f_{i} (\tilde{w}) (U) = \tilde{w} (U), \\ f_{i} (\tilde{w}) (U) = 0 \forall i \notin Supp (U), \end{matrix}$ where $f_{i} (\tilde{w}) (U)$ is the ith element of $f \in R_{+}^{n}$ .

Axiom 2. For $\tilde{w} \in G (N)$ , U ∈ F (N), $K \in C (U | \tilde{w})$ , we obtain $f_{i} (\tilde{w}) (U) = f_{i} (\tilde{w}) (K), \forall i \in N .$

Axiom 3. If $\tilde{w} \in G (N)$ , U ∈ F (N), $U_{ij}^{U} \in C (U | \tilde{w})$ and $\tilde{w} (K) = \tilde{w} (p_{ij} [K])$ for any $K \in F (U_{ij}^{U})$ . Then $f_{i} (\tilde{w}) (U) = f_{j} (\tilde{w}) (U) .$

Axiom 4. Let ${\tilde{w}}_{1}, {\tilde{w}}_{2} \in G (N)$ . Define ${\tilde{w}}_{1} + {\tilde{w}}_{2}$ by $({\tilde{w}}_{1} + {\tilde{w}}_{2}) (K) = ({\tilde{w}}_{1}) (K) + {\tilde{w}}_{2} (K)$ for any K ∈ F (N). If ${\tilde{w}}_{1} + {\tilde{w}}_{2} \in G (N)$ and U ∈ F (N) then $\begin{matrix} f_{i} ({\tilde{w}}_{1} + {\tilde{w}}_{2}) (U) = f_{i} ({\tilde{w}}_{1}) (U) + f_{i} ({\tilde{w}}_{2}) (U), \\ \forall i \in N . \end{matrix}$

The above Shapley axiom system can be applied to all kinds of cooperative games involving crisp games, fuzzy coalition game, fuzzy characteristic game, and game with both fuzzy coalition and characteristic function.

3 The generalized integral game

Before we define a new class of games with fuzzy coalition and fuzzy payoff value, we introduce a kind of fuzzy characteristic games as follows.

Definition 3.1. [13] Let (N, w) be fuzzy characteristic game. If (N, w) satisfy the following conditions:

$\begin{matrix} w_{λ} (S \cup T) \geq w_{λ} (S) + w_{λ} (T), \forall λ \in (0, 1], \\ S \cap T = \emptyset \forall S, T \in P (N) \end{matrix}$

$\begin{matrix} w_{λ}^{L} (I) - w_{λ}^{L} (D) \leq w_{β}^{L} (I) - w_{β}^{L} (D) \leq w_{β}^{R} (I) \\ - w_{β}^{R} (D) \leq w_{λ}^{R} (I) - w_{λ}^{R} (D), \\ D \subseteq I, \forall D, I \in P (N) \end{matrix}$

where λ, β ∈ (0, 1] such that β > λ. Then we call (N, w) Hukuhara fuzzy game, because w (I) - _Hw (D) exists by Theorem 2.1.

We denote by G_H (N) the set of all Hukuhara fuzzy games.

Definition 3.2. Let w ∈ G_H (N) be Hukuhara fuzzy game, then a game $(N, \tilde{w})$ is said to be generalized integral game if and only if the following holds: $\tilde{w} (S_{U}) ≜ \sum_{T \subseteq N} (a_{T}^{U} \cdot w (T \cap Supp (S_{U})))$ (5)

for any U ∈ F (N), S ∈ P (N), where $a_{T}^{U} \in ℝ_{+}$ , ∀T ⊆ N. We denote by G_L (N) the set of the generalized integral game.

Given the coefficients $a_{T}^{U} \geq 0$ for all U ∈ F (N) and all T ⊆ N, there is a one-to-one correspondence between Hukuhara fuzzy game (N, w) and the generalized integral game $\tilde{w}$ . We call Hukuhara fuzzy game to a the generalized integral game the associated Hukuhara fuzzy game, and the generalized integral game corresponding to Hukuhara fuzzy game the associated generalized integral game. Also, because the crisp game is a special kind of fuzzy characteristic game, the generalized integral game is transferred to the fuzzy coalition game when Hukuhara fuzzy game (N, w) is simplified to crisp game (N, v).

Remark 3.1. Let $\tilde{w} \in G_{L} (N)$ . Given U, K ∈ F (N) such that U∩ K = ∅, then the following hold: $\begin{matrix} \tilde{w} (U) = \sum_{T \subseteq N} (a_{T}^{U} \cdot w (T \cap Supp (U))) \\ = \sum_{T \subseteq N} (a_{T}^{U \cup K} \cdot w (T \cap Supp (U))) . \end{matrix}$

Let w ∈ G₀ (N), U ∈ F (N), Q (U) = {U (i) |U (i) >0, i ∈ N}, q (U) be the cardinality of Q (U), i.e. q (U) = |Q (U) |, and r_m (U) = {i|i ∈ N, U (i) = r_m}. The element in Q (U) are written in the increasing order as r₁ < … < r_q(U), and let r₀ = 0. For any T ⊆ N, we let $(a_{T}^{U})_{1} = {\begin{matrix} \prod_{i \in T} U (i) \prod_{i \notin T} (1 - U (i)), & if T \subseteq Supp (U), \\ 0, & otherwise, \end{matrix}$ $(a_{T}^{U})_{2} = {\begin{matrix} r_{m}, & if T = r_{m} (U), r_{m} \in Q (U), \\ 0, & otherwise, \end{matrix}$ $(a_{T}^{U})_{3} = {\begin{matrix} r_{m} - r_{m - 1}, & T = [U]_{r_{m}}, r_{m} \in Q (U), \\ 0, & otherwise, \end{matrix}$

then we obtain three special kinds of generalized integral games as follows $\begin{matrix} {\tilde{w}}^{1} (S_{U}) = \sum_{T \subseteq N} ({(a_{T}^{U})}_{1} \cdot w (T \cap Supp (S_{U}))) \\ = \sum_{T \subseteq N} \prod_{i \in T} U (i) \prod_{i \notin T} (1 - U (i)) \cdot w (T \cap Supp (S_{U})) \end{matrix}$ (6) $\begin{matrix} {\tilde{w}}^{2} (S_{U}) = \sum_{T \subseteq N} ({(a_{T}^{U})}_{2} \cdot w (T \cap Supp (S_{U}))) \\ = \sum_{m = 1}^{q (U)} w (S_{r_{m} (U)}) \cdot r_{m} \end{matrix}$ (7) $\begin{matrix} {\tilde{w}}^{3} (S_{U}) = \sum_{T \subseteq N} ({(a_{T}^{U})}_{3} \cdot w (T \cap Supp (S_{U}))) \\ = \sum_{m = 1}^{q (U)} w (S_{[U]_{r_{m}}}) \cdot (r_{m} - r_{m - 1}) \end{matrix}$ (8)

for any U ∈ F (N).

Theorem 3.1.LetS ∈ P (N), U ∈ F (N). Then Equation (6) is equivalent to the following: ${\tilde{w}}^{1} (S_{U}) = \sum_{T \subseteq N} \prod_{i \in T} S_{U} (i) \prod_{i \notin T} (1 - S_{U} (i)) \cdot w (T)$ (9)

Proof. For any T ∈ P (N), we let T₁ = T ∩ Supp (S_U) and T₂ = T ∖ Supp (S_U). Then we get that T = T₁ ∪ T₂ and T₁∩ T₂ = ∅. Therefore, the following holds: $\begin{matrix} {\tilde{w}}^{1} (S_{U}) \\ = \sum_{T \subseteq N} \prod_{i \in T} U (i) \prod_{i \notin T} (1 - U (i)) \cdot w (T \cap Supp (S_{U})) \\ = \sum_{T_{1} \subseteq Supp (S_{U}), T_{2} \subseteq N ∖ Supp (S_{U})} \begin{matrix} \prod_{i \in T_{1} \cup T_{2}} U (i) \prod_{i \in N ∖ (T_{1} \cup T_{2})} (1 - U (i)) \\ \times w (T_{1}) \end{matrix} \end{matrix} ∥$ $\begin{matrix} = \sum_{T_{1} \subseteq Supp (S_{U})} \begin{matrix} (\prod_{i \in T_{1}} U (i) \prod_{i \in Supp (S_{U}) ∖ T_{1}} (1 - U (i)) \cdot w (T_{1})) \times \\ \sum_{T_{2} \subseteq N ∖ Supp (S_{U})} (\prod_{i \in T_{2}} U (i) \prod_{i \in N ∖ (Supp (S_{U}) \cup T_{2})} (1 - U (i))) \end{matrix} \\ = \sum_{T_{1} \subseteq Supp (S_{U})} (\prod_{i \in T_{1}} U (i) \prod_{i \in Supp (S_{U}) ∖ T_{1}} (1 - U (i)) \cdot w (T_{1})) \\ = \sum_{T \subseteq N} (\prod_{i \in T} S_{U} (i) \prod_{i \in Supp (S_{U}) ∖ T} (1 - S_{U} (i)) \cdot w (T)), \end{matrix}$ where in the last three lines we use coprod _i∈TS_U (i) coprod _{i∈Supp(S_U)∖T} (1 - S_U (i)) = 0, ∀ T ⊄ Supp (S_U) and $\sum_{T_{2} \subseteq N ∖ Supp (S_{U})} (\prod_{i \in T_{2}} U (i) \prod_{i \in N ∖ (Supp (S_{U}) \cup T_{2})} (1 - U (i))) = 1$ . The proof is completed.

Remark 3.2. Let S ∈ P (N), U ∈ F (N), Q (S_U) = {U (i) |U (i) >0, i ∈ N}, q (S_U) be the cardinality of Q (S_U), i.e.q (S_U) = |Q (S_U) |. The element in Q (S_U) are written in the increasing order as k₁ < … < k_{q(S_U)}, and let k₀ = 0. Then the following holds:

$\begin{matrix} (1) Q (S_{U}) \subseteq Q (U) \\ (2) \begin{matrix} {\tilde{w}}^{2} (S_{U}) = \sum_{m = 1}^{q (U)} w (S_{r_{m} (U)}) \cdot r_{m} \\ = \sum_{m = 1}^{q (S_{U})} w (S_{k_{m} (U)}) \cdot k_{m} \end{matrix} \end{matrix}$ (10)

$(3) \begin{matrix} {\tilde{w}}^{3} (S_{U}) = \sum_{m = 1}^{q (U)} w (S_{[U]_{r_{m}}}) \cdot (r_{m} - r_{m - 1}) \\ = \sum_{k = 1}^{q (S_{U})} w (S_{[U]_{k_{m}}}) \cdot (k_{m} - k_{m - 1}) \end{matrix}$ (11)

Remark 3.3. When Hukuhara fuzzy game (N, w) is restricted to crisp game (N, v), the generalized integral game $(N, \tilde{w})$ is transferred to the fuzzy coalition game $(N, \tilde{v})$ , i.e., $(1) {\tilde{v}}^{1} (S_{U}) = \sum_{T \subseteq N} \prod_{i \in T} S_{U} (i) \prod_{i \notin T} (1 - S_{u} (i)) \cdot v (T)$ (12) $(2) {\tilde{v}}^{2} (S_{U}) = \sum_{m = 1}^{q (S_{U})} v (S_{k_{m} (U)}) \cdot k_{m}$ (13) $(3) {\tilde{v}}^{3} (S_{U}) = \sum_{k = 1}^{q (S_{U})} v (S_{[U]_{k_{m}}}) \cdot (k_{m} - k_{m - 1})$ (14)

Hence, it is not hard to see that the generalized integral games in Equation (9) are Owen’s multilinear extensions game [9] the generalized integral games in Equation (10) are the games with proportional value given by Butnariu [4], and the ones in Equation (11) are the games with Choquet integral form proposed by Tsurumi et al. in [6]. It is obvious that the generalized integral games introduced in this paper can be seen as a generalized form of games with fuzzy coalitions.

Ref. [13] has also defined a kind of game with fuzzy coalition and fuzzy payoff value, named fuzzy game with “indeterminate integral form”.

Definition 3.3. [13] Let any U ∈ F (N). Then a fuzzy game $(N, \tilde{w})$ is said to be fuzzy game with “indeterminate integral form” if and if only it satisfies: $\tilde{w} (K) = \sum_{l = 1}^{q (U)} w ([U]_{h_{l}}) \cdot (h_{l} - h_{l - 1}),$ (15)

where w ∈ G_H (N) and h_o = 0. The set of all fuzzy game with indeterminate integral form is represented by G_F (N).

It is not hard to see that the fuzzy game with indeterminate integral form in Equation (15) is just the generalized integral games defined in Equation (14). Hence, the fuzzy game with indeterminate integral form is a special kind of generalized integral games, so G_F (N) ⊆ G_L (N).

We define the fuzzy characteristic game introduced by the generalized integral game $\tilde{w} \in G_{F} (N)$ in the fuzzy coalition U ∈ F (N) as follows.

Definition 3.4. The Hukuhara fuzzy game introduced by the generalized integral game $\tilde{w} \in G_{F} (N)$ in the fuzzy coalition U ∈ F (N) is (N, w_U) where $w_{U} : P (N) \to R_{+}$ is defined by $w_{U} (S) = \tilde{w} (S_{U})$ , ∀S ∈ P (N).

Theorem 3.2.Let $\tilde{w} \in G_{L} (N)$ andU ∈ F (N). If the associated Hukuhara fuzzy gamew ∈ G_H (N) is supperadditive, then the generalized integral game $\tilde{w} \in G_{L} (N)$ is supperadditive.

Proof. Let ∀U, K ∈ F (N) such that U∩ K = ∅. Assume w ∈ G_H (N), then we have $\begin{matrix} \begin{matrix} \tilde{w} (K \cup U) = \sum_{T \subseteq N} (a_{T}^{K \cup U} \cdot w (T \cap Supp (K \cup U))) \\ = \sum_{T \subseteq N} (a_{T}^{K \cup U} \cdot w ((T \cap Supp (K)) \cup (T \cap Supp (U)))) \end{matrix} \\ \begin{matrix} \geq \sum_{T \subseteq N} (a_{T}^{K \cup U} \cdot w (T \cap Supp (K))) \\ + \sum_{T \subseteq N} (a_{T}^{K \cup U} \cdot w (T \cap Supp (U))) \\ = \tilde{w} (K) + \tilde{w} (U), \end{matrix} \end{matrix}$

where in the last line we use Remark 3.1. Hence, $\tilde{w} \in G_{L} (N)$ is supperadditive.

Example 3.1. Let N = {1, 2, 3}, w be a Hukuhara fuzzy game on N, whose generalized integral game is denoted by triangular fuzzy number as shown in Table 1.

Table 1

Fuzzy characteristic function for Hukuhara fuzzy game (N, w)

{T}	{1}	{2}	{3}	{1,2}	{1,3}	{2,3}	{1,2,3}
w(T)	(1, 1.2, 1.4)	(2, 2.2, 2.4)	(3, 3.2, 3.4)	(4, 4.6, 5.2)	(6, 6.8, 7.6)	(8, 9, 10)	(14, 16, 18)

Let U (1) =0.1 and U (2) = U (3) =0.2. From Equation (9), Equation (10) and Equation (11), we get generalized integral game $(N, {\tilde{w}}^{1})$ , $(N, {\tilde{w}}^{2})$ and $(N, {\tilde{w}}^{3})$ as shown in Table 2. Also, we define a new generalized integral game $(N, {\tilde{w}}^{4})$ by $(a_{T}^{U})_{4} = 0.5 (a_{T}^{U})_{2} + 0.5 (a_{T}^{U})_{3}$ , for any ∀T ⊆ N.

Table 2

The coefficients of generalized integral game $(N, {\tilde{w}}^{1})$ , $(N, {\tilde{w}}^{2})$ , $(N, {\tilde{w}}^{3})$ and $(N, {\tilde{w}}^{4})$

T	$(a_{T}^{U})_{1}$	$(a_{T}^{U})_{2}$	$(a_{T}^{U})_{3}$	$(a_{T}^{U})_{4}$
{1}	0.064	0.1	0	0.05
{2}	0.144	0	0	0
{3}	0.144	0	0	0
{1,2}	0.016	0	0	0
{1,3}	0.016	0	0	0
{2,3}	0.036	0.2	0.1	0.15
{1,2,3}	0.004	0	0.1	0.05

It is obvious that $(N, {\tilde{w}}^{1})$ , $(N, {\tilde{w}}^{2})$ , $(N, {\tilde{w}}^{3})$ and $(N, {\tilde{w}}^{4})$ is supperadditive.

Obviously, the generalized integral game can be seen as a generalized form of games with fuzzy coalitions, which contain not only the Choquet integral game, the games with proportional value, and Owen’s multilinear extensions game, but also some new kinds of generalized integral game such as the game $(N, {\tilde{w}}^{4})$ in Example 3.1.

4 The fuzzy Shapley value for the generalized integral game

As mentioned above, the fuzzy Shapley axiom system in Definition 2.9 can be applied to all kinds of cooperative game involving crisp game, fuzzy coalition game, fuzzy characteristic game and game with both fuzzy coalition and payoff value. Hence, we do not need to propose new fuzzy Shapley value for generalized integral game, and we only need to give the explicit form of fuzzy Shapley value for generalized integral game. Preparatory to the proof, we give the following lemma.

Lemma 4.1.[13] Letw ∈ G_H (N). Define a function $f : G_{H} (N) \to (R_{+}^{n})^{P (N)}$ by $\begin{matrix} f_{i} (w) (S) \\ = {\begin{matrix} \sum_{T \subseteq S : i \in T} β (| T |; | S |) \cdot [w (T) -_{H} w (T ∖ {i})] & if i \in S, \\ 0 & otherwise, \end{matrix} \end{matrix}$ (16)

where β (|T| ; |S|) = (|T|-1) ! (|S| - |T|) !/ - |S| !, then f is the unique fuzzy Shapley function on w ∈ G_H (N).

Theorem 4.1. Define a function $\tilde{f} : G_{L} (N) \to (R_{+}^{n})^{F (U)}$ by ${\tilde{f}}_{i} (\tilde{w}) (U) = \sum_{T \subseteq N : i \in T} γ_{T} \cdot {\tilde{w} (T_{U}) -_{H} \tilde{w} ((T ∖ {i})_{U})}$ (17)

for ∀i ∈ N, where γ_T = (|T|-1) ! (N - |T|) !/ - n !. Then the function $\tilde{f}$ is the Shapley value on G_L (N).

Proof. We shall prove that the function f given in Equation (17) satisfies Axiom F₁-F₄.

Axiom F₁. It is obvious that the function $\tilde{f}$ can be seen as the Shapley value for the characteristic fuzzy game (N, w_U) induced by the generalized integral game $\tilde{w}$ in fuzzy coalition U. By the efficiency property of the fuzzy Shapley value, we obtain that $\sum_{i \in N} {\tilde{f}}_{i} (\tilde{w}) (U) = w_{U} (N) = \tilde{w} (N_{U}) = \tilde{w} (U) .$

Next, consider any i ∉ Supp (U). Then for any T ⊆ N, we get that $\tilde{w} (T_{U}) -_{H} \tilde{w} ((T ∖ {i})_{U}) = 0 .$

Hence, we get that ${\tilde{f}}_{i} (\tilde{w}) (U) = 0$ .

Axiom F₂. Let $K \in FC (U | \tilde{w})$ . Given any T ⊆ N, if i ∈ T, then we get that (T_U ∩ K) (i) = K (i), otherwise (T_U ∩ K) (i) =0. Thus, we obtain that T_U ∩ K = T_K. Consider any i ∈ N, the following holds: $\begin{matrix} {\tilde{f}}_{i} (\tilde{w}) (U) \\ = \sum_{T \subseteq N : i \in T} γ_{T} \cdot {\tilde{w} (T_{U}) -_{H} \tilde{w} ((T ∖ {i})_{U})} \\ = \sum_{T \subseteq N : i \in T} γ_{T} \cdot {\tilde{w} (T_{U} \cap K) -_{H} \tilde{w} ((T ∖ {i})_{U} \cap K)} \\ = \sum_{T \subseteq N : i \in T} γ_{T} \cdot {\tilde{w} (T_{K}) -_{H} \tilde{w} ((T ∖ {i})_{K})} = {\tilde{f}}_{i} (\tilde{w}) (K) . \end{matrix}$

Axiom F₃. Let $U_{ij}^{U} \in FC (U | \tilde{w})$ and let $\tilde{w} (K) = \tilde{w} (p_{ij})$ for any $K \in F (U_{ij}^{U})$ . Note that $U_{ij}^{U} (i) = U_{ij}^{U} (j)$ . Consider any T ⊆ N ∖ {i, j}, we obtain that $\tilde{w} (T_{U_{ij}^{U}} \cup {i}_{U_{ij}^{U}}) = \tilde{w} (T_{U_{ij}^{U}} \cup {j}_{U_{ij}^{U}})$ . Thus, thefollowing holds: $\begin{matrix} {\tilde{f}}_{i} (\tilde{w}) (U_{ij}^{U}) \\ = \sum_{T \subseteq N : i \in T} γ_{T} \cdot {\tilde{w} (T_{U_{ij}^{U}}) -_{H} \tilde{w} ((T ∖ {i})_{U_{ij}^{U}})} \\ = \sum_{T \subseteq N : j \in T} γ_{T} \cdot {\tilde{w} (T_{U_{ij}^{U}}) -_{H} \tilde{w} ((T ∖ {j})_{U_{ij}^{U}})} \\ = {\tilde{f}}_{j} (\tilde{w}) (U_{ij}^{U}) . \end{matrix}$

From Axiom F₂ proved above, we have that ${\tilde{f}}_{i} (\tilde{w}) (U) = {\tilde{f}}_{i} (\tilde{w}) (U_{ij}^{U})$ , ∀i ∈ N. Consequently, we get that ${\tilde{f}}_{i} (\tilde{w}) (U) = {\tilde{f}}_{i} (\tilde{w}) (U_{ij}^{U}) = {\tilde{f}}_{j} (\tilde{w}) (U_{ij}^{U}) = {\tilde{f}}_{j} (\tilde{w}) (U) .$

Axiom F₄. Let ${\tilde{w}}_{1}, {\tilde{w}}_{2} \in G_{L} (N)$ , For any U, K ∈ F (N) satisfying U∩ K = ∅, we obtain that $\begin{matrix} ({\tilde{w}}_{1} + {\tilde{w}}_{2}) (U \cup K) \\ = {\tilde{w}}_{1} (U \cup K) + {\tilde{w}}_{2} (U \cup K) \\ \geq {\tilde{w}}_{1} (U) + {\tilde{w}}_{2} (U) + {\tilde{w}}_{1} (K) + {\tilde{w}}_{2} (K) \\ = ({\tilde{w}}_{1} + {\tilde{w}}_{2}) (U) + ({\tilde{w}}_{1} + {\tilde{w}}_{2}) (K), \end{matrix}$

which implies that ${\tilde{w}}_{1} + {\tilde{w}}_{2} \in G_{L} (N)$ . Then the following holds: $\begin{matrix} {\tilde{f}}_{i} ({\tilde{w}}_{1} + {\tilde{w}}_{2}) (U) \\ = \sum_{T \subseteq N : i \in T} γ_{T} \cdot {({\tilde{w}}_{1} + {\tilde{w}}_{2}) (T_{U}) -_{H} ({\tilde{w}}_{1} + {\tilde{w}}_{2}) ((T ∖ {i})_{U})} \\ = \sum_{T \subseteq N : i \in T} γ_{T} \cdot {\begin{matrix} ({\tilde{w}}_{1} (T_{U}) -_{H} {\tilde{w}}_{1} ((T ∖ {i})_{U})) \\ + ({\tilde{w}}_{2} (T_{U}) -_{H} {\tilde{w}}_{2} ((T ∖ {i})_{U})) \end{matrix}} \\ = {\tilde{f}}_{i} ({\tilde{w}}_{1}) (U) + {\tilde{f}}_{i} ({\tilde{w}}_{2}) (U) . \end{matrix}$

The proof is completed.

Yu et al. [13] showed that the explicit form of Shapley value on fuzzy game with indeterminate integral form, which is an extension of fuzzy coalition game with Choquet integral form in Ref. [6]. In other words, the Shapley value for the games with Choquet integral form can be written in Choquet integral form of crisp Shapley value. Hence, because the generalized integral game introduced in this paper can be seen as a generalized form of games with fuzzy coalitions, which contains the fuzzy game with indeterminate integral form in Ref. [13], it is an interesting problem about the relationship between the generalized integral game and the Shapley value for the associated crisp games? Next, we attempt to solve thisquestion.

Let w ∈ G_H (N). Given any T ⊆ N, we define the game $(N^{T}, \bar{w})$ by $\bar{w} (S) = w (T \cap S), \forall S \subseteq N .$

Firstly, we introduce the following lemma proved by Li et al. [7].

Lemma 4.2.[7] Let (N, v) be crisp cooperative game, and f(v) be the Shapley function. For anyT ⊆ N, we define $(N^{T}, \bar{v})$ by $\bar{v} (S) = v (T \cap S)$ , for ∀S ⊆ N, then $f_{i} (v) (T) = f_{i} (\bar{v}) (N^{T})$ , ∀i ∈ N.

Lemma 4.3.Letw ∈ G_H (N) be Hukuhara fuzzy game, and f is the fuzzy Shapley function onw. Then for anyT ⊆ N, $f_{i} (w) (T) = f_{i} (\bar{w}) (N^{T})$ , ∀i ∈ N.

Proof. If i ∉ T, then $\begin{matrix} \bar{w} (S) = w (T \cap S) \\ = w (T \cap (S \cup {i})) = \bar{w} (S \cup {i}) \end{matrix}$ Hence, $f_{i} (\bar{w}) (N^{T}) = f_{i} (w) (T) = 0$ . On the other hand, if i ∈ T, then for any λ ∈ (0, 1], by Proposition 2.1, we have that $\begin{matrix} [f_{i} (w) (T)]_{λ} \\ = [\sum_{M \subseteq T ∖ {i}} β (| T |; | S |) \cdot [w (M \cup {i}) -_{H} w (M)]]_{λ} \\ = \sum_{M \subseteq T ∖ {i}} \begin{matrix} β (| T |; | S |) \times \\ [\begin{matrix} w_{λ}^{L} (M \cup {i}) -_{H} w_{λ}^{L} (M), \\ w_{λ}^{R} (M \cup {i}) -_{H} w_{λ}^{R} (M) \end{matrix}] \end{matrix} \\ = [f_{i} (w_{λ}^{L}) (T), f_{i} (w_{λ}^{R}) (T)] \end{matrix}$

where in the last line, the Hukuhara-difference exsit because of w ∈ G_H (N).

By Lemma 4.2, for any λ ∈ (0, 1], we further have that $\begin{matrix} [f_{i} (w) (T)]_{λ} = [f_{i} (w_{λ}^{L}) (T), f_{i} (w_{λ}^{R}) (T)] \\ = [f_{i} ({\bar{w}}_{λ}^{L}) (N^{T}), f_{i} ({\bar{w}}_{λ}^{R}) (N^{T})] \end{matrix}$

which means $f_{i} (w) (T) = f_{i} (\bar{w}) (N^{T})$ . The proof is completed.

Theorem 4.2.Let $\tilde{w} \in G_{L} (N)$ andU ∈ F (N). Then the fuzzy Shapley value f on $\tilde{w}$ in U can be represented by the Shapley value of the associated fuzzy characteristic games as follow: ${\tilde{f}}_{i} (\tilde{w}) (U) = \sum_{T \subseteq N} (a_{T}^{U} \cdot f_{i} (w) (T \cap Supp (U)))$ (18)

for ∀i ∈ N.

Proof. By Theorem 4.1 and Remark 3.1, we get that $\begin{matrix} {\tilde{f}}_{i} (\tilde{w}) (U) \\ = \sum_{S \subseteq N : i \in S} γ_{S} \cdot {\tilde{w} (S_{U}) -_{H} \tilde{w} ((S ∖ {i})_{U})} \\ = \sum_{S \subseteq N : i \in S} γ_{S} \times {\begin{matrix} \sum_{T \subseteq N} (a_{T}^{U} \cdot w (T \cap Supp (S_{U}))) \\ -_{H} \sum_{T \subseteq N} (a_{T}^{U} \cdot w (T \cap Supp ((S ∖ {i})_{U}))) \end{matrix}} \end{matrix}$ $\begin{matrix} = \sum_{T \subseteq N} (a_{T}^{U} \cdot \sum_{S \subseteq N : i \in S} γ_{S} \cdot {\begin{matrix} w (T \cap Supp (S_{U})) \\ -_{H} w (T \cap Supp ((S ∖ {i})_{U})) \end{matrix}}) \\ = \sum_{T \subseteq N} (a_{T}^{U} \cdot f_{i} (\bar{w}) (N^{T \cap Supp (U)})) \\ = \sum_{T \subseteq N} (a_{T}^{U} \cdot f_{i} (w) (T \cap Supp (U))), \end{matrix}$

where in the last line we use the Lemma 4.3. The proof is completed.

Remark 4.1. Let the game $\tilde{w} \in G_{L} (N)$ be defined by Equation (15), then the fuzzy Shapley value is as follows, ${\tilde{f}}_{i} (\tilde{w}) (U) = \sum_{l = 1}^{p (U)} f_{i} (w) ([U]_{h_{l}}) \cdot (h_{l} - h_{l - 1})$ (19)

It is not hard to see that the Shapley value in Equation (19) coincides with the results in Ref. [13].

Corollary 4.1.The Shapley function defined by Equation (18) is an imputation of a fuzzy game $(N, \tilde{w})$ .

Proof. By Theorem 4.1, it is obvious that $\sum_{i \in Supp (U)} {\tilde{f}}_{i} (U) = \tilde{w} (U)$ . For any U ∈ F (N), if i ∉ Supp (U), then f_i (w) (T ∩ Supp (U)) =0 because f (w) is an imputation on (N, w). Hence, ${\tilde{f}}_{i} (\tilde{w}) (U) = 0$ . Next, for ∀i ∈ Supp (U), we will prove that ${\tilde{f}}_{i} (U) \geq \tilde{w} ({i}_{U})$ . $\begin{matrix} {\tilde{f}}_{i} (\tilde{w}) (U) = \sum_{T \subseteq N} (a_{T}^{U} \cdot f_{i} (w) (T \cap Supp (U))) \\ \geq \sum_{T \subseteq N} (a_{T}^{U} \cdot w ({i})) = x_{i} (U) = \tilde{w} ({i}_{U}) \end{matrix}$

The proof is completed.

Example 4.1. Consider the game in Example 3.1. Let the coefficients of generalized integral game be shown in Table 4.1, U (1) =0.1 and U (2) = U (3) =0.2. Now we compute fuzzy Shapley value for four kinds of generalized integral games.

Table 3

The Shapley value of the associated fuzzy characteristic games (T, w)

T	f₁ (w) (T)	f₂ (w) (T)	f₃ (w) (T)
{1}	(1, 1.2, 1.4)	0	0
{2}	0	(2, 2.2, 2.4)	0
{3}	0	0	(3, 3.2, 3.4)
{1,2}	(1.5, 1.8, 2.6)	(2.5, 2.8, 3.1)	0
{1,3}	(2, 2.4, 2.9)	0	(4, 4.4, 4.7)
{2,3}	0	(3.5, 3.9, 4.5)	(4.5, 5.1, 5.5)
{1,2,3}	(2.833, 3.367, 3.900)	(4.333, 5.322, 5.600)	(6.834, 7.311, 8.500)

Table 4

The fuzzy Shapley value of generalized integral game

i	$(N, {\tilde{w}}^{1})$	$(N, {\tilde{w}}^{2})$	$(N, {\tilde{w}}^{3})$	$(N, {\tilde{w}}^{4})$
1	(0.143, 0.157, 0.192)	(0.1, 0 . 12, 0.14)	(0.283, 0.337, 0.390)	(0.192, 0.229, 0.265)
2	(0.471, 0.527, 0.580)	(0.7, 0 . 78, 0.9)	(0.633, 0.752, 0.800)	(0.667, 0.766, 0.850)
3	(0.674, 0.741, 1.319)	(0.9, 1 . 02, 1.1)	(1.111, 1.411, 1.61)	(1.091, 1.215, 1.355)

(1) From Table 2, we get the payoff value for generalized integral game as follows, $\begin{matrix} {\tilde{w}}^{1} (U) \\ = 0.064 w ({1}) + 0.144 [w ({2}) + w ({3})] \\ + 0.016 [w ({1, 2}) + w ({1, 3})] \\ + 0.036 w ({2, 3}) + 0.004 w ({1, 2, 3}) \\ = (1.288, 1.4248, 2.08128) \\ \approx (1.288, 1.425, 2.081), \end{matrix}$ $\begin{matrix} {\tilde{w}}^{2} (U) = 0.1 w ({1}) + 0.2 w ({2, 3}) \\ = (1.7, 1.92, 2.14) \end{matrix},$ $\begin{matrix} {\tilde{w}}^{3} (U) = 0.1 w ({1, 2, 3}) + 0.1 w ({2, 3}) \\ = (2.2, 2.5, 2.8), \end{matrix}$ $\begin{matrix} {\tilde{w}}^{4} (U) = 0.05 w ({1, 2, 3}) + 0.15 w ({2, 3}) \\ + 0.05 w ({1}) \\ = (1.95, 2.21, 2.47) . \end{matrix}$

(2) We need compute the Shapley value of the associated Hukuhara fuzzy games (N, w) before the fuzzy Shapley value for the generalized integral game. By Equation (16), we get that $\begin{matrix} f_{1} (w) (N) \\ = \frac{1}{6} w ({1}) + \frac{1}{3} {w ({1, 2}) -_{H} w ({2})} + \\ \frac{1}{3} {w ({1, 3}) -_{H} w ({3})} + \frac{1}{6} {w ({1, 2, 3}) -_{H} w ({2, 3})} \\ = (2.833, 3.367, 3.900) \end{matrix}$

Similarly, we get that $f_{2} (w) (N) = (4.333, 5.322, 5.600),$ $f_{3} (w) (N) = (6.834, 7.311, 8.500) .$

In the similar way, we compute Shapley value of the (T, w), for any T ⊆ N as shown in Table 3.

(3) By Equation (18), we compute the fuzzy Shapley value for $(N, {\tilde{w}}^{1})$ based the coefficients of generalized integral fuzzy game. $\begin{matrix} {\tilde{f}}_{1} ({\tilde{w}}^{1}) (U) \\ = 0.064 f_{1} (w) ({1}) + 0.004 f_{1} (w) (N) \\ + 0.016 [f_{1} (w) ({1, 2}) + f_{1} (w) ({1, 3})] \\ = (0.143, 0.157, 0.192), \end{matrix}$

Similarly, we get that $\begin{matrix} {\tilde{f}}_{2} ({\tilde{w}}^{1}) (U) = \\ 0.144 f_{2} (w) ({2}) + 0.016 f_{1} (w) ({1, 2}) \\ + 0.036 f_{2} (w) ({2, 3}) + 0.004 f_{2} (w) (N) \\ = (0.471, 0.527, 0.580), \end{matrix}$ $\begin{matrix} {\tilde{f}}_{3} ({\tilde{w}}^{1}) (U) \\ = 0.144 f_{3} (w) ({3}) + 0.016 f_{3} (w) ({1, 3}) \\ + 0.036 f_{3} (w) ({2, 3}) + 0.004 f_{3} (w) (N) \\ = (0.674, 0.741, 1.319) . \end{matrix}$

In the similar way, we compute the fuzzy Shapley value for the generalized integral game $(N, {\tilde{w}}^{2})$ , $(N, {\tilde{w}}^{3})$ and $(N, {\tilde{w}}^{4})$ as shown in Table 4.

Obviously, we can see that the generalized integral fuzzy game use the same formula to express many kinds of fuzzy games, which is defined by the characteristic fuzzy game. The only difference between this kinds of fuzzy games lies in the coefficients before Hukuhara fuzzy game. Hence, it is also proved that the Shapley value of the generalized integral game may be expressed as the same formula. Just as showed in the above example, if the generalized integral fuzzy game’s coefficients is Equation (9), then the generalized integral fuzzy games an extension of multilinear extension defined by Owen, as so on. However, no matter what kind of generalized integral fuzzy game, it Shapley value can be computed by Equation (17).

5 Conclusions

A new fuzzy game named generalized integral game is proposed, which contains the fuzzy characteristic game, Owen’s multilinear extension, Butnariu’s games in propositional value and Tsurumiet al.’s game with Choquet integral. The proposed fuzzy game is generalized form of fuzzy game in Ref. [13], which is further extension of fuzzycharacteristic game and the fuzzy coalition game. Also, the relationship between the fuzzy characteristic solution concepts and the fuzzy solution for the generalized integral games is shown, respectively. Although the Shapley value of the generalized integral game can be represented by the same formula, but the coefficients of generalized integral game is different. Hence, by controlling the coefficients of generalized integral game, we could get different kinds of fuzzy games, and their relationship of fuzzy Shapley also lie in the coefficients before the associated fuzzy characteristic game.

Footnotes

Acknowledgments

The authors gratefully thank the associate editor and referees for their valuable comments, which have much improved the paper. This work was supported by the youth fund project for Humanities and social sciences research of Ministry of Education(Study on Coordination profit allocation strategy of supply chain under fuzzy information, NO:17YJC630203), the National Natural Science Foundation of China (71771025, 71371030, 71401003, 71271029), the Natural Science Foundation of Beijing under the Grant (9152002), and Scientific and technological cooperative research team construction project of Beijing Wuzi University(2017GG06).

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