A Fuzzy Owen function on games with coalition structure and fuzzy coalitions

Abstract

In this paper, a game with coalition structure and fuzzy coalitions is introduced, which can be regarded as an extension of the game with coalition structure. Firstly, an Owen function is defined. Then, the unique explicit form of the Owen function is given by extending the Owen value. Secondly, as an extension of Owen function, a fuzzy Owen function is proposed. In addition, an explicit form of the fuzzy Owen function is given by considering the fuzzy intermediate game with Owen extension form. Finally, an illustrate example is provided to illustrate the allocation method of gains.

Keywords

Cooperative games fuzzy coalition coalition structure Owen function

1 Introduction

Coalition structure is defined as a partition of the set of players into disjoint coalitions that are called a priori unions. A priori unions represent firms or societies formed as a consequence of prior agreements among the players. Owen value is one of the successful solution concepts for games with coalition structure that can be regarded as an expansion of the Shapley value for the situation when a coalition structure is involved. It is introduced by Owen [19] and based on the assumption that there are two bargaining levels: (a) between a priori unions; (b) between players within each a priori union.

Owen value has been widely studied in literatures. Hart and Kurz [12] provided an alternative characterization of the Owen value for the case in which there was an infinite universe of players. Peleg [20] gave an alternative characterization of the Owen value. In this case coalitional symmetry was removed from Owen’s characterization and the intermediate game property was added. Vázquez-Brage et al. [23] analyzed the determination of aircraft landing fees by using the Owen value and provided an axiomatic characterization of the Owen value by balanced contributions and quotient game property. Hamiache [11] provided associated consistency axioms and other axioms such as positivity, efficiency to characterize the Owen value. Khmelnitskaya and Yanovskaya [13] characterized Owen value without the additivity axiom similarly as it was done by Young for the Shapley value. Alibizuri [2] employed a simple axiom to provide three characterizations of the Owen value without efficiency. Bergantiños and Vidal-Puga [5] described a value which coincides with the Owen value for transferable utility (TU) games with coalition structure. Alibizuri et al. [1] generalized the Owen value to coalition configuration.

For games with coalition structure, it is assumed that each player participates a coalition with two possibilitites: non-cooperation (participation level 0) and full cooperation (participation level 1). However, more freedom may be given to players by considering fuzzy cooperation, that is participation at any level between noncooperation and full cooperation. For example, in a class of production games, partial participation in a coalition means to offer a part of the resources while full participation means to offer all the resources.

With regard to the transfer utility games, fuzzy coalition is introduced by Aubin [3, 4] together with his solution concept of core. In the meantime, many solution concepts have been developed in the framework of games with fuzzy coalitions. Butnariu [8, 9] defined a Shapley value and showed the explicit form of the Shapley function on a limited class of fuzzy games. Butnariu and Kroupa [10] extended the fuzzy games with proportional values to fuzzy games with weighted function, and the corresponding Shapley function also is given. Tsurumi et al. [22] defined new Shapley axioms and a new class of fuzzy games with Choquet integral form. This class of fuzzy games is both monotone nondecreasing and continuous with respect to players participation. Sakawa and Nishizaki [21] introduced lexicographical solution as a solution concept for transfer utility games and extend it to fuzzy games. Further, Molina and Tejada [16] analyzed the lexicographical solution for fuzzy games and studied its properties and obtained a characterization. Branzei et al. [6, 7] showed a detailed characterization of the class of convex fuzzy games and studied the related solution concepts. Li and Zhang [14] proposed a simplified expression of the Shapley function for games with fuzzy coalitions. Yu and Zhang [24] studied the fuzzy core of games with fuzzy coalition, which can be regarded as the generalization of crisp core. Liu and Liu [15] introduced the concept of fuzzy bargaining sets and proved existence theorems for fuzzy bargaining sets.

Fuzzy coalitions in games with coalition structure is different from the games with fuzzy coalition, on one hand, the participation level of player and a priori unions should both be considered in the first scenario, on the other hand, the relationship between players and a priori unions should also be considered in the first scenario because the participation level of a priori unions is influenced by players. This paper has two aims. Firstly, it will give the definition of a fuzzy Owen function on games with coalition structure and fuzzy coalitions, which will be called fuzzy games with coalition structure for the sake of simplicity. Secondly, it will give an explicit form of the fuzzy Owen function by considering the fuzzy intermediate game with Owen extension form.

The structure of the paper is as follows. In Section 2, some basic definitions are introduced. Then, a Owen value function is discussed. In Section 3, the definitions of symmetric and null player are extended to fuzzy games with coalition structure. A fuzzy Owen function is studied. Furthermore, the explicit form of the fuzzy Owen function is given. In Section 4, an applicable illustrative example is provided. Finally, some conclusions will be discussed in Section 5.

2 Three classes of cooperative games and the Owen function

2.1 Crisp cooperative games with coalition structure and Owen function

A crisp cooperative game with transferable utility game is a pair (N, v), where N is a finite set (the set of players) and v is the characteristic function, defined from 𝒫 (N) = {S : S ⊆ N} to $ℝ_{+}$ and satisfying v (∅) =0. For any S ∈ 𝒫 (N), v (S) represents an evaluation of the potential utility of coalition S. We mainly discuss the supperadditive crisp games in this paper, i.e., $\begin{matrix} v (S \cup T) \geq v (S) + v (T), \forall S, T \in 𝒫 (N), \\ s . t . S \cap T = \emptyset . \end{matrix}$

The set of all supperadditive crisp games on N is denoted by G^N. A coalition structure ℬ is a finite partition ℬ = {B₁, B₂, ⋯, B_m} of N, i.e., $⋃_{i = 1}^{m} B_{i} = N$ and B_i∩ B_j = ∅ for i ≠ j. M = {1, 2, ⋯, m} is the set of subscripts of elements in ℬ. The set of all coalition structures of N is denoted by C^N. Let GC^N = G^N × C^N denote all supperadditive crisp games with coalition structures.

Players i and j are symmetric with respect to the game (N, v, ℬ) ∈ GC^N if they make the same marginal contribution to any coalition not containing them, i.e., v (S ∪ {i}) = v (S ∪ {j}) for any S ⊆ N ∖ {i, j}. Player i is a null player if v (S ∪ {i}) = v (S) for any S ⊆ N ∖ {i}. Let (M, v_ℬ) be the intermediate game induced by (N, v, ℬ) ∈ GC^N by considering each coalition B_k of ℬ as a player. (M, v_ℬ) is defined as v_ℬ (K) = v (⋃ _k∈KB_k) (∀ K ⊆ M).

An Owen function based on GC^N is defined as follows.

Definition 2.1. A function $φ : {GC}^{N} \to (ℝ_{+}^{n})^{𝒫 (M)}$ is said to be an Owen function if it satisfies the following five axioms.

Axiom C₁. Let (N, v, ℬ) ∈ GC^N and W ∈ 𝒫 (M). Then $\begin{matrix} \sum_{i \in N} φ_{i} (N, v, ℬ) (W) = v (⋃_{k \in W} B_{k}), \\ φ_{i} (N, v, ℬ) (W) = 0 \forall i \notin ⋃_{k \in W} B_{k}, \end{matrix}$ where φ_i (N, v, ℬ) (W) is the ith element of $φ (N, v, ℬ) (W) \in ℝ_{+}^{n}$ .

Axiom C₂. Let (N, v, ℬ) ∈ GC^N and W ∈ 𝒫 (M). If i ∈ B_t (t ∈ W) is a null player, then $φ_{i} (N, v, ℬ) (W) = 0 .$

Axiom C₃. Let (N, v, ℬ) ∈ GC^N and W ∈ 𝒫 (M). If i, j ∈ B_p (p ∈ W) are symmetric, then $φ_{i} (N, v, ℬ) (W) = φ_{j} (N, v, ℬ) (W) .$

Axiom C₄. Let (N, v, ℬ) ∈ GC^N and W ∈ 𝒫 (M). For any p, q ∈ W, if p and q are symmetric with respect to the intermediate game v_ℬ, then $\sum_{i \in B_{p}} φ_{i} (N, v, ℬ) (W) = \sum_{i \in B_{q}} φ_{i} (N, v, ℬ) (W) .$

Axiom C₅. For any (N, v₁, ℬ), (N, v₂, ℬ) ∈ GC^N, and α, $β \in ℝ$ , define a game (αv₁ + βv₂) (S) = αv₁ (S) + βv₂ (S) for any S ⊆ N. If (N, v₁, ℬ), (N, v₂, ℬ) ∈ GC^N and W ∈ 𝒫 (M), then $\begin{matrix} φ_{i} (N, α v_{1} + β v_{2}, ℬ) (W) \\ = α φ_{i} (N, v_{1}, ℬ) (W) + β φ_{i} (N, v_{2}, ℬ) (W) . \end{matrix}$

The unique explicit form of an Owen function on GC^N is obtained by extending the Owen value.

Theorem 2.1. Define the function $Ow : {GC}^{N} \to (ℝ_{+}^{n})^{𝒫 (M)}$ by

$\begin{matrix} {Ow}_{i} (N, v, ℬ) (W) \\ = {\begin{matrix} \sum_{\binom{K \subseteq W}{t \notin K}} \sum_{\binom{S \subseteq B_{t}}{i \notin S}} σ (| K |; | W |) \cdot δ (| S |; | B_{t} |) \\ \times [v (B \cup {i}) - v (B)] & if i \in ⋃_{k \in W} B_{k}, \\ 0 & otherwise . \end{matrix} \end{matrix}$ (1)

where $σ (| K |; | W |) = \frac{| K |! (| W | - | K | - 1)!}{| W |!}$ , $δ (| S |; | B_{t} |) = \frac{| S |! (| B_{t} | - | S | - 1)!}{| B_{t} |!}$ and B = ⋃ _k∈KB_k ∪ S. Then the function Ow is the unique Owen function.

Proof. Let (N, v, ℬ) ∈ GC^N and W ∈ 𝒫 (M). Then Ow_i (N, v, ℬ) (W) can be obtained in the same manner as the usual Owen value, namely Ow_i (N, v, ℬ) (M), which has been proved in [19].

2.2 Cooperative games with fuzzy coalition

Let N be a finite set of players. A fuzzy coalition s is a fuzzy subset of N, which is a vector s = (s (1), s (2), ⋯, s (n)). The number s (i) ∈ [0, 1] is a constant which denotes the participation level of player i. The set of fuzzy coalitions is denoted by ℱ (N). For a fuzzy coalition s, the support set is denoted by supp (s) = {i ∈ N | s (i) >0}.

Definition 2.2. For any u, s ∈ ℱ (N), u and s are n-dimensional vector. The ordering “≤” between u and s is defined as $u_{i} \leq s_{i}, \forall i \in N .$

For all S ⊆ N, e^s = (s₁, s₂, ⋯, s_n) satisfying s_i = 1 for i ∈ S and s_i = 0 for i ∈ N ∖ S denotes a crisp-like coalition, which corresponding to the situation where the players within S fully cooperate with each other, that is, they have participation level 1, and the players outside S are not involved at all, that is, they have participation level 0. e^N = (1, 1, ⋯, 1) is called the grand coalition. The empty coalition in a fuzzy set is denoted by e^∅ = (0, 0, ⋯, 0). For simplicity, we write eⁱ instead of e^{i}. The fuzzy coalition s_T corresponding to T is denote by s_T = ∑_i∈Ts_ieⁱ. We will denote s_T = ∑_i∈Ts_ieⁱ by t = ∑_i∈Ts_ieⁱ.

Remark 2.1. For any T ⊆ N and a fuzzy coalition s, the fuzzy coalition t = ∑_i∈Ts_ieⁱ denotes a n-dimensional vector t = (t₁, t₂, ⋯, t_n) with t_i = s_i for i ∈ T and t_i = 0 for i ∈ N ∖ T.

For r = ∑_i∈Rs_ieⁱ and t = ∑_i∈Ts_ieⁱ, where R ⊆ N and T ⊆ N, r ∨ t and r ∧ t denote elements of ℱ (N) with the ith coordinate equal to max{r_i, t_i} and min{r_i, t_i}, respectively. The operations ∨ and ∧ play the same role for the fuzzy coalitions as the union and intersection for crisp coalitions. A cooperative game with fuzzy coalition is a pair $(N, \tilde{v})$ in which the characteristic function $\tilde{v} : ℱ (N) \to ℝ_{+}$ such that $\tilde{v} (\emptyset) = 0$ .The supperaddtive fuzzy games is defined as following. $\tilde{v} (s \lor t) \geq \tilde{v} (s) + \tilde{v} (t), \forall s, t \in ℱ (N), s . t . s \land t = e^{\emptyset} .$

The set of all the supperaddtive fuzzy games is denoted by F^N.

In general, it is difficult to identify a characteristic function of a game with fuzzy coalitions in practice. Hence, a fuzzy game is often specified by a crisp game when a decision maker tries to incorporate fuzzy coalitions in a model. Extending the crisp game to the game with fuzzy coalition can be represented by a mapping from the characteristic function of the crisp game to that of the game with fuzzy coalitions, such as the Owen extension [17, 18], Butnariu extension [9] and the Tsurumi et al. extension [22]. The Owen extension is defined as follows.

Definition 2.3. A fuzzy game with fuzzy coalition $(N, \tilde{v}) \in F^{N}$ is said to be a fuzzy game ’with Owen extension form’ if and only if it satisfies $\tilde{v} (s) = \sum_{T \subseteq N} {\prod_{i \in T} s (i) \prod_{i \notin T} (1 - s (i))} \cdot v (T) .$ (2)

The set of all fuzzy games with Owen extension form is denoted by $F_{O}^{N}$ .

There is a one-to-one correspondence between crisp game and a fuzzy game with Owen extension form. For the sake of simplicity, a crisp game corresponding to a game with fuzzy coalitions is called a associated crisp game.

2.3 Fuzzy games with coalition structure

A fuzzy game with coalition structure is a triple $(N, \tilde{v}, ℬ)$ , where $(N, \tilde{v}) \in F^{N}$ and ℬ ∈ C^N. The set of all fuzzy games with coalition structures is denoted by FGC^N.

The crisp game with coalition structure is based on the assumption that all players and a priori unions fully participate in a coalition. But this assumption is not realistic because there are many uncertain factors during negotiation and coalition formation. In many situations, the player in each a priori union do not fully participate in a coalition, but to a certain extent. Therefore, the participation level of the players will influence the participation level of a priori unions. We will use a function to show the influence relationship between players and a priori unions as follows.

Definition 2.4. A function f : ℱ (N) → ℱ (M) is called influence function if for any u, s ∈ ℱ (N) and u ≤ s, then f (u) ≤ f (s), where f (u) = (f (u) ₁, f (u) ₂, ⋯, f (u) _m) and f (s) = (f (s) ₁, f (s) ₂, ⋯, f (s) _m).

Remark 2.2. The influence function f (s) is a non-negative and increasing function on ℱ (N). It expresses the relationship of participation level between players and a priori unions. Suppose that $f (s) = m^{s} = (\sum_{i \in B_{1}} λ_{i}^{1} s (i), \sum_{i \in B_{2}} λ_{i}^{2} s (i), \dots, \sum_{i \in B_{m}} λ_{i}^{m} s (i))$ , where $\sum_{i \in B_{k}} λ_{i}^{k} = 1$ for all i ∈ B_k, k ∈ M. $λ_{i}^{k}$ can be considered as a weighting factor. In actual applications, the weighting factor can be chosen according to the actual situation. For example, let N = {1, 2, 3, 4, 5}, ℬ = {{1, 2}, {3, 4, 5}}, s = (0.5, 0.6, 0.4, 0.8, 0.6). Suppose that $λ_{i}^{k} = λ_{j}^{k} = \frac{1}{| B_{k} |}$ (i ≠ j) for any i, j ∈ B_k and k ∈ M, then f (s) = (0.55, 0.6) = m^s.

Given a fuzzy coalition s ∈ ℱ (N), assume that there exists a fuzzy coalition f (s) = m^s ∈ ℱ (M) corresponding to s. Therefore, similar to the representation of fuzzy coalition in ℱ (N), $t = \sum_{i \in T} m_{i}^{s} e_{M}^{i}$ for any t ∈ ℱ (M) and T ⊆ M, where $e_{M}^{s}$ denotes a crisp-like coalition and is a m-dimensional vector.

For any Q ⊆ M and $q = \sum_{i \in Q} m_{i}^{s} e_{M}^{i}$ . Let $(M, {\tilde{v}}_{ℬ})$ be the fuzzy intermediate game given by ${\tilde{v}}_{ℬ} (q) = \tilde{v} (\sum_{j \in ⋃_{i \in Q} B_{i}} s_{j} e^{j}) .$

The fuzzy intermediate game is induced by the fuzzy game $(N, \tilde{v}, ℬ) \in {FGC}^{N}$ when a priori unions of ℬ are considered as players. The associated crisp intermediate game (M, v_ℬ) ∈ G^M can be written by $v_{ℬ} (Q) = v (supp (\sum_{j \in ⋃_{i \in Q} B_{i}} s_{j} e^{j})) .$ (3)

Remark 2.3. The associated crisp intermediate game (M, v_ℬ) defined by Equation (3) is consistent with the intermediate game defined by Owen [19]. For example, N = {1, 2, 3, 4, 5, 6, 7}, ℬ = {{1, 2}, {3, 4, 5}, {6, 7}}, M = {a, b, c}, where a = {1, 2}, b = {3, 4, 5}, c = {6, 7}. Suppose that s = (0.7, 0.5, 0.6, 0.8, 0, 0.8, 0) ∈ ℱ (N) and m^s = (0.6, 0.7, 0.8). Let Q = {b, c} ⊆ M, then q = (0, 0.7, 0.8) ∈ ℱ (M). On one hand, from Equation (3), v (supp (∑_{j∈⋃_i∈QB_i}s_je^j)) = v ({{3, 4}, {6}}). On the other hand, player 5 and 7 do not participation any coalition because the participation level of player 5 and 7 is zero, then ℬ can be rewritten as ℬ^′ = {{1, 2}, {3, 4}, {6}}. Therefore, the intermediate game defined by Owen can be written as v (⋃ _k∈QB_k) = v ({{3, 4}, {6}}).

Remark 2.4. Let ℬ (R) = {B₁, B₂, ⋯, B_t-1, R, B_t+1, ⋯, B_m} and M_R = {1, 2, ⋯, t - 1, t_R, t + 1, ⋯, m}, where t_R denotes the subscript of R in ℬ (R). Then ${\tilde{v}}_{ℬ (R)} (t_{R}) = \tilde{v} (\sum_{j \in R} s_{j} e^{j})$ .

Definition 2.5. A function $x : ℱ (N) \to ℝ_{+}^{n}$ is said to be an imputation for a fuzzy game $(N, \tilde{v}, ℬ) \in {FGC}^{N}$ in the fuzzy coalition s ∈ ℱ (N) if it satisfies

x_i (s) =0, ∀ i ∉ supp (s),

$\sum_{i \in supp (s)} x_{i} (s) = \tilde{v} (s)$ ,

$x_{i} (s) \geq \tilde{v} (s_{i} e^{i})$ , ∀ i ∈ supp (s),

where x (s) = (x₁ (s), x₂ (s), ⋯, x_n (s)).

3 A Fuzzy Owen function on fuzzy games with coalition structure

3.1 Fuzzy Owen function

In this section, the definition of fuzzy Owen value on classes of fuzzy games with coalition structure is given. In order to give the definition, we firstly extend the definitions of symmetric and null player to fuzzy games with coalition structure.

Let $\bar{s} = {t \in ℝ^{n} | t_{i} = s_{i} or t_{i} = 0}$ be the class of fuzzy coalitions created by s.

Definition 3.1. Let $(N, \tilde{v}, ℬ) \in {FGC}^{N}$ and s ∈ ℱ (N). Players i, j ∈ N are symmetric for $(N, \tilde{v}, ℬ)$ in fuzzy coalition s. if for any $u \in \bar{s - \sum_{k \in {i, j}} s_{k} e^{k}}$ , $\tilde{v} (u \lor {s_{i} e^{i}}) = \tilde{v} (u \lor {s_{j} e^{j}}) .$

Definition 3.2. Let $(N, \tilde{v}, ℬ) \in {FGC}^{N}$ and s ∈ ℱ (N). Players i ∈ N is a null player if for any $u \in \bar{s - s_{i} e^{i}}$ , $\tilde{v} (u \lor s_{i} e^{i}) = \tilde{v} (u) .$

Remark 3.1. It is easy to see that Definitions 3.1 and 3.2 can be applicable to crisp game with coalition structure. In the framework of fuzzy games with coalition structure, while two players in the same a priori unions have different participation, they may also be symmetric. For example, Let $(N, \tilde{v}, ℬ) \in {FGC}^{N}$ , N = {1, 2, 3}, $\tilde{v} (s) = s_{1} + s_{1} s_{2} + 3 s_{2} s_{3} + 3 s_{3}$ (s₁ ≠ s₃) for any s ∈ ℱ (N). Then player 1 and player 3 are symmetric while s₁ = 3s₃. In fact, for any $u \in \bar{s - s_{1} e^{1} - s_{3} e^{3}}$ , u = (0, s₂, 0), then $\begin{matrix} \tilde{v} (u \lor s_{1} e^{1}) = \tilde{v} ((s_{1}, s_{2}, 0)) = s_{1} + s_{1} s_{2}, \\ \tilde{v} (u \lor s_{3} e^{3}) = \tilde{v} ((0, s_{2}, s_{3})) = 3 s_{2} s_{3} + 3 s_{3}, \end{matrix}$

Therefore, $\tilde{v} (u \lor s_{1} e^{1}) = \tilde{v} (u \lor s_{3} e^{3})$ while s₁ = 3s₃.

In the following, we will give the definition of a fuzzy Owen function.

Definition 3.3. A function $φ : {FGC}^{N} \to (ℝ_{+}^{n})^{ℱ (M)}$ is said to be a fuzzy Owen function on FGC^N if it satisfies the following five axioms.

Axiom F₁. Let $(N, \tilde{v}, ℬ) \in {FGC}^{N}$ and s ∈ ℱ (N), then ${\begin{matrix} \sum_{i \in N} φ_{i} (N, \tilde{v}, ℬ) (m^{s}) = v (s) \\ φ_{i} (N, \tilde{v}, ℬ) (m^{s}) = 0 i \notin supp (s), \end{matrix}$ where $φ_{i} (N, \tilde{v}, ℬ) (m^{s})$ is the ith element of $φ (N, \tilde{v}, ℬ) (m^{s}) \in ℝ_{+}^{n}$ .

Axiom F₂. Let $(N, \tilde{v}, ℬ) \in {FGC}^{N}$ . For any i ∈ B_t (t ∈ supp (m^s)), if i is a null player, then $φ_{i} (N, \tilde{v}, ℬ) (m^{s}) = 0 .$

Axiom F₃. Let $(N, \tilde{v}, ℬ) \in {FGC}^{N}$ and s ∈ ℱ (N). If i, j ∈ B_p (p ∈ supp (m^s)) are symmetric with respect to $\tilde{v}$ , then $φ_{i} (N, \tilde{v}, ℬ) (m^{s}) = φ_{j} (N, \tilde{v}, ℬ) (m^{s}) .$

Axiom F₄. Let $(N, \tilde{v}, ℬ) \in {FGC}^{N}$ and s ∈ ℱ (N). For any p, q ∈ supp (m^s), if p and q are symmetric with respect to ${\tilde{v}}_{ℬ}$ , then $\sum_{i \in B_{p}} φ_{i} (N, \tilde{v}, ℬ) (m^{s}) = \sum_{i \in B_{q}} φ_{i} (N, \tilde{v}, ℬ) (m^{s}) .$

Axiom F₅. For any $(N, {\tilde{v}}_{1}, ℬ), (N, {\tilde{v}}_{2}, ℬ) \in {FGC}^{N}$ and α, $β \in ℝ$ , define a game $(α {\tilde{v}}_{1} + β {\tilde{v}}_{2}) (s) = α {\tilde{v}}_{1} (s) + β {\tilde{v}}_{2} (s)$ for any s ∈ ℱ (N), then $\begin{matrix} φ_{i} (N, α {\tilde{v}}_{1} + β {\tilde{v}}_{2}, ℬ) (m^{s}) \\ = φ φ_{i} (N, {\tilde{v}}_{1}, ℬ) (m^{s}) + β φ_{i} (N, {\tilde{v}}_{2}, ℬ) (m^{s}), \forall i \in N . \end{matrix}$

Remark 3.2. If the participation level of players is equal to 1, then the fuzzy Owen function is identical to Definition 2.1. Therefore, the above axioms of the fuzzy Owen function can be applied to crisp games with coalition structure. Hence, it is an extension of the crisp Owen function.

3.2 Fuzzy Owen value with Owen extension

In this section, based on the relationship between the fuzzy Owen value and the crisp Owen value, an explicit form of fuzzy Owen function is given by considering the fuzzy intermediate game $(M, {\tilde{v}}_{ℬ})$ with Owen extension form.

Lemma 3.1. Let (M, v_ℬ) ∈ G^M is the associated crisp game of $(M, {\tilde{v}}_{ℬ}) \in {FG}_{O}^{M}$ . Then for any s ∈ ℱ (N) and $u = \sum_{i \in U} m_{i}^{s} e_{M}^{i}$ (∀ U ⊆ M), $\begin{matrix} {\tilde{v}}_{ℬ} (u) & = & \sum_{R \subseteq supp (m^{s})} {\prod_{i \in R} m^{s} (i) \prod_{i \notin R} (1 - m^{s} (i))} \\ \cdot v_{ℬ} (R \cap U) \end{matrix}$

Proof. Let s ∈ ℱ (N) and $u = \sum_{i \in U} m_{i}^{s} e_{M}^{i}$ (∀U ⊆ M). If i ∉ supp (u), then u (i) =0. If i ∈ supp (u), then i ∈ supp (m^s). Hence, u (i) = m^s (i) for any i ∈ T ⊆ supp (u). $\begin{matrix} {\tilde{v}}_{ℬ} (u) = \sum_{T \subseteq M} {\prod_{i \in T} u (i) \prod_{i \in M ∖ T} (1 - u (i))} \cdot v_{ℬ} (T) \\ = \sum_{T \in 𝒫 (supp (u))} {\prod_{i \in T} u (i) \prod_{i \in M ∖ T} (1 - u (i))} \cdot v_{ℬ} (T) \\ + \sum_{T \in 𝒫 (M) ∖ 𝒫 (supp (u))} {\prod_{i \in T} u (i) \prod_{i \in M ∖ T} (1 - u (i))} \cdot v_{ℬ} (T) \\ = \sum_{T \in 𝒫 (supp (u))} {\prod_{i \in T} u (i) \prod_{i \in M ∖ T} (1 - u (i))} \cdot v_{ℬ} (T) \\ = \sum_{T \in 𝒫 (supp (u))} {\prod_{i \in T} u (i) \prod_{i \in M ∖ supp (u)} (1 - u (i)) \\ \prod_{i \in supp (u) ∖ T} (1 - u (i))} \cdot v_{ℬ} (T) \\ = \sum_{T \in 𝒫 (supp (u))} {\prod_{i \in T} m^{s} (i) \prod_{i \in supp (u) ∖ T} (1 - m^{s} (i))} \cdot v_{ℬ} (T) . \end{matrix}$

For any R ∈ 𝒫 (supp (m^s)), let T = R ∩ U, Q = R \ T. Then T ∈ 𝒫 (U ∩ supp (m^s)), Q ∈ 𝒫 (supp (m^s) \ U), R = T ∪ Q, T∩ Q = ∅, and $\begin{matrix} \sum_{R \subseteq supp (m^{s})} {\prod_{i \in R} m^{s} (i) \prod_{i \notin R} (1 - m^{s} (i))} \cdot v_{ℬ} (R \cap U) \\ = \sum_{T \in 𝒫 (U \cap supp (m^{s}))} \sum_{Q \in 𝒫 (supp (m^{s}) \ U)} {\prod_{i \in T} m^{s} (i) \prod_{i \in Q} m^{s} (i) \\ \prod_{i \in U \ T} (1 - m^{s} (i)) \prod_{i \in supp (m^{s}) \ (U \cup Q)} (1 - m^{s} (i)) \\ \prod_{i \in M \ supp (m^{s})} (1 - m^{s} (i))} \cdot v_{ℬ} (T) \\ = \sum_{T \in 𝒫 (U \cap supp (m^{s}))} ({\prod_{i \in T} m^{s} (i) \prod_{i \in U \ T} (1 - m^{s} (i))} \cdot v_{ℬ} (T) \\ \sum_{Q \in 𝒫 (supp (m^{s}) \ U)} \prod_{i \in Q} m^{s} (i) \prod_{i \in supp (m^{s}) \ (U \cup Q)} (1 - m^{s} (i))) \\ \prod_{i \in M \ supp (m^{s})} (1 - m^{s} (i)) \\ = \sum_{T \in 𝒫 (U \cap supp (m^{s}))} {\prod_{i \in T} m^{s} (i) \prod_{i \in U \ T} (1 - m^{s} (i))} \cdot v_{ℬ} (T) \\ = \sum_{T \in 𝒫 (supp (u))} {\prod_{i \in T} m^{s} (i) \prod_{i \in supp (u) \ T} (1 - m^{s} (i))} \cdot v_{ℬ} (T) \\ = {\tilde{v}}_{ℬ} (u) \end{matrix}$

Let ${FGC}_{O}^{N}$ denote the set of the fuzzy game $(N, \tilde{v}, ℬ) \in {FGC}^{N}$ in which the associate fuzzy intermediate game $(M, {\tilde{v}}_{ℬ}) \in F_{O}^{M}$ .

Remark 3.3. If $(N, \tilde{v}, ℬ) \in {FGC}_{O}^{N}$ , s ∈ ℱ (N) and i ∈ B_t, then the condition (3) in Definition 2.3 can be rewritten as $\begin{matrix} x_{i} (s) & \geq & \tilde{v} (s_{i} e^{i}) = {\tilde{v}}_{ℬ ({i})} (m_{t}^{s} e_{M}^{t}) \\ = & m^{s} (t) v_{ℬ ({i})} (t) = m^{s} (t) v ({i}) . \end{matrix}$

Lemma 3.2. For any s ∈ ℱ (N) and t ∈ M, $\begin{matrix} \sum_{\binom{L \subseteq supp (m^{s})}{t \in L}} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} = m^{s} (t) . \end{matrix}$

Proof. For any s ∈ ℱ (N) and t ∈ M, $\begin{matrix} 1 & = & \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ = & \sum_{\binom{L \subseteq supp (m^{s})}{t \in L}} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ + \sum_{L \subseteq supp (m^{s}) \ {t}} {\prod_{l \in L} m^{s} (l) \\ \prod_{l \in supp (m^{s}) ∖ L \ {t}} (1 - m^{s} (l)) (1 - m^{s} (t))} \\ = & \sum_{\binom{L \subseteq supp (m^{s})}{t \in L}} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ + (1 - m^{s} (t)) \end{matrix}$

Hence $\sum_{\binom{L \subseteq supp (m^{s})}{t \in L}} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} = m^{s} (t) .$

Theorem 3.1. Let $(N, \tilde{v}, ℬ) \in {FGC}_{O}^{N}$ and s ∈ ℱ (N). The function defined as follow is a fuzzy Owen function on ${FGC}_{O}^{N}$ .

$\begin{matrix} {\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) \\ = \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ \times {Ow}_{i} (N, v, ℬ) (L) . \end{matrix}$ (4)

Proof. We shall prove the function Ow defined by Equation (4) satisfies Axioms F1 - F5.

Axiom F₁. Let (N, v, ℬ) is the associated crisp game of $(N, \tilde{v}, ℬ) \in {FGC}_{O}^{N}$ and s ∈ ℱ (N). Since ∑_i∈NOw_i (N, v, ℬ) (L) = v (⋃ _k∈LB_k) = v_ℬ (L) holds for any L ⊆ supp (m^s) from Axiom C₁, then $\begin{matrix} \sum_{i \in N} {\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) \\ = \sum_{i \in N} \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ \times {Ow}_{i} (N, v, ℬ) (L) \\ = \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ \times \sum_{i \in N} {Ow}_{i} (N, v, ℬ) (L) \\ = \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \cdot v_{ℬ} (L) \\ = {\tilde{v}}_{ℬ} (m^{s}) = \tilde{v} (s) . \end{matrix}$

For any i ∉ supp (s), then i ∉ ⋃ _k∈LB_k for any L ⊆ supp (m^s). Thus, Ow_i (N, v, ℬ) (L) =0. Therefore ${\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) = 0 .$

Axiom F₂. For any i ∈ B_t (t ∈ supp (m^s)), i is a null player. Then $\tilde{v} (u \lor s_{i} e^{i}) = \tilde{v} (u)$ for any $u \in \bar{s - s_{i} e^{i}}$ . Let L, V ∈ 𝒫 (M) and L ⊆ V, then $\begin{matrix} {Ow}_{i} (N, v, ℬ) (L) \\ = \sum_{\binom{R \subseteq L}{t \notin R}} σ (| R |; | L |) \sum_{\binom{S \subseteq B_{t}}{i \notin S}} δ (| S |; | B_{t} |) \\ \times [v (S \cup {i} \cup ⋃_{k \in R} B_{k}) - v (S \cup ⋃_{k \in R} B_{k})] \\ = \sum_{\binom{R \subseteq L}{t \notin R}} σ (| R |; | L |) \sum_{\binom{S \subseteq B_{t}}{i \notin S}} δ (| S |; | B_{t} |) \\ \times [v ((S \cup {i} \cup ⋃_{k \in R} B_{k}) \cap ⋂_{k \in V} B_{k}) \\ - v ((S \cup ⋃_{k \in R} B_{k}) \cap ⋂_{k \in V} B_{k})] \\ = \sum_{\binom{R \subseteq L}{t \notin R}} σ (| R |; | L |) \sum_{\binom{S \subseteq B_{t}}{i \notin S}} δ (| S |; | B_{t} |) \\ \times [v ((S \cup ⋃_{k \in R \cap V} B_{k}) \cap (⋂_{k \in V} B_{k} \cap {i})) \\ - v (S \cup ⋃_{k \in R \cap V} B_{k})] . \end{matrix}$

If t ∉ V, then Ow_i (N, v, ℬ) (L) =0.

For t ∈ V, let K = R ∩ V, Q = R \ T. Then K ∈ 𝒫 (V \ {i}), Q ∈ 𝒫 (L \ V), R = K ∪ Q, K∩ Q = ∅, and $\begin{matrix} {Ow}_{i} (N, v, ℬ) (L) \\ = \sum_{\binom{R \subseteq L}{t \notin R}} σ (| R |; | L |) \sum_{\binom{S \subseteq B_{t}}{i \notin S}} δ (| S |; | B_{t} |) \\ \times [v (S \cup ⋃_{k \in R \cap V} B_{k} \cup {i}) - v (S \cup ⋃_{k \in R \cap V} B_{k})] \\ = \sum_{\binom{K \subseteq V}{t \notin R}} \sum_{Q \subseteq L \ V} σ (| K \cup Q |; | L |) \sum_{\binom{S \subseteq B_{t}}{i \notin S}} δ (| S |; | B_{t} |) \\ \times [v (S \cup ⋃_{k \in K} B_{k} \cup {i}) - v (S \cup ⋃_{k \in K} B_{k})] \\ = \sum_{\binom{K \subseteq V}{t \notin R}} σ (| K |; | V |) \sum_{\binom{S \subseteq B_{t}}{i \notin S}} δ (| S |; | B_{t} |) \\ \times [v (S \cup ⋃_{k \in K} B_{k} \cup {i}) - v (S \cup ⋃_{k \in K} B_{k})] \end{matrix}$ where in the last line we use $\begin{matrix} \sum_{Q \subseteq L \ V} σ (| K \cup Q |; | L |) \\ = \sum_{Q \subseteq L \ V} \frac{| K \cup Q |! (| L | - | K \cup Q | - 1)!}{| L |!} \\ = \sum_{| Q | = 0}^{| L | - | V |} (| L | - | V \binom{|}{|} Q |) \frac{(| K | + | Q |)! (| L | - | K | - | Q | - 1)!}{| L |!} \\ = \frac{| K |! (| V | - | K | - 1)!}{| V |!} \\ = σ (| K |; | V |) . \end{matrix}$

Then $\begin{matrix} {\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) \\ = \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ \times {Ow}_{i} (N, v, ℬ) (L) \\ = \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ \times (\sum_{\binom{K \subseteq V}{t \notin K}} σ (| K |; | V |) \sum_{\binom{S \subseteq B_{t}}{i \notin S}} δ (| S |; | B_{t} |) \\ \times [v ((S \cup {i} \cup ⋃_{k \in K} B_{k})) - v ((S \cup ⋃_{k \in K} B_{k})]) \\ = \sum_{\binom{K \subseteq V}{t \notin K}} σ (| K |; | V |) \sum_{\binom{S \subseteq B_{t}}{i \notin S}} δ (| S |; | B_{t} |) \\ \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ \times [v_{ℬ (S \cup {i})} (K \cup {t}) - v_{ℬ (S)} (K \cup {t})] \\ = \sum_{\binom{K \subseteq V}{t \notin K}} σ (| K |; | V |) \sum_{\binom{S \subseteq B_{t}}{i \notin S}} δ (| S |; | B_{t} |) \\ \times [{\tilde{v}}_{ℬ (S \cup {i})} (\sum_{j \in K \cup {t}} m_{j}^{s} e_{M}^{j}) - {\tilde{v}}_{ℬ (S)} (\sum_{j \in K \cup {t}} m_{j}^{s} e_{M}^{j})] \\ = \sum_{\binom{K \subseteq V}{t \notin K}} σ (| K |; | V |) \sum_{\binom{S \subseteq B_{t}}{i \notin S}} δ (| S |; | B_{t} |) \\ \times [\tilde{v} (\sum_{j \in ⋃_{k \in K} B_{k} \cup S} s_{j} e^{j} \lor s_{i} e^{i}) - \tilde{v} (\sum_{j \in ⋃_{k \in K} B_{k} \cup S} s_{j} e^{j})] \\ = 0 . \end{matrix}$ (5)

Axiom F₃. For any i, j ∈ B_t (t ∈ supp (m^s)), i, j are symmetric. Then $\tilde{v} (u \lor s_{i} e^{i}) = \tilde{v} (u \lor s_{j} e^{j})$ for any $u \in \bar{s - s_{i} e^{i} - s_{j} e^{j}}$ . From Equation (5), $\begin{matrix} {\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) \\ = \sum_{\binom{K \subseteq V}{t \notin K}} σ (| K |; | V |) \sum_{\binom{S \subseteq B_{t}}{i \notin S}} δ (| S |; | B_{t} |) \\ \times [\tilde{v} (\sum_{h \in ⋃_{k \in K} B_{k} \cup S} s_{h} e^{h} \lor s_{i} e^{i}) - \tilde{v} (\sum_{h \in ⋃_{k \in K} B_{k} \cup S} s_{h} e^{h})] \\ = \sum_{\binom{K \subseteq V}{t \notin K}} σ (| K |; | V |) (\sum_{\binom{S \subseteq B_{t}}{i, j \notin S}} δ (| S |; | B_{t} |) \\ \times [\tilde{v} (\sum_{h \in ⋃_{k \in K} B_{k} \cup S} s_{h} e^{h} \lor s_{i} e^{i}) - \tilde{v} (\sum_{h \in ⋃_{k \in K} B_{k} \cup S} s_{h} e^{h})] \\ + \sum_{\binom{S \subseteq B_{t}}{i \notin S, j \in S}} δ (| S |; | B_{t} |) \\ \times [\tilde{v} (\sum_{h \in ⋃_{k \in K} B_{k} \cup S} s_{h} e^{h} \lor s_{i} e^{i}) - \tilde{v} (\sum_{h \in ⋃_{k \in K} B_{k} \cup S} s_{h} e^{h})]) \\ = \sum_{\binom{K \subseteq V}{t \notin K}} σ (| K |; | V |) (\sum_{\binom{S \subseteq B_{t}}{i, j \notin S}} δ (| S |; | B_{t} |) \\ \times [\tilde{v} (\sum_{h \in ⋃_{k \in K} B_{k} \cup S} s_{h} e^{h} \lor s_{i} e^{i}) - \tilde{v} (\sum_{h \in ⋃_{k \in K} B_{k} \cup S} s_{h} e^{h})] \\ + \sum_{\binom{S \subseteq B_{t}}{j \notin S, i \in S}} δ (| S |; | B_{t} |) \\ \times [\tilde{v} (\sum_{h \in ⋃_{k \in K} B_{k} \cup S} s_{h} e^{h} \lor s_{i} e^{i}) - \tilde{v} (\sum_{h \in ⋃_{k \in K} B_{k} \cup S} s_{h} e^{h})]) \\ = {\tilde{Ow}}_{j} (N, \tilde{v}, ℬ) (m^{s}) . \end{matrix}$

Axiom F₄. For any p, q ∈ supp (m^s), p and q are symmetric with respect to the fuzzy intermediate game ${\tilde{v}}_{ℬ}$ , then ${\tilde{v}}_{ℬ} (u \lor m_{p}^{s} e_{M}^{p}) = {\tilde{v}}_{ℬ} (u \lor m_{q}^{s} e_{M}^{q})$ for any $u \in \bar{m^{s} - m_{p}^{s} e_{M}^{p} - m_{q}^{s} e_{M}^{q}}$ . Then from Lemma 1, $\begin{matrix} \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \notin L} (1 - m^{s} (l))} \\ \cdot v_{ℬ} (L \cap (U \cup {p})) \\ = \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \notin L} (1 - m^{s} (l))} \\ \cdot v_{ℬ} (L \cap (U \cup {q})) . \end{matrix}$

Hence $v_{ℬ} (L \cap (U \cup {p})) = v (L \cap (U \cup {q})) .$

Case 1. If p ∈ L and q ∈ L, then Ow_i (N, v, ℬ) (L) =0 for any i ∈ B_p ∪ B_q from Axiom C₁. Hence $\sum_{i \in B_{p}} {Ow}_{i} (N, v, ℬ) (L) = \sum_{i \in B_{q}} {Ow}_{i} (N, v, ℬ) (L) .$

Case 2. If p ∈ L and q ∈ L, then v_ℬ ((L ∩ U) ∪ {p}) = v_ℬ ((L ∩ U) ∪ {q}) for any L ∩ U ⊆ L ∖ {p, q} . From Axiom C₃, $\sum_{i \in B_{p}} {Ow}_{i} (N, v, ℬ) (L) = \sum_{i \in B_{q}} {Ow}_{i} (N, v, ℬ) (L) .$

Case 3. If p ∉ L and q ∈ L, then v_ℬ ((L ∩ U)) = v_ℬ ((L ∩ U) ∪ {q}) for any L ∩ U ⊆ M ∖ {p, q} . Hence v (⋃ _k∈L∩UB_k) = v (⋃ _k∈L∩UB_k ∪ B_q) .

Since (N, v, ℬ) is supperadditive game, then for any i ∈ B_q, $\begin{matrix} v (⋃_{k \in L \cap U} B_{k}) \leq v (⋃_{k \in L \cap U} B_{k} \cup {i}) \\ \leq v (⋃_{k \in L \cap U} B_{k} \cup B_{q}) = v (⋃_{k \in L \cap U} B_{k}) \end{matrix}$

Hence, i is a null player. Then Ow_i (N, v, ℬ) (L) =0 from Axiom C₂.

On the other hand, since p ∉ L, then Ow_i (N, v, ℬ) (L) =0 for any i ∈ B_p. Hence $\sum_{i \in B_{p}} {Ow}_{i} (N, v, ℬ) (L) = \sum_{i \in B_{q}} {Ow}_{i} (N, v, ℬ) (L) .$

Case 4. If p ∈ L and q ∉ L, then similar to the proof process of case 2, $\sum_{i \in B_{p}} {Ow}_{i} (N, v, ℬ) (L) = \sum_{i \in B_{q}} {Ow}_{i} (N, v, ℬ) (L) .$

Therefore $\begin{matrix} \sum_{i \in B_{p}} {\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) \\ = \sum_{i \in B_{p}} \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ \times {Ow}_{i} (N, v, ℬ) (L) \\ = \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ \times \sum_{i \in B_{p}} {Ow}_{i} (N, v, ℬ) (L) \\ = \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ \times \sum_{i \in B_{q}} {Ow}_{i} (N, v, ℬ) (L) \\ = \sum_{i \in B_{q}} {\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) . \end{matrix}$

Axiom F5. Let s ∈ ℱ (N) and $(N, {\tilde{v}}_{1}, ℬ), (N, {\tilde{v}}_{2}, ℬ) \in {FGC}_{O}^{N}$ , and α, $β \in ℝ$ , it is clear that $(α {\tilde{v}}_{1} + β {\tilde{v}}_{2}) \in {FGC}_{O}^{N}$ from the definition of ${FGC}_{O}^{N}$ . By Axiom C₅, the following is valid. $\begin{matrix} {\tilde{Ow}}_{i} (N, α {\tilde{v}}_{1} + β {\tilde{v}}_{2}, ℬ) (m^{s}) \\ = \sum_{L \subseteq supp (m^{s})} {\prod_{l \in T} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ T} (1 - m^{s} (l))} \\ \times {Ow}_{i} (N, α v_{1} + β v_{2}, ℬ) (L) \\ = \sum_{L \subseteq supp (m^{s})} {\prod_{l \in T} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ T} (1 - m^{s} (l))} \\ \times (α {Ow}_{i} (N, v_{1}, ℬ) (L) + β {Ow}_{i} (N, v_{2}, ℬ) (L)) \\ = α \sum_{L \subseteq supp (m^{s})} {\prod_{l \in T} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ T} (1 - m^{s} (l))} \\ \times {Ow}_{i} (N, v_{1}, ℬ) (L) \\ + β \sum_{L \subseteq supp (m^{s})} {\prod_{l \in T} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ T} (1 - m^{s} (l))} \\ \times {Ow}_{i} (N, v_{2}, ℬ) (L) \\ = α {\tilde{Ow}}_{i} (N, {\tilde{v}}_{1}, ℬ) (m^{s}) + β {\tilde{Ow}}_{i} (N, {\tilde{v}}_{2}, ℬ) (m^{s}) . \end{matrix}$

Theorem 3.2. Let $(N, \tilde{v}, ℬ) \in {FGC}_{O}^{N}$ and s ∈ ℱ (N), then ${\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s})$ is an imputation of $\tilde{v}$ in fuzzy coalition s.

Proof. Let s ∈ ℱ (N). From Axiom F₁, it is apparent that ${\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) = 0$ for any i ∉ supp (s) and $\sum_{i \in N} {\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) = v (s)$ . Hence, $\sum_{i \in supp (s)} {\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) = \tilde{v} (s) .$

It remains to show that ${\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) \geq \tilde{v} (s_{i} e^{i})$ , i.e., ${\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) \geq m^{s} (t) v ({i})$ for any i ∈ supp (s). By Lemma 3.2, $\begin{matrix} {\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) \\ = \sum_{L \subseteq supp (m^{s})} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ \times {Ow}_{i} (N, v, ℬ) (L) \\ = \sum_{\binom{L \in \subseteq supp (m^{s})}{t \in L}} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \\ \times {Ow}_{i} (N, v, ℬ) (L) \\ \geq \sum_{\binom{L \in \subseteq supp (m^{s})}{t \in L}} {\prod_{l \in L} m^{s} (l) \prod_{l \in supp (m^{s}) ∖ L} (1 - m^{s} (l))} \cdot v ({i}) \\ = m^{s} (t) v ({i}) . \end{matrix}$

4 An illustrative example

Consider a joint production model in which three investors pool three resources to produce seven products. Three investors, named 1, 2 and 3, possess three different initial resources. Investor i has 100 units of resource R_i and can produce n_i units product P_i, i = 1, 2, 3. Now, investors make decide to undertake a joint project: if investors i and j cooperate, they will produce n_i units of product P_ij, and if all three cooperate, n₁₂₃ units of product P₁₂₃ can be produced. If investors i and j make up a full cooperative relationship, i.e., a crisp coalition {i, j}, then they can obtain gains v ({i, j}) by P_ij, and if all three cooperate by a crisp coalition {1, 2, 3}, they can obtain gains v ({1, 2, 3}). The gains of each crisp coalition are shown in Table 1.

Table 1
The gains of each crisp coalition

Product P ₁ P ₂ P ₃ P ₁₂ P ₁₃ P ₂₃ P ₁₂₃

Coalition {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}

Gains 100 60 70 200 200 300 600

Product	P ₁	P ₂	P ₃	P ₁₂	P ₁₃	P ₂₃	P ₁₂₃
Coalition	{1}	{2}	{3}	{1, 2}	{1, 3}	{2, 3}	{1, 2, 3}
Gains	100	60	70	200	200	300	600

Due to some common interests, players 2 and 3 are prefer to cooperative together. Then players 2 and 3 form one coalition and player 1 forms another one so that they form a coalition structure ({2, 3}, {1}). In other words, two a priori unions are formed, named a and b, where a = {1} and b = {2, 3}. Then the Owen values in crisp game with coalition structure are obtained as in Table 2.

Table 2

The Owen value in games with coalition structure

	Investors
Coalition	1	2	3
{1}	100	0	0
{2}	0	60	0
{3}	0	0	70
{1}, {2}	120	80	0
{1}, {3}	115	0	85
{2, 3}	0	145	155
{1}, {2, 3}	200	197.5	202.5

As is in the real life, every investor has to consider how many resources he or she should provide in the cooperation. As we all know, each investor does not need to supply all of his or her resources to cooperate in real life. It depends on many factors, such as individual preference so on. Thus, we have to consider a fuzzy game. Suppose that investor 1 can supply 40 units of R₁ to the cooperation, while investor 2 can supply 20 units of R₂, and investor 3 can supply 50 units of R₃. Because investor 1 supplies 40 units of R₁, we regard the rate of participation of investor 1 as $0.4 = \frac{40}{100}$ . Similarly, the participation level of investor 2 is 0.2,and that of investor 3 is 0.5. Therefore, a fuzzy coalition s has been formed. i.e., s = (0.4, (0.2, 0.5)) .

Because investors do not fully participate the coalition, but to a certain extent. Hence, a priori unions also participate in some level. The participation level of a priori unions depends on the participation level of players in this unions. Suppose that 2 and 3 have the same influence factor in the cooperation. Hence, the participation level of the two priori unions is 0.4, 0.35, respectively. i.e., m^s = (0.4, 0.35). The value of this fuzzy coalition is evaluated by (2), $\begin{matrix} {\tilde{v}}_{ℬ} (m^{s}) & = & 0.4 \times 0.65 \cdot v_{ℬ} ({a}) + 0.6 \times 0.35 \cdot v_{ℬ} ({b}) \\ + 0.4 \times 0.35 \cdot v_{ℬ} ({a, b}) \\ = & 0.4 \times 0.65 \cdot v ({1}) + 0.6 \times 0.35 \cdot v ({2, 3}) \\ + 0.4 \times 0.35 \cdot v ({1, 2, 3}) \\ = & 173 . \end{matrix}$

By the Own value in games with coalition structure as shown in Table 2. The fuzzy Owen function for fuzzy game with coalition structure can be calculated as $\begin{matrix} {\tilde{Ow}}_{i} (N, \tilde{v}, ℬ) (m^{s}) \\ = 0.4 \times 0.65 \cdot {Ow}_{i} (N, v, ℬ) ({a}) \\ + 0.6 \times 0.35 \cdot {Ow}_{i} (N, v, ℬ) ({b}) \\ + 0.4 \times 0.35 \cdot {Ow}_{i} (N, v, ℬ) ({a, b}) \end{matrix}$

The profit share of investor 1 is calculated as $\begin{matrix} {\tilde{Ow}}_{1} (N, \tilde{v}, ℬ) (m^{s}) \\ = 0.4 \times 0.65 \cdot {Ow}_{1} (N, v, ℬ) ({a}) \\ + 0.6 \times 0.35 \cdot {Ow}_{1} (N, v, ℬ) ({b}) \\ + 0.4 \times 0.35 \cdot {Ow}_{1} (N, v, ℬ) ({a, b}) \\ = 0.4 \times 0.65 \times 100 + 0.6 \times 0.35 \times 0 \\ + 0.4 \times 0.35 \times 200 = 54 . \end{matrix}$

In the same way, the profit share of investor 2 and 3 can be calculated as $\begin{matrix} {\tilde{Ow}}_{2} (N, \tilde{v}, ℬ) (m^{s}) = 58.1, \\ {Ow}_{3} (N, v, ℬ) (m^{s}) = 60.9 . \end{matrix}$

5 Conclusions

Game theoretic approaches to cooperative situations in fuzzy environments have given rise to several kinds of fuzzy games. We have discussed a fuzzy function on fuzzy games with coalition structure. Owen value is an extension of Shapley value, if considering fuzzy coalitions in the framework of games with coalition structure, it is different from games with fuzzy coalitions. Therefore, we have proposed influence function, which can reflect the relationship of the participation between a priori union and players. The concepts related to a Owen function have been extended to the case of fuzzy games with coalition structure, then a fuzzy Owen function is given, which is an extension of crisp games with coalition structure. In order to discuss the explicit form of a fuzzy Owen function, we considering the fuzzy intermediate game with Owen extension form. After giving a concrete form of the Owen function, we have shown its rational properties. For the purpose of bridging the results to a real world problem, we have givenan illustrate example.

As is in real life, we should use the explicit form of fuzzy Owen function to get the payoff of each players. That how to give an suitable expression of fuzzy game is a key question. In this paper, we only considered the fuzzy game with Owen extension form, which is a kind of game with fuzzy coalitions. It will be interesting to consider the other class of games with fuzzy coalitions.

Footnotes

Acknowledgments

The work on this paper is supported by National Natural Science Foundation of China (No. 71401003).

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A Fuzzy Owen function on games with coalition structure and fuzzy coalitions

Abstract

Keywords

1 Introduction

2 Three classes of cooperative games and the Owen function

2.1 Crisp cooperative games with coalition structure and Owen function

3.1 Fuzzy Owen function

3.2 Fuzzy Owen value with Owen extension

Table 1 The gains of each crisp coalition Product P 1 P 2 P 3 P 12 P 13 P 23 P 123 Coalition {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3} Gains 100 60 70 200 200 300 600

Footnotes

Acknowledgments

References

Table 1
The gains of each crisp coalition

Product P ₁ P ₂ P ₃ P ₁₂ P ₁₃ P ₂₃ P ₁₂₃

Coalition {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}

Gains 100 60 70 200 200 300 600