A further discussion on fuzzy interval cooperative games

Abstract

The $F$ -core and the $F$ -balancedness have been introduced for fuzzy interval cooperative games by Mallozzi et al. (2011). It was pointed out that the $F$ -balancedness was a necessary but not sufficient condition for the non-emptiness of the $F$ -core. In this paper, a numerical example is given to show that the $F$ -balancedness can not guarantee the non-emptiness of the $F$ -core. Furthermore, the $F$ -cores of trapezoidal fuzzy interval games are discussed in detail. Some properties of the $F$ -core are obtained and three sufficient conditions for the non-emptiness of the $F$ -core are given. These conclusions are generalizations of the corresponding results in both interval cooperative game theory and classical cooperative game theory. Lastly, an application example is also provided.

Keywords

Fuzzy interval cooperative games convex games core

1 Introduction

The model of fuzzy interval cooperative games can be used to analyze and handle cooperative situations in fuzzy environments. This model was introduced by Mares [11] in 1999 and later researched by many scholars [5 , 13].

In [8], Mallozzi et al. introduced the fuzzy interval core ( $F$ -core in short) and the $F$ -balancedness for fuzzy interval cooperative games. They proved that the $F$ -balancedness was a necessary condition for the non-emptiness of the $F$ -core and gave an example to show that it was not a sufficient condition. However, their example is inappropriate for this problem. In this paper, a new example will be given which show that their conclusions are true, i.e. the $F$ -balancedness is only a necessary condition for the existence of elements of the $F$ -core. Moreover, the properties of the $F$ -core are discussed and three sufficient conditions for the non-emptiness of the $F$ -core are given.

The structure of the paper is as follows. In Section 2, we recall some needful notations and concepts of fuzzy intervals and fuzzy interval games. Section 3 presents our results, a new example for the $F$ -balanced condition and some properties of the $F$ -core. Section 4 is an example for application. Finally, Section 5 gives conclusions and remarks.

2 Preliminaries

In this section, we recall some basic definitions and results about fuzzy intervals and fuzzy interval cooperative games, which are used in [8].

2.1 Fuzzy intervals

By I (R), we denote the set of all closed intervals in R. Let I, J ∈ I (R), with $I = [\underline{I}, \bar{I}], J = [\underline{J}, \bar{J}]$ , the addition and the order relation on I (R) are defined as follows:

$I + J = [\underline{I} + \underline{J}, \bar{I} + \bar{J}]$ ;

I ≥ J if and only if $\underline{I} \geq \underline{J}$ and $\bar{I} \geq \bar{J}$ .

A fuzzy set F in R is a mapping u_F : R → [0, 1] where u_F assigns to each point in R a degree of membership [7]. For any α ∈ [0, 1], the α-cut set (α-level set) of F is

[u_F] ^α = {x ∈ R : u_F (x) ≥ α} , if α> 0 ;

$[u_{F}]^{0} = \bar{{x \in R : u_{F} (x) > 0}} = supp (u_{F})$ ,

here $\bar{{x \in R : u_{F} (x) > 0}}$ is the closure of the set {x ∈ R : u_F (x) >0}.

Definition 1. [2] A fuzzy set F in R is said to be a fuzzy interval, if the following conditions are satisfied:

[u_F] ^α is compact for any α ∈ [0, 1];

[u_F] ^α is convex for any α ∈ [0, 1];

there exists x ∈ R such that u_F (x) =1.

We denote the set of all fuzzy intervals by $F (R)$ .

Let F₁, F₂ be two fuzzy intervals, and * be a binary operation in R, the operation and the order relation in $F (R)$ are defined as follows:

F₁ * F₂ is a fuzzy interval with membership function $u_{F_{1} * F_{2}} (z) = sup_{z = x * y} {u_{F_{1}} (x) \land u_{F_{2}} (y)}$ ;

F₁ ⊵ F₂ ⇔ [u_{F
₁}] ^α ≥ [u_{F
₂}] ^α, ∀ α ∈ [0, 1].

Let $F \in F (R)$ , F is said to be a trapezoidal fuzzy interval if its membership function has the form $u_{F} (x) = {\begin{matrix} \frac{x - a_{1}}{a_{2} - a_{1}}, & if a_{1} \leq x < a_{2} \\ 1, & if a_{2} \leq x \leq a_{3} \\ \frac{a_{4} - x}{a_{4} - a_{3}}, & if a_{3} < x \leq a_{4} \\ 0, & otherwise \end{matrix}$ and this trapezoidal fuzzy interval is denoted by (a₁, a₂, a₃, a₄). Specially, when a₂ = a₃, F degenerates to a triangular fuzzy interval; when a₁ = a₂ and a₃ = a₄, F degenerates to a interval number; when a₁ = a₂ = a₃ = a₄, F degenerates to real number. We denote the set of all trapezoidal fuzzy intervals by $T F (R)$ .

Let F₁, F₂ be two trapezoidal fuzzy intervals, with F₁ = (a₁, a₂, a₃, a₄) , F₂ = (b₁, b₂, b₃, b₄), and λ be a non-negative real number, it can be verified that

F₁ + F₂ = (a₁ + b₁, a₂ + b₂, a₃ + b₃, a₄ + b₄);

λF₁ = (λa₁, λa₂, λa₃, λa₄);

$F_{1} ⊵ F_{2} \Leftrightarrow a_{1} \geq b_{1}, a_{2} \geq b_{2}, a_{3} \geq b_{3}, a_{4} \geq b_{4}$ .

2.2 Fuzzy interval cooperative games

Definition 2. [8] A fuzzy interval game is a pair $(N, U)$ , where N = {1, 2, …, n} is the set of players, and $U : 2^{N} \to F (R)$ is the characteristic function such that $U (\emptyset) = 0$ , where 0 is a fuzzy interval with membership function $u_{0} (x) = {\begin{matrix} 1, & x = 0, \\ 0, & x \neq 0 . \end{matrix}$

Definition 3. [8] Let $(N, U)$ be a fuzzy interval game. The fuzzy interval core (shortly $F$ -core) is defined by: $\begin{matrix} C^{F} (U) = {(F_{1}, F_{2}, \dots, F_{n}) \in F (R)^{n} : \sum_{i = 1}^{n} F_{i} \\ = U (N) and \sum_{i \in S} F_{i} ⊵ U (S), \forall S \subset N} \end{matrix}$

The fuzzy interval games are generalizations of the interval games [16, 17] and the classical TU-games. If all the characteristic function values of a fuzzy interval game degenerate to closed intervals, the $F$ -core coincides with the interval core [17]. And if all the characteristic function values of a fuzzy interval game degenerate to real numbers, the $F$ -core coincides with the classical core [1].

Proposition 1. [8] If $(N, U)$ is a fuzzy interval game such that the $F$ -core is non-empty and let $(F_{1}, F_{2}, \dots, F_{n}) \in C^{F} (U)$ . Then, the interval core of anyα-cut game is non-empty and $([u_{F_{1}}]^{α}, \dots, [u_{F_{n}}]^{α}) \in C (u^{α})$ for anyα ∈ [0, 1], where theα-cut game is defined byu^α (S) = [u_U(S)] ^α.

For trapezoidal fuzzy interval games, Mallozzi et al. [8] defined the balanced condition which generalizes the balancedness in classical TU-games and interval games.

Definition 4. [8] A trapezoidal fuzzy interval game $(N, U)$ is said to be $F$ -balanced if $U (N) ⊵ \sum_{S \subseteq N} λ (S) U (S)$ for any map λ : S ⊆ N → λ (S) ∈ R₊ such that $\sum_{\begin{matrix} S \subseteq N \\ i \in S \end{matrix}} λ (S) = 1, \forall i \in N$ (λ is called a balanced map).

3 Some results on fuzzy interval cooperative games

In this section, we shall give our main results, including two parts. Firstly, a new numerical example is given to show that the $F$ -balanced condition is not sufficient for the non-emptiness of the $F$ -core. Then, some properties of the $F$ -core are given.

3.1 A new example for the $F$ -balanced condition

For trapezoidal fuzzy interval games, Mallozzi et al. [8] proved that the $F$ -balanced condition is necessary for the non-emptiness of the $F$ -core, and gave an example to show that the $F$ -balanced condition is not sufficient for the existence of $F$ -core elements. However, their example fails to demonstrate this conclusion.

Example 1. [8] Let $(N, U)$ be a trapezoidal fuzzy interval game with N = {1, 2, 3} and $U$ defined as below: $\begin{matrix} U ({1}) = \frac{1}{2}, U ({2}) = U ({3}) = 0, \\ U ({1, 2}) = U ({1, 3}) = [0, 1], \\ U ({2, 3}) = (\frac{1}{2}, \frac{3}{2}, \frac{3}{2}, \frac{5}{2}), \\ U ({1, 2, 3}) = (\frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \frac{7}{2}) . \end{matrix}$

It has been verified that the $F$ -core of $(N, U)$ is empty in [8], however, this game is not $F$ -balanced. Thus, this example is unable to show that the $F$ -balancedness is not sufficient for the existence of elements of $F$ -core. In fact, for a balanced map λ : S ⊆ N → R₊ with $λ (S) = {\begin{matrix} 1, & S = {1}, {2, 3} \\ 0, & otherwise \end{matrix}$ we have $\begin{matrix} 1 • U ({1}) + 1 • U ({2, 3}) \\ = (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}) + (\frac{1}{2}, \frac{3}{2}, \frac{3}{2}, \frac{5}{2}) \\ = (1, 2, 2, 3) ⋬ U (N) = (\frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \frac{7}{2}) . \end{matrix}$

Therefore, $(N, U)$ is not $F$ -balanced. □

Note that the balanced condition for a TU-game can be checked only for minimal balanced collections [4]. Besides { {1, 2} , {1, 3} , {2, 3}}, there are four minimal balanced collections of a three-person game, which are partitions of N including { {{1} , {2} , {3}} , 1pt {{1} , {2, 3}} , {{2} , {1, 3}} , {{3} , {1, 2}}. In [8], the authors only check for the balanced collections {{1, 2} , {1, 3} , {2, 3}}.

Next, we give a new example, which is inspired by Zhao et al. [18], to show that the $F$ -balanced condition is not sufficient for the non-emptiness of the $F$ -core.

Example 2. Let $(N, U)$ be a trapezoidal fuzzy interval game with N = {1, 2, 3} and $U$ defined as below: $\begin{matrix} U ({1}) = U ({2}) = U ({3}) = 0, \\ U ({1, 2}) = U ({2, 3}) = [8, 9], \\ U ({1, 3}) = (4, 6, 10, 14), \\ U ({1, 2, 3}) = (10, 12, 14, 16) . \end{matrix}$

We check all the minimal balanced collections: $\begin{matrix} 1 • U ({1}) + 1 • U ({2}) + 1 • U ({3}) = (0, 0, 0, 0) \\ ⊴ U (N) = (10, 12, 14, 16); \\ 1 • U ({1}) + 1 • U ({2, 3}) = (8, 8, 9, 9) \\ ⊴ U (N) = (10, 12, 14, 16); \\ 1 • U ({3}) + 1 • U ({1, 2}) = (8, 8, 9, 9) \\ ⊴ U (N) = (10, 12, 14, 16); \\ 1 • U ({2}) + 1 • U ({1, 3}) = (4, 6, 10, 14) \\ ⊴ U (N) = (10, 12, 14, 16); \\ \frac{1}{2} U ({1, 2}) + \frac{1}{2} U ({1, 3}) + \frac{1}{2} U ({2, 3}) \\ = \frac{1}{2} (8, 8, 9, 9) + \frac{1}{2} (4, 6, 10, 14) + \frac{1}{2} (8, 8, 9, 9) \\ = (10, 11, 14, 16) ⊴ U (N) = (10, 12, 14, 16); \end{matrix}$

So, $(N, U)$ is $F$ -balanced. Next, we show that the $F$ -core of this game is empty. Consider the 0-cut game of $(N, U)$ , denoted by (N, u⁰), which is an interval game with $\begin{matrix} u^{0} ({i}) = 0, (i = 1, 2, 3), \\ u^{0} ({1, 2}) = u^{0} ({2, 3}) = [8, 9], \\ u^{0} ({1, 3}) = [4, 14], \\ u^{0} ({1, 2, 3}) = [10, 16] \end{matrix}$

It can be verified that the interval core of (N, u⁰) is empty [18]. By Proposition 1, the $F$ -core of $(N, U)$ is empty. □

3.2 Some properties of the $F$ -core

The $F$ -cores of trapezoidal fuzzy interval games are further discussed in this subsection. Some properties of the $F$ -core are obtained and three sufficient conditions for the non-emptiness of the $F$ -core are given.

Proposition 2.Let $(N, U)$ be a trapezoidal fuzzy interval game. The $F$ -core is nonempty if and only if $C (u^{0}) \neq \emptyset, C (u^{1}) \neq \emptyset$ and there exist $\begin{matrix} ([a_{1}^{0}, b_{1}^{0}], \dots, [a_{n}^{0}, b_{n}^{0}]) \in C (u^{0}), \\ ([a_{1}^{1}, b_{1}^{1}], \dots, [a_{n}^{1}, b_{n}^{1}]) \in C (u^{1}) \end{matrix}$ such that $a_{i}^{0} \leq a_{i}^{1}, b_{i}^{1} \leq b_{i}^{0}, i = 1, \dots, n$ .

Proof. The necessity is immediately obtained by Proposition 1 and Definition 3.

To show the sufficiency, take $\begin{matrix} ([a_{1}^{0}, b_{1}^{0}], \dots, [a_{n}^{0}, b_{n}^{0}]) \in C (u^{0}), \\ ([a_{1}^{1}, b_{1}^{1}], \dots, [a_{n}^{1}, b_{n}^{1}]) \in C (u^{1}) \end{matrix}$ such that $a_{i}^{0} \leq a_{i}^{1}, b_{i}^{1} \leq b_{i}^{0}, i = 1, \dots, n$ , then let $F_{i} = (a_{i}^{0}, a_{i}^{1}, b_{i}^{1}, b_{i}^{0})$ (i = 1, …, n) be trapezoidal fuzzy intervals. It can be verified that $(F_{1}, F_{2}, \dots, F_{n}) \in C^{F} (U)$ . □

The conditions in Proposition 2 guarantee that there exists $([a_{1}^{α}, b_{1}^{α}], \dots, [a_{n}^{α}, b_{n}^{α}]) \in C (u^{α})$ for any α ∈ [0, 1] and $[a_{i}^{α_{1}}, b_{i}^{α_{1}}] \supseteq [a_{i}^{α_{2}}, b_{i}^{α_{2}}]$ (i = 1, …, n) for any α₁ ≤ α₂. Thus one can generate $(F_{1}, F_{2}, \dots, F_{n}) \in C^{F} (U)$ by the representation theorem of fuzzy sets.

Consider functions $γ_{k} : TF (R) \to R$ , whereγ_k (F) = a_k(k = 1, 2, 3, 4), for any F = (a₁, a₂, $a_{3}, a_{4}) \in TF (R)$ . When we research a trapezoidal fuzzy interval game $(N, U)$ , four related classical games (N, u_{γ
_k}) , k = 1, 2, 3, 4, will play important roles, which are defined by $u_{γ_{k}} (S) = γ_{k} (U (S)), \forall S$ ⊆N.

Proposition 3.Let $(N, U)$ be a trapezoidal fuzzy interval game. The $F$ -core is nonempty if and only ifC (u_{γ
_k}) ≠ ∅ , k = 1, 2, 3, 4 and there exist $(x_{1}^{k}, \dots, x_{n}^{k}) \in C (u_{γ_{k}})$ (k = 1, 2, 3, 4) such that $x_{i}^{1} \leq x_{i}^{2} \leq x_{i}^{3} \leq x_{i}^{4}$ , i = 1, …, n.

The proof is similar to Proposition 2 and so omitted. Moreover, Proposition 3 can be seen as a corollary of Theorem 3.2 in [6].

By using interval games and classical games in which data only taken from 0-cut game and 1-cut game of a trapezoidal fuzzy interval game, Proposition 2 and Proposition 3 have given two characterizations of the $F$ -core. We can utilize these two propositions to look for $F$ -core elements.

Note that the non-emptiness of the $F$ -core of a trapezoidal fuzzy interval game $(N, U)$ is not equivalent to the condition that C (u_{γ
_k}) ≠ ∅ , k = 1, 2, 3, 4. For example, the game in Example 2 satisfies this condition, however, the $F$ -core is empty. So it may be not easy to judge the non-emptiness of the $F$ -core by Proposition 2 or Proposition 3. In the following, we shall give some sufficient conditions which can guarantee the non-emptiness of the $F$ -core and are easy to be checked.

Definition 5. A trapezoidal fuzzy interval game $(N, U)$ is strongly balanced if for each balanced map λ : 2^N ∖ {∅} → R⁺ it holds that $\sum_{S \subseteq N} λ (S) γ_{4} (U (S)) \leq γ_{1} (U (N)) .$

Remark that if all the characteristic function values of a fuzzy interval game degenerate to closed intervals, strongly balanced condition coincides with strongly balancedness in interval game theory [16]. And if all the characteristic function values of a fuzzy interval game degenerate to real numbers, this strongly balanced condition coincides with balancedness in classical game theory [9, 14].

Proposition 4.If a trapezoidal fuzzy interval game $(N, U)$ is strongly balanced, then $C^{F} (U) \neq \emptyset$ .

Proof. Consider a classical TU-game (N, v) with $v (S) = {\begin{matrix} γ_{4} (U (S)), S \subset N, \\ γ_{1} (U (N)), S = N . \end{matrix}$

Since $(N, U)$ is strongly balanced, by Definition 5, (N, v) is a classical balanced game. According to Bondareva-Shapley theorem [9, 14] for classical TU-games, C (v)≠ ∅. Take $(x_{1}^{1}, \dots, x_{n}^{1}) \in C (v)$ , then ${\begin{matrix} \sum_{i \in N} x_{i}^{1} = v (N), \\ \sum_{i \in S} x_{i}^{1} \geq v (S), \forall S \subset N, \end{matrix}$ i.e. ${\begin{matrix} \sum_{i \in N} x_{i}^{1} = γ_{1} (U (N)), \\ \sum_{i \in S} x_{i}^{1} \geq γ_{4} (U (S)), \forall S \subset N \end{matrix}$ (1)

By Equation (1) and note that γ₄ (U (S)) ≥ γ₁ $(U (S))$ , we obtain that $\sum_{i \in S} x_{i}^{1} \geq γ_{1} (U (S)),$ ∀S ⊂ N, which means $(x_{1}^{1}, \dots, x_{n}^{1}) \in C (u_{γ_{1}})$ .

Let $d_{1} = (γ_{2} (U (N)) - γ_{1} (U (N)) / n$ , then d₁ ≥ 0. Let $x_{i}^{2} = x_{i}^{1} + d_{1}$ , then $x_{i}^{2} \geq x_{i}^{1}$ (i = 1, 2, …, n) and $\sum_{i \in N} x_{i}^{2} = γ_{2} (U (N)) .$ (2)

By Equation (1), we obtain that $\sum_{i \in S} x_{i}^{2} \geq \sum_{i \in S} x_{i}^{1} \geq γ_{4} (U (S)) \geq γ_{2} (U (S)), \forall S \subset N$ (3)

Equations (2) and (3) imply that $(x_{1}^{2}, \dots, x_{n}^{2}) \in C (u_{γ_{2}})$ .

Similarly, let $d_{2} = (γ_{3} (U (N)) - γ_{2} (U (N)) / n$ , d₃ $= (γ_{4} (U (N)) - γ_{3} (U (N)) / n$ and $x_{i}^{3} = x_{i}^{2} + d_{2}$ , $x_{i}^{4} = x_{i}^{3} + d_{3}$ , we can prove that $(x_{1}^{3}, \dots, x_{n}^{3}) \in C (u_{γ_{3}})$ , $(x_{1}^{4}, \dots, x_{n}^{4}) \in C (u_{γ_{4}})$ and $x_{i}^{3} \geq x_{i}^{2}, x_{i}^{4} \geq x_{i}^{3} (i = 1, 2, \dots, n)$ .

In summary, we prove that $(x_{1}^{k}, \dots, x_{n}^{k}) \in C (u_{γ_{k}})$ (k = 1, 2, 3, 4) and $x_{i}^{1} \leq x_{i}^{2} \leq x_{i}^{3} \leq x_{i}^{4}$ , i = 1, …, n. According to Proposition 3, $C^{F} (U) \neq \emptyset$ . □

Example 3. Let $(N, U)$ be a trapezoidal fuzzy interval game with N = {1, 2, 3} and $U$ defined as below: $\begin{matrix} U ({1}) = U ({2}) = U ({3}) = 0, \\ U ({1, 2}) = U ({2, 3}) = [8, 9], \\ U ({1, 3}) = (4, 6, 10, 14), \\ U ({1, 2, 3}) = (20, 21, 22, 23) . \end{matrix}$

$(N, U)$ is strongly balanced, according to Proposition 4, then $C^{F} (U) \neq \emptyset$ . For example, ((7, 7, 7, 7) , $(6, 6, 6, 6), (7, 8, 9, 10)) \in C^{F} (U)$ .□

Consider functions $d_{k} : TF (R) \to R$ , whered_k (F) = a_k+1 - a_k(k = 1, 2, 3), for any F = (a₁, $a_{2}, a_{3}, a_{4}) \in TF (R)$ . Given a trapezoidal fuzzy interval game $(N, U)$ , three related classical games (N, u_{d
_k}) , k = 1, 2, 3, will be used in the following, which are defined by $u_{d_{k}} (S) = d_{k} (U (S)),$ ∀S ⊆ N.

Proposition 5.Let $(N, U)$ be a trapezoidal fuzzy interval game. If (N, u_{γ
₁}) and (N, u_{d
_k}) , k = 1, 2, 3 are balanced, then $C^{F} (U) \neq \emptyset$ .

Proof. (N, u_{γ
₁}) and (N, u_{d
_k}) , k = 1, 2, 3, are classical TU-games. If they are balanced, then C (u_{γ
₁})≠ ∅ and C (u_{d
_k}) ≠ ∅ , k = 1, 2, 3 . Take $(x_{1}^{1}, \dots, x_{n}^{1}) \in C (u_{γ_{1}})$ and $(d_{1}^{k}, \dots, d_{n}^{k}) \in C (u_{d_{k}}), k = 1, 2, 3$ , then ${\begin{matrix} \sum_{i \in N} x_{i}^{1} = γ_{1} (U (N)), \\ \sum_{i \in S} x_{i}^{1} \geq γ_{1} (U (S)), \forall S \subset N \end{matrix}$ (4) ${\begin{matrix} \sum_{i \in N} d_{i}^{1} = d_{1} (U (N)), \\ \sum_{i \in S} d_{i}^{1} \geq d_{1} (U (S)), \forall S \subset N . \end{matrix}$ (5)

Let $x_{i}^{2} = x_{i}^{1} + d_{i}^{1}$ , i = 1, …, n. Sum Equations (4) and (5) and note that $γ_{1} (U (S)) + d_{1} (U (S)) = γ_{2} (U (S)),$ ∀S ∈ 2^N, we have ${\begin{matrix} \sum_{i \in N} x_{i}^{2} = γ_{2} (U (N)), \\ \sum_{i \in S} x_{i}^{2} \geq γ_{2} (U (S)), \forall S \subset N \end{matrix}$ implying that $(x_{1}^{2}, \dots, x_{n}^{2}) \in C (u_{γ_{2}})$ .

Let $x_{i}^{3} = x_{i}^{2} + d_{i}^{2}$ , $x_{i}^{4} = x_{i}^{3} + d_{i}^{3}$ . Using the similar proof method, we can obtain that $(x_{1}^{3}, \dots, x_{n}^{3}) \in C (u_{γ_{3}})$ , $(x_{1}^{4}, \dots, x_{n}^{4}) \in C (u_{γ_{4}})$ and $x_{i}^{3} \geq x_{i}^{2}, x_{i}^{4} \geq x_{i}^{3} (i = 1, 2, \dots, n)$ . According to Proposition 3, $C^{F} (U) \neq \emptyset$ . □

Remark that if all the characteristic function values of a fuzzy interval game degenerate to real numbers, the condition in Proposition 5 corresponds to the classical balanced game. Therefore, Proposition 5 generalizes the conclusion that a balanced game has a nonempty core in classical game theory. In addition, the conditions in Proposition 4 and Proposition 5 are easy to be checked, since only some classical TU-games are involved.

Example 4. Let $(N, U)$ be a trapezoidal fuzzy interval game with N = {1, 2, 3} and $U$ defined as below: $\begin{matrix} U ({1}) = U ({2}) = U ({3}) = 0, \\ U ({1, 2}) = U ({2, 3}) = [8, 9], \\ U ({1, 3}) = (4, 6, 10, 14), \\ U ({1, 2, 3}) = (10, 12, 16, 20), \end{matrix}$ then $(N, U)$ satisfies the condition in Proposition 5. Therefore, $C^{F} (U) \neq \emptyset$ . For example, ((2, 3, 5, 7) , $(6, 6, 6, 6), (2, 3, 5, 7)) \in C^{F} (U)$ . □

Definition 6. A fuzzy interval game $(N, U)$ is convex if for all S, T ∈ 2^N, $U (S) + U (T) ⊴ U (S \cup T) + U (S \cap T) .$

In classical game theory, the core of a convex game is nonempty. In the following, we attempt to extend this result to trapezoidal fuzzy interval cooperative games. For a trapezoidal fuzzy interval cooperative game $(N, U)$ , we prove that if $(N, U)$ is size monotonic and convex, then its $F$ -core is nonempty.

Definition 7. A trapezoidal fuzzy interval game $(N, U)$ is size monotonic if (N, u_{d
_k}) , k = 1, 2, 3, are all monotonic, which means that u_{d
_k} (S) ≤ u_{d
_k} (T)(k = 1, 2, 3) for all S ⊆ T, S, T ∈ 2^N.

This concept is inspired by Branzei et al. [15], which coincides with a size monotonic interval game when all the characteristic function values degenerate to closed intervals.

Definition 8. [3] Let $F_{1}, F_{2} \in F (R)$ . If there exists $F_{3} \in F (R)$ such that F₁ = F₂ + F₃, then F₃ is said to be the Hukuhara difference between F₁ and F₂, denoted by F₃ = F₁ - _HF₂.

Lemma 1.Let $F_{1}, F_{2} \in T F (R)$ . The Hukuhara differenceF₁ - _HF₂exists if and only ifd_k (F₁) ≥ d_k (F₂), k = 1, 2, 3 .

Lemma 2.Let $F_{1}, F_{2} \in T F (R)$ withF₁ = (a₁, a₂, a₃, a₄) , F₂ = (b₁, b₂, b₃, b₄). IfF₁ - _HF₂exists, thenF₁ - _HF₂ = (a₁ - b₁, a₂ - b₂, a₃ - b₃, a₄ - b₄).

Proposition 6.Let $(N, U)$ be a trapezoidal fuzzy interval game. If $(N, U)$ is size monotonic and convex, then $C^{F} (U) \neq \emptyset$ .

Proof.Let $F_{1} = U ({1}),$ $F_{i} = U ({1, 2, \dots, i}) -_{H} U ({1, 2, \dots, i - 1}),$ i = 2, …, n .

Since $(N, U)$ is size monotonic, by Definition 7 and Lemma 1, we have that F_i (i = 1, …, n) exist and $\begin{matrix} γ_{k} (F_{1}) = γ_{k} (U ({1})), \\ γ_{k} (F_{i}) = γ_{k} (U ({1, 2, \dots, i})) - γ_{k} \\ (U ({1, 2, \dots, i - 1})), i = 2, \dots, n, k = 1, 2, 3, 4 . \end{matrix}$

Next, we show that $(F_{1}, \dots, F_{n}) \in C^{F} (U)$ . From the convexity of $(N, U)$ , we have (N, u_{γ
_k}) , (k = 1, 2, 3, 4) are convex classical games. According to the classical convex game theory [10], we have (γ_k (F₁) , …, γ_k (F_n)) ∈ C (u_{γ
_k}) k = 1, 2, 3, 4. Then by Proposition 3, we get that $(F_{1}, \dots, F_{n}) \in C^{F} (U)$ . □

The Hukuhara-Shapley value of cooperative game with fuzzy characteristic function was introduced in [19]. As a byproduct of Proposition 6, we have the following results.

Corollary 1.Let $(N, U)$ be a trapezoidal fuzzy interval game. If $(N, U)$ is size monotonic and convex, then the Hukuhara-Shapley value of $(N, U)$ , which is defined by

$f (U) = (f_{1} (U), \dots, f_{n} (U))$ with

$f_{i} (U) = \sum_{S \subset N ∖ {i}} \frac{| S |! (n - | S | - 1)!}{n!} [U (S \cup {i}) -_{H} U (S)]$ is in the $F$ -core.

4 An application example

In this section, we shall give an application examples, which is a joint production problem used in [19].

Example 5. Yu et al. [19]. Consider a joint production problem with three decision makers, denoted by player 1, 2 and 3, pooling three kinds of resources to manufacture seven kinds of products. Let N = {1, 2, 3} be the player set. For any S ⊂ N, $U (S)$ represents the revenue of coalition S if the coalition is formed to manufacture the product P_S. However, the revenues are uncertain in normal conditions. Here the wealth of each coalition is represented by triangular fuzzy numbers, shown in Table 1.

If the coalition {1,2,3} is formed, we try to look for the allocations which can stabilize the grand coalition.

Firstly, this fuzzy interval game is convex. In fact, it is easy to see that this game is superadditive. To show the convexity, we only need to check part of expressions in Definition 6. $\begin{matrix} U ({1, 2}) + U ({1, 3}) = (87.2, 96.05, 104.9) \\ ⊴ U ({1, 2, 3}) + U ({1}) = (104, 114, 124), \\ U ({1, 2}) + U ({2, 3}) = (106.2, 113.4, 120.6) \\ ⊴ U ({1, 2, 3}) + U ({2}) = (115.7, 125, 134.3), \\ U ({1, 3}) + U ({2, 3}) = (89, 97.85, 106.7) \\ ⊴ U ({1, 2, 3}) + U ({3}) = (98.6, 108, 118.4) . \end{matrix}$

Secondly, we calculate the characteristic function values of some related classical cooperative games. See Table 2.

We can see that this fuzzy interval game is size monotonic. According to Proposition 6 and Corollary 1, the $F$ -core is non-empty and the Hukuhara-Shapley value is in the $F$ -core. So the Hukuhara-Shapley value can give the grand coalition stability.

Thirdly, calculate the Hukuhara-Shapley value. Therefore, the allocation scheme can be taken as (25.35, 28.64, 31.77) for player 1, (40.70, 42.82, 44.77) for player 2 and (23.55, 26.54, 29.86) for player 3.

Besides, this game also satisfies the conditions in Proposition 3 and Proposition 5. We can use these two propositions to look for other allocation schemes with stability.

5 Concluding remarks

The $F$ -core and the $F$ -balancedness for fuzzy interval cooperative games have been discussed in detail in this paper. A new example is given to show that the $F$ -balancedness is not a sufficient condition for the non-emptiness of the $F$ -core. In addition, some properties of the $F$ -core are also discussed. These conclusions will help to judge the non-emptiness of the $F$ -core and look for elements in the $F$ -core.

We have given three sufficient conditions for the existence of $F$ -core elements. Different sufficient conditions could be considered for further research. Apart from that, we could consider fuzzy interval cooperative games with different fuzzy interval payoffs; for instance, LR-fuzzy intervals.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China and Specialized Research Fund for the Doctoral Program of Higher Education (No. 71071018, 71271029, 71371030, 20111101110036).

References

Gillies

D.B.

, Some theorems on n-person games, Princeton University, 1954.

Dubois

, Fuzzy sets and systems: Theory and applications, Academic press, 1980.

Dubois

, Kerre

, Mesiar

and Prade

, Fuzzy interval analysis, Fundamentals of fuzzy sets, Springer US, 2000.

Owen

, Game Theory, Academic Press, NY1995.

Nishizaki

and Sakawa

, Fuzzy cooperative games arising from linear production programming problems with fuzzy parameters, Fuzzy Sets and Systems114(1) (2000), 11–21.

Puerto

, Fernández

F.R.

and Hinojosa

, Partially ordered cooperative games: Extended core and Shapley value, Annals of Operations Research158(1) (2008), 143–159.

Zadeh

L.A.

, Fuzzy sets, Information and control8(3) (1965), 338–353.

Mallozzi

, Scalzo

and Tijs

, Fuzzy interval cooperative games, Fuzzy Sets and Systems165(1) (2011), 98–105.

Shapley

L.S.

, On balanced sets and cores, Naval Research Logistics Quarterly14(4) (1967), 453–460.

10.

Shapley

L.S.

, Cores of convex games, International Journal of Game Theory1(1) (1971), 11–26.

11.

Mareš

, Additivities in fuzzy coalition games with side-payments, Kybernetika35(2) (1999), 149–166.

12.

Mareš

and Vlach

, Linear coalitional games and their fuzzy extensions, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems9(3) (2001), 341–354.

13.

Mares

, Fuzzy cooperative games: Cooperation with vague expectations, Physica, 2013.

14.

Bondareva

O.N.

, Some applications of linear programming methods to the theory of cooperative games, Problemy Kibernetiki10 (1963), 119–139.

15.

Branzei

, Branzei

, Alparslan Gök

S.Z.

and Tijs

, Cooperative interval games: A survey, Central European Journal of Operations Research18(3) (2010), 397–411.

16.

Alparslan Gök

S.Z.

, Miquel

and Tijs

, Cooperation under interval uncertainty, Mathematical Methods of Operations Research69(1) (2009), 99–109.

17.

Alparslan Gök

S.Z.

, Branzei

and Tijs

, Set-valued solution concepts using interval-type payoffs for interval games, Journal of Mathematical Economics47(4) (2011), 621–626.

18.

Zhao

and Zhang

, A further discussion on cores for interval-valued cooperative game, Knowledge Engineering and Management, Springer, Berlin Heidelberg, pp, 2014, 673–682.

19.

X.H.

and Zhang

, An extension of cooperative fuzzy games, Fuzzy Sets and Systems161(11) (2010), 1614–1634.

A further discussion on fuzzy interval cooperative games

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Fuzzy intervals

2.2 Fuzzy interval cooperative games

3 Some results on fuzzy interval cooperative games

3.1 A new example for the F -balanced condition

3.2 Some properties of the F -core

5 Concluding remarks

Footnotes

Acknowledgments

References

3.1 A new example for the $F$ -balanced condition

3.2 Some properties of the $F$ -core