A simplified method of interval-valued solidarity values for a special class of interval-valued cooperative games

Abstract

The aim of this paper is to extend the concept of the solidarity value to interval-valued (IV) cooperative games and develop a fast and simplified method for computing the IV solidarity values for a special class of IV cooperative games under some weaker conditions. In the method, we find a size (or coalition) monotonicity-like condition so that we can compute the IV solidarity values of a special class of IV cooperative games via determining their lower and upper bounds by utilizing the lower and upper bounds of the IV coalitions’ values, respectively. In addition, some important and useful properties of IV solidarity values are discussed and the feasibility and applicability of the method proposed in this paper are verified with numerical examples as well as comparison analysis with existing methods is conducted.

Keywords

Cooperative game interval-valued cooperative game solidarity value interval uncertainty

1 Introduction

Cooperative game is an important part of game theory. The coalitions’ values of the classic cooperative games are expressed as real numbers. However, due to uncertainty and complexity in some real situations, we often only can predict the ranges of the coalitions’ values rather than exact values. Thus, it seems to be suitable for employing intervals to estimate coalitions’ values and hereby there appear interval-valued (IV) cooperative games [1], which are a generalization of cooperative games. Currently, many researchers pay attention on the solution concepts and related properties of IV cooperative games. Branzei et al. [2] updated the results about IV cooperative games and reviewed various existing and potential applications of IV cooperative games in management situations. Han et al. [3] proposed the IV core and the IV Shapley-like values of IV cooperative games based on a new defined order relation and the Moore’s subtraction of intervals. Guan et al. [4] defined a new ranking method of intervals and hereby investigated the existence of the IV core and proved the relations between the IV core and the IV dominance core. Based on a partial subtraction operator, Palanci et al. [5] studied the IV Shapley value and its properties and also introduced the IV Banzhaf value and the IV egalitarian rule. Palanci et al. [6] studied the IV Shapley value and IV core of a transportation IV game based on the partial subtraction operator. Branzei et al. [7] defined the IV core of IV cooperative games through discussing the IV square dominance and IV dominance imputations. Alparslan Gök et al. [8] introduced some set-valued solution concepts of IV cooperative games, such as the IV core, the IV dominance core, and the IV stable sets. Based on the extended generalized Hukuhara difference, Zou and Zhang [9] considered a type of IV Shapley value for IV cooperative games. Clearly, most of the aforementioned works used the partial subtraction operator and/or the Moore’s interval subtraction [10] which is not invertible and usually enlarges uncertainty of the resulted interval. In order to avoid the above issues, Li and Ye [11] investigated the IV least square prenucleolus without using interval subtraction. Hong and Li [12] developed a nonlinear programming method for computing the IV core.

Recently, several researchers paid attention on the solidarity values of cooperative games, which were firstly introduced by Nowak and Radzik [13]. Casajus and Huettner [14] investigated a new one-parameter family of solidarity values whose center is the solidarity value of a cooperative game. Using the balanced cycle contributions property and invariance axioms, Kamijo and Kongo [15] gave an axiomatization of the egalitarian and solidarity values of cooperative games. Xu et al. [16] characterized the solidarity values of cooperative games from the view of egalitarianism in three approaches which include potential function, axiomatization, and non-cooperative mechanism. Béal et al. [17] introduced a new class of solidarity values which rely on both marginalistic and egalitarian principles. Radzik and Driessen [18] discussed three properties of cooperative games’ values: general probabilistic consistency, λ - standardness and some modifications of the null player property, and hereby used 1/2-standardness and probabilistic consistency to characterize the solidarity value. Calvo and Gutiérrez-López [19] characterized the family of weighted solidarity values of cooperative games by two axiomatizations. Hu and Li [20] presented a new axiomatization of the Shapley-solidarity value for games with a coalition structure. As far as we know, however, there is no research on IV solidarity values of IV cooperative games. Therefore, the primary goal of this paper is to extend the solidarity values to IV cooperative games and hereby develop a simplified and an effective method for computing IV solidarity values of a special class of IV cooperative games. With the help of associated cooperative games of the IV cooperative game, we prove that the solidarity value of the associated cooperative game is a non-decreasing function under some weaker size monotonicity-like condition. Hence, according to the monotonicity condition, we can directly and explicitly determine the lower and upper bounds of the IV solidarity by the lower and upper bounds of the coalitions’ values. The main difference of the proposed method from the aforementioned methods is that it computes IV solidarity values of IV cooperative games without using the interval subtraction and the ranking of intervals and hereby can effectively avoid the irrational issues resulted from noninvertible interval subtraction.

The rest of this paper is organized as follows. Section 2 briefly introduces some concepts of intervals and IV cooperative games. In Section 3, we introduce the concept of IV solidarity values of IV cooperative games and develop a fast and simplified method for computing IV solidarity values of IV cooperative games under some conditions. Some useful and important properties are studied. Section 4 uses numerical examples to demonstrate the feasibility and applicability of the method proposed in this paper and conducts comparison analysis with existing methods. A short conclusion is given in Section 5.

2 Concepts and notations of intervals and interval-valued cooperative games

2.1 Intervals and arithmetic operations

Usually, $\bar{a} = [a_{L}, a_{R}] = {a | a \in R, a_{L} \leq a \leq a_{R}}$ is called an interval, where R is the set of real numbers, a_L ∈ R and a_R ∈ R are called the lower bound and the upper bound of the interval $\bar{a}$ , respectively. Let $\bar{R}$ be the set of intervals on the set R. Some interval arithmetic operations are given as follows [10].

Definition 1. Assume that $\bar{a} = [a_{L}, a_{R}]$ and $\bar{b} = [b_{L}, b_{R}]$ are two intervals on the set $\bar{R}$ . Then, 1) Interval equality: $\bar{a} = \bar{b} \Leftrightarrow a_{L} = b_{L} and a_{R} = b_{R}$ ; 2) Interval addition (or sum): $\bar{a} + \bar{b} = [a_{L} + b_{L}, a_{R} + b_{R}]$ ; 3) Scalar multiplication: $γ \bar{a} = [γ a_{L}, γ a_{R}]$ for any nonnegative real number γ.

2.2 Interval-valued cooperative games

An n-person IV cooperative game $\bar{υ}$ is an ordered-pair $< N, \bar{υ} >$ , where N = {1, 2, …, n} is the set of players and $\bar{υ} : 2^{N} \mapsto \bar{R}$ is the IV characteristic function, $\bar{υ} (S)$ is called the value of the coalition S and $\bar{υ} (\emptyset) = [0, 0]$ . Generally, for any coalition S ⊆ N, $\bar{υ} (S)$ is denoted by the interval $\bar{υ} (S) = [υ_{L} (S), υ_{R} (S)]$ , where υ_L (S) ≤ υ_R (S). We usually write $\bar{υ} (S ∖ i)$ , $\bar{υ} (S \cup i)$ , $\bar{υ} (i)$ , and $\bar{υ} (i, j)$ instead of $\bar{υ} (S ∖ {i})$ , $\bar{υ} (S \cup {i})$ , $\bar{υ} ({i})$ , and $\bar{υ} ({i, j})$ , respectively. In the sequent, an n-person IV cooperative game $< N, \bar{υ} >$ is simply called the IV cooperative game $\bar{υ}$ . The set of n-person IV cooperative games is denoted by ${\bar{G}}^{n}$ . Obviously, if υ_L (S) = υ_R (S) for any coalition S ⊆ N, then the IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ is reduced to a (crisp) cooperative game, denoted by υ, the set of cooperative games is denoted by Gⁿ.

For any pair of IV cooperative games $\bar{υ} \in {\bar{G}}^{n}$ and $\bar{ν} \in {\bar{G}}^{n}$ , according to the case 2) of Definition 1, $< N, \bar{υ} + \bar{ν} >$ is defined as an IV cooperative game with the IV characteristic function $\bar{υ} + \bar{ν}$ , where $\begin{matrix} (\bar{υ} + \bar{ν}) (S) = \bar{υ} (S) + \bar{ν} (S) \\ = [υ_{L} (S) + ν_{L} (S), υ_{R} (S) + ν_{R} (S)] \end{matrix}$ (1) for any coalition S ⊆ N. Usually, $\bar{υ} + \bar{ν}$ is called the sum of the IV cooperative games $\bar{υ} \in {\bar{G}}^{n}$ and $\bar{ν} \in {\bar{G}}^{n}$ . Obviously, $\bar{υ} + \bar{ν}$ is also an IV cooperative game belonging to ${\bar{G}}^{n}$ , i.e., $(\bar{υ} + \bar{ν}) \in {\bar{G}}^{n}$ .

For any IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ , it is easy to see that each player should receive an IV payoff from the cooperation due to the fact that each coalition’s value is an interval. In fact, the IV payoff is the worth (profit or cost) which is allocated to the player i (i ∈ N) when he/she participates in the IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ under the condition that the grand coalition N is reached. Let ${\bar{x}}_{i} (\bar{υ}) = [x_{Li} (\bar{υ}), x_{Ri} (\bar{υ})]$ be an IV payoff, and denote the vector of the IV payoffs for all n players in the grand coalition N by $\bar{x} (\bar{υ}) = ({\bar{x}}_{1} (\bar{υ}), {\bar{x}}_{2} (\bar{υ}), \dots, {\bar{x}}_{n} (\bar{υ}))$ . For an IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ , the efficiency and individual rationality of an IV payoff vector $\bar{x} (\bar{υ}) = ({\bar{x}}_{1} (\bar{υ}), {\bar{x}}_{2} (\bar{υ}), \dots, {\bar{x}}_{n} (\bar{υ}))$ can be expressed as $\sum_{i = 1}^{n} {\bar{x}}_{i} (\bar{υ}) = \bar{υ} (N)$ and ${\bar{x}}_{i} (\bar{υ}) \geq \bar{υ} (i)$ (i = 1, 2, …, n), respectively.

Definition 2. (Symmetric players) For two players i ∈ N and k ∈ N (i ≠ k), if $\bar{υ} (S \cup i) = \bar{υ} (S \cup k)$ for any coalition S ⊆ N ∖ {i, k}, then the players i and k are said to be symmetric in the IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ .

Assume that σ is any permutation on the player set N, we can define a new IV cooperative game ${\bar{υ}}^{σ} \in {\bar{G}}^{n}$ of the IV cooperative games $\bar{υ} \in {\bar{G}}^{n}$ , where the IV characteristic function ${\bar{υ}}^{σ} (S) = \bar{υ} (σ^{- 1} (S))$ for any coalition S ⊆ N. Let σ ^# : Rⁿ ↦ Rⁿ be a mapping so that $σ_{σ (i)}^{#} (z) = z_{i}$ for any vector z = (z₁, z₂, …, z_n) ^T and i ∈ N, where $σ^{#} (z) = (σ_{σ (1)}^{#} (z), σ_{σ (2)}^{#} (z), \dots, σ_{σ (n)}^{#} (z))^{T}$ .

3 Interval-valued solidarity values and properties

For any IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ , we can easily construct an associated cooperative game υ (α) ∈ Gⁿ (α ∈ [0, 1]), where N = {1, 2, …, n} is the set of players and the characteristic function υ (α) of players’ coalitions is given as follows: $υ (α) (S) = (1 - α) υ_{L} (S) + α υ_{R} (S) (S \subseteq N)$ (2) and υ (α) (∅) =0, where the parameter α ∈ [0, 1] is any real number.

According to the concept of the solidarity value [13] of a cooperative game, we can obtain the solidarity value

$ρ^{SV} (υ (α)) = (ρ_{1}^{SV} (υ (α)), ρ_{2}^{SV} (υ (α)), \dots, ρ_{n}^{SV} (υ (α)))^{T}$ of the cooperative game υ (α), whose components are given as follows:

$ρ_{i}^{SV} (υ (α)) = \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ (α) (S) - υ (α) (S \ j)) (i = 1, 2, \dots, n) .$

Combining with Equation (2), the solidarity value of any player i = 1, 2, …, n is obtained as follows: $\begin{matrix} ρ_{i}^{SV} (υ (α)) = & \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \\ \times \sum_{j \in S} {[(1 - α) υ_{L} (S) + α υ_{R} (S)] \\ - [(1 - α) υ_{L} (S ∖ j) + α υ_{R} (S ∖ j)]} \end{matrix}$

Thus, $ρ_{i}^{SV} (υ (α))$ (i = 1, 2, …, n) can be further rewritten as follows: $\begin{matrix} ρ_{i}^{SV} (υ (α)) = \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} [(1 - α) (υ_{L} (S) \\ - υ_{L} (S ∖ j)) + α (υ_{R} (S) - υ_{R} (S ∖ j))] \\ (i = 1, 2, \dots, n) . \end{matrix}$ (3)

Apparently, the solidarity value $ρ_{i}^{SV} (υ (α))$ (i = 1, 2, …, n) of the cooperative game υ (α) is a continuous function of the parameter α ∈ [0, 1].

Lemma 1. For any IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ ,if the following system of inequalities

$\begin{matrix} \sum_{j \in S} (υ_{R} (S) - υ_{L} (S)) \\ \geq \sum_{j \in S} (υ_{R} (S ∖ j) - υ_{L} (S ∖ j)) (\emptyset \neq S \subseteq N) \end{matrix}$ (4) is satisfied, then the solidarity value $ρ_{i}^{SV} (υ (α))$ (i = 1, 2, ⋯ , n) of the cooperative game υ (α) is a monotonic and non-decreasing function of the parameter α ∈ [0, 1].

Proof. For any parameters α ∈ [0, 1] and α′ ∈ [0, 1], it is derived from Equation (3) that $\begin{matrix} ρ_{i}^{SV} (υ (α)) - ρ_{i}^{SV} (υ (α^{'})) \\ = \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (α - α^{'}) [(υ_{R} (S) \\ - υ_{R} (S ∖ j)) - (υ_{L} (S) - υ_{L} (S ∖ j))] \\ (i = 1, 2, \dots, n) . \end{matrix}$

If α ≥ α′, then it easily follows from Equation (4) that $ρ_{i}^{SV} (υ (α)) - ρ_{i}^{SV} (υ (α^{'})) \geq 0$ (i = 1, 2, …, n), which infer that $ρ_{i}^{SV} (υ (α)) \geq ρ_{i}^{SV} (υ (α^{'}))$ (i = 1, 2, …, n).

Accordingly, the solidarity value $ρ_{i}^{SV} (υ (α))$ (i = 1, 2, …, n) is a monotonic and non-decreasing function of the parameter α ∈ [0, 1]. We have completed the proof of Lemma 1.

Equation (4) can be rewritten as follows: $\sum_{j \in S} l (S) \geq \sum_{j \in S} l (S ∖ j) (\emptyset \neq S \subseteq N),$ where l (S) is the value (i.e., interval) length of the coalition S. Thus, if an IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ satisfies Equation (4), then it is said to be size (or coalition) monotonicity-like.

Thus, for any IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ , if it satisfies Equation (4), then according to Lemma 1, the lower and upper bounds of the IV solidarity value ${\bar{ρ}}^{SV} (\bar{υ}) = ({\bar{ρ}}_{1}^{SV} (\bar{υ}), {\bar{ρ}}_{2}^{SV} (\bar{υ}), \dots, {\bar{ρ}}_{n}^{SV} (\bar{υ}))$ can be attained at the lower and upper bounds of the interval [0, 1], respectively, i.e., $\begin{matrix} {\bar{ρ}}_{i}^{SV} (\bar{υ}) = [ρ_{Li}^{SV} (\bar{υ}), ρ_{Ri}^{SV} (\bar{υ})] = [ρ_{i}^{SV} (υ (0)), ρ_{i}^{SV} (υ (1))] \\ (i = 1, 2, \dots, n) . \end{matrix}$

Therefore, according to Equation (3), we can directly and explicitly define the IV solidarity value ${\bar{ρ}}^{SV} (\bar{υ})$ of any IV cooperative game with the size monotonicity-like, whose components are given as follows:

$\begin{matrix} \begin{matrix} {\bar{ρ}}_{i}^{SV} (\bar{υ}) = [\sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{L} (S) - υ_{L} (S ∖ j)), \\ \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{R} (S) - υ_{R} (S ∖ j))] \end{matrix} \\ (i = 1, 2, \dots, n) . \end{matrix}$ (5)

In the sequent, we study some important properties of IV solidarity values of IV cooperative games, which are summarized as in Theorem 1.

Theorem 1. For an y IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ , if it satisfies Equation (4), then there always exists a unique IV solidarity value ${\bar{ρ}}^{SV} (\bar{υ})$ , which satisfies the efficiency, additivity, symmetry, and anonymity properties.

Proof. 1) Existence and uniqueness. It is obvious from the above discussion that ${\bar{ρ}}^{SV} (\bar{υ})$ given by Equation (5) is the IV solidarity value of the IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ which satisfies Equation (4). Moreover, the IV solidarity value ${\bar{ρ}}^{SV} (\bar{υ})$ is unique due to the size monotonicity-like and continuous. Therefore, any IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ which satisfies Equation (4) always has a unique IV solidarity value ${\bar{ρ}}^{SV} (\bar{υ})$ , which is given by Equation (5).

2) Efficiency. It follows from Equation (5) that $\begin{matrix} \sum_{i \in N} ρ_{Li}^{SV} (\bar{υ}) \\ = \sum_{i \in N} \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n!} \frac{1}{s} \sum_{j \in S} (υ_{L} (S) - υ_{L} (S ∖ j)) \\ = \sum_{S \subseteq N : S \neq \emptyset} \frac{(s - 1)! (n - s)!}{n!} \sum_{j \in S} (υ_{L} (S) - υ_{L} (S ∖ j)) \\ = \frac{(n - 1)! (n - n)!}{n!} \sum_{j \in N} (υ_{L} (N) - υ_{L} (N ∖ j)) \\ + \sum_{S \subseteq N : S \neq \emptyset, N} \frac{(s - 1)! (n - s)!}{n!} \sum_{j \in S} (υ_{L} (S) - υ_{L} (S ∖ j)) \\ = \frac{(n - 1)! (n - n)!}{n!} n υ_{L} (N) - \frac{(n - 1)! (n - n)!}{n!} \sum_{j \in N} υ_{L} (N ∖ j) \\ + \sum_{S \subseteq N : S \neq \emptyset, N} \frac{(s - 1)! (n - s)!}{n!} [s υ_{L} (S) - \sum_{j \in S} υ_{L} (S ∖ j)] \\ = υ_{L} (N) - \frac{(n - 1)! (n - n)!}{n!} \sum_{j \in N} υ_{L} (N ∖ j) \\ + \sum_{S \subseteq N : S \neq \emptyset, N} \frac{s! (n - s)!}{n!} υ_{L} (S) - \\ \sum_{S \subseteq N : S \neq \emptyset, N} \frac{(s - 1)! (n - s)!}{n!} \sum_{j \in S} υ_{L} (S ∖ j) \\ = υ_{L} (N) + \sum_{S \subseteq N : S \neq \emptyset, N} \frac{s! (n - s)!}{n!} υ_{L} (S) \\ - \sum_{S \subseteq N : S \neq \emptyset} \frac{(s - 1)! (n - s)!}{n!} \sum_{j \in S} υ_{L} (S ∖ j) \\ = υ_{L} (N) . \end{matrix}$

Analogously, we can prove that $\begin{matrix} \sum_{i \in N} ρ_{Ri}^{SV} (\bar{υ}) \\ = \sum_{i \in N} \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n!} \frac{1}{s} \sum_{j \in S} (υ_{R} (S) - υ_{R} (S ∖ j)) \\ = υ_{R} (N) \end{matrix}$

Hence, according to Definition 1, we obtain $\sum_{i \in N} {\bar{ρ}}_{i}^{SV} (\bar{υ}) = \bar{υ} (N)$ . Namely, the IV solidarity value ${\bar{ρ}}^{SV} (\bar{υ})$ of any IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ which satisfies Equation (4) always possesses the efficiency.

3) Additivity. It follows from Equations (5) and (1) that $\begin{matrix} {\bar{ρ}}_{i}^{SV} (\bar{υ} + \bar{ν}) \\ = [\sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} ((υ_{L} (S) + ν_{L} (S)) - (υ_{L} (S ∖ j) + ν_{L} (S ∖ j))), \\ \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} ((υ_{R} (S) + ν_{R} (S)) - (υ_{R} (S ∖ j) + ν_{L} (S ∖ j)))] \\ = [\sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{L} (S) - υ_{L} (S ∖ j)), \\ \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{R} (S) - υ_{R} (S ∖ j))] + \\ [\sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (ν_{L} (S) - ν_{L} (S ∖ j)), \\ \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (ν_{R} (S) - ν_{L} (S ∖ j))] \\ = {\bar{ρ}}_{i}^{SV} (\bar{υ}) + {\bar{ρ}}_{i}^{SV} (\bar{ν}), \end{matrix}$ i.e., ${\bar{ρ}}_{i}^{SV} (\bar{υ} + \bar{ν}) = {\bar{ρ}}_{i}^{SV} (\bar{υ}) + {\bar{ρ}}_{i}^{SV} (\bar{ν}) (i = 1, 2, \dots, n)$ .

Thus, we can obtain ${\bar{ρ}}^{SV} (\bar{υ} + \bar{ν}) = {\bar{ρ}}^{SV} (\bar{υ}) + {\bar{ρ}}^{SV} (\bar{ν}),$ i.e., the IV solidarity value ${\bar{ρ}}^{SV} (\bar{υ})$ of any IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ which satisfies Equation (4) always possesses the additivity.

4) Symmetry. It easily follows from the assumption and Definition 2 that $\bar{υ} (S \cup i) = \bar{υ} (S \cup k)$ , i.e., $υ_{L} (S \cup i) = υ_{L} (S \cup k)$ and $υ_{R} (S \cup i) = υ_{R} (S \cup k) .$

Accordingly, it is easy to see from Equation (5) that ${\bar{ρ}}_{i}^{SV} (\bar{υ}) = {\bar{ρ}}_{k}^{SV} (\bar{υ})$ . Namely, the IV solidarity value ${\bar{ρ}}^{SV} (\bar{υ})$ of any IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ which satisfies Equation (4) always possesses thesymmetry.

5) Anonymity. For any IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ and any permutation σ on the set N, if $\bar{υ}$ satisfies Equation (4), then it follows from Equation (5)that $\begin{matrix} {\bar{ρ}}_{σ (i)}^{SV} ({\bar{υ}}^{σ}) \\ = [\sum_{S \subseteq N : σ (i) \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{L}^{σ} (S) - υ_{L}^{σ} (S ∖ j)), \\ \sum_{S \subseteq N : σ (i) \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{R}^{σ} (S) - υ_{R}^{σ} (S ∖ j))] \\ = [\sum_{S \subseteq N : σ (i) \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{L} (σ^{- 1} (S)) - υ_{L} (σ^{- 1} (S ∖ j))), \\ \sum_{S \subseteq N : σ (i) \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{R} (σ^{- 1} (S)) - υ_{R} (σ^{- 1} (S ∖ j)))] \\ = [\sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{L} (S) - υ_{L} (S ∖ j)), \\ \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{R} (S) - υ_{R} (S ∖ j))] \\ = {\bar{ρ}}_{i}^{SV} (\bar{υ}), \end{matrix}$ i.e., ${\bar{ρ}}_{σ (i)}^{SV} ({\bar{υ}}^{σ}) = {\bar{ρ}}_{i}^{SV} (\bar{υ})$ (i = 1, 2, …, n). That is to say, the IV solidarity value ${\bar{ρ}}^{SV} (\bar{υ})$ of any IV cooperative game $\bar{υ} \in {\bar{G}}^{n}$ which satisfies Equation (4) always possesses the anonymity.

Thus, we have completed the proof of Theorem 1.

Note that IV solidarity values of IV cooperative games do not always satisfy the individual rationality even if the IV cooperative games satisfyEquation (4).

4 Two numerical examples and computational results and analysis

Example 1. There are three departments in a company, numbered by 1, 2, and 3, respectively. Department 1 cannot make profit if working alone. It makes no contribution to any coalition, either. Departments 2 and 3 make profits when they work alone. They also make some contribution to the coalitions and the profits are usually estimated by using intervals. Hence, the situation may be regarded as a three-person IV cooperative game ${\bar{υ}}^{'}$ , where the departments 1, 2, and 3 are regarded as the players 1, 2, and 3, respectively, the grand coalition is N′ = {1, 2, 3}, the IV characteristic function ${\bar{υ}}^{'}$ is defined as follows: ${\bar{υ}}^{'} (1) = [0, 0]$ , ${\bar{υ}}^{'} (2) = {\bar{υ}}^{'} (1, 2) = [100, 113]$ , ${\bar{υ}}^{'} (3) = {\bar{υ}}^{'} (1, 3) = [135, 143]$ , ${\bar{υ}}^{'} (2, 3) = {\bar{υ}}^{'} (N^{'}) = [291, 306]$ . Let us compute the IV solidarity value ${\bar{ρ}}^{SV} ({\bar{υ}}^{'})$ of the IV cooperative game ${\bar{υ}}^{'}$ .

4.1 Computational results obtained by the method proposed in this paper

According to the above coalitions’ values, we easily have $\begin{matrix} \sum_{j \in {1}} (υ_{R}^{'} (1) - υ_{L}^{'} (1)) = 0 = υ_{R}^{'} (\emptyset) - υ_{L}^{'} (\emptyset), \\ \sum_{j \in {2}} (υ_{R}^{'} (2) - υ_{L}^{'} (2)) = 13 > υ_{R}^{'} (\emptyset) - υ_{L}^{'} (\emptyset) = 0, \\ \sum_{j \in {3}} (υ_{R}^{'} (3) - υ_{L}^{'} (3)) = 8 > υ_{R}^{'} (\emptyset) - υ_{L}^{'} (\emptyset) = 0, \\ \sum_{j \in {1, 2}} (υ_{R}^{'} (1, 2) - υ_{L}^{'} (1, 2)) = 26 > (υ_{R}^{'} (2) - υ_{L}^{'} (2)) \\ + (υ_{R}^{'} (1) - υ_{L}^{'} (1)) = 13, \\ \sum_{j \in {1, 3}} (υ_{R}^{'} (1, 3) - υ_{L}^{'} (1, 3)) = 16 > (υ_{R}^{'} (3) - υ_{L}^{'} (3)) \\ + (υ_{R}^{'} (1) - υ_{L}^{'} (1)) = 8, \\ \sum_{j \in {2, 3}} (υ_{R}^{'} (2, 3) - υ_{L}^{'} (2, 3)) = 30 > (υ_{R}^{'} (3) - υ_{L}^{'} (3)) \\ + (υ_{R}^{'} (2) - υ_{L}^{'} (2)) = 21, \\ and \\ \sum_{j \in N^{'}} (υ_{R}^{'} (N^{'}) - υ_{L}^{'} (N^{'})) \\ = 45 > \sum_{j \in N^{'}} (υ_{R}^{'} (N^{'} ∖ j) - υ_{L}^{'} (N^{'} ∖ j)) = 36, \end{matrix}$ i.e., the IV cooperative game ${\bar{υ}}^{'}$ satisfies Equation (4). Thus, according to Equation (5), we can have $\begin{matrix} ρ_{L 1}^{SV} ({\bar{υ}}^{'}) \\ = \sum_{S \subseteq N^{'} : 1 \in S} \frac{(s - 1)! (3 - s)!}{3! s} \sum_{j \in S} (υ_{L}^{'} (S) - υ_{L}^{'} (S ∖ j)) \\ = \frac{0! 2!}{3! \times 1} (υ_{L}^{'} (1) - υ_{L}^{'} (\emptyset)) + \\ \frac{1! 1!}{3! \times 2} [(υ_{L}^{'} (1, 2) - υ_{L}^{'} (2)) + (υ_{L}^{'} (1, 2) - υ_{L}^{'} (1))] + \\ \frac{1! 1!}{3! \times 2} [(υ_{L}^{'} (1, 3) - υ_{L}^{'} (3)) + (υ_{L}^{'} (1, 3) - υ_{L}^{'} (1))] + \\ \frac{2! 0!}{3! \times 3} [(υ_{L}^{'} (N^{'}) - υ_{L}^{'} (2, 3)) + (υ_{L}^{'} (N^{'}) - υ_{L}^{'} (1, 3)) + \\ (υ_{L}^{'} (N^{'}) - υ_{L}^{'} (1, 2))] \\ = \frac{1}{3} (0 - 0) + \frac{1}{12} [(100 - 100) + (100 - 0)] + \\ \frac{1}{12} [(135 - 135) + (135 - 0)] + \\ \frac{1}{9} [(291 - 291) + (291 - 135) + (291 - 100)] \\ \approx 58.14 \end{matrix}$ and $\begin{matrix} ρ_{R 1}^{SV} ({\bar{υ}}^{'}) \\ = \sum_{S \subseteq N^{'} : 1 \in S} \frac{(s - 1)! (3 - s)!}{3! s} \sum_{j \in S} (υ_{R}^{'} (S) - υ_{R}^{'} (S ∖ j)) \\ = \frac{0! 2!}{3! \times 1} (υ_{R}^{'} (1) - υ_{R}^{'} (\emptyset)) + \\ \frac{1! 1!}{3! \times 2} [(υ_{R}^{'} (1, 2) - υ_{R}^{'} (2)) + (υ_{R}^{'} (1, 2) - υ_{R}^{'} (1))] + \\ \frac{1! 1!}{3! \times 2} [(υ_{R}^{'} (1, 3) - υ_{R}^{'} (3)) + (υ_{R}^{'} (1, 3) - υ_{R}^{'} (1))] + \\ \frac{2! 0!}{3! \times 3} [(υ_{R}^{'} (N^{'}) - υ_{R}^{'} (2, 3)) + (υ_{R}^{'} (N^{'}) - υ_{R}^{'} (1, 3)) + \\ (υ_{R}^{'} (N^{'}) - υ_{R}^{'} (1, 2))] \\ = \frac{1}{3} (0 - 0) + \frac{1}{12} [(113 - 113) + (113 - 0)] + \\ \frac{1}{12} [(143 - 143) + (143 - 0)] + \\ \frac{1}{9} [(306 - 306) + (306 - 143) + (306 - 113)] \\ \approx 60.89 . \end{matrix}$

Analogously, we have $\begin{matrix} ρ_{L 2}^{SV} ({\bar{υ}}^{'}) \approx 109.14, ρ_{R 2}^{SV} ({\bar{υ}}^{'}) \approx 116.31, \\ ρ_{L 3}^{SV} ({\bar{υ}}^{'}) \approx 123.72, ρ_{R 3}^{SV} ({\bar{υ}}^{'}) \approx 128.8 . \end{matrix}$

Therefore, the IV solidarity value ${\bar{ρ}}^{SV} ({\bar{υ}}^{'})$ of the IV cooperative game ${\bar{υ}}^{'}$ can be obtained as follows: ${\bar{ρ}}^{SV} ({\bar{υ}}^{'}) = ([58.14, 60.89], [109.14, 116.31], [123.72, 128.8])$ (6) which means that the player 1 (i.e., Department 1) gets the profit [58.14, 60.89], the player 2 (i.e., Department 2) gets the profit [109.14, 116.31], and the player 3 (i.e., Department 3) gets the profit [123.72, 128.8] from the cooperation.

4.2 Computational results obtained by using the Moore’s interval subtraction

According to the Moore’s interval subtraction [10], i.e.,

$\bar{a} - \bar{b} = [a_{L} - b_{R}, a_{R} - b_{L}]$ , we have $\begin{matrix} {\bar{ρ}}_{i}^{MSV} ({\bar{υ}}^{'}) = \sum_{S \subseteq N : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} ({\bar{υ}}^{'} (S) - {\bar{υ}}^{'} (S ∖ j)) \\ = [\sum_{S \subseteq N^{'} : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{L}^{'} (S) - υ_{R}^{'} (S ∖ j)), \\ \sum_{S \subseteq N^{'} : i \in S} \frac{(s - 1)! (n - s)!}{n! s} \sum_{j \in S} (υ_{R}^{'} (S) - υ_{L}^{'} (S ∖ j))] . \end{matrix}$

Combining with the coalitions’ values given in example 1, we obtain the IV solidarity value ${\bar{ρ}}^{MSV} ({\bar{υ}}^{'})$ based on the Moore’s interval subtraction of the IV cooperative game ${\bar{υ}}^{'}$ as follows: ${\bar{ρ}}^{MSV} ({\bar{υ}}^{'}) = ([52.39, 66.64], [102.31, 123.14], [117.31, 135.22]) .$ (7)

Compare Equation (6) with Equation (7), we find that, on the one hand, Equation (7) enlarges the uncertainty (i.e., the lengths of the profit intervals) of the results, on the other hand, the IV solidarity value ${\bar{ρ}}^{MSV} ({\bar{υ}}^{'})$ based on the Moore’s interval subtraction does not satisfy the efficiency. So it cannot distribute the profit made by the grand coalition wholly.

4.3 Computational results obtained by Zou and Zhang’s method

Zou and Zhang [9] developed an IV Shapley value based on the extended generalized Hukuhara difference, i.e., $\begin{matrix} {\bar{φ}}_{i}^{SH} ({\bar{υ}}^{'}) = \sum_{S \subseteq N^{'} ∖ i} \frac{s! (n - s - 1)!}{n!} ({\bar{υ}}^{'} (S \cup i) \underset{egH}{⊖} {\bar{υ}}^{'} (S)) \\ = [\sum_{S \subseteq N^{'} ∖ i} \frac{s! (n - s - 1)!}{n!} (υ_{L}^{'} (S \cup i) - υ_{L}^{'} (S)), \\ \sum_{S \subseteq N^{'} ∖ i} \frac{s! (n - s - 1)!}{n!} (υ_{R}^{'} (S \cup i) - υ_{R}^{'} (S))] . \end{matrix}$

Hence we can obtain the IV Shapley value ${\bar{Φ}}^{SH} ({\bar{υ}}^{'}) = ({\bar{φ}}_{1}^{SH} ({\bar{υ}}^{'}), {\bar{φ}}_{2}^{SH} ({\bar{υ}}^{'}), \dots, {\bar{φ}}_{n}^{SH} ({\bar{υ}}^{'}))$ of the above IV cooperative game ${\bar{υ}}^{'}$ as follows: ${\bar{Φ}}^{SH} ({\bar{υ}}^{'}) = ([0, 0], [128, 138], [163, 168]),$ (8) which is remarkably different from the above IV solidarity value ${\bar{ρ}}^{SV} ({\bar{υ}}^{'})$ obtained by our method. The main difference is that the player 1’s profit obtained from the IV Shapley value ${\bar{Φ}}^{SH} ({\bar{υ}}^{'})$ is 0 and the players 2 and 3 obtain more profits from the IV Shapley value ${\bar{Φ}}^{SH} ({\bar{υ}}^{'})$ than that from the IV solidarity value ${\bar{ρ}}^{SV} ({\bar{υ}}^{'})$ . Therefore, if the player 1 (i.e., Department 1) is particularly important for the cooperation of all three players from some other considerations/reasons rather than profit, then the player 1 does not participate in the grand coalition according to the IV Shapley value ${\bar{Φ}}^{SH} ({\bar{υ}}^{'})$ . However, the player 1 may participate in the grand coalition according to the IV solidarity value ${\bar{ρ}}^{SV} ({\bar{υ}}^{'})$ . This is just the reason and motivation why we should introduce the IV solidarity value.

Example 2. Let us consider a modified version ${\bar{υ}}^{″} \in {\bar{G}}^{3}$ of the IV cooperative game ${\bar{υ}}^{'}$ given in Example 1. The IV characteristic functions of the IV cooperative game ${\bar{υ}}^{″}$ are given as follows:

${\bar{υ}}^{″} (2) = {\bar{υ}}^{″} (1, 2) = [100, 116]$ , ${\bar{υ}}^{″} (S) = {\bar{υ}}^{'} (S) (S \subseteq N^{″})$ , where N″ = N′ = {1, 2, 3}. Let us discuss the IV solidarity value ${\bar{ρ}}^{SV} ({\bar{υ}}^{″})$ of the IV cooperative game ${\bar{υ}}^{″}$ .

1) According to the above coalitions’ values, we can easily know that the IV cooperative game ${\bar{υ}}^{″}$ satisfies Equation (4), hence according to Equation (5), we have $\begin{matrix} {\bar{ρ}}^{SV} ({\bar{υ}}^{″}) & = & ([58.14, 60.81], [109.14, 116.97], \\ [123.72, 128.22]) . \end{matrix}$

2) If we try to use the partial subtraction operator [2], i.e., $\bar{a} - \bar{b} = [a_{L} - b_{L}, a_{R} - b_{R}] if a_{R} - a_{L} \geq b_{R} - b_{L},$ to calculate the IV solidarity value, we may obtain $\begin{matrix} ρ_{3}^{SV} ({\bar{υ}}^{″}) \\ = \sum_{S \subseteq N^{″} : 3 \in S} \frac{(s - 1)! (3 - s)!}{3! s} \sum_{j \in S} ({\bar{υ}}^{″} (S) - {\bar{υ}}^{″} (S ∖ j)) \\ = \frac{0! 2!}{3! \times 1} ({\bar{υ}}^{″} (3) - {\bar{υ}}^{″} (\emptyset)) + \\ \frac{1! 1!}{3! \times 2} [({\bar{υ}}^{″} (1, 3) - {\bar{υ}}^{″} (3)) + ({\bar{υ}}^{″} (1, 3) - {\bar{υ}}^{″} (1))] + \\ \frac{1! 1!}{3! \times 2} [({\bar{υ}}^{″} (2, 3) - {\bar{υ}}^{″} (3)) + ({\bar{υ}}^{″} (2, 3) - {\bar{υ}}^{″} (2))] + \\ \frac{2! 0!}{3! \times 3} [({\bar{υ}}^{″} (N^{″}) - {\bar{υ}}^{″} (2, 3)) + ({\bar{υ}}^{″} (N^{″}) - {\bar{υ}}^{″} (1, 3)) + \\ ({\bar{υ}}^{″} (N^{″}) - {\bar{υ}}^{″} (1, 2))], \end{matrix}$

However, it is easy to discover that $\begin{matrix} υ_{R}^{″} (2, 3) - υ_{L}^{″} (2, 3) = 15 < υ_{R}^{″} (2) - υ_{L}^{″} (2) = 16, \\ υ_{R}^{″} (1, 2, 3) - υ_{L}^{″} (1, 2, 3) \\ = 15 < υ_{R}^{″} (1, 2) - υ_{L}^{″} (1, 2) = 16 . \end{matrix}$ (9)

Namely, we cannot use the partial subtraction operator [2] to calculate the IV solidarity value of IV cooperative game ${\bar{υ}}^{″}$ .

5 Conclusions

We first introduced the concept of the IV solidarity values of IV cooperative games, which is a natural generalization of that of cooperative games. A simplified and fast method is developed to compute the IV solidarity value of any IV cooperative game with the size monotonicity-like which is first defined and proposed in this paper. Unlike most of existing methods, the proposed method does not utilize the interval subtraction or partial subtraction operator so that it can effectively avoid the resulted irrational issues. Furthermore, we prove that the IV solidarity values of IV cooperative games have some important and useful properties which are desirable as crisp cooperative games. In the future, we will further investigate the solidarity values of cooperative games under other uncertain situations and/or in fuzzy settings.

Footnotes

Acknowledgments

This work was supported by the Key Program of the National Natural Science Foundation of China [grant No. 71231003].

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