Axiomatic representations for nonadditivity and nonmodularity indices: describing interactions of fuzzy measures

Abstract

Both the nonadditivity index and nonmodularity index have emerged as valuable indicators for characterizing the interaction phenomenon within the realm of fuzzy measures. The axiomatic representation plays a crucial role in distinguishing and elucidating the relationship and distinctions between these two interaction indices. In this paper, we employ a set of fundamental and intuitive properties related to interactions, such as equality, additivity, maximality, and minimality, to establish a comprehensive axiom system that facilitates a clear comprehension of the interaction indices. To clarify the impact of new elements’ participation on the type and density of interactions within an initial coalition, we investigate and confirm the existence of proportional and linear effects in relation to null and dummy partnerships, specifically concerning the nonadditivity and nonmodularity indices. Furthermore, we propose the concept of the t-interaction index to depict a finer granularity for the interaction situations within a coalition, which involves subsets at different levels and takes the nonadditivity index and nonmodularity index as special cases. Finally, we establish and discuss the axiomatic theorems and empirical examples of this refined interaction index. In summary, the contributions of this work shed light on the axiomatic characteristics of the t-interaction indices, making it a useful reference for comprehending and selecting appropriate indices within this category of interactions.

1 Introduction

Nonadditivity and nonmodularity are the two fundamental characteristics of fuzzy measures [15] or capacities [6] that distinguish them from probabilistic or additive measures and play a crucial role in capturing and explaining the interaction among coalitions [11, 14]. The nonadditivity index and nonmodularity index have been successively proposed in recent studies [17, 18] as metrics to quantify the nature and intensity of interaction among elements or variables within a coalition.

The two interaction indices share several common properties [1]. For instance, they both condense diverse types of interactions into a unified range of values between -1 and 1. They attain their extreme values of 1 and -1 in unanimity games and their dual counterparts, while they achieve a middle value of 0 for additive measures. Additionally, these indices consistently maintain the same sign for superadditive and subadditive measures [19]. These similarities often result in confusion when selecting the appropriate index. Consequently, the axiomatic representation of these interaction indices becomes crucial from both theoretical and practical standpoints [8, 12].

In this paper, we establish an axiomatic representation system based on intuitive and fundamental properties of interactions. The first property, referred to as the equality property, asserts that all or a portion of subsets within a coalition contribute equally to the interaction index. The second property, called additivity, states that the interaction index of additive measures is always zero, indicating independence among the elements or variables. Additionally, we consider the properties of maximality and minimality, which indicate that the extreme values of the interaction index should only be achieved by the minimum and maximum fuzzy measures. We observe that nonadditivity and nonmodularity indices share common characteristics. Both indices are linear functions of fuzzy measures and adhere to the principles of additivity, maximality, and minimality. The primary distinction lies in the equality property, where nonadditivity requires equality among all subsets, while nonmodularity requires equality between the top and bottom levels of subsets.

Indeed, in reference [2], a specific axiomatic representation investigation has been carried out, focusing on the nonmodularity index within the context of dummy or independent partnerships. The consideration of null and dummy coalitions or partnerships [12] is highly valuable for effectively exploring and evaluating the impact of new elements’ participation on the type and density of interactions within an initial coalition. Therefore, the second contribution of this paper involves confirming the presence of proportional and linear effects in relation to null and dummy partnerships, specifically with regard to the nonadditivity and nonmodularity indices, see Theorems 3, 4, 5 and 6.

The third and final objective of this paper is to achieve a higher level of granularity in describing interactions. To accomplish this, we introduce the concept of the t-interaction index, which serves as an intermediary index between the nonadditivity and nonmodularity indices. By incorporating different levels of subsets in the interaction indices, we enable a more comprehensive analysis of various intermediate interaction situations.

The structure of the remaining sections in this paper is as follows: Section 2 provides an brief overview of fuzzy measures, nonadditivity index, and nonmodularity index which lays the foundation for the subsequent discussions. In Section 3, we introduce a set of axiomatic theorems and engage in detailed discussions concerning their implications. The investigation into the null and dummy partnerships regarding both the nonadditivity and nonmodularity indices is presented in Section 4. Section 5 introduces the concept of the t-interaction index, discussing its properties, axiomatic representations, and presenting some empirical analyses. Finally, the paper is concluded in Section 6.

2 Preliminaries

Let N = {1, …, n}, n ≥ 2, be the multiple elements set, $\mathcal{P}\left(\mathcal{N}\right)$ be its power set, and |S| denote the cardinality of subset S ⊆ N. The definitions listed in this section can be found in references [1, 16].

Definition 1. A set function $\mu :\mathcal{P}\left(N\right)\to [0,1]$ is called as a fuzzy measure on N if

Boundary condition: μ (∅) =0, μ (N) =1 ;

Monotonicity condition: ∀A, B ⊆ N, if A ⊆ B then μ (A) ≤ μ (B).

The nonadditivity and modularity indices serve as the explicit representations of monotonicity in a fuzzy measure, providing intuitive insights into the diverse range of interaction phenomena observed among elements.

Definition 2. A fuzzy measure μ on N is additive, if μ (A ∪ B) = μ (A) + μ (B) for arbitrary pair of nonempty disjoint subsets A, B of N. Further, μ is called superadditive, subadditive, strictly superadditive, and strictly subadditive if the direction "=" in former equation changes to ≥, ≤, > and <, respectively.

Definition 3. A fuzzy measure μ on a set N is called modular if it satisfies the equation: μ (A ∪ B) + μ (A ∩ B) = μ (A) + μ (B), for all subsets A, B ⊆ N where A ∩ B ≠ A, B.

Moreover, μ is classified as supermodular, submodular, strictly supermodular, or strictly submodular based on the direction of the inequality in the aforementioned equation, where the equality "=" is replaced by "≥", "≤", "> ", or "< " respectively.

Nonmodularity is a broader concept compared to nonadditivity. Nonmodularity implies nonadditivity, but the reverse is not necessarily true. Typically, an additive or modular fuzzy measure indicates a degree of independence among all elements. Similarly, a strictly superadditive/supermodular, strictly subadditive/submodular, superadditive/supermodular, or subadditive/submodular fuzzy measure represents varying degrees of complementarity, substitutability, non-substitutability, and non-complementarity among the elements, respectively.

For convenience, we will utilize the notation " $\stackrel{★}{=}$ " to represent "= (resp . ≥ , ≤ , > , and, <)". Additionally, the term "★-additive" will be used to denote "additive (resp. superadditive, subadditive, strictly superadditive, and strictly subadditive)", while "★-modular" will indicate "modular (resp. supermodular, submodular, strictly supermodular, and strictly submodular)".

A fuzzy measure μ on N is called ★-additive within S ⊆ N, if $\mu (A\cup B)\stackrel{★}{=}\mu \left(A\right)+\mu \left(B\right),\forall A,B\subseteq S,A,B\ne \varnothing ,A\cap B=\varnothing .$ A fuzzy measure μ on N is called ★-modular within S ⊆ N if $\mu (A\cup B)\stackrel{★}{=}\mu \left(A\right)+\mu \left(B\right)-\mu (A\cap B),\forall A,B\subseteq S,A\cap B\ne A,B$ .

Definition 4. The unanimity game focused on a subset A ⊆ N, denoted as ${\varepsilon }_A^{+}$ , is a fuzzy measure such that ${\varepsilon }_A^{+}\left(B\right)=1$ if and only if B ⊇ A, B ⊆ N, and ${\varepsilon }_A^{+}\left(B\right)=0$ otherwise. The dual of the unanimity game, denoted as ${\varepsilon }_A^{-}$ , is a fuzzy measure such that ${\varepsilon }_A^{-}\left(B\right)=1$ if and only if B∩ A ≠ ∅, and ${\varepsilon }_A^{-}\left(B\right)=0$ otherwise.

To explicitly capture the nature and intensity of interactions among elements, nonadditivity and nonmodularity indices have been sequentially introduced [1, 18]. These indices provide a framework for quantifying and characterizing different types of interactions.

Definition 5. The nonadditivity index of subset A with respect to a fuzzy measure μ on N is given as $n_{\mu }\left(A\right)=\mu \left(A\right)-\frac{1}{2^{\left|A\right|-1}-1}\sum _{C\subset A}\mu \left(C\right).$(1)

Definition 6. The nonmodularity index of subset A with respect to a fuzzy measure μ on N is given as $d_{\mu }\left(A\right)=\mu \left(A\right)-\frac{1}{\left|A\right|}\sum _{\left\{i\right\}\subset A}\left[\mu \right(\left\{i\right\})+\mu (A\setminus \left\{i\right\}\left)\right].$(2)

Nonadditivity and nonmodularity indices possess several desirable properties, including additivity, uniform range, maximality, and minimality [1, 18].

★-modularity and ★-additivity: If a fuzzy measure μ on N is ★-modular or ★-additive, then n _μ (A) , d _μ (A) overset ★ =0, ∀A ⊆ N, |A|≥2. If a fuzzy measure μ on N is ★-modular or ★-additive, then n _μ (A) , d _μ (A) overset ★ =0, ∀A ⊆ N, |A|≥2.

Uniform Range: -1 ≤ n _μ (A) , d _μ (A) ≤1, ∀A ⊆ N.

Maximality and Minimality: n _μ (A) , d _μ (A) =1   ⇔   $\mu \left(S\right)={\varepsilon }_A^{+}\left(S\right)$ , S ⊆ A, and n _μ (A) , d _μ (A) = -1   ⇔   $\mu \left(S\right)={\varepsilon }_A^{-}\left(S\right),S\cap A\ne \varnothing$ .

Indeed, we can designate ${\varepsilon }_A^{+}$ as the minimum fuzzy measure on A, and ${\varepsilon }_A^{-}$ as the maximum fuzzy measure on A. This pair of fuzzy measures represents the extreme scenarios of interaction within subset A [13]: the former characterizes a fully complementary case, while the latter denotes a fully redundant case. These properties of nonadditivity and nonmodularity indices play a important role in establishing theoretical frameworks and facilitating practical applications.

3 The axiomatization of two types of interaction indices

We first confine that nonadditivity index and nonmodularity index as linear function of fuzzy measure. For a linear function of μ, $l_{\mu }\left(A\right)=\sum _{B\subseteq A}{\alpha }_B\mu \left(B\right),$(3) we can define the following properties:

Equality for All Subsets (EAS), i.e., all the proper subsets’ coefficients in linear function l _μ (A) are same and not equal to zero: ${\alpha }_B={\alpha }_C\ne 0,\forall B,C\subset A.$(4)

Additivity (Add), i.e., the linear function l of A, |A|>1, is zero if the fuzzy measure is additive within A: $\begin{array}{cc}\hfill \mu \left(A\right)=& \mu \left(B\right)+\mu \left(C\right)\forall B,C\subset A,\hfill \\ \hfill & B\cap C=\varnothing ,B\cup C=A\hfill \\ \hfill \Rightarrow & l_{\mu }\left(A\right)=0.\hfill \end{array}$(5)

Maximality (Max), i.e., the maximum value of the linear function l of A, |A|≥1, is obtained by the minimum fuzzy measure on A: $\begin{array}{cc}\hfill \mu \left(A\right)& =1\mathrm{and}\mu \left(B\right)=0,\forall B\subseteq A\hfill \\ \hfill & \Rightarrow l_{\mu }\left(A\right)=1.\hfill \end{array}$(6)

Minimality (Min), i.e., the minimum value of the linear function l of A, |A|>1, is obtained by the maximum fuzzy measure on A: $\begin{array}{cc}\hfill \mu \left(A\right)& =1\mathrm{and}\mu \left(B\right)=1,\forall \varnothing \ne B\subset A\hfill \\ \hfill & \Rightarrow l_{\mu }\left(A\right)=-1.\hfill \end{array}$(7)

3.1 The axioms of nonadditivity index

Theorem 1. The linear function l _μ (A) is the nonadditivity index n _μ (A) if and only if one of following conditions holds:

l _μ (A) has properties (EAS), (Max) and (Add)

l _μ (A) has properties (EAS), (Max) and (Min)

Proof. From the properties of nonadditivity index, the sufficient conditions of this theorem hold. And now we focus on the necessity conditions.

Since l _μ (A) has properties (EAS), we can let α _B  = y ≠ 0, ∀ B ⊂ A and α _A  = x ≠ 0, Then from the property (Max), if μ is the minimum fuzzy measure on A, we have $\begin{array}{cc}\hfill l_{\mu }\left(A\right)=& x\mu \left(A\right)+\sum _{B\subset A}y\mu \left(B\right)=x\mu \left(A\right)=1\hfill \\ \hfill & \Rightarrow x=1.\hfill \end{array}$ Then from the property (Add), if μ is additive within A, we have

$\begin{array}{cc}\hfill l_{\mu }\left(A\right)=& \mu \left(A\right)+\sum _{B\subset A}y\mu \left(B\right)\hfill \\ \hfill =& \mu \left(A\right)+\sum _{B,A\backslash B,B\ne \varnothing }y\left[\mu \right(B)+\mu (A\backslash B]\hfill \\ \hfill =& \mu \left(A\right)+(2^{\left|A\right|-1}-1)y\mu \left(A\right)\hfill \\ \hfill =& 0\hfill \end{array}$

We have $y=-\frac{1}{2^{\left|A\right|-1}-1}$ . That is, l _μ (A) is nonadditivity index.

From above, we have x = 1 from the properties of (EAS) and (Max). Then from the property (Min), if μ is the maximum fuzzy measure on A, we have $\begin{array}{cc}\hfill l_{\mu }\left(A\right)=& \mu \left(A\right)+\sum _{B\subset A}y\mu \left(B\right)\hfill \\ \hfill =& 1+\sum _{B\subset A}y\hfill \\ \hfill =& 1+(2^{\left|A\right|}-2)y\hfill \\ \hfill =& -1\hfill \end{array}$

We have $y=-\frac{1}{2^{\left|A\right|-1}-1}$ . That is, l _μ (A) is nonadditivity index. □

Actually, the Singleton Value Unchanged (SVU) property will hold usually for a linear function of μ, i.e., the value of the linear function l of singleton equals to its fuzzy measure value: $l_{\mu }\left(\right\{i\left\}\right)=\mu \left(\right\{i\left\}\right).$(8) Then in order to coincide with the one dimensional representation and also without loss of generality, we can define another linear function of μ as: ${\overline{l}}_{\mu }\left(A\right)=\mu \left(A\right)+\sum _{B\subset A}{\alpha }_B\mu \left(B\right).$(9)

Corollary 1. The linear function ${\overline{l}}_{\mu }\left(A\right)$ is the nonadditivity index n _μ (A) if and only if one of following conditions holds:

${\overline{l}}_{\mu }\left(A\right)$ has properties (EAS) and (Add)

${\overline{l}}_{\mu }\left(A\right)$ has properties (EAS) and (Min)

Proof. From the proof of Theorem 1, one can easily get the results. □

3.2 The axioms of nonmodularity index

We notice that in Equation (2), only one dimensional sets as well as their complementary sets are involved in the nonmodularity index. Here we can introduce another property of linear function of fuzzy measure as:

1-dimension sets and their complements involved with equality (1-DCI), i.e., besides the subset itself, only the one dimensional subsets and their complement sets, the |A|-1 dimensional subsets, are equally involved in the linear function l _μ (A) , |A|>1: $\begin{array}{cc}\hfill & {\alpha }_B={\alpha }_C\ne 0\mathrm{if}\left|B\right|,\left|C\right|=1,\left|A\right|-1;\hfill \\ \hfill & {\alpha }_A\ne 0;\hfill \\ \hfill & {\alpha }_D=0,\mathrm{if}\left|D\right|\ne 1,\left|A\right|-1.\hfill \end{array}$(10)

Theorem 2. The linear function l _μ (A) is the nonmodularity index d _μ (A) if and only if one of following conditions holds:

l _μ (A) has properties (1-DCI), (Max) and (Add)

l _μ (A) has properties (1-DCI), (Max) and (Min)

Proof. From the properties of nonmodularity index, the sufficient conditions of this theorem hold. We focus on the necessity conditions. From the property (Max), we have α _A  = 1

From the properties (1-DCI) and (Add), if μ is additive within A, we have $\begin{array}{cc}\hfill l_{\mu }\left(A\right)=& \mu \left(A\right)+\sum _{B\subset A,\left|B\right|=1,\left|A\right|-1}y\mu \left(B\right)\hfill \\ \hfill =& \mu \left(A\right)+\sum _{i\in A}y\left[\mu \right(\left\{i\right\})+\mu (A\backslash i\left)\right]\hfill \\ \hfill =& \mu \left(A\right)+\left|A\right|y\mu \left(A\right)\hfill \\ \hfill =& 0\hfill \end{array}$ We have $y=-\frac{1}{\left|A\right|}$ . That is, l _μ (A) is nonmodularity index.

From the properties (1-DCI) and (Min), if μ is the maximum fuzzy measure on A, we have $\begin{array}{cc}\hfill l_{\mu }\left(A\right)=& \mu \left(A\right)+\sum _{B\subset A,\left|B\right|=1,\left|A\right|-1}y\mu \left(B\right)\hfill \\ \hfill =& 1+2\left|A\right|y=-1\hfill \end{array}$

We have $y=-\frac{1}{\left|A\right|}$ . That is, l _μ (A) is nonmodularity index.

Corollary 2. The linear function ${\overline{l}}_{\mu }\left(A\right)$ is the nonmodularity index d _μ (A) if and only if one of following conditions holds:

${\overline{l}}_{\mu }\left(A\right)$ has properties (1-DCI) and (Add)

${\overline{l}}_{\mu }\left(A\right)$ has properties (1-DCI) and (Min)

Proof. From the proof of Theorem 2, one can easily get the results. □

4 The interaction indices with respect to null and dummy subset

4.1 The properties about null elements

Definition 7. [1, 11] A subset B is called null for subset C, C ⊆ N \ B with respect to fuzzy measure μ on N if μ (D∪ B) = μ (D) , ∀ D ⊂ C, D ≠ ∅.

In the above definition, if let A = C ∪ B, we have μ (A) = μ (A \ B), i.e., B is null for A \ B.

Theorem 3. If B is null for A \ B with respect to fuzzy measure μ on N, then

$\begin{array}{cc}\hfill n_{\mu }\left(A\right)=& \frac{2^{\left|A\right|-1}-2^{\left|B\right|}}{2^{\left|A\right|-1}-1}{n}_{\mu }(A\backslash B)\hfill \end{array}$(11)

Proof.

From the definition of null elements and Equation (1). nonaddi of fuzzy measure, we get $\begin{array}{cc}\hfill n_{\mu }\left(A\right)=& \mu (A\backslash B)-\frac{1}{2^{\left|A\right|-1}-1}\sum _{C\subseteq A}\mu (C\backslash B).\hfill \end{array}$ One can figure out that set A \ B appears 2^|B| - 1 times in the set {C \ B} _C⊂A and set D ⊂ A \ B appears 2^|B| times in the set {C \ B} _C⊂A. Hence, the coefficient of μ (A \ B) should be $1-\frac{2^{\left|B\right|}-1}{2^{\left|A\right|-1}-1}$ and the coefficient of μ (D|D ⊂ A \ B) should be $-\frac{2^{\left|B\right|}}{2^{\left|A\right|-1}-1}$ . That is,

$\begin{array}{cc}\hfill n_{\mu }\left(A\right)=& (1-\frac{2^{\left|B\right|}-1}{2^{\left|A\right|-1}-1})\mu (A\backslash B)\hfill \\ \hfill & -\frac{2^{\left|B\right|}}{2^{\left|A\right|-1}-1}\sum _{D\subset A\backslash B}\mu \left(D\right)\hfill \\ \hfill =& \left(\frac{2^{\left|A\right|-1}-1-2^{\left|B\right|}+1}{2^{\left|A\right|-1}-1}\right)\mu (A\backslash B)\hfill \\ \hfill & -\frac{2^{\left|B\right|}}{2^{\left|A\right|-1}-1}\sum _{D\subset A\backslash B}\mu \left(D\right)\hfill \\ \hfill & \hfill \end{array}$ $\begin{array}{cc}\hfill =& \left(\frac{2^{\left|A\right|-1}-2^{\left|B\right|}}{2^{\left|A\right|-1}-1}\right)\mu (A\backslash B)\hfill \\ \hfill & -\frac{2^{\left|B\right|}}{2^{\left|A\right|-1}-1}\sum _{D\subset A\backslash B}\mu \left(D\right)\hfill \\ \hfill =& \left(\frac{2^{\left|A\right|-1}-2^{\left|B\right|}}{2^{\left|A\right|-1}-1}\right)\left(\mu \right(A\backslash B)\hfill \\ \hfill & -\frac{2^{\left|B\right|}}{2^{\left|A\right|-1}-2^{\left|B\right|}}\sum _{D\subset A\backslash B}\mu \left(D\right))\hfill \\ \hfill =& \left(\frac{2^{\left|A\right|-1}-2^{\left|B\right|}}{2^{\left|A\right|-1}-1}\right)\left(\mu \right(A\backslash B)\hfill \\ \hfill & -\frac{1}{2^{\left|A\right|-\left|B\right|-1}-1}\sum _{D\subset A\backslash B}\mu \left(D\right))\hfill \\ \hfill & \hfill \end{array}$ Considering the following fact, $\begin{array}{cc}\hfill & n_{\mu }(A\backslash B)\hfill \\ \hfill & =\mu (A\backslash B)-\frac{1}{2^{\left|A\right|-\left|B\right|-1}-1}\sum _{D\subset A\backslash B}\mu \left(D\right)\hfill \end{array}$ we can have the result.□

Theorem 4. If B is null for A \ B with respect to fuzzy measure μ on N, then $\begin{array}{cc}\hfill d_{\mu }\left(A\right)=& \frac{\left|A\right|-\left|B\right|}{\left|A\right|}{d}_{\mu }(A\backslash B)\hfill \end{array}$(12)

Proof.

From the definition of null elements and Equation (2), we get $\begin{array}{cc}\hfill d_{\mu }\left(A\right)=& \mu (A\backslash B)\hfill \\ \hfill & -\frac{1}{\left|A\right|}\sum _{\left\{i\right\}\subset A}\left[\mu \right(\left\{i\right\}\backslash B)+\mu (A\setminus \left\{i\right\}\backslash B\left)\right].\hfill \end{array}$ One can figure out that set A \ B appears |B| times in the set {A ∖ {i} \ B} _{i}⊂A and set {i} ⊂ A \ B appears once in the set {{i} \ B} _{i}⊂A. Hence, the coefficient of μ (A \ B) should be 1 - |B| and the coefficient of [μ ({i} \ B) + μ (A ∖ {i} \ B)] for {i} ⊂ A \ B should be $-\frac{1}{\left|A\right|}$ . That is, $\begin{array}{cc}\hfill d_{\mu }\left(A\right)=& (1-\frac{\left|B\right|}{\left|A\right|})\mu (A\backslash B)\hfill \\ \hfill & -\frac{1}{\left|A\right|}\sum _{\left\{i\right\}\subset A\backslash B}\left[\mu \right(\left\{i\right\})+\mu (A\setminus B\backslash \left\{i\right\}\left)\right]\hfill \\ \hfill =& \left(\frac{\left|A\right|-\left|B\right|}{\left|A\right|}\right)\mu (A\backslash B)\hfill \\ \hfill & -\frac{1}{\left|A\right|}\sum _{\left\{i\right\}\subset A\backslash B}\left[\mu \right(\left\{i\right\})+\mu (A\setminus B\backslash \left\{i\right\}\left)\right]\hfill \\ \hfill & \hfill \end{array}$ $\begin{array}{cc}\hfill =& \left(\frac{\left|A\right|-\left|B\right|}{\left|A\right|}\right)\left(\mu \right(A\backslash B)-\frac{1}{\left|A\right|-\left|B\right|}\hfill \\ \hfill & \sum _{\left\{i\right\}\subset A\backslash B}\left[\mu \right(\left\{i\right\})+\mu (A\setminus B\backslash \left\{i\right\}\left)\right])\hfill \\ \hfill & \hfill \end{array}$ Considering the expression of d _μ (A \ B), we can have the result. □

4.2 The properties about dummy elements and mutually independent coalitions

Definition 8. [7] The criterion r is called dummy criterion if μ (A ∪ {r}) = μ (A) + μ ({r}) for all A ⊂ N, r ∉ A.

Definition 9. Two subset A and B, A∩ B = ∅ are mutually independent with respect to μ if for any S ⊆ A we have μ (S ∪ B) = μ (S) + μ (B) and for any T ⊆ B we have μ (A ∪ T) = μ (A) + μ (T).

Theorem 5. if A and B, A∩ B = ∅ are mutually independent with respect to μ, then for the nonadditivity index, we have

$\begin{array}{cc}\hfill n_{\mu }(A\cup B)=& \frac{2^{\left|A\right|+\left|B\right|-1}-2^{\left|B\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}{n}_{\mu }\left(A\right)\hfill \\ \hfill & +\frac{2^{\left|A\right|+\left|B\right|-1}-2^{\left|A\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}{n}_{\mu }\left(B\right).\hfill \end{array}$(13)

Proof. $\begin{array}{cc}\hfill & n_{\mu }(A\cup B)\hfill \\ \hfill =& \mu (A\cup B)-\frac{1}{2^{\left|A\right|+\left|B\right|-1}-1}\sum _{C\subset A\cup B}\mu \left(C\right)\hfill \\ \hfill =& \mu \left(A\right)+\mu \left(B\right)-\frac{2^{\left|B\right|}-1}{2^{\left|A\right|+\left|B\right|-1}-1}\mu \left(A\right)\hfill \\ \hfill & -\frac{2^{\left|A\right|}-1}{2^{\left|A\right|+\left|B\right|-1}-1}\mu \left(B\right)\hfill \\ \hfill & -\frac{2^{\left|B\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}\sum _{C\subset A}\mu \left(C\right)\hfill \\ \hfill & -\frac{2^{\left|A\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}\sum _{D\subset B}\mu \left(D\right)\hfill \\ \hfill =& (1-\frac{2^{\left|B\right|}-1}{2^{\left|A\right|+\left|B\right|-1}-1})\mu \left(A\right)\hfill \\ \hfill & -\frac{2^{\left|B\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}\sum _{C\subset A}\mu \left(C\right)\hfill \\ \hfill & +(1-\frac{2^{\left|A\right|}-1}{2^{\left|A\right|+\left|B\right|-1}-1})\mu \left(B\right)\hfill \\ \hfill & -\frac{2^{\left|A\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}\sum _{D\subset B}\mu \left(D\right)\hfill \\ \hfill =& \left(\frac{2^{\left|A\right|+\left|B\right|-1}-1-2^{\left|B\right|}+1}{2^{\left|A\right|+\left|B\right|-1}-1}\right)\mu \left(A\right)\hfill \\ \hfill & -\frac{2^{\left|B\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}\sum _{C\subset A}\mu \left(C\right)\hfill \\ \hfill & +\left(\frac{2^{\left|A\right|+\left|B\right|-1}-1-2^{\left|A\right|}+1}{2^{\left|A\right|+\left|B\right|-1}-1}\right)\mu \left(B\right)\hfill \\ \hfill & -\frac{2^{\left|A\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}\sum _{D\subset B}\mu \left(D\right)\hfill \\ \hfill =& \left(\frac{2^{\left|A\right|+\left|B\right|-1}-2^{\left|B\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}\right)\hfill \\ \hfill & \hfill \end{array}$

$\begin{array}{cc}\hfill & \left(\mu \right(A)-\frac{2^{\left|B\right|}}{2^{\left|A\right|+\left|B\right|-1}-2^{\left|B\right|}}\sum _{C\subset A}\mu \left(C\right))\hfill \\ \hfill & +\left(\frac{2^{\left|A\right|+\left|B\right|-1}-2^{\left|A\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}\right)\hfill \\ \hfill & \left(\mu \right(B)-\frac{2^{\left|A\right|}}{2^{\left|A\right|+\left|B\right|-1}-2^{\left|A\right|}}\sum _{D\subset B}\mu \left(D\right))\hfill \\ \hfill =& \left(\frac{2^{\left|A\right|+\left|B\right|-1}-2^{\left|B\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}\right)n_{\mu }\left(A\right)\hfill \\ \hfill & +\left(\frac{2^{\left|A\right|+\left|B\right|-1}-2^{\left|A\right|}}{2^{\left|A\right|+\left|B\right|-1}-1}\right)n_{\mu }\left(B\right)\hfill \end{array}$

Theorem 6. if A and B, A∩ B = ∅ are mutually independent with respect to μ, then for the nonmodularity index, we have $\begin{array}{cc}\hfill d_{\mu }(A\cup B)=& \frac{\left|A\right|}{\left|A\right|+\left|B\right|}{d}_{\mu }\left(A\right)+\frac{\left|B\right|}{\left|A\right|+\left|B\right|}{d}_{\mu }\left(B\right).\hfill \end{array}$(14)

Proof. $\begin{array}{cc}\hfill d_{\mu }(A\cup B)=& \mu (A\cup B)-\frac{1}{\left|A\right|+\left|B\right|}\hfill \\ \hfill & \sum _{\left\{i\right\}\subset A\cup B}\left[\mu \right(\left\{i\right\})+\mu (A\cup B\setminus \left\{i\right\}\left)\right]\hfill \\ \hfill =& \mu \left(A\right)+\mu \left(B\right)-\frac{\left|B\right|}{\left|A\right|+\left|B\right|}\mu \left(A\right)\hfill \\ \hfill & -\frac{\left|A\right|}{\left|A\right|+\left|B\right|}\mu \left(B\right)\hfill \\ \hfill & -\frac{1}{\left|A\right|+\left|B\right|}\sum _{\left\{i\right\}\subset A}\left[\mu \right(\left\{i\right\})+\mu (A\setminus \left\{i\right\}\left)\right]\hfill \\ \hfill & \hfill \end{array}$ $\begin{array}{cc}\hfill & -\frac{1}{\left|A\right|+\left|B\right|}\sum _{\left\{j\right\}\subset B}\left[\mu \right(\left\{j\right\})+\mu (B\setminus \left\{i\right\}\left)\right]\hfill \\ \hfill =& \frac{\left|A\right|}{\left|A\right|+\left|B\right|}\left(\mu \right(A)-\frac{1}{\left|A\right|}\hfill \\ \hfill & \sum _{\left\{i\right\}\subset A}\left[\mu \right(\left\{i\right\})+\mu (A\setminus \left\{i\right\}\left)\right])\hfill \\ \hfill & +\frac{\left|B\right|}{\left|A\right|+\left|B\right|}\left(\mu \right(B)-\frac{1}{\left|B\right|}\hfill \\ \hfill & \sum _{\left\{j\right\}\subset B}\left[\mu \right(\left\{j\right\})+\mu (B\setminus \left\{j\right\}\left)\right])\hfill \end{array}$ That is, we have the result. □

The above Theorem 6 has been extensively investigated in reference [2]. In fact, the nonmodularity index exhibits various other desirable properties in different scenarios, including dummy coalition, dummy partnership, and non-interacting cases. For a comprehensive understanding of these properties and their implications, we recommend referring to the detailed discussions in [2].

5 Interaction granularity and t -interaction indices

5.1 Nonadditivity index and nonmodularity index as special cases of t-interaction index

Upon examining the mathematical structures of the aforementioned indices, it becomes apparent that nonadditivity encompasses all subsets within A, while the nonmodularity index only involves subsets at two levels: singleton subsets and subsets of size |A|-1. By considering nonadditivity and nonmodularity as two extreme cases that encompass all subsets of A, we can establish intermediate cases of interaction indices, such as those solely involving subsets at a specific level, such as t-level subsets. This allows for the exploration of various levels of granularity when describing and analyzing the interactions within A.

Definition 10. The t-interaction index of subset A with respect to a fuzzy measure μ on N is defined as $\begin{array}{cc}\hfill t_{\mu }\left(A\right)=& \mu \left(A\right)-\frac{1}{{\sum }_{k=1}^t\left(\left|A\right|k\right)}\hfill \\ \hfill & \sum _{B\subset A,0<\left|B\right|\le t}\left[\mu \right(B)+\mu (A\setminus B\left)\right].\hfill \end{array}$(15) where t ≤ |A|/2.

When t = 1, the t-interaction index collapses into nonmodularity index. When t = |A|/2, the t-interaction index is just the nonadditivity index.

Theorem 7. If a fuzzy measure μ on N is ★-modular or ★-additive, then t _μ (A) overset ★ =0, ∀A ⊆ N, |A|≥2.

Proof. From Equation (15), we have $\begin{array}{cc}\hfill t_{\mu }& \left(A\right)=\frac{1}{{\sum }_{k=1}^t\left(\left|A\right|k\right)}\hfill \\ \hfill & \sum _{B\subset A,0<\left|B\right|\le t}\left(\mu \right(A)-[\mu \left(B\right)+\mu (A\setminus B)\left]\right).\hfill \end{array}$(16) Since ★-modular case implies ★-additive case, then μ (A) - [μ (B) + μ (A ∖ B)] overset ★ =0 and we can get the result. □ Corollary 3. If a fuzzy measure μ on N is ★-modular or ★-additive within S ⊆ N, then t _μ (A) overset ★ =0, ∀A ⊆ S, |A|≥2.

Theorem 8. -1 ≤ t _μ (A) ≤1, ∀A ⊆ N.

Proof. From Equation (15) or (16) and the definition of fuzzy measure, we can conclude that t _μ (A) get the maximum value of 1 when μ (A) =1 and μ (B) =0, ∀ B ⊂ A; on the other hand, t _μ (A) get the minimum value of -1 when μ (A) =1 and μ (B) =1, ∀ B ⊂ A.

□

Theorem 9. t _μ (A) =1   ⇔   $\mu \left(S\right)={\varepsilon }_A^{+}\left(S\right)$ , S ⊆ A, and t _μ (A) = -1   ⇔   $\mu \left(S\right)={\varepsilon }_A^{-}\left(S\right),S\cap A\ne \varnothing$ .

Proof. From the above proof of Theorem 8 and the definition of maximum and minimum fuzzy measures, see 4, we have the result. □ It is evident that nonadditivity index, nonmodularity index, and their general form t-interaction index share several similar or identical properties. However, to gain a deeper understanding, we need to differentiate and analyze these families of interaction indices from both theoretical and empirical perspectives. To begin, let us examine them from an axiomatic standpoint.

5.2 The axioms of t-level interaction index

Here we can introduce another property of linear function of fuzzy measure fro t-level interaction index as:

t-dimension sets and their complements involved with equality (t-DCI), i.e., besides the subset itself, only the t and less dimensional subsets and their complement sets are equally involved in the linear function l _μ (A) , |A|≥2t: $\begin{array}{cc}\hfill {\alpha }_B={\alpha }_C\ne 0& \mathrm{if}\left|B\right|,\left|C\right|\in \hfill \\ \hfill & \{0,\cdots ,t,|A|-t,\cdots ,|A|-1\}\hfill \\ \hfill & \hfill \end{array}$ $\begin{array}{cc}\hfill {\alpha }_A\ne 0;& {\alpha }_D=0,\mathrm{if}\left|B\right|,\left|C\right|\notin \hfill \\ \hfill & \{0,\cdots ,t,|A|-t,\cdots ,|A|-1\}.\hfill \end{array}$(17)

Theorem 10. The linear function l _μ (A) is the t-level interaction index index n _μ (A) if and only if one of following conditions holds:

l _μ (A) has properties (t-DCI), (Max) and (Add)

l _μ (A) has properties (t-DCI), (Max) and (Min)

Proof. From the properties of t-level interaction index, the sufficient conditions of this theorem hold. We head on the necessity conditions. From the property (Max), we have α _A  = 1.

From the properties (t-DCI) and (Add), if μ is additive within A, we have

$\begin{array}{cc}\hfill l_{\mu }\left(A\right)=& \mu \left(A\right)+\sum B\subset A,\left|B\right|\in \{0,\cdots ,t,\hfill \\ \hfill \left|A\right|-t,\cdots ,\left|A\right|-1\left\}y\mu \right(B)& \hfill \\ \hfill =& \mu \left(A\right)+\sum _{B\subset A,0<\left|B\right|\le t}y\left[\mu \right(B)+\mu (A\setminus B\left)\right]\hfill \\ \hfill =& \mu \left(A\right)+\sum _{B\subset A,0<\left|B\right|\le t}y\mu \left(A\right)\hfill \\ \hfill =& 0\hfill \end{array}$

We have $y=-\frac{1}{{\sum }_{k=1}^t\left(\left|A\right|k\right)}$ . That is, l _μ (A) is t-level interaction index.

From the properties (t-DCI) and (Min), if μ is the maximum fuzzy measure on A, we have $\begin{array}{cc}\hfill l_{\mu }\left(A\right)=& \mu \left(A\right)+\sum B\subset A,\left|B\right|\in \{0,\cdots ,t,\hfill \\ \hfill \left|A\right|-t,\cdots ,\left|A\right|-1\left\}y\mu \right(B)& \hfill \\ \hfill =& \mu \left(A\right)-2\sum _{B\subset A,0<\left|B\right|\le t}y\hfill \\ \hfill =& -1\hfill \end{array}$ We have $y=-\frac{1}{{\sum }_{k=1}^t\left(\left|A\right|k\right)}$ . That is, l _μ (A) is t-level interaction index.

□ Corollary 4. The linear function ${\overline{l}}_{\mu }\left(A\right)$ is the nonmodularity index t _μ (A) if and only if one of following conditions holds:

1. ${\overline{l}}_{\mu }\left(A\right)$ has properties (t-DCI) and (Add)

2. ${\overline{l}}_{\mu }\left(A\right)$ has properties (t-DCI) and (Min)

Proof. From the proof of Theorem 10, one can easily get the results. □ Based on the previous theorem system, we can conclude that the main theoretical difference among nonadditivity index, nonmodularity index and any level of interaction index lie in properties of (EAS), (1-DCI), and (t-DCI), as presented in Theorems 1, 2, and 10.

In the following subsection, we present empirical examples to provide a more detailed description of the interaction among subsets with different t-level of interaction indices.

5.3 Empirical comparisons of t-interaction indices on granularity

Using the random generation methods in [3], we obtain fuzzy measures on sets of 6 and 8 elements respectively and then select 10 representative examples for each. Tables 1 and 2 present empirical results for the nonadditivity index (with columns for t = 3 and t = 4), the nonmodularity index (with columns for t = 1), and the t-interaction indices.

Table 1

Some instances of t-interaction indices about 6 elements

	t= 1	t= 2	t= 3
1	–0.1190	–0.3533	–0.4638
2	–0.1755	–0.3491	–0.4557
3	–0.1564	–0.1235	–0.1142
4	0.0204	–0.0745	–0.1347
5	–0.0070	0.0114	–0.0058
6	–0.0317	0.0582	0.1250
7	–0.0198	0.0717	0.1813
8	0.0217	0.0289	0.0158
9	0.0888	0.2158	0.2867
10	0.3198	0.5072	0.6017

Table 2

Some instances of t-interaction indices about 8 elements

	t= 1	t= 2	t= 3	t= 4
1	–0.0330	–0.0585	–0.0656	–0.0572
2	–0.0870	–0.2103	–0.3006	–0.3470
3	0.0162	–0.0034	–0.1143	–0.1970
4	0.0011	–0.0225	–0.0361	–0.0381
5	–0.0136	–0.0190	0.0047	0.0342
6	–0.0010	0.0271	0.1143	0.1758
7	0.0002	0.0208	0.0439	0.0555
8	0.0127	0.0644	0.1377	0.1813
9	0.0087	0.0290	0.0657	0.0887
10	0.1226	0.2125	0.2526	0.2564

First, in general, the signs of the different t-level indices can exhibit consistency, as observed in Tables 1 (rows 1–3, 8–10) and 2 (rows 1–2, 7–10). However, it is important to note that this consistency does not apply to all rows in these tables, as there may be variations in the signs of the indices for other rows.

Second, a lower value of t generally corresponds to a smaller absolute value of the index, showing relatively minor fluctuations. Specifically, in Tables 1 and 2, the absolute value of the nonmodularity index (with t = 1) is typically lower compared to that of the nonadditivity index (with t = 3 and t = 4).

Finally, it is important to note that the magnitude difference between neighboring level interaction indices can vary significantly. These variations among the different level indices reflect the complexity of the interaction situations when different levels of element subsets are involved. The varying magnitudes highlight the diverse and intricate nature of the interactions within the coalition.

6 Conclusions

In this paper, we established the axiomatic representation of the t-interaction index, which encompasses nonadditivity and nonmodularity as two extreme cases. We achieved this by exploring the linearity, equality, maximality, and minimality properties with respect to fuzzy measures of associated subsets. While the null and dummy partnership yielded proportional and linear formula results for nonadditivity and nonmodularity indices, it is important to note that these results do not encompass all t-interaction indices. For more details and approximate results, please refer to the appendix.

In the domain of multicriteria decision making [9] and multimodel ensemble learning [4], the decision maker or trainer often desires to express their subjective preferences regarding the dependencies or interactions among individual criteria and models. In this context, the following recommendations hold practical value for choosing these types of interaction indices:

The t-interaction index provides fine granularity to investigate different levels of interaction within a coalition, allowing for a more nuanced analysis.

The nonadditivity index considers all subsets and captures the overall interaction among members, providing equality among members and their various combinations. It tends to exhibit more drastic interaction values, leading to a high discrimination ability for interaction cases.

The nonmodularity index focuses on the least combination of members to assess and differentiate interaction cases. It possesses desirable axiomatic properties and is particularly suitable for handling large-scale coalitions and sparse structures of fuzzy measures.

These insights can assist researchers and practitioners in selecting the appropriate interaction indices based on their specific needs and contexts.

Indeed, the nonadditivity index, nonmodularity index, and t-interaction index all pertain to the internal type of interactions, which solely consider the interactions within the coalition and disregards the presence and influence of elements or variables outside the coalition. Consequently, future research can concentrate on establishing the axiomatic representation of comprehensive interaction indices, such as the probabilistic bipartition interaction index [19] and comprehensive nonmodularity index [20]. These endeavors would involve the development of an expanded axiom system that encompasses a broader scope, taking into account the interactions between the coalitions and the effects of external elements or variables. This would provide a more comprehensive understanding of the complex interaction situations.

Acknowledgements

The work was supported by the Australian Research Council Discovery project DP210100227.

Appendix

A The t-interaction index with null elements

If B is null for A \ B with respect to fuzzy measure μ on N, |A \ B|≥2t, then from the definition of null elements and Equation (15), we get $\begin{array}{cc}\hfill t_{\mu }\left(A\right)=& \mu (A\backslash B)-\frac{1}{{\sum }_{k=1}^t\left(\left|A\right|k\right)}\hfill \\ \hfill & \sum _{C\subset A,0<\left|C\right|\le t}\left[\mu \right(C\backslash B)+\mu (A\setminus C\backslash B\left)\right].\hfill \end{array}$ Considering the relationship between |B| and t, one can figure out that set A \ B only appears ${\sum }_{k_1=1}^{min\{t,|B\left|\right\}}(\left|B\right|)k_1$ times in the set {A ∖ C \ B} _{C⊂A,0<|C|≤t} and set D ⊂ A \ B, |D| ≤ t appears ${\sum }_{k_2=1}^{min\{t,|B\left|\right\}-\left|D\right|}(\left|B\right|)k_2$ , so same times for the complement set of D in A \ B, i.e., A \ B \ D, hence we have

$\begin{array}{cc}\hfill t_{\mu }\left(A\right)=& \mu (A\backslash B)-\frac{{\sum }_{k_1=1}^{min\{t,|B\left|\right\}}(\left|B\right|)k_1}{{\sum }_{k=1}^t(\left|A\right|)k}\mu (A\backslash B)\hfill \\ \hfill & -\frac{1}{{\sum }_{k=1}^t(\left|A\right|)k}\sum _{D\subset A\backslash B,0<\left|D\right|\le t}\hfill \\ \hfill & \sum _{k_2=1}^{min\{t,|B\left|\right\}-\left|D\right|}\left(\left|B\right|)k_2\right[\mu \left(D\right)+\mu (A\setminus D\backslash B)]\hfill \end{array}$ $\begin{array}{cc}\hfill & \hfill \\ \hfill =& \frac{{\sum }_{k=1}^t(\left|A\right|)k-{\sum }_{k_1=1}^{min\{t,|B\left|\right\}}(\left|B\right|)k_1}{{\sum }_{k=1}^t(\left|A\right|)k}\left(\mu \right(A\backslash B)-\hfill \\ \hfill & \frac{1}{{\sum }_{k=1}^t(\left|A\right|)k-{\sum }_{k_1=1}^{min\{t,|B\left|\right\}}(\left|B\right|)k_1}\sum _{D\subset A\backslash B,0<\left|D\right|\le t}\hfill \\ \hfill & \sum _{k_2=1}^{min\{t,|B\left|\right\}-\left|D\right|}\left(\left|B\right|)k_2\right[\mu \left(D\right)+\mu (A\setminus B\backslash D)\left]\right)\hfill \end{array}$ Further considering the expression of t _μ (A \ B), we can give the following approximate result: $\begin{array}{cc}\hfill t_{\mu }\left(A\right)\approx \frac{{\sum }_{k=1}^t(\left|A\right|)k-{\sum }_{k_1=1}^{min\{t,|B\left|\right\}}(\left|B\right|)k_1}{{\sum }_{k=1}^t(\left|A\right|)k}{t}_{\mu }(A\backslash B)& \hfill \end{array}$(18) One can see the above approximation aims to seek the similar formulas with Equations (11) and (12).

B The t-interaction index with dummy elements

If A and B, A∩ B = ∅ are mutually independent with respect to μ, then

since |A|, |B|>2t

$\begin{array}{cc}\hfill t_{\mu }(A\cup B)=& \mu (A\cup B)-\frac{1}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}\hfill \\ \hfill & \sum _{C\subset A\cup B,0<\left|C\right|\le t}\left[\mu \right(C)+\mu (A\cup B\setminus C\left)\right]\hfill \\ \hfill =& \mu \left(A\right)+\mu \left(B\right)-\frac{{\sum }_{k_1=1}^t(\left|B\right|)k_1}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}\mu \left(A\right)\hfill \\ \hfill & -\frac{{\sum }_{k_2=1}^t(\left|A\right|)k_2}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}\mu \left(B\right)\hfill \\ \hfill & -\frac{1}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}\sum _{C\subset A,0<\left|C\right|\le t}\hfill \\ \hfill & \sum _{k_3=1}^{t-\left|C\right|}\left(\left|B\right|)k_3\right[\mu \left(C\right)+\mu (A\setminus C)]\hfill \\ \hfill & -\frac{1}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}\sum _{D\subset B,0<\left|D\right|\le t}\hfill \\ \hfill & \sum _{k_4=1}^{t-\left|D\right|}\left(\left|B\right|)k_4\right[\mu \left(D\right)+\mu (B\setminus D)]\hfill \\ \hfill =& \frac{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k-{\sum }_{k_1=1}^t(\left|B\right|)k_1}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}\mu \left(A\right)\hfill \\ \hfill & -\frac{1}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}\sum _{C\subset A,0<\left|C\right|\le t}\sum _{k_3=1}^{t-\left|C\right|}\hfill \\ \hfill & \left(\left|B\right|)k_3\right[\mu \left(C\right)+\mu (A\setminus C)]\hfill \\ \hfill & +\frac{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k-{\sum }_{k_2=1}^t(\left|A\right|)k_2}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}\mu \left(B\right)\hfill \\ \hfill & -\frac{1}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}\sum _{D\subset B,0<\left|D\right|\le t}\sum _{k_4=1}^{t-\left|D\right|}\hfill \\ \hfill & \left(\left|B\right|)k_4\right[\mu \left(D\right)+\mu (B\setminus D)]\hfill \\ \hfill & \hfill \end{array}$

$\begin{array}{cc}\hfill =& \frac{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k-{\sum }_{k_1=1}^t(\left|B\right|)k_1}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}\left(\mu \right(A)\hfill \\ \hfill & -\frac{1}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k-{\sum }_{k_1=1}^t(\left|B\right|)k_1}\hfill \\ \hfill & \sum _{C\subset A,0<\left|C\right|\le t}\sum _{k_3=1}^{t-\left|C\right|}\left(\left|B\right|)k_3\right[\mu \left(C\right)+\mu (A\setminus C)\left]\right)\hfill \\ \hfill & +\frac{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k-{\sum }_{k_2=1}^t(\left|A\right|)k_2}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}\left(\mu \right(B)\hfill \\ \hfill & -\frac{1}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k-{\sum }_{k_2=1}^t(\left|A\right|)k_2}\hfill \\ \hfill & \sum _{D\subset B,0<\left|D\right|\le t}\sum _{k_4=1}^{t-\left|D\right|}\left(\left|B\right|)k_4\right[\mu \left(D\right)+\mu (B\setminus D)\left]\right)\hfill \end{array}$ Further considering the expression of t _μ (A ∪ B), we can give

$\begin{array}{cc}\hfill t_{\mu }(A\cup B)& \approx \frac{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k-{\sum }_{k_1=1}^t(\left|B\right|)k_1}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}{t}_{\mu }\left(A\right)\hfill \\ \hfill & +\frac{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k-{\sum }_{k_2=1}^t(\left|A\right|)k_2}{{\sum }_{k=1}^t(\left|A\right|+\left|B\right|)k}{t}_{\mu }\left(B\right)\hfill \end{array}$(19)

Similarly, one can conclude that the above approximation aims to seek the similar formulas with Equations (13) and (14).