Some existence conditions for decomposable k-monotone set functions having no k
-monotone decompositions Katsushige FUJIMOTO1) Toshiaki MUROFUSHI2) , Yoshinari SAWATA2) 1)

2)

College of Symbiotic Systems Science, Fukushima University, 1 Kanayagawa, Fukushima, 960-1296, Japan [email protected] Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, 4259-G3-47, Nagatsuta, Midori-ku, Yokohama 226-8502, Japan [email protected], [email protected]

Abstract— It has been already known that a set function µ defined on the power set 2 X of a finite set X and/or the Choquet integral w.r.t. µ have a C-decomposition iff C is an inclusionexclusion covering of X. This paper shows that for any two positive integers k and k
such that k
≤ k, there exist a finite set X, a k-monotone set function µ on 2X , and an inclusionexclusion covering C of X w.r.t. µ such that µ has no k
-monotone C-decompositions.

I. Introduction So far, we have obtained a necessary and sufficient condition for the Choquet integral to be decomposable into a sum of Choquet integrals [7]:

 (C) f dµ = (C) f dµC ∀ f ∈ RX , (1) X

C∈C

C

where µ is a set function defined on the power set 2 X of a finite set X, C ⊆ 2X is a covering of X, µC is a set function defined on 2C for each C ∈ C, R X is the family of real-valued functions on X, and (C) X f dµ is the Choquet integral of f over X with respect to µ. However, in the case of µ being monotone, i.e., µ(A) ≤ µ(B) whenever A ⊆ B, even if the condition is satisfied, µC ’s are not necessarily monotone. The monotonicity of µ is equivalent  to that of the Choquet integral [7] : (C) X f dµ ≤ (C) X g dµ whenever f (x) ≤ g(x) for all x ∈ X. The monotonicity plays a key role in applications to decision making, so that several existence conditions for monotone decomposition of µ have been studied and provided to date [8]. On the other hand, the following question has arisen: “In what situations/conditions does a decomposable monotone set function have no monotone decompositions? Would these conditions be determined depending on monotonicity of set functions, or on structures of decomposition, or on both of them?” Several answers to this question will be provided in Section IV. The paper does not deal with the Choquet integral at all since the problem can be described without the Choquet integral; indeed, Eq. (1) is equivalent to the following [7]:  µ(A) = µC (A ∩ C) ∀A ∈ 2X . (2) C∈C

II. Preliminaries Throughout the paper, X is assumed to be a nonempty finite   set. |A| denotes the cardinality of the set A, Xl the set of all subsets of X whose cardinality is l, and R the set of real numbers. µ is assumed to be a set function on 2 X , which means that µ is a function µ : 2 X → R vanishing at ∅. For B ⊆ D, we write [B, D] := {A | B ⊆ A ⊆ D}, [B, D[ := {A | B ⊆ A
D} = [B, D] \ {D}. Definition 1: (e.g., [5]) The M¨obius transform µ M of µ is defined as  (−1)|A\B| µ(B) for A ⊆ X. µM (A) := B⊆A

Conversely, it holds that for every A ⊆ X  µ(A) = µM (B).

(3)

B⊆A

 For A ⊆ 2X , we write µM (A) = A∈A µM (A); in other words, we treat µM not only as a set function µ M : 2X → R but also X as an additive set function µ M : 2(2 ) → R. In this notation, (3) is written as µ(A) = µM ([∅, A]). According to Shafer [6], a set F such that µM (F)
0 is called a focal element of µ. We denote the family of focal elements by F (µ), i.e., F (µ) := {F | µM (F)
0}. Definition 2: [1], [2] 1) µ is called 1-monotone, if µ is monotone, that is, µ(A) ≤ µ(B) whenever A ⊆ B. 2) For a positive integer k ≥ 2, µ is called k-monotone if µ is monotone and    µ( A) ≥ (−1)|B|+1 µ( B) ∅
B⊆A

whenever A ⊆ 2 X and |A| ≤ k.

Hereafter, we denote by M k (X) the set of all k-monotone set functions on 2 X . For k ≥ 2, the k-monotonicity defined in [1] does not assume monotonicity. We adopt, however, the original definition of kmonotonicity in [2], that is, every k-monotone set function has monotonicity; note that the definition of the k-monotonicity, in [1], without the assumption of monotonicity yields essentially the same results, however, and makes the following propositions (Propositions 1 and 2) slightly more complex. Proposition 1: [1] If µ is k-monotone, then µ is k
monotone for any positive integer k
≤ k, i.e.,

Proposition 4: [3], [7], [8] Let C be a covering of X. The following three conditions are equivalent to each other. 1) C ∈ C µ (X). 2) A(C) ∈ C µ (X). 3) H(C) ∈ C µ (X). By Proposition 4, it is sufficient to consider only antichain coverings as inclusion-exclusion coverings. Therefore, hereafter we consider only antichain coverings as coverings and denote by A (X) the set of all antichain coverings of X, and C µ (X) is redefined as the set of all inclusion-exclusion antichain coverings of X with respect to µ.




k ≥ k
≥ 1 ⇒ Mk (X) ⊆ Mk (X). Proposition 2: [8] For every positive integer k, the following two conditions are equivalent to each other. 1) µ is k-monotone. 2) µM (A) ≥ 0 whenever A ⊆ X and 0 < |A| < k, and µM ([B, D]) ≥ 0 whenever B ⊆ D ⊆ X and |B| = k. Definition 3: (e.g., [8]) A family C ⊆ 2 X is called a covering of X if C = X. A family A ⊆ 2 X is called an antichain if A ⊆ A
, {A, A
} ⊆ A ⇒ A = A
. A family H ⊆ 2 X is called hereditary if H
⊆ H ∈ H ⇒ H
∈ H. For a family C ⊆ 2 X , we denote by A(C) the antichain consisting of all the maximal elements of C with respect to set inclusion ⊆, and by H(C) the hereditary family generated by C. i.e.,   [∅, C[ , H(C) := [∅, C]. A(C) := C \ C∈C

C∈C

In [7], an antichain covering is called an irreducible covering. Note that, if C is a covering of X, then so are A(C) and H(C). Definition 4: [3], [7] A covering C of X is called an inclusion-exclusion covering with respect to µ if    A∩A ∀A ⊆ X. (−1)|A|+1 µ µ(A) = A⊆C A
∅

Clearly, every set function on 2 X has {X} as a trivial inclusionexclusion covering of X. We denote by C µ (X) the set of all inclusion-exclusion coverings of X with respect to µ. Proposition 3: [3], [7] C ∈ C µ (X) iff F (µ) ⊆ H(C).

Definition 5: (e.g., [7]) Let C and D be two families of subsets of X. It is said that C is finer than D if C ⊆ H(D). Then, conversely, D is said to be coarser than C. Proposition 5: [3], [7] If C ∈ C µ (X) then D ∈ C µ (X) for any D ⊆ 2 X which is coarser than C. i.e., C ∈ C µ (X), C ⊆ H(D) ⇒ D ∈ C µ (X). III. Conditions to have a k
-monotone decomposition Definition 6: Let C be a covering of X. A family {µ C }C∈C is called a C-decomposition of µ if each µ C is a set function on 2C and (2) holds. Then µ is said to be C-decomposable (merely decomposable). If every µ C is k-monotone, then {µ C }C∈C is said to be k-monotone. Proposition 6: [7] µ has a C-decomposition iff C ∈ C µ (X). In Proposition 6, we cannot add “1-monotone” to “Cdecomposition” even if µ is 1-monotone; this will be shown in Proposition 7. In other words, C being an inclusion-exclusion covering is merely a necessary condition for a 1-monotone set function to have a 1-monotone C-decomposition. Based on this fact, more generally, the monotone decomposition problem, which is to determine, for a given k-monotone set function µ and a given C ∈ C µ (X), whether µ has a k-monotone Cdecomposition or not, has been investigated [8]. Proposition 7: [8] For every positive integer k, the following two conditions are equivalent to each other. 1) Every k-monotone set function whose inclusionexclusion covering is C has a k-monotone Cdecomposition. 2) |C1 ∩ C2 | ≤ k for every pair C 1 and C 2 of distinct members of C.

By Proposition 1, as a corollary to Proposition 7 we obtain the following sufficient condition to have k
-monotone Cdecomposition.


Corollary 1: [8] Let k and k be two positive integers such that k
≤ k. If µ has an inclusion-exclusion covering C, if |C1 ∩ C2 | ≤ k for every pair C 1 and C 2 of distinct members of C, and if µ is k-monotone, then µ has a k
-monotone Cdecomposition. Proposition 8 below gives another sufficient condition to have k
-monotone C-decomposition. Proposition 8: [9] Let k and k
be two positive integers such that k
≤ k. If µ is k-monotone, if µ has an inclusion
exclusion  X  covering C, and if |{C | B ⊆ C ∈ C}| ≤ k−k +1 for all B ∈ k
, then µ has a k
-monotone C-decomposition. IV. Conditions to have no k
-monotone decompositions In this section, we will show several propositions and examples supporting the following assertion. Assertion 1: For any two positive integers k and k
such that k
≤ k, there exist a finite set X, a k-monotone set function µ on 2X , and an inclusion-exclusion covering C ∈ C µ (X) such that µ has no k
-monotone C-decompositions. A. Motivations Note the fact that, in the assertion above, X is determined depending on k and k
. Indeed, if an n-element set X is fixed first, from Corollary 1 and Proposition 8, every C-decomposable n-monotone set function always has a k
-monotone C-decomposition for any k
≤ n. Furthermore, from Corollary 1, the following proposition is obtained immediately.


Proposition 9: Let k and k be two positive integers such that k
≤ k and X an n-element set. If n ≤ k + 2, then for any C ∈ A (X), every k-monotone set function whose inclusion-exclusion covering is C has a k
-monotone C-decomposition. Indeed, if C ∈ A (X) is a non-trivial covering of X (i.e., C
{X}), then |C 1 ∩ C2 | ≤ |X| − 2 = n − 2 ≤ k for every pair C 1 and C 2 of distinct members of C. On the other hand, there exists a C-decomposable k-monotone set function having no k
-monotone C-decompositions for a positive integer k
< k.

Example 1: Let k = 2, k
= 1, and |X| = n = 7. µ is defined via its M¨obius transform µ M as follows: ⎧ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨−1 µM (E) = ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎩ 0

if |E| = 4, if |E| = 3, if |E| = 2, otherwise.

Then, it follows from Propositions 2 and 3 that µ is a   2-monotone set function with X4 as an inclusion-exclusion   covering. However µ has no 1-monotone X4 -decompositions. Proposition 9 and Example 1 illustrate that whether a C-decomposable 2-monotone set function has a 1-monotone C-decomposition or not depends on the cardinality of X. More generally, in the next subsection (Corollary 2), we have a condition for the existence of C-decomposable k-monotone set functions that have no k
-monotone C-decompositions. Next, we will discuss another indecomposability condition from a different viewpoint. It follows from Proposition 5 that a set function µ on 2 X with C as a non-trivial inclusionexclusion covering of X has generally multiple  X non-trivial  inclusion-exclusion coverings of X except C is |X|−1 . Indeed, X  for l < m < |X|, m is an inclusion-exclusion covering of X   whenever so is Xl . Example 2: Let k = 2, k
= 1, |X| = n = 7, and µ be the set function   given in Example 1. As shown in Example 1, µ is a X4 -decomposable 2-monotone set function that has   no 1-monotone X4 -decompositions. On the other hand, X  6 is also an inclusion-exclusion covering with respect to µ. Moreover, {µC }C∈(X) defined via {µCM }C∈(X) as follows becomes 6  6 a 1-monotone X6 -decomposition:   For each C ∈ X6 , ⎧ 1 ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎨− 4 M µC (E) = ⎪ ⎪ 1 ⎪ ⎪ ⎪ 5 ⎪ ⎪ ⎪ ⎩ 0

if |E| = 4, if |E| = 3, if |E| = 2, otherwise.

That X  is, µ is a set function thatX has a 1-monotone 6 -decomposition but no 1-monotone 4 -decompositions. Example 2 illustrates that monotone-decomposability of a decomposable set function µ depends not only on values of µ but also on what inclusion-exclusion covering is taken. In the next subsection (Corollary 3), we will give a condition for the existence of C-decomposable k-monotone set functions having no non-trivial k
-monotone decompositions at all.

B. Indecomposability conditions In this subsection, a proposition and its corollaries supporting Assertion 1 will be given. Here, we focus on set functions whose values for each E ∈ 2 X are determined depending only on its cardinality, –that is, the finest inclusion-exclusion covering with respect to such a set function is expressed as X  for an l ≤ |X| = n. l
The next proposition gives a relation between X  k, k , l, and |X| = n in the case where there exists a l -decomposable   k-monotone set function that has no k
-monotone Xl decompositions.

Proposition 10: Let k, k
, l, and n be positive integers such that k
≤ k and k + 2 ≤ l < n, and X an n-element set. If l−k−1>

2(l − k
− 1) , n−l+2

X  there exists a k-monotone set function µ on 2 such that then X with respect to µ l is the finest inclusion-exclusion covering   and the set function µ has no k
-monotone Xl -decompositions.

Note that, since k
≤ k, every k-monotone set function constitutes a trivial k
-monotone decomposition of itself, –that is, C = {X}. Corollary 3 is summarized as follows: Case II’ (|X| > 3k − 2k
+ 2) : ∃µ ∈ Mk (X) s.t. ∀C(
{X}) ∈ C µ (X), µ has no k
-monotone C-decompositions.

V. Concluding remarks Proposition 10 gives a condition where there exists a decomposable k-monotone set function that has no k
-monotone decompositions. We have dealt only with the case where inclusion-exclusion coverings are expressed as Xl for some l and besides, only with indecomposability. Therefore we have had no results on decomposability and/or indecomposability √ in the case where k + 2 < |X| ≤ k + 1 + 8(k − k
) + 1. To investigate in the case is an important subject for future research. Acknowledgments

Corollary 2: Let k and k
be two positive integers such that
k ≤ k, and X an n-element set. If  n > k + 1 + 8(k − k
) + 1,

Partial financial support from the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aids for Young Scientists (B), 16710105, 2006, and for the 21st Century COE Program “Creation of Agent-Based Social Systems Sciences” is gratefully acknowledged.

then there exist a covering C of X and a k-monotone set function µ on 2 X with C as an inclusion-exclusion covering of X such that µ has no k
-monotone C-decompositions.

References

Proposition 9 and Corollary 2 are summarized as follows: Let k and k
be two positive integers such that k
≤ k. Case I (|X| ≤ k + 2) : ∀µ ∈ Mk (X), ∀C ∈ C µ (X), µ has a k
-monotone C-decomposition. √ Case II (|X| > k + 1 + 8(k − k
) + 1) : ∃µ ∈ Mk (X), ∃C ∈ C µ (X) s.t. µ has no k
-monotone C-decompositions.

In a subcase, (|X| > 3k−2k
+2), of Case II above, Corollary 2 can be strengthened as follows: Corollary 3: Let k and k
be two positive integers such that k ≤ k, and X an n-element set. If


n > 3k − 2k
+ 2, then there exists a k-monotone set function on 2 X that has no non-trivial k
-monotone decompositions at all.

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