k-Additivity and C-decomposability

of bi-capacities and its integral

Katsushige Fujimoto,a Toshiaki Murofushi,b Michio Sugenoc

aCollege of Symbiotic Systems Science, Fukushima University,

1 Kanayagawa, Fukushima 960-1296, Japan

b Department of Computational Intelligence and Systems Science,

Tokyo Institute of Technology,

4259-G3-47 Nagatsuta, Midori-ku, Yokohama 226-8502, Japan

cFaculty of Culture and Information Science, Doshisha University,

1-3 Miyakodani Tatara, Kyotanabe-shi, Kyoto 610-0394, Japan

Abstract

k-Additivity is a convenient way to have less complex (bi-)capacities. This paper

gives a new characterization of k-additivity, introduced by Grabisch and Labreuche,

of bi-capacities and contrasts between the existing characterization and the new one,

that differs from the one of Saminger and Mesiar. Besides, in the same way for ca-

pacities, a concept of C-decomposability, distinct from the proposal of Saminger and

Mesiar, but closely-linked to k-additivity, is introduced for bi-capacities. Moreover,

the concept of C-decomposability applies to the Choquet integral with respect to

bi-capacities.

Key words: Bi-capacity, the bipolar Mobius transform, k-additivity,

C-decomposability

Preprint submitted to Elsevier Science 2 March 2007

1 Introduction

Let N = {1, . . . , n} be a set of criteria describing the preference of a decision

maker over a set U of objects in a multicriteria decision problem, or a set of

players in a cooperative game, etc. Capacities [5], set functions which vanish

at the empty set and are monotone with respect to set inclusion, have become

a fundamental tool in decision making, especially decision under uncertainty

and under multiple criteria. Characteristic functions [22], in cooperative game

theory, are set functions vanishing at the empty set. These concepts have been

defined on the set of all subsets of N (i.e., on 2N := {A | A ⊆ N}). Recently,

there have been some attempts to define more general concepts both in game

theory and multicriteria decision theory. In game theory, Aubin [1] has pro-

posed the concept of generalized coalition as a function c : N → [−1, 1] which

associates each player i ∈ N with his/her level of participation c(i) ∈ [−1, 1].

Later, Bilbao et. al. [3] have proposed what they call bi-cooperative games,

which generalize the idea of ternary voting games proposed by Felsenthal

and Machover [7] on the set of all signed coalitions T : N → {−1, 0, 1}(recall that the power set 2N is isomorphic to the set of all functions c :

N → {0, 1}. Then, we have a family {T : N → {−1, 0, 1}}, which is iso-

morphic to 3N := {(S1, S2) | S1, S2 ⊆ N, S1 ∩ S2 = ∅}, as a natural ex-

tension of {c : N → {0, 1}}). In multicriteria decision theory, Grabisch and

Labreuche [11–13] have proposed what they call bi-capacities on the lattice

(3N ,�), which vanish at (∅, ∅) and are monotone with respect to the order

� on 3N , where (S1, S2) � (T1, T2) if and only if S1 ⊆ T1 and S2 ⊇ T2 (re-

call that the power set 2N can be regarded as the Boolean lattice (2N ,⊆)

equipped with the order ⊆, and ordinary capacities as monotone functions

2

with respect to ⊆ on the lattice (2N ,⊆)). However, bi-capacities generally re-

quire 3n − 1 parameters. In order to reduce the number of these parameters,

(i.e., to reduce complexity of bi-capacities), Grabisch and Labreuche [11,12]

have proposed the notion of k-additivity of bi-capacities by using the Mobius

transform via the order � on 3N (note that Saminger and Mesiar [25] also

have generalized and characterized the notion of k-additivity by taking a

different approach). On the other hand, as another approach to the prob-

lem, Saminger and Mesiar [25] have proposed the notion of decomposable

bi-capacities (see, Note 2 in section 3.2). In this paper, we give a new character-

ization of k-additivity of bi-capacities and propose another concept to reduce

complexity of bi-capacities, C-decomposability, which is closely-linked to k-

additivity but not to decomposable bi-capacities. Moreover, the concept of C-

decomposability applies to the Choquet integral with respect to bi-capacities

which have been proposed axiomatically by Grabisch and Labreuche [16]

(note that Greco, Matarazzo, and Slowinski [14,15] also have introduced and

discussed the Choquet integral with respect to bipolar measures).

In the remainder of this introduction, we shall indicate the contents of four

sections as follows: Section 2 introduces the concept of a bi-capacity and

its equivalent representations, the Mobius transform and the bipolar Mobius

transform. Section 3 discusses the notions to avoid/reduce complexity of bi-

capacities, k-additivity and C-decomposability of bi-capacities. Section 4 dis-

cusses C-decomposition of the Choquet integral with respect to bi-capacities.

Section 5 concludes and presents ideas for future works.

Throughout the paper, we denote R as the set of all real numbers and R+ the

set of all non-negative real numbers; for a real number x ∈ R, we denote x+ :=

max{x, 0} and x− := (−x)+; for a set of non-negative real numbers {xi} ⊆ R+,

3

we adopt the convention that∧

∅ xi = ∞ and 0 · ∧∅ xi = 0. In order to avoid

heavy notation, we will often omit braces for singletons, e.g., by writing v(i),

U \ i instead of v({i}), U \ {i}, respectively. Similarly, we will often write ij,

ijk instead of {i, j}, {i, j, k}, respectively.

2 Bi-capacities

We denote P(N) := 2N = {S ⊆ N} and Q(N) := 3N = {(A1, A2) ∈ P(N) ×P(N) | A1∩A2 = ∅}. When equipped with the following order � : for arbitrary

(A1, A2), (B1, B2) ∈ Q(N)

(A1, A2) � (B1, B2) iff A1 ⊆ B1 and A2 ⊇ B2,

(Q(N),�) becomes a lattice. Sup and inf are denoted by , �, and are given by

(A1, A2) (B1, B2) = (A1 ∪ B1, A2 ∩ B2),

(A1, A2) � (B1, B2) = (A1 ∩ B1, A2 ∪ B2),

respectively; the top and bottom are respectively (N, ∅) and (∅, N).

Definition 1 (irreducible elements (e.g., [6])) Let (L,≤,∨,∧, �,⊥) be a

lattice, where ∨,∧,�,⊥ denote sup, inf, the top and bottom element, respec-

tively. An element x ∈ L is ∨-irreducible if x �= ⊥ and x = a∨ b implies x = a

or x = b for any a, b ∈ L.

Proposition 2 [11] The -irreducible elements of Q(N) are (∅, ic) and (i, ic)

for all i ∈ N . Moreover, for any (A1, A2) ∈ Q(N),

(A1, A2) =⊔

i∈A1

(i, ic) ⊔j∈N\(A1∪A2)

(∅, jc). (1)

Eq. (1) is called the minimal decomposition of (A1, A2).

4

Note 1 [11] -irreducible elements permit to define layers in Q(N) as fol-

lows: (∅, N) is the bottom layer (layer 0 : the black square on Fig.1), the set

of all -irreducible elements forms layer 1, black circles on Fig.1, and layer

k, for k = 2, . . . n, consists of all elements whose minimal decomposition

contains exactly k -irreducible elements. In other words, layer k consists

of all elements (A1, A2) ∈ Q(N) such that |A2c| = k, for k = 2, . . . , n. On

the other hand, let us consider the Boolean lattice (P(N),⊆, ∪,∩, N, ∅).Then, the empty set is the bottom layer; all singletons are ∪-irreducible

elements, (i.e., in layer 1); the set of all A ∈ P(N) whose cardinality is k

forms layer k, for k = 2, . . . , n.

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�(3, 1)

�(∅, 2)

�(1, ∅)

�(12, ∅)

�(123, ∅)

�(13, ∅)

�(3, ∅)

�(23, ∅)

(∅, 123)

�(2, 13)

�(1, 23)

�(3, 12)

�(12, 3)

�(23, 1)

�(13, 2)

�(∅, 23)

�(∅, 12)

�(1, 2)�

(3, 2)

�(2, 3)

�(1, 3)

�(∅, 13)

�(∅, 3)

�(2, 1)

�(∅, 1)

�(2, ∅)

�(∅, ∅)

Fig. 1. The lattice Q(123): the element in layer 0 is indicated by the black square

and elements in layer 1 black circles.

Definition 3 (capacity (e.g., [4])) A function μ : P(N) → R is a capacity

on P(N) if it satisfies:

(i) μ(∅) = 0,

(ii) A, B ∈ P(N), A ⊆ B implies μ(A) ≤ μ(B).

5

Definition 4 (Mobius transform of a capacity (e.g., [23])) To any capac-

ity μ on P(N), another function mμ : P(N) → R can be associated by

μ(A) =∑B⊆A

mμ(B) ∀A ∈ P(N).

In combinatorics [23], the function mμ is called the Mobius transform of μ and

is given by

mμ(A) :=∑B⊆A

(−1)|A\B|μ(B) for A ∈ P(N).

Definition 5 (bi-capacity [11]) A function v : Q(N) → R is a bi-capacity

on Q(N) if it satisfies:

(i) v(∅, ∅) = 0,

(ii) (A1, A2), (B1, B2) ∈ Q(N), (A1, A2) � (B1, B2) implies v(A1, A2) ≤ v(B1, B2).

In addition, v is called additive if it satisfies

v(A1, A2) =∑i∈A1

v(i, ∅) +∑

j∈A2

v(∅, j) ∀(A1, A2) ∈ Q(N);

A function v : Q(N) → R is said to be a bipolar set function on Q(N) if it

satisfies the condition (i) above.

Definition 6 (Mobius transform of a bi-capacity [11]) To any bi-capacity

v on Q(N), another function m : Q(N) → R can be associated by

v(A1, A2) =∑

(B1,B2)�(A1,A2)

m(B1, B2) ∀(A1, A2) ∈ Q(N).

This correspondence proves to be one-to-one, since conversely

m(A1, A2) :=∑

B1⊆A1A2⊆B2⊆A1

c

(−1)|A1\B1|+|B2\A2|v(B1, B2) for (A1, A2) ∈ Q(N).

(2)

6

The validity of Eq. (2) is proved by Grabisch and Labreuche [11] who call m

the Mobius transform of a bi-capacity v. Indeed, it is the Mobius transform in

the sense of Rota [23] for a function on the partially ordered set (3N ,�).

Another equivalent representation of bi-capacities has been proposed, to facil-

itate the expression of the Choquet integral with respect to bi-capacities (see

Proposition 26), by Fujimoto [8] as follows:

Definition 7 (bipolar Mobius transform of a bi-capacity [8]) To any

bi-capacity v on Q(N), another function b : Q(N) → R can be associated by

b(A1, A2) :=∑

B1⊆A1B2⊆A2

(−1)|A1\B1|+|A2\B2|v(B1, B2) (3)

=∑

(∅,A2)�(B1,B2)�(A1,∅)(−1)|A1\B1|+|A2\B2|v(B1, B2) (4)

for (A1, A2) ∈ Q(N). Then, the function defined by Eq. (3) is called the bipolar

Mobius transform of v. Conversely, it holds that

v(A1, A2) =∑

B1⊆A1B2⊆A2

b(B1, B2) ∀(A1, A2) ∈ Q(N). (5)

The next proposition shows a correspondence between the Mobius transform

and the bipolar Mobius transform.

Proposition 8 [9] Let v be a bi-capacity on Q(N), m the Mobius transform of

v, and b the bipolar Mobius transform of v. Then, it holds, for any (A1, A2) ∈Q(N), that

m(A1, A2) = (−1)|A1c\A2| ∑

A1c\A2⊆C2⊆A1

c

b(A1, C2) (6)

and

b(A1, A2) = (−1)|A2| ∑C2⊆A1

c\A2

m(A1, C2). (7)

7

Proposition 9 [9] Let v be a bi-capacity on Q(N), m the Mobius transform

of v, and b the bipolar Mobius transform of v. If v is additive, then for any

(A1, A2) ∈ Q(N),

v(A1, A2) =∑i∈A1

m(i, ic) +∑

j∈A2

m(∅, jc)

=∑i∈A1

b(i, ∅) +∑

j∈A2

b(∅, j).

3 k-Additivity and C-decomposability of bi-capacities

3.1 k-Additivity

Bi-capacities on Q(N) generally require 3n − 1 parameters. In order to reduce

the number of these parameters, (i.e., to reduce complexity of bi-capacities),

Grabisch and Labreuche [11] have proposed the notion of k-additivity of bi-

capacities, similar to the way they have done for capacities (see also [10]).

Definition 10 (k-additivity of capacities (e.g., [10])) Given a positive in-

teger k < n, a capacity μ is said to be k-additive if mμ(A) = 0 whenever

|A| > k. In addition, a k-additive capacity μ is said to be purely k-additive if

there exists some A ∈ P(N) such that |A| = k and mμ(A) �= 0.

Definition 11 (k-additivity of bi-capacities (e.g., [11])) Given a positive

integer k < n, a bi-capacity is said to be k-additive if its Mobius transform

vanishes for all elements in layer l whenever l > k. Equivalently, a bi-capacity

v is k-additive if and only if m(A1, A2) = 0 whenever |A2c| > k. In addition,

a k-additive bi-capacity v is said to be purely k-additive if there exists some

(A1, A2) ∈ Q(N) such that |A2c| = k and m(A1, A2) �= 0.

8

1-Additivity coincides with ordinary additivity. If v is k-additive, then v is

k′-additive for any n > k′ ≥ k.

Lemma 12 Let v be a bi-capacity on Q(N), m the Mobius transform of v, b

the bipolar Mobius transform of v, and S a subset of N . Then,

m(A1, A2) = 0 whenever A2c ⊇ S iff b(B1, B2) = 0 whenever B1 ∪B2 ⊇ S.

Proof. In order to prove the necessity, assume that b(B1, B2) = 0 whenever

B1∪B2 ⊇ S. For any (A1, A2) ∈ Q(N) such that A2c ⊇ S, from Proposition 8,

m(A1, A2) = (−1)|A1c\A2| ∑

A1c\A2⊆C2⊆A1

c

b(A1, C2) = 0

since A1c \ A2 ⊆ C2 implies A1 ∪ C2 ⊇ A1 ∪ (A1

c \ A2) = A2c ⊇ S.

For showing the sufficiency, we assume that m(A1, A2) = 0 whenever A2c ⊇ S.

For any (B1, B2) ∈ Q(N) such that B1 ∪ B2 ⊇ S, from Proposition 8,

b(B1, B2) = (−1)|B2| ∑C2⊆B1

c\B2

m(B1, C2) = 0

since C2 ⊆ B1c \ B2 implies C2

c ⊇ (B1c \ B2)

c = B1 ∪ B2 ⊇ S. �

Proposition 13 Given a positive integer k < n, a bi-capacity is k-additive if

and only if b(A1, A2) = 0 whenever |A1 ∪ A2| > k.

Proof. We denote by(

Nk+1

)the set of all subsets of N whose cardinalities

are k + 1, i.e.,(

Nk+1

):= {S ⊆ N | |S| = k + 1}. Then,

v is k-additive

⇐⇒ m(B1, B2) = 0 whenever |B2c| > k

⇐⇒ m(B1, B2) = 0 whenever B2c ⊇ S for some S ∈

(N

k + 1

)

9

⇐⇒ b(A1, A2) = 0 whenever A1 ∪ A2 ⊇ S for some S ∈(

N

k + 1

)

⇐⇒ b(A1, A2) = 0 whenever |A1 ∪ A2| > k;

the third equivalence follows from Lemma 12. �

Now we provide an intuitive illustration of the above two representations (Def-

inition 11 and Proposition 13 ) of k-additivity of bi-capacities as follows.

Let us introduce the following two coordinate-like systems for the graphic as

displayed in Fig. 1.

Coordinate-like system I : It has the origin (∅, 123) and three coordinate

axes:

Axis 1) through (∅, 123) and (1, 23),

Axis 2) through (∅, 123) and (2, 13),

Axis 3) through (∅, 123) and (3, 12).

Coordinate-like system II : It has the origin (∅, ∅) and three coordinate

axes:

Axis 1) through (∅, 1) and (1, ∅),Axis 2) through (∅, 2) and (2, ∅),Axis 3) through (∅, 3) and (3, ∅).

Then, if v is 1-additive, m(A1, A2) = 0 (resp. b(A1, A2) = 0) for all points

(A1, A2) ∈ Q(123) which are not on any axis in coordinate-like system I (resp.

II); if v is 2-additive, m(A1, A2) = 0 (resp. b(A1, A2) = 0) for all points

(A1, A2) ∈ Q(123) which are not on any plane spanned by two among three

axes in coordinate-like system I (resp. II).

10

3.2 C-decomposability

Decomposition of bi-capacities into subdomains is another approach to re-

duce complexity of defining its values. In this subsection, the notion of C-

decomposability of bi-capacities is proposed and discussed.

Definition 14 (e.g., [24]) A family C ⊆ P(N) is called a covering of N

if⋃ C = N . A family A ⊆ P(N) is called an antichain if

A ⊆ A′, {A, A′} ⊆ A ⇒ A = A′.

A family H ⊆ P(N) is called hereditary (or a downset in the poset (P(N),⊆))

if

H ′ ⊆ H ∈ H ⇒ H ′ ∈ H.

For a family C ⊆ P(N), 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., A(C) := C \ ⋃C∈C

{A ∈ P(N) | A � C}and H(C) :=

⋃C∈C

{H ∈ P(N) | H ⊆ C}. Note that A(C) ⊆ C ⊆ H(C) and

H(A(C)) = H(C) = H(H(C)).

Definition 15 Let C be a family of subsets of N and v a bi-capacity on Q(N).

A family {vC}C∈C is called a C-decomposition of v if each vC is a bipolar set

function on Q(C) and the following holds:

v(A1, A2) =∑C∈C

vC(A1 ∩ C, A2 ∩ C) ∀(A1, A2) ∈ Q(N). (8)

Then, v is said to be C-decomposable.

Table 1 shows the values of a C-decomposable bi-capacity v, where N =

{1, 2, 3} and C = {{1, 2}, {2, 3}}. The values of a C-decomposition {v12, v23}

11

of v are displayed in Table 2.

Table 1. {{1, 2}, {2, 3}}-decomposable bi-capacity v.

v(A1, A2) ∅ {1} {2} {3} {1, 2} {2, 3} {1, 3} N

∅ 0 −2 −2 −2 −6 −6 −4 −10

{1} 2 – 0 0 – −4 – –

{2} 2 −1 – −1 – – −4 –

{3} 2 0 0 – −4 – – –

{1, 2} 6 – – 3 – – – –

{2, 3} 6 3 – – – – – –

{1, 3} 4 – 2 – – – – –

N 10 – – – – – – –

Table 2. {{1, 2}, {2, 3}}-decomposition of v defined in Table 1.

v12(A1, A2) ∅ {1} {2} {1, 2}∅ 0 −2 −1 −5

{1} 2 – 1 –

{2} 1 −2 – –

{1, 2} 5 – – –

v23(A1, A2) ∅ {2} {3} {2, 3}∅ 0 −1 −2 −5

{2} 1 – −2 –

{3} 2 1 – –

{2, 3} 5 – – –

Note 2 We recall another approach to decomposition (U-decomposition)

of bi-capacities, introduced by Saminger and Mesiar [25] and show the dif-

ferences with respect to our approach.

Definition 16 (U-decomposable bi-capacites) [25] For an arbitrary

interval I ⊇ [−1, 1], consider some uninorm U : I2 → I with natural el-

ement 0, i.e., U is an associative and commutative mapping with U(x, 0) =

U(0, x) = x. A (normalized) bi-capacity v : Q(N) → [−1, 1] is called de-

composable (U-decomposable) if for all (A1, A2), (B1, B2) ∈ Q(N),

U(v(A1, A2), v(B1, B2)) = U(v((A1, A2) (B1, B2)), v((A1, A2)� (B1, B2))).

12

Therefore, for any (A1, A2) ∈ Q(N),

v(A1, A2) = U(Ui∈A1v({i}, ∅), Uj∈A2v(∅, {j}). (9)

Eq. (9) means that the value of a bi-capacity v(A1, A2) can be decomposed

into the values v({i}, ∅) and v(∅, {j}) for i ∈ A1, j ∈ A2. That is, de-

composability in the sense of Saminger and Mesiar [25] means that a (U)-

decomposable bi-capacity can be constructed, through a uninorm U , by fix-

ing the values of v({i}, ∅) and v(∅, {j}) for all i, j ∈ N . On the other hand,

decomposability in our sense means that a (C)-decomposable bi-capacity can

be constructed by a sum of bi-capacities on subdomains {Q(C)}C∈C.

Proposition 17 Let v be a bi-capacity on Q(N), m the Mobius transform of

v, b the bipolar Mobius transform of v, and C a family of subsets of N . Then,

the following three conditions are equivalent to each other.

(i) v has a C-decomposition.

(ii) m(A1, A2) = 0 if A2c �∈ H(C).

(iii) b(A1, A2) = 0 if (A1 ∪ A2) �∈ H(C).

Proof. (i) ⇒ (iii). Let v be a bi-capacity on Q(N) which has a C-decomposition

{vC}C∈C, i.e., each vC is a bipolar set function on Q(C) and the following holds:

v(A1, A2) =∑C∈C

vC(A1 ∩ C, A2 ∩ C) ∀(A1, A2) ∈ Q(N).

13

For any (S1, S2) ∈ Q(N) such that S1 ∪ S2 �∈ H(C),

b(S1, S2) =∑

T1⊆S1T2⊆S2

(−1)|S1\T1|+|S2\T2|v(T1, T2)

= (−1)|S1∪S2| ∑T1⊆S1T2⊆S2

(−1)|T1∪T2|v(T1, T2)

= (−1)|S1∪S2| ∑T1⊆S1T2⊆S2

(−1)|T1∪T2| ∑C∈C

vC(T1 ∩ C, T2 ∩ C)

= (−1)|S1∪S2| ∑C∈C

∑T1⊆S1T2⊆S2

(−1)|(T1∪T2)∩C|+|(T1∪T2)\C|vC(T1 ∩ C, T2 ∩ C)

= (−1)|S1∪S2| ∑C∈C

∑U1⊆S1U2⊆S2

U1∪U2⊆C

∑V1⊆S1V2⊆S2

V1∪V2⊆(S1∪S2)\C

(−1)|U1∪U2|+|V1∪V2|vC(U1, U2)

= (−1)|S1∪S2| ∑C∈C

∑U1⊆S1U2⊆S2

U1∪U2⊆C

(−1)|U1∪U2|vC(U1, U2)∑

V ⊆(S1∪S2)\C(−1)|V |

= 0;

the last equality follows from the assumption S1 ∪S2 �∈ H(C), i.e., (S1 ∪S2) \C �= ∅ for all C ∈ C.

(iii) ⇒ (i). Put, for (A1, A2) ∈ Q(N),

C(A1, A2) := {C ∈ C | C ⊇ (A1 ∪ A2)}.

For each C ∈ C, we define bC and vC on Q(C) as follows:

bC(S1, S2) :=1

| C(S1, S2) | b(S1, S2) for (S1, S2) ∈ Q(C),

vC(S1, S2) :=∑

T1⊆S1T2⊆S2

bC (T1, T2) for (S1, S2) ∈ Q(C).

14

Then, for any (S1, S2) ∈ Q(N),

v(S1, S2) =∑

T1⊆S1T2⊆S2

b(T1, T2)

=∑

T1⊆S1T2⊆S2

T1∪T2⊆C for some C∈C

b(T1, T2)

=∑C∈C

∑T1⊆S1∩CT2⊆S2∩C

1

|C(T1, T2) | b(T1, T2)

=∑C∈C

∑T1⊆S1∩CT2⊆S2∩C

bC(T1, T2)

=∑C∈C

vC(S1 ∩ C, S2 ∩ C);

the second equality follows from the assumption, b(T1, T2) = 0 if (T1 ∪ T2) �∈H(C), i.e., b(T1, T2) = 0 if (T1 ∪ T2) �⊆ C for all C ∈ C.

(ii) ⇔ (iii).

m(A1, A2) = 0 if A2c �∈ H(C)

⇐⇒ m(A1, A2) = 0 if A2c ⊇ H for some H �∈ H(C)

⇐⇒ b(B1, B2) = 0 if (B1 ∪ B2) ⊇ H for some H �∈ H(C)

⇐⇒ b(B1, B2) = 0 if (B1 ∪ B2) �∈ H(C);

the second equivalence follows from Lemma 12. �

The following corollary is immediately obtained from Proposition 17 and the

fact that H(C) = H(A(C)) = H(H(C)).

Corollary 18 Let C be a family of subsets of N and v a bi-capacity on Q(N).

Then, the following three conditions are equivalent to each other.

(i) v has a C-decomposition.

(ii) v has an A(C)-decomposition.

(iii) v has an H(C)-decomposition.

15

The following proposition shows that C-decomposability and k-additivity are

closely-linked to each other.

Proposition 19 Let v be a bi-capacity on Q(N). Then, for any positive in-

teger k, the following three conditions are equivalent to each other.

(i) v is k-additive.

(ii) v has an(

Nk

)-decomposition, where

(Nk

)= {S ⊆ N | |S| = k}.

(iii) v has a C-decomposition for some C ⊆ P(N) such that

k ≥ maxC∈C |C|.

Proof. (i) ⇒ (ii). Assume that v is k-additive, that is, m(A1, A2) = 0

whenever |A2c| > k. Then, m(A1, A2) = 0 whenever A2

c �∈ H((

Nk

)) = {A ⊆

N | |A| ≤ k}. Therefore, from (ii) ⇒ (i) in Proposition 17, v has an(

Nk

)-

decomposition.

(ii) ⇒ (iii). It is trivial, indeed,(

Nk

)is a covering of N , i.e.,

(Nk

)⊆ P(N),

such that k ≥ maxC∈(Nk)

|C|.(iii) ⇒ (i). Assume that v has a C-decomposition for some C ⊆ P(N) such

that k ≥ maxC∈C |C|. Then, B �∈ H(C) if |B| > k. Therefore, from (i) ⇒ (ii) in

Proposition 17, m(A1, A2) = 0 whenever |A2c| > k. Hence, v is k-additive. �

Proposition 20 Let C and D be two families of subsets of N and v a bi-

capacity on Q(N). If C ⊆ H(D) (i.e., for any C ∈ C there exists D ∈ D such

that C ⊆ D) and if v has a C-decomposition, then v has a D-decomposition.

Proof. Assume that C ⊆ H(D), then H(C) ⊆ H(H(D)) = H(D). If v has

a C-decomposition, then, from (i) ⇒ (ii) in Proposition 17, m(A1, A2) = 0

whenever A2c �∈ H(C). Therefore, m(A1, A2) = 0 whenever A2

c �∈ H(D).

Indeed, A2c �∈ H(D) implies A2

c �∈ H(C). Hence, from (ii) ⇒ (i) in Proposition

17, v has a D-decomposition. �

16

By Corollary 18, it is sufficient to consider only antichain as families of subsets

of N . The next corollary is obtained immediately from the proofs of Proposi-

tion 17 and Corollary 18.

Corollary 21 Let v be a bi-capacity on Q(N), b the bipolar Mobius transform

of v, and F(v) := {A1 ∪ A2 | b(A1, A2) �= 0}. Then for each F ∈ A(F(v))

there exists a bipolar set function vF : Q(F ) → R such that

v(A1, A2) =∑

F∈A(F(v))

vF (A1 ∩ F, A2 ∩ F ) ∀(A1, A2) ∈ Q(N). (10)

Table 3 shows the values of the bipolar Mobius transform of the {{1, 2}, {2, 3}}-decomposable bi-capacity v defined in Table 1. Indeed, b(A1, A2) = 0 when-

ever A1 ∪ A2 �∈ H({1, 2}, {2, 3}) = {∅, {1}, {2}, {3}, {1, 2}, {2, 3}} and/or

A(F(v)) = A({{1}, {2}, {3}, {1, 2}, {2, 3}}) = {{1, 2}, {2, 3}}.

Table 3. Bipolar Mobius transform of v defined in Table 1.

b(A1, A2) ∅ {1} {2} {3} {1, 2} {2, 3} {1, 3} N

∅ 0 −2 −2 −2 −2 −2 0 0

{1} 2 – 0 0 – 0 – –

{2} 2 −1 – −1 – – 0 –

{3} 2 0 0 – 0 – – –

{1, 2} 2 – – 0 – – – –

{2, 3} 2 0 – – – – – –

{1, 3} 0 – 0 – – – – –

N 0 – – – – – – –

Proposition 22 Let v be a bi-capacity on Q(N) and C a family of subsets of

N . Then, the following two conditions are equivalent to each other.

(i) v has a C-decomposition.

(ii) v(A1, A2) =∑D⊆CD=∅

(−1)|D|+1v

( ⋂D∈D

D ∩ A1,⋂

D∈DD ∩ A2

)

for any (A1, A2) ∈ Q(N).

17

Proof. (i) ⇒ (ii). Assume that v has a C-decomposition {vC}C∈C.

Then, for any (A1, A2) ∈ Q(N),

∑D⊆CD=∅

(−1)|D|+1v

( ⋂D∈D

D ∩ A1,⋂

D∈DD ∩ A2

)

=∑D⊆CD=∅

(−1)|D|+1∑C∈C

vC

( ⋂D∈D

D ∩ A1 ∩ C,⋂

D∈DD ∩ A2 ∩ C

)

=∑C∈C

⎛⎜⎜⎝∑

D⊆CDC

(−1)|D|+1vC

( ⋂D∈D

D ∩ A1 ∩ C,⋂

D∈DD ∩ A2 ∩ C

)

+∑D⊆C

DC,D=∅

(−1)|D|+1vC

( ⋂D∈D

D ∩ A1 ∩ C,⋂

D∈DD ∩ A2 ∩ C

)⎞⎟⎟⎠

=∑C∈C

⎛⎜⎜⎝∑

D⊆CDC

(−1)|D|+2vC

( ⋂D∈D

D ∩ A1 ∩ C,⋂

D∈DD ∩ A2 ∩ C

)

+∑D⊆C

DC,D=∅

(−1)|D|+1vC

( ⋂D∈D

D ∩ A1 ∩ C,⋂

D∈DD ∩ A2 ∩ C

)⎞⎟⎟⎠

=∑C∈C

vC(A1 ∩ C, A2 ∩ C)

= v(A1, A2);

the first and last equalities above follow from the assumption.

(ii) ⇒ (i). Suppose that

v(A1, A2) =∑D⊆CD=∅

(−1)|D|+1v

( ⋂D∈D

D ∩ A1,⋂

D∈DD ∩ A2

)∀(A1, A2) ∈ Q(N).

Now we put, for each C ∈ C,

vC(A1 ∩ C, A2 ∩ C) :=∑D⊆CDC

(−1)|D|+1

v

( ⋂D∈D

D ∩ A1,⋂

D∈DD ∩ A2

)

|D|

18

for all (A1, A2) ∈ Q(N). Then, for any (A1, A2) ∈ Q(N),

∑C∈C

vC(A1 ∩ C, A2 ∩ C)

=∑C∈C

∑D⊆CDC

(−1)|D|+1

v

( ⋂D∈D

D ∩ A1,⋂

D∈DD ∩ A2

)

|D|

=∑D⊆CD=∅

(−1)|D|+1v

( ⋂D∈D

D ∩ A1,⋂

D∈DD ∩ A2

)

= v(A1, A2).

Hence, v has a C-decomposition. �

4 C-decomposition of the Choquet integral with respect to bi-

capacities

The Choquet integral with respect to capacities has been well-known as an

aggregation operator in decision problems under uncertainty (see e.g., [2]).

The Choquet integral with respect to bi-capacities has been introduced ax-

iomatically by Labreuche and Grabisch [16]. In this section, we discuss C-

decomposability of the Choquet integral with respect to bi-capacities. In order

to simplify the notations we denote, for a function f over N , by fi the value

f(i) for each i ∈ N .

Definition 23 (Choquet integral w.r.t. capacities (e.g., [5])) Let μ be a

capacity on Q(N) and f : N → R+ a non-negative function on N . The Choquet

integral I μN(f) of f over N with respect to μ is defined by

I μN(f) :=

n∑i=1

fσ(i)

(μ(Aσ(i)) − μ(Aσ(i+1))

),

19

where σ is a permutation on N such that fσ(1) ≤ · · · ≤ fσ(n), and Aσ(i) :=

{σ(i), . . . , σ(n)} and Aσ(n+1) := ∅.

An important remark is that μ needs not to be a capacity (i.e., monotone) in

order to define the Choquet integral properly; any set function vanishing at

the empty set could do [21].

Definition 24 (Choquet integral w.r.t. bi-capacities [16]) Let v be a bi-

capacity on Q(N) and f : N → R a real-valued function on N . The Choquet

integral I vN(f) of f over N with respect to v is given by

I vN(f) := I

μNf

N (|f |),

where μNf is a real-valued set function on P(N) defined by

μNf (S) = v(S ∩ N+

f , S ∩ N−f ), (11)

and N+f := {i ∈ N | fi ≥ 0}, N−

f := N \ N+f .

It should be noticed that μNf is not a capacity in general, since it may be not

monotone, and even take negative values. Therefore, v also needs not to be a bi-

capacity (i.e., monotone) in order to define the Choquet integral properly; any

bipolar set function could do. Observe that we have I vN(1(A1,A2)) = v(A1, A2)

for any (A1, A2) ∈ Q(N), where 1(A1,A2) denotes the characteristic function of

(A1, A2) which is the function on N whose value for i ∈ N is 1 if i ∈ A1, −1

if i ∈ A2, and 0 otherwise. Hence, the Choquet integral is indeed an extension

of v.

The next proposition gives a necessary and sufficient condition for the Cho-

quet integral with respect to bi-capacities to be decomposable into a sum of

Choquet integrals as shown in Fig.2.

20

f1

f2

f3

f4

f5

f6

I vN I v

N(f)

f1

f2

f3

f4

f5

f6

IvC1C1

I vN(f)

C3

C1

C2

IvC2C2

IvC3C3

Σ⇔

Fig. 2. Decomposition of the Choquet integral via C = {C1, C2, C3}, where

N = {1, 2, 3, 4, 5, 6}, C1 = {1, 2, 3}, C2 = {3, 4, 5}, and C3 = {4, 5, 6}.

Proposition 25 Let C be a covering of N and v a bi-capacity on Q(N). Then,

for each C ∈ C there exists a bipolar set function vC on Q(C) such that

I vN(f) =

∑C∈C

I vCC (f |C) (12)

for all functions f : N → R, where f |C denotes the restriction of f to C, if

and only if v has a C-decomposition.

Proof. It is easily verified from the definition of the Choquet integral with

respect to bi-capacities that the Choquet integral is additive with respect to

bi-capacities, that is,

I∑n

i=1vi

N (f) =n∑

i=1

I viN (f)

for any f : N → R. Suppose that v has a C-decomposition. Then for each

C ∈ C there exists a bipolar set function vC on Q(C) such that

v(S1, S2) =∑C∈C

vC(S1 ∩ C, S2 ∩ C)

for any (S1, S2) ∈ Q(N). For each C ∈ C, we define a bipolar set function v′C

21

on Q(N) such that

v′C(S1, S2) := vC(S1 ∩ C, S2 ∩ C)

for (S1, S2) ∈ Q(N). Then we have that v(S1, S2) =∑

C∈C v′C(S1, S2) for any

(S1, S2) ∈ Q(N). Therefore,

I vN(f) =

∑C∈C

Iv′CN (f) =

∑C∈C

I vCC (f |C)

for any f : N → R (for details on the last equality, see Appendix A1).

In order to prove the sufficiency, we show that Eq.(12) implies that v has a

C-decomposition. For any (A1, A2) ∈ Q(N), by considering the Choquet inte-

gral of 1(A1,A2) with respect to v, we obtain that

v(A1, A2) = I vN(1(A1,A2)) =

∑C∈C

I vCC (1(A1∩C,A2∩C)) =

∑C∈C

vC(A1 ∩ C, A2 ∩ C).

�

The Choquet integral has an equivalent representation by means of the bipo-

lar Mobius transform as shown in the following proposition. By using this

proposition, we can give another proof of Proposition 25 above (see Appendix

A2).

Proposition 26 [9] Let v be a bi-capacity on Q(N). The Choquet integral

I vN(f) of a function f over N with respect to v can be represented as

I vN(f) =

∑A∈P(N)

b(A ∩ N+f , A ∩ N−

f )∧i∈A

|fi| (13)

=∑

(S1,S2)∈Q(N)

b(S1, S2)

⎛⎝ ∧

i∈S1

f+i ∧ ∧

j∈S2

f−j

⎞⎠ . (14)

Through the discussion in sections 3 and 4, we obtain the following corollary.

Corollary 27 Let C be a family of subsets of N and v a bi-capacity on Q(N).

22

Then, the following four conditions are each equivalent to the C-decomposability

of v.

(i) For each C ∈ C there exists a bipolar set function vC on Q(C)

such that I vN(f) =

∑C∈C

I vCC (f |C) for any function f : N → R.

(ii) m(A1, A2) = 0 if A2c �∈ H(C).

(iii) b(A1, A2) = 0 if (A1 ∪ A2) �∈ H(C).

(iv) v(A1, A2) =∑D⊆CD=∅

(−1)|D|+1v(⋂

D∈DD ∩ A1,

⋂D∈D

D ∩ A2)

for any (A1, A2) ∈ Q(N).

5 Concluding remarks and future works

This paper discusses notions to reduce complexity of bi-capacities, k-additivity

and C-decomposability of that. These notions are closely-linked to each

other via the Mobius and bipolar Mobius transforms. C-decomposability of

bi-capacities is naturally extended to that of the Choquet integral with re-

spect to bi-capacities. It should be noticed that monotonicity of bi-capacities

do not play an essential role in this paper, indeed, almost all notions and re-

sults for bi-capacities given in the paper hold for merely bipolar set functions.

However, monotonicity of bi-capacities and of the Choquet integral is one of

the most important properties in decision problems. Therefore, to investigate

monotone C-decomposition (i.e., every vC in C-decomposition {vC}C∈C is a

bi-capacity) of bi-capacities is an important subject for future work.

23

Acknowledgements

The authors would like to express their appreciation to Michel Grabisch for

his useful comments. 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.

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Appendix

A1 Supplement to the proof of Proposition 25

Proof of Iv′CN (f) = I vC

C (f |C) :

26

Since for any S ⊆ N , using Eq.(11),

(μ′C)N

f (S) = v′C(S ∩ N+

f , S ∩ N−f )

= vC(S ∩ N+f ∩ C, S ∩ N−

f ∩ C)

= vC((S ∩ C) ∩ N+f , (S ∩ C) ∩ N−

f )

= vC((S ∩ C) ∩ C+f |C , (S ∩ C) ∩ C−

f |C)

= (μC)Cf |C (S ∩ C) ( = (μ′

C)Nf (S ∩ C) ),

it follows that N \ C is a null set (see e.g., [20]) with respect to (μ′C)N

f , and

hence from a well-known property (see e.g., [20]) of the Choquet integral that

for any function g : N → R

I(μ′

C )Nf

N (g) = I(μC)C

f |CC (g|C).

Therefore, we have

Iv′CN (f) = I

(μ′C )N

f

N (|f |) = I(μC )C

f |CC (|f ||C) = I

(μC )Cf |C

C (|f |C|) = I vCC (f |C).

A2 Another proof of the “necessity” part of Proposition 25

Suppose that v has a C-decomposition. Put

I(C) := {(A1, A2) ∈ Q(N) | (A1 ∪ A2) ∈ H(C)},

O(C) := {(A1, A2) ∈ Q(N) | (A1 ∪ A2) �∈ H(C)},

and for (A1, A2) ∈ Q(N)

C(A1, A2) := {C ∈ C | C ⊇ (A1 ∪ A2)}.

27

Then, clearly, {I(C),O(C)} is a partition of Q(N). For each C ∈ C, we define

two functions on Q(C), bC and vC , as follows:

bC(A1, A2) :=1

|C(A1, A2)| b(A1, A2) for (A1, A2) ∈ Q(C),

vC(A1, A2) :=∑

B1⊆A1B2⊆A2

bC(B1, B2) for (A1, A2) ∈ Q(C).

Then vC(∅, ∅) = 0 for any C ∈ C, and b(A1, A2) = 0 for any (A1, A2) ∈ O(C)

from (i) ⇒ (iii) in Proposition 17. Therefore, for any f : N → R,

I vN(f) =

∑(A1,A2)∈Q(N)

b(A1, A2)

⎛⎝ ∧

i∈A1

f+i ∧ ∧

j∈A2

f−j

⎞⎠

=∑

(A1,A2)∈I(C)∪O(C)

b(A1, A2)

⎛⎝ ∧

i∈A1

f+i ∧ ∧

j∈A2

f−j

⎞⎠

=∑

(A1,A2)∈I(C)

b(A1, A2)

⎛⎝ ∧

i∈A1

f+i ∧ ∧

j∈A2

f−j

⎞⎠

=∑C∈C

∑(A1,A2)∈Q(C)

b(A1, A2)

|C(A1, A2)|

⎛⎝ ∧

i∈A1

(f |C)+i ∧ ∧

j∈A2

(f |C)−j

⎞⎠

=∑C∈C

∑(A1,A2)∈Q(C)

bC(A1, A2)

⎛⎝ ∧

i∈A1

(f |C)+i ∧ ∧

j∈A2

(f |C)−j

⎞⎠

=∑C∈C

I vCC (f |C). �

28