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International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems c World Scientific Publishing Company 

SOME RELATIONS AMONG VALUES, INTERACTIONS, AND DECOMPOSABILITY OF NON-ADDITIVE MEASURES

Katsushige FUJIMOTO Faculty of Economics, Fukushima University, 1 Kanayagawa Fukushima, 960-1296, JAPAN [email protected] Toshiaki MUROFUSHI Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, 4259 G3 47 Nagatsuta, Midori-ku Yokohama, 226-8502, JAPAN [email protected] Received July 2001 Revised January 2007 We will discuss some relations among values, i.e., global/overall importance of each single element of the object to be considered, interaction indices, i.e., simultaneous interaction among the elements in a coalition, and inclusion-exclusion coverings, i.e., decomposability of non-additive measures and/or of the Choquet integral. So, we show that inclusion-exclusion coverings are characterized by the Shapley interaction index. Then, inclusion-exclusion coverings are invariant under the transformations which characterize the Shapley value and/or the Shapley interaction index, and preserve the Shapley value. Keywords: The Shapley value, interaction index, inclusion-exclusion covering, Sequivalence, duality.

1. Introduction Non-additive measures (e.g., fuzzy measures, characteristic functions in cooperative game theory, belief and plausibility functions in evidence theory, and capacities in potential theory, etc.) have a high potential for representing interactions among the elements of the object to be considered, especially in decision making problems and cooperative game situations, where they have been already successfully applied. In this paper, we discuss the non-additive measure on a finite set N as a general tool for modeling importance/worth of coalitions (i.e., subsets of N ). Elements of N could be players in a cooperative game, criteria in a multi-criteria decision problem, experts or voters in an opinion pooling problem, etc. However, the complexity of the models using non-additive measures increases as the number of elements goes to 4, 5, · · ·. In practical applications, one often en1

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counters the following two problems: 1. The model requires too many parameters, —in fact, for a set N with n elements, the definition of a non-additive measure requires 2n − 1 parameters. 2. It is difficult to interpret values of the identified nonadditive measures. As an answer to the second problem, the concept of value (solution/allocation function) has been proposed and investigated in cooperative game theory for many years as a representation of global/overall importance/contribution of each player. The Shapley value defined by Shapley(1953)20 is the most famous one. Grabisch(1996)9 proposed an interaction index, –so-called the Shapley interaction index–, which is an extension of value as a representation of simultaneous interaction among the elements in a coalition. As an answer to both the first and second problems, Sugeno et.al.(1995)24 proposed inclusion-exclusion coverings with respect to a non-additive measure as a concept for decomposability of non-additive measures, which offer a trade-off between richness and complexity. It is interesting that all these concepts, the Shapley value, the Shapley interaction index, and inclusion-exclusion coverings, are characterized by using the M¨ obius transforms of non-additive measures. Our aims in this paper are: to investigate and to clarify relations among these concepts, the Shapley value, the Shapley interaction index, inclusion-exclusion coverings. In the remainder of this introduction, we shall indicate the contents of the 5 sections as follows: Section 2 recalls some basic definitions we will use in this paper. Section 3 is devoted to the concepts of value and interaction index. Basic reasonable axioms are recalled, and axiomatizations of value and interaction index are provided. Section 4 recalls some basic definitions and results relating to decomposability of non-additive measures. Section 5 discusses the invariance and stability of decomposability of non-additive measures under some transformations which characterize the Shapley value and interaction index. Section 6 concludes this paper. Throughout the paper, we shall work in discrete case, denoting a finite space N with n elements. In a similar way, s, t, . . . will denote the cardinality of subsets S, T, . . . of N . A game v on N is a set function (non-additive measure) on N vanishing at ∅. A monotonic game on N , i.e., v(A) ≤ v(B) whenever A ⊆ B ⊆ N , is especially called a fuzzy measure I . A game v on N is said to be full-domain if v(S ∪ T ) = v(T ) for any T ⊆ N implies S = ∅. In order to avoid inessential or technical complexity, throughout the paper we assume that all games are full-domain. We denote the set of all set functions on N by F N , the set of all game on N by G N , and the set of all additive measures/games on N by AN .

I In this paper, we shall not use the term “fuzzy measure”, because the concept of “game” is wider concept than that of fuzzy measure. That is, the term “game” can be replaced by “fuzzy measure” in almost all discussions and results in this paper.

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2. Preliminaries obius transform m is a mapping from G N to G N Definition 1. Let v ∈ G N . The M¨ defined by:  (−1)s−t v(T ), for S ⊆ N. (1) m(v)(S) := T ⊆S

This correspondence proves21 to be one-to-one, since conversely  v(S) = m(v)(T ), ∀S ⊆ N.

(2)

T ⊆S

In cooperative game theory, {m(v)(S)}S⊆N is called dividends12 of a game v. In evidence theory, this corresponds also to the basic probability assignment 22 . Now, we shall introduce some notions: S-equivalence, duality, reduction, and partnership, which are well-known in game theory (e.g., see Friedman(1986)4, Chun(1989)3 , Funaki(1996)8, and Hart and Mas-Colell(1989)14). Definition 2. Let v,w ∈ G N . v and w are called S-equivalent if w is represented as an affine transform of v, i.e., there exists a real constant α and an additive measure λ ∈ AN such that  w(S) = α · v(S) + λ({i}) = α · v(S) + λ(S), ∀S ⊆ N. (3) i∈S

We denote v ∼ w if v and w are S-equivalent. Indeed, the relation ∼ is an equivalence relation. v and w are said to be positive S-equivalent if the equation (3) holds with (α,λ)

a positive real constant α. Especially, we denote v ∼ w if v and w are S-equivalent with respect to the relation: w(S) = α · v(S) + λ(S), ∀S ⊆ N. S-equivalence requires, first, that a change in scale common to all elements should affect the value accordingly; and second, that adding a fixed amount, whenever an element i appears, should lead to just adding this amount to its final value.

For example, the degrees of the temperature by Celsius and Fahrenheit scales are (positive) S-equivalent. Consequently, methods of measurement are the only difference between v and w if v and w are S-equivalent. That is, these two games may differ and be essentially the same. Definition 3. For any v ∈ G N , the dual game of v, denoted as v ∗ , is the game defined by v ∗ (S) := v(N ) − v(N \ S), ∀S ⊆ N.

(4)

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In another representation using the M¨ obius transform:   v(S) = m(v)(T ) = m(v ∗ )(T ), ∀S ⊆ N. T :T ⊆S

v ∗ (S) =

(5)

T :S∩T =∅



m(v)(T ) =

T :S∩T =∅



m(v ∗ )(T ), ∀S ⊆ N.

(6)

T :T ⊆S

Funaki(1996)8 provided the following interpretation of duality: The dual game is an induced game in which a coalition (i.e, set of players/criteria) gets the rest of the grand coalition (i.e., the set of all players/criteria) importance/worth after the complement of the coalition gets its importance/worth of the original game. It is considered as an optimistic valuation of the game situation if the original game is considered as a pessimistic valuation like maxmin standard.

For example, the dual function of a belief function in evidence theory is a plausibility function. Consequently, attitudes toward evaluations are the only difference between v and v ∗ . Definition 4. 13 Let v ∈ G N , T be a non-empty subset of N . The reduced game with respect to T , denoted as v[T ] , is a game on (N \ T ) ∪ [T ], where [T ] indicates a single hypothetical element considered to be the representative of the elements in T . It is defined by, for any S ⊆ N \ T , v[T ] (S) := v(S), v[T ] (S ∪ [T ]) := v(S ∪ T ). Definition 5. Let v ∈ G N . A non-empty coalition P ⊆ N (i.e., non-empty subset of N ) is called partnership with respect to v if, for any T ⊆ N \ P , v(S ∪ T ) = v(T ),

∀S  P.

The concept of partnership is well-known concept proposed by Kalai and Samet(1988)15 in the framework of game theory. We quote from Kalai and Samet(1988)15 : the partnership is an agreement among a set of players P that none of them will establish any coalitional agreement with the players outside P unless all of them do. By forming such an agreement, the players in a partnership P restrict cooperation with players outside P .

Thus, a partnership P behaves like a single hypothetical player [P ], that is, the game v and its reduced version v[P ] can be considered as essentially the same. Clearly, every single element is a partnership. On the other hand, a partnership is a special case of semiatom defined by Murofushi, et. al.19 .

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Lastly, we recall the definition of the Choquet integral. Definition 6. 2 The Choquet integral of a function f : N → R over A ⊆ N with respect to v ∈ G N is defined by   0  ∞ (C) f dv := {v({f > r} ∩ A) − v(A)} dr + v({f > r}) dr. (7) −∞

A

0

3. Value as an overall importance of each single element in a game We consider at first the problem of measuring the global/overall importance of a single element (player/criterion) of the object to be considered/evaluated. We could not say that the importance of an element xi ∈ N is simply expressed by v({xi }) alone. In fact, all v(A) such that A  xi must also be taken into account. In game theory, the concept of value has been proposed and investigated as a representation of global/overall importance of each player in a game. So, the payoff for each player is determined or allocated based on the value of the game. Especially, Shapley has proposed the so-called Shapley value based on a set of reasonable and simple axioms (See Shapley(1953)23 ). Definition 7. 1 A value on G N is a linear mapping φ from G N into AN such that φ(v)(N ) = v(N ) for all v ∈ G N . The reduced game of the original game with respect to a value φ is defined by removing one player from the original game under a premise that the removed player gets a payoff prescribed by value φ as a compensation for the cooperation to the rest of the players (e.g., see Hart and Mas-Colell(1988)14). Definition 8. 14 Let v ∈ G N and φ be a value. The reduced game of v to T ⊆ N with respect to φ is denoted as vTφ and defined by vTφ (S) := v(S ∪ T c) − φ(v|S∪T c )(T c ) for S ⊆ T,

(8)

where v|S∪T c is the restriction of v to S∪T c, –that is, v|S∪T c (A) = v(A) if A ⊆ S∪T c and 0 otherwise. Definition 9.

23

The Shapley value φ is the value defined by  (n − s − 1)! · t! [v(T ∪ {i}) − v(T )] for i ∈ N. φ(v)({i}) := n!

(9)

T ⊆N \{i}

This formula is rewritten using the M¨ obius transform as  1 m(v)(T ) ∀i ∈ N. φ(v)({i}) = t

(10)

{i}⊆T ⊆N

The Shapley value φ satisfies the following three reasonable axioms wellknown in game theory: (e.g., see Chun(1989)3 , Funaki(1996)8 , and Hart and MasColell(1989)14 .)

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Axiom 1 (Measurement invariance) : Axiom 2 (Self-duality) :

(α,λ)

(α,λ)

v ∼ w =⇒ φ(v) ∼ φ(w).

φ(v ∗ ) = φ(v).

Axiom 3 (Internal consistency) :

φ(vTφ ) = φ(v)|T .

Axiom 1 leads to that φ(v) ∼ φ(w) whenever v ∼ w, –that is, global/overall importance should be independent of methods of measurement. Axiom 2 indicates that global/overall importance should be independent of attitudes toward evaluations. Axiom 3 indicates that global/overall importance in the reduced game with respect to φ always should yield the same global/overall importance as in the original game (e.g., see, in detail, Chun(1989) 3 , Funaki(1996) 8 , and Hart and Mas-Colell(1989) 14 ). Theorem 1. Let φ ∈ AN . φ satisfies Axioms 1,2, and 3, (i.e., measurement invariance, self-duality, and internal consistency) if and only if φ is the Shapley value. Proof. This theorem is immediately verified from the following Lemmas 1 and 2.

Lemma 1. 14 Let φ ∈ AN . φ is a standardII for two-person game satisfying Axiom 3 if and only if φ is the Shapley value. Lemma 2. Let φ ∈ AN . φ is standard for two-person game if φ satisfies Axioms 1 and 2. Proof. Let v ∈ G {i,j} and v ∗ be the dual game of v. Define λ ∈ A{i,j} as λ(k) := v({i, j}) + v({k}) − v({i, j} \ {k}) for k ∈ {i, j}. Then, we have v ∗ = (−1) · v + λ. Hence, v and v ∗ are S-equivalent with −1 ∈ R and λ ∈ A{i,j} . If φ satisfies Axioms 1 and 2, φ(v)({k}) = φ(v ∗ )({k}) = φ((−1) · v + λ)({k}) = −φ(v)({k}) + λ({k}) = −φ(v)({k}) + v({i, j}) + v({k}) − v({i, j} \ {k}). Hence, φ(v)({k}) = v({k}) + 1 2 [v({i, j}) − v({k}) − v({i, j} \ {k})] for any k ∈ {i, j}. The Shapley value φ can be regarded as a function on N since φ is an additive measure on N . In order to enable us to represent interaction phenomena modeled by a game on a set of players/criteria, Grabisch(1996)9 proposed an interaction index I : G N → F N , —the so-called Shapley interaction index—, as an extension of the Shapley value. ∈ AN is said to be standard for two-person game if φ(v)({k}) = v({k})+ 12 [v({i, j})− v({k})− v({i, j} \ {k})] for any k ∈ {i, j} and any v ∈ G {i,j} . II φ

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Definition 10. The Shapley interaction index I is a mapping from G N into F N defined by, for any S ⊆ N and v ∈ G N , I(v)(S) :=

 (n − t − s)! · t!  (−1)s−l v(L ∪ T ). (n − s + 1)!

T ⊆N \S

(11)

L⊆S

The Shapley interaction index was characterized axiomatically based on reasonable axioms,III which are natural extensions of the axioms characterizing the Shapley value and the following Axiom 4 (see e.g., Kalai and Samet(1988)15 , Grabisch and Roubens(1999)11 , Fujimoto, Kojadinovic, and Marichal6,7 ). Axiom 4 (Reduced partnership consistency) : If P ⊆ N is a partnership with respect to v ∈ G N , then I(v[P ] )([P ]) = I(v)(P ). Recall that a partnership P can be considered as behaving as a single hypothetical element (player/criterion) [P ], –that is, P and [P ] are essentially the same. In other words, when we measure the interaction among the elements in a partnership P , it is as if we were measuring the global/overall importance of a hypothetical element [P ]. Axiom 4 then simply states that the interaction among the elements in a partnership P in a game v should be regarded as the global/overall importance of the reduced partnership [P ] in the corresponding reduced game v[P ] . 4. Decomposability of non-additive measures To flexibly represent complex interaction phenomena among criteria, the Choquet integral has been widely used as an aggregation operator in multicriteria decision making problems. However, interactions do not always exist among criteria in all coalitions. Therefore, it is practical to use the Choquet integral on only coalitions/sub-domains where interactions among criteria exist. For this purpose, an approach to hierarchical decompositions of Choquet integral models has been proposed and applied5,24 .  Definition 11. 24 Let v ∈ G N . A covering C of N , (i.e., C = N and C ⊆ 2N ), is called an inclusion-exclusion covering with respect to v if   (−1)|D|+1 v( D ∩ S) ∀S ⊆ N. (12) v(S) = D⊆C D=∅

III In Kalai and Samet(1988)15 , the Shapley value is characterized by 5 axioms, Additivity, Positivity(monotonicity), Dummy player, Symmetry, and Efficiency, while, in Fujimoto, Kojadinovic, and Marichal6,7 , the Shapley interaction index is characterized by 6 axioms, Additivity, k-Positivity(monotonicity), Dummy partnership, Symmetry, Efficiency, and Reduced partnership consistency (i.e., Axiom 4).

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Clearly, any covering of N is an inclusion-exclusion covering with respect to v whenever v ∈ AN . Moreover, C = {N } is always an inclusion-exclusion covering of N for any v ∈ G N . Theorem 2. 24 Let v ∈ G N and C be a covering of N . Then, there exists vC ∈ G C for each C ∈ C such that  v(S) = vC (S ∩ C) ∀S ⊆ N, (13) C∈C

if and only if, C is an inclusion-exclusion covering with respect to v. Corollary 1. 24 Let v ∈ G N and C be a covering of N . Then there exists vC ∈ G C for each C ∈ C such that    f dv = (C) f dvC ∀f : N → R, (14) (C) N

C∈C

C

if and only if, C is an inclusion-exclusion covering with respect to v. Theorem 2 and Corollary 1IV provide a necessary and sufficient condition for a game (resp. the Choquet integral) to be decomposable into a sum of games (resp. Choquet integrals) over subdomains. Definition 12. 17,18 Let v ∈ G N , C ⊆ 2N . A family {vC }C∈C is called a Cdecomposition of v if each vC is a game on C ∈ C and (13) holds, –that is, v has a C-decomposition iff C is an inclusion-exclusion covering with respect to v. Then, v is said to be C-decomposable. Definition 13.

17,18,24

Let A ⊆ 2N . A is called an antichain on 2N if A ⊆ A , {A, A } ⊆ A ⇒ A = A .

That is, antichain coverings are coverings containing no nested elements. For example, let N := {1, 2, 3, 4, 5}. Then, {{1, 2, 3}, {2, 3, 4}, {3, 4, 5}} is an antichain covering of N , but {{1, 2, 3}, {2, 3, 4}, {2, 3, 4, 5}} is not. Because, {2, 3, 4} ⊆ {2, 3, 4, 5}. Definition 14. 24 Let C and D be two coverings of N . C is called coarser (resp. finer) than D if, for any D ∈ D (resp. C ∈ C), there exists C ∈ C (resp. D ∈ D) such that C ⊇ D (resp. C ⊆ D). Then, we denote C  D (resp. C  D) if C is coarser (resp. finer) than D. Proposition 1. 24 For any covering C of N , there uniquely exists an antichain covering D of N such that C  D and C  D. IV In Theorem 2 and Corollary 1, game v C can not be replaced by fuzzy measure, even if v is a fuzzy measure.

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Then, the D is called the antichain covering induced by C and obtained by D = C \ {C ∈ C|∃D ∈ C s.t C  D}. For example, {{1, 2, 3}, {2, 3, 4, 5}} is the antichain covering of {1, 2, 3, 4, 5} induced by {{1, 2, 3}, {2, 3, 4}, {2, 3, 4, 5}}. Proposition 2. 24 Let C and D be two covering of N . If v has a C-decomposition and D  C, then v has a D-decomposition, –that is, if C is an inclusion-exclusion covering, then D is an inclusion-exclusion covering whenever D  C. Theorem 3. 5 Let v ∈ G N , I : G N → F N be the Shapley interaction index, E N (I) := {A ⊆ N |I(v)(A) = 0}, and C be the antichain induced by E N (I). Then, v has a C-decomposition, –that is, C is an inclusion-exclusion covering. Corollary 2. 5 Let v ∈ G N , I : G N → F N be the Shapley interaction index, E N (I) := {A ⊆ N |I(v)(A) = 0}, and C be the antichain induced by E N (I). v has a D-decomposition if and only if D  C, –that is, C is the finest inclusion-exclusion covering, in other words, any covering which is coarser than C is an inclusionexclusion covering. For example, if v has a C-decomposition, and if A ⊆ N has no C ∈ C such as A ⊆ C, then I(v)(A) = 0. That is, the concept of decomposability of v ∈ G N is characterized by the Shapley interaction index which is a generalization of the Shapley value. Therefore, we can say that decompositions of games and/or of Choquet integrals can be expressed from a point of view of interaction. 5. Invariance of decomposability In sections 2 and 3, we showed that the Shapley value φ and/or the Shapley interaction index I, are essentially invariant under the following transformations: T1 : v → w (v ∼ w), T2 : v → v ∗ , T3 : v → vTφ , and T4 : v → v[P ] , and characterized based on these transformations. In section 4, we showed that inclusion-exclusion coverings are characterized by the Shapley interaction index, –that is, the concepts of decomposability of games and/or of Choquet integrals can be expressed from a point of view of interaction. In this section, we shall discuss invariance of the structures of inclusion-exclusion coverings (i.e., decomposability of games and/or of Choquet integrals) under the above transformations. Theorem 4. Let v, w ∈ G N and C be an inclusion-exclusion covering of N with respect to v. If v and w are S-equivalent, then C is also an inclusion-exclusion covering with respect to w, –that is, if v is C-decomposable, then w is also C-decomposable whenever v ∼ w. Proof. It is easily verified from substituting the equation (3) to (12). Lemma 3. 5 Let C be an inclusion-exclusion covering of N with respect to v ∈ G N and R := {R ⊆ N |∃C ∈ C such that R ⊆ C}. Then, m(v)(S) = 0

∀S ⊆ N, S ∈ R.

(15)

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Lemma 4.

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Let v ∈ G N and v ∗ be the dual game of v. Then,  m(v ∗ )(S) = (−1)s+1 m(v)(T ) ∀S ⊆ N.

(16)

T ⊇S

Theorem 5. Let v ∈ G N , v ∗ ∈ G N be the dual game of v, and C an inclusionexclusion covering of N with respect to v. Then, C is also an inclusion-exclusion covering with respect to v ∗ , –that is, if v is C-decomposable, then v ∗ is also Cdecomposable. Proof. Let R := {R ⊆ N |∃C ∈ C such that R ⊆ C}. For any S ∈ R,  m(v ∗ )(S) = (−1)s+1 m(v)(T ), from Lemma 4, T ⊇S

= 0, Here, for any U ⊆ N , v ∗ (U ) =

from Lemma 3. 

m(v ∗ )(V ), from (1),

V ⊆U

=



m(v ∗ )(V ) +

V ⊆U V ∈R

=



V ⊆U V ∈R

=



(17)



m(v ∗ )(V )

V ⊆U V ∈R

m(v ∗ )(V ), from (17) 

C∈C V ⊆C∩U

m(v ∗ )(V ) . |{C ∈ C|C ⊇ V }|

(18)

Hence, it follows, from the equation (18) and Theorem 2, that C is an inclusionexclusion covering with respect to v ∗ . Theorem 6. Let v ∈ G N , φ be the Shapley value, vTφ the reduced game of v to T ⊆ N with respect to φ, and C an inclusion-exclusion covering of N with respect to v. Then, C ∩ T is an inclusion-exclusion covering of T with respect to vTφ , –that is, if v is C-decomposable, then the reduced game vTφ on T is C ∩ T -decomposable. Proof. See Appendix A. Lemma 5. Let v ∈ G N , P ⊆ N be a partnership with respect to v, and C an inclusion-exclusion covering of N with respect to v. Then, there exists C ∈ C such that P ⊆ C. Proof. See Appendix B. The following lemma is easily verified. Lemma 6. Let v ∈ G N , P ⊆ N be a partnership with respect to v, C = {C1 , . . . , Cm } an inclusion-exclusion covering of N , and Di := Ci \P ∪[P ] if P ⊆ Ci

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and Di := Ci \ P if P ⊆ Ci . Then, {D1 , . . . , Dm } is a covering of N \ P ∪ [P ]. Moreover, for any S ⊆ N \ P and any i ∈ {1, . . . , m}, S ⊆ Ci ⇔ S ⊆ Di , S ∪ P ⊆ Ci ⇔ S ∪ [P ] ⊆ Di . Here, we call this {D1 , . . . , Dm } the covering of N \ P ∪ [P ] induced by inclusionexclusion covering C and partnership P with respect to v ∈ G N and denote it by C[P ] . Theorem 7. Let v ∈ G N , P ⊆ N be a partnership with respect to v, v[P ] the reduced game with respect to partnership P , and C an inclusion-exclusion covering of N with respect to v. Then, C[P ] is an inclusion-exclusion covering of N \ P ∪ [P ] with respect to v[P ] , –that is, if v is C-decomposable, then the reduced game v[P ] is C[P ] -decomposable. Proof. See Appendix B. 6. Conclusions We discussed some relations among values, (i.e., global/overall importance of each single element), interaction indices, (i.e., simultaneous interactions among the elements in a coalition), and inclusion-exclusion coverings, (i.e., decomposability of non-additive measures and/or of Choquet integrals). So, we showed that inclusionexclusion coverings are characterized by the Shapley interaction index. Then, inclusion-exclusion coverings are invariant under the transformations which characterize and preserve the Shapley value and the Shapley interaction index.

References 1. J. Aumann and L. S. Shapley, Values of Non-Atomic Games, Princeton Univ. Press, 1974. 2. G. Choquet, “Theory of capacities”, Annales de l’Institut Fourier, Vol.5, pp.131-295, 1953. 3. Y. Chun, “A new axiomatization of the Shapley value”, Games and Economic Behavior, Vol.1, No.2, pp.119-130, 1989. 4. J.W. Friedman, Game theory with applications to economics, Oxford University Press, 1986. 5. K. Fujimoto and T. Murofushi, “Some characterizations of the systems represented by Choquet and multi-linear functionals through the use of M¨ obius inversion”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.5, No.5, pp.547-561, 1997. 6. K. Fujimoto, I. Kojadinovic, and J.-L. Marichal, “Characterizations of probabilistic and cardinal-probabilistic interaction indices”, In Proc. of Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU2004), Perugia, Italy, July 2004.

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7. K. Fujimoto, I. Kojadinovic, and J-L. Marichal, “Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices”, submitted. 8. Y. Funaki, “Dual axiomatizations of solutions of cooperative games”, In Proc. of International Conference on Game Theory and Economic Applications, Bangalore, India, January 1996. 9. M. Grabisch, “k-order additive fuzzy measures”, In Proc. of 6th International Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems(IPMU96), pp.1345-1350, Granada, Spain, July 1996. 10. M. Grabisch, “Alternative representations of discrete fuzzy measures for Decision Making”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 5, No. 5, pp.587-507, 1997. 11. M. Grabisch and M. Roubens, “An axiomatic approach to the concept of interaction among players in cooperative games”, International Journal of Game Theory, Vol. 28, pp.547-565, 1999. 12. J. C. Harsanyi, “A bargaining model for cooperative n-person games”, In: A.W.Tucker and Luce (eds.), Contributions to the Theory of Games, pp.325-335, Princeton Univ. Press., 1959. 13. J. C. Harsanyi, “A simplified bargaining model for n-person cooperative game, International Economic Review, Vol.4, pp.194-220, 1963. 14. S. Hart and A. Mas-Colell, “Potential, value, and consistency”, Econometrica, Vol. 57, No. 3, pp.589-614, 1989. 15. E. Kalai and D. Samet, “Weighted Shapley values”, In: A. E. Roth(ed.),The Shapley Value, Cambridge University Press 1988. 16. H.A. Michener and W.T. Au, “A probabilistic theory of coalition formation in nperson side payment games”, Journal of Mathematical Sociology, Vol. 19, pp.165-188, 1994. 17. T. Murofushi, Y. Sawata, and K. Fujimoto, “ Decomposition of fuzzy measures into a sum of fuzzy measures on subdomains”, In Proc. 10th International Fuzzy System Association World Congress (IFSA2003), Istanbul, Turkey, June 2003. 18. Y. Sawata, K. Fujimoto, and T. Murofushi, “ Decomposition of monotone set functions into a sum of monotone set functions on subdomains”, submitted. 19. T. Murofushi, M. Sugeno, and K. Fujimoto, “Separated hierarchical decomposition of the Choquet integral”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 5, No. 5, pp. 563-585, 1997. 20. L.S. Shapley, “A value for n-person game”, In: A. W. Tucker and H. W. Kuhn (eds.), Contributions to the Theory of Games II, Princeton University Press 1953. 21. G.C. Rota, “On the foundation of combinatorial theory I: Theory of M¨ obius functions”, Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol.2, pp.340-368, 1964. 22. G.A. Shafer, A Mathematical Theory of Evidence, Princeton Univ. Press., 1976. 23. L.S. Shapley, “A value for n-person games”, In: H.W. Kuhn and A.W. Tucker (eds.), Contributions to the Theory of Games, II, Princeton Univ. Press, pp.307-317, 1953. 24. M. Sugeno, K. Fujimoto, and T. Murofushi, “A hierarchical decomposition of Choquet integral model”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 3, No.1, pp.1-15, 1995.

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13

Appendix A. Proof of Theorem 6 Lemma 7. Let v ∈ G N , φ be the Shapley value, and vTφ the reduced game to T ⊆ N of v with respect to φ. Then, ⎧  s ⎪ m(v)(S ∪ R) if ∅ = S ⊆ T, ⎪ ⎨ s + r R⊆T c φ (A.1) m(vT )(S) = ⎪ ⎪ ⎩ 0 if S = ∅. Proof. At first, for any S ⊆ T (S = ∅),   (−1)s−u v(U ∪ T c ) = (−1)s−u U⊆S



(−1)s−u

 

 

(−1)s−u

R⊆T c U⊆S

=



m(v)(X ∪ R)

X⊆U R⊆T c

U⊆S

=

m(v)(W )

W ⊆U∪T c

U⊆S

=





m(v)(X ∪ R)

X⊆U

m(v)(S ∪ R).

(A.2)

R⊆T c

Next, for any S ⊆ T (S = ∅), 

(−1)s−u φ(v|U∪T c )(T c ) =

U⊆S



(−1)s−u

= =

{i}⊆W ⊆U∪T c





(−1)s−u

i∈T c U⊆S

=



(−1)s−u

i∈T c U⊆S

 



i∈T c {i}⊆W ⊆U∪T c

U⊆S

 





=



i∈T c {i}⊆R⊆T c

=

m(v)(W ) w



X⊆U {i}⊆R⊆T c



(−1)s−u

i∈T c {i}⊆R⊆T c U⊆S



m(v)(W ) w

m(v)(X ∪ R) x+r

 m(v)(X ∪ R) x+r

X⊆U

m(v)(S ∪ R) s+r

 r · m(v)(S ∪ R) . s+r c

R⊆T

It follows, from the equation (A.2) and (A.3), that, for any S ⊆ T (S = ∅),  m(vTφ )(S) = (−1)s−u vTφ (U ) U⊆S

=



U⊆S

(−1)s−u [v(U ∪ T c ) − φ(v|U∪T c )(T c )]

(A.3)

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Katsushige FUJIMOTO and Toshiaki MUROFUSHI

 

r · m(v)(S ∪ R) = m(v)(S ∪ R) − s+r R⊆T c  s · m(v)(S ∪ R). = s+r c R⊆T

Proof of Theorem 6. Let C be an inclusion-exclusion covering of N , T ⊆ N , P := {P ⊆ T |∃C ∈ C s.t. P ⊆ C ∩ T }, Q := 2T \ P, and R := {R ⊆ N |∃C ∈ C s.t. R ⊆ C ∩ T }. Then, S ∪ R ∈ R for any S ∈ Q and any R ⊆ T c since S ∈ R and S ∩ T c = ∅. Hence, it follows from Lemma 3 and 7 that m(vTφ )(S) = 0 ∀S ∈ Q. Here, for any S ⊆ T , vTφ (S) = =

 R⊆S

m(vTφ )(R)



m(vTφ )(R) +

R⊆S R∈P

=

(A.4)





m(vTφ )(R)

R⊆S R∈Q

m(vTφ )(R), from (A.4),

R⊆S R∈P

=





CT ∈C∩T R⊆CT ∩S

m(vTφ )(R) |{CT ∈ C ∩ R|CT ⊇ R}|

(A.5)

Hence, it follows from Theorem 2 and the equation (A.5) that C ∩ T is an inclusionexclusion covering of T with respect to vTφ . Appendix B. Proof of Theorem 7 Lemma 8. Let v ∈ G N , P be a partnership with respect to v, and v[P ] the reduced game of v on N \ P ∪ [P ] with respect to P . Then, m(v)(S ∪ R) = 0 for any S ⊆ N \ P and R  P (R = ∅).

(B.1)

Moreover, for any S ⊆ N \ P : m(v[P ] )(S) = m(v)(S),

(B.2)

m(v[P ] )(S ∪ [P ]) = m(v)(S ∪ P ).

(B.3)

Proof. First, for any S ⊆ N \ P and R  P (R = ∅),  m(v)(S ∪ R) = (−1)s+r−t v(T ) T ⊆S∪R

=

 

T1 ⊆S T2 ⊆R

(−1)s−t1 (−1)r−t2 v(T1 ∪ T2 )

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Some Relations among Values, Interactions, and Decomposability of Non-Additive Measures



=

⎛ (−1)r−t2 ⎝

T2 ⊆R

15

⎞



(−1)s−t1 v(T1 )⎠

T1 ⊆S

= 0.

(B.4)

Next, the equation (B.2) is easily verified. Last, for any S ⊆ N \ P ,  (−1)s+1−t v[P ] (T ) m(v[P ] )(S ∪ [P ]) = T ⊆S∪[P ]



=

T ⊆S∪[P ] T [P ]



=

T ⊆S

=

(−1)s+1−t v[P ] (T )

T ⊆S∪[P ] T [P ]

(−1)s−t v[P ] (T ∪ [P ]) +





(−1)s+1−t v[P ] (T ) +



(−1)s+1−t v[P ] (T )

T ⊆S

s−t

(−1)

  v[P ] (T ∪ [P ]) − v[P ] (T )

T ⊆S



=

(−1)s−t (v(T ∪ P ) − v(T ))

(B.5)

T ⊆S

and m(v)(S ∪ P ) =



(−1)s+p−t v(T )

T ⊆S∪P

=



(−1)s+p−t v(T ) +

T ⊆S∪P T ⊇P

=



=

(−1)s−t v(T ∪ P ) +

=

(−1)

v(T ∪ P ) +

=



T2 P s−t

(−1)

v(T ∪ P ) + (−1)

T ⊆S



 

(−1)s+p−t1 −t2 v(T1 ∪ T2 )

T1 ⊆S T2 P s−t

T ⊆S



(−1)s+p−t v(T )

T ⊆S∪P T ⊇P

T ⊆S





(−1)p−t2 



(−1)s−t1 v(T1 )

T1 ⊆S

(−1)s−t1 v(T1 )

T1 ⊆S

(−1)s−t (v(T ∪ P ) − v(T )) .

(B.6)

T ⊆S

It follows, from the equation (B.5) and (B.6), that the equation (B.3) holds. Proof of Lemma 5. By contradiction. Let v ∈ G N , P ⊆ N be a partnership with respect to v, and C be an inclusion-exclusion covering of N with respect to v. Now, we assume that P ⊆ C for any C ∈ C. Then, m(v)(R ∪ S) = 0,

∅ = ∀R  P, ∀S ⊆ N \ P

(B.7)

from Lemma 8 and m(v)(P ∪ S) = 0,

∀S ⊆ N \ P

(B.8)

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Katsushige FUJIMOTO and Toshiaki MUROFUSHI

from Theorem 2. For any S ⊆ N \ P ,  m(v)(T ) v(P ∪ S) = T ⊆P ∪S

=

 

m(v)(T1 ∪ T2 )

T1 ⊆P T2 ⊆S

=



m(v)(∅ ∪ T2 ) +

T2 ⊆S

=





m(v)(P ∪ T2 ), from (B.7),

T2 ⊆S

m(v)(T2 ), from (B.8),

T2 ⊆S

= v(S).

(B.9)

Hence, P = ∅ since v is full-domain. This contradiction shows that there exists C ∈ C such that P ⊆ C. Proof of Theorem 7. Let v ∈ G N , P ⊆ N be a partnership with respect to v, v[P ] the reduced game with respect to partnership P , C an inclusion-exclusion covering of N with respect to v, C[P ] the covering of N \P ∪[P ] induced by inclusion-exclusion covering C and partnership P with respect to v, and R := {R ⊆ N |∃C ∈ Cs.t. R ⊆ C}. For any S ⊆ N \ P , v[P ] (S) = v(S), from Definition 4,  = m(v)(T ) T ⊆S

=



m(v)(T ) from Lemma 3,

T ⊆S T ∈R

=

  C∈C T ⊆S

1 m(v)(T ) |{C ∈ C|C ⊇ T }|

 

=

C∈C[P ] T ⊆S

1 m(v[P ] )(T ) |{C ∈ C[P ] |C ⊇ T }|

(B.10)

from Lemma 6 and 8, and v[P ] (S ∪ [P ]) = v(S ∪ P ), from Definition 4,  m(v)(T ) = T ⊆S∪P

=

 

m(v)(T1 ∪ T2 )

T1 ⊆S T2 ⊆P

=



m(v)(T1 ∪ ∅) +

T1 ⊆S

=



T1 ⊆S T1 ∈R



m(v)(T1 ∪ P ) from Lemma 8,

T1 ⊆S

m(v)(T1 ) +



T1 ⊆S T1 ∪P ∈R

m(v)(T1 ∪ P ) from Lemma 3,

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Some Relations among Values, Interactions, and Decomposability of Non-Additive Measures

=

 C∈C T ⊆S

+

1 m(v)(T ) |{C ∈ C|C ⊇ T }|



C∈C T ⊆S

=

1 m(v)(T ∪ P ) |{C ∈ C|C ⊇ T ∪ P }|

⎧  ⎨

1

⎩

|{C ∈ C[P ] |C ⊇ T }|

C∈C[P ]

+

 T ⊆S

17

T ⊆S

m(v[P ] )(T )

⎫ ⎬ 1 m(v[P ] )(T ∪ [P ]) ⎭ |{C ∈ C[P ] |C ⊇ T ∪ [P ]}|

(B.11)

from Lemma 6 and 8. It follows, from equations (B.10), (B.11) and Theorem 2, that C[P ] is an inclusionexclusion covering of N \ P ∪ [P ] with respect to v[P ] .