1

An Axiomatic approach to the probabilistic interaction representations

Katsushige FUJIMOTO Faculty of Economics, Fukushima University, 1 Kanayagawa Fukushima, 960-1296, Japan, [email protected]

This research was partially supported by the Japan Society for the Promotion of Science, Japan, Grant-in-Aid for Encouragement of Young Scientists, 13780349, 2001-2002. August 12, 2002

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Abstract This paper provides an axiomatic characterization for cardinal-probabilistic interaction indices which represent the interaction among elements which could be players in a cooperative game, criteria in a multi-criteria decision problem, attributes in a multi-attributes decision problem, experts or voters in an opinion pooling problem, etc. These indices are characterized by four axioms: additivity, k-monotonicity, neutral partnership and symmetry axioms. Moreover, the Shapley, Banzhaf, and chaining interaction indices which are typical cardinal-probabilistic interaction indices are also characterized. Each of these indices is based on above four axioms and characterized by using two axioms among three axioms, respectively: efficiency, partnershipreduction-consistency, and partnership-allocation-equivalence. Keywords Shapley and Banzhaf power indices, Partnership, Interaction index, Probabilistic interaction index, Balanced-probabilistic interaction index.

I. Introduction Non-additive measures, (e.g. fuzzy measures, characteristic functions in game theory, belief and plausibility functions in evidence theory, etc...), have a high potential for representing interactions among elements of considered objects, especially in multi-attributes decision making, where they have been already successfully applied. In this paper, we will discuss non-additive measures as a general tool for modeling the worth/importance of a set of elements which could be players in a cooperative game, criteria in a multi-criteria decision problem, attributes in a multi-attributes decision problem, experts or voters in an opinion pooling problem, etc. Our aims, in this paper, are: to provide new axiomatic characterizations of interaction representations for a set of elements and to investigate and to clarify relations among them. Considering the interaction among elements in the set S, we should take not only the worth of the set S but also the worth of all sets containing S into account. The notion on the interaction of a single element has been proposed/investigated for many years as the solution concept in the framework of game theory. As an example of solution concepts, the Shapley value/power index[14] is a most famous one. Murofushi and Soneda[12] proposed an interaction index for a pair of elements, which was a generalization of the Shapley value, based on multi-attributes utility theory. Grabisch[4] generalized this index to any August 12, 2002

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set of elements. Later, Grabisch and Roubens[5] provided an axiomatic approach of the so-called Shapley interaction index which is a generalization of both the Shapley value and the Murofushi and Soneda’s interaction index, and of the so-called Banzhaf interaction index, which is a generalization of the Banzhaf value. Marichal[10] proposed the chaining interaction index which is generalization of the Shapley value but not of the Murofushi and Soneda’s interaction index. This paper will provide new axiomatic characterizations of the Shapley, Banzhaf, and chaining interaction indices, and show that any of these three interaction indices is situated in the intermediate/neutral position, in a certain sense, of the two remaining indices. In the remainder of this introduction, we shall indicate the contents of the 4 sections as follows: Section II provides some basic definitions on interaction indices and shows some preliminary propositions. Section III provides axioms and some relations among them. Section IV provides and states new axiomatic characterizations of interaction indices and shows some relations among them. Section V 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.

II. Preliminaries A. Worth Functions Definition 1 (worth function) Let N be a non-empty set with n elements. The worth function on N is a real valued set function v N : 2N → R that satisfies v N (∅) = 0. The set of all worth functions on N is denoted by W N . Definition 2 (k-monotone)

[1]

A worth function v N : 2N → R is 1-monotone whenever

v N (A) ≤ v N (B) for any A, B ⊆ N such that A ⊆ B; furthermore, given an integer k ≥ 2, a worth function v N : 2N → R is k-monotone, if and only if vN (

k  i=1

Si ) ≥

 I⊆{1,...,k} I=∅

(−1)|I|+1v N (

 i∈I

Si )

(1)

for any Si ⊆ N, 1 ≤ i ≤ k. The worth function which is k-monotone for any integer k ≥ 1 August 12, 2002

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is called ∞-monotone. If the Si ’s⊆ N are distinct, the formula (1) is expressed as N

v (

k  i=1

Si ) ≥

k  i=1

v N (Si ).

(2)

A fuzzy measure is a 1-monotonic worth function. A convex game, in game theory, is 2-monotone. A belief function, in evidence theory, is ∞-monotone. By using the fact that the Si ’s are not necessarily distinct, v N is k  -monotone for any integer 2 ≤ k  ≤ k if v N is k-monotone for some integer k ≥ 2, –that is, k-monotonic worth functions, for any integer k ≥ 2, are fuzzy measures. k-monotonic worth functions represent the situation in which synergy effects exist on the combination of at most k sets of elements. Therefore, there are synergy effects among elements in any S ⊆ N such that |S| ≤ k if v N is k-monotone. Definition 3 (reduced worth function) Let T be a non-empty subset of N. The reduced worth function with respect to T of v N ∈ W N is a worth function on 2N \T ∪{[T ]} denoted by N N N N N v[T ] and defined for any S ⊆ N \T as follows: v[T ] (S) = v (S), v[T ] (S ∪{[T ]}) = v (S ∪T ),

where [T ] indicates a single hypothetical element, which seems to be the representative (or macro element) of the elements in T . Definition 4 (Harsanyi’s reduced worth function)

[7]

Let T be a non-empty subset of N,

φ a real valued function on W N × N. The Harsanyi’s reduced worth function with respect to φ of v N ∈ W N to T is a worth function on 2T denoted by vφT and defined for any S ⊆ T as follows: vφT (S) = v N (S ∪ T c ) −

 i∈T c

c

φ(v S∪T , i),

c

where v S∪T (U) = v(U ∩ (S ∪ T c )) for any U ⊆ N. The interpretation of this reduced worth function is provided by Hart and Mas-Colell[8] , in the framework of game theory. We quote from Hart and Mas-Colell[8] : Given the solution function φ, a game (N, v) and a coalition T ⊆ N, every subcoalition of T needs to consider the total payoff remaining after paying the members August 12, 2002

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of T c according to φ. To compute the worth of a coalition S ⊆ T (in the reduced game), we assume that the members of T \ S are not present; in other words, one considers the game (S ∪ T c , v), in which payoff are distributed according to φ. B. The Sets of Elements Definition 5 (neutral and null elements) An element i ∈ N is said to be neutral with respect to v N if v N (S ∪ {i}) = v N (S) + v N ({i}) for any S ⊆ N \ {i}. A neutral element whose worth is zero is said to be null.

The notion of neutrality means that (i) a neutral element has no interaction for any other sets, and (ii) the overall effect of a neutral element is its own worth. Thus, a neutral element i has neither synergy nor overlapped effect for any set of elements S  i, and a null element has no meaningful/value for any set.

Definition 6 (partnership)

[9]

A set of elements P ⊆ N is said to be a partnership with

respect to v N if v(R ∪ T ) = v(R) for any T ⊆ P , T = P and any R ⊆ N \ P . Any proper subset T of partnership P is worthless and generates no new effect by forming the set U ⊇ T , or by amalgamating elements within the set U ⊇ T , unless U ⊇ P . That is, the partnership P behaves like a single hypothetical element [P ]. Thus, the worth function v N and its reduced worth function v (N \P )∪{[P ]} are essentially the same when P is a partnership with respect to v N . Clearly, any single element is a partnership. In fuzzy measure theoretical sense, a partnership is a special case of semi-atoms[11] . Definition 7 (neutral partnership) A set of elements P ⊆ N is said to be a neutral partnership with respect to v N if (i) P is a partnership with respect to v N , and (ii) v N (S ∪ P ) = v N (S) + v N (P ) for any S ⊆ N \ P . Considering the reduction of a partnership P to the single hypothetical element [P ], the neutral partnership P can be regarded as the neutral element [P ] with respect to v (N \P )∪{[P ]} . August 12, 2002

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C. Interaction Indices We shall interpret interactions among elements of a set of elements S as the overall effects which appear by forming S or by amalgamating elements within S (i.e. the overall effect of S). Hence, representing the overall effect of S, we should take not only the worth of S (i.e. v N (S)) but also the worth of all sets containing S (i.e. v N (T ) for any T ⊇ S) into consideration. The marginal worth/contribution of a single element i to the set T ⊆ N, T i is expressed as v N (T )−v N (T \{i}). Murofushi and Soneda[12] proposed a marginal interaction between elements i and j in the presence of elements in T ⊆ N \ {i, j} as a generalization of the marginal worth, which was defined by v N (T ∪ {i, j}) − v N (T ∪ {i}) − v N (T ∪ {j}) + v N (T ). This notion is represented as the difference between “the marginal worth of j to (T ∪{i}) ∪{j}” and “the marginal worth of j to T ∪{j}”. That is, the interaction between i and j is represented as positive value when the marginal worth of j to every subset that contains i is greater than the marginal worth of j to the same subset when i is excluded. Later, Grabisch and Roubens[5] generalize this marginal interaction between two elements in {i, j}, in the presence of elements in T ⊆ N \ {i, j}, to the interaction among elements in any S ⊆ N, in the presence of elements in T ⊆ N \ S, as the s-th order derivative of v N N

N

N

N

at S defined recursively by Δv{i} (T ) := v N (T ) − v N (T \ {i}), Δv{i,j} (T ) := Δv{i} (Δv{j} (T )) = v N (T ) − v N (T \ {i}) − v N (T \ {j}) + v N (T \ {i, j}), etc..., for all T ⊆ N. It is easy to show by induction over s that N

ΔvS (T ∪ S) :=



(−1)s−k v N (K ∪ T ).

(3)

K⊆S

for any S ⊆ N and any T ⊆ N \ S. The following proposition shows that the marginal interaction represents synergy effects as positive values. N

Proposition 1: A worth function v N : 2N → R is k-monotone, if and only if ΔvS (T ∪S) ≥ 0 for any S ⊆ N, s ≤ k and any T ⊆ N \ S. Here, the marginal interaction among elements in the set S in which effects to the August 12, 2002

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outside of S are not taken into consideration (i.e. ΔvS (S) = =



s−k N v (K) K⊆S (−1)



s−k N v (K K⊆S (−1)

∪ ∅)

) coincides with the M¨obius transform of S. Thus, this marginal

interaction (i.e. the M¨obius transform) expresses the internal interaction among elements in S with respect to v N . We define an interaction index to be a function I : W N × 2N → R. Thus, I(v N , S) expresses the amount of interaction among elements in S ⊆ N for the worth function v N . I(v N , {i}) represents the game-theoretical value related to the element i. Now, we will show the notion of probabilistic interactions as the average of the marginal interactions.

Definition 8 (probabilistic interaction indices)

[6]

Let v N be a worth function on N.

Probabilistic interaction indices I(v N , S) are interaction indices represented as an expected marginal interaction of S over all sets T ⊆ N \ S, i.e. I(v N , S) are represented as follows: there exists a probability distribution {pN S (T )}T ⊆N \S on N \ S such that I(v N , S) =

 T ⊆N \S

N

v pN S (T ) · ΔS (T ∪ S)

for any S ⊆ N and any v N ∈ W N . Especially, probabilistic interaction indices are called cardinal-probabilistic interaction indices when probability distributions depend only on cardinalities of S and T (i.e. {pN S (T )} can be expressed as {pns (t)} ). Here, we can see that the internal interaction is a one of the probabilistic interaction index with pN S (T ) = 1 if T = ∅ and = 0 otherwise. Thus, this internal interaction is called the internal interaction index for S with respect to v N and denoted as Iint (v N , S). From the above viewpoints, Grabisch and Roubens[5] , and Marichal[10] have proposed the following interaction indices:

Definition 9 (Typical probabilistic interaction indices)

[5][10]

Let v N be a worth function

on N. The Shapley, Banzhaf, and chaining interaction index for S ⊆ N with respect to v N is an interaction index (i.e. a real valued set function on W N × 2N ) denoted by ISh (v N , S), August 12, 2002

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IB (v N , S), Ich (v N , S), and defined by 

ISh (v N , S) :=

T ⊆N \S

IB (v N , S) :=

(n − t − s)!t! vN Δ (T ∪ S), (n − s + 1)! S

 T ⊆N \S

Ich (v N , S) :=

 T ⊆N \S

1 2n−s

(4)

N

ΔvS (T ∪ S),

(5)

s(n − t − s)!(t + s − 1)! vN ΔS (T ∪ S), n!

(6)

respectively. Both Shapley and chaining interaction indices for any {i} ⊆ N coincides with the Shapley power index, the Banzhaf interaction index with the Banzhaf power index, and the Shapley interaction index for any {i, j} ⊆ N with the Murofushi and Soneda’s interaction index Iij [12] . All of the above three interaction indices: ISh (v N , S), IB (v N , S), and Ich (v N , S), are cardinal-probabilistic interaction indices. Indeed, 

1 T ⊆N \S 2n−s

=



T ⊆N \S

s·(n−t−s)!(t+s−1)! n!



(n−t−s)!t! T ⊆N \S (n−s+1)!

=

= 1 (i.e. these are probability distributions on

N \ S). That is, these three interaction indices are characterized by the probability distributions on N \ S. Definition 10 (Balanced-probabilitistic interaction indices) Let I : W N × 2N → R be a cardinal-probabilistic interaction index associated with a probability distribution pns (t). Then, I is said to be the balanced-probabilistic interaction index if pns (t) is expressed by: pns (t)

 1

=

0

xt (1 − x)n−s−t dFs (x)

(7)

for some probability measures/cumulative distribution functions Fs (x) in [0, 1]. Thus, it is straightforward to see that the {pns (t)} verify the relation: m m−1 (t) pm s (t) + ps (t + 1) = ps

for 0 < m ≤ n and any 0 ≤ s < m, 0 ≤ t < m − s. The balanced interaction indices for singletons satisfy the balanced contribution axiom[13] : I(v S , {i}) − I(v S\{j} , {i}) = I(v S , {j}) − I(v N \{i} , {j}) August 12, 2002

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for any S ⊆ N and any i, j ∈ S. Proposition 2: All of the Shapley, Banzhaf, chaining and internal interaction indices are balanced-probabilistic interaction indices.

In fact, these interaction indices are expressed by using following probabilistic measures Fs (x) in [0, 1]: Shapley interaction index : FsSh (x) = x, ∀x ∈ [0, 1], 

Banzhaf interaction index :

FsB (x)

=

0 if x < 1/2

,

1 if x ≥ 1/2

chaining interaction index : Fsch (x) = xs , ∀x ∈ [0, 1], internal interaction index : Fsint (x) = 1, ∀x ∈ [0, 1]. As we can see above, the Fs (x)’s in the Shapley, Banzhaf and internal interaction indices are independent of s, while, the only Fsch (x) in the chaining interaction index depends on s. Now, we consider the probability measure Fsα (x) in [0, 1] which stands at the opposite extreme from Fsint (x):



i.e.

Fsα (x)

=

0 if x < 1,

1 if x = 1. Then, we obtain the following balanced-probability interaction index Iα (v N , S): N

Iα (v N , S) := ΔvS (N)

(8)

for any S ⊆ N and any v N ∈ W N . Moreover, we provide an interaction index Iβ characterized by using probability measure Fsβ (x) in [0, 1] which stands symmetrical position of FsB (x) with respect to FsSh (x) = x as follows: 

Fsβ (x)

=

1/2 if x < 1 1

if x = 1

,

1 Iβ (v N , S) = (Iint (v N , S) + Iα (v N , S)) 2 August 12, 2002

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for any S ⊆ N and any v N ∈ W N . Note 1: Above interaction indices can be represented, through the use of M¨obius transN

form ΔvS (S), as follows: ISh (v N , S) = Ich (v N , S) = N

ΔvS (S) +

1 2





s vN T ⊇S t ΔT (T ),

T ⊇S T =S



1 vN T ⊇S t−s+1 ΔT (T ),

IB (v N , S) =

N

Iint (v N , S) = ΔvT (T ), Iα (v N , S) =



T ⊇S



1 vN T ⊇S 2t−s ΔT (T ),

N

ΔvT (T ) and Iβ (v N , S) =

N

ΔvT (T ).

Note 2: As we showed above, any balanced interaction index I satisfies the following formula: I(v N , {i}) − I(v N \{j} , {i}) = I(v N , {j}) − I(v N \{i} , {j})

(9)

for any i, j ∈ S. Especially, if I is ISh , IB , or Iα , the equation (9) gives the interaction between i and j (i.e. I(v N , {i, j})), –that is, in ISh , IB , and Iα , the interactions between i and j are represented as the difference between “the overall effect of i in the original worth function” and “that of i in the absence of j” (i.e. j’s contribution to the overall effect of i). Proposition 3: Let φ be a real valued function on W N × N and defined by φ(v N , i) := Ich (v N , {i}) (i.e. the Shapley value). Then, for any S ⊆ N, the chaining interaction index Ich with respect to v N can be represented by the internal interaction index Iint with respect to the Harsanyi’s reduced worth function vφS as follows: Ich (v N , S) = Iint (vφS , S) ∀S ⊆ N. Corollary 1: Let φ be a real valued function on W N × N and defined by φ(v N , i) := Ich (v N , {i}) (i.e. the Shapley value). Then, for any S ⊆ N, the chaining interaction index Ich with respect to v N can be represented as follows: Ich (v N , S) = Ich (vφS , S) = ISh (vφS , S) = IB (vφS , S) = Iα (vφS , S) = Iβ (vφS , S), ∀S ⊆ N.

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III. System of Axioms In order to characterize interaction indices, we will provide some axioms, and show some relation among these axioms. A. Axioms Additivity axiom : I(v N , ·) is an additive function on W N , –that is, I(v N + w N , ·) = I(v N , ·) + I(w N , ·) for any v N , w N ∈ W N . The additivity axiom says that interaction indices should be decomposable additively whenever worth functions are decomposable additively. k-positivity axiom : If v N ∈ W N is k-monotone, then I(v N , S) ≥ 0 for any S ⊆ N such that |S| ≤ k. The k-positivity axiom says that synergy effects should be represented as positive interactions. Neutrality axiom : If i is a neutral element with respect to v N ∈ W N , then (i) I(v N , {i}) = v N ({i}), (ii) I(v N , S ∪ {i}) = 0 for any non-empty set S ⊆ N \ {i}. Neutral partnership axiom : If P is a neutral partnership with respect to v N ∈ W N , then (i) I(v N , P ) = v N (P ), (ii) I(v N , S ∪ P ) = 0 for any non-empty set S ⊆ N \ P .

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The neutrality and neutral partnership axioms say that (i) the effects of neutral elements/partnerships should be theirs own worth, and (ii) there should be no interaction between a neutral element/partnership and its outside. Symmetry axiom : For any v N ∈ W N and any permutation π on N, I(v N , S) = I(πv N , πS), where πv N (π(S)) := v N (S) and π(S) := {π(i)|i ∈ S}. The symmetry axiom says that the names of the elements should play no role in determining the effects and interactions. Efficiency axiom : For any v N ∈ W N , I(v N , ·) satisfies the following property: 

I(v N , {i}) = v N (N)

i∈N

for any v N ∈ W N . The efficiency axiom is a kind of normalization condition. In game theory, this axiom states that the players/elements share the whole worth v N (N). Reduced-partnership-consistency axiom : If P ⊆ N is a partnership with respect to v N ∈ W N , then N I(v N , P ) = I(v[P ] , {[P ]}).

As we said above, a partnership P functions/behaves as a hypothetical single element [P ] which seems to be the representative of the elements in P . The reduced-partnershipconsistency axiom says that the overall effect of the reduced partnership [P ] should be the same as that of the original partnership P . Partnership-allocation-equivalence (I) axiom : If P ⊆ N is a partnership with respect to both v N ∈ W N and w N ∈ W N , then I(v N , P ) · I(w N , {i}) = I(w N , P ) · I(v N , {i}) August 12, 2002

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for any i ∈ P . Partnership-allocation-equivalence (II) axiom : If P ⊆ N is a partnership with respect to v N ∈ W N , then I(uP , {i}) · I(v N , P ) = I(v N , {i}) ∀i ∈ P , where uP is a unanimity game for P (i.e. uP (S) = 1 if S ⊇ P and = 0 otherwise). Any element in the partnership is worthless and effectless unless all of them do act simultaneously. Thus, we can see that each element in the partnership has the same role and the same overall effect. The partnership-allocation-equivalence axiom says that the ratio of “the overall effect of each element in partnership P ” to “the overall effect of P ” should be equivalently fixed and independent of methods of determining worths. B. Properties among Axioms Proposition 4: If I(v N , ·) satisfies the additivity and k-positivity axioms, then it is a linear function on W N . Proposition 5: If I(v N , ·) satisfies the neutrality and reduced-partnership-consistency axioms, then it satisfies the neutral partnership axiom.

Proposition 6: Under neutral partnership axiom, then the partnership-allocation-equivalence (I) and partnership-allocation-equivalence (II) axioms are equivalent. IV. Axiomatic Characterizations Let I : W N × 2N → R be an interaction index. Thus, I(v N , S) expresses the amount of interaction among elements in the set S for the worth function v N . In this section, we will state new axiomatic characterizations for the Shapley, Banzhaf, and chaining interaction indices. Lemma 1: If I(v N , ·) satisfies additivity and k-positivity axioms, then for every S ⊆ N,

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there exists a family of real constants {aST }T ⊆N such that 

I(v N , S) =

T ⊆N

aST · v(T ).

Lemma 2: If I(v N , ·) satisfies additivity, k-positivity and neutrality axioms, then for every S ⊆ N, there exists a family of non-negative real constants {aST }T ⊆N \S such that I(v N , S) =



N

T ⊆N \S

aST · ΔvS (T ∪ S).

Theorem 1 (Probabilistic interaction index) (i) If I(v N , ·) satisfies additivity, k-positivity and neutral partnership axioms, then for every S ⊆ N, there exists a probability distribution {pN S (T )}T ⊆N \S on N \ S such that 

I(v N , S) =

T ⊆N \S

N

v pN S (T ) · ΔS (T ∪ S).

(ii) If I(v N , ·) satisfies additivity, k-positivity, neutral partnership and symmetry axioms, then for every S ⊆ N, there exists a probability distribution {pns (t)}T ⊆N \S on N \ S, which depend only on cardinalities s and t of S and T , such that I(v N , S) =

 T ⊆N \S

N

pns (t) · ΔvS (T ∪ S).

That is, (i) I(v N , ·) is a probabilistic interaction index if and only if it satisfies additivity, k-positivity and neutral partnership axioms, (ii) I(v N , ·) is a cardinal-probabilistic interaction index if and only if it satisfies additivity, k-positivity, neutral partnership and symmetry axioms.

Theorem 2 (Shapley interaction index) The Shapley interaction index is the unique interaction index I(v N , ·) satisfying the following six axioms: additivity, k-positivity, neutral partnership, symmetry, reduced-partnership-consistency and efficiency axioms.

Theorem 3 (Banzhaf interaction index) The Banzhaf interaction index is the unique interaction index I(v N , ·) satisfying the following six axioms: additivity, k-positivity, neuAugust 12, 2002

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tral partnership, symmetry, reduced-partnership-consistency and partnership-allocationequivalence axioms.

Theorem 4 (chaining interaction index) The chaining interaction index is the unique interaction index I(v N , ·) satisfying the following six axioms: additivity, k-positivity, neutral partnership, symmetry, partnership-allocation-equivalence and efficiency axioms.

Here, “the neutral partnership” axiom in theorem 2, 3, and 4 can be replaced by “the neutrality” axiom. Corollary 2: I(v N , ·) is the internal interaction index if and only if it satisfies additivity, k-positivity, neutral partnership, symmetry, and (partnership-allocation-equivalence or reduced-partnership-consistency) axioms, and I(v N , {i}) corresponds to its worth (i.e the worth function v N ({i})) for any i ∈ N. Theorem 2, 3, and 4 indicate that each of the Shapley, Banzhaf, and chaining interaction indices is the cardinal-probabilistic interaction index determined/characterized by which axiom is rejected among three axioms: efficiency, reduced-partnership-consistency, and partnership-allocation-equivalence axioms. –that is, we can say that any of these three interaction indices is situated in the intermediate/neutral position, in a certain sense, of the two remaining indices. V. Conclusions In this paper, we stated the new axiomatic characterizations of cardinal-probabilistic interaction indices and its typical instances: the Shapley, Banzhaf, chaining and internal interaction indices. and provide some relations among them. Moreover, we provided the notion of balanced-probabilistic interaction indices which include all above interaction indices and some new balanced-probabilistic interaction indices. The class of balancedprobabilistic interaction indices is smaller class than that of cardinal-probabilistic interaction indices. However, in this paper, no characterization of balanced-probabilistic interaction indices is provided. Researches of this point would be matter for further work. August 12, 2002

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Acknowledgments The Author is grateful to Michel GRABISCH for his valuable advises and comments. References [1] A.Chateauneuf and J-Y.Jaffray, “Some Characterization of Lower Probabilities and Other Monotone Capacities through the Use of M¨ obius Inversion”, Mathematical Social Sciences, Vol.17, pp.263-283,1989. [2] K.Fujimoto, “Interactions and Decomposable Structures of Fuzzy Measures”, Proc. of Sixth International Conference on Soft Computing, pp.1002-1007, 2000. [3] K.Fujimoto, “Representation of Interactions among attributes, and its Axiomatization”, Proc. of IEEE 2002 International Conference on Fuzzy Systems, 2002. [4] M.Grabisch, “k-order additive fuzzy measures,” Proc. of 6th International Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems(IPMU), pp.1345-1350,Granada, Spain,1996. [5] 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. [6] .Grabisch and M.Roubens, “Probbilistic Interactions among plyers of cooperative game”, In: M.J.Machina and B.Munier (eds.) Beliefs, Interactions and Preferences, Kluwer Academic,1999. [7] 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. [8] S.Hart and A.MasColell, “Potential, Value, and Consistency”, Econometrica, Vol.57, No.3, pp.589-614, 1989. [9] E.Kalai and D.Samet, “Weighted Shapley values”, In: A.E.Roth(Ed.),The Shapley Value, Cambridge University Press 1988. [10] J-L.Marichal, “Aggregation Operators for Malticriteria Decision Aid”, Doctoral Thesis, Universite de Liege, 1999. [11] T.Murofushi, K.Fujimoto and M.Sugeno, “Canonical Separated Hierarchical Decomposition of Choquet Integral over a Finite Set,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.6, No.3, pp.547-562, 1998. [12] T.Murofushi and S.Soneda, “Technique for reading fuzzy measure (III): interaction index,” Proc. of Ninth Fuzzy System Symposium, Sapporo, Japan, pp.693-696,1993. In Japanese. [13] F.S.S` anchez “Balanced Contributions Axiom in the Solution of Cooporative Games”, Games and Economic Behavior, Vol.20, pp.161-168,1997. [14] L.S.Shapley, “A value for n-person games”, H.W.Kuhn and A.W.Tucker eds., Contributions to the Theory of Games, II, Prinston Univ. Press, pp.307-317, 1953.

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