Interaction and Decomposable Structures of Fuzzy Measures Katsushige FUJIMOTO Department of Industory-Support Computer Engineering, Faculty of Economics, Fukushima University. 1 Kanayagawa Fukushima, 960-1296, JAPAN [email protected] Key Words: Inclusion-Exclusion Covering, k-additive measure, Interaction representation Abstract— The inclusion-exclusion covering (IEC) is one of the most important concepts for considering the decomposable coalitional structures of the model, represented by fuzzy measures, while the M¨obius/interaction representation is one of the most useful indices to interpret values of fuzzy measures. This paper investigates the behavior of inclusion-exclusion coverings and of M¨obius and interaction representations under game-theoretical rational transforms of fuzzy measures – strategic-equivalency, self-duality, and reduced game consistency. The fact that the IECs are invariant under these rational transforms is shown.

1

Introduction

Fuzzy measures, with its non-additivity, have a high potential for representing interactions among attributes of the considering object, especially in decision making, where they have been already successfully applied. In this paper, we discuss the fuzzy measure as a general tool for modeling importance of coalitions. However, the complexity of the model, represented by a fuzzy measure, increases as the number n of attributes/elements of the considering object goes to 4, 5, · · ·. In practical applications, one often encounters the following two problems: 1.“The fuzzy measure requires too many parameters”: in fact, for a set N with n elements, the definition of a fuzzy measure requires 2n − 1 parameters. 2.“It is difficult to interpret values of the identified fuzzy measure.” As an answer to these problems, Grabisch[4] proposed the concept of k-additive measure and interaction representation, and Sugeno et.al.[15] inclusion-exclusion covering (IEC), which offers a trade-off between richness and complexity. The concept of k-additive measures is designed quantitatively: k-additive measures are defined only on any subset with at most k elements. While, the concept of inclusion-exclusion coverings is designed qualitatively: fuzzy measures with inclusion-exclusion coverings are defined through decomposable coalitional structures. Both of these concepts can be characterized by using M¨obius and interaction representations which are very important indices for interpreting the values of fuzzy measures.

Our aims, in this paper, are: to investigate the behavior of inclusion-exclusion coverings and of M¨obius and interaction representations under game-theoretical rational transforms of fuzzy measures – strategic equivalency, self-duality, and reduced game consistency. In consequence, it is shown that inclusion-exclusion coverings are invariant for these rational transforms. 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 . Set functions from the power set P(N ) to [0, ∞), vanishing on the empty set are called games, while fuzzy measures v are games satisfying v(A) ≤ v(B) whenever A ⊆ B.

2

Representations of interaction, k-additive measure and IEC

In this section, we discuss the M¨obius and interaction representations as representaions of interaction. The interpretation of k-additive measures and of inclusionexclusion coverings through the use of these representations of interaction will be provided.

2.1

Basic Definitions

Definition 2.1 (M¨ obius representation) Let v be a game on N . The M¨ obius transform of v is a set function on N defined by: X mv (S) := (−1)s−t v(T ), ∀S ⊆ N. (1) T ⊆S

This correspondence proves to be one-to-one, since conversely X v(S) = mv (T ), ∀S ⊆ N. (2) T ⊆S

v

m is called the M¨ obius representation of v. In game v theory, m is called dividends of game v [8] . Definition 2.2 (interaction representation) [6] Let v be a game on N . The interaction transform of v is a

set function on N defined by: X X v

(−1)s−k v(K ∪ T ), ∀S ⊆ N,

Ξ(n, s, t)

I (S) :=

T ⊆N \S

K⊆S

(3) (n − t − s)!t! with Ξ(n, s, t) = . (n − s + 1)!

I v is called the interaction representation of v. The interaction representation is a generalization of both the Shapley value φi (N, v) for player i ∈ N of the game v and the interaction index Iij for {i, j} ⊆ N of Murofushi and Soneda[12] , since φi (N, v) coincides with I v ({i}) and Iij with I v ({ij}). This complicated definition has a simple representation using its M¨obius representation as follows: I(S) =

X

U ⊇S

1 mv (U ) ∀S ⊆ N. u−s+1

(4)

Generally, mv (∅) = 0 for any game v, but I v (∅) = 6 0. Inverse transform is expressed by[6] : X mv (S) = Bt−s I v (T ), (5) T ⊇S

where the coefficient Bk are Bernoulli numbers, defined by   k−1 X k Bl , k > 0, (6) Bk := − k−l+1 l l=0

and B0 = 1, –that is, B|T \S| =

X

T ⊇U ⊇S U 6=∅

B|T \U | , ∀T, S ⊆ N, S ⊆ T 6= ∅, |U \ S| + 1 (7)

and B|∅| = 1.

2.2

interpretation of representations of interaction

We consider the interpretation of the fuzzy measure, the M¨obius representation, and, the interaction representation. Following the interpretation in Introduction, we first discuss/interpret the fuzzy measure/game v(S) as a representation of the importance or the strength of a coalition S. Next, the M¨obius representation mv (S) of v can be interpreted as an incentive to form a coalition S for each players within S. This interpretation is compatible with the fact that it is named “dividends” in cooperative game theory[8] . From above interpretations, both the fuzzy measure and the M¨obius representation represent a kind of values for only coalitions inside S. On the other hand, representing the global worth of a coalition S, we should consider not only v(S) or mv (S) but also fuzzy measures or M¨obius representations of all coalitions containing S. The equation (3) shows that the interaction representation is defined as a kind of average

values of the added value given by putting S, all coalitions being considered, –that is, the interaction representation I v (S) is interpreted as a global worth of S brought by making the coalition S. Following the interpretation in Grabisch and Roubens[7] , we call this global worth, brought by making the coalition S, interaction amomg players in S. In summary, we discuss, in this paper, v(S) as the strength of a coalition S, mv (S) as the incentive to form a coalition S, and I v (S) as the interaction among players in a coalition S.

Now, we consider the concept of partnership[10] : the partnership is an agreement among a set of players U that none of them will establish any coalitional agreement, the players outside U unless all of them do. By forming such an agreement, the players in a partnership U restrict cooperation with players outside U . Expressed by a formula: v(S) = v(S \ U ) whenever S ∩ U 6= U.

(8)

In other formula: mv (S ∪ R) = 0 f or any S ⊆ N \ U and R

⊂ U, 6=

(9)

–that is, the players in a partnership U have no incentive to form any coalition with players outside U unless all of them do. However, the same thing is not true of the interaction among players in a partnership U . The reduced game with respect to U [7] is a game denoted v[U ] defined on (N \ U ) ∪ [U ] as follows for any S ⊆N \U : v[U ] (S) = v(S), (10) v[U ] (S ∪ [U ]) = v(S ∪ U ).

(11)

Property 2.2.1 Let v be a game on N , U be a partnership, and v[U ] be the reduced game with respect to the partnership U . Then
I v (U ) = φ[U ] (N \ U ) ∪ [U ], v[U ] , (12)
where φ[U ] (N \ U ) ∪ [U ], v[U ] is the Shapley value for the partnership [U ] of the game v[U ] .

2.3

k-additive measure and IEC

Definition 2.3 (k-additive measure) [5] A fuzzy measure v is said to be k-additive if its M¨obius representation satisfies mv (S) = 0 for any S such that s > k, and there exists at least one subset S of N of exactly k elements such that mv (S) 6= 0. Definition 2.4 (Inclusion-Exclusion covering) [15] A covering {Ci }i∈I of N is said to be an inclusionexclusion covering(IEC) with respect to a fuzzy measure v if v satisfies   X \ v(S) = (−1)|J|+1 v  Cj ∩ S  , ∀S ⊆ N. (13) J⊆I J6=∅

j∈J

An IEC C is said to be irreducible if there exists no C, D ∈ C such that C ⊂ 6= D. Proposition 2.1 There exists the only one irreducible IEC C such that C ⊑ D for any IEC D, where C ⊑ D denotes that for any C ∈ C there exists D ∈ D such that C ⊆ D. This C is said to be the finest inclusion-exclusion covering. Conversely, let C be the finest inclusionexclusion covering. Then any covering D is an IEC whenever C ⊑ D. Proposition 2.2 Let v be a fuzzy measure with the finest IEC C. Then, there exists a family of games {vC }C∈C on C ∈ C such that v(S) =

X

vC (S ∩ C), ∀S ⊆ N.

(14)

Both the k-additive measure and the finest IEC have a kind of boundaries. In k-additive measures, the boundaries are determined by the cardinality, –that is, the cardinality of subset S of N is less than or equal to k, or not. While, in fuzzy measures with the finest IEC, the boundaries are delineated by decomposable coalitional structures, –that is, the subset S of N is within the elements C in the finest IEC C, or not. Property 2.3.2 and 2.3.3 express that there are no incentive to form any coalition, and no interaction among players in any coalition accross the boundaries, –that is, there are no attraction or no force acting across the boundaries. In summary, k-order additive measures determine interaction structures/boundaries quantitatively, based on cardinality, while fuzzy measures with finest IEC qualitatively, based on decomposable coalitional structures.

C∈C

If v is l-monotone, each games vC can be a fuzzy measure, where l := maxi∈N |{C ∈ C|i ∈ C}| and l-monotone if for all families of l subsets S1 , . . . , Sl of N , v(

l [

i=1

Si ) ≤

X

I⊆{1,...,l} I6=∅

(−1)|I|+1 v(

\

Si ).

(15)

i∈I

This proposition indicates that the finest IEC can be interpreted as a decomposable coalitional structure of N with respect to a fuzzy measure/game v. Above two concepts, –that is, the concept of k-order additive measure and of IEC have arisen independently. The relation between these concepts is very interesting. We have in addition the following properties for any kadditive measure and any IEC.

3

Some properties on interaction and decomposable Structure

In this section, we discuss the relations among some game theoretical rational transforms, M¨obius and interaction representations, and IECs.

3.1

strategic-equivalence and decomposable structure

We shall discuss that two fuzzy measures/games may differ and be essentially the same.

Property 2.3.1 Let v be a k-additive measure and w a fuzzy measure with the finest IEC C. Then,

Definition 3.1 (strategic equivalence) Let v and w be two fuzzy measures/games. v and w are said to be generalized strategic-equivalent if w is represented as an affine transform of v, i.e. there exists a real constant a and an additive game λ on N such that

I v (S) = mv (S) for any S ⊆ N such that s = k.[6]

w(S) = a · v(S) + λ(S), ∀S ⊆ N.

I w (S) = mw (S) for any S ∈ C.

v and w are said to be strategic-equivalent if the equation (16) holds with a positive real constant a.

(16)

Property 2.3.2 Let v be a k-additive measure and w a fuzzy measure with the finest IEC C. Then,

For instance, the difference between the values of the strength of a coalition measured by v and w is the same as between the degrees of the temperature by Celsius and I v (S) = mv (S) 6= 0, for some S ⊆ N such that s = k.[6] . Fahrenheit scales. Consequently, v and w in strategic equivalence are essentially no difference. As the followI v (S) = mv (S) 6= 0, for any non − null S ∈ C. ing property shows, considering the incentive to form a coalition and/or the interaction among players in a coaliProperty 2.3.3 Let v be a k-additive measure, w a tion, with 2 elements or more, the graduation of scale is fuzzy measure with the finest IEC C. Then, the only difference between v and w. I v (S) = mv (S) = 0 for any S ⊆ N such that s > k.[6] I w (S) = mw (S) = 0 for any S ⊆ N such that S 6∈ A, where A := {S ⊆ N | ∃C ∈ C such that S ⊆ C}.

Property 3.1.1 Let v and w be strategic-equivalent with the relation: w(S) = a · v(S) + λ(S), ∀S ⊆ N . Then  a · mv (S) + λ(S) if s = 1, w m (S) = (17) a · mv (S) if s ≥ 2,

  a · I v (S) + λ(N ) w I (S) = a · I v (S) + λ(S)  a · I v (S)

if s = 0, if s = 1, if s ≥ 2.

(18)

Property 3.1.2 Let v and w be (generalized) strategicequivalent. If v is k-additive, then w is also k-additive. As the following proposition indicates, decomposable coalitional structures are invariant under strategic equivalent transforms.

Property 3.2.2 Let v be a fuzzy measure/game and v ∗ the dual fuzzy measure/game of v. If v is k-additive, then v ∗ is also k-additive. Proposition 3.2 Let v be a fuzzy measure/game and v ∗ the dual fuzzy measure/game of v. If C is an IEC with respect to v, then C is also an IEC with respect to v ∗ . In other words, if v is a fuzzy measure with the finest IEC C, then v ∗ is also a fuzzy measure with the finest IEC C.

Proposition 3.1 Let v and w be (generalized) strategicequivalent. If C is an IEC with respect to v, then C is also an IEC with respect to w. In other words, if v is a fuzzy measure/game with the finest IEC C, then w is also a fuzzy measure/game with the finest IEC C.

This indicates that the decomposable coalitional structures also hold the self-duality property.

3.2

In this section, we discuss reductions of a fuzzy measure/game to R ⊆ N . Then, it is desirable that reductions should preserve any of the strength of a coalition or the incentive to form a coalition, or the interaction among players in a coalition. As discussed in section 2.2, both the strength of a coalition and the incentive to form a coalition are defined by some values for only inside the coalition. Therefore, both the fuzzy measurepreserving and the M¨obius representation-preserving reductions are just the restrictions of the original one to R. On the other hand, the interaction among players in a coalition I v (S) is defined under considering not only v(S), mv (S) but also fuzzy measures or its M¨obius representations of all coalitions containing S. Hence, the interaction representation/interaction-preserving reduction is very interesting.

duality and decomposable structure

Definition 3.2 For any fuzzy measure/game v, the dual fuzzy measure/game of v is defined by v ∗ (S) := v(N ) − v(N \ S), ∀S ⊆ N.

(19)

In other representation using the M¨ obius representation: X v ∗ (S) = mv (T ), ∀S ⊆ N. (20) T :S∩T 6=∅

Now, we consider the case where the incentive to form a coalition mv (S), for any S ⊆ N , is given on P(N ). As the equation (2) and (20) indicate, v ∗ is determined by taking a more positive attitude toward the incentives that v. As the following property indicates, M¨obius and interaction representations have interesting properties under the dual transform, especially the interaction representation is invariant for a coalition with even elements, while a sign reversal for that with odd elements. Regretfully, the author cannot provide appropriate interpretations for this property. Property 3.2.1 [4] Let v be a fuzzy measure/game and v ∗ the dual fuzzy measure/game of v. Then  if S = ∅, 0 X ∗ v (21) mv (S) = (−1)s+1 m (T ) otherwise.  T ⊇S

∗

I v (S) =



v(N ) − I(∅) (−1)s+1 I v (S)

if S = ∅, otherwise.

(22)

For any singleton {i}, we obtain that ∗

I v ({i}) = I v ({i}).

(23)

This indicates that the Shapley value is invariant under the dual transform. This property is called self-duality[3] of the Shapley value.

3.3

3.3.1

reduction and decomposable structure

fuzzy measure and M¨ obius representationpreserving reduction

Proposition 3.3 Let v be a fuzzy measure/game and vR the restriction of v to R ⊆ N . If C is an IEC with respect to v, then C ∩ R is also an IEC with respect to vR , where C ∩ R := {R ∩ C|C ∈ C}. 3.3.2

Shapley value-preserving reduction

We discuss, here, the relations among fuzzy measures/games called the reduced game with respect to the Shapley value in the game theory[9] and decomposable coalitional structures. This reduced fuzzy measure/game is defined as a Shapley value-preserving reduction, namely especially one of interaction-preserving reductions. Definition 3.3 [9] Let v be a fuzzy measure/game. For any R ⊆ N , the reduced fuzzy measure/game with respect to the Shapley value on R is defined by X φi (S ∪ Rc , v), ∀S ⊆ R. (24) vR (S) := v(S ∪ Rc ) − i∈Rc

In other representation: X vR (S) = φi (S ∪ Rc , v), ∀S ⊆ R.

∀S 6= ∅ ⊆ R. Then,

(25)

X X X

mvR (S) =

V ⊆Rc

i∈R

Property 3.3.1 [9] Let v be a fuzzy measure/game and vR the reduced fuzzy measure/game with respect to the Shapley value on R ⊆ N . Then,

Bt−s

W ⊇S T ⊇S W ⊆R T ⊆W

1 mv (W ∪V ), w−t+1+v (30)

∀S 6= ∅ ⊆ R.

(26)

We will call this vR the interaction representationpreserving reduction of v to R. For this vR , the decomposable coalitional structures are invariant.

This property is called reduced game consistency[9] with respect to the Shapley value, and we will call this vR the Shapley value-preserving reduction of v to R.

Proposition 3.5 Let v be a fuzzy measure/game and vR its interaction representation-preserving reduction to R. If C is an IEC with respect to v, then C ∩ R is also an IEC with respect to vR , where C ∩ R := {R ∩ C|C ∈ C}.

Property 3.3.2 Let v be a fuzzy measure/game and vR its Shapley value-preserving reduction to R ⊆ N . Then,

3.3.4

φi (N, v) = φi (R, vR ), ∀i ∈ R.

mvR (S) =

X

V ⊆Rc

s mv (S ∪ V ), ∀S ⊆ R. s+v

(27)

This indicates that the incentive to form a coalition S, in the reduced fuzzy measure/game, is obtained as the total sum of proportional distributions of each incentives to form the coalition which is the combination of S and outside the R, i.e. V ⊆ Rc , in the original game. Proposition 3.4 Let v be a fuzzy measure/game and vR its Shapley value preserving-reduction to R ⊆ N . If C is an IEC with respect to v, then C ∩ R is also an IEC with respect to vR , where C ∩ R := {R ∩ C|C ∈ C}. This indicates that the decomposable coalitional structures are invariant under the Shapley value-preserving reduction. 3.3.3

Interaction representation-preserving reduction

In this section, we discuss the interaction representationpreserving reduction/reduced game, –that is, the game obtained by the restriction of I v to R. Property 3.3.3 Let v be a fuzzy measure/game and vR its reduced game on R ⊆ N . Suppose that the interaction representations of them are satisfying the following condition: I vR (S) = I v (S), ∀S 6= ∅ ⊆ R.

(28)

That is, the M¨obius representation mvR of vR is necessarily/uniquely determined, by the equation (4) and (28), recursively as follows: mvR (S) =

X

T ⊇S

X 1 mv (T ) − t−s+1

R⊇T ⊇S T 6=S

1 mvR (T ), t−s+1 (29)

k-order interaction-preserving reduction

In this section, we discuss the k-order interactionpreserving reduction as a generalization of both the Shapley value-preserving one and the interaction representation-preserving one. Definition 3.4 Let v be a fuzzy measure/game. For any R ⊆ N , the k-order interaction-preserving reduced fuzzy k on R is defined by the following promeasure/game vR cedure: (i)

k

mvR (S) :=

s−k+1 mv (S ∪ V ), s−k+1+v

X

V ⊆Rc

(31)

∀S ⊆ R such that s ≥ k. k (ii) For any S ⊆ R such that k > s ≥ 1, mvR (S) is defined recursively as follows: X X k k 1 1 vR v vR (S) :=

m

T ⊇S

t−s+1

m (T ) −

R⊇T ⊇S T 6=S

t−s+1

m

(T ), (32)

and m

k vR

(∅) = 0.

(iii) k (S) := vR

X

k

mvR (T ), ∀S ⊆ R.

(33)

T ⊆S k Property 3.3.4 Let v be a fuzzy measure/game and vR its k-order interaction-preserving reduction to R ⊆ N . Then k

I vR (S) = I v (S), ∀S ⊆ R, such that 1 ≤ s ≤ k. (34) In the definition 3.4, the equation (31) is a generalization of the equation (27) and (30), and the equation (32) is necessarily/uniquely led in the same way as the equation (29) by the condition (34). The k-order interactionpreserving reduced fuzzy measure/game is a generalization of both the Shapley value-preserving one and the interaction representation-preserving one, since 1-order interaction-preserving reduction to R coincides with the Shapley value-preserving one and r-order interactionpreserving reduction with the interaction representationpreserving one.

k Proposition 3.6 Let v be a fuzzy measure/game and vR its k-order interaction-preserving reduction to R ⊆ N . If C is an IEC with respect to v, then C ∩ R is also an IEC with respect to vR , where C ∩ R := {R ∩ C|C ∈ C}.

This indicates that decomposable coalitional structures are invariant under the k-order interaction-preserving reduction for any k. Moreover, for these game-theoretical rational transforms, discussed above, the following proposition holds: 1 , and w its Proposition 3.7 Let v be a game, v ∗ , vR dual game, 1-order interaction-preserving reduction to R ⊆ N , and the generalized strategic-equivalent game with the relation: w = a · v + λ, respectively. Then, φi (N, v) is the Shapley value for the player i ∈ N of v, if and only if it satisfies the following properties: (1) self-duality: φi (N, v) = φi (N, v ∗ ) for any i ∈ N , 1 (2) reduced game consistency: φi (N, v) = φi (R, vR ) for any i ∈ R, (3) (generalized) strategic-equivalency: φi (N, w) = a · φi (N, v) + λ({i}) for any i ∈ N.

4

Conclusions

We have interpreted the fuzzy measure as the strength of a coalition, the M¨obius representation as the incentive to form a coalition, and the interaction representation as the interaction among players in a coalition. We have shown that inclusion-exclusion coverings, i.e. the decomposable coalitional structures, are invariant under the all of strategic equivalent transforms, the dual transform, and the interaction-preserving reduction. In a certain sense, we can say that inclusion-exclusion coverings provide a kind of game-theoretical rational decomposable coalitional structures. Conversely, these gametheoretical transforms are what preserve a kind of boundaries discussed in 2.2.

Acknowledgement This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists ,11780320 , 19992000.

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