Interaction Transform of Set Functions over a Finite Set - CiteSeerX

Interaction Transform of Set Functions over a Finite Set Dieter Denneberg FB 3, Universit¨at Bremen Postfach 33 04 40, D-28334 Bremen, Germany email: [email protected]

Michel Grabisch Thomson-CSF, Laboratoire Central de Recherches Domaine de Corbeville, F-91404 Orsay Cedex, France email: [email protected]

September 1997

Abstract The paper introduces a new transform of set functions over a finite set, which is linear and invertible as the well known M¨obius transform in combinatorics. This transform leads to the interaction index, a central concept in multicriteria decision making. The interaction index of a singleton happens to be the Shapley value of the set function or, in terms of cooperative game theory, of the value function of the game. Properties of this new transform are studied in detail, and some illustrative examples are given.

1

Introduction

The set function over a finite set is a central concept in many fields of applied mathematics, as cooperative game theory, multicriteria decision making or decision under risk and uncertainty. In cooperative game theory, set functions are called games, and are used to model the worth of any coalition (i.e. a subset of the set of players), see e.g. (Shapley, 1953). The only requirement on games is that they vanish on the empty set. In multicriteria decision making, set functions are used to model the importance (w.r.t decision making) of groups of criteria (Grabisch 1996). Usually, it is required that such set functions take only non negative values, vanish 1

on the empty set, and are monotonic with respect to set inclusion. This kind of set functions is called usually non-additive measure, capacity or fuzzy measure, due to the analogy with classical measures. However, dealing with set functions over a finite set refers more to combinatorics than to measure theory, and in this paper we will adopt clearly a combinatorial approach. One of the important notions in combinatorics of partial orders is the M¨obius transform (see e.g. (Rota 1964)). In our case the partial order is given by set inclusion and the M¨obius transform gives another representation of the set function. This representation is an example of a linear invertible transformation, and of course one may imagine many others, provided they are meaningful in some application context, and possess remarkable mathematical properties. This is clearly the case of the M¨obius transform which is very useful e.g. in cooperative game theory and in the Dempster-Shafer theory of evidence (Shafer 1976). The main objective of this paper is to present a second linear invertible transform of set functions, which appears to be important at least in multicriteria decision making and in decision under uncertainty. It is called the interaction transform, and has been originally introduced by Grabisch (Grabisch 1996a, 1996b). The motivation behind this transform and its history is the following. For a cooperative game, which is a set function on the set of players, the Shapley value (Shapley 1953) is one of the most important solution concepts, it assigns to every player his/her prospect from playing the game. In multicriteria decision making, the Shapley value of a criterion expresses the importance of the criterion to make a decision. However, the interaction between criteria (or between players), i.e. the fact that for example putting two weak criteria (players) together make them strong or important, has always been felt to be a crucial notion, but has never been defined properly. However, Murofushi and Soneda proposed in 1993 a convincing “index of interaction” between two criteria, very similar to the Shapley value (Murofushi and Soneda 1993). Later, Grabisch extended this index to any subset of criteria, thus defining implicitly a set function transform, which happened to be linear and invertible, and moreover seeming to have remarkable properties, in particular in connection with the M¨obius transform (Grabisch 1996a, 1996b). Moreover, Grabisch and Roubens found an axiomatic basis for this interaction index, similar in spirit to the axiomatics of the Shapley value (Grabisch and Roubens 1997). This paper reformulates the construction of the interaction transform in a combinatorial point of view, thus giving a much more concise and organized view than in the original works (Grabisch 1996a, 1996b), and extending the results from fuzzy measures to arbitrary set functions. This convenient and rigorous framework permits us to find new interesting symmetry properties (equation (12), Theorem 5.1), which are closely related to the Bernoulli numbers and are interesting for decision under uncertainty. The paper is organized as follows. Section 2 lays down the framework of transformation of set functions, a transformation being a set function of two variables. 2

Here we introduce the transformation which generates the new interaction representation of a set function. In Section 3 we show that this transformation, like the M¨obius transform, depends only on the cardinality of the set difference of the two variables, which simplifies the expressions of the transformations. Only triangular n × n−matrices are needed to describe the relevant transformations of set functions, i.e. of 2n -vectors, n = |Ω| being the cardinality of our finite set Ω. In Section 4 we list the symmetries of the interaction operator and its inverse operator, which are symmetries of the corresponding n × n−matrices. Here our new proof of the central Theorem 4.1 is presented. In Section 5 it is shown that the interaction index of a set function and of its conjugate differ at most in the sign. We get nice corollaries for the Shapley value and for the decomposition of a non-additive measure in its symmetric part and its uncertainty (or ambiguity) part. Finally, Section 6 illustrates our results with examples of particular fuzzy measures computing their M¨obius and interaction representations.

2

Transformations of set functions

We recall the definitions of the zeta function and the M¨obius function for the partially ordered set (or poset for short) of subsets of a finite set and introduce the inverse Bernoulli function which is needed to transform a game or fuzzy measure into its interaction representation comprising the Shapley value. Let Ω be a finite set. Here we consider real functions on the power set P = 2Ω of Ω in one and two variables, ν : P −→ R ,

Φ : P × P −→ R ,

which we always write with small and capital greek letters, respectively. The functions of two variables will play the role of transformations applied to the functions of one variable. For this purpose we introduce a multiplication ? between functions of two variables and between a function of one variable and a function of two variables. For A, B ∈ P we define (Φ ? Ψ)(A, B) :=

X

Φ(A, C)Ψ(C, B) ,

C∈P

(Φ ? ν)(A) :=

X

Φ(A, C)ν(C) ,

C∈P

(ν ? Ψ)(B) :=

X

ν(C)Ψ(C, B) .

C∈P

If we fix a linear order on P we can identify P with {1, 2, ..., 2n } and the operation ? becomes ordinary multiplication of square matrices or of a vector with a matrix. This shows that the operation ? is distributive with respect to the usual sum of functions. ? is also associative, but with the restriction that a function of one variable 3

is not allowed between two functions of two variables, i.e. in general (Φ ? ν) ? Ψ 6= Φ ? (ν ? Ψ) like for matrices and vectors, where one of the products is not defined. Furthermore Kronecker’s delta (

∆(A, B) :=

1 if A = B 0 else

is the unique neutral element from the left and from the right. If Φ is invertible we denote the inverse with Φ−1 , i.e. Φ ? Φ−1 = ∆, Φ−1 ? Φ = ∆. It is an elementary fact that triangular matrices with non zero entries on the diagonal are invertible. A similar result holds with respect to the partial order ⊂. For simplicity we suppose that the elements on the diagonal are all 1. Proposition 2.1 The family G := {Φ : P × P → R | Φ(A, A) = 1 ∀A ∈ P, Φ(A, B) = 0 if A 6⊂ B} of functions of two variables together with the operation ? forms a group. The inverse Φ−1 ∈ G of Φ ∈ G computes recursively through Φ−1 (A, A) = 1, Φ−1 (A, B) = −

Φ−1 (A, C)Φ(C, B)

X

⊂

if A 6= B.

C ⊂ A⊂C 6= B

Proof. Straightforward (see Berge 1968 Chap. 3 §2).

2

Functions in G of great importance are the zeta function or better zeta operator ( 1 if A ⊂ B Z(A, B) := 0 else and its inverse Z−1 , the M¨ obius operator. It is well known that ( −1

Z (A, B) =

(−1)|B\A| if A ⊂ B 0 else .

We get a proof in Corollary 3.2 below. Now let be given a set function ν : P −→ R . The function µ := ν ? Z−1 is called the M¨ obius transform of ν. Explicitly µ(B) =

X

(−1)|B\A| ν(A).

A⊂B

4

(1)

Conversely (multiply with Z from the right) ν is the zeta transform of µ ν = µ ? Z, or explicitly ν(B) =

X

µ(A)Z(A, B) =

A

X

µ(A) .

(2)

A⊂B

In terms of cooperative game theory this is the representation of the game ν as linear combination of the unanimity games νA := Z(A, · ) (cf. Example 6.1). Example 2.1 Let p be a probability measure on P = 2Ω with the property p(A) = 0 iff A = ∅. Sundberg and Wagner 1992 multiply the function (

Π(A, B) :=

0 if A = ∅ p(B|A) else

with the M¨obius transform µ of ν, supposed ν(∅) = 0, µ ? Π(B) =

X

µ(A)p(B|A)

A6=∅

and call it the p-smear of ν. Since p(·|A) is additive (in fact a probability measure on P) µ ? Π is additive, too, but µ ? Π may assume negative values. If p is the 1 for ω ∈ Ω, then the p-smear uniform distribution on Ω , i.e. p(ω) = |Ω| µ ? Π(ω) =

X 1 A3ω

|A|

µ(A)

of ν at ω ∈ Ω is just the Shapley value of the ’cooperative game’ ν for ’player’ ω (Shapley 1953). The usual representation of the Shapley value in terms of ν will be obtained in Theorem 4.1. Sundberg and Wagner give conditions for ν in terms of the p-smears of ν to be monotone and to be supermodular (convex). 2 A central role in this article is played by the function Γ ∈ G, (

Γ(A, B) :=

1 |B\A|+1

0

if A ⊂ B else

.

which we call the inverse Bernoulli function. This name will be justified in Proposition 3.3. Before proceeding we discuss a generalization of Γ which seems to be important for the applications to multi-criteria decision, which we have in mind. The cardinality or counting measure on P = 2Ω can be written |A| = |Ω|p(A) where p is the uniform distribution. Employing, like in Example 2.1, another probability distribu1 tion p on P, one could define more generally Γ(A, B) = |Ω|p(B\A)+1 if A ⊂ B. This function has property (6) below, but not (7). Also for infinite Ω one can hope to find an appropriate generalization employing a probability measure p on a σ-algebra in 5

2Ω . Notice that the zeta and M¨obius transforms have also been defined for general Ω (see the discussion in Denneberg 1996). Back to Γ as defined above, it can be interpreted as follows. Γ(A, B) 6= 0 iff A ⊂ B and Γ(A, B) is the higher the closer A, B are. Here A, B are understood to be closer the smaller the cardinality of B \ A is. Γ assumes its maximal value Γ(A, B) = 1 iff A = B . Given a set function ν : P → R with M¨obius transform µ = ν ? Z−1 , we call ι := Γ ? µ the interaction representation of ν and ι(A) =

1 µ(B) B⊃A |B \ A| + 1 X

(3)

the interaction index of A with respect to ν (see equations (6) in Grabisch 1996). We are interpreting ι(A) as an average interaction of A with the other sets in the following sense. In Dempster-Shafer theory µ(B)Z(B, C) is regarded as the contribution of the set B to the value ν(C) (see (2)). Now the interaction index ι(A) is the weighted sum of the values Γ(A, B) of A with all sets B the weight being µ(B), which is the contribution of B to the value of ν at all sets containing B. For a singleton A = {ω} we get ι(ω) =

1 µ(B), B3ω |B| X

(4)

which is the Shapley value of ν for ω. Thus our interaction representation ι of ν completes in a natural way the Shapley value (ι(ω))ω∈Ω to a system (ι(A))A∈P of parameters determining ν uniquely. Using (4) and (2) we easily get the well known property X ι(ω) = ν(Ω) , (5) ω∈Ω

which, in game theory, is called efficiency of the allocation given by the Shapley value. Fuzzy measures ν and their integrals can be applied to multi-criteria decision making. In certain practical situations one has information which allows to assign values to ι rather than to ν directly (Grabisch 1996a). Therefore it is necessary to compute µ and then ν from a given ι. For this purpose we have to invert Γ, which is done in the next section.

3

Cardinality operators

In this section we employ a special operation in the poset P = 2Ω , namely set differences. Then each of the functions ∆, Z and Γ of two variables can be represented 6

through a function of one variable and the operation ? becomes convolution. Even more, these one variable functions on 2Ω depend only on the cardinality of the sets in Ω and so do their inverse functions. To any function Φ of two variables we associate a function ϕ(A) := Φ(∅, A),

A ∈ P,

which we always denote with the corresponding small greek letter. If Φ ∈ G has the property Φ(A, B) = Φ(∅, B \ A) for A ⊂ B, (6) then ϕ determines Φ uniquely, (

Φ(A, B) =

ϕ(B \ A) for A ⊂ B , 0 else

and Φ−1 has property (6), too (Proposition 2.1). Hence g := {ϕ | Φ ∈ G with property (6)} = {ϕ : P → R | ϕ(∅) = 1} is a group with operation ? defined by ϕ ? ψ(A) := Φ ? Ψ(∅, A) ,

A ∈ P.

ϕ ? ψ is the convolution of ϕ, ψ ∈ g, ϕ ? ψ(A) =

X

Φ(∅, C)Ψ(C, A) =

X

ϕ(C)ψ(A \ C).

C⊂A

C⊂A

One easily checks (sum over D := A \ C) that the convolution ? is a commuting operation, hence g is an abelian group. The neutral element ∆ of G becomes the neutral element δ of g, (

δ(A) :=

if A = ∅ . else

1 0

The inverse of ϕ ∈ g will be denoted ϕ?−1 as it is common for convolutions. Also Z and Γ have property (6) and they correspond to for all A ∈ P, 1 γ(A) = , A ∈ P, |A| + 1

ζ(A) = 1

which we also call zeta function and inverse Bernoulli function, respectively, in g. We remark that the classical Riemann zeta function is obtained in a similar manner if one starts with the poset N of the natural numbers where the partial 7

order is given by divisibility. Then the Dirichlet series expansion of the zeta function has all coefficients equal to 1 and multiplication of Dirichlet series corresponds to convolution with respect to divisibility. An arbitrary function ν : 2Ω → R will be called a cardinality function if ν(A) depends only on the cardinality |A| for each set A ∈ 2Ω . Observe that the functions δ, ζ and γ are cardinality functions. For such functions the inverse is computed as follows. Proposition 3.1 Let ϕ ∈ g and suppose ϕ(A) = f (|A|) ,

A ∈ 2Ω ,

(7)

with a sequence f : N 0 −→ R . Then ϕ?−1 (A) = g(|A|) ,

A ∈ 2Ω ,

with a sequence g, and g computes recursively through g(0) = 1 , g(m) = −

m−1 X

!

m f (m − k)g(k) , k

k=0

m ∈ N.

(8)

N 0 denotes the natural numbers including zero. In fact, only the |Ω| + 1 values f (0), f (1), · · · , f (|Ω|) of the sequence f matter, similarly for g. We will call f a car1 is a cardinal representation dinal representation of ϕ. For example, f (k) = k+1 of the inverse Bernoulli function γ. Proof. We apply the recursion formula in Proposition 2.1 with Φ ∈ G defined by Φ(A, B) = ϕ(B \ A) for A ⊂ B and get the recursion ϕ?−1 (∅) = 1 , X ϕ?−1 (C)ϕ(A \ C) ϕ?−1 (A) = −

for A 6= ∅.

⊂

C 6=A

Then, using (7), ϕ

?−1

(A) = −

m−1 X k=0

X

ϕ?−1 (C)f (m − k)

for |A| = m > 0.

C⊂A |C|=k

We show by induction that ϕ?−1 (A) depends only on |A|. For |A| = 0 nothing has to be proven since |A| = 0 only for A = ∅. Supposing that ϕ?−1 (C) = g(|C|) for |C| < m we get ϕ

?−1

(A) = −

m−1 X k=0

!

m g(k)f (m − k) k 8

for |A| = m.

Hence ϕ?−1 (A) is already determined by |A| = m. Setting g(m) = ϕ?−1 (A) where m = |A|, our assertion is proved. 2 The M¨ obius function ζ ?−1 in g is now easily computed.

Corollary 3.2 ζ ?−1 (A) = (−1)|A| ,

A ∈ 2Ω .

Proof. With the notations of Proposition 3.1 and ϕ = ζ we have f (m) = 1 for all m. Then the binomial formula for (1 + (−1))m tells us that equations (8) are solved for g with g(m) = (−1)m . 2 Next we show that the cardinal representation of the inverse of γ is given by the Bernoulli numbers. Therefore we call γ ?−1 the Bernoulli function which explains our former name inverse Bernoulli function for Γ and γ.

Proposition 3.3 The inverse of the function γ in the group g is γ ?−1 (A) = g(|A|) ,

A ∈ 2Ω ,

where g : N 0 −→ R is the sequence of Bernoulli numbers defined recursively by g(0) = 1 , g(m) = −

m−1 X k=0

!

m 1 g(k), k m−k+1

m∈ N

(9)

X m+1 1 m−1 = − g(k) . m + 1 k=0 k !

The M¨ obius transform µ of a set function ν can be computed from the interaction representation ι of ν by the following formula µ(A) =

X

g(|C \ A|)ι(C)

(10)

C⊃A |Ac |

=

X

g(m)

m=0

X

ι(C) ,

A ∈ 2Ω .

C⊃A |C\A|=m

The last expression for µ(A) is formula (7) in Grabisch 1996a. The sequence g of 1 , · · · and it is well known that Bernoulli numbers starts with 1, − 12 , 61 , 0, − 30 g(m) = 0

for m ≥ 3 odd. 9

(11)

Proof. The first assertion derives from Proposition 3.1. with ϕ = γ and f (k) = Then µ(A) = Γ−1 ? ι(A) =

1 k+1

.

Γ−1 (A, C)ι(C)

X C⊃A

=

X

γ

?−1

(C \ A)ι(C)

C⊃A |Ac |

=

X

X

g(m)

m=0

ι(C).

C⊃A |C\A|=m

2 If the interaction representation ι of a set function ν is known, we can now compute ν. Corollary 3.4 X

ν(A) =

|D|

b|D∩A| ι(D)

D∈P

where bdm

:=

m X

!

m g(d − k) k

k=0

for m ≤ d

and g is the sequence of Bernoulli numbers given by (9). Proof. With the M¨obius transform µ of ν and (10) we get X

ν(A) =

µ(C)

C⊂A

X X

=

g(|D \ C|)ι(D)

C⊂A D⊃C

X

=

D

X

(

g(|D \ C|))ι(D)

C⊂A∩D

! X |A∩D| X |A ∩ D|

=

(

D

k=0

k

g(|D| − k))ι(D) 2

The first values of bdm are m\d 0 0 1 2 3 4

1

2

1 − 21

1 6 − 13 1 6

1 2

3

4

1 0 − 30 1 1 − 30 6

2 − 16 15 1 0 − 30 1 − 30

10

Using the recursion for the binomial numbers one easily derives the recursion d d+1 bd+1 m+1 = bm + bm ,

0 ≤ m ≤ d.

Denoting with bd the function bdm as depending on the variable m, the recursion can also be put in the form bd = ∆bd+1 , where the difference operator ∆f (n) := f (n + 1) − f (n) for functions f : N 0 −→ R applies to the lower index. The coefficients bdm show the following symmetry bdm = (−1)d bdd−m ,

0 ≤ m ≤ d.

(12)

The numbers bd0 in the first row are the Bernoulli numbers so that by (12) and (11) also the diagonal elements bdd = (−1)d bd0 = (−1)d g(d) are the Bernoulli numbers except for d = 1. Furthermore the columns sum to zero, d X

bdm = 0 ,

for d > 0 .

m=0

The symmetry (12) can be proved by induction on d. The induction hypothesis for an odd d implies that bd is an odd function with center m = d2 so that, by the recursion in the difference equation form, the function bd+1 is even with center . A similar argument applies if one starts with an even d. That the columns m = d+1 2 sum to zero is, for odd d, a direct consequence of (12). Then for even d we get, using bd+1 d+1 = −g(d + 1) = 0, d X m=0

bdm =

d X

∆bd+1 m = ∆

d X

bd+1 m = ∆

m=0

m=0

d+1 X

bd+1 m = ∆0 = 0 .

m=0

In most parts of the paper we do not suppose ν(∅) = 0, but in applications this is often done. By (1) ν(∅) = 0 iff µ(∅) = 0 and by (10) ι(∅) does not affect µ(A) except for µ(∅). Hence if one has assigned all values of ι except for ι(∅) then ι(∅) is uniquely determined through the requirement ν(∅) = 0. Since Grabisch 1996, 1996a and 1996b does not mention ι(∅) explicitly, the transformation from ι to ν (Corollary 3.4) is not treated there. For applications one is further interested in conditions on ι in order to get a fuzzy measure ν, i.e. a monotone set function with ν(∅) = 0. Grabisch 1996b solved this problem in Theorem 4. We present this result in a slightly different form with a simple proof. Corollary 3.5 A set function ν is monotone iff for all ω ∈ Ω and all A ∈ Pω := 2Ω\{ω} X |D| b|D∩A| ι(D ∪ {ω}) ≥ 0 . D∈Pω

11

Proof. By (2) ν(A ∪ {ω}) =

X B⊂A

µ(B) +

X

µ(B ∪ {ω})

for ω ∈ / A.

B⊂A

Then ν is monotone iff the second sum is ≥ 0 for all ω ∈ Ω and all A ∈ Pω (noted already in Chateauneuf and Jaffray 1989). Transforming this sum like in the proof of Corollary 3.4 results in the desired formula. 2

4

The symmetries of the interaction operator

Here we compute the interaction indices directly from the set function, generalizing the common formula for the Shapley value. The formula reveals a symmetry which will be further exploited within the next section. Like above let be given a set function ν : P → R and let µ be the M¨obius transform of ν and ι the interaction representation of ν. Using ι = Γ?µ = Γ?(ν?Z−1 ) one can immediately derive a formula of type ι = ν ? I, but notice that I : P × P → R does not belong to the group G. We call I the interaction operator. Like the operators in G the interaction operator I has as domain and range the linear space V := {ν : P −→ R } , of set functions defined on P = 2Ω . Furthermore I is a linear operator since the operators in G are linear and the multiplication ? is distributive so that ν ? I = Γ ? (ν ? Z−1 ) is linear in ν, too. A first symmetry property of the interaction operator I is its compatibility with permutations p of Ω, (ν ◦ p) ? I = (ν ? I) ◦ p ,

ν ∈V.

This fact is well known for the Shapley value. It is obvious since ζ ?−1 and γ are cardinality functions and |A| = |p(A)|, A ∈ P. The inverse operator of I has already been computed in Corollary 3.4. |B|

I−1 (B, A) = b|B∩A| , and our symmetry (12) implies I−1 (B, A) = (−1)|B| I−1 (B, Ac ) . 12

Grabisch (1996a equation (5)) found a simple expression for I which generalizes the common formula for the Shapley value and reveals the symmetry I(Dc , A) = (−1)|A| I(D, A)

(13)

m or, in terms of Theorem 4.1, am k = am−k . Of course, (13) can also be derived from the above symmetry of I−1 .

Theorem 4.1 |Ω\A|

I(D, A) = (−1)|A\D| a|D\A| , where am k :=

(m − k)!k! , (m + 1)!

A, D ∈ P,

k ≤ m.

ι = ν ? I can be written in the form ι(A) =

|Ac |

X

a|C|

C⊂Ac

X

(−1)|A\B| ν(B ∪ C).

B⊂A

For |A| = 1 this is the original formula for the Shapley value of ν in Shapley 1953, ι(ω) =

(|C c | − 1)!|C|! (ν(C ∪ {ω}) − ν(C)), |Ω|! C⊂Ω\{ω} X

ω ∈ Ω.

The theorem implies that ι(A) = 0 if ’coalition’ A is a dummy with respect to the ’game’ ν, i.e. ν(B ∪ C) = ν(C) for all C ⊂ Ac and all B ⊂ A. For |A| = 1 this property is one of the axioms which characterize the Shapley value. Proof. Applying ι = Γ ? (ν ? Z−1 ) we get ι(A) =

1 (−1)|B\C| ν(C) |B \ A| + 1 B⊃A C⊂B X X

substituting D := B \ A, CA := C ∩ A, CD := C ∩ D X X X 1 = (−1)|D\CD | (−1)|A\CA | ν(CA ∪ CD ) |D| + 1 D⊂Ac CD ⊂D CA ⊂A substituting C := CD , E := D \ C, B := CA X X X 1 = (−1)|E| (−1)|A\B| ν(B ∪ C) |C| + |E| + 1 c c C⊂A E⊂A \C B⊂A letting m := |Ac | =

m X m−k X X k=0

C⊂Ac |C|=k

j=0

X E⊂Ac \C |E|=j

(−1)j X (−1)|A\B| ν(B ∪ C) . k + j + 1 B⊂A 



Since |Ac \ C| = m − k, the fourth sum has m−k terms so that the assertion follows j from the subsequent lemma with n = m − k. 2 13

Lemma 4.2 n X j=0

!

1 n n!k! (−1)j = , j k+j+1 (n + k + 1)!

n, k ≥ 0.

Proof. Regard the shift and identity operators Sf (k) := f (k + 1),

Ef (k) := f (k)

for functions on N 0 and the special function f (k) =

1 , k+1

k ≥ 0,

which we encountered in the cardinal representation of γ. Since S and E commute, i.e. SE = ES, the binomial formula applies (cf. Berge 1968 Chap.1 §8) n

((E − S) f )(k) =

n X j=0

Since S j f (k) = f (k + j) =

1 k+j+1

!

n (−1)j (S j f )(k) j

, it remains to prove that

(E − S)n f (k) =

n!k! . (n + k + 1)!

(14)

We do it by induction on n. The case n = 0 is obvious. Now suppose (14). Applying E − S to (14) gives equation (14) for n + 1, n!(k + 1)! n!k! − (n + k + 1)! (n + (k + 1) + 1)! n!k!((n + 1) + k + 1) − n!(k + 1)! = ((n + 1) + k + 1)! (n + 1)!k! = ((n + 1) + k + 1)!

(E − S)n+1 f (k) =

2

5

The interaction operator and conjugation

The interaction indices of conjugate pairs ν and ν of set functions coincide on 2Ω \{∅} except for the sign which depends on the cardinality mod 2 of the sets. Hence the interaction representation of a fuzzy measure ν reflects the decomposition of ν in its symmetric part and its uncertainty (or ambiguity) part. For the sake of brevity we write ιν := ν ? I , 14

ν ∈V,

for the interaction representation of ν. Let be given ν ∈ V. We associate with ν some other set functions in V, ν c (A) := ν(Ac ) , A ∈ 2Ω , ν(A) := ν(Ω) − ν(Ac ) , A ∈ 2Ω , ν+ν ν−ν ρ := , σ := , 2 2 and call ν c the complementary set function of ν, ν the conjugate set function of ν, ρ the symmetric part of the pair (ν, ν) and σ the uncertainty in the pair (ν, ν) (cf. Example 4.3 in Denneberg 1996). ρ is called the symmetric part since ρ + ρc = ν(Ω) and this = ρ(Ω) if ν(∅) = 0. Hence, supposing ν(∅) = 0, ρ is symmetric, i.e. additive on complementary sets, ρ(A) + ρ(Ac ) = ρ(Ω), A ∈ 2Ω , or ρ = ρ, i.e. ρ is self conjugate. The definition of σ is arbitrary with respect to the sign and to normalisation. In fact, in applications to (Nash) equilibria of ambiguous games (Marinacci 1996) or to incomplete contracts (Mukerji 1996), where ν is supposed to be a normed convex fuzzy measure or a belief function, the set function −2σ = ν − ν = 1 − ν − ν c is called vagueness or ambiguity. The uncertainty σ is self complementary, σ = σc. If ν is subadditive then σ ≥ 0. Furthermore the pair (ρ, σ) determines the pair (ν, ν) uniquely, ν = ρ−σ.

ν = ρ+σ,

These very natural decompositions of ν and ν have a nice counterpart in the corresponding interaction representations. Since the interaction operator is linear we have ιν = ιρ + ισ , ιν = ι ρ − ι σ and, most important, ιρ and ισ have disjoint support on 2Ω \ {∅}. This is a corollary of the next theorem. Theorem 5.1 For ν ∈ V ιν (∅) = ν(Ω) − ιν (∅) , ιν (A) = −(−1)|A| ιν (A)

for A 6= ∅.

For |A| = 1 Theorem 5.1 is known in cooperative game theory where ν(∅) = 0 is supposed: The Shapley values of ν and ν coincide. Hence the Shapley values of ν and ρ are identical, too, and the Shapley value of σ is zero. 15

Proof. First we regard a constant set function ν. Its M¨obius transform is ν(Ω)δ. Recall that δ(∅) = 1 and δ(A) = 0 for A 6= ∅. Hence ιν = ν(Ω)δ, too. Then we get for arbitrary ν ιν = (ν(Ω) − ν c ) ? I = ν(Ω)δ − ν c ? I and ν c ? I(A) =

X

=

X

ν(C c )I(C, A)

C

ν(D)I(Dc , A)

D

= (−1)|A|

X

ν(D)I(D, A)

by symmetry (13)

D

= (−1)|A| ν ? I(A) . 2 Corollary 5.2 Given ν ∈ V let ρ be the symmetric part and σ the uncertainty part of the pair (ν, ν), then for A 6= ∅ ( ρ

ι (A) =

(

ιν (A) for |A| odd , 0 else

σ

ι (A) =

0 for |A| odd . ν ι (A) else

Proof. 1 1 ν (ι (A) + ιν (A)) = (1 − (−1)|A| )ιν (A) 2 2 ( ιν (A) for |A| odd = . 0 else

ιρ (A) =

2

The second assertion proves similarly. A set function ν ∈ V is said to be of order k, ord ν = k ,

iff there is a set A ∈ 2Ω , |A| = k, so that ιν (A) 6= 0 and ιν (B) = 0 for all B ∈ 2Ω , |B| > k. Grabisch 1996a used the M¨obius transform µν of ν to define ”k-order additivity” of ν but this amounts to the definition above. Proposition 5.3 If ν ∈ V is of order k then ιν (A) = µν (A) for |A| ≥ k and both = 0 for |A| > k. 16

2

Proof. Equation (10)

If ν(∅) = 0 then ord ν ≤ 1 is equivalent to additivity of ν. In this case µν = ιν . Finally we prove another implication of Theorem 5.1. Corollary 5.4 Let ν ∈ V and suppose ν(∅) = 0. Then the symmetric part ρ of the pair (ν, ν) is additive iff ιν (A) = 0

for |A| ≥ 3 odd .

(15)

Especially, (15) is valid for set functions ν of order 2. Proof. In view of Corollary 5.2 condition (15) is equivalent to ord ρ ≤ 1 which, in turn, is equivalent to additivity of ρ. 2 Under the equivalent conditions of Corollary 5.4 one has a simple formula for calculating the Shapley value (ιν (ω))ω∈Ω of ν, ιν (ω) =

ν(ω) + ν(ω) . 2

It follows from ιν (ω) = ιρ (ω) = µρ (ω) = ρ(ω).

6

Some examples

In this section, we give some examples of particular set functions, borrowed from the field of fuzzy measures, and compute their M¨obius transform and interaction representation. As always in this paper we will work on a finite set Ω. Example 6.1 We begin by considering the unanimity game νA = Z(A, ·) for coalition A ⊂ Ω, i.e. νA (B) = 1 if B ⊃ A, and 0 otherwise. Let us compute its M¨obius transform, denoted µA . Considering that µA is the unique solution of νA = µA ? Z, we see that µA , viewed as function of two variables, is Kronecker’s delta, ( 1 if A = B µA (B) = ∆(A, B) := 0 else Now the interaction representation ιA of νA computes by ιA := Γ ? µA or formula (3), ( 1 , if A ⊃ B, ιA (B) = Γ(B, A) = |A\B|+1 0 otherwise. 1 In particular, the Shapley value is ιA (ω) = |A| if ω ∈ A, and 0 otherwise, that is, the value νA (Ω) of the game is evenly distributed among the players of A. Note that νA is a fuzzy measure of order |A|, and also a belief function, since its M¨obius

17

transform is positive (see Chateauneuf and Jaffray, 1989). Also, ιA is a positive set function. 2 Example 6.2 Let us now turn to cardinality measures, i.e. fuzzy measures such that ν(A) = f (|A|), f being non decreasing and f (0) = 0, f (n) = 1, where n := |Ω| (for cardinality functions in Section 3 there was no restriction on f ). In another terminology ν is a distortion of the uniform distribution on the finite set Ω. Cardinality measures lead to ordered weighted aggregation (OWA) operators in multi-criteria decision making, and weighted order filters in signal processing via the Choquet integral, so that they represent an interesting class of fuzzy measures on the point of view of applications (cf. Yager 1988, Maragos and Schafer 1987). Since ν(A) depends only on the cardinality of |A|, so will be the M¨obius transform µ and the interaction representation ι as well. Viewing ν as a game, this means that every coalition with the same number of players has the same importance. Especially for a single player we get by (5) the Shapley value ι(ω) =

1 , n

ω ∈ Ω.

Before computing the other interaction indices we write down the M¨obius transform in terms of f , using (1), µ(A) =

|A| X k=0

!

|A| (−1)|A|−k f (k) , k

A ⊂ Ω.

Let us compute now the interaction index of a set A = {ω, η} of cardinality 2 using Theorem 4.1, ι({ω, η}) =

X A⊂{ω,η}c

(n − |A| − 2)!|A|! (ν(A ∪ {ω, η}) − ν(A ∪ {ω}) − ν(A ∪ {η}) + ν(A)) (n − 1)!

n−2 X

!

(n − k − 2)!k! n − 2 2 = ∇ f (k + 2) (n − 1)! k k=0 X 1 n−2 ∇2 f (k + 2) n − 1 k=0 1 = (∇f (n) − ∆f (0)) n−1

=

Here we used the difference operators ∇h(k) := h(k) − h(k − 1) ,

∆h(k) := h(k + 1) − h(k)

for functions h on the integers. For arbitrary sets we get ι(A) =

1 (∇|A|−1 f (n) − ∆|A|−1 f (0)) , n − (|A| − 1) 18

A ⊂ Ω.

For |A| = 1 we apply the conventions ∇0 = id = ∆0 and for |A| = 0 we apply P P ∇−1 h(k) = ki=0 h(i) , ∆−1 h(k) = k−1 i=0 h(i) . Explicitly we get for A = ∅ k 1 X f (i) . n + 1 i=0

ι(∅) =

We see that the interaction index of A, |A| ≥ 2, depends on the |A| first and last values of f , showing that f (0), f (1) and f (n−1), f (n) are the most important values etc. 2 For the last example it is convenient to apply the notion of inner and outer set function ν? and ν ? , respectively, of a set function ν on a set system S ⊂ 2Ω , ∅, Ω ∈ S, defined by ν? (B) :=

_

ν ? (B) :=

ν(A),

W

ν(A) for B ∈ 2Ω .

A∈S A⊃B

A∈S A⊂B

Here

^

denotes the maximum and

V

the minimum operator.

Example 6.3. This example concerns possibility measures, usually denoted by Π and the conjugate measures N, called necessity measures (Shafer 1976, Zadeh 1978). In using issues of non-additive measure theory (cf. Denneberg 1994) the most simple way to define them, is to start with a maximal chain of nested subsets of Ω, ∅ = A0 ⊂ A1 ⊂ A 2 ⊂ · · · ⊂ An = Ω so that |Ak | = k. On the family A := {A0 , A1 , · · · , An } of these so called focal subsets let further be given a monotone set function ν : A −→ [0, 1] with ν(∅) = 0, ν(Ω) = 1. On the complementary chain B := {Acn , · · · , Ac0 } we introduce the conjugate set function ν : B → [0, 1],

ν(Ack ) := 1 − ν(Ak ),

of ν which is monotone, too. Now the necessity measure induced by ν is defined as N = ν? or explicitly N(B) =

_

ν(Ak ) ,

B ∈ 2Ω .

Ak ⊂B

The possibility measure Π := N is easily seen to be (ν)? or explicitly Π(B) =

^

ν(Ack ) ,

B ∈ 2Ω .

Ack ⊃B

Enumerating the elements of Ω so that Acn−k = {ω1 , · · · , ωk }, k = 1, · · · , n, we define a function π : Ω → [0, 1] , π(ωk ) := ν(Acn−k ), 19

which gives us the usual definition of Π, Π(B) =

_

π(ω).

ω∈B

Let us compute the M¨obius transform µN of N. Since A is a chain we get N(B) =

X

(ν(Ak ) − ν(Ak−1 )),

B ∈ 2Ω .

Ak ⊂B

Comparing this equation with formula (2) we conclude, using invertibility of Z, that ( N

µ (A) =

ν(Ak ) − ν(Ak−1 ) if A = Ak for some k ∈ {1, · · · , |Ω|} 0 else

Equivalently one can write µN (Ak ) = π(ωn−k+1 ) − π(ωn−k ). Since µN ≥ 0 any necessity measure is a belief function. We mention without proof that Π has the M¨obius transform µΠ (A) = (−1)|A|+1

^

π(ω).

ω∈A

These results are known from a long time in possibility theory and evidence theory (Shafer, 1976). Let us now compute ιΠ and ιN . We know from Theorem 5.1 that it is sufficient to compute one of them to get the other one by a simple sign shift. Also it seems much simpler to compute ιN using formula (3) since µN has a very simple form. For a subset B ⊂ Ω letting ^ k b := Ak ⊃B

we get ιN (B) = =

1 µN (A) |A \ B| + 1 A⊃B X

n X

1 µN (Ak ) k=b k − |B| + 1

or in terms of π ιN (B) =

n−b X i=0

1 (π(ωi+1 ) − π(ωi )) n − i− | B | +1

with the convention π(ωo ) := 0. Remark that ιN is a non negative function, whereas, due to Theorem 5.1, ιΠ (B) ≤ 0 for |B| > 0 even. 20

As a particular case, the identical Shapley values of N and Π (cf. Theorem 5.1) are: ιN (ωk ) = ιΠ (ωk ) =

k−1 X i=0

1 (π(ωi+1 ) − π(ωi )). n−i

Pn

Since Π(Ω) = 1, we have i=1 ιΠ (ωi ) = 1 (cf. (5)) so that the Shapley value, what is well known in game theory, can be viewed as a probability distribution on Ω, and the above equation can be viewed as a transformation from possibility measures to probability measures. In fact, this transformation has been already introduced by Dubois and Prade in the field of possibility theory (Dubois and Prade, 1993). 2 In the different examples above the fuzzy measure needs much less than 2|Ω| non zero real numbers to be represented, and some transformations between the different representations of the fuzzy measure simplify. The last example shows that it may pay to conjugate a fuzzy measure in order to find a simple representation. References Berge, C. (1968): Principes de Combinatoire. Dunod, Paris. (English translation (1971): Principles of Combinatorics. Academic Press, New York.) Chateauneuf, A. and Jaffray, J.Y. (1989): Some Characterizations of Lower Probabilities and Other Monotone Capacities Through the Use of M¨obius Inversion, Mathematical Social Sciences 17, 263-283. Denneberg, D. (1994): Non-additive Measure and Integral. Theory and Decision Library: Series B, Vol 27. Kluwer Academic, Dordrecht, Boston. Denneberg, D. (1996): Representation of the Choquet integral with the σ-additive M¨obius transform. To appear in Fuzzy Sets and Systems. Dubois, D. and Prade, H. (1993): On Possibility/Probability Transformations. In Fuzzy Logic, State of the Art, R. Lowen and M. Roubens (eds), Kluwer Academic, Dordrecht, Boston. Grabisch, M. (1996): The Application of Fuzzy Integrals in Multicriteria Decision Making. European Journal of Operational Research 89, 445-456. Grabisch, M. (1996a): k-Order Additive Discrete Fuzzy Measures. 6th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), pages 1345-1350, Granada, Spain, july 1996. Grabisch, M. (1996b): k-Order Additive Discrete Fuzzy Measures and Their Representation. To appear in Fuzzy Sets and Systems. Grabisch, M. and M. Roubens (1997): An axiomatic approach to the concept of interaction among players in cooperative games. Submitted to International J. of Game Theory. Maragos, P. and Schafer R.W. (1987): Morphological filters — Part I,II. IEEE Tr. ASSP 36, 1153-1184.

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Marinacci, M. (1996): Ambiguous Games. Paper presented at the Workshop ”Decision Making Under Uncertainty with Non-additive Beliefs: Economic and Game-theoretic Applications”, Saarbr¨ ucken. Mukerji, S. (1996): Ambiguity Aversion and Incompleteness of Contractual Form. American Economic Review, forthcoming. Murofushi, T. and S. Soneda (1993): Techniques for Reading Fuzzy Measures (iii): Interaction Index. In 9th Fuzzy System Symposium. 693-696, Sapporo (in Japanese). Rota, G.C. (1963): On the foundations of combinatorial theory I. Theory of M¨obius functions. Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und Verwandte Gebiete 2, 340-368. Shafer, G. (1976): A Mathematical Theory of Evidence. Princeton University Press. Shapley, L.S. (1953): A Value for n-Person Games. In H.W. Kuhn and A.W. Tucker (editors), Contributions to the Theory of Games, Vol. II, number 28 in Annals of Mathematics Studies, 307-317. Princeton University Press, Princeton. Sundberg, C. and C. Wagner (1992): Characterization of Monotone and 2-Monotone Capacities. Journal of Theoretical Probability 5, 159-167. Zadeh, L.A. (1978): Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets & Systems 1, 3-28. Yager, R.R. (1988): On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. Systems, Man & Cybern. 18, 183-190.

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