Approaches to Criteria Combination - Probabilistic Composition of Preferences, Theory and Applications - page 14

ne the 2 n coef cients cor-

responding to the capacities of the 2 n subsets of S. Modeling the capacity by means
of its Moebius transform may simplify this task.
For

To compute a capacity µ , the modeler needs to de

µ a capacity on S, the Moebius transform of µ is the function ʽ : P(S) R

→

de ned by

X

ðÞ A B

j j

2 S

v ð A Þ ¼ 1 l ð B Þ; 8 A 2

B A

The Moebius transform determines the capacity by:

X

l ð A Þ ¼ m ð B Þ:

B A

The determination of the capacity may employ the Penrose-Banzhaf or Shapley
interaction indices (Grabisch and Roubens 1999 ) for limited levels of iteration.
Given a capacity

µ ↆ

on S, the Penrose-Banzhaf joint index for any subset A Sis

given by (Penrose 1946 ; Banzhaf 1965 )

2 #ð S n A Þ X
K S n A

X

Þ #ð A L Þ l ð K [ L Þ; Banzhaf A ðÞ ¼ ð 1

L A

for # the cardinality function, i.e., the function that associates to each set the number
of elements in it.
Analogously the Shapley joint index is de

ned by

X X

ð 1 Þ #A n L

ð Þ

Sl ð A Þ ¼

½ ð # ð S n A n K ÞÞ!ð # ð K ÞÞ!=ð # ð S n A Þþ 1 Þ!

l ð K [ L Þ

ð K S n A Þ L A

For an isolated criterion i, Sµ({i}) µ ({i}) is called the Shapley value (Shapley 1953 ).
The capacity µ is said to be k-additive, for a positive integer k, if its Moebius

transform ʽ satis es (Grabisch 1997 ):

2 S ; m

(1) 8 T 2 ðÞ ¼ T 0if # ðÞ [ T k ;

2 S such that

(2) 9 B 2 # ðÞ ¼ B k and m ðÞ6 ¼ B 0 :

By assuming 2-additivity, the complexity of the problem of determining the
capacity is reduced. The capacity can then be determined employing only the
coef cients µ ({i}) and µ ({i, j}) for i and j ∈ S.

Necessary and suf cient conditions for 2-additivity are:

(1) P
i ; fg S

Þ P i 2 S l fi; l fi; i ; j gÞ ð m 2 i gÞ ¼ 1 ð normality Þ;

(2) l fi; ðÞ i 0 ; 8 i 2 S nonnegativity ð Þ and

8 A S with # ðÞ 2 ; 8 k 2 A ; P i 2 A n fg l fi; i ; k g

(3)

ð ð Þ l fi; i ðÞ Þ # ðÞ 2 ð Þ l fi; k ðÞ

ð monotonicity Þ: