Approaches to Criteria Combination - Probabilistic Composition of Preferences, Theory and Applications - page 13

l ð A [ B Þ ¼l ðÞþ l A ðÞ B

for all disjoint subsets A and B of S.
Capacities generalize probabilities in the sense that an additive capacity is a
probability.
The Choquet integral of x = (x 1 , ,x m ), an R m+ valued function, with respect to …

the capacity µ on S = {1, … ,m}isde ned as:

X

m

C l x ðÞ ¼ ð x s ðÞ j x s ð j 1 Þ Þ l ð ð f j Þ; ...; s ðÞ m g Þ;

j¼ 1

for τ , a permutation on S such that

ðÞ ðÞ ð Þ ðÞ ðÞ ¼ :

x s 1 x s 2 x s m 1 x s m and x s 0 0

R + and

Let x: S µ a capacity. The Choquet integral of x with respect to µ

→

satis es

X

m

Cl ð x Þ ¼ x ð s ð i ÞÞ ½l ð A s ð i ÞÞ l ð A s ð i þ 1 ÞÞ

i¼1

for A τ (i) = { τ (i), … , τ (m)} for every i from 1 to m, and A s ð m þ 1 Þ / .

A fundamental property of the Choquet integral is that

C l ð 1 A Þ ¼l ð A Þ; 8 A S ;

for 1 A , the indicator of A, the function x de ned by

xi ðÞ ¼ 1ifi 2 A and x i ðÞ ¼ 0 otherwise :

The expected value of a function x with domain S with respect to a probability P
in the

finite space S is the weighted average
X

xi ðÞ Pi ðÞ:

i 2 S

Thus, the Choquet integral with respect to a capacity extends the expected value
with respect to a probability.
But this de nition makes sense only if xτ(i) τ if and x τ (j) , for the different possible

values of i and j are commensurable. Commensurability of the measures of pref-
erence according to different criteria means that they make us able to compare the
results of the evaluations according to the different criteria. This property holds for
the case of evaluations according to the criteria in S given in terms of probabilities
of being the best.