The Probabilistic Approach to Preferences Measurement - Probabilistic Composition of Preferences, Theory and Applications - page 26

Let us take for this distribution the triangular form. To make the translation
precise, the exact value 1 will be replaced by a triangular probability distribution on
[0, 1] with mode 1 and the exact value 0 by a triangular probability distribution on
[0, 1] with mode 0. The probability of a car of the

first group presenting the highest
value in a draw of evaluations of the 20 cars by this criterion will then be the
probability of [X a ≥ a from 1 to 20] for X a and X b independent

random variables. For triangular distributions on [0, 1], corresponding to the 19
values of b, eight of them with mode 1 and eleven with mode 0, this value is
0.1054.

X b for all b ≠

3.4 Combination of Probabilities of Preference

To combine the probabilities of preference according to multiple criteria into a
unique score of preference, different computations of probabilities of events may be
employed.
The most common way is through weighted averages. Taking the probabilities
of preference according to multiple criteria as conditional probabilities and
assuming a distribution of preferences among the criteria, the Theorem of Total
Probability gives a

final probability as a linear combination of the probabilities of
choice of the alternative by the criteria with weights given by the probabilities of
choice of the criteria.
The formula for the final probability of preference by alternative A will then be

X

m

pB j ;

pA ðÞ ¼ p j ð A Þ

j¼ 1

where p j (A) denotes the probability of the choice of A according to the j-th criterion
and p(B j ) denotes the probability of the choice of the j-th among the m criteria.
To better account for the interactions between the criteria, instead of the
weighted average, the Choquet integral may be employed. Determining the pref-
erences among the criteria by a capacity l , the final score will be given by

X

m

C l A ðÞ ¼ p s

ðÞ ð

A Þ p s j 1

Þ ð

A Þ l ðf s ð j Þ ; ...; s ð m ÞgÞ

ð

j¼1

for ˄ , a permutation of {1, … , m} such that

p s

ðÞ ð

A Þ [ p s j 1 ðÞ A for every j

ð Þ