The Probabilistic Approach to Preferences Measurement - Probabilistic Composition of Preferences, Theory and Applications - page 27

and

p s ðÞ ð A Þ ¼ 0

:

Other more intuitive forms of compositions of the preferences do not depend on
weighting neither the isolated criteria nor sets of them. For instance, the global
score may be given by the probability of being the preferred alternative according to
every criterion or by the probability of being preferred by at least one criterion.
The event of being the preferred alternative according to every criterion is the
intersection of the events of being the best according to each criterion and the event
of being the preferred according to at least one criterion is the complement of the
union of the events of not being preferred.
Assuming independence, the

final score for alternative A is then given, in the rst

case, by the product ˀ j p j (A) of the probabilities of choice according to isolated criteria.

In the second case, treating the union as the complement of the intersection of the
complements, the

p j (A)).
To check the importance of the assumption of independence, these scores may
be compared to those obtained by replacing the product by the minimum, what
corresponds to assuming the hypothesis of maximum dependence or, equivalently,
to the fuzzy operation of intersection described in the previous section.
Since min j (1

final score under independence will be given by 1 − ˀ j (1 −

max j p j (A), in the case of employing the
probability of being the best according to at least one, the score for maximal
dependence will be max j p j (A), what also corresponds to the operation of fuzzy
union of the last section.

− p j (A)) equals 1 −

3.5 Example of Choice of a Car

The transformation into probabilities of being the best may be employed to replace
the indicator of presence and absence of the attributes for the cars with values 0 and
1 in Table 2.4 by fractionary numbers in such a way that the sum of the values in
any column equals 1.
This will make those criteria contemplating rarer attributes assign higher values
for the presence of that attribute. For instance, to the presence in nine models of low
gas consumption, assuming the independent triangular distributions with modes
zero and one, corresponds a value of 0.1054, larger than the value of 0.096 assigned
to the presence of low acquisition price, what occurs in ten models.