Computation of Probabilities of Preference - Probabilistic Composition of Preferences, Theory and Applications - page 33

4.4 Transformation in Probabilities of Being the Best

The association of distributions of possible values to exact measurements makes it
possible to replace, by means of a simple computation, the vector of measurements
of an attribute in a set of alternatives with a vector of probabilities of each of these
alternatives being the best.
After associated a probability distribution for the evaluation of each of a set of n
alternatives, the measure of preference for the a-th of these alternatives is given by
the probability of, in a sample formed by taking a random realization of each of the
n distributions, the value obtained for the a-th distribution being larger than all the
other n 1 values.

This development assumes that an alternative is better than other if its value is
higher. If the opposite occurs, i.e., the criterion associates a larger value of the
attribute with a lower preference for the alternative, instead of the score being the
probability of presenting the highest value in the sample, it will be the probability of
presenting the lowest one.
This may be put more formally, with e j ¼ e 1j ; ... ; e nj representing again the
vector of measurements of the j-th attribute in the n alternatives and p j ¼

−

p 1j ; ... ; p nj representing the vector of probabilities of preference. In the triangular
model, the probabilistic preference p aj

is given by the probability that, in an

with inde-

eventual vector of observations of n random variables E 1j ; ... ; E nj

pendent triangular distributions with extremes e 0j and e n þ 1j and with modes

respectively at e 1j ; ... ; e nj ,

E aj E bj for all b from1 to n :

0 and e n +1j = 1 and
assuming independence between the disturbances that affect the evaluation of dif-
ferent alternatives, the probability of maximization, for that alternative with eval-
uation e aj

Assuming, without loss of generality, the extremes e 0j ¼

is directly given by

( ) f a x

Z

e l þ 1j

h i Y

X Y

n l 1 n

2

x 2 1 ð 1 x Þ =ð 1 e bj Þ = e bj ðÞ dx :

l ¼ 0 b ¼ 1 b ¼ l

e lj

rst product
if l = 0 or 1 and the second if l = n or n + 1. The integration is with respect to the
triangular density f a given by

In these products, replace by 1 the factors with b = a, as well as the

f a x ðÞ¼ 21 ð x Þ= 1 e aj for a \ l