The Probabilistic Approach to Preferences Measurement - Probabilistic Composition of Preferences, Theory and Applications - page 25

the alternative is evaluated in successive assessments of the preference based on
that attribute.
For this notion to make sense, it is necessary to take into account the imprecision
that follows from the subjectivity implicit in the application of any criterion of
choice. The application of the criterion is based on the measurement of the attribute
and on the possibility of an error in the transposition of that measurement into an
evaluation of the satisfaction of the evaluator with the presence of the attribute at
the level that the measurement indicates.

3.3.1 Computation of the Probability of Being the Best

After associating the measurements of the attribute for the n alternatives with a
vector of n random variables X1, , Xn, the next step is to derive from this vector

the probabilities of being the best. From a joint distribution of n random variables,
software like R (R

...

Core Team 2014 ) or MATLAB (Mathworks 2014 ) immedi-
ately produces the probability of each X a presenting a value higher than the value of
any other X b .
A few general assumptions help to determine the joint distributions of the ran-
dom variables informing the evaluations according to n criteria. One assumption
that considerably simpli

©

es the modeling of joint distribution is that of indepen-
dence. It is reasonable to consider as independent the errors in the measurement of
the satisfaction of the criterion resulting from each of the observed values of the
attribute considered.
Other assumptions have to do with shape. Continuous distributions avoid jumps
in isolated points, which are usually dif cult to address. In addition to indepen-

dence, the limitation to continuous distributions is recommended at this point.
Among the continuous distributions, the normal distribution is employed to
represent the effect of subjective measurement deviations. With easier forms to
visualize shape and allowing for simpler computation,

triangular and uniform

distributions may replace the normal (Sant Anna

2002 ). For numerical evaluations on a scale of few alternatives, a lognormal dis-
tribution for the absolute deviations has also been recommended (Martino 1970 ).

' Anna and Sant ' Anna 2001 ; Sant '

3.3.2 Example of Probability of Being the Best

To fix ideas consider the evaluation of gas consumption by the 20 cars of the

previous chapter. Nine of them were evaluated as presenting the attribute of low gas
consumption and eleven not presenting it. The randomized criterion will assign to
each car measure of satisfaction by gas consumption a probability distribution that
will have a mode of one for the

first nine cars and of zero for the other eleven. The
important point is that this will be a probability distribution, not a constant.