Computation of Probabilities of Preference - Probabilistic Composition of Preferences, Theory and Applications - page 31

This chapter is directed to the discussion of practical aspects of each of these
three stages.

4.2 The Initial Quantification of Attributes

The preference for an alternative is given by the probability of its choice. However,
the measurement of preferences need not be made directly in probabilistic terms.
For each criterion, there is a most natural initial form of explaining the position of
each alternative with respect to the others. This natural form is sometimes based on
the value of an attribute measured objectively in terms of quantity, weight, cost,
speed, etc. For other criteria, it will be based on evaluations of abstract concepts and
expressed in ordinary language, in terms of small, moderate or large preference.
Any form of measurement may be employed to obtain initial evaluations.
However, some properties of the initial data may improve the effectiveness of the
probabilistic transformation. For instance, initial standardization procedures may
simplify the computation and the future interpretation of the results.
Handling data measured on the same scale facilitates comparisons. Although the
probabilistic transformation standardizes the scale, a general rule for the formula-
tion of the initial assessments also facilitates uniformization. To establish a general
rule, the initial evaluations may be set on a Likert ( 1932 ) scale of five linguistic
positions: , represented

in the sequence by the numbers 1, 2, 3, 4 and 5. This scale may be ampli

very small , small , moderate , large and very large

“ ” “ ” “ ” “ ” “ ”

ed to a
scale of nine positions with inclusion of a position close to each extreme position
and two positions

flanking the central position. When the attribute is measured by a
continuous variable, a discretization to such a Likert scale may be realized.
Otherwise, a starting point may be ranking the alternatives, ties allowed, and
reducing subsequently the number of classes by deciding to either consider some
alternatives in adjacent ranks as tied or increase distances between some others. A
systematic procedure to make experts in each attribute revise the distances between
adjacent alternatives, possibly reducing them to zero, may be made available to
improve the quality of the initial evaluations.

4.3 Modeling the Probability Distribution

To obtain probabilistic distributions, randomness is introduced by adding distur-
bances with null expected values to the exact initial values. This makes the initial
value a central measure, a location parameter of the distribution. The determination
of the other parameters is simpli

ed by assuming independence between distur-
bances affecting different alternatives and equal dispersion.
As errors in measurement due to the conceptual distance between the concrete
attributes observed and the subjective satisfaction extracted from them are the