Computation of Probabilities of Preference - Probabilistic Composition of Preferences, Theory and Applications - page 32

source of the random disturbances, it is reasonable to model the distributions as
normal distributions. However, simpli cations may be considered.

If the attributes that give rise to the preferences are represented by discrete
variables, the approach of Fuzzy Sets Theory, of replacing exact values by mem-
bership functions, is naturally taken. A triangular distribution will then be able to
model the randomness.
However, even if continuous distributions are assumed from the beginning and
no discretization of the initial values is performed, triangular distributions may be
assumed instead of normal distributions.
Discretization may be avoided to explore the accuracy of the measurements
when the alternatives are compared by weight, volume or other physical features.
Imprecision in the preference derived from such accurate measures is still due to the
subjectivity of the evaluator in evaluating the bene

t or cost that results from them.
In that case, as the centers of distinct distributions may be closer than in the discrete
case, to maintain small the distance between distributions with close centers, a
normal distribution is more adequate than a triangular distribution.
Another point to take into account in the choice of the shape of the distributions
is the dependence or not of the disturbance on the value observed for the attribute.
The basic approach is modeling the random component as resulting from identically
distributed measurement errors, as in classical statistical models. A symmetric
distribution around the location parameter will then appear.
It may be more realistic, however, to make the dispersions depend on the
location, to compensate for a possible excessive deviation in the initial measure-
ment towards the preferred side. The distributions centered closer to the maximum
or the minimum observed value can spread more to the side where there is a larger
range of other possible values.
To obey this principle, the modeler may employ asymmetric triangular distri-
butions with the steepest slope for the side where the extreme is closer and a milder
decline for the side where the extreme is more distant. This may be done, for
instance, in the case of the Likert scale of nine levels from 1 to 9 by adopting
extremes of 0 and 10 for all distributions.
Alternatively, to reduce the importance of unobserved values in one of the
possible extremes, different bounds may be used for the different criteria. If the
vector of observed values for the j-th attribute is

, extremes for the

e 1j ; ... ; e nj

triangular distributions will then be xed at e 0j ¼ min e 1j ; ... ; e nj 1 and

e n þ 1j ¼ 1.

In the symmetric case, if a normal distribution is assumed, the observed values
for the other alternatives may also be taken as a basis to model a dispersion
parameter, which is enough to determine a normal distribution after the mean is
known. To give a reasonable chance of occurrence in any distribution for all the
observed values,

max e 1j ; ... ; e nj þ

, given by

the standard deviation of the vector e 1j ; ... ; e nj

1 = 2

2

P a ¼ 1 e aj P b ¼ 1 e bj = n

, or some transformation of it, may be used
as a common dispersion parameter for all of them.

= n