Computation of Probabilities of Preference - Probabilistic Composition of Preferences, Theory and Applications - page 34

and

f a x ðÞ¼ 2x = e aj for a [ l :

The transformation in probabilities of maximizing the preference allows for
translating from the linear scale of ranks to a more realistic scale with concentration
of signi

cant preferences on a small number of alternatives. While a few most-
preferred alternatives receive clearly distinct probabilistic evaluations, probabilities
of preference close to zero are given to the other alternatives.
Thus, the transformation from ranks to probabilities of being the best or the
worst alternative brings an additional bene

t, besides those advantages inherent to
taking uncertainty into account: it increases the distance between the most impor-
tant alternatives. This answers to the practical need of the evaluators to distinguish
the most important alternatives with higher accuracy, while distinctions between the
least preferred ones may be less reliable. Barzilai et al. ( 1987 ), Tryantaphilou et al.
( 1994 ) and Brugha ( 2000 ), among others, present good reasons to prefer nonlinear
scales with such form.

4.5 Comparison to a Sample

The calculation of probabilities of being the most preferred alternative involves
comparison to all the competing alternatives, even to those with the worst ratings.
This ensures greater resistance to the in

uence of errors in the evaluation of some
alternatives, but, if the number of alternatives is too large, the probabilities of
preference become too small. If the comparison were made to a representative
sample with a small number of alternatives, the computation would be simpli ed.

An undesirable side effect of this sampling approach is that then the sum of the
scores will no longer be 1, as will happen if they are given by the probabilities of
being the best in the whole population of alternatives. In fact, as a result of com-
parison to a smaller sample, all the scores become higher. But a comparatively
higher score still means higher preference.
The simpli

cation may be performed without losing the desired robustness. For
that, the small sample must be built in a representative way. It may be constituted,
for instance, of the deciles or the quartiles of the set of evaluations observed.

Table 4.1 Sample of three reference alternatives
Quartiles

Beauty Comfort Consumption Power Price Reliability Safety

1st
quartile

0 0 0 0 0 1 1

Median 1 1 0 1 0.5 1 1

3rd
quartile

1 1 1 1 1 1 1