Capacities Determination - Probabilistic Composition of Preferences, Theory and Applications - page 95

for a varying along the set of alternatives and P aj denoting the probability of
alternative a maximizing the preference according to criterion j,
and

finally compute

C ð f C 1 ; ... ; C s g Þ ¼ P ð f C 1 ; ... ; C s g Þ= PU ðÞ;

for U denoting the set of all the criteria.
Thus, for subsets of more than one criterion, the capacity will be proportional to
the maximum, along the set of observed alternatives, of the complement to 1 of the
product of the probabilities of not maximizing the preference according to each
criterion in the subset.
By giving more importance in the composition of the preferences to the criteria
or sets of criteria with higher probabilities of choice of some alternative, this rule
guarantees the highest final preferences for the alternatives eventually chosen. This

agrees with the principle of Bayesian statistical inference (Box and Tiao 1973 )of
assigning to the parameters a probability distribution that maximizes the posterior
probability of observing whatever has been observed.
This algorithm assumes independence between the disturbances affecting the
evaluations according to each criterion. The choice of the independence assumption
has the advantage of maximizing the attention given, in the estimation process, to
the numerical evaluation according to each criterion. It does not mean, as already
pointed out in preceding chapters, assuming unsubstitutability of criteria.

9.2 Use of the Capacity to Evaluate the Alternatives

With this construction of the capacity, to compute the aggregate probability of
maximizing the preference for a given alternative with respect to a set of criteria by
the Choquet integral (Choquet 1953 ), the following procedure may be followed.
First, to the lowest between the probabilities of maximizing the preference for
the alternative according to a single criterion, add the product of the difference
between the second lowest of such probabilities and the lowest by the capacity
derived from the maximum along all the alternatives of the probabilities of being of
the best according to at least one of those criteria different from that for which the
alternative presents its lowest probability. Then, to this sum add the product of the
difference between the third and the second probabilities by the capacity derived
from the maximum along all the alternatives of the probabilities of being of the best
according to at least one of the criteria in the set complementary to that formed by
those with the two lowest probabilities. And so on.
To make clearer the procedure, let us consider, for instance, the case of four
criteria and let us denote by P j the probability of the alternative being that one
maximizing the probability of preference by the j-th criterion and by the per-

τ

mutation of {1, 2, 3, 4} such that