Composition by Joint Probabilities - Probabilistic Composition of Preferences, Theory and Applications - page 42

These formulae follow from the fact that the event of being preferred according
to at least one criterion is the complement of the intersection of the events of not
being preferred and, in the case of maximum dependence,

¼ ¼

1 min j 1 1 max j M j max j M j :

1 M j

5.2.3 Conservative and Pessimistic Point of View

;

independence 1 m j

: p j

maximal dependence 1 max j m j :

:

Notice that there are two equivalent forms of computation under the assumption of
maximal dependence, as the intersection of the probabilities of not being the worst
can be given by min j (1

- m j ), as well as by 1 - max j m j .

5.2.4 Conservative and Optimistic Point of View

independence 1 p j m j ;

:

maximal dependence 1 min j m j :

:

This follows from the fact that not being the worst by at least one criterion is the
complement of being the worst by all of them.
To compare in a more practical context the effect of the two assumptions of
independence and maximal dependence, let us formally represent the application of
them to the model derived from ANP. Denoting by P aj the probability of preference
by the a-th alternative according to the j-th criterion and indexing by b, o, c and r,
respectively, the criteria of the four types, bene

ts, opportunities, costs and risks,
the joint probability of preference by the a-th alternative would be given, assuming
independence between events of maximizing preference according to different
criteria, by:

p b P ab 1 ð p o 1 ð P ao Þ Þ p c 1 ð P ac Þ ð 1 p r P ar Þ

and, assuming maximal positive dependence, by

min b P ab max o P ao min c 1 ð P ac Þ ð 1 min r P ar Þ ;

for b varying along all bene ts, o varying along all opportunities, c varying along

all costs and r varying along all risks.