Approaches to Criteria Combination - Probabilistic Composition of Preferences, Theory and Applications - page 12

according to each criterion with weights given by the probabilities of the criteria
being chosen in the rst stage.

In this case, to build the composition algorithm, possible correlations between
the events corresponding to being preferred by the different criteria need only be
taken into account in determining the distribution of weights among the criteria.
That is, this correlation must be taken into account in the initial stage of weighting
the criteria: the probability of each one being chosen must reduce the likelihood of
choosing all others positively correlated with it.
However, the problem cannot always be formulated in this manner and in
general the determination of weights for the weighted average is inef

cient by not
taking into account these correlations. A more general form of composition that
draws attention to the need to consider the possible presence of correlations is to
replace the weighted average of a probability distribution by the Choquet integral
with respect to a capacity (Choquet 1953 ).
To use this new form of composition of preferences the criteria must be com-
parable, i.e., the preference measurement according to the various criteria must
employ the same scale, or scales between which a precise relationship is known.
This problem of comparability is eliminated if the preferences are given as prob-
abilities of being the best alternative, the scale, then, being always that of the
probability of being the best.

2.3.1 Choquet Integral

To make expected utility models more flexible, additive subjective probabilities are
replaced by non-additive probabilities, or capacities.
Capacities may be used to model different types of behavior. Most decision
makers, for example, overestimate small and underestimate large probabilities.
Furthermore, most decision makers prefer decisions where more criteria are com-
bined rather than decisions based on less available information. These behaviors
cannot be expressed through an additive model.
A (normalized) capacity on the

:2 S

finite set of criteria S is a set function µ [0,

→

1] satisfying the three properties:

(1) l ðÞ ¼ 0 (a set function satisfying this property is also called a cooperative

game),

(2) µ (S) = 1 (normality),

2 N ; (3) 8 A ; B 2 ½ A B ) l ðÞ l A ðÞ B (monotonicity).

Thus, a capacity is a monotonic (normalized) cooperative game.
The capacity

µ

on S is said to be additive if