Approaches to Criteria Combination - Probabilistic Composition of Preferences, Theory and Applications - page 6

The set of criteria must be both exhaustive, in the sense of enabling the decision
maker to take into account all relevant aspects of the alternatives, and non-redundant,
with each one adding some relevant aspect to discriminate between alternatives.
Roy ( 1996 , 2005 ) adds to these two properties a third, called cohesiveness, as a
necessary condition to have a coherent family of criteria. A family of criteria would
present cohesion if a move that is not for the better in the evaluation according to
any particular criteria would never lead to a move for a better general evaluation.
These and other properties are required of the set of criteria, and all have to do
with the general fact that the model must represent as approximately as possible the
reality, no model being able to ever completely cover all features of the decision
problem. A gamma of different approaches to reduce the reality to a multi-criteria
model is developed, for instance, in Greco and Ehrgott ( 2005 ) and in Ehrgott et al.
( 2010 ).
However, besides that, the form of combining the criteria depends on the goal
that the decision maker has in mind. This adds to the dif

culty of adequately
making clear the composition rules. Modeling should not only enhance the ability
to conduct the evaluation to produce an outcome that achieves the requirements of
the evaluator but also the ability to explain how the values declared by the evaluator
lead to the

final outcome. The composition rules should allow for relating as clearly
as possible the final preferences to previously exposed motives.

2.2 Weighted Averages

The classic criteria composition form, developed precisely in Keeney and Raiffa
( 1976 ), is conducted by assigning weights to the criteria and obtaining final scores

as weighted averages of the measurements of preferences according to each of
them. These final scores are sometimes called expected utilities.

From the point of view of making the rules clear, this form of composition has
the advantage of simplicity. The model is built by de

ning the criteria to be taken
into account and, through the weights, the importance assigned to each of them.
The idea behind this form of composition is that the decision maker starts by
choosing one objective from among multiple options. Here, choosing an objective
means preferring one among the multiple criteria. This choice may be not univocal
but randomized, i.e., it can be given by a probability distribution of preference
among the criteria.
The concept of a probability distribution will be formulated more clearly in the
Appendix. This term is used here in the context of a lottery where each possible
prize has a different chance of being won. In the present case, the criteria will be
thought of as the prizes and the weights as their chances. This corresponds to the
decision maker running a possibly biased roulette game to pick the preferred
criterion.
Another way to look at this form of composition is by associating each criterion
with a different evaluator and considering the averaging as a rule to satisfy the