Mathematics and Operations Research department
Faculté Polytechnique de Mons
9, rue de Houdain, B-7000 Mons, Belgique
Two distinct traditions underlie most of the work done in the area of decision analysis with multiple attributes. The conjoint measurement tradition has deep roots both in Mathematical Psychology and Decision Theory. Starting with a binary relation defined on a product set, it aims at finding conditions under which it is possible to build a convenient numerical representation of the relation. The model that has been most studied in this framework is the additive utility. Besides their theoretical interest, many conjoint measurement results are constructive in nature and, therefore, give hints on how to devise assessment procedures of utility functions and, thus, preferences. A more pragmatic tradition starts with alternatives evaluated along several attributes. Along each attribute, alternatives are supposed to be compared using a well-behaved preference relation. The central problem is then to build a preference relation between alternatives taking all attributes into account, i.e. a global preference relation, based on the preference relations on each attribute and ''inter-attribute'' information such as weights or trade-offs.
The notion of dominance plays a crucial role here. An alternative is said to dominate another if the former is judged ``at least as good as'' the latter on all attributes. It is usually considered that global preference should be compatible with dominance, i.e. that the dominance relation is included in the global preference relation. When a global preference relation is compatible with dominance, it makes sense to limit the search for ``good'' alternatives in the set of efficient alternatives, i.e. alternatives that are undominated. These two lines of thought seem to coexist since the beginning of decision analysis with multiple attributes, in the late '60s. Both have generated important theoretical and practical achievements but the two traditions have remained largely unrelated. Their setting differ significantly. The conjoint measurement tradition starts with a well- behaved preference relation taking all attributes into account. The pragmatic one starts with a well-behaved preference relation defined on each attribute and derives a global preference relation using the notion of dominance and inter-attribute information. The principles used in order to build the global preference relation do not always guarantee that this relation will be transitive or complete, e.g. if a qualified weighted majority of attributes is used.
In this work, we attempt to establish connections between these two traditions. In order to do so, we adopt a classical conjoint measurement setting, while not requiring transitivity or completeness. We provide a simple axiomatic characterization of preference relations compatible with dominance and show that all such relations admit a nontrivial numerical representation. This extends the traditional scope of conjoint measurement to include binary relations that are not ``well- behaved''. Furthermore this shows that many techniques developed in the pragmatic tradition can usefully be analyzed in a conjoint measurement framework. The key tool for the analysis of such preference relations is the consideration of various kinds of traces on coordinates induced by the original relation.