Conjointness as a derived property

Abstract

In this paper we investigate the question of which conditions imposed on a weak order ( A, ≾) allow for a representation of ( A, ≾) as a Cartesian product structure ( X 1, X 2, ≾′) with independent components X 1, X 2 and a weak order ≾′ on X 1 × X 2. Two possibilities are analyzed: first, two weak orders on A are supposed to exist in addition to ≾ and a compatibility condition between all three is shown to yield the decomposition; second, one additional weak order, together with a relation which reflects the compensatory trade-off components typically exhibit in an additive representation, is demonstrated to give essentially the same result. Furthermore, the paper deals with the automorphism groups of these structures and relates them to the important concept of a factorizable automorphism of a conjoint structure. Next, a generalization to more than two components is presented and it is shown that the compatibility between the weak orders gives rise to a partial order or, if additional assumptions are met, even to a lattice. The final section of the paper discusses an application of one of our theorems to the theory of psychological tests.