Two Factor Additive Conjoint Measurement with One Solvable Component

This paper addresses conditions for the existence of additive separable utilities. It considers mainly two-dimensional Cartesian products in which restricted solvability holds w.r.t. one component, but some results are extended to n-dimensional spaces. The main result shows that, in general, cancellation axioms of any order are required to ensure additive representability. More precisely, a generic family of counterexamples is provided, proving that the (m+1)st order cancellation axiom cannot be derived from the mth order cancellation axiom when m is even. However, a special case is considered in which the existence of additive representations can be derived from the independence axiom alone. Unlike the classical representation theorems, these representations are not unique up to strictly positive affine transformations, but follow Fishburn's (1981) uniqueness property. Copyright 2000 Academic Press.