Unique finite conjoint measurement

This paper studies one of the most important types of measurement that has arisen from the social sciences, the additive conjoint measurement introduced by Debreu (1960) and Luce and Tukey (1964). It also considers the variant we call additive conjoint extensive measurement. Both types of measurement are based on qualitative comparisons between multiattribute alternatives in Cartesian products of sets. This paper initiates a study of their uniqueness for the case in which all sets are finite. It considers uniqueness up to similar positive affine transformations for additive conjoint measurement, and uniqueness up to similar proportionality transformations for additive conjoint extensive measurement. Both types of uniqueness are related to sets of 'indifference' comparisons that correspond to sets of linearly independent equations for the measurement representation. After we explicate necessary and sufficient conditions for uniqueness, we explore specific aspects of sets of unique solutions for two-factor (two-set) additive conjoint measurement and two-factor additive conjoint extensive measurement.