Classification of real measurement representations by scale type

Abstract

The scale type (M, N) of an ordered relational structure is defined in terms of two properties, called M-point homogeneity and N-point uniqueness, of the automorphism group of the structure. For real structures on an open interval, scale types (1, 1) and (2, 2) correspond to ratio and interval representations, respectively. Accepting certain key properties, such as transitivity of the ordering relation and, in the case of a binary operation, monotonicity, and assuming that a real representation exists, then for each scale type whose real transformation group is known, the possible forms for the representation can be derived. For structures with a monotonic, binary operation, this is done completely for the ratio and interval cases, and incompletely in what is shown to be the only other interesting case exhibiting substantial symmetry, (1, 2). These results are then used to gain a better understanding of the psychological theory of utility of gambles and the possible generalisations of multiplicative conjoint structures, which are of importance in dimensional analysis.