Classification of concatenation measurement structures according to scale type

Abstract

A relational structure is said to be of scale type ( M, N) iff M is the largest degree of homogeneity and N the least degree of uniqueness ( Narens, Theory and Decision, 1981, 13, 1–70; Journal of Mathematical Psychology, 1981, 24, 249–275 ) of its automorphism group. Roberts (in Proceedings of the first Hoboken Symposium on graph theory, New York: Wiley, 1984; in Proceedings of the fifth international conference on graph theory and its applications, New York: Wiley, 1984 ) has shown that such a structure on the reals is either ordinal or M is less than the order of at least one defining relation (Theorem 1.2). A scheme for characterizing N is outlined in Theorem 1.3. The remainder of the paper studies the scale type of concatenation structures 〈 X, ≿, ∘ 〉, where ≿ is a total ordering and ∘ is a monotonic operation. Section 2 establishes that for concatenation structures with M>0 and N<∞ the only scale types are (1,1), (1,2), and (2,2), and the structures for the last two are always idempotent. Section 3 is concerned with such structures on the real numbers (i.e., candidates for representations), and it uses general results of Narens for real relational structures of scale type ( M, M) (Theorem 3.1) and of Alper ( Journal of Mathematical Psychology, 1985 , 29, 73–81) for scale type (1, 2) (Theorem 3.2). For M>0, concatenation structures are all isomorphic to numerical ones for which the operation can be written x∘y = yf( x y ) , where f is strictly increasing and f(x) x is strictly decreasing (unit structures). The equation f( x ϱ )= f( x) ϱ is satisfied for all x as follows: for and only for ϱ = 1 in the (1,1) case; for and only for ϱ= k n , k > 0 fixed, and n ranging over the integers, in the (1, 2) case; and for all ϱ>0 in the (2, 2) case (Theorems 3.9, 3.12, and 3.13). Section 4 examines relations between concatenation catenation and conjoint structures, including the operation induced on one component by the ordering of a conjoint structure and the concept of an operation on one component being distributive in a conjoint structure. The results, which are mainly of interest in proving other results, are mostly formulated in terms of the set of right translations of the induced operation. In Section 5 we consider the existence of representations of concatenation structures. The case of positive ones was dealt with earlier ( Narens & Luce ( Journal of Pure & Applied Algebra 27, 1983 , 197–233). For idempotent ones, closure, density, solvability, and Archimedean are shown to be sufficient (Theorem 5.1). The rest of the section is concerned with incomplete results having to do with the representation of cases with M>0. A variety of special conditions, many suggested by the conjoint equivalent of a concatenation structure, are studied in Section 6. The major result (Theorem 6.4) is that most of these concepts are equivalent to bisymmetry for idempotent structures that are closed, dense, solvable, and Dedekind complete. This result is important in Section 7, which is devoted to a general theory of scale type (2, 2) for the utility of gambles. The representation is a generalization of the usual SEU model which embodies a distinctly bounded form of rationality; by the results of Section 6 it reduces to the fully rational SEU model when rationality is extended beyond the simplest equivalences. Theorem 7.3 establishes that under plausible smoothness conditions, the ratio scale case does not introduce anything different from the (2, 2) case. It is shown that this theory is closely related to, but somewhat more general, than Kahneman and Tversky's ( Econometrica 47, 1979 , 263–291) prospect theory.