R. Duncan Luce

Simultaneous conjoint measurement: A new type of fundamental measurement

The essential character of what is classically considered, e.g., by N. R. Campbell, the fundamental measurement of extensive quantities is described by an axiomatization for the comparision of effects of (or responses to) arbitrary combinations of “quantities” of a single specified kind. For example, the effect of placing one arbitrary combination of masses on a pan of a beam balance is compared with another arbitrary combination on the other pan. Measurement on a ratio scale follows from such axioms. In this paper, the essential character of simultaneous conjoint measurement is described by an axiomatization for the comparision of effects of (or responses to) pairs formed from two specified kinds of “quantities”. The axioms apply when, for example, the effect of a pair consisting of one mass and one difference in gravitational potential on a device that responds to momentum is compared with the effect of another such pair. Measurement on interval scales which have a common unit follows from these axioms; usually these scales can be converted in a natural way into ratio scales. A close relation exists between conjoint measurement and the establishment of response measures in a two-way table, or other analysis-of-variance situations, for which the “effects of columns” and the “effects of rows” are additive. Indeed, the discovery of such measures, which are well known to have important practical advantages, may be viewed as the discovery, via conjoint measurement, of fundamental measures of the row and column variables. From this point of view it is natural to regard conjoint measurement as factorial measurement.

Semiorders and a Theory of Utility Discrimination

In the theory of preferences underlying utility theory i t is generally assumed that the indifference relation is transitive, and this leads t o equivalence classes of indifferent elements or, equally, t o indifference curves. It has been pointed out that this assumption is contrary to experience and that utility is not perfectly discriminable, as such a theory necessitates. In this paper intransitive indifference relations are admitted and a class of them are axiomatized. This class is shown t o be substantially equivalent t o a utility theory in which there are just noticeable difference functions which state for any value of utility the change in utility so that the change is just noticeable. In the case of risk represented by a linear utility function over a mixture space, the precise form of the function is examined in detail.

Contemporary Developments in Mathematical Psychology

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The Choice Axiom after Twenty Years

This survey is divided into three major sections. The first concerns mathematical results about the choice axiom and the choice models that devoIve from it. For example, its relationship to Thurstonian theory is satisfyingly understood; much is known about how choice and ranking probabilities may relate, although little of this knowledge seems empirically useful; and there are certain interesting statistical facts. The second section describes attempts that have been made to test and apply these models. The testing has been done mostly, though not exclusively, by psychologists; the applications have been mostly in economics and sociology. Although it is clear from many experiments that the conditions under which the choice axiom holds are surely delicate, the need for simple, rational underpinnings in complex theories, as in economics and sociology, leads one to accept assumptions that are at best approximate. And the third section concerns alternative, more general theories which, in spirit, are much like the choice axiom. Perhaps I had best admit at the outset that, as a commentator on this scene, I am qualified no better than many others and rather less well than some who have been working in this area recently, which I have not been. My pursuits have led me along other, somewhat related routes. On the one hand, I have contributed to some of the recent, purely algebraic aspects of fundamental measurement (for a survey of some of this material, see Krantz, Lute, Suppes, & Tversky, 1971). And on the other hand, I have worked in the highly probabilistic area of psychophysical theory; but the empirical materials have led me away from axiomatic structures, such as the choice axiom, to more structural, neural models which are not readily axiomatized at the present time. After some attempts to apply choice models to psychophysical phenomena (discussed below in its proper place), I was led to conclude that it is not a very promising approach to, these data, and so I have not been actively studying any aspect of the choice axiom in over 12 years. With that understood, let us begin.

Rank- and Sign-Dependent Linear Utility Models for Finite First-Order Gambles

Finite first-order gambles are axiomatized. The representation combines features of prospect and rank-dependent theories. What is novel are distinctions between gains and losses and the inclusion of a binary operation of joint receipt. In addition to many of the usual structural and rationality axioms, joint receipt forms an ordered concatenation structure with special features for gains and losses. Pfanzagls (1959) consistency principle is assumed for gains and losses separately. The nonrational assumption is that a gamble of gains and losses is indifferent to the joint receipt of its gains pitted against the status quo and of its losses against the status quo.

Connectivity and generalized cliques in sociometric group structure

By using the concepts of antimetry andn-chain it is possible to define and to investigate some properties of connectivity in a sociometric group. It is shown that the number of elements in a group, the number of antimetries, and the degree of connectivity must satisfy certain inequalities. Using the ideas of connectivity, a generalized concept of clique, called ann-clique, is introduced.n-cliques are shown to have a very close relationship to the existence of cliques in an artificial structure defined on the same set of elements, thus permitting the determination ofn-cliques by means of the same simple matrix procedures used to obtain the clique structures. The presence of two or morem-cliques, wherem is the number of elements in the group, is proved to mean an almost complete splitting of the group.

The effect on the preference-reversal phenomenon of using choice indifferences

Abstract Preference reversal appears to be non-transitive choice behavior, inasmuch as subjects choosing between two gambles with similar expected values typically select the one with a larger chance of winning (a P-bet), yet they place a higher certainty equivalent, or judged indifference point (JIP), on the one with the larger amount to win (the

Classification of concatenation measurement structures according to scale type

-bet). In the present experiments, we explore a pair of methods for determining money indifference points that are procedurally closer than is common in this research to the choice phase of the preference reversal experiment. The two new approaches for determining money indifference points for gambles (called choice indifference points, or CIP) are as follows: a version of the classical psychophysical up-down method and a variant called PEST (i.e., Parameter Estimation by Sequential Testing). With both methods, CIPs reduced the number of reversals, as compared to JIPs, to the point with PEST that the remaining ones may be due primarily to chance fluctuations in choices between gambles. Subjects who reversed frequently usually judged the gambles to have similar values; those who did not reverse at all, judged them as different. The typical subjects exhibits JIP>CIP for

Operant matching is not a logical consequence of maximizing reinforcement rate

-bets, but not for P-bets. This bias appears to underlie the observed intransitivity when JIP is used.

Sequential effects in judgments of loudness.

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 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 Tverskys ( Econometrica 47 , 1979 , 263–291) prospect theory.