In formal theories of measurement meaningfulness is usually formulated in terms of numerical statements that are invariant under admissible transformations of the numerical representation. This is equivalent to qualitative relations that are invariant under automorphisms of the measurement structure. This concept of meaningfulness, appropriately generalized, is studied in spaces constructed from a number of conjoint and extensive structures some of which are suitably interrelated by distribution laws. Such spaces model the dimensional structures of classical physics. It is shown that this (...) qualitative concept corresponds exactly with the numerical concept of dimensionally invariant laws of physics. (shrink)

Suppose that entities composed of two independent components are qualitatively ordered by a relation that satisfies the axioms of conjoint measurement. Suppose, in addition, that each component has a concatenation operation that, together either with the ordering induced on the component by the conjoint ordering or with its converse, satisfies the axioms of extensive measurement. Without further assumptions, nothing can be said about the relation between the numerical scales constructed from the two measurement theories except that they are strictly monotonic. (...) An axiom is stated that relates the two types of measurement theories, seems to cover all cases of interest in physics, and is sufficient to establish that the conjoint measurement scales are power functions of the extensive measurement scales. (shrink)

One aspect of the utility of gambling may evidence itself in failures of idempotence, i.e., when all chance outcomes give rise to the same consequence the `gamble' may not be indifferent to its common consequence. Under the assumption of segregation, such gambles can be expressed as the joint receipt of the common consequence and what we call `an element of chance', namely, the same gamble with the common consequence replaced by the status quo. Generalizing, any gamble is indifferent to the (...) joint receipt of its element of chance and a certain consequence, which is called the `kernel equivalent' of the gamble. Under idempotence, the kernel equivalent equals the certainty equivalent. Conditions are reported (Theorem 4) that are sufficient for the kernel equivalents to have the kind of utility representation first discussed by Luce and Fishburn (1991), including being idempotent. This utility representation of the kernel equivalents together with the derived form of utility over joint receipts yields a utility representation of the original structure. Possible forms for the utility of an element of chance are developed. (shrink)

Using H. Whitney's algebra of physical quantities and his definition of a similarity transformation, a family of similar systems (R. L. Causey [3] and [4]) is any maximal collection of subsets of a Cartesian product of dimensions for which every pair of subsets is related by a similarity transformation. We show that such families are characterized by dimensionally invariant laws (in Whitney's sense, [10], not Causey's). Dimensional constants play a crucial role in the formulation of such laws. They are represented (...) as a function g, known as a system measure, from the family into a certain Cartesian product of dimensions and having the property gφ =φ g for every similarity φ . The dimensions involved in g are related to the family by means of certain stability groups of similarities. A one-to-one system measure is a proportional representing function, which plays an analogous role in Causey's theory, but not conversely. The present results simplify and clarify those of Causey. (shrink)

One aspect of the utility of gambling may evidence itself in failures of idempotence, i.e., when all chance outcomes give rise to the same consequence the `gamble' may not be indifferent to its common consequence. Under the assumption of segregation, such gambles can be expressed as the joint receipt of the common consequence and what we call `an element of chance', namely, the same gamble with the common consequence replaced by the status quo. Generalizing, any gamble is indifferent to the (...) joint receipt of its element of chance and a certain consequence, which is called the `kernel equivalent' of the gamble. Under idempotence, the kernel equivalent equals the certainty equivalent. Conditions are reported (Theorem 4) that are sufficient for the kernel equivalents to have the kind of utility representation first discussed by Luce and Fishburn (1991), including being idempotent. This utility representation of the kernel equivalents together with the derived form of utility over joint receipts yields a utility representation of the original structure. Possible forms for the utility of an element of chance are developed. (shrink)

Empirical evidence from both utility and psychophysical experiments suggests that people respond quite differently—perhaps discontinuously—to stimulus pairs when one consequence or signal is set to `zero.' Such stimuli are called unitary. The author's earlier theories assumed otherwise. In particular, the key property of segregation relating gambles and joint receipts (or presentations) involves unitary stimuli. Also, the representation of unitary stimuli was assumed to be separable (i.e., multiplicative). The theories developed here do not invoke separability. Four general cases based on two (...) distinctions are explored. The first distinction is between commutative joint receipts, which are relevant to utility, and the non-commutative ones, which are relevant to psychophysics. The second distinction concerns how stimuli of the form (x, C; y) and the operation of joint receipt are linked: by segregation, which mixes stimuli and unitary ones, and by distributivity, which does not involve any unitary stimuli. A class of representations more general than rank-dependent utility (RDU) is found in which monotonic functions of increments U(x)-U(y), where U is an order preseving representation of gambles, and joint receipt play a role. This form and its natural generalization to gambles with n > 2 consequences, which is also axiomatized, appear to encompass models of configural weights and decision affect. When joint receipts are not commutative, somewhat similar representations of stimuli arise, and joint receipts are shown to have a conjoint additive representation and in some cases a constant bias independent of signal intensity is predicted. (shrink)

A research program is announced, and initial, exciting progress described. Many inference problems, poorly modeled by some traditional approaches, are surprisingly well handled by kinds of simple-minded Bayesian approximations. Fuller Bayesian approaches are typically more accurate but rarely are they either fast or frugal. Open issues include codifying when to use which heuristic and to give detailed evolutionary explanations.

This article argues that there is a natural solution to carry out interpersonal comparisons of utility when the theory of gambles is supplemented with a group operation of joint receipts. If so, three types of people can exist, and the two types having multiplicative representations of joint receipt have, in contrast to most utility theories, absolute scales of utility. This makes possible, at least in principle, meaningful interpersonal comparisons of utility with desirable properties, thus resolving a long standing philosophical problem (...) and having potentially important implications in economics. Two behavioral criteria are given for the three classes of people. At this point the relative class sizes are unknown. (shrink)

This note addresses 3 issues that seem to pervade much of economic thought about individual decisions among uncertain alternatives: (1) Restricting primitives to just orderings of first-order gambles and not admitting, e.g., compound acts or joint receipt of consequences and gambles. (2) Great resistance to experimental findings that strongly suggest that most current theories fail descriptively. (3) Taking for granted the innocence of some assumptions when, in fact, they are not innocent, e.g., that constant acts are idempotent. My conclusion is (...) that these restrictions have greatly inhibited developments in utility theory. (shrink)

The present theory leads to a set of subjective weights such that the utility of an uncertain alternative (gamble) is partitioned into three terms involving those weights—a conventional subjectively weighted utility function over pure consequences, a subjectively weighted value function over events, and a subjectively weighted function of the subjective weights. Under several assumptions, this becomes one of several standard utility representations, plus a weighted value function over events, plus an entropy term of the weights. In the finitely additive case, (...) the latter is the Shannon entropy; in all other cases it is entropy of degree not 1. The primary mathematical tool is the theory of inset entropy. (shrink)

A detailed theoretical analysis is presented of what five utility representations – subjective expected utility (SEU), rank-dependent (cumulative or Choquet) utility (RDU), gains decomposition utility (GDU), rank weighted utility (RWU), and a configural-weight model (TAX) that we show to be equivalent to RWU – say about a series of independence properties, many of which were suggested by M. H. Birnbaum and his coauthors. The goal is to clarify what implications to draw about the descriptive aspects of the representations from data (...) concerning these properties. The upshot is a sharp rejection of SEU and RDU and no clear choice between GDU and TAX, but a list of 8 properties is given that should receive more attention to discriminate between the latter two models. (shrink)