‘Additive difference’ models without additivity and subtractivity

This paper studies conjoint measurement models tolerating intransitivities that closely resemble Tversky's additive difference model while replacing additivity and subtractivity by mere decomposability requirements. We offer a complete axiomatic characterisation of these models without having recourse to unnecessary structural assumptions on the set of objects. This shows the pure consequences of several cancellation conditions that have often been used in the analysis of more traditional conjoint measurement models. Our models contain as particular cases many aggregation rules that have been proposed in the literature.     

Combining additive representations on subsets into an overall representation

Many traditional conjoint representations of binary preferences are additively decomposable, or additive for short. An important generalization arises under rank-dependence, when additivity is restricted to cones with a fixed ranking of components from best to worst (comonotonicity), leading to configural weighting, rank-dependent utility, and rank- and sign-dependent utility (prospect theory). This paper provides a general result showing how additive representations on an arbitrary collection of comonotonic cones can be combined into one overall representation that applies to the union of all cones considered. The result is applied to a new paradigm for decision under uncertainty developed by Duncan Luce and others, which allows for violations of basic rationality properties such as the coalescing of events and other framing conditions. Through our result, a complete preference foundation of a number of new models by Luce and others can be obtained. We also show how additive representations on different full product sets can be combined into a representation on the union of these different product sets.     

A two-stage algorithm for assessing violations of additivity via axiomatic and numerical conjoint analysis

An algorithm for assessing additivity conjunctively via both axiomatic conjoint analysis and numerical conjoint scaling is described. The algorithm first assesses the degree of individual differences among sets of rankings of stimuli, and subsequently examines either individual or averaged data for violations of axioms necessary for an additive model. The axioms are examined at a more detailed level than has been previously done. Violations of the axioms are broken down into different types. Finally, a nonmetric scaling of the data can be done based on either or both of two different badness-of-fit scaling measures. The advantages of combining all of these features into one algorithm for improving the diagnostic value of axiomatic conjoint measurement in evaluating additivity are discussed.     

Theory and tests of the conjoint commutativity axiom for additive conjoint measurement

The empirical study of the axioms underlying additive conjoint measurement initially focused mostly on the double cancellation axiom. That axiom was shown to exhibit redundant features that made its statistical evaluation a major challenge. The special case of double cancellation where inequalities are replaced by indifferences–the Thomsen condition–turned out in the full axiomatic context to be equivalent to the double cancellation property but without exhibiting the redundancies of double cancellation. However, it too has some undesirable features when it comes to its empirical evaluation, the chief among them being a certain statistical asymmetry in estimates used to evaluate it, namely two interlocked hypotheses and a single conclusion. Nevertheless, thinking we had no choice, we evaluated the Thomsen condition for both loudness and brightness and, in agreement with other lines of research, we found more support for conjoint additivity than not. However, we commented on the difficulties we had encountered in evaluating it. Thus we sought a more symmetric replacement, which as Gigerenzer and Strube (1983) first noted, is found in the conjoint commutativity axiom proposed by Falmagne (1976, who called it the “commutative rule”). It turns out that, in the presence of the usual structural and other necessary assumptions of additive conjoint measurement, we can show that conjoint commutativity is equivalent to the Thomsen condition, a result that seems to have been overlooked in the literature. We subjected this property to empirical evaluation for both loudness and brightness. In contrast to Gigerenzer and Strube (1983), our data show support for the conjoint commutativity in both domains and thus for conjoint additivity.     

Preferences for multi-attributed alternatives: Traces, dominance, and numerical representations

This paper analyzes conjoint measurement models allowing for intransitive and/or incomplete preferences. This analysis is based on the study of marginal traces induced on coordinates by the preference relation and uses conditions guaranteeing that these marginal traces are complete.Within the framework of these models, we propose a simple axiomatic characterization of preference relations compatible with the notion of dominance. We show that all such relations have a nontrivial numerical representation.Our results allow us to establish useful connections between two lines of thought in the area of decision analysis with multiple attributes that have largely remained unrelated: the one based on conjoint measurement and the one emphasizing the idea of dominance.     

An algorithm for finite conjoint additivity

A complete solution to the problem of finite conjoint additivity is provided, not only to answer the question of whether a particular order is conjointly additive, but also to detail the possible value assignments which demonstrate conjoint additivity for the order. In the process, a complete solution is given to the problem of solving a finite number of homogeneous linear inequalities in a finite number of variables. While others have reduced the problem to the solution of systems of linear inequalities, the algorithm presented here is based on “basic inequalities,” and it provides considerable computational simplicity over previous algorithms.     

Additivity and expected utility in risky multiattribute preferences

This experiment analyzed human preferences among even chance gambles for commodity bundles. The purpose of the experiment was to test several independence assumptions that distinguish between models for risky multiattribute preferences. In particular, the additivity and the expected utility part of the additive expected utility model were tested symmetrically. The degree and form of model violations were established, and the effects of instructions and of response modes were examined. All independence assumptions were violated by a bias to prefer a gamble or a commodity bundle that was previously matched against a standard. Systematic and strong violations that superseded this bias questioned the validity of the additive expected utility model. In violation of the additivity part of that model subjects consistently preferred the gamble with more balanced outcomes when comparing gambles with identical marginal probability distributions. This trend, called multiattribute risk aversion, was independent of subjects' single attribute risk attitude. Instructions and response modes had no noticeable impacts on these response patterns.     

Utility independence of multiattribute utility theory is equivalent to standard sequence invariance of conjoint measurement

Utility independence is a central condition in multiattribute utility theory, where attributes of outcomes are aggregated in the context of risk. The aggregation of attributes in the absence of risk is studied in conjoint measurement. In conjoint measurement, standard sequences have been widely used to empirically measure and test utility functions, and to theoretically analyze them. This paper shows that utility independence and standard sequences are closely related: utility independence is equivalent to a standard sequence invariance condition when applied to risk. This simple relation between two widely used conditions in adjacent fields of research is surprising and useful. It facilitates the testing of utility independence because standard sequences are flexible and can avoid cancelation biases that affect direct tests of utility independence. Extensions of our results to nonexpected utility models can now be provided easily. We discuss applications to the measurement of quality-adjusted life-years (QALY) in the health domain.     

Axiomatic and numerical conjoint measurement: An evaluation of diagnostic efficacy

Synthetic data are used to examine how well axiomatic and numerical conjoint measurement methods, individually and comparatively, recover simple polynomial generators in three dimensions. The study illustrates extensions of numerical conjoint measurement (NCM) to identify and model distributive and dual-distributive, in addition to the usual additive, data structures. It was found that while minimum STRESS was the criterion of fit, another statistic, predictive capability, provided a better diagnosis of the known generating model. That NCM methods were able to better identify generating models conflicts with Krantz and Tversky's assertion that, in general, the direct axiom tests provide a more powerful diagnostic test between alternative composition rules than does evaluation of numerical correspondence. For all methods, dual-distributive models are most difficult to recover, while consistent with past studies, the additive model is the most robust of the fitted models.Douglas Emery is now at the Krannert Graduate School of Management, Purdue University, West Lafayette, IN, on leave from the University of Calgary.     

Evaluating the Equal-Interval Hypothesis with Test Score Scales

The axioms of additive conjoint measurement provide a means of testing the hypothesis that testing data can be placed onto a scale with equal-interval properties. However, the axioms are difficult to verify given that item responses may be subject to measurement error. A Bayesian method exists for imposing order restrictions from additive conjoint measurement while estimating the probability of a correct response. In this study an improved version of that methodology is evaluated via simulation. The approach is then applied to data from a reading assessment intentionally designed to support an equal-interval scaling.     

Additivity of loudness across critical bands: A critical test

The use of magnitude estimation as well as axiomatic measurement theory has led to the suggestion that loudness adds across critical bands. In the present paper, we challenge this postulate by applying a more sensitive methodology, based on Falmagne’s (1976) random conjoint measurement procedure. A necessary condition for additivity of loudness was investigated in tone complexes consisting of 2-kHz and 5-kHz (resp. 3-kHz) components; the results showed systematic deviations from additivity. We argue that these deviations are due to asymmetric masking of the higher component by the lower one, and we propose a tentative quantitative model to account for the data. Such a model is in line with results from tone-on-tone masking, which show masking to be effective over a range of several critical bands.     

Cross-modal additive conjoint structures and psychophysical scale convergence

I investigated a variety of issues related to the measurement of the magnitude of psychological experience, especially the magnitude of sensations. Different groups of subjects made pair comparisons, magnitude estimations, and category judgments of the "total sensory magnitude" of light and sound stimuli presented conjointly. Another group judged the dissimilarity of pairs of conjoint stimuli. Various axioms, especially double cancellation, were tested on the resulting rank orders of conjoint stimuli. Judgements of the total magnitude of conjoint combinations of sound and light stimuli formed an additive conjoint structure. Dissimilarity judgments gave rise to a closely related lattice structure. Moreover, various scales of the individual attributes (loudness and brightness) calculated from the two types of judgments of the conjoint stimuli displayed substantial convergence, each scale for a given modality being linear with all other scales for that modality.     

An empirical test for Fishburn's additivity axiom

Three experiments were done to test the empirical relevance of Fishburn's (1967) additivity axiom, which says that people should be indifferent between pairs of gambles which satisfy certain conditions specified in the axiom. Each of the experiments consisted of two parts. In the first part, subjects had to evaluate consequences which were used in the second part as possible outcomes in a gamble. In the second part, subjects had to make choices among pairs of gambles. The experiments differed in respect to kinds of consequences and kinds of subjects used. Additivity analysis was applied to the data of the first part of each experiment, using a conjoint measurement model. A Monte Carlo study is included, which provides some hints for the evaluation of the stress coefficients obtained after applying additivity analysis to the empirical data matrices. The data of the second part of each experiment are discussed in respect to their relevance for Fishburn's (1967) additivity axiom. It was not strongly supported by the data, unless for a very restricted situation.     

Folded additive structures: A nonpolynomial model of some factorial interactions

A nonpolynomial measurement model for representing nonadditive compositions of stimulus dimensions is presented. This representation, which is referred to as a folded additive structure, permits the decomposition of a behavioral ordering into independent factors in some of the cases where the behavior itself does not directly reflect this independence. Psychologically, this representation corresponds to a model of behavior which assumes that (1) the effect stimulus attributes have on behavior is mediated by a single relevent psychological dimension, (2) the behavioral ordering is single-peaked over this mediating dimension and (3) the mediating dimension has a conjoint additive ordering over a set of relevent stimulus dimensions. Necessary and sufficient tests for folded additive structures are given. In addition, five psychological theories which make these assumptions are discussed.     

A conjoint measurement approach to color harmony

The aim of this study was to find a rule in color harmony that could reveal the mutual influences of the three basic harmonies, those of hue, lightness, and saturation. One hundred and four color pairs were judged for harmony by five subjects five times. The results were analyzed by the conjoint measurement algorithm of Krantz and Tversky (1971). It was concluded that there is an interaction between saturation harmony and hue and lightness harmony together, and an additivity between hue harmony and lightness harmony. This implies the rule: color harmony = saturation harmony * (hue harmony + lightness harmony).     

A MARKET RESEARCH APPROACH TO DETERMINING LOCAL LABOR MARKET AVAILABILITY FOR NON-MANAGEMENT JOBS

As an alternative approach to setting affirmative action hiring targets, preferences for non-management jobs were explored using a research technique known as conjoint measurement or trade-off analysis. A general population sample and an applicant sample were surveyed in two cities. Results showed sharp differences in job preferences between men and women. It was also found that women were less interested in non-traditional telephone work than men—probably because the jobs that are non-traditional for women were outdoors, dirty, and had a higher chance of injury. The research technique also permitted the identification of the demographic characteristics of those people who are interested in non-traditional work.     

Preference as fulfilment of desires

We consider preferences as fulfillment of conditional desires, which can be either positive or negative, or both. We go beyond the standard multi-attributive additive utility theory in the sense that we separate the data given by the preference relation over an unstructured space from the property structure representing (conditional) attributes or desires. The model accounts for the psychologically motivated and empirically confirmed asymmetry between desire fulfillment and disappointment (loss aversion). The only restriction on the set of desires is, loosely speaking, a kind of mutual logical independence. We formulate a representation theorem characterising when a weak order (i.e. complete and transitive) preference is compatible with the logical structure of desires and has an additive representation over it. It is unique in the sense that each utility function representing the preferences has at most one such additive decomposition.     

Solutions to some problems in the implementation of conjoint analysis

Methodological problems encountered in implementing conjoint analysis include (1) the impractically large set of multiattribute choice alternatives created by the factorial combination of more than a few attributes, (2) the hypothetical nature of the alternatives in the choice set, and (3) the assumption that each individual’s preferences can be described by the same composition rule. The techniques of tailoring, belief matching, and axiom testing are suggested as solutions to these problems, and their use is demonstrated in a conjoint analysis study of individuals’ contraceptive preferences. It is noted that tailoring and belief matching can also be used as methodological enhancements in functional measurement studies.