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 Systematic and strong violations that superseded this bias questioned the validity of the additive expected utility model.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.     

Models of best–worst choice and ranking among multiattribute options (profiles)

We develop and characterize new representations for the maxdiff model (Marley & Louviere, 2005) for best–worst choice between multiattribute options; in particular, we state conditions under which the scale value of a multiattribute option is a product of independent ratio scales on each attribute. We show that there is a vector of simple “scores” that are sufficient for the model, with each score a closed-form function of the maximum likelihood estimates of the model’s parameters. Several related models are developed for rank orders obtained by repeated best and/or worst choices, and two of these models are tested on stated preferences between mobile phones.     

Decision making under conflict: Decision time as a measure of conflict strength

Conflict and choice are closely related in that choice produces conflict and conflict is resolved by making a choice. Although conflict was invoked in psychological approaches to decision making early on (Lewin, 1931/1964), no generally accepted measure of conflict strength has been established (Tversky & Shafir, 1992). The present study introduces a model (multiattribute decision field theory) that predicts a decision time pattern depending on the conflict situation. In a risky decision-making experiment with multiattribute choice alternatives, decision time is investigated as a possible measure of conflict strength. It is shown that the model can be fitted to a complex choice pattern.     

MDFT account of decision making under time pressure

In this paper, decision making under time pressure for multiattribute choice alternatives in a risky environment is investigated. A model, multiattribute decision field theory (MDFT), is introduced that describes both the dynamic and the stochastic nature of decision making and accounts for the observed changes in choice probabilities, including preference reversals as a function of time limit. An experiment in which five different time limits were imposed on the decision maker is presented to test the predictions of the model. It is shown that MDFT is able to account for the complex decision behavior observed in the data. Furthermore, MDFT is compared with the predictions of decision field theory (Busemeyer & Townsend, 1993; Roe, Busemeyer, & Townsend, 2001).     

Overlapping stage models and reaction time additivity: Effects of the activation equation

Many experimental factors have been found to affect mean reaction time (RT) additively in factorial experiments. What sorts of RT models are compatible with this fact? Sternberg (1969) showed that serial, discrete-stage models are consistent with additivity, and as a result additivity has sometimes been regarded as evidence in favor of such models. However, McClelland (1979) showed that an alternative “cascade” model, which violates crucial assumptions of discrete-stage models, also predicts RT additivity in many cases. This article examines various modified versions of the cascade model, and shows that mean RT additivity arises from many overlapping stage models, including some with thresholds, information quantization, and nonlinear activation transformations. This suggests that other aspects of the data besides mean RT additivity should be examined to distinguish between serial and overlapping stage models.     

Functional measurement approach to self-estimation in multiattribute evaluation

Multiattribute analysis depends on measurement of values and weights. Unless these measures reflect the decision maker's true values and weights, the multiattribute formula may put a less-preferred alternative in first place. To avoid such disordinality requires stringent measurement conditions: First, the values and weights must be on linear (equal interval) or ratio (known zero) scales. Second, these scales must satisfy a condition of common unit across disparate attribute dimensions. Most methods of range adjustment beg both of these measurement questions. Functional measurement theory can solve both problems and so can be useful in multiattribute analysis. Past work has established the operation of a general cognitive algebra as an empirical reality. The averaging model, in particular, makes possible the definition and estimation of weights and values as distinct psychological parameters. It can also solve the problem of common unit. Cognitive algebra thus provides a grounded theoretical foundation on which to develop self-estimation methodology, in which decision makers provide direct estimates of their values and weights. The logic is straightforward. Functional measurement can analyze global judgments to obtain validated psychological scales. These scales may then be used as validational criteria for the self-estimates. Procedures to eliminate biases in the self-estimates can thus be tested and refined in well-learned multiattribute tasks, such as judgments of meals, in which global judgments are trustworthy. Once developed, such self-estimation procedures may be used with some confidence for general multiattribute analysis. A number of studies from 20-odd years of work on the theory of information integration are summarized to show good, although not unmixed promise for self-estimation.     

The influence of decision heuristics and overconfidence on multiattribute choice: A process-tracing study

In the present study it was shown that decision heuristics and confidence judgements play important roles in the building of preferences. Based on a dual-process account of thinking, the study compared people who did well versus poorly on a series of decision heuristics and overconfidence judgement tasks. The two groups were found to differ with regard to their information search behaviour in introduced multiattribute choice tasks. High performers on the judgemental tasks were less influenced in their decision processes by numerical information format (probabilities vs. frequencies) compared to low performers. They also looked at more attributes and spent more time on the multiattribute choice tasks. The results reveal that performance on decision heuristics and overconfidence tasks has a bearing both on heuristic and analytic processes in multiattribute decision making.     

Measurement foundations for multiattribute psychophysical theories based on first order polynomials

The class of first order polynomial measurement representations is defined, and a method for proving the existence of such representations is described. The method is used to prove the existence of first order polynomial generalizations of expected utility theory, difference measurement, and additive conjoint measurement. It is then argued that first order polynomial representations provide a deep and far reaching characterization of psychological invariance for subjective magnitudes of multiattributed stimuli. To substantiate this point, two applications of first order polynomial representation theory to the foundations of psychophysics are described. First, Relation theory, a theory of subjective magnitude proposed by Shepard (Journal of Mathematical Psychology, 1981, 24, 21–57) and Krantz (Journal of Mathematical Psychology, 1972, 9, 168–199), is generalized to a theory of magnitude for multiattributed stimuli. The generalization is based on a postulate of context invariance for the constituent uniattribute magnitudes of a multiattribute magnitude. Second, the power law for subjective magnitude is generalized to a multiattribute version of the power law. Finally, it is argued that a common logical pattern underlies multiattribute generalizations of psychological theories to first order polynomial representations. This abstract pattern suggests a strategy for theory construction in multiattribute psychophysics.     

Positioning models in marketing: Toward a normative-integrated model

In this paper we argue that the optimal use of positioning models employed in marketing communication (e.g., multiattribute attitude models, self-image congruence models) is a function of situational factors such as product utilitarianism and product value-expressiveness. A normative model is introduced to show how positioning models can be used effectively.     

Random error in additive decompositions of multiattribute utility

Decomposition is a common strategy for dealing with the complexity of multiattribute decision problems. A cognitively demanding task is broken down into tasks requiring simpler, perhaps easier, judgments which can then be aggregated. But individual judgments can be inconsistent in systematic or random fashion and when aggregated there is the possibility of propagation of this inconsistency. In this paper inconsistency in the form of random error is investigated in the context of additive decomposition of multiattribute utility. The process of aggregation of random error is studied and a comparison made with random error in holistic estimates of multiattribute utility. Conditions under which decomposition improves the consistency of the multiattribute utility estimate are presented and discussed.     

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.     

On the measurement of preferences in the analytic hierarchy process

In this paper we apply multiattribute value theory as a framework for examining the use of pairwise comparisons in the analytic hierarchy process (AHP). On one hand our analysis indicates that pairwise comparisons should be understood in terms of preference differences between pairs of alternatives. On the other hand it points out undesirable effects caused by the upper bound and the discretization of any given ratio scale. Both these observations apply equally well to the SMART procedure which also uses estimates of weight ratios. Furthermore, we demonstrate that the AHP can be modified so as to produce results similar to those of multiattribute value measurement; we also propose new balanced scales to improve the sensitivity of the AHP ratio scales. Finally we show that the so-called supermatrix technique does not eliminate the rank reversal phenomenon which can be attributed to the normalizations in the AHP. © 1997 John Wiley & Sons, Ltd.     

Probability summation model for heterochromatic luminance additivity failure at absolute visual threshold

Vos and Wabraven recently derived a new set of foveal primaries. These primaries, in conjunction with the revised zone fluctuation theory of color vision, account for more psychophysical color vision data than any other model to date. The new primaries depend heavily on Abney’s law, and their success has been challenged by Guth’s observations of additivity failure at absolute threshold. The present paper develops a probability summation model which accounts for threshold additivity failure within the framework of Vos’s and Walraven’s assumptions.     

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.     

Modelling and describing human judgement processes: The multiattribute evaluation case

In this article we describe research methods that are used for the study of individual multiattribute evaluation processes. First we explain that a multiattribute evaluation problem involves the evaluation of a set of alternatives, described by their values on a number of alternatives. We discuss a number of evaluation strategies that may be applied to arrive at a conclusion about the attractiveness or suitability of the alternatives, and next introduce two main research paradigms in this area, structural modelling and process tracing. We argue that the techniques developed within these paradigms all have their advantages and disadvantages, and conclude that the most promising technique to detect the true nature of the evaluation strategy used by a judge seems to be the analysis of verbal protocols. At the same time we think it is wise not to rely on just one technique, but to use a multimethod approach to the study of multiattribute evaluation processes whenever that is possible.     

Simple matrix methods for analyzing diffusion models of choice probability, choice response time, and simple response time

Diffusion processes (e.g., Wiener process, Ornstein-Uhlenbeck process) are powerful approaches to model human information processes in a variety of psychological tasks. Lack of mathematical tractability, however, has prevented broad applications of these models to empirical data. This tutorial explains step by step, using a matrix approach, how to construct these models, how to implement them on a computer, and how to calculate the predictions made by these models. In particular, we present models for binaries choices for unidimensional and multiattribute choice alternatives; for simple reaction time tasks; and for three alternatives choice problems.     

Random error in holistic evaluations and additive decompositions of multiattribute utility — an empirical comparison

This paper details the results of an empirical investigation of the random errors associated with decomposition estimates of multiattribute utility. In a riskless setting, two groups of subjects were asked to evaluate multiattribute alternatives both holistically and with the use of an additive decomposition. For one group, the alternatives were described in terms of three attributes, and for the other in terms of five. Estimates of random error associated with the various elicitations (holistic, single-attribute utility, scaling constants, or weights) were obtained using a test-retest format. It was found for both groups that the additive decomposition had significantly smaller levels of random error than the holistic evaluation. However, the number of attributes did not seem to make a significant difference to the amount of random error associated with the decomposition estimates. The levels of error found in the various elicitations were consistent with theoretical bounds that have recently been proposed in the literature. These results show that the structure imposed on the problem through decomposition results in measurable improvement in quality of the multiattribute utility judgements, and contribute to a greater understanding of the decomposition method in decision analysis.     

Anticipations of Effort and Accuracy in Multiattribute Choice

Research on multiattribute choice strategies suggests that decision makers adapt to variations in task features by switching decision strategies. One perspective is that strategy selection is determined by a trade-off between the goals of minimizing effort and maximizing accuracy. However, since strategy selection is based on anticipated effort and accuracy, empirical evidence is needed on the nature of these anticipations. An experiment examines how well decision makers anticipate the influence of two particular task features, information display organization and number of alternatives, on effort and accuracy. Subjects chose the best alternative from sets of loan applications described by six relevant attributes. Prior to each choice, they prospectively judged the effort required and the accuracy that would result. Results suggest that anticipated consequences deviate substantially from experienced consequences for both effort and accuracy. Neither task experience nor explicit feedback brought anticipations and experiences closer together. These findings underscore the importance of anticipation judgments in multiattribute choice.     

A feature model of immediate memory

A feature model of immediate memory is presented, and simulations are described. List items are characterized as multiattribute vectors that can be selectively overwritten by subsequent external events and by the ongoing stream of internal activity. Degraded primary memory vectors are compared with intact secondary memory vectors, and retrieval likelihood is computed as the ratio of similarities. The model is shown to account for the major modality-based phenomena of the immediate serial recall literature, including modality-based temporal grouping effects and the negative effects of phonological similarity.     

Are there limits to binaural additivity of loudness?

In recent years there has been notable interest in additive models of sensory integration. Binaural additivity has emerged as a main hypothesis in the loudness-scaling literature and has recently been asserted by authors using an axiomatic approach to psychophysics. Restrictions of the range of stimuli used in the majority of former experiments, and inherent weaknesses of the axiomatic study by Levelt, Riemersma, and Bunt (1972) are discussed as providing reasons for the present investigation. A limited binaural additivity (LBA) model is proposed that assumes contralateral binaural inhibition for interaural intensity differences that exceed a critical level. Experimental data are reported for 12 subjects in a loudness-matching task designed to test the axioms of cancellation and of commutativity, both necessary to the existence of strict binaural additivity. In a 2 X 2 design, frequencies of 200 Hz and 2 kHz were used, and mean intensity levels were 20 dB apart. Additivity was found violated in 33 out of 48 possible tests. The LBA model is shown to predict the systematic nonadditivity in the loudness judgments and to conform to results from other studies.