Additive Difference Model Research Articles

Intransitivity and transitivity of preferences: Dimensional processing in decision making.

Transitive preference, that is, if you prefer apples to bananas and bananas to cherries, you also prefer apples to cherries, is a basic property of some influential rational choice models. Contrary to this, Tversky, in his seminal 1969 article, presented evidence of intransitive preferences in two contexts, one of which involved choices between simple monetary lotteries. While early replications corroborated his findings, more recent research cast doubt on the strength of evidence of intransitive preferences in this task. Here, from Tversky’s extended additive difference model we develop a simplified additive difference (SAD) model that corresponds to alternative dimensional processing strategies. This predicts transitive or intransitive preferences, depending on its parameter values. We review six replications of Tversky’s lottery task and fit variants of the model to the choice data. We estimate the SAD model’s parameters for each individual data set using maximum likelihood estimation, examine the goodness of fit of the model, and use likelihood ratio tests to evaluate specific variants. The model has a very good fit to most individual choice data sets reviewed, with many predictably violating weak stochastic transitivity. We also find that many transitive patterns correspond to the application of simple, one-dimensional “take the best” heuristics. The findings support the view that human decision making is often based on dimensional processing in such a way that evaluations of decision alternatives are relative to the set under consideration, resulting in intransitivity of preferences. (PsycInfo Database Record (c) 2020 APA, all rights reserved)

Ambiguous comparative judgment: Fuzzy set model and data analysis

Abstract: This paper proposed two types of fuzzy set models for ambiguous comparative judgments, which did not always hold transitivity and comparability properties. The first type of model was a fuzzy theoretical extension of the additive difference model for preference that was used to explain ambiguous preference strength. The second was a fuzzy logic model for explaining ambiguous preference in which preference strength was bounded, such as a probability measure. In both models, multi‐attribute weighting parameters and all attribute values were assumed to be asymmetric fuzzy L‐R numbers. For each model, a method of parameter estimation using fuzzy regression analysis was proposed. Numerical examples were also provided for comparison. Finally, the theoretical and practical implications of the proposed models were discussed.

‘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.

Measuring credibility of compensatory preference statements when trade-offs are interval determined

This paper studies how an overall fuzzy preference relation can be constructed in the compensatory context of the ‘simple additive difference model’, when imprecision on the trade-offs has to be taken into account. Three credibility indices of preferences are analysed and illustrated by a numerical example. Arguments are presented supporting the use of the third index, for which an interesting transitivity property (which was an open problem) is proved.

Additive differences and simple preference comparisons

Two axiomatizations of the multiattribute additive difference representation of preferences are presented. One uses topological assumptions to obtain a continuous representation; the other uses a more general algebraic structure. Both use only simple preference comparisons and require three or more attributes. Proofs of the representation theorems are based on recent work in nontransitive additive conjoint measurement. Specializations of the additive difference model for homogeneous product sets are also axiomatized. Contexts for these specializations include time streams and finite-states decision under uncertainty.

Efficient sets, decision heuristics, and single-peaked preferences

The origins of single-peaked preferences in multiattribute choice behavior are analyzed. The starting point is Coombs and Avrunin's ( Psychological Review, 1977, 84, 216–230; Journal of Mathematical Psychology, 1977 , 16, 261–266) theory which explains single-peakedness as the joint effect of task and behavior characteristics. Whereas the task conditions—so-called efficient sets—are retained, and their importance is even more stressed, it is argued that Coombs and Avrunin's behavioral assumptions—classical marginally decreasing preference functions—may not be representative of actually displayed choice behavior. Rather, the information processing approach to decision research has shown that multiattribute choices are often based on simplifying heuristic rules which, due to their possible intransitivity, are inconsistent with Coombs and Avrunin's behavioral requirements. Tversky's ( Psychological Review, 1969 , 76, 31–48) additive difference model is used as an algebraic framework for analyzing such so-called attribute-based choice rules. Sufficient (and in the continuous case also necessary) conditions are presented which guarantee that the additive difference model will also be single-peaked on efficient sets. These conditions show that the behavioral principles yielding single-peaked preferences are far more general than Coombs and Avrunin assumed and include most of the empirically substantiated forms of attribute-based choice rules. Further, a necessary and sufficient condition is derived which insures that elimination rules, like the conjunctive rule, for instance, and generalizations thereof are single-peaked. Finally, making use of a result by Edgeworth, ( Mathematical Psychics, London, Paul, 1881 ), it is shown that efficient sets per se may originate in quite different ways. It is concluded that the phenomenon of single-peakedness is mainly a function of the task rather than of some specific behavior.

Lexicographic additive differences

Several authors have identified sets of axioms for a preference relation ≻ on a two-factor set A × X which imply that ≻ can be represented by specific types of numerical structures. Perhaps the two best-known of these are the additive representation, for which there are real valued functions f A on A and f X on X such that ( a, x) ≻ ( b, y) if and only if f A ( a) + f X ( x) > f A ( b) + f X ( y), and the lexicographic representation which, with A as the dominant factor, has ( a, x) ≻ ( b, y) if and only if f A ( a) > f A ( b) or { f A ( a) = f A ( b) and f X ( x) > f X ( y)}. Recently, Duncan Luce has combined the additive and lexicographic notions in a model for which A is the dominant factor if the difference between a and b is sufficiently large but which adheres to the additive representation when the difference between a and b lies within what might be referred to as a lexicographic threshold. The present paper specifies axioms for ≻ which lead to a numerical model which also has a lexicographic component but whose local tradeoff structure is governed by the additive-difference model instead of the additive model. Although the additive-difference model includes the additive model as a special case, the new lexicographic additive-difference model is not more general than Luce's model since the former has a “constant” lexicographic threshold whereas Luce's model has a “variable” lexicographic threshold. Realizations of the new model range from the completely lexicographic representation to the regular additive-difference model with no genuine lexicographic component. Axioms for the latter model are obtained from the general axioms with one slight modification.

A theoretical framework for testing the additive difference model for dissimilarities data: Representing gambles as multidimensional stimuli

Implicit within the acceptance of most multidimensional scaling models as accurate representations of an individual's cognitive structure for a set of complex stimuli, is the acceptance of the more general Additive Difference Model (ADM). A theoretical framework for testing the ordinal properties of the ADM for dissimilarities data is presented and is illustrated for a set of three-outcome gambles. Paired comparison dissimilarity judgments were obtained for two sets of gambles. Judgments from one set were first analyzed using the ALSCAL individual differences scaling model. Based on four highly interpretable dimensions derived from this analysis, a predicted set of dimensions were obtained for each subject for the second set of gambles. The ordinal properties of the ADM necessary for interdimensional additivity and intradimensional subtractivity were then tested for each subject's second set of data via a new computer-based algorithm, ADDIMOD. The tests indicated that the ADM was rejected. Although violations of the axioms were significantly less than what would be expected by chance, for only one subject was the model clearly supported. It is argued that while multidimensional scaling models may be useful as data reduction techniques, they do not reflect the perceptual processes used by individuals to form judgments of similarity. Implications for further study of multidimensional scaling models are offered and discussed.

Cortical potential correlates of selective attention in multidimensional scaling

Cortical evoked potentials were recorded from the vertex of subjects performing a similarity rating task. Proximity data were analysed according to INDSCAL (Carroll and Chang, 1970). Brain activity was averaged separately for different stimulus attributes and different stages of the judgement process. The amplitudes of the two positive deflections of the averaged evoked potential, peaking around 160 and 330 msec after stimulus onset respectively, were systematically influenced by two factors: The weight, which the evoking attribute received in the overall dissimilarity judgement, and the position of the evoking attribute in the stimulus train. The results support the hypothesis that differences in the perception of similarity are due to differences in attentional set. Reference is made to the ‘additive-difference model’ of scaling behaviour (Tversky and Krantz, 1970) and Broadbent's (1970) distinction between stimulus set and response set attention.

ADDIMOD: A Program to Test the Axioms of the Additive Difference Model for Multidimensional Scaling

ADDIMOD: A Program to Test the Axioms of the Additive Difference Model for Multidimensional Scaling

Strong ordinal properties of an additive model for the sequential processing of probabilistic information

A general theory of probabilistic information processing, together with its formal representation as an additive-difference model, is specialized to account for behavior when information is observed sequentially. Both the basic assumptions and the derived additive model are testable separately with ordinal data from individual subjects. Furthermore, scale values for the sequential information can be derived with the ordinal data and the model. Eight subjects were run extensively in probabilistic information processing tasks, in each problem sequentially receiving two samples of information in order to decide which of two hypotheses was more likely correct. The additive model was supported by the data of all but one subject; for those subjects described by this model, the prior assumptions received moderate to strong support. The pattern of results was employed to discuss additive versus averaging processes, sequential effects, and the general processing theory.

Using conjoint-measurement models to investigate a theory about probabilistic information processing

A theory of how opinions are formed and revised on the basis of probabilistic evidence is presented. The decision maker is viewed as a limited information processor who attends to the dimensions of a sample of equivocal information in a sequential fashion, beginning with the most salient dimension and continuing in decreasing order of salience. The contribution of each dimension level to the final opinion depends on the strength of association between that level and each of the two hypotheses under consideration. This theory is represented formally as an additive-difference model, various special cases of which correspond to other algebraic models in the literature. Two of the special cases were empirically investigated under varying levels of experience and payoffs, using the techniques of conjoint-measurement theory and ordinal data from individual subjects. The data provided reasonably good support for the models under all conditions investigated, and in addition showed interesting effects on salience and on processing of the independent variables.

The additive difference model and a metric and an ordering on sets

The additive difference model and a metric and an ordering on sets

AN INVESTIGATION OF INTRANSITIVE CHOICE BEHAVIOR

ABSTRACTIndividuals are often asked to make choices from all possible pairs of a set of stimuli. For each pair they may have to select the alternative which is heavier, more preferable, more favorable to Blacks, etc. Intransitive or circular choice behavior occurs when an individual selects stimulus A over B, B over C, and then C over A. Such a pattern of choices is referred to as a circular triad. Since most theories of choice assume transitivity as a basic principle of human choice behavior, explanation of the occurrence of intransitivities is a crucial problem in the behavioral sciences.Three types of explanation have been posited for the existence of circular choices. One suggests that the subject is not taking the necessary effort to make careful choices; that is, intransitivities occur due to the individual's sloppiness or carelessness. Since subjects typically seem to make choices in a conscientious manner, this outlook is not very plausible. Another explanation is that intransitivities arise from the close proximity of stimuli on an underlying linear continuum. Thus, even though the subject is conscientious, the stimuli may in some cases be too similar to distinguish. This explanation implies that there is really no difference between intransitivity and inconsistency ‐ the reversal of responses by a subject over two administrations of the same pairs of choices.The third type of explanation concerns the development of a choice model which might account for intransitive behavior. Some attempts at such a model focus on the multidimensional nature of the stimuli, circularities being caused by the way in which a subject combines ratings on the separate dimensions to yield an overall choice between the pair of alternatives. Specifically, the additive difference model developed by Tversky (1969), drawing on earlier work by May (1954) and Morrison (1962) suggests that stable circular triads (i.e., intransitivities occurring in exactly the same way) would be expected from two administrations of the same choice pairs.Accordingly, five questionnaires were administered to a group of subjects on two separate occasions. In each case the subjects for the second administration were the same individuals who had completed the questionnaire the first time, so that data could be obtained on the responses of the same subject to the same pairs of choices on two distinct administrations. The principal concerns were the following: To find evidence for the existence of stable (i.e., over testings) circular triads. To investigate the appropriateness of the additive difference model, if such stable circularities are found. If such evidence is not found, to investigate the intrasitivity‐as‐inconsistency explanation by exploring the relationship between circular triads and changes in response to individual items over the two administrations. With the exception of one subject on one questionnaire, no such evidence for stable intransitivities was found.Consequently, the inconsistency explanation was investigated by a factor analysis of nine variables derived from the subjects' choices. These variables were measures of consistency, transitivity and discrimination among stimuli. Results of this analysis suggest that the assumption of circular choices as synonymous with inconsistency is upheld for three of the five questionnaires.