The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data

In analogy with the wandering vector model, a probabilistic multidimensional unfolding model is proposed for representing paired comparisons data. It is shown that contrary to other stochastic multidimensional unfolding models, the present model does not imply strong stochastic transitivity, only moderate stochastic transitivity. A maximum likelihood parameter estimation procedure is developed and an illustrative application is presented.     

Evidence against Rank-Dependent Utility Theories: Tests of Cumulative Independence,Interval Independence,Stochastic Dominance,and Transitivity

This study tests between two modern theories of decision making. Rank- and sign-dependent utility (RSDU) models, including cumulative prospect theory (CPT), imply stochastic dominance and two cumulative independence conditions. Configural weight models, with parameters estimated in previous research, predict systematic violations of these properties for certain choices. Experimental data systematically violate all three properties, contrary to RSDU but consistent with configural weight models. This study also tests whether violations of stochastic dominance can be explained by violations of transitivity. Violations of transitivity may be evidence of a dominance detecting mechanism. Although some transitivity violations were observed, most choice triads violated stochastic dominance without violating transitivity. Judged differences between gambles were not consistent with the CPT model. Data were not consistent with the editing principles of cancellation and combination. The main findings are interpreted in terms of coalescing, the principle that equal outcomes can be combined in a gamble by adding their probabilities. RSDU models imply coalescing but configural weight models violate it, allowing configural weighting to explain violations of stochastic dominance and cumulative independence.     

The exchangeable multinomial model as an approach to testing deterministic axioms of choice and measurement

The multinomial (Dirichlet) model, derived from de Finetti's concept of exchangeability, is proposed as a general Bayesian framework to test axioms on data, in particular, deterministic axioms characterizing theories of choice or measurement. For testing, the proposed framework does not require a deterministic axiom to be cast in a probabilistic form (e.g., casting deterministic transitivity as weak stochastic transitivity). The generality of this framework is demonstrated through empirical tests of 16 different axioms, including transitivity, consequence monotonicity, segregation, additivity of joint receipt, stochastic dominance, coalescing, restricted branch independence, double cancellation, triple cancellation, and the Thomsen condition. The model generalizes many previously proposed methods of axiom testing under measurement error, is analytically tractable, and provides a Bayesian framework for the random relation approach to probabilistic measurement (J. Math. Psychol. 40 (1996) 219). A hierarchical and nonparametric generalization of the model is discussed.     

The Development of Children's Concepts of Hardness

This study was conducted for three purposes. First, we assessed the development of children's concepts of hardness as an intensive physical quantity. We hypothesized a U-shaped behavioral growth curve for children's judgments on tasks measuring intensive physical quantity, but the hypothesis was not confirmed. The justifications accompanying correct and incorrect judgments for hardness were similar to those where U-shaped behavioral growth was found for other intensive physical quantities (temperature and sweetness). The second purpose was to determine how children solve seriation and transitivity tasks where hardness was the content. More older than younger children solved these tasks, and the transitivity task was correctly solved after the seriation task was correctly solved. One pattern of solutions for transitivity and intensive physical quantity allowed for differentiation between younger and older children who solved the intensivity tasks correctly. The third purpose was to determine whether children think that objects' hardnesses are labile (i.e., hard objects can become soft and vice versa) by simply adding more of that substance. More younger than older children thought that this was true     

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.     

Laterality effects in risk preference: A test of portfolio theory

A central assumption in portfolio theory of risk preference is that for games with expected value constant the preference ordering of an individual should exhibit a single peaked preference function. This assumption was tested by determining the level of stochastic transitivity satisfied by triples of pairwise preference proportions among eight gambles. In only a small percentage of cases were the predicted levels of stochastic transitivity not satisfied. It was concluded that the results provided strong support for portfolio theory.     

Making trade-offs: a probabilistic and context-sensitive model of choice behavior

The stochastic difference model assumes that decision makers trade normalized attribute value differences when making choices. The model is stochastic, with choice probabilities depending on the normalized difference variable, d, and a decision threshold, delta. The decision threshold indexes a person's sensitivity to attribute value differences and is a free estimated parameter of the model. Depending on the choice context, a person may be more or less sensitive to attribute value differences, and hence delta may be used to measure context effects. With proportional difference used as the normalization, the proportional difference model (PD) was tested with 9 data sets, including published data (e.g., J. L. Myers, M. M. Suydam, & B. Gambino, 1965; A. Tversky, 1969). The model accounted for individual and group data well and described violations of stochastic dominance, independence, and weak and strong stochastic transitivity.     

Testable conditions for triads of paired comparison choices

Forty-two choice models, each representing stimuli by one-dimensional probability distributions, are obtained by relaxing the assumptions of Thurstone's Case V Law of Comparative Judgment. The models which imply or fail to imply each of nine testable probabilistic conditions are determined. Stochastic transitivity is vulnerable in most of these models. The results suggest discarding weak stochastic transitivity, and in its place using the conjunction of weak stochastic transitivity and the triangular condition. However, unless it is possible to predict which stimuli will produce violations of the conditions, none of the conditions can be rejected on the basis of too frequent intransitive triads of choices.I am grateful to the referees, and to J. H. Griesmer and C. H. Coombs for a number of helpful criticisms and comments. J. H. Griesmer suggested a proof of Theorem 9, and a referee suggested a more direct proof of Theorem 5. A proof of Theorem 5 and the result concerning a maximally intransitive judge were given in [18].     

Mixed models,random utilities,and the triangle inequality

A new probabilistic binary choice model, which we call the mixed model is proposed. This model is used to give an interpretation of probabilistic choice behavior in terms of complete and partial orders. The model is used as a guide to disprove Jacob Marschak's conjecture that the binary random utility model is equivalent to the triangle inequality. Another result is presented concerning the relation between moderate stochastic transitivity and the triangle inequality.     

Violations of stochastic transitivity on concurrent chains: Implications for theories of choice

The concurrent-chains procedure has been used to measure how choice depends on various aspects of reinforcement, such as its delay and its magnitude. Navarick and Fantino (1972, 1974, 1975) have found that choice in this procedure can violate the condition of stochastic transitivity that is required if a unidimensional scale for reinforcements is to be possible. It is shown in this paper that two simple unidimensional models of choice on concurrent chains can produce violations of stochastic transitivity. It is argued that such violations may result from the complex contingencies of the concurrent-chains procedure.     

Metacognition in psychophysical judgment: an unfolding view of comparative judgments of mental workload.

An experiment is reported in which it was found that when subjects were required to indicate which of two visual extents was more difficult to categorize as "long" or "short," they executed these categorizations and then measured the distance of the representation of each stimulus from the long-short category boundary; the stimulus nearer the boundary was judged to be the more difficult. When they were requested to indicate which was easier to categorize, they selected the alternative that was farther. Coombs's theory of data (1952, 1964) and his unfolding theory of preferential choice (1950, 1964) provided the conceptualization of metacognition in this psychophysical task context. Strong support for the probabilistic version of unfolding theory was obtained from the observed selective effects of laterality on the levels of stochastic transitivity attained for various classes of triples and the reliably longer times for comparisons with bilateral pairs than with unilateral pairs. The semantic congruity effects obtained, together with the changes in the form of the relationship between probability and response time as a function of practice, can be best accounted for by an evidence accrual theory in which the distances from the active reference point are measured and compared with a criterion on each evidence accrual. No support is provided for the view that propositionally based semantic "ease"-"difficulty" codes serve as the basis for these metacognitive comparative judgments of ease and difficulty.     

Metacognition in psychophysical judgment: An unfolding view of comparative judgments of mental workload

An experiment is reported in which it was found that when subjects were required to indicate which of two visual extents was more difficult to categorize as “long” or “short,” they executed these categorizations and then measured the distance of the representation of each stimulus from the long-short category boundary; the stimulus nearer the boundary was judged to be the more difficult. When they were requested to indicate which was easier to categorize, they selected the alternative that was farther. Coombs’s theory of data (1952, 1964) and his unfolding theory of preferential choice (1950, 1964) provided the conceptualization of metacognition in this psychophysical task context. Strong support for the probabilisitic version of unfolding theory was obtained from the observed selective effects of laterality on the levels of stochastic transitivity attained for various classes of triples and the reliably longer times for comparisons with bilateral pairs than with unilateral pairs. The semantic congruity effects obtained, together with the changes in the form of the relationship between probability and response time as a function of practice, can be best accounted for by an evidence accrual theory in which the distances from the active reference point are measured and compared with a criterion on each evidence accrual. No support is provided for the view that propositionally based semantic “ease”- “difficulty” codes serve as the basis for these metacognitive comparative judgments of ease and difficulty.     

On the importance of distinguishing between within- and between-subject effects in intransitive intertemporal choice

Most probabilistic paired comparison models treat inconsistent choices as caused by independent and random errors in the pairwise judgments. In this paper, we argue that this assumption is too restrictive for the analysis of paired comparison data obtained from multiple judges when transitivity violations are systematic. We present a new framework that contains the random error assumption as a special case but also allows for systematic changes in an option's utility assessments over the pairwise comparisons. Accounting for both between- and within-judge sources of variability, we demonstrate in an application on intertemporal choice that the proposed framework can capture systematic transitivity violations as well as individual taste differences.     

Contrasting stochastic and support theory accounts of subadditivity

Numerous studies have found that likelihood judgment typically exhibits subadditivity in which judged probabilities of events are less than the sum of judged probabilities of constituent events. Whereas traditional accounts of subadditivity attribute this phenomenon to deterministic sources, this paper demonstrates both formally and empirically that subadditivity is systematically influenced by the stochastic variability of judged probabilities. First, making rather weak assumptions, we prove that regressive error (or variability) in mapping covert probability judgments to overt responses is sufficient to produce subadditive judgments. Experiments follow in which participants provided repeated probability estimates. The results support our model assumption that stochastic variability is regressive in probability estimation tasks and show the contribution of such variability to subadditivity. The theorems and the experiments focus on within-respondent variability, but most studies use between-respondent designs. Numerical simulations extend the work to contrast within- and between-respondent measures of subadditivity. Methodological implications of all the results are discussed, emphasizing the importance of taking stochastic variability into account when estimating the role of other factors (such as the availability bias) in producing subadditive judgments.     

Violations of transitivity: Implications for a theory of contextual choice

Violations of strong stochastic transitivity in concurrent-chains choice were first reported by Navarick and Fantino. In a series of articles, Navarick and Fantino concluded that neither a unidimensional model capable of predicting exact choice probabilities nor a fixed-variable equivalence rule was possible for the concurrent-chains procedure. I show that when choice is modeled contextually (i.e., when preference for a schedule is affected by factors other than the schedule itself, e.g., aspects of the alternative schedule), a unidimensional, exact-choice probability model is possible that both predicts the intransitivities reported by Navarick and Fantino and provides a fixed-variable equivalence rule for the concurrent-chains procedure. The contextual model is an extension of the generalized matching law and violates a key assumption underlying traditional choice models—simple scalability—because of (a) schedule interdependence and (b) bias from procedural contingencies. Therefore, strong stochastic transitivity cannot be expected to hold. Contextual scalability is analyzed to reveal a hierarchy of context effects in choice. Navarick and Fantino's intransitivities can be satisfactorily explained by bias. If attribute sensitivity is context dependent, however, and if there are similarity structures among choice alternatives, the contextual model is shown to be able to predict violations of ordinal preference. Therefore, it may be possible to formulate a deterministic, general psychophysical model of choice as a behavioral alternative to probabilistic, multidimensional choice theories.     

The Assessment of Children's Understanding of Inclusion Relations: Transitivity,Asymmetry, and Quantification

This study investigated the development of the understanding of class inclusion in children age 5, 7, and 9 years, whose performance on a qualitative class-inference task assessing their appreciation of the transitive and asymmetrical nature of inclusive relations within the animal domain was compared with their ability to make quantitative inferences in Piagetian class-inclusion problems. Results showed that, although 5-year-olds demonstrate a fair knowledge of the transitivity of inclusion relations, this notion is not fully understood until the age of 7. In contrast, the process leading to the acquisition of asymmetry understanding appears relatively slowly, and is not yet completed by 9 years of age. Whereas the ability to make qualitative inferences requiring the understanding of transitivity is acquired well before the ability to make quantitative inferences, making qualitative inferences requiring a knowledge of asymmetry is as difficult as making quantified judgments. Methodological considerations about the complementarity of the two kinds of tasks, along with the theoretical implications of our findings for Blewitt's developmental model of hierarchical knowledge, are discussed.     

Binding of multidimensional context information as a distinctive characteristic of remember judgments

This research investigated the cognitive processes underlying remember-know judgments in terms of contextual binding in multidimensional source memory. Stochastic dependence between the retrieval of different context attributes, which formed the empirical criterion of binding, was observed for remembered items but not for known items. Experiment 1 showed that the qualitative difference in the stochastic relation holds even if quantitative source-memory performance is equated for items with remember and know judgments. Experiment 2 generalized the findings to context information from different modalities, and Experiment 3 ruled out a spurious stochastic dependence due to interindividual differences. Supporting recent dual-process models of remember-know judgments, the findings show that remember and know judgments differ with respect to binding processes that correspond to episodic recollection.     

Choice theories for differentially comparable alternatives

Several binary choice theories represent choice probabilities as a function of a utility difference divided by a comparability index which is a distance metric. All such theories have the property of moderate stochastic transitivity.     

Modelling intransitive preferences: A random-effects approach

Current models for individual-level paired comparison data are based on the three assumptions that (1) pairwise judgments are independent, (2) the utility of an item remains invariant across trials, and (3) pair-specific variability can account for intransitive choice behaviour. All three assumptions seem strong and likely to be violated in empirical applications. This paper introduces a new framework for the analysis of paired comparison data which relaxes these three assumptions and considers the utilities associated with the same item across trials to be neither independent nor identical, but related. The proposed approach provides new insights about the reliability and consistency of paired comparison judgments and can account for systematic violations of transitivity. An application, based on a replication of Tversky's [(1969). Intransitivity of preference. Psychological Review, 76, 31-48] gamble study, illustrates the usefulness of the new approach in modelling both transitive and intransitive preferences.     

Note on lexicographic‐order preservation and stochastic dominance

This note concerns two issues left unresolved in our study of lexicographic‐order preservation and stochastic dominance in settings where preferences are represented by utility vectors, ordered lexicographically, and judgements emerge as matrices that premultiply utility vectors in expected utility sums. First, a generalization of the ‘Conjecture Σ’, which implied transitivity of a stochastic dominance relation under non‐vacuous resolution‐level information, is proved. Second, this paper comments on using resolution‐level information in higher as well as in first degree stochastic dominance analysis. Copyright © 1999 John Wiley & Sons, Ltd.