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.     

Stochastic transitivity and the unidimensional control of choice

A concurrent-chains paradigm was used to test three models of preference for schedules of reinforcement. Each model assumes choice to be a monotonic function of values on a single dimension (“simple scalability”). For choices involving variable-interval and fixed-interval schedules, this dimension is assumed to be an harmonic average of the interreinforcement intervals; for choices involving fixedinterval and fixed-ratio schedules, the dimension is the time to reinforcement irrespective of response rate; for choices involving different combinations of rate and duration of reinforcement, the dimension is the rate of “reinforcement-time”, or the product of the two parameters. Each model implies a strong form of transitivity (“functional equivalence”), which held for half the subjects (that half exclusively sensitive to the reinforcement variable specified by the model). An alternative unidimensional model which implies a weaker form of transitivity (“weak binary utility model”) was consistent with the preferences of all subjects. This model states that a single factor controls the direction of preferences but not necessarily the exact choice probability, as simple scalability assumes.     

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.     

Prediction outcome probabilities as determinants of choice reaction time

In a two-stimulus two-response choice reaction time (RT) task in which Ss made stimulus predictions, the probability of a correct prediction was manipulated between Ss. The magnitude of the difference in RT to correctly and incorrectly predicted stimuli (i.e., the prediction outcome effect) was an increasing function of the probability of a correct prediction This finding was primarily due to a reliable decrease in RT to correctly predicted stimuli as the probability of a correct prediction increased, since RT to incorrectly predicted stimuli was not affected by prediction outcome probability. These results were interpreted as partially supporting a continuous expectancy notion which involves facilitory and inhibitory mechanisms winch are differentially influenced by the probability of a correct prediction.     

Binary choice, subset choice, random utility, and ranking: A unified perspective using the permutahedron

The d-permutahedron Π_d−1⊂R^d is defined as the convex hull of all d-dimensional permutation vectors, namely, vectors whose components are distinct values of a d-element set of integers [d]≡{1,2,…,d}. By construction, Π_d−1 is a convex polytope with d! vertices, each representing a linear order (ranking) on [d], and has dimension dim(Π_d−1)=d−1. This paper provides a review of some well-known properties of a permutahedron, applies the geometric-combinatoric insights to the investigation of the various popular choice paradigms and models by emphasizing their inter-connections, and presents a few new results along this line.Permutahedron provides a natural representation of ranking probability; in fact it is shown here to be the space of all Borda scores on ranking probabilities (also called “voters profiles” in the social choice literature). The following relations are immediate consequences of this identification. First, as all d! vertices of Π_d−1 are equidistant to its barycenter, Π_d−1 is circumscribed by a sphere S^d−2 in a (d−1)-dimensional space, with each spherical point representing an equivalent class of vectors whose components are defined on an interval scale. This property provides a natural expression of the random utility model of ranking probabilities, including the condition of Block and Marschak. Second, Π_d−1 can be realized as the image of an affine projection from the unit cube of dimension d(d−1)/2. As the latter is the space of all binary choice vectors describing probabilities of pairwise comparisons within d objects, Borda scores can be defined on binary choice probabilities through this projective mapping. The result is the Young's formula, now applicable to any arbitrary binary choice vector. Third, Π_d−1 can be realized as a “monotone path polytope” as induced from the lift-up of the projection of the cube onto the line segment [0,d]⊂R¹. As the 2^d vertices of the d-cube are in one-to-one correspondence to all subsets of [d], a connection between the subset choice paradigm and ranking probability is established. Specifically, it is shown here that, in the case of approval voting (AV) with the standard tally procedure (Amer. Pol. Sci. Rev. 72 (1978) 831), under the assumption that the choice of a subset indicates an approval (with equal probability) of all linear orders consistent with that chosen subset, the Brams-Fishburn score is then equivalent to the Borda score on the induced profile. Requiring this induced profile (ranking probability) to be also consistent with the size-independent model of subset choice (J. Math. Psychol. 40 (1996) 15) defines the “core” of the AV Polytope. Finally, Π_d−1 can be realized as a canonical projection from the so-called Birkhoff polytope, the space of rank-position probabilities arising out of the rank-matching paradigm; thus Borda scores can be defined on rank-position probabilities. To summarize, the many realizations of a permutahedron afford a unified framework for describing and relating various ranking and choice paradigms.     

On estimating inter-subject variability of choice probabilities under observability constraints

Methods are presented for estimating inter-subject variability of the probability of a given event defined in terms of subject's behavior (e.g., probability of a given choice in a discrimination experiment). The constraints consist of using no more than two independent observations for each subject. Estimators are provided for assessing the inter-subject “variance” of the analyzed probabilities; also, a method is given for testing the hypothesis that the average probability is the same for two groups of subjects.     

Binary choice reaction time as a criterion of motor equivalence: Further evidence

In a previous study Heuer (1982) found shorter binary choice RTs for finger movements of the right hand when the alternative movement with the left hand was of the same form that in case of different forms. Contrary to expectation there was no corresponding effect if the fingers used were the same of different ones. This result was replicated for right hand RTs as well as left hand RTs and for male and female subjects, using a different experimental design. It is discussed in respect to different preparatory processes in the foreperiod.     

Global and local control of choice behavior by cyclically varying outcome probabilities

Subjects asked to simulate decision makers in a competitive bidding situation chose repeatedly between two alternatives; reward probability varied according to a sine wave function of time for one alternative but was held constant over time for the other. Learning functions for choice probability exhibited the wavelike pattern predicted by a statistical learning model. However, on later transfer trials, when success probability was independent of subjects' choices, their choice behavior continued to follow a wavelike function rather than approaching the constant .5 level predicted by the learning model. A possible basis in memory for the transfer performance was revealed in subjects' sketches of the remembered pattern of variation in reward probabilities. It is concluded that choice performance is controlled by a mixture of local feedback, in a manner described by the learning model, and more global information encoded in an abstract memory representation, with the balance of the influence shifting toward the latter when local feedback becomes uninformative.     

Binary choice reaction time as a criterion of motor equivalence

In a binary choice situation the two alternative movements may vary in the degree in which they are equivalent in respect to controlling processes and/or structures. With a higher degree of motor equivalence a shorter RT is to be expected. A series of five experiments shows that a movement of a finger of one hand has a shorter latency if the alternative movement with the other hand is of the same form than in case of different forms. There is no evidence of corrsponding effect of using the same or different fingers with both hands. These results indicate that programming of the form of a movement is at least partly independent of the muscles involved in movement execution.     

A note on Putnam and the transitivity of the real

Professor Putnam's argument that future happenings are already real is clarified and examined. It is argued that, contrary to other claims in the literature, Putnam's premises are consistent, but that the field of the relation R on which he bases his conclusion is such that the conclusion does not in fact follow.     

On the basis of early transitivity judgments

This research had two aims. The first was to test three explanations of performance on N-term series tasks by young children: the labeling model of B.DeBoysson-Bardies and K. O'Regan (1973), Nature (London), 246, 531–534, the sequential-contiguity model of L. Breslow (1981, Psychological Bulletin, 89, 325–351), and the ordered array or image model of C. A. Riley and T. Trabasso (1974, Journal of Experimental Child Psychology, 17, 187–202). In the first experiment, 5-year-old children were taught additional premises which would interfere with labeling and sequential-contiguity processes, but not with forming an ordered array. Reasoning performance was essentially comparable to previous results with the paradigm, thus supporting the ordered array model. The second aim was to reexamine children's ability to learn sets of premises which can be assembled into an ordered array, since there was reason to believe that previous studies had created false positives. In the second experiment, 3- to 7-year-old children were taught either overlapping (a > b, b > c, …) or nonoverlapping (a > b, c > d, …) premises. Overlapping premises can be integrated into an ordered array (a, b, c, d, e), but nonoverlapping premises cannot. However, the overlapping condition proved more difficult, and the success rate for preschoolers ($\text{3}\text{1}\text{2}\text{-}\text{to 4}\text{1}\text{2}\text{-year-olds}$) was of zero order. This raises doubts about their ability to learn a set of premises of the kind required for transitive inference. These doubts were strengthened by the third experiment which showed that when premises were not presented in serial order, preschool ($\text{3}\text{1}\text{2}\text{-}\text{to 4}\text{1}\text{2}\text{-year-old}$) children could not learn the premises of an N-term series task.     

Interpersonal comparison of subjective probabilities: Toward translating linguistic probabilities

Interpersonal variability in understanding linguistic probabilities can adversely affect decision making. Using the fact that everyone judges canonical probability events similarly in a manner consistent with axiom systems that yield a probability measure, we developed and tested a method for comparing the meanings of probability phrases across individuals. An experiment demonstrated that despite extreme heterogeneity in participants' linguistic probability lexicons, interpersonal similarity in phrase meaning is well predicted by phrase rank order within the lexicons. Thus, equally ranked phrases have similar meanings, and individual differences in linguistic probabilities may simply be explained by the phrases people use at each rank.     

The varieties of religious development in adulthood: a longitudinal investigation of religion and rational choice

The authors used growth mixture models to study religious development during adulthood (ages 27-80) in a sample of individuals who were identified during childhood as intellectually gifted. The authors identified 3 discrete trajectories of religious development: (a) 40% of participants belonged to a trajectory class characterized by increases in religiousness until midlife and declines in later adulthood; (b) 41% of participants belonged to a trajectory class characterized by very low religiousness in early adulthood and age-related decline; and (c) 19% of participants belonged to a trajectory class characterized by high religiousness in early adulthood and age-related increases. Gender, strength of religious upbringing, number of children, marrying, and agreeableness predicted membership in the trajectory classes. Results were largely consistent with the rational choice theory of religious involvement.     

Binary choice reaction time as a function of the relationship between durations and forms of responses

In a choice between responding with the left or right hand, some kinds of differences between the movements increase RT (Reaction Time) while others do not. Of the first kind are differences in form, while differences in the finger used are of the latter kind. In previous experiments differences in form were confounded with differences in duration. Since there is some indication that a difference in duration is sufficient to lengthen RT, both characteristics were varied separately. It turned out that a difference in form (duration being constant) has essentially the same effects as a difference in duration (form being constant): Mean RT is longer, variability of RT and MT (movement time) is larger, and frequency of choice errors is smaller than in choice between identical movements. These effects, which seem to be associated with choice between movements of different temporal patterns, are interpreted in terms of advance specification of movement parameters. Additional results on the relationship between response duration and RT suggest that RT does not depend on duration (or velocity) per se, but on how much the duration deviates from quickest performance.