The optimal number of alternatives at a choice point with travel time considered

The problem of optimal design is considered for tasks that can be represented as n choice points with a_i alternatives per choice point. Examples are mazes or multiple choice tests. Some fixed time t′ elapses between choice points, say traveling from one to the other or reading a question, and additional time t is spent considering each alternative at the choice point. For fixed total time, the goal is to find the optimum number of choice points and the optimum number of alternatives at each choice point. In general, the longer the time spent between choice points, the greater the optimum number of alternatives at each point.     

Intensity and number of alternatives in hue identification: Piéron’s law and choice reaction time

Two experiments investigated the effects of intensity and the number of response alternatives in hue identification. Three questions were addressed. First, what is the relationship between these effects? If intensity and number of alternatives affect different processes in series, these factors would have additive effects on reaction time (RT). Other models predict a multiplicative interaction. These models assume that when a stimulus is presented, evidence accrues over time for each response alternative, and that when the evidence for a particular alternative exceeds its criterion, the response is made. A natural auxiliary assumption is that intensity affects the rate of evidence accrual, while the number of alternatives affects the criterion. The second question addressed is, how does choice RT change as a function of intensity? Piéron used a power function to describe the effect of intensity on simple RT. This paper considers three candidates for intensity’s effect on choice RT: the power, logistic, and Michaelis functions. Each candidate function was tested in an additive model and a multiplicative model. The best account was given by the power function in an additive model. The last question is, does each process produce an output that can be classified as correct or incorrect? Schweickert (1985) proposed that factors that selectively influenced such processes would have additive effects on log percent correct. Instead, an interaction was found. Perhaps the assumption that the process affected by intensity produces a single output is wrong.     

Quasi-dynamic choice models: Melioration and ratio invariance

There is continuing controversy about the behavioral process or processes that underlie the major regularities of free-operant choice such as molar matching and systematic deviations therefrom. A recent interchange between Vaughan and Silberberg and Ziriax concerned the relative merits of melioration, and a computer simulation of molecular maximizing. There are difficulties in evaluating theories expressed as computer programs because many arbitrary decisions must often be made in order to get the programs to operate. I therefore propose an alternative form of model that I term quasi-dynamic as a useful intermediate form of theory appropriate to our current state of knowledge about free-operant choice. Quasi-dynamic models resemble the game-theoretic analyses now commonplace in biology in that they can predict stable and unstable equilibria but not dynamic properties such as learning curves. It is possible to interpret melioration as a quasi-dynamic model. An alternative quasi-dynamic model for probabilistic choice, ratio invariance, has been proposed by Horner and Staddon. The present paper compares the predictions of melioration and ratio invariance for five experimental situations: concurrent variable-interval variable-interval schedules, concurrent variable-interval variable-ratio schedules, the two-armed bandit (concurrent random-ratio schedules), and two types of frequency-dependent schedule. Neither approach easily explains all the data, but ratio invariance seems to provide a better picture of pigeons' response to probabilistic choice procedures. Ratio invariance is also more adaptive (less susceptible to “traps”) and closer to the original expression of the law of effect than pure hill-climbing processes such as momentary maximizing and melioration, although such processes may come in to play on more complex procedures that provide opportunities for temporal discrimination.     

Estimating group size from human speech: Three's a conversation,but four's a crowd

Much previous research has examined various aspects of auditory processing, including the localization of sounds, and the influence of lexical and indexical information on language processing. In the present set of experiments we explored the ability of listeners to estimate the number of speakers in a group solely from the information in an auditory signal. The bound on accurately estimating the number of simultaneous speakers is 3. We suggest that subitization—the ability to estimate numerosity of visual and auditory elements without explicitly counting these elements—rather than the capacity of short-term memory, may underlie this limitation. The cognitive constraint on estimating the number of simultaneous speakers may have implications for a wide variety of seemingly unrelated psychological phenomena.     

Distinguishing between stochastic models of heterogeneity and contagion

Historically, the Poisson process has been the “benchmark” model for many stochastic processes. When event counts from a particular process fail to be described adequately by a Poisson distribution, a researcher may turn to generalized Poisson processes to model the empirical data more accurately. Two common generalizations are the (1) heterogeneous Poisson process, in which the process rate is a Gamma random variable, and (2) contagious Poisson process, in which the process rate depends linearly on the current state of the process. Paradoxically, both the heterogeneous and contagious processes yield the same theoretical distribution for the number of events that occur in a single interval of time. Consequently, distinguishing between these two using event counts can be difficult. This situation is discussed, first reviewing the models and distribution, then giving strategies for choosing between them using event count data, and demonstrating these techniques on episodes of hospitalization for schizophrenia.     

Discrete dynamic choice: An extension of the choice models of thurstone and luce

The present paper deals with the extension of two well-known static discrete choice theories to the dynamic situation in which individuals make choices at several points in (continuous) time. A dynamic version of Luce's Axiom, “independence from irrelevant alternatives”, is proposed and some of its implications are derived. In the static case Yellott (J. Math. Psych. 1977, 15, 109–146) and others have demonstrated that an independent random utility model generated from the extreme value distribution exp(?e^?ax?b) becomes equivalent to Luce's Axiom. Yellott also introduced an axiom called “invariance under uniform expansions of the choice set”, and he proved that within the class of random utility models with independent identically distributed utilities (apart from a location shift) this axiom is equivalent to Luce's Axiom. These results are extended to the dynamic situation and it is shown that if the utility processes are expressed by so-called extremal processes the corresponding choice model is Markovian. A nonstationary generalization is proposed which is a substantial interest in applications where the parameters of the choice process are influenced by previous choice experience or by time-varying exogenous variables. In particular, it is demonstrated that the nonstationary model is Markovian if and only if the joint choice probabilities at two points in time have a particular form. Thus, the paper provides a rationale for applying a specific class of Markov models as the point of departure when modelling mobility processes that involve individual discrete decisions over time.     

From eye movement to cognition: Toward a general framework of inference comment on Liechty et al., 2003

Liechty, Pieters & Wedel (2003) developed a hidden Markov Model (HMM) to identify the states of an attentional process in an advertisement viewing task. This work is significant because it demonstrates the benefits of stochastic modeling and Bayesian estimation in making inferences about cognitive processes based on eye movement data. One limitation of the proposed approach is that attention is conceptualized as an autonomous random process that is affected neither by the overall layout of the stimulus nor by the visual information perceived during the current fixation. An alternative model based on the input-output hidden Markov model (IOHMM; Bengio, 1999) is suggested as an extension of the HMM. The need for further studies that validate the HMM classification results is also discussed.     

The effects of response mode and importance on decision-making strategies: Judgment versus choice

Judgment and choice have not always been clearly distinguished, and there is little direct research on their effects on the decision process. As compared to choice, judgment is predicted to lead to more information searched, a less variable pattern of search, and a greater amount of interdimensional search. In essence, search processes that imply elimination by aspects will occur more often under choice than judgment. It is further argued that a number of variables may moderate this response mode effect. The potential moderator examined here is decision importance, which is predicted to produce analytical judgment processes prior to choice. Using an information search board, 104 subjects made decisions (choices or judgments) on candidates for residence hall advisor, under high or low decision importance. Response mode produced strong effects on all measures of the decision process. Importance had fewer effects; it affected the frequency of an intradimensional/variable search mode, but moderated the response mode effect only for amount of information searched. Given these results, several implications follow. Research which uses ajudgment response (e.g., policy capturing) should not be generalized to choice tasks. Also, requiring an explict judgment even when only the final choice matters may be a useful procedure. Further research is needed to examine the effects of response mode on additional outcomes, such as how the information is organized in memory and which alternative is actually selected.     

Multitasking versus multiplexing: Toward a normative account of limitations in the simultaneous execution of control-demanding behaviors

Why is it that behaviors that rely on control, so striking in their diversity and flexibility, are also subject to such striking limitations? Typically, people cannot engage in more than a few—and usually only a single—control-demanding task at a time. This limitation was a defining element in the earliest conceptualizations of controlled processing; it remains one of the most widely accepted axioms of cognitive psychology, and is even the basis for some laws (e.g., against the use of mobile devices while driving). Remarkably, however, the source of this limitation is still not understood. Here, we examine one potential source of this limitation, in terms of a trade-off between the flexibility and efficiency of representation (“multiplexing”) and the simultaneous engagement of different processing pathways (“multitasking”). We show that even a modest amount of multiplexing rapidly introduces cross-talk among processing pathways, thereby constraining the number that can be productively engaged at once. We propose that, given the large number of advantages of efficient coding, the human brain has favored this over the capacity for multitasking of control-demanding processes.     

On data-limited and resource-limited processes

This paper analyzes the effect on performance when several active processes compete for limited processing resources. The principles discussed show that conclusions about the interactions among psychological processes must be made with caution, and some existing assumptions may be unwarranted. When two (or more) processes use the same resources at the same time, they may both interfere with one another, neither may interfere with the other, or one may interfere with a second without any interference from the second process to the first. The important principles are that a process can be limited in its performance either by limits in the amount of available processing resources (such as memory or processing effort) or by limits in the quality of the data available to it. Competition among processes can affect a resource-limited process, but not a data-limited one. If a process continually makes preliminary results available even before it has completed all its operations, then it is possible to compute performance-resource operating characteristics that show how processes interact. A number of experiments from the psychological literature are examined according to these processing principles, resulting in some new interpretations of interactions among competing psychological processes.     

Dual-task interference with equal task emphasis: graded capacity sharing or central postponement?

Most studies using the psychological refractory period (PRP) design suggest that dual-task performance is limited by a central bottleneck. Because subjects are usually told to emphasize Task 1, however, the bottleneck might reflect a strategic choice rather than a structural limitation. To evaluate the possibility that central operations can proceed in parallel, albeit with capacity limitations, we conducted two dual-task experiments with equal task emphasis. In both experiments, subjects tended to either group responses together or respond to one task well before the other. In addition, stimulus-response compatibility effects were roughly constant across stimulus onset asynchronies (SOAs). At the short SOA, compatibility effects also carried over onto response times for the other task. This pattern of results is difficult to reconcile with the possibility that subjects share capacity roughly equally between simultaneous central operations. However, this pattern is consistent with the existence of a structural central bottleneck.     

Time discounting and time preference in animals: A critical review

Animals are an important model for studies of impulsivity and self-control. Many studies have made use of the intertemporal choice task, which pits small rewards available sooner against larger rewards available later (typically several seconds), repeated over many trials. Preference for the sooner reward is often taken to indicate impulsivity and/or a failure of self-control. This review shows that very little evidence supports this assumption; on the contrary, ostensible discounting behavior may reflect a boundedly rational but not necessarily impulsive reward-maximizing strategy. Specifically, animals may discount weakly, or even adopt a long-term rate-maximizing strategy, but fail to fully incorporate postreward delays into their choices. This failure may reflect learning biases. Consequently, tasks that measure animal discounting may greatly overestimate the true discounting and may be confounded by processes unrelated to time preferences. If so, animals may be much more patient than is widely believed; human and animal intertemporal choices may reflect unrelated mental operations; and the shared hyperbolic shape of the human and animal discount curves, which is used to justify cross-species comparisons, may be coincidental. The discussion concludes with a consideration of alternative ways to measure self-control in animals.     

Face-specific capacity limits under perceptual load do not depend on holistic processing

Previous observations that face recognition may proceed automatically, without drawing on attentional resources, have been challenged by recent demonstrations that only a few faces can be processed at one time. However, a question remains about the nature of the stimulus properties that underlie face-specific capacity limits. Two experiments showed that speeded categorization of a famous face (such as a politician or pop star) is facilitated when it is congruent with a peripheral distractor face. This congruency effect is eliminated if the visual search is loaded with more than one face, unlike previous demonstrations of speeded classification using semantic information. Importantly, congruency effects are also eliminated when the search task is loaded with nontarget faces that are shown in an inverted orientation. These results indicate that face-specific capacity limits are not determined by the configural (“holistic”) properties of face recognition.     

Decision rules and signal detectability in a reinforcement-density discrimination

Two probabilistic schedules of reinforcement, one richer in reinforcement, the other leaner, were overlapping stimuli to be discriminated in a choice situation. One of two schedules was in effect for 12 seconds. Then, during a 6-second choice period, the first left-key peck was reinforced if the richer schedule had been in effect, and the first right-key peck was reinforced if the leaner schedule had been in effect. The two schedule stimuli may be viewed as two binomial distributions of the number of reinforcement opportunities. Each schedule yielded different frequencies of 16 substimuli. Each substimulus had a particular type of outcome pattern for the 12 seconds during which a schedule was in effect, and consisted of four consecutive light-cued 3-second T-cycles, each having 0 or 1 reinforced center-key pecks. Substimuli therefore contained 0 to 4 reinforcers. On any 3-second cycle, the first center-key peck darkened that key and was reinforced with probability .75 or .25 in the richer or leaner schedules, respectively. In terms of the theory of signal detection, detectability neared the maximum possible d′ for all four pigeons. Left-key peck probability increased when number of reinforcers in a substimulus increased, when these occurred closer to choice, or when pellets were larger for correct left-key pecks than for correct right-key pecks. Averaged over different temporal patterns of reinforcement in a substimulus, substimuli with the same number of reinforcers produced choice probabilities that matched relative expected payoff rather than maximized one alternative.     

Probabilistic Subset Conjunction

The authors introduce subset conjunction as a classification rule by which an acceptable alternative must satisfy some minimum number of criteria. The rule subsumes conjunctive and disjunctive decision strategies as special cases. Subset conjunction can be represented in a binary-response model, for example, in a logistic regression, using only main effects or only interaction effects. This results in a confounding of the main and interaction effects when there is little or no response error. With greater response error, a logistic regression, even if it gives a good fit to data, can produce parameter estimates that do not reflect the underlying decision process. The authors propose a model in which the binary classification of alternatives into acceptable/unacceptable categories is based on a probabilistic implementation of a subset-conjunctive process. The satisfaction of decision criteria biases the odds toward one outcome or the other. The authors then describe a two-stage choice model in which a (possibly large) set of alternatives is first reduced using a subset-conjunctive rule, after which an alternative is selected from this reduced set of items. They describe methods for estimating the unobserved consideration probabilities from classification and choice data, and illustrate the use of the models for cancer diagnosis and consumer choice. They report the results of simulations investigating estimation accuracy, incidence of local optima, and model fit. The authors thank the Editor, the Associate Editor, and three anonymous reviewers for their constructive suggestions, and also thank Asim Ansari and Raghuram Iyengar for their helpful comments. They also thank Sawtooth Software, McKinsey and Company, and Intelliquest for providing the PC choice data, and the University of Wisconsin for making the breast-cancer data available at the machine learning archives.     

Divided attention between simultaneous auditory and visual signals

Past studies of simultaneous attention to pairs of visual stimuli have used the “dual-task” paradigm to show that identification of the direction of a change in luminance, whether incremental or decremental, is “capacity-limited,” while simple detection of these changes is governed by “capacity-free” processes. On the basis of that finding, it has been suggested that the contrast between identification and detection reflects different processes in the sensory periphery, namely the responses of magno- and parvocellular receptors. The present study questions that assertion and investigates the contribution of central processing in resource limitation by applying the dual task to a situation in which one stimulus is auditory and one is visual. The results are much the same as before, with identification demonstrating the tradeoff in performance generally attributed to a limited capacity but detection showing no loss compared with single-task controls. This implies that limitations on resources operate at a central level of processing rather than in the auditory and visual peripheries.     

Temporal brightness enhancement: Studies of individual differences

We have previously identified categorical individual differences in the occurrence of temporal brightness enhancement (TBE) by using a simultaneous brightness discrimination paradigm (Bowen & Markell, 1980).TBE is a nonmonotonic relation between brightness and pulse duration, pulses of intermediate duration (75–125 msec) can appear brighter than longer or shorter pulses of the same luminance. Three classes of observers can be defined based on whether they perceive TBE under one of two conditions of temporal asynchrony between a short test pulse and a longer (500 msec) comparison pulse:simultaneous onset of the pulses orsimultaneous offset. Type A observers show TBE for both asynchrony conditions; Type B observers show the effect for simultaneous offset but not simultaneous onset; Type C observers do not show TBE for either asynchrony. In the present study, we show that Type A and Type C observers maintain a constant brightness-duration relation as the asynchrony between test and comparison pulses is varied from simultaneous onset to simultaneous offset. Type B observers show a gradual shift in the brightness-duration relation as asynchrony changes. In a separate experiment, we find that practice has little effect on Type A and Type B observers but that Type C observers may change in classification to Types A and B over as few as five experimental sessions. The hypothesis that individual differences are due to differential “weighting” of chromatic (sustained) and achromatic (transient) visual channels is discussed.     

On choosing between movement sequences: comments on Rose (1988)

One method of investigating human motor programming is to determine how the choice reaction time for a memorized response sequence depends on the composition of that sequence as well as the other sequence that may be required. Using this method, Rose (1988) found that the total number of responses in the two possible response sequences predicts the choice reaction time to initiate either one. On the basis of this result, Rose claimed that the hierarchical editor (HED) model of motor programming, developed by Rosenbaum, Inhoff, and Gordon (1984), may have to be reevaluated. In this commentary I argue that Rose's results are inconsistent with a precursor of the HED model, not with the HED model itself, that the HED model actually provides a better fit to Rose's data than her total-number-of-responses model, that in general, choice reaction time does not increase with the total number of possible responses, and that structural relations between alternative movement sequences are the main determinants of choice reaction time. Taken as a whole, the results suggest that possible responses are not held in completely readied form before being selected for execution. A further implication is that the storage capacity of the motor output buffer (the MOB) is extremely limited.