Choice with delayed and probabilistic reinforcers: effects of prereinforcer and postreinforcer stimuli.

In Experiment 1, pigeons' pecks on a green key led to a 5-s delay with green houselights, and then food was delivered on 20% (or, in other conditions, 50%) of the trials. Pecks on a red key led to an adjusting delay with red houselights, and then food was delivered on every trial. The adjusting delay was used to estimate indifference points: delays at which the two alternatives were chosen about equally often. Varying the presence or absence of green houselights during the delays that preceded possible food deliveries had large effects on choice. In contrast, varying the presence of the green or red houselights in the intertrial intervals had no effects on choice. In Experiment 2, pecks on the green key led to delays of either 5 s or 30 s with green houselights, and then food was delivered on 20% of the trials. Varying the duration of the green houselights on nonreinforced trials had no effect on choice. The results suggest that the green houselights served as a conditioned reinforcer at some times but not at others, depending on whether or not there was a possibility that a primary reinforcer might be delivered. Given this interpretation of what constitutes a conditioned reinforcer, most of the results were consistent with the view that the strength of a conditioned reinforcer is inversely related to its duration.     

Second-order schedules of token reinforcement: effects of varying the schedule of food presentation

In the initial link of a complex schedule, one discriminative stimulus was presented and lever pressing produced tokens on fixed-ratio schedules. In the terminal link, signalled by a second discriminative stimulus, deposits of the tokens produced food. With two rats, the terminal link was presented after each sixth component schedule of token reinforcement was completed. With the other two rats, the terminal link was presented following the first component schedule completed after a fixed interval. During the terminal link, each token deposit initially produced food. The schedule of food presentation was subsequently increased such that an increasing number of token deposits in the terminal link was required for each food presentation. Rates of lever pressing in the initial link were inversely related to the schedule of food presentation in the terminal link. These results are similar to those of experiments that have varied schedules of food presentation in chained schedules. Rates and patterns of responding controlled throughout the initial link were more similar to those ordinarily controlled by second-order brief-stimulus schedules than to those controlled by comparable extended chained schedules.     

Dealing with Multiple Solutions in Structural Vector Autoregressive Models

Structural vector autoregressive models (VARs) hold great potential for psychological science, particularly for time series data analysis. They capture the magnitude, direction of influence, and temporal (lagged and contemporaneous) nature of relations among variables. Unified structural equation modeling (uSEM) is an optimal structural VAR instantiation, according to large-scale simulation studies, and it is implemented within an SEM framework. However, little is known about the uniqueness of uSEM results. Thus, the goal of this study was to investigate whether multiple solutions result from uSEM analysis and, if so, to demonstrate ways to select an optimal solution. This was accomplished with two simulated data sets, an empirical data set concerning children's dyadic play, and modifications to the group iterative multiple model estimation (GIMME) program, which implements uSEMs with group- and individual-level relations in a data-driven manner. Results revealed multiple solutions when there were large contemporaneous relations among variables. Results also verified several ways to select the correct solution when the complete solution set was generated, such as the use of cross-validation, maximum standardized residuals, and information criteria. This work has immediate and direct implications for the analysis of time series data and for the inferences drawn from those data concerning human behavior.     

Rewording the Rules on Disjunctive Probability

Logic is a central and highly useful part of philosophy. Its value is particularly evident when it comes to keeping our thinking about disjunctive probabilities clear. Because of the two meanings of “or” (“just one of these statements is true,” “at least one of these statements is true”), logic can show how the likelihood of a disjunction being true can be determined quite easily. To gauge the chance that one of two or more exclusive alternatives is true, one need only sum up their respective likelihoods. And to know the chance that at least one of two or more compatible alternatives is true, one simply has to figure the chance that it is false that all of them are false!     

Estimation of the Coefficient of Variation with Minimum Risk: A Sequential Method for Minimizing Sampling Error and Study Cost

The coefficient of variation is an effect size measure with many potential uses in psychology and related disciplines. We propose a general theory for a sequential estimation of the population coefficient of variation that considers both the sampling error and the study cost, importantly without specific distributional assumptions. Fixed sample size planning methods, commonly used in psychology and related fields, cannot simultaneously minimize both the sampling error and the study cost. The sequential procedure we develop is the first sequential sampling procedure developed for estimating the coefficient of variation. We first present a method of planning a pilot sample size after the research goals are specified by the researcher. Then, after collecting a sample size as large as the estimated pilot sample size, a check is performed to assess whether the conditions necessary to stop the data collection have been satisfied. If not an additional observation is collected and the check is performed again. This process continues, sequentially, until a stopping rule involving a risk function is satisfied. Our method ensures that the sampling error and the study costs are considered simultaneously so that the cost is not higher than necessary for the tolerable sampling error. We also demonstrate a variety of properties of the distribution of the final sample size for five different distributions under a variety of conditions with a Monte Carlo simulation study. In addition, we provide freely available functions via the MBESS package in R to implement the methods discussed.     

Probability of reciprocation in repeated prisoner's dilemma games

Both social cooperation and self‐control require reciprocation. In social cooperation situations a single person's cooperative act, if not reciprocated by others, would be unreinforced both immediately and in the long term. Similarly, a single act of self‐control (refusing a single cigarette, for example), if not followed up by other such acts, would also be unreinforced immediately and in the long term. The present experiments varied probability of reciprocation in a prisoner's dilemma‐like game that could resemble a social cooperation or a self‐control situation. Participants earned points by playing against a computer. The computer offered the participant a choice between 5 or 6 points (‘cooperation’ by the computer) or between 1 or 2 points (‘defection’ by the computer). The participant's choice of the lesser alternative (‘cooperation’) or the greater alternative (‘defection’) on the current trial led, respectively, to cooperation or defection by the computer on the next trial with a probability (probability of reciprocation, PR) which could vary. When PR was greater than 63%, consistent cooperation maximized the participant's earnings. When PR was less than 63%, defection maximized earnings. Three conditions were studied: PR signaled by spinners; PR unsignaled; PR unsignaled with participants believing that they were playing with another person, not the computer. With PR = 100% (‘tit‐for‐tat’) and PR = 50%, the modal participant maximized earnings under all three conditions. With PR = 75%, participants maximized earnings only when PR was signaled. These results indicate that differences between people's tendency to cooperate with other people (social cooperation) and their tendency to cooperate with their own future interests (self‐control) may lie in differences in subjective PR. Copyright © 2001 John Wiley & Sons, Ltd.     

The ‘conjunction fallacy’ revisited: how intelligent inferences look like reasoning errors

Findings in recent research on the ‘conjunction fallacy’ have been taken as evidence that our minds are not designed to work by the rules of probability. This conclusion springs from the idea that norms should be content‐blind—in the present case, the assumption that sound reasoning requires following the conjunction rule of probability theory. But content‐blind norms overlook some of the intelligent ways in which humans deal with uncertainty, for instance, when drawing semantic and pragmatic inferences. In a series of studies, we first show that people infer nonmathematical meanings of the polysemous term ‘probability’ in the classic Linda conjunction problem. We then demonstrate that one can design contexts in which people infer mathematical meanings of the term and are therefore more likely to conform to the conjunction rule. Finally, we report evidence that the term ‘frequency’ narrows the spectrum of possible interpretations of ‘probability’ down to its mathematical meanings, and that this fact—rather than the presence or absence of ‘extensional cues’—accounts for the low proportion of violations of the conjunction rule when people are asked for frequency judgments. We conclude that a failure to recognize the human capacity for semantic and pragmatic inference can lead rational responses to be misclassified as fallacies. Copyright © 1999 John Wiley & Sons, Ltd.     

Patterns of Preference for Numerical and Verbal Probabilities

We report results of an experiment designed to test a principle formulated by Budescu and Wallsten (1995), that, when communicating uncertainty information, mode choices are sensitive to sources and degrees of vagueness. In addition, we examined subjects’ efficacy in using such uncertainty information as a function of communication mode, source, and vagueness. In phase one of the experiment, subjects in a dyad used precise (numerical) or imprecise (verbal) expressions to communicate to a remote partner precise or vague uncertainty about the likelihoods of events. Spinner outcomes were used to generate precise uncertainty while answers to almanac questions were used to elicit vague uncertainty. In phase two, subjects saw the events paired with their partners’ estimates of similar events, and were asked to gamble on one event from each pair. Communication mode preferences were measured as the relative frequency that subjects chose the numerical mode to either express or receive uncertainty information regarding the events. Efficacy was measured as the relative frequency that subjects choose from the pair the event associated with the objectively more probable uncertainty expression. Underlying uncertainty interacted with direction of communication to affect preferences for modes of expression of the probabilities. Subjects preferred precise (numerical) information, especially for precise events (spinners). For vague events (questions), their preference for precise (numerical) information was stronger when receiving than when communicating information. Similar preferences were reflected in the efficiency of subsequent gamble decisions based on the probability estimates. Specifically, decisions were more efficacious (i.e. consistent with Expected Utility) when degrees of precision in events and estimates matched. © 1997 by John Wiley & Sons, Ltd.     

Flexible representations in visual working memory and interactions with long‐term learning: Commentary on the special issue

This special issue of the British Journal of Psychology brings together cutting edge research on a range of topics in visual working memory (VWM). In this commentary, we attempt to summarize common themes in current VWM research exemplified in this issue. The articles include several reviews of important topics as well as empirical papers covering three main themes. The first concerns the nature of mental representations of memoranda in the commonly used delayed estimation task, where both fine‐grained and broad categorical details appear to be represented, and their susceptibility to interference. The second concerns interactions between VWM representations, both those that produce individuation of representations and those that create an overarching ensemble structure. Finally, the third main topic concerns the use of VWM during visual search and in the learning of repeated configurations in search displays. The work presented here, and other work in the field, points to a rich interplay between representations in VWM but also between VWM and information in long‐term memory. Opportunities for further investigation are highlighted throughout.     

The influence of number line estimation precision and numeracy on risky financial decision making

This study examined whether different aspects of mathematical proficiency influence one's ability to make adaptive financial decisions. “Numeracy” refers to the ability to process numerical and probabilistic information and is commonly reported as an important factor which contributes to financial decision‐making ability. The precision of mental number representation (MNR), measured with the number line estimation (NLE) task has been reported to be another critical factor. This study aimed to examine the contribution of these mathematical proficiencies while controlling for the influence of fluid intelligence, math anxiety and personality factors. In our decision‐making task, participants chose between two options offering probabilistic monetary gain or loss. Sensitivity to expected value was measured as an index for the ability to discriminate between optimal versus suboptimal options. Partial correlation and hierarchical regression analyses revealed that NLE precision better explained EV sensitivity compared to numeracy, after controlling for all covariates. These results suggest that individuals with more precise MNR are capable of making more rational financial decisions. We also propose that the measurement of “numeracy,” which is commonly used interchangeably with general mathematical proficiency, should include more diverse aspects of mathematical cognition including basic understanding of number magnitude.