A unified fixed point approach to study the existence and uniqueness of solutions to the generalized stochastic functional equation emerging in the psychological theory of learning

<abstract><p>The model of decision practice reflects the evolution of moral judgment in mathematical psychology, which is concerned with determining the significance of different options and choosing one of them to utilize. Most studies on animals behavior, especially in a two-choice situation, divide such circumstances into two events. Their approach to dividing these behaviors into two events is mainly based on the movement of the animals towards a specific choice. However, such situations can generally be divided into four events depending on the chosen side and placement of the food. This article aims to fill such gaps by proposing a generic stochastic functional equation that can be used to describe several psychological and learning theory experiments. The existence, uniqueness, and stability analysis of the suggested stochastic equation are examined by utilizing the notable fixed point theory tools. Finally, we offer two examples to substantiate our key findings.</p></abstract>

Unbiased post-error slowing in interference tasks: A confound and a simple solution

We typically slow down after committing an error, an effect termed post-error slowing (PES). Traditionally, PES has been calculated by subtracting post-correct from post-error RTs. Dutilh et al. (Journal of Mathematical Psychology, 56(3), 208-216, 2012), however, showed PES values calculated in this way are potentially biased. Therefore, they proposed to compute robust PES scores by subtracting pre-error RTs from post-error RTs. Based on data from a large-scale study using the flanker task, we show that both traditional and robust PES estimates can be biased. The source of the bias are differential imbalances in the percentage of congruent vs. incongruent post-correct, pre-error, and post-error trials. Specifically, we found that post-correct, pre-error, and post-error trials were more likely to be congruent than incongruent, with the size of the imbalance depending on the trial type as well as the length of the response-stimulus interval (RSI). In our study, for trials preceded by a 700-ms RSI, the percentages of congruent trials were 62% for post-correct trials, 66% for pre-error trials, and 56% for post-error trials. Relative to unbiased estimates, these imbalances inflated traditional PES estimates by 37% (9 ms) and robust PES estimates by 42% (16 ms) when individual-participant means were calculated. When individual-participant medians were calculated, the biases were even more pronounced (40% and 50% inflation, respectively). To obtain unbiased PES scores for interference tasks, we propose to compute unweighted individual-participant means by initially calculating mean RTs for congruent and incongruent trials separately, before averaging congruent and incongruent mean RTs to calculate means for post-correct, pre-error and post-error trials.

Preregistration in diverse contexts: a preregistration template for the application of cognitive models

In recent years, open science practices have become increasingly popular in psychology and related sciences. These practices aim to increase rigour and transparency in science as a potential response to the challenges posed by the replication crisis. Many of these reforms—including the increasingly used preregistration —have been designed for purely experimental work that tests straightforward hypotheses with standard inferential statistical analyses, such as assessing whether an experimental manipulation has an effect on a variable of interest. But psychology is a diverse field of research. The somewhat narrow focus of the prevalent discussions surrounding and templates for preregistration has led to debates on how appropriate these reforms are for areas of research with more diverse hypotheses and more intricate methods of analysis, such as cognitive modelling research within mathematical psychology. Our article attempts to bridge the gap between open science and mathematical psychology, focusing on the type of cognitive modelling that Crüwell et al. (Crüwell S, Stefan AM, Evans NJ. 2019 Robust standards in cognitive science. Comput. Brain Behav. 2 , 255–265) labelled model application , where researchers apply a cognitive model as a measurement tool to test hypotheses about parameters of the cognitive model. Specifically, we (i) discuss several potential researcher degrees of freedom within model application, (ii) provide the first preregistration template for model application and (iii) provide an example of a preregistered model application using our preregistration template. More broadly, we hope that our discussions and concrete proposals constructively advance the mostly abstract current debate surrounding preregistration in cognitive modelling, and provide a guide for how preregistration templates may be developed in other diverse or intricate research contexts.

No Envy

The important ‘no-envy’ fairness criterion has typically been attributed to Foley (1967) and sometimes to Tinbergen (1946, 1953). We reveal that Jan Tinbergen introduced ‘no-envy’ as a fairness criterion in his article “Mathematiese Psychologie” published in 1930 in the Dutch journal Mens en Maatschappij and translated as “Mathematical Psychology” in 2021 in the Erasmus Journal for Philosophy and Economics. Our article accompanies the translation: we introduce Tinbergen’s 1930 formulation of the ‘no-envy’ criterion, compare it to other formulations, and comment on its significance for the fairness literature in philosophy and economics.

Mathematical Psychology

This article appeared originally in 1930, in Dutch, under the title “Mathematiese Psychologie” in Mens en Maatschappij. Translated and annotated by Conrad Heilmann, Stefan Wintein, Ruth Hinz, and Erwin Dekker, it is accompanied—in the present issue—by the article “No Envy: Jan Tinbergen on Fairness” written by Conrad Heilmann and Stefan Wintein.

Risk and emotion: measuring the effect of emotions and other visceral factors on decision making under risk

The science of modelling choice preferences has evolved into an interdisciplinary field contributing to several branches of microeconomics and mathematical psychology. As theories in decision science and related fields mature, descriptive theories have emerged to explain systematic violations of rationality through cognitive mechanisms underlying the thought processes that guide human behaviour. Cognitive limitations are not, however, solely responsible for systematic deviations from rationality and there is a growing body of literature exploring the effect of visceral factors as the more dominant drivers. This study builds on the existing literature by investigating the impact of anger, sadness, happiness, anxiety, hunger, energy, tiredness and stress on three distinct elements that define risk preference: utility, decision weights and loss aversion. By decomposing the impact of visceral factors on risk preference, I am able to provide evidence supporting the proposition that a portion of the variability in individual choice preferences can be explained by interacting visceral states. My findings suggest that visceral factors have the strongest effect on loss aversion, which is a major factor in how people code and evaluate financial outcomes. Anger, sadness, happiness, anxiety, energy and tiredness each affect five or more of the model parameters, while hunger and stress are significant only in their interaction with other visceral factors. I also provide evidence to show that the generalized approaches to characterizing visceral factors and risk preference are too broad to be descriptively meaningful. The results of this study show that emotions and other drive states effect the way people process and interpret information, which is crucial in informing decision-makers of the sources and consequences of irrational behaviour. These findings will be of immediate interest to wealth management specialists, public relations advisers as well as to engineers in designing socially intelligent machines capable of interacting more effectively with humans.

Risk and emotion: measuring the effect of emotions and other visceral factors on decision making under risk

The science of modelling choice preferences has evolved into an interdisciplinary field contributing to several branches of microeconomics and mathematical psychology. As theories in decision science and related fields mature, descriptive theories have emerged to explain systematic violations of rationality through cognitive mechanisms underlying the thought processes that guide human behaviour. Cognitive limitations are not, however, solely responsible for systematic deviations from rationality and there is a growing body of literature exploring the effect of visceral factors as the more dominant drivers. This study builds on the existing literature by investigating the impact of anger, sadness, happiness, anxiety, hunger, energy, tiredness and stress on three distinct elements that define risk preference: utility, decision weights and loss aversion. By decomposing the impact of visceral factors on risk preference, I am able to provide evidence supporting the proposition that a portion of the variability in individual choice preferences can be explained by interacting visceral states. My findings suggest that visceral factors have the strongest effect on loss aversion, which is a major factor in how people code and evaluate financial outcomes. Anger, sadness, happiness, anxiety, energy and tiredness each affect five or more of the model parameters, while hunger and stress are significant only in their interaction with other visceral factors. I also provide evidence to show that the generalized approaches to characterizing visceral factors and risk preference are too broad to be descriptively meaningful. The results of this study show that emotions and other drive states effect the way people process and interpret information, which is crucial in informing decision-makers of the sources and consequences of irrational behaviour. These findings will be of immediate interest to wealth management specialists, public relations advisers as well as to engineers in designing socially intelligent machines capable of interacting more effectively with humans.

A Neurocognitive Psychometrics Account of Individual Differences in Attentional Control

Attention control processes play an important role in many substantial psychological theories, but are hard to reliably and validly measure on the subject-level. Therefore, associations between individual differences in attentional control and other variables are often inconsistent. Here we propose a novel neurocognitive psychometrics account of attentional control that integrates model parameters from the dual-stage two-phase model (Hübner et al., 2010), a mathematical model of selective attention, with neural correlates of conflict processing in a multi-layer structural equation model framework. We analyzed data from 150 participants who completed the Eriksen Flanker task while their EEG was recorded and used the neurocognitive psychometric approach to distinguish between two sequential stages of information-processing – target selection and response selection. Model parameters and neural correlates showed convergent validity and could be meaningfully related to each other. Together, these neurocognitive process parameters jointly predicted 37 % of the variance in individual differences in higher-order cognitive abilities. Individuals with greater cognitive abilities were not only better at focusing their attention on the target stimulus, but also at subsequent response-selection. All in all, our results support the idea that individual differences in attentional control processes contribute to individual differences in cognitive abilities. Moreover, they provide hope that the measurement crisis of individual differences in attentional control can be overcome by integrating measurement approaches from related disciplines such as mathematical psychology and cognitive neuroscience.

If Mathematical Psychology Did Not Exist We Might Need to Invent It: A Comment on Theory Building in Psychology

It is commonplace, when discussing the subject of psychological theory, to write articles from the assumption that psychology differs from the physical sciences in that we have no theories that would support cumulative, incremental science. In this brief article I discuss one counterexample: Shepard’s law of generalization and the various Bayesian extensions that it inspired over the past 3 decades. Using Shepard’s law as a running example, I argue that psychological theory building is not a statistical problem, mathematical formalism is beneficial to theory, measurement and theory have a complex relationship, rewriting old theory can yield new insights, and theory growth can drive empirical work. Although I generally suggest that the tools of mathematical psychology are valuable to psychological theorists, I also comment on some limitations to this approach.