Mathematical Psychology Research Articles

Modify the mathematical behavior and its influence on the player level of The view of mathematical psychology

The summary in English The current study finds the reality of using selection from a face Consider the clubs' mathematical psychology sport in Palestine, southern governorate, Ascale has been created that includes two axes : Importance disabilities , and applied a sample of ( 1200 ) players in ( 40 ) clubs .The results revealed possibilities for using mathematical psychology in sports clubs on the other hand , the results showed that there are some disabilities in use mathematical psychology with players and clubs. The most important of these are : The refusal of the players and clubs to subject the players to psychological treatment or preparation, some players and sports clubs find it difficult use Mathematical psychology. The researcher recommended that players and sports clubs should be set up before the beginning During the sports season and friendly and official matches.

Numerical approximation of the first-passage time distribution of time-varying diffusion decision models: A mesh-free approach

Sequential sampling theory is the dominant theoretical framework for explaining human decision making behavior. In this theory, the decision process is determined by a stochastic process in a bounded domain, with the likelihood function of the model being the first passage-time distribution of this process. However, many sequential sampling models have no analytic formulation for the first passage-time distribution, making these models difficult and computationally taxing to fit. Thus, developing an efficient and general approximation procedure for the first-passage time distribution of sequential sampling models is crucial for the field of mathematical psychology. Here, we have developed a numerical algorithm based on the mesh-free techniques for numerically solving the Fokker–Planck differential equation, which allows us to approximate the first passage-time distribution of some important classes of sequential sampling models, including the urgency signal model, Ornstein–Uhlenbeck model, and collapsing threshold model. Since the considered equation is a singular partial differential equation in a moving boundary domain, the proposed numerical method first transforms the equation to a regular domain, and then the solution is approximated based on the collocation method. However, since satisfying the boundary conditions in the collocation algorithms can be very tricky, we have imposed the boundary conditions in the basis functions. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. The performance of the proposed method is illustrated by solving several examples.

Rational inattention when decisions take time

Decisions take time, and the time taken to reach a decision is likely to be informative about the cost of more precise judgments. We formalize this insight using a dynamic model of optimal evidence accumulation. We provide conditions under which the resulting belief dynamics resemble either diffusion processes or processes with large jumps. We then consider the in which discounting effects are small relative to the opportunity cost of time, and show that the state-contingent choice probabilities predicted by our model are identical to those predicted by a static rational inattention model, providing a micro-foundation for such models. In the diffusion case, our model provides a normative foundation for a variant of the drift-diffusion model from mathematical psychology.

Convite à Psicologia Matemática: Modelos e Benefícios da Teorização Formal

ABSTRACT In most areas, psychological phenomena tend to be explained only through textual constructions. Several authors, however, point to the need for theories that have a more formal nature, based on mathematical reasoning. In order to encourage broader access to its applications, we present the models and advantages of a mathematical psychology approach to the study of behavior. We review the limitations of verbal theorizing, then a common taxonomy in mathematical psychology follows, that classifies formal models as descriptive, process characterization, and explanatory. As well succeeded cases, we examine the mathematical psychology of decision making, of helping behavior, of memory, and of romantic relationships. Finally, we discuss the potential benefits and uses of this approach. Welcome to mathematical psychology.

Invitation to Mathematical Psychology: Models and Benefits of Formal Theorizing

RESUMO Na maior parte das áreas os fenômenos psicológicos tendem a ser explicados apenas por meio de construções textuais. Diversos autores, no entanto, apontam para a necessidade de teorias que tenham uma natureza mais formal, baseada em raciocínio matemático. A fim de incentivar acesso mais amplo às suas aplicações, apresentamos os modelos e vantagens da abordagem da psicologia matemática para o estudo do comportamento. Revisamos as limitações da teorização verbal, apresentando em seguida uma taxonomia, comum na psicologia matemática, que classifica os modelos de dados como descritivos, explicativos e de caracterização. Como casos bem sucedidos, examinamos a psicologia matemática da tomada de decisão, do comportamento de ajuda, da memória e dos relacionamentos românticos. Por fim, discutimos os benefícios e usos potenciais da abordagem. Bem-vindo(a) à psicologia matemática.

On the solution of the generalized functional equation arising in mathematical psychology and theory of learning approached by the Banach fixed point theorem

In mathematical psychology, the model of decision practice represents the development of moral judgment that deals with the time to decide the meaning of the various choices and selecting one of them for use. Most animal behavior research classifies such situations as two distinct phenomena. On the other hand, reward plays a big part in this kind of study since, based on the selected side and food location, such circumstances may be classified into four categories. This paper intends to investigate such types of behavior and establish a general functional equation for it. The proposed functional equation can be used to describe several psychological and learning theory models in the existing literature. By using the fixed point theory tools, we obtain the results related to the existence, uniqueness, and stability of a solution to the proposed functional equation. Finally, we give two examples to support our main results.

Algorithms for the adaptive assessment of procedural knowledge and skills.

Procedural knowledge space theory (PKST) was recently proposed by Stefanutti (British Journal of Mathematical and Statistical Psychology, 72(2) 185-218, 2019) for the assessment of human problem-solving skills. In PKST, the problem space formally represents how a family of problems can be solved and the knowledge space represents the skills required for solving those problems. The Markov solution process model (MSPM) by Stefanutti et al. (Journal of Mathematical Psychology, 103, 102552, 2021) provides a probabilistic framework for modeling the solution process of a task, via PKST. In this article, three adaptive procedures for the assessment of problem-solving skills are proposed that are based on the MSPM. Beside execution correctness, they also consider the sequence of moves observed in the solution of a problem with the aim of increasing efficiency and accuracy of assessments. The three procedures differ from one another in the assumption underlying the solution process, named pre-planning, interim-planning, and mixed-planning. In two simulation studies, the three adaptive procedures have been compared to one another and to the continuous Markov procedure (CMP) by Doignon and Falmagne (1988a). The last one accounts for dichotomous correct/wrong answers only. Results show that all the MSP-based adaptive procedures outperform the CMP in both accuracy and efficiency. These results have been obtained in the framework of the Tower of London test but the procedures can also be applied to all psychological and neuropsychological tests that have a problem space. Thus, the adaptive procedures presented in this paper pave the way to the adaptive assessment in the area of neuropsychological tests.

The Psychology of Mathematics: A Journey of Personal Mathematical Empowerment for Educators and Curious Minds by Anderson Norton

Reviewed by: The Psychology of Mathematics: A Journey of Personal Mathematical Empowerment for Educators and Curious Minds by Anderson Norton Firdous Ahmad Mala Anderson Norton The Psychology of Mathematics: A Journey of Personal Mathematical Empowerment for Educators and Curious Minds Oxfordshire, UK. Routledge. 2022. 238 pages. USING AN APPROACH that empowers students by fostering their hidden constructions of mathematics structures, The Psychology of Mathematics offers an innovative introduction to the often-ignored and less-discussed psychology of mathematics. The author—an award-winning mathematician, educator, and writer—uses an engaging and accessible style, thereby making the reading experience both enriching and revealing. The book, by virtue of its style and simplicity, is accessible to a nonmathematical audience, too. In a dozen chapters, readers are exposed to the gorgeous world of mathematics. Combatting the common misnomer that mathematics is boring and hard, Norton offers exciting discussions on various mathematical themes, such as the power of symbols, the geometry of numbers, and the wonderful gift of mathematics. The book discusses a plethora of engaging and enriching discussions, such as the unreasonable effectiveness of numbers in our diurnal lives, the ineradicable interplay of language and mathematics, and the innate ability—found in babies and some animals—to immediately apprehend small numbers. Figures receive more appreciation and attention than numbers do. A figure speaks more than a thousand words. By focusing on space as a product of our own psychology, we make geometry more intuitive. Observing that mathematical symbols provide a way for recording the products of our mental actions, we are no longer required to keep them in the mind at all. Mathematics is a wonderful gift. In the author's words, "It is a gift that keeps on giving, having given us everything from numbers to Neptune." The author does not attempt to make mathematics easier; he only envisions making it dearer and more valuable to the reader. He does so, and successfully, by cutting down on jargon and by enhancing conversational communication of otherwise difficult concepts of geometry and other branches of mathematics. The motivation for writing this book is not simply to marvel at mathematics but also to promote mathematics as personally enriching and empowering. The book is an excellent exposition of the incessant wonder that mathematicians and philosophers have experienced and expressed for millennia. It draws on what Galileo is reported to have remarked in his 1623 masterpiece, The Assayer: "Philosophy is written in this grand book—I mean the universe—that stands continuously open to our gaze, but it cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics." It is not an exaggeration to proclaim that mathematics is the language in which the universe has been written, and Psychology attempts to bring the same fact to the fore for a nonmathematical audience. [End Page 79] Firdous Ahmad Mala Government Degree College Sopore, Jammu and Kashmir Copyright © 2022 World Literature Today and the Board of Regents of the University of Oklahoma

Virtual social networking increases the individual's economic predictability

Forecasting economic choice is hard because today we still do not know enough about human motivation. A fundamental problem is the lack of knowledge about how the neural networks in the brain give rise to thinking and decision making. One way to address the issue has been to develop simplified economic experiments, in which participants need skills of little complexity and their minds employ cognitive mechanisms, already well understood by mathematical psychology and neuroscience. Here we take a neural model for rudimentary emotion generation and memorizing and use it as a guiding theory to understand decision making in an experimental oligopoly market. For the first time in that line of research, participants are put in a lab virtual social network serving to exchange opinions about deals with companies. On average, choices become significantly more predictable when people participate in the network, in contrast to working alone with expert information. Calibrating the model for each person, we find that some people are predicted with startling precision.

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 (i.e., latencies of the stimulus-locked lateralized readiness potential) in a multilayer 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 explained 37 % of the variance in individual differences in higher-order cognitive abilities that were assessed with a battery of intelligence tests and working memory tasks. 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 are related 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. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

The Intermediate Value Theorem and Decision-Making in Psychology and Economics: An Expositional Consolidation

On taking the intermediate value theorem (IVT) and its converse as a point of departure, this paper connects the intermediate value property (IVP) to the continuity postulate typically assumed in mathematical economics, and to the solvability axiom typically assumed in mathematical psychology. This connection takes the form of four portmanteau theorems, two for functions and the other two for binary relations, that give a synthetic and novel overview of the subject. In supplementation, the paper also surveys the antecedent literature both on the IVT itself, as well as its applications in economic and decision theory. The work underscores how a humble theorem, when viewed in a broad historical frame, bears the weight of many far-reaching consequences; and testifies to a point of view that the apparently complicated can sometimes be under-girded by a most basic and simple execution.

Commentary on "Extending the Basic Local Independence Model to Polytomous Data" by Stefanutti, de Chiusole, Anselmi, and Spoto.

The Polytomous Local Independence Model (PoLIM) by Stefanutti, de Chiusole, Anselmi, and Spoto, is an extension of the Basic Local Independence Model (BLIM) to accommodate polytomous items. BLIM, a model for analyzing responses to binary items, is based on Knowledge Space Theory, a framework developed by cognitive scientists and mathematical psychologists for modeling human knowledge acquisition and representation. The purpose of this commentary is to show that PoLIM is simply a paraphrase of a DINA model in cognitive diagnosis for polytomous items. Specifically, BLIM is shown to be equivalent to the DINA model when the BLIM-items are conceived as binary single-attribute items, each with a distinct attribute; thus, PoLIM is equivalent to the DINA for polytomous single-attribute items, each with a distinct attribute.

Continuity postulates and solvability axioms in economic theory and in mathematical psychology: a consolidation of the theory of individual choice

This paper presents four theorems that connect continuity postulates in mathematical economics to solvability axioms in mathematical psychology, and ranks them under alternative supplementary assumptions. Theorem 1 connects notions of continuity (full, separate, Wold, weak Wold, Archimedean, mixture) with those of solvability (restricted, unrestricted) under the completeness and transitivity of a binary relation. Theorem 2 uses the primitive notion of a separately continuous function to answer the question when an analogous property on a relation is fully continuous. Theorem 3 provides a portmanteau theorem on the equivalence between restricted solvability and various notions of continuity under weak monotonicity. Finally, Theorem 4 presents a variant of Theorem 3 that follows Theorem 1 in dispensing with the dimensionality requirement and in providing partial equivalences between solvability and continuity notions. These theorems are motivated for their potential use in representation theorems.

On Solutions and Stability of Stochastic Functional Equations Emerging in Psychological Theory of Learning

We show how to apply the well-known fixed-point approach in the study of the existence, uniqueness, and stability of solutions to some particular types of functional equations. Moreover, we also obtain the Ulam stability result for them. The functional equations that we consider can be used to explain various experiments in mathematical psychology and arise in a natural way in the stochastic approach to the processes of perception, learning, reasoning, and cognition.

A distribution-free, Bayesian goodness-of-fit method for assessing similar scientific prediction equations

In this paper a new Bayesian distribution-free goodness-of-fit method is described. Unlike other goodness-of-fit tests, the new analysis does not examine the statistical structure of the residuals; rather it is based on an examination of the concordance or discordance of the experimental observations from the expectations of the scientific theory. The novel goodness-of-fit procedure is used to assess four highly similar mathematical theories for the probability weighting function from the risky choice literature. Prior research in mathematical psychology and behavioral economics has demonstrated the importance of a nonlinear weighting function for the probabilities that are associated with risky choice, but it has been a challenge to discriminate among a set of different nonlinear functions. However, a reanalysis of the data from a study by Barch and Chechile (2016) with the new goodness-of-fit procedure shows that each of the previously established leading models are sharply discriminated from one another. This analysis is thus an example of how scientific theories, which propose a specific mathematical function, can be assessed with the Bayesian distribution-free goodness-of-fit test. The method is quite general, and it can also be used to assess scientific models that predict multivariate outcomes.

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>

Эпистемологическая модель И.Ф. Гербарта и ее влияние на возникновение неокантианской теоретико-познавательной программы

The theoretical heritage of J.F. Herbart is still an underinvestigated page in the history of world philosophical thought. The epistemological component of his works has not been the object of close attention until now, and the appeal to Herbart’s texts was carried out mainly in the context of Herbertarian pedagogy. The article is devoted to the epistemological model of Herbart and its influence on the formation of the neo-Kantian program of philosophy at the end of the 19th century. It is suggested that Herbart anticipated neo-Kantian attempts to sci­entize philosophy by proposing modeling as a method of constructing a philo­sophical theory: Herbart’s program was based on the mathematical method and consistent deterministic construction. The article also emphasizes that Herbart’s concept provided a reliable basis for the central problems of neo-Kantian thought: discussions about extensive and intensive quantities, their measurabil­ity, infinitesimal magnitude, mechanistic theory of perception, the requirement to translate philosophical concepts into the language of mathematics as a source of verification of the entire theory. Herbart identified a significant array of factual material, concentrated around the mechanics of cognition, and demonstrated a new for his time way of working with a text, the analysis of which was based not on a reference to the “historical” Kant, his vision or language, but modeling and the corresponding redefinition of the basic provisions of the entire system. Even though neo-Kantians criticized Herbart’s conclusions, his mathematical psychology became an essential consolidating factor for the emergence of the Marburg school.

How to enhance military research using mathematical psychology

Military research encompasses a wide variety of topics that would benefit from the advanced analytical techniques of mathematical psychology. This tutorial provides important context and guidance on how to best prepare and present experimental findings for a military audience. The first discussion points cover how decisions are made by end users and funders in the military. Next, several simple approaches are described to convert inferential statistics and effect size information into data more readily appreciable by military end users. Finally, the tutorial concludes with simple guidance that should help anyone interested in military research better convey their findings to funding agencies and for practical applications.

An integrated theory of deciding and acting.

This article presents a theory in which motor execution in perceptual decision-making tasks is determined by the same evolving decision variable that drives response time. The theory builds upon recent insights from the neuroscience of decision-making and motor control. It is formalized as an extension of Ratcliff's diffusion model, and assumes that two thresholds operate on the evidence accumulation decision variable. The first threshold, referred to as electromyographic (EMG) threshold, marks the onset of electrical activity in the response-relevant muscle and the beginning of force production. The second threshold corresponds to the response. The theory makes several benchmark predictions. Notably, the mean duration of motor execution, as quantified by the mean latency between EMG onset and the response, should depend on the rate of evidence accumulation, and should thus increase as the perceptual difficulty of the task increases. We tested these predictions in a paradigmatic perceptual decision-making task, the random dot motion task, and recorded the EMG activity of response-relevant muscles. The behavioral and EMG data provide very strong evidence for each prediction. A final quantitative evaluation of the model showed good fits to these data. The theory resolves conflicting findings in the fields of mathematical psychology, motor control, and decision neurosciences. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

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.