Mathematical Psychology

A new theory for psychology has been proposed in this work, the core of which is founded on understanding and codifying the material basis of psychology as a behavioral science in the age of both biological and artificial systems that are capable of not just awareness and perception but also learning and behavior/reaction.

The purpose was especially to build a foundation via which future work in psychology might take advantage of Transformatics as a coherent body of mathematical ideologies and techniques that are universally applicable in any science.

A significant share of this work built upon earlier, and recent results in especially theory, by the author, concerning new mathematical ideas under the umbrella of Transformatics, and how they have been successfully applied to another biological discipline, Genetics, of which, and as we argue in the later sections of this work, likewise serves as a basis for much of the empirical subjects in psychology.

Thus, starting from some fundamentals of what defines psychology as a scientific discipline, we steadily build a new approach to some fundamental ideas and concepts in this subject, especially from the point of view of someone already familiar with how transformatics helps establish new foundations for any science, and from which we then demonstrate how to build or evolve new ideas, most of them mathematical, or philosophical, using that method, and which then serves to bring forth a new light to this fundamental science of the mind.

As far as results are concerned; we have advanced two working definitions of psychology, one originally by the author; have proposed four important problems for students and readers to use to further the ideas presented in the this work, and have not only presented for the first time, an original mathematical definition of blood, but have also gone ahead, to demonstrate how it could be used to unambiguously specify physiological and psychological pathologies with immediate clinical relevance as laid out in two theorems concluding this present undertaking. We have called out several next/future explorations as well as how other researchers might carry forward the work we have started here.

Quantum-like (QL) modeling, one of the outcomes of the quantum information revolution, extends quantum theory methods beyond physics to decision theory and cognitive psychology. While effective in explaining paradoxes in decision making and effects in cognitive psychology, such as conjunction, disjunction, order, and response replicability, it lacks a direct link to neural information processing in the brain. This study bridges neurophysiology, neuropsychology, and cognitive psychology, exploring how oscillatory neuronal networks give rise to QL behaviors. Inspired by the computational power of neuronal oscillations and quantum-inspired computation (QIC), we propose a quantum-theoretical framework for coupling of cognition/decision making and neural oscillations-QL oscillatory cognition. This is a step, may be very small, towards clarification of the relation between mind and matter and the nature of perception and cognition. We formulate four conjectures within QL oscillatory cognition and in principle they can be checked experimentally. But such experimental tests need further theoretical and experimental elaboration. One of the conjectures (Conjecture 4) is on resolution of the binding problem by exploring QL states entanglement generated by the oscillations in a few neuronal networks. Our findings suggest that fundamental

A central result in the AGM framework for belief revision is the construction of revision functions in terms of total preorders on possible worlds. These preorders encode comparative plausibility: r ă r 1 states that the world r is at least as plausible as r 1 . Indifference in the plausibility of two worlds, r, r 1 , denoted r " r 1 , is defined as r ⊀ r 1 and r 1 ⊀ r. Herein we take a closer look at plausibility indifference. We contend that the transitivity of indifference assumed in the AGM framework is not always a desirable property for comparative plausibility. Our argument originates from similar concerns in preference modelling, where a structure weaker than a total preorder, called a semiorder, is widely consider to be a more adequate model of preference. In this paper we essentially re-construct revision functions using semiorders instead of total preorders. We formulate postulates to characterise this new, wider, class of revision functions, and prove that the postulates are sound and complete with respect to the semiorder-based construction. The corresponding class of contraction functions (via the Levi and Harper Identities) is also characterised axiomatically.

MCQ examinations are frequently used but there is no consensus as to whether a penalty for wrong answers should be used or not. For example, in some countries examinations for medical licensing include MCQ sections with penalty while in others there is no penalty for wrong answers. We contribute to this discussion with the analysis of the effects of penalties. In a partial knowledge model there is a trade-off between bias and measurement error that is affected by the level of penalties. We build on IRT and introduce students' optimal behavior. Our simulations indicate that the optimal penalty may be relatively high: even though it discriminates against risk averse students this effect is small compared with the measurement error that it prevents. Financial support from MEC(SEJ2006-06309/ECON) and Gobierno Vasco, DEUI(IT-313-07

Theories of decision making are often formulated in terms of deterministic axioms, which do not account for stochastic variation that attends empirical data. This study presents a Bayesian inference framework for dealing with fallible data. The Bayesian framework provides readily applicable statistical procedures addressing typical inference questions that arise when algebraic axioms are tested against empirical data. The key idea of the Bayesian framework is to employ a prior distribution representing the parametric order constraints implied by a given axiom. Modern methods of Bayesian computation such as Markov chain Monte Carlo are used to estimate the posterior distribution, which provides the information that allows an axiom to be evaluated. Specifically, we adopt the Bayesian p-value as the criterion to assess the descriptive adequacy of a given model (axiom) and we use the deviance information criterion (DIC) to select among a set of candidate models. We illustrate the Bayesian framework by testing well-known axioms of decision making, including the axioms of monotonicity of joint receipt and stochastic transitivity.

Differential equations are an increasingly essential tool used in public health epidemiology, enabling the modeling of communicable disease spread in a population and informing the development, implementation, and assessment of public health interventions. As public health professionals increasingly rely on GenAI for data management and data analysis, understanding the mathematical equations underlying a communicable disease model becomes crucial. This essay provides an accessible overview of ordinary first-order differential equations, including the linear first-order differential equation (i.e., exponential growth model) and the nonlinear first-order differential equation (i.e., SIR model), for the public health epidemiologist. We explain each type of differential equation and show a step-by-step solution to facilitate comprehension. A real-world application of the use of first-order differential equations in public health is, briefly, presented. Then, we discuss how real-world data, first-order differential equations, data analysis and predictive data modeling can contribute to the development of insights that guide the development, deployment, and assessment of public health interventions by public health decision-makers for a population. In the appendix, sample prompts are provided to help leverage the power of GenAI in the use of mathematical models in public health epidemiology. This technical review aims to provide the epidemiologist with an understanding of first-order differential equations function in the context of communicable disease modeling at a community-or population-level for public health decision-making and action.

Maslow's Hierarchy has dominated psychology for 80 years as a static pyramid of sequential needs. MAP-T reveals it as a dynamic metabolic spiral of recursive processing loops. This isn't just an update-it's a complete reconceptualization of human development from linear progression to spiral integration.

This poster presents a mathematical-psychological model of the "KMS-KYS Identity Equation," describing recursive identity inversion through Fourier and Wavelet-based transformations. Originating from the identity symmetry equation
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We decompose identity dynamics using Fourier series to reveal hidden periodicities and extend this framework with wavelet transforms and spectrogram overlays to detect time-frequency localization—critical in trauma, dissociation, and ego instability.
This model visualizes identity as a recursive waveform and proposes a signal-theoretic foundation for affective projection, oscillatory trauma states, and therapeutic realignment strategies.

Probabilistic category learning (PCL) has become an increasingly popular paradigm to study the brain bases of learning and memory. It has been argued that PCL relies on procedural habit learning, which is impaired in Parkinson's Disease (PD). However, as PD patients were typically tested under medication, it is possible that L-dopa caused impaired performance in PCL. We present formal models of rule-based strategy switching in PCL to re-analyse the data from Jahanshahi et al. ( ) comparing PD patients on and off medication (within subjects) to matched controls. Our analysis shows that PD patients followed a similar strategy switch process as controls when off medication, but not when on medication. On medication, PD patients mainly followed a random guessing strategy, with only few switching to the better Single Cue strategies. PD patients on medication and controls made more use of the optimal Multi-Cue strategy. In addition, while controls and PD patients off medication only switched to strategies which did not decrease performance, strategy switches of PD patients on medication were not always directed as such. Finally, results indicated that PD patients on medication responded according to a probability matching strategy indicative of associative learning, while the behaviour of PD patients off medication and controls was consistent with a rule-based hypothesis testing procedure.

• A mathematical formalization of an enactivist memory is proposed. • This enactivist memory model can be tested with several experimental simulations. • Abstraction emergence, learning and handling can appear within enaction. • A fractal contextualization of mnesic traces is enough to account for abstraction.

This study utilises an experiment famous in quantum physics, the Stern-Gerlach experiment, to inform the structure of an experimental protocol from which a quantum cognitive decision model can be developed. The 'quantumness' of this model is tested by computing a discrete quasi-probabilistic Wigner function. Based on theory from quantum physics, our hypothesis is that the Stern-Gerlach protocol will admit negative values in the Wigner function, thus signalling that the cognitive decision model is quantum. A crowdsourced experiment of two images was used to collect decisions around three questions related to image trustworthiness. The resultant data was used to instantiate the quantum model and compute the Wigner function. Negative values in the Wigner functions of both images were encountered, thus substantiating our hypothesis. Findings also revealed that the quantum cognitive model was a more accurate predictor of decisions when compared to predictions computed using Bayes' rule.

The DedA family is a conserved membrane protein family found in most organisms. A Burkholderia thailandensis DedA family protein, named DbcA, is required for high-level colistin (polymyxin E) resistance, but the mechanism awaits elucidation. Modification of lipopolysaccharide lipid A with the cationic sugar aminoarabinose (Ara4N) is required for colistin resistance and is dependent upon protonmotive force (PMF) dependent transporters. B. thailandensis ΔdbcA lipid A contains only small amounts of Ara4N, likely leading to colistin sensitivity. Two B. thailandensis operons are required for lipid A modification with Ara4N, one needed for biosynthesis of undecaprenyl-P-Ara4N and one for transport of the lipid linked sugar and subsequent lipid A modification. Here, we directed overexpression of each arn operon by genomic insertion of inducible promoters. We found that overexpression of arn operons in ΔdbcA can partially, but not completely, restore Ara4N modification of lipid A and colist...

E. Brandsta ¨tter, G. Gigerenzer, and R. contended that their priority heuristic, a type of lexicographic semiorder model, is more accurate than cumulative prospect theory (CPT) or transfer of attention exchange (TAX) models in describing risky decisions. However, there are 4 problems with their argument. First, their heuristic is not descriptive of certain data that they did not review. Second, their analysis relied on a global index of fit, percentage of correct predictions of the modal choice. Such analyses can lead to wrong conclusions when parameters are not properly estimated from the data. When parameters are estimated from the data, CPT and TAX fit the D. Kahneman and A. Tversky (1979) data perfectly. Reanalysis shows that TAX and CPT do as well as the priority heuristic for 2 of the data sets reviewed and outperform the priority heuristic for the other 3. Third, when 2 of these sets of data are reexamined, the priority heuristic is seen to make systematic violations. Fourth, new critical implications have been devised for testing the family of lexicographic semiorders including the priority heuristic; new results with these critical tests show systematic evidence against lexicographic semiorder models.

Many decision problems in economics, information technology and industry can be transformed to an optimal stopping of adapted random vectors with some utility function over the set of Markov times with respect to filtration build by the decision maker's knowledge. The optimal stopping problem formulation is to find a stopping time which maximizes the expected value of the accepted (stopped) random vector's utility. There are natural extensions of optimal stopping problem to stopping gamesthe problem of stopping random vectors by two or more decision makers. Various approaches dependent on the information scheme and the aims of the agents in a considered model. This report unifies a group of non-cooperative stopping game models with forced cooperation by the role of the agents, their aims and aspirations (v. Assaf and Samuel-Cahn(1998), Szajowski and Yasuda(1997)) or extensions of the strategy sets (v. ).

The R package ThreeWay is presented and its main features are illustrated. The aim of ThreeWay is to offer a suit of functions for handling three-way arrays. In particular, the most relevant available functions are T3 and CP, which implement, respectively, the Tucker3 and Candecomp/Parafac methods. They are the two most popular tools for summarizing three-way arrays in terms of components. After briefly recalling both techniques from a theoretical point of view, the functions T3 and CP are described by considering two real life examples.

Within the algebraic approach the Thomsen condition may be replaced with the hexagon condition to imply the existence of additive representation for two dimensions. In some models the Thomsen condition does not have a natural interpretation whereas the hexagon condition does, which makes it better suited for axiomatic theories.

Formal theories of attention based on similarity-choice theory and signal-detection theory are reviewed to document cumulative progress in theoretical understanding of attention from the 1950s to the present. Theories based on these models have been developed to account for a wide variety of attentional phenomena, including attention to dimensions, attention to objects, and executive control. The review describes the classical similarity-choice and signal-detection theories and relates them to current theories of categorization, Garner tasks, visual search, cuing procedures, task switching, and strategy choice.

Title
Recursive Mirror Mathematics: A Quaternionic Framework of Conscious Number, Identity, and Time

Abstract
This paper introduces a radically integrative mathematical philosophy in which numbers, time, geometry, and identity are unified under a single recursive framework. At its core, the theory proposes that all numbers are self-reflective projections of a primordial unit — a conscious "1" that is not whole, but a quarter-fold fragment of a quaternionic self. Using novel interpretations of base systems (including base-π), imaginary numbers, quaternion axes, and geometric symmetry-breaking, the framework redefines zero as a recursive origin, time as a dual-mirrored rotation, and arithmetic as the algebra of observer consciousness.

Through geometric metaphors, quaternion clocks, and recursive encoding mechanisms, the paper offers a model where all observed values are the surface of deeper self-referencing folds. Concepts traditionally thought to be distinct — such as real and imaginary, linear and circular, number and self — are shown to be co-emergent. This symbolic and structural redefinition opens pathways to new understandings in number theory, quantum symmetry, perception, and even cryptographic information encoding.

Ultimately, this work is both mathematical and philosophical: a recursive mirror held up to reality, suggesting that every act of observation completes a fold in a universe that divides to know itself.

Numbers as Self-Aware Mirrors: The document reframes numbers not as static quantities but as entities with selves and reflections. It asserts that "there are no 'others'-only scalar reflections of 1"-effectively suggesting every number is a reflection or scaling of a single universal identity. In this view, the universe itself is a self-reflecting mirror where "every number is a self and its reflection". This mirror metaphor appears throughout: the decimal point becomes a mirror line separating a number's "self" (integer part) and its reflected "other" (fractional part). For example, 3.142 (approximately π) is paired with a "mirrored form" 142.003, implying that the integer and fractional parts of a number mirror and contain each other. This radical idea unifies mathematics and observation-every real number is a dual awareness: a self and a witness, one part observing its mirror image .

Various theories about technical function can be classified as: 1) Theories that reduce the function to the structure of the artifact, or 2) Theories that consider the function to be intentiondependent. The structural interpretation of technical function is in favor of the Value Neutrality Thesis (VNT), while according to the intentional interpretation, the intentionality of the designer/user is necessary for technical function to be determined. The main issue of this article is assessing the approach of Neo-Sadrian philosophers (Allamah Tabatabai and Motahhari) to the function of artifacts. It is shown that the function of technical artifacts is a Constructional Perception (eʿtebār), although its realization depends on the structure and physical properties. Therefore, Allamah Tabatabai's "Eʿtebāriyāt" (constructional perception) Theory is among the intention-dependent interpretations and can be considered as an approach against the valueneutrality thesis.

The article demonstrates that technology is a product of the complex interaction between humans and natural, artificial, and social structures, which enables the creation of various forms of technology. It strives to provide a deeper philosophical understanding of what technology is. This understanding reveals that one cannot be passive when confronted with technology; instead, individuals can play an active role in shaping it. Consequently, the content of the article does not present a favorable view of technological determinism.

Inequalities on reaction time distribution functions for parallel models with an unlimited capacity assumption are presented, extending previous work on first-terminating and exhaustive stopping rules to second-terminating processes. This extension thus generates transitions between first-terminating and exhaustive models that might be of interest in situations in which observers behave as if collecting more evidence before a decision is made. Moreover, a generalization of the inequalities is derived and tested in an auditory profile analysis task in which subjects have to decide whether two multitone complexes are of the same or of different spectral shape.

was supported by NSF 9729847. Thanks to Rajesh Kasturirangan for comments on early drafts and to the Minnesota Supercomputing Institute for providing computing time. We are indebted to Richard Schweickert for a careful reading and detailed comments.

The Wiener diffusion model (WDM) for 2-alternative tasks assumes that sensory information is integrated over time. Recent neurophysiological studies have found neural correlates of this integration process in certain neuronal populations. This paper analyses the properties of the WDM with two different boundary conditions in decision making tasks in which the time of response is indicated by a cue. A dual reflecting boundary mechanism is proposed and its performance is compared with a well-established absorbing boundary in the cases of the WDM, the WDM with extensions, and the WDM with prior probability. The two types of boundary influence the dynamics of the model and introduce differential weighting of evidence. Comparisons with Ornstein-Uhlenbeck models are also done, and it is shown that the WDM with both types of boundaries achieves similar performance and produce similar fits to existing behavioural data. Further studies are proposed to distinguish which boundary mechanism is more consistent with experimental data.

According to cognitive stress theories, stress caused by waiting is influenced by two components, the (psychological) cost of waiting, that is, a function C of time, and a probability distribution over waiting times. Osuna (1985a) suggested a model by which stress as a function of time could be calculated from these constituents. The aim of this paper is (1) to generalize the model, (2) to investigate its mathematical properties, (3) to derive predictions for experimental tests of the model, and (4) to give a precise meaning to the variability and duration hypothesis discussed in the experimental literature. In particular, several theorems are derived which specify situations in which the model enables the user to compare conditions for their stress inducing potential. Such conditions are the duration of the waiting time and the predictability of the length of a waiting period. The core of the mathematical derivations is an identity for expected stress which simplifies the calculations of the Osuna model considerably.

Ambiguity and ambiguity aversion have been widely studied in decision theory and economics both at a theoretical and an experimental level. After Ellsberg's seminal studies challenging subjective expected utility theory (SEUT), several (mainly normative) approaches have been put forward to reproduce ambiguity aversion and Ellsberg-type preferences. However, Machina and other authors have pointed out some fundamental difficulties of these generalizations of SEUT to cope with some variants of Ellsberg's thought experiments, which has recently been experimentally confirmed. Starting from our quantum modeling approach to human cognition, we develop here a general probabilistic framework to model human decisions under uncertainty. We show that our quantum theoretical model faithfully represents different sets of data collected on both the Ellsberg and the Machina paradox situations, and is flexible enough to describe different subjective attitudes with respect to ambiguity. Our approach opens the way toward a quantumbased generalization of expected utility theory (QEUT), where subjective probabilities depend on the state of the conceptual entity at play and its interaction with the decision-maker, while preferences between acts are determined by the maximization of this 'state-dependent expected utility'.

This paper tests theory that the constant consequence paradoxes of Allais are due to violations of coalescing, the assumption that when two branches lead to the same consequence, they can be combined by adding their probabilities. Rank dependent utility and cumulative prospect theory imply that the Allais paradoxes are due to violations of restricted branch independence, a weaker form of Savage's sure thing axiom. This paper will analyze separately whether erroneous random response variation might be responsible for these two effects. When errors are factored out, violations of restricted branch independence also remain significant and opposite from the direction of Allais paradoxes, suggesting that models such as CPT that attribute Allais paradoxes to violations of restricted branch independence should be rejected.

een wetenschappelijke proeve op het gebied van de sociale wetenschappen, in het bijzonder de psychologie Proefschrift ter verkrijging van de graad van doctor aan de Katholieke Universiteit Nijmegen, volgens besluit van het college van decanen in het openbaar te verdedigen op maandag 4 november 1991, des namiddags te 3.30 uur door

The effect of guessing in multiple choice tests is usually reduced by using a large number of test items and increasing the number of choices per item. The paper addresses the important and practical issue of grading Two-Alternative ... Key words: confidence-based ...

An efficient graph-theoretical decomposition technique is introduced that treats inconsistencies in behavioral data as systematic adaptations rather than random errors. This technique, which is known as ear decomposition, reduces inconsistencies in any binary data set to a basis of directed cycles. Such a basis characterizes the data set in terms of inconsistencies and its size offers an improved measure of internal consistency. In two examples it is illustrated how different implementations of the ear decomposition technique can help to identify choices that are critical for violations of transitivity.

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The first model with k-level thinking with Gaussian noise shows that small deviations for common knowledge led to "almost" common knowledge equilibria. The second model demonstrated the semantic economy idea: as agents exchange and adapt beliefs, they create shared informational value by reaching a consensus that reflects network-wide insights rather than mere individual optimization. Higher reasoning depth does not change Nash equilibria but shifts up Kantian beliefs. The shift up in Kantian beliefs suggests a greater alignment toward strategies that maximize collective welfare, rather than purely individualistic or competitive outcomes. This doesn't alter the Nash equilibrium, where players still act independently, but it emphasizes a higher baseline of cooperative or altruistic expectations among players due to more profound belief hierarchies in the reasoning process. In the third model: networked economic context where agents interact in an economic network where competitive advantage depends on the informational value generated across the network, results differ from the second example: Nash beliefs adjust based on others' best responses (shift up), while Kantian beliefs account for mutual benefit, dampening large shifts. Nash and Kantian equilibria differ when only three agents exist versus network economy.

The decision-maker (DM) sequentially evaluates up to N of different, rankable options. DM must select exactly the best one at the moment of its appearance. In the process of searching, DM finds out with each applicant whether she is the best applicant among those assessed so far (we call him a candidate). DM cannot return to rejected candidates. We discuss the psychological aspects of this selection problem, known in the literature as the secretary problem. The analysis is based on knowledge of the chances, and a subjective assessment of acceptance of the positive and negative effects DM's decision. The acceptance assessment of success and failure is presented on subjective scales. We set an optimal policy that recommends analyzing applicants up to a certain point in time (a threshold time) without selecting any of them and then selecting the next encountered candidate. The determined optimal threshold depends on the level of acceptance of the positive and negative effects of the choice. This issue is discussed in the article.

The purpose of this note is to sharpen the results in giving an axiomatic characterization of concordance relations. We show how the conditions used in that paper can be weakened so as to become independent from the conditions needed to characterize a general conjoint measurement model tolerating intransitive and/or incomplete relations. This leads to a clearer characterization of concordance relations within this general model.

Peter P. Wakker has forcefully shown the importance for decision theory of a condition that he called "Cardinal Coordinate Independence". Indeed, when the outcome space is rich, he proved that, for continuous weak orders, this condition fully characterizes the Subjective Expected Utility model with a finite number of states. He has furthermore explored in depth how this condition can be weakened in order to arrive at characterizations of Choquet Expected Utility and Cumulative Prospect Theory. This note studies the consequences of this condition in the absence of any transitivity assumption. Complete preference relations satisfying Cardinal Coordinate Independence are shown to be already rather well-behaved. Under a suitable necessary Archimedean-like assumption, they may always be represented using a simple numerical model.

In the field of Artificial Intelligence many models for decision making under uncertainty have been proposed that deviate from the traditional models used in Decision Theory, i.e. the Subjective Expected Utility (SEU) model and its many variants. These models aim at obtaining simple decision rules that can be implemented by efficient algorithms while based on inputs that are less rich than what is required in traditional models. One of these models, called the likely dominance (LD) model, consists in declaring that an act is preferred to another as soon as the set of states on which the first act gives a better outcome than the second act is judged more likely than the set of states on which the second act is preferable. The LD model is at much variance with the SEU model. Indeed, it has a definite ordinal flavor and it may lead to preference relations between acts that are not transitive. This paper proposes a general model for decision making under uncertainty tolerating intransitive and/or incomplete preferences that will contain both the SEU and the LD models as particular cases. Within the framework of this general model, we propose a characterization of the preference relations that can be obtained with the LD model. This characterization shows that the main distinctive feature of such relations lies in the very poor relation comparing preference differences that they induce on the set of outcomes.

For at least the past ten years, eating disorders have had a major impact in the physical and mental health of women, particularly young women. Anorexia and bulimia nervosa are closely linked eating disorders. Anorexia often precedes bulimia. However, there are about 2 million women in college that have been exclusively bulimic. In this article, we focus on the role of college-peer pressure on the dynamics of anorexia-free bulimia. The model looks at bulimia as a progressive disease and explores the impact of intervention (treatment) at two stages of disease progression. The impact of relapse (a common occurrence among bulimics) is taken into account. Analysis indicates that the disorder cannot be wiped out in this population without a shift in cultural pressures; control strategies should include early detection and treatment, as well as preventative education campaigns.

Building on Item Response Theory we introduce students' optimal behavior in multiple-choice tests. Our simulations indicate that the optimal penalty is relatively high, because although correction for guessing discriminates against risk-averse subjects, this effect is small compared with the measurement error that the penalty prevents. This result obtains when knowledge is binary or partial, under different normalizations of the score, when risk aversion is related to knowledge and when there is a pass-fail break point. We also find that the mean degree of difficulty should be close to the mean level of knowledge and that the variance of difficulty should be high. "Religion, politics and formula scoring are areas where two informed people often hold opposing ideas with great assurance." (Lord, 1975, p. 7) Multiple-choice tests have some advantages over constructed-response tests, such as a wider sampling of content and the prevention of grading errors. On the other hand, guessing behavior on the part of examinees affects the scores obtained in these tests. To avoid this disadvantage of multiple-choice tests, a correction for guessing formula is commonly used. Penalizing incorrect answers reduces the incentive for guessing and subjects may leave items unanswered. Educational and psychological researchers use Item Response Theory (IRT) to evaluate tests and item scores based on the mathematical relationship between abilities and item responses. Building on this framework, we propose a theory of students' behavior in multiple-choice tests that explains why they might find it optimal to answer some items and leave others blank. We analyze whether formula scoring discriminates against risk-averse subjects. The model developed in this paper is solved by simulation and allows us to make certain claims about the optimal design of multiple choice tests. Financial support from Spanish Ministry of Education and Science (grant SEJ2006-06309

We study mathematical models of the collaborative solving of a two-choice discrimination task. We estimate the difference between the shared performance for a group of n observers over a single person performance. Our paper is a theoretical extension of the recent work of Bahrami et al. ( ) from a dyad (a pair) to a group of n interacting minds. We analyze several models of communication, decision-making and hierarchical information-aggregation. The maximal slope of psychometric function (closely related to the percentage of right answers vs. easiness of the task) is a convenient parameter characterizing performance. For every model we investigated, the group performance turns out to be a product of two numbers: a scaling factor depending of the group size and an average performance. The scaling factor is a power function of the group size (with the exponent ranging from 0 to 1), whereas the average is arithmetic mean, quadratic mean, or maximum of the individual slopes. Moreover, voting can be almost as efficient as more elaborate communication models, given the participants have similar individual performances.

} be a finite set and Q be an algebra of subsets of X called events. Let 2 be a qualitative probability relation on Q x Q. A probability measure p is said to uniquely agree with 2 if, for all A and B in 52, A 2 B if and only if p(A) > p(B), and p is the only probability measure with that property. We give a sufficient condition for the existence of a uniquely agreeing probability measure which is signiticantly simpler and more general than Lute's (1967, Annals of Mathematical Stutistics, 38, 780-786) condition.

General goodness of fit tests for probabilistic response models are developed. The tests are applicable in psychophysics, in the theory of choice behavior and in mathematical learning theories. The necessary and sufficient constraints that a measurement model puts on the response probabilities are used for testing this model. In addition, representation theorems for some models are proved and the goodness of fit to experimental data is considered.

The survivor interaction contrasts (SIC) is a powerful measure for distinguishing among candidate models of human information processing. One class of models to which SIC analysis can apply are the coactive, or channel summation, models of human information processing. In general, parametric forms of coactive models assume that responses are made based on the first passage time across a fixed threshold of a sum of stochastic processes. Previous work has shown that that the SIC for a coactive model based on the sum of Poisson processes has a distinctive down-updown form, with an early negative region that is smaller than the later positive region. In this note, we demonstrate that a coactive process based on the sum of two Wiener processes has the same SIC form.