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A wealth of empirical evidence shows that people display opposite behaviors when deciding whether to rely on an algorithm, even if it is inexpensive to do so and using the algorithm should enhance their own performance. This paper develops a formal theory to explain some of these conflicting facts and submit new testable predictions. Drawing from decision analysis, I invoke two key notions: the 'value of information' and the 'value of control'. The value of information matters to users of algorithms like recommender systems and prediction machines, which essentially provide information. I find that ambiguity aversion or a subjective cost of employing an algorithm will tend to decrease the value of algorithmic information, while repeated exposure to an algorithm might not always increase this value. The value of control matters to users who may delegate decision making to an algorithm. I model how, under partial delegation, imperfect understanding of what the algorithm actually does (so the algorithm is in fact a black box) can cause algorithm aversion. Some possible remedies are formulated and discussed. • This paper initiates a formal decision-theoretic approach to make sense of the empirical evidence concerning people's attitudes towards algorithms. • This approach exploits two fundamental notions: the value of information and the value of control. • Ambiguity aversion will tend to decrease the value of algorithmic information; repeated exposure to algorithms may not increase it. • A first model of 'black box' algorithms is developed to analyze the value of keeping versus delegating control. [ABSTRACT FROM AUTHOR]

This paper demonstrates how averaging over individual learning and forgetting curves gives rise to transformed averaged curves. In an earlier paper (Murre and Chessa, 2011), we already showed that averaging over exponential functions tends to give a power function. The present paper expands on the analyses with exponential functions. Also, it is shown that averaging over power functions tends to give a log power function. Moreover, a general proof is given how averaging over logarithmic functions retains that shape in a specific manner. The analyses assume that the learning rate has a specific statistical distribution, such as a beta, gamma, uniform, or half-normal distribution. Shifting these distributions to the right, so that there are no low learning rates (censoring), is analyzed as well and some general results are given. Finally, geometric averaging is analyzed, and its limits are discussed in remedying averaging artefacts. • Averaging over individual learning (or forgetting, etc.) curves changes their shape, where exponential functions may change into power functions and power functions into log power functions. • Averaging over logarithmic functions retains the shape. • For exponential and power functions, geometric averaging retains the shape. • Geometric averaging over logarithmic functions does not retain that shape. [ABSTRACT FROM AUTHOR]

Self-determination theory is a well-established theory of motivation. This theory provides for fundamental concepts related to human motivation, including self-determination. The mathematization of this theory has been envisaged in a series of two papers by the author. The first paper entitled "Mathematical self-determination theory I: Real representation" addressed the representation of the theory in reals. This second paper is in continuation of it. The representation of the first part allows to abstract the results in more general mathematical structures, namely, affine spaces. The simpler real representation is reobtained as a special instance. We take convexity as the pivotal starting point to generalize the whole exposition and represent self-determination theory in abstract affine spaces. This includes the affine space analogs of the notions of internal locus, external locus, and impersonal locus, of regulated and graded motivation, and self-determination. We also introduce polar coordinates in Euclidean affine motivation spaces to study self-determination on radial and angular line segments. We prove the distributivity of the lattice of general self-determination in the affine space formulation. The representation in an affine space is free in the choice of primitives. However, the different representations, in reals or affine, are shown to be unique up to canonical isomorphism. The aim of this paper is to extend on the results obtained in the first paper, thereby to further lay the mathematical foundations of self-determination motivation theory. • Mathematization of self-determination theory. • Combinatorial definitions and characterizations of self-determination. • Algebraic self-determination. • Self-determination theory in reals. • Self-determination theory in affine space. [ABSTRACT FROM AUTHOR]

Doignon and Falmagne (1985) introduced a surmise system, which generalized the precedence relation, allowing multiple possible learning paths for an item. Heller (2021) took into account precedence relations on an extended set of (virtual) items and further generalized quasi-ordinal knowledge spaces to polytomous items. Wang et al. (2022) proposed CD-polytomous knowledge space and provided its corresponding polytomous surmise system. Following these developments and drawing upon the so-called extended polytomous knowledge structure, this paper presents two concepts: weak polytomous structure and extended surmise system. Via setting up a Galois connection, a one-to-one correspondence is established between the collection of all extended surmise functions and the collection of certain weak polytomous structures. This paper also comprehensively discusses the relationships among the precedence relations, the polytomous surmise systems, and the extended surmise systems. • A new type of polytomous surmise system is proposed. • The concept of weak polytomous knowledge structure is provided. • The Galois connection is established. • The concepts associated with the extended surmise system are discussed. [ABSTRACT FROM AUTHOR]

In two parts, MSDT1 this paper and MSDT2 the follow-up paper, we treat the topic of mathematical self-determination theory. MSDT1 considers the real representation, MSDT2 the affine space representation. The aim of the two papers is to lay the mathematical foundations of self-determination motivation theory. Self-determination theory was proposed by Deci and Ryan, which is a popular theory of motivation. The fundamental concepts are extrinsic and intrinsic motivation, amotivation, their type of regulation, locus of causality, and especially, self-determination. First, we give a geometric description of its concepts for the regulated case (no amotivation), as the unit 1-simplex. Thereby, we derive a symmetric definition of self-determination. Second, we extend the geometric description to the regulated and unregulated case, based on a more general ternary model, in internal motivation, external motivation, and amotivation. We define gradations of amotivation (and motivation), as 1-simplexes parallel to the unit 1-simplex. The ternary representation implies the types of strong, weak, and general self-determination, as partial orders on the motivation space. Third, we study the order, lattice, and algebraic properties of self-determination. In a version of polar coordinates, strong self-determination turns out to be a complete lattice on angular line segments, weak self-determination is a complete lattice on radial line segments, and general self-determination entails a complete lattice on the entire motivation space. In addition, the modified polar coordinates are employed to obtain necessary and sufficient conditions for strong, weak, and general self-determination. We propose measures for the strength of an ordinal dependency in self-determination, which are partial metrics on the motivation space. [ABSTRACT FROM AUTHOR]

In this paper, we establish a formal connection between two dynamic modeling approaches that are often taken to study affect dynamics. More specifically, we show that the exponential discounting model can be rewritten to a specific case of the VARMAX, thereby shedding light on the underlying similarities and assumptions of the two models. This derivation has some important consequences for research. First, it allows researchers who use discounting models in their studies to use the tools established within the broader time series literature to evaluate the applicability of their models. Second, it lays bare some of the implicit restrictions discounting models put on their parameters and, therefore, provides a foundation for empirical testing and validation of these models. One of these restrictions concerns the exponential shape of the discounting function that is often assumed in the affect dynamical literature. As an alternative, we briefly introduce the quasi-hyperbolic discounting function. • When investigating dynamical systems, it is important to understand how different models relate to each other. • We analytically demonstrate that discounting models are nested within autoregressive models. • This demonstration exposes some assumptions of the discounting model regarding the dynamic structure of the data. • These assumptions should be tested in empirical studies to assess the viability of discounting models. • One assumption to test is exponential decay, for which we propose a quasi-hyperbolic alternative. [ABSTRACT FROM AUTHOR]

Recent crowdsourcing experiments have shown that true contextuality of the kind found in quantum mechanics can also be present in human behavior. In these experiments simple human choices were aggregated over large numbers of respondents, with each respondent dealing with a single context (set of questions asked). In this paper we present experimental evidence of contextuality in individual human behavior, in a psychophysical experiment with repeated presentations of visual stimuli in randomly varying conteXts (arrangements of stimuli). The analysis is based on the Contextuality-by-Default (CbD) theory whose relevant aspects are reviewed in the paper. CbD allows one to detect contextuality in the presence of direct influences, i.e., when responses to the same stimuli have different distributions in different contexts. The experiment presented is also the first one in which contextuality is demonstrated for responses that are not dichotomous, with five options to choose among. CbD requires that random variables representing such responses be dichotomized before they are subjected to contextuality analysis. A theorem says that a system consisting of all possible dichotomizations of responses has to be contextual if these responses violate a certain condition, called nominal dominance. In our experiment nominal dominance was violated in all data sets, with very high statistical reliability established by bootstrapping. • Contextuality is established in an experiment with a within-subject design. • The choices made in this experiment were 5-optional. • The analysis is based on the Contextuality-by-Default theory. • A necessary condition for noncontextuality of such systems is nominal dominance. • This condition was violated for all three participants. [ABSTRACT FROM AUTHOR]

Based on the notion of an assessment structure introduced by Falmagne (2015) the present paper characterizes the so-called quasi ordinal assessment spaces. It establishes a one-to-one correspondence to certain binary relations on an extended item set, containing a positive instance and a negative instance for each of the items. This result allows for building quasi ordinal assessment spaces from data through a generalized version of a well-established procedure, known as item tree analysis. Analyzing data from psychological testing constitutes a prototypical application that calls for this generalization of knowledge spaces, as it allows for describing dependencies between positive and negative answers to the items in a questionnaire. Empirical application of the approach is illustrated for a published data set on the Woodworth Psychoneurotic Inventory. • Assessment structures generalize knowledge structures. • They form structures appropriate for application in psychological diagnostics. • The paper provides a theoretical basis for building these structures from data. • The development focuses on quasi ordinal assessment structures. • The main result generalizes the well-known Birkhoff Theorem. [ABSTRACT FROM AUTHOR]

Multiple-choice and continuous-response tasks pose a number of challenges for models of the decision process, from empirical challenges like context effects to computational demands imposed by choice sets with a large number of outcomes. This paper develops a general framework for constructing models of the cognitive processes underlying both inferential and preferential choice among any arbitrarily large number of alternatives. This geometric approach represents the alternatives in a choice set along with a decision maker's beliefs or preferences in a "decision space," simultaneously capturing the support for different alternatives along with the similarity relations between the alternatives in the choice set. Support for the alternatives (represented as vectors) shifts over time according to the dynamics of the belief / preference state (represented as a point) until a stopping rule is met (state crosses a hyperplane) and the corresponding selection is made. This paper presents stopping rules that guarantee optimality in multi-alternative inferential choice, minimizing response time for a desired level of accuracy, as well as methods for constructing the decision space, which can result in context effects when applied to preferential choice. [ABSTRACT FROM AUTHOR]

A state-trace model for a natural phenomenon proposes that there is a causal "bottleneck" that makes the joint causal effects of two independent variables look like the effect of a single independent variable. Imagine that a state-trace model has been formulated, but the model seems implausible and we wish to find empirical evidence that it is wrong. How can we do that? Given reasonable monotonicity assumptions, a state-trace model is wrong if a point on one state trace is delta-discordant with a point on another state trace. [We say that two points (x , y) and (x ′ , y ′) in the plane are delta-discordant if Δ x = x ′ − x and Δ y = y ′ − y are nonzero and have opposite signs.] So, to show the model wrong, we need to find a pair of delta-discordant points lying on different state traces. One method for finding two such points is to observe multiple points on each of two state traces with the hope that, among those points, there will be a delta-discordant pair. But this approach can have bad luck and fail to find delta-discordant pairs even though they exist. A new approach to "locating" a delta-discordant pair of points, an approach that is much less dependent on luck, is formulated in this paper. In addition, the paper describes a statistical test for evaluating whether the "located" points are really delta-discordant. • Imagine that we want to discredit a particular state-trace model. • We can do that by finding delta-discordant points on different state traces. • But, even if such points exist, we may fail to find them. • We need a systematic method for finding delta-discordant points. • This paper describes such a method. [ABSTRACT FROM AUTHOR]

This paper contributes to quantum-like modeling of decision making (DM) under uncertainty through application of Heisenberg’s uncertainty principle (in the form of the Robertson inequality). In this paper we apply this instrument to quantify uncertainty in DM performed by quantum-like agents. As an example, we apply the Heisenberg uncertainty principle to the determination of mutual interrelation of uncertainties for “incompatible questions” used to be asked in political opinion pools. We also consider the problem of representation of decision problems, e.g., in the form of questions, by Hermitian operators, commuting and noncommuting, corresponding to compatible and incompatible questions respectively. Our construction unifies the two different situations (compatible versus incompatible mental observables), by means of a single Hilbert space and of a deformation parameter which can be tuned to describe these opposite cases. One of the main foundational consequences of this paper for cognitive psychology is formalization of the mutual uncertainty about incompatible questions with the aid of Heisenberg’s uncertainty principle implying the mental state dependence of (in)compatibility of questions. [ABSTRACT FROM AUTHOR]

Enhancing learning effectiveness and comprehension, well-gradedness plays a crucial role in knowledge structure theory by establishing a systematic and progressive knowledge system. Extensive research has been conducted in this domain, resulting in significant findings. This paper explores the properties of well-gradedness in polytomous knowledge structures, shedding light on both classical confirmations and exceptional cases. A key characteristic of well-gradedness is the presence of adjacent elements within a non-empty family that exhibit a distance of 1. The study investigates various manifestations of well-gradedness, including its discriminative properties and its manifestation in discriminative factorial polytomous structures. Furthermore, intriguing deviations from classical standards in minimal polytomous states are uncovered, revealing unexpected behaviors. • Provide a characterization of well-gradedness for general families. • Introduce the concept of strong discriminative property. • Manifestation of well-gradedness in polytomous knowledge structures. • Demonstrate that factorial polytomous structures are well-graded. [ABSTRACT FROM AUTHOR]

This paper illustrates applications of a new, modern version of quantum logic in quantum cognition. The new logic uses ‘effects’ as predicates, instead of the more restricted interpretation of predicates as projections — which is used so far in this area. Effect logic involves states and predicates, validity and conditioning, and also state and predicate transformation via channels. The main aim of this paper is to demonstrate the usefulness of this effect logic in quantum cognition, via many high-level reformulations of standard examples. The usefulness of the logic is greatly increased by its implementation in the programming language Python . [ABSTRACT FROM AUTHOR]

By modifying the concept of polytomous surmise functions, this paper introduces polytomous surmising functions. Then, it is shown that there is a one-to-one correspondence f between granular polytomous spaces and polytomous surmising functions where polytomous surmising functions cannot be replaced with polytomous surmise functions. This result gives a correction for a correspondence between granular polytomous spaces and polytomous surmise functions. As an application of the correspondence f , this paper demonstrates that the pair (f , f − 1) of mappings forms a Galois connection where all granular polytomous spaces and all polytomous surmising functions are closed elements of this Galois connection. • The polytomous surmising function is introduced. • Set up a one-to-one correspondence defined on granular polytomous spaces. • Set up a Galois connection around granular polytomous spaces. [ABSTRACT FROM AUTHOR]