Journal of Mathematical Psychology最新文献

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.

Although knowledge is extremely high-dimensional, human episodic memory performance appears extremely low-dimensional, focused largely on stimulus-features that distinguish list items from one another. A cognitively plausible way this tension could be addressed is if selective attention selects a small number of features from each item. I consider an ongoing debate about whether stronger items (better encoded) interfere more than weaker items (less well encoded) with probe items during old/new episodic recognition judgements. This is called the list-strength effect, concerning whether or not effects of encoding strength are larger in lists of mixed strengths than in pure lists of a single strength. Analytic derivations with Anderson’s (1970) matched filter model show how storing only a small subset of features within high-dimensional representations, and assuming those same subsets tend to reiterate themselves item-wise at test, can support high recognition performance. In the sparse regime, the model produces a list-strength effect that is small in magnitude, resembling previous findings of so-called null list-strength effects. When the attended feature space is compact, such as for phonological features, attentional subsetting cannot be sparse. This introduces non-negligible cross-talk from other list items, producing a large-magnitude list-strength effect, similar to what is observed for the production effect (better recognition when reading aloud). This continuum-based account implies the existence of a continuous range of magnitudes of list-composition effects, including occasional inverted list-strength effects. This lays the foundation for propagating effects of task-relevant attention to sparse subsets of features through a broad range of models of memory behaviour.

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.

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.

Partial orders defined on a nonempty set $X$ admitting a two-agent Pareto representation are characterized. The characterization is based upon the fulfillment of two axioms. The first one entails the existence, for any point $x\in X$, of a very particular decomposition of the points which are incomparable to $x$. The second one encodes a separability condition. Our approach is then applied to show that if the cardinality of $X$ is, at most, 5, then a two-agent Pareto representation always exists whereas this need not be the case otherwise. The connection with the concept of the dimension of a poset is also discussed. Certain examples are also presented that illustrate the scope of our tools.

This paper generalizes subjective expected utility by incorporating signed threshold, whose positive (respectively, negative) value enhances (respectively, reduces) subjective expected utility of chosen alternative against unchosen one. It can be interpreted, for example, that positivity of the signed threshold reflects domination of rejoicing feeling against regret feeling. Since the signed threshold representation is a special case of skew-symmetric additive (SSA) representation, we prove that in addition to SSA axiomatization, restriction of probabilistic sophistication to pairs of acts which are regret-free separates subjective expected utility and signed threshold. It is assumed that regret-freeness is measured by monetary differences or ex post strength of preferences.

Many decision making theories assume a principle of sequentially sampling decision-relevant evidence from the stimulus environment, where sampled evidence is dynamically accumulated toward a threshold to trigger a decision in favour of the threshold-crossing option. A core prediction of sequential sampling models is that options more likely to be chosen are chosen more quickly. This result has been empirically supported hundreds of times for low-level speeded perceptual decisions — the traditional domain of sequential sampling models. More recently, sequential sampling models have been generalised and applied to higher-level preferential, or value-based, decisions — decisions for which there is no objectively correct option. Preferential options are typically composed of multiple attributes, like a phone defined by its price, camera quality, memory capacity, and so on. Here, we show that decisions for such multi-attribute preferential options with defined features violate the core prediction of sequential sampling models: options more likely to be chosen are not chosen more quickly. We find this invariance across 4 data sets spanning multi-attribute choices made in unconstrained conditions, under time pressure, and for multi-attribute options with artificial or marketplace compositions. The result remains whether the relationship between choice frequency and choice time is inspected at the lower level of component attributes or the higher level of whole options. Our finding places critical constraints on the capacity to generalise sequential sampling models from low-level perceptual decisions to high-level multi-attribute preferential choice.

This study aims to clarify sufficient conditions for weak orders on the existing and null alternatives to make leximax and leximin rules over the power set satisfy extensibility. Each null alternative indicates ‘choosing not to choose the corresponding existing alternative’. Extensibility requires that a preference order of any two alternatives is equivalent to that of their singleton sets. Then, the leximax (alternatively, leximin) rule ranks any two subsets by comparing the same-ranked (null) alternatives in the two transformed subsets (which include the existing alternatives in each subset and the null alternatives of other existing alternatives) from top to bottom (alternatively, bottom to top). We then introduce the following two new properties: Semi-inversion desirability requires that a preference of any two null alternatives is not identical to that of their existing alternatives. Consistent desirability requires that a preference order of ‘a null alternative and a non-paired existing alternative’ is not identical to that of their paired (null) alternatives. We show that semi-inversion desirability implies extensibility, and the combination of semi-inversion desirability and consistent desirability is weaker than a traditional property, namely self-reflecting. Furthermore, we clarify the sufficient condition to make the leximax and leximin rules equivalent.

The present article lays out the foundations of an axiomatic theory of attribute maps, an extension of skill maps to polytomous knowledge structure theory. A deterministic relationship between the available attributes and the observable item responses is established by means of two mappings denoted attribute map and item–response function. The attribute map assigns to each item–response pair the set of attributes that are instrumental for observing that particular response to the item. The item–response function assigns to each set of attributes the set of item responses that, according to the attribute map, can be obtained with those attributes. The proposed approach is shown to be rather general and capable of handling a multitude of polytomous items that can be encountered in practice. Examples are provided that cover the analysis of responses on Likert scales, responses awarded partial credits, and the investigation of misconceptions.

In response time (RT) research, it is common to instruct participants to respond as fast and as accurately as possible, which is easily conceived as a contradiction. Participants may overcome this dilemma using a two-fold strategy, with (A) delaying their response until they feel confident that enough information has been sampled, and (B) scheduling the response right before the end of the response window to avoid omissions. The purpose of this strategy is to satisfy the contradictory requirements of the task instructions, but both (A) and (B) may yield a distorted picture of the processing times under investigation. We asked participants to discriminate random dot motion with fixed and variable deadlines for responding. With the exponentially distributed variable deadline, strategic responding is useless because it is impossible to schedule an optimal time point for the response. We present two analyses, a model-free approach that investigates the effect of an unpredictable deadline on standard RT measures, and the fit of an RT model testing for effects of the deadline on specific parameters. Compared to the fixed deadline, faster responses that were less variable across participants were observed under the variable deadline, suggesting that the new paradigm can reduce strategic responding. We demonstrate how to deal with omitted responses and conclude that variable deadlines are a promising tool to exert time pressure in RT experiments and potentially yield better estimates of the underlying processing times.