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.

Dynamical models are crucial for developing process-oriented, quantitative theories in cognition and behavior. Due to the impressive progress in cognitive theory, domain-specific dynamical models are complex, which typically creates challenges in statistical inference. Mathematical models of eye-movement control might be looked upon as a representative case study. In this tutorial, we introduce and analyze the SWIFT model (Engbert et al., 2002; Engbert et al., 2005), a dynamical modeling framework for eye-movement control in reading that was developed to explain all types of saccades observed in experiments from an activation-based approach. We provide an introduction to dynamical modeling, which explains the basic concepts of SWIFT and its statistical inference. We discuss the likelihood function of a simplified version of the SWIFT model as a key foundation for Bayesian parameter estimation (Rabe et al., 2021; Seelig et al., 2019). In posterior predictive checks, we demonstrate that the simplified model can reproduce interindividual differences via parameter variation. All computations in this tutorial are implemented in the R-Language for Statistical Computing and are made publicly available. We expect that the tutorial might be helpful for advancing dynamical models in other areas of cognitive science.

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.

We introduce a context-dependent theory for choice under risk, called range utility theory. It builds on Parducci’s range principle from psychophysics and modifies expected utility by positing that risky prospects are evaluated relative to the range of consequences of all prospects in the decision context. When the context is fixed, choices typically exhibit the four-fold pattern of risk preferences, yet are fully consistent with expected utility (linear in probabilities) without invoking rank-principles. We illustrate this advantage in game theory contexts. As the same time, when the context varies, the relative value of an alternative also does, yielding different forms or preference reversals, some of which have been robustly documented.

In this paper, we propose a dynamical model for the best–worst choice task. The proposed model is a modification of the multi-episode Cube model proposed and studied in so-called (Mallahi-Karai and Diederich, 2019, 2021). This model postulates that best–worst choice (or more generally, ranking) task is the outcome of sequential choices made in a number of episodes. The underlying model is a multivariate Wiener process with drift issued from a point in the unit cube, where episodes are defined in terms of a sequence of stopping times. This model can also be extended to an attention-switching framework in a standard way.

Psychological theories are often formulated at the level of latent, not directly observable, variables. Empirical measurement of latent variables ought to be valid. Classical psychometric validity indices can be difficult to apply in experimental contexts. A complementary validity index, termed retrodictive validity, is the correlation of theory-derived predicted scores with actually measured scores, in specifically designed calibration experiments. In the current note, I analyse how calibration experiments can be designed to maximise the information garnered and specifically, how to minimise the sample variance of retrodictive validity estimators. First, I harness asymptotic limits to analytically derive different distribution features that impact on estimator variance. Then, I numerically simulate various distributions with combinations of feature values. This allows deriving recommendations for the distribution of predicted values, and for resource investment, in calibration experiments. Finally, I highlight cases in which a misspecified theory is particularly problematic.

Forward-graded and backward-graded structures of knowledge are two important classes of knowledge structures. Spoto and Stefanutti (2020) establish necessary and sufficient conditions for skill maps to delineate these structures. We introduce fuzzy skills to describe varying levels of proficiency in skills and extend the theoretical results of Spoto and Stefanutti (2020) for delineating forward- and backward-graded knowledge structures using fuzzy skill maps. The paper establishes necessary and sufficient conditions for fuzzy skill maps to delineate a backward-graded simple closure space, a forward-graded knowledge space, and a forward-graded simple closure space. Furthermore, the competence-based local independence model (CBLIM) with fuzzy skills is introduced and its unidentifiability is discussed.

The paper studies a variety of domains of preference orders that are closely related to single-peaked preferences. We develop recursive formulas for the number of single-peaked preference profiles and the number of preference profiles that are single-peaked on a circle. The number of Arrow’s single-peaked preference profiles is found for three, four, and five alternatives. Random sampling applications are discussed. For restricted tier preference profiles, a forbidden subprofiles characterization and an exact enumeration formula are obtained. It is also shown that each Fishburn’s preference profile is single-peaked on a circle preference profile, and Fishburn’s preference profiles cannot be characterized by forbidden subprofiles.

We introduce a new approach to modelling decision confidence, with the aim of enabling computationally cheap predictions while taking into account, and thereby exploiting, trial-by-trial variability in stochastically fluctuating stimuli. Using the framework of the drift diffusion model of decision making, along with time-dependent thresholds and the idea of a Bayesian confidence readout, we derive expressions for the probability distribution over confidence reports. In line with current models of confidence, the derivations allow for the accumulation of “pipeline” evidence that has been received but not processed by the time of response, the effect of drift rate variability, and metacognitive noise. The expressions are valid for stimuli that change over the course of a trial with normally-distributed fluctuations in the evidence they provide. A number of approximations are made to arrive at the final expressions, and we test all approximations via simulation. The derived expressions contain only a small number of standard functions, and require evaluating only once per trial, making trial-by-trial modelling of confidence data in stochastically fluctuating stimuli tasks more feasible. We conclude by using the expressions to gain insight into the confidence of optimal observers, and empirically observed patterns.

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.