Journal of Mathematical Psychology最新文献

The elaboration of preference relations and their representations trace their source to early economic theory. Classical representations of preferences theorems rely on Debreu–Eilenberg’s theorems in the topological setting. An important class of preferences consists of interval orders. A natural question is to achieve a bi-utility representation for interval orders. We suggest to introduce a condition reminiscent of N. Wiener’s early works on the relativeness of positions. We obtain a bi-utility representation through the precedence and succession relations.

Mathematical models of association memory (study AB, given A, recall B) either predict that knowledge for constituent order of a word pair (AB vs. BA) is perfectly unrelated, or completely dependent on knowledge of the pairing itself. Data contradict both predictions; when a pair is remembered, constituent-order is above chance, but still fairly low. Convolution-based models are inherently symmetric and can explain associative symmetry, but cannot discriminate AB from BA. We evaluated four extensions of convolution, where order is incorporated as item features, partial permutations of features, item-position associations, or by adding item and position vectors. All approaches could discriminate order within behaviourally observed ranges, without compromising associative symmetry. Only the permutation model could disambiguate AB from BC in double-function lists, as humans can do. It is possible that each of our proposed mechanisms might apply to a different, particular task setting. However, the partial permutation model can thus far explain the broadest set of empirical benchmarks.

Decisions that involve bundling or unbundling a large number of objects, such as deciding on the bundle structure or optimizing bundle prices, are based on underlying valuation function over the set of all possible bundles. Given that the number of possible bundles (i.e., subsets of the given set of objects) is exponential in the number of objects, it is important for the decision-maker to be able to represent this valuation function succinctly. Identifying all structural sources of synergy in subset valuations might point to simple and concise representation of the valuation function. We characterize additive and multiplicative representations of synergies in subset valuations and subset utility, which in turn points to necessary and sufficient conditions for a succinct representation of subset valuations to exist.

Three context effects pertaining to stochastic discrete choice have attracted a lot of attention in Psychology, Economics and Marketing: the similarity effect, the compromise effect and the asymmetric dominance effect. Despite this attention, the existing literature is rife with conflicting definitions and misconceptions. We provide theorems relating different variants of each of the three context effects, and theorems relating the context effects to conditions on discrete choice probabilities, such as random utility, regularity, the constant ratio rule, and simple scalability, that may or may not hold for any given discrete choice model. We show how context effects at the individual level may or may not aggregate to context effects at the population level. Importantly, we offer this work as a guide for researchers to sharpen empirical tests and aid future development of choice models.

Understanding how attentional resources are deployed in visual processing is a fundamental and highly debated topic. As an alternative to theoretical models of visual search that propose sequences of separate serial or parallel stages of processing, we suggest a queueing processing structure that entails a serial transition between parallel processing stages. We develop a continuous-time queueing model for standard visual search tasks to formalize and implement this notion. Specified as a finite-time, single-line, multiserver queueing system, the model accounts for both accuracy and response time (RT) data in visual search on a distributional level. It assumes two stages of processing. Visual stimuli first go through a massively parallel preattentive stage of feature encoding. They wait if necessary and then enter a limited-capacity attentive stage serially where multiple processing channels (“servers”) integrate features of several stimuli in parallel. A core feature of our model is the serial transition from the unlimited-capacity preattentive processing stage to the limited-capacity attentive processing stage. It enables asynchronous attentive processing of multiple stimuli in parallel and is more efficient than a simple chain of two successive, strictly parallel processing stages. The model accounts for response errors by means of two underlying mechanisms, namely, imperfect processing of the servers and, in addition, incomplete search adopted by the observer to maximize search efficiency under an accuracy constraint. For statistical inference, we develop a Monte-Carlo-based parameter estimation procedure, using maximum likelihood (ML) estimation for accuracy-related parameters and minimum distance (MD) estimation for RT-related parameters. We fit the model to two large empirical data sets from two types of visual search tasks. The model captures the accuracy rates almost perfectly and the observed RT distributions quite well, indicating a high explanatory power. The number of independent parallel processing channels that explain both data sets best was five. We also perform a Monte-Carlo model uncertainty analysis and show that the model with the correct number of parallel channels is selected for more than 90% of the simulated samples.

A total preorder is a transitive and complete binary relation on a set. A modal preference structure of rank $n$ is a string composed of 2 to the exponent $n$ binary relations on a set such that there is a family of total preorders that gives all relations by taking intersections and unions. Total preorders are structures of rank zero, NaP-preferences (Giarlotta and Greco, 2013) are structures of rank one, and GNaP-preferences (Carpentiere et al., 2022) are structures of rank two. We characterize modal preference structures of any rank by properties of transitive coherence and mixed completeness. Moreover, we show how to construct structures of a given rank from others of lower rank. Modal preference structures arise in economics and psychology, in the process of aggregating hierarchical judgements of groups of agents, where each of the $n$ coordinates represents a feature/stage of the decision procedure.

Drift-diffusion models have become valuable tools in many fields of contemporary psychology and the neurosciences. The present study compares and analyzes different methods (i.e., stochastic differential equation, integral method, Kolmogorov equations, and matrix method) to derive the first-passage time distribution predicted by these models. First, these methods are compared in their accuracy and efficiency. In particular, we address non-standard problems, for example, models with time-dependent drift rates or time-dependent thresholds. Second, a mathematical analysis and a classification of these methods is provided. Finally, we discuss their strengths and caveats.

Heller (2021) and Stefanutti et al. (2020) provided the mathematical foundation for the generalization of knowledge structure theory (KST) to polytomous items. Based on their works, the well-gradedness can be extended to polytomous knowledge structures. We propose the concepts of discriminative polytomous knowledge structure and well-graded polytomous knowledge structure. Then we show that every well-graded polytomous knowledge structure is discriminative. The basis of any polytomous knowledge space is formed by the collection of all the atoms. We discuss the sufficient and necessary conditions of polytomous knowledge structures to be well-graded polytomous knowledge spaces. Moreover, we provide an example to illustrate that a well-graded polytomous knowledge space is not necessarily a polytomous closure space.

The main aim of the current paper is to propose Item Response Theory (IRT) models derived from the nondecomposable measurement theories presented in Fishburn (1974). More specifically, we aim to: (i) present the theoretical basis of the Rasch model and its relations to psychophysics’ models of utility; (ii) give a brief exposition on the measurement theories presented in Fishburn (1974, 1975), some of which do not require an additive structure; and (iii) derive IRT models from these measurement theories, as well as Bayesian implementations of these models. We also present two empirical examples to compare how well these IRT models fit to real data. In addition to deriving new IRT models, we also discuss theoretical interpretations regarding the models’ capability of generating fundamental measures of the true scores of the respondents. The manuscript ends with prospects for future studies and practical implications.