Journal of Mathematical Psychology最新文献

Several papers co-authored by A.A.J. Marley helped in popularizing the best-worst-choice paradigm due to Finn and Louviere (1992). Inspired by Block and Marschak (1960), Marley conceived a random utility model for the choice frequencies of the best and worst alternatives in any proposed set of alternatives (Marley and Louviere, 2005). He then asked for a characterization of the prediction range of the model. The range being a convex polytope, an affine description of this polytope would provide a solution to Marley problem. For four alternatives, we show that a minimal such description consists in 26 affine equalities and 144 affine inequalities. The result derives from the Gale transform of the set of polytope vertices: the transform being a family of 24 vectors in a one-dimensional vector space, it plainly reveals the affine structure of the polytope. As far as we know, Marley problem is still open when the number of alternatives exceeds 4.

A novel and easy-to-compute measure is proposed that compares the relative contribution of each parameter of a mathematical model to the model’s mathematical flexibility or complexity, with respect to accounting for the results of some specific experiment. When the data space is a two-dimensional plot of the type used in standard state-trace analysis, then the model complexity contributed by a single parameter equals the length of the state trace (LOST) that results when that parameter is varied and all other parameters are held constant. For the normal, equal-variance, signal-detection model, the average LOST when the response-criterion parameter $X_C$ is varied is about four times greater than the average LOST when the sensitivity parameter $d^{\prime }$ is varied. As a result, applying the signal-detection model to random data almost always leads to the conclusion that all the points share the same value of $d^{\prime }$ but were generated under different values of $X_C$. Parameters that have non-monotonic effects on performance, such as the attention-weight parameter that is used in popular exemplar and prototype models of categorization, tend to have large LOSTs, and therefore contribute to model flexibility more than parameters that have monotonic effects on performance. Comparing LOSTs for exemplar and prototype models also leads to some deep new insights into the structure of both models.

Properties of a binary choice probability function $p$ defined on multiattributed outcomes are studied to represent $p$ as a transformation of additive difference evaluations of chosen and unchosen outcomes into the unit interval. We use an algebraic assumption to obtain an additive difference representation, but allow for restricting strict increasingness of the transformation to the subset of the domain on which transformed values are strictly between 0 and 1. We also apply a topological assumption to axiomatize the cases of homogeneous product sets in the context of finite-state decision making under uncertainty.

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 problem of finding a utility function for a semiorder has been studied since 1956, when the notion of semiorder was introduced by Luce. But few results on continuity and no result like Debreu’s Open Gap Lemma, but for semiorders, was found. In the present paper, we characterize semiorders that accept a continuous representation (in the sense of Scott–Suppes). Two weaker theorems are also proved, which provide a programmable approach to Open Gap Lemma, yield a Debreu’s Lemma for semiorders, and enable us to remove the open-closed and closed-open gaps of a set of reals while keeping the threshold.

We report a “Double Decoy” experiment designed to separate two competing accounts of the asymmetric dominance effect. The experiment places an additional decoy alternative within the range of existing alternatives, which should leave choice behaviour unaltered if attributes are weighted by their range. Instead, we observe a decrease in the relative proportion of targets chosen, particularly for subjects who exhibited an initial decoy effect. We also observe considerably more variation in individual behaviour than expected. We therefore consider an alternative theory in which attributes values are compared with diminishing sensitivity (via divisive normalization) and assess its performance in an additional discrete choice experiment previously used in the discrete choice literature. We find that divisive normalization captures behaviour better than range normalization and the linear additive Logit model typically used in applied settings. We therefore propose divisive normalization as both a neuro-computational explanation for context effects and a useful empirical tool for applied researchers.

Cultural consensus theory is a model-based approach for analyzing responses of informants when correct answers are unknown. The model provides aggregate estimates of the latent consensus knowledge at the group level while accounting for heterogeneity in informant competence and item difficulty. We develop a new version of cultural consensus theory for two-dimensional continuous judgments which are obtained when asking informants to locate a set of unknown sites on a geographic map. The new model is fitted using hierarchical Bayesian modeling. A simulation study shows satisfactory parameter recovery for realistic numbers of informants and items. We also assess the accuracy of the aggregate location estimates by comparing the new model against simply computing the unweighted average of the informants’ judgments. A simulation study shows that, due to weighing judgments by the inferred competence of the informants, cultural consensus theory provides more accurate location estimates than unweighted averaging. The new model also showed a higher accuracy in an empirical study in which individuals judged the location of 57 European cities on maps.

According to the theory of derived attention, organisms attend to cues with strong associations. Prior work has shown that – combined with a Rescorla–Wagner style learning mechanism – derived attention explains phenomena such as learned predictiveness, inattention to blocked cues, and value-based salience. We introduce a Bayesian derived attention model that explains a wider array of results than previous models and gives further insight into the principle of derived attention. Our approach combines Bayesian linear regression with the assumption that the associations of any cue with various outcomes share the same prior variance, which can be thought of as the inherent importance of that cue. The new model simultaneously estimates cue–outcome associations and prior variance through approximate Bayesian learning. A significant cue will develop large associations, leading the model to estimate a high prior variance and hence develop larger associations from that cue to novel outcomes. This provides a normative, statistical explanation for derived attention. Through simulation, we show that this Bayesian derived attention model not only explains the same phenomena as previous versions, but also retrospective revaluation. It also makes a novel prediction: inattention after backward blocking. We hope that further development of the Bayesian derived attention model will shed light on the complex relationship between uncertainty and predictiveness effects on attention.

The aim of this paper is to promote quantum logic as one of the basic tools for analyzing human reasoning. We compare it with classical (Boolean) logic and highlight the role of violation of the distributive law for conjunction and disjunction. It is well known that nondistributivity is equivalent to incompatibility of logical variables — the impossibility to assign jointly the two-valued truth values to these variables. A natural question arises as to whether quantum logical nondistributivity in human logic can be tested experimentally. We show that testing the response replicability effect (RRE) in cognitive psychology is equivalent to testing nondistributivity — under the prevailing conjecture that the mental state update generated by observation is described as orthogonal projection of the mental state vector (the projective update conjecture of Wang and Busemeyer). A simple test of RRE is suggested. In contrast to the previous works in quantum-like modeling, we proceed in the state-dependent framework; in particular, distributivity, compatibility, and RRE are considered in a fixed mental state. In this framework, we improve the previous result on the impossibility to combine question order and response replicability effects by using (von Neumann–Lüders) projective measurements.