Journal of Mathematical Psychology最新文献

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.

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.