In decision making under risk, adults tend to overestimate small and underestimate large probabilities (Tversky & Kahneman, 1992). This inverse S-shaped distortion pattern is similar to that observed in a wide variety of proportion judgment tasks (see Hollands & Dyre, 2000, for review). In proportion judgment tasks, distortion patterns tend not to be fixed but rather to depend on the reference points to which the targets are compared. Here, we tested the novel hypothesis that probability distortion in decision making under risk might also be influenced by reference points in this case, references implied by the probability range. Adult participants were assigned to either a full-range (probabilities from 0-100%), upper-range (50-100%), or lower-range (0-50%) condition, where they indicated certainty equivalents for 176 hypothetical monetary gambles (e.g., a 50% chance of $100, otherwise $0). Using a modified cumulative prospect theory model, we found only minimal differences in probability distortion as a function of condition, suggesting no differences in use of reference points by condition, and broadly demonstrating the robustness of distortion pattern across contexts. However, we also observed deviations from the curve across all conditions that warrant further research.

What do numerical estimates tell us about developing an understanding of number? One theory is that bounded number line estimation (NLE) tasks reveal a representational shift from logarithmically to linearly organized mental representations of number over development. According to a different theoretical framework, developmental change in estimation reflects changes in children's numerical knowledge and their ability to make appropriate relative judgments. Empirical support for this proportion estimation framework includes the fact that quantitative models of proportion estimation describe signature patterns of estimation bias. A recent study argued against this latter theory by suggesting that patterns of curvature in number line placements are simply artifacts of a task procedure in which participants receive explicit information about the location of the numerical midpoint. We tested this claim in two experiments with children aged 6-8 years (Experiment 1: N = 47; Experiment 2: N = 104). Results demonstrated that the proportion estimation framework provides a good explanation of children's number line placement in the absence of explicit midpoint cues, that explicit cues to the midpoint are associated with more frequent use of middle reference points in young children, and that children can use a middle reference point spontaneously in the absence of explicit cues (with this tendency increasing with age). These findings provide novel support for the idea that psychophysical models of proportion estimation successfully account for numerical estimates across development regardless of whether spatial and numerical midpoint cues are provided as part of the NLE task. (C) 2019 Elsevier Inc. All rights reserved.

Similar estimation biases appear in a wide range of quantitative judgments, across many tasks and domains. Often, these biases (those that occur, for example, when adults or children indicate remembered locations of objects in bounded spaces) are believed to provide evidence of Bayesian or rational cognitive processing, and are explained in terms of relatively complex Bayesian models (e.g., the Category Adjustment Model). Here, we suggest that some of these phenomena may be accounted for instead within a simpler alternative theoretical framework that has previously been found to explain bias in common numerical estimation tasks across development. We report data from university undergraduate students and 7- through 10-year-olds completing a speeded linear position reproduction task. Bias in both adults' and children's responses was effectively explained in terms of a relatively simple psychophysical model of proportion estimation. These data clearly show that the proportion estimation framework is a viable alternative to theories that explain biases as the result of a Bayesian cognitive adjustment process. We also discuss our view that these data are not easily reconciled with the requirements of the more complex Category Adjustment Model that assumes estimates should exhibit a central tendency bias.

The grouping of options into arbitrary categories influences adults' decisions about allocating choices or resources among those options; this is called partition dependence. Partition dependence has been demonstrated in a wide range of contexts in adults and is often presented as a technique for designing choice architectures that nudge people towards better decisions. Whether children also make partition dependent decisions is unknown, as are potential patterns of developmental change. In this experiment (N = 159), we examined whether children exhibit partition dependence using a novel resource allocation task. This novel task, distributing food tokens to zoo animals, did elicit partition dependence in our developmental sample. Both older children (ages 7-10 years) and younger children (ages 3-6 years) made partition dependent allocations, and younger children exhibited a larger partition dependence effect than did older children. This work provides the first evidence that children's decisions, like adults', are influenced by the arbitrary grouping of the options, and suggests that younger children may be more susceptible to this influence, at least in the context explored here.

Learning the meanings of Arabic numerals involves mapping the number symbols to mental representations of their corresponding, approximate numerical quantities. It is often assumed that performance on numerical tasks, such as number line estimation (NLE), is primarily driven by translating from a presented numeral to a mental representation of its overall magnitude. Part of this assumption is that the overall numerical magnitude of the presented numeral, not the specific digits that comprise it, is what matters for task performance. Here we ask whether the magnitudes of the presented target numerals drive symbolic number line performance, or whether specific digits influence estimates. If the former is true, estimates of numerals with very similar magnitudes but different hundreds digits (such as 399 and 402) should be placed in similar locations. However, if the latter is true, these placements will differ significantly. In two studies (N = 262), children aged 7-11 and adults completed 0-1000 NLE tasks with target values drawn from a set of paired numerals that fell on either side of Hundreds boundaries (e.g., 698 and 701) and Fifties boundaries (e.g., 749 and 752). Study 1 used an atypical speeded NLE task, while Study 2 used a standard non-speeded NLE task. Under both speeded and non-speeded conditions, specific hundreds digits in the target numerals exerted a strong influence on estimates, with large effect sizes at all ages, showing that the magnitudes of target numerals are not the primary influence shaping children's or adults' placements. We discuss patterns of developmental change and individual difference revealed by planned and exploratory analyses.

The partitioning of options into arbitrary categories has been shown to influence decisions about allocating choices or resources among those options; this phenomenon is called partition dependence. While we do not call into question the validity of the partition dependence phenomenon in the present work, we do examine the robustness of one of the experimental paradigms reported by Fox, Ratner, and Lieb (Journal of Experimental Psychology: General, 134, 538-551, 2005, Study 4). In three experiments (N = 300) conducted here, participants chose from a menu of perceptually partitioned options (varieties of candy distributed across bowls). We found no clear evidence of partition dependent choice in children (Experiment 1) and no evidence at all of partition dependence in adults' choices (Experiments 1-3). This was true even when methods were closely matched to those of Fox et al.'s Study 4 (Experiment 3). We conclude that the candy-bowl choice task does not reliably elicit partition dependence and propose possible explanations for the discrepancy between these findings and prior reports. Future work will explore the conditions under which partition dependence in consumer choice does reliably arise.