Pizzas: π or Square? Psychophysical Biases in Area Comparisons
Why this work is in the frame
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Bibliographic record
Abstract
Many product categories, from pizzas to real estate, present buyers with purchase decisions involving complex area judgments. Does a square look larger or smaller than a circle? How much smaller does a circle of 8-inch diameter look when compared to one with a 10-inch diameter? In this paper, we propose a psychophysical model of how consumers make area comparison judgments. The model involves consumers making effort-accuracy trade-offs that lead to heuristic processing of area judgments and systematic shape- and size-related biases. The model is based on four propositions: P1. Consumers make an initial comparison between two figures based on a single dimension; P2. The dimension of initial comparison—the primary dimension—is the one that is most salient to consumers, where salience is figure and context dependent; P3. Consumers insufficiently adjust an initial comparison using a secondary dimension, which we assume to be orthogonal to the primary dimension used for the initial comparison; and P4. The magnitude by which the initial comparison is adjusted is directly related to the relative salience of the secondary dimension versus the primary dimension. The model predicts that a single linear dimension inappropriately dominates the two-dimensional area comparison task and that contextual factors affect which linear dimension dominates the task. The relative use of the second dimension depends on its relative salience, which can be influenced in a variety of ways. The model extends the area estimation literature in cognitive psychology by exploring new biases in area estimation and is able to resolve controversial effects regarding which shape is perceived to be “bigger,” the square or the circle, by incorporating contextual factors into model specifications. A set of six studies—five laboratory experiments and one field experiment—systematically test model predictions. Study 1 is a process study that shows that when two dimensions are available to make an area comparison judgment, people choose one of those to be the primary dimension, with the other being the secondary dimension. Furthermore, it shows that the choice of the primary dimension is dependent on its relative salience that can be contextually manipulated via manner of visual presentation. Studies 2 and 3 show how the use of a diagonal versus the side of a square (contextually determined) can affect whether a square is perceived to be smaller or larger than a circle of the same area. Study 3 extends the investigation to the domain of the price people are willing to pay for “pizzas” of different shapes, presented differently. Study 4, a field study, demonstrates external validity by showing that purchase quantities are greater when a circular package is expected to contain less than a rectangular package of the same volume in a domain where consumption goal is constant (cream cheese with a bagel). Studies 5 and 6 examine ways in which one can increase the salience of the secondary dimension, in a size estimation task, i.e., judging the rate of increase of area. While Study 5 does so via contextual visual cues (incorporating lines that draw one's attention to the underused dimension), Study 6 does the same using semantic cues that direct attention to a single dimension (e.g., diameter) or the total area and comparing these with a visual presentation of the figure. Overall, results suggest that the manner in which information is presented affects the relative salience of dimensions used to judge areas, and can influence the price consumers are willing to pay. Underlining the external validity of these findings, container shape can significantly affect quantity purchased and overall sales. The paper highlights biases in area comparison judgments as a function of area shape and size. The model is parsimonious, demonstrates good predictive ability, and explains seemingly contradictory results in the cognitive psychology literature. Implications for pricing, product design, packaging, and retailing are suggested.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it