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Selective overweighting of larger magnitudes during noisy numerical comparison

Nature Human Behaviour volume 1, Article number: 0145 (2017) Cite this article

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Abstract

Humans are often required to compare average magnitudes in numerical data; for example, when comparing product prices on two rival consumer websites. However, the neural and computational mechanisms by which numbers are weighted, integrated and compared during categorical decisions are largely unknown1,2,3,4,5. Here, we show a systematic deviation from ‘optimality’ in both visual and auditory tasks requiring averaging of symbolic numbers. Participants comparing numbers drawn from two categories selectively overweighted larger numbers when making a decision, and larger numbers evoked disproportionately stronger decision-related neural signals over the parietal cortex. A representational similarity analysis6 showed that neural (dis)similarity in patterns of electroencephalogram activity reflected numerical distance, but that encoding of number in neural data was systematically distorted in a way predicted by the behavioural weighting profiles, with greater neural distance between adjacent larger numbers. Finally, using a simple computational model, we show that although it is suboptimal for a lossless observer, this selective overweighting policy paradoxically maximizes expected accuracy by making decisions more robust to noise arising during approximate numerical integration2. In other words, although selective overweighting discards decision information, it can be beneficial for limited-capacity agents engaging in rapid numerical averaging.

We used a simple psychophysical model to understand the rational policy for performing noisy numerical averaging in our task (see Methods). Model input Xi was the number occurring on each sample, i, normalized (for convenience) within the range −1 to 1. The model was parameterized to allow two potential sources of loss during averaging. The first, kappa (k), encoded a potential compression of the number line, allowing numbers to carry different weights in the decision: each sample Xi was transformed to a momentary decision value via a sign-preserving exponential function of the form (X + b)k, where b is an additive offset parameter. When k < 1, the transfer function has a sigmoidal form that compresses outlying values (Xi ≫ 0 or Xi ≪ 0) relative to inliers (Xi ≈ 0; Fig. 1b, light grey lines). The converse is true when k > 1 (see Fig. 1b, dark grey lines). The second source of loss was assumed to occur after the processing of each sample; that is, during numerical averaging or at the response itself7,8. To generate simulated model choices, we passed the difference in cumulative decision values for each category through a sigmoidal function with inverse slope sigma (s), where higher values of s (that is, low slopes) indicate more noise in neural computation.

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Figure 1: Task, model simulations and human behaviour.
Figure 2: Overview of model results and simulations for each experiment and condition.
Figure 3: CPP analysis.
Figure 4: Representational similarity analysis.

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Acknowledgements

This work was supported by grants from the German Research Foundation to B.S. (DFG SP 1510/1-1 and DFG SP 1510/2-1) and a European Research Council Starter Grant (281628) to C.S. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript. We thank V. Li, T. Flesch and H. Nili for helpful suggestions and scripts, F. Blankenburg for resources, A. Epure for help with data acquisition and R. Kievit for helpful comments on a previous version of the manuscript.

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Authors and Affiliations

  1. Department of Experimental Psychology, University of Oxford, Oxford, OX1 3UD, UK

    Bernhard Spitzer & Christopher Summerfield

  2. Department of Education and Psychology, Freie Universität Berlin, Habelschwerdter Allee 45, Berlin, 14195, Germany

    Bernhard Spitzer

  3. Department of Psychology, University of Lübeck, Lübeck, 23562, Germany

    Leonhard Waschke

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  1. Bernhard Spitzer
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Contributions

B.S. designed the experiments with contributions from L.W. and C.S. L.W. and B.S. conducted the experiments. B.S. and C.S. developed the analysis approach. B.S. analysed the data with contributions from C.S. B.S. and C.S. wrote the paper.

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Correspondence to Bernhard Spitzer.

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The authors declare no competing interests.

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Supplementary Information

Supplementary Methods, Supplementary Figures 1–3

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Spitzer, B., Waschke, L. & Summerfield, C. Selective overweighting of larger magnitudes during noisy numerical comparison. Nat Hum Behav 1, 0145 (2017). https://doi.org/10.1038/s41562-017-0145

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