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Individual risk attitudes arise from noise in neurocognitive magnitude representations

Nature Human Behaviour volume 7, pages 1551–1567 (2023)Cite this article

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Abstract

Humans are generally risk averse, preferring smaller certain over larger uncertain outcomes. Economic theories usually explain this by assuming concave utility functions. Here, we provide evidence that risk aversion can also arise from relative underestimation of larger monetary payoffs, a perceptual bias rooted in the noisy logarithmic coding of numerical magnitudes. We confirmed this with psychophysics and functional magnetic resonance imaging, by measuring behavioural and neural acuity of magnitude representations during a magnitude perception task and relating these measures to risk attitudes during separate risky financial decisions. Computational modelling indicated that participants use similar mental magnitude representations in both tasks, with correlated precision across perceptual and risky choices. Participants with more precise magnitude representations in parietal cortex showed less variable behaviour and less risk aversion. Our results highlight that at least some individual characteristics of economic behaviour can reflect capacity limitations in perceptual processing rather than processes that assign subjective values to monetary outcomes.

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Fig. 1: The NLC model, experimental model and behavioural results.
Fig. 2: Neural magnitude representations in parietal cortex.
Fig. 3: Domain-generality of scale invariance across perceptual and risky-choice tasks.
Fig. 4: Noise in mental magnitude representations is individually robust across different types of visual display of risky prospects.
Fig. 5: The precision of mental magnitude representations systematically relates to individual risk attitudes.
Fig. 6: Relationship between neural measures of the precision of magnitude representations and risk aversion.
Fig. 7: The mediating effect of behavioural magnitude precision reveals the association between neural magnitude precision and risk aversion.

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Data availability

The behavioural data used in Figs. 2–7 and Extended Data Figs. 2–4, 6, 8 and 9 are available at https://doi.org/10.5281/zenodo.7966313. The neuroimaging data are available at Open Neuro: https://openneuro.org/datasets/ds004259.

Code availability

The codes are available at https://doi.org/10.5281/zenodo.7966313.

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Acknowledgements

We are grateful to C. Schnyder, K. Treiber and M. Moisa at the Zurich Center for Neuroeconomics for their excellent assistance in recruitment and participant facilitation. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 725355, ERC consolidator grant BRAINCODES). M.B.G. was funded by the Marlene Porsche Graduate School in Neuroeconomics and C.C.R. received funding from the University Research Priority Program ‘Adaptive Brain Circuits in Development and Learning’ (grant no. URPP AdaBD) at the University of Zurich and the Swiss National Science Foundation (grant no. 100019L-173248). G.D.H. was funded by the Dutch Research Council NWO (Rubicon grant no. 019.183SG.017/8O3B). We are thankful for the relevant feedback that we received during the 2019 Sloan-NOMIS Workshop for the Attentional and Perceptual Foundations of Economic Behavior as well as the 2020 virtual Annual Meeting of the Society for Neuroeconomics. The funders had no role in the study design, data collection and analysis, decision to publish or preparation of the manuscript.

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Author notes

  1. These authors contributed equally: Miguel Barretto-García, Gilles de Hollander.

Authors and Affiliations

  1. Zurich Center for Neuroeconomics, Department of Economics, University of Zürich, Zurich, Switzerland

    Miguel Barretto-García, Gilles de Hollander, Marcus Grueschow & Christian C. Ruff

  2. Department of Neuroscience, School of Medicine, Washington University in St Louis, St. Louis, MO, USA

    Miguel Barretto-García

  3. University Research Priority Program ‘Adaptive Brain Circuits in Development and Learning’ (URPP AdaBD), University of Zurich, Zurich, Switzerland

    Gilles de Hollander & Christian C. Ruff

  4. Decision Neuroscience Laboratory, Department of Health Sciences and Technology, ETH Zürich, Zurich, Switzerland

    Rafael Polanía

  5. Department of Economics, Columbia University, New York, NY, USA

    Michael Woodford

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Contributions

All authors developed the experimental design and procedures, and contributed to the analysis pipeline of the behavioural data. M.B.G. collected the data. M.B.G. and G.D.H. analysed the data. G.D.H. set up the analysis pipeline for the fMRI decoding analysis. M.B.G., G.D.H. and C.C.R. wrote and revised the manuscript, with input from M.G., R.P. and M.W.

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Correspondence to Miguel Barretto-García or Christian C. Ruff.

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Extended data

Extended Data Fig. 1 Model comparison between the NLC model and competing economic choice models.

DIC difference between the best model (the NLC model in all cases) and the competing economic choice models: constant relative risk aversion (CRRA), cumulative prospect theory (CPT), and salience theory (ST) in both (a) non-symbolic and (b) symbolic visual display formats. We fitted each of these economic choice models using the Logit (1) and Probit (2) model specifications.

Extended Data Fig. 2 Individual behavioural differences.

Observed choice probabilities of each individual participant were plotted against the log-ratio of monetary offers or magnitudes for (a) non-symbolic risky choice (orange), (b) symbolic risky choice (blue), and (c) perceptual magnitudes (purple). The NLC model was used to fit the psychometric curves of each individual (black solid line). The numbers above each plot denotes participant ID.

Extended Data Fig. 3 Individual differences in choices for extreme offer magnitudes.

Histograms of choice probabilities (across subjects) for coin clouds (left, purple) or risky offers in nonsymbolic (middle, orange) and symbolic (right, blue) presentation formats, for trials with (a) the highest and (b) lowest magnitudes/offers. The distributions are drawn from the extreme value points of the individual choice data in Extended Data Fig. 2. The more the distributions are skewed towards 1 (for a) and 0 (for b), the less evidence for lapses across individual subjects.

Extended Data Fig. 4 The NLC model accounts for perceptual bias and risk aversion arising from noisy magnitude representations.

(a) Choice probabilities plotted in linear (left) and logarithmic (right) space as the ratio of the second- and first-coin cloud magnitudes (perceptual task, red) or risky and sure payoffs (risky-choice task, blue). The NLC model predicts that the indifference point in the perceptual task is \({\theta }_{{perceptual}}\approx 1\) regardless of noisy magnitude representations while \({\theta }_{{risky}}\) in risky choice depends on magnitude noise. The indifference point—the ratio of magnitudes where the individual is indifferent between \(X\) or \(C\), expressed as \(\theta ={\left(\frac{1}{p}\right)}^{\frac{1}{\beta }}\)—is represented as the intersection of the black horizontal dashed line and the points of the psychometric curve. In perceptual magnitude, there is no outcome uncertainty (\(p=1\)) in both dot clouds; hence, \({\theta }_{{perceptual}}\approx 1\) regardless of whether magnitude representations are noisy (\(\beta < 0\) and \(\nu > 0\), solid pink psychometric curve) or not (\(\beta =1\) and \(\nu =0\), dashed pink stepwise function). This implies that the intercept is \({\delta }_{{perceptual}}\approx 0\). In risky choice, outcome probability is fixed at \(p=0.55\) for the risky payoff. In the absence of noise (\(\beta =1\) and \(\nu =0\)), the indifference point reflects the relative value of payoffs, \({\theta }_{{risky}}=\frac{X}{C}=\frac{1}{0.55}\) (intersection of the black dashed horizontal line and blue dashed stepwise function). With magnitude noise (\(\beta < 1\) and \(\nu > 0\)), \({\theta }_{{risky}}\) is larger than risk-neutral indifference of \(\frac{1}{0.55}\), \({{\theta }_{{risky}}=\left(\frac{1}{0.55}\right)}^{\frac{1}{\beta }} > \frac{1}{0.55}\) (intersection of the black dashed horizontal line and blue solid psychometric curve). This is reflected as a shift of the psychometric curve to the right relative to the risk-neutral stepwise function. Risk aversion is quantified as the magnitude of the rightward shift of the psychometric function compared to the risk-neutral indifference point, \({\theta }_{{risky}}=\frac{1}{0.55}\). Risk-neutral probability, \({\pi }_{{risky}}\), is the reciprocal of the indifference point, \({\pi }_{{risky}}={0.55}^{\frac{1}{\beta }}\). (b) Decreasing the width of the prior, \(\sigma\), shifts the psychometric curve to the right, and thus increases risk aversion. The indifference point can also be expressed as \(\theta ={\left(\frac{1}{p}\right)}^{\frac{1}{\beta }}={\left(\frac{1}{p}\right)}^{\frac{{\sigma }^{2}+{\nu }^{2}}{{\sigma }^{2}}}\) to explicitly show the link between \(\sigma\) and \(\theta\), and why decreasing the prior increases the indifference point and shifts the psychometric curve rightward. The psychometric curves are plotted in linear (left) and logarithmic (right) space and the different colours represent different prior widths.

Extended Data Fig. 5 Behavioural effects of the presentation format of monetary magnitudes.

(a) Population posterior distributions of the (a, b) intercept, ẟ, as well as (c, d) the indifference point, θ, for both (a,c) perceptual magnitude and (b, d) risky choice tasks. The intercept during perceptual magnitude is no different from zero (indicated here by the vertical dashed line) while it is significantly larger than zero during risky choice in both visual displays. Similarly, the indifference point in perceptual magnitude is no different from one while in risky choice, it is significantly larger than the threshold, \(\frac{1}{0.55}\) (the vertical dashed line). Distributions in pink represent data from the perceptual magnitude task, in blue represent data from risky symbolic payoffs, and in yellow-orange for risky nonsymbolic payoffs. One-sided Bayesian ‘p-values’ were calculated: The light pink-shaded mass of the highest density interval (HDI) covers 95% of the posterior distribution while the dark-shaded tail-ends represent the most extreme 5% probability mass of the posterior distribution. Bayesian comparison between posteriors reveal that the posterior distribution is significantly different from zero (represented here as a vertical dashed line) if the light-shaded mass does not cross zero. (e) Individual measures of the indifference point for nonsymbolic and symbolic payoffs are positively correlated (n = 62). The shaded area around the regression line represents 95% confidence intervals. The black dashed line represents the identity line. p-values were estimated from one-sided Pearson correlations.

Extended Data Fig. 6 Behavioural and neural measures of representation acuity and risky choice variability.

(a) The estimated precision of mental magnitude representations employed for the perceptual task, γperceptual, and the risky decision-making task, γsymbolic and γnonsymbolic, are related (n = 64), for both types of visual displays, as predicted by the NLC model. The diagonal dashed line represents the identity line. Shaded area are error bands corresponding to 95% confidence interval for the regression line. p-values were estimated from one-sided Pearson correlations. (b) Group-level posterior distributions of non-symbolic risk (γnonsymbolic, orange), symbolic risk (γsymbolic, blue), and perceptual magnitude (γperceptual, pink) precision. Top plots the posterior distributions while the bottom plots are distributions of differences between precision measurements (in pink). One-sided Bayesian ‘p-values’ were calculated: The light-shaded mass of the highest density interval (HDI) covers 95% of the posterior distribution while the dark-shaded tail-ends represent 5% of the posterior distribution. The vertical dashed line represents zero. (c) The correlations between the neural precision parameter and the risky choice precision parameter γ were not statistically significant (n = 64), but in the hypothesised direction: the higher the neural precision, the less variable the behaviour. (d) Neural diminishing sensitivity was significantly correlated with the risky choice precision parameter γ (n = 64) for the non-symbolic presentation format and marginally significant for the symbolic presentation format. The shaded area around the regression line represents 95% confidence intervals. p-values were estimated from one-sided Pearson correlations.

Extended Data Fig. 7 Individual risk measures relate systematically to choice accuracy in the perceptual task.

(a) Raw measures of individual perceptual choice accuracy and the frequency of choosing the risky payoff (n = 64) are related across all visual displays. Perceptual accuracy is also related (n = 64) to the more precise model-defined measurements of (b) the estimated precision of potential payoffs and (c) the index of risk aversion across visual display types. (d) These correlations reflect that magnitude precision in the perceptual task relates strongly to choice accuracy (but is not affected by response biases and the response noise contained in pure choice accuracy measures). Circular dots represent subjects. The shaded area around the regression line represents 95% confidence intervals. p-values were estimated from one-sided Pearson correlations.

Extended Data Fig. 8 Perceptual choice precision mediates the association between neural precision and risky choice precision.

(a) The effect of neural precision on risky choice precision for the task using non-symbolic numbers (n = 64) is mediated by perceptual choice variability. There is no significant direct or total effect. (b) The effect of neural precision on risky choice precision for the task using symbolic presentation format is mediated by perceptual choice variability. There is no significant direct or total effect. The effect of neural diminishing sensitivity on risky choice precision for the task using (c) non-symbolic numbers is mediated by perceptual choice precision, but less so for (d) symbolic numbers. One-sided Bayesian ‘p-values’ were calculated using hierarchical Bayesian mediation analysis (see Methods and Extended Data Fig. 9b).

Extended Data Fig. 9 Hierarchical Bayesian models.

Graphical representations of the hierarchical Bayesian (a) noisy logarithmic coding model and (b) mediation analysis. Clear circles represent latent variables while filled circles are observed variables, such as trialwise choice (rcsi) data, subject-wise behavioural and neural measurements (yj), and numerosity/payoff inputs (X,C). See Supplementary Note for more details.

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Barretto-García, M., de Hollander, G., Grueschow, M. et al. Individual risk attitudes arise from noise in neurocognitive magnitude representations. Nat Hum Behav 7, 1551–1567 (2023). https://doi.org/10.1038/s41562-023-01643-4

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