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Discrete confidence levels revealed by sequential decisions


Humans can meaningfully express their confidence about uncertain events. Normatively, these beliefs should correspond to Bayesian probabilities. However, it is unclear whether the normative theory provides an accurate description of the human sense of confidence, partly because the self-report measures used in most studies hinder quantitative comparison with normative predictions. To measure confidence objectively, we developed a dual-decision task in which the correctness of a first decision determines the correct answer of a second decision, thus mimicking real-life situations in which confidence guides future choices. While participants were able to use confidence to improve performance, they fell short of the ideal Bayesian strategy. Instead, behaviour was better explained by a model with a few discrete confidence levels. These findings question the descriptive validity of normative accounts, and suggest that confidence judgments might be based on point estimates of the relevant variables, rather than on their full probability distributions.

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Fig. 1: Dual-decision paradigm and results.
Fig. 2: Ideal Bayesian model and comparison with human behaviour.
Fig. 3: Alternative sub-optimal models.

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This work was partially supported by funding from the French National Research Agency (grant ANR-12-BSH2-0005 to A.G.). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript. We thank J. A. Solomon and M. J. Morgan for providing facilities. This paper is dedicated to the memory of Andrei Gorea, whose creative, untameable and questioning mind inspired this project and made it possible.

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Authors and Affiliations



M.L., A.G. and G. Mongillo conceived the study. M.L. and G. Mongillo developed the computational models. M.L. programmed the experiments. M.L. and G. Milne collected the data. M.L analysed the data. M.L, G. Mongillo and T.D. interpreted the results. M.L. wrote the paper with feedback from all authors.

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Correspondence to Matteo Lisi.

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The authors declare no competing interests.

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Peer review information Primary Handling Editor: Marike Schiffer

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Differences between observed and predicted choice probabilities.

These plots represent for each participant, the difference between the observed and predicted probability of choosing ‘right’ in the second decision, expressed as standardized Pearson residuals. Positive values indicate that the model underestimates the probability, and negative values indicate overestimation. Panel a shows the average residuals after a wrong first decision and panel b after a correct first decision. Blue bars represent the the discrete model (with 2 confidence levels) and the orange bars represent the biased-Bayesian model. Error bands represent bootstrapped standard errors. To facilitate interpretation, blue dots on the bottom denote participants for which the average residuals of the discrete model are smaller than that of the biased-Bayesian model, indicating that the discrete model made predictions that were on average closer to the observed probability. Note that individual data display the same pattern seen in the group data (Fig. 2, Main text), in which the biased-Bayesian model provides a poorer fit to the data, because it tends to more severely underestimate the probability of choosing ‘right’ after a wrong response.

Extended Data Fig. 2 Individual AIC values.

AIC differences (alternative model minus discrete model) are plotted for each participant. In all panels positive values (coloured in blue) indicate that the discrete model with two confidence levels provide a better fit to the data.

Extended Data Fig. 3 Model recovery analysis.

In order to ensure that the models were distinguishable we performed a model recovery analysis for our three main computational models: ideal Bayesian, biased-Bayesian and discrete. We generated synthetic data (20 simulated observers, for 600 trials each), with parameters randomly sampled from the (multivariate) Gaussian distribution of the parameters fitted to our empirical data. Each panel indicate a different generative model, while different lines represent the mean and standard error of the models fit to the synthetic dataset. In each case the model with the highest relative likelihood (computed by transforming the AIC onto a likelihood scale) is the one that generated the data, indicating that the model was correctly recovered and confirming that they are distinguishable.

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Supplementary Methods, Supplementary Results and Supplementary Figs. 1–3.

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Lisi, M., Mongillo, G., Milne, G. et al. Discrete confidence levels revealed by sequential decisions. Nat Hum Behav 5, 273–280 (2021).

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