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The computational cost of active information sampling before decision-making under uncertainty

Abstract

Humans often seek information to minimize the pervasive effect of uncertainty on decisions. Current theories explain how much knowledge people should gather before a decision, based on the cost–benefit structure of the problem at hand. Here, we demonstrate that this framework omits a crucial agent-related factor: the cognitive effort expended while collecting information. Using an active sampling model, we unveil a speed–efficiency trade-off whereby more informative samples take longer to find. Crucially, under sufficient time pressure, humans can break this trade-off, sampling both faster and more efficiently. Computational modelling demonstrates the existence of a cost of cognitive effort which, when incorporated into theoretical models, provides a better account of people’s behaviour and also relates to self-reported fatigue accumulated during active sampling. Thus, the way people seek knowledge to guide their decisions is shaped not only by task-related costs and benefits, but also crucially by the quantifiable computational costs incurred.

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Fig. 1: Active information-gathering task, with large and complex sampling space.
Fig. 2: A passive counterpart of the game shows that participants are sensitive to reward and uncertainty.
Fig. 3: Economic and time constraints influence the extent, speed and efficiency of active information sampling.
Fig. 4: Choosing how quickly, efficiently and extensively to sample.
Fig. 5: A model comparison robustly shows, across five experiments, that a model with a quadratic cognitive effort cost best explains active information-sampling behaviour.
Fig. 6: The cost of efficiency is associated with fatigue development.
Fig. 7: A breakable speed–efficiency trade-off governs active information sampling.

Data availability

Anonymized participant data have been deposited on the Open Science Framework platform and can be found at https://osf.io/25wkh/.

Code availability

Code for replicating the main results reported in the manuscript has been deposited on the Open Science Framework platform and can be found at https://osf.io/25wkh/. Additional analysis codes are available on request from the corresponding authors.

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Acknowledgements

We thank L. Acerbi for help designing an optimal search algorithm (infoMax) and M. Serrano Bonilla for help with figure design. We also thank all of the members of the Cognitive Neurology Research Group for assistance as experimental confederates. This work was supported by the Wellcome Trust (206330/Z/17/Z). P.P. and M.H. were funded by the Wellcome Trust (206330/Z/17/Z). B.A. was funded by a Rhodes Scholarship. S.G.M. was funded by an MRC Clinician Scientist Fellowship (MR/P00878/X). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

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P.P., B.A. and M.H. designed the study. P.P. and B.A. collected the data. P.P., B.A. and S.G.M. analysed the data. P.P. and M.H. wrote the paper.

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Correspondence to Pierre Petitet or Masud Husain.

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The authors declare no competing interests.

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Peer review information Nature Human Behaviour thanks Jacqueline Gottlieb and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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

Extended data

Extended Data Fig. 1 Demographic and questionnaire measures.

The table shows group means with standard deviation in parenthesis. F: Female; M: Male; YOE: years of education (since kindergarten); BDI: Beck Depression Inventory-II; AMI: Apathy Motivation Index; BIS-11: Barratt Impulsiveness Scale-11; IUS: Intolerance of Uncertainty Scale.

Extended Data Fig. 2 Experimental design.

All experiments included four types of trial (24 repetitions per trial type; 96 trials in total). The cost-benefit and temporal structure of these trial types is depicted here. R0: initial credit; ηs: sampling cost; ηe: error cost; \({{\rm{t}}}_{\max }\): time allocated to the sampling phase.

Extended Data Fig. 3 A visual illustration of efficient versus inefficient sampling.

A map of the posterior belief (top row) and of the expected change in expected error (EE) (bottom row) is plotted for the two example trials plotted in Fig. 1c (panel a: High α trial; panel b: Low α trial). The posterior belief map represents (in blue) all pixels that may be the centre of the hidden circle given the information on the screen (grey: search space; purple dots: observations inside the hidden circle; white dots: observations outside the hidden circle). The smaller this area, the lower the overall uncertainty (EE). For each sample, the associated map represents the change in EE expected to occur for every possible candidate sampling location. An efficient sampler systematically samples where the expected ΔEE is the lowest. a. During efficient sampling, the area covered by the posterior belief decays because the participant samples where the expected ΔEE is the lowest. b. By contrast, when sampling is inefficient, the area covered by the posterior belief decays less sharply because the participant does not sample where the expected ΔEE is the lowest.

Extended Data Fig. 4 On average, the expected error decreases exponentially over successive samples.

Grey lines represent, for each participant (n = 127 across all five experiments), the average Expected Error as a function of sample index. The black line shows the average across all participants ( ± s.d.).

Extended Data Fig. 5 Free parameters included in our models of speed, efficiency and extent of active information sampling.

Models were defined using the equation reported in the Methods. Tick: parameter included in the model and estimated through the fitting procedure; Cross: parameter not included in the model (effectively set to zero). *: Winning model (Model comparison in Fig. 5, Extended Data Fig. 7& Supplementary Fig. 2).

Extended Data Fig. 6 Economic and time constraints influence the extent, speed and efficiency of active information sampling.

a. Individuals sampled more with large initial reward reserves (Exp. 2, n = 20), when acquiring data was cheap (Exp. 3, n = 20), when final error was strongly penalised (Exp. 4, n = 20), and when they had time to search (Exp. 5, n = 19). Hand icon adapted from Freepik: macrovector. b-c. The initial credit and severity of error discounting did not influence how quickly or efficiently participants resolved localisation uncertainty (Exps. 2 & 4). Players sampled slower but more efficiently when acquiring data was more expensive (Exp. 3). By contrast, they speeded up but maintained a high level of efficiency (indexed by information extraction rate) under tighter time pressure (Exp. 5). Points and error bars show group mean ± s.e.m. Lines show condition averages for each participant. Positive/negative main effect of the experimental variable manipulated is shown in red/blue. Full statistical details are reported in Supplementary Tables 3-5.

Extended Data Fig. 7 Model comparison from Experiments 2-5.

The same models as in Fig. 5 are compared here. Model 1 and 2 included no cognitive effort cost (Model 1: rational specification; Model 2: nonlinear utility function; in grey) and were used as a reference. The other 12 models included a cognitive effort cost specified in terms of either speed (Models 3, 6, 9, 12), efficiency (Models 4, 7, 10, 13) or both (Models 5, 8, 11, 14). They differed with respect to the shape of the cost function (green: linear; orange: quadratic; blue: exponential; pink: sigmoid). Across all four experiments, participants’ choices of sampling speed, efficiency and extent were best explained by a model penalising both speed and efficiency with a quadratic cost function (Model 8, lowest Bayesian Information Criterion, BIC).

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Supplementary Methods, Supplementary Results, Supplementary Figs. 1–9 and Supplementary Tables 1–15.

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Petitet, P., Attaallah, B., Manohar, S.G. et al. The computational cost of active information sampling before decision-making under uncertainty. Nat Hum Behav (2021). https://doi.org/10.1038/s41562-021-01116-6

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