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A solution to the single-question crowd wisdom problem

Abstract

Once considered provocative1, the notion that the wisdom of the crowd is superior to any individual has become itself a piece of crowd wisdom, leading to speculation that online voting may soon put credentialed experts out of business2,3. Recent applications include political and economic forecasting4,5, evaluating nuclear safety6, public policy7, the quality of chemical probes8, and possible responses to a restless volcano9. Algorithms for extracting wisdom from the crowd are typically based on a democratic voting procedure. They are simple to apply and preserve the independence of personal judgment10. However, democratic methods have serious limitations. They are biased for shallow, lowest common denominator information, at the expense of novel or specialized knowledge that is not widely shared11,12. Adjustments based on measuring confidence do not solve this problem reliably13. Here we propose the following alternative to a democratic vote: select the answer that is more popular than people predict. We show that this principle yields the best answer under reasonable assumptions about voter behaviour, while the standard ‘most popular’ or ‘most confident’ principles fail under exactly those same assumptions. Like traditional voting, the principle accepts unique problems, such as panel decisions about scientific or artistic merit, and legal or historical disputes. The potential application domain is thus broader than that covered by machine learning and psychometric methods, which require data across multiple questions14,15,16,17,18,19,20.

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Figure 1: Two example questions from Study 1c, described in text.
Figure 2: Why ‘surprisingly popular’ answers should be correct, illustrated by simple models of Philadelphia and Columbia questions with Bayesian respondents.
Figure 3: Selection of stimuli from Study 4 in which respondents judged the market price of 20th century artworks.
Figure 4: Results of aggregation algorithms on studies discussed in the text.
Figure 5: Logistic regressions showing the probability that an artwork is judged expensive (above $30,000) as function of actual market price.

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Acknowledgements

We thank M. Alam, A. Huang and D. Mijovic-Prelec for help with designing and conducting Study 3, and D. Suh with designing and conducting Study 4b. Supported by NSF SES-0519141, Institute for Advanced Study (Prelec), and Intelligence Advanced Research Projects Activity (IARPA) via the Department of Interior National Business Center contract number D11PC20058. The US Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright annotation thereon. The views and conclusions expressed herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/NBC, or the US Government.

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All authors contributed extensively to the work presented in this paper.

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Correspondence to Dražen Prelec.

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Reviewer Information Nature thanks A. Baillon, D. Helbing and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Figure 1 Performance of all methods across all studies, shown with respect to the Matthews correlation coefficient.

Error bars are bootstrapped standard errors. Details of studies are given in Fig. 4 of the main text.

Extended Data Figure 2 Performance of all methods across all studies, shown with respect to the macro-averaged F1 score.

Error bars are bootstrapped standard errors. Details of studies are given in Fig. 4 of the main text.

Extended Data Figure 3 Performance of all methods across all studies, shown with respect to percentage of questions correct.

Error bars are bootstrapped standard errors. Details of studies are given in Fig. 4 of the main text.

Extended Data Figure 4 Performance of aggregation methods on simulated datasets of binary questions, under uniform sampling assumptions.

One draws a pair of coin biases (that is, signal distribution parameters), and a prior over worlds, each from independent uniform distributions. Combinations of coin biases and prior that result in recipients of both coin tosses voting for the same answer are discarded. An actual coin is sampled according to the prior, and tossed a finite number of times to produce the votes, confidences, and vote predictions required by different methods (see Supplementary Information for simulation details). As well as showing how sample size affects different aggregation methods the simulations also show that majorities become more reliable as consensus increases. A majority of 90% is correct about 90% of the time, while a majority of 55% is not much better than chance. This is not due to sampling error, but reflects the structure of the model and simulation assumptions. According to the model, an answer with x% endorsements is incorrect if counterfactual endorsements for that answer exceed x% (Theorem 2), and the chance of sampling such a problem diminishes with x.

Supplementary information

Supplementary Information

This file contains Supplementary Text and Data sections 1-3 – see contents page for details. (PDF 207 kb)

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Prelec, D., Seung, H. & McCoy, J. A solution to the single-question crowd wisdom problem. Nature 541, 532–535 (2017). https://doi.org/10.1038/nature21054

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