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Where’s Waldo, Ohio? Using Cognitive Models to Improve the Aggregation of Spatial Knowledge

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We apply cognitive modeling to improve the wisdom of the crowd in a spatial knowledge task. Participants provided point estimates for where 48 US cities are located and then, using the point estimate as a center point, chose a radius large enough that they believed the resulting circle was certain to contain the city’s location. Simple and radius-weighted arithmetic averages of the individuals’ point estimates produced more accurate group answers than the majority of individuals. These statistical aggregates, however, assume there are no differences in individual expertise nor in the difficulty of locating different cities. Accordingly, we develop a set of cognitive models to infer group estimates that make various assumptions about individual expertise and differences in city difficulty. The model-based estimates generally outperform the statistical averages. The models are especially accurate if they allow for individual differences in expertise that can vary city by city. We replicate this finding by applying the same cognitive models to data reported by Mayer and Heck (2023) in which participants provided point estimates for the locations of European cities.

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Data and Code Availability

The US cities’ data set is available on an Open Science Framework project page at The Mayer and Heck (2023) data set is available on an Open Science Framework project page at The JAGS code, code for any analyses presented in this paper, and other supplementary materials are available on an Open Science Framework project page at


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We thank the Bayesian Cognitive Modeling lab at the University of California, Irvine, for their feedback, and Daniel Heck, Brandon Turner, and an anonymous reviewer for useful comments.


This research was supported by the US Air Force Research Laboratory’s Continuous Learning Branch. LEM’s collaboration was enabled through an appointment to the Oak Ridge Institute for Science and Education (ORISE) Summer Research Internship Program. MDL’s collaboration was enabled through an appointment to the ORISE Faculty Research Program. The views expressed in this paper are those of the authors and do not reflect the official policy or position of the US Air Force, Department of Defense, or the US Government.

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Authors and Affiliations



LEM and MDL conceived and designed the study. LEM, CMB, and MDL designed the experiment, and CMB programmed the experiment. LEM collected the experimental data. LEM, MDL, and JV developed the model. LEM performed the modeling analysis. CMB wrote the first draft of the Stimuli and Procedure subsections. LEM wrote the rest of the first draft. LEM and MDL contributed to revisions of the paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Lauren E. Montgomery.

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Ethics Approval

This project was approved via exempt self-determination by the University of California Irvine (UCI) Institutional Review Board (IRB). This project also made use of de-identified archival data that was obtained from

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Informed consent was obtained from all individuals who participated in the study.

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Not applicable

Competing Interests

The authors declare no competing interests.

Additional information

Parts of this research were presented at the 2022 Annual Joint Meeting of the Society for Mathematical Psychology and the International Conference on Cognitive Modeling, the Meeting of the European Mathematical Psychology Group in 2022, and the 58th Edwards Bayesian Research Conference held in 2023.

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Montgomery, L.E., Baldini, C.M., Vandekerckhove, J. et al. Where’s Waldo, Ohio? Using Cognitive Models to Improve the Aggregation of Spatial Knowledge. Comput Brain Behav 7, 242–254 (2024).

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