Overcoming Individual Limitations Through Distributed Computation: Rational Information Accumulation in Multigenerational Populations..

Topics in Cognitive Science 14 (2022) 550–573
© 2022 The Authors. Topics in Cognitive Science published by Wiley Periodicals LLC on behalf of Cognitive
Science Society.
ISSN: 1756-8765 online
DOI: 10.1111/tops.12596
This article is part of the topic “Extending Rationality,” Emmanuel M. Pothos and Timothy
J. Pleskac (Topic Editors).
Overcoming Individual Limitations Through Distributed
Computation: Rational Information Accumulation in
Multigenerational Populations
Mathew D. Hardy,aPeaks M. Krafft,bBill Thompson,a,c
Thomas L. Grifﬁthsa,c
aDepartment of Psychology, Princeton University
bCreative Computing Institute, University of the Arts London
cDepartment of Computer Science, Princeton University
Received 28 February 2021; received in revised form 29 November 2021; accepted 30 November 2021
Abstract
Many of the computational problems people face are difﬁcult to solve under the limited time and
cognitive resources available to them. Overcoming these limitations through social interaction is one of
the most distinctive features of human intelligence. In this paper, we show that information accumula-
tion in multigenerational social networks can be produced by a form of distributed Bayesian inference
that allows individuals to beneﬁt from the experience of previous generations while expending little
cognitive effort. In doing so, we provide a criterion for assessing the rationality of a population that
extends traditional analyses of the rationality of individuals. We tested the predictions of this analysis in
two highly controlled behavioral experiments where the social transmission structure closely matched
the assumptions of our model. Participants made decisions on simple categorization tasks that relied on
and contributed to accumulated knowledge. Success required these microsocieties to accumulate infor-
mation distributed across people and time. Our ﬁndings illustrate how in certain settings, distributed
Mathew D. Hardy and P. M. Krafft contributed equally.
Correspondence should be sent to Thomas L. Grifﬁths, Department of Psychology, Princeton University,
Princeton, NJ 08540, USA. Email: tomg@princeton.edu
This is an open access article under the terms of the Creative Commons Attribution License, which permits
use, distribution and reproduction in any medium, provided the original work is properly cited.
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M. D. Hardy et al. / Topics in Cognitive Science 14 (2022) 551
computation at the group level can pool information and resources, allowing limited individuals to
perform effectively on complex tasks.
Keywords: Bayesian inference; Cultural evolution; Collective intelligence; Distributed computation;
Group rationality; Social learning
1. Introduction
Humans’ success and survival depends on our ability to make intelligent choices and judg-
ments. Like other animals, we must do so under hard constraints on the resources available tomake our decisions (Grifﬁths, 2020). Not only is our cognition constrained by our restricted
brain capacity, but we also must learn from limited and often incomplete data. This is true
not only in our ﬁnite childhoods and lifespans but also in the limited opportunities we haveto learn many crucial facts—to survive, we need to learn about what life-threatening dangersexist in our immediate environments (from chemicals to electric outlets to poisonous insectsand animals).
Given these constraints, understanding how people routinely form accurate beliefs on com-
plex topics is a central focus of research in cognitive science, psychology, and economics(Simon, 1990). However, people rarely develop beliefs alone—rather, they learn from theknowledge, experiences, and opinions of other people. By doing so, individuals can obtainuseful information while expending little physical or cognitive effort. A particularly impor-tant focus of previous research has been on the effects of social learning when repeated oversuccessive groups of individuals, such as child–parent learning, formal education, and otherdomains where knowledge is transmitted from older members of a popvvvulation to youngerlearners. In these multigenerational settings, knowledge can accumulate over time in a pop-ulation, allowing individuals to extend their cognitive skills by learning from others (see,e.g., Almaatouq, Alsobay, Yin, & Watts, 2021; Almaatouq et al., 2020; Belikov, Rzhetsky, &Evans, 2020; Caldwell, Atkinson, & Renner, 2016; Frey & Goldstone, 2018; Galesic, Olsson,Dalege, van der Does, & Stein, 2021; Goldstone, Wisdom, Roberts, & Frey, 2013; Hazła,Jadbabaie, Mossel, & Rahimian, 2021; Kempe & Mesoudi, 2014b; Mesoudi, 2016; Mesoudi& Thornton, 2018; Miton & Charbonneau, 2018; Riedl, Kim, Gupta, Malone, & Woolley,
2021; Rzhetsky, Foster, Foster, & Evans, 2015; Salhab, Ajorlou, & Jadbabaie, 2020; Wis-
dom, Song, & Goldstone, 2013; Wojtowicz & DeDeo, 2020, for recent overviews and relatedstudies). Crucially, the constraints and structure of interpersonal transmission often lead col-lective knowledge and learning to differ from individual outcomes (Kirby, Tamariz, Cornish,& Smith, 2015; Ravignani, Thompson, Grossi, Delgado, & Kirby, 2018; Silvey, Kirby, &Smith, 2019).
The accumulation of collective knowledge through sequential social learning is known
ascultural transmission (Boyd & Richerson, 1985) and is thought to underpin cumulative
cultural evolution (Mesoudi, 2011). Cultural transmission has been studied experimentallyin a number of paradigms, including the evolution of simple technologies such as knots(Muthukrishna, Shulman, Vasilescu, & Henrich, 2014), virtual ﬁshing nets (Derex, Beugin,
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552 M. D. Hardy et al. / Topics in Cognitive Science 14 (2022)
Godelle, & Raymond, 2013), stone tools (Morgan et al., 2015), or arrowheads (Mesoudi &
O’Brien, 2008); artiﬁcial languages (Kirby, Cornish, & Smith, 2008); jigsaw puzzles (Kempe
& Mesoudi, 2014a); and social phenomena like stereotypes (Martin et al., 2014).
Theories of cultural evolution have primarily been underpinned by a Darwinian framework
grounded in parallels and disanalogies with biological evolution (Boyd & Richerson, 1985;Mesoudi & Whiten, 2008; Nettle, 2020; Smolla et al., 2020). These frameworks have shownthat the accumulation of knowledge and technology can be understood as a form of evolution,helping to situate cultural evolution within the biological and evolutionary sciences (Laland,Sterelny, Odling-Smee, Hoppitt, & Uller, 2011). While successful, evolutionary frameworkshave been difﬁcult to connect to psychological theory (Heyes, 2018) and are therefore difﬁcultto connect with the concepts of computation that are central to the study of cognition andintelligence (Nettle, 2020). Similarly, while research on collective intelligence has extensivelydocumented the advantages of groups over individuals, the computational structure of thesocial processes that lead to long-term, open-ended collective intelligence remain unclear(Krafft et al., 2016).
One way to measure the accumulation of knowledge in populations is by viewing collec-
tive behavior as distributed computation, a process that allows groups to “store and processthe cumulative innovations and collaborations of generations of individuals” (Smaldino &Richerson, 2013). From this perspective, knowledge accumulation becomes a problem ofdistributed Bayesian inference, extending probabilistic models of inference in individuals to
the group setting (Chater, Tenenbaum, & Yuille, 2006; Grifﬁths, Chater, Kemp, Perfors, &Tenenbaum, 2010; Harper, 2009). This perspective establishes a formal connection between
cultural evolution and statistical models of social learning used in cognitive science (Cushman
& Gershman, 2019) and economics (Acemoglu, Dahleh, Lobel, & Ozdaglar, 2011).
In this paper, we build on this work and show how social learning can facilitate rational
action that goes beyond the direct experience of individuals. That is, we show how socialinteractions allow limited individuals to improve their cognition without modifying their timeand resource constraints. We do this in part by offering a formal criterion for population
rationality : the probability that any individual in a network makes a particular decision is the
same as the probability of that decision under a Bayesian posterior distribution conditionedonallthe information observed by the population (Foster, 2018, cf.).
To develop our account, we ﬁrst formulate a model of individual social learning based
on a simple heuristic that requires only limited social observation. We then show that undercertain conditions, groups of individuals following this heuristic will accumulate informationthrough distributed Bayesian inference. This model offers insight into how social learning canextend the cognitive abilities of limited individuals—while conditioning on all accumulatedinformation would be too complex for any individual, social interactions allow individualsto beneﬁt from this information while expending little cognitive effort. That is, populationrationality can be achieved by networks of highly limited individuals.
We tested our model in two large-scale experiments where participants made basic catego-
rization decisions in simple multigenerational networks. In both experiments, the decisionsof participants at one generation were propagated to those in the next generation, allowingus to study belief accumulation and transmission in a controlled laboratory setting. We found
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M. D. Hardy et al. / Topics in Cognitive Science 14 (2022) 553
that a substantial proportion of the data from these experiments are well-approximated by
our model. Furthermore, by comparing participants’ choices with the Bayesian posterior dis-
tribution conditioned on the information observed by the entire population, we were able to
quantify accumulation in our experimental networks relative to the Bayesian ideal. Our ﬁnd-
ings thus offer a clear demonstration of how limited individuals can use simple social learning
heuristics to make intelligent inferences.
2. Information accumulation through social sampling
Here, we describe a simple social learning heuristic we call social sampling that yields
distributed Bayesian inference at the population level. That is, populations of individuals fol-
lowing social sampling will accumulate information in a way that is consistent with Bayesian
inference, allowing individuals to ofﬂoad computation to the group and make accurate infer-
ences with little cognitive effort. We focus on multigenerational settings, where individu-
als are organized in discrete “batches” and learn from those in the previous generation. We
also assume individuals observe others’ true beliefs and do not need to discard or modify
any observations.
2.1. Model
Information accumulation by rational agents is speciﬁed by Bayes’ rule, which indicates
how a probability distribution over hypotheses θ(known as the prior distribution) should be
updated (to the posterior distribution) in light of evidence D:
p(θ|D)=p(D|θ)·p(θ)
p(D). (1)
In this paper, we extend this characterization of optimal belief updating from individuals to
groups (see Chater, Oaksford, Hahn, & Heit, 2010; Grifﬁths, Kemp, & Tenenbaum, 2008;
Tenenbaum, Kemp, Grifﬁths, & Goodman, 2011, for reviews of Bayesian models of individ-
ual cognition). Viewing a population as a single agent is a perspective with roots in many tra-
ditions, including philosophy (Easwaran, 2019), economics (Gale & Kariv, 2003), economics
(Hayek, 1945), organization science (Argote, 2012), cognitive science (Krafft, Shmueli, Grif-
ﬁths, Tenenbaum, & Pentland, 2021), anthropology (Hutchins, 1995), collective intelligence
(Engel et al., 2015; Woolley, Chabris, Pentland, Hashmi, & Malone, 2010), ethology (Sasaki
& Pratt, 2011), and computer science (Lynch, 1996; Shoham & Leyton-Brown, 2009). Cru-
cially, as we will show, rational action at the population level requires only limited compu-
tation by individuals—while each individual agent may follow a simple heuristic, the infor-
mation accumulated by the population as a whole makes it possible for those individuals to
act rationally.
We consider the simple case where individuals reason about a set of that can be either true
or false, represented as a binary feature vector x. Individuals beliefs are shaped by pieces
of evidence Y=(y1,..., yT) that arrive over time. At each time t, individuals observe J
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554 M. D. Hardy et al. / Topics in Cognitive Science 14 (2022)
pieces of information about feature i, with Pi=P(yij t=1|xi=0)=θi0ifxi=0a n d Pi=
P(yij t=1|xi=1)=θi1ifxi=1. For simplicity, we let Sit=/summationtextJ
j=11(yijt=1) be the number
of positive observations made about feature iat time t.
We assume that the popularity of a belief among members of the population acts as a
prior distribution over beliefs for new individuals drawing inferences about the environ-
ment. Speciﬁcally, an individual aﬁrst chooses a member of the population a/primeto learn from
uniformly at random.1The learner athen accepts or rejects their companion’s decision da/prime,i,t−1
with probability proportional to ( θi,da/prime,i,t−1)Sit(1−θi,da/prime,i,t−1)J−Sit, the likelihood of the evidence
thataobserves at time tbased on the beliefs of a/prime.
In an inﬁnite population of individuals following this strategy, the probability that an indi-
vidual makes a categorization decision about feature iis equal to the posterior probability of
that decision conditioned on the evidence observed by the entire population:
pi,t+1=P(xi=1|yi,·,≤t). (2)
This result establishes that social sampling is a valid algorithm for distributed Bayesian infer-
ence in inﬁnitely large populations, satisfying our criterion for population rationality (see
Supplementary Electronic Material (SEM) for proof). The ﬁnite population case is more com-
plex, but social sampling can be seen to be formally equivalent to a class of sequential Monte
Carlo algorithms known as particle ﬁlters (Crisan & Doucet, 2002; Murphy, 2012). Social
sampling can thus support Bayesian inference in expectation in ﬁnite populations, allowing
groups of bounded individuals to accumulate knowledge over time.
2.2. Measuring accumulation relative to optimal inference
The model we have outlined illustrates how a simple social decision-making heuristic
can lead to optimal information aggregation over generations. This suggests that in certain
contexts the Bayesian posterior distribution can be used to construct a valid upper limit on
information accumulation, because posterior distributions are information-theoretically and
decision-theoretically optimal belief representations (Bernardo & Smith, 2000; Jaynes, 2003;
Ortega, 2011). To quantify ideal information accumulation, we use the sufﬁcient statistics of
the Bayesian posterior distribution over the environment features given the evidence observed
by the entire population. We assume a uniform prior, and so the posterior probability that a
feature categorization is correct given the evidence received by the group up to generation t
is
P(xit=1|yi,·,≤t)=(θi1)Si,≤t(1−θi1)t·J−Si,≤t
(θi1)Si,≤t(1−θi1)t·J−Si,≤t+(θi0)Si,≤t(1−θi0)t·J−Si,≤t, (3)
where Si,≤t=/summationtextt
k=1/summationtextJ
j=11(yijk=1) is the total positive evidence observed for feature iup to
time t.
1This assumption may seem implausible in large populations, but it is satisﬁed if each person chooses from a
small number of others and those others are an unbiased sample from the full population.
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M. D. Hardy et al. / Topics in Cognitive Science 14 (2022) 555
The sufﬁcient statistics of this posterior distribution areSi,≤t
t·J, or the proportion of pieces
of evidence favoring a feature categorization. Because these statistics can be used to exactly
compute the posterior distribution, we can measure accumulation in observed networks by
comparing this statistic with the popularity of a belief, or proportion of people who believe it.
We express this formally by letting pit=1
N/summationtext
a1(da,i,t−1=1) be the proportion of individuals
who believe xi=1 at time t−1, with daitindicating whether individual aat generation t
chooses category i,a n d Nrepresenting the number of individuals in each generation. If the
popularity is close toSi,≤t
t·Jat time t, we can conclude that information is being effectively
accumulated in the population.
2.3. Summary
This analysis demonstrates that individuals in multigenerational networks can improve
their inferences by following a simple social learning heuristic. To perform social sampling,
an individual ﬁrst randomly selects a person in the previous generation to learn from. They
then evaluate their companion’s belief against observed evidence—the more aligned the belief
with the evidence, the higher the chance the individual will accept it as their own, otherwise
they will continue searching and sample another person. While similar social learning models
have been explored in multiarmed bandit problems (Celis, Krafft, & Vishnoi, 2017; Krafft,
2017; Krafft et al., 2021), they have not been tested in controlled laboratory settings.
Social sampling is simple to perform and only requires limited computation by individuals.
More precisely, at any time the popularity of a hypothesis will approximate the true Bayesian
posterior probability conditioned on all the evidence observed by the population, and so the
population itself performs distributed Bayesian inference. Because this distributed inference
matches the inference problems individuals face, our model allows us to be precise about how
social learning extends people’s cognitive abilities.
In our model, individuals make unbiased judgments and cannot choose which information
to transmit. Instead, every person’s true beliefs can be observed by those in the next gen-
eration. While this setup—and assumptions of uniform sampling of others’ true beliefs—is
highly simpliﬁed, it reﬂects a common approach to modeling individual cognition as a two-
stage decision-making process (Howard & Sheth, 1969; Krumme, Cebrian, Pickard, & Pent-
land, 2012; Payne, 1976; Pratt, Sumpter, Mallon, & Franks, 2005; Seeley & Buhrman, 1999;
Vul, Goodman, Grifﬁths, & Tenenbaum, 2014) that has also been applied to modeling iterated
learning in populations (Bonawitz, Denison, Gopnik, & Grifﬁths, 2014; Kalish, Grifﬁths, &
Lewandowsky, 2007; Mozer, Pashler, & Homaei, 2008; Sanborn & Grifﬁths, 2008; Vul &
Pashler, 2008) and allows us to make progress in understanding knowledge accumulation in
a way that can be studied empirically and extended to more naturalistic settings.
While our model shows how group-level Bayesian inference is possible, can we identify
and quantify accumulation in real populations? To investigate, we ran two behavioral exper-
iments where participants made categorization decisions in multigenerational networks. In
both experiments, we assessed population rationality by comparing observed popularity with
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556 M. D. Hardy et al. / Topics in Cognitive Science 14 (2022)
the Bayesian ideal and evaluated our social sampling model by comparing it with several
alternative models.
3. Experiment 1: Gem classiﬁcation
In Experiment 1, participants completed a categorization task that was framed as gem-
stone classiﬁcation. This experiment was designed to closely mirror an idealized setting in
which optimal information aggregation is possible through our proposed Bayesian social sam-
pling mechanism.
In this experiment, participants played the role of technicians classifying gemstones in a
certain shift (i.e., generation).2Each gemstone could have up to eight potential classiﬁcations,
with each classiﬁcation being a randomized nonsense word (e.g., “pesho,” “ivil,” “thyun”).
On each trial, participants observed social information in the form of classiﬁcation judgments
drawn from the previous shift of technicians, as well as non-social information in the form
of a set of four new laboratory results. The laboratory results presented stochastic positive or
negative evidence for each potential gemstone classiﬁcation. We recruited three independent
networks of participants, with each network including 10 shifts of 20 technicians (i.e., 200
participants per network). The interface for the experiment is illustrated in Fig. 1.
3.1. Methods
3.1.1. Participants
We recruited 600 participants from Amazon Mechanical Turk. Recruitment was restricted
to participants in the United States with an Amazon Mechanical Turk approval rating of 95 or
above. We paid participants $1.75 as compensation, plus a performance-based bonus payment
of up to $0.50. The task typically took participants under 10 min to complete. Prior to starting
the experiment, participants completed an attention and comprehension check which included
questions about details of the study, including the probability of different kinds of evidence
appearing in individual tests. Participants who failed to answer these questions correctly in
three attempts were excluded from taking part in the study.
3.1.2. Stimuli
On each round of the experiment, participants observed a cartoon gemstone positioned at
the top of the screen on a white background. Each gem had eight possible binary classiﬁca-
tions, and participants performed lab tests to make these classiﬁcations. Each test resulted in
either positive or negative evidence for each classiﬁcation. Fig. 1 shows an example partici-
pant view.
Classiﬁcations were given labels from an artiﬁcial vocabulary. These labels were differ-
ent on each trial a participant completed (i.e., none of the labels recurred across trials for a
2See https://github.com/pkrafft/Overcoming-individual-limitations-through-distributed-computation for
experiment code and data.
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M. D. Hardy et al. / Topics in Cognitive Science 14 (2022) 557
Fig. 1. Participant interface for Experiment 1. Each categorization task consisted of a different set of gem clas-
siﬁcations that were given random nonsense words. Each participant belonged to a “generation” of participants,
called a “shift” in this experiment. In each generation after the ﬁrst, participants could view the gem classiﬁcationsmade by participants in the last generation. Each participant made their own classiﬁcation decisions for a gem four
times after observing four pieces of non-social evidence.
participant) and were presented in randomized order. After each test, participants indicated
which of the eight classiﬁcations they thought were true for this gem and could modify theseclassiﬁcations throughout the trial.
Participants in generations 2–10 observed the ﬁnal classiﬁcation decisions of a randomly
sampled participant from the previous generation of the same network. Participants couldchoose to resample from the previous generation at any time during the trial. When partici-pants chose to resample, the new sample was selected at random with replacement from the20 participants in the previous generation of the same network. Social information (an earlierparticipant’s classiﬁcation decisions) was presented in a feedback table below the participant’s
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558 M. D. Hardy et al. / Topics in Cognitive Science 14 (2022)
Fig. 2. Illustration of the evidence correlation structure used in Experiment 1. If a gem classiﬁcation test was
positive for one participant, it was positive for all participants in that generation on that classiﬁcation task. Positive
evidence about a classiﬁcation is illustrated here using turquoise circles and negative evidence with red circles. Theclassiﬁcation judgments made by participants are shown as checkmarks and x-marks. Participants could change
their classiﬁcations after each new test.
own feedback table. After completing the fourth lab test, the results of the test were displayed
and participants moved on to the next trial.
3.1.3. Procedure
Participant completed four trials (i.e., gem classiﬁcations). The ﬁrst two trials were practice
rounds and were presented in the same order for all participants. The last two tasks wereexperimental trials and were ordered using simple randomization. We limited our analysis todata from the two experimental trials.
Mirroring the social sampling model, participants were organized into discrete shifts, or
“generations.” We recruited three independent networks of participants, with each networkconsisting of 10 generations with 20 participants per generation (see Fig. 2). Participants ingenerations 2–10 observed the classiﬁcations of a randomly sampled member of the previous
generation and could choose to draw additional samples as many times as they wished.
At the beginning of the experiment, the true classiﬁcations for each task were chosen uni-
formly at random. To reduce variance between networks, the lab test results for a given clas-siﬁcation were the same for all participants in a single generation (see Fig. 2). The probabilityof positive evidence in a lab test result was 0.6 for true classiﬁcations and 0.3 for false classi-ﬁcations (participants were informed of both probabilities).
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M. D. Hardy et al. / Topics in Cognitive Science 14 (2022) 559
Fig. 3. Experiment 1 results. The graph on the left shows the relationship between the total evidence available for
each gem classiﬁcation and the proportion of participants who selected that classiﬁcation (i.e., the popularity of
the classiﬁcation).The graph on the right plots the relationship between the most recently observed evidence and
the popularity. Each point represents a gem classiﬁcation for a single shift, trial, and replication.
3.2. Results and discussion
We limited our analysis to participants in generations 2–10 and preregistered our statistical
analyses and sample size before the experiment.3These analyses, however, were adjusted and
expanded over the course of revisions. This included adding model comparisons, and using all
the test data in the regression described below (rather than only one decision in generations
6–10) for consistency with our analysis for Experiment 2. Limiting our regression to the
preregistered subset does not change our ﬁndings.
As predicted by our social sampling model, we observed a strong correlation between the
popularity of each gem classiﬁcation and both the total evidence ( r(430)=0.6,p<.0001)
and the most recent evidence ( r(430)=0.88,p<.0001; see Fig. 3). Furthermore, total evi-
dence was signiﬁcantly predictive of popularity in a regression that included the total evi-
dence, most recent evidence, and ﬁxed effects for each network ( t(424)=4.54,p<.0001;
see Table 1 for full regression results). The correlation between popularity and total evidence
shows that the proportion of social sampling was high enough to facilitate signiﬁcant infor-
mation aggregation over time.
We then compared the performance of our Bayesian social sampling model to three aso-
cial models and two social models—a probability matching social sampling model and naive
copying model (models are described in detail in the SEM). In the probability matching social
sampling model, individuals accept candidate hypotheses with probability equal to the frac-
tion of most recent positive evidence and in the naive copying model, individuals simply
copy others’ categorizations. To account for non-social learners, both social sampling models
include a term capturing the proportion of social decision making. We estimated this value as
the proportion of trials where participants chose to resample at least once and found partici-
3Preregistration available at https://osf.io/7qx8e/?view_only=53781c6aa4c742e2a9843fcef533dfd7.
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560 M. D. Hardy et al. / Topics in Cognitive Science 14 (2022)
Table 1
Regression results for Experiments 1 and 2
Regression Model
Variable Experiment 1 Experiment 2
Intercept −0.10**0.29***
(−3.15) (12.69)
Total evidence 0.24***0.05*
(4.54) (2.44)
Last evidence 0.91***0.25***
(29.26) (18.74)
Dependent variable Popularity Popularity
Degrees of freedom 424 2270
Observations 432 2304
tstatistics are in parentheses.
*p<.05, ** p<.01, *** p<.001.
Note that ﬁxed effects for each network (Experiments 1 and 2) and each condition (Experiment 2) were included
in the regressions but are not shown.
Table 2
Mean squared error of predicted and observed popularity for each model in Experiment 1
Model Mean Squared Error
Non-social problem matching 0.0213
Non-social Bayesian problem matching 0.0220
Non-social utility maximizing 0.1220
Naive copying 0.1260
Social sampling (problem matching) 0.0181
Social sampling (Bayesian) 0.0182
The best performing model is shown in bold.
pants did so on 19% of trials. This is a lower bound on the true proportion of social sampling,
since participants may have used the initial social information without resampling. However,
it is close to both a separate estimate of the asocial learning proportion based on a qualitative
coding of participants’ strategy descriptions, and the estimated regression coefﬁcient on total
evidence available, which should correspond to the level of social sampling (see Table 1).
No free parameters are used in any of these models, and so we evaluate each model using
their performance on all the experimental data. We found that the probability matching social
sampling model achieved the lowest mean squared error (MSE) in predicting the popularity of
each gem classiﬁcation (MSE : 0 .0.0181), followed closely by the Bayesian social sampling
model (MSE : 0 .0.0182; see Fig. 4 and Table 2 for results). Because both social sampling
models predict accumulation, these results support our other analyses from Experiment 1,
suggesting that networks of participants accumulated information about each categorization
across time.
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M. D. Hardy et al. / Topics in Cognitive Science 14 (2022) 561
Fig. 4. Model predictions for the three non-social models and three social models we compared in Experiment 1.
As in Fig. 3, 432 points are shown in each plot.
4. Experiment 2: Spaceship construction
We constructed Experiment 1 to have a high correspondence with the assumptions of our
social sampling model. Experiment 2 was designed to test our model in a less idealized con-text. Most notably, in Experiment 2l all information was social—participants received feed-back on the previous generation’s decisions, rather than their own. Furthermore, we did not
allow participants to resample from the previous generation, and feedback was censored based
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562 M. D. Hardy et al. / Topics in Cognitive Science 14 (2022)
Fig. 5. Example of the participant interface for the Experiment 2 spaceship design task. Each trial was associated
with a different set of spaceship parts (using different visual icons). Participants were organized into discrete
generations, or “turns.” In each generation after the ﬁrst, a participant started by viewing an example spaceship
design from a participant of the last generation. The participant then chose their own parts for that task and
observed the part failures on each ﬂight of that design.
on the observed participant’s choices. That is, participants did not always observe evidence
on every possible categorization.
In Experiment 2, participants designed spaceships by choosing which components to
include from an inventory of alternatives (see Fig. 5).4Each component was either a good
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M. D. Hardy et al. / Topics in Cognitive Science 14 (2022) 563
Table 3
The parameter settings associated with each condition of Experiment 2
Experiment Reps NJ θ
Condition 1 2 20 4 0.6
Condition 2 1 20 1 0.6
Condition 3 1 20 4 0.8
Condition 4 2 5 4 0.6
Condition 5 2 5 1 0.6
Note. The repetitions (reps) is the number of repetitions of the parameter settings we ran. Nis the number
of participants per generation. Jis the number of ﬂights, that is the amount of evidence, shown per spaceship
design trial. θgives the probability of success of good parts, and the probability of failure of bad parts—that is,
the strength of evidence.
part that rarely failed or a bad part that failed often. On each trial, participants observed
spaceships designed in the previous generation, along with the success or failure of each of
the included components on one or more ﬂights. We organized participants into ﬁve condi-
tions, varying the number of participants in each generation (large vs. small), the amount of
evidence presented to each participant (high vs. low), and the strength of that evidence (high
v s .l o w )( s e eT a b l e3 ) .
4.1. Methods
4.1.1. Participants
We recruited 1000 participants from Amazon Mechanical Turk. Recruitment was restricted
to participants living in the United States. The task typically took participants around 5–10
min to complete, and participants earned $1.80 as compensation.
4.1.2. Stimuli
Experiment 2 consisted of a spaceship design categorization task inspired by prior experi-
ments on social learning (Brand, Brown, & Cross, 2018). Example stimuli from this catego-
rization task are shown in Fig. 5. In this task, participants designed spaceships by selecting
which parts to include from a set of available options. A different set of eight parts was avail-
able on each trial. Each of the eight items in a trial’s set of spaceship parts could be included
in that trial’s spaceship design. Participants could select or remove a part in a trial’s design by
clicking on an icon of the part.
Participants were organized into 10 discrete generations, with each generation composed
of a different set of participants. In the ﬁrst generation, each participant had no information
about part failures and had to simply guess which parts might be good or bad. After selecting
which parts to include, a participant was given feedback from a number of “ﬂights” of their
constructed spaceship. On each ﬂight, spaceships parts could either succeed or fail. Each
4See https://github.com/pkrafft/Overcoming-individual-limitations-through-distributed-computation for
experiment code and data.
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564 M. D. Hardy et al. / Topics in Cognitive Science 14 (2022)
ﬂight was shown in sequence, and we did not allow participants to change the spaceship
design between ﬂights.
In all generations except the ﬁrst, participants viewed the spaceship design and ﬂight out-
comes from a spaceship selected uniformly at random from the designs made in the previousgeneration. Participants viewing another participant’s design could select which of the priorﬂights to examine, as different parts may have failed on different ﬂights. A prior design wasdisplayed in the same way that a participant’s own spaceship design was displayed, exceptthat parts could not be modiﬁed on a previous design and part failures were shown at thestart of the trial. Unlike Experiment 1, participants could not resample a choice from theprevious generation.
4.1.3. Procedure
Participants completed eight categorization tasks. The ﬁrst four trials participants com-
pleted were practice trials, and the last four were test trials. Practice trials were displayed inthe same order for all participants, and test trials were ordered using simple randomization.We did not inform participants of this practice-test distinction, and we limited our analysis todata from the four test trials.
In order to mirror the structure of the social sampling model, we recruited participants
in discrete generations. Generation twas recruited after all participants in generation t−1
completed the experiment. In the ﬁrst generation, good and bad parts were chosen at randomwith probability .5. As in Experiment 1, spaceship failures were perfectly correlated acrossdesigns for a single ﬂight in a given generation to reduce variance between networks. That
is, if two different participants both used the same part in a certain generation, then the part
would either succeed or fail for both participants on a given ﬂight (see Fig. 6).
4.1.4. Results and discussion
As in Experiment 1, all statistical tests were two-tailed with an alpha level of 0.05 and we
excluded the initial generation of participants from our analysis. We ran experiment condi-tions separately, and so no between-subject randomization into conditions was used. Instead,participants could only participate in the experiment once and thus could not complete multi-ple experimental conditions.
Replicating our ﬁndings from Experiment 1, we found that total evidence was signiﬁcantly
predictive of popularity, controlling for the most recent evidence observed in a regressionthat included the total evidence, most recent evidence, and ﬁxed effects for each networkand condition ( t(2270) =2.44,p=.015; full results in Table 1). While we found signiﬁ-
cant positive correlations between popularity and total evidence in each of our conditions,the strength of this correlation varied between 0.16 in Condition 5 and 0.65 in Condition3 (see Table 4 and Fig. 7). Indeed, popularity tended to be less correlated with total evi-dence in networks with small generation sizes, low evidence strength, and low evidenceamount.
All decisions in Experiment 2 were social, and so we limited model comparison to the three
social models used in Experiment 1 and set the proportion of asocial learners to zero. As in
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M. D. Hardy et al. / Topics in Cognitive Science 14 (2022) 565
Fig. 6. Illustration of the evidence constraints in Experiment 2. As in Experiment 1, evidence was correlated across
participants: If a part failed on one ﬂight, it failed for all participants in that generation on that ﬂight (part failures
are displayed in the ﬁgure as red circles). In contrast to Experiment 1, participants could not change their spaceship
designs in between ﬂights, and evidence about a particular spaceship part was only shown if the part was chosen
in the sample design shown to the participant.
Table 4
Overview of the ﬁve conditions of Experiment 2 and the correlation between popularity and total evidence within
each condition
Condition RepsPopulation
SizeEvidence
AmountEvidence
StrengthCorrelation Between Popu-
larity and Total Evidence
Condition 1 2 Large High Low .43***
Condition 2 1 Large Low Low .39***
Condition 3 1 Large High High .65***
Condition 4 2 Small High Low .29***
Condition 5 2 Small Low Low .16***
*p<.05, ** p<.01, *** p<.001.
Experiment 1, each model was assessed by comparing the popularity predicted by the model
at each generation with the observed popularity. We found that in four of the ﬁve experimental
conditions, the Bayesian social sampling model achieved the lowest MSE in pre-
dicting observed popularity. However, in the remaining condition, where the evidence
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566 M. D. Hardy et al. / Topics in Cognitive Science 14 (2022)
Fig. 7. Plots showing the relationship between the total evidence available for each particular spaceship part and
the proportion of participants who selected that part in Experiment 2. Each point represents one part (of eight) in
one social generation (of nine) on one trial (of four) in a single repetition.
strength was high, the naive copying model outperformed both the Bayesian and
probability matching social sampling models (Bayesian social sampling MSE: 0.1188;probability matching social sampling MSE: 0.0825; naive copying MSE: 0.0371; seeTable 5).
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M. D. Hardy et al. / Topics in Cognitive Science 14 (2022) 567
Table 5
Mean squared error of predicted and observed popularity for the three social models we compared for each of the
conditions (C1–C5) in Experiment 2
Mean Squared Error
M o d e l C 1C 2C 3C 4C 5
Naive copying 0.0433 0.0424 0.0371 0.1078 0.1030
Social sampling (matching) 0.0542 0.1703 0.0825 0.1069 0.2196
Social sampling (Bayesian) 0.0366 0.0333 0.1188 0.0969 0.0960
The best performing model in each column is shown in bold.
5. General discussion
Understanding how people make intelligent decisions under limited time and cognitive
resources is a central focus of research in cognitive science, psychology, and economics.Social interactions offer a way for people to overcome these limitations by distributing
computation across a group. By observing and learning from others, individuals do not need
to perform complex computations or to condition on large amounts of data to make rationalinferences. Instead, they can use simple heuristics that leverage accumulated social informa-tion.
To show how distributed inference can emerge in populations, we derived a social sampling
model of individual decision making in multigenerational networks. We then showed thatsocial sampling at the individual level can lead to distributed Bayesian inference at the popu-lation level. We tested the predictions of our social sampling model in two highly controlledbehavioral experiments where participants made simple categorization decisions. While thetransmission structure we used in both experiments was idealized and highly simpliﬁed com-pared to real-world social networks, this allowed us to directly quantify the degree to whichinformation accumulated across time relative to the Bayesian ideal.
Although we observe information accumulation in both experiments, the performance of
the Bayesian social sampling model varied in different conditions. Most notably, in Exper-iment 1 the probability social sampling model achieved a slightly lower MSE than theBayesian social sampling model (see Table 2). Furthermore, in Experiment 2 the naive copy-ing model had a lower MSE than both social sampling models in large networks with highlevels of strong evidence (see Tables 4 and 5). These results suggest that individuals mayadapt their social learning strategies to different domains and may be more likely to use socialsampling in noisy, low-information environments (Toyokawa, Whalen, & Laland, 2018).
It is important to emphasize that for information aggregation to occur, the precise details of
how social sampling takes place are less important than that individuals’ decisions incorporate
social information and new evidence using a probabilistic rule. Indeed, our social samplingmodels in Experiment 1 included a mixture of social and non-social decision makers, repli-cating related ﬁndings on underexploitation of social information in behavioral experiments(Mercier & Morin, 2019). Our analyses are thus not intended to show that social sampling isa deﬁnitive description of human behavior, but that a social decision-making model derived
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568 M. D. Hardy et al. / Topics in Cognitive Science 14 (2022)
from a normative Bayesian standard can help make sense of the extent of information aggre-
gation we observe in speciﬁc experimental contexts.
While we ﬁnd that information aggregation can be robust to individual differences, accu-
mulation can fail if people’s decision-making differs systematically from a probabilistic beliefadoption–rejection strategy. Indeed, related research on the wisdom of crowds has shown thatin certain contexts social interactions and observations can actually decrease group perfor-mance by reducing the diversity and independence of people’s beliefs (Jenness, 1932; Lorenz,Rauhut, Schweitzer, & Helbing, 2011; Myers & Bishop, 1971). These dynamics—that is,whether social interactions improve or worsen group outcomes—appear to vary dependingon people’s social learning strategies (Toyokawa et al., 2018). For example, accumulation canbe disrupted if people are utility-maximizing rather than probabilistic in their responses to evi-dence (Anderson & Holt, 1997; Bikhchandani, Hirshleifer, & Welch, 1992). While probabilitymatching in individual decision-making is observed in certain domains (Shanks, Tunney, &McCarthy, 2002; Vul et al., 2014; Vulkan, 2000), our experiments show that this strategycan be extended to social settings: when a substantial proportion of individuals incorporatesocial information into their probability matching behavior, information accumulates acrosspeople and time. However, people may be less likely to follow a social sampling strategy indomains with low levels of probability matching, such as tasks with large ﬁnancial incentivesor consistent feedback (Shanks et al., 2002).
Our ﬁndings complement other approaches to understanding rationality in limited indi-
viduals. For example, previous work on adaptive heuristics (Gigerenzer & Goldstein, 1996)has shown that people use simple decision-making rules to exploit the structure of the envi-
ronment and make decisions on complex tasks. Our work suggests that this framework can
be extended to social decision making. Furthermore, by providing a rational motivation forpeople’s social heuristics, our work offers a way to connect adaptive heuristics with resource-rational analysis (Grifﬁths, Lieder, & Goodman, 2015) in simple multigenerational popula-tions. Integrating our model and ﬁndings with other frameworks for studying bounded ratio-nality, such as models that utilize quantum probability theory (Pothos et al., 2021), should beaddressed in future work.
We tested our model on a simple binary-choice decision-making task where the transmis-
sion structure was explicitly designed to match our model assumptions. These simpliﬁcationslimit our ability to draw general conclusions about information accumulation in natural pop-ulations. For example, while enforcing uniform sampling of the previous generation gave ustight control over the transmission dynamics, people’s sampling, and exploration strategiesmay depend on the task (Oaksford & Chater, 1994) and are likely considerably more complexin naturalistic domains and networks (Latora, Nicosia, & Russo, 2017). Similarly, people maynot always transmit their true beliefs—instead, they may give noisy, incomplete, or mislead-ing accounts of their opinions and decisions to others (Xu & Grifﬁths, 2010).
Despite these limitations, the general framework we have outlined and the principle of
identifying the computational structure of population dynamics could therefore be extendedto more complex and naturalistic domains in future work. For example, individuals could be
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M. D. Hardy et al. / Topics in Cognitive Science 14 (2022) 569
given greater control over which (if any) beliefs to transmit or sample from others. Mod-
els of learning in structured representational domains, such as language of thought models
(Goodman, Tenenbaum, & Gerstenberg, 2015) or posterior sampling in general Markov deci-sion processes (Agrawal & Jia, 2017; Osband, Van Roy, & Russo, 2013), could also beextended to the population setting in a way that is analogous to our extension of simple cat-egorization decisions. Testing our model and quantifying knowledge accumulation in morenaturalistic domains is an exciting challenge for future work.
Acknowledgments
This work was funded in part by NSF grant 1456709, DARPA Cooperative Agreement
D17AC00004, and the support of the NOMIS Foundation.
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Hardy
M
D
Krafft
P
M
Thompson
B
& Griffiths
T
L