# The impact of incorrect social information
on collective wisdom in human groups

Bertrand Jayles1,2,3,∗††Corresponding author –<EMAIL_ADDRESS>Ramón Escobedo3, Stéphane Cezera4, Adrien Blanchet4,5,
Tatsuya Kameda6, Clément Sire2, Guy Theraulaz3,5

(1Center for Adaptive Rationality, Max Planck Institute for Human Development,
Berlin, Germany
2Laboratoire de Physique Théorique, CNRS, Université de Toulouse, France
3Centre de Recherches sur la Cognition Animale, CNRS, Université de Toulouse,
France
4Toulouse School of Economics, INRA, Université de Toulouse (Capitole), 31000
Toulouse, France
5Institute for Advanced Study in Toulouse, 31015 Toulouse, France
6Department of Social Psychology, The University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-0033, Japan
)

###### Abstract

A major problem that results from the massive use of social media networks is
the possible spread of incorrect information. However, very few studies have
investigated the impact of incorrect information on individual and collective
decisions. We performed experiments in which participants had to estimate a
series of quantities before and after receiving social information.
Unbeknownst to them, we controlled the degree of inaccuracy of the social
information through “virtual influencers”, who provided some incorrect
information. We find that a large proportion of individuals only partially
follow the social information, thus resisting incorrect information. Moreover,
we find that incorrect social information can help a group perform better when
it overestimates the true value, by partly compensating a human
underestimation bias. Overall, our results suggest that incorrect information
does not necessarily impair the collective wisdom of groups, and can even be
used to dampen the negative effects of known cognitive biases.

The digital revolution has changed the way people access and share
information. In particular, the past few decades have seen an exponential
increase of media sources and amount of available information [1]. Moreover, a
growing distrust in traditional media has given an increasing share of news
consumption to social networks and other pathways to relay information. This
facilitated and more diverse access to information may arguably enhance
people’s ability to make informed decisions, but at the same time such an
information overload dramatically increases the difficulty to verify
information, understand an issue, or make efficient decisions [2, 3]. In
certain cases, it has also disrupted the relationship between citizens and the
truth [4, 5], leading to polarized communities unable to listen to each other
[6]. Recently, the effects of large scale diffusion of incorrect information
and fake news on the behavior of crowds have gained increasing interest,
because of their major social and political impact [7]. The propagation of
false information is also reinforced by the use of social bots simulating the
behavior of Internet users [8]. In particular, there has been recent evidence
that fake news can propagate faster and affect people deeper than true
information on Twitter, especially when they carry political content [9]. In
this context, there is a strong need to understand how the diffusion of
incorrect information among group members affects individual and collective
decisions.

To address this issue, we use the experimental framework of estimation tasks,
which is highly suitable for quantitative studies on social influenceability
[10, 11, 12, 13, 14, 15, 16]. We performed experiments in which subjects had
to estimate a series of quantities with varying levels of demonstrability,
before and after having received social information. The demonstrability of a
quantity can be interpreted as the amount of prior information a group has
about it. To put it in simple terms, it represents the “difficulty” to
determining the actual value of a quantity, a notion which will be made
explicit and quantitative hereafter. Knowing the individuals’ estimates before
and after receiving the social information, as well as the value of the social
information, we can deduce their sensitivity to social influence. Moreover, by
introducing _“virtual influencers”_ providing either the true value or some
incorrect information – without the subjects being aware of it – we control
the _quality_ of the information provided to the subjects, and quantify its
resulting impact on individual and collective accuracy.

To compare estimates of different quantities, it is usually necessary to
normalize them. A natural and commonly used normalization consists in dividing
estimates by the true value of the quantity of interest [17, 18]. Here, we
show that this normalization is insufficient for comparing quantities of a
very different nature, and that the dispersion of estimates must be included
in the normalization process, which has hitherto largely been neglected [11,
12, 16, 17, 18, 19]. We provide an adequate normalization procedure, and
discuss its implications in terms of distributions of estimates. Moreover, we
demonstrate that providing a moderate amount of incorrect information to
individuals can counterbalance a human tendency to underestimate quantities
[20, 21, 22, 23], and thereby improve estimation accuracy. We also find that
when social information contradicts the underestimation bias, a strategy which
consists in compromising between one’s opinion and the group’s opinion
increases the performance of groups, even when the social information is
substantially inaccurate. However, compromising with inaccurate social
information that does not contradict the bias may amplify its deleterious
effects, and hamper collective performances.

Finally, we use a modified version of an agent-based model developed in [18]
to better understand the present results, and to analyze the collective
response of human groups to information of which levels of inaccuracy go
beyond the values tested in our experiments. The model quantitatively
reproduces the experimental results, and confirms the counter-intuitive
observation that incorrect information can improve a group’s performance, in
particular when the group underestimates the true value and when the
information compensates this bias.

## Experimental design

180 subjects participated in our experiment. 20 sessions were organized, in
each of which 9 subjects were asked to estimate 32 quantities. Each quantity
was estimated twice: subjects first provided their personal/prior estimate
$E_{\rm p}$. Next, they received as social information the geometric mean $G$
of the $\tau$ previous estimate(s) in the sequence ($\tau=1$ or 3), and were
then asked to provide a second/final estimate $E_{\rm s}$. The value of $\tau$
was unknown to the subjects and so was the exact nature of the mean provided.
Moreover, this second estimate $E_{\rm s}$ was used to update the social
information for the corresponding subject in the next session. Hence, our
experiment produced $9\times 32\times 20=5760$ personal and second estimates,
adding up to a total of 11520 estimates.

We controlled the quality of the social information provided to the subjects,
without them being aware of it. To that end, we inserted in the sequence of 20
final estimates given by the subjects – unbeknownst to them – $n=0$, 5, or 15
artificial estimates. These additional estimates correspond to a fraction
$\rho=\frac{n}{20+n}=0\,\%$, $20\,\%$, or $43\,\%$ of “virtual influencers”.
Each sequence thus consisted of $N=20+n=20$, 25, or 35 estimates overall,
among which 20 were estimates given by 20 actual participants, one per
session. The influencers’ estimates were introduced at random locations in the
sequences. The value $T_{\rm I}$ of the influencers’ estimates provided to the
participants was controlled through a parameter $\alpha$, which represents a
distance to the true value quantifying the (in)correctness of the influencers’
estimates ($\alpha=-2$, $-1$, $-0.5$, $0.5$, 1, $1.5$, 2, 3). $\alpha$ was
normalized so as to make it independent of the questions and its computation
will be described in the Results section. In each session and for each
question, a subject was thus assigned a value of $\rho$, $\tau$ and $\alpha$,
and his/her second estimate was a single step in a sequence of 20, 25, or 35
estimates (Supplementary Figure S1 provides a graphic representation of the
protocol). Note that the estimates of the virtual influencers are also used to
update the social information which is then provided to the next subjects in
the sequence.

The quantities to estimate were grouped into four categories: visual
perception (number or length of objects in an image), population of large
cities in the world, daily life facts, and extreme (astronomical or
biological/geological) events. As we will see, the separation into these
loosely defined categories is reflected in the collected data. Three
additional questions were asked, which cannot be assigned to any of these
categories (see the list of questions in Supplementary Information). All
experimental details are given in the Materials and Methods section.

## Results

### Comparing quantities of very different nature

Because humans perceive numbers roughly as their order of magnitude [24, 25],
the logarithm of estimates is the natural quantity to consider in estimation
tasks, especially for large quantities, rather than the actual estimates
themselves. Distributions of estimates have indeed often been found highly
right-skewed, while the distribution of their common logarithm is generally
much more symmetric [11, 13, 17]. An important issue in estimation tasks is to
find a proper way to normalize and aggregate estimates arising from questions
with very different quantitative answers. Within studies, how can one
aggregate estimates of quantities that differ by several orders of magnitude?
Between studies, how can we compare findings coming from different sets of
quantities?

Figure 1: (a) Median and (b) mean dispersion $\langle|\log(E_{\rm p})-{\rm
Median}(\log(E_{\rm p}))|\rangle$ of the logarithms of estimates $E_{\rm p}$,
for the 32 questions asked in the experiment. Median($\log(E_{\rm p})$) scales
linearly with the log of the true value $T$. The red dashed line is the linear
regression, which slope is lower than 1 (blue dotted line), revealing the
human tendency to underestimate quantities.

In line with other works [26, 27], we find that the median log-estimate scales
linearly with the logarithm $\log(T)$ of the true value (Figure 1a), which
leads to the natural normalization: $X_{\rm p}=\log\left(\frac{E_{\rm
p}}{T}\right)$. $X_{\rm p}$ represents the deviation of an estimate from the
true value in orders of magnitude, and is often used as the quantity of
interest in estimation tasks [11, 12, 17, 18]. However, this normalization
does not take into account the dispersion of the log-estimates
$\langle|\log(E_{\rm p})-{\rm Median}(\log(E_{\rm p}))|\rangle$ (where
$\langle x\rangle$ refers to the mean of $x$) which can vary considerably for
different questions (Figure 1b). In the following, we simply refer to $X$ as
the “estimates”, dropping the “log-” prefix.

Figure 2: (a) Median $m_{\rm p}$ and (b) dispersion $\sigma_{\rm
p}=\langle|X_{\rm p}-m_{\rm p}|\rangle$ of estimates $X_{\rm
p}=\log(\frac{E_{\rm p}}{T})$, for the 32 questions asked in the experiment,
whose ID are ranked according to their $\sigma_{\rm p}$ which also reflects
their demonstrability. The 4 categories of questions (from left to right:
visual perception, population of large cities in the world, daily life facts,
extreme events), plus the three additional questions, are separated by dashed
lines. The categories are well distinct in panel (b), indicating that
$\sigma_{\rm p}$ is characteristic of the type of quantity to estimate, and
more precisely of a question’s demonstrability. In panel (a), the correlation
with demonstrability is much less clear, although $m_{\rm p}$ tends to grow on
average when the demonstrability decreases (i.e., when the question ID and
$\sigma_{\rm p}$ increase). The blue and red dashed lines in (a) are
respectively the average value of $m_{\rm p}$, and the quantity
$-\sqrt{\langle m_{\rm p}^{2}\rangle}$, in each category.

Figure 2a presents the median $m_{\rm p}$ and Figure 2b the dispersion
$\sigma_{\rm p}=\langle|X_{\rm p}-m_{\rm p}|\rangle$ of the personal estimates
$X_{\rm p}$, for all questions asked in this experiment (sorted by category of
questions). One can notice the extreme variation of both quantities depending
on the question, suggesting that including $m_{\rm p}$ and $\sigma_{\rm p}$ in
the normalization process is crucial to compare quantities of a different
nature. Figure 2b shows that the category of a question is clearly
identifiable by the dispersion of estimates $\sigma_{\rm p}$ (but not by the
median $m_{\rm p}$, see Figure 2a). The natural classification that we have
chosen a priori is thus reflected in the experimental data. Moreover, we see
that the less demonstrable a question is, the higher the dispersion of
estimates. This is further supported by the three unclassified questions (30
to 32): one could have predicted that they had a low demonstrability (i.e.,
that people have little prior information about them), and that they would
therefore be closer to the “extreme events” category than to the other
categories, as observed.

### Full normalization of estimates

In a previous study, we found and justified that the estimates $X_{\rm p}$ for
low demonstrability questions have a probability distribution function (PDF)
close to the Cauchy distribution [18]. This property can be explained by a
simple probabilistic argument: if two people provide estimates $X_{1}$ and
$X_{2}$ of a quantity about which they have no information at all, then the
average $\frac{X_{1}+X_{2}}{2}$ of both estimates cannot be a statistically
better estimation of the correct answer $T$. Hence, this average has
necessarily the same probability distribution as $X_{1}$ and $X_{2}$, and the
only distribution that satisfies such a property is the Cauchy distribution
(see also Material and Methods). Our model based on Cauchy distributions
convincingly reproduced the experimental data, and in particular fitted well
with the experimental distribution of estimates $X_{\rm p}$ [18].

However, as we pointed out above, $m_{\rm p}$ and $\sigma_{\rm p}$ have to be
considered to compare estimates for questions with answers spanning several
orders of magnitude. Hence, for each question characterized by its intrinsic
median $m_{\rm p}$ and width $\sigma_{\rm p}$, we normalize the estimate as
$Z_{\rm p}=\frac{X_{\rm p}-m_{\rm p}}{\sigma_{\rm p}}$. Figure 3 shows that
the normalized estimates $Z_{\rm p}$ follow the standard Laplace distribution
(i.e., with center 0 and width 1), $f(Z)=\exp(-|Z|)/2$, implying that the
$X_{\rm p}$ are also Laplace distributed for individual questions. It is only
when different questions with arbitrary widths $\sigma_{p}$ are aggregated
without our normalization that an overall Cauchy-like distribution for the
$X_{\rm p}$ emerges. Similarly, note that after social influence (red dots),
the $Z_{\rm s}=\frac{X_{\rm s}-m_{\rm s}}{\sigma_{\rm s}}$, with $m_{\rm
s}={\rm Median}(X_{\rm s})$ and $\sigma_{\rm s}=\langle|X_{\rm s}-m_{\rm
s}|\rangle$ also follow the standard Laplace distribution, implying that the
$X_{\rm s}$ also follow a Laplace distribution for each question. We will
therefore slightly modify the model developed previously [18], to replace
Cauchy distributions by Laplace distributions (see Model section).

Figure 3:  Distribution of fully normalized estimates $Z=\frac{X-m}{\sigma}$,
before (blue) and after (red) social influence. $m$ and $\sigma$ are
respectively the median of the estimates $X=\log(\frac{E}{T})$ and their
dispersion, for each corresponding question. $E$ are the actual estimates and
$T$ the true value for each corresponding quantity. The black lines are the
standard (center 0 and width 1) Laplace distribution (full line), the Cauchy
distribution (dashed line) and the Gaussian distribution (dotted line) of same
width. The Laplace distribution fits the experimental data the best. Red lines
(overlapping blue lines) are model simulations.

By measuring $m_{\rm p}$ and $\sigma_{\rm p}$ and using them in the
normalization process, we fix the quantity $\langle|Z_{\rm p}|\rangle=1$, and
therefore have some information about the distribution, instead of none for
the Cauchy distributions argument presented above. As shown in the Material
and Methods by exploiting the principle of maximum entropy, the most likely
distribution satisfying such a constraint is the Laplace distribution.

This constraint on the dispersion of estimates can be understood as an
intrinsic property of the system {group of individuals, question}: the
dispersion is characteristic of a given group of individuals estimating a
given quantity, and gives the typical range of answers that would seem
reasonable to most people in the group for that question. The lower the
demonstrability of a question (i.e., the lower the amount of prior information
held by individuals in a group about that question), the larger this range.
This is intuitive when considering the following example: an estimate three
orders of magnitude from the true value would seem absurd if one considers the
age of death of a celebrity, while it would seem perfectly plausible if one
considers the number of stars in the universe. While the normalization by
$m_{\rm p}$ is somewhat trivial (it simply shifts the center of the
distribution of $X$ to 0 for every question), the normalization by
$\sigma_{\rm p}$ is crucial in order to be able to properly compare and
aggregate estimates from different questions (and possibly, from different
studies). We wish to insist on the fact that this prescription is not a mere
methodological detail and that it should be adopted by future works in the
field.

In Figure S2, we show the distribution of $Z$ for the four categories of
questions. One can notice that for very large quantities (Figures S2c and d),
the left side of the distribution collapses faster than the right side,
suggesting that people have an intuition that such quantities must be large,
even though they know little about them, such that very small estimates are
less frequent. Such asymmetric Laplace distributions can also be derived from
the principle of maximum entropy, by adding a constraint that penalizes small
or large estimates (see Materials and Methods).

### Model

In [18], we have introduced an agent-based model to better understand the
effects of individual sensitivity to social influence, and of the quantity of
information delivered to the individuals, on collective performance and
accuracy observed at the group level in estimation tasks. Estimates $X_{\rm
p}$ are drawn from Laplace distributions, the center and width of which are
respectively the median $m_{\rm p}$ and dispersion $\sigma_{\rm
p}=\langle|X_{\rm p}-m_{\rm p}|\rangle$ of the experimental personal estimates
$X_{\rm p}$ for each question. Figure 4a presents the distribution of
estimates $X$ for all questions combined, before (blue) and after social
influence (red), as well as the corresponding distributions generated by our
model, when the $X_{\rm p}$ are generated from Cauchy distributions (as in our
previous research [18], dashed lines) and Laplace distributions (full lines).

Figure 4: (a) Probability density function (PDF) of individual estimates,
before ($X_{\rm p}$, blue) and after social influence ($X_{\rm s}$, red). Dots
show experimental data, dashed lines are model simulations based on Cauchy
distributions and full lines model simulations based on Laplace distributions.
Note the sharp decay on the left side of the distribution, well reproduced by
the model. The questions asked in our experiment imposed answers higher than
one, which translates into $X>-\log(T)$. (b) PDF of the sensitivity to social
influence $S$. The fractions of the five behavioral categories are shown, from
left to right: contradicters (“Cont”, $S<0$), keepers (“Ke”, $S=0$),
compromisers (“Comp”, $0<S<1$), adopters (“Ad”, $S=1$) and overreacters
(“Over”, $S>1$). Experimental data are shown in black, and model simulations
in red. (c) Average sensitivity to social influence $S$ against the distance
$D=M-X_{\rm p}$ between the social information $M$ and the personal estimate
$X_{\rm p}$. Because the average is sensitive to extreme values, we excluded
the values such that $|S|>100$, which represent less than $1\%$ of the data.
Black dots correspond to the experimental data, and red empty circles to the
model simulations. The dashed line shows the fraction of data for each dot.
Beyond three orders of magnitude ($|D|>3$) are only about $7.6\,\%$ of the
data, such that we neglect, in the model, the asymmetric weighting of social
information observed in this range of values.

The Laplace distribution is able to capture the estimates far from the truth
($X_{\rm{p,s}}>5$) better than the Cauchy distribution. It is important to
mention that in our previous study [18], the range of possible answers were
limited to plus or minus 3, 5, or 7 orders of magnitude from the true value,
depending on the question. By not allowing extreme answers, we probably
increased artificially the probability of estimates in the interval [5,7],
making the distribution even closer to a Cauchy distribution.

After providing its personal (log-)estimate $X_{\rm p}$, each agent receives
as social information the arithmetic mean $M$ of the $\tau$ previous final
estimates in the sequence, among which some information $V$ (provided by the
virtual influencers) is introduced with probability $\rho$. Note that the
actual participants where provided the geometrical mean $G$ of the $\tau$
previous estimates. In terms of log-estimates, the social information
$M=\log(G)$ indeed transforms into the standard arithmetic mean. The agent
then provides a second estimate $X_{\rm s}$, defined as the weighted average
of its personal estimate $X_{\rm p}$ and the social information $M$: $X_{\rm
s}=(1-S)\,X_{\rm p}+S\,M$, where $S$ is the weight given to the social
information, that we call sensitivity to social influence. $S$ can thus be
expressed as $S=\frac{X_{\rm s}-X_{\rm p}}{M-X_{\rm p}}$. In Figure 4b, we
show the distribution of $S$ from which five natural behavioral categories can
be identified: subjects keep their opinion (“keepers”, $S=0$), compromise with
the social information (“compromisers”, $0<S<1$), adopt the social information
(“adopters”, $S=1$), contradict it (“contradicters”, $S<0$) or overreact to it
(“overreacters”, $S>1$). In the model, after receiving the social information,
an agent keeps its personal estimate ($S=0$) with probability $P_{0}$, adopts
the social information ($S=1$) with probability $P_{1}$, or draws an $S$ in a
Gaussian distribution of center $m_{\rm g}$ and width $\sigma_{\rm g}$ with
probability $P_{\rm g}=1-P_{0}-P_{1}$.

Figure 4c shows that the average sensitivity to social influence $S$ increases
linearly with the distance $D=M-X_{\rm p}$ between the average social
information $M$ and the personal estimate $X_{\rm p}$. This is implemented in
the model by making the probability $P_{\rm g}$ increase linearly with $D$,
according to the equation: $\langle S\rangle=P_{1}+P_{\rm g}\,m_{\rm
g}=a+b\,|D|$, where the intercept $a$ and the slope $b$ characterize the
linear cusp observed in Figure 4c. More details can be found in the Materials
and Methods section. Notice the subjects’ tendency to give more weight to
social information that is much lower than their personal estimate ($D<-3$),
than to social information that is much higher ($D>3$). Since this concerns
only about $7.6\,\%$ of the data, we neglect this effect in the model.

The model, originally developed in [18], predicted that by providing subjects
with information that overestimates the truth, it was possible to improve
individual and collective accuracy more than by providing them with the truth
itself, by partly compensating the underestimation bias. We provide empirical
evidence for this prediction in the next section.

Note that the distribution of $X$ narrows after social influence (red dots and
lines in Figure 4a), implying that estimates have overall gotten closer to the
truth, all conditions mixed. This may seem counter-intuitive, since in most
conditions, incorrect information was provided into the sequence of estimates.
To understand this result, we next investigate the impact of incorrect
information on estimation accuracy for each condition separately.

### Impact of incorrect information on estimation accuracy

As explained above, we controlled the quality of the social information
received by the individuals, by introducing $n=0$, 5, or 15 virtual
influencers providing artificial estimates of value $T_{\rm I}$ randomly
inserted in the sequences of 20 estimates provided by the participants, and
hence affecting the social information delivered to them. Since we are looking
for an information parameter that is independent of the questions, we define,
consistently with the previous discussion on the normalization procedure, the
normalized (log) deviation from the truth $\alpha=\frac{\log\left(\frac{T_{\rm
I}}{T}\right)}{{\sigma_{\rm p}}_{\rm exp}}=\frac{V}{{\sigma_{\rm p}}_{\rm
exp}}$ as an indicator of information quality, where ${\sigma_{\rm p}}_{\rm
exp}$ is an expected value of the dispersion of personal estimates $X_{\rm p}$
(the values of the ${\sigma_{\rm p}}_{\rm exp}$ are given in Supplementary
Table S1), and $V$ the (log) deviation from the truth of the virtual
influencers estimates $T_{\rm I}$. We obviously did not know the dispersion of
estimates before running the experiment. Yet, since the questions were similar
to others used in a previous study [18], we could formulate reasonable
expectations. Indeed, Supplementary Figure S3 shows that ${\sigma_{\rm
p}}_{\rm exp}$ scale more or less linearly with the actual dispersion of
estimates $\sigma_{\rm p}$, although it tends to underestimate it.

$\alpha$ represents the deviation of $T_{\rm I}$ from the truth $T$ in the
(expected) natural scale of each question, which is the dispersion
${\sigma_{\rm p}}_{\rm exp}$ of individual errors for that question ($X_{\rm
p}$ are deviations from the truth or errors). The value $T_{\rm I}$ introduced
in the sequence of estimates is hence $T_{\rm I}=T.10^{\alpha{\sigma_{\rm
p}}_{\rm exp}}$, and equals the true value $T$ when $\alpha=0$. Subsequently,
to study the impact of information quality on the group performance, we
introduce the variable $Y=\frac{X}{\sigma_{\rm p}}$, where $\sigma_{\rm p}$ is
the dispersion of $X_{\rm p}$ for a given question, and define:

  * •

individual accuracy as the median of the absolute values of the $Y$ of all
individuals $i$, averaged over all questions $q$: $\langle{\rm
Median}_{i}(|Y_{i,q}|)\rangle_{q}$, and

  * •

collective accuracy as the absolute value of the median of the $Y$ of all
individuals $i$, averaged over all questions $q$: $\langle|{\rm
Median}_{i}(Y_{i,q})|\rangle_{q}$.

The individual accuracy measures how close individual estimates are to the
truth (i.e., close to 0 in terms of log variables $X$) on average, while the
collective accuracy measures how close the median estimate is to the truth.
Figure 5 shows that both measures improve after social influence (i.e., red
dots are closer to 0 than blue dots), over almost the whole range of the
considered values of $\alpha$, suggesting that incorrect information can,
counterintuitively, be beneficial to the performance of groups. Our results
also suggest that individual accuracy slightly improves after social influence
when $\rho=0\,\%$ (i.e., no virtual influencers), but not collective accuracy,
confirming previous findings [18].

Figure 5: Collective (a and b) and individual (c and d) accuracy, as a
function of the quantifier of information quality $\alpha$, before (blue) and
after (red) social influence, for $\rho=20\,\%$ (a and c) and $\rho=43\,\%$ (b
and d) of influencers in the sequence of estimates. Dots are experimental data
from the experiment presented here, while squares at $\alpha=0$ are
experimental data from a previous study, in which the same percentage of
virtual influencers provided some perfectly accurate information [18]. Full
lines are model simulations. Surprisingly, incorrect information can be
beneficial to collective and individual accuracy, which optima are reached for
positive values of $\alpha$, i.e., for incorrect information that
overestimates the truth. The computation of the error bars is explained in the
Materials and Methods.

Moreover, the optimum value $\alpha_{\rm opt}$ of $\alpha$ at which collective
or individual accuracy improves the most is strictly positive, confirming the
model prediction that such improvement is maximized not by providing perfectly
accurate information to individuals, but information that overestimates the
true value. Such incorrect information partly compensates the underestimation
bias, thus bringing second estimates closer to the truth.

Collective accuracy before social influence (blue dots and lines) represents
the absolute value of the collective bias of the group, averaged over all
questions, i.e., the distance between the median estimate and the truth. If
the value of the collective bias is $\alpha_{0}\approx-0.72$, one may naively
expect that $\alpha_{\rm opt}=-\alpha_{0}$ in order to compensate the
collective bias and thus optimize collective accuracy. But, since not all
subjects follow the social information fully, one should rather expect
$\alpha_{\rm opt}>-\alpha_{0}$, as supported by the data and model.

The fraction $\rho$ of virtual influencers has no significant effect on
collective accuracy in the data in Figure 5. However, the simulations of the
model predict that collective accuracy degrades after social influence either
when $\alpha<\alpha_{\rm min}\approx-1.2$ (for both $\rho=20\,\%$ and
$\rho=43\,\%$) or when $\alpha>\alpha_{\rm max}\approx 13.4$ for $\rho=20\,\%$
and $\alpha_{\rm max}\approx 7.2$ for $\rho=43\,\%$, which corresponds
respectively to $\langle{\sigma_{{\rm p}_{\rm
exp}}}_{q}\rangle_{q}\times\alpha_{\rm max}\approx 0.46\times 13.4\approx 6.2$
and $\langle{\sigma_{{\rm p}_{\rm exp}}}_{q}\rangle_{q}\times\alpha_{\rm
max}\approx 0.46\times 7.2\approx 3.3$ orders of magnitude beyond the true
value (see Figure S4). The impact of $\alpha$ is therefore not symmetric with
respect to its optimum $\alpha_{\rm opt}$: incorrect information that largely
overestimates the truth can still be beneficial to collective accuracy, while
incorrect information that only moderately underestimate the true value is
enough to damage collective accuracy. The same analysis remains true for
individual accuracy, only with different values of $\alpha_{\rm opt}$,
$\alpha_{\rm min}$, $\alpha_{\rm max}$, and $\alpha_{0}$.

### Incorrect information and sensitivity to social influence

It has been shown that estimation accuracy strongly depends on the sensitivity
to social influence of individuals in groups [18]. Analyzing the above results
in the light of the five behavioral categories of sensitivity to social
influence (Figure 4b) helps us to understand the mechanisms underlying them.
They cannot be explained by contradicters ($S<0$), adopters ($S=1$) or
overreacters ($S>1$), who only represent a small percentage of the population.
Figure 6 shows collective and individual accuracy as a function of $\alpha$,
for the keepers (Figure 6a) and compromisers (Figure 6b), which each
represents a substantial fraction of the population ($\sim 91\%$). Note that
the effects are clearer when this separation into behavioral categories is
made.

Figure 6: Collective (a, b, c, d) and individual accuracy (e, f, g, h) as a
function of the quantifier of information quality $\alpha$, before (blue) and
after (red) social influence, for $\rho=20\,\%$ (a, c, e, g) and $\rho=43\,\%$
(b, d, f, h) of influencers in the sequence of estimates. Keepers ($S=0$) are
shown in (a, b, e, f) and compromisers ($0<S<1$) in (c, d, g, h). Dots are
experimental data from the experiment presented here, while squares at
$\alpha=0$ are experimental data from a previous study, in which the same
percentage of influencers provided some perfectly accurate information [18].
Full lines are model simulations. By disregarding social information, keepers
are unable to improve in individual and collective accuracy after social
influence. Compromisers however, by partly following social information,
improve in individual and collective accuracy after social influence, even
when influencers provide some incorrect information.

Since keepers disregard social information, we observe no improvement in
individual or collective accuracy after social influence (Figure 6a, b, e, and
f). However, compromisers (Figure 6c, d, g, and h), who partly follow the
social information, significantly improve their performance over the whole
range of incorrect information tested here (except for $\alpha=-2$ and
$\rho=43\%$ of virtual influencers). Indeed, because subjects in general, and
compromisers in particular, tend to substantially underestimate quantities,
they can improve their estimates by following incorrect social information
that is closer to the true value than their own personal estimate. Moreover,
partially following the social information that overestimates the truth allows
their second estimates to reach more accurate values, even when the
overestimation is quite pronounced. Contrariwise, individual accuracy degrades
quickly when subjects are given incorrect social information which reinforces
their natural cognitive bias by underestimating the true value. Compromising
thus allows group members to resist incorrect information, as long as this
information goes against their cognitive bias. Similar conclusions can be
drawn for collective accuracy, but the patterns are slightly less pronounced.

Figure 7 shows the equivalent graphs for the “isolated” subjects of our
experiment (see Materials and Methods). Isolated subjects received as social
information for each question, an estimate $T_{\rm I}$ generated from a random
value of $\alpha$ uniformly distributed in the interval $[-3,3]$.

Figure 7: Collective (a, b) and individual (c, d) accuracy against the
quantifier of information quality $\alpha$, before (blue) and after (red)
social influence, in the separated experiment with the isolated subjects. (a
and c) keepers ($S=0$); (b and d) Compromisers ($0<S<1$). Dots are
experimental data, and full lines are model simulations.

Figure 7 confirms the above conclusions, but displays sharper patterns, due to
a discretization effect: social information in the main experiment was
generated from a discrete set of values of $\alpha$, whereas for isolated
subjects, it was drawn from a continuous distribution.

Before social influence (blue), we find that keepers are slightly more
accurate than compromisers (average collective accuracy: $0.98$ versus $1.07$;
average individual accuracy: $1.08$ versus $1.28$). This was already observed
in [18], and justified by the fact that a higher tendency to disregard social
information is usually associated with a higher average confidence of the
subjects in their own estimates, which often comes with a higher prior
knowledge about the quantity to estimate.

Note the slight U-shaped curve for keepers in Figure 7a and c. This effect is
a direct consequence of people’s tendency to stick to their personal estimate
more when the social information is closer to it (Figure 4c): when
participants receive inaccurate information and retain their opinion, it is
often because they were close to it and therefore relatively inaccurate too.
Conversely, when participants receive accurate information and keep their
opinion, it is often because they were close to it and therefore quite
accurate too. Both effects can be observed in Figure 6, but are less
pronounced there.

### Influence of the fraction of virtual influencers on individual behavior

We have seen that compromisers, by partially following social information,
were able to improve their accuracy over a wide range of incorrect social
information. Figure 8 shows the fraction of keepers and compromisers as a
function of $\alpha$, when $\rho=20\%$ (Figure 8a) and $\rho=43\%$ (Figure 8b)
of virtual influencers are introduced in the sequence of estimates.

Figure 8: Proportion of compromisers (orange) and keepers (brown) as a
function of the quantifier of social information quality $\alpha$, for
$\rho=20\,\%$ (a) and $\rho=43\,\%$ (b) of influencers in the sequence of
estimates. Dots are experimental data from the experiment presented here,
while squares at $\alpha=0$ are experimental data from a previous study, in
which the same percentage of virtual influencers provided some perfectly
accurate information [18]. When $\rho=43\%$, the fraction of compromisers
increases with $\alpha$ at the expense of the fraction of keepers, which
decreases with $\alpha$.

When $\rho=20\%$, both fractions of compromisers and keepers remain more or
less independent of $\alpha$ (Figure 8a). However, when the proportion of
virtual influencers providing incorrect information is doubled ($\rho=43\%$,
Figure 8b), the fraction of compromisers gradually increases (from 0.5 to
0.68, orange line) with $\alpha$ (from $-2$ to 3), at the expense of the
fraction of keepers which decreases (from 0.38 to 0.25, brown line). For this
to happen, this increasing transition from keeping to compromising behavior as
$\alpha$ increases thus necessitates a significant proportion of subjects to
be provided with incorrect social information. Moreover, this result also
suggests that subjects not only adapt their behavior to the degree of
incorrectness of the social information they receive, but also tend to
compromise more with some social information that overestimates the truth,
than with some social information that underestimates it. The model predicts
this increased behavioral transfer with $\alpha$, even when $\rho=20\%$. This
is a direct consequence of the cusp relationship between the sensitivity to
social influence $S$ and the distance to the social information $D$ (Figure
4c): people tend to compromise with the social information more as it gets
farther from their personal estimates (and from the truth) on average.
However, this effect is significantly stronger in the data, suggesting that
other mechanisms exist, that are not implemented in the model.

Supplementary Figure S5 demonstrates that this increasing fraction of
compromisers with $\alpha$ when $\rho=43\%$ leads to an increased improvement
in individual and collective accuracy after social influence (but not when
$\rho=20\%$).

## Discussion

Understanding the effects of incorrect information on individual and
collective decisions is crucial in modern digital societies, where social
networks and other vectors of information transmission allow a fast and deep
flow of information, the accuracy of which is increasingly hard to verify
[28]. Here, we rigorously controlled the quality of the information delivered
to subjects in estimation tasks, by means of “virtual influencers”, i.e.,
virtual agents inserted into the sequence of estimations – unbeknownst to the
subjects – and providing a value whose level of inaccuracy was controlled. We
were thus able to precisely quantify the impact of information quality on
individual and collective accuracy in those tasks.

We demonstrated that a proper normalization of estimates must take into
account their dispersion, which gives the natural range of “reasonable”
estimates of a given quantity for a given group. This normalization process
led to the conclusion that estimates follow a Laplace distribution when
subjects have little prior information about a quantity to estimate. Early
research showed that in many data sets, estimates $X$ (i.e., deviations from
the truth) were often close either to Gaussian distributed or to Laplace
distributed [29, 30]. Later work have encompassed Laplace and Gaussian
distributions into a broader family of exponential distributions, the
Generalized Normal Distribution (GND) family [31, 32], of PDF:

$\displaystyle f(X,m,\sigma,n)=\frac{1}{2\,\sigma\,\Gamma(1+1/n)}\,{\rm
exp}\left\\{-\left|\frac{X-m}{\sigma}\right|^{n}\right\\},$ (1)

where $m$ is the center of the distribution (often called location parameter),
$\sigma$ is the width of the distribution (often called scale parameter) and
$n$ is the tailedness (often called shape parameter), which controls the
thickness of the tails. The fatter the tails of a distribution, the higher the
probability to find outliers (i.e., estimates that are very far from the
distribution center). More recent work has studied various data sets of
estimates and forecasts in the light of Generalized Normal Distributions, and
showed that the tailedness of distributions ranged from $n=1$ (Laplace
distribution) to $n=1.6$, $n$ being equal to 2 for Gaussian distributions
[33]. They concluded that most distributions of estimates for usual quantities
are actually closer to Laplace distributions than to Gaussian distributions.
This discussion can be related to the amount of prior information held by a
group about a certain quantity. We found that when a quantity is “hard” to
estimate (i.e., low demonstrability, corresponding to a low amount of prior
information about the quantity in the group), the expected distribution of
estimates is very close to a Laplace distribution. When a quantity is “easy”
to estimate (i.e., high demonstrability, corresponding to a high amount of
prior information about the quantity in the group), few outliers are expected,
such that the distribution of estimates could be expected to be closer to a
Gaussian distribution. However, our results show that regardless of the
questions demonstrability, distributions of estimates are significantly closer
to Laplace distributions than to Gaussian distributions when properly
normalized, in agreement with [33]. In any case, we consider that future
studies involving estimation tasks should apply the normalization procedure
presented here when comparing and aggregating the estimates of different
quantities, for which the width $\sigma_{\rm p}$ should be used to quantify
their demonstrability.

We then studied the impact of incorrect information on individual and
collective accuracy, and found that providing incorrect information that
overestimates the true value can help a group perform better than providing
the correct value itself, by partly compensating for the human underestimation
bias. Moreover, collective and individual accuracy can improve after social
influence over a surprisingly wide range of incorrect information. This
counter-intuitive result is a consequence of a large proportion of individuals
compromising with the social information, i.e., partially following it. By
doing so, subjects are able to benefit not only from relatively accurate
social information, but also from incorrect information that goes against
their cognitive bias. Indeed, because of the human tendency to underestimate
quantities, partially following an overestimation of the truth – even a large
one – can bring second estimates closer to the truth, thus improving accuracy.
However, incorrect social information can also harm accuracy if it amplifies
the bias. This may be related to some deleterious effects of social
information observed at times, for instance, how the spread of misinformation
can deeply affect the behavior of crowds as well as public opinion [34, 35,
36].

In a former study [18], we showed that adopting the social information was the
best strategy in order to improve accuracy, if virtual influencers provide
perfectly accurate information in the sequence of estimates. However, while
adopting can lead to higher performance than compromising in this particular
case, our results show that compromising offers more resilience when the
information provided is potentially less accurate.

We also found that subjects were sensitive to the degree of incorrectness of
the social information they received. They adapted their behavior to the
social information, by compromising more with the social information as it
overestimated the truth more, and compromising less as it underestimated the
truth more. This asymmetric strategy is surprisingly well adapted to counter
the human underestimation bias. Indeed, as explained above, following (even
partly) social information that underestimates the truth may increase the
bias, while following social information that overestimates the truth may
decrease it. Following less in the former case, and more in the latter is thus
bound to increase the performance of groups. Former studies have already
observed this subjects’ tendency to rely more on social information that is
higher than their personal estimate, than on social information that is lower,
and showed that it had valuable consequences for collective performance in
estimation tasks [26, 27]. In [27], it is suggested that people can more
easily assess the validity of small numbers compared to large numbers, because
they have no direct experience with events related to those large numbers
[37], and as a consequence reject more often small numbers provided by the
social information.

We then used a modified version of a model of collective estimation developed
in [18]. The predictions of the model are in good agreement with the
experimental data, and confirm that to optimize collective accuracy, social
information must do more than compensate the initial collective bias, as most
individuals only partly follow social information. In addition, the impact of
the quality of information is not symmetric with respect to its optimum:
collective accuracy can be improved by delivering incorrect information which
overestimates the true value by up to several orders of magnitude, whereas it
decays fast if the information delivered only slightly underestimates it. In
other words, social information reinforcing the bias of the group has a strong
negative impact on its accuracy.

Overall, we found that incorrect social information does not necessarily
impair the collective wisdom of groups, and can even be used to counter some
deleterious effects of cognitive biases. Individuals demonstrated an ability
to discriminate the validity of social information, depending on its distance
from their personal estimates, and thus to benefit from accurate social
information, while at the same time resisting inaccurate social information.
These results suggest that people may be more resilient to malicious
information than is often thought, and at the same time that the negative
effects of identified biases can be dampened by exchanging relevant social
information, thus improving collective decisions.

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Data: The data supporting the findings of this study are available at
figshare:
https://doi.org/10.6084/m9.figshare.11903421

Funding: This work was supported by Agence Nationale de la Recherche project
11-IDEX-0002-02–Transversalité–Multi-Disciplinary Study of Emergence
Phenomena, a grant from the CNRS Mission for Interdisciplinarity (project
SmartCrowd, AMI S2C3), and by Program Investissements d’Avenir under Agence
Nationale de la Recherche program 11-IDEX-0002-02, reference
ANR-10-LABX-0037-NEXT. B.J. was supported by a doctoral fellowship from the
CNRS, and R.E. was supported by Marie Curie Core/Program Grant Funding Grant
655235–SmartMass. T.K. was supported by Japan Society for the Promotion of
Science Grant-in-Aid for Scientific Research JP16H06324 and JP25118004.

Author Contributions: B.J., C.S., and G.T. designed research; B.J., R.E.,
S.C., A.B., T.K., C.S., and G.T. performed research; B.J., C.S., T.K., and
G.T. analyzed data; and B.J., C.S., and G.T. wrote the article with critical
input from all other authors.

Competing Interests: The authors declare that they have no competing
interests.

## Materials and methods

### Ethics statement

The aims and procedures of the experiments conformed to the Toulouse School of
Economics Ethical Rules. All subjects provided written consent for their
participation.

### Experimental design

180 subjects – mostly students from the University of Toulouse – participated
in our experiments. 20 sessions were organized, in each of which 9 subjects
had to answer 32 questions for which they had first to give a prior/personal
estimate and then a second/final estimate, the latter after being confronted
to social information. Each final estimate in each session constituted a step
in a sequence of 20 estimates. Hence, our main experiment (see the experiment
with “isolated subjects” hereafter) produced a total of 9 subjects $\times$ 32
questions $\times$ 20 sessions $=5760$ final estimates and the same number of
prior estimates. For each question, subjects first provided their
prior/personal estimate $E_{\rm p}$. Next, they received as social information
the geometric mean $G$ of the $\tau$ previous estimate(s) in the sequence
($\tau=1$, 3), and were then asked to provide a second/final estimate $E_{\rm
s}$ (see illustration in Supplementary Figure S1). Subjects were oblivious to
the actual value of $\tau$, only being told that the social information was
the average of the final answers of some previous participants. The time
allowed to provide an answer was limited to 40 s per estimate, after which a
blinking text urging the subjects to answer quickly would appear on their
computer screen.

The specificity of our experiment lies in the design of a system aimed to
control the quantity and quality of the social information provided to the
subjects, without them being aware of it. To that end we inserted, at random
locations in the sequence of estimates (and unbeknownst to the subjects),
$n=0$, 5, or 15 artificial estimates, corresponding to a fraction
$\rho=n/(20+n)=0\,\%$, $20\,\%$, or $43\,\%$ of virtual influencers. Each
sequence thus consisted of a total of $N=20+n=20$, 25, or 35 estimates and the
estimates of the virtual influencers were also used to compute the updated
social information for the following participants in each sequence. We
controlled the value $T_{\rm I}$ of the artificial estimates, through a
parameter $\alpha$ ($\alpha=$ -2, -1, -0.5, 0.5, 1, 1.5, 2, 3) defined such
that $T_{\rm I}=T.10^{\alpha{\sigma_{\rm p}}_{\rm exp}}$, where $T$ is the
true value of a quantity to estimate, and ${\sigma_{\rm p}}_{\rm exp}$ an
expected value of the dispersion of estimates $\sigma_{\rm p}$ to be obtained.
The closer $T_{\rm I}$ to $T$ (i.e., the closer $\alpha$ to 0), the higher the
information quality. In particular, when $\alpha=0$, artificial estimates are
equal to the true value ($T_{\rm I}=T$). A proper justification of the choice
of $\alpha$ as the quantifier of information quality, and a detailed
explanation of its physical interpretation, are provided in the main text. The
range of values of $\alpha$ used in the experiment was chosen by exploiting
preliminary simulations of the model developed in [18], that we describe in
detail below.

Each participant in each session was associated a value of $\rho$ ($\rho=0\%$
for subject 1, $\rho=20\,\%$ for subjects 2 to 5 and $\rho=43\,\%$ for
subjects 6 to 9). For subjects 2 to 9 ($\rho\neq 0\%$; no $\alpha$ for
$\rho=0\%$), one particular value of $\alpha$ was associated to each question.
We distributed the values of $\alpha$ such that at the end of each session (20
sessions), each combination of a fraction $\rho$ of influencers ($\rho\neq 0$)
and information quality $\alpha$ was repeated 16 times, such that the
experimental points in Figures 5, 6 and 8 are all constructed from $16\times
20=320$ values. For $\rho=0\,\%$, there were 32 sequences per session (640
values). Although our previous study did not show evidence for a significant
effect of $\tau$ on the dynamics [18], we decided to perform an additional
check. A value of $\tau$ (1 or 3) was thus associated to each question,
independent of the values of $\rho$ and $\alpha$. Preliminary results
confirmed that the effect of $\tau$ can be neglected, and we thus combined the
data from both values of $\tau$ in all graphs presented in this paper.
Supplementary Table S1 summarizes the conditions $(\rho,\alpha,\tau)$ for each
subject and each question in a session. In all sessions, these conditions were
repeated and the order of the questions was randomized. In the first session
(i.e., the first step of all sequences), subjects received as social
information (initial condition) the value $T_{\rm I}$ provided by the
influencers. In the particular case $\rho=0\,\%$ (no influencers), the initial
social information provided was the true answer to the question $T$. Anyway,
note that our previous study showed that the choice of the initial condition
had little impact on the subsequent dynamics [18]. At the end of each session,
subjects were rewarded according to their overall performance/accuracy: 20€
for the two first ones, 15€ for the next four ones and 10€ for the three last
ones.

As a safeguard against subjects not showing up, 3 additional subjects were
recruited in each session. Hence, when more than nine subjects were present in
a session, the additional subjects (“isolated subjects”; 51 in total) were
given a special treatment, unbeknownst to them: they were not part of a
sequence or associated to any specific condition ($\rho$, $\alpha$, $\tau$),
and received instead, as social information, a value $T_{\rm
I}=T.10^{\alpha{\sigma_{\rm p}}_{\rm exp}}$ generated from a random value of
$\alpha$ uniformly distributed in the interval $[-3,3]$. These “isolated
subjects” were paid 10€, independently of their performance.

### Cauchy and Laplace distributions

In the context of estimation tasks, the Cauchy and Laplace distributions
emerge naturally.

#### Cauchy distribution

The Cauchy distribution centered at $m$ (the median) and of width $\sigma$ is
given by

$f(X,m,\sigma)=\frac{1}{\pi}\frac{\sigma}{(X-m)^{2}+\sigma^{2}}.$ (2)

The Cauchy distribution has fat tails and an infinite variance. One of the
notable and unique properties of the Cauchy distribution is that, if
$X_{1},...,X_{N}$ are independently drawn from the same Cauchy distribution,
then the average $\bar{X}_{n}=\sum_{i}^{n}X_{i}/N$ (or any weighted average of
the $X_{i}$) has exactly the same Cauchy distribution as the $X_{i}$, i.e.,
with the same center and width. In particular, the distribution of the average
$\bar{X}_{n}$ does not have a smaller width than the distribution of the
$X_{i}$, as one would obtain if the $X_{i}$ were drawn from a distribution
with a finite variance. For instance, if the $X_{i}$ are drawn in the same
Gaussian distribution with standard deviation $\sigma$, the average
$\bar{X}_{n}$ is also Gaussian distributed, but with a reduced standard
deviation $\sigma_{n}=\sigma/\sqrt{N}$, a basic property leading to the law of
large numbers.

In the context of estimation tasks, if $N$ subjects having absolutely no clue
about the actual answer to a question draw their personal estimates
$X_{1},...,X_{N}$ from some random distribution, one cannot expect the average
of the estimates $\bar{X}_{n}$ to provide more information than the individual
estimates, and $\bar{X}_{n}$ should then have the same distribution as the
$X_{i}$. The unique probability distribution satisfying this property is the
Cauchy distribution.

#### Laplace distribution

Now, let us assume that the distribution for a random variable $X$ is known to
be symmetric and centered in $m$ with a known width given by
$\sigma=\int_{-\infty}^{\infty}|X-m|f(X)\,dX$. What is the most likely
distribution satisfying this property? To answer this question, one has to
maximize the entropy $S(f)=-\int_{-\infty}^{\infty}f(X)\log[f(X)]\,dX$,
subject to the above constraint for $\sigma$ along with the normalization
constraint $\int_{-\infty}^{\infty}f(X)\,dX=1$. This optimization problem
(maximum entropy principle) with constraints is solved by introducing two
Lagrange multipliers $\alpha$ and $\beta$, and maximizing with respect to
$f(X)$ the functional $F(f)$ given by

$F(f)=-\int_{-\infty}^{\infty}f(X)\log[f(X)]\,dX+\alpha\int_{-\infty}^{\infty}f(X)\,dX+\beta\int_{-\infty}^{\infty}|X-m|f(X)\,dX.$
(3)

Expressing that the functional derivative of $F(f)$ with respect to $f(X)$
vanishes (maximum of $F(f)$), one obtains

$-\log[f(X)]-1+\alpha+\beta|X-m|=0,$ (4)

leading to $f(X)=\exp(1-\alpha-\beta|X-m|)$. Finally, $\alpha$ and $\beta$ are
computed by expressing the normalization and width constraints leading to the
symmetric Laplace distribution

$f(X,m,\sigma)=\frac{{\rm e}^{-\frac{|X-m|}{\sigma}}}{2\sigma}.$ (5)

The normalized variable $Z=\frac{X-m}{\sigma}$ has the standard Laplace
distribution (i.e., with center 0 and width 1), $f(Z)=\exp(-|Z|)/2$.

Note that if the assumption of a symmetric distribution is relaxed and that
one wants for instance to penalize large or small values of $X$ (in the
context of human estimations, this is related to the bias against large
answers), one can add a term $\gamma\int_{-\infty}^{\infty}Xf(X)\,dX$ in
$F(f)$, adding an independent constraint on the mean of $X$. The same
calculation as above leads to an asymmetric Laplace distribution with
different exponential tails below and above the maximum of the distribution,
reminiscent of what is observed experimentally in Supplementary Figure S2c and
S2d for estimates about questions with very large answers.

In physics, when the variable $X$ is the energy of the considered system, the
only constraint in addition to the normalization is the knowledge of its
average energy $E=\int_{-\infty}^{\infty}Xf(X)\,dX$. Repeating the same
optimization procedure as above ($F(f)$ in Eq. (3) is then proportional to the
opposite of the free energy, to be maximized), one finds that the most likely
distribution for the energy of the system is the famous Boltzmann distribution

$f(X)=\frac{{\rm e}^{-\beta X}}{\cal Z},$ (6)

where $\cal Z$ is a normalization constant, and the Lagrange multiplier
$\beta$ can be ultimately shown to be proportional to the inverse of the
temperature $\cal T$, $\beta=(k_{\rm B}{\cal T})^{-1}$, where $k_{\rm B}$ is
the Boltzmann constant.

### Model

#### Main experiment

Our model simulates a sequence of $N=20+n$ successive estimates performed by
20 agents and $n$ virtual influencers for a given question. The model directly
implements the log deviations from the truth $X=\log\left(\frac{E}{T}\right)$.
A typical run of the model consists of the following steps, for an agent
estimating a given quantity in a given condition $(\rho,\alpha,\tau)$:

  1. 1.

An agent’s personal estimate $X_{\rm p}$ is drawn from the Laplace
distribution $f(X_{\rm p},m_{\rm p},\sigma_{\rm p})$, where $m_{\rm p}$ and
$\sigma_{\rm p}$ are respectively the median and average absolute deviation
from the median of the experimental distribution of estimates for each
question. We impose that estimates $E_{\rm p}$ are greater than 1, i.e.,
$X_{\rm p}>-\log(T)$. This condition explains the fast decay on the left side
of the distribution in Figure 4a;

  2. 2.

With probability $\rho=\frac{n}{n+20}$, a virtual influencer plays and
provides the value $V=\alpha\,{\sigma_{\rm p}}_{\rm exp}$ (the values of
${\sigma_{\rm p}}_{\rm exp}$ for each question are given in Supplementary
Table S1). With probability $(1-\rho)$, the agent receives as social
information the average $M$ of the $\tau$ previous estimates (possibly
including estimates from virtual influencers). $M_{0}=V$ is used as initial
condition for the first agent;

  3. 3.

The agent’s sensitivity to social influence $S$ is drawn from a Gaussian
distribution of mean $m_{\rm g}=0.58$ and standard deviation $\sigma_{\rm
g}=0.3$ with probability $P_{\rm g}$, or takes the value $S=0$ or $S=1$ with
probability $P_{0}$ and $P_{1}=1-P_{0}-P_{\rm g}$, respectively (Figure 4b).
$P_{\rm g}$ and $P_{0}$ have a linear cusp relationship with the distance
$D=M-X_{\rm p}$ between the social information $M$ and the personal estimate
$X_{\rm p}$ (Figure 4c), while $P_{1}=0.012$ is kept independent of $D$.

For a given distance $D$, the average sensitivity to social influence is
$\langle S\rangle=P_{0}\times 0+P_{1}\times 1+P_{\rm g}\times m_{\rm
g}=a+b\,|D|$, where the intercept $a=0.34$ and the slope $b=0.09$ are the
coefficients of the linear cusp relationship extracted from Figure 4c. $P_{\rm
g}$ is hence given by $P_{\rm g}=(a+b\,|D|-P_{1})/m_{\rm g}$. A plateau is
added at $|D|>3$ by setting an upper limit ${P_{\rm g}}_{\rm max}=0.85$ on
$P_{\rm g}$.

  4. 4.

The agent’s final estimate ${X_{\rm s}}$ is the weighted average of the
personal estimate and the social information: $X_{\rm s}=(1-S)\,X_{\rm
p}+S\,M$. The condition $X_{\rm s}>-\log(T)$ is also imposed on the $X_{\rm
s}$.

  5. 5.

One starts again from step 1 for the next agent.

Apart from $\alpha$ over which we loop in order to make it pseudo-continuous
and to explore values beyond those tested in the experiment, our model closely
follows the experimental structure: one subject is associated to $\rho=0\%$,
four subjects to $\rho=20\%$ ($5/25$) and four subjects to $\rho\approx 43\%$
($15/25$). Moreover, each question is associated with the corresponding value
$\tau$ used experimentally (see Supplementary Table S1). One simulation of the
model thus mimics an experimental run, and results are calculated for each
simulation, then averaged over 10000 simulations.

#### Isolated subjects

In a parallel experiment, “isolated” subjects received as social information
an estimate generated from a random value of $\alpha$, uniformly distributed
between $-3$ and 3. The model for isolated subjects is very similar to the one
presented above, and also sticks closely to the experimental structure:
instead of receiving the average of $\tau$ previous estimates, agents receive
a value of $\alpha$ randomly generated from a uniform distribution. There is
no loop over $\alpha$ here, and the variables $\rho$ and $\tau$ do not exist
in this separate experiment. All the rest is the same as in the model of the
main experiment. 10000 simulations were run with 200 agents in each.

### Computation of the error bars

The error bars indicate the variability of our results depending on the
$N_{\rm Q}=32$ questions presented to the subjects. We call $x_{0}$ the actual
measurement of a quantity appearing in the figures by considering all $N_{\rm
Q}$ questions asked. Then, we generate the results of $N_{\rm exp}=1000$ new
effective experiments. For each effective experiment indexed by
$j=1,...,N_{\rm exp}$, we randomly draw $N_{\rm Q}^{\prime}=N_{\rm Q}$
questions among the $N_{\rm Q}$ questions asked (so that some questions can
appear several times, and others may not appear) and recompute the quantity of
interest which now takes the value $x_{j}$. The upper error bar $b_{+}$ for
$x_{0}$ is defined so that $C=68,3\,$% (by analogy with the usual standard
deviation for a normal distribution) of the $x_{j}$ greater than $x_{0}$ are
between $x_{0}$ and $x_{0}+b_{+}$. Similarly, the lower error bar $b_{-}$ is
defined so that $C=68,3\,$% of the $x_{j}$ lower than $x_{0}$ are between
$x_{0}-b_{-}$ and $x_{0}$. The introduction of these upper and lower
confidence intervals is adapted in cases where the distribution of the $x_{j}$
is unknown and potentially not symmetric.