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A Castro Consensus:
Understanding the Role of Dependence in Consensus Formation
Jarred Allen
AMISTAD Lab
Dept. of Computer Science
Harvey Mudd College
Claremont, CA, USA
jaallen@hmc.edu
Cindy Lay
AMISTAD Lab
Dept. of Mathematical Sciences
Claremont McKenna College
Claremont, CA, USA
clay22@cmc.edu
George D. Monta˜
nez
AMISTAD Lab
Dept. of Computer Science
Harvey Mudd College
Claremont, CA, USA
gmontanez@hmc.edu
Abstract
Consensus is viewed as a proxy for truth in
many discussions of science. When a consen-
sus is formed by the independent and free de-
liberations of many, it is indeed a strong indi-
cator of truth. Yet not all consensuses are in-
dependent and freely formed. We investigate
the role of dependence and pressure in the for-
mation of consensus, showing that strong po-
larization, external pressure, and dependence
among individuals can force consensus around
an issue, regardless of the underlying truth of
the affirmed position. Dependence breaks con-
sensus, often rendering it meaningless; a con-
sensus can only be trusted to the extent that
individuals are free to disagree with it.
1 Introduction
Water freezes at 0 ◦C, the earth revolves around
the sun, and cigarettes can cause lung cancer. You
likely agree with these statements, as most do.
Furthermore, sound scientific evidence supports
each of these claims and many experts agree with
that evidence. A consensus surrounds each of
the claims, seemingly formed from the research
and testimony of many independent expert parties.
However, complete independence is rare. When
independence is compromised, the reliability of
consensus as a marker of truth suffers.
Consider cases of political consensus. In Fi-
del Castro’s Cuba, superficially democratic elec-
tions unfailingly produced overwhelming support
for the Communist Party of Cuba. However, a lack
of choices left no other option. In both municipal
and national elections, potential candidates had
to pass specific requirements and secure backing
from government-influenced organizations to be
placed on a ballot (Shugerman,2018;Foundation
for Human Rights in Cuba,2019). Thus, the over-
whelming consensus supporting the Cuban Com-
munist Party in elections resulted from political
pressure on voters and government restrictions on
candidates. The consensus was neither indepen-
dent nor meaningful.
Dependence can compromise consensus in
other ways. Psychologists have found that eyewit-
ness accounts can become distorted, often subcon-
sciously, when witnesses unwittingly influence the
testimony of other witnesses (Memon et al.,2008).
Using mathematical modeling, we explore the
influence of dependence on the formation of con-
sensus. We show that dependence among a pop-
ulation often leads to consensus regardless of the
truth of the underlying position. We find that while
a consensus of independent individuals strongly
correlates with truth even under adverse condi-
tions, dependence greatly weakens the evidential
value of any consensus formed. Centuries ago,
the Reverend Thomas Bayes derived his name-
sake theorem to answer the question of whether
the testimony of a large number of independent
eyewitness could ever establish the probable oc-
currence of an a priori highly unlikely event. He
showed that even highly unlikely events become
plausible once the testimony of many independent
witnesses is taken into account. Bayes’ theorem
has found many real-world uses, including mili-
tary and medical applications (Cepelewicz,2016).
In agreement with its historical purpose, we use
Bayes’ theorem to model how consensus within
a population can provide evidence for or against
a position. We extend our analysis to dependent
models, allowing us to consider the effects of dif-
fering levels of dependence, external pressure, and
hyper-polarization on the formation of consensus
within our framework.
2 Related Work
Bayes’ theorem is widely used in the data sci-
ences (Lock and Dunson,2013;Sambasivan et al.,
2018;Rudner and Liang,2002), often under in-
dependent data assumptions, which may not be
met in practice (Gammerman and Thatcher,1991;
Domingos and Pazzani,1996). Some have in-
vestigated use of Bayes’ theorem with dependent
data, either by reformulating the data to be inde-
pendent (Gammerman and Thatcher,1991) or ad-
justing the math to account for dependence (Brune
and Pierce,1974). Others have analyzed errors
introduced into Bayes’ theorem by incorrect in-
dependent data assumptions (Domingos and Paz-
zani,1996;Russek et al.,1983), finding that such
approaches usually identify the correct hypothesis
as the most likely, even if incorrectly computing
posterior probabilities.
Researchers have investigated means by which
witnesses can become dependent. Two approaches
have been taken to explain this phenomenon: peer
pressure and the spread of misinformation. Many
peer pressure models exist. Some model peer pres-
sure from axioms observed in the behavior of peo-
ple (Estrada and Vargas-Estrada,2013;Castillo-
Garsow et al.,1997), while others derive peer pres-
sure from considerations of utility maximization
(Bishop,2006;Yang et al.,2015). Misinformation
spread has also been modeled, in the context of so-
cial networks (Kempe et al.,2003). More recent
research uses sophisticated techniques to model
the concurrent spread of both misinformation and
correct information (Nguyen et al.,2012;Tambus-
cio et al.,2015;Abdullah et al.,2015), for both
homogenous and heterogenous populations.
Decades ago, Gold discussed the dangers that
herd behavior poses for scientists, potentially
leading to an inertia-driven persistence of false
consensus opinion within the sciences (Gold,
1989). Gold’s prescient essay stands as a warning.
More recently, Kahneman highlighted the statis-
tical advantages of averaging independent errors,
and the dangers of dependent corroboration (Kah-
neman,2011). Foreshadowing the mathematical
analysis presented here, Kahneman states, “How-
ever, the magic of error reduction only works well
when the observations are independent and their
errors uncorrelated. If the observers share a bias,
the aggregation of judgments will not reduce it.”
(Kahneman,2011).
3 Models
We introduce our models, first in the fully inde-
pendent setting, followed by a series of modifi-
cations that allow us to account for polarization,
time-varying dependence, external pressure, and
partial dependence. We begin by reviewing when
consensus performs as expected, allowing us to in-
fer the probable truth of a position based on the
strength of consensus affirming it. In what fol-
lows, we refer to individuals who vote to either
affirm or deny a position as witnesses, referenc-
ing both our previous political voting discussion
and Bayes’ original use of the theorem (while thor-
oughly mixing metaphors).
3.1 Independence: When Consensus Works
We begin with fully independent witnesses, where
each witness has some probability pfof fallibly
affirming a hypothesis when it is not true. Let E
represent our observed evidence, namely, the event
of observing a given set of affirmations and de-
nials. We then let pt:= P(E|H)be the fixed
probability of the evidence emerging if the hypoth-
esis is true. We also let the hypothesis have some
prior probability, ph. Lastly, we let Nbe the num-
ber of witnesses who have reported.
Letting Hdenote the event that a hypothesis is
true, and letting Ebe the event that there is an
observed consensus of all Nindividuals affirming
the hypothesis, we can use Bayes’ theorem to de-
termine the posterior probability of the hypothesis
being true given the affirming consensus, namely
P(H|E). Under our independence assumption we
have P(E|¬H) = pN
f, and applying Bayes’s the-
orem, we obtain
P(H|E) = P(E|H)P(H)
P(E|H)P(H) + P(E|¬H)P(¬H)
=ptph
ptph+pN
f(1 −ph)
with P(¬H|E) = 1 −P(H|E)as expected.
Let us suppose that a consensus is unlikely to
emerge when the hypothesis holds (pt= 0.1) and
that people are each often mistaken (pf= 0.95).
Additionally, let us assume the prior probability of
the hypothesis is low (ph= 10−9) and that the
number of people is small (N= 500). We find
that P(H|E)≈0.932. Thus, despite highly un-
favorable assumptions, an independently-formed
consensus still provides strong evidence that the
hypothesis is true, and a consensus is unlikely to
emerge around a false claim. When individuals are
fully independent, P(H|E)rapidly approaches 1
as we increase the number of individuals, imply-
ing the likely truth of an affirmed position given a
consensus around it.
3.2 Dependence: Compromising Consensus
While a consensus of fully independent individ-
uals can provide strong evidence in favor of the
affirmed position, a forced consensus is epistem-
ically much less meaningful; it can result from
either truth or coercion, and we have no way of
knowing which is the case based on the strength
of the consensus alone. We next explore models
of increasing complexity that allow us to control
the level of dependence and observe its effect on
consensus formation.
3.3 Polarized Majority Vote Model
We can view eyewitness accounts as coming in se-
quentially, with people being aware of the total
fraction of prior testimonies which affirm or re-
ject the hypothesis. Witness ireceives a numerical
representation of this fraction:
Mi=ni+ 1
Ni+ 2 (1)
where Miis the witness’s perception of the prior
accounts (which could be viewed as a propor-
tion of “votes” in favor, hence our name for this
model), niis the number of witnesses who af-
firmed the hypothesis before the i-th witness, and
Niis the total number of witnesses who either af-
firmed or rejected the hypothesis before the i-th
witness. We use Laplace smoothing (i.e., pseu-
docounts) to ensure that Miis defined for all wit-
nesses and to prevent the second witness from see-
ing a pure consensus around whichever position
the first witness happens to take.
Next, we define the probability of a witness in-
correctly affirming a false hypothesis. Let
f(θ, Mi) = 1
1 + e−θ(Mi−0.5)
where θ > 0is a polarization parameter con-
trolling the sensitivity of witnesses to slight ma-
jorities. Letting Xidenote the random outcome
of witness iand defining γθ,min := f(θ, 0) and
γθ,max := f(θ, 1), the probability that the i-th wit-
ness affirms a false hypothesis can be modeled by
gθ(Mi) = P(Xi= 1|Mi) := f(θ, Mi)−γθ,min
γθ,max −γθ,min
.
Note we have rescaled gθto the range [0,1], en-
suring that only the sensitivity of each witness to a
change in the majority response among witnesses
is adjusted as the value of θchanges. Figure 1
shows the effect of θon the shape of the gθ.
Figure 1: Effect of polarization parameter θon the
probability of affirming a false hypothesis versus Mi.
Once witness ieither affirms or rejects the hy-
pothesis, the updated value Mi+1 is computed, and
then the cycle repeats for witness i+ 1. In mod-
eling gθ(Mi)as we have, we have assumed that
Miis sufficient for determining the conditional
probability of Xigiven X1, . . . , Xi−1. Within
our Bayesian framework from Section 3.1, we can
then model P(E|¬H)using gθ(Mi)as
P(E|¬H) =
n
Y
i=1
P(Xi|X1, . . . , Xi−1)
=
n
Y
i=1
P(Xi|Mi)
=
n
Y
i=1
gθ(Mi)Xi(1 −gθ(Mi))1−Xi.
For a unanimous consensus, this simplifies to
P(E|¬H) = Qn
i=1 gθ(Mi). Note that because
Miis affected by all prior outcomes, depen-
dence among witnesses is introduced through the
chained conditioning on statistic Mi.
3.3.1 Polarized Majority Vote Model Results
As shown in Figure 2, witnesses are likely to reach
an agreement about their claims of whether or not
the event happened, at a rate which depends on the
polarization θ. Thus, under this model a consen-
sus usually emerges. Because this model is sym-
metric under switching affirming and denying, a
consensus is as likely to form affirming an idea as
it is to form denying it. In this case, the amount of
Figure 2: The probability of consecutive witness agree-
ment over time, estimated through simulation using the
polarized majority vote model. The shaded regions in-
dicate 95% confidence intervals.
evidence gained by observing a consensus among
witnesses is greatly diminished, since the forma-
tion of a consensus is not strongly affected by the
truth or falsity of the position affirmed.
3.3.2 Polarized Majority Model Limitations
There are a few limitations of this model. First,
the behavior of witnesses is completely symmetric
to the hypothesis being true or false. This does
not reflect many real-world scenarios, where there
may be external pressures to bias responses, such
as the aforementioned political coercion present in
one-party states, which severely punish dissenters.
We address this issue in Section 3.5.
Second, this model is also slow to adapt to a
changing consensus opinion. If there is a change
in the local majority opinion, then it is likely that
people would perceive the new local majority as
the actual majority. However, in this model, the
shift to the new local majority being perceived as
the majority will not occur until after it reaches an
absolute majority, which may take much longer.
We address this issue in Section 3.4.
Third, while our model controls for polarization
and introduces a level of dependence among wit-
nesses, it does not allow us to fine-tune for par-
tial dependence or interpolate between a fully in-
dependent and fully dependent model. We explore
such an extension in Section 3.6, though the model
presented there is not exhaustive and leaves room
for future work.
Figure 3: Plot of (1−r)vs. number of witnesses needed
to overcome an existing consensus, if W1= 0 and all
subsequent witnesses affirm the hypothesis. As (1 −r)
increases, indicating a stronger reliance on historically
entrenched outcomes, the number of witnesses needed
to overturn the existing consensus surges.
3.4 Recent Majority Vote Model
In the polarized majority vote model, Miwas
taken as the smoothed average of the proportion
of witnesses who affirmed a position prior to i,
with each outcome equally weighted. Instead of
computing the simple uniform average, we can in-
troduce a notion of time-varying dependence by
weighting more recent votes more heavily. We do
so by computing a new proportion statistic, Wi,
defined recursively as
Wi= (1 −r)·Wi−1+r·xi−1(2)
where W1is the perception of the initial witness
(with W1= 0.5representing an even split, W1=
0representing a consensus against the hypothesis,
and W1= 1 representing a consensus affirming
the hypothesis), ris a parameter of the system
which represents the strength of the bias towards
recent responses, and xiis 1 if witness number i
affirmed the hypothesis and 0 if witness number i
rejected it.
Equation 2is equivalent to the summation ex-
pression
Wi=
i−1
X
j=0
w(r, i −j)·xj,
w(r, δ) = (1 −r)δ−1r
(3)
with x0defined equal to W1/r, and all other xide-
fined the same as in Equation 2. Plotted in Figure
4, the weighting function w(r, δ)determines the
Figure 4: Weight versus temporal distance for various
values of r. Larger values of rassign higher weight to
more recent witnesses.
contribution of previous witnesses based on tem-
poral proximity, where ris a parameter control-
ling the strength of temporal decay and δis the
temporal distance between the two witnesses. Us-
ing w(r, δ), Figure 3plots the number of new wit-
nesses needed to overcome an existing consensus
as a function of r, showing that as temporal prox-
imity matters less and more weight is placed on
older outcomes (namely, as rdecreases), the num-
ber of contrary witnesses needed to overcome an
existing consensus grows exponentially. In such
cases, consensus around a false position becomes
extremely difficult to dislodge; unless one is open
to new ideas and giving nonzero weight to con-
trary opinions, they might find themselves trapped
in a false consensus for an extremely long time.
Such a dynamic is closely related to herd behavior
as noted by Gold (Gold,1989).
3.5 External Pressure Model
The three models defined thus far do not consider
external pressures placed on witnesses, as often
arise in repressive political situations. To model
this behavior, we add two new parameters, αand
βwith 0≤α≤β≤1, to encode external pres-
sures influencing witness responses. We then de-
fine the probability of a given witness affirming
the hypothesis, gθαβ (Wi), as
gθαβ (Wi)=(β−α)gθ(Wi) + α. (4)
Thus, we rescale the function to output a value be-
tween the parameters αand β. The Recent Major-
ity Vote Model presented in Section 3.4 is a special
case of this model with α= 0, β = 1.
Wiis determined by the method defined in
Equation 2. Thus, this model resolves two issues
with the Polarized Majority Vote Model presented
in Section 3.3, by considering external pressures
and adapting to a recent majority.
To model an external pressure against affirming,
we can set βless than one, to decrease the likeli-
hood of affirmative responses. Likewise, to model
an external pressure towards affirming, αcan be
set greater than zero, which decreases the likeli-
hood of negative responses.
3.5.1 External Pressure Model Results
Figure 5shows the influence of αon the likelihood
of witnesses affirming. This result, along with
the setup being symmetric around affirming vs. re-
jecting the hypothesis, demonstrates that external
pressure will likely push the group to a consensus
on whichever position they are being pressured to-
wards. However, in the presence of a weak exter-
nal pressure, it is possible, albeit unlikely, that a
near-consensus may emerge against that position.
Accordingly, a consensus position towards
which witnesses are pressured supplies little evi-
dence in favor of the consensus position, as such a
consensus is highly likely to emerge regardless of
the truth of the position around which it emerges.
3.6 A Spectrum of Dependence
Finally, we introduce a model which allows us to
modify the level of dependence among witnesses.
We add two parameters to the model, λand pf,
where λ∈[0,1] controls the dependence of each
witness on others and pf∈[0,1] is the probabil-
ity that a witness will fallibly affirm a false hy-
pothesis, independent of the influence of other wit-
nesses. We then define the probability that a given
witness will incorrectly affirm a hypothesis as
gθ,λ(Wi) = (1 −λ)pf+λgθ(Wi).(5)
The independent model from Section 3.1 is a spe-
cial case of this model, with λ= 0. The dependent
model from Section 3.4 is another special case,
with λ= 1. Furthermore, this model can be emu-
lated by Equation 4from Section 3.5, by setting α
and βas
α= (1 −λ)pf
β=pf+ (1 −pf)λ. (6)
In the simulations for this model, we run the
model until it reaches an equilibrium of Wival-
(a) (b)
Figure 5: Evolution of the probability of a witness affirming for various values of α. Shaded regions denote 95%
confidence intervals. Simulations were run with θ= 10, β = 1, r = 0.035. In (a), W1= 0.5. In (b), W1is chosen
such that the first witness has a 50% chance of affirming.
ues, and then observe the probability of a consen-
sus among the Nsubsequent witnesses. We are
concerned with factors that lead to the eventual
formation of a consensus, rather than only consid-
ering consensuses that emerge immediately.
3.6.1 A Spectrum of Dependence Results
Considering the case when individual witnesses
are each highly likely to mistakenly affirm a false
hypothesis (as in Section 3.1), Figure 6shows the
result of applying Bayes’ theorem to find the pos-
terior probability that a hypothesis is true, given
that a consensus has formed affirming it. It shows
that the strength of evidence from a consensus de-
creases rapidly as independence among the wit-
nesses becomes compromised, and the consensus
no longer protects against the fallibility of individ-
ual witnesses, in stark contrast to the full indepen-
dence case from Section 3.1.
From Figure 7, we can see that the probability
of a consensus affirming a hypothesis grows ex-
ponentially on the level of dependence, reaching
a probability of half at full dependence. We only
show data for one set of parameters, because all
values of the parameters we investigated resulted
in similar plots.
Figure 8shows the results of the same calcu-
lation, but allowing for a few dissenters from the
clear consensus position, for two different sets of
parameters. Figure 8a shows that, if individual
witnesses are likely to be reliable, small amounts
of dependence do not change the probability of
consensus substantially, but larger values of de-
Figure 6: The posterior probability of a hypothesis
given different values of λand different priors, with
pf= 0.95, pt= 0.5, N = 150, θ = 10, r = 0.035.
Figure 7: The probability of a consensus affirming for
different values of λ, found by simulation with param-
eters pf= 0.5, N = 13, θ = 10, r = 0.035, averaged
over 8,000 independent trials. Note the log scale. 95%
confidence intervals are plotted, but are not visible.
(a) (b)
Figure 8: The probability of a near-consensus affirming for values of λ, found by simulation with θ= 10, r =
0.035. (a) was run with pf= 0.35, N = 20, and a consensus was defined as ≥90%. (b) was run with pf=
0.5, N = 30, and a consensus was defined as ≥90% agreement on the hypothesis. The shaded regions indicate
95% confidence intervals.
pendence result in exponential growth of the prob-
ability, causing the evidence from the consensus
to become compromised. Figure 8b shows that,
when an individual witness is equally likely to be
right as wrong, the probability of a near-consensus
forming grows exponentially on the value of λ.
3.6.2 Overall Consensus Probability
Section 3.6.1 presents the probability of a con-
sensus forming affirming a given hypothesis. If
one observes a consensus around a given position,
that probability would be useful to determine the
strength of the evidence for that position provided
by the observed consensus. However, it may also
be useful to know the overall probability of any
consensus forming (for or against a hypothesis).
Figure 9shows the overall probability of a con-
sensus in either direction, based on λand pf. It
shows that the probability of consensus formation
increases with λ, equaling 1 when λ= 1, and it
decreases as pfapproaches 0.5, being symmetric
around individuals being either reliable or error-
prone. With strong dependence a consensus is al-
most certain to arise, whether or not the position
being held is true. Thus, observing a consensus
in highly dependent cases tells us almost nothing
concerning the truth status of the consensus posi-
tion. Such a consensus lacks any real meaning.
4 Conclusion
ACastro Consensus is a near-unanimous show
of agreement brought about by means other than
the honest and uncoerced judgements of individ-
uals. Using mathematical modeling, we demon-
strate how dependence, polarization, and external
pressure compromise the relation between truth
and consensus. When individuals are fully inde-
pendent, even under highly unfavorable circum-
stances a consensus provides strong evidence for
the correctness of the affirmed position. This no
longer remains the case once dependence, polar-
ization, and external pressure are introduced. With
such interventions, the probability of a false con-
sensus increases dramatically.
All models except the one presented in Section
3.6 tend to approach a consensus around one po-
sition. In Section 3.5, we find that strong pressure
against a position will eventually result in a con-
sensus against that position. This suggests that in
cases where holding certain contrary positions are
punished, consensus becomes an unreliable indi-
cator of truth. In Section 3.6, we find that the prob-
ability of a consensus or near-consensus forming
grows rapidly as dependence among the witnesses
increases. We find that even slight amounts of de-
pendence between witnesses can greatly decrease
the strength of evidence provided by a consensus
for the truth of a position.
There are many ways to model agreement and
consensus formation, of which our chosen mod-
els only represent one small set of possible op-
tions. Let us be clear that we make no sociolog-
ical claims concerning our models’ ability to fully
capture and adequately model the dynamics of hu-
(a) (b)
Figure 9: The probability of a near consensus forming, found via simulation with θ= 10, r = 0.035, N = 20.
A near-consensus was defined as ≥90% agreement on the majority position. Subfigure (a) plots the consensus
probability as a function of pffor several values of λ, while (b) plots the consensus probability as a function of λ
for several values of pf. The shaded regions indicate 95% confidence intervals.
man agreement and testimony. The primary value
of this work is not in presenting any particular set
of models but in demonstrating how those models
change in response to external factors like depen-
dence and pressure. As such, we present an initial
model for which consensus does strongly and reli-
ably give evidence of the truth of the affirmed po-
sition, and proceed to show how the introduction
of dependence, polarization, and pressure destroys
the evidential value of the consensus in that same
model. We demonstrate how dependence, pres-
sure, and polarization can force a consensus, mak-
ing reliance on consensus as an indicator of truth
unreliable. As a result, a consensus can only be
trusted to the extent that individuals are free to dis-
agree with it, without repression or reprisal. Simi-
larly, when strong incentives favor affirmation of a
position, a consensus affirming it becomes almost
inevitable, and therefore all but meaningless.
Acknowledgements
The authors would like to thank Caroline Chou,
Jessica Winssinger, and Corbin Bethurem, whose
early classroom collaboration on a similar project
helped inspire the work presented here. This re-
search was supported in part by the generous sup-
port of Harvey Mudd College and the National
Science Foundation under Grant No. 1950885.
Any opinions, findings or conclusions expressed
are the authors’ alone, and do not necessarily re-
flect the views of Harvey Mudd College or the Na-
tional Science Foundation.
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