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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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