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Stubborn extremism as a potential pathway to group polarization

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Abstract

Group polarization is the widely-observed phenomenon in which the opinions held by members of a small group become more extreme after the group discusses a topic. For example, conservative individuals become even more conservative, while liberal individuals become even more liberal. Social psychologists have offered competing explanations for this phenomenon. These typically require questionable assumptions about human psychology. Here, we posit a more parsimonious explanation: the stubbornness of extreme opinions. Using agent-based modeling, we demonstrate that such "stubborn extremism" gives rise to group polarization, as well as other trends observed across the literature on polarization. Our study revealed a further methodological problem for the study of group polarization: reporting opinions as categories (e.g. on a Likert scale) inflates the observed increase in opinion extremity. We conclude with a call for deeper integration of opinion dynamics modeling with the cognitive science of communication and influence.
Stubborn extremism as a potential pathway to group polarization
Matthew A. Turner and Paul E. Smaldino
Abstract
Group polarization is the widely-observed phenomenon
in which the opinions held by members of a small group
become more extreme after the group discusses a topic.
For example, conservative individuals become even more
conservative, while liberal individuals become even more
liberal. Social psychologists have offered competing ex-
planations for this phenomenon. These typically re-
quire questionable assumptions about human psychol-
ogy. Here, we posit a more parsimonious explanation:
the stubbornness of extreme opinions. Using agent-
based modeling, we demonstrate that such “stubborn
extremism” gives rise to group polarization, as well as
other trends observed across the literature on polariza-
tion. Our study revealed a further methodological prob-
lem for the study of group polarization: reporting opin-
ions as categories (e.g. on a Likert scale) inflates the
observed increase in opinion extremity. We conclude
with a call for deeper integration of opinion dynamics
modeling with the cognitive science of communication
and influence.
Keywords: opinion dynamics; polarization; social in-
fluence; agent-based modeling
Introduction
Group polarization is a phenomenon in which the opin-
ions held by members of a small group become more
extreme after the group discusses a topic (Myers, 1982;
Brown, 1986; Isenberg, 1986; Sunstein, 2002; Sieber
& Ziegler, 2019). This phenomenon is socially impor-
tant for many reasons. First, small groups of advisers
often influence executive decisions in government and
business. At the “grass roots” level in politics, individ-
uals discuss important issues first in small groups be-
fore they vote. Second, group polarization at the local
level increases overall polarization at the societal level.
Polarization, commonly understood, increases whenever
either of two opposed groups moves to a greater ex-
treme, whatever the polarization measure (Bramson et
al., 2016). Most studies of political polarization frame
the issue in terms of intergroup conflict (Mason, 2018;
Klein, 2020). However, we also must understand how
group polarization can exacerbate political polarization
through increased in-group extremism. Understanding
the cognitive mechanisms supporting group polarization
is therefore a matter of concern.
Social psychologists have typically offered one of two
explanations for group polarization, sometimes combin-
ing the two (Sieber & Ziegler, 2019). Social compari-
son theory posits that individuals’ privately-held opin-
ions tend be more extreme than those they express pub-
licly, and exposure to consonant opinions gives them
confidence to express their true opinions openly (Myers,
1982). Persuasive arguments theory posits that when in-
dividuals discuss a topic within an already-biased group,
they accumulate more persuasive arguments supporting
those biases, leading to a more extreme version (Bishop
& Myers, 1974; Vinokur & Burstein, 1974). These ex-
planations may explain the empirical phenomenon of
group polarization, though more formal modeling will
be required to bring precision to the underlying the-
ories (Smaldino, 2017, 2019). Nevertheless, each pre-
sumes either that opinions are intrinsically extreme (so-
cial comparisons theory) or that moderate opinions exist
only due to uncertainty concerning the state of the world
(persuasive arguments theory).
We present an alternative explanation for group polar-
ization that, while not mutually exclusive with the other
theories discussed, manages to explain the phenomenon
of group polarization without assuming anything about
the intrinsic distribution of extreme opinions in human
groups. We do so by appealing to a property of human
psychology we call stubborn extremism: as a person’s
opinion on some topic becomes more extreme, that opin-
ion also becomes more stubborn, i.e. less susceptible to
social influence. We support this explanation using a
computational model of group polarization. Our model
was originally developed for explaining how polariza-
tion emerges where two groups become more extremely
opposed (Flache & Macy, 2011; Turner & Smaldino,
2018)—it incorporates both negative, repulsive social in-
fluence (Cikara & Van Bavel, 2014), assimilative influ-
ence, and the stubborn extremism assumption, though
repulsive influence is not at work in group polarization
because all opinions start out similarly valenced.
Group polarization can emerge computationally by
simply assuming agents hold discrete opinions on a mul-
titude of topics (Mueller & Tan, 2018; Banisch & Ol-
brich, 2019). However, most group polarization stud-
ies do not measure participants’ binary opinions (e.g.,
for vs. opposed) on a multitude of topics, but rather
measure opinions as falling on a range between strongly
for and strongly opposed. Furthermore, the assumption
of discrete opinions is problematic from a psychological
perspective, since it is rare for quantum leaps in opinion
to occur—more often we are influenced gradually over
the course of many interactions (Baldassarri & Bear-
man, 2007, p.793). Our model is most similar to that
of Baldassarri and Bearman (2007) in that stubbornness
is a function of opinion extremity directly. Martins and
Galam (2013) allow for agents to become more or less
stubborn, but assume discrete opinions and a separate,
continuous measure of open-mindedness/stubbornness.
Most other opinion dynamics models that link stubborn-
ness to extremism assume infinitely stubborn extreme
agents (sometimes called “zealots”) whose opinions are
static and whose existence is specified a priori by the
modeler (Galam & Jacobs, 2007; Mobilia, Petersen, &
Redner, 2007; Arendt & Blaha, 2015; Mueller & Tan,
2018). Baldassarri and Bearman (2007) nearly make the
connection between stubborn extremism and group po-
larization, but they mischaracterize group polarization
and discuss it in terms of negative influence, saying “in-
teraction with dissimilar others may increase distance,
leading to group polarization” (p. 792). Group polar-
ization experiments are designed so that this never oc-
curs. Instead, it is only interaction among relatively like-
minded individuals that leads to the group polarization
opinion shift.
Current empirical support for the stubborn extremism
explanation is positive, though not uniformly so. Zaller
(1992) and Converse (2006) established that, at least
at the time of their studies, most of the United States
electorate, for example, were relatively ignorant of real
political issues and easily swayed by momentary predilec-
tions and the framing of questions. Guazzini, Cini, Bag-
noli, and Ramasco (2015) found that stubborn extrem-
ists drove the opinions in groups discussing the use of
animals in laboratory experiments, and Lewandowsky,
Pilditch, Madsen, Oreskes, and Risbey (2019) found that
stubborn extremists have an outsized influence in the
perpetuation of scientific misinformation regarding cli-
mate change. Group polarization opinion shifts have
been observed to increase with the group’s initial ex-
tremity (Teger & Pruitt, 1967; Myers & Arenson, 1972;
Myers, 1982; Brown, 1986). This has only been tested in
detail by Teger and Pruitt (1967) and Myers and Aren-
son (1972), apparently, and has not been established for
political opinions. This could cause acceleration of polit-
ical polarization. Some researchers have suggested that
stubbornness is an attribute found generally among peo-
ple, and is not limited to those with extreme opinions.
However, support for this view often comes from studies
in which opinions are operationalized as answers to gen-
eral knowledge tests (such as found in a pub quiz), and
not on opinions with political or ethical components in
which subjective judgment plays a larger role (Moussa¨ıd,
ammer, Analytis, & Neth, 2013; Chacoma & Zanette,
2015). More direct empirical tests of the stubborn ex-
tremism explanation for group polarization are needed.
Here we investigate whether assuming stubborn ex-
tremism can explain and predict observed empirical pat-
terns of group polarization. We do this with an eye to-
wards future empirical experiments. In doing so, we also
pay close attention to the scales used to measure opinion
extremity. There has recently been scrutiny of the use
and analysis of Likert-scaled data in social psychology,
which indicates that using metric statistical models on
Likert-scaled data can distort effect sizes (Liddell & Kr-
uschke, 2018). We therefore also examine the effect of
using a Likert scale in which simulated agents bin their
continuous opinions.
The rest of the paper is organized as follows. We
first briefly review the empirical evidence for group po-
larization upon which we will based our analyses. We
will then introduce an agent-based model of opinion dy-
namics with stubborn extremists, which is adapted from
previous work by Flache and Macy (2011), and we will
demonstrate how the model supports the stubborn ex-
tremism hypothesis. We will then compare our model
to the persuasive arguments model of M¨as and Flache
(2013), and show how our model can yield a fit to the
empirical dataset they test that is at least as congruent.
We conclude with limitations of our model’s assump-
tions, and suggestions for future work.
Group polarization studies
A prototypical experiment involving group polarization
works as follows. First, participants are pre-screened for
their opinions about some issue or set of issues. They
give their opinions on an ordinal scale, such as a 7-point
Likert scale that ranges from -3, which would indicate
“strongly disagree” to +3 which would indicate “strongly
agree. Then the participants are sorted into groups
with similarly valenced views. The participants are then
asked to publicly give their initial opinion on some issue,
discuss the issue within their group for a time, and then
report their opinion again to the experiments. It is reg-
ularly observed that individuals’ opinions shift towards
greater extremity following group discussion.
Moscovici and Zavalloni (1969) studied group polar-
ization in the context of national (French) and global
politics. In their study, they first asked participants the
degree to which they agreed or disagreed with the claim
that Charles de Gaulle, then president of France, was
“too old to carry out such a difficult political task. Sec-
ond, participants were asked the degree to which they
agreed or disagreed that “American economic aid is al-
ways used for political pressure. Participants responded
on a 7-point Likert scale where -3 indicated total dis-
agreement, 0 represented a neutral psychological posi-
tion of neither agree nor disagree, and 3 indicated total
agreement. Forty individuals in ten groups answered,
discussed, then answered again the de Gaulle question,
and twenty individuals in five groups did the same for
the Americans question, with four per group in each
case. In reporting the shifts in opinions, only the mean
shifts for the entire subject pool were reported. For the
de Gaulle question, a shift was observed from a mean
pre-discussion opinion of 1.36 to a mean of 1.82 post-
discussion, a shift of 0.46. On the Americans question,
the initial pre-discussion mean opinion was 0.88 and the
post-discussion mean opinion was 1.69, a shift of 0.81.
Myers and Bishop (1970) asked participants about
their attitudes on eight items, where responses varied
from -9 to 9, an 18-point scale. -9 indicated maximal
racial prejudice and 9 indicated minimal prejudice. Par-
ticipants were grouped into low, medium, and high prej-
udice supergroups, from which discussion groups of 4-
7 members were formed. The average shift of the low
prejudice group was 0.47, while the average shifts of the
medium and high prejudice groups were -0.64 and -1.30,
respectively. In other words, individuals in the low prej-
udice group decreased their expressed prejudice levels,
while prejudiced groups became more prejudiced.
Myers and Lamm (1975) binned attitudes about the
role of women in society from -3 (conservative) to +3
(liberal), again with 0 a neutral opinion. Groups of
“chauvinists” showed no significant opinion shift, while
“feminists” showed a significant average shift of 0.95.
Discussions occurred in groups with four or five mem-
bers from 95 total participants.
as and Flache (2013) developed a model of opinion
change based on persuasive arguments theory and con-
ducted a laboratory study of opinion change similar to
the ones described above, concerning the better of two
locations (town A or town B) to build a new leisure cen-
ter. Participants were asked their preference among the
two hypothetical towns (“A” or “B”). Participants were
sorted into groups “A” and “B,” and given a number of
pro-A and pro-B arguments they could exchange with
other agents. Members of the “A” group (A-Type) were
provided with two pro-A arguments and one pro-B ar-
gument, while members of the “B” group (B-Type) were
provided with one pro-A argument and two pro-B argu-
ments. All members of the “A” group all received the
same pro-B argument, and all members of the “B” group
received the same pro-A argument.
All participants participated in seven interaction
rounds. In each round, participants interacted through
a computer interface with a single interaction partner,
wherein each participant selected one of their pre-written
arguments and sent it to their partner. Participants first
interacted with the three other members of their own
group, and then interacted with the four members of
the other group. After all seven rounds of interaction,
participants again reported their opinions.
as and Flache (2013) observed that following within-
group interactions, the group’s average opinion became
more extreme in accordance with the general predic-
tion of group polarization. Their explanation was based
on persuasive arguments theory, which also predicts—
on the basis of a computational model analyzed by the
authors—that when opposing groups interact they will
become more similar in opinion. This happens because
when an individual from one group interacts with a mem-
ber of the opposing group, they will be exposed to new
counterarguments that reduce their extremism towards
a more moderate opinion. During the four rounds of
between-group interactions in M¨as and Flache’s study,
the average opinion in both groups converged towards
zero. This is predicted by both persuasive arguments
and stubborn extremism models, which we demonstrate
in computational experiments below.
The model
We developed an agent-based model to demonstrate the
stubborn extremism model predicts the empirical results
reviewed above. Our goal is to demonstrate that assum-
ing stubborn extremism can lead to group polarization
opinion shifts as reliably as social comparisons or per-
suasive arguments models. The model is similar to a
Hopfield network model (Hopfield, 1982), in which node
values change based on neighboring node values, and me-
diated by network weights. Here, these weights are de-
termined by the distance in opinion space between two
agents. This model allows for both positive and neg-
ative influence, wherein initially similar agents become
more similar after interacting, while initially dissimilar
agents become more polarized. The model is identical to
that studied previously in Flache and Macy (2011) and
Turner and Smaldino (2018), but is analyzed here with
a different focus than was used in those studies.
We consider a population of Nagents, who each
have opinions on Kdifferent cultural topics. Agent i’s
opinion on topic kat time tis written oik,t (1,1)
and changes after ihas interacted with its Ninetwork
neighbors. The weight of social influence with each
neighbor jis wij,t, with zero direct influence over non-
neighbors. Weights depend on the Manhattan distance
between agents iand j, normalized over cultural top-
ics: dij,t =1
KPK
k=1 |oik,t ojk,t|. The specific operation
of these social influence mechanisms is defined by the
following dynamical equation
oik,t =oik,t1+ ∆oik,t (1 − |oik,t1|α) (1)
where
oik,t =1
2NiX
j
wij,t(oj k,t oik,t ) (2)
and
wij,t = 1 dij,t.(3)
Our model includes both positive and negative influence.
Positive influence is when agents become increasingly
similar to their dyad partner if the pair are sufficiently
similar to begin with (dij <1). Negative influence is
when interaction causes a dyad to become more differ-
ent, to be repulsed away from one another toward more
extreme regions of opinion space if the pair are suffi-
ciently dissimilar to begin with (dij >1). The parame-
ter αdetermines the degree to which extreme opinions
are stubborn. In the analyses presented here, we use
α= 1. Stubborn extremism emerges in our model due
to the smoothing factor (1 − |oik,t1|), which is smaller
when |oik,t1|is larger. Therefore, more extreme opin-
ions (larger |oik,t1|) are less susceptible to social influ-
ence than less extreme opinions (smaller |oik,t1|).
Our model generates a number of empirically-observed
outcomes. First, we show that our model yields group
polarization in an idealized generic case that resembles
the studies of Moscovici and Zavalloni (1969), Myers and
Bishop (1970), and Myers and Lamm (1975). For our
computational experiments, we set the number of agents
in the population to N= 25 and the number of relevant
opinions K= 1. The social network for this first ex-
periment was fully connected, meaning all agents could
potentially influence all other agents. Second, we rep-
resent the M¨as and Flache (2013) empirical experiment
with our model and show our model predicts their em-
pirical observations as accurately as their computational
model of persuasive arguments theory.
Computational experiments
Our first experiment examined the correlation between
initial mean opinion and shift magnitude. This also es-
tablishes that our model generates group polarization.
We set N= 25 and K= 1. Initial agent opinions were
drawn from a normal distribution with σ= 0.25. In or-
der to demonstrate that our model predicts the corre-
lation between opinion shift and initial opinion extrem-
ity, we ran the model with seven different experimental
conditions. Each of the seven conditions specified a dif-
ferent mean for the normal distribution from which ini-
tial opinions were drawn, µ∈ {0.2,0.3,...,0.8}. For each
condition we ran 100 trials. Since opinions are bounded
between ±1 and group polarization experiments force
group members to have opinions of the same valence, we
re-mapped any drawn opinions greater than 1 to be +1 if
the drawn opinion was greater than 1, and 0 if the drawn
value was less than 0. Each model run consisted of 100
rounds of agent interactions. In one round of agent inter-
action, Nagents are selected at random to update their
opinions according to Equation 1. To model a typical
group polarization experiment with open discussion, we
assume a fully-connected network, so all agents influence
one another.
Our second experiment was designed to examine how
opinion shifts and the pattern of shift versus initial ex-
tremity may be distorted when agents are forced to re-
port their opinions on a seven-point Likert scale. This
was done by first transforming the continuous opinions
on (1,1) to a continuous opinion on (3.5,3.5), rep-
resenting seven equally-sized bins of width 1, and then
rounding each agent’s continuous opinion to the closest
integer, e.g. 2.9 becomes 3 and 1.49 becomes 1.
Our third and final experiment was designed to gen-
erate the results of M¨as and Flache (2013). Here we
utilized the multidimensionality of opinions to represent
different “persuasive arguments” that participants held.
To do this, we set K= 12, the total number of persuasive
arguments available to each agent in M¨as and Flache’s
study, and initialized three of the twelve opinions to be
non-zero. Recall that in their study, M¨as and Flache
provided individuals with one of twelve pre-defined “ar-
guments” they were to share with others to advocate for
their opinion. Six of the twelve were chosen as pro-A
arguments and six of the twelve were chosen as pro-B
arguments. The pro-A arguments were given initial val-
ues of 1/3 and pro-B arguments given initial values of
1/3. In our adaptation of this experimental setup, we
are using each of Kelements of agent i’s opinion vec-
tor to represent the presence or absence of an argument.
As in the M¨as and Flache study, group “A” members
all received the same initial pro-B argument, and vice
versa. To calculate each agent’s scalar opinion based
on its K= 12 “persuasive argument” components, we
first normalize opinions so their absolute values sum to
1, and then averaged over all opinions. This is similar
to the persuasive argument model that assumes an indi-
vidual’s opinion is an aggregate of the arguments they
know for their position. This computational experiment
mirrors M¨as and Flache’s persuasive arguments model,
but includes stubborn extremism. Furthermore, in our
formulation, agents can partially agree or disagree with
a given argument, unlike persuasive arguments which as-
sumes an agent either knows an argument or not. For our
computational experiment’s outcome measure, we calcu-
lated the average over all agent opinions in each group at
each timestep, and then averaged those averages across
100 trials at each timestep, identical to M¨as and Flache’s
procedure for obtaining their results (Figures 5 and 6 of
their paper).
Implementation
The model was implemented as an agent-based model
written in plain Python with user-defined Agent,Model,
and Experiment classes. We use NumPy and SciPy
for numerical and scientific routines and functions. For
full implementation details including instructions for in-
stalling and running model code and reproducing our
results, please visit the GitHub repository, https://
github.com/mt-digital/group-polarization. Our
computational experiments easily run on a laptop.
0.20 0.30 0.40 0.50 0.60 0.70 0.80
Initial mean opinion
0.1
0.0
0.1
0.2
0.3
Opinion shift
(a) Group opinion shift when individuals’ initial and final opin-
ions are given on a continuous scale.
0.20 0.30 0.40 0.50 0.60 0.70 0.80
Initial mean opinion
0.1
0.0
0.1
0.2
0.3
Opinion shift
(b) Group opinion shift when individuals’ initial and final opin-
ions are given on a 7-point Likert scale.
Figure 1: Demonstration of the trend that opinion shift
is positively correlated with the mean initial group opin-
ion. The trend is distored by binning into Likert scale
responses. Boxes enclose the first and third quartile of
the data. 100 trials shown for each condition.
Analysis
Our model predicts that more extreme initial group opin-
ion results in larger shifts up to a certain extremity
where the trend reverses (Figure 1a). In terms of stub-
born extremism, this general trend is expected because
there will be more extremists when the initial mean is
greater. These initial extremists exert a greater pull to-
wards extremism when they are more numerous. How-
ever, when many agents are extreme and there are few
neutral agents to be shifted to more extreme views, the
shift begins to decrease in magnitude compared to the
maximum shift over initial mean (occurs at initial mean
of 0.8 in Figure 1a).
Binning disrupts the positive linear relationship be-
tween opinion shift and initial mean group opinion (Fig-
01234567
Interaction round
0.4
0.2
0.0
0.2
0.4
Mean Opinion
B-Type
A-Type
Figure 2: Our model’s prediction of group opinions in the
as and Flache (2013) study. Within-group interactions
are rounds 1-3, intergroup interactions are rounds 4-7.
ure 1b). This is because, in our model, if enough agents
are neutral and not too many agents are extreme, then
some agents with an opinion of +3 will shift to +2, and
enough agents with opinions of +1 or less do not shift
their opinions, the sign of the shift may be negative, and
group polarization will not emerge.
Our model predicts group polarization as observed by
as and Flache (2013), but via the assumption of stub-
born extremists instead of persuasive arguments. Our
model predicts the same initial increase in the extremity
of the average group opinion for both A- and B-Type
agents as predicted and observed in M¨as and Flache
(2013). Then when A-Types and B-Types interact with
one another, our model predicts consensus emerges, as
was observed by M¨as and Flache’s experiments and pre-
dicted by their model (Figure 2 above; compare with Fig-
ure 6 M¨as and Flache (2013)). Note that, in our model,
no explicit persuasive arguments are exchanged. Instead,
each argument is represented as an opinion on a certain
cultural topic. Influence occurs on all cultural topics,
and similar group members draw one another closer in
hypothetical 12-dimensional opinion space through at-
tractive social influence and stubborn extremism, result-
ing in group polarization.
Discussion
We have shown that stubborn extremists are a feasible
explanation for group polarization. Our model that in-
corporates this simple mechanism predicts behavior ob-
served in a number of empirical studies. These empirical
studies have often considered two alternative pathways
to group polarization: persuasive arguments and social
comparisons. The persuasive argument theory explains
that group polarization occurs because individuals are
exposed to more arguments supporting their initial po-
sition in contrast with the opposing opinions, thereby
strengthening that opinion. At the group level, this leads
the average opinion to shift towards an extreme. Al-
ternatively, social comparison theory posits that group
polarization is due to group members calculating some
optimal opinion to express publicly that takes into ac-
count both their private opinion and the perceived so-
cial consequences of expressing that opinion. The the-
ory posits that, following group discussion, this optimal
public opinion is usually judged to be more extreme than
individuals’ initially stated opinions.
Neither of these theories are necessarily wrong; indeed,
they appear to be both psychologically and sociologically
plausible. What we have done is to identify another
mechanism, independent of these, which is at least as
plausible and which generates known empirical results
when formalized in a computational model. Moreover,
our model is the only one to assume neither that ex-
treme opinions are the default state (as social compari-
son theory does), nor that moderate opinions result only
from lack of arguments for more extreme positions (as
persuasive arguments theory does). Our theory requires
only that extreme opinions, once reached, are more stub-
bornly held than more moderate opinions.
We also demonstrated that, if we assume individu-
als’ opinions are continuous, forced binning of opinions
into Likert-style point values leads to distortions in opin-
ion shifts compared to the underlying continuous opin-
ion shifts. Depending on the final distribution of agent
opinions, these distortions could be either an over- or
under-estimate of the opinion shift. In extreme cases the
sign of the opinion shift reversed, making the shift ap-
pear to be towards a more moderate group opinion, not
more extreme. This puts existing group polarization re-
sults in question since most empirical group polarization
studies used a Likert scale. This is further compounded
by similar effects that could be caused by using metric
statistical models on ordinal data, which apparently all
existing group polarization studies have done (Liddell &
Kruschke, 2018).
Although there is evidence supporting the hypoth-
esis that extreme opinions are more stubbornly held,
we are aware of no research specifically investigating
the relationship between stubbornly held opinions and
group polarization. Future empirical work should evalu-
ate the stubborn extremism hypothesis using a statistical
model to detect correlation between opinion extremity
and stubbornness. Such work should use both Likert
and continuous scales due to the distortions in opinion
shift we identified. In the Likert condition, one must use
the appropriate statistical model (Liddell & Kruschke,
2018). We advocate the use of both scales for two rea-
sons. First, it is not clear that a continuous model of
opinions really is superior to a discrete model of opinions.
The empirical situation may determine which opinion
model is appropriate. The second reason is that, in prac-
tice, responses given on Likert-type scales may be more
reliable than those given on continuous scales (Toepoel
& Funke, 2018).
Models of opinion dynamics should be able to explain
a number of empirical phenomena, including but not lim-
ited to group polarization. Another program of future
work could be to perform similar computational exper-
iments shown here using alternative, influential models
of political polarization, such as Bayesian/information-
theoretic models (e.g. Dixit and Weibull (2007)) or algo-
rithmic models (e.g. Dandekar, Goel, and Lee (2013)).
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