Opinion Dynamics with Confirmation Bias

Abstract

Background
Confirmation bias is the tendency to acquire or evaluate new information in a way that is consistent with one's preexisting beliefs. It is omnipresent in psychology, economics, and even scientific practices. Prior theoretical research of this phenomenon has mainly focused on its economic implications possibly missing its potential connections with broader notions of cognitive science.

Methodology/Principal Findings
We formulate a (non-Bayesian) model for revising subjective probabilistic opinion of a confirmationally-biased agent in the light of a persuasive opinion. The revision rule ensures that the agent does not react to persuasion that is either far from his current opinion or coincides with it. We demonstrate that the model accounts for the basic phenomenology of the social judgment theory, and allows to study various phenomena such as cognitive dissonance and boomerang effect. The model also displays the order of presentation effect–when consecutively exposed to two opinions, the preference is given to the last opinion (recency) or the first opinion (primacy) –and relates recency to confirmation bias. Finally, we study the model in the case of repeated persuasion and analyze its convergence properties.

Conclusions
The standard Bayesian approach to probabilistic opinion revision is inadequate for describing the observed phenomenology of persuasion process. The simple non-Bayesian model proposed here does agree with this phenomenology and is capable of reproducing a spectrum of effects observed in psychology: primacy-recency phenomenon, boomerang effect and cognitive dissonance. We point out several limitations of the model that should motivate its future development.

Full text

Opinion Dynamics with Confirmation Bias
Armen E. Allahverdyan
1
*, Aram Galstyan
2
1Department of Theoretical Physics, Yerevan Physics Institute, Yerevan, Armenia, 2USC Information Sciences Institute, Marina del Rey, California, United States of America
Abstract
Background:
Confirmation bias is the tendency to acquire or evaluate new information in a way that is consistent with one’s
preexisting beliefs. It is omnipresent in psychology, economics, and even scientific practices. Prior theoretical research of
this phenomenon has mainly focused on its economic implications possibly missing its potential connections with broader
notions of cognitive science.
Methodology/Principal Findings:
We formulate a (non-Bayesian) model for revising subjective probabilistic opinion of a
confirmationally-biased agent in the light of a persuasive opinion. The revision rule ensures that the agent does not react to
persuasion that is either far from his current opinion or coincides with it. We demonstrate that the model accounts for the
basic phenomenology of the social judgment theory, and allows to study various phenomena such as cognitive dissonance
and boomerang effect. The model also displays the order of presentation effect–when consecutively exposed to two
opinions, the preference is given to the last opinion (recency) or the first opinion (primacy) –and relates recency to
confirmation bias. Finally, we study the model in the case of repeated persuasion and analyze its convergence properties.
Conclusions:
The standard Bayesian approach to probabilistic opinion revision is inadequate for describing the observed
phenomenology of persuasion process. The simple non-Bayesian model proposed here does agree with this
phenomenology and is capable of reproducing a spectrum of effects observed in psychology: primacy-recency
phenomenon, boomerang effect and cognitive dissonance. We point out several limitations of the model that should
motivate its future development.
Citation: Allahverdyan AE, Galstyan A (2014) Opinion Dynamics with Confirmation Bias. PLoS ONE 9(7): e99557. doi:10.1371/journal.pone.0099557
Editor: Pablo Branas-Garza, Middlesex University London, United Kingdom
Received December 8, 2013; Accepted May 15, 2014; Published July 9, 2014
Copyright: ß2014 Allahverdyan, Galstyan. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This research was supported by DARPA grant No. W911NF-12-1-0034 and AFOSR MURI grant No. FA9550-10-1-0569. The funders had no role in study
design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* Email: armen.allahverdyan@gmail.com
Introduction
Confirmation bias is the tendency to acquire or process new
information in a way that confirms one’s preconceptions and
avoids contradiction with prior beliefs [52]. Various manifestations
of this bias have been reported in cognitive psychology [5,67],
social psychology [24,54], politics [46] and (media) economics
[31,51,57,73]. Recent evidence suggests that scientific practices
too are susceptible to various forms of confirmation bias
[12,38,43,44,52], even though the imperative of avoiding precisely
this bias is frequently presented as one of the pillars of the scientific
method.
Here we are interested in the opinion revision of an agent P
who is persuaded (or advised) by another agent Q[10,13,52].
(Below we use the terms opinion and belief interchangeably.) We
follow the known framework for representing uncertain opinions
of both agents via the subjective probability theory [13]. Within
this framework, the opinion of an agent about propositions (events)
is described by probabilities that quantify his degree of confidence
in the truth of these propositions [13]. As we argue in the next
section, the standard Bayesian approach to opinion revision is
inadequate for describing persuasion. Instead, here we study
confirmationally-biased persuasion within the opinion combina-
tion approach developed in statistics; see [21,30] for reviews.
We suggest a set of conditions that model cognitive aspects of
confirmation bias. Essentially, those conditions formalize the
intuition that the agent Pdoes not change his opinion if the
persuasion is either far away or identical with his existing opinion
[15,60]. We then propose a simple opinion revision rule that
satisfies those conditions and is consistent with the ordinary
probability theory. The rule consists of two elementary operations:
averaging the initial opinion with the persuading opinion via linear
combination, and then projecting it onto the initial opinion. The
actual existence of these two operations has an experimental
support [8,9,18,72,].
We demonstrate that the proposed revision rule is consistent
with the social judgment theory [10], and reproduces the so called
change-discrepancy relationship [10,35,40,45,69]. Furthermore, the
well-studied weighted average approach [9,27] for opinion revision is
shown to be a particular case of our model.
Our analysis of the revision rule also reveals novel effects. In
particular, it is shown that within the proposed approach, the
recency effect is related to confirmation bias. Also, repeated
persuasions are shown to hold certain monotonicity features, but
do not obey the law of diminishing returns. We also demonstrate
that the rule reproduces several basic features of the cognitive
dissonance phenomenon and predicts new scenarios of its emer-
gence. Finally, the so called boomerang (backfire) effect can emerge as
an extreme form of confirmation bias. The effect is given a
straightforward mathematical description in qualitative agreement
with experiments.
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The rest of this paper is organized as follows. In the next section
we introduce the problem setup and provide a brief survey of
relevant work, specifically focusing on inadequacy of the standard
Bayesian approach to opinion revision under persuasion. In the
third section we define our axioms and introduce the confirma-
tionally biased opinion revision rule. The fourth section relates our
setup to the social judgment theory. Next two sections describe
how our model accounts for two basic phenomena of experimental
social psychology: opinion change versus discrepancy and the
order of presentation effect. The seventh section shows how our
model formalizes features of cognitive dissonance, followed by
analysis of opinion change under repeated persuasion. Then we
study the boomerang effect–the agent changes his opinion not towards
the persuasion, but against it– as a particular case of our approach.
We summarize and conclude in the last section.
The Set-Up and Previous Research
Consider two agents Pand Q. They are given an uncertain
quantity (random variable) Xwith values k~1,:::,N, e.g.
k~rain,norainðÞ, if this is a weather forecast. Xconstitutes the
state of the world for Pand Q. The opinions of the agents are
quantified via probabilities
p~pk
fg
N
k~1and q~qk
fg
N
k~1,X
N
k~1
pk~X
N
k~1
qk~1ð1Þ
for Pand Qrespectively.
Let us now assume that Pis persuaded (or advised) by Q.
(Persuasion and advising are not completely equivalent [71].
However, in the context of our discussion it will be useful to
employ both terms simultaneously stressing their commmon
aspects.) Throughout this paper we assume that the state of the
world does not change, and that the agents are aware of this fact.
Hence, Pis going to change his opinion only under influence of
the opinion of Q, and not due to any additional knowledge about
X(For more details on this point see [3,41] and the second section
of File S1.)
The normative standard for opinion revision is related to the
Bayesian approach. Below we discuss the main elements of the
Bayesian approach, and outline certain limitations that motivates
the non-Bayesian revision rule suggested in this work.
Within the Bayesian approach, the agent Ptreats his own
probabilistic opinion p~fpkgN
k~1as a prior, and the probabilistic
opinion q~fqkgN
k~1of Qas an evidence [28,30,47]. Next, it is
assumed that Pis endowed with conditional probability densities
P(qDk), which statistically relate qto the world state k. Upon
receiving the evidence from Q, agent Pmodifies his opinion from
p
k
to p(kDq)via the Bayes rule:
p(kDq)~P(qDk)pk=X
N
l~1
P(qDl)pl:ð2Þ
One issue with the Bayesian approach is that the assumption on
the existence and availability of P(qDk)may be too strong
[13,25,30]. Another issue is that existing empirical evidence
suggests that people do not behave according to the Bayesian
approach [13,61], e.g. they demonstrate the order of presentation
effect, which is generally absent within the Bayesian framework.
In the context of persuasion, the Bayesian approach (2) has two
additional (and more serious) drawbacks. To explain the first
drawback, let us make a generic assumption that there is a unique
index ^
kk for which P(qDk)is maximized as a function of k(for a
given q): P(qD^
kk)wP(qDk)for ^
kk 6~k.
Now consider repeated application of (2), which corresponds to
the usual practice of repeated persuasion under the same opinion q
of Q. The opinion of the agent then tends to be completely
polarized, i.e. p^
kk?1and pk?0for k6~^
kk. In the context of
persuasion or advising, we would rather expect that under
repeated persuasion the opinion of Pwill converge to that of Q.
The second issue is that, according to (2), Pwill change his
opinion even if he has the same opinion as Q:p=q. This feature
may not be realistic: we do not expect Pto change his opinion, if
he is persuaded towards the same opinion he has already. This
drawback of (2) was noted in [28]. (Ref. [28] offers a modification
of the Bayesian approach that complies with this point, as shown
in [28] on one particular example. However, that modification
betrays the spirit of the normative Bayesianism, because it makes
conditional probabilities depending on the prior probability.)
It is worthwhile to note that researchers have studied several
aspects of confirmation bias by looking at certain deviations from
the Bayes rule, e.g. when the conditional probability are available,
but the agent does not apply the proper Bayes rule deviating from
it in certain aspects [31,51,57,73]. One example of this is when the
(functional) form of the conditional probability is changed
depending on the evidence received or on the prior probabilities.
Another example is when the agent does not employ the full
available evidence and selects only the evidence that can
potentially confirm his prior expectations [39,48,67]. More
generally, one has to differentiate between two aspects of the
confirmation bias that can be displayed either with respect to
information acquiring, or information assimilation (or both) [52].
Our study will concentrate on information assimilation aspect;
first, because this aspect is not studied sufficiently well, and second,
because because it seems to be more directly linked to cognitive
limitations [52]. We also stress that we focus on the belief revision,
and not on actions an agent might perform based on those beliefs.
Opinion Revision Rule
We propose the following conditions that the opinion revision
rule should satisfy.
1. The revised opinion ~
ppkof Pis represented as
~
ppk~F½pk,qk=XN
l~1F½pl,ql,ð3Þ
where F½x1,x2is defined over x1[½0,?)and x2[½0,?).We
enlarged the natural range x1[½0,1and x2[½0,1, since below we
plan to consider probabilities that are not necessarily normalized
to 1. There are at least two reasons for doing so: First,
experimental studies of opinion elicitation and revision use more
general normalizations [8,9]. For example, if the probability is
elicited in percents, the overall normalization is 100. Second, and
more importantly, the axioms defining subjective (or logical)
probabilities leave the overall normalization as a free parameter
[22].
We require that F½x1,x2is continuous for x1[½0,?)and
x2[½0,?)and infinitely differentiable for x1[(0,?)and x2[(0,?).
Such (or similar) conditions are needed for features that are
established for certain limiting values of the arguments of F(cf. (5,
6)) to hold approximately whenever the arguments are close to
those limiting values. Fcan also depend on model parameters, as
seen below.
Eq. (3) means that Pfirst evaluates the (non-normalized) weight
F½pk,qkfor the event kbased solely on the values of p
k
and q
k
, and
then applies overall normalization. A related feature of (3) is that it
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is local: assume that N§3and only the probability q
1
is
communicated by Qto P. This suffices for Pto revise his
probability from p
1
to ~
pp1, and then adjust other probabilities via
renormalization:
~
pp1~F½p1,q1=(F½p1,q1z1{p1),
e
pk
pk~pk=Fp
1,q1:
½z1{p1
ðÞk§2:ð4Þ
Eq. (3) can be considered as a succession of such local processes.
2. If pk~0for some k, then ~
ppk~0:
F½0,y~0:ð5Þ
The rationale of this condition is that if Psets the probability of
a certain event strictly to zero, then he sees logical (or factual)
reasons for prohibiting the occurrence of this event. Hence Pis
not going to change this zero probability under persuasion.
3. If pkqk~0for all k, then ~
ppk~pk:Pcannot be persuaded by
Qif their opinions have no overlap.
4. If Q’s and P’s opinions are identical, then the latter will not
change his opinion: fpkgN
k~1~fqkgN
k~1(for all k) leads to
f~
ppkgN
k~1~fpkgN
k~1. This can be written as
F½x,x~x:ð6Þ
Conditions 3and 4are motivated by experimental results in
social psychology, which state that people are not persuaded by
opinions that are either very far, or very close to their initial
opinion [10,17,69].
(Recall that we do not allow the uncertain quantity Xto change
during the persuasion or advising. If such a change is allowed, 4
may not be natural as the following example shows. Assume that P
holds a probabilistic opinion (0:1,0:9) on a binary X. Let Plearns
that Xchanged, but he does not know in which specific way it did.
Now Pmeets Qwho has the same opinion (0:1,0:9). Provided
that Qdoes not echo the opinion of P, the agent Pshould perhaps
change his opinion by decreasing the first probability (0.1) towards
a smaller value, because it is likely that Xchanged in that
direction.)
5. Fis a homogeneous function of order one:
Fcx,cy½~cFx,y½for c§0:ð7Þ
The rationale for this condition comes from the fact that
(depending on the experimental situation) the subjective proba-
bility may be expressed not in normalization one (i.e. not with
XN
k~1pk~XN
k~1qk~1), but with a different overall normal-
ization (e.g. XN
k~1pk~XN
k~1qk~c) [8,9,22]; cf. 1. In this
light, (7) simply states that any choice of the overall normalization
is consistent with the sought rule provided that it is the same for P
and Q. Any rescaling of the overall normalization by the factor c
will rescale the non-normalized probability by the same factor c;
cf. (7).
6. Now we assume that the opinion assimilation by Pconsists of
two sub-processes. Both are related to heuristics of human
judgement.
6.1 Pcombines his opinion linearly with the opinion of Q
[8,9,18,29,30]:
^
ppk~Epkz(1{E)qk,0ƒEƒ1, ð8Þ
where Eis a weight. Several mathematical interpretations of the
weight Ewere given in statistics, where (8) emerged as one of the
basic rules of probabilistic opinion combination [16,29]; see
section I of File S1. One interpretation suggested by this approach
is that Eand 1{Eare the probabilities–from the subjective
viewpoint of P–for, respectively, pand qto be the true description
of states of the world [29]: it is not known to Pwhich one of these
probabilities (por q) conveys a more accurate reflection of the
world state. Then f^
ppkgN
k~1is just the marginal probability for the
states of the world. There is also an alternative (normative) way of
deriving (8) from maximization of an average utility that under
certain natural assumptions can be shown to be the (negated)
average information loss [16]; see section I of File S1 for more
details.
Several qualitative factors contribute to the subjective assess-
ment of E. For instance, one interpretation is to relate Eto
credibility of Q(as perceived by P): more credible Qleads to a
larger 1{E[18]. Several other factors might affect E: egocentric
attitude of Pthat tends to discount opinions, simply because they
do not belong to him; or the fact that Phas access to internal
reasons for choosing his opinion, while he is not aware of the
internal reasons of Qetc [18]. Taking into account various factors
that contribute to the interpretation of E, we will treat it as a free
model parameter.
6.2 Note that (8) does not satisfy conditions 2and 3above. We
turn to the last ingredient of the sought rule, which, in particular,
should achieve consistency with conditions 2and 3.
Toward this goal, we assume that Pprojects the linearly
combined opinion ^
pp (see (8)) onto his original opinion p. Owing to
(3), we write this transformation as
~
ppk~w½pk,^
ppk=XN
l~1w½pl,^
ppl,ð9Þ
where the function wis to be determined.
The above projection operation relates to trimming [18,72], a
human cognitive heuristics, where Ptends to neglect those aspects
of Q’s opinion that deviate from a certain reference. In the
simplest case this reference will be the existing opinion of P.
To make the projection process (more) objective, we shall
assume that it commutes with the probabilistic revision: whenever
p’
k~ckpk
PN
l~1clpl
,b
pk
pk
’~ckb
pk
pk
PN
l~1clb
pl
pl
,1ƒkƒN,ð10Þ
where ck~Pr(:::Dk)w0are certain conditional probabilities, ~
pp is
revised via the same rule (10):
~
pp’
k:w½p’k,^
pp’
k
PN
l~1w½p’
l,^
pp’
l~ckw½pk,^
ppk
PN
l~1clw½pl,^
ppl:ð11Þ
This feature means that the projection is consistent with
probability theory: it does not matter whether (3) is applied before
or after (10).
It is known that (9) together with (10, 11) selects a unique
function [30]:
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wpk,k
½ ~pm
k
1{m
k,0vmƒ1ð12Þ
where mquantifies the projection strength: for m~1the projection
is so strong that Pdoes not change his opinion at all
(conservatism), while for m?0,Pfully accepts ^
pp (provided that
pkw0for all k). (The above commutativity is formally valid also for
mƒ0or mw1, but both these cases are in conflict with (5).) In
particular, E?0and m?0is a limiting case of a fully credulous
agent that blindly follows persuasion provided that all his
probabilities are non-zero. (For a sufficiently small m, a small p
k
is less effective in decreasing the final probability ~
ppk; see (12). This
is because pm
k~emln pktends to zero for a fixed mand pk?0, while
it tends to one for a fixed p
k
and m?0. This interpaly between
pk?0and m?0is not unnatural, since the initial opinion of a
credulous agent is expected to be less relevant. The case of
credulous agent is of an intrinsic interest and it does warrant
further studies. However, since our main focus is confirmation
bias, below we set m~1=2and analyze the opinion dynamics for
varying E.)
The final opinion revision rule reads from (12, 8, 9):
~
ppk~ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pk½Epkz(1{E)qk
p
PN
l~1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pl½Eplz(1{E)ql
p,0ƒEƒ1:ð13Þ
It is seen to satisfy conditions 1–5.
(Note that the analogue of (11), p’
k!ckpk,q’
k!ckqkdoes not
leave invariant the linear function (8). First averaging,
Epkz(1{E)qkand then applying p’
k!ckpk,q’
k!ckqkis equiva-
lent to first applying the latter rules and then averaging with a
different weight E’. This is natural: once Ecan be (in principle)
interpreted as a probability it should also change under
probabilistic revision process.)
The two processes were applied above in the specific order: first
averaging (8), and then projection (9). We do not have any strong
objective justifications for this order, although certain experiments
on advising indicate on the order that led to (13) [72]. Thus, it is
not excluded that the two sub-processes can be applied in the
reverse order: first projection and then averaging. Then instead of
(13) we get (3) with:
Fp
k,qk;E,m½~Epkz1{EðÞpm
kq1{m
k,0vmv1:ð14Þ
Our analysis indicates that both revision rules (13) and (14)
(taken with m~1=2) produce qualitatively similar results. Hence,
we focus on (13) for the remainder of this paper.
Returning to (1), we note that k=xcan be a continuous
variable, if (for example) the forecast concerns the chance of
having rain or the amount of rain. Then the respective probability
densities are:
pxðÞand qxðÞ,ðdxpxðÞ~ðdxq xðÞ~1:ð15Þ
Since the revision rule (13) is continuous and differentiable (in
the sense defined after (3)), it supports a smooth transition between
discrete probabilities and continuous and differentiable probability
densities. In particular, (13) can be written directly for densities: for
pk^p(xk)dxwe obtain from (13)
~
pp(x)~ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p(x)½Ep(x)z(1{E)q(x)
p
Ðdx’ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p(x’)½Ep(x’)z(1{E)q(x’)
p:ð16Þ
Social Judgment Theory and Gaussian Opinions
Opinion latitudes
Here we discuss our model in the context of the social judgment
theory [59,10], and consider several basic scenarios of opinion
change under the rule (16).
According to the social judgment theory, an agent who is
exposed to persuasion perceives and evaluates the presented
information by comparing it with his existing attitudes (opinions).
The theory further postulates that an attitude is composed of three
zones, or latitudes: acceptance,non-commitment and rejection [10,59].
The opinion that is most acceptable to P, or the anchor, is located
at the center of the latitude of acceptance. The theory states that
persuasion does not change the opinion much, if the persuasive
message is either very close to the anchor or falls within the
latitude of rejection [10,59]. The social judgment theory is
popular, but its quantitative modeling has been rather scarce. In
particular, to the best of our knowledge, there has been no attempt
to develop a consistent probabilistic framework for the theory.
(The literature on the social judgment theory offers some formal
mathematical expressions that could be fitted to experimental data
[45]. There is also a more quantitative theory [34] whose content
is briefly reminded in section III of File S1.)
Let us assume that k=xis a continuous variable (cf. (15)) and
that p(x) and q(x) are Gaussian with mean mland dispersion vl
(l~P,Q):
p(x)~e{(x{mP)2
2vP
ffiffiffiffiffiffiffiffiffiffi
2pvP
p,q(x)~e
{
(x{mQ)2
2vQ
ffiffiffiffiffiffiffiffiffiffiffi
2pvQ
p:ð17Þ
Effectively, Gaussian probabilistic opinions are produced in
experiments, when the subjects are asked to generate an opinion
with &95% confidence in a certain interval [18]. Now we can
identify the anchor with the most probable opinion ml, while v{1
l
quantifies the opinion uncertainty.
The latitude of acceptance amounts to opinions not far from the
anchor, while the latitude of rejection contains close-to-zero
probability events, since Pdoes not change his opinion on them;
recall point 2from the previous section. One can also identify the
three latitudes with appropriately chosen zones in the distribution.
For instance, it is plausible to define the latitudes of acceptance
and rejection by, respectively, the following formulas of the 3srule
known in statistics
x[½mP{2ffiffiffiffiffi
vP
p,mlz2ffiffiffiffiffi
vP
p,ð18Þ
x[({?,mP{3ffiffiffiffiffi
vP
p|½mPz3ffiffiffiffiffi
vP
p,?), ð19Þ
where the latitude of non-commitment contains whatever is left
out from (18, 19). Recall that the latitudes of acceptance, non-
commitment and rejection carry (respectively) 95.4, 4.3 and 0.3%
of probability.
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b
p
b
p

While the definitions (18, 19) are to some extent arbitrary, they
work well with the rule (16), e.g. if the opinions of Pand Qoverlap
only within their rejection latitudes, then neither of them can
effectively change the opinion of another. Also, Pis persuaded
most strongly, if the anchor of the persuasion falls into the non-
commitment latitude of P. This is seen below when studying
change-discrepancy relations.
Weighted average of anchors
Next, we demonstrate that the main quantitative theory of
persuasion and opinion change–the weighted average approach
[9,27] –is a particular case of our model. We assume that the
opinions p(x) and q(x) are given as
p(x)~f(x{mP), p(x)~g(x{mQ), ð20Þ
f’(0):df=dxDx~0~g’(0)~0,
f’’(0)v0, g’’(0)v0, ð21Þ
where both f(x) and g(x) have a unique maximum at x= 0. Hence
p(x) (resp. q(x)) has a single anchor (maximally probable opinion)
mP(resp. mQ); see (17) for concrete examples.
If DmP{mQDis sufficiently small, ~
pp(x)given by (20, 16) has a
single anchor which is shifted towards that of q(x); see Fig. 1(a). We
now look for the maximum me
PP of ~
pp(x)by using (20) in (16). We
neglect factors of order O½(mP{mQ)2=vPand
O½(mP{mQ)2=vQand deduce:
me
PP~(1{aQ)mPzaQmQ,ð22Þ
aQ:
(1{E)Dg’’(0)D
g(0)
(1{E)Dg’’(0)D
g(0) zDf’’(0)D
f(0) 2Ef(0)
g(0) z1{E

:ð23Þ
Eq. (22) is the main postulate of the weighted average approach;
see [9,27] for reviews. Here aQand 1{aQare the weights of Q
and of P, respectively. For the Gaussian case (17), we have
aQ~1zy1z2Effiffiffi
y
p
1{E

{1
,y:vQ
vP
:ð24Þ
Furthermore, we have
LaQ
Ly
~{a2
Q1z3Effiffiffi
y
p
1{E

,LaQ
LE
~{
2y3=2a2
Q
1{EðÞ
,ð25Þ
Thus, aQ’s dependence on the involved parameters is intuitively
correct: it increases with the confidence 1=vQof Q, and decreases
with the confidence 1=vPof P. Note also that aQdecreases with E.
Now let p(x) and q(x) (and hence ~
pp(x)) have the same maximum
mP~mQ, but vP&vQ; see (17). Expanding (16, 17) over vP{vQ
and keeping the first-order term only we get
Figure 1. Opinions described via Gaussian densities (17). The
initial opinion of Pis described by Gaussian probability density p(x)
(blue curve) centered at zero; see (17). The opinion of Qamounts to
Gaussian probability density q(x) (purple curve) centered at a positive
value. For all three figures continuous density f(x)(f~p,q,~
pp) were
approximated by 100 points ff(xk)g100
k~1,xkz1{xk~0:1. The resulting
opinion ~
pp(x)of Pis given by (16) with E~0:5(olive curve). (a) The
opinion of Pmoves towards that of Q;mP~0,sP~1,mQ~1,sQ~0:5.
(b) The maximally probable opinion of Pis reinforced; mP~0,sP~1,
mQ~0,sQ~0:25. (c) The change of the opinion of Pis relatively small
provided that the Gaussian densities overlap only in the region of non-
commitment; cf. (18), (19). Whenever the densities overlap only within
the rejection range the difference between p(x) and ~
pp(x)is not visible
by eyes. For example, if p(x) and q(x) are Gaussian with, respectively,
mP~{3,mQ~3,vP~vQ~1, the Hellinger distance (see (30) for
definition) h½p,q~0:99 is close to maximally far, while the opinion
change is small: h½p,~
pp~3:48|10{2.
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2

ve
PP~1{E
2vQz1zE
2vP,ð26Þ
where ve
PP is the dispersion of (non-Gaussian) ~
pp(x). Eq. (26) implies
(ve
PP{vP)(vQ{vP)~1{E
2(vQ{vP)2§0, ð27Þ
i.e. if 1=vQw1=vP(resp. 1=vQv1=vP), the final opinion of P
becomes more (resp. less) narrow than his initial opinion. Fig. 1(b)
shows that (ve
PP{vP)(vQ{vP)§0holds more generally.
Thus, the weighted average approach is a particular case of our
model, where the agent Pis persuaded by a slightly different
opinion. Note also that our model suggests a parameter structure
of the weighted average approach.
Opinions and bump-densities
Gaussian densities (with three latitudes) do correspond to the
phenomenology of social psychology. However, in certain
scenarios one might need other forms of densities, e.g., when the
probability is strictly zero outside of a finite support. Such opinions
can be represented by bump-functions
Xx;bðÞ~bðÞexp b
x2{1

for DxDv1ð28Þ
~0forxƒ{1 and x§1:
where bw0is a parameter, N(b)is the normalization and the
support of the bump function was chosen to be ½{1,1for
concretness. The advantage of the bump function that is infinitely
differentiable despite of having a finite support.
For sufficiently large b,x(x;b)is close to a Gaussian, while for
small b,x(x;b)represents an opinion that is (nearly) homogeneous
on the interval ½{1,1; see Fig. 2. The opinion revision with bump
densities follows to the general intuition of rule (16); see Fig. 2.
Opinion Change vs Discrepancy
One of extensively studied questions in social psychology is how
the opinion change is related to the discrepancy between the initial
opinion and the position conveyed by the persuasive message
[10,35,40,45,69]. Initial studies suggested a linear relationship
between discrepancy and the opinion change [35], which agreed
with the prediction of the weighted average model. Indeed, (22)
yields the following linear relationship between the change in the
anchor and the initial opinion discrepancy of Pand Q:
me
PP{mP~aQ(mQ{mP):ð29Þ
However, consequent experiments revealed that the linear
regime is restricted to small discrepancies only and that the actual
behavior of the opinion change as a function of the discrepancy is
non-monotonic: the opinion change reaches its maximal value at
some discrepancy and decreases afterward [10,40,45,69].
To address this issue within our model, we need to define
distance h½p,qbetween two probability densities p(x) and q(x).
Several such distances are known and standardly employed [32].
Here we select the Hellinger distance (metric)
h½p,q:1ffiffiffi
2
pðdx½ffiffiffiffiffiffiffiffiffi
p(x)
p{ffiffiffiffiffiffiffiffiffi
q(x)
p2

1=2
,ð30Þ
~1{ðdxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p(x)q(x)
p

1=2
:ð31Þ
Since ffiffiffiffiffiffiffiffiffi
p(x)
pis a unit vector in the ‘2norm, Eq. (30) relates to
the Euclidean (‘2-norm) distance. It is applicable to discrete
probabilities by changing the integral in (30, 31) to sum. For
Gaussian opinions (17) we obtain
h½p,q~1{(vQvP)1=2
vQzvP
2
2
6
43
7
5
1=2
e
{
(mQ{mP)2
4(vQzvP)
2
6
43
7
5
1=2
:ð32Þ
A virtue of the Hellinger distance is that it is a measure of
overlap between the two densities; see (31). We stress, however,
that there are other well-known distances measures in statistics
[32]. All results obtained below via the Hellinger distance will be
checked with one additional metric, the total variation (‘1-norm
distance):
d½p,q~1
2ðdxDp(x){q(x)D:ð33Þ
(To motivate the choice of (33), let us recall two important
variational features of this distance [32]: (1) d½p,q~
maxV[R1DÐVdx(p(x){q(x)) D. (2) Define two (generally dependent)
random variables X,Ywith joint probability density g(x,y)such
that Ðdxg(x,y)~q(y),Ðdyg(x,y)~p(x). Now it holds that
d½p,q~min Pr(X6~Y)½, where Pr(X6~Y)~1{Ðdxg(x,x),
and the minimization is taken over all g(x,y)with fixed marginals
equal to p(x) and q(y), respectively.)
Figure 2. Opinions described via bump densities (28). Blue curve:
the initial opinion of Pgiven by (28) with b= 1. Purple curve: the
opinion of Qdescribed by (28) with b~0:001. Olive curve: the resulting
opinion of Pobtained via (16) with E~0:5.
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N

The opinion change is characterized by the Hellinger distance
h½p,~
ppbetween the initial and final opinion of P, while the
discrepancy is quantified by the Hellinger distance h½p,qbetween
the initial opinion of Pand the persuading opinion. For
concreteness we assume that the opinion strengths 1=vPand
1=vQare fixed. Then h½p,qreduces to the distance m~DmP{mQD
between the anchors (peaks of p(x) and q(x)); see (32).
Fig. 3(a) shows that the change h½p,~
ppis maximal at m~mch;it
decreases for mwmch, since the densities of Pand Qhave a
smaller overlap. The same behavior is shown by the total variation
d½p,~
ppthat maximizes at m~mcd; see Fig. 3(a).
The dependence of mch (and of mcd)onEis also non-
monotonic; Fig. 3(b). This is a new prediction of the model. Also,
mch and mcdare located within the latitude of non-commitment of
P(this statement does not apply to mch, when Eis close to 1 or 0);
cf. (18, 19). This point agrees with experiments [10,69].
Note that experiments in social psychology are typically carried
out by asking the subjects to express one preferred opinion under
given experimental conditions [10,35,40,45,69]. It is this single
opinion that is supposed to change under persuasion. It seems
reasonable to relate this single opinion to the maximally probable
one (anchor) in the probabilistic set-up. Thus, in addition to
calculating distances, we show in Fig. 3(c) how the final anchor me
PP
of Pdeviates from his initial anchor mP.
Fig. 3(c) shows that for Ew0:25, the behavior of Dm~Dme
PP{mPD
as a function of m~DmP{mQDhas an inverted-U shape, as
expected. It is seen that Dmsaturates to zero much faster
compared to the distance h½p,~
pp. In other words, the full
probability ~
pp keeps changing even when the anchor does not
show any change; cf. Fig. 3(c) with Fig. 3(a).
A curious phenomenon occurs for a sufficiently small E; see
Fig. 3(c) with E~0:1. Here Dmdrops suddenly to a small value
when mpasses certain crticial point; Fig. 3(c). The mechanism
behind this sudden change is as follows: when the main peak of p(x)
shifts towards mQ, a second, sub-dominant peak of ~
pp(x)appears at
a value smaller than mP. This second peak grows with mand at
some critical value it overcomes the first peak, leading to a
bistability region and an abrupt change of Dm. The latter arises
due to a subtle interplay between the high credibility of Q(as
expressed by a relatively small value of E) and sufficiently large
discrepancy between Pand Q(as expressed by a relatively large
value of m). Recall, however, that the distance h½p,~
ppcalculated via
the full probability does not show any abrupt change.
The abrupt change of Dmis widely discussed (and experimen-
tally confirmed) in the attitude change literature; see [49] for a
recent review. There the control variables for the attitude change–
information and involvement [49]–differ from Eand m. However,
one notes that the weight Ecan be related to the involvement:
more Pis involved into his existing attitude, larger is E, while the
discrepancy mconnects to the (new) information contained in the
persuasion (m= 0 naturally means zero information).
Let us finally consider a scenario where the change-discrepancy
relationship is monotonic. It is realized for mP~mQ(coinciding
anchors), where the distance (32) between p(x) and q(x) is controlled
by vQ(for a fixed vP). In this case, vthe change h½p,~
ppis a
Figure 3. Opinion change versus discrepancy. (a) The opinion
change is quantified via the Hellinger distance h~h½p,~
ppbetween the
old and new opinion of P(blue curves); see (30) for the definition. For
comparison we also include the total variance distance d~d½p,~
pp
(purple curves); see (33). These two distances are plotted versus the
discrepancy m~DmP{mQD. The initial opinion of the agent Pis
Gaussian with mP~0and vP~1; see (17). The opinion of Qis Gaussian
with mQ~mand vQ~1. Thus mquantifies the initial distance between
the opinions of Pand Q. The final opinion ~
pp(x)is given by (13).
Different curves correspond to different E. Blue curves: h(m)~h½p,~
ppfor
E~0:1(upper curve) and E~0:5(lower curve). Purple curves:
d(m)~d½p,~
ppfor E~0:1(upper curve) and E~0:5(lower curve). The
maximum of h(m)(d(m)) is reached at mch (mcd). (b) mch (mcd) is the
point where h(m)(d(m)) achieves its maximum as a function of m. Blues
points: mch(E)versus Efor same parameters as in (a). mch(E)grows both
for E?1and E?0, e.g. mch(0:01)~3:29972,mch(0:0001)~4:53052,
mch(0:9)~2:94933,mch(0:999)~4:12861. Purple points: mcd(E)versus E
for same parameters as in (a). (c) The difference of the anchors
(maximally probable values) Dm~me
PP{mPversus mQ~mfor the initial
opinions of Pand Qgiven by (17) under mP~0,vP~1,mQ~mand
vQ~1. The final opinion ~
pp(x)of P(and its maximally probable value
me
PP) if found from (13) under E~0:1(black points), E~0:25 (blue points)
and E~0:5(red points).
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monotonic function of discrepancy h½p,q: a larger discrepancy
produces larger change. This example is interesting, but we are
not aware of experiments that have studied the change-discrep-
ancy relation in the case of two identical anchors.
Order of Presentation
Recency versus primacy
When an agent is consecutively presented with two persuasive
opinions, his final opinion is sensitive to the order of presentation
[10,13,25,34,35,50,52]. While the existence of this effect is largely
established, its direction is a more convoluted matter. (Note that
the order of presentation effect is not predicted by the Bayesian
approach; see (2).) Some studies suggest that the first opinion
matters more (primacy effect), whereas other studies advocate that
the last interaction is more important (recency effect). While it is
not completely clear which experimentally (un)controlled factors
are responsible for primacy and recency, there is a widespread
tendency of relating the primacy effect to confirmation bias
[13,52]. This relation involves a qualitative argument that we
scrutinize below.
We now define the order of presentation effect in our situation.
The agent Pinteracts first with Q(with probability density q(x)),
then with Q’with probability density q’(x). To ensure that we
compare only the order of Qand Q’and not different magnitudes
of influences coming from them, we take both interactions to have
the same parameter 0vEv1. Moreover, we make Qand Q’
symmetric with respect to each other and with respect to P, e.g. if
p(x), q(x) and q’(x)are given by (17) we assume
v’~v,m’{m~m{m:ð34Þ
We would like to know whether the final opinion p(xDq,q’)of P
is closer to q(x) (primacy) or to q’(x)(recency).
In the present model (and for 0ƒEv1), the final opinion
p(xDq,q’)is always closer to the last opinion q’(x), both in terms of
maximally probable value and distance. In other words, the model
unequivocally predicts the recency effect. In terms of the Hellinger
distance (30)
h½p(xDq,q’),q’vh½p(xDq,q’),q:ð35Þ
See Fig. 4 for an example (In our model primacy effect exists in
the boomerang regime Ew1; see below.)
To illustrate (35) analytically on a specific example, consider the
following (binary) probabilistic opinion of P,Qand Q’
p~1=2,1=2ðÞ,q~0,1ðÞ,q’~1,0ðÞ:ð36Þ
Pis completely ignorant about the value of the binary variable,
while Qand Q’are fully convinced in their opposite beliefs. If P
interacts first with Qand then with Q’(both interactions are given
by (13) with E~1
2), the opinion of Pbecomes (0:52727, 0:47273).
This is closer to the last opinion (that of Q’).
The predicted recency effect in our model seems rather
counterintuitive. Indeed, since the first interaction shifts the
opinion of Ptowards that of Q, one would think that the second
interaction with Q’should influences P’s opinion less, due to a
smaller overlap between the opinions of Q’and Pbefore the
second interaction. In fact, this is the standard argument that
relates primacy effect to the confirmation bias [13,52]: the first
interaction shapes the opinion of Pand makes him confirmation-
ally biased against the second opinion. This argument does not
apply to the present model due to the following reason: even
though the first interaction shifts P’s anchor towards Q’s opinion,
it also deforms the shape of the opinion; see Fig. 1(a). And the
deformation produced by our revision rule happens to favor the
second interaction more.
To get a deeper understanding of the recency effect, let us
expand (13) for small g:1{E:
~
ppk~pkzg
2(qk{pk)
zg2
8(pk{1) X
l
(ql{pl)2
pl
zO½g3:ð37Þ
If now Pinteracts with an agent Q’having opinion q’, the
resulting opinion p(q,q’)reads from (37):
pk(q,q’)~pk
zg
2(qk{pk)zg2
8(pk{1) X
l
(ql{pl)2
pl
zg
2(q’
k{pk)zg2
8(pk{1) X
l
(q’l{pl)2
pl
Figure 4. Order of presentation effect. Blue curve: The initial
opinion of Pis described by Gaussian probability density p(x) with
mP~0and vP~1; see (17). Purple (resp. olive) curve: the initial opinion
of Q(resp. Q’) are given by (17) with mQ~1:5(resp. mQ’~{1:5) and
vQ~0:5(resp. vQ’~0:5). Green curve: the resulting opinion of Pafter
interacting first with Qand then with Q’. Both interactions use E~0:5.
The final opinion of Pis inclined to the most recent opinion (that of Q’)
both with respect to its maximally probable value and distance. The
final opinion of Phas a larger width than the initial one.
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Q
P
QQ
P
Q

zg2
4(pk{qk)zO½g3:ð38Þ
Hence in this limit pk(q,q’){pk(q’,q)depends only on q’k{qk
(and not e.g. on ql6~k):
pk(q,q’){pk(q’,q)~g2½q’
k{qk=4zO½g3:ð39Þ
It is seen that the more probable persuasive opinion (e.g. the
opinion of Q’if q’
kwqk) changes the opinion of Pif it comes later.
This implies the recency effect. Indeed, due to symmetry
conditions for checking the order of presentation effect we can
also look at h½p(q,q’),q{h½p(q’,q),q. Using (39) we get for this
quantity: g2
16h½p(q’,q),qX
k½qk{q’
kffiffiffiffiffiffiffiffiffiffiffiffi
qk=pk
pw0, again due to
symmetry conditions.
Note that this argument on recency directly extends to more
general situations, where the agent is exposed to different opinions
multiple times. For instance, consider an exposure sequence
qqq’q’and its reverse q’q’qq. It can be shown that the model
predicts a recency effect in this scenario as well. For this case, we
get instead of (39): pk(q,q’){pk(q’,q)~g2½q’
k{qkzO½g3.
Note that the primacy-recency effect is only one (though
important!) instance of contextual and non-commutative phenom-
ena in psychology; see [11,66] and references therein. Hence in
section IV of File S1 we study a related (though somewhat less
interesting) order of presentation effect, while below we discuss our
findings in the context of experimental results.
Experimental studies of order of presentation effect
We now discuss our findings in this section in the context of
experimental results on primacy and recency. The latter can be
roughly divided into several group: persuasion tasks [10,50],
symbol recalling [70], inference tasks [34], and impression
formation [7,9]. In all those situations one generally observes
both primacy and recency, though in different proportions and
under different conditions [34]. Generally, the recency effect is
observed whenever the retention time (the time between the last
stimulus and the data taking) is short. If this time is sufficiently
long, however, the recency effect changes to the primacy effect
[10,50,62,70]. The general interpretation of these results is that
there are two different processes involved, which operate on
different time-scales. These processes can be conventionally
related to short-term and long-term memory [70], with the
primacy effect related to the long-term memory. In our model the
longer time process is absent. Hence, it is natural that we see only
the recency effect. The prevalence of recency effects is also seen in
inference tasks, where the analogue of the short retention time is
the incremental (step-by-step) opinion revision strategy [34].
At this point, let us remind the importance of symmetry
conditions [such as (34)] for observing a genuine order of
presentation effect. Indeed, several experimental studies–in
particular those on impression formation–suggest that the order
of presentation exists due to different conditions in the first versus
the second interaction [7,10,34,68,]. (In our context, this means
different parameters Eand E’for each interaction). For instance,
Refs. [7,10] argue that the primacy effect is frequently caused by
attention decrement (the first action/interaction gets more
attention); see also [68] in this context. This effect is trivially
described by our model, if we assume Eto be sufficiently smaller
than E’. In related experiments, it was shown that if the attention
devoted to two interactions is balanced, the recency effect results
[33], which is consistent with the prediction of our model.
At the same time, in another interesting study based on
subjective probability revision, where the authors had taken special
measures for minimizing the attention decrement, the results
indicated a primacy effect [55].
We close this section by underlining the advantages and
drawbacks of the present model concerning the primacy-recency
effect: the main advantage is that it demonstrates the recency effect
and shows that the well-known argument on relating confirmation
bias to primacy does not hold generally. The main drawback is
that the model does not involve processes that are supposedly
responsible for the experimentally observed interplay between
recency and primacy. In the concluding section we discuss possible
extensions of the model that can account for this interplay.
Cognitive Dissonance
Consider an agent whose opinion probability density has two
peaks on widely separated events. Such a density–with the most
probable opinion being different from the average–is indicative of
cognitive dissonance, where the agent believes in mutually
conflicting things [10,26].
The main qualitative scenario for the emergence of cognitive
dissonance is when an agent–who initially holds a probabilistic
opinion with a single peak–is exposed to a conflicting information
coming from a sufficiently credible source [10,26]. We now
describe this scenario quantitatively.
Consider again the opinion revision model (16, 17), and assume
that DmP{mQDis neither very large nor very small (in both these
cases no serious opinion change is expected), vQ=vPv1(self-
assured persuasive opinion) and 0vEv1. In this case, we get two
peaks (anchors) for the final density ~
pp(x). The first peak is very
close to the initial anchor of p(x), while the second closer to the
anchor of q(x); see Fig. 5(a). Thus, persuasion from Qwhose
opinion is sufficiently narrow and is centered sufficiently close (but
not too close) to P’s initial anchor leads to cognitive dissonance: P
holds simultaneously two different anchors, the old one and the
one induced by Q.
There are 3 options for reducing cognitive dissonance:
(i) Increase Emaking it closer to 1, i.e. making Qless credible;
see Fig. 5(b).
(ii) Decrease the width of the initial opinion of P.
(iii) Decrease Emaking Qmore credible. In this last case, the
second peak of ~
pp(x)(the one close to the anchor of Q) will be
dominant; see Fig. 5(c).
To understand the mechanism of the cognitive dissonance as
described by this model, let us start from (1) and assume for
simplicity that the opinion of Qis certain: qk~0for k6~land
ql~1. We get from (13):
e
pk
pk~pk
1{plzplffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ez1{EðÞp{1
l
qfor k=l,ð40Þ
~
ppl~
plffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ez(1{E)p{1
l
q
1{plzplffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ez(1{E)p{1
l
q:ð41Þ
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Now ~
ppl=plw1w~
ppk=pk, where k6~l; hence even if lwas on the
tail of fpkgN
k~1, it is possible to make it a local (or even the global)
maximum of f~
ppkgN
k~1provided that Eis not close to 1.
The existence of at least two widely different probable opinions
is only one aspect of cognitive dissonance [10,26]. Another aspect
(sometimes called Freud-Festinger’s law) is that people tend to
avoid cognitive dissonance: if in their action they choose one of the
two options (i.e. one of two peaks of the subjective probability),
they re-write the history of their opinion revision so that the
chosen option becomes the most probable one [10,26]. This aspect
of cognitive dissonance found applications in economics and
decision making [2,73]. The above points (i)–(iii) provide concrete
scenarios for a such re-writing.
Repeated Persuasion
Here we analyze the opinion dynamics under repeated
persuasion attempts. Our motivation for studying this problem is
that repeated exposure to the same opinion is generally believed to
be more persuasive than a single exposure.
Under certain conditions (pkqk=0, for all kand 1wEw0)we
show that the target opinion converges to the persuading opinion
after sufficient number of repetition. Below we also examine how
exactly this convergence takes place.
Assume that Previses his opinion repeatedly with the same
opinion of Q. Eq. (13) implies (1ƒkƒN)
p½nz1
k~ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p½n
k½Ep½n
kz(1{E)qk)
q
PN
l~1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p½n
l½Ep½n
lz(1{E)ql)
q,ð42Þ
where 1wEw0, and n~1,2,::: is the discrete time. For simplicity,
we assume
p1½
k:pkw0, qkw0 for 1ƒkƒN:ð43Þ
Eq. (42) admits only one fixed point q~fqkgN
k~1. Section VI of
File S1 shows that for any convex, d2f(y)
dy2§0, function f(y) one has
W½p½nz1;qƒW½p½n;q,ð44Þ
W½p;q:XN
k~1qkf(pk=qk):ð45Þ
Hence W½p;qis a Lyapunov function of (42). Since W½p;qis a
convex function of p,W½p;q§f(1)~W½q;qand f(1) is the unique
global minimum of W½p;q. Section VI of File S1 shows that the
equality sign in (45) holds ony for p½nz1~p½n. Thus W½p½n;q
monotonically decays to f(1)~W½q;qshowing that the fixed point
qis globally stable. More generally, the convergence reads:
p½n
k?f½p½1
kqk=XN
l~1f½p½1
lql, where f(xw0)~1and f(0)~0.
To illustrate (44, 45), one can take f(y)~{ ffiffiffi
y
p. Then (44)
amounts to decaying Hellinger distance (30). Many other
reasonable measures of distance are obtained under various
choices of f. For instance, f(y)~Dy{1Damounts to decaying total
variation distance (33), while f(y)~{ ln yleads to the decaying
relative entropy (Kullback-Leibler entropy).
As expected, 0vEv1influences the convergence time. We
checked that this time is an increasing function of E, as expected.
Figure 5. Cognitive dissonance. (a) Blue (resp. purple) curve: the
initial opinion of agent P(resp. Q) described by probability density p(x)
(resp. q(x)). Olive curve: the final opinion ~
pp(x)of Pas given by (16) with
E~0:35. Here p(x) and q(x) are defined by (17) with mP~0,vP~1,
mQ~2,vQ~0:1. The final opinion develops two peaks of comparable
height (cognitive dissonance). (b) Avoiding the cognitive dissonance
due to a larger E~0:75: the second peak is much smaller (other
parameters are those of (a)). (c) Avoiding the cognitive dissonance due
to a smaller E~0:05: the first peak is much smaller (other parameters
are those of (a)).
doi:10.1371/journal.pone.0099557.g005
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In section VI of File S1 we also show that the convergence to the
fixed point respects the Le Chatelier principle known in
thermodynamics [4]: the probabilities of those events that are
overestimated from the viewpoint of Q(i.e. p½1
kwqk) tend to decay
in the discrete time. Likewise, probabilities of the underestimated
events (i.e. p½1
kvqk) increase in time.
Let us consider the Hellinger distance hn~h½p½nz1,p½nbetween
two consecutive opinions of Pevolving as in (42). It is now possible
that
max1ƒnv?½hn~hm6~h1,ð46Þ
i.e. the largest change of the opinion of Pcomes not from the first,
but from one of intermediate persuasions. A simple example of this
situation is realized for N= 3, an initial probability vector
p~(0:98,0:01,0:01) and q~(0:01,0:01,0:98) in (43). We then
apply (42) under E~0:5. The consecutive Hellinger distances read
h1~0:1456vh2~0:1567wh3~0:1295wh4:::. Hence the second
persuasion changes the opinion more than others. For this to hold,
the initial opinion pof Phas to be far from the opinion qof Q.
Otherwise, we get a more expected behavior h1wh2wh3wh4:::
meaning that the first persuasion leads to the largest change.
(The message of (46) is confirmed by using the discrete version
d½p,q~1
2XkDpk{qkDof the distance (33). Define
dn~d½p½nz1,p½n. Then with p~(0:98,0:01,0:01) and
q~(0:01,0:01,0:98) we get d1~0:0834,d2~0:1636,
d3~0:1717,d4~0:1444.)
We conclude by stressing that while repeated persuasions drive
the opinion to its fixed point monotonically in the number of
repetitions, it is generally not true that the first persuasion causes
the largest opinion change, i.e. the law of diminishing returns does
not hold. To obtain the largest opinion change, one should
carefully choose the number of repetitions.
Finally, note that the framework of (42) can be applied to
studying mutual persuasion (consensus reaching). This is described
in Section VII of File S1; see also [23] in this context.
Boomerang (Backfire) Effect
Definition of the effect
The boomerang or backfire effect refers to the empirical observation
that sometimes persuasion yields the opposite effect: the persuaded
agent Pmoves his opinion away from the opinion of the
persuading agent, Q, i.e. he enforces his old opinion [53,58,64,69].
Early literature on social psychology proposed that the boomerang
effect may be due to persuading opinions placed in the latitude of
rejection [69], but this was not confirmed experimentally [40].
Experimental studies indicate that the boomerang effect is
frequently related with opinion formation in an affective state,
where there are emotional reasons for (not) changing the opinion.
For example, a clear evidence of the boomerang effect is observed
when the persuasion contains insulting language [1]. Another
interesting example is when the subjects had already announced
their opinion publicly, and were not only reluctant to change it (as
for the usual conservatism), but even enforced it on the light of the
contrary evidence [64] (in these experiments, the subjects who did
not make their opinion public behaved without the boomerang
effect). A similar situation is realized for voters who decided to
support a certain candidate. After hearing that the candidate is
criticized, the voters display a boomerang response to this criticism
and thereby increase their support [53,58].
Opinion revision rule
We now suggest a simple modification of our model that
accounts for the basic phenomenology of the boomerang effect.
Recall our discussion (around (8)) of various psychological and
social factors that can contribute into the weight E. In particular,
increasing the credibility of Qleads to a larger 1{E. Imagine now
that Qhas such a low credibility that
Ew1:ð47Þ
Recall that E~1means a special point, where no change of
opinion of Pis possible whatsoever; cf. (13).
After analytical continuation of (13) for Ew1, the opinion
revision rule reads
~
ppk~ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pkDEpkz(1{E)qkD
p
PN
l~1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
plDEplz(1{E)qlD
p,ð48Þ
with obvious generalization to probability densities. The
absolute values in (48) are necessary to ensure the positivity of
probabilities.
It is possible to derive (rather simply postulate) (48). Toward this
end, let us return to the point 6.1 and (8). During the opinion
combination step, Pforms Epkz(1{E)qkwhich in view of Ew1
can take negative values and hence is a signed measure. Signed
measures have all formal features of probability besides positivity
[6,14,19,65]; see section V of File S1 for details. There is no
generally accepted probabilistic interpretation of signed measures,
but in section V of File S1 we make a step towards such an
interpretaion. There we propose to look at a signed measure as a
partial expectation value defined via joint probability of the
world’s states and certain hidden degrees of freedom (e.g.
emotional states). After plausible assumptions, the marginal
probability of the world’s states is deduced to be
Figure 6. Opinion change in the boomerang regime. Blue (resp.
purple) curve: the initial opinion of agent P(resp. Q) described by
probability density p(x) (resp. q(x)). Olive curve: the final opinion ~
pp(x)of
Pgiven by (16) with E~2. Here p(x) and q(x) are given by (17) with
mP~0and vP~mQ~vQ~1. The anchor (maximally probable opinion)
of Pnot only moves away from the anchor of Q; but it is also enhanced:
the (biggest) peak of ~
pp(x)is larger than that of p(x). The second (smaller)
peak of ~
pp(x)arises because the initial probability of Plocated to the
right from the anchor mQof Q, moves away from mQ;~
pp(x)gets a local
minimum close to mQ.
doi:10.1371/journal.pone.0099557.g006
Opinion Dynamics with Confirmation Bias
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^
ppk~DEpkz(1{E)qkD=X
N
l~1
DEpkz(1{E)qkD,ð49Þ
We obtain (48) after applying (9, 10) to (49).
Scenarios of opinion change
According to (47, 48) those opinions of Pwhich are within the
overlap between pand q(i.e pkqk6&0) get their probability
decreased if pk=qk&(E{1)=Ev1, i.e. if the initial p
k
was already
smaller than q
k
. In this sense, Pmoves his opinion away from that
of Q. Hence for continuous densities p(x) and q(x) there will be a
point x
0
, where ~
pp(x0)is close to 0. This point is seen in Figs. 6 and
7.
Fig. 6 illustrates the shape of ~
pp(x)produced by (48) for initially
Gaussian opinions (17) of Pand Q. It is seen that P’s anchor
moves away from Q’s anchor, while the width of ~
pp(x)around the
anchor is more narrow than that of p(x); cf. with Fig. 4. To
illustrate these points analytically, we return to (29, 24, 24) that for
vP&vQand mP&mQpredict me
PP{mP~1{E
2(mQ{mP): for
Ew1,P’s anchor drifts away from Q’s anchor.
Likewise, whenever the two anchors are equal, mP~mQ,
inequality (27) is reversed in the boomerang regime (47).
Let us now consider the impact of the presentation order under
this settings. We saw that for 0vEv1the model predicts recency
effect. For 1
*
Ewe expect the recency effect is still effective as
implied by the argument (39). However, the situation changes
drastically for Esufficiently larger than 1, as indicated in Fig. 7.
Now the primacy effect dominates, i.e. instead of (35) we get the
opposite inequality. Fig. 7 also shows that interaction with two
contradicting opinions (in the boomerang regime) enforces the
initial anchor of P.
To understand the primacy-recency effect analytically, consider
the example (36), and recall that Pinteracts first with Qand then
with Q’with the same parameter E. The resulting opinion p(q,q’)
of Preads:
p(q,q’)~g(E)
g(E)z1,1
g(E)z1

,ð50Þ
g(E)~ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiE
pz(1{E)ffiffiffiffiffiffiffiffiffiffiffiffi
D2{ED
p
ffiffiE
p(2{E)

sð51Þ
Fig. 8 shows how p1(q,q’)~g(E)
g(E)z1behaves as a function of E.
The recency effect holds for Ev2zffiffiffi
2
p; for Ew2zffiffiffi
2
pwe get
primacy. Similar results are obtained for initially Gaussian
opinions.
Thus, in the present model, the primacy effect (relevance of the
first opinion) can be related to the boomerang effect.
We now examine the emergence of cognitive dissonance in the
boomerang regime Ew1. Our results indicate that in this regime
the agent is more susceptible to cognitive dissonance; cf. Fig. 6
with Figs. 1. The mechanism of the increased susceptibility is
explained in Fig. 6: P’s opinion splits easier, since the probability
mass moves away (in different directions) from the anchor of Q.
Let us now assume that Prepeatedly interacts with the same
opinion of Q[cf. (42)]:
p½nz1(x)~ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p½n(x)DEp½n(x)z(1{E)q(x)D
p
Ðdx’ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p½n(x’)DEp½n(x’)z(1{E)q(x’)D
p,ð52Þ
where n~1,2,::: is the discrete time. Starting from initially
Gaussian opinion, Pdevelops two well-separated peaks, which is
another manifestation of cognitive dissonance: the smaller peak
moves towards the anchor of Qand finally places itself within the
acceptance latitude of Q, where the larger peak becomes more
narrow and drifts away from q(x); see Fig. 9. After many iterations
(^103for parameters of Fig. 9) the larger peak places itself within
the rejection latitude of Q, at which point p½n(x)stops changing
(stationary opinion). The above scenario suggests that in the
boomerang regime there is a finite probability that the target agent
Figure 7. Order of presentation effect in the boomerang
regime. The same as in Fig. 4but for E~1:5(boomerang regime).
Now the final opinion of Pis inclined to the first opinion (that of Q)
with respect to the distance. The initial maximally probable opinion of
Pis still maximally probable. Moreover, its probability has increased
and the width around it has decreased. The final opinion has 3 peaks.
doi:10.1371/journal.pone.0099557.g007
Figure 8. Illustration of the order of presentation effect in the
boomerang regime. p1(q,q’)~g(E)
g(E)z1given by (50, 51) versus E.
doi:10.1371/journal.pone.0099557.g008
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v

will eventually be persuaded after repeated exposure to the same
opinion.
Let us mention an experimental work that is relevant to our
discussion above. Ref. [58] carried out experiments with subjects
displaying boomerang effect, where each subject was exposed to
sufficiently many different (but still similar) persuasive opinions. It
was found that, sooner or later, the subjects exit the boomerang
regime, i.e. they start to follow the persuasion [58]. Our set-up is
somewhat different in that the subject (P) is repeatedly exposed to
the same persuading opinion. Modulo this difference, our
conclusion is similar to the experimental finding: the agent starts
following the persuasion with a certain probability.
Discussion
We presented a new model for opinion revision in the presence
of confirmation bias. The model has three inputs: the subjective
probabilistic opinions of the target agent Pand a persuading
(advising) agent Q, and the weight of Qas perceived by P.
The basic idea of the opinion revision rule is that no opinion
change is expected if the persuasion is either too far or too close to
the already existing opinion [15,36,60]. The opinion revision rule
is not Bayesian, because the standard Bayesian approach does not
apply to processes of persuasion and advising; see the second
section for more details.
The model accounts for several key empirical observations
reported in social psychology and quantitatively interpreted within
the social judgment theory. In particular, the model allows to
formalize the concept of opinion latitudes, explains the structure of
the weighted average approach to opinion formation, and relates
the initial discrepancy (between the opinions of Pand Q) to the
magnitude of the opinion change (shown by P). In all these cases
our model extends and clarifies previous empiric results, e.g. it
elucidates the difference between monotonic and non-monotonic
change-discrepancy relations, identifies conditions under which
the opinion change is sudden, as well as provides a deeper
perspective on the weighted average approach.
New effects predicted by the model are summarized as follows.
(i) For the order of presentation set-up (and outside of the
boomerang regime) the model displays recency effect. We
suggested that the standard argument that relates confirmation
bias to the primacy effect does not work in this model. In this
context we recall a widespread viewpoint that both recency and
primacy relate to (normative) irrationality; see e.g. [13]. However,
the information which came later is generally more relevant for
predicting future. Hence recency can be more rational than
primacy.
In many experimental set-ups the recency changes to primacy
upon increasing the retention time; see e.g. [70]. Our model
demonstrates the primacy effect only in the boomerang regime (i.e.
only in the special case). Hence, in future it needs to be extended
by involving additional mechanisms, e.g. those related to ‘‘long-
term memory’’ processes which could be responsible for the above
experimental fact. Recall in this context there are several other
theoretical approaches that address the primacy-recency difference
[11,34,42,56,66].
(ii) The model can be used to describe the phenomenon of
cognitive dissonance and to formalize the main scenario of its
emergence.
(iii) Repeated persuasions display several features implying
monotonous change of the target opinion towards the persuading
opinion. However, the opinion changes