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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,x2is defined over x1[½0,?)and x2[½0,?).We

enlarged the natural range x1[½0,1and 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,x2is 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,qkfor 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,q1z1{p1),

e

pk

pk~pk=Fp

1,q1:

½z1{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,y½for 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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PLOS ONE | www.plosone.org 3 July 2014 | Volume 9 | Issue 7 | e99557

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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PLOS ONE | www.plosone.org 4 July 2014 | Volume 9 | Issue 7 | e99557

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=vPand

O½(mP{mQ)2=vQand 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{1,1for

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,qbetween 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)

p2

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,~

ppbetween the initial and final opinion of P, while the

discrepancy is quantified by the Hellinger distance h½p,qbetween

the initial opinion of Pand the persuading opinion. For

concreteness we assume that the opinion strengths 1=vPand

1=vQare fixed. Then h½p,qreduces 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,~

ppis 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,~

ppthat 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,~

ppcalculated 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,~

ppis a

Figure 3. Opinion change versus discrepancy. (a) The opinion

change is quantified via the Hellinger distance h~h½p,~

ppbetween 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,~

ppfor

E~0:1(upper curve) and E~0:5(lower curve). Purple curves:

d(m)~d½p,~

ppfor 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),qX

k½qk{q’

kﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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{qkzO½g3.

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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PLOS ONE | www.plosone.org 9 July 2014 | Volume 9 | Issue 7 | e99557

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;qƒW½p½n;q,ð44Þ

W½p;q:XN

k~1qkf(pk=qk):ð45Þ

Hence W½p;qis a Lyapunov function of (42). Since W½p;qis a

convex function of p,W½p;q§f(1)~W½q;qand 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;qshowing that the fixed point

qis globally stable. More generally, the convergence reads:

p½n

k?f½p½1

kqk=XN

l~1f½p½1

lql, 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)).

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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,p½nbetween

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.

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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.

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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