doi: 10.1371/journal.pone.0099557.
eCollection 2014.
Opinion dynamics with confirmation bias
Affiliations
- PMID: 25007078
- PMCID: PMC4090078
- DOI: 10.1371/journal.pone.0099557
Item in Clipboard
Opinion dynamics with confirmation bias
PLoS One.
.
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.
Conflict of interest statement
Figures
The initial opinion of 
is described by Gaussian probability density p(x) (blue curve) centered at zero; see (17). The opinion of 
amounts to Gaussian probability density q(x) (purple curve) centered at a positive value. For all three figures continuous density f(x) (
) were approximated by 100 points 
, 
. The resulting opinion 
of 
is given by (16) with 
(olive curve). (a) The opinion of 
moves towards that of 
; 
, 
, 
, 
. (b) The maximally probable opinion of 
is reinforced; 
, 
, 
, 
. (c) The change of the opinion of 
is 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 
is not visible by eyes. For example, if p(x) and q(x) are Gaussian with, respectively, 
, 
, 
, the Hellinger distance (see (30) for definition) 
is close to maximally far, while the opinion change is small: 
.
is described by Gaussian probability density p(x) (blue curve) centered at zero; see (17). The opinion of amounts to Gaussian probability density q(x) (purple curve) centered at a positive value. For all three figures continuous density f(x) () were approximated by 100 points , . The resulting opinion of is given by (16) with (olive curve). (a) The opinion of moves towards that of ; , , , . (b) The maximally probable opinion of is reinforced; , , , . (c) The change of the opinion of is 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 is not visible by eyes. For example, if p(x) and q(x) are Gaussian with, respectively, , , , the Hellinger distance (see (30) for definition) is close to maximally far, while the opinion change is small: .
Blue curve: the initial opinion of 
given by (28) with b = 1. Purple curve: the opinion of 
described by (28) with 
. Olive curve: the resulting opinion of 
obtained via (16) with 
.
given by (28) with b = 1. Purple curve: the opinion of described by (28) with . Olive curve: the resulting opinion of obtained via (16) with .
(a) The opinion change is quantified via the Hellinger distance 
between the old and new opinion of 
(blue curves); see (30) for the definition. For comparison we also include the total variance distance 
(purple curves); see (33). These two distances are plotted versus the discrepancy 
. The initial opinion of the agent 
is Gaussian with 
and 
; see (17). The opinion of 
is Gaussian with 
and 
. Thus m quantifies the initial distance between the opinions of 
and 
. The final opinion 
is given by (13). Different curves correspond to different 
. Blue curves: 
for 
(upper curve) and 
(lower curve). Purple curves: 
for 
(upper curve) and 
(lower curve). The maximum of h(m) (
) is reached at 
(
). (b) 
(
) is the point where h(m) (
) achieves its maximum as a function of m. Blues points: 
versus 
for same parameters as in (a). 
grows both for 
and 
, e.g. 
, 
, 
, 
. Purple points: 
versus 
for same parameters as in (a). (c) The difference of the anchors (maximally probable values) 
versus 
for the initial opinions of 
and 
given by (17) under 
, 
, 
and 
. The final opinion 
of 
(and its maximally probable value 
) if found from (13) under 
(black points), 
(blue points) and 
(red points).
between the old and new opinion of (blue curves); see (30) for the definition. For comparison we also include the total variance distance (purple curves); see (33). These two distances are plotted versus the discrepancy . The initial opinion of the agent is Gaussian with and ; see (17). The opinion of is Gaussian with and . Thus m quantifies the initial distance between the opinions of and . The final opinion is given by (13). Different curves correspond to different . Blue curves: for (upper curve) and (lower curve). Purple curves: for (upper curve) and (lower curve). The maximum of h(m) () is reached at (). (b) () is the point where h(m) () achieves its maximum as a function of m. Blues points: versus for same parameters as in (a). grows both for and , e.g. , , , . Purple points: versus for same parameters as in (a). (c) The difference of the anchors (maximally probable values) versus for the initial opinions of and given by (17) under , , and . The final opinion of (and its maximally probable value ) if found from (13) under (black points), (blue points) and (red points).
Blue curve: The initial opinion of 
is described by Gaussian probability density p(x) with 
and 
; see (17). Purple (resp. olive) curve: the initial opinion of 
(resp. 
) are given by (17) with 
(resp. 
) and 
(resp. 
). Green curve: the resulting opinion of 
after interacting first with 
and then with 
. Both interactions use 
. The final opinion of 
is inclined to the most recent opinion (that of 
) both with respect to its maximally probable value and distance. The final opinion of 
has a larger width than the initial one.
is described by Gaussian probability density p(x) with and ; see (17). Purple (resp. olive) curve: the initial opinion of (resp. ) are given by (17) with (resp. ) and (resp. ). Green curve: the resulting opinion of after interacting first with and then with . Both interactions use . The final opinion of is inclined to the most recent opinion (that of ) both with respect to its maximally probable value and distance. The final opinion of has a larger width than the initial one.
(a) Blue (resp. purple) curve: the initial opinion of agent 
(resp. 
) described by probability density p(x) (resp. q(x)). Olive curve: the final opinion 
of 
as given by (16) with 
. Here p(x) and q(x) are defined by (17) with 
, 
, 
, 
. The final opinion develops two peaks of comparable height (cognitive dissonance). (b) Avoiding the cognitive dissonance due to a larger 
: the second peak is much smaller (other parameters are those of (a)). (c) Avoiding the cognitive dissonance due to a smaller 
: the first peak is much smaller (other parameters are those of (a)).
(resp. ) described by probability density p(x) (resp. q(x)). Olive curve: the final opinion of as given by (16) with . Here p(x) and q(x) are defined by (17) with , , , . The final opinion develops two peaks of comparable height (cognitive dissonance). (b) Avoiding the cognitive dissonance due to a larger : the second peak is much smaller (other parameters are those of (a)). (c) Avoiding the cognitive dissonance due to a smaller : the first peak is much smaller (other parameters are those of (a)).
Blue (resp. purple) curve: the initial opinion of agent 
(resp. 
) described by probability density p(x) (resp. q(x)). Olive curve: the final opinion 
of 
given by (16) with 
. Here p(x) and q(x) are given by (17) with 
and 
. The anchor (maximally probable opinion) of 
not only moves away from the anchor of 
; but it is also enhanced: the (biggest) peak of 
is larger than that of p(x). The second (smaller) peak of 
arises because the initial probability of 
located to the right from the anchor 
of 
, moves away from 
; 
gets a local minimum close to 
.
(resp. ) described by probability density p(x) (resp. q(x)). Olive curve: the final opinion of given by (16) with . Here p(x) and q(x) are given by (17) with and . The anchor (maximally probable opinion) of not only moves away from the anchor of ; but it is also enhanced: the (biggest) peak of is larger than that of p(x). The second (smaller) peak of arises because the initial probability of located to the right from the anchor of , moves away from ; gets a local minimum close to .
The same as in Fig. 4
but for 
(boomerang regime). Now the final opinion of 
is inclined to the first opinion (that of 
) with respect to the distance. The initial maximally probable opinion of 
is still maximally probable. Moreover, its probability has increased and the width around it has decreased. The final opinion has 3 peaks.
(boomerang regime). Now the final opinion of is inclined to the first opinion (that of ) with respect to the distance. The initial maximally probable opinion of is still maximally probable. Moreover, its probability has increased and the width around it has decreased. The final opinion has 3 peaks.
given by (50, 51) versus 
.
Blue (resp. purple) curve: the initial opinion of agent 
(resp. 
) described by probability density p(x) (resp. q(x)) as given by (17) with 
, 
, 
. Olive curve: the opinion of 
after 50 iterations (52) with 
.
(resp. ) described by probability density p(x) (resp. q(x)) as given by (17) with , , . Olive curve: the opinion of after 50 iterations (52) with .Similar articles
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