Willful Ignorance in a Biased Information World
David Bjerk
Robert Day School of Economics and Finance
Claremont McKenna College
[email protected]
Preliminary Draft, please do not cite
May 23, 2013
Abstract
Suppose individuals face a claim stating they can incur some bene…t by
taking a costly action, but individuals are uncertain of the validity of the
claim and can only obtain more information from biased sources. In such
situations, optimal behavior for expected utility maximizers is to select the
source biased toward their current behavior, even if that source is more biased
than the alternatively biased one. Moreover, if bias is inevitable, usually a
source on one side of the claim will optimally report with high bias, while the
source on the other side will report with minimal bias.
Thanks to seminar participants at UC-Irvine, Claremont McKenna College, Claremont Graduate University, and participants at the 2013 American Law and Economics Association Meetings
for helpful comments on earlier versions of this work. Thanks to the Lowe Institute for Political
Economy for …nancial help on this project.
1
In order to maintain an untenable position, you have to be actively
ignorant. -Stephen Colbert
All I know is just what I read in the papers, and that’s my alibi for
my ignorance. -Will Rogers
1
Introduction
When deciding whether to act on a claim of uncertain validity, individuals often
choose to inform themselves about the claim from a source whose perceived or
measured bias is consistent with their own and avoid sources that may be less
biased but in the opposite direction. For example, news readers tend to seek out
news sources that slant stories toward their existing views (Gentzkow and Shapiro
2010). More broadly, managers and politicians are often perceived to seek advice
from “yes”men who they know tend to o¤er advice biased toward the way they are
already leaning; investors seek out advisors who they know have a …nancial incentive
to steer them toward certain actions and investments; and on issues ranging from
homeopathic cold medicine to climate change, people often appear to avoid the
consensus of the scienti…c community when that consensus is in con‡ict with their
own current beliefs or actions.
What causes people to behave in this manner? In particular, why do individuals
tend to inform themselves from sources they know to be biased toward the way
they are already behaving, especially in cases when the decisions made with such
information have real consequences and when it is possible to access less biased
sources of information at a similar cost? In other words, why do individuals often
seem to …nd it optimal to be willfully ignorant?
This issue of willful ignorance is particularly well illustrated with respect to
the claim that vaccines increase the likelihood of autism. In 2001, over 17,000
kids went unvaccinated. Almost 50% of the parents who chose not to vaccinate
their children cited a belief that vaccines increase the likelihood of autism as the
reason for not vaccinating (Smith, Chu, and Barker 2004). The primary support for
these beliefs come from notably non-scienti…c sources that clearly state their vested
interest in getting parents to question vaccinating their children. Wolfe, Sharpe,
and Lipsky (2002) found that of the twenty-two English language anti-vaccination
websites they identi…ed, 100 percent claimed vaccinations cause ideopathic illness,
2
primarily autism. However, the only truly scienti…c source supporting this claim was
a 1998 Lancet article by Andrew Wake…eld that was later shown to be fraudulent
and was fully retracted by the Lancet’s editors by 2010 (Wallis 2010). Despite this,
one of the leading information sources promoting the claim that vaccines increase
the risk of autism, Generation Rescue, still cites and supports Wake…eld’s study and
conclusions (McCarthy 2012).
What is also notable however, is that roughly 90 percent of the anti-vaccination
websites identi…ed by Wolfe, Sharpe, and Lipsky (2002) present some version of
the claim that the information coming from government sources is biased due to
in‡uence from drug companies. Despite this claim, it would be hard to argue that
any of these anti-vaccination information sources could be thought to be less biased
and contain more accurate information than publications coming the Centers for
Disease Control. Yet, people who are sympathetic to this claim continue to choose
to inform themselves about this claim via such highly biased, arguably low-quality
information sources such as Generation Rescue rather than the CDC.
This notion of willful ignorance arising in an environment where individuals
believe information sources on both sides of a claim to be biased (at least to some
degree) is the key aspect of the analysis that follows. In particular, this paper shows
that in a world where all information sources are biased in their reporting on the
validity of a given claim, it is actually optimal for an expected utility maximizing
individual to inform himself about that claim from the source that is biased in the
direction he is already leaning or behaving, even if that individual knows this source
to be more biased than an alternative one on the other side of the issue.
The speci…c environment I consider is one in which individuals face a claim that
says by taking a costly action they can increase the likelihood of experiencing a
positive utility shock. However, individuals do not know whether or not the claim
is true at the time they must choose whether or not to act on the claim. Rather,
they simply have a belief regarding whether or not the claim is true. Before making
their decision in any particular time period however, they can consult one of two
information sources, both of whom observe unbiased informative signals regarding
the validity of the claim each period. However, both information sources are biased
(at least to some extent) in their reporting of these signals to individuals— one biased
toward the claim being true, and the other biased toward the claim being false. The
one biased toward the claim being true reports the actual underlying signal with
3
some probability less than one, and a …ctitious signal in support of the claim with
some probability greater than zero. The one biased against the claim reports the
actual underlying signal with some probability less than one, and a …ctitious signal
in opposition to the claim with some probability greater than zero.
In this rather general environment, the …rst part of this paper shows that it
turns out to be optimal for expected utility maximizing individuals to choose the
information source biased toward how he would initially choose to act, even if that
source is known to be more biased (in some cases much more biased) than the other.
The intuition for the above result rests on the key insight that information is only
important to an expected utility maximizer to the extent to which it can alter his or
her beliefs enough to change his or her actions. The bias of an information source
is important in that it a¤ects the degree to which information from that source can
alter a person’s beliefs enough to actually a¤ect behavior. Information against the
claim from the source biased towards the claim being true will have a greater impact
on a person’s belief, and potentially their behavior, than similar information from
the source biased towards the claim being false.
For example, consider an individual who buys organic food because he believes
it will make his family healthier, and suppose this person perceives Fox News to be
biased against organic food, but National Public Radio to be biased toward organic
food. Hearing a report suggesting that eating organic food has little impact on child
health likely has a bigger impact on the individual’s beliefs if it is heard on NPR
than on Fox News, since he will discount such information coming from Fox News
because of their perceived bias against organic food. In this way, such a report on
NPR may have a bigger in‡uence on the individual’s behavior regarding whether
to continue buying organic food than the same report on Fox News. Alternatively,
information in support of organic food coming from either source will not a¤ect this
person’s behavior, as such information will never cause the individual to stop buying
organic food. Therefore, information from a source biased toward the way one is
already acting can be more valuable than even a less biased source that is biased in
the opposite direction, as it will have a greater instrumental value with respect to
changing behavior.
The second part of this paper considers the behavior of information sources.
Using simulations, I show two key results relating to what bias an information source
should choose to employ given the bias chosen by the alternately biased source. First,
4
if one side chooses the minimal possible bias, but this minimal possible bias is not
negligible, then the other side actually does better in the long run by choosing to
report information with a relatively high amount of bias. Alternatively, if one side
chooses to report with a very high amount of bias, the alternately biased source
should not respond in kind, but rather report with the minimal possible bias. These
results suggest that in a world where bias in reporting is inevitable, the equilibrium
will generally have one side reporting with minimal bias while the other reports with
very high bias.
In general, this model shows that even in a world of completely rational expected
utility maximizers, a sizeable fraction of the population can continue to believe in
and act on a claim, and inform themselves about the claim from a source known to be
very biased, for extended periods of time even when the rest of the population is quite
certain that the claim is untrue. However, while choosing to inform oneself from a
more biased information source may be optimal for an individual, and o¤ering very
biased information may be optimal for an information source, neither will generally
be optimal for society at large.
2
Related Literature
The perception that information sources are often biased is quite widespread. For
example, only 26 percent of Americans say that news organizations are careful that
their reporting is not politically biased (Pew Research Center 2009). Evidence that
these perceptions have some basis in reality is presented Grosclose and Milyo (2005),
who develop a measure that suggests most major news outlets indeed have a bias.
Moreover, DellaVigna and Kaplan (2007) present evidence that the introduction of
a news source with a known bias can impact behavior.
This paper builds on the relatively recent theoretical literature on media bias,
developed in part by the novel work of Baron (2006), Mullainathan and Shleifer
(2003), and Gentzkow and Shapiro (2006). Baron (2006) considers a supply side
explanation, where personal preferences or career prospect concerns may cause journalists to accept a lower wage in return for less oversight and greater tolerance of
bias from their superiors.1 This contrasts with the more demand side explanations
1
In some sense, Crawford and Sobel’s (1982) “Cheap Talk”model also has some aspect of biased
information from the supply side, where a biased information source may choose to “coarsen” the
5
posed by Mullainathan and Shleifer (2003) and Gentzkow and Shapiro (2006). In
Mullainathan and Shleifer (2003), there exist individuals who are assumed to want
to know the truth and individuals who care about the truth but also have signi…cant
preferences for not hearing information inconsistent with their own prior beliefs. The
presence of these news consumers with a preference for some bias can cause news
sources to o¤er only biased news in the case of a monopoly, or polarizing viewpoints
in a duopoly setting. In Gentzkow and Shapiro (2006), news sources want to build a
reputation for quality reporting of the truth, but quality is not directly observable.
However, rational consumers will tend to place more faith in news sources whose
reports are generally consistent with their prior beliefs. This allows a demand for
biased news to arise endogenously, causing news sources to slant their coverage to
some degree to build a reputation as a higher quality source, especially concerning
issues that will take a long time to reveal their true state.
As will be seen, the model developed below contains elements of all these approaches. Like in Baron (2006), information providers are allowed ideologies or
preferences beyond pro…t maximization. Like in Gentzkow and Shapiro (2006) and
Mullainathan and Shleifer (2003), consumers may end up demanding more biased
information. However, in contrast to Baron, the model below does not hinge on
it being costly to an information source to reduce bias. Unlike Mullainathan and
Shleifer (2003), the model below does not require the assumption that some readers
incur substantial disutility from reading information inconsistent with their beliefs.
Finally, unlike Gentzkow and Shapiro (2006), consumers know the bias/quality of
each information source prior to choosing it. This is not to say that the frictions
assumed in these papers are somehow ‡awed, as all of these seem quite reasonable.
Rather this paper simply focuses on a di¤erent, potentially additional friction.
Biased information reporting also arises in Prendergast’s (1993) theory of “Yes
Men,” which is somewhat of a precursor to Gentzkow and Shapiro’s (2006) model,
in that workers may shade the information they uncover toward their manager’s
…ndings in an e¤ort to be perceived as providing more accurate information. Again,
this di¤ers from the model below in that the consumers of information in the model
developed below will know exactly the bias, and therefore quality, of each of the
available sources.
The model arguably most closely related to this paper is developed in a very
underlying information it delivers to the recipient.
6
interesting paper by Suen (2004). In this paper, individuals are uncertain about a
certain claim. Each period noisy signals regarding the veracity of the claim arise, but
individuals cannot observe this new information directly, rather they must choose
an intermediary (e.g. a newspaper) to “interpret” new information, where this
intermediary reveals whether the emitted signal of information exceeded a given
threshold and di¤erent intermediaries o¤er di¤erent thresholds. The model reveals
that individuals will generally choose an intermediary that has a threshold biased
towards the individual’s prior.
Like Suen’s (2004) model, in the model developed below, individuals cannot
observe the available information about a claim directly, but rather they must go
through an information intermediary. However, the model developed below di¤ers
from Suen’s (2004) in several ways, most notably the bias of information sources
relates speci…cally to their objective quality— more bias is assumed to imply a higher
likelihood of misrepresenting or failing to report new information regarding the
claim. By contrast, in Suen, more bias does not necessarily mean it is objectively
“worse” information in the sense of being truly less informative. Rather, in the
context of Suen’s model, individuals face a trade-o¤ with respect to bias. Less bias
means a source will be more likely to deliver some information any given period,
but more bias will mean any information that is given will be more informative
regarding the strength of the underlying evidence.2
This facet of the model where individuals potentially choose an information
source that they know is objectively less informative than another, or where individuals choose to be willfully ignorant, arises to some extent in several papers in
the economics literature in somewhat di¤erent contexts. For example, Benabou and
Tirole (2002) develop a model where individuals choose coarser information about
their ability in order to keep their self-con…dence high enough to overcome their tendency to procrastinate or fail to undertake potentially bene…cial actions. Carrillo
and Mariotti (2000) also consider a model where individuals weight current payo¤s
disproportionately high relative to future payo¤s. Such dynamic inconsistency may
mean that plans that are optimal for the current “self” may no longer be optimal
for “future”selves which becomes a problem when individuals are unable to commit
to a given consumption path. Therefore, individuals may choose to forgo better
2
Suen’s (2004) model is similar in many ways to Calvert (1985). Like in Suen’s model, bias
in Calvert’s model manifests itself in how stong (or weak) underlying evidence has to be before a
source reports that new evidence supports (or does not support) a given claim.
7
information regarding the likelihood of di¤erent outcomes from current choices because the current self cannot hide that information from his future self and therefore
cannot trust how that future self will act on that information.
A certain amount of willful ignorance also arises in Dal Bo and Tervio’s (2008)
model of individual corruption. Namely, individuals may try to stay willfully ignorant of whether they are a good type or a bad type by choosing to resist temptations,
even though they will immediately learn if they are bad type if they do not resist,
since only bad types will succumb to temptations. In Koszegi (2006), individuals may avoid certain potentially productive tasks in order to avoid learning bad
information about one’s ability, and the associated costs to one’s ego. Somewhat
similarly, Karlsson, Loewenstein, and Seppi (2009) model what they term the “ostrich e¤ect” in the context of investors. Speci…cally, under some parameterizations
of their model, investors may choose to put o¤ learning important information about
their returns until a later period, even though such learning would be costless.3
3
Model of Willful Ignorance
Suppose individuals will incur a bene…t of size v in any given period with probability
p0 2 (0; 1): They also encounter a claim stating that if they take some action at a
cost c, they will increase the likelihood of incurring the bene…t in any given period
to p1 2 (0; 1); where p1 > p0 : However, when choosing whether to act on the claim
the individual does not know whether the claim is true or false. Rather, at the
beginning of a period each person has a belief 2 (0; 1) that the claim is true.
Given these beliefs, each person must then choose whether or not to act on that
claim.4
3
There are also several related models where individuals do not choose to be underinformed
or misinformed, but rather become misinformed due to their underlying pscyhological tendencies.
For example, Blomberg and Harrington (2000) consider a model where individuals update their
beliefs regarding a certain issue, but some are a¤ected by this new information more than others.
Kopczuk and Slemrod (2005) develop a model where individuals willingly repress (i.e., forget)
relevant information regarding the likelihood of their own mortality in order to reduce fear of
death. Experimental work by Eil and Rao (2011) shows that subjects often discounted negative
information about themselves, and subjects avoided learning potentially useful information when
that information might be costly to their self-image.
4
Note that all the results are also consistent with a set-up where individuals incur a negative
shock of size v each period with probability p0 ; but encounter a claim stating that if they take
some action at a cost c, they will decrease the likelihood of incurring the shock each period to p1 ;
where p1 < p0 ; which may be a set-up more consistent with some motivating examples. However,
8
3.1
Choosing an Action
Given this set-up, an individual who has beliefs at the beginning of a period will
incur an expected utility of (p1 v) + (1
)(p0 v) c if he acts on the claim, and
an expected utility of simply p0 v if he does not act on the claim. Therefore, if an
individual is an expected utility maximizer, he will act on the claim if an only if
(p1 v) + (1
)(p0 v) c > p0 v; or if and only if > c=(p1 v p0 v): If we de…ne
to equal p1 v p0 v (i.e., equals the expected gross bene…t of acting on the claim
if it is indeed true), then each person’s optimal action with respect to the claim is
summarized in Proposition 1 below:
Proposition 1 An individual will act on the claim in a given period if and only if
his beliefs that period exceed
= c= :
The above proposition is quite intuitive and straightforward. The remainder
of this section analyzes an individual’s decision regarding choosing an information
source regarding the validity of the claim.
3.2
Choosing an Information Source Regarding the Validity
of the Claim
Suppose there are two information sources reporting on the validity of the claim.
Each period, both information sources observe a signal from nature regarding whether
or not the claim is true. In particular, both observe a signal 2 fP; N g each period, where Pr( = P jT rue) = (in words, the probability of observing a P signal
given the claim is actually true equals ); for some
2 (0:5; 1).5 Continuing,
Pr( = N jT rue) = 1
, Pr( = P jF alse) = 1
; and Pr( = N jF alse) = .
Intuitively, if the claim is true, then there is a higher likelihood of observing a P (or
“positive”) signal than an N (or “negative”) signal, while if the claim is false, there
is a higher likelihood of observing an N signal than a P signal. The parameter
is simply the precision of the information nature can o¤er regarding the validity of
the claim.
this latter way requires accounting for and keeping track of numerous negative signs which provides
needless complexity to the math below.
5
Note, it need not be the case that both sources observe the same signal in any given period.
9
Assume that individuals cannot observe the true underlying signal from nature.
Rather, they must rely on the signal reported by of one of the information sources.
While both information sources observe a true underlying signal from nature, suppose both are biased in their reporting of the underlying signal they observe. One of
the sources is positively biased ( hereafter also referred to as the positive bias source),
while the other is negatively biased ( hereafter also referred to as the negative bias
source). In practice, bias here refers to the likelihood that instead of reporting the
true observed signal , a source simply reports a signal corresponding to its bias.
Therefore, if we let the parameter bP capture the bias of the positive bias source,
then the reported signal from the positive bias source is a P with probability bP ,
and equals with probability 1 bP : Similarly, if we let the parameter bN capture
the bias of the negative bias source, then the reported signal from the negative bias
source is an N with probability bN , and equals with probability 1 bN .
There are several interpretations of this bias. One can interpret the bias of
each source as the likelihood the source outright misrepresents or ignores newly
available evidence. However, there are also more nuanced views. For example, one
can consider bi to be a parameter that captures the depth of reporting for source i,
where that source uncovers new information about the claim with probability 1 bi
and simply repackages old information consistent with their bias with probability
bi . Or, we can assume reporters all have their own biases a la Baron (2006), and
managers of any given information source cannot fully eradicate such biases in their
editing and oversight. Finally, one can presume news sources su¤er from cognitive
bias of a form similar to that described by Rabin and Schrag (1999). Namely,
each information source su¤ers from a cognitive bias that impedes their ability to
correctly evaluate newly available evidence regarding the claim, and instead causes
them to interpret it in a manner consistent with their own bias.
While any of these interpretations work for the analysis of individuals that follows, only the …rst is consistent with the later analysis of information sources choosing their level of bias. However, as will be discussed later, the preferred interpretation is that limitations on a source’s ability to uncover new information, an inability
to eradicate reporter bias, and a cognitive bias among information sources, cause
information sources to always su¤er from at least some minimal level of bias. However, information sources can also choose to report with a higher bias if they so
desire.
10
For the interest of this paper, I will assume that individuals know the bias of each
source, and that one information source is more biased than the other— in particular
assume bP > bN : Note that this means that the positive bias source is objectively of
lower quality than the negative bias source in the sense that it is less likely to o¤er
the individual new real information that arises regarding the underlying validity
of the claim. As stated in the introduction, the question of interest is when (if
ever) will it be optimal for an individual to choose such a lower quality/higher bias
information source when a higher quality/less biased is available?
3.2.1
Belief Updating
To analyze the key question stated above, let us …rst consider how individuals update their beliefs after observing the signal reported by a given information source.
To isolate the impact of bias among information sources, let us suppose individuals
update their beliefs optimally and unbiasedly using Bayes’rule and perfect information regarding the underlying signal process (as well as the extent of each source’s
biased treatment of the underlying signals). Given this assumption, let us …rst consider an individual with initial beliefs at the beginning of a period who obtains
information from the positive bias source. If he observes a P signal from this source,
his updated beliefs will equal
bP (P ) =
((1 bP ) + bP )
bP ) + bP ) + (1
)((1 bP )(1
((1
) + bP )
:
(1)
Similarly, if this individual observes an N signal from this positive bias source, his
updated beliefs will equal
or simplifying,
bP (N ) =
(1
(1
bP )(1
bP (N ) =
(1
bP )(1
) + (1
)
)(1
(1
)
) + (1
)
:
bP )
;
(2)
As can be seen from equations (1) and (2) above, the positive bias (bP ) a¤ects an
individual’s beliefs upon observing a P signal from the positive bias source since he
knows there is some chance that it is a “false”signal, but does not a¤ect his beliefs
upon observing an N signal from the positive bias source since he knows that must
11
indeed correspond to the true underlying signal.
Next consider an individual with initial beliefs at the beginning of a period
who obtains information from the negative bias source. If he observes a P signal
from this source, his updated beliefs will equal
bN (P ) =
+ (1
)(1
)
(3)
:
Similarly, if this individual observes an N signal from this negative bias source, his
updated beliefs will equal
bN (N ) =
((1
((1 bN )(1
) + bN )
bN )(1
) + bN ) + (1
)((1
bN ) + bN )
:
(4)
So with respect to the negative bias source, the bias a¤ects an individual’s beliefs
upon observing an N signal (since such a signal might be “false”), but does not
a¤ect an individual’s beliefs upon observing a P signal since such a signal must be
indicative of an underlying P signal from nature.
Given the above equations, one can easily con…rm that the following Proposition
holds:
Proposition 2 For any initial beliefs ; bP (N ) < bN (N ) <
< bP (P ) < bN (P ):
Intuitively, given the individual knows the bias of each source, observing an N
(or “negative”) signal from the positive bias source will cause him to downward
adjust his beliefs more than he would if he observed an N signal from the negative
bias source, since he knows that an N signal from the negative bias source might
be the result of bias/misinformation. Similarly, observing a P (or “positive”) signal
from the positive bias source will cause him to upward adjust his beliefs by less than
he would if he observed a P signal from the negative bias source, since he knows that
a P signal from the positive bias source might be the result of bias/misinformation.
3.2.2
Reacting to New Information
Given the belief updating speci…ed above, we can now consider how individuals’
behavior will respond to di¤erent sorts of new information from the two information
sources. In doing so, we will only consider individuals whose beliefs entering a
12
period are such that > , or in words, individuals whose beliefs are such that
they must have found it optimal to act on the claim last period. This is without loss
of generality as one could make an argument essentially analogous to what follows
for those with < :
First note that for individuals acting on the claim, observing a P signal from
either news source will strengthen their beliefs that the claim is true. However, this
will not have any impact on their behavior, as they will simply continue to act on
the claim as they were before. Next consider the consequence of observing an N
signal from the positive bias source. Note that an individual acting on the claim
will only change his behavior in response to observing such information if his initial
beliefs are such that when he updates his beliefs after seeing this signal, bP (N ), are
less than : Applying equation (2) and the de…nition of
from Proposition 1, the
aforementioned condition amounts to the following:
(1
)
) + (1
(1
)
<
c
:
Re-arranging and simplifying the above expression we get
<
(
c
c)(1
)+c
:
So, we can say that an individual who is acting on the claim will stop acting on the
claim if he observes an N , or “negative,”signal from the positive bias source and his
current beliefs are such that < 1 ; where 1 is de…ned by the following equation:
1
=
(
c
c)(1
)+c
(5)
:
Next, consider the consequence of observing an N signal from the negative bias
source. An individual acting on the claim will change his behavior if and only if his
beliefs going into the period are such that his updated beliefs bN (N ) upon observing
such a signal are less than : Again, applying equation (4) and the de…nition of
from Proposition 1, this condition amounts to if an only if
((1
((1 bN )(1
) + bN )
bN )(1
) + bN ) + (1
)((1
bN ) + bN )
Re-arranging and simplifying the above expression we get
13
<
c
:
<
(
c((1 bN ) + bN )
bN )(1
) + bN ) + c((1
c)((1
bN ) + bN )
:
So, we can say that an individual who is acting on the claim will stop acting on
the claim if he observes an N signal from the negative bias source and his current
beliefs are such that < 2 ; where 2 is de…ned by the following equation:
2
=
(
c((1 bN ) + bN )
bN )(1
) + bN ) + c((1
c)((1
bN ) + bN )
:
(6)
Simple comparisons between equation (5), equation (6), and the de…nition of
in Proposition 1 will con…rm the following proposition:
Proposition 3
<
2
<
1
for all 0 < bN < 1:
Given the above de…nitions and Proposition 2, we can now stratify those acting
on the claim into three groups: (i) the True Believers are individuals whose beliefs
coming into a period exceed 1 , meaning they act on the claim and would not switch
behavior this period regardless of what signal they observe from either information
source; (ii) the Strong Believers are individuals whose beliefs coming into a period
are between 2 and 1 , meaning they act on the claim, but would switch their
behavior if they observed an N signal from the positive bias source but not if they
observed a similar N signal from the negative bias source; (iii) …nally, the Tentative
Believers are individuals whose beliefs coming into a period exceed
but are less
than 2 ; meaning they act on the claim but they would switch their behavior upon
observing an N signal from either information source. Figure 1 summarizes this
information.
3.2.3
Choosing an Information Source
We can now analyze which information source would be optimal for a person to
choose depending on his beliefs coming into a period, assuming individuals can
consume only one source. To do so, we will analyze the three groups de…ned above
separately.
True Believers (
1
< )
14
These individuals will not change their behavior regardless of what they observe
from either information source. Therefore, at the beginning of a period before
observing any new information, their expected utility from choosing the negative
bias source is EU (N egative) = (p1 v c) + (1
)(p0 v c): Recalling that we
earlier de…ned = p1 v p0 v, the previous expression simpli…es to
EU (N egative) = p0 v +
c:
However, since these individuals will not change their behavior even if they observe
an N signal from the positive bias source, their expected utility from choosing the
positive bias source is also EU (P ositive) = (p1 v c) + (1
)(p0 v c); or more
simply
EU (P ositive) = p0 v +
c:
Given the expected utility from choosing the positive bias source is identical to the
expected utility from choosing the negative bias source, True Believers are indifferent between information sources regardless of how much more biased one is than
the other. This is actually quite obvious as there is no instrumental value of information for True Believers since they will continue to do this period what they did
last period regardless of any information they observe. However, if one makes the
assumption that individuals incur even the slightest bit of utility from observing
information consistent with their current behavior, or incur the slightest bit of disutility from observing information in con‡ict with their current behavior, then True
Believers would always choose the positive bias information source regardless of its
relative bias to the negative bias source (i.e., regardless of the di¤erence between bP
and bN ):
Strong Believers ( 2 < < 1 )
These individuals will change their behavior if they observe an N signal from
the positive bias source, but not if they observe an N signal from the negative bias
source. Therefore, like the True Believers above, at the beginning of a period their
expected utility from choosing the negative bias source is
EU (N egative) = p0 v +
c:
(7)
However, given these individuals will stop acting on the claim if they observe an N
15
signal from the positive bias source, but otherwise continue what they are doing,
their expected utility from choosing the positive bias source is
EU (P ositive) =
[((1
+(1
Again noting
= p1 v
bP ) + bP )(p1 v
c) + (1
)[((1
) + bP )(p0 v
bP )(1
bP )(1
)p0 v]
c) + (1
bP ) p0 v]:
p0 v, we can simplify the previous expression to
EU (P ositive) = p0 v + ((1 bP ) +bP )
((1 bP )(
+(1
)(1
))+bP )c: (8)
Clearly, an individual will have higher expected utility by consuming the positive
bias source if and only if EU (P ositive) EU (N egative) > 0: Given equations (7)
and (8) above this will be true if and only if
[ ((1
bP ) + bP )
]
[(1
bP )(
+ (1
)(1
)) + bP
1)c > 0:
(9)
Simplifying the above expression we get
(1
bP )[(1
(1
)(1
))c
(1
) ] > 0:
Since bP < 1; the above expression only holds if (1
(1 )(1 ))c (1
0: Simplifying this once more, we can say that equation (9) will only hold if
<
(
c
c)(1
)+ c
) >
:
Now, note that from equation (5) above we know 1 = ( c)(1c )+ c : So the above
argument implies that EU (P ositive) EU (N egative) > 0 for anyone with beliefs
< 1 , which in turn is true by de…nition for the group of Strong Believers. Therefore, Strong Believers strictly prefer to obtain information from the positive bias
information source over the negative bias information source regardless of how large
the bias of the positive bias source is relative to that of the negative bias source
(i.e., regardless of how large bP is relative to bN ).
The intuition for this result is that individuals in this group have strong enough
16
beliefs that the claim is true such that no information from the negative bias source
can be su¢ cient to sway their behavior. Hence, information from the negative bias
source simply doesn’t have any instrumental value. On the other hand, even if the
positive bias source is known to be extremely biased, meaning it regularly fails to
report true information about the claim (indeed possibly much more often than
the negative bias source does), such information can still be valuable to the Strong
Believers, since there is still a possibility of observing a “negative” signal (i.e., N
signal) from this source, which would indeed be valuable as it would be su¢ cient to
change their behavior.
While Strong Believers will …nd it optimal to choose the positive bias information
source over the negative bias information source regardless of how much more biased
the positive bias source is than the negative bias source, it is still worth considering
how the expected utility of the Strong Believers is a¤ected by the extent of the bias
from the positive bias source. Intuition suggests that more bias from the positive
bias source cannot be helpful for Strong Believers, as this information is of value to
them, so making this information less accurate cannot be helpful. We can con…rm
this intuition by taking the derivative of equation (8) with respect to bP and checking
whether or not it is negative, or con…rming that
@EU (P ositive)
= (1
@bP
)
(1
(
+ (1
)(1
))c < 0:
With some manipulation, the above expression will be true as long as
<
(
c
c)(1
)+ c
:
will be negative as long as < 1 , which
The above expression implies @EU (P@bositive)
P
is again true by de…nition for Strong Believers.
So indeed, while Strong Believers are willing to choose the positive bias source
over the negative bias source no matter the relative bias of the positive bias source
versus the negative bias source, but Strong Believers are better o¤ the lower the
bias from the positive bias source.
Tentative Believers ( < < 2 )
These individuals will change their behavior if they observe an N signal from
either the positive bias source or the negative bias source. Therefore, at the beginning of a period before observing any new information, their expected utility from
17
choosing the negative bias source is
EU (N egative) =
[(1
+(1
bN ) (p1 v
)[(1
c) + (bN + (1
bP )(1
)(p0 v
bN )(1
)p0 v]
c) + (bN + (1
bN ) p0 v]:
+ (1
))c:
Simplifying, the above expression becomes
EU (N egative) = p0 v + (1
bN )
(1
bN )(
)(1
(10)
Similarly, at the beginning of a period before observing any new information,
their expected utility from choosing the positive bias source is
EU (P ositive) =
[((1
+(1
bP ) + bP )(p1 v
c) + (1
)[((1
) + bP )(p0 v
bP )(1
bP )(1
)p0 v]
c) + (1
bP ) p0 v];
)(1
))+bP )c: (11)
which can again be simpli…ed to
EU (P ositive) = p0 v+ ((1 bP ) +bP )
((1 bP )(
+(1
Like before, an individual will have higher expected utility by consuming the
positive bias source if and only if EU (P ositive) EU (N egative) > 0;which given
equations (10) and (11) above will only be true if
[((1 bP ) +bP ) (1 bN ) ]
[(1 bP )(
+(1
)(1
))+bP (1 bN )(1
Simplifying this equation we get
bP (
c) + (bN
bP )[(
c) + ( (1
) + (1
) )c] > 0:
Manipulating the above expression a bit further we get the condition
18
)(1
)]c > 0:
bP
bN
bN
<
(
( (1
c)
) + (1
) )c
:
Therefore, if we de…ne Acceptable Excess Bias (AEB) by the following equation
AEB( ) =
(
( (1
c)
) + (1
) )c
;
(12)
then Tentative Believers are better o¤ choosing the positive bias information source
as long as the percentage di¤erence in bias between the positive bias source and the
negative bias source is less than AEB( ); otherwise they should choose the negative
bias source.
A couple of things to note here. First, since
c > 0 for all those acting on the
claim (i.e., all those for whom > ) as shown in Proposition 1, it must be true
that AEB( ) > 0 for all Tentative Believers. This implies that even the Tentative
Believers will …nd it optimal to choose a more biased positive bias information
source than the less biased negative bias information source, as long as it isn’t “too”
much more biased.
The second thing to note is that lim AEB( ) = 0 (since
= c= ) and
@AEB( )
@
!
> 0: This implies that those whose beliefs are such that they act on the
claim, but are almost indi¤erent between acting and not acting on the claim, are
willing accept only a negligible amount of “excess bias” from the positive bias information source. However, as beliefs in the validity of the claim become stronger,
even tentative believers are willing to accept more an more “excess bias” from the
positive bias information source relative to the negative bias information source. In
other words, among the Tentative Believers, the stronger their beliefs that the claim
is true, the more misinformation they are willing to accept from the positive bias
information source relative to the negative bias information source.
The intuition for the results with respect to the Tentative Believers is a bit
subtle. Essentially, it comes down to Type I versus Type II errors. Given these
individuals are choosing to act upon the claim, they are more worried about not
acting on the claim if it is actually true than acting on the claim though it is actually
false. Therefore, they are slightly more worried about observing a “false” negative
signal that would cause them to cease acting on the claim when it is actually true,
than a “false” positive signal that would cause them to keep acting on the claim
even though it is not true. This means such individuals are willing to accept a bit
19
more misinformation from the source that tilts toward the claim being true than
the source that tilts toward the claim being false. The relative balance of these tips
more and more toward the former as the individual’s belief that the claim is true is
stronger.
Finally, given equation (11) and equation (8), we know the expression for EU (P ositive)
is the same for Tentative Believers as it was for Strong Believers. Therefore, since
we know that < 1 for all Tentative Believers, once again it will be true that
@EU (P ositive)
will be negative for all Tentative Believers. In words, Tentative Believ@bP
ers will also be made worse o¤ the greater the bias from the positive bias source.
3.2.4
Summarizing Behavior with Respect to Choosing an Information
Source
The above argument shows that as long as all information sources are biased, everyone who chooses to act on the claim will actually …nd it optimal to choose the
information source biased toward the claim, even if that source is known to be more
biased than an alternatively biased source in the sense of having a higher likelihood
of misrepresenting and/or failing to report new information regarding a claim. In
other words, in a world of biased information sources, it is rational from an individual perspective to be willfully ignorant.
While all those acting on the claim are willing to accept some excess bias from
the positive bias source, they are not all the same. True Believers …nd the positive
bias source superior no matter how much more biased it is than the negative bias
source, and moreover, are actually made better o¤ the stronger the bias from the
positive bias source if they get even the smallest amount of disutility from hearing
information in con‡ict with their current behavior (a la Mullainathan and Shleifer
(2003)). Strong Believers also …nd the positive bias source superior no matter how
much more biased it is than the negative bias source, but would prefer less bias
to more bias from the positive bias source. Alternatively, Tentative Believers have
their limits, only …nding the positive bias source superior if its bias isn’t “too”much
larger than the negative bias source, and are made worse o¤ the greater the bias
from the positive bias source. However; this willingness to accept more bias from
the positive bias source among the Tentative Believers increases in the strength of
their own beliefs regarding the validity of the claim.
Since True Believers and Strong Believers choose the positive bias source regard20
less of its relative bias, and Tentative Believers choose the positive bias source as
long as its relative bias is within their Acceptable Excess Bias (AEB) range (which
itself is increasing in beliefs ), we can state the following proposition that fully
characterizes choice regarding information source:
Proposition 4 For any given information source biases bP and bN , there exists a
@ 3 (bP )
threshold belief 3 (bP ) >
such that @b
> 0 and:
P
1. If 3 (bP )
2 , then an expected utility maximizing individual will choose the
positive bias information source if and only if his beliefs
3 (bP ):
2. If 3 (bP ) > 2 , then an expected utility maximizing individual will choose the
positive bias information source if and only if his beliefs
2:
Proof. In Appendix.
Propositions 1 - 4 are summarized graphically in Figure 2.
The above results also let us consider the evolution of each individual’s tolerance
for bias. In particular, they imply that the longer an individual continues to act on
a claim, the more bias he is willing to tolerate from the information source biased
toward the claim relative to a source biased against a claim.
To see why, consider an individual who has acted on the claim for n 1 periods.
If he was a True Believer at the beginning of the n 1th period, then he is either still
a True Believer or at least a Strong Believer at the beginning of the nth period.
Either way, he will tolerate more bias from the the positively biased information
source since he will not switch to the alternately biased source regardless of the bias
from the positive bias source. Alternatively, if he was a Strong Believer or Tentative
Believer at the beginning of the n 1th period, then for him to have still acted on
the claim in the n 1th period, he must have observed a positive signal from the
positively biased source. Such information will cause his beliefs to strengthen. This
may cause him to become a Strong Believer, or simply cause him to continue to be
a Strong Believer. In either case he will tolerate more bias from the the positively
biased information source since he will again not switch to the alternately biased
source this period. Finally, if he was a Tentative Believer in the n 1th period,
then the strengthening of his beliefs due to observing a positive signal (even from
the positive biased source) will cause him to also be willing to accept more bias
21
from the positive bias source since the Acceptable Excess Bias function, AEB( );
is increasing in beliefs :
4
Behavior of Information Sources
As discussed above, among the individuals for whom information is valuable (i.e.,
Strong Believers and Tentative Believers), less bias is better. Indeed, if information
sources could o¤er truly unbiased information, the above results would go away for
these two groups. But, more interesting implications arise if one assumes some level
of bias, q > 0; is inevitable from information sources. Given one source always
reports information with some bias, how much bias would the other want to choose?
On some dimensions, the answer to this question depends on the objective of
the information source. On the one hand, the information source could be a pro…t
maximizer, in which case it would want to choose its bias to maximize its number
of consumers. On the other hand, the information source may have a di¤erent
goal, such as maximizing the number of people acting in accordance with a given
claim. As it turns out, within this model, choice of bias interacts with both of these
objectives in similar ways.
Consider …rst the question of what an information source should do if the alternative source is reporting with the minimal possible bias. For example, suppose the
negative information source reports information with the minimum possible bias,
how should the positive bias source respond? To answer this, let us think about the
trade-o¤ the positive bias source is making by increasing its bias. From Proposition 4 we know that as the bias of the positive bias source, bP , rises, so will 3 (bP ):
Therefore, as the positive bias source increases its bias, it will attract fewer Tentative
Believers. Intuitively, as the positive bias source increases its bias, it will surpass
the Acceptable Excess Bias threshold for more and more Tentative Believers. On the
other hand, by increasing its bias, it is more likely to deliver a “P”signal, meaning it
will be more likely to strengthen the beliefs of those who do choose to use it, making
them more likely to continue to use it the next period. In a sense, by increasing its
bias, it is trading o¤ the number of initial users for the loyalty of those initial users.
Which of these two forces will dominate depends on the parameterization of
the model. However, the following simulation results indicate that the latter will
generally dominate the former under relatively generic parameterizations.
22
The following simulation results show the mean results of 10 repetitions of populations of 300 individuals, where initial beliefs for each individual are randomly
drawn from a uniform distribution over [0,1]. In each case, = 2 and c = 1, implying that individuals act on the claim as long as their beliefs that the claim is true in
that period exceed 0.5 (see Proposition 1). Moreover, it is assumed that the claim
is false in actuality, and set equal to 0.75. This means that the likelihood of an
information source observing a “positive” signal in any given period is 0.25, while
the likelihood of an information source observing a “negative” signal in any given
period is 0.75. Finally, the bias of the negative information source is assumed to be
bN = 0:15, meaning that instead of reporting the true underlying signal they simply
report a “negative”signal 15% of the time, which is assumed to be the lower bound
on bias.
Figure 3 shows the evolution of the fraction of the population choosing the
positive bias information source under three di¤erent levels of bias for the positive
bias source— low (bP = 0:16); medium (bP = 0:50), and high (bP = 0:80): As can
be seen in early rounds of Figure 3, the positive bias information source is initially
consumed far more often when it is known to report with low bias compared to a
higher bias. However, as the truth starts getting out due to this relatively low bias,
more and more individuals switch to the negative bias source. On the other hand,
when reporting with the very high bias, the fraction of individuals who consume
the positive bias source stays relatively constant, so that after about …ve rounds of
reporting, the fraction of individuals choosing to consume the positive bias source
is roughly the same whether it reports with a low bias or a very high bias, and in
later rounds a higher fraction of individuals choose the positive bias source when it
is known to report with a very high bias than with a relatively low bias.
This point is made even stronger in Figure 4, which shows the evolution of the
fraction of the population acting on the claim under the three di¤erent levels of bias
from the positive bias information source. As can be seen, up through about three
rounds of reporting, the fraction of individuals acting on the claim is about the same
regardless of the bias of the positive bias source. Intuitively, by reporting with a
low bias, the positive bias source gets more consumers, but this is o¤set by the fact
that these consumers are then more likely to hear information in con‡ict with the
claim, causing them to switch sources, relative to what would have happened if the
reported with more bias. However, after round 4, the fraction of the population
23
acting on the claim is increasing in the amount of bias from the positive bias source.
Indeed, after nine rounds of reporting, only about 15% of individuals continue to
act on the (false) claim when the positive bias source chooses to report with close
to the minimal bias. On the other hand, by reporting with a very high bias, the
positive bias information source is able to keep almost 30% of the population acting
on the claim (even though it is actually false).
These results suggest that if an information source biased toward the correct side
of the claim is reporting with minimal bias, but this minimal bias is still signi…cant,
it is generally optimal for an information source biased toward the other side of the
claim to respond by reporting with a very high amount of bias, as such behavior will
maximize the long-run number of individuals consuming that source and the number
of individuals who continue to act in accordance with that information source’s bias.
As a point of comparison, the dotted grey line in Figure 4 shows the evolution of
the fraction of the population acting on the claim if individuals themselves could
simply access and interpret the raw unbiased information rather than go through a
biased intermediary. As can be seen, after 10 rounds, just over 5% of the population
would continue to act on the claim. This is less than one-…fth of the fraction that
would be acting on the claim when the negative bias source reports with a minimal
possible bias that is still signi…cant (bN = 0:15) and the positive bias source reports
with high bias (bP = 0:80):
A second question of interest is what an information source should do if the
source on the other side of the issue (the incorrect side) is reporting with a very
large bias? For example, if the positive bias source reports with a very high amount
of bias, should the negative bias source respond by trying to be as unbiased as
possible as in the simulation above, or should …ght …re with …re by also reporting
with high bias? The simulations summarized in Figures 5 and 6 strongly suggest
they should not get into a bias arms race. In particular, the simulations underlying
Figures 4 and 5 are similar to those described above, but the bias of the positive bias
source is …xed at bP = 0:8, and three di¤erent levels of bias are considered for the
negative bias source— low (bN = 0:15); medium (bN = 0:50), and high (bN = 0:75):
As can be seen in Figure 5, the fraction of the population consuming the positive
bias source (and therefore not consuming the negative bias source) is always higher
when the negative bias source chooses high bias rather than attempts to minimize its
bias. In other words, by responding to highly biased information from the positive
24
bias side with a strong bias of their own, the negative bias side loses consumers
relative to if they had tried to report as unbiasedly as possible. Similarly, as can
be seen in Figure 6, when the positive bias source reports with very high bias, the
fraction of the population acting on the claim is always higher when the negative
bias source also chooses a high level of bias (bN = 0:75) than when it chooses the
lowest level of bias (bN = 0:15).
Moreover, again comparing these results in Figure 6 to the dotted grey line showing the fraction of the population acting on the claim if individuals themselves could
simply access and interpret the raw unbiased information themselves rather than go
through a biased intermediary, we can see that the bias in reporting from available
information sources can lead to up to seven times more individuals continuing to
act on the (false) claim after 10 periods.
So again, when the positive bias information source reports with a lot of bias (and
it is on the incorrect side of the claim), the negative bias source actually induces
more people to continue acting on the claim if it were to respond by reporting
with a lot of bias as well. Therefore, regardless of whether its goal is to maximize
consumer usage, or to minimize the fraction of the population acting on the claim,
if an information source biased toward the incorrect side of the claim is reporting
with a lot of bias, the information source biased toward the correct side of the claim
should always try to report as unbiasedly as possible.
Finally, Figure 7 con…rms the relatively intuitive result that the presence of bias
from information sources can exacerbate the divergence in the beliefs between those
who act on the claim and those who do not. Speci…cally, the solid lines in Figure 7
show the average beliefs for those who do and do not act on the claim in simulations
where the positive biased source reports with high bias (bP = 0:80) and the negative
bias source reports with low bias (bN = 0:15): These lines can be compared to
the dashed lines in Figure 6 which show the average beliefs for those who do and
do not act on the claim in simulations where individuals can directly observe and
interpret the raw unbiased information from nature themselves rather than having
to go through one of the biased intermediary sources. While average beliefs from
these two di¤erent groups diverge under both simulations, this divergence is greater
in the world with only biased information available.
In general, the above results suggest that in an equilibrium with two information
sources reporting on a given claim, where neither can actually report with no bias,
25
there will be a bifurcation where one source attempts to report with the minimal
possible bias, while the other (the one on the wrong side of the claim) will report on
the claim with a lot of bias. Notably, as shown in Figures 3-7, in such an equilibrium,
a relatively large fraction of the population may choose to obtain information from
an information source that is known to be very biased and continue to act on a
false claim for an arguably long period of time, especially relative to a world where
individuals could obtain unbiased information. Moreover, as time passes, those
acting on each side of the claim will become more and more convinced their actions
are correct and the other actions of the other side are wrongheaded and hard to
fathom.6
5
Summary and Conclusion
Why would someone choose to inform himself about an important issue from an
information source he knew lied to him more often than other available sources? Or,
as termed in this paper, why would someone be willfully ignorant? As this paper
argues, such behavior is actually optimal for standard expected utility maximizing
individuals when all information sources report with at least some bias. Intuitively,
either an individual’s beliefs are such that further information simply isn’t valuable
in the sense of a¤ecting behavior (the True Believers), or information from the
oppositely biased source is just less valued because either it would be su¢ ciently
discounted so as not to be able to change behavior (Strong Believers), or because
the individual is more concerned about false information causing him to not act on
the claim when it is true than causing him to act on the claim when it is actually
false (Tentative Believers).
One thing that comes out of this analysis is that optimal behavior for rational
individuals and information sources may not be optimal for society as a whole.
Speci…cally, if a claim is false, and acting on the claim is costly to individuals
and/or society at large, then the greater the bias of the information source biased
6
One issue that comes up here is the fact that if biases are known, and inidividuals are aware
enough of the equilibrium forces, individuals will know that the side the reports with the higher
bias is on the incorrect side of the debate. However, if one were to think more broadly, and assume
each side of the debate doesn’t know or care whether it is on the correct side, but rather is just
in the business of attracting customers and/or getting people to act in accordance with its bias,
and individuals aren’t able to fully understand the equilibrium forces, then this issue doesn’t really
arise.
26
toward the claim, the greater will be the number of people who continue to act on
the claim, thereby increasing the total societal ine¢ ciency. In other words, “free
speech”is not free.
By concealing or misrepresenting the scienti…c …ndings regarding vaccines, antivaccination information sources increase the unvaccinated and thereby threaten the
health of others. By overstating the harmful e¤ects of marijuana, anti-drug advocates may be inducing more individuals to use it. By overstating the bene…ts of
mammograms, breast cancer awareness groups may be causing excessive medical
interventions with few bene…ts. By surrounding himself with advisors who long
advocated for removing Saddam Hussein from power, President George W. Bush
may have developed an overly strong belief in the claim that Hussein had acquired
weapons of mass destruction, giving him cause to invade Iraq. While some may
argue that the war in Iraq was in fact the correct foreign policy decision, such an
assessment certainly cannot be made on the grounds of con…scating weapons of mass
destruction.
A …nal thing to come out of this analysis is a consideration for how information
sources should determine their bias, when a little bias is inevitable. On the one
hand, the results suggest the rather unhopeful conclusion that if some bias in the
reporting of information is unavoidable, then even if one side tries to be as unbiased
as possible, advocates of the other side my …nd it optimal to respond with a lot of
bias (leading to the types of situations described in the preceding paragraph). On
the other hand, the analysis also suggests that it is generally not helpful to …ght gross
misinformation from one side with further misinformation of your own. Rather than
engaging in such an arms race, it is better to try to deliver information as unbiasedly
as possible (if you are indeed on the correct side). However, the analysis also shows
that in a world of biased information, as time goes on, the beliefs of those acting
on each side of the claim will become stronger and stronger that they are right and
those on the other side are wrong. This will mean that as time passes, those acting
on one side of a controversial claim will become more an more convinced that the
actions and beliefs of those acting on the other side of the claim are simply “crazy.”
In general, this paper shows that there are signi…cant costs associated with the
reality, or even perception, that unbiased information is not possible. To the extent
to which actual and perceived bias can be mitigated, society will likely engage in
less ine¢ cient behavior.
27
6
Appendix - Proof of Proposition 4
As shown in the text, all those whose beliefs exceed 2 (i.e., True Believers and
Strong Believers) will …nd it optimal choose the positive bias information source
regardless of bP relative to bN . Moreover, as shown in the text, those whose beliefs
are between
and 2 (i.e., Tentative Believers) will …nd it optimal to choose the
positive bias source as long as the di¤erence between bP and bN is less than or equal
to their Acceptable Excess Bias (AEB( )). As can be con…rmed from equation (12),
as beliefs approach
from above, AEB( ) converges to zero. However, as also
can be con…rmed from equation (12), AEB( ) is continuous and strictly increasing
in . Therefore, for any given bN and bP such that 0 < bN < bP < 1 (meaning
bP bN
< 1); the intermediate value theorem implies that there must exist a 3 (bP )
bN
such that AEB( 3 (bP )) = bPbNbN and AEB( ) > bPbNbN for all > 3 (bP ):
This in turn means that if for a given bP and bN ; 3 (bP )
2 , then for all
those whose beliefs exceed 3 (bP ) the percentage di¤erence between bP and bN
will be less than their Acceptable Excess Bias (AEB( )), meaning they will also
…nd it optimal to choose the positive bias source, while those whose beliefs are
less than 3 (bP ) the percentage di¤erence between bP and bN will be greater than
their Acceptable Excess Bias (AEB( )), meaning they will …nd it optimal to choose
the negative bias source.
On the other hand, if for a given bP and bN , 3 (bP ) > 2 , then percentage di¤erence between bP and bN will be greater than the Acceptable Excess Bias (AEB( ))
for all those for whom
< < 2 ; meaning all Tentative Believers will …nd it
optimal to choose the negative bias source.
Finally, recall that 3 (bP ) was de…ned to be the value of such that bPbNbN =
AEB( 3 (bP )); which we can re-write as bN = AEB( bP(bP ))+1 . Again noting that
3
AEB( ) is strictly increasing in , we know from this equation that for any …xed
bN , it will be true that 3 (bP ) must increase with bP :
28
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30
Fig 1: Summary of "types"
Possible Beliefs entering a period (π)
π*
0 π2*
Will not act on claim
π1*
1 Will act on claim
Tentative
Believers
Strong
Believers
True
Believers
Tentative Believers: Will cease acting period on the claim this period upon hearing a negative signal from either source. Strong Believers: Will cease acting on the claim this period upon hearing a negative signal from positive bias source but not negative bias source. True Believers: Will not cease acting upon the claim this period regardless of what information they hear from either source. Fig 2: Graphical Summary of Propositions 1 - 4 (for bP > bN)
If π3*(bp) < π2*:
π*
0 π3*(bp)
Will not act on claim
π2*
π1*
1 Will act on claim
Tentative
Believers
Strong
Believers
True
Believers
Choose Negative Bias source
Choose Positive Bias source
If π3*(bp) > π2*:
π2*
π*
0 Will not act on claim
1 Will act on claim
Tentative
Believers
π1*
Strong
Believers
True
Believers
Choose Negative Bias source
Choose Positive Bias source
Fig 3: Fraction of Population Consuming Positive Bias Source (negative bias source with low bias: bn = 0.15)
0.60
0.50
Bias of positive
bias source:
0.40
Low (bp = 0.16)
0.30
Medium (bp = 0.5)
0.20
high (bp = 0.8)
0.10
0.00
1
2
3
4
5
6
7
8
9
Round
Fig 4: Fraction of Population Acting on Claim
(negative bias source with low bias: bn = 0.15)
0.60
0.50
Bias of positive
bias source:
0.40
Low (bp = 0.16)
0.30
Medium (bp = 0.5)
0.20
high (bp = 0.8)
0.10
Unbiased source
0.00
1
2
3
4
5
6
7
8
9
10
Round
Fig 5: Fraction of Population Consuming Positive Bias Source (positive bias source with high bias: bp = 0.80)
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
Bias of negative
bias source:
Low (bn = 0.15)
Medium (bn = 0.5)
High (bn = 0.75)
1
2
3
4
5
6
7
8
9
Round
Fig 6: Fraction of Population Acting on Claim
(positive bias source with high bias: bp = 0.80)
0.60
0.50
Bias of negative
bias source:
0.40
Low (bn = 0.15)
0.30
Medium (bn = 0.5)
0.20
High (bn = 0.75)
0.10
Unbiased source
0.00
1
2
3
4
5
6
7
8
9
10
Round
Fig 7: Evolution of Beliefs
Average Belived 1.00
Those acting on claim (biased world)
0.80
0.60
Those not acting on claim (biased world)
0.40
Those acting on claim (unbiased world)
0.20
0.00
1
2
3
4
5
6
7
8
9
10
Those not acting on claim (unbiased world)
Round