DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA 91125
Fake News, Information Herds, Cascades and Economic
Knowledge
Lazarina Butkovich
Nina Butkovich
Charles Plott
Han Seo
California
Institute of Technology
SOCIAL SCIENCE WORKING PAPER 1442
July 2018
2
Fake News, Information Herds, Cascades and
Economic Knowledge
L. Butkovich, N. Butkovich, C. Plott and H. Seo
California Institute of Technology
Abstract
The paper addresses the issue of “fake news” through a well
-known and widely studied
experiment that illustrates a possible science behind the phenomenon. Public news is viewed
as an aggregation of decentralized pieces of valuable information about comple
x events. Such
systems rely on accumulated investment in trust in news sources. In the case of fake news,
news source reliability is not known. The experiment demonstrates how fake news can destroy
both the investment in trust and also the benefits that news provides.
A. Introduction
1
While fake news, propaganda and misinformation are not newly emergent phenomena the
public
’s recognition of
their
growth
is found in many sources. Each of the major news
networks, CNN, NBC, and Fox News, cons
istently
publish remarks about a competitor’s lack of
reliability. Different sides of the political divide accuse each other of promoting fake news
, and
the “main stream” media is accused of taking sides. Analysis finds different reasons for growth
including
underly
ing economics that reward
s “click baits”
, an erosion of moral fiber
and even
espionage
. Against that background, sources express concerns about the consequences of the
tendency. Bloomberg suggest
s direct damages to wealth through an effect of fake new
s on the
stock market.
2
Others have expressed concerns about long
-term consequences. For example,
politician
Hillary Clinton points to the possibility of systemic damage due to a leveling effect or
“false equivalency” in which the unreliable news sources
and reliable sources are regarded as
equals
.
3
The underlying concerns about the consequences of fake news rest on more than intuition
or
impressions
. Scientific support appears to exist. Simple experimental exercises illustrate that
basic principles of
information processes that are known to be features of markets can also
operate for public sources of information and broader decisions influenced by news. News
from public sources can enhance efficiency and create economic benefits for the same reasons
that improved information enhances market efficiency. The experiments also illustrate that the
benefits that emerge from reliable news can be systematically disrupted by fake
news. The
1
The help of B. Atsavapranee, is gratefully acknowledged. All experimental programs were designed by Travis
Maron. The technical contributions of Hsing Yang Lee were very helpful. The financial support of the John
Te
mpleton Foundation is gratefully acknowledged.
2
(Cite
: https://www.bloomberg.com/view/articles/2017-
10-
23/why
-fake
-news
-is -so
-harmful
-to-investors
)
3
The comments are her assessment of academic research as contained in her 2018 Author Miller Freedom to
Write Lecture at the 14th Annual Pen World Voices Festival, April 14, 2018.
3
experiments explore a specific basic principle that suggests how and w
hy associated benefits
are lost.
The experiment
s are
narrowly focused and thus do not address many of the important issues.
Of course, news is a deep part of social fabric, and policies that shape its role, such as freedom
of the press, are widely recog
nized as a cornerstone of a free society.
4
The experiment
s reflect the perspective of F. Hayek, “The Use of Knowledge in Society”
5
, which
is now integrated with economic theory and finance as well as political theory. The basic
principles apply to a world in which information is produced from decentralized sources of
observation.
Information originates locally as a product of the actions of those close to the
facts and with interests aligned with the use of the facts. The information becomes transferre
d
and aggregated through publicly observed actions, such as trades and trading prices in which
good information and incentives to act quickly are closely related. In a statistical sense, the
information contained in scattered observations can become aggre
gated, pooled, and
processed.
6
The resulting knowledge is a resource
, a type of “public good”,
that can be used
multiple times and grants economic value by preventing costly mistakes.
Trust in the reliability of information sources is fundamental to the process. Typically, trust is
derived from an alignment between incentives and the information revealing actions. For
example, people normally visit a restaurant because they like the food
, and thus, the number of
customers can be a source of information about the quality of the restaurant. Or, the location
of fishermen suggests information about the location of fish. Analogously, in markets, the
upward movement of a stock price suggests the possibility of favorable earnings because those
who have uncover
ed favorable facts have an incentive to buy the stock before others do. When
incentives and information revealing actions are aligned, the actions can reveal information
that can be trusted and used. On the other hand, if incentives and actions are not al
igned, the
information revealed can become a type of “fake news” that is misleading and can lead to
costly mistakes.
Experience contributes to an understanding about the reliability of
information. For instance, the diners and fishermen will ignore information if it is known to be
unreliable. That is, if customers are known to be paid by the owner
to sit at a restaurant, or if
boats are known to be populated by
sightseers
as opposed to fishermen, the diners and
fishermen will ignore the actions knowing tha
t no information is carried by observed behavior.
4
The founders were very clear in their thinking about the matter. “Freedom of speech is a principal pillar of a free
government: When this support is taken away, the constitution of a free society is dissolved,” wrote Benjamin
Franklin in
The Pennsylvania Gazette
. John Adams felt that “The liberty of the press is essential to the security of
the state” and Thomas Jefferson held similar opinions holding that “Our liberty depends on the freedom of the
press, and that cannot be limited without being lost.”
5
Hayek, F. A. (1945). The use of knowledge in society. The American economic review, 35(4), 519-
530.
; Hayek, F. A.
(1948). Individualism and economic order. University of Chicago Press.
6
The
information aggregation
phenomen
on
was first demonstrated in market experiments by Plott and Sunder
(1982, 1988).
Recent applications of the basic principles to complex experiments and field settings can be found at
Court, McKenzie, Plott, and
Gillen (2018).
4
If incentives and actions are not aligned to produce reliable information, the use and value of
public news suffer.
A key feature of the model is a principle of “information revealing choices/behavior”. If an
observer knows the motivations of a decision maker, then the observer can use the decisions
made to make an inference about what the decision maker knew at the time of the decision.
That is, if a decision maker’s incentives are known, then their cho
ice of actions can give insights
about the information on which the choice was made
– a type of “invertability”
. The fact that
people can make this deduction, even though not always successfully, suggests the processes
through which fake news finds its wa
y to making an impact. If the bias of a source is known, an
observer can compensate for the bias and extract the hidden information as if the bias did not
exist. On the other hand, if the incentives are not known, then a foundation for collecting and
usin
g the information does not exist. The news carries no information aside from the fact that
it cannot be trusted.
B. Behavioral P
rinciples
A simple experiment is used to demonstrate four fundamental points. First and most basic are
the facts that (i) tr
usted
news is a key feature of information aggregation and (ii) the
aggregated information creates economic value. The improved information leads to better
decision
-making that becomes directly translated into improved income, which can be
measured in the exper
iment. (iii)
Successful aggregation depends
on the confidence placed by
decision makers on the reliability of an information source. The confidence is derived from the
decision makers’ understanding of the alignment of the source’s incentives and reports
derived
from actions taken by the source. Do the reports from the source accurately reflect the
information held by the source? Do the actions taken by the source, possibly adjusted for
known biases, accurately reflect what the source knows
? (iv)
Finally, if the confidence in the
information source or the aggregation process is damaged then the value created by the
information is lost. The confidence reflects a building process, a type of social investment
in a
“public good”
resulting from time and expe
rience.
Together the experiments illustrate how fake news can destabilize and erode the foundation of
the information building process and cause a loss of the potential economic value. The
experiments demonstrate how fake news can be economically
damagin
g through an erosion of
a major resource –
knowledge. Such erosion could be damaging in other ways as well, e.g.
socially.
C. The Experiment and Overview
Three experimental conditions are studied. The first condition is a case in which news sources
can
be trusted, i.e. no fake news. The second condition is a condition in which both the sources
of inaccurate news and associated biases are
known to all, i.e. known fake news. The third
condition is one in which all know about the possibility of fake news b
ut it cannot be objectively
5
identified as such. The reader
of the news
cannot reliably differentiate unreliable news from
reliable news, i.e. unknown fake news.
Naturally, the experiment is a very simple case where the principles can be observed. The three
conditions studied are all based on the same set experimental setting.
7
Two urns have three
balls each. One of the urns has two red balls and one white. Call this the red urn. The other urn
has two white balls and one red. Call this the white urn. One
of these urns
, called the chosen
urn
, is selected
at random at the beginning of a period (50:50). The subject
s do not know which
urn was chosen but each subject is
privately
shown a random draw of one ball
from the chosen
urn. The subject gives a (public
) report
reflecting the subject’s guess about the color of the
chosen urn,
red urn or white urn. T
he earnings of the subject depend on the report
and the
actual color of the chosen urn. Different conditions involve different reporting incentives
that
wil
l be called “normal” or “reverse” as will be described next
. All subjects know their own
reporting incentives. Subjects draw
from the chosen urn
and report in sequence and a
ll subjects
observe all previous reports
. T
he ball is replaced after each subject draws.
In the
no fake news
condition, subjects have incentive
s to report the actual urn used to make
the draw
, called “normal incentives”
. Of course, they do not know definitively but they have
incentives to make a correct report and
the incentives are c
ommon knowledge
. In the no fake
news
condition,
the subject earns a monetary reward (+$1.50 in the experiment) if the report is
correct and loses money (
-$0.50) if the report is incorrect. Since
the urns are chosen with equal
probability
, the best report
based on a
single,
isolated draw is the color of the revealed ball.
That is, subjects with
normal incentives
report red urn if the ball is red and report white urn if
the ball is white
.
In the known fake news
condition some subjects have normal
incenti
ve
s to report the actual
urn used, and three
randomly chosen subjects have incentive
s to report the urn NOT used.
These three have “reverse” incentives and all subjects know which reports were made by
subjects with reverse incentives. Given their informat
ion, those with reverse incentives can
make their own determination about the actual urn and use that determination to form a
report. Those with reverse incentives earn a monetary reward (+$1.50 in the experiment) if the
report is the urn not used and los
e money (
-$0.50) if the report is the actual urn. Again, since
the choice of the urn is 50/50
, in the absence of additional information the best report is the
color of the revealed ball for those with normal incentives (draw x and report x) but the
opposi
te color for those with reverse incentives (draw x and report y).
In the
unknown fake news
condition subjects know that some subjects might have reverse
incentives but they do not know the number or location of subjects with reverse incentives. All
7
The experiment was first introduced by Anderson and Holt (1997). Information and efficiency measures were
developed by Hung and Plott (2001) who also replicated the Anderson and Holt results. The experiments are based
on theoretical mod
els develop
ed
by Banerjee (1992) and by Bikhchandani, Hirshleifer,
and Welch (1992).
Willinger
and Ziegelmeyer (1998) and Ziegelmeyer, Koessler,
Bracht and
Winter
(2010) study cases in which participants
receive different qualities of information and show
that subjects with more accurate private signals correct
inaccurate information aggregation.
6
subjects know their own incentives but no
t the incentives of others.
Other features of the
experiment are exactly the same as the other two conditions.
In all conditions, all other subjects observe the report of the first subject but not the color of
the
ball that was shown to the first subject, and they do not know if
the first subject reported
correctly or not. A second subject is chosen at random. The experimenter uses the same urn
as before, draws one ball at random and reveals it to the second subje
ct who makes a guess
about the urn and makes a public report on the urn. All other subjects observe the report. The
ball is replaced in the urn. A third subject is chosen at random and shown a ball drawn at
random from the same urn as the previous two s
ubjects. The third subject reports on the urn,
and all other subjects observe the report. The process continues until all subjects have made a
report about the actual urn. Again, all subjects know that
unless the incentives are reverse a
subject earns $1
.50 if the subject reported the actual urn but loses $0.50 if the subject reported
the other urn. If the subject has reverse incentives the incentives are the opposite and in that
sense the subject has an incentive to report “fake news”.
Now, notice the i
nformation the third
subject
in the sequence has available as a result of the
“news” of the previous two subject reports. If subject three believes the previous two subjects
are rewarded by accuracy in the sense of reporting the correct urn and that the t
wo other
subjects want to make as much money as possible, then the reports carry information about
the color of the drawn balls revealed to them. Thus, at the time of decision in the no fake news
condition the third subject has information about three dra
ws, the two previous subjects and
his/her own draw. The optimal choice is dictated by the proportion of colors in the sample. If
two or three are of the same
color,
then the best choice of urn is that color.
The value and productivity of the “news” can be theoretically computed and compared with the
observed. If an individual has no private information and no news, then a natural model of
choice would be random with 50% being correct and 50% being incorrect. The expected payoff
would be $0.50 per perio
d. If each individual has a private source of information, the private
draw, then by using it as the basis of a decision
they
will choose the correct urn with 67%
probability and the incorrect urn 33% of the time with an expected value of $0.83 (purely
theoretical). If the calculation is based on the actual draws that were used in the experiment as
opposed to the probabilities,
then
on average the value
is $.79.
If an individual has access to news such as the reports of others, the information can be add
ed
to the information provided by the individual’s private draw. In the case of no fake news if all
individuals use the data from reports (the “news”) then “herds” or “cascades” will be observed
in which all decisions are eventually the same, which
would
lead to all making the correct
choice approximately 75% of the times and incorrect 25% producing a per person expected
value of, $1.01.
8
I t is important to realize that such conformity i
s not accidental and is, indeed,
8
The perfect cascading result was calculated by considering if each subject reports based on their draw unless the
majority public information plus the private dr
aw contradicts the private draw, in which case the subject would
7
beneficial
as this value is higher t
han $0.79 that would be obtained if all subjects disregarded
the public information.
D. Experimental Design and Procedures
Four experimental sessions were conducted on four different days. Each session used ten
subjects for a total of forty subjects. Each session consisted of unpaid practice periods ranging
from 5 to 10
and three experimental conditions. Subjects made fifteen reports
under
each of
the conditions. The major features for all experiments are in Table
1.
Table
1
Experimental design: dates conditions, periods,
experiment
20180207
20180228
20180305
20180411
Number of subjects
10
10
10
10
Average Total Earnings per
person
($)
36.3
40.1
35.5
36.3
Experiment length
1.5 hours
1.5 hours
1.5 hours
1.5 hours
Subjects
EEPS lab
Caltech
students
Caltech
students
Caltech
students
Caltech
students
Condition
periods
periods
periods
periods
Unpaid Practice
fake news
1
-
5
1
-
5
1
-
5
1
-
5
No Fake News
Paid Practice (unused)
Data used
*
6
-
20
6-10
11
-
25
6-10
11
-
25
6-10
11
-
25
Fake known
21
-
35
26
-
40
26
-
40
26
-
40
Unknown fake
36
-
50
41
-
55
41
-
55
41
-
55
*For experiments 20180228, 20180305, and 20180411, the No Fake News condition has 5 extra
practice rounds (rounds 6
-10), and the data from these rounds were not used in the data
analysis (they were treated as the other practice rounds).
Subjects we
re Caltech students recruited using the recruiting system of the Caltech Laboratory
for Experimental Economics and Political Science (EEPS) and reported to the Caltech EEPS
laboratory. Upon arriving at the laboratory, subjects were randomly seated at a st
ation with
screening partitions and a computer and were instructed to not talk or communicate.
All experiments were conducted in the same manner. When participants walked into the room,
they were given colored PowerPoint instructions (see Appendix), a table to fill out during the
experiment (about information such as their incentive type and per
-round payoffs), and a
report based on the public information. All subjects consider the incentives behind each report. Over 15 rounds,
the 10 subjects earn $151 total using this method, so the per person expected
value per round is $1.01,
approx
imately
75% correct and 25% incorrect.
8
writing utensil. Each participant was guided to a seat with a computer, without view of any
other computers or individuals; no communication
of any kind was allowed except for
questions to supervisors. After all 10 people read instructions, the instructions were
summarized (specifically, incentive types and how to use the program), and initial questions
were answered individually. Then, several simple examples were shown on the board in the
style of the program to be used. Examples included: [If I have normal incentives] (I) If the first 2
people chose red, and I drew red, then I should choose red; (II) If the first person chose white,
the seco
nd chose red, and I drew red, then I should choose red; (III) If the first person chose
red, and I drew white second, then
given a slight confidence in my choice over someone else,
I
should choose white; (IV) If the first 2 people chose red, but the first two people have reverse
incentives, and I drew white, then I should choose white. (V) If I have reverse incentives, and
the first 2 people (normal incentives) chose white, and I drew white, then I should choose red.
After these examples, individual questi
ons were answered. All programs were initiated for the
practice rounds, during which people could ask final questions. After the practice rounds, no
further questions were answered. People were reminded if parameters changed at the start of
new periods. At
the end of the experiment, participants calculated their total earnings,
excluding the practice rounds, and were presented cash accordingly.
E. Measurements
Subjects’ earnings and thus efficiency of the news system depended on the incentive structure
and the decisions subjects made. The subject had incentives to make a correct report about
either which urn was used to make the draw or the opposite, the urn that was NOT used to
make the draw. The incentives differed across experimental conditions.
Normal incentive
: If the subject reports the “correct urn”, the urn from which the ball was
drawn, the subject earns $1.50 and loses $0.50 if the report is not the correct urn.
Reverse incentive
: If the subject reports the “incorrect urn”, the urn from which the ball was
not drawn, the subject earns $1.50 and loses $0.50 if the subject reports the correct urn (the
urn from which the ball was drawn).
E.1
. Efficiency
Measurements
Typically,
efficiency reflects the wealth produced by a process. In these types o
f experiments,
it
is the money earned by participants relative to the maximum that could have been earned.
When earning depends on information the measurement must be adjusted to the information
possibilities.
Complete information standard
: The complete
information standard reports efficiency relative
to the hypothetical case in which there is complete public sharing of all draws before any
choices are made. From the actual draws used in the experiment, it was calculated that if all
subjects are informed
of all draws before making a choice, the expected value for a condition
9
(15 rounds) is $16.
5 per person.
9
This means that in 12 out of 15 rounds, aggregating all the 10
private signals would lead to the choice of correct state of the world. For reference,
there is
79% probability that six or more signals would be from the correct urn.
Thus, 100%
efficiency
according
to this measure is based on all available information even though potentially
impossible to use due to the timing or incentives.
10
F. RESULTS
Four classes of results are reported. The first result demonstrates that the principles work by
measuring the wealth created by a news delivery system to an identical economic environment
in which no news system exists. The second result is a demonstra
tion that a news system that
carries fake news can function well if the sources of fake news and the nature of the fake news
are known in the sense that the motivation of those reporting the news is known. An
understanding of the incentives of those repor
ting the news works as part of a correct
interpretation. The third result demonstrates that fake new has a negative impact on the value
of an information reporting process. If fake news sources cannot be distinguished from reliable
news sources, then the
benefits of news are lost.
11
The system tends to revert to the case
where no news is available. The final result focuses on the process by illustrating that the value
of news is a product of an investment in learning that builds with experience
, and in th
e
presence of fake news the investment value becomes eroded.
A comparison between reliable news and no news
at all
is made possible by experiment 1
The
appropriate measures from a technical computation when no news exists are in Table
2 and are
presented
with measures
from reliable news
produced by experiment 1.
The comparison is
reported as Result 1
.
9
From the actual draws used in the experiment, 1 of the 15 rounds had a tie between the two colors. To account
for this in the hypothetical complete information standard, half of the s
ubjects gained money and half lost money.
10
All efficiencies were calculated by analyzing the actual draws people were given.
Other efficiency measures are possible including
efficiency relative to “completely rational” behavior of others.
The measure is based on the assumption that all other individuals use statistics properly and all assume that all
others do as well. This means that in the absence of tied reports all individuals after the first 3 choose according to
the majority signal of the first 3 people. If the first two choices disagree with the third person’s draw, then the
third person also chooses according to the first 2 people. If the first two choices disagree then
the third person
follows own signal. The fourth person can be placed in the same position as the third and in this case the analysis
is repeated. Because this measure is sensitive to the first three draws it can exhibit variability of performance and
efficiencies above 100%. Each such instance requires additional explanation and thus the measure has deficiencies
as an explanatory and comparison tool.
11
Hung and Plott (2001) study a case with social pressures for conformity in which people are punished for reports
that diverge from the reports of others. The results demonstrate that incentives for conformity have the capacity
to remove almost all benefits of news. The very first report has an impact of blocking the information content of all
additional
reports as subsequent reports acquire conformity and avoid punishment by matching the content of the
first report.
10
Result 1. Reliable news creates
additional
value compared to the case
when only private
information exists. This occurs through a reporting and information aggregation process. If
news from a public source is reliable (the absence of fake news) information becomes
aggregated through a reporting process and the resulting knowledge becomes an additional,
productive source of value creation.
Support. Table 2
demonstrates that in comparison with the identical system that has only
private news sources, no public news, the existence of a reliable news source (no fake news)
gene
rates benefits in all dimensions of comparison. Total income goes up by 4.62%, the percent
of correct (income earning) decisions goes up by 1.8
%
and
the
system efficiency goes up
by 3%
.
The second result compares behavior when everyone knows that
no sourc
es of fake news exist
and when sources of fake new
s do
exist and
everyone knows
both the sources of fake news and
the
biases.
Table
3
Comparison of reliable (no fake) news condition and known fake news sources
condition
condition
Total earnings
$
675
maximum
Earnings per
person
Percentage
correct
Efficiency
No fake news
$498
$0.83
66.5%
80%
Known fake
news sources
$52
9
$0.88
69
.0
%
85%
Result 2. If news sources have known biases, individuals adjust for the biases and information
aggregation
tends
to
operate as if there were no bias. Biased news can be processed and the
unbiased news
content
extracted if the biases are fully recognized and understood.
Support. The support for the Result 2 is found in Table
3. The existence of fake news has
no
impact if people know the sources that are fake. If seeming irrationality or motives are
understood, observers just modify the information according to the reliability of the
information source. Here the comparisons are substantially along the lines of
#1.
i. With known fake news numbers of correct decisions as comparable to no fake news Compare
69% (fake news) vs. 66.5% (no fake news) (Z score 0.927 p value 0.177)
Table 2 Comparison of private information condition and
the no fake news condition
condition
Total
earnings
Earnings per person
per round
Percentage
correct
Efficiency
Private Information Only
$476
$0.79
64.7%
77%
No fake news
$498
$0.83
66.5%
80%
11
ii. earnings are comparable between no fake news and known fake news
- no statistical
dif
ference between earnings per person of no fake and known fake. compare –
not different
with 83% confidence, z=0.9374
iii. information use in decisions is comparable –
comparison of the Bayesian use of other’s
decisions as opposed to private information sug
gests similarities between no fake news and
known fake news environments
. Compared
to
the
theoretically
perfect
information aggregation
(cascade
), no fake news/known fake news achieve 82.5% and 87.4% efficiencies
Of course
, issues exist regarding
the nature of the bias
. Bias can exist in many forms including
use of the language, illustrations, opinions advanced as reported facts, exposure frequency,
location in a new source, etc. The question posed here is if a bias has an impact when it is
obvi
ously a bias
. Is there a natural tendency to translate the message
; remove the
obvious
bias
and extract the accurate content?
Result 2 demonstrates that bias removal is clearly within a
natural
scope of individual human capabilities
.
The third result studies the case in which bias
es are known to exist but the sources of fake news
are unknown. The public information carries no obvious method of determining reports that
represent fake news from reports that are accurate.
The consequences are reductions in
earnings and efficiency.
Table
4
Comparison of reliable (no fake) news condition and fake news (fake news sources
unknown) condition
condition
Total earnings
$X maximum
Earnings per
person
Percentage
correct
Efficiency
No fake news
$498
$0.83
66.5%
80%
Unknown fake
news sources
$456
$0.76
63
.0
%
74%
Result 3. The effect of fake news is to substantially diminish or to terminate information
aggregation and remove the benefits of the use of available information.
Basically
, the impact
of fake news is to
destroy the benefits of the news system. The system performance returns to
the base condition in which a public news process does not exist referenced in Table 1.
Individual decisions do not benefit from the information held by others.
Support. The support for Result 3 is found in Table 4
. When compared to a system of reliable
news (no fake news) the performance becomes degraded in all dimensions. Total earnings and
average earnings fall.
The percentage of correct decisions falls as does system effici
ency.
(Z
score 1.29 p
-value 0.099) Additionally, compared to the baseline case of no public news, the
earnings and the percentage of correct choices are lower.
The next result, Result 4
, addresses the use of news sources
and demonstrates that the impact
of fake news works through a shift from decisions based on the aggregation of information
12
found in the news to the less informative
, private sources of information
.
12
The result illustrates
that in the absence of fake news the public news is used for decisi
ons. However, in the
presence of unidentifiable
fake news the public news sources are abandoned in favor of news
produced by private sources
.
Bayes Law is used as a model of how
decision
data are incorporated into
individual decisions
(Grether, 1980, 199
2). The following definitions are needed.
Let A be defined as the event that urn A is the actual urn, and let B be defined as the event that
urn B is the actual urn. Let x
it
= (a
it
, d
it
) be the information of individual i at position t, such that
a
it
is defined as the information (the number of A and B choices made by those ahead) that
individual i has observed from individuals at positions previous to t, and d
it
is defined as the
private draw of individual i at position t. While a
it
and d
it
are corre
lated with each other, they
are conditionally independent given a particular state of the world (A or B).
Using Bayes Law,
we obtain the following.
푃푃
(
퐴퐴
|
푥푥
푖푖푖푖
)
푃푃
(
퐵퐵
|
푥푥
푖푖푖푖
)
=
푃푃
(
푥푥
푖푖푖푖
|
퐴퐴
)
푃푃
(
퐴퐴
)
푃푃
(
푥푥
푖푖푖푖
|
퐵퐵
)
푃푃
(
퐵퐵
)
=
푃푃
(
푎푎
푖푖푖푖
|
퐴퐴
)
푃푃
(
푑푑
푖푖푖푖
|
퐴퐴
)
푃푃
(
퐴퐴
)
푃푃
(
푎푎
푖푖푖푖
|
퐵퐵
)
푃푃
(
푑푑
푖푖푖푖
|
퐴퐴
)
푃푃
(
퐵퐵
)
Taking logs and rearranging:
(1)
푌푌
푖푖푖푖
≡
ln
�
푃푃
(
퐴퐴
|
푥푥
푖푖푖푖
)
푃푃
(
퐵퐵
|
푥푥
푖푖푖푖
)
�
=
훼훼
+
훽훽
ln
�
푃푃
(
푎푎
푖푖푖푖
|
퐴퐴
)
푃푃
(
푎푎
푖푖푖푖
|
퐵퐵
)
�
+
훾훾
ln
�
푃푃
(
푑푑
푖푖푖푖
|
퐴퐴
)
푃푃
(
푑푑
푖푖푖푖
|
퐵퐵
)
�
+
푢푢
푖푖푖푖
Where
푌푌
푖푖푖푖
is the belief about the state of the world given
푥푥
푖푖푖푖
. Note that
푙푙푙푙�
푃푃
(
퐴퐴
)
푃푃
(
퐵퐵
)
�
=
0
is
canceled out
since P(A) = P(B) = ½ from the initial priors. From the data, we can apply equation
(1) and
find
훼훼
,
훽훽
,
and
훾훾
.
We want to determine if
훽훽
and
훾훾
coefficients differ under different trial conditions (specifically
comparing the
earnings of news conditions “no reverse” with the “known reverse” and
“unknown reverse”).
The key variable,
푑푑
푖푖푖푖
, the private information, is measured as +1 or
-1
depending on the signal the subjects receive.
Similarly
푎푎
푖푖푖푖
, the public information in r
eports
available to subject i at position t is measured by the difference in observed actions
measured
as (1 or -
1) depending on the report.
Define
푠푠
푖푖푖푖
as the private signals each individual i receives at position t (1 or -
1) and
푆푆
푖푖푖푖
=
(
푠푠
푖푖1
,
푠푠
푖푖2
,
⋯푠푠
푖푖
(
푖푖−1
)
)
as the signals all individual from position 1 to t
-1 received. Then
푃푃
(
푆푆
푖푖푖푖
|
퐴퐴
)
푃푃
(
푆푆
푖푖푖푖
|
퐵퐵
)
=
푃푃
�
∑
푠푠
푘푘
푖푖−1
푘푘=1
|
퐴퐴
�
푃푃
�
∑
푠푠
푘푘
푖푖−1
푘푘=1
|
퐵퐵
�
. In other words, sufficient statistic for calculating the posterior odds of the state of
the wor
ld is simply the difference between the number of signals received and the order of the
signals do not matter.
With the additional assumption that the subjects believe that previous choices were made in
accordance with private signals, we can use the diff
erence in the number of publicly observed
12
Goeree et al (2007) demonstrate the agents tend to overweigh their private signals
.
13
choices as a proxy for
푎푎
푖푖푖푖
. Given such measurements
, the variables are bounded:
ln
�
푃푃
(
푑푑
푖푖푖푖
|
퐴퐴
)
푃푃
(
푑푑
푖푖푖푖
|
퐵퐵
)
�
would either be
±0.693 depending on the private signal and
ln
�
푃푃
(
푎푎
푖푖푖푖
|
퐴퐴
)
푃푃
(
푎푎
푖푖푖푖
|
퐵퐵
)
�
would range from -
6.9
31 to 6.931. So we can perform a linear regression after appropriate change in variables.
Result 4. (i) Individuals always place more decision weight on the private sources of information
than on the public sources of information. (ii) The relative weig
ht on public information is
reduced if the condition is changed from either no fake news or known fake news to the
condition of unknown fake news.
Support.
Table 5 contains the
result of the regression using Bayes’ law as expressed in (1) as a
model.
Table
5: Bayes’ linear regression
Variable
Coefficient
s.e.
Z
-
test
p
-
value
Logit 1: No reverse
Intercept
-
0.0897
0.0257
-
3.49
0.0005
Public Information
0.1773
0.0091
19.48
<2e
-
16
Private Signal
0.7159
0.0369
19.41
<2e
-
16
Ratio
4.0380
Logit 2:
Known reverse
Intercept
-
0.0216
0.0261
-
0.83
0.4070
Public Information
0.1808
0.0092
19.62
<2e
-
16
Private Signal
0.6822
0.0382
17.86
<2e
-
16
Ratio
3.7727
Logit 3: Unknown reverse
Intercept
-
0.0210
0.0193
-
1.09
0.2770
Public Information
0.1229
0.0119
10.35
<2e
-
16
Private Signal
1.1914
0.0282
42.33
<2e
-
16
Ratio
9.6981
Overall, we observe that the regression coefficients stated in (1) are similar for ‘no reverse’ and
‘known reverse’ condition
meaning that under those two conditions individual
s give
public and
private information about the same
weight
when making their own decision
. Specifically
,
훽훽
1
and
훽훽
2
are statistically equivalent to each other
(Z score 0.270, p value 0.787)
and
훾훾
1
and
훾훾
2
are
statistically equivalent to each other
(Z score 0.635 p value 0.526)
. However, in the ‘unknown
reverse’ condition the data
measurements from (1)
demonstrates
both a decreased influence
of public information and an increased influence of private signal.
훽훽
1
≈훽훽
2
>
훽훽
3
(One tailed test
between
훽훽
2
and
훽훽
3
leads to Z score of 3.8439 and p value 0.00006) and
훾훾
1
≈훾훾
2
<
훾훾
3
(One
tailed test between
훾훾
2
and
훾훾
3
leads to Z score of 10.7242 and p value<10
-10
).This suggests that
14
in the ‘unknown reverse’ condition, subjects tend to ignore the public information and stick
with their private signal
when making a decision
.
13
To check for robustness,
of the Bayesian model we use a “Direct
Measurement” method
for
comparing the relative importance of public information and private signal.
Basically, we count
the number of reports consistent and inconsistent with the information in an individual’s own
draw
. Let
u be the number of reports that a subject i observes which support
s d,
the pri
vate
draw of i
. Conversely, v is the number of reports observed by i that are
the opposite of i's
private draw
. Then we l
et
the public information a(i,
u,v ) =
u-v be
a measure of the difference
between
reports in accordance with the private
signal and against it.
In the Direct Measurement model the public information is based on the difference of observed
choices between red(+1) and white(
-1). The order is deemed to be irrelevant.
The numbers are
applied in a model to produce the probability of the col
or ball the individual will choose given
the numbers the individual observes at the time of choice.
The measurement is taken from a logistic
regression on the differences where we choose
β
and
γ
that best fits
(2)
y =
�
1
푖푖푖푖
β
∙
a +
γ
∙
d +
ε
>
0
0
푒푒푙푙푠푠 푒푒
where a is the public information available to the individual at the time of decision, d is the
private signal and y is the choice variable.
ε
is the error term distributed by standard
logistic
distribution and
y
′
=
β
∙
a +
γ
∙
d +
ε
is the latent variable
which acts as an intermediate step
towards
measuring
subjects’ choices
.
Note that d={
-1,1} and a={x|
-9
≤
x
≤
9, x is an integer}. The restriction follows from the number
of subjects and the choices sequences used in the experiment. The individual in the kth choice
slot has k
-1 reports to view and there were 10 subjects in the experiment.
In the (2) t
he key variable of interest is
γ
/
β
,which is the relative weight of the private signal
relative to on
e unit of public information.
13
T
he intercept (α) is statistically equivalent to zero for both ‘known reverse’ and ‘unknown reverse’ condition.
This is to be expected if
the subjects do
not have a bias for Red or White once public information and private
signals are taken into consideration. The fact that the intercept is non-
zero for the ‘no reverse’ condition is quite
odd but the effect size is not that large since the bias corresponds
to about 4% swing in probability.
15
Table 6: Direct Measurement logit regression model
The
completed regression
produces
a binary prediction of the subjects’ choice based on the
coefficients.
In other words, the estimated parameters for
γ
and
β
are
given as
β
�
and
γ
�
. Then
for all sets of private signal and publicly observed information (a,d), if
β
�
∙
a +
γ
�
∙
d
>0, subjects
are predicted to choose red (+1), if
β
�
∙
a +
γ
�
∙
d
<0, then subjects are predicted to choose
white(
-1).
Figure 1
demonstrates that
the direct measurement model’
s prediction agrees
with
the actual
subjects
’ behavior. The horizontal axis displays the net public information. i.e., [
{the number of
observed public
reports
that agree with the subject’s private signal
} minus
{the number of
observed public
reports
that disagree with the subject’s private signal}
]. The stacked bar graph
shows the subjects’ choices that were correctly predicted
by
the model
in blue and that were
incorrectly predicted in
grey
.
Variable
Coefficient
s.e.
Z
-
test
p
-
value
Logit 1: No reverse
Public
Information
1.1605
0.1097
10.58
<2e
-
16
Private Signal
3.313
0.2948
11.24
<2e
-
16
Ratio
2.855
Logit 2: Known reverse
Public
Information
1.0852
0.1035
10.49
<2e
-
16
Private Signal
2.9449
0.2613
11.27
<2e
-
16
Ratio
2.714
Logit 3: Unknown reverse
Public
Information
1.1717
0.1475
7.946
1.92e
-
15
Private Signal
5.0822
0.422
12.043
<2e
-
16
Ratio
4.337
16
Figure 1: Count of correctly and incorrectly predicted actions according to model
The model
correctly predicts 92.8% of subjects’ choices pooled across all settings.
Most of the
“incorrect” prediction occurs when the subjects observe a net of
-2 or -
3 public information
that goes against their private signal.
Furthermore,
the model can be used to augment the binary prediction by adding
measurements of
how the choice probability changes according to the
different
levels of
public
information encountered. The
figures show
the actual proportion of subjects choosing
according to their private signals (blue
solid
line)
in
contrast to
the choice probability model
derived from
the logit regression (
pink dashed line).
The horizontal axis displays the net public
information as be
fore.
0
50
100
150
200
250
300
350
400
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
Count of correct and incorrect predictions
Net Public Information
Model acccuracy when predicting subject's actions
Correct
Incorrect
17
Figure
2: Probability of choosing in accordance with private signal (no reverse) –
Data and Logit
model prediction
18
Figure
3: Probability of choosing in accordance with private signal (known reverse) –
Data and
Logit model prediction
19
Figure
4: Probability of choosing in accordance with private signal (Unknown reverse) –
Data
and Logit model prediction
We can observe that the regression model fits the data well across all settings.
This provides
additional validity to our usage of ‘direct meas
urement’ model.
Specifically, the ratio
of
γ
/
β
tells us the answer to the question “How strong should the public
information be in order for the majority of subjects to go against their private signals?”. In the
‘no reverse’ and ‘known reverse’ case, the answer is somewhere between 2 and 3 while for the
‘unknown reverse’ case, a signal strength of 4~5 is required.
While the result itself is intuitive in that people would put more emphasis on their private signal
in the face of uncertainty, the difference between the settings is more pronounced than the
numbers simply suggest.
In the ‘no reverse’ and ‘known reverse’ case, 21% of the subjects chose to go against their
private signal while in the ‘unknown reverse’ case, that number drops to 7%, 1/3 of the value.
20
The reason for this big change is that the ‘unknown reverse’ setting
has two effects. The first
one, as mentioned, is that subjects put more emphasis on their private signals. The second one
is that the introduction of reverse incentives itself inherently reduces the probability of a strong
public information, even by chanc
e. In the ‘no reverse’ case even if everybody were naïve and
relied on their own private signals, the fourth subject would observe
-3 public signal with 12%
probability. In the ‘unknown’ reverse case, this is reduced to 4%.
The lack of cascade in the
‘unknown reverse’ case is a crucial feature.
G.
Summary of Conclusions
Current news providers appear to be engaged in a war with each devoting resources to
illustrate that the other side is guilty of producing fake news. Each provides evidence that the
othe
r side does not produce reliable news and that the
other side
use
s subtle tools to avoid
being detected. Conflicts of this sort have properties that game theorists term a “war of
attrition” that is wasteful as are all wars and ends only when a “winner” eme
rges and collects
all
resources that survived
. In the case of fake news
, the damage might be to the news system
itself as the public loses confidence in the reliability of information delivered through the news.
Basically
, the experiment suggests that the fears and negative prognostications appear to be
justified and that the damage will not be repaired immediately.
The experiment reported here draws on research found in the information aggregation
literature from economic
s and finance. Information dispersed across many observers becomes
aggregated in the form of a signal that can be valuable in worlds of decision making under
uncertainty.
The aggregation can be mistaken but on average it represents and improvement
and wh
en source reliability is known automatic recovery is possible (Goeree, Palfrey, Rogers
and McKelvey (2007).
The experiments demonstrate that fake news can undermine the
foundation of the process when source reliability is unknown.
Three experiments were
conducted. The first involved no fake news and in this case the
experiments demonstrated that subjects learned to rely on public news sources because such
reliance improved their income. The second experiment introduced reverse incentives for
some news so
urces that gave the incentive to produce false reports. In this experiment
subjects knew which reporters had reverse incentives and as a consequence subjects adjusted
and translated the report so the proper news was extracted. The report from a source kn
own
to be biased toward x was properly translated to y. The result was that fake news had no
effect.
The third experiment removed information about source reliability. Incentives of all individuals
were unknown but the possible existence of reverse ince
ntives was known. Information use
shifted away from public sources to private sources. The advantage of information aggregation
was lost. As a result, this third experiment removed the advantages of public news sources and
information aggregation. The pr
ofits made by participants decreased and were comparable to if
they had only private information.
21
The lesson here is that some of the advantages of public news are derived from known
principles of behavior found operating in many places in the economy. I
n part these principles
depend on a trusted connection between reporters' incentives and the information they are
capable of reporting. Fake news destroys that relationship and consequently carries
implications beyond the fact that some people lie.
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23
SECTION I APPENDICIES
INSTRUCTIONS
DATA APPENDIX
0=normal
INSTRUCTIONS POWERPOINT: