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DEWEY,
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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
PUBLIC DISAGREEMENT
Rajiv Sethi
Muhamet
Yildiz
Working Paper 09-03
February 15, 2009
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
This
http:/7ssm.com/abstract= 1345484
Public Disagreement*
Muhamet
Rajiv Sethi 1
February
15,
Yildiz*
2009
Abstract
Members
of different social groups often hold widely divergent public beliefs regarding the
nature of the world
in
disagreement, and use
which they
mation and the consequences of
private information,
show that when
and repeated communication
beliefs are identical to those arising
disagreement
is
some
greater
ber of individuals
is
when
large,
until beliefs
private information
under observable
is
priors.
approximately equal to disagreement
terms of the observability of
priors,
is
involves heterogeneous priors,
become public information.
When
priors are independently dis-
we show how
when they
are observable.
If
the num-
entirely in the sense that disagreement
in prior beliefs.
Interpreting integration
increases in social integration can give rise to
divergent public beliefs on average. Communication in segregated societes can cause
biases to be amplified,
and new biases
announcements are public and
disadvantage
in
all
to
We
eventually aggregated and public
never revealed and the expected value of public
priors are unobservable than
in
all
The model
communication breaks down
public beliefs
less
all
private information
in
is
develop a model that can accommodate such public
social integration.
priors are correlated,
tributed, however,
We
live.
to explore questions concerning the aggregation of distributed infor-
it
initial
emerge where none previously existed. Even though
signals equally precise, minority groups
the interpretation of public information that results in
members
medium run
face a
beliefs that
are less closely aligned with the true state,
*We thank
the Institute for Advanced Study for financial support and hospitality, and Danielle Allen, Roland
Benabou, Sylvain Chassang, Eric Maskin, Debraj Ray, and seminar participants at IAS and
NYU
for helpful
com-
ments.
'Department
of
Economics,
Barnard
College,
Columbia University
and
Institute
(rs328@columbia.edu).
'Department
of
Economics,
MIT
and Institute
for
Advanced Study (myildiz@mit.edu).
for
Advanced Study
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/publicdisagreemeOOseth
Introduction
1
Members
of different social groups often hold widely divergent beliefs regarding the nature of the
world in which they
many
In
live.
instances such beliefs are not openly expressed, and hence
knowledge of the disparity remains confined to a
however,
when
inescapable.
A
the criminal
trial of
dramatic example of
moments
one, heads
verdict
all
was
this occurred
on October
O.J. Simpson tuned in to hear the
report describes the scene in
"In the
relatively small
set.
a high profile event triggers public reactions that
New
1995,
3,
York's Times Square (Allen et
of the divergence
when a nation
verdict.
upwards, eyes trained on the giant video screen.
delivered, the
crowd
transfixed by
The
following
1995):
al.,
before the O.J. Simpson verdict was announced, the crowd
tilted
Many
make knowledge
announcement of the
moved
as
But when the
two distinct camps one predominantly black,
split into
the other white and each with a vastly different response.
jubilation.
of observers. There are times,
Many
with
blacks... reacted
whites wore faces of shock and anger directed not only at the verdict,
but at the reaction from blacks... Throughout the country, the scene was similar. In
Wall Street
offices, college
campuses, stores, train stations and outside the Los Angeles
County Courthouse, the Simpson
verdict
drew reactions that
split
along racial lines."
Differences in reaction to the verdict reflected substantial racial differences in beliefs regarding
the likelihood that Simpson was guilty. Brigham and
Wasserman (1999) tracked such
beliefs over
the course of a year, starting with the period of jury selection in 1994 and ending three weeks after
the announcement of the verdict. During jury selection
54%
sample thought that Simpson was "guilty" or "probably
were concluded these numbers had risen to 70%
larger racial gap.
The
reaction had been
made
final
for
record.
(as
A
public,
12%
showed modest convergence but a
in
number
for blacks, reflecting
after the verdict
it
it
of whites) considered
or possibly true that the
available in poor black neighborhoods,"
and
respectively.
These
guilt.
found that
29%
of black respondents
to be true or possibly true that the
60%
AIDS
virus
was
of blacks believed
government "deliberately makes sure that drugs are
easily
and 77% gave credence to the claim that "the government
investigates black elected officials in order to discredit
doesn't do with white officials."
34%
initial
of other dimensions on which stark differences are a matter of public
5%
deliberately singles out
and
visible manifestation of racial differences
New York Times and WCBS
was true
an even
remaining disparity,
significant
"deliberately created in a laboratory in order to infect black people." Almost
that
of blacks in their
the time closing arguments
probable or certain
1990 survey by the
compared with
By
round of the survey, taken several days
While reactions to the Simpson verdict may be the most
in beliefs, there are a
guilty".
whites and
with 63%) of whites and 15% of blacks declaring a belief
and 10%
of whites
The corresponding numbers
for
them
in
a way
it
white respondents were 16% and
differences cannot be attributed to differences in socioeconomic status or
demographic characteristics (Crocker
et
al.,
1999).
To take
a
more recent example, a June 2008
survey found that while
5%
corresponding figure was
12%
for
And
Research Center, 2008).
38%
of black respondents believed that Barack
in
a poll conducted just a few days after the presidential election,
8%
of whites stated that racial discrimination against blacks in
the United States continues to be "a very serious problem"
(CNN/Opinion Research
2008).
persistence of such public disagreement appears to conflict with the standard hypothesis
economic theory that differences across individuals
in
a Muslim, the
white respondents, and 19% for white evangelical protestants (Pew
of black respondents but only
The
Obama was
information.
If this
revised beliefs
in beliefs are
view were correct, then disagreement
itself
due
would be informative and lead to
and eventual convergence (Geanakoplos and Polemarchakis, 1982). This
underlying Aumann's (1976) theorem, which states that two individuals
and are commonly known
themselves
Aumann
common
to be rational
must have
the insight
is
who have common
identical posterior beliefs
knowledge, no matter how different their information
may
priors
if
these beliefs are
be.
As suggested by
(1976), the widespread public disagreement that one observes in practice can be attributed
either to differences of priors
biases in
solely to differences in
computing
on the underlying parameter that
probabilities,
i.e.,
is
being estimated or to systematic
on the broader state space
to differences of priors
in
which
individuals update their beliefs.
In this paper,
we develop a
tractable framework which allows for public disagreement and
can be used to explore questions concerning the aggregation of distributed information and the
consequences of social integration.
We
consider a finite population of individuals
who
respect to both their priors and their information about the state of the world.
signals are
assumed
to be normally distributed; priors
are independent.
The
we consider both
cases.
beliefs are publicly
update their
also
We
and
beliefs
be informative.
occurs.
profile of priors in the population
Given
their priors
and
truthfully announced.
The sequence
At the end of the process,
of
are interested in whether or not
all
and
or
may
not be correlated, and signals
may
or
may
not
The announcements
in a further
itself
be commonly known;
form
beliefs
are informative,
and these
and individuals
round of announcements, which may
announcements continues
all beliefs
All priors
with
may
their information, individuals
based on them. This results
differ
until
no further
become public information; we
distributed information
is
call
belief revision
these public
beliefs.
incorporated into public beliefs
through the process of communication, and the manner in which the extent of disagreement in
public beliefs
is
affected
by patterns of
Given the heterogeneity of
if
social integration.
priors, public beliefs
would involve some
priors were observable, so that each person's signal could
When
level of
disagreement even
be deduced from his announcement.
priors are unobservable, the possibility arises that the process of
communication may
fail
to aggregate all distributed information, resulting in different levels of disagreement relative to the
case of observable priors.
signal-jamming problem.
and
his signal.
This happens because unobservability of priors gives
An
individual's
first
announcement
is
rise to
a natural
a convex combination of his prior
Since other individuals observe neither the prior nor the signal, they can only
extract partial information about each of these from the announcement.
round of communication, therefore,
beliefs
do not
At the end of the
reflect all distributed information.
We
first
show that
when
As a
none of the subsequent announcements has any informational value.
priors are uncorrelated,
result,
some distributed information remains uncommunicated,
despite potentially unlimited
rounds of communication. Public disagreement now stems not only from heterogeneous priors but
from informational differences that are induced by the
also
The problem becomes
of information
to a
breakdown
uals
is
in
We
especially acute in a large society.
among
distributed
is
fact that priors are privately observed.
a large
number
a fixed
amount
of individuals, unobservability of priors leads
communication: the difference between the public
approximately equal to the difference
been received and communicated. Hence,
beliefs of
any two individ-
though no information had
in their prior beliefs, as
public disagreement
in a large society,
unobservable priors than under observable priors at almost
On
show that when
is
greater under
realizations of priors
all
and
signals.
the other hand, in small groups, unobservability of priors can lead to smaller levels of public
disagreement at some realizations, simply because a more optimistic person
pessimistic signal
and cannot communicate
show that the expected value
Even
his information fully.
of public disagreement
must be
receive a
in this case,
when
larger
may
more
however, we
priors are unobservable
and uncorrelated.
With
correlated priors the situation
more complex.
is
correlated with that of at least one other individual,
that
if
is
generically satisfied)
priors are unobservable.
precisely
all
As long
we show that
distributed information
is
fully
as each individual's prior
(subject to a regularity condition
incorporated into public beliefs even
While individuals may agree to disagree,
what they would be
is
their eventual beliefs are
they had been able to observe each other's signals. This happens
if
because the manner in which an individual responds to the announced beliefs of others reveals
his beliefs
about the priors of others, which
in
turn reveals his
own
prior.
As
a consequence,
public beliefs in the case of unobserved (but correlated) priors are identical to those resulting from
observable priors. However, convergence to public beliefs takes longer
and involves
levels of statistical sophistication that far
when
priors are unobserved,
exceed those required for convergence under
observable priors.
Hence, we view this result as a benchmark, suggesting that the beliefs that
emerge
run are invariant to the manner
in the long
and the pattern of observability of
priors.
which we interpret as medium run
The
beliefs,
in
which information
beliefs held before
exhibit
all
is
distributed in society
convergence has been attained,
of the properties of public beliefs under
independently distributed priors.
One
interpretation of the assumption that priors are observable
the thought processes and perspectives of others, even
if
is
that individuals understand
they do not share them.
Such under-
standing could arise through social integration and mutual understanding that goes beyond the
announcement of posterior
beliefs.
Under
this interpretation, our results enable us to investigate
the relationship between social integration and public disagreement.
structures.
We
say that society
is
fragmented
if
do
this
by exploring a
no priors are observable, segregated
vidual observes only the priors of those within his
are observed.
We
model with uncorrelated priors, two social groups and three possible information
variant of the
Our
earlier results
own
social group,
imply that expected disagreement
is
and integrated
if
each indi-
if all
priors
greater under fragmentation
than under integration.
A
segregated society with uncorrelated priors behaves in a manner similar
to a fragmented society with correlated priors:
medium
run. In the
run, however,
distributed information
all
some novel
We
effects arise.
of public disagreement held by individuals in a minority group
held by
ment
members
in the
of a majority group.
medium run
is
is
is
aggregated in the long
show that the ex-ante expectation
smaller than the
same expectation
Furthermore, the expected magnitude of public disagree-
greater under segregation than under integration,
and greater under
fragmentation than under segregation.
When
the population size
is
large,
medium run
intriguing characteristics. First,- the bias
difference across groups in prior beliefs.
communication under segregation. In
is
beliefs
state dependent,
Hence
fact,
differences in priors can
even
there will be a difference in beliefs after the
under segregation exhibit a number of
and can be much
there
if
first
is
larger than the ex-ante
become
amplified through
no ex-ante difference
round of communication.
in prior beliefs,
Second, individuals
belonging to a minority group face a disadvantage under segregation even though
receive equally precise signals
and have access
arises in the interpretation of public
segment of the
priors of a smaller
to the
same
belief
are
more
of minority group
individuals
announcements. The disadvantage
announcements. Since minorities (by definition) observe the
total population, their inability to extract full information
the announcements of others can be very costly. As a result,
members
all
medium run
beliefs of majority
closely aligned with reality (interpreted as the true state)
than are the
from
group
beliefs
members.
Our work contributes
to a growing literature that allows for heterogeneity in prior beliefs. 1
particular, Banerjee and Somanathan (2001) and Che and Kartik (2008) explore strategic
In
commu-
nication under observable heterogeneous priors. Since heterogeneous priors lead to heterogeneous
preferences,
some information cannot be communicated
model
communication
in our
communication
in order to focus
their opinions.) In this
truthful, non-strategic
is
we have
we
on the
in
adopt, the
same model
follow Geanakoplos
of sequential
arise
consider non-strategic
communication. (More-
and Polemarchakis (1982), who show how the agreethrough a sequence of truthful belief announcements.
announcement introduced
Freedman
(e.g.
We
mind, individuals do not face strong incentives to misrepresent
heterogeneous and possibly unobserved priors. Our work
with heterogeneous priors
and two-sided.
role of unobservability of priors in
ment predicted by Aumann (1976) could
We
Crawford and Sobel, 1982). Our work
not only to be heterogeneous, but also to be unobserved. Furthermore,
differs in allowing priors
over, in the applications
(as in
1965;
is
Acemoglu
there, but apply
it
to the case of
also related to the literature
et
al.
on learning
2008), which focuses on learning
from external sources rather than from communication.
Our work may
also
be seen as part of a literature dating back to Loury (1977) on the economic
effects of social integration; see
Heterogeneous priors play a role
in
Chaudhuri and Sethi (2008) and Bowles
many
applications, including
et
al.
(2009) for recent
work on asset pricing (Harrison and Kreps 1978;
Morris 1994; Scheinkman and Xiong 2003), political economy (Harrington 1993), bargaining (Yildiz 2003, 2004),
organizational performance (Van den Steen 2005), and mechanism design (Morris 1994; Eliaz and Spiegler 2007;
Adrian and Westerfield 2007).
contributions. While this literature examines the effects of integration on income differences, our
on disparities
focus here
is
Crocker et
al.
Such disparities can have significant welfare consequences. As
(1999) note, blacks and whites "exist in very different subjective worlds" and "a chasm
ways they understand and think about
remains... in the
in beliefs
in beliefs.
and events." Such differences
racial issues
can make "communication and interaction across racial
and
lines painful
difficult", as
blacks find "their construal of reality flatly denied" and whites feel hurt or outraged that blacks
give credence to conspiracy theories that they find bizarre or outlandish. In addition, beliefs affect
responses to government policies such as public health initiatives aimed at reducing the spread of
communicable diseases or the promotion
about the fairness of the justice system or the
corrosive effects on the functioning of a
is
differences in beliefs
can have
extent, of racial discrimination in daily life
democracy and erode confidence and participation
While a serious analysis of such welfare
political process.
our analysis
Most fundamentally,
of birth control.
effects is
beyond the scope
in the
of this paper,
motivated in part by the sense that persistent public disagreement can be welfare
reducing in subtle but substantial respects.
The remainder
of the paper
is
structured as follows.
explore a special two-person case in Section
priors result in the
commonly known
same
When
3.
limiting beliefs (and hence the
The
priors.
general case
examined
is
We
introduce the model in Section
in Section
The
5,
and
there are just two individuals, correlated
same
levels of
in Section 4,
expected disagreement) as
where
it
is
shown that the
irrelevance of observability result continues to hold as long as the primitives of the
a genericity condition.
2,
case of uncorrelated priors (which
fails this
model
condition)
satisfy
explored
is
where we identify conditions under which observability of priors lowers expected
disagreement relative to unobservability.
Section 6 uses our results to explore the relationship
between social integration and public disagreement, and Section
7 concludes.
The Model
2
There are n individuals
we
call
i
E
N
=
{1,2, ... ,n}
and an unknown
real- valued
the state of the world. Individuals differ with respect to both
parameter
9,
which
and
their prior beliefs
their
private information about the state of the world. Before the receipt of any information, individual
i
believes that 9
is
normally distributed with mean fa and unit variance:
9~,N(fi,,l).
Given these (possibly heterogeneous and privately observed) prior
a private signal X{ that
is
use the subscript
i
to denote the belief of
conditional- expectation operators under
X~
/V (0, 1)
means that
expectation operator
i's beliefs.
l
i.
We
9
+
all
individuals agree; e.g.,
For example, E, and
6
observes
e,.
£
omit the subscript when
E [X]
i
e^.
t
[-|-]
all
denote the ex-ante- and the
individuals agree. For example,
individuals agree that A' has the standard normal distribution. Likewise,
all
when
each individual
informative about 9 with additive idiosyncratic noise
x
"We
beliefs,
means that E{ [X]
=
Ej [X]
= E [X]
E
denotes the
for all i,j
6 N.
All individuals agree that 6, e\,:..,e n are
independently distributed, and that
s1
Observing x
z
individual
,
updates 3
i
~N{0,t 2
about 9 to a normal distribution with mean
his belief
= am +
A'i i
t
).
(1
- a)xi
(2)
and variance
°
Hence, one can think of
manner
as the
\i\
= TT^-
0)
which individual
in
processes his information X{, about
i
which other individuals are uncertain. One can also think of x as the component of the belief of
i
l
that
is
which
type,
in
perceived to be informative about 9 by other individuals, and
is
assuming that
m
which case
The
priors
£
matrix
i
perceived by others to contain no information about
.
.
.
,
privately
is
common
be
will
(/^j,
,x l )
{(i l
known by
we
unless
i
for i,j
£ N.
A
crucial
believes that the state 9, the others' priors /z^ 2
That
stochastically independent.
We
9.
refer to the pair (/ij,x,) as i's
explicitly specify that
mean
is,
player
assumption
=
Within
.
.
.
,
any information about
made
received, beliefs are
A
lt \
After observing
his
own
and variance-covariance
/t t
ej, j
individual
,
£ N, are
all
some uncertainty about how each
is
manner
in
which j updates
9.
public in period
denotes player
prior
i's
1
and Polemarchakis (1982). Once
/ij,
£ N,
i
expectation of 9 conditional on the prior
own
his
in
period
Here
2.
A
lt
2
and the others'
signal x lt
denotes
initial
announcements
is
denoted
A
l
^
ca for
i
.
emphasizing the fact that this belief becomes public information
end of the communication process.
We
and the
signal %{.
expectation of 9 conditional
i's
announcements A-^i
N We
£
/i,
A h \,
and simultaneously announce
beliefs
Individuals continue to update and announce their beliefs indefinitely.
the sequence of
signals are
by simultaneous (and truthful) announcements
announcements, individuals update their
all
these updated beliefs A^-,
on
observable,
is
framework, we consider a model of deliberation involving truthful communication
this
£ N, where
p, n )
and the noise terms
(f-ij)j^i,
of beliefs in a sequence of stages, as in Geanakoplos
i
m
that conditional on
is
thinks that there
i
(fi\,
individual j processes his information Xj, but does not think that the
his beliefs reflects
component,
as the residual
knowledge.
are distributed normally with
/i n )
with entries a XJ
]i x
call
(i.e.
The
A l}00
=
(^4?, 1)7/1-
limiting values of
the public belief of
common
i,
knowledge) at the
assume that everything we have described to
this point
is
common knowledge.
Remark
and
9,
1.
but
Since (^1,
/i_
;
and
.
.
/J.
.
,
n
)
may
be correlated,'
i
may
independent conditional on
9 are
think that
pt,.
=
8
+
e,
is
+
v
2
\.
correlated with both /i_j
If
9
~
N(fi,o~) and e
~
beliefs arise
N(0,v
),
then
normally distributed with mean
£[0| s
and variance a 2 v 2 / (a 2
is
Such seemingly inconsistent
''Throughout the paper, we use the following well-known formula.
conditional on signal s
p, x
]
= -j£_^
-I-
-^—^s
(1)
Suppose that
naturally as follows.
family
{Xm } m( M
random
of
-
potentially relevant historical facts are represented by a
all
variables.
Each individual
variables to be relevant for understanding 9 for
variables
which
on
fi t
Xm
is all
/Li_i
,
m$R
with
l
does not affect his beliefs about
considers
i
Remark
2.
and
/i,
Our model
information, or the
although they can
p.j
{Xm } m€M
/i^,
{Xm
t
of
}
random
me
|
R,}
is
m,
Consequently, conditional
i.
and
9 to
R
t
be independent.
DRj ^
of deliberation presumes that individuals cannot directly
manner
fully
R
6
some
for
On
^
j
i,
to be stochastically dependent.
communicate
which they have incorporated their information into
in
communicate
incorporated into one's final opinion
their resulting beliefs. This
is itself
bits
and
is
their beliefs,
because we think of information
and the manner
pieces,
their
in
which these are
a complex process that involves interpretation in light of
one's upbringing and experience. For simplicity,
Xi,
according to
he considers
assigns positive probability to
if i
m
\
some R^ C M; he considers the remaining random
8; i.e.,
complex object consisting of many small
as a
{Xm
considers a set
irrelevant. His conditional expectation of 8 given
the relevant information about 9 in
the other hand, at the ex-ante stage,
then
i
which incorporates everything that others
we represent
find relevant,
this process using
and
/i,,
two
real
numbers:
which represents everything
that others find irrelevant.
We
conclude this section by describing the two environments that we
that priors are observable
distribution).
We
if (/ij,
.
.
.
,/i n
)
is
common knowledge
say that priors are unobservable
if /i,
say
(although drawn from an ex-ante
privately
is
We
will investigate.
known by
for
i
each
i.
We
use superscripts ck and u to denote variables in the observable and unobservable priors cases,
respectively. For example,
we write A^ kk
or
Af k
for the
on whether priors are observable or unobservable,
announcement
of
i
at
round
depending
respectively.
Examples
3
Before proceeding to more general results, we consider the case of two individuals,
with the simplest environment
generality that
m
Consider the case
i
>
in
which priors are
We
i
and
j,
starting
common
knowledge.
common
knowledge. Denote the announcement
assume without
loss of
fij.
Observable Priors
3.1
of
k,
at
on x
L
in
which the priors
round k by Afi. In the
first
^
and
fij
are
round, each individual announces his expectation of 9 conditional
:
A$ =am + (l
Since
i
knows
fij,
he correctly infers from j's
first
-a)x,.
announcement that
j
signal
-a {Af
l
1
-a x )={l + r-i )Af -r 2 H
l
j
1
.
must have observed the
Hence player
t2 )
+
(1
i,
Af x
whose
r
2
first
round
was given by
belief
which has expected value
fj,j
6
~;
N(A^,a),
and variance r
8
2
.
receives an additional signal
From
and
(1)
(3),
he therefore
updates his belief to a normal distribution with mean
a
+ T*
l'
1
a + T*\
i^'
y
rJ
J
2
+ t 3 :^'
1
J,1/
2
+
r
and variance
Vh2
Note that
i
much weight on
puts as
j's
oTT 2 ~ JT^
~
announcement
4
the exact information that led to j's announcement.
still differ
because each also adjusts
for
'
2
as he puts on his own, since he can extract
The updated
beliefs of the
two individuals
the other's "bias".
Since both individuals can predict what each will announce in the second round, they do not
update their
beliefs after hearing the
nouncements are foreseen ahead
process stops at round
This
is
The
round. That
== A* = A* = i±l!
is, all
distributed information
difference in public beliefs
{Ai,i
+ AjA ) - t^-sH-,
-
AfTC =
Note that although each individual's public
is
is
common knowledge
at the
end
aggregated in one round of communication.
is
^
difference in beliefs
subsequent an-
and do not provide any further information; the updating
of time
simply because each individual's private information becomes
first
all
2:
A&o
of the
second round announcements. Likewise,
independent of both
2
-—
r
j
belief
initial
(m-n).
(4)
depends on the other's
initial
announcement, the
announcements, and the individuals agree on the
distribution of this difference.
Unobservable Independent Priors
3.2
Next consider the case
in
which the priors
distributed, each with variance
round
beliefs
a2
and announcements
.
We
write
all
i
can
infer
is
to variation in xj.
More
(1
that a/ij
4
(x\
The
+
precisely,
+
r
resulting beliefs are precisely
12) /2
=
9
+
(e\
+
£2) /2 with
A* k
=oim +
+ (1 —
values of each variable. Hence, he attributes
some
and
are not observed,
jXj
announcement
for the
and are independently
of
i
at
round
k.
First
are exactly as in the case of observable priors:
i4J|i
Observing A^j,
\i{
2
)
oi)
(1
-a)xi.
Xj
is
some
equal to
Aj„ and
of the variation in
cannot know the specific
A" x
to variation in
\ij
and
he observes an additional signal
2
Al, - r ft
=
6
+
t
2
(
N
what they would have been
if
-
ft)
+
e,
(5)
each player had observed a normal signal
normal noise having mean zero and'variance r
2
/2.
with additive noise r 2
updates
his beliefs to a
2
CT
M
ci
+
+ £j. The
fij)
j.
Here, the
first
2
T4
+T 2 u
+ (1+<7
equality
using the signal
in (5),
T 2 )(1
2
a+
CT
2
the signal Xj, motivating
become
that of j. This
to increase his
i
after the
•? '
He then
.
is
by
is
because
some
TVfyl^j,
When j
j's prior.
or)
and
announces
probability j has obtained a higher value of
of 9 too.
He
his expectation of
9.
larger
(i.e.
fij),
in
Consequently, each player's beliefs
the other's announcement than in the case of
does not
with some
also thinks that,
to a bias towards higher values
i
~
puts greater weight on his
i
know
does not
own expectation
second round announcements,
from 9
(1) starting
Note that
(3).
i
+ r 2 )Al 1 -r%)
(l
know
commonly known
Xj, so there
priors.
remains some relevant
asymmetric information. In other words, some of the distributed information
the
r2
^Jy
1
tl
believes that with
i
would not want to increase
i
;
+ a^)(l + r 2 )Ar +
((l
announcement may be due
less sensitive to
Even
T2 )
+r 2U
and the second equality
a higher expectation AJi,
which case
+
r4
obtained by updating according to
is
own announcement than on
probability, the high
+
^2
1
1
and variance o 2 t a
mean
noise term has
normal distribution with mean
r4
cr
-
(fij
is
not aggregated at
round. One might hope that further announcements communicate more private informa-
first
aggregation of the remaining distributed information. This
tion, resulting in the
however. Since A^ Y and A^j are sufficient statistics for
Af 2 an d A%,
not the case,
is
the second round announce-
ments provide no additional information, and
AU
^1,2
The
difference in public beliefs
Atoo - A?J,oo
=
_
—
AU
2
,
;
,
I
J-
Kr
2
W1
(l+aU 2 r2)(l4
+„
~[~
• —-
,
The
+
(1
(/2;
—
difference in information (e ;
+ U2 T2
)
(1
+
T2 )
[(lM
ftj)
—
,
+
t
2
)
(Aitl -
Ej)
.
information play a larger role in affecting
common knowledge
)
2
T (ft
-
ft))
- Hj) + a
t {ft-fij)
means
+a
(ei-Ej)).
(6)
of the distributions
(/i,
—
fij)
,
from
and the
Since communication never completely eliminates informational
Communication
of differential information as the coefficient of (A,j
in the
Ajtl +
the difference in the realized values of the priors
differences, these differences affect public beliefs.
As
(1
difference of opinion has three sources: the difference in the
which priors are drawn
in
2
|
T2
1
Au
^i.oo-
is
*
-
l
JL
_
—
^z.3
case,
all
initial
— Ajj)
is
does, however, decrease the role
strictly less
announcements than
than
1.
That
is,
differences
in affecting public beliefs.
individuals agree on the distribution of the difference in
public beliefs.
Note from
priors
(m =
identical.
(6) that
fij)
,
the two individuals will generally agree to disagree even
if
they have identical
since they cannot deduce from the announcements that their priors are in fact
This makes transparent the obvious but sometimes overlooked fact that the standard
10
common
fact
prior assumption requires not only that the players have the
is itself
same
but also that this
prior,
commonly known.
In conclusion, uncertainty about the
manner
which other individuals process information
in
hinders the communication of relevant private information through the announcement of beliefs.
Consequently, individuals hold different beliefs both because they have (possibly) different priors
and because of
A
3.3
different information.
Comparison
of Belief Differences
— Aj tOC measures the amount that
Hence we call Ai >00 — Aj t00 public bias
Note that Ai >00
i
deliberation.
of
priors leads to less
bias.
This
communication
is not the case.
knowledge of
this
may
observed, by (6), any
the priors are
It
of information, one
may
so
overestimates 9 relative to j at the end of
i
relative to j. Since uncertainty regarding
may
think that
it
also leads to greater public
happen that the individuals have very
lead to a very large difference of opinion. Indeed,
amount
common
of public bias
knowledge, by
on the difference in realized
(4),
is
the
possible, including
amount
when
no bias at
of public bias
is
different priors,
and
all.
the priors are not
In contrast,
when
constant, depending only
priors.
Public Bias with Observable Priors
Figure
Figure
1
1.
Public Bias with Observable and Unobsevable Priors
plots the values of public bias under observed
a set of randomly drawn type realizations.
5
The
figure
is
based on 500 realizations of type
and unobserved
priors, respectively, for
Here, for each realization, the horizontal coordinate
profiles for
11
parameter values a
=
r
=
1,
\h
=
1,
and
\ij
=
-1.
A t°° ~ A foo and
is
the vertical coordinate
diagonal, public beliefs differ
that
making
more when the
priors observable
may
Af^ - A^<00
is
In the realizations that
.
lead to greater disagreement in
While observability of priors can
Hence the
priors are observable.
many
when
by
priors are observable,
r
when
o
If
jii
=
Aj t00 \. To see
this,
is
(^ ~ H)
the expected public bias
(6),
is
^(1+qV)
-A-
I
then the expected public bias
Jij
,
the priors are not observable, by
E\A±
E \A l>OQ -
2
[
the other hand,
figure demonstrates
cases.
the expected bias in public beliefs
(4),
E A i%> ~ A foo] =
On
below the
result in greater public bias for particular type realizations,
observability always lowers the ex-ante expected value of public bias,
note that
lie
is
zero in both cases.
If fa
>
however, then a 2
fij,
>
implies
E[Ai,co
That
is,
the expected public bias
observable. This
is
is
~ A loo\ > E[A?oo ~ A<jU >
when
higher
This result
between the average opinions of various groups.
groups
if
in beliefs
members
is
We
it
full
when they
are
aggregation of the
comparing the difference
implies that differences across
racial profiling
would narrow on average
and therefore understand how
return to this point in Section
their
6.
Unobservable Correlated Priors
Under the assumption that the
ity of priors
the
beliefs.
useful in
For example,
about the incidence of police brutality or
incorporated into
impedes the
is
of each group were to observe each other's priors
information
3.4
priors are not observable than
intuitive because unobservability of priors
distributed information through deliberation.
0.
may impede
amount
is
we have
so far illustrated that unobservabil-
the aggregation of distributed information through deliberation and affect
of public disagreement.
information
Assume
priors are uncorrelated,
We now show
that
when
priors are correlated,
aggregated and hence the observability of priors has no
that
\x r
and
ji
3
effect
on public
are correlated:
"
to
where p
^
0.
Observing
/i,,
i
believes that
[i
Eiltolto]
3
is
=
j
1
P
p
i
distributed normally with
to
and variance
12
+P(t* ~
M»)
all
mean
distributed
beliefs.
That
Ei[(j.j\m} is
is,
round
k.
As
a one-to-one function of m. Let
+
r
2
)
his belief, in the
2
AVj -
additive noise r 2
The
A^ = A
we have
(jl
T-
^^!^] =
K
is
second round
and L are known
strictly decreasing in
Player
prior.
j,
r
2
= A jA
and AVj
(^ -
- E^fi^p,^) +
(fj,j
.
the announcement of player
Now,
for
i,
=
+
2
announcement A]
1
the
,
i
at
of j
+
£,[ Mj |^])
e 3 has
x3
9
r
(
and variance a 2
mean
- £,[/;>,]) +
H
-
(l
2
p
)
rA
+
r
2
.
£j
(7)
.
Updating
announces
i
= KAl + LAl -aLE
Al 2
where
Af k denote
round yields an additional noisy signal
in the first
(1
before,
1
i [
1
strictly positive constants. 6
which
Ei{/j,j\fii\,
The
i },
X
that
A" 2
has observed his
own
crucial observation here
expectation of j's prior once
is i's
having observed Af^ and Aj
H \^
i
is
from the previous round, can therefore use
A{2
l
to
deduce that
Ei[
Moreover, since
and
E
l
Hence
[p,j\p,i\.
is,
at the
+
Jij
1
—
p(j.i t
^
and p
pi)
0,
there
is
W
a one-to-one
=
Pi
1
P' ((aLy {KAl,
1
+
end of second round,
in the
common knowledge
manner
in
^t,oo "
A'.oo
(^'•
'"'-'1
^2
t2
"*'
'
both individuals can
priors are correlated,
which they react
+
,
to the initial
in all
Ji
1+T
o
2
announcements
case:
,-2
=
—-
,<*
A,3 -
when
t
+ LAI, ~ A b
prior beliefs are revealed. Hence, the
all
subsequent rounds are precisely the same as
Accordingly,
mapping between p
j correctly infers that
fH
That
—
Ei[fj,j\fj,i]
(aLy {KA? + LAI, " A
A
N \m\ =
"
^J''"
1
J
infer
)
r
~ o
2
+
i.
^2^J'
7-2
each other's prior beliefs from the
announcements. All distributed information
aggregated through communication, and the resulting public bias
is fully
is
therefore
attributable to differences
in prior beliefs:
A,oo
We
show next that
this
is
~~
A j,oo
r2
~~
~
+ T2
'
true under broad conditions.
Aggregation of Distributed Information
4
We now return
to the general
model with unobservable
priors,
the cases in which the private information of an individual
'One applies
(1), starting
-fi
^.,2
The
=
;
a+
desired equation
is
fi
YTl
a*- (1
N
from 9'~
„2\ _4
- p
2\
2
)
r
revealed through deliberation.
(A"i,q) and using the signal
in (7), to
obtain
2
.
4
is
and provide a near-characterization
+
i
2' 4 '.'
t2
obtained by letting
K
+ a+
1
2» 4|
2
4
2
a 22Tl
(1 - p ) t
-f t
J
and L respectively denote the
13
+
T
I
A J,1 ~ T E
coefficients of
*
Mj Mi
A?tl and
.
J
.4"i.
of
We show
that, roughly speaking,
his private information
an individual's prior
if
is
correlated with the prior of any other individual,
is
revealed by the end of the second round; otherwise his information
revealed. Hence, except for certain knife-edge cases (as in the
example of independent
process of sequential belief announcements leads to the aggregation of
The
idea that an individual's private information
is
is
never
priors), the
distributed information.
all
revealed through communication
is
formal-
ized as follows.
Definition
k
i
say that the private information of individual
revealed by (the end of) round
is
i
measurable with respect to Mj,m}y eAfm
<f If tlie private information of individual
not revealed by round k for any fc, we say that his private information is never revealed.
if {(J.i,Xi)
is
We
1.
That
is,
is
the private information of
ments up
and including those
to
Aj,
To present our
m — Ej
9
=
covariance matrix of
{p. ] ) J
l
the identity matrix. Finally,
Mi=
M
The next
Definition
2.
^
Note that
cannot
infer
i
is
write
we
M =
lfc
x
^',//
'
(ct^j)
•
x
row vector Mi as
define the
i's
+T 2 I + T 4
if
i
,
isolated
is
S)
is
and only
o--
if
regular
if
if fi,
prior from the
way he
t
i
and some other knife-edge cases
=
is
(o^t)
signal x
x
In
.
k:
N)
,
lk
i
€ N, we define
for the variance
,
x
matrix with entries
1
and /
for
follows:
(E.
=
every
is
a-
£7_,:.;cr_
„•
therefore independent of
is
i
We
0.
is
say that
in
which
is
i
all
type
regular under (t
independent of
all
,
if
S).
other priors
own
his
2
2
regular under (r ,S)
prior.
reacts to their announcements,
The
of his private information.
all
=
announce-
all
and
>
following notation. For any
for the k x /-dimensional
;
m
(Vj e
and write £_;,_;
,
by observing
any round
e /v\{,;, 7 },/< r
[
any information about the priors of others from
that Afjcr_j i
at
j/
{
we introduce the
(7_
if,
his prior belief ^,
definition provides the terminology of the characterization.
isolated
cannot learn about
to uncover
common knowledge
be
llxn-l (al rl _ixn-l
We say that
We say that (r 2
0.
one can compute
k,
depends only on the primitives of the model and
%
realizations.
MiO-i^
round
fXi,Xi,flj,Xj,
\
^ and
We
/U_j.
revealed by the end of round k
is
will
characterization,
column vectors ^_;
Note that
in
information
that, case, his private
i
fij.
In this case,
i
Consequently, others
and
it is
not possible
regularity condition rules out this knife-edge case
MjCT-,,,
=
(without, requiring that o~^ lt
,
=
0).
2
Note
a non-trivial linear equality restriction on the variances (r ,E), and hence
is
i
satisfied only
on a lower-dimensional subspace of the space of
the set of regular parameters (t
Our
Proposition
is
,
E) has
full
all
variances (r
Lebesgue measure and
is
,
E). In particular,
open and dense.
characterization establishes that whether the private information of an individual
depends on whether he
mation
2
is
1.
is
Assume
is
revealed
regular or isolated.
that the priors are not observable.
revealed by the end of round
2.
Conversely,
never revealed.
14
if
i
If
i
is
regular, then his private infor-
is isolated,
then his private information
An immediate
implication of this
Corollary
If (r ,E) is regular, then all private
is:
2
1.
information
This result establishes the irrelevance of observability:
information
is
how
aggregated no matter
little
2.
public beliefs under unobservable priors
common knowledge
are identical to public beliefs under
revealed by the end of round
is
of priors as long as (r 2 E)
,
know about each
individuals
other's
regular.
is
way
All
of thinking.
Moreover, as in the two person case, this process requires just two rounds of communication.
In order to prove Proposition
After the
2).
announcements
of
all
and the prior
of
i
/Xj
the Appendix, we compute the announcements (see
in
1,
round, the announcement of an individual
first
himself. In the second
to MiCT-ij. Hence,
when
information and Ai
t
2-
^
Mjcr-^i
0,
—
ct-i_i
0,
the
of publicly available information,
of
i
private information
direct implication of Proposition
These
If (r
2.
2
,
E)
is
regular, then
Xj
is
revealed by the end of the second
is
/i 2
proportional to ct_ Zi ,. Hence,
it is
a function
is
never revealed.
2
E)
,
is
A" = Af^
regular:
for
fc
all
i
and k >
3.
have a particularly simple form. Under observable priors, each individual
beliefs
we have
revealed,
proportional
that public beliefs are identical under observability
1 is
deduce the signal Xj of any other individual from her
(2),
\i{ is
round
round announcements and the priors that have
first
and unobservability of priors provided that (r
Corollary
coefficient of
new information because
does not contain any
i's
i
the coefficient of
first,
namely the
already been revealed. In that case,
Another
round announcement, the
In that case, the private information of
announcement
first
whose information has been
other individuals can compute yn using the publicly available
round. Moreover, in any round after the
when
an afhne function of the
i is
individuals, the priors of the individuals
Lemma
=
(l
+
t
2
)
A
]t
\
-
(xi H
r
2
[ij.)
h
first
round announcement.
(Specifically,
i
can
from
Hence, individuals extract the entire relevant signal
x n ) jn
-
9
=
r /n.
+
he„) /n,
(ei H
where the noise has variance
f
Using this
signal,
T
Ack
n
difference
between the public
2
E)
regular, then
(t
is
also given
is,
,
is
by
(8)
(9).
2
by Corollary
^
2,
n
+r
beliefs of any'
C~If
2
they form their public beliefs as follows:
/tck
The
2
=
n+
A
n
V^
two individuals
rr
Xj
r* ^-^ n
i,j
€
N
is
therefore simply
2
^2 (W
0)
-Mi)-
the difference in public beliefs with unobservable priors
Note that holding constant f 2
,
this difference
is
independent of
n.
That
under the regularity assumption, regardless of whether priors are observable or unobservable,
15
differences in public beliefs are
to the precision 1/f
The
2
due only to differences
which are scaled down according
in priors,
of the distributed information.
regularity assumption
weaker than genericity and contains many interesting "non-
is
generic" cases, as the following example illustrates.
Example
an
=
a and
otherwise.
in his
N= BUW
Take
1.
2
consisting of two groups
for all distinct individuals
That
is,
from
his
own
own group, but he cannot
One can
regular.
prior,
oc 1
and
>
j, ov,
if
i
and
W
{1,2} and
+
distributed information
model
in
some
Nevertheless, (r 2 £)
,
When
incorporate
2
+ a 2 r 2 + a 2 r 4 + a 2 r 2 p + a 2 r4 p ±
incorporated into public
0.
beliefs.
We
in priors across
groups,
analyze this special case of the
implies that public beliefs are discontinuous with respect to the correlation of
1
the priors are correlated, no matter
all
is
detail below.
Proposition
priors.
is
i,
=
same group and a tJ
j are in the
This example illustrates that even in a segregated society with no correlation
all
For each
{3,4}.
6 N,
i
t
=
an individual can learn about the other individual's prior
learn anything about the other group.
check that, for any
Mi<T-ij
i
B =
private information.
When
how
small the correlation
may
be, public beliefs
amount
priors are independent, however, a substantial
of private information remains private. This
true even for the third round announcements.
is
discontinuity stems from our assumption that individuals can
communicate
The
their beliefs precisely.
In reality, individuals have only noisy information about the beliefs of others. For example, public
polls reveal only partial information
about the
beliefs of a
few randomly selected individuals. With
imperfect observability, beliefs at any round will be continuous with respect to the correlation
parameter.
amount
Accordingly, for sufficiently low levels of correlation between priors, a substantial
of private information will remain
Full aggregation after only
for tractability.
assumption
uncommunicated
two rounds
is
an
artifact of the
may
is
common
prior
take arbitrarily long to stabilize. Hence the complete aggregation of dis-
tributed information could take an arbitrarily large
information
two-dimensional model we use
Geanakoplos and Polemarchakis (1982) show that even under the
beliefs
general setting.
at each round.
Accordingly,
we view Proposition
1
number
to
aggregated under correlated priors. That
of
communication rounds
in a
more
be demonstrating that in the long run
is,
we
interpret third
all
round announcements
as corresponding to the long run.
Complete aggregation of distributed information
all
in the long
inferences based
on the
based on the manner
initial beliefs of others,
relies
they are also able to
which others adjust their
in
beliefs after
announcements. This requires that individuals assume that
and assume that
all
on the assumption that
make
make
.
.
up
to high orders.
When
to aggregate distributed information fully,
rational
rational inferences
hearing each successive round of
beliefs are as described in the
individuals assume that beliefs are as described in the model,
their beliefs accordingly.
fail
run
individuals have high levels of statistical sophistication. Not only are they able to
such strong assumptions
fail,
model,
and update
individuals
may
and long-run behavior may resemble the case of
16
independent
where individuals do not make inferences based on the manner
priors,
react to information.
and we explore
environment in
this
is
interesting in
own
its
right,
detail next.
Public Biases
5
we explore the impact
In this section
As we have seen
beliefs.
is
Furthermore, the case of independent priors
which others
in
of observability of priors on the degree of bias in public
when
in the previous section,
priors are correlated, all of the information
aggregated, and observability of priors does not play any role in the long run. Here we consider
only the case of uncorrected priors:
Assumption
That
The variance- covariance matrix for priors
1.
for all distinct pairs
is,
i
variances of priors are equal
(i.e.
Consider two individuals,
of 9
is
that
i
Aj i<x> He
.
i
For any i,j S
3.
A'',
Similarly, the ex-ante bias of
i
the priors
ou — a2
and
knows that
also
j,
i
for all
— Aj t00
j4j ]00
(9) that
relative to j
when
difference in realized priors,
lemma
Lemma
priors
amount
identifies the
of Section 3 to
1.
ft;
—
is /i,
is
priors are
which
of public bias
is
Under Assumption
'
(l
+
t
2
)
(l
+
Under unobservable
fij),
may
r
j
=
and the
0)
is
A,oo- Therefore, j thinks
/ij
—
A l<oc —
is
bias after
/j,j,
i
Aj i00
and
which we
.
j
have observed their
call
the prior bias of
and the public
bias
comes
i
to be
never be revealed.
knowledge, the only source of public bias
down through communication. The
scaled
when
is
The
to all players,
common
1,
for any
i
is
the
following
priors are unobservable, generalizing the analysis
andj, the public bias ofi
2
=
a
(i.e.
n individuals.
^oo -
(fij
p,j.
known
is
where 7
are independent
fj,j
relative to j
i
-
the prior bias
fij,
/.
This leads to our notion of public bias.
.
the public bias of
Note that the ex-ante bias
from
2
thinks that the expected value of 9
known through communication, but
We know
<j
i).
priors but before they observe any information
relative to j.
—
and
/i.4
£=
At the end of deliberation, j thinks that the expected value
j.
overestimates 9 by an amount
Definition
own
and
is
4",oo
J
i
t
2
2
ct
)
=
- (w ^y
+n—
—
4
ih)
+
r^j
2
difference (e —£j)-
without
under unobservable
2
j*V
te -
O
(io)
-
1.
z
full
ex-ante bias
The informational
bias because unobservability of priors impedes the
affects public bias because,
—
2
(w - h) +
priors, public bias has 'three sources:
and informational
relative to j
full
(ju l
—
flj),
prior bias
difference contributes to public
aggregation of information.
Ex-ante bias
aggregation, individuals use ex-ante information on priors
to estimate the information of others.
By Lemma
1,
the magnitude of public biases does not depend on
9.
Hence
all
individuals agree
on the distribution of these biases (although they disagree on the distribution of public
17
beliefs).
A
Our next
bias
are
is
result establishes that,
if
drawn from
the priors are
distinct distributions, the
necessarily larger under unobservable priors. (The expected bias
drawn from the same
Proposition
1,
if
and
FM"
E[A1% -
is
=
AfJ
a
for
=
T
4"
0,
i
+
(l
u
E\A^^~ A hO0
and
A^)
Suppose that
j.
although the actual prior
j,
2
1
we have E[A?<eo -
Consider two individuals
mates 9 relative to
then
p,j,
_?f_(fc- Ai
independent of a 2 while
2
>
>
A<;%}
0.
(10) that
EIA&.-Afc,)
ElAf^-Af^]
p^
AU > E[Af% -
E[Afi00 Proof. Note from (9)
always zero when priors
is
distribution.)
Under Assumption
2.
/i 2
)
r
2
q2 )
increasing in
is
\
> £[#;% -
>
fL L
.
a
for
2
>
E^^-
or
may
After each player k forms his prior and receives his information,
Proposition 2 establishes that
all
when
That
is,
]
i
overesti-
not turn out to be larger than
all
fij.
commu-
individuals deliberate,
U/J/l^ — A ^} <
=
D
become
individuals expect that, at the end of the deliberation,
priors are observable.
u
joo
0.
so that at the ex-ante stage
p,j
may
of?
AfJ
a 2 Since
nicating their beliefs as in our model. At the end of the deliberation, their beliefs
timates 6 less vis-a-vis j
expected
i
public.
overes-
E^Af^ — A^^}
according to each k £ N. Therefore, making priors observable decreases public biases on average.
This suggests that social integration, interpreted as an increased understanding of the manner
which other people think, should
explore these issues further in Section
We
We
on average.
result in lower levels of public disagreement
in
6.
conclude this section with a discussion of the manner in which increases in population size
number
When
aggregation of distributed information.
affect the
of individuals, unobservability of priors
end
so that the bias at the
has variance f
in society
becomes detrimental
of the deliberation process
Towards establishing
deliberation begins.
=
2
2
r /n.
one
If
this,
information
is
recall
fixes r
and
is
distributed
among
a large
communication, so much
for
approximately the same as the bias before
from
(8)
that the distributed information
varies n, as
n gets
large, the distributed
information becomes very precise. Consequently, the individuals approximately learn 9 from each
other. In order to disentangle the effect of group size n from the effect of the information available
to the group,
we now
In particular,
?i,
where
We
Jl n
the precision of the distributed information and
we consider a family
(/li,
.
.
.
,
jl n
)
is
of
models
the vector of
2
(T .a
means
2
,jl n
,
some
if
for the priors;
the variance of distributed information
fixed f
>
0.
Our next
result
is
/i z
N
~
as
vary.
number
(/i,, <7
n —
>
2
oo.
)
of individuals
for
A
each
priors,
when
i
<
n.
special case of
2
independent of n, in which case t
shows that, under unobservable
18
n
let
n), indexed by the
assume that the variance T 2 /n approaches some positive value f 2
this arises
for
=
fix
the
= nf 2
number n
of individuals
i
large, the public bias of
is
relative to j
i
approximately equal to the prior bias of
is
relative to j:
Proposition
individuals
i
Under Assumption
3.
and
j,
and for any
2
2
for any family (r ,er
1,
n) of models, for any distinct
/}„,
,
(£,-,£,-),
T*/n->f 2 >0
By Lemma
Proof.
1,
A
-
t,oo
^",00
= Tn^V {fM -
Mi)
+
V {{Pi -
V2
+
P-j)
(£i
-
6j-))
•
where
2
n(r2/n) a 2
—
As n
Mi
-
>
oo and r%/n
—
»
f2
>
n goes to
0,
0,
+ (r2/n)(l+a 2 +
)
while T^a 2 n goes to
l
Hence
1.
A^ - A"^
goes to
Mi-
Under observable
priors, individuals use all distributed information efficiently.
public beliefs and the bias in those beliefs do not depend on
priors are unobservable, however, even
announce
their beliefs sincerely, they
if
their priors. Recall
from
amount
is
distributed.
When
and
individuals have very precise information as a group
cannot communicate any significant information:
of the deliberation process, their beliefs are as they
individual has such a small
how information
Hence, their
were
at the outset.
of information that their
The
intuition
announcements reveal
at the
end
that each
is
little
more than
An
immediate
(9) that
f2
^,oo ~
Aj'.oo
= -1,-2
so the difference in beliefs under observable priors
implication of Proposition 3
a given
amount
is
of information
In contrast, as
observability of priors
6
may
is
~ W/
independent of n holding f
therefore the following: as the population size
is
public bias under unobservability
realizations.
is
(^ l
distributed
is
clear
among an
increasingly large
fixed.
becomes
number
large, so that
of individuals,
greater not only in expectation but also for almost
from Figure
increase the
amount
1,
in
all
type
a small group of well-informed individuals
of public bias in
some
instances.
Social Structure
As an
illustration of the theory developed in the previous sections,
we now analyze the amount
of expected bias between two groups under three alternative social structures, which
we
call frag-
mentation, integration, and segregation.
in
which no
Fragmentation corresponds to a structure
individual observes the prior of any other.
Under
19
integration, each individual observes the prior
And under
of every other individual.
)
.
.
segregation, each individual observes the priors of
those
all
belonging to the same group, but none of the priors of those in the other group.
More
N = BU
formally, let
members,
We
respectively.
distributed,
is,
and we assume that
ex-ante,
members
may
of B, of course,
of
W
and
are disjoint sets with
S =
=
B
for
ji b
some
and
fit,
>
=
fi
jLj
p.
w
nb >
prior.
We
2
and n w >
/,
(Vi
relative to
G B,j G
W)
members
WJ
of
An
individual
member
member
it
suffices to
W
of
assume that opinions are communicated by successive
as before. In this example,
2
so priors are independently
,
w
overestimate
er
2
turn out to have a higher expectation than an individual
once each observes his own
announcement
B
maintain the assumption that
JH
That
W, where
belief
announce the average opinion of each group,
namely
A b,k =
— Y] A
"
We
-
ieB
) A ]M
j£W
.
use the same superscript to denote other group averages as well, so
— X^jgvv
at
I
and A w<k =
itk
e J' e ^ c
'
e are interested in the extent to
k,
let
[}F
,
Pj.
and
which average opinion
in
B
A l i"
'
£l "
W
exceeds that in
defined as follows:
Pk
We
—
n X^es
=
b
^
any given round
jib
=
Ab,k
- A Wt k-
denote the values of this difference under fragmentation, integration and
/3j?
segregation respectively.
For reasons discussed at the end of Section
the long run.
Ai^k
=
A{
From
2 for all
k
4,
we regard k
=
2 as the
medium run and
k
=
3 as
the previous section, recall that in both fragmented and integrated societies,
>
2,
and the distinction between the medium and long run
is
not meaningful.
:
However, the distinction
is
meaningful
in a segregated society, in which individuals behave as
in
the correlated priors case (although the priors are in fact independent).
6.1
Fragmentation
In a fragmented society, individuals obtain information, form beliefs,
pollsters,
and observe the aggregate
belief distribution.
any other individual. Instead, he uses
From Lemma
1,
for
Pk
any round k
=
T
is
without
2,
TA
iV-b
-flw)+
loss of generality
2
(A&
7
even
if
B
overestimate with respect to the
-
M
T 2 CT 2
"^
(£fc
-
members
of
is
Ew)
(
1 1
7
the groups are of unequal
20
order to
the case of unobservable priors.
size,
because
can simply reverse the order on 9 by considering —8. Simply put, we are measuring the biases
ex ante, members of
in
the difference in average opinions across groups
2
7
'This assumption
>
is
beliefs to
individual observes the prior belief of
about the thinking of the others
his prior belief
extract the information revealed in the polls. This
No
communicate these
W
if fib
in
<
p-w,
then we
the direction that,
Hence, the bias has three sources: the ex-ante bias between groups
bias between groups
(p. b
-
p.
-
p.
w)
the average prior
,
w ), and the average informational difference between groups
Recalling the definition of 7 from
Lemma
E lPk] = n T
(1 +
6.2
(fit,
the expected value of the between-group bias
1,
l
T 2W1
) (1
+
,
/+ n - 7
T l2 G 2l )
{fib
-
- iw ).
therefore
(12)
v
;
fiw)
1
Integration
In an integrated society, each individual observes the priors of every other individual.
nicate directly, understanding the
is
is
(i b
manner
which information
in
They commu-
incorporated into
is
beliefs.
This
the case of observable priors.
From
(9), for
any round k >
2,
the difference in average opinions across groups
T2
H
Hence, the difference across groups
average priors, scaled
expected value of this
down by a
in
is
= -jrr—{h-Maverage opinion
(13)
the difference between their respective
is
factor that uses all of the distributed information efficiently.
The
is
E
and hence, from Proposition
W=
-
(w.
-rirr
t z + n.
fr»)
(
14 )
2,
Effi] < E\pl].
6.3
Segregation
Now we
consider a segregated society partitioned into two components, one for each group. Each
component
is
like
an integrated society that
closed to
is
members
of the other component; individ-
uals in different groups receive information about each other only through opinion polls. Formally,
we assume
that the prior of an individual
unobservable to the members of other group. That
knowledge among
B
Now, when any
i
and
£
B
fij is
members
observable to the
is
for
is,
each
i
6
B
of his
own group and
and j 6 W,
[n
is
common
common knowledge among W.
observes the
information the other members of
xb
=
B
first
round announcement A^i, he extracts
all
of the relevant
have, concluding correctly that
— Vii =
+
(1
t
2
)
i6 ,i
r
2
li b
(15)
.
rib-'—'
On
the other hand, he can extract only limited 'information from the
only relevant information for him
fi
w
.
(l
is
+
r
2
)
A w<
\
=
xw
+
Combining these two pieces of information, he updates
T
jl
announcement
w where he knows
,
his belief,
and
in the
neither x w nor
second round, he
announces
Af:2 =
c b {a b fj.i
+
(1
- ab
)
xb )
+
(1
-
c b ) ((1
21
+
2
t )A Wi1
-
r
2
p,
w)
(i
A w j. The
G B)
where
r
otb
2
= -s-
(16)
and
2
+ n )(l+r 2 a 2 )
+ r 2 a 2 )(r 2 + n )+n lu
(r
C6
(l
Hence the average opinion
A sb2 =
It
in
B
+
c b {a b ji b
{\
To
relevant information.
all
announcements of the other group,
round deduces
computing £b and
members
in
/}(,.
That
B know how
of
that
is
is,
1,
)x b )
+
-
(l
B
+
c b ) ((1
r
)^ - r
2
2
^).
(18)
round announcements, the second round announcements
see this, consider any j
j deduces that
(l
j can solve these
j does not
+
r
W
e
From
.
Ab^ =
2
)
+
xb
the average
T
need to know how members of
is
2
jl b
,
and
first
in the
round
second
two independent linear equations, thereby
B
think:
knowing that
relevant information from the
all
As a
react to each others' announcements.
distributed information
all
b
is
each other thinks, j can infer
which members of the
second round,
>
Since n b
(18).
-a
first
(17)
-
fc
at this stage
turns out that, together with the
reveal
fc
result,
aggregated and a segregated society
manner
by the end of the
is
identical to
one
integrated in the long run:
Proposition
For each
4.
i
G
and k >
TV
3,
Afk — A[ k and
=
/3 k
l3[.
This illustrates the power of the argument behind Proposition
When some
1.
information about other individuals (through correlation in Proposition
third parties can extract that information from the
manner
in
1
individuals have
and observation here),
which these individuals react
to each
other's announcements.
We now
turn to the
A-b,2
=
(1
medium run
- a b cb
6
)
beliefs in a segregated society.
+ a b c b fl b +
-
(1
c b )r
2
(fi
w -
p,
w) +
From
(15)
- a b )c b i b +
(1
and
(1
-
we obtain
(18),
c b )i w
.
(19)
ib
(20)
Similarly,
^S,2
=
(1
where a w and
If nt,
= nw
depend on
6,
_
ctwCw) 6
+ a w c w fi w +
c w are defined
then a b c b
and
all
.
easily verified that for
cw
)
t
2
(fi b
In that case, the
individuals have the
[Pi]J
L
-
-
fi b
)
+
(1
- aw
)
cwiw
+
(1
- cw
)
analogously to (16) and (17).
= a w cw
E
It is
(1
=
Mi.
(1 +
any n >
medium-run
bias, /3f
same expectation:
z
2 2W(t 2
r^a^j
+
i
/ON_L
10
n/2)
+ n/2
2,
E\p'2 < E[0$] < E\$).
\
22
^
b
~
^
'
= A b2 - A^ 2
>
does not
.
That
is,
when groups
.
are of equal size, they agree about the value of the
three information structures, and the bias
medium-run
bias under
all
greatest under fragmentation, least under integration,
is
and intermediate under segregation.
When
members
groups are of unequal
medium run
however, the
size,
bias does
of different groups will have different expectations about
members
that, in a segregated society, ex ante,
it.
depend on
Our next
9,
and hence the
result establishes
of a minority group will expect a smaller
run bias than the members of a majority group. Despite
this, it further establishes
agree that the expected medium-run bias under segregation
medium-
that they
all
higher than that under integration,
is
and lower than that under fragmentation:
Proposition
5.
If
< nw
rib
E{/3.{}
Proof. For any
i
e B,
E
L
then
,
< E^l] < Ej\p$) <
[fib]
=
Ei
[9]
=
Ei
Similarly, for
any j G
fib
and
[/8f]
{Mi
E[f3[]
E
L
[jj,
= aw ciu
w
]
{Jj. b
=
-
w
p,
.
£B,je W)
Hence, by (19) and
(20),
(21)
fin)
W,
Ej[p$]
=a
b cb
(flb-fiw)-
(22)
Now,
=
CVfcCfa
+ T 2z a^.2
+ t 2 2 + nbT 2 a 2
T 2 (l + r 2 a 2
+ r 2 2 + n T 2 a 2 +n'
t2
t 2 (1
\
1
<7
)
)
r 2 (l
Since
rib
< n w we have
,
CbQ-b
> cw a w showing
,
cr
LU
)
observe that (14) can be obtained from (21) by setting a
see that EjlP^)
is
decreasing in
In
<
< Ej
that Ei [P2]
2
=
[P2]
^
and aw Cu,
0,
see tna ^ E\i3f
2
is
)
<
£',
increasing in a
Elfy], observe that (12) can be obtained from (22) by setting
nb =
1,
,
[/5^],
2
and
.
To
ctbCb
CI
rib.
summary, expected biases are always highest under fragmentation.
higher under segregation than under integration in the
are identical in the long run. This
is
medium
Expected biases are
run, but the two social structures
intuitive, since individuals
have the least
ability to process
information under fragmentation and the greatest ability to process information under integration.
6.4
We
Large Societies
have so far compared the expected value of biases under three social structures
values of the population size n.
for arbitrary
In large societies idiosyncratic differences cancel each other out.
and we can compare the magnitudes
of actual biases under various social structures state by state.
Doing so reveals that our analysis of expectations misses an interesting and potentially disturbing
23
—
fact
about medium-run
'
segregation puts minorities at a disadvantage in processing public
beliefs:
information and consequently results in biases even
when groups
are formed from ex-ante identical
individuals.
In order to
compare
we consider a family
biases in large societies,
of models, indexed by n, such
that
for
2
some f >
and
e
r
(0, \).
That
lim
By
Pk =
—
fib
f
>
we adopt
is,
a large fragmented society, by (11). the bias
n—>oo
—
—> oo, T 2 /n
n
as
fi
2
and n^/n —>
(23)
B
the convention that
is
the minority group. In
approximately as great as the ex-ante
is
almost surely, for
w
(13), in a large integrated society, the bias
r
is
all
>
k
bias:
smaller, to a degree that
2.
depends on the precision
of the distributed information:
lim
71—»00
=
f3k
35
+
T
(fib
—
fi
almost surely, for
w)
all
k
>
2.
1
In a large segregated society, the bias
is
identical to that
under integration
in the long run, as
we
have seen above:
o
lim
n—>oo
=
j3 k
—r
however,
medium
mean
(fib
—
fi
almost surely, for
w)
run, under segregation, information
that the magnitude of the bias
beliefs in the
medium-run
k
>
3.
societies use all available information efficiently.
This does not,
not fully aggregated.
is
lies strictly
under fragmentation and integration respectively. To see
group
all
1
and integrated
In the long run, both segregated
In the
_
+
T
between the corresponding magnitudes
this,
note from (19) and (20) that average
are given by:
-2
A^ = rjz
n—»oo "'*2
lim
T
V
hm
AAS
2
=
+
fi b
T^
r
+
r
f2
l
-j—
—
1
-fi u
r
tz
+1—
r
Notice that neither group processes information as efficiently as in an integrated society. In
a representative
representative
member
member
of the minority group faces a noisy signal with variance f /r,
2
of the majority group faces a noisy signal with variance f / (1
integration, each individual obtains a noisy signal with variance f which
,
2
2
f /r and f /
(1
—
r).
reality.
>
f 2 /(l
—
r).
As a
result, the
r).
majority view
is
more
in
both
processing
closely aligned
more
is
closely aligned with the reality, the
medium-run
bias depends on
9:
n-.oo
Under
This disadvantage becomes more pronounced as group sizes become more unequal.
Since the majority view
lim
and a
clearly smaller than
Furthermore, under segregation, minorities are disadvantaged
public information, since f 2 /r
with
is
—
effect,
2
=
f2
35
t~
+
fb
l
r
~
f2
35A
r
+
1
—
r
fw -
f2
/
T5z
\t +
\
24
f2
5
r
t
z
+
I
\
—
^
r
I
almost surely.
Because of this dependence on
the bias can take any value. In particular, in the
9,
the difference in beliefs under segregation
and therefore exceed the
medium run
biases even
More
it.
ex-ante beliefs, the medium-run bias
if
9 turns out to be very
under segregation, there
interestingly,
the groups have identical ex-ante beliefs, fa
if
run,
increase (relative to the ex-ante belief difference)
difference under fragmentation. This will occur
from ex-ante expectations of
different
may
medium
= w=
p,
may be
With
p,.
large
identical
is
-2
?s
lim
=
n-^ /321
Hence there are differences
r
i
—5
+r
\f 2
^
f2
in opinion across
lu
)
+
-rj
1
groups
in
groups are formed from ex-ante identical individuals.
the
medium and
limn-.oo
Pk
=
long run bias
negligible
is
—
the
despite the fact that the
under fragmentation and integration: lim n _oo f3[
almost surely. Since the dependence of the bias on 9
it
is
=
caused by the disadvantage
become more
increases as group sizes
8
Conclusions
7
If
medium run
In contrast, with ex-ante identical beliefs,
faced by minorities in the processing of public information,
unequal.
almost surely.
J
9)
common
a group of individuals share a
(in the sense that
common
each
member
and are commonly known to be Bayesian rational
prior
of the group forms beliefs using Bayes' rule according to the
prior) then public disagreement cannot arise.
Accounting
requires a departure from one or both of these hypotheses.
We
for
such disagreement therefore
have chosen here to explore the
implications of heterogeneous priors, while maintaining highly stringent assumptions regarding
Bayesian rationality.
Two main
results follow
from
model's primitives, the extent of public disagreement
observable,
and public
Second, we find that
we
this. First,
is
find that for generic values of the
independent of whether or not priors are
beliefs involve the aggregation of all distributed
when
information in the long run.
priors are uncorrelated, the expected value of public bias
is
lower in an
integrated society than in a fragmented one. For large societies, a stronger result holds: public bias
is
greater in a fragmented society relative to an integrated one under almost
and information. This suggests that
all
social integration (in the sense of better
realizations of priors
understanding of the
priors of others) should result in diminished public disagreement, especially in large populations.
Our
ferences,
results
depend on the
ability of individuals to
based not only on the
beliefs are
initial beliefs of
make
highly sophisticated statistical in-
others but also on the
adjusted over time on the basis of earlier announcements.
If
manner
in
which these
cognitive limitations pre-
vent individuals from making inferences based on the manner in which one person responds to
another's announcement, then our
8
medium run
analysis applies,
and expected bias across
Individuals belonging to a minority within any population tend to have a smaller
networks (Currarini et
al.,
be endogenously increasing
2008), which should reinforce
in
extent of public disagreement
this effect.
On
number
social
of affiliates in friendship
the other hand, segregation itself tends to
the size of the minority group (Sethi and Somanathan, 2004), which suggests that the
may not
vary monotonically with the size of the minority group.
25
groups depend systematically on the extent of social integration. Expected public bias
mediate
smallest
(where priors are observable both within and between social groups) and
in integrated societies
largest in
is
fragmented societies (where priors are unobservable even within social groups). Inter-
levels of
across groups.
expected bias arise under segregation, when priors are observable within but not
Hence integration both within and across
social groups tends to reduce expected
levels of bias.
Communication
in
segregated societies can cause
to emerge where none previously
signals equally precise, minority groups
information that results
members
initial biases to
existed. Despite the fact that all
members
(or outside observers) fail to appreciate this effect,
factors such as the
them
all
face a disadvantage in the interpretation of public
in beliefs that are less closely aligned
as "bizarre" or "outlandish", attributing
be amplified, and new biases
announcements are public and
with the true state.
they
may
If
majority group
regard the views of minorities
to failures in reasoning rather than to structural
demographic composition and the constraints on information exchange induced
by the heterogeneity and unobservability of prior
beliefs.
26
A
Appendix
—Proofs
Aggregation of Distributed Information
A.l
we prove Proposition
In this subsection,
random
formula. For any two
X
X
and Y,
IUk
Y /
\
then conditional on Y,
vectors
The proof
1.
if
))'
Ek
J'Uv..y
N (E [X\Y]
distributed with
is
requires the use of the following well-known
Var (X\Y)) where
,
E \X\Y\ = p x + XxyZy (Y - fiy)
Var(X\Y) = E x - ExyVyEy*1
We
also
need to introduce some more notation. For any subset N'
denote the column vector obtained by stacking up
write
/.i
N =
>
((Mj)j^n',
and any matrix
A''
N",
i.e.,
E_i,_i
=
=
Xn'^ii
(cr^fc) .,
X
= A^k)^ N
A N tk
= {x{ j)
>
iWJ
(
We
.
eJV
i
i
{xij) l&N
We
identity matrix.
write
Q.
=
Wfl
=
Afi
=
R
H
and
2
2
2
2
r / (r
+
Lemma
in
+
1
\R\)
Ah =
N"
of
the submatrix with entries from N' and
=
e.g., /z_,1
and
(fJ.j)j^i
and I
for the
p—i
+ a^a-ij
-
S_,,_i
2 below,
we
+
r2/
+
llxn-l fQl„_l x „_l
T
+
t4 ^H,H
2
+
f /
in the following
N\ (R U
T
-
2
t
2
,
!
1
ctI !|1
.
also write
T
4
~
o~}vMr
llx R (lR
[a H
t HtR £~^R t RjH
S. lrl -
)
tV'm/,,)
For any
Y,
)
A jA +
(26)
i
£
N
is
and any round
k,
let
revealed by the end of
\
\
(\
+
t
2
)vM r A h
,,
j£RU{i]
/_
\R\
4
lemma.
2
i
T
Then,
{i}).
^\ R ^-yMR l mxl
'
,
whose private information
,
T2
o-~ (7_ 1
(25)
£k)
l),
llx|//| f^l|//|x|//|
H=
-
{/m
1
,
be the set of other individuals
let
For any subsets iV and
j€/v ,.
instead of 7V\ {i},
ylsswne that the priors are not observable.
and
For example, we
k x /-dimensional matrix with entries
= Var ((J.-i\lH) =
compute the announcements
N\ {i}
round k — 1,
R
—i
use subscript
for the
(oj,,)
A'yv'.iV" for
= E [fJ.-i\fM.] =
a = r /(r +
2.
and a W ',i
i-,
write
use subscript N' to
the values for j G N'.
all
=
C N, we
write
Using the definitions of
Lemma
We
lfc x /
£_,_,
v
we
,
jefjii-
f
Jx-i
We
(24)
~ r2{iM« (AW ~
\
^H,R^R (VR
- t H ,Rt R ^a Rtl {m j
~
Aft)
J
l
,i
27
fi t )
(27)
R^
when
and
A% k =
R=
w/ien
aAfiln-lxl)
AiA + r 2 M A- hl -
T
t
2
aM fi_
l
- T 2 o^aM o^i
l
t
A;.
true
(
-
W
(28)
fi t )
0.
We will use mathematical induction on We first compute Af 2
= a/ij + (1 - a) Xj,
for k = 2. For each j, since A
Proo/.
is
-
(1
J:
\
+
(\
showing that the statement
,
t
2
)Aji =
e
+
+
£:
2
T
^lj.
Hence,
+r
E, [(l
Now, the
first
+
r
2
A_i,i
)
91 n -\x\
=
+£-, +
2
normal
T ^_j
=
=
E [9\^x u
A hl + aMi
-
+
{l
l
(1
((1
+
aMT„_ lxl
where the second equality
is
by
suppose that the proposition
is
)
T
+
2
t I
r
R
defined for
k!
and
j
,
2
)A^
l
2
+
(24)
+
t
in
+r 2 a-
new
vector (l
1
CT_
!
,
(/x,
2
+ t 2 A)
-
ft,)
=
lt \
(29)
.
#l n _i x
i
+
e_,
+
the variance of
(Xj,/i,),
4
the second round
Ei [(1
i
announces
2
)
A^, A \^,x,])
Mu and the
definition of
<
true for rounds k'
~
r
- Qr M^_i - rV^aMjCr-^
r ^/^,,,!
and the
+
2
2
[crHj
- a~ a^ hl al
l
(£__>
}
-
A_i,i
)
Aj,!
Mr
for
a
i
2
r ^.,
Notice that, conditional on
noise.
Hence, updating his belief according to (24),
=
T Ei[^\in].
is
q1„_i x „_i
Al 2
+
ln-ixl^,!
round of announcements provides
of signals with additive
T")j,-j
2
+
A,,,
|
W ,Si]
|
^x;] =
A,-,i
)
we obtain
Substituting (25) in this equality,
Ei [(1
2
—
k
1
and
1
£//,fl£fl, tf cr flj)
no new information
is
at the
w
last equality is
Then,
for all j.
-
by
(30)
ft)
Now
(29).
if
=
revealed by the announcement
measurable with respect to the public information
(
end of round
k!
—
1.
A\, because
On
it is
the other hand,
if
Mr
then we can solve for
/ij
from
revealed by the end of round k
if
R=
(28)
is
0,
i
has not learned any
equivalent to (30).
Now
(cr/y,j
(27) for k'
—
1, i.e.,
- EH,R^fi R IJ
R,jJ /
l
and
That
j.
j 6 R, or
new information
suppose that
i
is,
0,
either the private information of j
knows only that Aj
\
t
=
afij
+
(1
after the first round. In that case,
R^
28
0. Individual
i
knows (m,Xi),
—
is
a) Xj. Now,
Af k = A™ 2
,
{(Xj,Xj) for j
a nd
6
R
and that
A
=
Jt \
conditioning on
a/i-j
+ (1 -
(ji t
x{),
,
then on
~
M-i
(fii,X{), (9, /i_,-,e_i)
x /?
on
xr =
l|fl|xi#
+
about
£fl
l\R\ x \9
+
£/{,
6>
and
2
)
AH =
l\
,i
AH =
a/j.
,\
+
H \xiS + e H
r
2
Nl
H+
(1
- a) x H
2
first
i.e.,
,
(31)
.
are independently and normally distributed with 9
~ N
N(/i_j, E_, _i), and e_ 2
=
r
conditional distributions sequentially,
xr), and finally on
((i,r,
+
(1
Conditional on
We compute
a) Xj for j £ R.
~ N (A
1t
\,a),
Then, from {^r,xr), he obtains a new signal
(0,t I).
new information about hh from
also potentially
Conditioning
/j,r.
he updates his belief about 9 to N(p,i,v) where
t2
*
T
2
1
1
+ 1 + 1^1^.1
+
r2
^
5Z
2
+1
+ l + |i?|
r
T2
+
1
+1+
|
r2
.
|^
|
r
A'M -
x| fi
1
jR
2
1
+
+
1
lx|/?
|^.|
|^
r2
T2
Conditioning on
Now,
i
fi/?,
A
l+j/?|'
he updates his belief about
/I//
=
/ttf
E//
=
E//,//
conditions on (31) starting from 9
+
(1
Using
+
(24),
=
2|fc
=
t
2
)A ha
+ Etf./jE^
~
E
+
r
Ai +wlix|/f| (ul|ff|x|ff|
+
(1
[9\/j, l
-
,x l ,id R ,x R
,
(1
fiMfll^ixi) Ai
2
+
'+ (1
2
) ^4//,!
4
t E/,
+
A/?)
Given the conditionings so
+r 2 ^//,i)l| W
+
-
(l
2
+
=
+
l|//|
7-
X
|
|
W |+T
4
E/y
+
far,
by
(31),
2
r /)
.
ff/
2
((1
/J
- r
2
+
+
-r>//]
2
)^ h -
t
,i
vMR
fi
Ail|//|xi
-
T
2
fi
H)
H
T2
jeKu{*}
2
T )i)M, R 4//,i
r vAffi
(/i//
+ a~>a H
is
by
,i
(Mz
-
Ai)
+ ^h.r^r
(24); the third is
the last by substituting the values of
fj, z
equality in the proposition.
2,
+
1
where the second equality
Lemma
x i^
vMrA ha
t2 )
1
Using
-
{ftR
- E//^E^ fl Sfl,//.
N(fii,v).
~'JV(/2il| H |xi
where
to N(jj,[{, E//)
he therefore obtains
T
Mr, and
fipj
we can now prove Proposition
1.
29
(pr -
A/?
- ^^R,i
(fM
-
pi)))
,
by arrangement of terms using the definition of
and
/}#.
By
rearranging terms, we obtain the
)
Proof of Proposition
private information
ft,
i.e.,
/Lij
=
+
(1
showing that x
tion of
is
i
m
x
r
)
is
A,i
=
(l-j
the coefficient of
fi,
is
-
It
and
Aj,i, /l_i,i,
x
^
^
Then, since no individual's
0.
(28),
Aii2
,
Moreover, since
/1 2 ,2-
=
j4 2 ,i
a/i,
+
-
(1
a,) x,,
=0.
%1
=
If /?
=
the other hand, since o"_ JiZ
lt
\,
l
=
%- Therefore, the private
isolated,
2, for
i.e.,
a^
any
/c
X]l
>
=
1,
(Note that,
0.
if
informa-
R
0, then
^=
Ehm^r^Hj) '=
[o Hj -
0, the coefficient
is
again r 2 a~
l
aM a^i
x
=
A
0.
Thus, A;^
is
end of round k — 1, revealing no new information.
at the
0, (xR,fj.f>)
is
i
x
Lemma
E-^-j.) Hence, by
vMR
A
and
A-i,\-,
Conversely, suppose that
2.
and £_;,_,
and or
A
-
:
_ llf
,
measurable with respect to the information
On
by
2,
^+
t
-T 2 a~
=
x
_
revealed by round
o~n,%
M O-
i.e.,
- aA/jln-xxiMi.i + T 2 MjA-i,i - T^aMip-i -
(1
,
measurable with respect to
in that case, /i_j
because
regular,
compute that
further
z,
is
i
H
measurable with respect to
is
we can
that
first
revealed by the end of round
is
—
Assume
1.
does not provide any information about
only reduces the variance of x x without revealing
/i x
Hence, the private information of
it.
either.
,
i
not
is
revealed at any round.
A. 2
Public Bias
Lemma 1. By Proposition 1, since the priors are independent, no
by Lemma 2, Af^ = A? 2 and Af 2 satisfies (28). To compute Af 2
Proof of
Hence,
,
^=(l +
T
2
)(l+T 2
information
from
is
revealed.
(28), first define
2
)
and note that
Mi
=
lixn-i (al n _i xn _!
=
a" llxn-1 (ln-lxn-1 +
a<p(<p
+
— lixn-1
+n-
(t
2
tV
)/)
+n-
1) /
+
2
<pl)
((p
- ln-lxn-1
1)
1
a(ip
Here, the
first
equality
is
+n-
(3'-
llxn-11)
obtained by substituting
£ = a2 I
simple algebra. In the third equality, we invert the matrix
in (26),
l n _i
xn _i
and the second equality
+ <pl.
It
1
+¥ ^)^ =
3
—
}
ip [ip
+n—
7T((V 3
1
30
1)
+ "-
!)
by
can be easily verified
that
(l„_lxn-l
is
- ln-lxn-1^),
yielding the third line.
we obtain
Finally,
(1
_
- QM,l n _ lxl
1
p+
n~l A
.
equality
of the value of
M
z
is
2
-r
^- A ^~
'
1
%
+
+
1
r
from
j4j,i
T2
T
+ n-l)
a(<p
^
2
=
1
^
2
-p + n -l^ A^-
=
A
+
1
'
+ n-l^
if
A
+
"' 1
T2 a 2
and hence are cancelled out
i,
+
(l
T
T2 )
4
-
(A.l
a2
(//,
-
- ^) +
fij)
+
T
Aj,l)
2
-
(ii
+
+
t
—
Xj))
-
Ej)
- ft)
(ft
+
T
—
-
fij)
+
r
—
(m -
fij)
r
2
)
A,i
=
:;i
t
2
(jM
7
7
by substitution of the definitions
(l
in the difference, yielding
2
a2
{e,
7
by
p + „.-l2^-
2
L
2
(lU
is
By adding and
2
(r
the third equality
by the substitution
r2
7
T 2 CT 2
1;
just
n-l^
+
tp
7
+n-
is
by straightforward algebra.
<z>-(l+r 2 )
if
1
'
and the second equality
0,
+n-
ip
t
last equality is
Terms with summations do not depend on
is
l n -ixn-i),
and m, we obtain
y + n-1
Here the second equality
-
t
n-l^'
The
(32).
1) I
M /U,i - dr^Mifl-i
simply (28) for o"-^,
subtracting new terms with
A
+
A,,!
llxn-lln-lxl ^ A
tp + n-1
J
f,
\
first
)
+n-
((ip
we then obtain
(32). Substituting (32) in (28),
K2 =
Here, the
by adding up the rows of the matrix
^+x
%
p>
=
and the
(l
+
last is
r2 )
(l
+
t 2 ct
by x l - Xj
—
)
and 7
=
ex
-
D
2
ej.
References
[1]
Acemoglu, Daron, Victor Chernozhukov, and Muhamet Yildiz (2008). "Fragility of Asymptotic
Agreement under Bayesian Learning", Unpublished Manuscript, MIT.
[2]
Adrian, Tobias and
Mark
Westerfield (2007). "Disagreement and Learning in a
Dynamic Con-
tracting Model", Review of Financial Studies, forthcoming.
[3]
Allen, Michael O., Patrice O'Shaughnessy,
New
vide."
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