Attitude Polarization: Theory and Evidence

Attitude Polarization: Theory and Evidence
Jean-Pierre Benoît
London Business School
Juan Dubra
Universidad de Montevideo
July 22, 2014
Abstract
Numerous experiments have demonstrated the possibility of attitude polarization.
For instance, Lord, Ross & Leper (1979) found that death penalty advocates became
more convinced of the deterrent e¤ect of the death penalty while opponents become
more convinced of the lack of a deterrent e¤ect, after being presented with the same
studies. However, there is an unclear understanding of just what these experiments
show and what their implications are. We argue that attitude polarization is consistent
with an unbiased evaluation of evidence. Moreover, attitude polarization is even to be
expected under many circumstances, in particular those under which experiments are
conducted. We also undertake a critical re-examination of several well-known papers.
Keywords: Attitude Polarization; Con…rmation Bias; Bayesian Decision Making.
Journal of Economic Literature Classi…cation Numbers: D11, D12, D82, D83
Take two individuals with priors p and q over and f and g over A: The …rst individual’s
prior over =
A is the product p f and the second has prior q g:
Take a signal s that has probability h a (s) in state ( ; a) 2 : For
= f 1 ; :::; n g
and i < i+1 for all i: Recall that sgn (x), the sign function, is 1; 0 or 1 according as
x > 0; x = 0 or x < 0: We say that s is unambiguous if for all i ; j and all a and a;
sgn h j a (s) h i a (s) = sgn h j a (s) h i a (s) : The property says that s is unambiguous
if j is more likely than i after s; given a; then the same must be true after a di¤erent a:
We say that there is polarization (after s) if p ( j s) p q q ( j s) (where p ( j s) is
the marginal of the posterior over after s).
The following Theorem provides a characterization of what it means for a signal to be
unambiguous, and what are its consequences. It is a generalization of Baliga et. al.
Theorem 1 If signal s is unambiguous, there is no polarization, otherwise there are p; q
(with p = q) and f; g such that polarization occurs.
We thank Gabriel Illanes and Oleg Rubanov. We also thank Vijay Krishna, Wolfgang Pesendorfer,
Debraj Ray and Jana Rodríguez-Hertz for valuable comments.
1
Proof. Suppose that after some unambiguous s there is polarization. In that case,
p ( n ) and q ( 1 ) q ( 1 j s) : That is,
n j s)
P
P
P
P
p ( n ) a f (a) h n a (s)
p ( n j s) = P P
p ( n ) , a f (a) h n a (s)
p ( ) a f (a) h a (s)
( ) f (a) h a (s)
a pP
P
P
q ( ) a g (a) h a (s)
a g (a) h 1 a (s)
p(
Similarly, from p (
1
j s) p ( 1 ) and q ( n ) q ( n
P
P
p(
a f (a) h 1 a (s)
P
P
q(
a g (a) h n a (s)
From the four inequalities
P
f (a) h n a (s)
Pa
a g (a) h 1 a (s)
j s) we obtain
P
) a f (a) h a (s)
P
) a g (a) h a (s)
P
P
p ( ) a f (a) h a (s)
P
P
q ( ) a g (a) h a (s)
P
f (a) h 1 a (s)
Pa
a g (a) h n a (s)
(1)
However, if h n a (s) > h 1 a (s)
the samePmust be true for
P
Pfor any a; by s unambiguous
all
P a; and would therefore imply a f (a) h n a (s) > a f (a) h 1 a (s) and a g (a) h n a (s) >
a g (a) h 1 a (s) ; which is a contradiction (similarly if h n a (s) < h 1 a (s) for any s). Hence,
we must have h n a (s) = h 1 a (s) for all a: This, in turn implies (in equation (1) the …rst and
third terms are equal)
P
P
P
P
p ( ) a f (a) h a (s) = a f (a) h 1 a (s)
a f (a) h n a (s) =
P
P
P
P
q ( ) a g (a) h a (s) = a g (a) h n a (s) :
a g (a) h 1 a (s) =
This implies p ( n j s) = p ( n ) ; q ( n j s) = q ( n ), p ( 1 j s) = p ( 1 ) and q ( 1 j s) = q ( 1 ) :
Assume now as an induction step that for i = 1; 2; :::; j; n j; :::; n we have p ( i j s) =
p ( i ) and q ( i j s) = q ( i ) : One can repeat the steps above to obtain the result for j + 1
and n j 1: This concludes the proof.
To show polarization assume s is not unambiguous, so that there exist H; L 2
and
gHh (s)
gHl (s)
h; l 2 A such that g (s) 1 g (s) with one inequality strict. Set
Lh
Ll
Ancillary distribution g A
Probability of each state in
A#
h
l
!
H
L
t # E ! (H; h) or (L; h) (H; l) or (L; l)
wz
(1 w) z
th
1
0
and
:
w (1 z) (1 w) (1 z)
tl
0
1
to obtain
p (H j th ; s) =
gHh zw
gHh zw + gLh z (1
Similarly, p (H j tl ; s) < w ,
w)
> w = p (H j th ) , gHh > gLh :
gHl (1 z)w
gHl (1 z)w+gLl (1 z)(1 w)
< w , gHl < gLl : Since one of the two
inequalities is strict, we obtain polarization. In this case it obtains with g A depending only
on a and p such that and A are independent.
2
Suppose types are
N (0; 1) and that A = fR; P g : If Hannah is Poor, the signal is
+ " where " N (0; 1) ; if Hannah is Rich, " N (0; 2 ) for < 1:
Individual 1 thinks the probability of R is r > 12 and individual 2 thinks it is q < 12 :
Fix any signal s: We have g j a (s) > g i a (s) , js
j j < js
i j ; which implies that
g j a (s) > g i a (s) for all other a: Hence, s is unambiguous.
In the previous proof, we need to check where we use that p and q have common support.
When priors are not independent, an unambiguous signal may lead to polarization.
Example 1 Consider the following two prior beliefs (where the prior beliefs of truth are
0:6875 and 58 = 0:625), and the unambiguous signal C
1
2
3
16
1
4
1
16
1
2
1
8
1
4
1
8
and C with likelihoods
1
10
1
2
11
16
=
1
20
9
20
The idea is that in both cases the signal will increase the posterior in each ancillary state,
but since the signal indicates that bottom state (L or "free") is so likely, in the second case
you are assigning a lot more weight to the "low" original distribution 21 ; 12 (that is the
distribution of 18 ; 18 conditional on the bottom state). The posteriors of Truth are
1 1
3
+ 12 16
10 2
1 1
9 1
1 1
3
+ 21 16
+ 20
+ 20
10 2
4
16
1 1
11
+
10 2
28
1 1
11
1 1
9 1
+
+
+ 20
10 2
28
20 4
8
=
46
11
>
59
16
=
18
5
<
29
8
So, bottom line, the characterization theorem is false with general beliefs. In particular, an
unambiguous signal can still generate polarization.
1
Experts
Our intuition is that if people disagree about how likely the truth of the statement is, and
they have observed more or less the same signals, then it must be because they disagree
about the likelihood of some ancillary state; then, when they are shown more information
like the previous one, those di¤erences in beliefs make them further polarize. The following
theorem, proves exactly this intuition, and con…rms a …nding in the experimental literature,
that “experts” are more likely to polarize than people who do not know much about the
issue.
People have prior (a probability distribution over the set ; where P; Q; R; T are numbers
that add to 1):
Prior over
useful not useful
selection
P
Q
free
R
T
We assume from the outset that the prior is independent:
as the product of a distribution on fu; ng fs; f g :
3
P
Q
=
R
T
(the prior can be written
Individuals observe a sample of signals “about”u or n: Let S be the (…nite) sample space
of signals S. For each signal Si 2 S we write its likelihood as
Likelihood of Si :J! (Si )
useful not useful
selection
pi
qi
free
ri
ti
(2)
In addition to the information S in S they observe signals about s or f , where the individual
observes the signal 2 (0; 1) with a density
Probability of
selection: in states us or ns signals drawn from
free: in states uf or nf signals drawn from
and
( )
( )
increasing in . We assume additionally that
( )
( )
! 1 is completely informative about
s (lim !1
= 1) and ! 0 is completely informative about f (lim !0 (( )) = 0):
After they observe this information, they observe a Common signal C with likelihoods
Likelihoods of C for ech !2
selection
free
useful not useful
p
q
:
r
t
(3)
We postulate the following assumption about signals S:
Assumption 1. Weak Ambiguity (WA). Signal Si satis…es Weak A1 if pi ti > qi ri :
Ambiguity (A). We say that C is ambiguous if p > q and t > r.
Theorem 2 Take two people who have observed the same S (say, two experts who know
the whole “body of evidence” about an issue). Assume the prior is independent. We know
C must satisfy ambiguity and we are told that C is a typical signal, so we also assume S
satis…es weak ambiguity.
There exists vS such that P (u j S; ; C) > P (u j S; ) , P (u j S; ) > vS :
Proof. Step 1. Individual increases belief after C i¤ high ; de…ne cuto¤ B .
For any probability distribution B over we have B (u j C; ) > B (u j ) (for C satisfying
Ambiguity) i¤
B (us j ) + B (uf j )
pB (us j ) + rB (uf j )
>
,
qB (ns j ) + tB (nf j )
B (ns j ) + B (nf j )
( )
( )
( )
( )
pB (us)
+ rB (uf )
B (ns)
+ B (nf )
>
B (us)
+ B (uf )
qB (ns)
+ tB (
( )
( )
( )
( )
Letting
f( )
B (ns) B (us)
( )
(p
( )
q)+B (us) B (nf ) p B (ns) B (uf ) q B (us) B (nf ) t+B (uf ) B (ns) r
4
equation (4) can be written as
( )
f ( ) > B (uf ) B (nf ) (t
( )
r)
We have that f ( ) is increasing in . As ! 0; f ( ) converges to a constant, so the lhs
converges to 0 < B (uf ) B (nf ) (t r) : As ! 1; (( )) f ( ) ! 1: Since (( )) and f ( ) are
increasing, there exists a unique B 2 (0; 1) such that B (u j C; ) > B (u j ) , > B :
For such a B ; B (u j C; B ) = B (u j B )
B:
From Step 1, there exists a S such that P (u j S; C; S ) = P (u j S; S ) and P (u j S; C; ) >
P (u j S; ) if and only if > S : De…ne vS as vS = P (u j S; S ) : Then, from Lemma 1 we
know if S is weakly ambiguous beliefs P (u j S; ) are increasing in ; so that P (u j S; ) >
vS , > S , P (u j S; C; ) > P (u j S; ) :
Lemma 1 Suppose the prior is independent. Si satis…es WA if and only if posteriors of u
increase with : In particular, pi ti > qi ri , P (u j S; ) is strictly increasing in (and < i¤
strictly decreasing): The e for which P (u j S; e ) = P (u j S) is de…ned by (( ee)) = 1:
Proof. We have that for a signal S with likelihoods pi ; qi ; ti ; ri
P (u j ; S) = P (us j ; S) + P (uf j ; S) =
which increases in
The derivative of this expression wrt
dX
d
( )
( )
+ Rri
+ Rri + T ti + Q
( )
q
( ) i
( )
( )
=
P pi T ti
T ti +
( )
( )
( )
P pi
( )
T ti +
+ Rri
( )
Qqi
( )
:
is
Qqi Rri
Qqi (( ))
2
> 0 , pi P ti T > qi Qri R:
(5)
= 1 we get
P (u j ; S) =
2
( )
p
( ) i
P
( )
p
( ) i
i¤ the following expression increases in
X=
When
P
P
P
( )
p
( ) i
( )
p
( ) i
+ Rri
+ Rri + T ti + Q
( )
q
( ) i
=
P pi + Rri
= P (u j S) :
P pi + Rri + T ti + Qqi
Di¤erent signals: counterexample
The question is then whether our results go through when people have observed di¤erent
signals. That is not necessarily the case. We now present an example to illustrate.
5
Consider the following signals.
3
7
2
7
s1
+"
+"
3
7
4
7
s2
4
7
3
7
"
and
"
"
"
2
7
3
7
s3
C
+"
0 2
and 2 7 and
+"
0
7
1
2
1
4
1
4
1
2
The prior is uniform, and people receive signals about Selection or not according to
distributions (when the state is selection) and (when it is no selection).
Notice …rst that signals s1 and s2 do not a¤ect the belief in Selection (which I call H
sometimes):
P (H j ; s1 ) =
3
7
1
4
+
3
7
3
7
1
4
1
4
+ 37
+ 27
1
4
1
4
+
4
7
1
4
=
+
=
1
4
+
1
4
1
4
+
+
1
4
1
4
+
1
4
= P (H j )
(and similarly for s2 ; the trick was having the rows add up to the same number).
With s1 “no one” believes in T with probability larger than 12 ; because you have to be
“certain”that the state is Selection.
With s2 the opposite is true: all believe in T with probability greater than 12 : I don’t
consider s3 because it has probability 0 in state T H:
Setting " = 0 for simplicity, we now …nd the cuto¤s for such that after the common
signal C individuals increase their beliefs.
P (T j si ; C; ) =
ppi ( ) 14 + rri ( ) 14
( ) 41 + rri ( ) 14 + qqi ( ) 41 + tti ( ) 14
ppi
P (T j si ; C; ) =
pi ( ) 14 + ri ( ) 14
( ) 14 + ri ( ) 14 + qi ( ) 14 + ti ( ) 14
pi
so
P (T j s1 ; C; ) =
P (T j s2 ; C; ) =
Letting
13
x
27
14
x
27
( )
( )
13 ( )
27 ( )
13 ( )
27 ( )
12
47
13 ( )
47 ( )
+
12
+
47
14 ( )
+ 41 37
27 ( )
14 ( )
+ 14 37 + 14 27 (( ))
27 ( )
+
+
14
27
+
13
27
and P (T j s1 ; ) =
and P (T j s2 ; C; )
3 ( )
7 ( )
2
7
3 ( )
2
3 ( )
+ 7 + 7 ( ) + 47
7 ( )
4 ( )
+ 37
7 ( )
= 4 ( ) 3 2 ( ) 3
+7+7 ( )+7
7 ( )
+
= x; we …nd the x that solves P (T j si ; C; ) = P (T j si ; ) :
13
x + 14 27
27
+ 14 27 + 41 37 x
14
x + 14 37
27
+ 14 37 + 41 27 x
+
14
27
+
13
27
=
=
3
x
7
4
x
7
3
x + 27
7
+ 27 + 37 x
4
+ 37
7
+ 37 + 27 x
+
4
7
+
3
7
2p
2 = 0:942 81
3
1p
1
, x2 =
541 +
= 1: 010 8
24
24
, x1 =
We then have:
those who believe in T with probability greater than 12 are those who observe s2 (no
one who received s1 ), who have a probability of 74 ; of those, those with (( )) > 1:01
increase their belief.
6
those who believe in T with probability less than 12 are those who observe s1 , who have
a probability of 73 ; of those, all who have (( )) > 0:94 increase their beliefs.
Hence, the proportion of those who increase their belief is less for those who believe in
T highly, than for those who do not believe in T : the probability of with (( )) > 1:01 is
lower than the prob of
3
with
( )
( )
> 0:94
Fixes
There are two reasons why the previous example doesn’t work. The …rst is quite simple: it is
not really a counterexample to our intuition, since there is not enough variation in the belief
in Selection (or in H). The following theorem shows that when there is enough variation in
that belief, then there is polarization.
People have prior (a probability distribution over the set ; where a; b 2 (0; 1)):
Prior over
True
False
High
ab
a (1 b)
Low (1 a) b (1 a) (1 b)
There is a set of signals S and a collection of likelihood functions f! for ! 2 such that
f! (S) is the probability that signal S 2 S will happen in state !: For each signal Si we
generally let pi = fT H (Si ) ; qi = fF H (Si ) ; ri = fT L (Si ) and ti = fF L (Si ) :
In addition to these signals, individuals also receive one of two signals fh; lg about the
ancillary state, where the probability of signal h is given by
if ! = T H; F H
if ! = T L; F L
P! (h) =
for
> :
We are interested in the informativeness of signals h or l : how are the beliefs about H
or L a¤ected by the signals. Thus, we analyze
P (H j S; h) = P (T H j h)+P (F H j h) =
p ab + q a (1 b)
b) + r (1 a) b + t (1
p ab + q a (1
This posterior is a monotone function (which converges to 1 as ; 1
a
1
pb + q (1
a rb + t (1
a) (1 b)
(6)
! 1) of
b)
:
b)
and similarly, P (H j S; l) is a monotone (decreasing) function of
a
1
1
a1
pb + q (1
rb + t (1
b)
:
b)
Therefore, signals about the ancillary issue are more informative when
decreases.
7
increases and
:
For the common signal, we assume that its likelihood in each state is
Likelihood of C for each state in
True False
High P
Q
Low
R
T
for P > Q; T > R (that is C is ambiguous).
We are interested in the following two quantities: the proportion of people with beliefs
greater than v who increase their beliefs (after C), and the proportion of people with beliefs
less than v who increase their beliefs; we want to show
P
P
P (Si ) P (a > aiv ; aiC )
P (Si ) P (aiv > a > aiC )
P
P
>
(7)
P (Si ) P (a > aiv )
P (Si ) P (a < aiv )
and we want to show that the expression on the left is greater than that on the right.
Theorem 3 If there is enough variation in the beliefs about the ancillary issue (if
is
su¢ ciently large and su¢ ciently low), and people have observed ambiguous signals, then
the proportion of people whose beliefs are larger than b = P (T ) and increase them after
observing C is larger than the proportion of people whose beliefs are lower than b and increase
them after observing C. In particular, for and 1
large enough, all those above v increase
and all those below b decrease their beliefs after C:
Proof. First notice that ambiguity of S implies that P (T j H; S) > b > P (T j L; S) :
Next, we know that if is large enough and is low enough, continuity of beliefs in and
ensure that all those who observe h will have beliefs larger than b :
P (T j S; h) =
p ab + r (1
p ab + r (1 a) b
a) b + q a (1 b) + t (1
a) (1
b)
!
;1
!1
pb
pb + q (1
Finally, note that the posterior belief after h; C converges (when ; 1
belief after C in state H :
P p ab + Rr (1 a) b
P (T j S; h; C) =
P p ab + Rr (1 a) b + Qq a (1 b) + T t (1 a) (1 b)
b)
= P (T j H; S) >
! 1) to the
!
;1
!1
P pb
P pb + Qq (1
pb
which is larger than pb+q(1
= P (T j H; S) : Hence, for ; 1
close to 1 we obtain
b)
P (T j S; h; C) > P (T j S; h) :
We conclude that those who observe signal h; if is large enough and small enough,
have beliefs larger than b and increase their beliefs after C: Those who observe signal l have
a belief lower than b and decrease their beliefs.
In the previous result, there are only two signals, and “enough variation”, but that is an
extreme case for the more general case that there are many signals, and those that are more
informative have enough probability.
But even without the “enough variation”, there is something else that makes the previous
example in Section 2 really not a counterexample: the signals are not really ambiguous.
Consider for example signal s1 with likelihoods:
3
7
2
7
3
7
4
7
+"
+"
8
"
:
"
b)
=P
It is not really ambiguous, since it is basically “bad news”: it is neutral when the state is
H; and bad news in state L: We therefore introduce the notion that there must be some
symmetry in that if the signal is good news in one state, it must be “comparably”bad news
in the other state. With likelihoods as in (2):
Weak Ambiguity. Signal Si satis…es Weak Ambiguity if pi ti > qi ri :
We now repeat some of the previous material, and show that when signals are weakly
symmetric, polarization holds.
People have prior (a probability distribution over the set ):
Prior over
T
F
H
ya
y (1 a)
L (1 y) a (1 y) (1 a)
and they observe a sample of signals “about” T or F: Let S be the (…nite) sample space of
signals S. For each signal S 2 S we write its likelihood as
Likelihood of Si :J! (Si )
useful not useful
selection
p
q
free
r
t
(8)
In addition to the information S in S they observe signals about s or f , where the individual
observes the signal 2 (0; 1) with a density
Probability of
selection: in states us or ns signals drawn from
free: in states uf or nf signals drawn from
and
( )
( )
increasing in
tionally that
(we might get rid of this which adds nothing). We assume addi-
! 1 is completely informative about s (lim
completely informative about f (lim !0
Note that after observing a signal
y
x = y+ (1
y)
ya + y (1
ya
a) + (1 y) a + (1
( )
( )
!1
( )
( )
= 1) and
! 0 is
= 0) **this is just to simplify**.
their beliefs become (for example, for T H), for
y) (1
a)
=
xa + x (1
xa
a) + (1 x) a + (1
x) (1
and similarly for the other states:
T
F
H
xa
x (1 a)
L (1 x) a (1 x) (1 a)
(9)
So from now on, we assume everybody has a prior as in (9), with the same a for everybody,
but a distribution of x; which is derived from the distribution of :
9
a)
= xa
Suppose v is a belief in T that can be attained when signal S (with likelihoods pq
), a
rt
pa
ra
value between the beliefs when H is known and when L is known: pa+q(1
>
v
>
.
a)
ra+t(1 a)
y
Then, the cuto¤ x = y+ (1 y) (that is, we are indirectly de…ning the cuto¤ ) for which
P (T j S; xS ) = v is de…ned by
xS =
tS 1 v v 1 a a
pS
rS +
v
rS
1 a
1 v a
(tS
qS )
tS g rS
gqS + tS g
pS
(10)
rS
After they observe this information, they observe a Common signal C with likelihoods
Likelihoods of C for ech !2
T F
H P Q
L R T
(11)
:
We postulate the following assumption about signals S and C:
Weak Ambiguity. Signal Si satis…es Weak Ambiguity if pi ti > qi ri :
Ambiguity. We say that C is ambiguous if p > q and t > r.
Weak Symmetry. We say that Si is weakly symmetric if pi = bqi and ti = bri :
Theorem 4 Assume S is WA, C is ambiguous, and both are weakly symmetric. Then, there
is attitude polarization:
P fS;
: P (T j S; ; C) > P (T j S; ) j P (T j S; ) > P (T )g > P fS;
: P (T j S; ; C) > P (T j S; ) j P
Proof. From equation (10), if S and C are weakly symmetric, and setting v = a
P (T j S; )
v,
xvS
tS 1 v v 1 a a
pS
rS +
rS
v
1 a
1 v a
(tS
If S with likelihoods pq
and C with likelihoods
qp
P (T j S; ) happens i¤
PQ
QP
qS )
=
bq
br r
r + br
q
=
r
:
q+r
(12)
are weakly symmetric, P (T j S; ; C) >
BQbqxa + Rra (1 x)
bqxa + ra (1 x)
>
BQbqxa + Rra (1 x) + Qqx (1 a) + BRbr (1 x) (1 a)
bqxa + ra (1 x) + qx (1 a) + br (1
(BQbqxa + Rra (1 x)) (qx (1 a) + br (1 x) (1 a)) > (bqxa + r (1 x) a) (Qqx (1 a) + BRb
(BQbqx + Rr (1 x)) (qx + br (1 x)) > (bqx + r (1 x)) (Qqx + BRbr (1 x))
C
It is easy to check that P (T j S; ; C) > P (T j S; ) , > C
S , x > xS 2 (0; 1) (there
is a cuto¤ for x or such that the individual revises upwards i¤ his belief in H; prior to
observing S is high enough).
Suppose Q > R; then an individual who believes in T exactly v = a = P (T ) ; revises
upwards: plugging xS from (12) in (13) we obtain
(BQbqr + Rrq) (qr + brq) > (bqr + rq) (Qqr + BRbrq) ,
(BQb + R) (1 + b) > (b + 1) (Q + BRb) , BQb + R > Q + BRb , Q > R:
10
This means that xvS > xC
S for all S: So all those who believe more than v (those that have
C
x > xvS ) also revise upward x > xvS > xC
S . At the same time, all those with x < xS believe in
T less than v; and revise downward after C: The inequality in the statement of the theorem
then satis…ed (as 1 > z for some positive z).
If Q < R; an individual who believes in T exactly v = P (T ) revises downward, which
means xvS < xC
S : Then all those with beliefs in T lower than v revise downward, while those
with x > xC
believe
in T more than v and revise upward, which establishes the inequality
S
in the theorem (as z > 0; for some z < 1).
Note the following generalization, that needs to be checked.
Theorem 5 Assume S is WA, C is ambiguous, and both are weakly symmetric. Then, there
is attitude polarization: for any v;
P fS;
: P (T j S; ; C) > P (T j S; ) j P (T j S; ) > vg > P fS;
Proof. Set g =
b2
v 1 a
1 v a
and plug xS =
(bg 1)r
bq r+g(br q)
: P (T j S; ; C) > P (T j S; ) j P (T j
from (12) in (??) to obtain
(BQbqx + Rr (1 x)) (qx + br (1 x))
(BQbq (bg 1) r + Rr (b g) q) (q (bg 1) r + br (b g) q)
(BQb (bg 1) + R (b g)) ((bg 1) + b (b g))
1 Bg (Q R) b2 + R BQ Qg 2 + BRg 2 b + (Q R) g
>
>
>
>
(bqx + r (1 x)) (Qqx + BRbr (1
(bq (bg 1) r + r (b g) q) (Qq (bg
(b (bg 1) + (b g)) (Q (bg 1) +
0
Note that if Q R > 0; the coe¢ cient in the quadratic term on b is positive, as is the
independent term, meaning to say that the equation is satis…ed for all b (that is, for every
signal S). This means that for all S; an individual who believes in T exactly v will revise
upwards, as will all those with beliefs larger than v ( Similarly, for Q R < 0; the equation
is satis…ed for no b; which means that those who believe in T exactly v will revise downwards
(while we know that those with high enough belief in T will revise upwards).
4
Plous: intuition for the basic step
A paper by Plous tests directly our intuition: he asks whether backup systems (checks) are
important in nuclear power safety, or whether having a low rate of potential accidents is
more important. He then checks that those who believe in backups are more likely (than
those who believe low rates are important) to increase their belief that nuclear power is safe
after receiving a report of another instance of a failure …xed by backup systems.
We now show that this intuition works in our model if we assume that the common signal
C is symmetric.
Symmetric. We say that the signal C is symmetric if its likelihoods are
Start with an independent prior,
True
False
High
xz
x (1 z)
Low (1 x) z (1 x) (1 z)
11
BQ;Q
.
Q;BQ
where z is the same for all, but x is not (because they have observed di¤erent s).
Their posteriors are then a constant times
True
False
High
bqxz
qx (1 z)
Low r (1 x) z br (1 x) (1 z)
A person revises up after C (with B > 1) if and only if
BQbqxz + Rr (1 x) z
Qqx (1 z) + BRbr (1 x) (1
Bbqx + r (1
qx + Bbr (1
bqxz + r (1 x) z
BQbqx + Rr (1
,
z)
qx (1 z) + br (1 x) (1 z)
Qqx + BRbr (1
x)
bqx + r (1 x)
>
, qx > r (1 x) :
x)
qx + br (1 x)
>
x)
bqx + r
>
x)
qx + br
We have that
1
, bqxz + qx (1
2
qx
z + b (1 z)
>
r (1 x)
bz + 1 z
P (H j S; ) >
z) > r (1
x) z + br (1
Theorem 6 Plous. Fix de…ne B = S; : P (H j S; ) >
is WS, C is symmetric and ambiguous (and S is similar)
1
2
x) (1
z) , qx (bz + 1
z) > r (1
and its complement B. If S
: P (T j S; ; C) > P (T j S; ) j B)
(16)
That is, those with higher belief in H (in “selection”) are more likely to update up after C.
P S;
: P (T j S; ; C) > P (T j S; ) j B > P (S;
1
1
; those who have P (H j S; )
have qx
r (1 x) (by (15) and
Proof. If z
2
2
b 1); so no one revises up (by 14), so the rhs of (16) is 0; while the rhs is positive (for x
close to 1; qx > r (1 x) ; which ensures that those individuals believe in H more than 12
and revise up).
If z < 21 ; those who have P (H j S; ) > 12 have qx > r (1 x) (by (15) and b 1); which
ensures that all revise up (by 14), so the lhs of (16) is 1; while the rhs is less than 1 (for x
close to 0; the individual believes in H less than 12 ; and revises down).
12
x) (z