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 priorspandqover andf andg overA:The …rst individual’s prior over = A is the product p f and the second has priorq g:
Take a signal s that has probability h a(s) in state ( ; a) 2 : For = f 1; :::; ng
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 ja(s) h ia(s) =sgn h ja(s) h ia(s) :The property says thatsis unambiguous
if j is more likely than i afters; given a; then the same must be true after a di¤erent a:
We say that there is polarization (after s)if p( js) p q q( js) (where p( js) 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.
Proof. Suppose that after some unambiguous s there is polarization. In that case,
p( n js) p( n) and q( 1) q( 1 js): That is,
p( njs) = p( n)Paf(a)h na(s) P P ap( )f(a)h a(s) p( n),Paf(a)h na(s) P p( )Paf(a)h a(s) P ag(a)h 1a(s) P q( )Pag(a)h a(s)
Similarly, fromp( 1 js) p( 1) and q( n) q( njs)we obtain
P af(a)h 1a(s) P p( )Paf(a)h a(s) P ag(a)h na(s) P q( )Pag(a)h a(s)
From the four inequalities
P af(a)h na(s) P p( )Paf(a)h a(s) P af(a)h 1a(s) (1) P ag(a)h 1a(s) P q( )Pag(a)h a(s) P ag(a)h na(s)
However, if h na(s) > h 1a(s) for any a; by s unambiguous the same must be true for alla;and would therefore implyPaf(a)h na(s)>
P
af(a)h 1a(s)and
P
ag(a)h na(s)>
P
ag(a)h 1a(s); which is a contradiction (similarly if h na(s)< h 1a(s)for any s). Hence, we must have h na(s) = h 1a(s)for all a:This, in turn implies (in equation (1) the …rst and third terms are equal)
P af(a)h na(s) = P p( )Paf(a)h a(s) = P af(a)h 1a(s) P ag(a)h 1a(s) = P q( )Pag(a)h a(s) =Pag(a)h na(s):
This implies p( njs) =p( n); q( n js) =q( n),p( 1 js) =p( 1)and q( 1 js) =q( 1):
Assume now as an induction step that for i = 1;2; :::; j; n j; :::; n we have p( i js) =
p( i) and q( i js) = 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
h; l 2A such that gHh(s)
gLh(s) 1
gHl(s)
gLl(s) with one inequality strict. Set
Probability of each state in
A# ! H L h wz (1 w)z l w(1 z) (1 w) (1 z) and Ancillary distributiongA t#E ! (H; h) or (L; h) (H; l) or (L; l) th 1 0 tl 0 1 : to obtain p(H jth; s) = gHhzw gHhzw+gLhz(1 w) > w=p(H jth),gHh > gLh: Similarly, p(H jtl; s) < w , 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 gA depending only
Suppose types are N(0;1) and that A = fR; Pg: 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 isq < 12:
Fix any signal s: We have g ja(s) > g ia(s) , js jj < js ij; which implies that
g
ja(s)> g ia(s)for all other a: Hence, s is unambiguous.
In the previous proof, we need to check where we use thatpandqhave 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 11 16 =
0:6875 and 58 = 0:625), and the unambiguous signal C
1 2 1 4 3 16 1 16 1 2 1 4 1 8 1 8
and C with likelihoods
1 10 1 20 1 2 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 12;12 (that is the distribution of 18;18 conditional on the bottom state). The posteriors of Truth are
1 10 1 2 + 1 2 3 16 1 10 1 2 + 1 2 3 16+ 1 20 1 4 + 9 20 1 16 = 46 59 > 11 16 1 10 1 2 + 1 2 1 8 1 10 1 2 + 1 2 1 8 + 1 20 1 4 + 9 20 1 8 = 18 29 < 5 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 ;whereP; 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: P Q =
R
T (the prior can be written
Individuals observe a sample of signals “about”uorn:LetS be the (…nite) sample space of signals S. For each signalSi 2 S we write its likelihood as
Likelihood ofSi:J!(Si)
useful not useful
selection pi qi
free ri ti
(2)
In addition to the informationS inS they observe signals aboutsorf, where the individual observes the signal 2(0;1)with a density
Probability of selection: in states us orns signals drawn from
free: in states uf ornf signals drawn from
and ( )( ) increasing in . We assume additionally that !1is completely informative about
s (lim !1 ( )( ) =1) and !0is completely informative aboutf (lim !0 ( )( ) = 0):
After they observe this information, they observe a Common signal C with likelihoods
Likelihoods ofCfor ech!2 useful not useful
selection p q
free r t
: (3)
We postulate the following assumption about signals S:
Assumption 1. Weak Ambiguity (WA). SignalSi satis…es Weak A1 if piti > qiri:
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 thatP (ujS; ; C)> P(ujS; ),P (ujS; )> vS:
Proof. Step 1. Individual increases belief after C i¤ high ; de…ne cuto¤ B.
For any probability distribution B over we have B(ujC; )> B(uj ) (for C satisfying Ambiguity) i¤ pB(usj ) +rB(uf j ) qB(nsj ) +tB(nf j ) > B(usj ) +B(uf j ) B(nsj ) +B(nf j ) , pB(us) ( ) ( ) +rB(uf) B(ns) ( ) ( ) +B(nf) > B(us) ( ) ( ) +B(uf) qB(ns) ( ) ( ) +tB(nf)(4): Letting f( ) B(ns)B(us) ( )
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(ujC; ) > B(uj ) , > B:
For such a B; B(ujC; B) = B(uj B) B:
From Step 1, there exists a Ssuch thatP (ujS; C; S) =P (ujS; S)andP (ujS; C; )>
P (ujS; ) if and only if > S: De…ne vS as vS =P (ujS; S): Then, from Lemma 1 we
know if S is weakly ambiguous beliefs P (ujS; )are increasing in ;so that P(ujS; )> vS , > S ,P (ujS; C; )> P(ujS; ):
Lemma 1 Suppose the prior is independent. Si satis…es WA if and only if posteriors of u
increase with :In particular, piti > qiri ,P (ujS; ) is strictly increasing in (and < i¤
strictly decreasing): The e for which P (ujS; e) =P (ujS) is de…ned by (( ee)) = 1:
Proof. We have that for a signal S with likelihoods pi; qi; ti; ri
P (uj ; S) =P (usj ; S) +P (uf j ; S) = P
( )
( )pi +Rri
P ( )( )pi+Rri+T ti+Q ( )( )qi
which increases in i¤ the following expression increases in
X =
( )
( )P pi +Rri
T ti+ ( )( )Qqi
:
The derivative of this expression wrt ( )( ) is
dX d ( )( ) = P piT ti QqiRri T ti+Qqi ( ) ( ) 2 >0,piP tiT > qiQriR: (5) When ( )( ) = 1 we get P (uj ; S) = P ( )( )pi+Rri P ( )( )pi+Rri +T ti+Q ( )( )qi = P pi+Rri P pi+Rri +T ti+Qqi =P (ujS):
2
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.
Consider the following signals. s1 3 7 +" 3 7 " 2 7 +" 4 7 " and s2 4 7 " 2 7 +" 3 7 " 3 7 +" and s3 0 2 7 2 7 0 and C 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 1 4 3 7 1 4 + 3 7 1 4 + 2 7 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
considers3 because it has probability0 in stateT 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 jsi; C; ) = ppi ( )14 +rri ( )14 ppi ( )14 +rri ( )14 +qqi ( )14 +tti ( )14 P (T jsi; C; ) = pi ( )14 +ri ( )14 pi ( )14 +ri ( )14 +qi ( )14 +ti ( )14 so P (T js1; C; ) = 1 2 3 7 ( ) ( ) + 1 4 2 7 1 2 3 7 ( ) ( ) + 1 4 2 7 + 1 4 3 7 ( ) ( ) + 1 2 4 7 and P (T js1; ) = 3 7 ( ) ( ) + 2 7 3 7 ( ) ( ) + 2 7 + 3 7 ( ) ( ) + 4 7 P (T js2; C; ) = 1 2 4 7 ( ) ( ) + 1 4 3 7 1 2 4 7 ( ) ( ) + 1 4 3 7 + 1 4 2 7 ( ) ( ) + 1 2 3 7 and P (T js2; C; ) = 4 7 ( ) ( ) + 3 7 4 7 ( ) ( ) + 3 7 + 2 7 ( ) ( ) + 3 7
Letting ( )( ) =x;we …nd thex that solves P (T jsi; C; ) =P (T jsi; ) :
1 2 3 7x+ 1 4 2 7 1 2 3 7x+ 1 4 2 7 + 1 4 3 7x+ 1 2 4 7 = 3 7x+ 2 7 3 7x+ 2 7 + 3 7x+ 4 7 ,x1 = 2 3 p 2 = 0:942 81 1 2 4 7x+ 1 4 3 7 1 2 4 7x+ 1 4 3 7 + 1 4 2 7x+ 1 2 3 7 = 4 7 + 3 7 4 7x+ 3 7 + 2 7x+ 3 7 ,x2 = 1 24 p 541 + 1 24 = 1:010 8 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 47; of those, those with ( )( ) > 1:01
those who believe inT with probability less than 12 are those who observes1, who have
a probability of 37; of those, all who have ( )( ) >0:94increase 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 with ( )( ) >0:94
3
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 inH). 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; b2(0;1)):
Prior over
True False
High ab a(1 b)
Low (1 a)b (1 a) (1 b)
There is a set of signalsS and a collection of likelihood functionsf! 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
P!(h) =
if !=T H; F H
if ! =T L; F L for > :
We are interested in the informativeness of signals h or l : how are the beliefs about H
orL a¤ected by the signals. Thus, we analyze
P (HjS; h) = P(T H jh)+P(F H jh) = p ab+q a(1 b)
p ab+q a(1 b) +r (1 a)b+t (1 a) (1 b):
(6) This posterior is a monotone function (which converges to 1 as ;1 !1) of
a 1 a
pb+q(1 b) rb+t(1 b):
and similarly,P (H jS; l) is a monotone (decreasing) function of
a 1 a 1 1 pb+q(1 b) rb+t(1 b):
Therefore, signals about the ancillary issue are more informative when increases and decreases.
For the common signal, we assume that its likelihood in each state is
Likelihood ofCfor each state in
True False
High P Q
Low R T
for P > Q; T > R (that isC is ambiguous).
We are interested in the following two quantities: the proportion of people with beliefs greater than v who increase their beliefs (afterC), and the proportion of people with beliefs less thanv who increase their beliefs; we want to show
P P(Si)P (a > aiv; aiC) P P(Si)P (a > aiv) > P P(Si)P (aiv > a > aiC) P P (Si)P(a < aiv) (7) 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 observingC is larger than the proportion of people whose beliefs are lower thanb and increase them after observingC. In particular, for and1 large enough, all those abovev increase and all those below b decrease their beliefs after C:
Proof. First notice that ambiguity of S implies that P (T jH; S) > b > P (T jL; 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 jS; h) = p ab+r (1 a)b
p ab+r (1 a)b+q a(1 b) +t (1 a) (1 b) ;1!!1
pb
pb+q(1 b) =P(T jH; S)> b:
Finally, note that the posterior belief after h; C converges (when ;1 ! 1) to the belief afterC in stateH :
P (T jS; h; C) = P p ab+Rr (1 a)b
P p ab+Rr (1 a)b+Qq a(1 b) +T t (1 a) (1 b) ;1!!1
P pb
P pb+Qq(1 b) =P (T jH; S; C)
which is larger than pb+qpb(1 b) = P (T jH; S): Hence, for ;1 close to 1 we obtain
P (T jS; h; C)> P(T jS; 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 afterC:Those who observe signall have a belief lower thanb 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 +" 3 7 " 2 7 +" 4 7 " :
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. SignalSi satis…es Weak Ambiguity if piti > qiri:
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: LetS be the (…nite) sample space of signals S. For each signalS 2 S we write its likelihood as
Likelihood ofSi:J!(Si)
useful not useful
selection p q
free r t
(8)
In addition to the informationS inS they observe signals aboutsorf, where the individual observes the signal 2(0;1)with a density
Probability of selection: in states us orns signals drawn from
free: in states uf ornf signals drawn from
and ( )( ) increasing in (we might get rid of this which adds nothing). We assume addi-tionally that ! 1 is completely informative about s (lim !1
( )
( ) = 1) and ! 0 is
completely informative about f (lim !0 ( )( ) = 0) **this is just to simplify**.
Note that after observing a signal their beliefs become (for example, for T H), for
x= y+ (1y y)
ya
ya+ y(1 a) + (1 y)a+ (1 y) (1 a) =
xa
xa+x(1 a) + (1 x)a+ (1 x) (1 a) =xa
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 samea for everybody, but a distribution ofx; which is derived from the distribution of :
Suppose v is a belief in T that can be attained when signal S (with likelihoods pqrt), a value between the beliefs whenH is known and whenL is known: pa+qpa(1 a) > v > ra+tra(1 a). Then, the cuto¤ x = y+ (1y y) (that is, we are indirectly de…ning the cuto¤ ) for which
P (T jS; xS) = v is de…ned by xS = tS1vv1aa rS pS rS+1vv1aa(tS qS) tSg rS pS gqS+tSg rS (10) After they observe this information, they observe a Common signal C with likelihoods
Likelihoods ofCfor ech!2 T F
H P Q
L R T
: (11)
We postulate the following assumption about signals S and C:
Weak Ambiguity. SignalSi satis…es Weak Ambiguity if piti > qiri:
Ambiguity. We say thatC is ambiguous if p > q and t > r.
Weak Symmetry. We say thatSi 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 jS; ; C)> P(T jS; )jP (T jS; )> P (T)g> P fS; :P(T jS; ; C)> P(T jS; )jP (T jS; )< P (T)g:
Proof. From equation (10), ifS and C are weakly symmetric, and settingv =a
P(T jS; ) v ,xvS tS v 1 v 1 a a rS pS rS+1vv1aa(tS qS) = br r bq r+br q = r q+r: (12)
If S with likelihoods pqqpand C with likelihoods P QQP are weakly symmetric, P (T jS; ; C)> P (T jS; )happens i¤
BQbqxa+Rra(1 x)
BQbqxa+Rra(1 x) +Qqx(1 a) +BRbr(1 x) (1 a) >
bqxa+ra(1 x)
bqxa+ra(1 x) +qx(1 a) +br(1 x) (1 a) , (BQbqxa+Rra(1 x)) (qx(1 a) +br(1 x) (1 a)) > (bqxa+r(1 x)a) (Qqx(1 a) +BRbr(1 x) (1 a)),
(BQbqx+Rr(1 x)) (qx+br(1 x)) > (bqx+r(1 x)) (Qqx+BRbr(1 x)): (13) It is easy to check thatP (T jS; ; C)> P (T jS; ), > CS ,x > xCS 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: pluggingxS from (12) in (13) we obtain
(BQbqr+Rrq) (qr+brq) > (bqr+rq) (Qqr+BRbrq),
This means that xvS > xCS for all S: So all those who believe more than v (those that have
x > xv
S) also revise upwardx > xvS > xCS. At the same time, all those withx < xCS believe in
T less than v; and revise downward afterC: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 xv
S < xCS: Then all those with beliefs inT lower than v revise downward, while those
with x > xCS believe in T more than v and revise upward, which establishes the inequality in the theorem (asz >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 jS; ; C)> P(T jS; )jP (T jS; )> vg> P fS; :P(T jS; ; C)> P(T jS; )jP (T jS; )< vg:
Proof. Setg = 1vv1aa and plug xS = bq r(bg+g(1)br qr ) from (12) in (??) to obtain
(BQbqx+Rr(1 x)) (qx+br(1 x)) > (bqx+r(1 x)) (Qqx+BRbr(1 x)),
(BQbq(bg 1)r+Rr(b g)q) (q(bg 1)r+br(b g)q) > (bq(bg 1)r+r(b g)q) (Qq(bg 1)r+BRbr(b g)q), (BQb(bg 1) +R(b g)) ((bg 1) +b(b g)) > (b(bg 1) + (b g)) (Q(bg 1) +BRb(b g)),
b2 1 Bg(Q R)b2+ R BQ Qg2+BRg2 b+ (Q R)g > 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 thanv ( Similarly, forQ R <0; the equation is satis…ed for nob; which means that those who believe inT exactlyv 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 signalC is symmetric if its likelihoods are BQ;QQ;BQ. Start with an independent prior,
True False
High xz x(1 z)
where z is the same for all, but xis 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 z) > bqxz+r(1 x)z qx(1 z) +br(1 x) (1 z) , BQbqx+Rr(1 x) Qqx+BRbr(1 x) > bqx+r(1 x) qx+br(1 x) Sym , Bbqx+r(1 x) qx+Bbr(1 x) > bqx+r(1 x) qx+br(1 x) ,qx > r(1 x): (14) We have that P (HjS; ) > 1 2 ,bqxz+qx(1 z)> r(1 x)z+br(1 x) (1 z),qx(bz+ 1 z)> r(1 x) (z+b(1 z)), qx r(1 x) > z+b(1 z) bz+ 1 z (15)
Theorem 6 Plous. Fix de…ne B = S; :P(H jS; )> 12 and its complement B. If S is WS, C is symmetric and ambiguous (and S is similar)
P S; :P (T jS; ; C)> P(T jS; )jB > P (S; :P (T jS; ; C)> P(T jS; )jB)
(16)
That is, those with higher belief in H (in “selection”) are more likely to update up after C.
Proof. If z 12; those who have P(H jS; ) 12 have qx r(1 x) (by (15) and
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).
Ifz < 1
2;those who haveP(H jS; )> 1
2 haveqx > r(1 x)(by (15) andb 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