∗
Stephen Morris
University of Pennsylvania
June 1997
,ALMK@BM
! " ! # $ % & # '
!"#$ % & ' ( ) &* + + ' &, '+- && &&- ' & '. &'&
& /* + "& $ &
0& $& & +&&- &+
1. Introduction
An uninformed decision maker repeatedly receives advice from an advisor. The
advisor is of one of two types. A “good” advisor is well (but not perfectly)
informed and cares only about the discounted future utility of the decision maker.
A “bad” advisor may or may not be informed and has different preferences from
the decision maker. Not surprisingly, the bad advisor will lie to the decision
maker. But the good advisor may have incentives to lie also. Although telling the
truth maximizes the current utility of the good advisor (since it leads to the best
decision for the decision maker), the good advisor will also be concerned about
her reputation (the probability the decision maker attaches to her being good).
Telling the truth need not maximize her reputation, as the following example
illustrates.
Consider a social scientist who (repeatedly) advises an uninformed policy
maker on alternative social policies. Suppose that the social scientist is not racist,
but, on one issue, her policy recommendation coincides with that of many racists.
Telling the truth will improve the policy maker’s current choice of action (since
the policy maker attaches positive probability to the social scientist not being
racist). But it would (in equilibrium) increase the policy maker’s probability that
the social scientist is racist. This is true . This means that less weight would be placed on her future
advice. Thus she might (in the interests of the policy maker) lie about her current
policy recommendation.
The same logic will apply in many contexts. Suppose an investment advisor
may either be exclusively concerned with her clients’ welfare (good) or may be
trying to off-load excess stocks (bad). The good advisor will forego recommending
profitable trades in order to enhance her reputation. An academic referee may
either be exclusively concerned with the intellectual merits of the work to be
refereed (good) or may seek only to enhance her research agenda (bad). The
good referee will lie about research that enhances her own agenda.
In all the above examples, the “bad” advisor has some bias in her preferences.
Consider instead cases where the bad advisor tells the truth but is stupid (i.e.,
has noisy signals). An investment advisor is either competent (good) or stupid
(bad). The good investment advisor hears good things (i.e., observes a positive
signal) about a crazy sounding scheme (i.e., an investment with a low ex ante
probability of success). Should she pass on this (useful) information to her client?
2
On average, people with noisier signals are more likely to hear good things about
crazy schemes. The investment advisor may forego passing on the good advice
in order to have more conservative future advice taken seriously. Again, this
reputational incentive effect works even if the client learns the outcome of the crazy
scheme. Finally, a scientist is either competent (good) or stupid (bad). Suppose
the competent scientist observes something really surprising in her laboratory. If
she reports it, her future observations may not be taken seriously. It is better to
establish a reputation with more conservative findings before revealing any radical
observations.
The purpose of this paper is to characterize in which circumstances truthtelling is optimal for a “good” advisor and in which circumstances the reputational
concerns cited above ensure that no information is conveyed in equilibrium. We
address these questions in a repeated cheap talk game, extending the framework
of Sobel (1985) and Benabou and Laroque (1992). In particular, suppose that
in each period a binary state of the world, 0 or 1, is realized. The advisor observes a noisy signal of that state and may (costlessly) announce that signal to
a decision maker. The decision maker chooses an action from a continuum. His
optimal action is a continuous increasing function of the probability he attaches
(in equilibrium) to state 1. The state is realized (and publicly observed) after the
decision maker’s action is chosen. In each period, there is a new, independent,
state and a new decision to be made. Another independent random variable in
each period determines the weight (importance) of the current decision problem;
the decision maker maximizes his discounted expected weighted utility from decisions. The good advisor maximizes the same expression. The paper considers
alternative specifications of the bad advisor.
I analyze in detail the case where bad advisor and good advisor are equally
well-informed but the bad advisor always prefers higher actions, independent of
the state of the world. (Recall that the good advisor, like the decision maker,
would like a high action in state 1 and a low action in state 0). It is useful to first
analyze what might happen in any individual decision problem if we exogenously
fix an increasing, continuous, value function in the probability of being good for
each advisor. It is possible to provide a general characterization of equilibria in
this reduced form game (propositions 1 and 2). There always exist “babbling”
equilibria, where no information about the state or the type of advisor is revealed.
In addition, there may exist informative equilibria. In such equilibria, the bad
advisor always announces signal 1 more often than the good advisor. So indepen3
dent of the state realized, announcing 1 decreases her reputation and announcing
0 increases her reputation. Thus each advisor always has a reputational incentive to announce 0; the bad advisor always has a current incentive to announce
1 (since she wants a high action realized); while the good advisor has a current
incentive to announce 1 only if she observes signal 1. One implication is that the
good advisor always announces 0 when she observes signal 0. Now consider what
happens to the set of equilibria as the importance of the current decision problem
(to the decision maker and thus the good advisor) is varied, holding fixed the
reputation value function of the good advisor (proposition 3). If the importance
of the current decision problem is sufficiently high, there is an equilibrium where
the good advisor always tells the truth. But if the importance of the current
decision problem is sufficiently low, no information is conveyed in any equilibrium
(i.e., only babbling equilibria exist).
Recall that the above results followed from the that players have
continuous increasing valuations of reputation. I show that such value functions
can arise endogenously from a purely instrumental concern for reputation. In
particular, there exists a Markov equilibrium of the infinitely repeated advise game
with continuous increasing value functions (proposition 4). All the properties
described in the previous paragraph are inherited by this equilibrium.
The above results concerned a particular type of bad advisor (informed but
with a specific bias in preferences). We would like to characterize (more generally)
which and of the bad advisor will lead the good advisor
to lie in equilibrium, and which are consistent with truth-telling by the good
advisor. However, it is hard to solve for all possible preferences and competences
of the bad advisors. Instead, there is a characterization (lemma 2) of which
for the bad advisor are consistent with equilibrium truth-telling by the
good advisor (however small the importance of the current decision problem). This
characterization can be used to solve for various types of bad advisor preferences
and competence:
1.
: a bad advisor who is informed but has biased preferences.
This is the case we already described. The good advisor will always have a
reputational incentive to lie given such a bad advisor.
2. : a bad advisor who is uninformed but always tells the truth.
The good advisor will have a reputational incentive to lie against such an
advisor only if his signal is ex ante very surprising.
4
3.
: a bad advisor who is uninformed has a systematic bias in
preferences. The good advisor may have a reputational incentive to always
tell the truth if value functions are sufficiently aligned in equilibrium.
4. : a bad advisor who is informed and has preferences
(with no particular bias) from the decision maker. The good advisor will
have a reputational incentive to tell the truth.
This paper builds on a number of earlier literatures and it will be useful to put
it in context. The static cheap talk literature (Crawford and Sobel 1982) showed
that an advisor (sender) may tell the truth if she is good (her preferences are close
to those of the receiver) but must babble is she is bad (her preferences are different
from the receiver). Sobel (1985) and Benabou and Laroque (1992) considered
cheap talk games with uncertainty about the type of the advisor. They
that the good advisor tells the truth. They showed that a bad advisor
will not always babble. Rather, the bad advisor will sometimes tell the truth
(investing in reputation) and sometimes lie (exploiting that reputation). This
paper endogenizes the behavior of the good advisor in Benabou and Laroque
(1992). Just as the bad advisor’s current interests are sometimes reversed by
reputational concerns, so too for the good advisor. Just as the bad advisor has an
incentive to tell the truth (despite a current incentive to lie) in order to enhance
her reputation, so the good advisor may have an incentive to lie (despite a current
incentive to tell the truth) in order to enhance her reputation. The purpose of
this paper is to characterize when this occurs.
Loury (1994) has explored the more general question of “self-censorship in
public discourse” for strategic reasons; the classic example is that of so-called
political correctness but Loury argues the phenomenon is quite pervasive. Loury’s
explanation is that participants in public debate are influenced by concerns about
how their statements will change listeners’ views of their type. In particular,
speakers want to be perceived to be respectful of social norms and therefore make
statements that respect social norms for expression (he suggests that a formal
model along the lines of Bernheim (1994) would be relevant here). There are
two problems with this approach. First, there is no explanation of the origin of
social norms concerning expression. Second, the cost of the self-censorship is not
examined. The model presented here can be seen as a formalization of some of
Loury’s ideas, which in addition addresses these two problems. In this paper,
there is real information content in expression, and therefore social value is lost
5
when self-censorship occurs. Speakers’ reputational concerns are not about some
reduced form feature such as “respect for social norms,” but rather about their
perceived preferences and ability. Furthermore, they care about the latter for
purely instrumental reasons. This foundation of the model in standard economic
and informational assumptions allows the possibility of the origin of
socially acceptable forms of expression and conducting welfare analysis of political
correctness.
Reputational concerns for competence have appeared in a number of papers;
Prendergast and Stole (1996) is especially relevant. They consider a manager
who takes an investment decision and is concerned both about the output of
the investment decision and her reputation for competence. Absent reputational
concerns, competent managers would take more extreme investment decisions,
because they have more accurate information. Given the reputational concerns,
all managers will initially take excessively extreme decisions in order to signal
competence. On the other hand, if there is correlation across optimal decisions
through time, managers who have been in place for some time will excessively
reduce variability, in order to signal their faith in their earlier decisions, and
thus their competence. A number of features lead to the different reputation
for competence results in this paper. First, there is no correlation in decision
problems through time. Second, because advice is not costly, there is no ability
to separate by choosing some sufficiently costly action.Q
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& strategy = i5 0j $ ?7 & & * G D+ E #
G G DG B E F
G H 0 B 0
?
F
S
0 H Q! + EE
S
7 7 0 F 7 H 0 7 0 7 5 F 0 7 0 )
(# * G DC E F C 0 F C 5 F 0 F 5 F 5 '
I; " 6 ( H C DG DC E E
2$
&&+ ++ &- '- 3 % &* &) && &- - + "- &&+ &- '& 7 * & -&
+ '- 3 '& '& &&&- * ' 8&- 3 -& & * &-+ &- ' '
&- - + 9
J
$?
5 G = D5 0E
0 *
B H B DG DG B E E
5 B ∗ 5 ∗ 0 B = D5 0E * G B K DG B E I;& (
0 0 ! !& H 0 B 0
K DG B E F 0 H 0 G 0
G
B 0 H G 5 H 0 B 5
)
(# * (
K DG B E F Q G 5 F B 5 F G 0 F B 0 F
5
G B ' & "(
# & G D G B E7 & (
K D G B E ' " & #
G B D [ ]
σ , σ , χ χ F + D : [0 1]
→a
D
[
G B ] () = D (Γ [G B ] ())
D 5 , a 0E qu( D 0 H 0 ( D 5 F 5
. "
u( a, ω F a ω ,
9
(0) = 0, 9 (1) = 1 and D () = .
Definition 1. σ+ , σ , χ 5
# [0? 1] + (0) = (0) = + (1) = (1) = .
F
0 F
Q
= 2
Proposition 1. * # ( ' ' +
, χ
0 5 χ 0 χ 5 ,σ
For any , any non-babbling equilibrium G B has
[1] the good advisor always telling the truth when she observes signal 5 (G 5 F
5); [2] strictly informative messages ( 0 5); and [3] a strict reputational
incentive for the advisor to announce 5 (G 5 0 G 5 5 G 0 0 G 0 5). More specifically, there exist three types of non-babbling equilibria:
= + 5 F 5 + 0 F 0
5 5 0 F 0 % ' G 5 0 F G 5 5 G 0 0 G 0 5
Truthful
0 F 5= + 5 F 5 + 0 5 0 5 F 5 0 + 0 % ' G 5 0 G 5 5 G 0 0 F G 0 5
+ 5 F 5 + 0 5 0 5 5 0 0 0 0 + 0 0 5 % ' G 5 0 G 5 5 G 0 0 G 0 5
Politically Correct
Full Support:
* 5 5 ! 0 0 5 ! 0 L
5 #
0 " ! * + '
) ' DQ D : (0? 1)
nn
nn
Q # $
' ≥ D # Di 9
Di !" 5 λ
0# !" xDQ ( )
0 0# # 5 99
( )
as 0; (iii) Di 5 5# !" as →
D
→
→
1
→
→
) (
C & #
(
nn7 M ) x.φ (x, y ) < ∞ ∞
[
[
X Y
Ex,y (x,y ) 5 0 ; × → D5 0E7 7
= {5 0} × 5 0 ×
0 = {5 0} × 5 0 × × → 7
& * !"# $ + + ? + , + # $ + '+ ! + " # % + + * &, !"# && ; ' " (
; ' $
% !
The reported prevalence of political correctness in academia might then be explained by
proposition 3 and the dictum (due to Bernard Shaw?) that academic disputes are so heated
precisely because there is so little at stake...
$
05
! 4' D011,E '
* : : B :
Now
we can construct the value function for the good advisor corresponding to these
strategies. For ( sufficiently small, the slope of the value function will be
determined by what happens next period. The good advisor will prefer the bad
advisor to have a low reputation (so she sometimes tells the truth) rather than a
high reputation (so she always lies).
Nonetheless, there do always also exist MM equilibria.
A monotonic Markov equilibrium always exists.
The intuition for existence is straightforward. Suppose some pair of valuations
( $
) * 0
* : ) ! :
; ; ' 0 , 9 * " 9 ≤ 9
+ 0 | F + 5 | F 0 | F 5 | Now complicate the model by first allowing bad advisors to be less accurate than
good advisors: Q + and Q 0 5 0
* ' % & 00
/ ( +
0
F 0 |
. (
+ $ * : ≥ / ⊆ D5 0E ( Lemma 2.
;
A
A
A
?
F A
A
A
=
[2]
0 1 =
[3]
0 γ+
+
> !γG
v+ nE
−
+
!λE!⁠Q!+
+
−> G
λ !γG nE
>
−
+ Q ! π
1!+
!v
β0
Q

+ nE!E!f -1
)
(
)
(Q
)
+ (1− )+(1−)0
" T λ, π 4
1! v+ λ F * D9E 0 +
+
0
+ 1
0 0 G 0 +
+
and π = ; figure 2
. In this particular
case, the region is independent of λ. But the qualitative properties of this example
generalize. Thus as π tends to 1, T (λ, π) becomes very narrow: more generally,
the characterization implies that if (β , β) # T (λ, π). As π 0, β γ; as
π 1, β 1 γ .
. 0 + F
illustrates the shape of this region when γ+ = and π =
! " ! # $%&
$
+ F F N 12 B# B 0 F
0,
0 B 0 5 0 1 1 F H
0 0 F 0 0 % 1 0 0 0 1 ( % ) 9 < " F 34 ' 12 109 3.3. Separating from an Honest Fool
% B G F 0 B F B * D0E D,E (
D9E # D0 G G E :
% ) 3 A " F ' ! " ( #
• Suppose =
Q
+
Q
* 0 5 ) - G F B 5 * B F Q G Q ' ) ' ∗ ∗
( ) ! "
)
See figures 7 and 8 for the example with
=
π ' 3.5. Separating from a Smart Enemy
* # $
" / π F γ+ = =
Q.
13
Thus suppose now that + (
) = + (
), where + is some strictly
concave function, while the bad advisor’s utility from action in state is
(
) = (
(1 )). Thus the decision maker (and good advisor) want
action 1 in state 1 and action 0 in state 0, while the bad advisor wants action 0
in state 1 and action 1 in state 0.
This is essentially the case studied by Benabou and Laroque (1982), building
on the perfect signals analysis of Sobel (1985). They consider what happens in
this setting assuming the good advisor always tells the truth (+ (1) = 1 and
+ (0) = 0), and focussing on equilibria where the bad advisor always behaves
symmetrically (+ (1 ) = 1 + (0 ) = ! ()). They show that there is a
unique monotonic Markov equilibrium within this class with ! () = 0 for all
9 ! # 5 0 9 / & * '
! 4' F Q2 ' G 0 0 F G 5 5 F n F
G 0 5
F
G 5 0 F
!=
0H
1+
1!λ
λ
Q!
0
! (>) + Q!
0
1
ξ λ H
Q!
0
!
<λ
ξ λ
γvG λn H 0 − >) + (− ), while the reputational value of lying is + (− ) +
(1 − ) + (n % ) !
D0E ! 8 B 4' 011, L @
* ;(
;= *
B $
Quarterly Journal of
Economics 05J 1,0(132
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-
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= ' 8 4 Journal of Political Economy
05< 0053(009<
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D05E % > 0123 $
Review of Economic Studies 3,
33J(3J9
# ' 7 | + "
(
) C 7 ( | C "
|
C 0 C F
|
C 5 C F
|
0 C F
|
5 C F
( | C F
( 0 H C DG DC E 0E
7
H 0 − ( 5 H C DG DC E 5E
0 − ( 0 H C DG DC E 0E
7
H ( 5 H C DG DC E 5E
0 H DG DC E 0E
7
H 0 − 5 H DG DC E 5E
0 − 0 H DG DC E 0E
7
H 5 H DG DC E 5E
K DC E ( 0 H 0 − K DC E ( 5 03
+ , σ , χ I
σ
0
σ7 s > 5
7 0 5 "
∈{01}
G, B G B "
m∈{,}
F
I
m s, σ+ , σ , χ #
χ m " DM + (M
,
. ' B @ % 0121
Some preliminary notation and results will be useful. Write + ( ) for expected value of ( for the good advisor if he has observed signal and the
decision maker believes the true state is 1 with probability ( (
( ) 1) + (1 ( D 5
C 5 F 0 ( 0 H ( 5 C ( 1) =
% "
0 7 =
F / C Lemma 3. u+ q, 0 # 0 # + 5 # 0 # # 0 B R "
0 5 R F D G K 0 G K 5 E RCB F
G K 0 G K 5E / RR
D
I "
0A
5 0 R 0 F
R
I
G 5 0
I G 0 0
I
G 5 5
H 0 I G 0 5
G 5 0
R 5 F 0 I G 0 0
I
R
I
I
H
G 5 5
I G 0 5
I
0 " RCI RRI 0
* 0 q "
a q.u( a, 0 H 0 − q u( a, 5 "
" a ' " q.u( 0 H 0 ( 5 F 5
2.
This will be proved via a pair of lemmas.
σG ? B (
G 5 5
G 0 5
G 5 0
G 0 0 Proof. / ' " 8 σ+ , σ , χ ' χ F χD Dσ+ , σ E χ 0 (0).
. Suppose that Λ (1 1) Λ (0 1) and Λ (1 0) Λ (0 0). Now Π () 0 and Π () 0 for each = 0 1, we must have (0) = (1) = 1. But now if
(0) = (1) = 1, Λ (1 1) = Λ (0 1) = Λ (1 0) = Λ (0 0) = , a contradiction.
But if (0) = 1 or (1) = 1, then Λ (0 1) = Λ (0 0) = 1, another contradiction.
Thus there is no such equilibrium.
17
. Suppose that Λ (1 1) Λ (0 1) and Λ (1 0) Λ (0 0). By definition
of Λ, we have
(1 1) (1 1)
(3.1)
(1 0) (1 0 )
(3.2)
φ
0 5 φ 0 5 9<
Observe first that Π (1) Π (0) and Π (1) Π (0) for = G I σ7 5 F 5 σ7 0 F 0 =
* σ 5 F σ 5 F 5 90 σ 0 > σ 0 9,
σ 0 σ 0 * σ 5 F 5 σ 0 F 0 90 σ 0 > 0 * σ 0 F 0 σ 5 F 5 9, σ 0 F 0 σ 5 F 5
φ 0 0 > φ 0 0 90
* σ 0 F σ 0 F 0 90 σ 5 > σ 5 9,
σ 5 σ 5 Case 3 % G 0, 0 G 5, 0 G 0, 5 > G 5, 5 ! )
G φ 0 0 > φ 0 0
99
* R 0 > R 5 R 0 F R 5 σ 0 F 5 σ 5 F 0 φ 0 0 φ 0 5 ! 99 9< φ 0 0 <
φ 0 5 ! D D+ E D0E Q D D+ E D5E 0
5, G 5 0 G 0 0
G 0 5, with one of the latter two inequalities holding as a strict
) In any non-babbling equilibrium, and
G 5 5
inequality.
4 < ' % ' 8 0 5 * 0 5 0 ! 0 F 5 '
% ' * 0 F 5 5 4 3 D,E , ! 3 R 5 F F 5 0 % R 5 5
"
02
5 F 5 D0E , 0
0 H 0 − 0 H 0 − 5
0
F
1! 1
0 H + (1) σ 0 H Q! 5
G 0 0 F
Q! λ
1
B 0 H Q! σ 5
λ
λ 0 − γ σ+ 0
F
λ 0 − γ σ+ 0 H 0 − λ 0 − γ σ 0 H γ σ 5
F G 0, 5
1+
Q
σ+ E
Case 1= G 5? 5 G 5? 07 G 5? 5 G 5? 0 G 0? 0 G0? 5 '(
R 5 5 R 0 5 5 F 5 0 F 0 !
+ , i.e., - ≤ ++((00 0)1) . But
G 5 5 G 5 0 + - S
S
S
S
S
S
S
S
+ (00)
(1 ) (1 + (1)) + =
+ (01)
(1 + (1)) + 1 1 Now if (0) = 0, then
(00)
(1 ) (1 (1)) + =
(01)
(1 (1)) + 1 which is less than or equal to G+ - 0 0 ! 0 0 F 0 , ! 0 F 0 S
S
55
0 5
F
F
50 0 0 5 0 ' φ++ - only if + 0 F 0 )
( ' + 5 F 5 + 0 F 0 0 F 0 *
S
S
01
5 F 5 0 + 0 F 0 , 0
G 0 0 F
1!
0H 1 + 1! 5
G 0 5
Λ (0 1)
G 5, 5
F
0H
=
1+
F
0H
Q!
0H
1
Q!
0
0
0
0
Q!
Q! (0)
ν
ν
G 5, 0 F G 5, 5 > λ > G 0, 0 > G 0, 5
= G 5? 0 G 5 5 5 F 5 * + 0 ≥ 0 0| ≤ + 0| F 0 , + 0 0 0
G 0 0 F
B E
Q−
0H G E
G 0? 5
G 5? 0
G 5? 5
F
F
F
0H
0H
0H
0
Q−λ
λ
1!λ
0
λ
Q!
σB (1)
σG (1)
1!γσ (1)
1!γσ+ (1)
0
Q!E! E
!E! + E
G 5? 0 G 5 5 G 0 0 F G 0 5
0)
Case 3: 5 5 G 5 0 G 5 5 ' B+ ((00 0)
i.e.,
S
S
+(0(0 1)1) ,
S
S
(1 ) (1 (1)) + (1 (0)) (00) + (00) (1 ) (1 + (1)) + =
=
(1 (1)) + (1 ) (1 (0)) (01)
+ (01)
(1 + (1)) + 1 This holds if and only if
H
(1 ) Q!! Q! - ! H 0 +
+
0
0
0
,5
0 H 0 0 H
0
0
0
5
0 + 0
0 0 0 + 0 0 5 PROOF OF PROPOSITION 9
D0E ' ! , ) + 5 F 5 + 0 F 0 5 F 5 0 F 0 F D + L
H 0 0 7
0 H 0 K 5 F 0 7
0
7
G 0 0 F
Q!
0H 0 H Q! K 0 F
G 0 5
G 5, 0
F
F
G 5 5 F
0H
0H
0H
Q!
(1
Q!
0
0
Q!
0H
0
Q!
0
7
ν
7
/ " 0 5
5 n
E!E! u 0 − γ H γv Qn Q! Qn
nE!
" F H 0
− γ v
Q
Q! Qn Q!
n 1
>
1+ Q!
> (1! Q! 

 .
This expression is decreasing in B − B 0 − 5 " " " 5 F 5 # F 5 F 0 ' '
/ ' - D0E → 5 → 5 D,E F 0 : D9E
"
"
,0
F 0 : 0 ! + C
0 − 0 − B
B
B D0E − B D5E
,
F 0 : 7 0 − 0 − ≤ D0E − D5E
B
B
B
B
,
0 0
!
1
D5E H , 0 H Q! 0 D 5
/
0
0
# F !Q min D0E D5E H ,
Q
> −D
Qn Q!
- >
0 5
D 0 & 0 ) 5 F 5 0 F 0 5 F D 0 F 0
F D % E!E! E nnE
,0 0 0
+
!E
Q
Q
H +
H
0
1
!
+
Q!> n
Q !> 1+
D
Qn
Qn
> Q! 5
> + Qn Q−>> QQ!D -
#
=
D ( )
; Q
A
A
? + Qn Q!> Q!
- 2
A
A
=
>
+
nE!E! E nE!
E
,,
,
0
0 +
+ 0 0
Q
-
>
Qn 1!
> 1+ 1! D
Q
Q− >
Qn Q−
-
> Qn D
F 0 : > Q
−
D 0 0 D # D > →
+
D 2
Q
D#B E
0
−
G D5E
+ 0 Q
5 2 >
;
A
A
? + D0E
AA=
+
?
0
Q
57 Q
γ
λ
Qn Q−
λ n Q−γ B (1 )
+ 0 0
−Q
D0E D5E H 0
G
5 +
F
G
5 0 5
E!E! + nnE
! 0 +
0 + 0
D G [0]
0 Q
Q! >
Qn Q!
> n 0
D,E ) 0 R 0 D 0 0 0E
, R
G B # E0? 1
1
(0 )
(1 )
1
1 + 1 + + (0 )
+ (1 )
Now let $ ( ) = (1 ) min + + Qn Q!>Q En ? n 1!>
)(
>
- Π (1) $ ( ) - Suppose and are not -close. Then +E
for some . So
|
"
|
Π (1) = (Λ (0 1))
(Λ (1 1)) + (1 )
If % (1) (0), let " ( %) =
the unique value of solving
1
% = Q! 1+
Q
Qn
>
Q
Qn )
0 H δ + (0S S
(Λ (0 0))
(Λ (1 0)) $ ( )
; if % (1) (0), let " ( %) be
,9
0H
0
Q! λ
λ
0
H
δ
− + . ' % 5 R 0 F / "
G 5 0
+
H 0
% G 0 0
"
G 5 5
G 0 5
%
K 0 nE!En Write & = + (1). Now:
+ (1|1) = &
+ (1|0) = (1 − ) &
&
(1 − ) &
≤
≤
(1|1) ≤ & (1 + )
(1|0) ≤ (1 − ) & (1 + )
+ (1 |1 ) + (1 − ) (1 |1)
+ (1 |1 ) + (1 − ) (1 |1) + + (1 |0) + (1 − ) (1 |0)
≥ & + (1 − ) & (&+ (1 − ) (1 + ))
=
+ (1 − ) ( + (1 − ) (1 + ))
≥ + (1 − ) (1 + )
Γ (1) =
Now define:
D F
$
1
2 −Q
2E− ? "

2
n Q
Q
+ 0 + 0 0
% Q (> ). The following claims must hold true of any nonbabbling equilibrium. By lemma 6,
,γ 0
0
γ
Π (1) ≤ $ λ, , g λ, y u
, 0 γ ,
γ H 12
,<
0 ,
! J
+ , 0
"
, 0 0
,
HQ
B
1 2
-close
Q
n Q − Q ! K 5 Q F
By lemma 8, Π (1) !Q
. Thus Π (1) Q
in
lemma
9,
Γ
(1)
n Q
+ Q
2E!
<
." x9 , y 9) and consider the following advisor strategy
σ+ (s λ, x, y ) =
σ s λ, x, y F
Q
y 9
s x, y F x9 , y9 Q
9
∗
F ,
F ? ∗
0 F ∗ ∗
; A
h 12 , if ( ) = (9 ? 9
A
? nE−
9 9
F
D
x, y F x , y m F 0 A
n2E−
A
= D
a 0 γ x, y F x9 , y 9 m F 5
v+ F + D+ E 1 1 1 1 + 2 , 0 H 2 u
+ 2 , 0 H δ+ v+ λ
0 − 2 u
! 1 +(1− Q
0 5
u
0 H + D+ E F 2 + n2E−
+
Hε "
! 1
Q
H +
+
+ (1 ) +
2
?
+1!
-
E! n!
H
F D E # $ 
2 + 0
0 Q2 H D E F nE!
-Q! Q
Q
H n2E! H 2 n! + E! n!
,3
C 7 (
D5 0E (
! 7 H F 7 H % !&
' ' (
)"
- +
0 + 5 0 +
G 0 0
1 H 2 + 0 5 + 5
Q
+ γ, 0
u
0 D 0 E / F 9 9 - "
0 5 ?
H 0 − >
R 0 F R 5 F > H , 0 − >
≥ 0, − 0 − − 0 − nE!λ , 0 u
0
γ,
0
G λn2E
!λ
R 0 F E!
G λγn
H 0 γ u
n2E! ? 5 + 0 5
Q
0
H 0+ ,0 u1G? 00 + 0 5
G 2
: Proof of Lemma ,
5 0 0 ,A
;

3 λ Q!γ+

A
G

A
+
E!+ nE!E!Q 

C

A

+
+ nE!E!+ 
?
+
+
(1
!
)
Q 
+
R 0 F A
.
+


v+ +nE

A
1!)(1!+ )
−
)(1
−
)
B

 
A

− + )

= H ++(1!)(1!+) + (1−+Q)+(1
−)B
-
(
-
The net reputational gain to the good advisor from announcing 0 (rather than 1)
if her signal were 0 is:








Π ( 0) = 






Q!+ )
+
!+ (1!+ )+(1!)(1!Q ) +
E!+ nE!+
-
-
v+ ++(1−)Q +
+
+ +nE!)(1!B ) H E!-+!nE
(1−γ+
!+ + λE−γ+ nE−λβB
F = R
5 ' ) H 0 0 +
and +
+ + (1 > Q
5 R 0 5
> 0 + +
0 + H 0 0
> 0 − + +
0 − + H 0 − 0 − Q
0 + 1 + Proof of Proposition 5
Assume the good advisor tells the truth and the bad advisor announces 1 with
probability (i.e., = Q (1) + Q (0)). The reputational gain to the bad
advisor of announcing 0 is
Q
Q
Qn( Q!>> QQ!! Qn Q−λλ µγ (
)=
Q
Q
0 − π v Q−> Q− 1!λ µ
'
'
# '
Qn > ,J
Qn λ 1!γ - ' # '
!
# 0 F "
0H
1!
 
h π, λ, γ F 0 π v 
*
0H
1!
0
Q!
#
!
$ "
Q!
1+
1
Q! #
!
$ vB "
Q
0 Qn Q!>> Q 0 F
1
− 0 − π vB 1+( Q!> )( Q ) # 0
0H
Q!
0
1!
#
$ 5
Q!
#
$ 5
D 0 E
Q !
>
' F 0 - ' ' #
' F D 0 E . 0 $ ' F
$
' 0
0 -Q!+ +
v
+ E−+ nE−E− + nE!
λγ
+
+
vG λγ+ nE!E! + λE!γ-+Q!
nE!λµ
Now $+ ('D ( 0)) = 1 and $+ ('D ( )) is continuously decreasing
in . So for
9
any given and , we can choose such that $+ ('D ( )) ∈ Q!++ ? 0 # D5 9 E
,2