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APR
working paper
department
of economics
CHEAP TALK WITH TWO AUDIENCES
Joseph Farrell
Robert Gibbons
No.
518
Revised
February 1989
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
4
1989
CHEAP TALK WITH TWO AUDIENCES
Joseph Farrell
Robert Gibbons
No.
518
Revised
February 1989
Cheap Talk with Two Audiences
by
Joseph
Farrell
and Robert Gibbons*
Berkeley and Hoover, and
MIT
Revised, February 1989
Abstract.
When
we show how
an informed party can engage
relationship with the other audience.
disclosure,
in
cheap talk with more than one audience,
may either discipline or subvert the speaker's
We ask how welfare is affected by public or private
the presence of one audience
and predict how much communication
Keyvt^ords. cheap
talk, signaling,
will take place.
communication, incentives.
,
Cheap Talk with Two Audiences
Joseph Farrell and Robert Gibbons*
When an informed party
how
may
the presence of one audience
We
with the other audience.
predict
Why
can engage in cheap talk with more than one audience, we show
ask
how
how much communication
are
some claims made
either discipline or subvert the speaker's relationship
welfare
is
affected by public or private disclosure,
and
will take place.
in public
and others
in private?
Are public announcements
always more credible than private ones? Should a politician meet with conservative and
liberal constituents separately or together?
its
profitability aff'ect
both
its
How
bond rating and
does
its
it
matter that a firm's claims about
labor negotiations?
When
promotion, could the boss improve the credibility of his claims about
two candidates
for a
their prospects
by talking to both candidates together rather than separately?
engagements and weddings public?
These problems have
in
Why
common
are letters of
beliefs; these actions affect
We
can
who
then take actions based on their
affect these beliefs
for truthful revelation to
then ask
how
are
recommendation private?
the sender as well as the receivers. In this paper
costless, nonverifiable claims [cheap talk)
and how the incentives
Why
a simple structure: an informed "sender" says some-
thing to two interested but uninformed "receivers,"
of the other.
there are
welfare
is
we study how
(and hence the actions),
one receiver are affected by the presence
affected by
whether claims axe made
in public or
in private.
As some
of the above examples suggest, public messages
private messages addressed to either audience.
One
may be more
possibility
is
that the presence of
one audience can discipline the sender's relationship with the other; we
discipline.
An
credible than
call this
one-sided
analogous idea (although with costly signals rather than with cheap talk)
'Berkeley and Hoover, and MIT, respectively. We thank David Austen-Smith, Jeremy Stein, and
workshop participants at Princeton and Hoover for helpful comments. This work was financially supported in
part by the National Science Foundation through Grants IRI 87-12238 (Farrell) and SES 88-09200 (Gibbons)
by the Hoover Institution, and by the Industrial Relations Section at Princeton University.
has been discussed in the finance literature by Sudipto Bhattacharya and Jay Ritter (1983)
and Robert Gertner, Gibbons and David Scharfstein (1988). But,
sided discipline
is
not the only interesting
instance, the presence of one audience
credible communication
the other:
one audience
may
in
we show below, one-
may be
impossible in public, though possible with
may be unable
credibly to
but able to communicate in public: we
in private,
For
a very simple model.
instead subvert the speaker's relationship with
in private. Or, the infornied party
with either audience
even
possibilitj',
as
communicate
mutual
call this
discipline.
In Section
we
we develop a
I,
simple model of cheap talk with two audiences. In Section
II,
characterize the equilibria of our model, including those discussed above,
and we give
we
consider the
we analyze
equilibriimi
examples of some of the more interesting kinds of equilibria. In Section
welfare implications of public or private disclosure. In Section IV,
and predict how much information
selection,
discuss (A)
some
III,
be revealed. Finally, in Section V,
will
we
further examples, (B) two extensions of our analysis, and (C) a possible
direction for further work.
I
A Model
.
We
develop the simplest possible model of cheap talk with two audiences.
A
sender, S,
observes the state of the world 5 € {^i, S2}; the (common-knowledge) prior probability of
5i
5
Qi
There are two audiences or
is TT.
to be a
or ^2
5
man and Q and R
2Ji<i
R
chooses
the state of the world,
or fj
r^
s:
to be
.
receivers,
Q
and R. (For pronominal
clarity,
we take
Q
chooses
women.) Each receiver can take an action:
Each
receiver's payoff
for simplicity
we suppose
that
depends on her own action and on
it
does not depend on the other's
action.
If
about
one audience had an action that was always optimal for her irrespective of her
s,
would be
then the sender would not have to consider her reactions. Consequently, there
in effect
only one audience, and this problem has already been well studied by
Vincent Crawford and Joel Sobel (1982). Without
we suppose
Q
beliefs
loss of interesting generality, therefore,
that the receivers' payoffs are such that
would choose
as in Table 1,
g^
and
R
would choose
and then our assumption
r^.
is
By
if
the state were
normalizing,
that Xi,X2,yi and
we can
j/j
are
known
to be
5^
then
display these payoffs
all
positive.
.
True State
Q's Action
Xl
9i
I2
92
Table lA: Q's Payoffs
(Assume
>
x^
0, I2
>
0)
0, yj
>
0)
True State
62
5i
yi
Action
i2's
y2
Table IB: R's Payoffs
Simplifying our problem further,
two components: one, u^
r
and
5.
We
write u^(s,,
that UQ{si,qj)
chooses
^2)
,
and
=
i2
we assume
=
g.)
=
fj
,
v,
and u^(s,, r^)
for j
^
i\
=
q,
and the
Wi for
see Table 2.
then 5's payoff
>
yj,
that the sender's payoff, u^
depending on Q's action,
u^(5, ,ry)
chooses
(Assume
+
is
lyj
,
is
True State
5i
Q's Action
9i
52
Vl
rj
92
Table 2 A: 5's Payoff With
Q
True State
Sj
52
t^l
i2's
Action
1,2,
W2
Table 2B: 5's Payoff With
R
the
0,
sum
u^
,
of
on
and we normalize so
Thus, for example,
Q's
is
state, s; the other,
=
i
,
and
i?'s is
if
j/j
5
=
5i,
Q
If
she has no information beyond her prior belief
tt,
Q
receiver
will take her pooling
action:
9l,
f
{ q^
Receiver
i?'s
pooling action
r''""' is
,
if
>
TTXi
-7r)i2,
(1
otherwise.
defined similarly.
may make
Before the receivers choose their actions, the sender
These claims do not
claims about the state
That
directly affect payoffs: they are "cheap talk."
verifiable, not binding,
and not
Formally, 5's claim
(directly) costly.^
any player's payoff function. Thus we need no notation
is
is,
s.
they are not
not an argument in
for the claims themselves; only their
information content matters. Since the meanings conveyed in pure-strategy equilibrium
can only be
"5
=
Sj", "s
=
no other messages are used
would give the same
II
.
Equilibria
We now
do
so,
Sj",
and "no information,"
in equilibrium.
it
is
convenient to assume that
Any message-space with
set of pure-strategy equilibrium
two messages
at least
outcomes.
With Two Receivers
describe the pure-strategy perfect Bayesian equilibria in our model. In order to
we must analyze
to Q, in private to R,
S speaking
three distinct cheap-talk games: the sender
and
in public to
In each of these games, and for
all
in private
both receivers at once.
values of the payoff parameters (ui Vj
,
pooling equilibrium exists. In such an equilibrium, the sender's talk
instance, whatever the true state, he says "no information."^
Thus
is
in a
,
u^i
,
11^2 )
>
3-
uninformative: for
pooling equilibrium
a receiver rationally ignores what the sender says: her posterior beliefs about s are identical
to her prior beliefs,^
But there can
and she takes her pooling
also be other equilibria, in
single-audience case for definiteness,
action.
which
let t/(Si,"5
=
talk does affect actions.
5y") be the sender's payoff
Taking a
when
the
We abstract from considerations such as reputation that might induce S to tell the truth even when it is
against his short-term interest. Similarly, we assume that it is impossible to promise side-payments contingent
on any of the players' realized payoffs: for instance, because those payoffs are not publicly observed.
Or, more generally, his message
is
uncorrelated with his private information
s.
^ To complete the formal description of equilibrium, we must specify what the receiver would believe if
she hecird an unexpected message. For simplicity, we can assume that she would maintain her prior beliefs.
.
true state
and the receiver believes the sender's claim that the state
s^
is
be equal or unequal to
Needless to say, such credulity
»').'*
separating equilibrium, however, the sender's claim
Formally, a separating equilibrium
in
"5
is
one
5
the most natural separating equilibrium,
=
when
^2"
posterior belief
the state
either that s
is
the equilibrium condition
and only
if
if
=
which 5's claim
says "5
is
may
not be warranted. In a
=
fully reveals the true state:
when the
61"
sure (so she takes her
is
Si
and says
=
s-2
either of these beliefs,
Thus a separating
lie.
,
action), or that s
first
5 can induce
that he has no incentive to
state
the receiver's rational
in a separating equilibrium,
Sj for
{j
credible.
second action).^ Given that
for sure (so she takes her
exists
Thus
53-
is
in
is
may
Sj
is
equilibriimi
both
(1)
U{s,,'^s
= sr)>U{s,,'^s =
s,^)
C/(52,"6
=
5i'').
and
(2)
From
with
Q
conditions
if
and only
in private
with
equilibrium
if
Proposition
tives for
if
R
Vi
if
honesty
,
Vj
(2),
>
0.
and only
and only
1.
and
(1)
if
Ui
+
there
52'')>C7(52,"5
is
a separating equilibrium
Likewise, there
Wi^w^ >
if
lOi
>
and
Uj
in public, the reverse is
is
a separating equilibrium
+
^2
in
^
twi
when he speaks
is
a separating
each relationship in private imply incen-
not true.
that the presence of one audience
if
and w^ are small and negative. Such one-sided
case, subversion)
in private
This proves
0-
tion with the other [one-sided discipline), for instance
while
when S speaks
Finally, in public, there
0.
While incentives for honesty
Our argument shows
=
may
Vi
discipline 5"s
and
V2 are large
communicaand
positive
discipline (as well as the opposite
can be illustrated in terms of the finance papers cited above,
in
which an
For example, if the receiver is R then £7(s,,"a = Si") = t^j and [/(si,''s = Sj") = 0. In the twoaudience case, U(si, "s = sy") is the sender's payoff when the true state is t and both receivers believe that
*
the state
As
is j;
thus, for instance, J7(si,''5
=
Sj")
=
and
=
J7(s2,''a
if
she heard an unexpected message. For simplicity,
=
we must
we assume
before, to complete the formal description of equilibrium,
would believe
Sj")
Vj
-i-
w^j.
what the receiver
believe s = Sj
would
she
that
also specify
informed firm signals to two uninformed audiences: a
the capital market. Suppose that credible cheap talk
firm and
but
is
is
uninformed
its
rival
rival firm
is
(contemplating entry) and
impossible between the informed
alone (the informed firm would always discourage entry),
possible between the informed firm and the capital market.
public, there can be either one-sided discipline or subversion.
relationship with the capital market
formed
rival,
is
more important than
its
When communication
If
the informed firm's
relationship with
its
unin-
then the latter can learn something from the informed firm's public messages,
because the presence of the capital market disciplines the communication. K, on the other
hand, the informed firm
more concerned with
is
potential entry than with
its
access to
outside capital, then the lurking potential entrant subverts public communication with the
capital market.®
Proposition
equilibriiun
1
may
A
also suggests another possibility, even in our simple model.
exist in public
and positive while
V2
because of mutual
discipline.
When
and w^ are
Vi
and Wi are small and negative, no separating equilibrium
either audience alone,
separating
large
exists with
but the presence of each disciplines 5's relationship with the other.
Possible examples of mutual discipline include the firm dealing with bond-raters and a
union, and a politician dealing with liberal and conservative constituents.
In a single-
audience problem in either of these examples, the sender's preferences over the receiver's
actions (and thus over her beliefs) are independent of his
cannot be informative.
Uue
type,
and so cheap
For example, the firm always wants the highest possible bond
rating and the lowest possible wage demands, so no (cheap-talk) claim
audience alone
is
credible.
directions, its claims
the politician
in private,
who
but
talk
But since these considerations tempt the firm
may be
credible
when
it
made
to
lie
to either
in opposite
speaks to both audiences in public. Similarly,
always wants votes cannot credibly claim to side with either audience
may be
be independent of the
able to do so in public, where his preferred image might no longer
truth.''
For other examples of subversion, consider, first, letters of recommendation. The writer of such a letter
less honest if he knows that the subject of the letter wiU read it. Second, consider treaty negotiations
or summit meetings between heads of state. K all negotiating statements were public, this might easily
eliminate all real content from negotiation, leaving only vacuous posturing. That is, the incentives for
dishonesty in playing to the public may subvert any incentives for honesty in dealing with other statesmen.
may be
The mechanism-design
literature provides several other
examples
in
which a privately informed agent
More
systematically,
we can
1.
No
2.
Full communication.
distinguish five cases, as follows:
communication. There
and hence
no separating equilibrium,
is
There
is
in
public or in private.
a separating equilibrium with each receiver in private,
both
also (by Proposition 1) with
in public.
There are no
credibility prob-
lems.
3.
One-sided discipline. There
is
not with the other, and there
4.
Mutual
5.
Subversion. There
There
Discipline.
a separating equilibrium in public.
is
no separating equilibriimi
is
but there
in private,
is
in public.
a separating equilibrium with one receiver in private, but not with
is
the other, and there
a separating equilibrium in private with one receiver but
is
none
(Notice that, by Proposition
in public.
1,
there cannot be
mutual subversion.)
These cases are shown
in the figures
below, which plot w^ against w^
figure corresponds to varying the nature
Figure
private with Q.
Figure
l)
If
5
>
the case ^1,^2
1 illustrates
also has
which there
no incentives to
R
to
lie
is
If
only small incentives to
lie
in
the upper-right part of
any forum,
there are strong incentives for
state or in both, then the presence of
R
S
to
lie
subverts 5's relationship with Q.
to R, then honest public
R.
a separating equilibrium in
(as in
then of course there can be honest communication
or in private with either receiver.
Moving around each
of 5's relationship with
and strength
0, in
.
i.e.,
in public
to R, in either
And
if
there are
communication can be achieved
as
Q
disciplines 5's statements to R. This suggests viewing one-sided discipline as intermediate
between subversion and
full
Figure 2 shows the case
but not
in state 2.
communication
If
5
communication.
Uj
<
<
^2, in
has strong incentives to
in public or in private.
state 2 then there
is
which 5 has an incentive to
If
lie
to
R
lie
to
Q
in state 1
then there can be no credible
those incentives are weaker and are limited to
a possibility of mutual discipline.
And
if
5 would
tell
the truth to
R
lie in opposite directions. See for instance Peter Cramton and Thomas Palfrey (1986),
Cramton, Gibbons and Paul Klemperer (1987), Tracy Lewis and David Sappington (1989), and Michael
Riordan and Sappington (1987). These papers show that such "countervailing incentives" can mitigate
the effect of the agent's private information, in much the same way that public communication improves
credibility in the case we call mutual discipline.
faces temptations to
Q's presence will subvert that unless 5's incentive to
in private,
state 1 overcomes his incentive to
discipline as intermediate cases
lie
to Q.
Thus Figure
tell
the truth to
2 displays subversion
R
and mutual
between one-sided discipline and no com^munication.
Finally, Figure 3, for the case Vi^V2
<
in
which S always wants to
lie
to Q, shows
subversion as intermediate between one-sided discipline and no communication: there
subversion
if
S
has incentives to
tell
the truth to
incentives are weaker than his incentives to
Ill
.
in
R
but
(in
is
one or both states) those
to Q.
lie
Welfare
In this section
we
More
discuss preferences over equilibria.
precisely,
we
ask whether
agents prefer pooling or separating equilibria (should the latter exist) in a given forum.
The
results are of
some
interest in themselves,
and may
also help us
understand
why an
agent, or an institution, might choose one forum over another.
When
get
a separating equilibrium exists, receivers always prefer
more information and that can only help them.* The
concern the sender's preferences.
It
it
to pooling, since they
interesting questions therefore
turns out that these preferences are very simple in
a single-audience version of our model, and also with two audiences
condition holds.
With one
More
receiver,
interesting effects arise
R,
in
when the coherence condition
R
to choose either of her
"expanded choice
audience case, as we now
set"
illustrate.
S
can, by
two actions: hence, he can induce
choice set in separating than in pooling equilibrium, so he
this
fails.
For recall that, in a separating equilibrium,
her to choose the action she would choose in the pooling equilibrium. Thus
But
a "coherence"
our two-action model, the sender also certainly prefers sep-
aration to pooling, even tx post.
choosing a message, induce
when
is
(at least
S has a
larger
weakly) better-off.®
argument cannot generally be extended to the twoSuppose
for
example that a boss S
is
communicating
IS a receiver had to take into account others' responses to the fact that she has been informed, she might
of course prefer to be kept ignorant, or
more precisely
to be thought to have been kept ignorant.
—
—
This result does not extend beyond our simple two-action model. For example, in a three-action, twoR may choose Tj (rj) when she thinks the state is 5i (sj), but choose rs in pooling equilibrium.
If S would always like her to choose r^ then he will prefer pooling to separation, even ex-post. In that Ccise,
however, a refusal by 5 to reveal the state would be a "credible neologism," which jirguably would destroy
state model,
the credibihty of the separating equilibrium. See Farrell (forthcoming) or the discussion in Section IV below.
with two candidates
get the promotion
to stay
prefer
on
if
in the
for a
(if
promotion. Let the state of the world
be "candidate
s^
now
she stays with the firm)." Each candidate must choose
will
:'
either
hope of the promotion, or to leave and take a new job (which she would
and only
if
In any separating equilibrium the
she will not get the promotion).
candidate with poor prospects leaves, but
if
(say)
tt
«
chance of promotion to leaving, then both would stay
^
and
in a
if
each candidate prefers her
K the
pooling equilibrium.
boss
prefers that both stay then he prefers pooling to a public separating equilibrium, even
if
the latter exists.^"
The key
difference
between
this
example and the single-receiver case
sender can "get the best of both worlds" by revealing nothing.
If
that here the
n were not near ^, then
one of the candidates would leave unless credibly assured that she was
while the other would stay unless given reason for pessimism.
is
And
in line for
if
promotion,
that were so then,
just as with one audience, the boss could always induce the pooling-equilibrium actions
using one of the two statements used
in
the separating equilibriimi: the expanded-choice
argument would apply and we could conclude again that he cannot prefer pooling
separating equilibrium exists. But
be promoted than about leaving
that
is,
if
if
each candidate cares more about staying
not, then the boss can "have
it
if
if
a
she will
both ways" by pooling:
by remaining inscrutable.
Formally,
she believed
the prior beliefs
if
Sj
"incoherent:"
,
and the other
it is
tt
would induce one audience to
to select her second, as
if
select her first action, as
she believed 53
easy to check that incoherence results whenever
tt
1
lies
we
call
the
2/2
.
Otherwise
— that
is, if
between
the receivers would both choose their
Vi -ry2
both choose their second actions, based on the prior
In the promotion example,
gj: "leave."
R
is
Q
is
the
"first"
belief
n
— the game
first
is
if
game
Xi
and
by
+
X2
actions, or
"coherent."
^^
audience, and her actions are gi:"stay" and
the second audience, and her actions are rj: "leave" and r2:"stay."
If
each
Similax considerations arise if a government is known to be planning to put a highway through one of
two neighborhoods. The residents' actions might be "continue to invest in the area" and "stop investing,"
and the government might prefer that residents of both neighborhoods continue to invest, as they might if
the government remains inscrutable.
It is worth remarking that there is no irrationality attached to incoherence:
care differently about the types of errors they might make.
it is
just that the receivers
is
concerned primarily not to leave when she would
2/2
>
game
!/n so the
incoherent
is
As our discussion
>
I2 ^.nd
close to ^.
if tt is
of the promotion
be promoted, then Zj
in fact
example suggests, the sender's preferences across
equilibria in the coherent case are like those in the one-audience case.
game
Proposition
2.
and therefore
also ex ante.
If the
is
coherent, the sender prefers separating to pooling, ex post
Proof. In a public separating equilibrium,
((/i,ri)
and
{q2,r2)-
S
action-pairs. So
By
Without
Then
Sj
is
has a larger choice set in the separating equilibriimi than in pooling. I
>
not so very different from the single-receiver case.
loss of generality,
the Si type of
if
is
much more complex,
S
and
u^i
(:)
3.
we now
>
if
and only
Vj,
If the
if lOj
>
Vj
+
^2
•
if
game
is
and only
Thus S
and prefers pooling ex ante
(2) for
if
and only
may
If a public separating equilibrium exists as a result of
(ti)
Vj
if
>
vi
+
[qi,
we
ttwi
+
(l
—
if
Tfjuj
and
<
0.
have:
prefer pooling to separating.
mutual
If there is 'i'uU
if
r2).
Wi, and the
prefers pooling ex-post
separating equilibrivim,
incoherent, the sender
pooling ex post (ajid so also ex ante),
But the
discuss.
suppose that the receivers' pooling actions are
Recalling the conditions (l) and
Proposition
as
prefers pooling to separating
type prefers pooling
only
the action-pairs
the definition of coherence, pooling induces one or other of those
Thus, the coherent case
incoherent case
5 can choose between inducing
discipline,
he
may prefer
communication," he prefers
separating to pooling ex post and ex ante, (tn) If a public separating equilibrium exists
as a result of one-sided discipline, then he
may prefer
either separating or pooling ex ante,
but cajinot prefer pooling ex post.
TV
.
Equilibrium Selection
In this section
we show
that, in the coherent case,
if
a separating equilibrium exists in
our two-action model then the pooling equilibrium generically
refinement criterion:
it
is
fails
to satisfy a natural
not "neologism-proof (Farrell, forthcoming).
10
This suggests
that information about the state not only could but will be conveyed whenever a separating equilibrium exists. In the incoherent case, as
we
shall see,
matters are again more
complicated.
A. The Coherent Case
We
consider for simplicity the one-receiver case: the coherent two-receiver (public com-
munication) case
is
the pooling equilibrium
the state
is
Si
.
Without
entirely similar.
is Tj
,
as
it is
[/(S2,"6
holds strictly, as
state 62
,
in separating equilibrium
Recall that for separating equilibrium to exist,
(2)
If (2)
loss of generality,
S has an
it
=
52'')
generically will
incentive to persuade
if it
R
>C/(s2,"5
holds at
that 52
=
all,
suppose that
when S persuades
action in
her that
we must have
5i'').
then in the pooling equilibrium, in
indeed the state.
is
i?'s
He cannot do
so by
using the message(s) used in pooling equilibrium, which will be interpreted as meaningless.
But he could deviate by saying something unexpected^ ^
"Although you were not expecting
state 62-
it is
I
really
in fact true,
mean
and
this.
that,
me
to reveal the state, please listen.
Notice that, by
by
(1), I
like:
(2), I
have a
have no such incentive
strict incentive to
if it is
The
state
is
actually
persuade you of this
in fact false. Therefore
if
you should
believe me."
To sustain the pooling equilibrium,
no evidence about the true
state.
R
must
interpret this speech as providing little or
Intuitively, this
is
implausible, since
5"s reminder about his incentives. Because such a speech can be
pooling equilibrium
is
R
cannot dispute
made, we say that the
not neologism-proof. Moreover, no such objection can be
made
to
the separating equilibrium: because both of the receiver's available actions are used in
equilibrium., neither type of
would get
Sender can
in separating equilibriimi.
strictly prefer
the pooling outcome to what he
Thus we have:
If in the pooling equilibrium S always says "no information" (as we assumed above), or says nothing, or
even uses meaninglessly simple statements such as ''s = Si' and "s = S3", then the following speech is an
unexpected message, or "neologism."
11
Proposition
rium then,
to
two actions,
generically, the pooling equilibrium
equilibrium
If
In the coherent case with
4.
is
if
there exists a separating equilib-
not neologisna- proof (and the separating
is).
the communication technology
make such a
speech.
is
very tightly controlled,
S may have no chance
But observe that the players unanimously prefer the separating
equilibrium to the pooling equilibrium, ex post (and hence also ex ante). Thus they have
no incentive to try
to prevent such speeches.
which a separating equilibrium
Let us
now
step for a
players prefer a
moment beyond our model
forum
in
if
is
one
that
S
made
exists,
instance, in the mutual discipline case,
we expect
in
which a separating equilibrium
that such a forum will be used,
case
—
forum
in
be played.
exists, that equilibrium will
be played), whether the choice
will
take this to imply that, in a
which the forum has been
— and informally consider the selection of a forum.
exogenously given
all
We
In the coherent case,
exists (and,
by Proposition
4,
ex post or ex ante. It therefore seems likely
and that communication
we expect
will
then occur. For
public separation, while in the subversion
will privately reveal the state to
one audience and (inevitably) pool
with the other.
B. The Incoherent Case
As the promotion example shows, Proposition
(Specifically, the boss gets a better payoff
4 does not extend to the incoherent case.
by pooling than by either separating message,
so he has no incentive to disrupt the pooling equilibrium
not only
may
— quite the
reverse.) Moreover,
pooling be neologism-proof in an incoherent game, but
S may
prefer
it
a separating equilibrium, especially in the mutual-discipline case (recall Proposition
to
3).
Hence, equilibriimi selection, whether by neologism-proofness or by players' preferences
across equilibria,
may
not rule out the pooling equilibrium even in a forum in which a
separating equilibrium exists. ^^
Moreover, we can no longer informally argue, as in the coherent case, that any "reasonable" mechanism to choose a forum will select a forum
13
By Proposition
4,
however, we do expect separation in the
12
"full
(if
one
exists) in
communication"
case.
which there
is
a separating equilibrium.
There
no longer unanimity.
is
chooses the forum (whether tx post or ex ante), he
Although there
is
willing to
V
.
we might expect
make
for instance, the sender
may choose
a pooling equilibrivun in every forum, choosing a
no separating equilibrium may help S
discipline case
If,
to see
to
remain inscrutable.
forum
in
which there
for instance in the
is
mutual
to
wear the mask, so that
S
refusing to speak in public, although he might be
(necessarily meaningless!) statements in private.
Discussion
In this section
we
discuss:
(A) three further examples, (B) two natural extensions of our
model, and (C) a possible direction
work.
for further
A. Further Examples
First,
Jeremy Stein (1989) has recently adapted Crawford and Sobel's one-audience model
to analyze cheap talk by the Federal Reserve
shows that
in
Board (about future monetary
policy).
He
equilibrium the Fed's claims can be credible but that they cannot be com-
pletely precise.^*
Our work suggests
that further results should follow from multi-audience
models:'^ the credibility of public claims should depend on the aggregate audience, but
private talk could be either
model mutual
discipline
more
or less credible than these public claim.s,
and subversion are both
much
as in our
possible.
Second, David Austen-Smith (I988a,b) has modeled legislative debate as (public) cheap
talk about the likely effects of a
Our model suggests that
voting.
He asked how such
bill.
it
talk affects agenda-setting
would be desirable to extend
this
work to include
private "back-room" or "closed-door" communication, which might sometimes be
credible.
and
more
Also in political science, Jeffrey Banks (1987) has asked what assumptions are
necessary to
make
candidates' non-binding electoral "promises" credible: these promises
can be modeled as cheap talk to many audiences
— the voters.
and Sobel's, the state of the world is drawn from an interval on the
and the most that can be credibly communicated is that the state lies in one of a certain collection
In Stein's model, as in Crawford
real line,
of sub-intervals that partition the state space.
In Stein's model, the single audience consists of speculators on the foreign-exchange market. Additional
audiences might include employers, labor unions, and investors. Also, one could develop analogous models
of the government's claims about future fiscal policy.
13
Third, an interesting two-audience communication problem arises in a firm's choice
among accounting
allov^'
conventions. To a considerable extent, different (legitimate) conventions
a firm to report higher or lower accounting
For example, a firm that carries
profits.^ ^
inventory for substantial periods of time in an inflationary era will report lower profits
it
uses the
LIFO
(last in, first out)
currently selling than
LIFO, the firm
(in
if it
uses the opposite
calculates profits as
nominal terms) recent
made
convention to determine the historic cost of units
if it
cost, while
is
FIFO
now
selling units
under FIFO
long ago at uninfiated prices. Consequently,
under FIFO. Many economists were puzzled
inflationary era of the 1970s.
as well as the Internal
A
(first in, first
it
made
its
tax liability
at firms' reluctance to
its
cheap talk to two audiences, since the firm's tax
bill
message), and not just on the IRS's posterior
(its
is
is,
if
Under
selling units
likely to
switch to
be higher
LIFO
in the
that the stock market
actions on reported profits,
the stock market to think that their profits are high. This
like
is
it is
at the relatively high
calculates profits as
two-audience view of the problem
Revenue Service bases
out) convention.
if
and that firms
however, not a case of
depends directly
its
reported profits
beliefs.
B. Extensions
Our
two-state, two-action
audiences. In
some
more than two
model
is
the simplest non-trivial model of cheap talk to two
settings, however,
even with only two states, a minimal model requires
actions, because a receiver's pooling action
may
differ
from both of her two
separating actions. In this case, the existence and characterization results in Section
unchanged, but even with just three actions the welfare
in Sections III
II
are
ajid equilibrium-selection ajialysis
and IV becomes much more complex, and. resembles the incoherent two-
action case.
It is
informative also to consider a model with more than two states of the world.
Suppose there are three. Then,
be partial-pooling equilibria
in
in addition to separating
and pooling
equilibria, there
may
which the Sender's claims reveal one of the three states
but pool the other two together. Suppose there exists a separating equilibrium in private
The Federal Accounting Standards Boiird (FASB), and the accounting profession in general, try to limit
such latitude, but a considerable amount remains.
14
with Q, only a pooling equilibrium
in private
with R, and a partial-pooling equilibrium
public. Thus, the distinction between one-sided discipline and subversion
is
in
sharply drawn
only in the two-state model; more generally, they are limiting cases of a single phenomenon.
C. Further
One
Work
natural question that
we have not addressed
the following:
is
if
the sender chooses
the forum ex-post, what can be inferred from his choice? For instance, what should you
infer
if
someone
tells
Such a choice might
he actually speaks.
you something
tell
in private that
a listener something about the sender's information even before
In our two-state model, however, a separating equilibriimi at the
forum-choice stage would leave nothing to be said
complex model
is
he could have told you in public?
needed
in
in
the chosen forum.
order to address this question.
15
A
richer
and more
References
Austen-Smith, David (1988a) "Credibility
in
Debate," mimeo, Department of Political
Science, University of Rochester, December.
Austen-Smith, David (1988b) "Information Transmission
in
Debate," mimeo, Department
of Political Science, University of Rochester, December.
"A Model of Electoral Competition with Incomplete Information,"
mimeo, Department of Political Science, University of Rochester, November.
Banks, Jeffrey
S. (1987)
Bhattacharya, Sudipto and Jay Ritter (1983) "Innovation and Communication: Signalling
with Partial Disclosure," Review of Economic Studies, 50, April, 331-346.
Cramton, Peter and Thomas Palfrey (1986) "Cartel Enforcement with Uncertainty about
Costs," Yale Discussion Paper, October.
Cramton, Peter, Robert Gibbons and Paul Klemperer (1987) "Dissolving a Partnership
Efficiently," Econometrica, 55, May, 615-632.
Crawford, Vincent and Joel Sobel (1982) "Strategic Information Transmission," Econometrica, 50, November, 1431-1451.
Farrell,
Joseph (forthcoming) "Meaning and Credibility in Cheap-Talk Games," in MathMichael Dempster. Oxford: Oxford University
ematical Models in Economics, ed.
Press.
Gertner, Robert, Robert Gibbons, and David Scharfstein (1988) "Simultaneous Signallmg
to the Capital and Product Markets," Rand Journal of Economics, 19, Summer,
173-190.
Lewis, Tracy and David Sappington (1989) "Inflexible Rules in Incentive Problems," American Economic Review, 79, March, forthcoming.
Riordan, Michael and David Sappington (1987) "Information, Incentives and Organizational Mode," Quarterly Journal of Economics, 102, May, 243-264.
Jeremy (1989) "Cheap Talk and the Fed: A Theory of Imprecise Policy Announcements," American Economic Review, 79, March, forthcoming.
Stein,
16
10,
Full
-^ Wi
*
(0,0)
One-Sided Discipline
i-Vi,~V2)
Subversion
Figure
1:
The Case
Vj
>
0, Vj
Communication
>
W2
Subversion
One-Sided Discipline
>
w.
(-1^1,0)
(0,0)
Mutual Discipline
(0,-^2)
No Communication
Figure
2:
The Case
v^
<
0,t;2
>
One-Sided Discipline
{-V,,-V2)
Subversion
-> w.
(0,0)
No Communication
Figure
3:
The Case
Uj
<
O.Vj
<
z u
3 c^oao
005 510
e-iH-e*^
BARCODE
ONNBXr
TO LAST
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