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CHEAP TALK, NEOLOGISMS, AND BARGAINING
Joseph Farrell
Robert Gibbons
No.
500
July 1988
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
CHEAP TALK, NEOLOGISMS, AND BARGAINING
Joseph Farrell
Robert Gibbons
No.
500
July 1988
«u?
^^^
AUG 2
Cheap Talk, Neologisms, and Bargaining*
by
Joseph Farrell
Berkeley
and
Roben Gibbons
MIT
and
NBER
July 1988
Abstract
This paper shows that in some bargaining games, cheap talk must matter: the unique
neologism-proof equilibrium involves informative talk. A simple tradeoff makes both
equilibrium talk and neologisms credible: by saying that he is interested in trading, the
buyer encourages the seller to continue to bargain but receives poorer terms of trade if trade
occurs. All tj^pes of both parties are (weakly) better off in the neologism-proof equilibrium
than in the equilibrium without talk. Thus, cheap talk before bargaining will happen and is
socially desirable.
* Financial support through
(Gibbons)
is
NSF grants
gratefully acknowledged.
IRI-8712238
(Farrell)
and xxx-88xxxxx
1
1
.
Introduction
Many game-theoretic models
on Spence's (1973) signaling model:
signal that he
is
of bargaining under asymmetric information are based
a bargainer incurs costs (t)q)ically
not desperate to trade and so should receive favorable terms of trade
occurs.! Outside bargaining theory, Crawford and Sobel (1982) have
communication
credible
that
shown
if
trade
that
possible without costly signals: costless messages (cheap talk) can be
is
if tliere is
some common
interest
between the
parties.
and Gibbons (fonhcoming) and Matthews and Postlewaite (forthcoming)
Farrcll
show
through delay) to
cheap
Gibbons show
talk
can matter
in a specific bargaining
game, a double auction. Farrell and
each bargainer faces a tradeoff that can make his talk credible: by
that
expressing an interest in trading, one bargainer encourages the other to continue
to
bargain
but receives poorer terms of trade if trade occurs; in equilibrium, only buyers (sellers) with
high (low) reservation prices are willing
and Postlewaite show
equilibria (each of
bargainers'
that talk
to
express such an interest
Mattheu's
can coordinate the bargainers on different double-auction
which would have been an equilibrium without
announced
in trading.
reser\'ation prices,
and
that
talk)
depending on the
such pre-play communication
dramatically enlarges the set of equilibrium outcomes of a double auction. Both papers
show
that talk
can matter by constructing equilibria
In this paper
must matter,
in the
we show
that in
encourage
his
could not be equilibria without
some bargaining games cheap
talk not
talk.
only can but
sense that the unique neologism-proof equilibrium (see Farrell
(forthcoming)) could not be an equilibrium without
credible for the
that
same reason we
opponent
talk.
In this equilibrium,
cheap
talk is
identified in our earlier paper: a bargainer can use talk to
to continue to bargain, but will receive
trade occurs. This tradeoff leads to
two
results in the
game we
poorer terms of trade
if
analyze: (1) buyers with
high reservation prices prefer to reveal themselves through talk rather than to masquerade
^The large
(1983).
literature
on
this subject
begins with Fudenberg and Tirole (1983) and Sobcl and Takaliashi
2
as buyers wdth
low reservation
prices,
reveal themselves rather than to pool.
and
(2)
The
first
buyers with high reservation prices prefer to
of these results implies that an informative
cheap-talk equilibrium exists, and the second establishes that equilibria in which talk
is
absent or uninformative are not neologism-proof.
We also consider the welfare implications of cheap talk before bargaining.
game we
analyze,
all
types of both parties are (weakly) better off in the neologism-proof
equilibrium than in the equilibrium without
only will happen but
auction that
some
In the
is
talk.
Thus, cheap talk before bargaining not
socially desirable. This contrasts with our findings for a
some types of both
parties
do
double
better in an equilibrium with informative talk, but
types do worse, and the ex-ante expected gains from trade are lower with talk than
without.
The body of the paper
on die familiar
is
organized as follows. In Section 2
take-it-or-leave-it bargaining model.
we
develop a variation
Our model may be of independent
interest as an analysis of trade in idiosyncratic goods, but
provide an uncluttered setting for our analysis of cheap
we have developed
talk.
In Section 3
we
it
mainly
to
characterize
the neologism-proof equilibria in our bargaining game. Section 4 concludes.
2.
The Model
We analyze a very simple bargaining game in
single take-it-or-leave-it offer to a buyer.
good
denoted by
in question is
seller's
payoff is p
-
s -
buyer rejects then the
makes no
c
s,
seller, at
seller's (privately
cost
c,
can
known) valuation
make
a
for the
the buyer's by b. If trade occurs at price p then the
and the buyer's
seller's
The
which a
payoff
is -c
is
b
-
p.
If the seller
and the buyer's
offer then each player's payoff is zero.
Both
is
makes an
zero.
offer that the
Finally, if the seller
parties are risk-neutral.-
Perry (1986) shows that the unique perfect Bayesian equilibrium of this game is identical to the unique
perfect Bayesian equilibrium of the infmiie-horizon altemaung-offers game with no discounting in which
(i)
3
The
willingness to
seller's
make an
depend on her valuation and on her
common knowledge that
[0,1].
We ask
whether cheap
talk
by the buyer can influence
We assume that
it is
make an
offer
belief about b, and
tlie seller's
and the price she demands
that, in a
if
she
double auction, the buyer faces a simple and
can make cheap talk matter
our take-it-or-leave-it bargaining game,
in a
an interval contained in
be well defined.) In order for
be credible
talk to
study this tradeoff
u([x,y],b) be the expected payoff to a
let
(Note that b need not be
[0,1].
To
double auction.
of valuation b when the seller believes that the buyer's valuation
to
she does so
offer.
intuitive tradeoff that
[x,y],
if
and b are independently drawn from a uniform distribution on
s
Our previous paper shows
in
demands
belief about the buyer's valuation.
so influence both the seller's decision to
makes an
offer and the price she
is
buyer
uniformly distributed on
in [x,y] for this
in equilibrium, there
must
expectation
exist a
valuation b* e (0,1) such that buyers with valuations above b* prefer to have the seller
believe that their valuation
is
high while those below b* prefer her to believe that
(1)
u([b*,l],b)
> u([0,b*],b)
forallbe(bM], and
(2)
u([0,b*],b)
> u([b*,l].b)
for all b e [0,b*).
In the take-it-or-leave-it bargaining
bargaining power makes
game, however, the
=
for
all
and
(iii)
opponuniiy
to
make
the buyer incurs cost
the first offer,
C>
c
^Formally, u(rb*,l],b*)
=
arbitrarily close to zero for
u([x,y],b)
is
because given
weakly increasing
this implies that u([0,b*],b)
=
(ii)
who do
in this
game, the only buyers
not trade following talk.
the seller incurs cost c each lime she
each time he makes an
Therefore, u([b*,I],b)
is
low.
fact that the seller has all the
b e [0,b*].3 Thus,
willing to admit to having low valuations are those
the seller has the
is
impossible to satisfy both of these inequalities, except in an
it
uninteresting way: u([0,b*],b)
offer,
it
makes an
offer.
ihis belief ihe seller will
not
b sufficiently close
make an
offer less than b*.
to (but greater than) b*.
Note
that
so u([0,b*],b*) = 0, else (1) fails for b slightly greater than b*. But
for all b < b*, so (2) cannot hold su-ictly.
in b,
4
many
In
other bargaining games,
expected payoff of
zero."*
become
will
It
take-it-or-leave-it bargaining
game
affect the bargaining. Rather than
we modify
therefore,
its
not true that the lowest- valuation buyer has an
it is
clear below, however, that the simplicity of the
enables us to derive crisp results about
abandon the
how
take-it-or-leave-it bargaining
talk will
game,
information structure so that the lowest-type buyer has a positive
expected payoff. In the process,
we develop
a model that
may
be of independent interest as
a description of an important class of bargaining problems that has not yet received
much
attention.
We assume that the buyer does not learn his exact willingness to pay for the seller's
good
until
she defines
complex or
it
Such imprecision
precisely.
idiosyncratic: if the seller
must carefully define
must write a contract covering many contingencies
that
relationship with the buyer), the buyer cannot precisely
Our assumption
is
and the
natural given that the buyer
when
arises
its
is
idiosyncratic, there
is
good
in question is
specifications (or
if
the seller
might arise during a long-term
know
his valuation beforehand.
seller are
trading in a competitive market at a market-determined price:
buyer's need)
the
bargaining rather than
when
the seller's
good
(or the
a matching problem as well as a pricing problem, so
competition disappears and bargaining results.
We model the buyer's imprecise knowledge of his valuation
nature draws the valuations
s
and the buyer learns what we
type
(t) is
either high (H) or
and
b, as
above. Second, the seller learns her valuation,
will call his type (as distinct
low
(L),
where
t
=
H
from
his valuation).
for b e [1/2,1] and
Thus, after learning his type, the buyer's belief about his valuation
t
= L and uniform on
[1/2,1] if
t
as follows. First,
is
t
= L
The
s,
buyer's
for b £ [0,1/2).
uniform on [0,1/2)
if
= H.
Chauerjee and Samuelson's (1983) linear equilibrium in a double auction, for instance, if s is
on [x,y] and b is (indcpendenily) distributed on [w,z], then the buyer with valuation w earns a
su-icily positive expected payoff provided w is sufficiently greater than x. Also, in infinite-horizon games
based on Rubinstein's (1982) model with discounting, such as Chatterjee and Samuelson (1987) and
"^In
distributed
Cramton (1984),
if
there are gains from trade then both parlies earn a positive expected payoff.
5
After learning his type, the buyer can send a costless message (cheap talk) to the
seller.
is
The message space
is
unrestricted, but there is not
sufficient to consider the space of equilibrium
The remainder of the game
chooses whether
to
make an
is
offer,
much
that
if
said.
In fact,
it
messages {"H", "L", "pool").^
a take-it-or-leave-it bargaining
and
can be
game. The
seller
makes an
so what price to demand. If she
offer,
the buyer learns his valuation, b, and then chooses to accept or reject the offer. If the seller
does not
make
an offer, the
game
ends, without the buyer learning
Tlie payoffs are
b.
exactly as described above because the messages introduced here are costless.
Two
special features of this
communicate anything other than
seller
makes
leave-it
the offer, there
game
there
model has moving
is
is
if
are worth noting. First,
the buyer's tj^e,
talk
by the
seller
3.
to affect the basic ideas presented
cheap
t
talk
cannot
because the buyer learns b after the
to talk about b,
and
about her valuation,
the buyer learns that
certain to exceed 1/2. Relaxing these assumptions
seems unlikely
t:
no scope for the buyer
no role for
support:
model
= H,
say, then
in this take-it-or-
s.^
Second, the
he knows that b
is
would complicate the analysis but
below.
Analysis
Our
analysis proceeds by comparing six expected payoffs, analogous to the
expected payoffs u([x,y],b) but based on the buyer's knowledge of his type but not his
valuation.
These
six
expected payoffs are denoted U("H",
t),
U("L",
t),
and U("pool",
t)
-This is in conirasl to our earlier paper, in which wc considered only the restricted message space {"keen",
"not keen") and so could noicompleicly characterize the equilibrium role of cheap talk in a double auction.
Also, noie well the distinction between the equilibrium messages ("H", "L", "pool") and the unrestricted
space of possible messages; Farrell describes the large set of potential out-of-cquilibrium messages involved
in the definition of neologism-proofness.
°In this take-it-or-leave-it game, talk by the seller about her valuation could matter only by inducing the
buyer to send a subsequent cheap-talk message that varies wiih ihe seller's talk. But no matter what her
valuation, the seller prefers to have the buyer reveal
more rather than less before she has to decide whether
Thus, independent of her valuation, the seller would send the message that induces the
buyer to reveal as much as possible, so there is no reason for the buyer's talk to varj' with the seller's: the
seller's talk will be uninformative.
to
make an
offer.
6
for
t
e
{
H,L)
The
.
first
and second of these are the expected payoff to a buyer of iypt
given that the seller believes that
payoff to a buyer of type
types
is
t
t
=
H and
t
=
L, respectively.
The
t
third is the expected
given that the seller holds the (prior) belief that each of the two
equally likely^
We are interested in two (perfect Bayesian) equilibria of our game: a separating
equilibrium
(in
which
pooling equilibrium
equilibrium,
tlie
tlie
(in
buyer's talk perfectly
which the buyer's
post-talk bargaining behavior
the take-it-or-leave-it bargaining
A
talk
game given
communicates
communicates nothing).
is
exists.
(3)
U("H", H) > U("L", H)
(4)
U("L", L) > U("H", L).
The refmement we use
games
is that
—
that
is,
satisfy a
refmement.
and
is Farrell's
and Banks and Sobel (1987), have no
notion of neologism-proof equilibrium.
(5)
U("H", H) > U("pooI", H)
(6)
U("H", L) < U("pool",
it
(Many
games by Cho and Kreps (1987)
effect in cheap-talk games.)
not neologism-proof (and so reject
'The payoff function
a pooling (or "babbling")
if
other refinements, such as those developed for signaling
We say that the pooling
as an equilibrium) if
and
L).
u([x,y],b) is of course related to lliese six expected payoffs: for instance,
E{u([l/2,l],b)IO<b<
outcome of
We tlierefore ask whether a separating equilibrium exists, and
Formally, a separating equilibrium exists
is
each kind of
the unique sequentially rational
also whether the equilibria that exist are reasonable
equilibrium
In
and a
the seller's belief.^
basic result in the theory of cheap-talk
equihbrium always
his type to the seller),
U("H", L) -
1/2).
'^Note that this uniqueness of scqueniially rational posi-talk bargaining behavior rules out equilibria in
which cheap talk matters for the coordination reason emphasized by Matthews and Postlewaiie.
The
idea behind these conditions can be phrased in terms of a speech like those popularized
by Cho and Kreps. Suppose
that his type is high,
t
the pooling equilibrium
= H. Then
is
the buyer should deviate
from the equilibrium by making
the following speech to the seller during the cheap-talk phase:
Note
that if
you believe
in equilibrium,
by
(5).
claim then
this
Noie also
I
that if
and the buyer learns
to be played
"I
claim
my
will be (strictly) better off than
my
t>pe were low then
1
I
type
is
high.
would have been
would have no
(strict)
incentive to get you to believe this claim, by (6)." In FarrcH's terminology, (5) and (6)
define a credible neologism by the high
t)'pe; if (5)
and
(6) hold,
we
say that a credible
neologism by the high buyer-type breaks the pooling equilibrium.
A
credible neologism by the low buyer-type also could break the pooling
equilibrium,
if
(7)
U("L", L) > UC'pool", L)
(8)
U("L", H) < UC'pool", H).
Proposition
1
below implies
and
that if a credible
neologism by
ilie
low buyer-type breaks the
pooling equilibrium then a credible neologism by the high buyer-tj'pe also breaks the
pooling equilibrium, so
we
ignore neologisms by the low buyer-type in what follows.
If a separating equilibrium exists, then
we
neologism-proof. Here the relevant neologism
am not going to reveal my
this deviation...").
The
is
also are interested in whether
(9)
UC'pool", H) > U("H", H)
(10)
UC'pool", L) > U("L", L).
and
is
is
pooling (the appropriate speech begins
type. Notice that regardless of
separating equilibrium
it
my type,
I
have reason
not neologism-proof
if
and only
to
if
"I
make
8
by analogy with
that both inequalities are strict,
Note
type cheap-talk
Lemma
1
If the
.
(7).
A
basic result for two-
is
proof then the separating equilibrium
Proof
and
a separating equilibrium exists and the pooling equilibrium
If
.
games
(5)
pooling equilibrium
must hold, hence
at least
one of
is
(9)
is
is
neologism-proof.
not neologism-proof then at least
and (10) must
fail.
one of
on a
Lemma
2
series of simple
.
distributed
Suppose the
on
[x,y].
lemmas
(5)
and
The
analysis
(proofs of which are relegated to the Appendix).
seller of valuation s believes that tlie buyer's valuation is
Then
(7)
Q.E.D.
We now characterize the neologism-proof equilibria of our model.
rests
not neologism-
if the seller
makes an
offer, the price
demanded
is
uniformly
p(s|x,y)
=
max{x,(y+s)/2}.
Lemma
3
.
For
all
Lemma
4
.
(a)
U("H", H) >
(b)
UC'pool", H) >
(c)
U("L", L) >
(d)
U("L", H) >
Lemma
5
.
values of
U("H", L) =
c,
if
if
if
and only
if
and UC'pool", L) =
if
c
and only
and only
and only
if
<
1/2.
if c
c
if c
<
<
0.
<
1/4.
1/8.
1/8.
For c e (0,1/2), U("H", H) > U("pool", H).
Combining
these
Lemmas
(and adding a small argument given in the Appendix)
yields the following characterization of the neologism-proof equilibria of our model.
Proposition
1
.
>
1/2 then the unique equilibrium is pooling and
(a)
If c
(b)
If c £ (0,1/2)
it is
neologism-proof.
then a credible neologism by the high buyer-t>'pe breaks the
pooling equilibrium. Funher, there exists a
c* e (0,1/8) such
critical cost level
e (c*,l/2) a separating equilibrium exists and
is
when
neologism-proof, but
tliat
when
c
c e (0,c*) a
separating equilibrium does not exist (and so there does not exist a neologism-proof
equilibrium).
Two
the
.seller
features of
result are
does not make an offer
Thus, by Proposidon
happens!
tliis
And
1
(a),
worth noting.
in the
First,
by
Lemma
the pooling equilibrium
high buyer-type, while
is
neologism-proof only
.
if
nothing
Lemmas
Lemma
5 establishes
3 and 4(c) do so for the low, and the seller
is
what her valuation) simply because she has more information. 9 Tlie
importance of Proposition
Proposition 2
1/4 then
second, assuming c e (c*,l/2) so that the separaung equilibrium exists, both
better off (no matter
real
>
pooling equiHbrium, no matter what her valuation.
parties are better off in the separating than in the pooling equiHbrium:
this for the
4(b), if c
1,
however,
is that
it
leads to our
main
For c E (c*,l/2), the post-talk bargaining behavior
proof equilibrium could not be equilibrium behavior without
The proof of this Proposition
is
result:
in the
unique neologism-
talk.
simple: the unique equilibrium without talk
unique neologism-proof equilibrium for c e (c*,l/2)
bargaining behavior in these two equilibria differs
is
in
is
pooling; the
separating; and the post-talk
many obvious ways.
We now turn
to
It IS of course not generally iruc in games that having more information makes a player better off. In our
bargaining game, however, the buyer's optimal action depends only on the price the seller demands and on
b, and so is mdcpendeni of what the buyer thinks the seller knows. In essence, this reduces the game to a
single-person decision problem for the seller, in which case it is true that more information is better.
1
a detailed discussion of these differences in post-talk bargaining behavior in order to
interpret the effects of cheap talk in our model.
There are two ways
which post-talk bargaining behavior can differ among
in
equilibria: in the seller's decision to
Our main
occurs
if trade
low buyer-type
think, less interesting)
instance, the
make
arguments suffice
is
demands
if
she makes an offer
in
We now identify the equilibria described in
Consider
valuation
s
first
makes an
intuition
cases, for
we have
the
is
same
in the
the probabilir\' that the
Proposition 2 that are
relatively uninteresting for either of these reasons; this leaves us with
embody the
some
we
indifferent about his message: (4) holds with equality.
the price the seller
offer.
however, simpler (and,
support the equilibrium. In
to
separating and pooling equilibria, so the equilibria differ only
makes an
pay a
this tradeoff.
equilibria considered in Proposition 2,
low buyer-type
some cases
precisely
(or both).
order to increase the probability that trade will occur, and the
in
strictly prefers not to
some of the
In
seller
demands
she
offer, or in the price
results involve both of these differences: the high buyer-type is willing to
higher price
Also, in
make an
tiie
equilibria that
described.
the pooling equilibrium.
offer then the price she
Lemma
demands
2 implies that
is
p(s|0,l)
=
if
the seller with
(l+s)/2.
Straightforward calculation then shows that (assuming c < 1/4) the sellers with valuations
s
<
2^/c
-
1
make
offers; as noted above, wb.en c
>
1/4
no seller-type makes an
Now consider the separating equilibrium. When
make
that
offers in the separating
make
equihbrium than
offers in both equilibria
make
the
in the
same
^^It is
only by accident that the price that a seller with valuation
=
(l+s)/2,
is
the
same
in the separating equilibrium is
claims to be the high type, p(s|l/2,l) =
b £
[b',1] for
when
some
b'
>
(l-i-s)/2.
demands
If
1/2, then the price a seller
the buyer claims to be the high type
price she
in the
demands
as the price that she
s
in the
p(s|b',l)
=
s
fhe
fact that
most apparent when
demands
in the
pooling equilibrium,
separating equilibrium
the high buyer-i>'pe consisted of
with valuation
would be
pooling equilibrium.
seller-tj^es
offer in each equilibrium. ^0
seller-types
p(s|0,l)
more
pooling equilibrium, but those types
more
make offers
c e (1/8,1/2),
offer.
demands
when
buyers with valuations
in the separating
max(b',(l-t-s)/2)
,
the buyer
which
differs
equilibrium
from the
1
c e (1/4,1/2): the seller
makes no
offer in the pooling equilibrium, but in the separating
equilibrium, with probability 1/2 the buyer announces that his type
the seller updates her belief about his valuation b, and,
analogous argument applies when c e
sellers
make
if s is
low,
(1/8,1/4), except that in this
is
high, in
makes an
which case
offer.
An
case ver)'-low-valuation
offers in both equilibria.
Consequently, when c £ (1/8,1/2), saying "H"
strictly
dominates saying "pool" or
"L" for the high buyer-type and weakJy dominates these alteniaiivcs for the low buyertype.
For these values of
critically
c,
therefore, the existence of a separating equilibrium
on the weak inequahty
A
in (4).
more
interesting equilibrium exists
depends
when
talk not
only determines which seller-types make offers but also what prices they demand. This
occurs
when
c e (c*,l/8), as the following table
1
1
Pooling
1
Equilibrium
Demanded by
Seller of Valuation
p(s|0,l)=-^
I.I
1
following "H"
Separating
1
following "L"
p(s|l/2,l)=^
demands
s*(cr'pool")= 1-2%^
1
p(sI0,l/2)
=
^^
1
1
seller, for instance,
makes
1
1
1
1
1
s*(c|"H")
= 1-V2c
1
1
1
1
Thus, a low-valuation
Highest Valuation
Seller to Make an Offer
1
1
Equilibrium
1
clear.
1
1
EquiUbrium
1
s
1
1
Separating
equilibria but
Price
makes
1
s*(cr'L")
=
^""^
'
1
offers in both the pooling and separating
a different price in the latter because the buyer's talk has influenced
her belief about the buyer's valuation,
b.
12
4.
Conclusion
The
in
which
first result
talk
of our analysis
does not matter)
high that no offer
is
made by a
is
is that
the pooling equilibrium (the only equilibrium
neologism-proof only
making an
strikingly: restricting attention to neologism-proof equilibria, if an offer (not to
trade)
is
observed, then talk must have played a
natural and credible
encourages the
and
this
seller to
make an
many
Our second
when
some
cost in
tlie
t
we
result is that
will
= H,
terms of trade
the buyer
if
trade occurs,
We expect analogous cheap-talk messages
pursue
when
mention
this idea in future
games
to
be
often
work.
c E (c*,l/2), the separating equilibrium not only
the unique neologism-proof equilibrium. In this equilibrium, talk affects
is
whether and
at
what terms trade takes
some bargaining games
that in
offer, at
so
is
this result is the
other bargaining games, so that pooling equilibria in these
be neologism-proof;
exists but
The force behind
to the high-type buyer:
breaks the pooling equilibrium.
credible in
will not
message available
role.
offer
We can rephrase this more
of any valuation.
seller
the cost of
if
(i.e.,
place.
for
Combined with our
some values of c)
talk not
first result, this
shows
only can matter but must
matter in the sense that the only neologism-proof equilibrium could not be an equilibrium
,
without talL
We find this result particularly appealing when
both buyer -types have a
strict
and both buyer-types have a
Our
in the
which case
incentive to reveal themselves in a separating equilibrium
strict
incentive to reveal themselves rather than to pool.
final result is that, in the
(weakly) better off
c e (c*,l/8), in
game we
analyze,
all
types of both parties are
neologism-proof equilibrium than
Thus, cheap talk before bargaining not only will happen but
in the
is
equilibrium without
socially desirable.
talk.
1
3
APPENDIX
Proof of Lemma 2
The
problem
choose p
is
to
x) for p e [x,y],
is
zero for p
order condition }delds p = (y+s)/2, which
is
a global
max
s
.
F(p)
seller's
+p
-
[1
to solve
F(p)],
P
where F(p) =
(p
x)/(y
-
-
Lemma
3
.
By Lemma
valuation seller and so
Thus,
in
U("H",
minimum
price
L),
and
distributed on [0,1], the
Lemma
valuation seller
in
(a) In
on
[1/2,1]
one for p >
maximizer provided
minimum
price
demanded comes from
is
minimum
price also
is 1/2.
F(p)
.
is
making an
U("H", H), the zero-valuation
zero-valuation seller
tlie
zero-
is
is
uniformly
does the low-type
In neither case
offer.
seller believes that
1/2 if she
makes an
b
offer.
is
uniformly distributed
This offer
indifferent about
making an
p(0|0,
1 )
=
1/2 if she
1/2.
accepted
Thus,
=
1/4, the zero- valuation seller is indifferent
if
c
offer.
UC'pool", H), the zero-valuation seller believes that b
demands
is
makes an
is
uniformly
offer.
This offer
accepted with probability 1/2 and so earns a profit for the zero- valuation seller of
if c
is
solved by computing the value of c such that the zero-
indifferent about
distributed on [0,1] and so
Thus,
first-
Q.E.D.
offer.
and so demands p(0|l/2,l) =
(b) In
The
y.
uniformly distributed on [1/2,1], the
with probability one and so earns a profit for the zero-valuation seller of
1/2, the
tliat
U("pool", L), where the seller believes that b
4 Each case
is
2, the
the seller believes that b
buyer receive an acceptable
Proof of
is
{x,y/2).
where
is 1/2,
and
is
= max
p(0|x,y)
x,
Q.E.D.
strictly positive at this price.
Proof of
<
about making an offer.
1/4.
is
=
14
U("L", L), the zero valuation
(c) In
[0, 1/2]
and so demands p(0|0, 1/2) = 1/4
if
seller believes tliat
b
is
uniformly distributed on
she makes an offer. This offer
is
accepted with
probability 1/2 and so earns a profit for the zero-valuation seller of 1/8. Thus,
the zero-valuation seller
is
U("L", H) the zero-valuation
(d) In
[0,1/2], as in (c). Tliis belief leads to the
=
p(0|0,l/2)
Proof of
offer
when
Lemma
5
c
The
.
argument
for c e (0,1/4)
valuation
s is
strictly
off.
=
lliis
same
one by
that b is
behavior as
1/8,
uniformly distributed on
in (c).
The demand
the high-type buyer, but the zero-
as follows.
from
Lemmas
is
indifferent about
By Lemma
H) and
in
4(a) and 4(b) for c e (1/4,1/2). TTie
2, the price
demanded by
the seller of
U("pool", H):
= p(s|0,l)=^.
difference between U("H", H) and U("pool", H)
more
=
Q.E.D.
result follows
is
beUeves
seller
c
offer.
behavior from the buyer and so
1/8.
identical in U("H",
p(s|l/2,l)
The only
seller
1/4 is accepted with probability
valuation seller does not expect
making an
making an
indifferent about
if
sellers
choose
to
make an
is
that, for
any given value of
offer in the former, so the high-type buyer
is
c,
better
Formally, simple computations show that in U("H", H) the highest-valuation seller to
make
an offer
former
is
s*(cl"H")
is strictly larger.
Proof of Proposition
(a)
1
=
1-V2c, while in U("pool", H)
= l-2Vc. The
.
not exist because U("H",
4(a), so (3) fails.
Similarly, the pooling equilibrium
Lemmas
4(c), (5)
and
s*(c|"pool")
Q.E.D.
The separating equihbrium does
4(a)
it is
and
(7) fail.
is
H) =
0,
by
Lemma
neologism-proof because by
15
(b)
By Lemma
5, (5) holds,
and by
Lemma
3, (6) holds.
Thus, a credible
neologism by the high buyer-type breaks the pooling equilibrium. Therefore, by
if the
separating equilibrium exists then
Suppose
e (1/8,1/2).
first that c
3 and 4(c), (4) holds weakly.
Now
remains
to
suppose
show
Hence
neologism-proof.
By Lemmas
4(a) and 4(d), (3) holds.
By Lemmas
and only
if
3
and
4(c), (4)
c E (c*,l/8), for
U( L
,
„,
H) =
and only
It
thus
E (0,1/8). Using the
in the text,
calculations
= 32x3
that f(l/4)
1/8 in
and
rr
that if c £ (0,1/8) then x e (0,1/4).
Thus, U("H", H) > U("L", H)
if
if
f(x)
=
strictly.
(l-4x)(3+4x)
where x = Vc72. Note
c
By Lemmas
that
,,,.,. „
Note
holds
some c*
values of p(s|0,l/2), s*(c|"L"), p(s|l/2,l), and s*(c|"H") given
show
1
a separating equilibrium exists.
that c e (0,1/8).
that (3) holds if
it is
Lemma
<
Lemmas
.
96x2
and
.
f(0)
24x + 5 <
>
0.
0.
(Both of these are obvious from
4(a) and 4(d) and c
=
0, respectively.)
first
principles: consider
Finally, note that f(x) is
decreasing for x e (0,1/4), so there exists a unique x* £ (0,1/4) that solves f(x) =
Therefore, c* solves X* = Vc72
.
Q.E.D.
0.
1
6
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2610 U27
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