Reserve Price Signaling
Hongbin Cai, John Riley and Lixin Ye*
Abstract
This paper studies an auction model in which the seller has private information about the
object’s characteristics that are valued by both the seller and potential buyers. We explore
the role of reserve prices in signaling this private information. We first characterize the
unique Pareto dominant separating equilibrium (“the Riley outcome”). Then we derive
comparative statics results and discuss an application to signaling in the Lemons Market.
The second part of the paper considers equilibrium refinements. The Cho-Kreps Intuitive
Criterion cannot be directly applied to our setup. Instead we apply a Local Credibility
Test (LCT) which is based on, but is slightly weaker than, Grossman and Perry’s
Strengthened Intuitive Criterion. For a more general signaling model in which reserve
price signaling is one special case, we identify necessary and sufficient conditions for the
LCT to be satisfied in equilibrium. In the reserve price signaling model, such conditions
require that the signaling “effectiveness” be sufficiently large.
*UCLA, UCLA, and Ohio State University. We would like to thank In-Koo Cho, David
Cooper, Massimo Morelli, James Peck, and seminar participants at Illinois Workshop on
Economic Theory, Ohio State University, Rutgers University, UC Riverside, and Case
Western Reserve University, for helpful comments and suggestions. All remaining errors
are our own.
1. Introduction
In this paper we consider an auction environment in which a seller of one object
has private information about the object’s characteristics. These characteristics determine
the seller’s expected valuation of the object and the expected common valuation for a
group of potential buyers, each of whom also has an independent private value for the
object. Since the characteristics can be multi-dimensional and the seller and the bidders
may place different weights on the relative importance of different dimensions, the
seller’s expected value and the buyers’ common value component are likely to be
positively but not perfectly correlated. For example, a seller of an artwork (e.g., an
auction house) may know its conditions (quality, rarity, history, etc.) as well as its
secondary market value better than potential buyers. While she is mostly concerned with
the artwork’s secondary market value, potential buyers (who buy for self consumption)
may also care about its conditions.
By the linkage principle (Milgrom and Webber, 1982), it is well known that the
seller can increase her expected revenue by truthfully revealing her private information
about the object’s characteristics. If direct verification of the seller’s information is
costless, it is indeed incentive compatible for the seller to truthfully reveal her
information for the following reason. Sellers with private information indicating high
common value for the buyers have an incentive to reveal their information since buyers
will then be willing to bid more for the object. Since the same argument holds for any set
of types, the sellers with high types within the set always have an incentive to reveal and
the only Nash equilibrium is full revelation of the seller’s private information. However,
in many auction settings, a costless revelation technology (e.g., a perfectly neutral and
objective evaluation method of a third party) may not be available to the seller. In such
cases, the seller’s announcing her information to the potential buyers may not be credible
as she faces the standard adverse selection problem, that is, she always wants to claim the
highest common value for the buyers. A natural way to credibly reveal the private
information is through signaling, and a natural signaling instrument in this environment is
the reserve price: a seller with a high type has an incentive to try to signal this to the
buyers by setting a high reserve price.
1
In section 2 we describe the reserve price signaling model that has the natural
interpretation of the auction settings described above. However, the model does not fit
into the standard signaling framework. We reformulate the model in two ways. First, we
redefine the seller’s type as her own valuation for the object, not her private information
about the object’s characteristics. Through variable transformation and by the positive
correlation between the seller’s value and the buyers’ common value, the seller’s
expected payoff function can be expressed in the standard form. We then reformulate the
model by focusing on the seller’s reserve markup as the key signaling variable, rather
than the reserve price itself. With this variable transformation, the reformulated model
belongs to the family of “screening” models examined by Riley (1979).
We start with the analysis of discrete types in Section 3. With the standard
technique, it is easy to characterize the separating equilibrium in which the lowest type
seller chooses her optimal reserve price under complete information (the Pareto dominant
equilibrium). It is well known that with only two types, this separating equilibrium is the
unique equilibrium satisfying the Cho-Kreps Intuitive Criterion. However, with more
than two types, and especially in our setting where the multidimensional characteristics
of the object can often lead to positive but not perfect correlations between the seller’s
value and the buyers’ common value, the Intuitive Criterion has no bite. The Intuitive
Criterion cannot be directly applied to the case of continuous type either. A stronger
equilibrium refinement concept based on Grossman and Perry (1986a,b) can be more
effective in selecting the Pareto dominant equilibrium under certain conditions. Here we
weaken slightly the Grossman-Perry Strong Intuitive Criterion, and propose a “Local
Credibility Test” in which a possible deviation is interpreted as coming from one or more
types whose equilibrium actions are nearby. This test works equally well whether there
are finite or a continuum of types.
In Section 4, we consider the case of a continuum of types and characterize the
unique Pareto dominant separating Nash equilibrium. In this equilibrium, the reserve
price schedule fully reveals the seller’s type (her own valuation for the object). Thus a
reserve price can play a more central role than perceived by the traditional literature. In
the standard private value auction model, the seller’s optimal reserve price is set to
capture additional revenue when there is only one buyer who has a valuation much higher
2
than her own. This optimal reserve price is independent of the number of bidders.
Therefore, unless the number of bidders is very small, the probability that the reserve
price is binding is small and hence the extra profit captured by setting a reserve price is
also low. In the Pareto dominant separating equilibrium in our model, we show that the
reserve price increases with the number of bidders and in the importance of buyers’
private information. At the end of Section 4, we also briefly discuss an application of our
results to signaling in the Lemons Market. Even though not in an auction setting, our
results can be readily applied to the well-known Akerlof Lemons Market (Akerlof, 1970)
to give rise to a separating equilibrium in which sellers with different qualities set
different prices and their private information is fully revealed to the market.
Section 5 studies equilibrium refinement in the continuous type model. We
consider a family of continuous type models of which the Spence education signaling and
the reserve price signaling are both members. We begin by formulating the concept of
“Local Credibility Test” (LCT) for the continuous type models. An equilibrium survives
LCT if no deviation-perception pair is credible in the following sense: for any possible
deviation signal (on- or off-equilibrium), if it is interpreted as from types of a small
neighborhood of the immediate equilibrium type, it is profitable for the types in this
neighborhood to deviate to this signal, but unprofitable for types out of this neighborhood
to do so. Clearly, if an equilibrium satisfies LCT, it must be the Pareto dominant
separating equilibrium. We identify necessary and sufficient conditions for the
separating equilibrium in our general signaling model to satisfy LCT. These conditions
are applied to the Spence education signaling model and the reserve price signaling
model. It is shown that the required conditions are intuitive and can be satisfied with
reasonable parameter values. In particular, as long as a measure of signaling
“effectiveness” is sufficiently high for every type, then the separating equilibrium can
survive our LCT test. We also note that one case in which no equilibrium can survive
LCT, both in finite and continuous type models, is when the marginal social cost of
choosing her optimal action is zero for the lowest type. If that is the case, the senders
with sufficiently low types will profit by pooling at the lowest signal when this is
correctly perceived by the receivers.
3
A paper by Jullien and Mariotti (2003) is closely related to ours. They solve for
the separating equilibrium of a signaling game similar to ours, and compare the
equilibrium outcome with the optimal mechanism for a monopoly broker who buys from
the seller and sells to the buyers. Without using the reformulation that makes the markup
the key variable and hence enables transforming the model into the standard framework,
their solution method is quite different from ours. Moreover, they only consider the case
with 2 bidders and assume effectively   1 in our perfect correlation formulation.
Finally, an important part of our paper is focused on equilibrium refinement, while
Jullien and Mariotti simply focus on the Pareto dominant separating equilibrium.
2. The Model
We consider a second-price sealed bid auction in which n buyers bid for a single,
indivisible object.1 The object’s characteristics are represented by   , where  can
be any finite dimensional space. Only the seller observes  prior to the auction, thus 
is the seller’s private information, or her “type”. The seller’s valuation for the object is
given by s( ) . Buyer j ’s valuation for the object is given by V j  t ( )  X j , where t ( )
is the common value component, and X j is the private value component which is only
observed by buyer j . Ex ante, X j is distributed with c.d.f. F () and support X  [ x, x ] ,
where F '( x )  0 for all x  [ x , x ] . We assume that  and X j ’s are all independent,
which implies that s( ) and X j ’s are all independent.
The above specification of information structure is rather general, including most
common existing models as special cases. For the simplest case,   is onedimensional, and s  t   ; that is, the seller observes her valuation for the object, which
coincides with the buyers’ common value component. An extension of this case is that
  is one-dimensional and s   , but t   where   0 ; that is, the buyers’ common
value component is proportional to the seller’s valuation. In a more general case, the
seller’s private information   is multidimensional, and both her valuation and the
1
By the Revenue Equivalence Theorem (Myerson 1981, Riley and Samuelson 1981), all our
results continue to hold if the object is sold in any other standard auction.
4
buyers’ common value component are stochastic functions of   . Formally, the
seller’s valuation is W ( , ) and the buyers’ common value is V0 ( ,  ) , where  and 
are random variables. Since the seller and the buyers are assumed to be risk neutral, only
their expected valuations matter in the analysis. The seller’s expected valuation
is s( )  E [W ( , )] , and the buyers’ expected common value component is
t ( )  EV0 ( ,  ) . However, since the buyers do not observe  , they have to infer from
the seller’s signal the expectation about the common value component. Upon observing
the seller’s signal and forming a belief that s ( )  s , the expected common value is
 ( s)  E{ , }[V0 ( ,  ) | s( )  s]  E [t ( ) | s( )  s] .
Throughout we will assume the following assumptions:
A1: Diminishing marginal revenue (regularity condition):
J ( x)  x 
1  F ( x)
is strictly increasing in x.
F ( x)
A2: Seller’s valuation s and the buyers’ common value component t are positively
correlated. Thus  ( s)  E[t | s  s] is an increasing function of s .
Note that Assumption A1 holds as long as the hazard rate function of X is increasing
(which is satisfied for almost all the common distributions, e.g., uniform, normal,
exponential, etc.). Assumption A2 is quite natural. A special case is when s and t are
perfectly correlated, t   s ,   0 , so  ( s)   s .
The seller moves first by announcing a reserve price. The buyers then submit
sealed bids. Suppose the buyers, upon observing a reserve price r , believe that the value
of the common value component is tˆ . Then a buyer of type x j has an expected value for
the item of tˆ  x j . Suppose that all the buyers follow their dominant strategies to bid their
valuations tˆ  x j , j  1,..., n . Then the possible outcomes of the auction are as follows:
If tˆ  x(1)  r , the object is not sold, which occurs with probability of F(1) (r  tˆ) ; if
tˆ  x(1)  r and tˆ  x(2)  r , the object is sold at the reserve price, which occurs with
5
probability of F(2) (r  tˆ)  F(1) (r  tˆ) ; if tˆ  x(2)  r , the item is sold at the price tˆ  x(2) .
Here the distributions of the first and second order statistics are given by
F(1) ( x)  Pr{X (1)  x}  F n ( x) and F(2) ( x)  Pr{X (2)  x}  F(1) ( x)  n(1  F ( x)) F n1 ( x) .
The seller’s expected payoff is therefore given by
u  sF(1) (r  tˆ)  r[ F(2) (r  tˆ)  F(1) (r  tˆ)] 
x
ˆ
(tˆ  x)dF(2) ( x )
(2.1)
r t
From the above expression, it is clear that even though the seller’s primitive type is  , all
that matters are her valuation (her true type), s  s( ) , and the buyers’ belief about the
common value component tˆ . In any equilibrium, sellers of different primitive types but a
same actual type s must choose the same signal and hence lead to the same tˆ ; otherwise
those sellers obtaining smaller expected payoff would change to the signal that leads to
higher expected payoff. In equilibrium, for a signaling strategy r ( s ) , the buyers’
perceived common value component is tˆ  E[t |  : r ( s( ))  r ] . With this reformulation,
the seller’s payoff function is in the standard form of signaling models, and is expressed
without direct reference to her private information about the object’s characteristics  .
As a function of the reserve price r , Equation (2.1) is difficult to analyze. In our
next reformulation, we define the seller’s reserve markup m  r  tˆ . Then we can rewrite
the seller’s expected payoff as follows.
u (s, tˆ, m)  sF(1) (m)  tˆ(1  F(1) (m))  B(m)
(2.2)
where
x
B(m)  m( F(2) (m)  F(1) (m))   xdF(2) ( x)
(2.3)
m
Since it will be useful below we note that
B( m)  F(2) ( m)  F(1) (m)  mf (1) (m)   f (1) (m) J (m)
If buyers believe that the seller’s type is ŝ , the expected common value tˆ   (sˆ) .
6
(2.4)
Define U ( s, sˆ, m)  u ( s, ( sˆ), m) , then the seller’s expected payoff can be
expressed in terms of the buyers’ perception about the type, ŝ , instead of their perception
about the common value component,  ( sˆ) .
Differentiating the seller’s expected payoff,
U 2 ( s, sˆ, m)   ( sˆ)(1  F(1) (m))
(2.5)
U3 (s, sˆ, m)  (s   (sˆ)) F(1) (m)  B(m)
 (s   (sˆ)  J (m)) F(1) (m)
(2.6)
Since U 2 is independent of s and U 3 is increasing in s , we have
 dsˆ
 U 
  3   0
s dm U s  U 2 
(2.7)
Therefore the single crossing property holds.
Benchmark: Full Information
If s were directly observable to buyers, their perception ŝ  s , so the seller will
choose her markup m to maximize U ( s, s, m) . Let m* ( s ) be the optimal full information
reserve markup, then by Assumption A1 and Equation (2.6), we have
s   (s )  J ( x ),
 x,
m* ( s)   1
 J ( s   ( s)),
s   (s)  J ( x )
(2.8)
In the perfect correlation case t   s , when   1 , the optimal markup is independent of
s ; when   1 the optimal markup is strictly decreasing in s . The intuition is clear.
With   1 , the bigger is s the smaller is the proportion of the private value component
out of the buyers’ total valuation. Hence the seller’s optimal strategy is to mark up the
reserve price less.
In the uniform case with support [ x , x ] , J (m)  2m  ( x  x) , and we have
7
m* ( s )  Max{ x ,
1
2
( x  x  (  1) s )}
3. Finite Type Case
Two type case
Suppose the object for sale has only two possible types of characteristics: either
“low” ( 1 ), or “high” (  2 ). Correspondingly, the seller’s valuations are s and s , and
s  s(1 )  s( 2 )  s ; the buyers’ common value components are t and t , and
t  t (1 )  t ( 2 )  t . With asymmetric information, a “high” type  2 seller must choose
a markup that is incentive compatible. Thus it cannot be in the preferred set for a type 1
(the shaded region in Figure 3.1).
m* ( s)
I1*
s, s
I 2*
I 2**
s
s
m
x
*
m ( s)
*
m ( s)
R
m ( s)
m
K
Fig: 3-1: Separating Nash Equilibria
8
S
m ( s)
Obviously there is a continuum of separating Nash equilibria, e.g.,
{(m* (s), s), (mS ( s ), s )} as depicted. Among them, there is a Pareto dominant separating
equilibrium (i.e., the Riley outcome), in which the low type chooses her full information
optimal markup while the high type chooses a lowest markup that makes the low type
just indifferent. This outcome is indicated in Figure 3.1 by the pair
{(m* (s), s), (mR ( s ), s )} . Cho and Kreps showed that for the standard two- type
signaling model, the Pareto dominant separating equilibrium is the only equilibrium for
which there is no credible out-of-equilibrium signal, where credibility is defined as
follows:
Intuitive Criterion (Cho and Kreps): Suppose that when some type  makes an out of
equilibrium choice
m K , her type is correctly perceived and, as a result, type  is better
off. If no other type    is better off mimicking type  , the perception of the buyers
is “credible.”
In Figure 3.1, it is easy to see why any Pareto inferior separating equilibrium such
as {(m* (s), s), (mS ( s ), s )} fails the Intuitive Criterion. If the high type  (or
equivalently s ) deviates to
m K from
m S ( s ), and it is correctly perceived by the buyers,
she is clearly better off. This deviation would not be profitable for the low type, thus it is
indeed credible for the high type to deviate, making {(m* (s), s), (mS ( s ), s )} fails the
Intuitive Criterion.
Three type case
While the Cho-Kreps Intuitive Criterion works well in the two-type case, it is well
known that it can be difficult to apply to cases when there are many types, and loses most
of its power in the continuous type case. We now argue that it runs into further problems
in settings like ours where the seller’s valuation is not perfectly correlated with the
buyers’ common value component.
9
Consider the case in which the object has three types of characteristics: 1 , 2 , 3 .
Let ( si , ti )  ( s(i ), t (i )) . Suppose the seller’s valuations and the buyers’ common value
components are given below:
Type Probability
1
2
3
p1
p2
p3
Seller’s
valuation
Common value component
of buyer’s valuation
s
t
s
s
t
t
Given any buyers’ perception about the common value component tˆ , the payoff
function for each type of seller is given by:
u(si , tˆ, m)  si F(1) (m)  tˆ(1  F(1) (m))  B(m)
(3.1)
Note that type 3 has the same common value component as type 1 but also the same
signaling cost as type  2 . This implies that types  2 and 3 have the same indifference
curves in m  tˆ space, that is, they are of the same actual type s . However, if one applies
the Intuitive Criterion to this three type example, it has no power: for any signalperception pair that is strictly preferred by type  2 , it will be preferred by type 3 as
well. Thus, no deviation is credible and so there is a continuum of separating equilibria
that survives the Intuitive Criterion.
Note that this three-type example is observationally equivalent to the two-type
case above because types  2 and 3 are behaviorally identical and can be treated as
equivalent. Formally, this can be seen by defining a grouped type  23 , such that
s( 23 )  s and t ( 23 )  E[t |  { 2 ,3}]  ( p2 t  p3 t ) ( p2  p3 ) . Thus it is highly
unsatisfactory that the Intuitive Criterion selects one equilibrium in the two-type case
while leaves in a continuum of separating equilibria in the observationally equivalent
three-type case. It is natural to seek a refinement that selects the same subset of equilibria
10
in either case. It can be verified that the Cho and Sobel (1990)’s refinement concept of
“divinity”, which is built on the idea of stability of Kohlberg and Mertens (1986) and can
be considered as a logic offspring of the Intuitive Criterion, does not have power either in
the above three-type example. Like the Intuitive Criterion, the divinity faces the same
problem of distinguishing types  2 and 3 to interpret a possible deviation, while these
types have the same incentives to deviate. Such situations are common when  is
multidimensional and s and t are positively correlated.2
The three-type example points out the need to consider deviations not only by a
single type but also by a pool of types. This idea is incorporated in the strengthened
Intuitive Criterion based on Grossman and Perry (1986a,b):
Strengthened Intuitive Criterion (Grossman and Perry): Suppose that when each type
 in a set 0   makes an out of equilibrium choice m , the buyers’ perception is that
the expected common value component is tˆ  E[t ( ) |   o ] and, as a result, each type
  o is better off. If no other type    is better off mimicking and choosing m , the
signal-perception pair (m, tˆ) is “credible.”
2
Riley (2001) discusses in greater details these and other refinement concepts. We will study equilibrium
refinement mostly in the continuous type case. Ramey (1996) extends the Cho and Sobel’s divinity
concept to the case of a continuum of types.
11
I1*
t
I2
t
t 23
Expected
Common value
Component for
Types 2 and 3
t
m
m
m
C
m
K
Fig. 3.2: Three type example
In our simple example it is clear how this can be done. Suppose buyers observe
the out-of-equilibrium signal m C . Knowing types  2 and  3 have identical preferences,
the perception is that the expected common value component is t23  E[t |  { 2 ,3}] .
Given this perception,
u( i , t23 , mC )  u( i , t23 , m K ), i  2,3 .
However u(1 , t23 , mC )  u(1 , t , m) so type 1 has no incentive to mimic. Thus the
Pareto dominated separating equilibrium in which types  2 and  3 choose mK fails the
strengthened Intuitive Criterion (SIC).
However, with the SIC, we may run into the problem of non-existence of
equilibrium as illustrated below:
12
I1*
t
I2
Expected
Common value
Component for
all 3 types
t 23
t123
t
m
m m̂
mS
Fig. 3.3: Separating equilibrium fails the Strengthened Intuitive Criterion
If t123  E[t ( ) |  {1 , 2 ,3}] is sufficiently high, then as depicted above all types would
have an incentive to deviate to an out-of-equilibrium signal m̂ and the pair (mˆ , t123 ) is
“credible” according to Grossman and Perry’s criterion. Thus no separating equilibrium
survives the SIC. Since no pooling equilibrium survives the weaker Cho-Kreps Intuitive
Criterion, no pooling equilibrium survives the SIC either.
We suggest a local credibility test (LCT) which weakens the pooling requirements
of the strengthened Intuitive Criterion. For any deviation, instead of interpreting it as
from any subset of types as in the SIC, the LCT suggests that the signal receivers
interpret it as only coming from one of the nearest types or both, and then check whether
it is credible for them to deviate. For example, in a three-type case with 1 , 2 , 3 , and
13
s1  s2  s3 , consider a separating equilibrium with m( s1 )  m( s2 )  m( s3 ) . To check
whether it satisfies the LCT, suppose there is an out-of-equilibrium signal
mc  (m(s1 ), m(s2 )) . The LCT is satisfied if neither (mc , t1 ) , (mc , t2 ) nor (mc , t23 ) is
credible. In contrast, the SIC requires to check all other possible pools of types (and a
single deviation by 3 ).
Naturally, the next step is to explore under what conditions an equilibrium
satisfies the LCT. Since our primary interest is on the continuous type model, we will
formulate the definition of LCT in the continuous type case and then characterize the
necessary and sufficient conditions for the existence of equilibrium satisfying the LCT.
Before doing that in Section 5, we characterize the Pareto dominant separating
equilibrium for the continuous type model and analyze its properties in Section 4.
4. Continuous Type Case
In the model with a continuum of types, we assume that, induced by the
distribution of  , ex ante s is distributed as c.d.f. G () with support [ s , s ] .
If there exists a separating equilibrium, denoted by the inverse markup schedule
s ( m ) , then it must satisfy the following condition:
s(m)  
U 3 ( s, s, m) ( J (m)   ( s)  s) f (1) (m)

U 2 ( s, s, m )
 ( s)(1  F(1) (m))
(4.1)
That is, given any separating equilibrium schedule, type s seller will optimally choose
reserve markup m according to the solution of (4.1) . This condition merely says that the
slope of the equilibrium schedule should equal the marginal rate of substitution between
the reserve markup and the market perception about the type.
14
s
Indifference curve for type s
z (s)
sˆ  s (mˆ )
MRS (m, s )  
'
U 3 ( s , s , m)
 s(m)
U 2 ( s , s , m)
 (m)
m̂
m
Fig. 4-1: Separating Equilibrium
Following arguments paralleling those in Riley (1979), there is a unique solution
through the full information optimum for the lowest seller type (m* ( s ), s) . Call this
s R (m) . We next show that this solution is incentive comparable and is hence a
separating equilibrium.
Suppose that the buyers’ perception is given by sˆ  s (m) , which is the solution to
(4.1) above. A seller of type s thus chooses m to maximize U ( s, s(m), m) .
Differentiating by m,
d
U ( s, s(m), m)  U 2 ( s, s(m), m) s(m)  U 3 ( s, s (m ), m )
dm
U ( s, s(m), m)
 U 2 (s , s (m ), m )[s(m )  3
]
U 2 ( s, s(m), m)
 U 2 ( s , s (m ), m )[
U 3 ( s( m), s( m), m) U 3 ( s, s(m), m)

]
U 2 ( s(m), s(m), m) U 2 ( s, s(m), m)
By the single crossing property, the terms in the bracket above only changes signs once
and U ( s, s(m), m) takes on its maximum at m where s(m)  s . Therefore we have
incentive compatibility. Letting m  m* ( s ) , we have the following result:
Proposition 1: The solution to the differential equation (4.1):
15
( J (m)   ( s)  s) f (1) (m)
 ( s)(1  F(1) (m))
s(m) 
through the full information optimum for the lowest seller type (m, s) characterizes the
unique Pareto dominant separating equilibrium (the Riley outcome).
Assuming an interior solution for the full information optimum, (4.1) can be
rewritten as
s(m) 
( J (m)  J (m* ( s)) f (1) (m)
 ( s)(1  F(1) (m))
This implies that m( s)  m* ( s) for all s  s .
Due to the reformulations in Section 2, the derivation of our characterization
result Proposition 1 is straightforward. Note that Proposition 1 holds for any increasing
function  () (i.e., positive correlation between s and t ). In the special case of perfect
correlation, tˆ  E[t | s]   ( s)   s , where   0 , (4.1) becomes:
s(m) 
[ J (m)  (1   ) s] f (1) (m)
(4.2)
 (1  F(1) (m))
which can be rewritten as:
(1  F(1) (m))
ds
1
1
 (1  ) f (1) (m) s  f (1) (m) J (m)
dm


Multiplying both sides by (1  F(1) (m))

1

, we have
1
1
1

d
1
[(1  F(1) (m))  s(m)]  f (1) (m)(1  F(1) (m))  J (m) .
dm

Integrating we obtain:
1 1
(1  F(1) (m))

1 1
s(m)  (1  F(1) (m))

s
1
m

m
16
f (1) ( y)(1  F(1) ( y))
1

J ( y)dy
Therefore, the inverse markup schedule in equilibrium can be written as
s(m)  (1  F(1) (m))
 (1 1 )

1 m
1
(1 1 ) 


s
  f (1) ( y)(1  F(1) ( y)) J ( y) dy  (1  F(1) ( m))

 m

(4.3)
When   1 , (4.3) completely characterizes the solution for the separating equilibrium.
As an example, when (i) X is uniform with support [0,1]; (ii) n  2 ; (iii)   1 ;
and (iv) s  0 ; we can integrate (4.3) analytically to obtain
1 m
1 m
s(m)  4(m  m)  3ln(
)  ln(
)
1 m
1 m
where m 
1
.
2
When 0    1 , the equilibrium reserve price schedule may be truncated at some
critical type, because the seller can be better off holding the item unsold as her own
valuation s gets sufficiently large to exceed the equilibrium reserve price. For the seller
to be willing to sell the item through signaling, the reserve price must be greater than s :
r ( s)  m( s)   s  s; or,
m( s)  (1   ) s
(4.4)
Taking this constraint into account explicitly, the equilibrium reserve price schedule can
be characterized more specifically in the case of perfect correlation:
Proposition 2: In the case of perfect correlation such that tˆ  E[t | s]   ( s)   s ,
Equation (4.3) characterizes the solution for the Pareto dominant separating equilibrium
when   1 or when   1 but x  (1   ) s . When   1 and x  (1   ) s , the equilibrium
schedule determined by (4.3) is truncated at (mc  x , s c  x (1   )) ; those types of
x
s [
, s ] will withdraw from the market.
1 
17
Proof: We have proved the proposition for   1 . When   1 but x  (1   ) s , clearly the
constraint of (4.4) does not bind since (1   ) s  (1   ) s  x . Now we consider the case
in which   1 and x  (1   ) s .
Define s c  inf{s : m( s)  (1   ) s} , and mc  m( s c ) . Then
J (mc )  (1   ) s c  mc 
1  F (mc )
1  F (mc )
c

(1


)
s


0
F (mc )
F (mc )
By inspecting (4.2), for s  m  to be an increasing equilibrium schedule, we must have
F (mc )  1 , i.e., mc  x . As a result, the schedule is truncated at s c  x (1   ) .
It remains to verify that this new endpoint condition (implied from the constraint
(4.4)) is satisfied in (4.3). First, as m  x , (1  F(1) (m))
 (1 1 )

(1  F (m))(1 1 ) s   0 .
(1)


Second, applying L’Hopital’s rule, we have
lim(1  F(1) ( m))
m x
m
 lim

m
1

m x
 lim
m x
m
 (1 1 )
1
m



f (1) ( y )(1  F(1) ( y ))
f (1) ( y )(1  F(1) ( y ))
(1  F(1) (m))
1


J ( y ) dy
J ( y )dy
(1 1 )
f (1) (m)(1  F(1) (m))
(  1)(1  F(1) (m))
1
1


1

J ( m)
(  f (1) ( m))
J ( m)
m x 1  
x

1 
 lim
Therefore, taking limits on both sides of equation (4.3), we have
s( x )  lim s(m)  s c  x /(1   ) , which confirms that for 0    1 , the separating
m x
equilibrium is given by (4.3) with a truncation at the second endpoint
(mc  x , s c  x (1   )) . Q.E.D.
18
Proposition 2 gives a complete characterization of the Pareto dominant separating
equilibrium for the case of perfect correlation. Jullien and Mariotti (2003) study a reserve
price signaling model in which the seller’s valuation is  and the buyers’ valuations are
  (1   )i , where   [0,1] . Their setup corresponds to the perfect correlation case in
our model with   1 .
Comparative Statics
We now derive the comparative statics results for the unique Pareto dominant
separating equilibrium. First we have:
Proposition 3: In the separating equilibrium, for every s  s , the markup and hence the
reserve price r ( s )   ( s )  m( s ) is higher for larger n.
Proof: See the Appendix.
This result is intuitive. When there are a larger number of bidders, the signaling
costs from higher reserve prices are smaller because the probability of no sale is lower.
As signaling costs go down, reserve prices will be higher in equilibrium. This result is in
contrast with the well known result that optimal reserve price is independent of n ; when
the signaling role is taken into account, a reserve price will in general depend on the
number of bidders.
The model can be readily extended to situations where the relative importance of
common value and private value components in bidders’ valuations can take on any
arbitrary degree. Let Vi  t   X i where   (0, ) measures the relative importance of
the private value component. Clearly   1 corresponds to our basic model. As before
let m  r  t but now call m  m /  the relative markup. The interesting question in this
case is how the relative markup m changes as  changes.
19
Proposition 4: Suppose  ( s )  s is increasing in s . For any s , the relative markup m is
increasing in  . Consequently, the reserve markup m and hence the reserve price
r ( s )   ( s )  m( s ) are increasing in  at an accelerating rate.
Proof: See the Appendix.
The intuition for this result is the following. When  increases, the private value
component becomes more important while the common value component becomes less
so. When  ( s )  s is increasing in s , and the common value component becomes
smaller, the relative markup for the lowest type actually increases, because the signaling
cost from no sale is relatively small. It can be shown that the relative reserve schedule
follows the same differential equation as before. As a result, a larger  implies a higher
initial condition, thus implies a higher relative markup schedule everywhere. Note that
the special case with t   s,  >1 satisfies the condition that  ( s )  s is increasing in s .3
Outside Certification
We now consider situations where in addition to signaling through reserve prices,
the seller can credibly reveal  to the bidders through an outside certification agency at a
fixed cost of c  0 . The question is when the seller is willing to pay for such a service.
For ease of analysis we consider the special case in which t   s,  >1 . Let
u* ( s)  U ( s, s, m* ( s)) be type s seller’s expected revenue under full information, and let
u( s )  U ( s, s, m( s )) be type s seller’s expected revenue in the separating equilibrium.
Also let W ( s)  u* ( s)  u( s). Immediately, W ( s)  0 and W ( s)  0 for all s .
To further simplify notation, let m*  m* ( s) and m  m( s) . By the Envelope
Theorem, we have
 ( s )  s is decreasing in s , no definite conclusion can be made about whether the relative
markup schedule will move down everywhere as  increases.
3
When
20
dW du* du


ds
ds ds
= F(1) (m* )   (1  F(1) (m* ))  F(1) (m)
  (1  F(1) (m))  (  1)( F(1) ( m)  F(1) ( m* ))
Since m  m* , we have dW / ds  0 . Clearly the seller is willing to pay for the
certification service if W ( s )  c . The following result is immediate.
Proposition 5: For any c  0 , there exists a cutoff type s*  s such that for
all s  [ s* , s ] , the seller hires the outside certification agency; for all s  [ s, s* ) , the seller
signals through reserve price r ( s)   s  m( s).
An Application to the Lemons Market
Even though our analysis so far has focused on auctions, our results can be readily
applied to studying signaling in the Lemons Market. Consider the following market
situation for a good (e.g., used cars). To keep things simple, suppose there is a unit mass
of buyers each with unit demand, and there is also a unit mass of sellers each with one
item to sell. Each seller knows  , the “quality” of the item for sale, which determines the
seller’s own valuation s . Suppose the common value component of the item is given by
 s,  >0 . However, the buyers do not observe the quality of the good, but know that the
population distribution of s (induced by the distribution of  ) is given by c.d.f. G ()
with support S  [ s , s ] . Buyer j ’s valuation for a good with quality  is V j   s  X j
where X j is a private value component only observable to buyer j . The population
distribution of X j is given by c.d.f F () with support [ x , x ] .
What is just described is a continuous type version of the Akerlof’s Lemons
Market model (Akerlof 1970). The fundamental idea of Akerlof’s analysis is that the
price-taking Walrasian equilibrium cannot achieve efficient resource allocation in the
presence of adverse selection problem. In the above model, absent the adverse selection
problem (i.e., if quality is known to the buyers), the first best allocation is easily achieved
by setting a price of s ( ) for the good with quality  . When  is not known to the
21
buyers, for any fixed price p chosen by the Walrasian auctioneer, only those sellers with
valuation s  p are willing to sell their goods, resulting in a total supply of G ( p ) .
Accordingly, the expected common value of the goods in the market is
  E[ s | s  s  p]   E[s | s  s  p] . Since only those buyers with valuation
V    x  p are willing to buy, the total demand is 1  F ( p   ) . For a market-clearing
price, we set 1  F ( p   )  G ( p ) . In general, the equilibrium price that clears the market
leads to less than efficient level of trade. For example, when both F (.) and G (.) are
uniform on [0,1] , the market-clearing price is p  2 /(4   ) if 0    2 , and 1 if  >2 ,
which implies a trade volume of min{2 /(4   ),1} . Trade is efficient only when   2 .
When   2 , equilibrium trade is less than the efficient level and is increasing in  .
The concept of Walrasian equilibrium assumes price-taking behavior on both
sides of the market and that price is public information. In many real life situations such
as the used car market, neither of these assumptions fits: sellers set prices for their goods
and buyers search for what they want. To model these features in the simplest way, we
consider the following situation: the sellers set prices for their goods, and without
knowing the prices in the market, each buyer randomly goes to one seller. In other words,
we consider a situation with pair-wise random matching in which the sellers set prices.
What is the equilibrium outcome in this market?
Observe that in our previous analysis of reserve price signaling in the auction
context, we can reinterpret the single seller with a type s drawn from the distribution
G (.) as a unit mass of sellers with unit supply whose types have a population distribution
of G (.) . Then it should be clear that our previous characterization result Proposition 1
applies to the current pair-wise matching market with n  1 . When n  1 , Equation (4.2)
becomes
ds F (m)[ J (m)  (  1) s ]

dm
 (1  F (m))
Accordingly, Equation (4.3) becomes
s(m)  (1  F (m))
 (1 1 )

1 m
1
(1 1 ) 


s
  F ( y )(1  F ( y )) J ( y )dy  (1  F ( m))
  m

22
(4.5)
where m  m* ( s) . Therefore, this characterizes a separating pricing equilibrium in which
a seller with valuation s chooses a posted price p   s  m( s ) and the buyer correctly
infers the true type s from this price schedule and decides whether to buy at this price.
When F (.) is uniform on [0,1] and s  0 (hence m  0.5 ), the equilibrium markup is
given by
1
 1 
2
 (1 1 )  
2 


1
(2
m

1)

(1

m
)

(1  m)



  ,   0,   1, 1 2
2
2  1 
1   
 

s( m )  1  2m  log(1  m)  log 2,
 1

  12
 2(2m  1)  4(1  m)[log(1  m)  log 2],


 
Again, for the case 0    1 , the equilibrium schedule is given by the above solution with
the understanding that it is truncated at s c  1/(1   ) if 1 (1   )  s .
In this specific example, it can be verified that both the social welfare and volume
of trade are greater in the Walrasian equilibrium than in the signaling equilibrium.4
However, this difference mainly results from the assumption that the searching
technology is extremely primitive and costly in the signaling equilibrium --- only one
round pair-wise matching is allowed --- while on the other hand, the searching cost is
zero in the Walrasian equilibrium. Note also that unlike in the Walrasian equilibrium, the
price schedule in the signaling equilibrium does not depend on the distribution function
G (.) , making the two equilibria more incomparable.
In the application to the Lemons Market, several extensions are desirable and
worth further research. One straightforward extension is to consider markets in which
each seller faces multiple buyers, e.g., housing market. In this case our previous results
directly apply. Another extension is to consider heterogeneous buyer preferences over
quality. For example, it may be reasonable to suppose that buyer j ’s valuation for a good
with quality  is V j  t ( ) Z j  X j where Z j is buyer j ’s preference for quality and is
Our computation results show that the seller’s expected revenue can be higher in the signaling
equilibrium.
4
23
only known to himself. In the preceding example we oversimplified situations by
assuming that each buyer can only sample one seller. It is desirable for future research to
study a more realistic model in which buyers can search more than one period and
perhaps have heterogeneous preferences for quality.
5. Equilibrium Refinement: Local Credibility Test
In this section we study equilibrium refinements for the continuous type case
using the concept of Local Credibility Test. We characterize necessary and sufficient
conditions under which the Pareto dominant separating equilibrium satisfies LCT in a
general signaling model that includes the reserve price signaling model and the wellknown Spence education signaling model as special cases. Then we apply the general
results to these two models.
A General Signaling Model
To begin, let us fix the notation. As before,   denotes the private
information of the sender. Let s() :   R, s  S  [ s , s ] , be the true type of the signal
sender. ŝ is the type of the sender perceived by the signal receiver(s). A signal chosen
by the sender is denoted by y  Y , where Y is the set of feasible signals. Let
z ( ) : S  Y be a strictly monotone signaling function that fully reveals the true type of
the sender. Let U ( s, sˆ, y ) denote the utility of the sender whose true type is s and who
sends out a signal of y and is perceived to be type ŝ . Accordingly, U* ( s)  U ( s, s, z( s))
is the utility of the sender of true type s in the separating equilibrium z ( s ) .
We maintain the following standard assumptions:
(a) U ( s, sˆ, y ) is third order differentiable in all its elements;
(b) U 2 ( s, sˆ, y)  0 ;
(c) The single crossing condition holds:
U U U U
 dsˆ
 U3

  13 2 2 12 3  0
s dy U
s U 2
U2
24
In addition, we make the following technical assumptions:
B1: U12  0 for all ( s, sˆ, y ) ;
B2: U 22  0 for all ( s, sˆ, y ) .
Example 1: The reserve price signaling model
In the model studied in previous sections, the seller’s expected payoff can be
expressed as U ( s, sˆ, y )  sy   ( sˆ)(1  y )  H ( y ) , where we adopt the transformation
y  F(1) (m)  [0,1] and H ( y )  B(m( y )) . Using (2.3) we can see that
 (m) J (m) F(1)
 (m)   J ( m( y ))
H ( y )  B(m) y (m)   F(1)
 ( m)
H ( y )   J (m) F(1)
With this transformation, the derivatives of U ( s, sˆ, y ) are
U1 ( s, sˆ, y )  y, U 2 ( s, sˆ, y )   ( sˆ)(1  y ), U 3 ( s, sˆ, y)  s   ( sˆ)  H ( y)
U11 ( s, sˆ, y )  0, U12 ( s, sˆ, y )  0, U13 ( s, sˆ, y)  1
U 22 ( s, sˆ, y )   ( sˆ)(1  y ), U 23 ( s, sˆ, y)   ( sˆ), U 33 ( s, sˆ, y)  H ( y)
The standard assumptions and B1 are all satisfied. B2 is satisfied when  ( s ) is
linear in s . By the standard results, when U3 ( s, s, z( s))  s   ( s)  H ( z( s))  0 , a
separating equilibrium satisfies
z ( s )  
U 2 ( s, s, z ( s ))
 ( s)(1  z ( s ))

U 3 ( s, s, z ( s )) s   ( s)  H ( z ( s ))
Example 2: The education signaling model
In a common formulation of the Spence education signaling model, a worker’s
expected payoff is U ( s, sˆ, y )  sˆ  C ( s, y ) , where s is the worker’s productivity unknown
to firms, ŝ is the worker’s productivity perceived by firms and hence is also the wage
offered to her by competing firms, and y is the education signal the worker can choose. It
is typically assumed that for all ( s, y ) (i) C1( s, y )  0 ; (ii) C2 ( s, y )  0 ; and (iii)
C12 ( s, y )  0 . The derivatives of U ( s, sˆ, y ) are
25
U1 ( s, sˆ, y )  C1 ( s, y ), U 2 ( s, sˆ, y)  1, U 3 ( s, sˆ, y)  C2 ( s, y)
U11 ( s, sˆ, y )  C11 ( s, y ), U12 ( s, sˆ, y)  0, U13 ( s, sˆ, y)  C12 ( s, y)
U 22 ( s, sˆ, y )  U 23 ( s, sˆ, y )  0, U 33 ( s, sˆ, y)  C22 ( s, y)
The standard assumptions and B1 and B2 are all satisfied. By the standard results,
and since by assumption U3 (s, sˆ, y)  C2 ( s, y)  0 for all ( s, sˆ, y ) , a separating
equilibrium satisfies
z ( s )  
U 2 ( s, s, z ( s ))
1

U 3 ( s, s, z ( s )) C2 ( s, z ( s ))
Local Credibility Test (LCT):
In Section 3, we proposed a refinement concept LCT for the finite type case. The
idea is to weaken the pooling requirement of the Grossman and Perry SIC by interpreting
a deviating signal as from a single type or a set of types that are “nearby” the signal. The
concept can be easily extended to the case of continuous types.
Local Credibility Test: Consider any separating equilibrium z ( s ) : S  [ z , z ]  Y .
Consider any signal ŷ  Y .
1. When ŷ  z , ( yˆ , s ) is a credible deviation if U ( s , s , z )  U ( s , s , yˆ ) .
2. When ŷ  z , ( yˆ , s ) is a credible deviation if U ( s , s , z )  U ( s , s , yˆ ) .
3. When yˆ  [ z , z ] , let z ( s0 )  yˆ and consider a small neighborhood of s0 ,
So  S . Let sˆ  E[s | s  S0 ] . If there exists   0 such that
(i) U (s, sˆ, yˆ )  U ( s, s, z( s))   , for all s  int So
(ii) U ( s, sˆ, yˆ )  U ( s, s, z ( s))   , for all s  So
then the out-of-equilibrium signal-perception ( yˆ , sˆ) is credible.
The above definition of LCT captures the idea that the receivers interpret
(potentially) deviations as from the nearby types. In particular, Part 1 says that any out26
of-equilibrium deviation below the lowest equilibrium signal is perceived as from the
lowest type. This implies
Lemma 2: If a separating equilibrium satisfies LCT, it is the Pareto dominant
equilibrium (the Riley outcome), that is, z ( s )  y* ( s ) , where y* ( s ) maximizes U ( s, s, y ) .
Proof: If z ( s )  y* ( s ) , then the out-of-equilibrium signal-perception pair ( y* ( s ), s ) is
credible, violating Part 1 of the LCT requirement. If z ( s )  y* ( s ) , by the single crossing
condition, the lowest type would want to deviate to y* ( s ) and be perceived as a higher
type, violating equilibrium condition. Q.E.D.
Part 2 of the LCT definition says that any out-of-equilibrium deviation above the
highest equilibrium signal is perceived as from the highest type. This credibility
requirement is also satisfied automatically by the Pareto dominant separating equilibrium.
Therefore, to check whether there exists an equilibrium satisfying LCT, we only need to
check whether the unique Pareto dominant separating equilibrium satisfies Part 3 of the
LCT definition. Consider any “on-equilibrium” signal, and the type of sender for this
signal in equilibrium. Suppose the nearby types all deviate to this signal, and this is
correctly perceived by the receivers, and all the deviating types can gain at least

relative to their equilibrium payoffs while all other types cannot. Then those nearby types
can credibly deviate to the particular “on-equilibrium” signal by throwing away

amount of money.5
The following result gives an equivalent, but more operational, requirement for
Part 3 of the LCT definition.
Proposition 6: Suppose that the single-crossing property holds.. For yˆ , S , sˆ as defined
0
in Part 3 of the LCT definition, if there is an s  sˆ such that
With a finite type space and continuous signal space Y , the set of signals that are not selected in
a separating equilibrium is dense in Y . Thus there is no need to signal by throwing money away.
5
27
(i ) U ( s, s, yˆ )  U ( s, s, z( s)), for all s  int So .
(ii ) U ( s, s, yˆ )  U ( s, s, z( s)), for all s  So .
then the out-of-equilibrium signal-perception ( yˆ , sˆ) is credible.
By Proposition 6, to check whether the Pareto dominant separating equilibrium
satisfies LCT, we only need to check whether there are any credible interior deviations
and any credible boundary deviations, in the sense that will be made precise below. The
equilibrium survives LCT if and only if there is no credible deviation of either kind.
Local Credibility Test: Interior Deviations
For any two types s  s , suppose those in the interval [ s, s] pool at a certain
signal y . Let v ( s, s) be the expected type of this pool, that is,
s
v ( s, s)  1 [G ( s)  G ( s )]   zdG ( z ) , where G (.) is the c.d.f. of s . Let v ( s, s)  [ s, s]
s
and y ( s, s)  Y be a solution to
U ( s, v, y )  U ( s, s, z ( s))

U ( s, v, y )  U ( s, s, z ( s))
(5.1)
The point ( y ( s, s), v ( s, s)) is depicted below in Figure 5.1. Given this signal-perception
pair, all those types in ( s, s) prefer the pool to their separating equilibrium payoff.
28
I*
s
I*
s
s
z (s)
( y, v( s, s))
( z ( s), s)
( z ( s ), s )
z
Fig. 5.1: Pool of types in [ s, s]
From Proposition 6, in order for the separating equilibrium characterized by z ( s )
to satisfy LCT, it must be that for s close to s , any such signal-perception pair of
( y ( s, s), v ( s, s)) is not credible. That is, for any s [s, s ) and s  s , v ( s, s)  v ( s, s) as
s  s . Note that for any s [s, s ) , v ( s, s)  v( s, s)  s . Thus, if v2 ( s, s)  v2 ( s, s)  0 as
s  s , then v ( s, s)  v ( s, s) as s  s .
Lemma 3: For any s [s, s ) , (i) v2 ( s, s)  1/ 2 ; (ii) v22 ( s, s) 
1 G ''( s)
.
6 G '( s)
Proof: See the Appendix.
Lemma 4: For any s such that U 3 ( s, s, z ( s))  0 , (i) under Assumption B1,
v2 ( s, s)  1 2 ; (ii) under Assumptions B1-B2,
v22 ( s, s) 
1 U 113 1  4U13  2U 23 U133 
1 U33
 

z '( s) 
[ z '( s)]2

6 U13 12 
U2
U13 
12 U 2
where all functions are evaluated at ( s, s, z ( s )) .
Proof: See the Appendix.
29
The proposition below follows immediately from Lemmas 3 and 4.
Proposition 7: For any s such that U 3 ( s, s, z ( s))  0 , the separating equilibrium
characterized by z ( s ) does not have credible local interior deviations as defined in LCT
if and only if
U 113 1  4U13  2U 23 U133 
1 U 33
G( s)
 

z( s) 
[ z( s)]2 

U13 2 
U2
U13 
2 U2
G( s)
where z( s )  
(5.2)
U 2 ( s, s, z ( s ))
.
U 3 ( s, s, z ( s ))
Proof: We know that v ( s, s)  v( s, s)  0 . From Lemmas 3 and 4, we have
v2 (s, s)  v2 (s, s)  0 . If the condition of the proposition is satisfied, then
v22 ( s, s)  v22 ( s, s) . Thus, for a neighborhood around s , it must be v2 ( s, s)  v2 ( s, s) ,
and so v ( s, s)  v ( s, s) . Q.E.D.
It is easier to understand the intuition for Proposition from the finite type case.
Consider the example depicted in Figure 3.3. Since the marginal rate of substitution
between m and t is similar for the different types in that example, the indifference maps
are similar and so indifference curves are close together. As a result, all types are better
off if buyers believe that all may be choosing to deviate, thus violating the requirement of
no credible deviation. However if the marginal rate of substitution declines sufficiently
rapidly with type, the difference in the slopes of the two indifference maps is greater as
depicted below. Now only type 1 is better off if the buyers think that all types may be
deviating. This belief is therefore no longer credible.
30
I1*
t
I2
Expected
Common value
Component for
all 3 types
t 23
t123
t
m
m m̂
mS
Fig. 5.2: Separating equilibrium satisfies the Local Credibility Test
This intuition is reflected in Equation (5.2). In the case of a continuum of types,
the slope of the indifference map is given by MRS ( s, s, z ( s )) 
First note that
U 3 ( s, s, z ( s ))
.
U 2 ( s, s, z ( s ))
U
U 23


by
MRS (s, s, z (s))  13 by B1 and
MRS (s, s, z (s))

s
U
U

s
ss
2
2
ss
B2. Thus the larger is U13 and U 23 the more rapidly the MRS declines. Moreover,

U 33U 2  U 3U 23
MRS ( s, s, z ) 
, hence the larger is U 23 and U 33 the more rapidly the
z
(U 2 )2
MRS declines as z increases. Clearly, the inequality of (5.2) is easier to be satisfied for
larger U13 , U 23 and U 33 . Intuitively, the rate at which the marginal rate of substitution
declines with s is a measure of signaling effectiveness. Thus Proposition 7 suggests that
when signaling effectiveness is sufficiently large, the separating equilibrium will survive
31
the LCT test. Figuratively, when the indifference curve I 2 is far from I1* in Figure 5.2,
there will be no credible deviation with the perception at t123 . The right hand side of
Equation (5.2) is the concavity of the distribution function of s , G ( s ) , normalized by its
density function. Intuitively, the more concave G ( s ) is (i.e., the smaller G is), the more
probability mass on smaller s in any set of types, thus the smaller the expected value of
any set of types. Consequently, the smaller G is, the less likely a deviation is credible.
Figuratively, when the expected value of t for all three types, t123 , is lower in Figure 5.2,
there will be no credible deviation.
Below we apply Proposition 7 to the two examples introduced above.
Example 1 (continued): The reserve price signaling model
We now analyze when the separating equilibrium characterized in Proposition 1
satisfies the condition of Proposition 7. We focus on the special case t   s,   1 . First
consider (1   )s  J ( x ) so that the separating equilibrium goes through m* ( s )  x at s .
Since U3 (s, s, z (s))  (1   ) s  J ( z ( s)) is decreasing in s ,
U3 ( s, s, z ( s))  (1   ) s  J ( x)  0 for all s [s, s ] .
In this model, since U113 ( s, v, y )  U133 ( s, v, y )  0 , (5.2) becomes
1
G( s)
,
[(4  2 ) z( s)  H ( z( s))( z( s)) 2 ] 
2 (1  z( s))
G( s)
where z( s)  
 (1  z ( s))
. When this condition holds, the separating
(1   ) s  H ( z ( s ))
equilibrium characterized in Proposition 1 does not allow credible interior deviations.
For the case (1   )s  J ( x) , then U 3 ( s, s, z( s))  0 , so Proposition 7 does not
apply. We will consider this case later.
Example 2 (continued): The education signaling model
32
In the common formulation of the model, U3 ( s, v, y )  C2 ( s, y )  0 for all
( s, v, y ) . From the derivatives of U ( s, v, y ) derived before, we have
U113 ( s, v, y )  C112 ( s, y ) and U133 ( s, v, y )  C122 ( s, y ) . Condition (5.2) becomes
C 112
C
1C
C  1 G "( s)
 2 12   22  122 

C12
C2 2  C2 C12  C2 G '( s)
where all functions are evaluated at ( s, z ( s )) . If the above condition holds, then the
separating equilibrium given by z( s )  
U 2 ( s, s, z ( s ))
1

and goes through
U 3 ( s, s, z ( s )) C2 ( s, z ( s ))
z( s)  0 does not allow credible interior deviations.
Local Credibility Test: Boundary Deviations
Finally, for the separating equilibrium characterized by z ( s ) to satisfy the LCT in
the general model, we need to consider the following kind of boundary deviations, in
addition to the interior deviations represented by (5.1). For any type s  s , suppose
those in the interval [s, s] all choose the signal y  z ( s ) , the equilibrium signal by s . Let
s
1
 zdG( z ) . Let
v ( s, s) be the expected type of this pool, that is, v ( s, s) 
G( s) s
v( s)  [s, s] be a solution to
U (s, v, y)  U (s, s, z (s '))
(5.3)
In order for the separating equilibrium characterized by z ( s ) to satisfy the LCT, it
must be that for s close to s , any such signal-perception pair of ( y, v ( s, s)) is not
credible. That is, v ( s , s)  v ( s) as s  s . Note that v ( s, s)  v( s)  s .
From (5.3), total differentiating gives
33
v( s) 
U1 ( s, s, z ( s))  U1 ( s, v, y )
U 2 ( s, v, y )
As s  s , v  s and z ( s)  y . Therefore, v( s)  0 . However, it is easy to show that
as s  s , v2 ( s, s)  0.5 . So in the neighborhood of s , v ( s, s)  v( s) . Therefore, there
is always a credible boundary deviation at the lowest signal y  z ( s ) .
The above “lower endpoint” problem can be overcome if we modify the model so
that given the signaling schedule z ( s ) , the sender of some lowest types does not actively
participate in the market because of some participation costs.
Proposition 8: There is no credible boundary deviation if there is a sufficiently large
c
subset of types [ s , s ] which do not signal in the separating equilibrium.
Let us suppose that due to some participation costs, the lowest type who
participates in the market is s c  s , so the sender of all types s  s c stays out of the
market. Now suppose for some s  sc those in the interval [ s c , s] all choose the
signal y c  z ( s c ) , the equilibrium signal by s c . If the signal receiver perceives their
expected type as higher than sc and thus pays them accordingly, then all those lowest
types s  s c will now find it profitable to participate in the market and join the pool of
[ s c , s] . But when the (probability or population) mass of types [ s c , s] is large, the
correct perception of the expanded pool for those who choose signal y c  z ( s c ) will be
smaller than sc , making it unattractive for types of [ s c , s] to deviate.
Let us consider the two specific examples studied earlier. In the auction model,
suppose the seller needs to invest a fixed cost of c to run the auction. With the
investment, the seller’s expected payoff from running the auction is
s
u( s)  c  u ( s)   F(1) (m(t ))dt  c;
*
s
34
(5.4)
while if the seller does not invest, her payoff is 0. Since the payoff in (5.4) is strictly
increasing in s , there is a unique cutoff type sc so that the seller of types smaller than
sc will not be willing to invest and participate in the game, while the seller of types
greater than sc will pay the investment cost and choose reserve prices according to m ( s ) .
When the auction cost c is sufficiently large, the mass of the types excluded [ s, s c ] will
be sufficiently large so that there is no credible boundary deviation. Moreover, when s c
is sufficiently large, then (1   ) s c  J ( x)  0 , hence U 3 ( s, s, z ( s))  0 for all s  [ s c , s ] .
Therefore, in the reserve price signaling model, as long as s c is sufficiently large, the
LCT only requires (5.2) for s  [ s c , s ] .
The analysis of the standard education signaling model is very similar. Introduce
a reservation wage function (alternative job opportunity) w0 ( s ) such that for sufficiently
small s , w0 ( s )  s . Suppose that the worker of types smaller than sc will not be willing
to participate in this market, since the highest payoff she can get in a separating
equilibrium is s  w0 ( s ) . By monotonicity, the worker of types greater than sc will
participate in this labor market and choose education according to z ( s ) starting
from z ( s c )  0 . When the outside opportunity sc is sufficiently large, the mass of the
types excluded [ s, s c ] will be sufficiently large so that the lower endpoint problem does
not arise.
In summary, when the condition of Proposition 8 holds, in order to check whether
a separating equilibrium satisfies the LCT, we only need to check whether (5.2) is
satisfied for s  [ s c , s ] .
6. Concluding Remarks
In this paper we consider auctions in which the seller’s valuation is correlated
with a common value component of each buyer’s valuation. Only the seller knows her
valuation and the common value component is not directly observable to anyone. We
characterize the unique Pareto dominant separating equilibrium in which a higher reserve
price set by the seller is a signal of her greater valuation.
35
Except in the special case of perfect correlation, standard refinements (Intuitive
Criterion, Divinity, Stability) are not applicable. We argue that to have any “bite” at all,
a refinement is needed in which the signal receivers take into account the way sender
types are distributed. We then propose a Local Credibility Test which is milder than, but
in the spirit of the Grossman-Perry Criterion. Only the Pareto dominant separating
equilibrium ever satisfies the LCT.
For a class of models which includes our model and the basic Spence model, we
provide necessary and sufficient conditions for this equilibrium to satisfy the LCT. These
conditions are the more likely to be met, (a) the less rapidly the density increases or the
more rapidly the density decreases with type, and (b) the more rapidly the marginal cost
of signaling decreases with type.
What is the “right” equilibrium when our conditions are not met? This is a
challenging question for which we have no satisfactory answer. However we conjecture
that pooling or partial pooling must be a part of any more complete analysis of signaling.
To make the point as starkly as possible, consider a standard 2 type case and let p be the
probability that a type is “bad.” Suppose this type has a signaling cost that is only
marginally higher than the cost for the “good” type. Then independently of p , in the
selected separating equilibrium the “good” type must undertake highly costly signaling.
With p  0 the good type does not have to signal at all. Thus the separating equilibrium
has an extreme discontinuity at p  0 . When p is close to zero, the pooling outcome
seems more reasonable than the highly inefficient separating equilibrium. That is,
“reasonable” out-of-equilibrium beliefs do not necessarily lead to reasonable outcomes.
36
Appendix
Proposition 3: In the separating equilibrium, for every s  s , the markup and hence the
reserve price r ( s )   ( s )  m( s ) is higher for larger n.
Proof: Rewrite (4.1) as the following:
 ( s)(1  F(1) (m))
dm
 b(m, s; ) 
ds
( J (m)   ( s)  s) f (1) (m)
(6.1)
where  is a parameter in the model (e.g.,  , n etc.). First we prove
Lemma 1: Suppose (i) b(m, s; ) /   0 for all ( m, s ) and (ii) m( s; ) is decreasing in
 , where m( s; )  m* ( s; ) . Then the solution to equation (6.1), m ( s;  ) is decreasing in
 , or m( s; ) /   0 for all s  s .
Proof: Differentiating (6.1) with respect to  gives
 m b

 0 by (i). So
s  
m( s;  ) /  is decreasing in s . By (ii) we have m( s;  ) /   0 for all s  s . Q.E.D.
Now to show Proposition 3, take   n . First note that the initial point (m, s) is
independent of n. In view of Lemma 1, it remains to show that b( m, s; n ) is increasing in
n, which is equivalent to showing that  (m, n) is decreasing in n where
 (m, n)  ln f(1) (m)  ln(1  F(1) (m))  ln n  (n 1) ln f (m)  ln(1  F n (m)) .
We have
 (m, n) 1
 F n (m) ln F (m)
  ln F (m) 
n
n
1  F n ( m)

1  F n (m)  n ln F (m)
n(1  F n (m))
37
Let  (m, n)  1  F n (m)  n ln F (m) . For any n,  (m, n)  0 at m  x . Furthermore, for
all m  x ,

nf (m)
 (m, n)  nF n1 (m) f (m) 
m
F ( m)

nf (m)
(1  F n (m))  0 .
F (m)
So it must be that  (m, n)  0 for all m  x and for all n. Therefore  (m, n) is
decreasing in n.
Q.E.D.
Proposition 4: Suppose  ( s )  s is increasing in s . For any s , the relative markup m is
increasing in  . Consequently, the reserve markup m and hence the reserve price
r   ( s )  m( s ) are increasing in  at an accelerating rate.
Proof: Let s  s /  and t  t /  , then the expected revenue to the seller with “relative
type” s , “relative perception” t and “relative markup” m is as follows:
u ( s , t , m)  sF(1) (m /  )  (m  t )( F(2) (m /  )  F(1) (m /  )) 
x

(t   x)dF(2) ( x)
m/ 
x
  [ sF(1) ( m)  ( m  t )( F(2) ( m)  F(1) ( m))   (t  x) dF(2) ( x)]
(6.2)
m
  u( s , t , m)
where u (, , ) is defined in (2.2).
So the problem can be viewed as a normalization from our basic model, and all
the analysis follows as before immediately. In particular, the differential equation (4.1)
(with the normalized variables) characterizes the separating equilibrium.
The only issue is how the initial condition for the differential equation is affected.
Notice that the lowest normalized type is now s  s /  . Under full information, by
Equation (2.8), we have
s   ( s)  J ( x)
 x,
m* ( s)   1
 J ( s   ( s)),
s   ( s)  J ( x)
38
(6.3)
Clearly, the full information relative markup m* ( s ) is increasing in s   ( s ) . When
s   ( s ) is decreasing in s (e.g., when  ( s)   s  s ), then m* ( s ) is decreasing in s
and hence increasing in  . Therefore, when  is larger, s is smaller but m* ( s ) is
greater. As a result, the equilibrium relative reserve schedule m () is higher everywhere.
Since the reserve markup is m   m , it is increasing in  at an accelerating rate.
Q.E.D.
Proposition 6: Suppose that the single-crossing property holds.. For yˆ, S0 , sˆ as defined
in Part 3 of the LCT definition, if there is an s  sˆ such that
(i ) U ( s, s, yˆ )  U ( s, s, z( s)), for all s  int So .
(ii ) U ( s, s, yˆ )  U ( s, s, z( s)), for all s  So .
then the out-of-equilibrium signal-perception ( yˆ , sˆ) is credible.
Proof: Given the single crossing property, if there is such a subset S 0 , it must be an
interval [ s1, s2 ] . This is depicted in Figure A.1 below.
I 2*
I1*
s
z (s)
( yˆ , sˆ)
( z ( s2 ), s2 )
( z ( s1 ), s1 )
z
Fig. A.1: Types s  [ s1 , s2 ] pooling
39
Suppose that s  s1  s2  s . For (i) and (ii) to hold it must be the case that
U ( s1, s1, z( s1 ))  U ( s1, s, yˆ ) and U ( s2 , s2 , z ( s2 ))  U ( s2 , s, yˆ ) .
Since s  sˆ  E{s | s [s1, s2 ]} , we have
U (si , si , z (si ))  U (si , sˆ, yˆ )  [U ( si , sˆ, yˆ )  U (si , s, yˆ )]
sˆ
 U ( si , sˆ, yˆ )  
s
sˆ
Define  ( s )  
s

U ( si , t , yˆ )dt , i  1, 2 .
t
(6.5)

U ( s, t , yˆ )dt . By hypothesis, U12 (s, sˆ, y)  0 , hence we may write
t
 ( s)  ˆ . From (6.5),
U (si , si , z (si ))  U (si , sˆ, yˆ )  ˆ, i  1, 2 .
By single crossing, for s  ( s1, s2 )
U ( s, s, z ( s))  U ( s, s, yˆ )  U ( s, sˆ, yˆ )  [U ( s, sˆ, yˆ )  U ( s, s , yˆ )]
sˆ
 U ( s, sˆ, yˆ )  
s

U ( s, t , yˆ )dt  U ( s, sˆ, yˆ )  ˆ
t
Thus
U (s, s, z (s))  U (s, sˆ, yˆ )  ˆ, if s  ( s1, s2 ) .
Also by single crossing, for s  [ s1, s2 ]
U ( s, s, z ( s))  U ( s, s, yˆ )  U ( s, sˆ, yˆ )  [U ( s, sˆ, yˆ )  U ( s, s, yˆ )]
sˆ
 U ( s, sˆ, yˆ )  
s

U ( s, t , yˆ )dt  U ( s, sˆ, yˆ )  ˆ
t
Thus conditions (i) – (ii) in Part 3 of the definition of LCT are satisfied.
The proof when S0  [ s , s1 ] or S0  [ s2 , s ] is almost identical. Q.E.D.
Lemma 3: For any s [s, s ) , (i) v2 ( s, s)  1/ 2 ; (ii) v22 ( s, s) 
40
1 G ''( s)
.
6 G '( s)
s
Proof: By definition, v ( s, s)  1 [G ( s)  G ( s )]   zdG ( z ) . Multiplying both sides by
s
G ( s)  G ( s ) and then differentiating by s , we have
v2 ( s, s)(G( s)  G( s))  v ( s, s)G( s)  sG( s) .
Differentiating by s again,
v22 ( s, s)(G( s)  G( s))  2v2 ( s, s)G( s ')  v ( s, s)G( s)  G( s)  sG( s)
(6.6)
Setting s  s , it follows immediately that v2 ( s, s)  1/ 2 .
Differentiating (6.6) by s again,
v222 ( s, s)(G( s)  G( s))  3v22 ( s, s)G( s)  3v2 ( s, s)G( s)  v ( s, s)G( s)  2G( s)  sG( s)
Since v ( s, s )  s and v2 ( s, s)  1/ 2 , setting s  s we obtain v22 ( s, s ) 
1 G( s )
. Q.E.D.
6 G( s )
Lemma 4: For any s such that U 3 ( s, s, z ( s))  0 , (i) under Assumption
B1, v2 ( s, s)  1/ 2 ; (ii) under Assumptions B1-B2,
v22 ( s, s) 
1 U 113 1  4U13  2U 23 U133 
1 U33
 

z( s) 
[ z( s)]2

6 U13 12 
U2
U13 
12 U 2
where all functions are evaluated at ( s, s, z ( s )) .
Proof: Total differentiating (5.1) gives
U1 ( s, v, y )ds  U 2 ( s, v, y )dv  U 3 ( s, v, y )dy  U1 ( s, s, z ( s))ds
U1 ( s, v, y )ds  U 2 ( s, v, y )dv  U 3 ( s, v, y )dy  U1 ( s, s, z ( s))ds
Solving the equations, we have
41
dv 
11

ds  12 ds;


dy 
  U 2 ( s, v, y )U 3 ( s, v, y )  U 2 ( s, v, y )U 3 ( s, v, y )
11  U 3 ( s, v, y )[U1 ( s, s, z ( s))  U1 ( s, v, y )],
 21  U 2 ( s, v, y )[U1 ( s, s, z ( s))  U1 ( s, v, y )],
 21

ds  22 ds


12  U 3 ( s, v, y )[U1 ( s, s, z( s))  U1( s, v, y )]
 22  U 2 ( s, v, y )[U1 ( s, s, z ( s ))  U1 ( s, v, y )]
Under Assumption B1, we have
dy  21 U 1 ( s, v, y )  U1 ( s, s, z ( s))


ds 
U 3 ( s, v, y )  U 3 ( s, v, y )
(6.7)
Fix any s, as s  s, it must be that v  s, z ( s)  z ( s ), and y  z ( s ) . For the
simplicity of notation, write v( s)  v2 ( s, s) and y( s)  y2 ( s, s) . Applying the
I’Hopital’s rule, as s  s, we get
dy
ds s s
U 11 ( s, v, y )  U12 ( s, v, y )v ( s )  U13 ( s , v, y ) y ( s )  U11 ( s , s , z ( s ))  U 12 ( s , s , z ( s ))  U 13 ( s , s , z ( s )) z ( s )
U 23 ( s, v, y )v ( s)  U 33 ( s, v, y ) y ( s)  U 23 ( s , v, y )v ( s )  U 33 ( s , v, y ) y ( s )  U 13 ( s , v, y )
U ( s, v, y ) y ( s)  U13 ( s, s, z ( s )) z ( s)
= lim 13
s  s
U13 ( s, v, y )
 lim
s  s
 z ( s ) 
dy
ds s s
Hence as s  s,
dy
 0.5z( s ) as long as z( s)  U 2 ( s, s, z( s)) U 3 ( s, s, z( s)) is defined
ds
at s , or U 3 ( s, s, z ( s))  0 at s .
Since
dv 11 11 dy
U ( s, v, y ) dy


 3
ds 
 21 ds
U 2 ( s, v, y ) ds
we have
dv
U ( s, v, y ) dy
U ( s, s, z ( s ))
dy
 lim  3
 3
lim


s

s
s

s
ds s s
U 2 ( s, v, y ) ds
U 2 ( s, s, z ( s ))
ds
=  0.5
U 3 ( s, s, z ( s ))
z( s)  0.5
U 2 ( s, s, z ( s ))
42
(6.8)
for any s such that U 3 ( s, s, z ( s))  0 . This proves part (i).
For part (ii), first note that from z( s)  U 2 ( s, s, z( s)) U3 ( s, s, z( s)) ,
z( s )  
U 22
U  2U 23  U 33 z ( s )
 z( s ) 13
U3
U3
From (6.7), and by Assumption B1, we have
dy
 U11 ( s, s, z ( s))  U13 ( s, s, z ( s)) z ( s)
ds
U 3 ( s, v, y )  U 3 ( s, v, y )
U 11 ( s, v, y )  U13 ( s, v, y )
2
d y

ds2



U 33 ( s, v, y )
dy
dy
 U 33 ( s, v, y )
 U13 ( s, v, y )
dy
ds
ds
U 3 ( s, v, y )  U 3 ( s, v, y )
ds
U 11 ( s, v, y )  U11 ( s, s, z ( s)) U 33 ( s, v, y )  U 33 ( s, v, y )  dy 



U 3 ( s, v, y )  U 3 ( s, v, y )
U 3 ( s, v, y )  U 3 ( s, v, y )  ds 
2
dy
 U13 ( s, s, z( s)) z( s)
ds
U 3 ( s, v, y )  U 3 ( s, v, y )
2U13 ( s, v, y )
(6.9)
Let Li ( s, s) be the ith term on the right hand side of the above equation. For any
s such that U 3 ( s, s, z ( s))  0 , it can be checked that
lim L1 
s s
1 U 113 ( s, s, z ( s )) z ( s )
2 U13 ( s, s, z ( s ))
lim L2  
s s
1 U 133 ( s, s, z ( s ))
[ z( s )]2
4 U13 ( s, s, z ( s ))
lim L3  z( s )  2
s s
d2y
ds2
 0.5
s s
U133 ( s, s, z ( s ))[ z ( s )]2
U13 ( s, s, z ( s ))
Therefore,
6
d2y
ds2
 2 z( s) 
ss
U 113 ( s, s, z ( s))  0.5U133 ( s, s, z ( s )) z ( s )
z( s)
U13 ( s, s, z ( s ))
From (6.8), and using Assumption B2, we have
43
d 2v
U 3 ( s, v, y ) d 2 y U 33 ( s, v, y ) 2U 3 ( s, v, y )U 23 ( s, v, y )   dy 




  ds 
ds2
U 2 ( s, v, y ) ds2  U 2 ( s, v, y )
U 22 ( s, v, y )
 
2
(6.10)
As s  s, we know that
d 2v
ds2

s s
U 3 ( s, s, z ( s )) d 2 y
U 2 ( s, s, z ( s )) ds2
dv
dy
 0.5 and U 3 ( s, v, y )
 0.5U 3 z ( s )  0.5U 2 . So,
ds
ds
 0.5
s s
U 23 ( s, s, z ( s ))  0.5U 33 ( s, s, z ( s )) z ( s )
z( s )
U 2 ( s, s, z ( s ))
2
d y
ds2 ss
U  0.5U 33 z( s )

 0.5 23
z( s)
z( s )
U2
1 U  2U 23  U 33 z( s ) 1 U 113 0.5U133 z ( s )
U  0.5U 33 z ( s )
 [ 13
]
 0.5 23
z( s )
3
U3
6
U13
U2
1 U  2U 23  U 33 z( s )
1 U 113 0.5U133 z ( s )
U  0.5U 33 z ( s )
 [ 13
z( s)] 
 0.5 23
z( s)
3
U2
6
U13
U2

1 U 113 1 4U13  2U 23 U133
1 U 33
 [

]z( s) 
[ z ( s)]2
6 U13 12
U2
U13
12 U 2
This proves part (ii).
Q.E.D.
44
References
Akerlof, George (1970), “The Market for “Lemons”: Quality Uncertainty and the Market
Mechanism,” Quarterly Journal of Economics, 84, 488-500.
Cho, In-Koo and Kreps, David M. (1987), “Signaling Games and Stable Equilibria,”
Quarterly Journal of Economics, 102, 179-221.
Cho, In-Koo and Sobel, Joel (1990), “Strategic Stability and Uniqueness in Signaling
Games,” Journal of Economic Theory, 50, 381-413.
Grossman, Sanford and Perry, Motty (1986a), “Sequential bargaining under Asymmetric
Information,” Journal of Economic Theory, 39, 120-154.
Grossman, Sanford J. and Perry, Motty (1986b), “Perfect Sequential Equilibrium,”
Journal of Economic Theory, 39, 97-119.
Jullien, B. and Mariotti, T. (2003), “Auction and the Informed Seller Problem,”
University of Toulouse Working Paper.
Kohlberg, Elon and Mertens, Jean-Franqis (1986), “On the Strategic Stability of
Equilibria,” Econometrica, 54, 1003-1037.
Milgrom, Paul and Robert Weber (1982), “A theory of Auctions and Competitive
Bidding,” Econometrica, 50, 1082-1122.
Myerson, Roger B. (1981), “Optimal Auction Design,” Mathematics of Operations
Research, 6, 58-73.
45
Ramey, Garey (1996), “D1 Signaling Equilibria with Multiple Signals and a Continuum
of Types,” Journal of Economic Theory, 69, 508-531.
Riley, John (1975), “Competitive Signaling,” Journal of Economic Theory, 10, 174-186.
Riley, John G. (1979), “Informational Equilibrium,” Econometrica, 47, 331-359.
Riley, John G. (2001), “Silver Signals: 25 years of Screening and Signaling,” Journal of
Economic Literature, 39, 432-478.
Riley, John G. “Weak and Strong Signals” (2002) Scandinavian Journal of Economics,
104, 213-236.
Riley, John G. and William F. Samuelson, (1981), “Optimal Auctions,” American
Economic Review, 71, 381-392.
Rothschild, Michael and Stiglitz, Joseph (1976), “Equilibrium in Competitive Insurance
Markets: An Essay on the Economics of Imperfect Information,” Quarterly Journal of
Economics, 90, 629-649.
Spence, A. Michael (1973), “Job Market Signaling,” Quarterly Journal of Economics,
87, 355-379.
46