Prejudice and Competitive Signaling
Sue H. Mialon∗
First Draft: November 20, 2013, This Version: June 17, 2014
Abstract
In a signaling game between a receiver and senders, prejudice occurs if
senders are pre-judged based on an index such as race without reference to
their qualifying signals. This paper investigates what makes the receiver ignore
senders’ informative signals in favor of the uninformative index. Prejudice arises
when competition between senders erodes their signaling incentive and reduces
the quality of signaling significantly. To minimize the decrease in quality, the
receiver may pre-screen the senders using the index, removing the impact of
competition. We advocate policies that enhance the quality of signals to ensure
the effectiveness of competitive signaling systems.
JEL Classification Numbers: J71, J78, D82, K31
Keywords: Competitive Strategic Signaling, Prejudice, Discrimination.
∗
Sue H. Mialon, Department of Economics, Emory University, Atlanta, GA 30322 (e-mail: smialon@emory.edu, phone: 404-712-8169). I am grateful to Daniel S. Hamermesh, R. Preston McAfee,
Hugo Mialon, Kaz Miyagiwa, Malathi Velamuri, and Seung Han Yoo for their helpful comments.
1
1
Introduction
Prejudice is "a preconceived judgment or opinion, an adverse opinion or learning
formed without just grounds or before sufficient knowledge" (Merriam Webster dictionary). Hence, prejudice implies decision-making without all relevant information.
Typically, this requires a situation in which the relevant information is private and
not readily available. Signaling theory predicts that, in such a situation, a person (a
sender) who has valuable, private information would have an incentive to reveal the
hidden information through an investment in signaling methods. What has not received much attention, however, is that successful signaling also requires the receiver
of the information to have an incentive to utilize the signaled information. Although
all existing signaling models assume that the receiver always fully utilizes the signaled
information, the existence of prejudice implies that this may not be the case in reality.
In this context, this paper develops a model of prejudice as an outcome of failed
signaling not because of a lack of signaling incentives by the sender, but because
of a lack of interest by the receiver in utilizing signals. This paper investigates the
conditions under which the receiver becomes less interested in fully utilizing the information (henceforth, full information). We also offer new insights about the potential
inefficiency of signaling under full information and the implications of related policies
and their effectiveness.
The model builds on a simple signaling game between a receiver and two senders.
The senders’ quality is private information, whereas the two senders are easily distinguishable by an irrelevant, but observable, characteristic, such as race, gender, or
physical beauty, which is termed an index, following Spence (1973). The senders can
signal their hidden quality to the receiver by investing in an effort.
To understand the model, for example, suppose that two candidates (senders),
a male and a female, are being interviewed for a job. In order to discover whether
they are qualified, the recruiter must review their application files thoroughly and
pay attention to the details of their responses to questions while being interviewed.
That is, the recruiter needs to process their signals. However, instead of acquiring
all of the information and signals from each candidate for their decision, the recruiter may decide to pay more attention to the male candidate’s information and
skim through the female candidate’s information. In this case, prejudice occurs, not
because gender was a factor in decision, but because some useful information about
2
female candidate’s qualification was intentionally ignored in favor of gender-based
screening. Whether the female candidate was truly qualified, or even more qualified
than the male candidate, would never be known.
We distinguish prejudice from discrimination in that prejudice refers to differentiating the two candidates by gender before the recruiter gets to know anything about
them (even though more relevant information is available), whereas discrimination
refers to treating them differently by gender after the recruiter have investigated all
of the available information about them.
Why would the recruiter want to rely on uninformative gender (index) first instead
of acquiring full information? To the receiver, prejudice is preferable only if the index
is more effective in identifying high quality candidates than full information. Then,
when and why is full information less efficient than prejudice? In our benchmark case,
considering a typical model of signaling in which each type of sender independently
invests in signals, we show that such a framework is inadequate to explain prejudice.
In general, it is never optimal for the receiver to discard any valuable information
from the senders in favor of less informative index in that framework.
However, the entire dynamics of the signaling game change if there is a shortage
of return from successful signaling, which we consider in Section 4. In the example
above, the shortage occurs if there is only one position available for the two candidates.
Then, there is a chance that senders’ signaling efforts will be wasted even without
prejudice, because there is insufficient room for all of the qualified senders. As one
sender’s chance of being chosen depends on whether the other sender has a qualifying
signal, two senders’ signaling decisions become interdependent. Also, since receiving a
return from signaling effort becomes uncertain, senders’ incentive to signal decreases.
Even fair competition under full information can be as discouraging as prejudice. The
only difference is that under prejudice, the effort is wasted if they do not possess the
favorable characteristic, whereas under competition for limited positions, the effort
is to no avail if the sender does not win the competition. As the senders’ signaling
efforts decrease, the value of full information to the receiver diminishes.
Noisy and imperfect signals reduce the value further. For example, if all senders
have equally qualifying signals (pooling), or if their signals are too similar to differentiate their hidden qualities, the receiver would have to rely on the index when
making a selection. We call such an index-based decision comparative discrimina-
3
tion to distinguish it from prejudice.1 The possibility of comparative discrimination
decreases the senders’ signaling incentives and thus reduces the value of signals to
the receiver. Such a chain reaction from strategic interactions between senders as
well as with the receiver can easily destroy the value of competitive signaling under
full information. When the receiver expects that quality of information under full
signaling deteriorates substantially, the receiver chooses prejudice.
The main role of prejudice is to pre-screen the senders in order to decrease the
number of senders to consider and have less information to process. Since the selection
is independent of senders’ effort choices, it lessens the impact of value-destroying
competition for the insufficient return between the senders. Thus, when overheated
competition between the senders lessens their signaling incentives, prejudice becomes
an alternative in order to moderate the negative impact of competition.
The efficiency of competitive signaling comes from fairly distributing signaling
incentives among the senders. The distribution is fair in the sense that ex ante
identical senders make symmetric efforts, although the efforts can be asymmetric if
the senders are not identical. In contrast, prejudice always results in the most biased
selection that favors one of them. Anticipating this, even ex ante identical senders
will have asymmetric efforts. Prejudice is inefficient if a fair distribution of incentives
creates an overall greater value of signaling than a biased distribution under prejudice.
However, competitive signaling can be as inefficient as prejudice, if its "fair"
distribution of incentives is not so different from the distribution under prejudice.
This occurs if signals are more correlated with race or gender than with the senders’
quality. For example, if signaling is prohibitively costly for a minority, there is no
essential difference between competitive signaling and prejudice. In either system, a
minority with high quality will never have sufficient incentive to signal, due either
to prejudice or to economic hardship. Antonovics and Backes (2013) report that, if
the admission standard in the University of California system had not changed after
the affirmative action ban, the rate of under-representation would have been twice as
great under the previous system that emphasized grades and test scores. This raises
doubts as to whether competitive signals like test scores are more correlated with the
quality of the senders than with their race.
In order to maintain the efficiency of full signaling under competition, we em1
Comparative discrimination is different from statistical discrimination. For detailed explanation
on the differences, see Section 4.2.1.
4
phasize the importance of policies that make the signals closely correlated with the
hidden quality but not with the senders’ current economic status, race, or gender.
For instance, the Head Start program or the No Child Left Behind policy may help
to reduce the racial gap due to an unfavorable signaling environment for minorities.
If these policies can make signaling functions from both races more homogenous independently of their economic status, any difference in their signaling outcomes could be
attributed to a difference in the hidden quality. In that case, a competition-oriented
admission policy would work efficiently. As Aristotle argued, treating "like cases
alike" is required for equality and "identical treatment is not equal treatment,[..] if
individuals are not similarly situated."2
The remainder of this paper is organized as follows. Section 2 briefly summarizes the contributions of this paper to the literature. Section 3 presents our basic
framework of a costly process to decipher signaled information and a benchmark case
when there is no competition between senders. In Section 4, we introduce a strategic/interactive signaling model with a possible shortage of the return from signaling
efforts. Section 5 derives equilibrium. In Section 6, we discuss the policy implications
of the results. In Section 7, we present our conclusions.
2
Contributions to the Literature
The novelty of this paper is its original model of prejudice that differentiates prejudice
from discrimination. The model highlights the role of a shortage of the return from
signaling in leading to prejudice in equilibrium. We advocate the importance of
enhancing the quality of signals under competition because prejudice arises as a result
of inefficient competitive signaling.
There is an abundance of theoretical literature that is devoted to an explanation
of discrimination. While some theories, such as in Becker, (1957) describe it as a
matter of preference, many others spurred by the seminal papers of Phelps (1972) and
Arrow (1973) explain it as a result of statistical discrimination. Fang and Moro (2011)
provide an excellent survey of the literature that deals with statistical discrimination.
In the statistical discrimination literature, Coate and Loury (1993), Moro and
Norman (2004), and Mialon and Yoo (2014) are closely related to the present paper.3
2
3
"Minority Report," The Economist (Oct. 23, 2013)
There are many other studies with interesting approaches to explain discrimination, although
5
Coate and Loury (1993) explain statistical discrimination as a result of coordination
failure between players in a self-fulfilling equilibrium and emphasize the importance
of affirmative action in shaping the players’ beliefs about non-discriminatory equilibrium. Moro and Norman (2004) are the first to explicitly model discrimination
as a "relative" advantage that induces a greater payoff for one group at the expense
of the other. In their general equilibrium model, the production technology requires
two complementary inputs. The marginal product of each input adjusts accordingly
if relatively too many workers of the population invest in the skills for one. Thus,
discrimination is an outcome of two groups specializing in different tasks. The present
paper and Mialon and Yoo (2014) pioneer in modeling interdependent signaling decisions of senders. Mialon and Yoo (2014) show that employers prefer a discriminatory
outcome because it reduces the overall risks of hiring. They show how employers can
actively influence the discriminatory outcome and that discrimination is thus likely
to be persistent.
In contrast, the present paper models prejudice, a more fundamentally inefficient
discrimination of senders, as a purposeful act of ignoring potentially valuable information about senders in favor of a pre-judgment based on the uninformative index.
It is a shortage of the return from signaling that motivates prejudice. The shortage
intensifies competition and deteriorates the efficiency of competitive signaling. Thus,
prejudice is more likely to occur when competition is intense.
Although the statistical discrimination literature explains how ex ante identical
senders can be discriminated in equilibrium, the present paper finds that such discrimination is unlikely to occur if more effective discrimination is possible by means
of prejudice. Since prejudice works as a most effective way of discriminating senders,
the receiver chooses full information only if fair distribution of incentives is desirable. Thus, the receiver would never choose full information to discriminate ex ante
identical senders.
they are not closely related to this paper. For example, Mailath et al (2000) model discrimination
as an outcome of search frictions, Chaudhuri and Sethi (2008) discuss the relationship between
integration and discrimination, and Klumpp and Su (2013a) show how self-fulfilling beliefs about
a low variance in female human capital distribution induce a low representation of female workers
in high-end jobs. Blume (2005) considers a dynamic model of statistical discrimination to explain
how a discriminatory equilibrium is selected instead of a non-discriminatory one. He explains that a
discriminatory equilibrium may occur as a result of firms’ learning about worker investment decisions.
6
3
The Costly Process of Extracting Information
from Signals
Consider a game that involves a receiver and two senders.4 Each sender is characterized by his/her quality, high () or low (), and an index  = 1 2. Although the
quality is the private information of senders,  is observable to the receiver  at no
cost. The two senders differ in . We assume that  is not subject to choice.  is
irrelevant in that ’s payoff depends only on the quality,  =  , of the sender and
not . The payoff from selecting a −type of sender is   0 whereas the payoff
from selecting a −type of sender is zero.
Let   0 be the probability that each sender has  quality. When a sender makes
an effort  , the sender receives   0 if selected, whereas the payoff for all other
cases are normalized to zero.5 Signaling is costly for senders. Let  ∈ [0 ∞) be the
cost of choosing  for a −type of sender,  = 1 1 2 2. This cost is also
private information. The receiver  has information only about the cost distribution
of each type. Let  and  be the cumulative distribution function (cdf) and
the probability distribution function (pdf) of type , respectively, on the support
[0 ∞). We assume that  is twice-differentiable and that the ratio of the two pdfs
 ()
, satisfies the property of  (1 ) ≥  (2 ) for 1 ≥ 2
for type  = 1 2,  () = 
()
(MLRP). Signals are intrinsically noisy. However, the MLRP ensures that the signals
are more likely to be correct than not in predicting an −type. The MLRP implies
that  () first order stochastically dominates (FOSD)  (). Thus, on average,
it is more costly for an −type to signal than it is for an −type, ceteris paribus.
The receiver  must incur a cost of   0 to extract information from signals
4
The model considers two senders only for simplicity. This game is equivalent to one in which
(i) there are many identical receivers and two types () of a continuum of senders of mass 1 on an
interval of [0 ] for  , and (ii) each receiver is randomly matched to the senders from each type.
Introducing   2 possible types of  makes the signaling decision of each type senders depend on
that of the remaining  − 1 types. Except for the increased dimension of strategic interactions, the
same qualitative results are expected to hold for   2 possible types of .
5
We assume that   0 is given exogenously for several reasons. First, in many cases, there is
prejudice in a non-market situation and the payoff  from acquiring the scarce position is often
a non-market price. For example, in the case of college admissions, there is no flexible market
price. Second, even if the payoff of competitive signaling is determined in the market, the outcomes
under prejudice are not determined by the market. For example, if  represents a wage differential
resulted from prejudice, it cannot be a completely market-determined price because, by definition,
a prejudiced group’s wage is not a competitive, market-clearing wage.
7
whereas the index  is costlessly observable.6 In some cases,  may wish to first make
a decision based on the observed  only. In that case,  observes signals only from
the selected type . In the benchmark case below, we first examine the role of the
cost  in determining prejudice in a model without competition between two types of
. Then, in Section 4, we introduce a shortage of returns from signaling that results
in an interaction between the senders.
Throughout the paper, regardless of whether or not there is an interaction between
the senders, the timing of the game between the receiver and the senders is as follows.
At stage 0, nature chooses type  and quality  for each sender. At stage 1, each
sender determines whether to make a signaling effort  or not. At stage 2, the
receiver  observes the index  of each sender and decides whether to make a decision
based on the index () first or to pursue full information from the signaling () of
each sender. If  is chosen, at stage 3,  observes all senders’ choice of effort. If  is
chosen instead,  selects type  with some probability, and only the selected sender’s
signals can be observed. After  or , how the signals are used for a decision differs
depending on whether or not the senders are competing.
3.1
Benchmark
For this benchmark case, we assume that there are unlimited positions as in the
statistical discrimination literature. Thus, there is no need for competition between
the senders, which makes each sender of type  signal independently. This feature
allows us to identify the effect of competition as we introduce limited positions in the
next section.
Given unlimited positions, as long as the signals are valuable,  is willing to
hire as many senders as possible upon observing the qualifying signal  for all .
Let e ∈ [0 1] be the probability that the receiver chooses  and e ∈ [0 1] be the
probability of observing type 0  signal if  is chosen. Since e and e are chosen
to reflect how much  values the signals in equilibrium, we assume that, if a sender
under consideration at stage 3 has a signal  , the sender is selected and receives  ,
6
For example, when hiring, recruiters must decipher the information that indicates candidates’
quality by examining the details of signals, such as academic records and evaluation letters. According to The Society for Human Resource Management (SHRM), in 2011, the estimated average
cost-per-hire for organizations with 1000 or more employees was $4,285 and $3,079 for organizations
with fewer than 1000 employees.
8
or nothing.7 Then, a sender of type  decides to invest in signaling effort ( ) if
and only if ( | ) = [e + (1 − e )e ] −   ( | ) = 0 or
  [e + (1 − e )e ] = e .
(1)
e =  ( −  ) − (1 − )  
(2)
Thus,  type makes a signaling effort with a probability of  =  (e ).
For given  and  , the receiver’s payoffs from  and  are () = e − ,
and () = e (e − ), respectively, where
for  = 1 2. Thus, as long as ()  0 ⇔ e  ,  prefers  and fully utilizes the
signals. e shows the value of signals from type . If e e  0, a higher incentive
e induces a higher signaling effort  , and increases ’s incentive to utilize signals.
Specifically,
 ( −  )
e
T0⇔
−  (e ) R 0
(3)
e
(1 − )


where  (e ) = 
. By the MLRP,  (e ) is increasing in e . Thus, there exists

 at which  = 0,   0 for e   and   0 for e   , where  =
³
´
 (− )
−1
, from (3). Thus, e is concave and maximized at  . Assumption
 (1− ) 
1 ensures that  is not so high that, at least in some cases, it is worthwhile to use all
of signaled information.
Assumption 1 (1)    , where  = e ( ). (2)    , where  is the lowest
 that satisfies  = e ( ).
To ensure that all signaling efforts are intrinsically valuable to the receiver, we
also assume that  is high enough to satisfy the following condition.
7
This is a behavioral assumption that we make in order to simplify the analysis. However, even
without this assumption, the same behavior is expected to arise in equilibrium. Given the MLRP,
any observed  is more likely from an −type than from an −type and thus,  is informative.
Then, for a given , using  improves 0  payoff. If  will not be used for selection, it is not
necessary to acquire all signals in the first place. Note that acquiring  is costly.  must incur
  0 to reach the information of  even if it is not used, whereas the cost  can be avoided if
 acquires only the information to be used. Thus, the receiver strictly prefers to reduce the set of
information to process rather than not use  . This implies that the receiver would never want to
waste  once the information has been obtained by costly efforts. Our assumption simply reflects
the nature of predicted equilibrium behavior of  who wants to prioritize the decision of   in
determining the value of  .
9
Assumption 2  ≥ 
When  ≥  , greater signaling efforts by senders increase the value of signals
for ,   0 Then, prejudice means ignoring intrinsically valuable information.
As we intend to explain prejudice as a decision to not pursue potentially valuable
information, we require this property of Assumption 2.8 Lemma 1 summarizes the
Perfect Bayesian Nash equilibria (PBE) in this benchmark case.
Lemma 1 If e is not too low, the unique equilibrium is that the receiver acquires all
of the signaled information (e = 1).
Proof. Proofs are provided in the Appendix.
As long as the senders expect that e is not too low, there is sufficient signaling,
and no information is wasted (e = 1). However, if senders make no effort at all as
a result of a self-fulfilling belief about a very low e ,  will not acquire any signaling
f1 = 1  f
 2 = 0, for instance,
information as it would be worthless. In particular, if 
the equilibrium outcome is discriminatory against type 2. This results in Lemma 1
resembles statistical discrimination in the literature.
However, like statistical discrimination models, this model does not explain prejudice. It does not explain why one group should expect a particularly lower e than
the other. Prejudice (like discrimination) is always a relative concept. Offering a
type 1 sender an opportunity to prove his qualification becomes prejudice because
the same opportunity is not available to a type 2 sender for no proper reason. That
is, prejudice is 0 s choice for an asymmetric distribution of the opportunities between
the two types of senders. This benchmark framework is inadequate to explain prejudice, because the framework treats a decision for one type in isolation from a decision
for the other. In the next section, we build a model in which the receiver considers
both types simultaneously in comparison.
4
Model of Strategic Signaling
For , the value of senders’ signals depends on the need. Let  be the parameter
that incorporates the need for signaled information. Roughly speaking,  represents
f,   0, and a higher incentive to signal e   lowers
If    , for a high enough level of 
 
the value of signals. Then,  is not always better off with greater signaling efforts. In this case,
ignoring information is not necessarily inefficient because signaling itself is ineffective and cannot be
considered to be prejudice.
8
10
the likelihood of an available return from successful signaling for senders, or the
available positions that  needs to fill with  quality senders. Given that we have
two candidates, we assume that  ≤ 2. If  = 2,  is willing to hire all senders
as long as they are qualified and, thus, there is no competition between the senders.
However, when   2,  needs to compare two senders to select the best candidate. To
be selected, a sender must have not only a qualifying signal, but also a better signal
than the competitor’s. Thus, senders’ signaling decisions become interdependent.
Below, we show that as  declines, the competition between senders intensifies and
prejudice is more likely to occur.
In this paper, prejudice differs from discrimination in the level of information
that  has, although in both situations,  refers to the index  in selecting senders.
There is prejudice when  decides to refuse an opportunity to learn more about the
quality of senders of one type, while being receptive to the information from senders
of the other type. Discrimination, on the other hand, is treating senders of two types
differently after acquiring all available information about them.
Let  denote the probability that  chooses  and acquires all of the signals from
all  = 1 2 in this new game of strategic signaling. Prejudice requires first that 
chooses  ( = 0). In that case,  also must decide what fraction of the acquired
information should be from each . Suppose that  acquires the information from
types 1 and 2 with probabilities b1 =  ∈ [0 1] and b2 = (1−) ∈ [0 1] respectively.
Prejudice occurs when  = 0 and b1 6= b2 .
 may need to use  in selection even after acquiring all of the signaling information ( = 1). If both senders have qualifying signals,( , ), given that   2, the
receiver will have to use  to choose between the two qualified candidates to fit . Let
b1 =  ∈ [0 1] (b2 = (1 − ) ∈ [0 1], respectively) be the probability that type 1
(2) sender is selected in this case. When b1  12 we say that there is comparative
discrimination in favor of type 1. Comparative discrimination occurs when  = 1
and b1 6= b2 .
If  is chosen and both senders have  ,  chooses type  with a b probability at
stage 3. If only one of them has  ,  selects the sender with  with a probability
of 0 = min{ 1}, 0 ≤ 1. If no sender has a qualifying signal, no one is selected. If
 chooses  instead,  selects type  with a probability of b . Only senders with
qualifying signals are selected for the position.9
9
For the same reason explained in Footnote 5, we make this behavioral assumption to simplify
11
4.1
Sender’s problem
Let Ψ2 be the probability that a type 2 sender invests in signaling. Then, for given
, b and b , a type 1 sender decides to make an effort to signal if and only if
( | ) = Ψ2 [ (b1  ) + (1 − ) (b1  )] + (1 − Ψ2 )[0  + (1 − )b1  ] − 1 
( | ) = 0, or
(4)
1  b1 = [1 + (1 − )b1 ] 
where 1 = 0 − Ψ2 (0 − b1 ), and Ψ2 = (2 + (1 − )2 ). Similarly, a type 2
sender decides to choose  if and only if 2  b2 = [2 + (1 − )b2 ]  , where
2 = 0 − Ψ1 (0 − b2 ), and Ψ1 = (1 + (1 − )1 ). Then, a type  sender makes
a signaling effort with  probability, where
 =  (b ) =  ([ + (1 − )b ]  )
(5)
b1 = , and b2 = (1 − ). (5) shows how the competition between the two senders
affects each sender’s best response function. From (1), when senders choose signaling
efforts independently, the cut-off for signaling incentive was e = [e + (1 − e )e ] .
To compare e and b , suppose that e = b , and e =   0. Since b ≤ 0 , and
0 ≤ 1, we find that b ≤ e always, with a strict inequality for at least one  if   2.
Therefore, other things being equal, the senders’ incentive to signal is lower for all 
in interactive signaling.
Proposition 1 (Strategic effect of competitive signaling) Other things being equal, if
  2, interaction between two senders reduces their incentives to signal.
When   2, one sender’s signaling effort imposes a negative externality on the
other sender. For a type 1 sender with a signal, the selection is guaranteed if type
2 does not signal. However, if type 2 does, the return from signaling is available
only with b1 ≤ 1 probability. As  declines, it becomes more difficult to expect a
return from their signaling effort (i.e., a lower  ). Thus, senders’ incentive to signal
decreases. If  is too small, no signaling would be worthwhile. To avoid such a case,
we assume that  has a lower bound ,  ∈ (, 2]. 10
the analysis.
10
Let  be the level of  at which  =  , where  satisfies  = e ( ) and    . Since
b = [ + (1 − )b ]  , when  = 0, b1 = b1  =  . In this case, even if  = 1, if   ,
b1 =     and the signal is too costly for  to use. Thus, we assume that  is greater than
this level.
12
On the other hand, (5) shows that the interaction occurs only if  chooses . If
 is chosen ( = 0), b = b  , a sender’s best response becomes independent of the
other, eliminating the effect of strategic interaction. This implies that one of the most
important roles of prejudice is to remove the impact of interaction and competition
between the two types. Thus, when the interaction between two types lowers the
value of signaling significantly,  has a greater incentive to choose  and to remove
the impact of interaction. We discuss this issue in detail in Section 5.2.
4.2
4.2.1
Receiver’s problem
Optimal Comparative Discrimination
First, consider ’s decision on the optimal   when  is chosen and  observes ( , )
from both types of senders. Upon observing ( , ),  favors a type 1 sender if and
[1 (b1 )2 (b2 ) − 2 (b2 )1 (b1 )]  0, where ∆ =  2 1 2 + (1 −
only if (1−)
∆
2 (2 )
1 (1 )
 1
⇔
)(1 2 + 2 1 ) + (1 − )2 1 2 . This condition holds iff 2
(2 )
(1 )
 ()
Φ2 (b2 )  Φ1 (b1 ), where Φ () =  () is the ratio of the probabilities that the two 
types signal,  =  . Then, the optimal   is characterized as follows.
⎧
⎪
⎨
 max
 =
[   ]
⎪ min max
⎩
 min

if Φ2 (b2 )  Φ1 (b1 )
if Φ2 (b2 ) = Φ1 (b1 ) 
if Φ2 (b2 )  Φ1 (b1 )
(6)
where  max = min{1 1 } and  min = max{0 1 − 1 }.
This shows that  sets the optimal   so as to minimize prediction errors. The
1 (1 )
increases as the signaling function becomes noisier
probability ratio Φ1 (b1 ) = 1
(1 )
and thus the size of prediction error for type 1 increases. Thus, if Φ2 (b2 )  Φ1 (b1 ),
it implies that type 2 has a greater prediction error, given the observation that the
two types had equally qualifying efforts. Thus, the optimal  is to favor type 1 with
a lower prediction error,   = max .
Comparative discrimination differs from statistical discrimination in several aspects. First, comparative discrimination requires competing senders who are compared in their "relative" strength, whereas in determining statistical discrimination,
senders are not compared. Second, comparative discrimination is required only when
the senders have equally qualifying signals, whereas statistical discrimination does
not require such a condition. When senders are in competition, since a sender with
13
a relatively stronger signal has an advantage, there is no need to use the index to
discriminate unless the senders’ signals are not differentiable. On the other hand, in
the case of statistical discrimination, what gives rise to discrimination is self-fulfilling
beliefs about each sender’s qualification, instead of a comparison based on signals.
At stage 2, the receiver ’s expected payoff from acquiring all the information is
(  ) = e1 1 (  ) + e2 2 (  ) − 2
(7)
where e is defined in (2), 1 = 0 − Ψ2 (0 −  0 ), 2 = 0 − Ψ1 (0 − (1 −   )), and
Ψ = ( + (1 − ) ). (7) shows how competition affects the value of signals
for . Without competition, () = e − . Now, the value of a signal from type 1
e1 is weighted by 1 which depends on type 2’s signaling effort Ψ2 . The weight 1
decreases if type 2’s signaling Ψ2 increases, and vice versa.
4.2.2
Optimal Level of Prejudice
The receiver’s payoff from  is
() =  [e1 + (1 − )e2 − ] 
(8)
From (8), for a given , () is merely a convex combination of signaling values
from the two senders, e1 and e2 . The weight of the signaling from type 1 (type
2, respectively) is  ((1 − )).  optimizes the weight considering whether type 1
is expected to generate a greater signaling value than type 2. If  expects that
e1  e2 , it is best to favor type 1,  = max , and vice versa. Thus, the optimal  is
characterized as follows.
⎧
⎪
max
if e1  e2
⎨

 =
(9)
[min  max ] if e1 = e2 
⎪
⎩
if e1  e2
min
where max = min{1 1 } and min = max{0 1 − 1 }. Note that if  = 2,  = 12, and
as  declines,  diverges from 12.
14
5
5.1
Equilibrium
Comparative Discrimination
To understand how the equilibrium  is determined under full information, , suppose
)  Φ2 (b
). In this
that signaling is much easier for type 1 for any given b
, i.e., Φ1 (b
situation, if ( , ) is observed after ,  rationally infers that the type 2 sender
is more likely to have  quality given that signaling was more difficult for type 2 to
do. Thus,  favors type 2 to minimize the prediction error. Expecting this, senders
adjust their signaling incentives accordingly. Expecting a favorable treatment (a low
) upon ( , ), type 2’s incentive to signal b2 increases, whereas b1 decreases. The

() is increasing in  (Eeckhoudt and Gollier [1995]).
MLRP implies that Φ () = 
That is, the quality of signals deteriorates as it becomes easier to signal. As a result,
Φ2 (b2 ) increases, whereas Φ1 (b1 ) decreases, which reduces ’s incentive to favor type
2. Proposition 2 describes the equilibrium .
Proposition 2 Suppose there exists a  ∗ ∈ [0 1] at which Φ2 (b2 ( ∗ )) = Φ1 (b1 ( ∗ ))
and  ∗  ∈ [0 1]. Then, in equilibrium, if  acquires all of the signaled information
and ( , ) is observed, the optimal comparative discrimination is ∗ . If such a  ∗
does not exist, the equilibrium  is min or  max .
) 6= Φ2 (b
),  ∗ 6= 12. Thus, in general, optimal comparative discrimination
If Φ1 (b
)  Φ2 (b
) at any
is discriminatory. In particular, Proposition 2 implies that, if Φ1 (b
b
, the condition Φ1 (b1 ) = Φ2 (b2 ) requires b1  b2 . Then, it must be that   1 − .
Therefore, the optimal  is set to counterbalance the effect of unequal signaling
conditions for the two senders.
)  Φ2 (b
) for a given b
. Then,  ∗  1 −  ∗ .
Corollary 1 Suppose Φ1 (b
Suppose that it is particularly difficult for minority students to pursue higher
education because of the high costs and that many of them are discouraged by the
large income disparity between minority and non-minority groups. Corollary 1 implies
that, in this case, if both a minority student and a non-minority student have the
same standardized test scores, it is optimal for college admission offices to favor the
minority student, as the probability that the student has  quality is higher than
that of the non-minority student. However, if the senders are ex ante identical, the
optimal  is to treat the two types equally.
15
Corollary 2 If the two senders are ex ante identical and they have equally qualifying
signals,  ∗ = 12 is optimal.
Since comparative discrimination counter-balances the effect of initial signaling
conditions for each sender, it reduces discriminatory outcomes in equilibrium.
5.2
Prejudice
Under the expectation of  = 0, b1 =  and b2 = (1 − ) . As long as a sender
 is selected in the first round (with a  or (1 − ) probability), the return from
signaling is guaranteed. Since  is independent of senders’ signaling decisions, their
only objective is to optimize their own signaling efforts independently for a given 
or (1 − ).
Proposition 3 For   2, if an index-based pre-screening  takes place ( = 0) in
equilibrium, the equilibrium ∗ is either ∗ = min or ∗ = max .
Proposition 3 states that if   2, 0  decision to use  to pre-screen ( = 0)
always leads to prejudice, b∗1 = max   b∗2 = min  (or b∗1  b∗2 ). For the rest of our
analysis, we assume that type 1 is favored if prejudice occurs, ∗ = max , without loss
of generality.
5.3
Equilibrium
At stage 2, the receiver’s expected payoffs are (   ) = e1 1 (  ) + e2 2 ( ) − 2
and (  ) =  [ e1 + (1 −  )e2 − ]  0.  chooses  if and only if (   ) ≤
(  ), or
(1 − Ψ2 )(0 −  )e1 + (1 − Ψ1 )(0 − (1 −  ))e2 + Γ(  −  )  (2 − ) (10)
|
|
|
{z
}
{z
}
{z
}
1
2
3
where Γ = (1 − ) [1 2 − 1 2 ] and   and  are given in (6) and (9).
The term  e shows the net gains in type ’s signal from switching to full information () if separating occurs (i.e., type  chooses  but the other type − does
not), whereas 3 shows the net gains from  when pooling occurs (i.e., both types
choose  ). To understand equilibrium conditions, suppose  expects e1 ≥ e2 and
thus plans to implement  = max if  is chosen. This results in (0 − max ) = 0, and
16
1 = 0, which implies that  never gains more from type 1’s signaling by switching
from  to . What matters to  is how much he gains from offering a type 2 sender
more opportunity to signal, 2 and 3 . The extra gain from type 2’s signal is not
large enough if
(1 − Ψ1 )(0 − min )e2 + Γ(  − max )  (2 − ) = .
|
|
{z
}
{z
}
2
(11)
3
Thus, if 2 e2 + 3 is very small, prejudice occurs ( = 0,  = max ) in equilibrium.
Proposition 4 summarizes the PBE when the two senders are ex ante identical.
Proposition 4 Suppose Φ1 = Φ2 = Φ. When there is prejudice,  = 0, and the
equilibrium signaling incentives are
b1 = max   b2 = min  
(12)
Under full information ( = 1),  ∗ = 12, and the equilibrium incentives are b∗1 =
b∗2 = b∗ , where
b∗ =  ∗ ( ∗ ) and
∗ ( ∗ ) = 0 − Ψ ( ∗ )(0 −  ∗ ).
(13)
(14)
Let ∗ and ∗ be the value of 2 e2 + 3 evaluated at (b∗  b∗ ) and (b1  b2 ), respectively.
1. If ∗  , and ∗  , the unique equilibrium involves full information and no
discrimination ( = 1,  ∗ = 12,  = max ).
2. If ∗  , and ∗  , the unique equilibrium involves prejudice. ( = 0,
 =  min ,  = max ).
3. If ∗    ∗ , there are multiple equilibria at ( = 1  ∗ = 12,  = max )
and ( = 0,  =  min , ∗ = max ).
4. If ∗    ∗ , there is a unique mixed strategy equilibrium at (0    =

 1,    12,  = max ).
+(2 −1 )
From Proposition 4, we can identify the source of prejudice. Prejudice arises in
equilibrium when competitive signaling cannot create a sufficiently high signaling
17
value and induce ∗   or ∗  . Corollary 3 shows when the conditions are likely
to be satisfied.
Corollary 3 Other things being equal, prejudice is likely to arise if  or  is low, or
Φ =   is high.
That is,  has a greater incentive to pre-judge senders without sufficient information if the added value of finding another −quality sender is not too high (a low )
or −quality is difficult to distinguish because it is too easy for any sender to signal
(a high Φ ). For the same reason,  is more likely to choose prejudice when  is low.
As  declines, signals become less valuable to . This reduces the senders’ incentive
to signal, b∗ or b2 , which in turn, lowers the value of full information and, thus,  is
more likely to choose prejudice.
6
Policy Implications
In this section, we discuss the implications of the main findings in this paper in the
context of affirmative action policies and the No-Child-Left-Behind policy. First, we
discuss the implication of Color-Blind affirmative action that bans any race-based
choice. Then, we discuss the implication of a purely competition-oriented college
admission policy when minorities face a severe disadvantage in signaling and the role
of the No-Child-Left-Behind policy in this case.
6.1
Color-Blind Affirmative Action
In the context of college admissions, color-sighted affirmative action means that the
admissions office is allowed to consider race as a factor when making a decision. In
contrast, affirmative action bans, or color-blind affirmative action, in contrast, denies
any race-based decision on the basis of a non-discriminatory principle, advocating
purely merit-based selection. The system of purely-merit based selection corresponds
to the competitive signaling under full information with no comparative discrimination ( = 12) in the current framework. Although there is merit in competitive
signaling given the inefficiency in disregarding informative signals under prejudice,
our findings concerning optimal comparative discrimination indicate that a purely
non-discriminatory policy is not always optimal.
18
From Proposition 2 and Corollary 1, we find that, if the senders are not ex ante
identical, the optimal level of comparative discrimination is to favor a disadvantaged
sender. This is because signaling efforts observed from senders facing greater difficulty
in signaling indicate a greater probability of high quality. In the context of college
admissions, this implies that, if two applicants have equally qualifying credentials,
allowing race-based decisions (color-sighted affirmative action) to favor minority indeed can be optimal.11 In this case, insisting on a purely non-discriminatory policy,
 = 12 6=  ∗ , would lower the value of signals under a competitive signaling system,
∗ , which would increase the receiver’s incentive to move to a more intensive indexbased decision, prejudice. Thus, such a purely non-discriminatory policy may indeed
backfire and promote prejudice.
In this paper, the need for an index-based decision exists even under competitive
signaling because both senders may have equally qualifying signals. Similarly, in
reality, even under a purely-merit based system, there may be a situation in which
the senders are not clearly distinguishable by their signals because the signals are
intrinsically imperfect. For example, highly qualified students can obtain a score of
88 and less qualified students can obtain 89 on a test. Because senders’ signals may
not be perfect in revealing their hidden quality, there is a need for another proxy
in decision-making. This paper finds that for that reason, even under a merit-based
system, allowing an index-based decision to improve the incentives of a disadvantaged
group can indeed improve the soundness of the merit-based system.
As a result of affirmative action bans for higher education, in California, Michigan,
and Texas, any race-based decision is now prohibited in the college admission process.
There have been some doubts that color-blind affirmative action actually promotes
efficiency. Chan and Eyster (2003) and Fryer et al (2008) argue that a color-blind
policy can be inefficient because strict enforcement of such a policy is impossible and
admissions offices may still use other race-related variables in deciding admissions
instead of race. Prior to the ban, although race could be a factor, the schools relied
primarily on standardized test scores and GPAs for admission. New admission policies
now give more weight to other traits, such as the student’s standing in his/her high
school as well as leadership and extra-curricular activities.12 Hence, in practice, color11
Fu (2006) makes a similar argument in a model of contest. He shows that affirmative action
favoring minorities improves the efficiency of a contest.
12
For example, in California and Texas, graduating high school seniors who are in the top 4% and
10%, respectively, of their class are guaranteed admission to the state’s university system.
19
blind affirmative action may simply replace race with other indices that are closely
related to race for admission criteria. The new admission criteria are often viewed as
more inefficient than race. This confirms the prediction in this paper that a purely
non-discriminatory policy may indeed promote more inefficiency.
6.2
The Effectiveness of Competitive Signaling
From the investigation on what promotes the receiver’s incentive to discard potentially valuable information, we find that the gist of the question lies in whether a
competition-oriented system is efficient. The immediate implication of Proposition 1
and Corollary 3 is that a competitive signaling mechanism may not always be efficient
if excessive competition among the senders destroys their signaling incentive.
However, Case 2 of Proposition 4 (or (11)) indicates that there is a more serious
issue when competitive outcomes are no different than that from prejudice, i.e., 2 e2 +
3 is very small. For example, suppose economic hardship significantly limits the
possibility of a high quality minority (type 2) to signal, e.g., Ψ2 ≈ 0, Φ1  Φ2 , and
even under competitive signaling, at b∗2 , 2 (b∗2 ) ≈ 0 In this case, e2 ≈ 0. Hence,
there is no essential difference between prejudice and competitive signaling in the
outcomes. In both cases, the signaling effort from a minority is substantially lower
than that from a majority. For the minority, there is no incentive to signal under
prejudice, and signaling is not affordable under competitive signaling. In this case,
there is no real gain from a competitive signaling system.
A competitive system is effective if competitive signaling can offer a meaningful
difference in signaling incentives for minority senders who would not have the same
opportunity under prejudice. This requires that signals are closely correlated with
the hidden quality, rather than an irrelevant factor like economic hardship. In an
effort to ensure that the signals are free from the influence of other irrelevant factors,
the Head Start program or the No-Child-Left-Behind policy can play an important
role. As the policies make the signaling environment more homogenous for both races,
any observed signaling outcomes would be correlated more with the hidden quality
for either race than the economic hardship. Thus, the policies enhance the value of
signaling under competition.
20
7
Conclusion
This paper models prejudice in a signaling game between a receiver and two senders
who differ in one irrelevant, but observable, index. Prejudice occurs when the receiver
chooses to (partially) ignore more informative signals from the senders in favor of less
informative indices from the senders. In equilibrium, prejudice is likely to arise if
competitive signaling is inefficient. Acquiring all of the signaled information under
competitive signaling may be too costly for the receiver if competition reduces senders’
signaling incentives substantially and worsens the quality of overall signals. In such
a case, prejudice removes the impact of competition by pre-screening senders on the
basis of their observable index. If signaling does not reveal enough information even
under competitive signaling, another index-based decision may be necessary. In such
a case, the optimal level of comparative discrimination will favor a disadvantaged
sender to minimize prediction errors.
The analysis of policy implications based on these results indicates that colorblind affirmative action is not always optimal and can actually increase the likelihood
of prejudice. We point out the importance of the Head Start program and the No
Child Left Behind policy in improving the quality of a signal itself. This is because
competitive equilibrium may not be effective if the signals are not better proxies for
the hidden quality than an index.
8
Appendix
1. Proof of Lemma 1
For any given  ∈ [0 1], if   = 1, b =  ,  =  ( ), and e (b )  ,
confirming that   = 1 is an equilibrium, for  = 1 2and  = 1 1 2 2. For
  = 0, it must be that ()  (), i.e., e (b )  , at equilibrium b . When
  = 0, b =   . Thus, if  is low enough, e (  )  . Let  be the level of
b that satisfies e ( ) =  in the range where b   . For e (  )  ,  needs
to be low enough that     . Let  be the level of  that satisfies   =  .
Then, for   , equilibrium involves   = 0. In this case, since () ≤ 0 for all
 ∈ [0 1], the optimal  = 0.
2. Proof of Proposition 2
21
Suppose   0. By construction, in equilibrium, for given  and ,
Φ (b∗ ) = Φ ( [ (0 − Ψ− (0 − b )) + (1 − )b ])
⇔ b∗1 =  [ (0 − Ψ2 (0 − )) + (1 − )] = Ω1 (b∗2 (b∗1 ))
⇔ b∗2 =  [ (0 − Ψ1 (0 − (1 − ))) + (1 − )(1 − )] = Ω2 (b∗1 (b∗2 )).
At min b2 = 0, b1  0 is at its maximum, and at max b2 , b1  0 is at its minimum.
Since Ω is continuous and monotone decreasing in the element, by Fixed Point
Theorem, such a b∗ exists and is unique. The stability condition requires that the
slope of the best response function for type 1 to be greater than that for type 2, i.e.,
¯
µ
¯
¯−Ψ−10
2
¯
¶¯
¯
1
¯  |−Ψ01  (0 − (1 −   ))|
 (0 −  ∗ ) ¯


⇔ 1  2 Ψ01 Ψ−10
2 (0 −  )(0 − (1 −  ))
If all the signaled information is acquired in equilibrium ( = 1), at b∗ , ∗  1 since
0 = min{ 1} ≤ 1, (0 −  )  0, Φ  0, and b∗   .
(1) An increase in  increases b1 but decreases b2 . When b is differentiable,
2
1
2
1
2
1
=  
 
=  
, 
= Ψ2 − Ψ2 (0 − ) = Ψ2 − 2 
(0 − ),




Ψ2
1
2
and  = −Ψ1 −  (0 − (1 − )) = −Ψ1 − 1  (0 − (1 − )) where
1 = 1 + (1 − )1 and 2 = 2 + (1 − )2 . Thus, by collecting terms, we
get
  (Ψ2 +  2 Ψ1 (0 − ))
b1
=
≥0


b2
  (Ψ1 +  1 Ψ2 (0 − (1 − )))
= −
≤0


(15)
(16)
where  = 1 −  2 2 1 2 (0 − )(0 − (1 − ))  0 by stability condition.
2
1
 0, 
 0 from (15) and
(2) Then, given that Φ is increasing in b , and 
∗
(16), Φ2 (b2 ) − Φ1 (b1 ) is decreasing in . Thus, for    , Φ2 (b2 )  Φ1 (b1 ), and thus
 is inclined to lower , and for   ∗ , Φ2 (b2 )  Φ1 (b1 ), and thus,  is inclined
to increase . Therefore, in equilibrium, the receiver randomizes at  ∗ such that
Φ2 (b2 ( ∗ )) = Φ1 (b1 ( ∗ )).
(3) Since Φ2 (b2 )−Φ1 (b1 ) is decreasing in ,  ∗ exists if Φ2 (b2 ( min ))−Φ1 (b1 ( min )) 
0 and Φ2 (b2 ( max )) − Φ1 (b1 ( max ))  0. Thus, if Φ2 (b2 ( min )) − Φ1 (b1 ( min ))  0 or
22
Φ2 (b2 ( max )) − Φ1 (b1 ( max ))  0,  ∗ does not exist. In the former case,  =  min ,
and in the latter case,  =  max . Both cases are trivial.
3. Proof of Corollary 2
When  = 1, if there is no difference in the distributions, Φ1 = Φ2 = Φ, in
equilibrium, it must be that Φ(b2 ) = Φ(b1 ), which occurs only if b2 (∗ ) = b1 ( ∗ ) ⇔
1 = 0 − Ψ (0 − ∗ ) = 0 − Ψ (0 − (1 −  ∗ )) = 2 . This requires that (0 −  ∗ ) =
(0 − (1 − ∗ )), i.e.,  ∗ = 12. Thus, Φ(b2 ) = Φ(b1 ) if and only if  ∗ = 12.
4. Proof of Proposition 3
A favorable treatment of type 1 ( = max ) based on the belief e1  e2 (from
(9)) actually results in e1  e2 because it gives greater incentives for type 1 to signal,
i.e., b1  b2 . Since e1 0  0, this confirms e1  e2 . Similarly, when  = min , this
confirms e1  e2 . If e1 = e2 , any  is optimal for . However, since e 0  0, for any
 ∈ (min  max ),
e1 (max  ) = e1 (max  ) + (1 − )e2 (max  )  e1 ( ) + (1 − )e2 ((1 − ) )
Thus, ∗ = max (or ∗ = min  respectively) is always a weakly dominant strategy if
e1 ≥ e2 (e1 ≤ e2 ) is expected.
5. Proof of Proposition 4
A. 3 ≥ 0.
If  = min ,    max , Γ  0, and thus, 3  0. If   =  max ,   − max = 0,
and thus, 3 = 0. If   =  ∗ , Γ = 0, and thus, 3 = 0.
B. General thresholds  and 
From Proposition 2, when   0, Φ1 (b1 ) = Φ2 (b2 ) in equilibrium. Plugging
  = ∗ from (6), we obtain 3 = 0 since Γ = 0 at  ∗ . Thus, (11) reduces
 = 2 e2 = (1 − Ψ1 )e2  (2 − )
(17)
where  =  if  ≤ 1 or (2 − ) if   1. On the other hand,  will not acquire full
information ( = 0) in equilibrium if and only if 2 and 3 are small around the
23
equilibrium signaling efforts. Evaluating (11) at  = 0, we obtain
 = 2 e2 + 3  (2 − ),
(18)
C. When Φ1 = Φ2 = Φ (ex ante identical senders)
In this case, any difference in signaling error or signaling value is due to different
incentives, b . Throughout the proof, without loss of generality, we assume that type
1 is the favored type. Then since  = max is a weakly dominant strategy, other things
being equal,  plans to favor type 1 with a max  probability, if  is ever chosen.
1. Suppose  expects Φ1 (b )  Φ2 (b ). (The argument is symmetric for the
case when( Φ1  Φ2 .) From (6),   =  min . At min , 1 = 0 − Ψ2 (0 −
if  ≤ 1
(1 − Ψ2 )
min ) =
, and 2 = 0 − Ψ1 (0 − (1 − min )) =
1 − Ψ2 (2 − ) if   1
(
 if  ≤ 1
. Thus, 1  2 .
1 if   1
(
 (1 − Ψ2 )
(a) Suppose (  min )  ( max ). In this case,  = 1, b1 =

 (1 − Ψ2 (2 − ))
(
 
, and thus, Φ1  Φ2 , which contradicts.
b2 =

b
(b) Suppose
(  min
(
( )  ( max ) and thus,  = 0. In this case, 1 =

0
 b2 =
, and thus, Φ1  Φ2 . Equilibrium occurs at

( − 1)
 = 0 if (11) is satisfied at the equilibrium incentives. Evaluating (11) at
b1 and b2 , we get ∗ = (1 − Ψ1 )e2 + 3 = (e2 (b2 ) − e1 (b1 )Ψ2 (b2 )) 
(2 − ).
(
[(1 − Ψ2 ) + (1 − )] 
(c) Suppose (  min ) = ( max ) . In this case, b1 =
[(1 − Ψ2 (2 − )) + (1 − )] 
(

and b2 =
. Thus, to have Φ1  Φ2 , it is required
( + (1 − )( − 1))
that b1  b2 , which holds if and only if   1 and   1+Ψ1 2 . When   1,
2 (−1)
), which is
however, in the range where  ∈ [0 1+Ψ1 2 ), b1 ∈ [1 1+Ψ
1+Ψ2
1
out of the range of consideration (where 1  0 by definition). Thus,
there is no mixed strategy equilibrium if Φ1  Φ2 .
24
2. Suppose  expects Φ1 = Φ2 . This requires that b1 = b2 = b∗ .
(a) Suppose (   )  ( max ). Given that  = 0, b1 =  = b2 =
(1 − ) can occur if and only if ∗ = 12, which is a weakly dominated
strategy by ∗ = max . When ∗ = max , b1  b2 , which contradicts.
(b) Suppose (   )  ( max ). When  = 1, b1 = b2 can occur only if at
 ∗ = 12. In this case, 1 = 0 − Ψ2 (0 −  ∗ ) = 2 = 0 − Ψ1 (0 − (1 −
 ∗ )) = ∗ . Let b1 = b2 = b∗ = ∗  . Equilibrium occurs at  = 1 if (17)
is satisfied at b∗ . That is,
∗ = (1 − Ψ1 (b∗ ))e2 (b∗ )  (2 − ),
(19)
where  =  if  ≤ 1, or (2 − ) if   1.
(c) Suppose (   ) = ( max ). Since b1 = [1 + (1 − )max ]  =
b2 = [2 + (1 − )min ]  = b . Then, for  ∈ [0 1] and   1, by
construction of b , we get
 =

.
 + (2 − 1 )
(20)
Such a   exists if 2  1 , which requires that     ∗ = 12. In
equilibrium,   and  (  ) must satisfy (20) and the following.
∗ = (1 − Ψ1 (  ))e2 (  ) = (2 − ).
From (19) and (21),
∗
 (2 − ) =
∗
and b =
(
2
if 
+(2 −1 )
2 −(−1)1
if
(2−)+(2 −1 )
(21)
≤1
.
1
Also, for the randomization strategy , it must be that   (2 − ) and
  (2 − ).
3. Then, the PBE are summarized as follows: Let ∗ = (1 − Ψ1 (b∗ ))e2 (b∗ )
and ∗ = (1 − Ψ1 (b ))e2 (b ) be the values of  evaluated at (b∗  b∗ ) and
(b  b ), respectively. Similarly, ∗ = (e2 (b2 ) − e1 (b1 )Ψ2 (b2 )) is the value of 
evaluated at (b1  b2 ).
(
(

0

 b2 =
,  = max ,  =  min .
1. If ∗  (2 − ),  = 0, b1 =

( − 1)
25
2. If ∗  (2 − ),  = 1, b1 = b2 = b∗ = ∗  ,  = max ,  = 12.
3. If ∗  ∗ = (2 − )  ∗ , the unique mixed strategy is   =
b1 = b2 = b ,  = max ,  =    12.

,
+(2 −1 )
6. Proof of Corollary 3
When   0, from Proposition 2,  is optimally chosen to make Γ = 0, and thus,
3 = 0. Thus, ∗ is determined by 2 e2 = (1 − Ψ1 )(0 − min )e2 . By construction
from (2), e2 is lower if  is low or Φ2 = 2 2 is high, ceteris paribus. An increase
in  directly increases 2 e2 since (0 − min ) is always an increasing function of . It
also, indirectly increases 2 e2 by increasing the senders’ incentive to signal. Around
b∗ =  [0 − Ψ (0 − )], the effect
h of a marginal increase
i in  on the incentive is

∗

1
∗
=   . Since  = −(0 − 2 ) (2 + (1 − )2 )  + 12 Ψ , collecting terms,

we obtain
1
 Ψ
b∗
2  
 0.
=
1

1 +  (0 − 2 ) [2 + (1 − )2 ]
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