Imperfect Signaling and the Local Credibility Test

By the LCT, it is easy to see that in any separating equilibrium a pooling deviation to some sufficiently low cost level of the signal is profitable to both types, so noseparating equili

Imperfect Signaling and the Local Credibility TestHongbin Cai, John Riley and Lixin Ye*

November, 2004

Abstract

In this paper we study equilibrium refinement in signaling models We propose a Local Credibility Test (LCT) that is somewhat stronger than the Cho and Kreps Intuitive Criterion but weaker than the “strong Intuitive Criterion” of Grossman and Perry

Allowing deviations by a pool of “nearby” types, the LCT gives consistent solutions for any positive, though not necessarily perfect, correlation between the signal sender’s true types (e.g., signaling cost) and the value to the signal receiver (e.g., marginal product) It also avoids ruling out reasonable pooling equilibria when separating equilibria do not make sense We identify conditions for the LCT to be satisfied in equilibrium for both thefinite type case and the continuous type case, and demonstrate that the results are

identical as we take the limit of the finite type case We then apply the characterization results to the Spence education signaling model and the Milgrom and Roberts advertisingsignaling model Intuitively, the conditions for a separating equilibrium to survive our LCT test require that a measure of signaling “effectiveness” is sufficiently high for every type and that the type distribution is not tilted upwards too much

*UCLA, UCLA, and Ohio State University We would like to thank In-Koo Cho, David Cooper, Massimo Morelli, James Peck, and seminar participants at Arizona University, Illinois Workshop on Economic Theory, Ohio State University, Penn State University, Rutgers University, UC Riverside, UC Santa Barbara, and Case Western Reserve

University, for helpful comments and suggestions All remaining errors are our own

unsatisfactory because the two cases are observationally equivalent

The reason for the inconsistent solutions in the above example is that the existing refinement concepts focus on deviations by a single type only and do not consider

deviations by a pool of types Grossman and Perry (1986a,b), in a bargaining context, propose an equilibrium refinement concept strengthening the Cho-Kreps Intuitive

criterion to allow pooling deviations In this paper, in a general signaling model, we weaken slightly the Grossman-Perry Strong Intuitive Criterion, and propose a “Local Credibility Test” (LCT) in which a possible deviation is interpreted as coming from one

or more types whose equilibrium actions are nearby We consider only local pooling deviations, first because they seem to us more natural, second because they have much ofthe power of global pooling deviations, and third because they are more easily analyzed

Moreover, the Local Credibility Test does not always rule out pooling equilibria infavor of separating equilibria We will argue that in some situations separating equilibria seem unreasonable while pooling equilibria can be rather appealing By allowing pooling deviations, the LCT avoids ruling out pooling equilibria in such situations Consider a simple two type education-signaling model, in which the high type must take a quite

type Now suppose there is only one low type agent in every 5 million high type agents

In such a situation separation seems highly unreasonable, because without taking the costly signaling action an agent should not be perceived much differently from being the high type By the LCT, it is easy to see that in any separating equilibrium a pooling deviation to some sufficiently low cost level of the signal is profitable to both types, so noseparating equilibrium satisfies the LCT

The thrust of our analysis is to derive conditions under which there exist

equilibria satisfying the LCT We study a family of continuous type models of which the Spence education signaling and the Milgrom and Roberts advertising signaling are both members We begin by formulating the concept of the LCT for the finite type models first, since the intuition is easier to present Then we consider a discretization of the continuous type model, and take the limit as the discretization becomes finer We

characterize conditions under which the Pareto dominant separating equilibrium of the model satisfies the LCT The required conditions are intuitive As long as a measure of signaling “effectiveness” is sufficiently high for every type and the type distribution is not tilted upwards too much, the separating equilibrium can survive our LCT test

In the continuous type case, the set of equilibrium signals is dense so that equilibrium signals can be only found outside the set of equilibrium signals However, thinking of the continuous type case as the limiting case of the finite type case with manyclose types, it is natural to generalize the concept of the LCT to the continuous type An equilibrium survives the 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 Another way of thinking about this credibility test in the continuous type case is the following If, for an on-equilibrium signal, there is such a deviation-perception pair, then those nearby types can credibly deviate to the particular on-equilibrium signal by throwing away ε amount of money

out-of-We derive conditions under which the LCT is satisfied in equilibrium in the continuous type case The conditions are exactly the same as in the limiting finite type case This is satisfactory, because models with continuous types and models with finitely

many types are theoretical tools for analyzing the same kind of real world problems Put differently, it would be highly unsatisfactory if an equilibrium refinement concept applies

to one case but not the other, or gives different answers for the two cases

The paper is structured as follows The next section uses simple examples to illustrate the basic idea of the LCT Then Section 3 presents the general signaling model

In Section 4, we formulate the concept of the LCT for the finite type case Then we derive conditions under which the LCT is satisfied by the Pareto dominant separating equilibrium in a discretized continuous type model as the discretization becomes finer Section 5 generalizes the formulation of the LCT to the continuous type case, and shows that the conditions for the LCT are exactly the same as in the discrete type case We discuss an issue of robustness in Section 6 Concluding remarks are in Section 7

A consultant ( , )s v has a signaling cost type i j s and a marginal product of i v , j

where s1< < <s2 s n and v1< < <v2 v m She can signal at level z at a cost of ( , ) C z s i

We suppose that C( , ) 0z s

s

∂ so that a higher type has a lower signaling cost If paid a

wage w her payoff is ( , , ) U z w s i = −w C z s( , )i In a competitive labor market for

consultants, her wage will be her marginal product perceived by the market Activity z is

a potential signal because the marginal cost of signaling, C( , )z s i

1 If ( , ) 0 π s v i j = for all i≠ j, the model reduces to the usual Spence model in which the negative

correlation between signaling cost and value to receivers is perfect While we assume ( , ) 0 π s v > for all

Initially we assume that each consultant observes her own signaling cost type but not her productivity We define ( )v s i =E v s{ | }i , and assume that types with lower

signaling costs have higher expected productivity , that is, v s( )1 <v s( ) 2 < <v s( )n

There is a continuum of separating Nash equilibria in this game A Nash

separating equilibrium with three signaling cost types is depicted below Each curve is anindifference curve for some signaling cost type A less heavy curve indicates a lower signaling cost type Note that the equilibrium choice for each type s (indicated by a i

shaded dot) is strictly preferred over the choices of the other types Such an equilibrium fails the Intuitive Criterion first proposed by Cho and Kreps (1987).2

To see this, suppose an individual chooses the signal ˆz and argues that she is type

2

s Is this credible? If the individual is believed, her wage will be bid up to v s so she ( )2

earns the same wage as in the Nash equilibrium but incurs a lower signaling cost

2 As noted by Cho and Kreps (1987), with more than two types, it is necessary to modify their original Intuitive Criterion or it loses much of its power For the modified Intuitive Criterion the question is whether any particular type is uniquely able to benefit from some out-of-equilibrium signal if the signal receivers correctly infer the signaler’s type

Fig: 2-1: Separating Nash Equilibria

1( )

v s

2( )

v s

3( )

However ( , ( ))z v sˆ 2 is strictly worse than ( , ( ))z v s1 1 for type s and strictly worse than1

3 3

( , ( ))z v s for types Thus the claim is indeed credible 3

Similar arguments rule out any Nash equilibrium where different signaling cost types are pooled Thus the only equilibrium that satisfies the Intuitive Criterion is the Pareto dominant separating equilibrium (i.e., the Riley outcome) in which each “local upward constraint” is binding

Next suppose that each consultant knows both her signaling cost type and her marginal product Again consider the Nash Equilibrium depicted above Suppose in this equilibrium three different types are pooled at each signal level Consider the three types

2 1 2 2 2 3

( , ), ( , ), ( , )s v s v s v pooled at z Suppose a consultant chooses ˆz and claims to be 2

type ( , )s v Is this credible? If the claim is believed, the consultant’s wage will rise 2 3

from v s to ( )2 v thus the consultant is indeed better off But any offer that makes type3

2 3

( , )s v better off also make types ( , ) and ( , )s v2 1 s v better off, since they have the 2 2

same signaling cost Thus there is no credible claim that type ( , )s v alone can make A 2 3

similar argument holds for each of the other types Thus any Nash separating equilibriumsatisfies the Intuitive Criterion An almost identical argument establishes that any Nash Equilibrium with (partial) pooling satisfies the Intuitive Criterion as well.3

Since all the types with the same signaling cost are observationally equivalent, it seems to us that any argument for ranking the equilibria in the first model (productivity unknown) should also be applicable to the second model as well There is a simple modification to the Intuitive Criterion that achieves this goal A consultant takes an out-

of-equilibrium action ˆz and argues that she is one of the types who, in the Nash

Equilibrium would have chosenz This being the case, applying the Bayes Rule, her 2

3 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 example Ramey (1996) extends the Cho and Sobel’s divinity concept to the case of a continuum of types Like the Intuitive Criterion, divinity faces the same problem of distinguishing types ( , ), ( , ), ( , )s v2 1 s v2 2 s v to interpret a possible deviation, while these 2 3

expected marginal product is v s Then once again, the unique Nash Equilibriums ( )2

satisfying the modified Intuitive Criterion is the Pareto Dominant separating equilibrium

We now argue that for some parameter values, the Pareto Dominant separating equilibrium defies common sense Consider the following example Suppose there are two signaling cost types For those with a high signaling cost (s s= 1), the cost of

signaling is c z with 1( ) c1(0) 0= and c1′ >0, and the mean marginal product is 100 For

those with a low signaling cost(s s= 2), the signaling cost is 2( ) (1 ) ( )1

100

c z = − ε c z

and the mean marginal product is 200 The Pareto dominant separating equilibrium is depicted below

The low type must be indifferent between (0, ( ))v s1 and the choice of types , that is2

2 ( )2 2( ) 200 (12 ) ( ) 1001 2

100

.Suppose that only 1 in 100 consultants is of types Then the unconditional mean 1

marginal product is 199 Thus essentially all the social surplus generated by the high types is dissipated by signaling and both types have an income which is approximately half the income that would have in the Nash pooling equilibrium! We believe a better criterion for ranking equilibria should not rule out pooling equilibria in such

circumstances We now introduce a further simple modification of the Intuitive Criterion that achieves this outcome

Local Credibility Test:

Suppose that an out-of-equilibrium signal ˆz is observed and that z− is the largest Nash

Equilibrium signal less than ˆz and z+ is the smallest Nash Equilibrium signal greater

than ˆz , if they exist Let ˆS be the subset of signaling cost types choosing z− or z+ withpositive probability For eachS⊂Sˆ, define ( )v S =E v s S{ | ∈ } Then the equilibrium passes the Local Credibility Test (LCT) if there is no S such that ˆ( , ( ))z v S is strictly

preferred over the Nash Equilibrium outcome if and only if s S∈

Note that if ˆz is smaller (greater) than all equilibrium signals, then z− ( z+) does

not exist and z+ ( z−) is the smallest (largest) equilibrium signal By the above definition,

ˆS is the subset of types choosing z+ ( z−) Also note that by considering a subset of ˆS

to be the singleton set of a single type choosing z+or z−, the definition of the Local Credibility Test allows deviations by single types It follows that only separating

equilibria can pass the Local Credibility Test

The idea of the Local Credibility Test is weaker than the Strong Intuitive Criterion

(SIC) proposed by Grossman and Perry (1986a,b) For any out-of-equilibrium signal ˆz ,

their criterion considers any subset of types as a potential deviating pool An equilibrium

fails the SIC if ˆz is credible for one subset of types Here we restrict attention to local

deviations This makes the analysis more tractable and, we believe, more plausible

Using the idea of the Local Credibility Test, we show that for the simple

consulting example above, the Pareto dominant separating Nash Equilibrium is robust to credible pooling deviations For concreteness, suppose c z1( ) 100= z and c z2( ) 99= z Then in the Pareto dominant separating equilibrium, z1=0 and z2 =1 Consider any convex combination zˆ (1= −λ)z1+λz2 =λz2 =λ Then by the definition of the LCT,

1 2

ˆ { , }

S = s s ConsiderS=Sˆ The average productivity of these two signaling cost types is

199 For ˆz=λ to be a credible deviation by S, both types must strictly prefer ˆ( ,199)z

Note that

1( ,199) 199 100ˆ

U z = − λ and U z2( ,199) 199 99ˆ = − λ,and the equilibrium payoffs are 100 and 101 Thus, for all λ = <zˆ 98 / 99, both types are

indeed better off choosing the out-of-equilibrium ˆz The Pareto dominant separating

Nash Equilibrium thus fails the LCT In fact no equilibrium passes the LCT, so the criterion fails to rank the different Nash Equilibria

We now show how the LCT can be applied when there are many types and, in the limit, a continuum of types Let the set of signaling cost types be S ={ , , }s1 s n , with probabilities { , , }f1 f where n ∑i f i =1 Suppose type s has a signaling cost i

product of those of type s have an expected marginal product of ( ) i v s i =s i

We seek conditions under which the Pareto dominant separating equilibrium passes the LCT In this equilibrium, the local upward constraints are binding Therefore,

as depicted below, those with signaling cost type s i−1 are indifferent between

Then choose z i+1 so that those with signal cost type s i+1 are indifferent between

1 1

1

2 2

( ,z s i+ s i+ ) and (z i+1,s i+1).

In Figure 2.3, these are the points C and C i+1 We will argue that type s must be i

indifferent between C i and C i+1 as depicted That is, C i+1 is the efficient separating contract for those with signaling cost types i+1.

A type s consultant is indifferent between ( , ) and ( , ) j z w z w′ ′ if and only if

Fig 2-3: Applying the LCT with many types

1 1 1

c z −c z =s− s + s+ −s = s− s+ −s Also, a type s i+1 consultant is indifferent between 1 1

( ,z z i + z i+ ) and (z i+,s i+ ) Again appealing to ,

c z+ −c z =s+ s+ − s − s+ = s+ s+ −s Adding equations and and noting that, by hypothesis, s i = 12s i−1+12s i+1, it follows that

( i ) ( )i i( i i)

c z+ −c z =s s+ −s Appealing, finally, to , it follows that type s is indeed indifferent between i

1 1

( , ) and (z s i i z i+,s i+ ) Thus the point C i+1 is the Pareto dominant separating Nash

Equilibrium contract for types i+1

Note that for any wage ˆw above12s i+12s i+1, there is a signal ˆz between z and

1

i

z+ that is strictly preferred by those with signaling cost type s i and s i+1 but is not

preferred by type s i−1 (or lower types) If f i < f i+1, the expected marginal product of

these two types, 1 1

++ exceeds 12s i +12s i+1 and, we can choose ˆw to be equal to the expected marginal product Then if types s and i s i+1 choose the out-of-equilibrium

signal ˆz , they can expect to be paid ˆw The out-of-equilibrium signal is therefore

credible and so the separating equilibrium fails the LCT Conversely if f i > f i+1, the expected marginal product of these two types is less than 1 1

f > f+ i= n− , the Pareto dominant separating equilibrium passes the LCT.

Note that this condition is independent of the step size δ between signaling cost types Treating the continuum of types in the limit as δ →0, it follows that the Pareto dominant separating equilibrium passes the LCT if the density function ( )f s is

everywhere decreasing

3 A General Signaling Model

We consider the following signaling environment A player, the sender, has privateinformation denoted byθ ∈Θ ⊆R n The sender’s (multidimensional) private informationaffects her expected payoff through an aggregated variable called her “true type” (e.g., marginal cost of signaling) Let ( ) :s g Θ →R s S, ∈ =[ , ]s s be the true type of the signal

sender The sender chooses an action (signal), y Y ∈ , where Y =[ , ]y y is the set of

feasible signals

Let ( ) :v g Θ →R v V, ∈ =[ , ]v v be the value of the sender’s product or service to areceiver We assume that the receiver’s payoff is linear in v so that it is only the

perceived expected value ˆv which determines the receiver’s response We can then write

the sender’s payoff as a function U s v y of her true type, the receiver’s expected value( , , )i ˆand the sender’s signal When the sender’s private information θ is multidimensional, it

is natural that the sender and the receiver may have similar preferences overθ, but place

different weights on its different dimensions In other words, the sender’s true type s and the receiver’s value v will be correlated, but not perfectly correlated.

Let ( ) :z g S →Y be a strictly monotone signaling function that fully reveals the

true type of the sender Accordingly, ( )U s =U s v s z s( , ( ), ( )) is the utility of the sender of

true type s in a separating equilibrium with the signaling schedule ( ) z s , where

( ) [ ( ) | ( ) ]

v s =E v q q s q =s

We maintain the following standard assumptions:

(a) U s v y is third order differentiable in all its elements; ( , , )ˆ

Under these assumptions, the standard result in the literature (Riley, 1979;

Mailath, 1987) shows that a separating equilibrium of the signaling game satisfies the following differential equation:

2 3

( , ( ), ( ))( )

In many signaling models, the signal is costly so that U s v s z s3( , ( ), ( )) 0< for all s In this

paper we make this assumption, and discuss what happens when this assumption is

relaxed Naturally, in a separating equilibrium the lowest type s gains nothing to choose any costly signal, so should just choose the least costly signal y The signaling schedule

given by and the initial condition ( , )s y give rise to the unique Pareto dominant

separating equilibrium of the game

This formulation of the signaling problem is quite general and includes many signaling models studied in the existing literature as special cases We illustrate this with two examples.4

Example 1: The Spence education signaling model

In the Spence education signaling model, a worker knows her own personal

characteristics related to productivity θ∈Θ (e.g., technical knowledge, discipline, learning ability, work habits, etc) These characteristics can be summarized into a single

dimensional variable: productivity, denoted by s Then a worker’s expected payoff is

( , , ) ( , )

U s s y = −s C s y , where s is her productivity unknown to firms, ˆs is her

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) C s y1( , ) 0< ; (ii) C s y2( , ) 0> ; and (iii) C s y12( , ) 0< It can be easily

4 Many other applications fit into the general signaling model In Appendix B, we show that with some variable transformations, the reserve price signaling model of Cai, Riley and Ye (2003) satisfies all the assumptions We then apply our characterization results to check when the Pareto dominant separating equilibrium in that model satisfies the LCT

verified that the single crossing and all other conditions are satisfied By the standard result, a separating equilibrium satisfies

2

( , , ( )) 1( )

Example 2: An advertising signaling model

This example is adopted from Milgrom and Roberts (1986) A monopolistic firm

can produce a good with a constant marginal cost c It sells to a unit mass of consumers The firm knows its product quality, denoted by s Among the consumers, α <1 are

informed about s The rest 1−α of consumers are uninformed about s , and their belief

is given by the distribution ( )G s on [ , ] s s For products of quality s , the consumers’ inverse demand function is p s bq= − , where b is a positive parameter and q is the quantity So, given price p , the demand of an informed consumer is q I = −(s p b) , and that of an uninformed consumer is q U = −s p bˆ ) , where ˆs is his perception of s The

total demand is thenq=αq I + −(1 α)q U

Suppose the firm spends z on advertising, which leads to a perception of ˆs by the uninformed consumers By choosing an optimal price given ˆs , the firm’s maximum

= Consider a possible separating signaling schedule ( )z s

The firm’s payoff function is

2

ˆ

ˆ( , , )

( , , ( )) (1 )( )( )

In the Pareto dominant separating equilibrium, the lowest type chooses zero advertising.

4 Finite Type Case

In this section we explore conditions under which an equilibrium satisfies the LCT in the finite type case

Specifically, the sender has N true types, denoted by s1= < < ×××<s s2 s N =s Let ( )G s i = prob s s.{ ≤ i} be the probability distribution of the sender’s true type Let ˆs

be the receiver’s perception of the sender’s type The sender’s expected payoff is

ˆ

( , , )

U s v y , which satisfies all the assumptions made in Section 2

Let z i =z s( )i , where z1 = < < ×××<z z2 z N =z , be an equilibrium signaling schedule Consider any signal ˆy Y y z∈ , ˆ≠ i, for all i

(L1): No credible deviation by the lowest type

When ˆy< z, ˆ( , ( ))y v s is a credible deviation if U s v s z( , ( ), )<U s v s y( , ( ), )ˆ

(L2): No credible deviation by the highest type

When ˆy z> , ˆ( , ( ))y v s is a credible deviation if U s v s z( , ( ), )<U s v s y( , ( ), )ˆ

(L3): No local pooling deviation

Definev s s( ,i i+1)=E v s s{ | = i ors s= i+1} When z s( )i < <y z sˆ ( i+1), ( , ( ,y v s sˆ i i+1))

is a credible deviation if U s v s s( , ( ,j i i+1), )yˆ >U s v s( , ( ), ),j j z j j i= and j i= +1and U s v s s( , ( ,j i i+1), )yˆ <U s v s( , ( ), ),j j z j j i≠ ori+1.

(L4): No local deviation by a single type

Whenz s( )i < <y z sˆ ( i+1), for l i i= , +1, ˆ( , ( ))y v s l is a credible deviation if

ˆ( , ( ), )l l

U s v s y >U s v s z( , ( ), ),l l l and, U s v s( , ( ), )j l yˆ <U s v s( , ( ), ),j j z j j¹ l

By the definition of the Local Credibility Test given in Section 2, if there does not exist any kind of credible out-of-equilibrium signal-perception, then the equilibrium {( , )}s z i i

survives the LCT

We say that an equilibrium “satisfies” a requirement among (L1)-(L4) if there is

no such credible deviation described in the requirement Requirements (L1), (L2) and

(L4) are all about deviations of one single type, which are the local conditions of the

Cho-Kreps Intuitive Criterion WhenN=2, they are exactly the Cho-Kreps Intuitive Criterion Requirement (L3) is the local condition of the Grossman and Perry criterion, allowing a deviation to be interpreted as from the two nearby types

By the argument similar to the Cho-Kreps’ Intuitive Criterion, it can be shown that if an equilibrium satisfies the LCT, it is the Pareto dominant separating equilibrium (the Riley outcome) Therefore, to check when an equilibrium satisfies the LCT, we only need to check when the Pareto dominant separating equilibrium satisfies (L3) Consider the Pareto dominant separating equilibrium depicted in Figure 4.1 below

Figure 4.1

For any interior point s with n n≥2, define ( , )v y be the intercept of indifference

curves I n-1and I n+1depicted in Figure 4.1 Then

For the lower boundary point s1 =s , define v as the solution to the following

equation: U s v z( , , )2 =U s s z( , , )2 2 2 The point v is depicted in Figure 4.2 below

there is no signal between z n and zn+1 (z1 and z2) to which both types ofs n and s n+1 (s1

and s2) are willing to deviate if the receiver has the perception of v So the LCT is satisfied On the other hand, if the LCT is satisfied, then it must be the case that v v≥

Otherwise, it is clear from Figures 4.1 and 4.2 that there are signals that both types of

The idea of Proposition 1 is simple For both interior points and boundary points,

v is the minimum perception by the receiver such that the two types s and n s n+1 can find

a common profitable deviation that is not attractive to any other types The receiver’s correct perception about the pool of s and n s n+1 is v If v <v for all n , then no pair of

types can find a local pooled deviation such that under the correct perception by the receiver, it is profitable only to them but not any other type If that is the case, the LCT is satisfied by the Pareto dominant separating equilibrium

Proposition 1 can be used to check the existence of equilibrium satisfying the LCT in any finite type model Let us return to the continuous type signaling model introduced in Section 2 in which the type space is S=[ , ]s s with a smooth distribution function ( )G s We study the following discrete type version where s1 =s s, i+1= +s i δ,

N

s =s and

1

s s N

− We let N =2k −1 fork≥2, that is, as k increases by one, each

interval is divided into two even ones We are interested in the limit case when k→ ∞, or0

δ → , as an approximation of the continuous type case

To simplify our analysis, we make the following technical assumptions, though these assumptions are not necessary for the general idea of our LCT to work

B1: U12 = 0 for all ( , , )s s y ; ˆ

B2: U22 =0 for all ( , , )s s y ˆ

It is easy to check that the Spence education signaling model satisfies both conditions, but the advertising signaling model does not However, in the next Section, we will show that our method can be easily applied to the advertising signaling model to get similar characterization results

Consider interior types first For any k and n∈{2,3, 2k−1}, fix s n = As s k

increases, let s n−1 and s n+1 be the nearest types to s , so s n-1= -s d( )k and

n

s + = +s d k Let v s( , )δ = =v E s s s[ | = n or s n+1], and ( , )v sδ be the solution to

Lemma 1: Whenδ →0, (i) ( , ) v s δ →s ; (ii) 2

1( , )

2

v s δ → ; (iii) 22

"( )( , )

Proof: See the Appendix

Lemma 2: Suppose conditions B1 and B2 hold When δ →0, (i) ( , ) v sδ →s ; (ii)

Proof: See the Appendix

For the boundary type s , let ( , ) v s δ be the solution to (U s+δ, ( , ), )v s δ z =

Thus, we suppose ( ) 0G s > We have

Lemma 3: Suppose conditions B1 and B2 hold When δ →0, (i) ( , ) v s δ → s ,

2( , ) 0

v s δ → , and 22

2 '( )( , )

3

2 ( , , )( , )

From Lemmas 1-3, we have our main characterization result for the limiting discrete type case

Theorem 1: Suppose conditions B1 and B2 hold When δ →0, the Pareto dominant separating equilibrium of the discrete type model satisfies the LCT if

(i) for any s∈( , ]s s ,

between signal z and perception v is similar for the different types in that example, the

indifference maps are similar and so indifference curves are close together As a result,

both types are better off deviating to ˆz if the receiver believes that both 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 indifference curve

2

I in Figure 4.1 will be everywhere above v , the average type Now only type s is better1

off deviating to ˆz if the receiver still thinks that both types may be deviating, making the

deviation-perception pair not credible

Intuitively, the rate at which the marginal rate of substitution declines with s is a

measure of signaling effectiveness Thus Theorem 1 suggests that when signaling

effectiveness is sufficiently large, the separating equilibrium will survive the LCT This intuition is reflected in conditions and Note that the slope of the indifference map is

( , , ( ))

( , , ( ))

U s s z s MRS s s z s

U s s z s

−

13 2

( , , ( ))

v s

U MRS s v z v

measures how rapidly the MRS declines with s at s (normalized at the level of MRS

itself) Figuratively, when the LHS is greater, the indifference curve I is far from 2 I in1Figure 4.1, hence there will be no credible deviation with the perception at v The RHS

of is the density function of s , normalized by the probability mass at s Clearly, the smaller this ratio, the smaller the perception of a pool involving s When the expected value of the two types, v , is lower in Figure 4.1, there will be no credible deviation.

The intuition for is similar, though less transparent The last term of the LHS

(over the denominator), 13

3

2U U

− , has exactly the same interpretation: a measure of how

rapidly the MRS declines with s From Figure 4.1., the critical value of v depends on

how rapidly the curve U n−1 increases with v and how slowly the curve U n+1 increases

2

( , , ( ))

v s

U MRS s v z s

33 2 3 23 2 2

U + is more flat (small MRS z∂ ∂ ), there will be no credible deviation with the

perception at v The RHS of 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

In summary, Theorem 1 says that the Pareto dominant separating equilibrium will satisfy the LCT if signals are effective in distinguishing types (i.e., MRS declining fast with types) and the type distribution is not too tilted upwards

In this section we derive conditions under which an equilibrium satisfies the LCT

in the continuous type case We show that the results here are exactly the same as those

in the limiting discrete type case studied in the preceding section

As in the discrete type case, we can focus on separating equilibria Consider any separating equilibrium ( ) :z s S →[ , ]z z ⊆Y Consider any signal ˆy Y∈

(L1): When ˆy< z, ˆ( , )y s is a credible deviation if U s s z( , , )<U s s y( , , )ˆ

(L2): When ˆy z> , ˆ( , )y s is a credible deviation if U s s z( , , )<U s s y( , , )ˆ

(L3c): When ˆ [ , ]y∈ z z , let z s( )0 = yˆ and consider a small neighborhood of s ,0 S0 ⊂S

Let s E s s Sˆ= [ | ∈ 0] If

( )i U s s y( , , )ˆ ˆ >U s s z s( , , ( )), for all s∈intS o

( )ii U s s y( , , )ˆ ˆ <U s s z s( , , ( )), for all s S∉ o

Then the signal-perception ˆ ˆ( , )y s is credible.

By the definition of the Local Credibility Test given in Section 2, if there does notexist any credible signal-perception, then the separating equilibrium ( )z s survives the

LCT

Requirements (L1) and (L2) are exactly the same as in the finite type case In the continuous type case, the results of Riley (1979) and Mailath (1987) show that there is a unique separating equilibrium if the lowest type sender chooses the signal that is optimal under complete information It is easy to see that this Pareto dominant separating

equilibrium is the only equilibrium that satisfies (L1) and (L2) Therefore, to check whether there exists an equilibrium satisfying the LCT, we only need to check whether this equilibrium satisfies Requirement (L3c)