Signaling Games and Applications

Fabrizio Adriani

April 2012. (Preliminary draft, please do not circulate.)

1 A simplified Spence-like model

Consider a Spence-type framework. There is a worker S and and employer R. The

worker can have two di↵erent types L, H. Type i = L, H has productivity ✓i with

✓L < ✓H . The worker has private information about his type and chooses a costly

signal s � 0 which may reveal information about his type to the employer (education).

The employer o↵ers a wage w that minimizes a quadratic loss function �(w � ✓i)2.

[Note: This implies that the employer o↵ers a wage w = E(✓i|s). (Results would not

change if we assumed many employers whose payo↵ is ✓i �w and competition among

employers)]. S’s payo↵ is w � cis where ci is the cost of the signal. We assume that

cL > cH (signaling is relatively less costly for higher types).

The timing of the game is as follows:

• t=0. Nature draws ✓ 2 {✓L, ✓H} from a distribution ↵. (↵i ⌘ Pr(✓i), i =

L, M, H).

• ✓ is observed by S who chooses s � 0.

• R observes s and chooses a wage w.

Equilibrium concept

The relevant equilibrium concept for this game is Perfect Bayesian Equilibrium

(PBE). A PBE is a strategy profile for S and R and beliefs for R which satisfy the

following two conditions

1. Both S and R play best replies given R’s beliefs.

2. R’s beliefs are determined from S’s strategy by using Bayes rule whenever pos-

sible.

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The first condition is straightforward. The second condition says that R uses S’s

equilibrium strategy to update his beliefs upon observing a signal. To see what this

means suppose that R’s prior is that the worker is type H with probability ↵H and

type L with probability ↵L = 1 � ↵H . Suppose that, in equilibrium, S’s strategy

prescribes choosing a signal s0 with probability �H(s0) when S is of type H and with

probability �L(s0) when he is of type L. Then, Bayes rule implies that

Pr(i = H|s0, ↵) =�H(s0)↵H

�H(s0)↵H + �L(s0)(1� ↵H)(1)

These are R’s (posterior) beliefs upon observing s0. A problem arises when s0 is never

played in equilibrium by S. If �L(s0) = �H(s0) = 0, Bayes rule breaks down. However,

in order to say whether a particular situation is an equilibrium, we need to say some-

thing about what happens out of equilibrium. What should R’s beliefs look like in this

case? The convention is that R is allowed to hold any beliefs (i.e. Bayes rule applies

only “whenever possible”). We will first characterize the standard cases of separating

equilibria and pooling equilibria. To start with, notice the following properties of the

sender’s (S) payo↵. Consider two wage-education combinations (w0, s0) and (w00, s00)

with w0 > w00 and s0 > s00. Then, whenever type L weakly prefers the high wage-high

education combination (w0, s0), i.e.

w0 � cLs0 � w00 � cLs00 (2)

type H strictly prefers the high wage-high education combination (w0, s0)

w0 � cHs0 > w00 � cHs00 (3)

This immediately follows from cH < cL. (To show that this is true, substitute w0 from

(2) into (3)). This property of payo↵s is called a sorting condition. A sorting condition

is a useful property of payo↵s which facilitates information transmission.

Separating equilibria

In a separating equilibrium (SE), type H chooses a di↵erent level of education from

type L. This implies that the level of education chosen is perfectly informative about

the type of worker. Denote with sH the level of education chosen by type H and with

sL the level of education chosen by type L. In a SE, sL 6= sH . This implies that

Pr(i = H|sH) = Pr(i = L|sL) = 1 while Pr(i = H|sL, ↵) = Pr(i = L|sH , ↵) = 0. As

a result, R will select a wage w = ✓H when observing sH and w = ✓L when observing

sL. In order to determine whether a given strategy (sH , sL) for S is an equilibrium

strategy, we should first check that each type has no incentive to mimic the other type.

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This reduces to checking that type H (L) cannot obtain higher payo↵ by sending signal

sL (sH). The two following constraints (aka incentive compatibility constraints) ensure

that this is the case,

✓H � cHsH � ✓L � cHsL (4)

✓L � cLsL � ✓H � cLsH (5)

The first inequality ensures that a type H is not better o↵ by “pretending” to be a

type L. The second inequality is the same for type L. The second thing we should

specify is what are the beliefs of the employer when the worker chooses a an education

level that is di↵erent from both sL and sH . Since in equilibrium the worker always

chooses either sL or sH , it is not clear at the outset how R would react to an out

of equilibrium signal s /2 {sL, sH}. Let ⇡(s) ⌘ Pr(i = H|s) denote the probability

that R assigns to the worker to be of type H when observing an out of equilibrium

signal s. Independently of R’s out of equilibrium beliefs, it is clear that in a SE type

L will always choose a signal sL = 0. If sL > 0, type L would profit by deviating to

s = 0. (Upon observing the deviation, R might attach either probability one to type

L and zero to type H or a positive probability to type H. In the first case type L only

benefits from “economizing” on the signal, in the second case he also benefits from

being o↵ered a higher wage). Hence, the incentive compatibility constraints can be

rewritten as

✓H � cHsH � ✓L (6)

✓L � ✓H � cLsH (7)

It is then immediate to check that any sH such that

✓H � ✓L

cL

sH ✓H � ✓L

cH

(8)

satisfies both incentive compatibility constraints. As a result, there is a continuum of

SE with sH in the above interval. To summarize, we have infinitely many equilibria.

Is there a way to reduce the indeterminacy? Riley argued that the most intuitive

equilibrium is the one where the cost of separation is minimized. This is the equilibrium

where sH = sminH ⌘ (✓H � ✓L)/cL. In this equilibrium, the level of education of

type H is the minimum necessary to prevent mimicking by type L. In other words,

the incentive compatibility constraint of type L holds with equality (which, from the

sorting condition, implies that the incentive compatibility of type H is satisfied with

strict inequality). Is there a formal argument that would select the Riley outcome?

Luckily, the Intuitive Criterion (Cho and Kreps 1987) is very helpful in this case.

The equilibria with sH > sminH are sustained by particular out of equilibrium beliefs.

Type H would always deviate to s < sH if a(s) were su�ciently high. In order to

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sustain the equilibrium, R must believe that the worker is of type L with su�ciently

large probability whenever observing a deviation to any s 2 (sminH , sH). These out of

equilibrium beliefs however violate the intuitive criterion. According to the intuitive

criterion, if a type can only lose (relative to his equilibrium payo↵) from a deviation

whereas other types might benefit, R’s out of equilibrium beliefs should assign zero

probability to the “losing” type. Consider any equilibrium with sH > sminH . In this

case, type L prefers his equilibrium payo↵ (i.e. sL = 0 and w = ✓L) to any deviation

s > sminH independently of the out of equilibrium beliefs that R holds. This immediately

follows from type L incentive compatibility constraint. On the other hand, type H,

who must choose an education level higher than sminH in equilibrium, could benefit

from reducing his education level if this would not change R’s beliefs, i.e. if R were to

believe that the deviation came from type H. As a result, out of equilibrium beliefs

⇡(s) should assign probability zero to type L for all deviations in the interval (sminH , sH).

But this implies that ⇡(s) = 1, so that type H would have incentive to deviate. As a

result, only the equilibrium with sH = sminH survives the intuitive criterion.

Pooling equilibria

In addition to the separating equilibria, there exist many pooling equilibria (PE).

In a PE, both types choose the same level of educations s̃. This implies that the

signal s is uninformative, i.e. Pr(i = H|s, ↵) = ↵H and Pr(i = L|s, ↵) = ↵L, where

↵H and ↵L = 1 � ↵H are the prior probabilities. In a PE, R chooses a wage w̃ =

↵H✓H + (1 � ↵H)✓L. In order to check that a strategy profile (s̃, s̃) for S is a PE,

we need first to check that S obtains a payo↵ larger than ✓L. This is because S can

always obtain at least ✓L by deviating to s = 0. Hence,

w̃ � cH s̃ � ✓L (9)

w̃ � cLs̃ � ✓L (10)

Clearly enough, cH < cL implies that (9) is always satisfied when (10) is satisfied. This

implies

s̃ w̃ � ✓L

cL

= ↵HsminH (11)

Second, we must specify out of equilibrium beliefs ⇡(s) for R upon observing a

signal s 6= s̃. Typically, a pooling equilibrium is sustained by out of equilibrium beliefs

that assign a large probability to type L. Consider the extreme case where, upon

observing any deviation s 6= s̃, R infers that S is of type L (and thus o↵ers w = ✓L).

Clearly enough, S has no incentive to deviate. In this case, we have a continuum of

PE with s̃ 2 [0, ↵HsminH ]. Are these out of equilibrium beliefs reasonable? Again, the

Intuitive Criterion is handy in this case. Consider a deviation to an education level

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sD 6= s̃. Independently of R’s out of equilibrium beliefs – a type L would always lose

from the deviation (relative to his equilibrium payo↵) if sD were such that

w̃ � s̃cL < ✓H � sDcL (12)

where the LHS is type L’s equilibrium payo↵, while the RHS is the maximum he can

obtain by deviating (i.e. if R believes that the deviation emanates from type H). A

type H might benefit from deviating to sD if

w̃ � s̃cH > ✓H � sDcH (13)

As a result, for deviations sD such that

s̃ + (1� ↵H)✓H � ✓L

cL

< sD < s̃ + (1� ↵H)✓H � ✓L

cH

(14)

type L would always lose while type H might benefit. As a result, R should infer

that such deviations emanate from type H. This would in turn give the incentive to

deviate to type JH. It then follows that no PE survives the intuitive criterion. With

only two types, the Riley outcome is the only intuitive PBE.

2 Countersignaling (Feltovich et al. 2002)

Several recent papers (one is Benabou and Tirole 2003 discussed in the next section)

consider the case in which the receiver, in addition to the endogenous signal sent by

the sender, has also access to exogenous private information.1. This opens the door

to interesting results. Feltovich, Harbaugh, and To (FHT) point out how people of

moderate ability show o↵ their credentials to impress employers or society, whereas the

very talented people often downplay their credentials. Similarly, the moderately rich

or powerful often engage in public displays of wealth or power while the extremely rich

or powerful prefer to avoid these displays. The list of examples is probably infinitely

long. In social relations, your acquaintances display good inclination by ignoring

yours flaws, while close friends show intimacy by highlighting them. FHT call this

behaviour countersignaling. Is it possible to rationalize countersignaling in the contest

of a signaling game? In what follows, I will modify our version of Spence’s model

to incorporate results by FHT. The basic intuition is that the receiver may have

access to noisy information about the sender’s type which is not available to the

sender himself. The model is thus characterized by two-sided private information. For

instance, with some probability the receiver may get information about the sender

1An example is Dessein 2006, se below.

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from a friend and the sender may be unsure about whether this has occurred or

not. The receiver’s private information creates scope for countersignaling. Again, the

original paper considers a much more general version. FHT also provide evidence of

countersignaling from a lab experiment.

2.1 Setup

We now extend our simple Spence-like framework to incorporate two ingredients. First,

now the worker can have three di↵erent types L, M, H. Type i = L, M, H has produc-

tivity ✓i with ✓L < ✓M < ✓H . Second, in addition to the endogenous signal s, R has

access to other sources of information (for instance references) that are not observed by

S. FHT consider a fairly general information structure. We will focus on an extreme

example. Assume that, whenever the worker is of type H, this will be exogenously

revealed to R with probability p. In other words, when the type is H, R observes a

signal � with probability p. Hence, when R observes the exogenous signal �, he will

know for sure that the worker is H. When he does not, he will remain unsure. The

timing of the game is as follows

• t=0. Nature draws ✓ 2 {✓L, ✓M , ✓H} from a distribution ↵. (↵i ⌘ Pr(✓i), i =

L, M, H).

• ✓ is observed by S who chooses s � 0.

• R observes s. When ✓ = ✓H observes a private exogenous signal � = H, revealing

the type with probability p. R then chooses a wage w.

In what follows we will focus on pure strategies. I will also ignore participation con-

straints by assuming that they are always satisfied. We will first characterize the

standard separating equilibrium and then discuss countersignaling equilibria.

Separating Equilibria In a SE, the worker chooses a signal si, i = L, M, H

according to his type with sL 6= sM , sM 6= sH and sL 6= sH . In a SE, R’s exogenous

information plays no role since all relevant information is already revealed by the choice

of the signal.

Before proceeding, we should note a technical problem. Suppose that type H

deviates and chooses a di↵erent signal, say sM , and at the same time R observes

� = H. What are R’s belief in this case? On the one hand, the endogenous signal sM

tells R that S is of type M with probability one. On the other hand, the exogenous

signal tells R that S is of type H with probability one. Clearly, both cannot be

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true. The problem is “solved” once we note that, given the separating equilibrium,

the joint event “S chooses sM and � = H” is a zero probability event. Hence, the

posterior probabilities Pr(✓H |� = H, sM) and Pr(✓M |� = H, sM) are not well defined

(Exercise: by applying Bayes rule, show that these probability are not well defined

when Pr(� = H, sM) = 0). As we have seen, in this case convention wants that R

can hold arbitrary beliefs. Therefore, we can assume that R discards his exogenous

signal and believes S to be of type M with probability one as the endogenous signal

suggests.2

We will now characterize the Riley outcome. Again, this is the separating equi-

librium in which signaling costs are minimized. Clearly enough, type L will always

choose sL = 0 since, given perfect separation, his type will be revealed. Hence, he has

no incentive to incur the cost of a positive signal. He will then receive a wage equal

to his productivity ✓L. Type M chooses a signal sM > 0 and obtains a wage equal to

✓M . His payo↵ is thus ✓M � cMsM . In the Riley outcome, type M chooses sM such

that type L is indi↵erent between choosing sM (mimicking) and sL = 0. Hence, sM

satisfies

✓L = ✓M � cLsM (15)

Type M ’s payo↵ is thus

✓M �cM

cL

(✓M � ✓L) (16)

Finally, type H chooses sH and receives a payo↵ ✓H � cHsH . Again, sH is the lowest

signal that prevents mimicking from type M (it is easy to show that if sH is not

mimicked by M , then it is not mimicked by L). Hence, sH solves

✓M �cM

cL

(✓M � ✓L) = ✓H � cMsH (17)

which implies that

sH =1

cM

(✓H � ✓M) +1

cL

(✓M � ✓L) (18)

Hence, type H payo↵ is

✓H �cH

cM

(✓H � ✓M)� cH

cL

(✓M � ✓L) (19)

So far, we have only considered “upward” incentive compatibility constraints. We

should also make sure that type H has no incentive to mimic type M and that type

M has no incentive to mimic type L. This is left as an exercise:

2To what extent these out of equilibrium beliefs are plausible is another matter. For instance, if

we allowed S to “tremble” (to make mistakes with arbitrary small probability) when choosing s, R

would reject the endogenous signal s in favour of the exogenous signal � = H. Intuitively, this follows

since R would rationally believe that S committed a mistake when choosing s.

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Exercise 1. 1) Show that ✓M � cM

cL(✓M � ✓L) > ✓L so that type M has no incentive to

mimic type L. 2) Show that

✓H �cH

cM

(✓H � ✓M)� cH

cL

(✓M � ✓L) > ✓M �cH

cL

(✓M � ✓L) (20)

so that type H has no incentive to mimic type M .

Finally, an example of out of equilibrium beliefs supporting this equilibrium are

beliefs such that R infers that S is of type L with probability one upon observing any

s > 0 such that s 6= sM and s 6= sH .

Countersignaling Equilibria So far, our results are perfectly in line with the

predictions of a standard Spence-like model. There are, however, other equilibria that

are interesting. A countersignaling equilibrium is an equilibrium in which type H and

type L are pooled together while type M reveals himself. Intuitively, a countersignaling

equilibrium would not be possible in a standard Spence-like framework since the sorting

condition cL > cM > cHwould prevent type H from being pooled only with type L

(one can however obtain equiibria with type H pooled with type M and type H pooled

with both). In our setting, however, the sorting condition is altered by the fact that

R receives an exogenous private signal.

Consider again, among all possible countersignaling equilibria, the one that involves

the lowest signaling costs. This is such that type L and type H choose s = 0 and

type M chooses s⇤M > 0. We want to find necessary and su�cient conditions for the

existence of this outcome.

Upon observing s = 0, R is uncertain on whether S is of type H or of type L. If

he receives � = H, he will believe that S is of type H with probability one. However,

what happens when he does not observe �? Let ↵L and ↵H be the prior probabilities

of type L and H. Denote with � = ? the event “R does not receive any (exogenous)

signal”. We want to determine Pr(✓i|s = 0, � = ?). From Bayes’ rule:

Pr(✓i|s = 0, � = ?) =Pr(s = 0, � = ?|✓i) Pr(✓i)P✓i

Pr(s = 0, � = ?|✓i) Pr(✓i)(21)

Note that, conditional on ✓i, the event s = 0 is independent of � since S does not

observe � when choosing s (intuitively, the only thing linking the two events is ✓i).

Hence, the above can be rewritten as:

Pr(s = 0|✓i) Pr(� = ?|✓i) Pr(✓i)P✓i

Pr(s = 0|✓i) Pr(� = ?|✓i) Pr(✓i)(22)

Since in equilibrium M never chooses s = 0, it follows that Pr(s = 0|✓M) = 0.

Similarly, since H and L always choose s = 0, Pr(s = 0|✓L) = Pr(s = 0|✓H) = 1. By

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assumption, Pr(� = ?|✓H) = 1 � p and Pr(� = ?|✓L) = Pr(� = ?|✓M) = 1. This

follows since R observes � = H with probability p when S is of type H and never

observes � = H when S is of a di↵erent type. By using these results we obtain

Pr(✓H |s = 0, � = ?) =↵H(1� p)

↵H(1� p) + ↵L

(23)

Pr(✓L|s = 0, � = ?) =↵L

↵H(1� p) + ↵L

(24)

Pr(✓M |s = 0, � = ?) = 0 (25)

The equilibrium payo↵s are as follows. Type M reveals himself and obtains

✓M � cMs⇤M (26)

Type L sends s = 0 and obtains

Pr(✓L|s = 0, � = ?)✓L + Pr(✓H |s = 0, � = ?)✓H = (27)

↵L

↵H(1� p) + ↵L

✓L +↵H(1� p)

↵H(1� p) + ↵L

✓H (28)

Type H obtains ✓H when � = H (with probability p) and

↵L

↵H(1� p) + ↵L

✓L +↵H(1� p)

↵H(1� p) + ↵L

✓H (29)

when � = ? (with probability 1� p). Hence, type H expected payo↵ is

p✓H + (1� p)

↵L

↵H(1� p) + ↵L

✓L +↵H(1� p)

↵H(1� p) + ↵L

✓H

�(30)

For a countersignaling equilibrium : 1) it is incentive compatible for type M to send

s⇤M > 0, 2) it is incentive compatible for type L and type H to send s = 0. We start

by checking incentive compatibility for type L

↵L

↵H(1� p) + ↵L

✓L +↵H(1� p)

↵H(1� p) + ↵L

✓H � ✓M � cLs⇤M (31)

Incentive compatibility for type M is given by

↵L

↵H(1� p) + ↵L

✓L +↵H(1� p)

↵H(1� p) + ↵L

✓H ✓M � cMs⇤M (32)

These two conditions can be rewritten as

↵L(✓M � ✓L)� (1� p)↵H(✓H � ✓M)

(↵H(1� p) + ↵L)cL

s⇤M ↵L(✓M � ✓L)� (1� p)↵H(✓H � ✓M)

(↵H(1� p) + ↵L)cM

(33)

Note that, since cM < cL, there always exists s⇤M satisfying the above whenever

↵L(✓M � ✓L)� (1� p)↵H(✓H � ✓M) > 0. Hence,

↵L(✓M � ✓L) > (1� p)↵H(✓H � ✓M) (34)

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is a necessary condition for a countersignaling equilibrium. Type H incentive compat-

ibility is

p✓H + (1� p)

↵L

↵H(1� p) + ↵L

✓L +↵H(1� p)

↵H(1� p) + ↵L

✓H

�� ✓M � cHs⇤M (35)

Solving for s⇤M yields

s⇤M �(1� p)↵L(✓M � ✓L)� (p↵L + (1� p)↵H)(✓H � ✓M)

(↵H(1� p) + ↵L)cH

(36)

A countersignaling equilibrium exists if and only if there exists s⇤M satisfying (33) and

(36). Assuming ↵L(✓M � ✓L) > (1� p)↵H(✓H � ✓M), this occurs if

(1� p)↵L(✓M � ✓L)� (p↵L + (1� p)↵H)(✓H � ✓M)

(↵H(1� p) + ↵L)cH

<

↵L(✓M � ✓L)� (1� p)↵H(✓H � ✓M)

(↵H(1� p) + ↵L)cM

(37)

or

[(1� p)↵L(✓M � ✓L)� (p↵L + (1� p)↵H)(✓H � ✓M)]cM <

[↵L(✓M � ✓L)� (1� p)↵H(✓H � ✓M)] cH (38)

Rearranging

↵L(✓M � ✓L)[(1� p)cM � cH ] < (✓H � ✓M)[p↵LcM + (1� p)↵H(cM � cH)] (39)

Summarizing, conditions (34) and (39) are necessary and su�cient for a countersig-

naling equilibrium. Intuitions can be obtained by looking at su�cient conditions.

Assuming that (34) is satisfied, a countersignaling equilibrium always exists if cM and

cH are relatively close. (The l.h.s of (39) becomes negative when cM and cH are close

while the r.h.s is always positive). Intuitively, when the cost of signaling is almost the

same for type H and type M , type H prefers to rely on the exogenous signal rather

than spending a large amount of resources in signaling his type. Also, both condi-

tions are satisfied when p is relatively large. When the probability of being directly

recognized as type H is large, type H does not bother to signal his type.

The e↵ect of di↵erences of productivity across types is more ambiguous. When the

ratio ✓H�✓M

✓M�✓Lis small, condition (34) is satisfied, but condition (39) may be violated.

The reverse happens when the ratio is large. Intuitively, if the productivity gap be-

tween type H and type M is too small relative to the gap between M and L, type H

may want to mimic type M . If it is too large, type M may want to mimic.

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3 Dissipative advertising as signals of product qual-

ity (Milgrom and Roberts 1986)

The idea that dissipative advertising (i.e. advertising with no direct information con-

tent) can be used as a signal of product quality is from Milgrom and Roberts (1986).

The model I am going to present is an (over)simplified version of their model.

Nelson (1970, 1974, 1978) di↵erentiates between “search” goods and “experience”

goods. Search goods are goods whose quality can be determined through inspection

of the good. Experience goods are goods whose quality can be assessed only after

consumption. All real world goods fall in between these two extreme categories but

some goods resemble experience goods more closely than others (e.g. a meal at a

restaurant, a new drink or a new drug).

In the case of search goods, the seller’s claim that the product is of high quality can

be verified before consuming the good. On the other hand, for an experience good, the

seller’s claims about the quality are unverifiable. One would accordingly be tempted

to conclude that consumers should not pay attention to such claims. Thus, it would

seem that advertising has no purpose in the case of experience goods. Why do we

observe advertisements for experience goods then?

If only high quality sellers advertise their products, consumers should infer, from

the very fact that the product is advertised, that the product is high quality. This

happens even if the advertisement by itself does not convey any meaningful information

about the product’s quality. The idea is that what is informative is not the content of

the advertisement but the fact that the seller chooses to “waste” resources to advertise

the product.

The natural question is then what prevents low quality sellers from advertising as

well (i.e. mimicking high quality). Formally, what is needed is a sorting condition

that makes advertisement relatively more costly to low quality sellers.

3.1 An Example

Consider the following lemon problem. There is a price setting seller S and a buyer

B living an infinite number of periods t = 1, ...,1. In each period, S is endowed

with a good and B demands the good. The good can be of two qualities: type H

with probability � and type L with probability 1� �. The quality of the good is the

same in all periods. S chooses the price in period 1 and cannot change it afterward. I

assume that if B does not purchase the good in period 1, he will leave the market and

will not purchase it in any of the following periods. S’s valuation for a good of quality

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✓ 2 {H, L} is v✓ > 0 (alternatively, v✓ > 0 can be interpreted as the cost of producing

the good). B’s evaluation for a good of quality ✓ is u✓ > v✓. I will also assume that

vH > uL so that a high quality seller is never willing to sell at the maximum price at

which B would buy a low quality good. Under full information, high quality goods

would trade at a price pH = uH and low quality goods would trade at a price pL = uL.

The discount factor is denoted with � 2 (0, 1). Profits for a type H seller are

1

1� �[uH � vH ] (40)

Profits for a type L seller are1

1� �[uL � vL] (41)

Notice that since S has all the bargaining power, under full information, B makes zero

surplus in each period. I will focus on separating equilibria in which both players play

pure strategies.

Assume now that, at time 1, only S knows the quality of the good while B relies on

the prior information that a type H good is drawn with probability �. If B consumes

the product at time 1, he will learn the quality and will benefit from this information

in all subsequent periods.

No Advertising Suppose that S can only convey information through the price.

In a separating equilibrium type L will set a price pL = uL in order to extract all the

surplus from B. He will then obtain his full information payo↵. What about type H?

Suppose he charges a price pH � vH (by charging pH < vH he would make a loss and,

therefore, we can ignore this case). If type L tries to mimic type H by charging pH

as well, he would be able to sell only in the first period (since, once B discovers the

quality at time 1, he will refuse to buy a product of quality L at a price greater than

uL < vH in the subsequent periods). For pH uH (B’s participation constraint), type

L’s incentive compatibility constraint then requires

1

1� �[uL � vL] � pH � vL (42)

Assume for the moment that 11��

[uL� vL] + vL � vH . The maximum price chargeable

by type H so that type L’s incentive compatibility and B’s participation constraint

are satisfied is

p⇤H = min

1

1� �[uL � vL] + vL, uH

�(43)

Hence, there is a separating equilibrium characterized by pL = uL and p⇤H . In general,

for � close enough to one, p⇤H = uH so that we obtain the full information outcome.

However, for � small enough, p⇤H < uH . The idea here is that, in order to prevent

12

mimicking, type H accepts lower profits (i.e. leaves a positive surplus to B) by setting

a price lower than the full information price. This is a credible signal and a separating

outcome can be sustained. Intuitively, since by mimicking type L makes profits only

in period one, the incentive to mimic is higher the lower the discount factor. Hence,

the lower the discount factor, the lower the price that type H must charge.

The equilibrium just characterized is not the only separating equilibrium. There is

a continuum of separating equilibria with pL = uL and pH in the interval [vH , p⇤H ]. To

see this, assume that pH = p 2 [vH , p⇤H). If type H deviates to p0 > p, B might infer

that S is of type L and chooses not to buy. Given these out of equilibrium beliefs,

type H is unwilling to increase the price. Hence, pH = p is indeed an equilibrium.

On the other hand, these out of equilibrium beliefs are not entirely convincing. As

it turns out, the separating equilibrium characterized by pL = uL and p⇤H is the only

separating equilibrium which survives the Intuitive Criterion (Cho and Kreps 1987).

Consider the equilibrium just described with pH = p < p⇤H and consider a deviation

to a price slightly lower than p⇤H . Clearly enough, type L has no incentive to deviate

to (a price slightly less than) p⇤H since, by the definition of p⇤H , he would make higher

profits by charging pL = uL. Di↵erently, type H could benefit from selling at a

higher price if B believed the deviation coming from type H. Hence, according to the

Intuitive Criterion, out of equilibrium beliefs should assign probability zero to type L

and probability 1 to type H deviating. Given these out of equilibrium beliefs, type

H would have an incentive to charge a higher price. The only separating equilibrium

surviving this argument is then pH = p⇤H .

If on the other hand, 11��

[uL � vL] + vL < vH , the only separating equilibrium is

such that type L charges pL = uL (and B buys) and type H charges pH > uH at which

B chooses not to buy. Intuitively, at any price at which type H would be willing to

sell, type L is willing to mimic. Hence, the only separating outcome is such that the

high quality is driven out of the market.

Exercise 2. Determine B’s out of equilibrium beliefs that support the equilibrium

characterized by pL = uL, pH > uH , and trade occurs only at pL.

Incidentally, note that 11��

[uL � vL] + vL < vH always holds when � ! 0. The

reason becomes clear once we note that, for � ! 0, the market converges to a standard

lemon market. Hence, Akerlof’s result applies.3 Notice that the outcome in which the

low quality trades at pL = uL and the high quality is not traded is an equilibrium also

when 11��

[uL � vL] + vL � vH but, in this case, it fails the Intuitive Criterion.

3I am however neglecting mixed strategies. This simplification is far from innocuous. If B were

allowed to randomize, quality H could be traded with positive probability even when � = 0.

13

Exercise 3. Show that the equilibrium characterized by pL = uL, pH > uH , and trade

occurring only at pL fails the Intuitive Criterion when 11��

[uL � vL] + vL � vH .

Dissipative Advertisements. Suppose now that the seller is allowed to spend

an amount A � 0 in dissipative advertisements in the introductory period. The basic

intuition is that type H may choose to “burn” some of his first period profits in order to

make mimicking unappealing to type L. To fix ideas, consider a separating equilibrium

in which type H sets a price pH = uH and chooses a level of advertisement AH > 0,

while type L sets pL = uL and chooses AL = 0. Incentive compatibility for type L

implies1

1� �[uL � vL] � uH � vL � AH (44)

The minimum advertisement expense preventing mimicking is thus

A⇤H = (uH � vL)� 1

1� �[uL � vL] (45)

Notice that A⇤H > 0 implies uL � vL < (1 � �)(uH � vL). Notice also that A⇤

H must

not violate type H participation constraint (type H should not make losses!). Type

H makes nonnegative profits if

(uH � vL)� 1

1� �[uL � vL] 1

1� �[uH � vH ] (46)

where the l.h.s. is A⇤H and the r.h.s. denotes type H’s discounted stream of profits

when charging pH = uH . The above can be rewritten as

1

1� �[uL � vL] � (uH � vL)� 1

1� �[uH � vH ] (47)

or1

1� �[uL � vL] � (vH � vL)� �

1� �[uH � vH ] (48)

and, hence, it may hold even when 11��

[uL� vL] < vH � vL (and the condition A⇤H > 0

always holds in this case). As mentioned, given 11��

[uL�vL] < vH�vL, in the absence

of advertising, the only equilibrium with no advertising would be one in which type H

does not trade. Hence, advertising allows type H to trade for parameter values such

that trade would no occur without advertising.

The intuition is that deterring mimicking through advertisement is more e�cient

than deterring mimicking through price reductions since advertising reduces only the

introductory phase profits (t = 1) while price reductions reduce profits in all subse-

quent periods. Mimicking generates profits only in the introductory phase when B

is not aware of the quality. Decreasing profits in all periods is a very costly way to

prevent mimicking. Hence, advertisement is a more e�cient way to “waste” resources

14

in signaling. Consider now the case 11��

[uL � vL] � vH � vL. As we have seen, in the

absence of advertisements, the intuitive equilibrium would be characterized by p⇤H and

pL = uL. Clearly enough, type L profits are the same independently of advertisement.

Type H is better o↵ in the equilibrium with advertisement if

1

1� �[uH � vH ]� A⇤

H �1

1� �[p⇤H � vH ] (49)

where the l.h.s. are type H’s profits with advertisement and the r.h.s. are his profits

without. Substituting A⇤H and p⇤H , the above becomes:

1

1� �[uL � vL] uH � vL (50)

Given uH > uL, type H always prefers advertising when � is relatively small (and

hence the incentive to mimic is stronger). More precisely, it can be shown that when

the condition above is violated so that

1

1� �[uL � vL] > uH � vL (51)

p⇤H is equal to uH and A⇤H is equal to zero so that advertising becomes irrelevant and

the two equilibria yield the same payo↵. For other parameter values, the equilibrium

with advertisement is preferred by type H (provided that the high quality is traded).

Some issues to note. First, the equilibrium characterized by a level A⇤H of adver-

tisements is not the only separating equilibrium with advertisement. However, it is

quite easy to show that the equilibrium characterized by A⇤H is the only separating

equilibrium that survives the Intuitive Criterion. Second, all equilibria with no adver-

tising (AL = AH = 0) discussed before are still equilibria when advertising is allowed.

It is su�cient that B, upon observing any A > 0, assumes that S is of type L. These

out of equilibrium beliefs are however not reasonable in some cases and (for param-

eter values) these equilibria fail the intuitive criterion once advertisement is allowed.

Third, I did not characterize pooling equilibria. This is left as an exercise.

A final (but to some extent more important) note is that B (the consumer) is

weakly worse o↵ when S uses advertisements. This is clear once we observe that, if

advertisement is not used, B may obtain positive surplus since p⇤H < uH if � is not too

large. By contrast, in the equilibrium with advertisement that we have characterized,

B always makes zero surplus. The reason is that, like education in Spence’s model,

advertisements are socially wasteful. The cost of advertisement is partially shifted on

the buyer.

15

3.2 The Original Paper and Other Applications

Some of the assumptions I made are not entirely plausible. For instance, the assump-

tion that the price cannot be changed after the introductory phase. The original paper

by Milgrom and Roberts (MR) considers a more realistic/general framework. MR also

consider a more general specification for the demand function (in my example demand

is perfectly inelastic). Interestingly, an e↵ect of relaxing these assumptions is that in

equilibrium the introductory price for type H may be higher than the full information

monopoly price. The intuition is that selling a lower quantity (charging a higher price)

is an alternative way to reduce first period profits and then can be used as a signal

of quality. The MR framework has been applied to a number of settings. A relatively

recent application to political economy is Prat 2002. He considers campaign adver-

tisements in a model of elections in which lobbies have private information about the

candidates’ ability.

4 Intrinsic and Extrinsic Motivation (Benabou and

Tirole 2003)

4.1 Overview

This model is related to the informed principal literature.4 The framework is similar to

standard principal-agent models. Di↵erently from those, however, here the principal

(P ) has private information. The principal’s problem is how to give incentives to

exert e↵ort to the agent (A). The principal moves first by o↵ering a contract that

the agent can accept or reject. Since P has private information, A will try to infer

P ’s information from the contract o↵ered. Hence, this is a signaling game. The fact

that the principal has private information can be interpreted in two di↵erent ways.

The principal may have private information about the agent’s ability (for instance, a

PhD supervisor may be able to assess his student’s potential better than the student

himself). Alternatively, the principal may have more information about the cost of

e↵ort (because he has private information about the type of task to be performed).

We will stick to the first interpretation but results would not change if we used the

second.

After accepting the contract, A chooses the level of e↵ort and a level of output is

realized. A is then rewarded according to the contract.

4See Maskin and Tirole 1992 Econometrica pp 1-42.

16

The bottom line of the model is that incentives may backfire. High powered incen-

tives may signal to the agent that the principal “does not trust him” (i.e. P believes

that A is low ability). This would lead A to revise his beliefs about his own abil-

ity. If the production function is such that ability and e↵orts are complement, high

powered incentives would lead to less e↵ort (which is the opposite of what standard

economic theory predicts). As a result, in equilibrium, P gives low powered incentives.

In Benabou and Tirole words, extrinsic (monetary) incentives may crowd out intrinsic

motivation.

4.2 Assumptions and Notation

There is a task that can be successful or unsuccessful. Both P and A experience a

direct private benefit (W > 0 and V > 0 respectively) when the task is successful. If A

does not exert e↵ort (e = 0), the task is always unsuccessful. If A exerts e↵ort (e = 1),

the task is successful with probability ✓ 2 (0, 1]. ✓ can be interpreted as A’s ability.

A has a (known) cost c of exerting e↵ort. P ’s problem is to o↵er a contract which

provides A with incentives to exert e↵ort. A’s e↵ort is not observed by the principal

and therefore cannot be contracted upon. We restrict attention to simple contracts

in which the principal pays a bonus b > 0 when the task is successful. Note that V

can be interpreted as A’s “intrinsic” motivation while the monetary reward b o↵ered

when the task is successful can be seen as A’s “extrinsic” incentive. The principal’s

expected payo↵ is thus

UP = ✓e(W � b) (52)

The agent’s expected payo↵ is

UA = e[✓(V + b)� c] (53)

Incidentally, note that ability ✓ and e↵ort e are complements. With perfect infor-

mation, A would exert e↵ort whenever ✓(V + b) � c. Under full information, P o↵ers

the minimum bonus that achieves e↵ort by A. The full information bonus b⇤ is thus

b⇤ = maxh c

✓� V, 0

i(54)

Information Structure

Ability ✓ can take two values: ✓H with probability ⇡ and ✓L < ✓H with probability

1 � ⇡. P knows the value of ✓. A observes a noisy private signal � 2 (�, �). �

is distributed according to the conditional density gK(�), where K 2 {L, H}. The

cumulative distribution is denoted by GK(�) (Intuitively, GH(�̂) is the probability

17

that the agent receives a signal lower than �̂ given that ✓ = ✓H . The same applies

when K = L.) The density gK(�) is assumed to satisfy the Monotonic Likelihood

Ratio Property (MLRP), i.e. the ratio

gH(�)

gL(�)(55)

is a continuous increasing function of �. This assumption ensures that the signal is

informative. Intuitively, higher signals are more likely to be sent when ✓ = ✓H than

when ✓ = ✓L.

Exercise 4. Show that the MLRP implies the following

• for all �, GH(�) < GL(�).

• for all � and �0 > �,gH(�0)

gH(�)>

gL(�0)

gL(�)(56)

• for all � and �0 > �,1�GH(�0)

1�GH(�)>

1�GL(�0)

1�GL(�)(57)

(Assume lim�!� gH(�)/gL(�) =1 and lim�!� gH(�)/gL(�) = 0. A possible way

to show the result is to rewrite the condition as

[1�GH(�0)][1�GL(�)]� [1�GL(�0)][1�GH(�)] > 0 (58)

and to show that the left hand side is an increasing/decreasing function of � for

� 2 (�, �0). Then show that it converges to zero when � goes to �0 or �. This

implies that the l.h.s. is positive in the relevant interval.)

Finally, we assume that lim�!� gH(�)/gL(�) = 1 and lim�!� gH(�)/gL(�) = 0.

To summarize, the timing of the game is the following

• t = 0. Nature draws ✓ 2 {✓L, ✓H} which is observed by P .

• t = 1. P o↵ers a bonus b to A.

• t = 2. A observes b, observes a private signal � and chooses whether to accept

or reject the o↵er.

If the o↵er is rejected the game ends with both parties obtaining payo↵ zero, if it is

accepted:

• t = 3. A chooses the level of e↵ort e 2 {0, 1}.

• t = 4. The outcome of the task (success/failure) is observed and payo↵s are

realized.

18

4.3 Self-confidence, Trust, and Incentives

Our aim is to prove proposition 3 in the Benabou and Tirole 2003 paper. Let b⇤L denote

the perfect information bonus when ✓ = ✓L, i.e.

b⇤L = max

c

✓L

� V, 0

�(59)

Proposition 1. (Proposition 3 in Benabou and Tirole 2003). In the two-type case:

(i) In any equilibrium, the principal o↵ers a low bonus b < b⇤L to a more able agent

(✓ = ✓H), and randomizes between the bonuses b and b⇤L when dealing with a less

able agent (✓ = ✓L ). (ii) There is a unique D1-refined equilibrium, and it is such

that b = 0. The probability of pooling (o↵ering b = 0 when ✓ = ✓L), x⇤ > 0, and

the unconditional probability of no bonus, ⇡ + (1� ⇡)x⇤, both increase with the agents

initial self-confidence, ⇡. The trust e↵ect thus forces the principal to adopt low-powered

incentives, and the more so the more self-confident the agent is.5

In every Perfect Bayesian Equilibrium, there are at most two bonuses (we do not

prove this but it is rather intuitive). A priori, equilibria can be either separating, or

pooling/hybrid. We first establish that there is no separating equilibrium.

Separation

Note that if P announces a bonus b only when observing ✓H , then A would know,

upon observing b, that he is of type H. Similarly, if P announces a bonus b0 only

when observing ✓L, then A would infer, upon observing b0, that he is of type L. As a

consequence, in a separating equilibrium, P completely discards his signal � (since the

bonus already reveals all information). Can an equilibrium have this feature? Note

that b must be lower than b0. If, at b0, the low type chooses to exert e↵ort ✓L(V +b0) � 0,

then the high type would also choose to exert e↵ort at b0 since

✓H(V + b0) > ✓L(V + b0) � 0 (60)

Hence, there is no reason why P would o↵er to the high type a bonus higher than b0

(since he could achieve e = 1 by o↵ering b0). Therefore, we focus on b < b0 (b = b0

implies pooling). As it turns out, b and b0 > b are incompatible with a separating

equilibrium. If b0 > b, P has an incentive to mislead A. When observing ✓L, he would

o↵er b and pretend that he observed ✓H . Upon being o↵ered b, A would (wrongly)

infer that he is of type H and exert e↵ort. In this way, P would be able to achieve

5Instead of D1, Benabou and Tirole use NWBR which is usually stronger (see Cho and Kreps

1987). However, in this application, NWBR and D1 are equivalent since the game is monotonic.

19

e = 1 by paying a lower bonus! This, however, cannot happen in equilibrium. As a

result, no separation is possible.

Pooling/Hybrid equilibria

In general, when P observes ✓H , P ’s information cannot be perfectly revealed in equi-

librium. This would create an incentive to mislead A. Hence, we can restrict attention

to situations in which P o↵ers b when observing ✓H and randomizes between b and

b0 > b when observing ✓L. (we neglect randomization upon observing ✓H). Suppose

then that P o↵ers b when observing ✓H . When observing ✓L, he o↵ers b with probability

x⇤ and b0 with probability 1� x⇤.

Upon observing b0, A infers that he is of type L with probability 1. In order to

achieve e = 1, P must o↵er a a bonus b0 such that A is willing to exert e↵ort knowing

that he is low ability for sure. Thus, b0 must be equal to the full information bonus

for a low type b⇤L. Things get more complicated when A observes b. In this case, he is

uncertain about his ability and will look at his signal �.

Exercise 5. By applying Bayes’ rule, show that A’s posterior probability of being of

type H upon being o↵ered b and observing a signal � is:

⇡gH(�)

⇡gH(�) + (1� ⇡)gL(�)x⇤(61)

Exercise 6. Given the result of exercise 5, show that: 1) A’s expected payo↵ from

exerting e↵ort upon being o↵ered b and observing a signal � is

✓⇡gH(�)

⇡gH(�) + (1� ⇡)gL(�)x⇤✓H +

(1� ⇡)gL(�)x⇤

⇡gH(�) + (1� ⇡)gL(�)x⇤✓L

◆(V + b)� c (62)

2) A’s expected payo↵ is strictly increasing in � (Hint: use the MLRP) , 3) there

always exists a threshold value of �, �⇤ 2 (�, �), such that A exerts e↵ort if and only

� � �⇤. (Hint: Remember that the payo↵ from no e↵ort is zero. Compare the expected

payo↵ from exerting e↵ort with the payo↵ from no e↵ort and use the MLRP).

As the last exercise implies, upon observing b, A follows a threshold strategy on his

signal. When he observes a high �, he exerts e↵ort. When he observes a low �, he does

not. Intuitively, he exerts e↵ort only if the information he receives about his ability is

good enough. We want to show that there exists a unique equilibrium surviving D1

and this is such that b = 0. The idea is to show that if b > 0, P has incentive to

deviate to a lower bonus. According to D1, this deviation should be interpreted by A

as emanating from type ✓H .

20

Suppose then that b > 0 and consider a deviation b̂ < b. Suppose that, upon

observing the deviation, A reacts by using a threshold �̂ on his signal. Type ✓L weakly

benefits from the deviation if

✓L[1�GL(�̂)](W � b̂) � ✓L[1�GL(�⇤)](W � b) (63)

where the left hand side is type ✓L expected payo↵ from the deviation and the right

hand side is type ✓H expected payo↵ in equilibrium (when o↵ering b). Type ✓H strictly

benefits if

✓H [1�GH(�̂)](W � b̂) > ✓H [1�GH(�⇤)](W � b) (64)

According to D1, type ✓L should be eliminated if, for any best response (read threshold)

that makes him weakly better o↵, type ✓H is strictly better o↵. The better/worse o↵

is as usual defined over the equilibrium payo↵s. Clearly enough, if �̂ < �⇤, type ✓H

would always strictly benefit since he would get a lower threshold by paying a lower

threshold (intuitively, more frequent e↵ort on average with less incentives). Hence, we

can restrict attention to the interesting cases in which �̂ > �. Condition (63) can be

rewritten as1�GL(�̂)

1�GL(�⇤)� W � b

W � b̂(65)

while condition (64) can be rewritten as

1�GH(�̂)

1�GH(�⇤)>

W � b

W � b̂(66)

From exercise 2, we know that

1�GH(�̂)

1�GH(�⇤)>

1�GL(�̂)

1�GL(�⇤)(67)

Hence, (64) holds with strict inequality whenever (63) holds. As a consequence, upon

observing b̂, A should believe that the deviation comes from type ✓H . Hence, upon

observing b̂, A would exert e↵ort with probability 1. Thus, when b > 0, P would

always benefit from deviating to a lower bonus. Since b can neither be greater nor

lower than zero, then it must be zero.

In summary, there is a unique D1-refined equilibrium and this is a hybrid in which P

randomizes between b⇤L and b = 0 when observing ✓L and o↵ers b = 0 with probability

one when observing ✓H . A always exerts e↵ort when observing b⇤L and uses a threshold

strategy �⇤ on his private signal when observing b = 0. In order to fully characterize

the equilibrium, we still have to check A’s threshold �⇤ and the probability x⇤ with

which P o↵ers b = 0. For � = �⇤, (62) holds with equality. Imposing b = 0, it can be

rewritten asgH(�⇤)

gL(�⇤)= x⇤

1� ⇡

⇡

c/V � ✓L

✓H � c/V(68)

21

The following exercise establishes that, given x⇤ there always exists a threshold satis-

fying the above equation.

Exercise 7. Given the MLRP and the assumptions lim�!� gH(�)/gL(�) = 1 and

lim�!� gH(�)/gL(�) = 0: 1) graphically show that for any x⇤ 2 (0, 1) there always

exists a unique �⇤(x⇤) 2 (�, �) satisfying equation (68); 2) show that �⇤(x⇤) is an

increasing function of x⇤.

For randomization to occur, P must be indi↵erent between b = 0 and b⇤L when

observing ✓L. This implies:

✓L[1�GL(�⇤(x⇤))]W = ✓L(W � b⇤L) (69)

If an x⇤ 2 (0, 1) satisfying the above equation exists, then the equilibrium is the one

discussed above. Since the lhs is a decreasing function of x⇤ and l.h.s.>r.h.s. for

x⇤ ! 0 (can you show this?), it follows that either there exists x⇤ 2 (0, 1) satisfying

the above equation or

✓L[1�GL(�⇤(x⇤))]W > ✓L(W � b⇤L) (70)

for all x⇤ 2 (0, 1). Clearly enough, in this case P does not randomize when observing

✓L but o↵ers b = 0 with probability one. Hence, the unique D1-refined equilibrium

would be a pure pooling in which P o↵ers no bonus independently of his information.

Whatever the equilibrium, A exerts e↵ort less frequently than under perfect infor-

mation. With full information, incentives can achieve e = 1 with probability 1. When

P has private information, by contrast, A exerts e↵ort with probability one only when

o↵ered b⇤L. When o↵ered b = 0, A exerts e↵ort only when � > �⇤(x⇤) which happens

with probability less than one.

4.4 Discussion

The idea of the model is that in the presence of asymmetric information, monetary

incentives may reduce the agent’s self confidence and hence induce a lower e↵ort in

equilibrium. Clearly enough, the principal cannot overcome this problem by o↵ering

“more money” since this would further undermine the agent’s self confidence. To what

extent this is a model of intrinsic motivation can be debated. Indeed, the agent’s direct

utility from performing the task (V ) does play a role in the model. Ideally, one would

like to have a framework in which more e↵ort can be achieved by o↵ering no money at

all relative to o↵ering a small amount of money. This is a type of discontinuity that

is typically observed in the real world (people often feel insulted when o↵ered a small

22

amount of money for something they would do for free). There are several papers

documenting how monetary incentives may crowd out intrinsic motivation. Notable

examples are Frey and Oberholzer-Gee (1997) and Gneezy and Rustichini (200).

5 Cheap Talk Games

5.1 Overview

Conventional wisdom suggests that “talk is cheap” meaning that, since words cost

nothing, they are not necessarily credible. So far, we have considered signaling games

in which signaling entails a cost for the sender. What happens when signals can be

sent at no cost (as in the case of words)? For some time, the presence of signaling costs

was regarded by economists as a prerequisite for meaningful communication whenever

sender and receiver’s interests were not perfectly aligned. However, we do observe in

reality that meaningful communication may occur through language or other means

which involve little or no cost.

How to reconcile these views? One possibility is reputation. If individual A chooses

to mislead individual B today, he will not be believed in the future. The threat of

future punishments is credible since A choosing to mislead and B choosing to ignore

the advice are mutual best responses. However, we will not discuss the case of repeated

interaction and we will instead focus on one shot games.

Another situation in which meaningful communication is likely to occur is when

there is no conflict of interest between the sender and the receiver. Suppose individual

S and individual R are going to Rome by car. Individual R is driving but does not

know the way. Individual S knows the way and can give advice to R. Assume that

both of them want to travel by taking the shortest route. Since their interests are

perfectly aligned, it is reasonable to conjecture that S will give accurate advice to

R and R will heed S’s advice. One can indeed imagine a game in which meaningful

communication is an equilibrium. However, from a game theoretic perspective there

is always also an equilibrium in which S sends random messages and R does not pay

attention. (Again, if S sends meaningless messages, R’s best response is to ignore

them. If R ignores S’s messages, a possible S’s best response is to send meaningless

messages). These equilibria are usually called “babbling equilibria” and exist in all

cheap talk games (see the example in Farrel and Rabin 1996).

A di↵erent problem is what happens when agents incentive are imperfectly aligned.

Suppose that S wants to be in Rome at 10:30 in the morning while R wants to be

there at 10:20. Will R always completely ignore S’s advice?

23

Crawford and Sobel (1982) (CS) give an answer to this problem. They consider

the case in which S’s preferences have a “bias” relative to R’s preferences. A good

example is the following. Suppose the government (S) has to report its current fiscal

policy (i.e. the total amount of new tax breaks included in the current budget) to

the central bank. Assume that the government knows the amount while the central

bank does not. The problem of the central banker is to choose the optimal interest

rate given the amount of tax breaks. In general, it is reasonable to assume that for

any level of breaks, the government preferred interest rate is lower than the central

bank’s preferred interest rate. Hence, there is a conflict of interests generated by the

government’s bias for lower interest rates. On the other hand, nor the government

neither the central bank want the interest rate to be excessively low (since it would

boost inflation) or excessively high (since it could cause a recession).

CS point out that meaningful communication is possible in this case. However,

communication is not perfect (the government never perfectly reveals the amount of

tax breaks) and the extent to which information is transferred is inversely related to

the government’s bias.

5.2 The Model by Crawford and Sobel

We will outline the general setup of Crawford and Sobel (1982), but we will not prove

their results. Instead, the next section considers a simple example.

The timing of the game is that of a standard signaling game:

• s = 0. Nature draws t 2 [0, 1] which is observed by S.

• s = 1. S sends a message m 2M to R.

• s = 2. R observes m and takes an action a 2 (�1, +1).

The state of the world (or equivalently, S’s type) t is drawn from a cumulative distri-

bution F (.) : [0, 1] ! [0, 1]. Payo↵s are given by the (continuous) functions UR(a, t)

for R and US(a, t, b) for S. Notice that payo↵s do not depend on the message m. In

this sense, the game considered is a cheap talk game. The parameter b in S’s payo↵

captures S’s bias. Other assumptions are that, for all t, there exists a unique action

aR(t) maximizing R’s payo↵ under perfect information. Similarly, given t, there exists

a unique action aS(b, t) - by R - which maximizes S’s payo↵.

Assume that b is such that aS(b, t) 6= aR(t) so that, known t, R’s optimal action is

di↵erent from the action preferred by S. Then

24

Result 1. There is no separating equilibrium in which S sends a di↵erent message for

every realization of t.

This result implies that perfect communication is not possible. We do not prove

this but the intuition is rather obvious. Consider a candidate separating equilibrium.

It is clear that R would always choose action aR(t) since S’s message perfectly reveals

t. But then, S would always benefit by misleading R. For instance, suppose that in

the candidate equilibrium S sends message m when t is realized. Assume also that

aR(t) > aS(b, t) and that there exists t0 such that aR(t0) = aS(b, t). Let m0 be the

equilibrium message associated with t0. Clearly enough, S would have incentive to

send message m0 (rather than m) when observing t. By doing this, he would induce

an action that maximizes his payo↵: US(aS(b, t), b, t) > US(aR(t), b, t). Hence, given

a separating equilibrium; S has incentive to deviate and mislead R.

Given that the equilibrium does not take the form of a separating equilibrium, one

wonders if there is any equilibrium in which information is transmitted. (We know

that there are always babbling equilibria but in these no information is transmitted).

As shown by CS, there are always equilibria characterized by a partition of [0, 1] - the

set of realizations of t. Denote with 0 = t0 < t1 < ... < tM = 1 the dividing points

of the partition. For every step of the partition (ti, ti+1), S sends a di↵erent message.

According to CS, we can wlog restrict attention to message spaces in which, whenever

t 2 (ti, ti+1), S announces a number in the interval (ti, ti+1) (always the same). For

example, suppose that [0, 1] is partitioned into the intervals [0, 0.2), [0.2, 0.5), and

[0.5, 1]. The idea is that we can restrict attention to equilibria in which S sends

messages m = 0.1 for all t 2 [0, 0.2), m = 0.4 for all t 2 [0.2, 0.5), and m = 0.9 for

all t 2 [0.5, 1]. Upon receiving message m = 0.1, R infers that t 2 [0, 0.2). Upon

receiving m = 0.4, R infers that t 2 [0.2, 0.5) and so on. Note that S could say

“red” when observing t 2 [0, 0.2), “blue” when observing t 2 [0.2, 0.5) and “yellow”

when observing t 2 [0.5, 1]. Since the message does not directly enter the payo↵s, this

behaviour is equivalent to the one just discussed (so long as R is able to understand

the meaning of the message). Clearly, the finer the partition, the more informative

the equilibrium. For instance, in an equilibrium characterized by the partition [0, 0.5),

and [0.5, 1] less information is transmitted than in an equilibrium with a partition

[0, 0.01), [0.01, 0.02),...,[0.99, 1]. Hence, it is interesting to know what determines the

“coarseness” of the partition. In the next section, we will argue that b, the extent of

the conflict of interest, plays an important role. In theorem 1 of the paper CS show

that all equilibria of a cheap talk game are economically equivalent to an equilibrium

25

characterized as follows:6

• Given a partition t0 = 0 < t1 < t2 < ... < tM = 1, S sends message mi whenever

t 2 (ti, ti+1) for all i. (Hence, the number of di↵erent messages is equal to the

number of steps in the partition).

• Denote with a⇤(mi) R’s optimal action when S sends mi, i.e.

a⇤(mi) = arg maxa

Z ti+1

ti

UR(t, a)dt (71)

the dividing point ti is determined as to satisfy

US(a⇤(mi), ti, b) = US(a⇤(mi�1), ti, b) (72)

where mi�1 is S’s message when t 2 (ti�1, ti).

CS also show that these equilibria always exist. The first point implies that mi is sent

with probability one when t 2 (ti, ti+1) and probability zero if t /2 [ti, ti+1].7 Intuitively,

the second point exploits the fact that, given a set of messages sent in equilibrium,

there are realizations of t for which incentive compatibility is satisfied with equality.

The division points of the partition are determined by these realizations. How this

result can be used to characterize the equilibrium is illustrated in the next section.

Finally, as we mentioned, the finer the partition, the more information is transmit-

ted. Among all possible equilibria, CS focus on the equilibrium in which information

transmission is maximized. This is the equilibrium with the largest number of mes-

sages M(b) sent in equilibrium. Their results show that M(b) depends on S’s bias.

Intuitively, the largest the conflict of interest between S and R, the coarser is the

partition and the less information is transmitted. This is illustrated in the following

example.

5.3 An Example (from Crawford and Sobel 1982)

Consider the following example: t is uniformly distributed in [0, 1]; US(a, t, b) = �[a�(t + b)]2; UR(a, t) = �[a � t]2. Note that, under full information, R’s optimal action

is a = t while S’s optimal action is a = t + b. Hence, when b > 0 (resp. < 0)

S’s preferences ar biased toward a higher (resp. lower) action. The equilibrium is

6The statement of the theorem is slightly more general. I am omitting some of the details for

simplicity.7In the original general statement, the probability of R receiving message mi given t is uniform

and has support (ti, ti+1).

26

characterized by a partition of [0,1] (the set of realizations of t) into M steps. Let

t0, t1, t2, ..., tM denote the dividing points between steps (with t0 = 0 < t1 < ... < tM =

1). In equilibrium, for all realizations t 2 (t0, t1), S sends the same message. Call it

m0. Similarly, for all realizations t 2 (ti, ti+1), S sends the same message which we can

denote with mi. From the theorem, upon observing mi, R infers that t is uniformly

distributed in the interval (ti, ti+1). Hence, his expected payo↵ is

Z ti+1

ti

� (a� t)2

ti+1 � tidt (73)

or

1

ti+1 � ti

�(a� ti+1)3

3+

(a� ti)3

3

�(74)

The first order condition then implies

a⇤(mi) =ti+1 + ti

2(75)

Consider now the sender. Upon observing a realization t = ti, he must be indi↵erent

between sending message mi (from which R infers that t 2 (ti, ti+1)) and sending

message mi�1 (from which R infers that t 2 (ti�1, ti)). He knows that, by sending

message mi, he will induce R to play a⇤(mi). If he sends mi�1, he will induce R to

play

a⇤(mi�1) =ti + ti�1

2(76)

Hence, he is indi↵erent if

US(a⇤(mi), ti, b) = US(a⇤(mi�1), ti, b), (77)ti+1 + ti

2� (ti + b)

�2

=

ti + ti�1

2� (ti + b)

�2

(78)

Solving for ti+1 yields two solutions. One is ti+1 = ti�1 which can be discarded (since

ti+1 > ti�1 by assumption). The other is

ti+1 = 2ti � ti�1 + 4b (79)

Solutions to this di↵erential equation are given by8

ti = it1 + 2i(i� 1)b (80)

This implies

tM = 1 = Mt1 + 2M(M � 1)b (81)

8Note that we have already fixed t0 = 0. Hence, we obtain a solution to the di↵erential equation

by choosing a value for t1 rather than for t0.

27

where M is the number of steps in the partition and hence the number of di↵erent

messages sent in equilibrium. Given that t1 must be greater than zero, M must

be an integer such that 2M(M � 1)b < 1. Let us focus on the most informative

equilibrium (the one characterized by the largest number M(b) of di↵erent messages).

In this equilibrium, the number of messages M(b) is the largest integer satisfying

2bM2 � 2bM � 1 < 0 or, equivalently, the largest integer less than

1

2+

1

2

r1 +

2

b(82)

From the above expression, note that as b goes to infinity, M goes to one (i.e. only

one message is sent). In other words, if S’s bias goes to infinity (interests are totally

misaligned), the equilibrium with maximum information is equivalent to a babbling

equilibrium: S sends the same message for any realization of t and therefore the

message is completely uninformative. For b converging to zero, M(b) goes to infinity.

Hence, the equilibrium approaches something similar to a separating equilibrium in

which information is almost fully revealed. For intermediate values of b, b determines

the quality of information transmitted in equilibrium. In general, as b becomes smaller

(interests become more aligned), the partition becomes finer and more information is

transmitted (or the information transmitted is more precise). As b becomes larger

(interests become less aligned), the partition becomes coarser and less information is

transmitted.

5.4 Neologisms (Farrel 1985)

As in (and even more than) signaling games, Cheap Talk games are plagued by multiple

equilibria. The problem is the usual problem of indeterminacy of beliefs out of the

equilibrium path. Hence, it is interesting to see whether an argument to refine the

equilibrium concept can be devised. Unfortunately, Cho and Kreps (1987) intuitive

criterion has no bite in Cheap Talk games. The intuition is clear. Since S’s utility does

not depend on the message, there is no type for which a particular out of equilibrium

message is always strictly dominated by the equilibrium payo↵ for any possible R’s

best responses.

Farrel (1985) devises an alternative approach. A neologism is defined as a message

that is not sent in equilibrium. One can think of a neologism as a word or phrase

that is never spoken in equilibrium. The idea behind the refinement is the following.

Suppose that there is a set of types (say T̂ ✓ T ) that could benefit from revealing

that they belong to T̂ . Suppose also that all other types t /2 T̂ would be worse o↵ by

pretending of being of a type in T̂ . (Note the di↵erence with the Intuitive Criterion.

28

Types not in T̂ must be worse o↵ only if believed of being in T̂ ). In this case, it is

reasonable to conjecture that types in T̂ could make the following speech: “My type

is in T̂ . If it were not, I would have no incentive to make this speech and pretend

otherwise since, if you believe me, I would benefit from the speech only if my type is

in T̂”. Intuitively, the speech is convincing and, therefore, R should believe it.

Formally, consider a generic cheap talk game

• s = 0. Nature draws t 2 T which is observed by S.

• s = 1. S sends a message m 2M to R.

• s = 2. R observes m and takes an action a 2 A.

Payo↵s are given by US(t, a) and UR(t, a) which do not depend on m.

Let T̂ be a nonempty subset of T , and denote with µ(t|T̂ ) the distribution over

types t when t is restricted to be in T̂ , i.e.

µ(t|T̂ ) =

(µ(t)/

P⌧2T̂ µ(⌧) if t 2 T̂

0 if t /2 T̂(83)

Let a⇤(T̂ ) = arg maxa2A

Pµ(t|T̂ )UR(t, a) be R’s best reply when beliefs are given

by µ(t|T̂ ).9 Assume that S sends the out of equilibrium message “My type is in T̂”.

Suppose also that R “believes” the message in the sense that his posterior beliefs are

given by µ(t|T̂ ). S’s payo↵ would then be US(a⇤(T̂ ), t). Define as K(T̂ |µ) as the set

of types who would benefit from this

K(T̂ |µ) ⌘ {t 2 T̂ |U⇤(t) < US(a⇤(T̂ ), t)} (84)

where U⇤(t) is S’s equilibrium payo↵. Given an equilibrium, the subset T̂ is said to

be self-signaling if K(T̂ |µ) = T̂ . In other words, a subset is self-signaling if only types

in the subset would benefit from revealing that they are in the subset. The neologism

“My type is in T̂” is then said to be credible if and only if T̂ is self-signaling. If there

exists a credible neologism, then the equilibrium is said to be not neologism proof and

should be eliminated.

Consider the following example. S can have two types: high ability (H) or low

ability (L) with equal probability. R can give S a demanding or an undemanding job.

S’s payo↵s are as follows

Demanding Undemanding

H 2 0

L 0 2

9I am assuming that a⇤(T̂ ) is unique. This is not necessarily the case.

29

So that S prefers the demanding job when high ability and the undemanding job

when low ability. R’s payo↵s are similar:

Demanding Undemanding

H 2 0

L 0 3So that R prefers to give S the demanding job when S is high ability and the unde-

manding job when low ability. Notice that there is no conflict of interests whatsoever

in this game. When S is high (low) ability, he prefers the demanding (undemanding)

job and R wants to assign him to the demanding (undemanding) job. Hence, it is sen-

sible to conjecture that S truthfully communicates his type to R and R assigns him to

the job more suited to the type. On the other hand, there is always an equilibrium in

which S babbles and R always assigns him to the undemanding job no matter what

he says. This equilibrium is sustained by out of equilibrium beliefs that assign a rela-

tively high probability to S being low ability whenever an out of equilibrium message

is received. This equilibrium does not fail the Intuitive Criterion since type L is not

strictly worse o↵ when sending an out of equilibrium message (If R believes him to

be low ability with large enough probability, his payo↵ is the same as in equilibrium).

However, this equilibrium is not neologism proof. To see this, consider a set T̂ = {H}formed only by type H sending the neologism “I am in T̂”. Clearly enough, only type

H benefits from this neologism. Hence, T̂ = {H} is self-signaling and the neologism

is credible.

Concluding note: Farrel’s neologism proof equilibria have inspired a solution con-

cept for signaling games known as perfect sequential equilibrium (Grossman and Perry

1986).

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