monopolist earns a profit of 4 if it is believed to be a high
type and a profit of 2 if it is believed to be a low type. The
low type monopolist earns a profit of 2 if it is mistakenly
believed to be the high type and a profit of 1 if it is believed
to be the low type. The price of an advertisement is 1.1 and
assume that there is no signal loss and no possibility for
celebrity endorsements. The following is a sequential
equilibrium M(H) =(2),M(L)=(0). The high type is known to be a
high type in equilibrium and earns a profit of 4-2(1.1)=1.8.
Consumer beliefs which are consistent with this equilibrium are
EQ(0) =EQ(1)=L and E£Q(2)=H. However, because in equilibrium the
consumer will never actually see one advertisement," #Q(1)=L is
referred to as an "out-of-equilibrium" belief.

An equilibrium refinement like the intuitive criterion would
argue that the previous equilibrium is implausible because the
out-of-equilibrium belief which sustains it (i.e. EQ(1)=L) is
implausible. In particular, why would the low-quality firm ever
purchase one advertisement? Even if after such a purchase the
low type was mistakenly believed to be a high type it would only
earn 0.9=2-(1.1). This is less than the low-type profits which
are guaranteed by not advertising (i.e. 1). On the other hand,
the high type would gladly defect to A,=1 if this would cause
consumers to believe it was the high type. Since only the high
type would ever have an incentive to purchase one ad it is argued
that consumer beliefs must be such that FQ(1)=H. Hence,

consumers will interpret a disequilibrium message (i.e. 1 ad,

16


product whose expected quality is 9<1 can be derived from this
by rotating the original demand curve counterclockwise around the
point Ron the quantity axis. The new demand curve will have
slope -Q. As R increases, the demand curve shifts upward and to
the right.

Consumers will purchase a maximum of one unit in each
period, whenever the expected value of the product exceeds the
market price. For simplicity, I assume that all consumers have
identical prior beliefs. Because there is a continuum of agents,
this assumption is equivalent to assuming that prior beliefs are
independent of an individual’s marginal valuations (i.e., their
address) .

I consider a very simple two-period game where the
monopolist is not allowed to adjust its strategy in the second
period. We can imagine that all the advertising takes place in
period 1, when the new product is initially introduced.
Consumers will then have an opportunity to again purchase the
product in period 2 at the same price as in period 1. In fact,
this would be the optimal response of a satisfied customer from
the first period. Hence, if the true quality is Q, the fraction
Q of the initial customers would now be repeat customers in
period 2. Given the expected quality £9, I define the profits
of the firm that selects the price P and the advertising

strategy (A,S) and whose actual quality is 9 as

(0, P, A, S, EQ) = (1+) (R-(P/EQ)) (P-Cp) - P,A-S. (4)

37


are listed below.

Equilibrium 1: {M(H)=(3,0), M(L)=(1,0) }, where EQ(1,0)=1/3,

EQ(2,0)=1, and EQ(A,S)=0 for all other (A,S).

Equilibrium 2: {M(H)=(0,0), M(L)=(0,0)}, where EQ(0,0)=p=1/2 and

EQ(A,S)=0 for all other (A,S).

Equilibrium 3: {M(H)=(1,2/3), M(L)=(0,0)}, where EQ(1,2/3) =H=1,

EQ(0,0)=L=0, and EQ(A,S)=0 for all other (A,S).

In equilibrium 1, the low type is actually indifferent between
advertising and not advertising since its profits are zero in
either case. The expected profits to the high type in
equilibrium 1 are: (5/3)=4[(1/2) (1) + (1/2) (1/3)]-3P,.

If neither type advertises then the expected quality must be
equal to the prior probability of facing a high quality firm
(recall p=1/2). The high type’s profits would always be 2,
while the low type’s profits would always be 1/2. A comparison
of equilibrium 1 and equilibrium 2 is shown in figure 2.

Finally, in the celebrity equilibrium, the low type is
deterred from advertising. Obviously EQ(0,0)=L=0, since the
observation (0,0) is consistent with the strategy of the low
type, but not consistent with the strategy of the high type.
Therefore, the profits of the low type (in equilibrium) are zero.

By theorem 6, S,=2/3 is set just high enough to deter the low

33


maximize the profits of the high type, since the low type makes
no sales. If B increases from 2 to 3, the LMSE will now include
the strategy M(H)=(2,2,1/2). Hence, the additional noise
results in a smaller celebrity endorsement and a greater

5

expenditure on air time. Profits to the high type decline from

2 to 1.7.

6. Conclusions

This paper proposes a plausible theory of celebrity
endorsements. I argue that because much advertising takes place
over electronic media, signal loss could be a serious problem.

In particular, signal loss interferes with the ability of
consumers to determine the true advertising expenditures of
firms. In an attempt to overcome this problem, firms may
"modulate" their signal by including celebrity endorsements (or
expensive special effects) in their commercials. This enables
the firm to pack more information into each commercial and
enables the consumer to receive a clearer signal. The end result
is that market performance can be improved from both the
standpoint of consumers and the firm.

For pedagogical reasons I have focussed on the special case
where C,#C, in order to rule out price separating equilibria.
However, in equilibrium, price signals of quality would not exist
simultaneously with stochastic advertising signals. Relaxing the

assumption that marginal cost is independent of quality would

42


Definition: A sequential equilibrium for this model is a

strategy M(Q) for the firm and system of beliefs for consumers

EQ(A,S) such that:

(1) M(H) =(A,,S,) maximizes En(H,A,S,EO(A,S))-AP,-S given
EQ(A,S) ;

(2) M(L)=(A,,S,) maximizes En(L,A,S,EQ(A,S))-AP,-S given
EQ(A, 8) ;

(3) EQ(A,S) is computed using Bayes’ rule along the equilibrium
path of play.

(4) Both A, and S, are nonnegative.

The exact nature of consumer expectations will depend upon
the stochastic process by which the strategy of the monopolist is
translated into a set of possible observations by consumers. In
this paper I view the process of viewing commercials as analogous
to the selection of balls from an urn without replacement. In
particular we imagine that the monopolist purchases a given
number of commercials and these commercials are placed in a large
urn with all the other commercials purchased by all the other
firms. B denotes the total number of commercials purchased by
other firms. I refer to these commercials as the irrelevant
commercials. I refer to the commercials or advertisements
purchased by the monopolist as the monopoly commercials. V
denotes the number of commercials randomly selected/viewed by
consumers. As stated earlier A, and A, represent the number of

commercials purchased by the high and low-quality type monopolist

10


if the urn is empty.)

11. The consumer will either observe zero ads or two ads depending
on whether nature selects the low-quality or high-quality
monopolist respectively.

12. These profit functions were chosen to illustrate the
equilibrium concept with the minimum possible effort. Later
examples utilize realistic payoff functions that are nonlinear in
expectations.

13. In order to deter mimicry by the low-type, it is always
necessary for the high-quality monopolist to be more profitable
than the low-quality monopolist.

14. Although there are other sequential equilibria with celebrity
endorsements, none of them can be Pareto Optimal with respect to
Sender Types since they increase the costs of the high type without
materially altering the induced set of consumer beliefs. Later in

the paper, examples of POST-equilibria with celebrity endorsements
will be presented.

15. I should point out here that consumers will not always wish to
overturn the same equilibria that the monopolist views as
inefficient. In the previous example, equilibrium 2 was better for
consumers since they have better information about quality when no
ads are viewed and also because they are more likely to discover
the high-quality monopolist.

16. Although, strictly speaking, they have proposed a refinement of
Bayesian Nash equilibria, there is no reason why their approach
cannot also be used to refine sequential equilibria.

17. In other words we are using Bayes’ rule along the equilibrium
path of play.

18. From now on, I will refer to this pooling equilibrium as the
canonical pooling equilibrium.

19. The reader should not be confused by the zero in £Q/(A,0). I

have introduced a zero so that S, can be defined explicitly rather
than implicitly.

49


[10]

[11]

[12]

[13]

[14]

Milgrom, Paul and John Roberts, "Price and Advertising
Signals of Product Quality," Journal of Political Economy,
Vol. 94 (1986), pp. 796-821.

Nelson, Phillip, "Information and Consumer Behavior,"

Journal Political Economy, Vol. 78 (1970), pp. 311-329.

- "Advertising as Information," Journal of
Political Economy, Vol. 81 (1974) pp. 729-754.

Overgaard, Per Baltzer, "Price as a signal of quality: A
discussion of equilibrium concepts in signaling games,"
European Journal of Political Economy, Vol. 9 (1993) pp.
483-504.

Spence, M., "Job Market Signaling," Quarterly Journal of

Economics, Vol. 87 (1973), pp. 355-374.

47


Pooling Equilibrium Celebrity Equilibrium

1 1

Ce) as Ce)

Ce) oe} - 22 ee ee

0.4 MMMM ee Cs

o.2HMRMm ~~~ -  s O.2f-- rts
° °

-0.2 -0.2



most attractive when £Q(1,S)=H=1 for small S. Consider
equilibrium 1. Given that the low type does not find it
desirable to defect to (1,0), it certainly would not find it
desirable to defect to (1,8), since both strategies induce the
same expectations, and the second strategy is costlier. The same
logic also applies to high type. Since the out-of-equilibrium
celebrity endorsements may be set arbitrarily, both of these
equilibria are "immune" to out-of-equilibrium belief refinements.

However, simple calculation reveals that equilibrium 1 is
the only POST equilibrium. If the high type purchases one ad
rather than two, then the expected quality (and also the profits
of the low type) upon viewing zero ads will be 1/3 rather 1/4.
The first equilibrium is, therefore, more profitable for the low
type. Although by purchasing two ads instead of one the high
type has a greater chance of being perceived as a high-quality
firm (2/3 chance as opposed to a 1/2 chance) this benefit is more
than offset by the additional advertising cost (and also by the
reduction in profits when zero ads are observed by consumers) .
Figure 1 indicates the relative profitability of equilibrium 1
and equilibrium 2.

As games have become more complicated, the stories
required to justify the implausibility of various out-of-
equilibrium beliefs have also become more complicated. As
discussed above, the introduction of noisy signaling is one such
complication. Even in those situations where there actually

exist out-of-equilibrium beliefs which are required to sustain

20


The (1+9) term reflects the additional profits from the second
period. The next term (R-(P/EQ)) is a measure of the initial
customers in period 1. The last two terms ~P,A-S reflect the
cost of advertising when A commercials are purchased at a price
of P, along with a celebrity endorsement S. Of course, as in
the previous section, expected quality £0, is actually a random
variable whose distribution depends on the strategy of the
monopolist. Knowing the distribution of expected quality enables
one to calculated expected profits.

By theorem 8 we can see that the introduction of price
signaling does not significantly complicate matters. Either all
the signaling will be done by price or all the signaling will be
done by advertising. Price signals and advertising signals will
not be simultaneously employed by the monopolist. This is the
same result as in Hertzendorf (1993).

This dichotomy between price and advertising signal creates
a special role for the equilibrium refinement. Under price
pooling or price separation it is the equilibrium refinement that
determines the equilibrium price.* Generally speaking, if
separating sequential equilibria exist, there will not be a
unique POST equilibrium since the high and low types cannot agree
on an optimal price. However, the unique LMSE will include the
price preferred by the high type in any equilibrium.

Whether or not the monopolist will signal quality
exclusively through price or advertising depends on the

relationship between quality and marginal cost. When marginal

38


1. Introduction

The notion that advertising can be used to signal quality
originates from Nelson (1970,1974). Nelson’s argument is based
on a distinction between search and experience goods. For
experience goods, the actual quality can only be verified after
purchase. Nelson’s key insight is that even though the content
of many advertisements conveys little or no direct information to
consumers, the very existence of an expensive advertising
campaign may convey information indirectly to consumers. As
Milgrom and Roberts (1986) put it, the key message of many
advertising campaigns seems to be: "We are spending an
astronomical amount of money on this ad campaign."

Because a firm with a high-quality product can expect more
repeat sales than a firm witha lower-quality product, such a
firm can also afford to spend more money on advertising. The
idea of signaling product quality through advertising is based
upon the notion that after having viewed an expensive advertising
campaign, the consumer is supposed to reason, "You must be a
high-quality firm that expects a lot of repeat business, for
otherwise, you could not possibly afford to spend all this money
on advertising." As long as consumers draw this inference from
advertising, the high-quality firms will, in fact, have an
incentive to advertise, and consumers will, therefore, have an
incentive to respond to advertising. In this way, a firm can

credibly signal the actual quality of its product indirectly to


alternative equilibrium increases the net profits of the high-
quality monopolist by (S,-Z). Hence, the original separating
equilibrium with S,>Z could not have been a POST equilibrium.
QED.

The previous two theorems have dealt with separating
equilibria where there is a celebrity endorsement. By theorems 1
and 4 we know that these equilibria must involve no advertising
by the low type. There may, however, also be a separating
equilibrium without a celebrity endorsement. By specifying the
relationship between the profits of low- and high-type

monopolist, we can show that these alternative equilibria will

have A,>A,.

Theorem 7: Suppose that HI(H,A,S, EQ) =cHI(L,A,S,EQ)TM where
o>1, then in any POST equilibrium with A,>0 and A,>0 we must

also have A,>A,.

Proof: By theorems 1 and 4, we know that in any POST equilibrium
with A,>0 there will not be any celebrity endorsement. Because
(Ay, 0) satisfies condition (1) of a sequential equilibrium we
know:

EDL (H, Ay, 0, EQ(A,0)) ~P,Ay> EIL(H, A,,0,E0(A,0))-P,A,. Using the
assumption about the relationship between the profit functions of
the high and low types and rearranging terms we discover:

(A) c[HIL(L, Ay, 0, HO(A, 0)) - HI(L, A, 0, £Q(A,0)]> P,Ay- PA, -

The fact that (A,,0) satisfies condition (2) of a sequential

31


instead of 2) as a signal that the monopolist is really a high
quality. This, however, would then give the high-quality
monopolist an incentive to defect and, thereby, overturn the
previous equilibrium. In doing so, the high type will now earn
2.9 = (4 - 1.1) instead of 1.8 as in the previous equilibrium.

Hence, the more common approach to equilibrium refinements
is to reduce the number of sequential equilibria by arguing that
certain out-of-equilibrium beliefs (necessary to sustain some of
the equilibria) are implausible. This approach will not work in
my model because with signal loss present there is no longer a
one-to-one correspondence between out-of-equilibrium strategies
and out-of-equilibrium observations. Consider the following
example where V=1 and B>1. In this situation consumers will
view either one monopoly commercial or they will view no monopoly
commercials. Regardless of what strategy either type selects,
observing zero observations is always a possibility, while it
also remains impossible for consumers to observe more than one
monopoly commercial. If the type-Q monopolist switches to a
strategy that involves a different number of commercials, the
consumers will have no way of knowing that they are observing an
out-of-equilibrium message that wasn't supposed to occur. Hence,
there is no out-of-equilibrium number of commercials that can be
observed. The use of a refinement that considers defections to
out-of-equilibrium strategies can, therefore, only be used to
limit the possible modulation (S,) that might exist in

equilibrium. However, as the next example shows, such a

17


of the ex-ante monopolist (i.e. the profits of the monopolist
before knowing its actual type).

In the next section, POST equilibria are characterized.
However, as will be shown in later examples there may be multiple
POST equilibria. I illustrate how the refinements discussed

above can be utilized to derive a unique outcome.

4. Analysis

I now present several theorems which attempt to characterize

the POST equilibria in my model.

Theorem 2: All Pooling POST Equilibria are of the form

M(H) =M(L)=(0,0) and £Q(0,0) =pH+(1~-p)L.

Proof: Clearly if both firms are playing the same strategy then
any observation consistent with that strategy reveals no
information to consumers. Hence, the expected quality must be
the same as the ex ante expected quality pH+(1-p)L. Consider
first the case where VsB. In this situation, observing zero
commercials is "in-equilibrium." I employ a proof by
contradiction. Suppose that M(H) =(A,S) and M(L)=(A,S) are part
of a pooling equilibrium. Then because (0,0) is an observation
consistent with the equilibrium FQ(0,0) =pH+(1-p)L." But then
M(Q)=(0,0) decreases the costs to type Q of signaling, while

leaving the realized expectations of quality intact. This means

23


advertising. Purchasing one advertising will result in profits
of (1-P,) =(1-17/36), 50% of the time (i.e. when the
advertisement is viewed by consumers, recall that V=B=1) and
profits of (1/3~-17/36) the other 50% of the time. Therefore,
the expected profit to the low type from mimicking the strategy
of the high type is #l(L,1,0,£0(A,0)) =7/36<1/3. It can
similarly be shown that any higher level of advertising will
likewise result in expected profits of. less than 1/3. Therefore,
the low type is maximizing its expected profits by not
advertising.

The high type earns an expected profit in equilibrium 1 of
24/9 - 17/36 (i.e. 4[(1/2) (1/3) + (1/2) (1)1-P,). Tf the high type
decides not to advertise then no commercials will ever be seen
and its profits are guaranteed to be 4(1/3)=12/9. If on the
other hand, the high type purchases 2 ads instead of one, there
is now a 2/3 chance that one of them will be viewed and so its
expected profits are 28/9 - 34/36 (4[(2/3) (1) + (1/3) (1/3)] -2P,) -
Simple calculation reveals that the high type will maximize its
profits at M(H)=(1,0). In a similar manner the conditions for
equilibrium 2 can also be verified.

We cannot discard either equilibrium because of implausible
out-of-equilibrium beliefs. First, since Ae[0,1], there are no
out-of-equilibrium number of ads. There are only out-of-
equilibrium celebrity endorsements. It is easy to see, however,
that regardless of how we set E£Q(1,5), neither type would wish

to ever defect from the equilibrium strategy. Defection would be

19


equilibrium implies that:

EIL(L, A, 0, £0(A,0)) ~P,A, > BOL, Ay, 0, EQ(A, 0) -P,Ay, or that

EIl(L,A,,0,EQ(A,0)) -BM(L, Ay, 0,E0(A,0)) > P,A,-P,Ay- Multiplying

through by -1 yields

(B) —-EI(L, Ay, 0, EQ(A, 0) ) - EIL(L, A,, 0, EO(A,0)) < P,Ay-P,A,- Putting

(A) and (B) together implies that

PalAuAy) ¢ ori(1, Ag, 0, BOA, 0)) -EIM(L, A,,0,BQ(A,0)) <P,(Ay-A,). This
c

last inequality tells us that c>1 implies that A,>A,. Hence, in

any POST equilibrium where both types advertise, the high type

will purchase more commercials than the low type. QED.

Example 2: Consider a model with the following parameters: L-=0,
H=1, p=1/2, B=l, V=2 and P,=1/3. The profit function for the
high- and low-type monopolist are respectively:

EIl(H, A, S, EQO(A, S)) =4EOQ(A,S) - (A) P,-S

Ell (L, A, S, EQ(A,S)) =EQ(A,S) - (A) Px-S

A key feature of this example is that V=2 and B=1. Hence, if the
high type purchases one or more advertisements, at least one of
these ads will be seen by consumers. Zero advertisements would
no longer be an in-equilibrium observation. If the high type
advertises, then the low type will have zero profits unless it
also advertises. Given the previous analysis in this section,
there are three different kinds of "potential" POST

equilibria: a separating equilibrium (without any celebrity
endorsement); a pooling equilibrium where neither type

advertises; and a celebrity equilibrium. These three equilibria

32


observations of the strategy do not reveal anything new to
consumers and so the expected quality of consumers must be
EQ(A,S) =pH+(1-p)L (for all “in-equilibrium" observations.) In
a separating equilibrium the high and low-type monopolist pick
different strategies. In games were there is no signal loss,
this would result in consumers knowing with certainty whether or
not the quality is high or low. However, in the current game
where there is signal loss this need not be the case.
Observations by consumers may reveal imperfect information about
quality. Only in the special case where an observation is
consistent with the strategy of the high type M(H) =(A,,S,) but
inconsistent with the strategy of the low type M(L)=(A,,S,) will
the consumers know with certainty that they are facing a high-

quality monopolist.

3. Pareto Optimal Equilibria

In general, there will not be a unique sequential
equilibrium. This hinders one’s ability to make testable
predictions and also makes it difficult to do comparative
statistics. However, many of the possible sequential equilibria
are absurdly inefficient. The equilibria are inefficient in the
sense that there are other sequential equilibria which reduce the
cost of separating without affecting the "in-equilibrium" beliefs
of consumers. These other equilibria are preferred by both the

high type and the low-type, because they result in the same gross

14


purpose of this refinement concept is to eliminate wildly

implausible sequential equilibria. Section 4 provides a

characterization of the refined equilibria, as well as examples.

In Section 5 I show how the model can be easily extended to

encompass price signaling. Section 6 concludes.


low type would not be able to improve its profits by spending
money on advertising unless it was prepared to select S,, since
any other modulation will be perceived as coming from the low
type. Condition (1) just states that the strategy M(L)=(0,0) is
preferred by the low type to any strategy of the form
M(L)=(A,S,) where A>O. If this condition is met, then the low
type is maximizing profits, subject to the beliefs of consumers.
For the same reason the high type would never have an incentive
to defect from M(H) =(A,,S,) to M/(H)=(A,S) where s#S, since this
would only result in consumers believing that the monopolist had
a low-quality product. In other words, if the monopolist is
considering a defection to another strategy with positive
advertising it should only consider those defections where it
maintains the in-equilibrium modulation level S,. Condition (2)
just says that the type-H has higher profits at M(H)=(A,,S,) then
at any strategy M'(H) =(A, Sy) where A#A,. The only other
defection that the high type might consider is a defection to
M(H)=(0,0). Although this will reduce consumer expectations of
quality it would also reduce advertising expenditures. Condition
(3) just says that the high-quality monopolist prefers to
separate. This proves sufficiency.

Of course if condition (1) was not met, then the low type
would start to mimic the strategy of the high type. If (2) was
not met then the high type would switch its advertising level
(but not its modulation) to some other level that increases

profits. Finally, if (3) was not met, then the high type would

29


likewise, the best possible profits for the low type. In other
words, when we compare the expected profits in the canonical
pooling equilibrium to the profits from some arbitrary sequential
equilibrium we must have:

EIL(H, 0, 0,H) > EL (H, Ay, Sy, EO(A, S)) -P,Ay-Sy and

EU(L,0,0,H) > EM (L,A,,S5,,£0(A,S)) -P,A,-S,

Hence, for p sufficiently close to 1, M(H)=(Ay,Sy), M(L)=(A,,S;)
cannot be part of a POST equilibrium since the canonical pooling
equilibrium with M(Q)=(0,0) is more profitable for both types.
Roughly speaking, when consumers think that a high quality
monopolist is sufficiently likely then it doesn’t pay for either
type to incur the expense of signaling product quality. This is
because the alternative is to sell goods under the consumers’

prior expectations of quality, which are already quite high.

QED.

Theorem 4: There are no POST equilibria of the form

M(H) =(Ay, 3), M(L)=(A,,3) where S>0.

Proof: Suppose instead that M(H) = (Ay, 3) and M(L) =(A,,S) are
part of a sequential equilibrium. Without loss of generality we
may assume that EQ(A,S)=L for all out-of-equilibrium
observations. We will show that M(H)=(A,,0), M(L)=(A,,0) are
part of a sequential equilibrium that is more profitable. By
condition (1) of a sequential equilibrium we know that

FIL (H, Ay, S, EO(A, 8)) - P,H,- 5 2 EIL(H, A, S, EQ(A, S)) -P,A-S for all

26


purchase many ads during one show or during one evening. As
part of a $45 million dollar campaign to introduce its new
Mercury Mystique, Lincoln-Mercury bought 24 commercial spots
during one evening (freezing out all automobile competitors that
night) in the fall of 1994. This helps ensure that many
consumers will see the commercial at one time and also increases
the chances that a company’s advertising will be noticed above
the clutter of media message assaulting consumers each day.!
Another approach to increasing the impact of commercials is to
make them so controversial that, not only can they not be
forgotten, but they also generate additional publicity for free.
The Italian clothing manufacturer Benetton, for example, has
developed controversial commercials that have included: an oil-
soaked sea-bird, a dying AIDS patient, and most recently the
bloody uniform of a Croatian soldier.’

Another effective method of overcoming signal loss is to
advertise when the signal loss is low. This undoubtably explains
why advertising during the Super Bowl is so attractive. In 1995
firms paid an average of 1 million dollars for each 30 second
commercial aired during the Super Bowl. These commercials were
viewed by about one-half of all Americans (130 million). Not
only may advertisements during the Super Bowl be watched by a
greater audience, it is also possible that they are more likely
to be remembered.*

Of course, different approaches to overcoming signal loss

may be combined. Advertising during the Super Bowl does not


consumers.

One limitation of previous models of advertising is that
they often tacitly assumed that all] dollars spent on advertising
are equivalent. Such an assumption hinders any analysis of how
firms divide up their advertising dollars. For example, why does
one firm purchase an expensive celebrity endorsement from a
famous rock star, while another firm spends almost all of its
advertising dollars on air time? In Hertzendorf (1993) I argued
that for advertising over various electronic media (i.e.
television and radio) signal loss is an important factor which
should not be ignored. Because most commercials will not
actually be seen by consumers, consumers can only make imprecise
estimates as to the firm’s total expenditure on advertising.
This inability of firms to convey directly to consumers their
actual total expenditures on advertising hinders the firm’s
potential to credibly signal their high-quality. In fact, as
shown in Hertzendorf (1993), the signal loss may provide an
opportunity for a low-quality firm to trick consumers into
thinking that it is high-quality firm by partially mimicking the
strategy of such a firm.

Confronted with the problem of signal loss, firms have a
strong incentive to increase the impact of each commercial
actually viewed by consumers. This might be accomplished via
extravagant production costs or by purchasing a celebrity
endorsement. There are other approaches to increasing the impact

of advertising. A firm might employ "saturation" advertising and


profits, but reduce the cost of advertising. To the extent that
one has some faith in the ability of market forces to resolve the
problem of incomplete information efficiently one would want to
focus on this subset of the sequential equilibria. Let SE

denote the set of all Sequential Equilibria.

Definition: Let M(L)*=(Az,SZ) and M(H)*=(Ay,S_), then
{M(L)*,M(H)*, £0*(A,S)} €SE is Pareto Optimal with respect to
Sender Types ("POST") if there does not exist another sequential

equilibrium {M(L)’,M(H)’, £Q/(A,S)} €SE such that:

(A) Ew (H, Aj, Sh, EO(A, S)) - AP, - Sh 2 En (H, Ay, Sy, EQ(A, S)) - Ag P,- Sp
(B) Em (L, AL, $1, EQ(A,S)) -ALP,- Si 2 En(L, AL, Si, HO(A,S)) -ALP,- Si

with one of the inequalities holding strictly.

This equilibrium refinement will enable me to rule out
inefficient signaling by the high-quality monopolist. The notion
of focusing on Pareto-dominant elements of the equilibrium set is
not new. This approach was originally proposed by Spence in the
1970's. The reader may, however, wonder why I choose to use this
approach rather than one of the more popular refinements that
consider the plausibility of various out-of-equilibrium beliefs
(i.e. Cho and Krep’s (1987) intuitive criterion or Banks and
Sobel’s (1987) universal divinity.) In most signaling games the
equilibrium is sustained by a collection of "out-of-equilibrium"

beliefs. Consider the following example. The high-type

15


In the game I develop, consumers attempt to infer a monopolist’s
total advertising expenditure from a stochastic advertising
signal. The advertising signal is stochastic because consumers
typically view fewer commercials than were purchased due to
random signal loss. However, once at least one commercial is
viewed by consumers, I assume that production costs are perfectly
observed.

I view celebrity endorsements (as well as the production
costs associated with elaborate commercials) as an attempt by
firms to overcome signal loss. Without a celebrity endorsement,
a consumer viewing one commercial might conclude with certainty,
that the firm has spent tens of thousands of dollars to air the
commercial that is being viewed.* However, if the same
commercial includes Elton John, the consumer might, after, viewing
one commercial, conclude that the firm is spending millions of
dollars on advertising, because that is how much Elton John
costs. Hence, celebrity endorsements are a way of packing more
information into each commercial, so that inferences about
quality can be drawn from viewing fewer commercials.

The goal of this paper is show how a monopolist would
attempt to minimize the cost of signaling product quality to
consumers by exploiting the tradeoff between air time and high
production costs (e.g. celebrity endorsements or special
effects). The paper proceeds in 6 sections. Section 2 presents
the basic model and the definition of a sequential equilibrium.

In Section 3 I introduce a modest equilibrium refinement. The


the equilibrium, there will not in general be a one-to-one
correspondence between out-of-equilibrium beliefs and out-of-
equilibrium strategies. Although a POST equilibrium is no
panacea, it does offer one simple framework within which to
consider the plausibility of sequential equilibria.

The whole concept of a signaling game is that there are
market mechanisms in place that can resolve incomplete
information and, thereby, improve market performance.
Advertising need not be viewed as an unproductive and
manipulative attempt to increase profits. Rather advertising
expenditures can be viewed as an important and efficient
mechanism by which information is exchanged, even if it is the
expenditure itself and not the content of the commercials that
conveys the information. Although there may be numerous
sequential equilibria there are obviously incentives for the
monopolist to pick a reasonably efficient one. At a minimum, we
should rule out those sequential equilibria which are not Pareto
optimal with respect to the two types.

Although the consumers may also have an incentive to
overturn inefficient equilibria, the benefits to each individual
consumer are likely to be small. Unless consumers are somehow
organized, it seems unlikely that they would provide the impetus
for overturning an inefficient equilibrium (at least in this
game) .*

As in other recently proposed refinements, the POST

equilibrium of a game will often be a pooling equilibrium rather

21


A Game Theory Model of

Celebrity Endorsements

Mark N. Hertzendorf’

Revised: February 1996

‘Federal Trade Commission

Bureau of Economics, Washington DC. 20580

This article reflects the views of the author and does not
necessarily reflect the views of the Federal Trade Commission or

the views of any individual Commissioner.


advertising simply because they can remember it, but they will
only remember those commercials that were expensive to produce.
Hence, the fact that no individual consumer is actually
attempting to infer the monopolist’s advertising expenditure is
irrelevant. As Nelson put it, "Whatever their explicit reasons,
the consumers’ ultimate reason for responding to advertising is
their self-interest in doing so... If it were not in consumer
self-interest to respond to advertising, then the consumers’
sloppy thinking about advertising would cost enough that they

would reform their ways."

44


consumers, this expenditure has been perfectly observed. The
carrier signal is chosen from the set of natural numbers N (i.e.
N=0,1,2,3...), while the modulation is chosen from the positive
reals R*. Consumers are initially unaware of the type of
monopolist they face, but form an expectation after observing one
or more of the commercials and any associated celebrity
endorsement. Hence, consumers only enter into the model
indirectly through the function EQ(A,S): (NxR*) + [0,1] which
translates their observations into an expectation of quality.
The complete map which consumers use to translate an observation
into an expectation of quality is sometimes referred to as the
"beliefs of consumers." The exact nature of consumer beliefs
cannot be specified until I introduce a specific stochastic
process that will translate a monopolist’s advertising
expenditure into a probability density function for possible
observations. The quality, which is bounded between 0 and 1, may
be thought of as the probability that the good in question will
perform satisfactorily for a given consumer. The number of
advertisements observed by consumers will not, in general, be the
same as the number of advertisements purchased by the monopolist.
It will typically be less. This reflects the possibility that
consumers will fail to view certain advertisements because of
signal loss.

Throughout the paper, I use the convention of denoting
a random variable in bold face and realizations of this random

variable in regular type. Hence, A is the random variable


2. The Model

I assume that a monopolist manufactures either a high-
quality or a low-quality good. Nature randomly selects between
two possible quality values H and L according to some prior
probability distribution (12H2L20). I use p to denote the
prior probability of Nature selecting a high-quality monopolist.
Hence, following Harsanyi (1967), I am modeling a game of
incomplete information as a game of imperfect information. A
strategy for the monopolist is a function, M(Q):[H,L]~(A,S),
that translates the actual quality of the firm into a nonnegative
advertising pair. The first component of the advertising pair
represents the number of advertisements purchased by the firm at
price P,. The second component of the advertising pair
represents a dissipative advertising expenditure that is observed
whenever at least one commercial is observed by consumers. We
might think about this second advertising expenditure as the cost
of a celebrity endorsement.

Since the "celebrity endorsement" expenditure is only
observed when a commercial is seen, I draw an analogy to
electronic communications and sometimes refer to the commercials
as the "carrier signal" and the other expenditure as the
"modulation." I assume that the carrier signal is stochastic
due to the possibility of signal loss (not all the commercials
will typically be seen by consumers) . The modulation is

essentially deterministic; once one commercial is seen by


Equilibrium 3: {M(H)=(2, 2, 5/3), M(L)=(2,0,0)} with

EQ(2, 1, 5/3)= 1 and EQ(P,A,S)< 1/4 for all other (P,A,S) .

Equilibrium 1 is the canonical pooling equilibrium. Since there
are no advertising or price signals of quality, consumer
expectations of quality are fixed at their prior beliefs. Given
that consumers are expecting quality to be 1/4, the Pareto
optimal response of both types must be charge a price of 1/2. At
this price the high type earns a profit of 2, the low type a
profit of 1. This equilibrium is preferred by the low type.
Equilibria 2 and 3 are celebrity equilibria. The only
difference between them is whether or not the high type purchases
one ad or two ads. In both equilibria the low type is deterred
from purchasing one commercial because of the celebrity
endorsement (S,=5/3). Because observing no commercials suggests
to consumers that a low type monopolist is more likely, the
corresponding expected quality (computed using Bayes’ rule) must
be less than 1/4 (the consumers’ prior expectation of quality) .
Given P=2, no sales will be made if no commercials are seen.
Since the low type purchases no commercials, its payoff is zero.
When P=2 the expected quality must be at least 1/2 before any
sales are made. Hence, we can think of EQ=1/2 as the
"reservation expectation." The POST equilibrium price is chosen
so as to maximize the expected profits to the high type when
expected quality is 1 (i.e. after viewing one or more ads).

Although, in principle, the high type could select a price low

40


an alternative equilibrium where both types are better off.
However, when both types adopt exactly the same modulation
neither type achieves any advantage over the other. If both
types can agree to suspend all celebrity endorsements each type
gains S, and consumer beliefs are unaffected. Combining theorem
1 and theorem 3 it is clear that all POST equilibria that involve
celebrity endorsements involve no advertising signals by the low-
type firm.

I now present two theorems that deal with separating
equilibria that include celebrity endorsements (i.e. signal
modulation). I will refer to these equilibria as "celebrity"
equilibria. The previous theorems have shown that all celebrity
equilibria are also separating equilibria. The next two theorems

provide a further characterization of celebrity equilibria.

Theorem 5: A necessary and sufficient condition for there to
exist a celebrity (separating) sequential equilibrium with

M(H) =(Ay, Sy), M(L)=(0,0) is that there must exist (A,,S,) such
that the following conditions (1), (2), and (3) hold.

(1) Max , (EIL(L,A, Sy, HO(A, S)) - P,A~S;) < EI(L,0,0,£0(0,0))

(2) Max , (HIL(H,A, Sy, EQ(A, S)) - P,A- Sq) = ED (H, Ay, Sy EO(A, S)) -PAy-Sy

(3) BO(H, Ay, Sy, EO(A, S) ) - PyAy- Sy2 EN (H, 0,0, EQ(0,0))

Proof: Conditions (1)-(3) are sufficient: for a sequential
equilibrium provided that £Q(A,S)=L for all out-of-equilibrium

observations. For example, given these out-of-equilibrium the

28


A Note On Figures 1-3

Figures 1-3 are bar graphs designed to facilitate a comparison
among various equilibria. Graphs were drawn to similar scales to
facilitate comparisons, even when some components on the
advertising axis are irrelevant. For example, in figure 1,
expectations for two, three and four advertisements are omitted
since, as I point out in the paper, these expectations can be
defined in any manner whatsoever. Also in figure 1, the negative
profit regions for three and four advertisements were

intentionally omitted.

In figure 2, on the left-hand side (separating equilibrium),
absence of a bar graph indicates a value of zero, rather than an
intentional omission. On the right-hand side (pooling
equilibrium), only the equilibrium strategies are plotted since

other strategies are irrelevant.

In figure 3, on the right-hand side (celebrity equilibrium), the
bar graphs are drawn under the assumption that $)=2/3, as is the
case for this equilibrium. The lack of a bar indicates a value

of zero, except for two, three and four advertisements on the

graph of the low-type profits (they would all be negative) .

45


(A, S)2(0,0), given EO(A,S). Next define £Q/(A,0)=EQ(A4,S) for
all AjinSASAggx and £Q/(A,S)=L for all other (A,S). By
construction we have #il(H,A,0,50'(4,3)) =EIL(H,A,S,EQ(4,S)) for
all A>O. Substituting into condition (1) reveals
EI (H, Ay, 0, EO!(A, S)) ~P,Ay-S2 EM (H, A, 0, EO'(A, S))-P,A-S. But since
EQ(A,S)=L for all out-of-equilibrium beliefs where S#0 we can
also be sure that
EM (H, Ay, 0, BO! (A, S)) -P,A,-S> Ell (H, A, S, EO'(A, S)) -P,A-S for all
(A, S)2(0,0). Adding S to the left side can only make it larger
so we must also have
EI (H, Ay, 0, EQ! (A, S)) - P,A,2 EIL(H, A, S, £Q!(A,S)) -P,A-S for all (A4,S)
holding fixed the beliefs of consumers. This implies that
M(H) =(Ay,0) satisfies condition (1) of a sequential equilibrium.
Notice also that by construction
FIL (H, Ay, 0, EQ!(A, S)) ~P,Ay2 EIL(H, A, S, HQ(A,S)) -P,Ay-S. We can do
the same analysis for type-L and discover that
FIL(L,A,,0,£0'(A, S)) ~P,A,2 EM (H,A, S, EO'(A, S)) ~P,A-S for all
(A, S8)2(0,0) and that
FIL (L,A,, 0, £0'(A, $)) -P,A,> BO (L, A,, S, £0(A, S)) -P,A,-S. All of this
implies that {M(H) =(Ay,0) ,M(L)=(A,,0) , £OQ/(A,S)} is a sequential
equilibrium that is preferred by both types to the sequential
equilibrium (M(H) =(A,,3),M(L)=(A,,S),50(A,S)}. Hence, the
conjectured sequential equilibrium could not have been a POST
equilibrium. QED.

Although technically challenging, the intuition behind this

theorem is obvious. A POST equilibrium requires the absence of

27


preclude extravagant production costs, rather it encourages it.
Any firm that is spending a million dollars for 30 seconds of air
time wants to be sure that its commercial will have an impact.
Wilson Sporting Goods, for example, spent $1 million dollars to
produce a 60 second commercial for Super Bowl XXIX (it cost $2
million to air it). The company has no plans to air the
commercial on TV at any time in the future.‘ Nike, on the other
hand, spent $3 million dollars to air one 90 second commercial
whose production costs were reported to be in six figures.°

At some point it becomes difficult to distinguish between
similar mechanisms. Celebrities, for example, might be used to
reduce signal loss directly. A special one-hour episode of the
television show "Friends" appeared immediately after Super Bowl
XXX and drew 66.3 million viewers. On that same day a monthlong
$30 million dollar promotion for Diet Coke was concluded. This
commercial included the entire cast of "Friends," each of whom
was paid an estimated $250,000 to $500,000.°

Although these examples highlight the tradeoff between
various approaches to advertising, the same tradeoffs exist in
all advertising campaigns. Each firm must decide to how to
allocate advertising dollars to produce the maximum effect.

This paper uses a signalling game to study the tradeoff
between expenditures on ad production and air time. Rather than
focussing on the psychological or emotional impact of
advertising, I treat high production costs and celebrity

endorsements as components of a multifaceted advertising signal.’


refinement is of little use when trying to decide between two
equilibria, especially when neither includes a celebrity
endorsement.

Example 1: Consider a model with the following parameters: L=0,
H=1, p=1/2, B=1, V=1 and P,=17/36. The profit function for the
high- and low-type monopolist are respectively:
EU(H,A,S,EO(A,S)) =4E0(A,S) - (A) P,-S

FU(L,A,S,EO(A,S)) =EQ(A, 8) - (A) P,-S

Since L=0, the low-type monopolist is selling junk which is sure
to dissatisfy consumers. Because p=(1-p) =1/2, L=0, and H=1,
the formula for expected quality (equation 2) is greatly
simplified: EQ(A,S,) =g(A:A,) /(g(A:A,) +g(A:A,)). There are two

separating equilibrium in this game with So=0:¥

Equilibrium 1: {M(H)=(1,0),M(L)=(0,0)} with EQ(0,0)=1/3,
EFQ(1,0)=1, and EQ(A,S) for all other (A,S) can be arbitrarily

defined.

Equilibrium 2: {M(H)=(2,0), M(L)=(0,0)} with EQ(0,0)=1/4,
EQ(1,0)=1, and EQ(A,S) for all other (A,S) can be arbitrarily

defined.

The conditions for a sequential equilibrium can be easily
verified. Consider equilibrium 1. Consumer expectations can be
verified given the monopolist’s strategy and the simplified

formula above. The low type is earning a profit of 1/3 by not

18


sequential equilibrium preferred by the high type.

5, Price Signaling

In this section I consider an extension of the previous
model to include price signaling. I limit my analysis to the
case where V<B. Hence, the strategy of the monopolist is now a
function, M(Q): [H,L]-(P,A,S), that translates the actual quality
of the firm into a nonnegative tuple. Gross profits and
expectations of quality will likewise be a function of price,
advertising and celebrity endorsements. Price is assumed to be
completely deterministic and always observable. For the first

result it is not necessary to specify the exact relationship

between price and profits.

Theorem 8: Suppose that V<B. Let M(H) =(Py,Ay,Sy) and

M(L)=(P,,A,,S,) be part of a sequential equilibrium with A,#0 or

A,#0, then P,=P,.

Proof: I employ a proof by contradiction. Let teé{L,H} be the
type with nonzero advertising, A,#0. Suppose that P,#P,. If
consumer beliefs are consistent with Bayes’ rule then
EQ(P,,0,0)=t, since the observation (P,,0,0) is consistent with
the strategy of type t, but not with the strategy of type s.
Given these beliefs, however, the strategy M(t) =(P,,0,0) will

induce the same beliefs but increase type t’s profits by

35


P,A,+S,. This proves that A,>0 could not have been an optimal
strategy for type t. Hence, it could not have been part of a
sequential equilibrium. The assumption V<B is required to
ensure that viewing zero commercials is a possible in-equilibrium
observation by consumers. Q.E.D.

By theorem 8 and theorem 1 we know that if advertising
exists P,=P, and that S,=S, if A,>O. Hence, equations (1)-(3),
which describe consumer expectations, can be extended in the
obvious manner. I now discuss a more explicit model.

I assume that there is a continuum of consumers uniformly
distributed with mass R, along the interval [0,R]. A consumer’s
address tells us how much he/she would value a product which
performs satisfactorily. The quality of the product is
operationalized as the probability that the good in question will
perform satisfactorily in each period, for a randomly selected
consumers. Throughout I make the simplifying assumption that
consumers get no utility from an unsatisfactory product. I also
assume that one purchase is sufficient (and necessary)”! for each
consumer to determine whether or not the good is satisfactory.
The assumption that consumers become completely informed after
the initial purchase enables a simplified analysis of repeat
business. These assumption are similar to those made by Milgrom
and Roberts (1986).

The end result of the above assumptions is a linear demand
curve for a satisfactory product whose slope is -1 and whose

price and quality intercepts are both R. The demand curve for a

36


1. "A Monopoly on Mystique, Lincoln-Mercury Uses Advertising Blitz

to Market New Model," The Washington Post, September 1994.
2. "Benetton Touches a Nerve and Germans Protest," The New York

Times, Friday 3, 1995.

3. See "Super Sunday: A Shared American Experience," The
Washington Post, January 29, 1995.

4. See "Wilson, Once a Sports Goliath, Plays David," in The Wall
Street Journal, January 1995.

5. See "Super Sunday: A Shared American Experience," in The
Washington Post, January, 29, 1995.

6. See "Friends for Hire," in The New York Times, Friday, February
2, 1996.

7. The game I develop does not provide for any formal distinction
between celebrity endorsements and high production costs. Rather
I cite celebrity endorsements as an example of high production
costs.

8. The cost of air time various considerably depending on the show.
Air time during David Letterman goes for about $40-$50 thousand
dollars for each commercial. Prime time shows sometimes charge
hundreds of thousands of dollars per commercial. (See "Late-Night
TV Becomes a Crowded Arena, The Wall Street Journal, Wednesday,
August 31, 1994.)

9. Although the costs of advertising are explicitly modeled for
obvious reasons, no attempt is made at this time to deal explicitly
with the costs associated with production, nor with the
determination of the optimal price. In specific examples, later in
the paper, these other factors will be fleshed out in greater
detail. All examples prior to section 5 are merely included for
the purpose of illustrating important concepts.

10. In the model I consider, I do not rule out the possibility that
V>B+min[A,,A,]. V is simply the number of times that the consumer

will stick his hand in the urn and attempt to remove a commercial.
I allow the possibility that the consumer will come up empty-handed

48


that a defection by type-Q to the strategy M(Q)=(0,0) will
reduce the costs associated with signaling, while leaving the
gross profits to the type-Q monopolist unaffected. Hence, a
defection to M(Q)=(0,0) increases the net profits of type-Q.
This proves that conditions (1) and (2) of a sequential
equilibrium are not satisfied. Hence M(H) =(A,S) and M(L) =(A,S)
cannot be part of any pooling sequential equilibrium whenever
VsB.

Next, assume that WB. Again I employ a proof by
contradiction. Suppose that M(H) =(A,S) and M(L)=(A,S) are part
of a pooling equilibrium. Consider an alternative sequential
equilibrium where £Q(A,S)=L for all (4,85) #(0,0). Clearly,

M(H) =M(L)=(0,0) and £Q(0,0)=pH+(1-p)L form the only sequential
equilibrium given these beliefs. Neither type has an incentive
to defect because signaling is costly and can only worsen
consumer expectations of quality. Hence, (1),(2) and (3) of a
sequential equilibrium are satisfied. However, we will also have
EM(H,0,0,£0(0,0)) > EI (H,A, 5, pH+(1-p) L) -AP,-S and

EM (L,0,0,80(0,0)) > BI(L,A, S, pH+(1-p) L) ~AP,-S.

Hence, the pooling equilibrium with M(Q)=(0,0) is Pareto
preferred by both types to our conjectured equilibrium with
M(H) =(A,8) and M(L)=(A,S). Roughly speaking, its mutually
beneficial for both types to simultaneously abandon all

advertising. This proves that all pooling POST equilibria are of

24






simply introduce a fourth signaling option. The three different
kinds of equilibria discussed in this paper (pooling, separating,
and celebrity) would have to be compared to an equilibrium where
the high type separates in price. A determination would then be
made as to which equilibrium maximizes the profits of the high-
quality monopolist. Although in the current article I treat
quality as something that is exogenously determined, there is no
reason to think that the importance of celebrity endorsements is
confined to these models. Ippolito (1990) argues that in many
markets where quality is endogenously determined, advertising has
the capacity to "bond" performance. In this situation the
message sent by advertising expenditures is not "This is a high-
quality firm," but rather "This will continue to be a high-
quality firm." Although in Ippolito’s model the exact role of
advertising has been altered, certainly the same consumer
inference problems remain. The upshot of the matter is that
celebrity endorsements can also provide a more efficient bonding
mechanism by enabling consumers to get a clearer picture of the
size of the bond that has actually been posted.

It is important to keep in mind that this is a model of
advertising and not a depiction of it. A commercial which is
quickly forgotten cannot influence consumer behavior. Special
effects and celebrity endorsements are included in commercials to
help -make them memorable. If we assume that making unforgettable
commercials is more expensive than making mundane commercials,

the basic idea is still valid. Consumers may be responding to

43


enough so that sales will be made even when no commercials are
seen, it will not find such a strategy optimal given the
parameters above. Simple calculation reveals that equilibrium 3
is the LMSE and satisfies Overgaard’s Criterion A.

If B increases from 2 to 3 so that there is more signal
loss, the LMSE will be identical except that a celebrity
endorsement of 1 (as opposed to 5/3) will now be sufficient to
deter mimicry by the low type. This is because the extra noise
reduces the value of mimicry. In particular, the extra noise
implies that the likelihood that any low-type commercials will be

seen is lower.

Example 4: Consider the following model with V=2, B=2, H=1, L=0,
P,=1, p=1/4. Payoff functions to the low- and high-type
monopolist are described in equation (4) with C,=C,=1/2 and R=4.

There is one POST equilibrium.

Equilibrium 1: {M(H)=(2, 1, 1), M(L)=(2,0,0)} with

EQ(2, 1, 1)= 1 and EQ(P,A,S)< 1/4 for all other (P,A,S).

The canonical pooling equilibrium is no longer a POST
equilibrium. Because C,=C,=1/2, the price must be set above 1/2
to create positive profits. However, given that the prior
expected quality is only 1/4, this price is too high to result in
sales. Expected profits to the high type are 2, expected profits

to the low type are 0. As before, the price is chosen to

41


respectively. This setup is similar to Hertzendorf (1993).

We might imagine the urn to be shaped like a television set.
The process of selecting balls from the urn without replacement
would be equivalent to turning on the television at a randomly
selected time, to a randomly selected channel. Since the same
exact commercial cannot be observed twice (unless of course there
is video tape involved), it seems appropriate to assume a
stochastic process without replacement.

Clearly the consumers cannot view more commercials then were
purchased by either type monopolist, nor can the number of
monopoly commercials viewed exceed the total number of
commercials viewed. Similarly, if the total number of
commercials viewed, V, exceeds the number of irrelevant
commercials, then the consumer must view at least (V-B) of the
monopoly commercials, unless of course, A,<(V-B)." Let

Apin=Max{O,min{V-B,A,}} and A,,,=min{V,A,}, then for A.i,sA<Ayy we

can define the probability of consumers viewing A monopoly ads

using the hypergeometric distribution.

Ag B
— ~ _ ally ~
Prob(A=A:Ag) = g(A:Ag) = for AjinsASAna- (1)
BtAg
Vv
For all other A, g(A:Ag)=0 . We are now ready for the first

result.

Theorem 1: In any sequential equilibrium A,=0 implies that

S,=0 while A,>0 implies that S,=S,

11


20. I am treating expected quality as a scaler rather than a random
variable at this point. Although the linear relationship between
the profit functions may seem a bit implausible, it is satisfied by
examples illustrated later in the paper.

21. In other words, I am also assuming that consumers do not
observe the quality level of the goods purchased by their
neighbors.

22. I am speaking loosely here. More precisely, different
equilibrium refinements determine different equilibria and these
equilibria include different equilibrium prices.

SE (i, P, 0,0, BO) «23% (L, P, 0,0, BQ) .

23. This is because
ap

24. Copies of the Mathematica programs used to calculate the
following equilibria are available via email. Please send your
request to: "MHertzendorf@ftc.gov"

25. The fact that increasing noise results in a smaller celebrity
endorsement may depend on the specific parameters in these
examples. In particular, when the low type makes sales in the
celebrity equilibrium this outcome might be reversed.

50


use the same modulation (denoted by S). This fact enables us to
utilize Bayes’ rule to define consumer expectations and the
expected monopoly profits. Let the monopolist’s strategy be

given by [M(H) = (Ay, 8) , M(L) =(A,,8)] . Then we have

pHg(A:Ay) +(1-p) Lg(A:A,)
pg(A:Ay) +(1-p) g(A:A,)

EQ(A, 8) = for Ac<A,, Ay (2)

When A,<A<A,; EOQ(A,S)=H and when a,<A<A,; FQ(A,S) =D.

Ann = — —
Em(Q,Ag,S,EQ(A,S)) = g(A:Ag)[1(Q, Ag, S, EQ(A, S)) ~AgP,-S | (3)
AoA,

Hence to compute expected profits we just multiply the
probability that A commercials will be observed by the profits
that will result from this realization of A, and sum up over all
possible realizations. The expected quality incorporates Bayes’
rule in the relevant region. Intuitively, the consumer is just
comparing the relative likelihood that the realization of A came
from either of the two types. In doing so the consumer considers
the strategy of each type and the prior probabilities p and
(1-p) that the firm is a high- or low-quality monopolist,
respectively.

It is common practice to distinguish between two different
kinds of equilibria. In a pooling equilibrium both the high and

low-type monopolist pick identical strategies. In this situation

13


Abstract

Since Milgrom and Roberts (1986) game theorists studying
advertising have generally assumed that aggregate advertising
expenditures are perfectly observed by consumers. In the real
world, however, consumers see only a_ small fraction of the
commercials aired by a given firm and typically do not view the
firm’s total advertising expenditure. This gives the firm an
incentive to make sure that each commercial has as much impact as
possible. The impact of a commercial may be enhanced through
extravagant production costs, or by purchasing a celebrity

endorsement.

Using a signalling game this paper shows how a monopolist may
attempt to balance the cost of production against the cost of air
time to send a credible signal to consumers at the minimum possible
cost. Several examples illustrate the extent to which extravagant
production costs (or expensive celebrity endorsements) can
substitute for additional spending on air time. Paradoxically,
although it is the existence of signal loss (i.e. consumers viewing
fewer commercials than were actually purchased) that makes a
multifaceted advertising signal attractive, greater signal loss
does not necessarily lead to greater production or endorsement
expenditures. Rather as signal loss increases, the monopolist has
a tendency to substitute expenditures on air time for expenditures

on production or celebrity endorsements.


type from playing M(L)=(1, 2/3). Hence, when Sy=2/3, the
expected profits to the low type from playing M(L)=(1,2/3) must
also be zero.

The high type knows with certainty that exactly one of its
advertisements will be seen by consumers along with the
associated celebrity endorsement. This will be sufficient to
convince consumers that it is the high type. Hence, the profits
of the high type are #Il(H,1, (2/3) ,£0(1,2/3)) =4-(1/3) -(2/3) =3.
A comparison between the pooling equilibrium and the celebrity
equilibrium is shown in figure 3.

Using this example we can also see theorem 3 in action.

In the celebrity equilibrium the high type gets 3, while the low
type gets a payoff of 0. In the canonical pooling equilibrium
the high type always receives a payoff of 4p, the low type
receives a payoff of p.. A demarcation between the (celebrity)
separating equilibrium and the pooling equilibrium is created at
p=3/4. For p<3/4 both the canonical pooling equilibrium and the
celebrity equilibrium are Pareto optimal with respect to sender
types. This is because the low type prefers the pooling
equilibrium, while the high type prefers the separating
equilibrium. For p>3/4 both types prefer the pooling equilibria
and this will be the unique Post equilibrium.

For p<3/4 the lexicographically maximum sequential
equilibrium ("LMSE") is the celebrity equilibrium. For p>3/4
the LMSE is the canonical pooling equilibrium. Recall that the

LMSE satisfies Overgaard’s Criterion A and is simply the

34


than a separating equilibrium. In particular, I show in the next
section that when the prior probability of facing a high-quality
monopolist is sufficiently high, only a pooling equilibrium is
possible. This is because as the prior probability increases,
the consumer’s (prior) expected quality also increases. This in
turn reduces the incentives for both types of monopolist to incur
the expense of advertising.

Other authors have also recently stressed the importance of
pooling equilibria and critiqued the use of some refinements that
restrict out-of-equilibrium beliefs. Mailath, Okuno-Fujiwara and
Postlewaite (1993) propose a refinement concept called the
lexicographically maximum sequential equilibrium ("LMSE"). By
construction the LMSE is pareto optimal with respect to the
sender types ("POST"). In the current game where there are only
two types, the LMSE is simply the sequential equilibrium that is
preferred by the high type. The idea of selecting the
equilibrium preferred by the high type has also been proposed by
Overgaard (1993) in his Criterion A. According to Overgaard, "At
any point in the game the type L firm is deciding whether or not
to mimic the H firm, hence, deciding whether to accept an
equilibrium profile proposed by type H. When we take this
viewpoint, the emphasis should be on the decision of type H..."
Maskin and Laffont (1987), however, propose a different selection
from the set of POST equilibria which they call the best perfect
6

Bayesian equilibrium.” In particular, they argue in favor of

selecting that equilibrium which maximizes the expected profits

22




Separating Equilibrium

Pooling Equilibrium

4

0.6

1

0.6

High-Type Profits

Profits

-0.2



switch to M’(H)=(0,0). This proves necessity. QED.

Theorem 6: A necessary condition for a separating celebrity
sequential equilibrium of the form M(H) =(Ay,S,), M(L)=(0,0) to
also be a POST equilibrium is that

S,=Max , [EIL(L,A,0, £0/(A,0)) -P,A]-EI(L,0,0,£0(0,0)) =Z>0 where

EQ'(A, 0) =EQ(A, Sy) .°

Proof: Suppose that this theorem is false and that in our
proposed POST equilibrium we had S,>Z. Consider the following
proposed alternative sequential equilibrium with M(H) =(A,,2Z),
M(L)=(0,0) and BOQ”(A,Z)=EQ(A,S,). Out-of-equilibrium
observations are associated with the low type. Let A* be the
maximizer in the definition of Z. Then by construction we have
EN(L,A*,0,£0'(A,0)) -P,A*-Z=EI(L,0,0,£0(0,0)) and for all other
A we must have ED(L,A,0,£Q'(A,0)) -P,A-ZsEII(L,0,0,E0(0,0)) . Now
by construction we know that

ElL(L,A,Z, EQ" (A, Z)) =EL(L, A,0,EQ'(A,0)) so we can also say that
FIL(L,A,Z, EO" (A, Z)) -P,A-Zs HI(L,0,0,£0(0,0)) for any arbitrary A.
This demonstrates that condition (1) of the previous theorem is
satisfied.

Condition (2) of the previous theorem will clearly be
satisfied with the modulation Z if it was satisfied at S,>Z,
since S, is just being replaced by Z on both sides of condition
(2). Finally condition (3) will be satisfied when s, is

replaced by Z<S, on the left side of condition (3). This

30


cost is independent of quality (i.e. C,=C,) it is easy to show
that price separation is impossible. This is because the profit
functions of the high- and low-type monopolist will be maximized
at the same price.” Roughly speaking, there aren’t any pricing
strategies that the low type finds too costly to mimic. In fact,
when the difference between C, and C, is sufficiently small
price signaling will not occur in equilibrium. When |Cy-C¢,| is
sufficiently great the profit functions of the two types are
adequately differentiated and price signaling becomes cheaper for
the high type. (Please see Hertzendorf (1990) for a detailed
discussion.)

These concepts are best illustrated with some examples. The
program Mathematica was used to calculate equilibria in the

following examples .*

Example 3: Consider the following model with V=2, B=2, H=1, L=0,
P,=1, p=1/4. Payoff functions to the low- and high-type
monopolist are described above in equation (4) with C,=C,=0 and

R=4. There are three potential POST equilibria.

Equilibrium 1: {M(H)=(1/2, 0, 0}, M(L)=(1/2, 0, 0)} with

FQ(1/4, 0, 0)= 1/4 and EQ(P,A,S)=0 for all other (P,A,S).

.Equilibrium 2: {M(H)=(2, 1, 5/3), M(L)=(2,0,0)} with

EQ(2, 1, 5/3)= 1 and EQ(P,A,S)< 1/4 for all other (P,A,S).

39


the form M(Q)=(0,0). Notice that in the first part of the proof
(V<B) we exploited the definition of a sequential equilibrium,
while in the second part of the proof (V2B) we exploited the

additional requirements of a POST equilibrium. QED.

Theorem 3: For p sufficiently close to 1, all POST equilibria
are pooling equilibria of the form M(Q)=(0,0), Q€{L,H} and

EQ(0,0) =pH+(1-p)L."8

Proof: Clearly M(Q) =(0,0) and E£Q(0,0)=pH+(1-p)L is always a
sequential equilibrium provided that out-of-equilibrium beliefs
are defined such that £Q(A,S)=L for all (A,S)#(0,0). The
expected profits to type-Q in the canonical pooling equilibrium
are HIl(Q,0,0,pH+(1-p)L). Consider any other sequential
equilibrium where M(H) =(A,,S,), M(L)=(A,,S,) and consumer beliefs
are EQ(A,S). We will compare the profits of this arbitrary
sequential equilibrium with the profits that both types will
obtain in the canonical pooling equilibrium as p approaches 1.
As p71, EQ(0,0)=pH+(1-p)L approaches H. By the continuity of
expected profits in expected quality, as p71, we also know that
the expected profits to the type-H monopolist will approach
EIL(H,0,0,H). These are the best possible profits for type-H,
since it has spent nothing on signaling, and yet consumers will
perceive the product as being of the highest possible quality.
Similarly, as p~1, the profits to the low type in the canonical

pooling equilibrium will approach HI(L,0,0,H) and these are,

25


observed by consumers and A is a particular realization of this
random variable. I use A, to denote the advertisements
purchased by the type-Q monopolist (in equilibrium) and g (A: Ap)
to denote the probability density function of A. Hence A isa
random variable whose distribution depends on Ap. Occasionally,
I use a regular type “A” to indicate the argument of various
functions.

The monopolist is only concerned about the expectations of
consumers, because higher expectations of quality translate into
greater profits for the monopolist. Given the beliefs of
consumers EQ(A,S), I represent the expected profits to the type-
Q monopolist which purchases the advertising pair (Ag, Sp) as
Em (Q, Ag, Sg, EQ(A, S)) ~P,Ag-S, . The advertising strategy (Ap, Sp)
affects profits in two ways. First, holding fixed the beliefs of
consumers, changing the advertising strategy will change the
expectations that consumers are likely to have because the
distribution of A will be altered. This in turn will change the
monopolist’s expected gross profits. Second, changing the
advertising strategy will affect the advertising costs
experienced by the monopolist. I assume that expected profits
are increasing and continuous in expected quality so that,
holding advertising expenditures fixed, the monopolist always
wants consumer expectations of quality to be as high as possible.

We are now ready to define the first equilibrium concept .?


Equilibrium 1 Equilibrium 2

1.2



[1]

[2]

3]

[4]

[5]

[6]

[7]

[8]

[9]

References

Banks, J. and J. Sobel, "Equilibrium selection in signaling
games," Econometrica, Vol. 55 (1987), pp. 647-661.

Cho, I.K. and D. Kreps, "Signaling games and stable
equilibria," Econometrica, Vol. 55 (1987), pp. 1367-1389.
Harsanyi, J., "Games with Incomplete Information Played by
Bayesian Players," Management Science, Vol. 14 (1967-68),
pp. 59-82,320-34, and 486-502.

Hertzendorf, Mark N., "I’m not a high-quality firm---but I
play one on TV," RAND Journal of Economics, Vol. 24, No. 2,
Summer 1993, pp. 236-247.

alae tei tata! , "I’m not a high-quality firm---but I
play one on TV," University of Rochester Working Paper, No.
216, January, 1990.

Ippolito, Pauline M., "Bonding and Nonbonding Signals of
Product Quality," Journal of Business, Vol. 63 (1990), pp.
41-60.

Kihlstrom, Richard and Michael Riordan, "Advertising as a
Signal," Journal of Political Economy, Vol. 92 (1984), pp.
427-450.

Laffont, Jean-Jacques and Eric Maskin, "Signaling, Pooling
and Efficiency," Harvard Working Paper, 1989.

Mailath, George J. and Masahiro Okuno-Fujiwara, "Belief
Based Refinements in Signaling Games," Journal of Economic

Theory, Vol. 60 (1993), pp. 241-276.

46


Proof: We employ a proof by contradiction. Suppose instead
there was a sequential equilibrium with A,>0 and S,#S,. If
consumer beliefs are consistent with Bayes’ rule then
EQ(A,S,)=L for all AetAyn,.-++Anax} while EQ(A,S,)=OzL if A=0.
However, given these beliefs the strategy M(L) =(0,0) will induce
a set of beliefs (in equilibrium) that are at least as favorable,
possibly more favorable than the strategy M(L) =(A,,S,). Because
the strategy M(L)=(0,0) is also less costly than the strategy
M(L)=(A,,S,), this later strategy could not have been optimal for
type L and, therefore, could not have been part of a sequential
equilibrium. The proof that M(L)=(0,S,) with S,>0 cannot be
part of an equilibrium is even easier. The alternative strategy
M(L)=(0,0) does not change any of the observations by consumers
since the celebrity endorsement cannot be seen when A,=0.,
Hence, the alternative strategy reduces advertising expenditures
of the low-type without affecting gross profits. This proves
that M(L)=(0,S,) could not have been part of a sequential
equilibrium. We might summarize the proceeding argument as
follows: (1) if the low-quality monopolist isn’t going to
advertise, then it’s certainly not going to waste money producing
a commercial and (2) if advertising by the low-type monopolist
doesn't trick consumers into believing its quality is higher than
it really is, then advertising can’t be worthwhile (hence,
S,=Sy)- Q-E.D.

The importance of the previous result is that it tells us

that if both types of monopolist advertise, they will also both

12