A simple model of herd behavior

A Simple Model of Herd BehaviorAuthor(s): Abhijit V. BanerjeeSource:

The Quarterly Journal of Economics,

Vol. 107, No. 3, (Aug., 1992), pp. 797-817Published by: The MIT PressStable URL: /stable/2118364Accessed: 27/05/2008 02:04Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at/page/info/about/policies/. JSTOR's Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and youmay use content in the JSTOR archive only for your personal, non-commercial contact the publisher regarding any further use of this work. Publisher contact information may be obtained at/action/showPublisher?publisherCode= copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We enable thescholarly community to preserve their work and the materials they rely upon, and to build a common research platform thatpromotes the discovery and use of these resources. For more information about JSTOR, please contact support@://

THE

QUARTERLY JOURNAL

OF

Vol. CVII

ECONOMICS

August 1992

Issue 3

A SIMPLE MODEL OF HERD BEHAVIOR*

ABHIJIT

V.

BANERJEE

We analyze a sequential decision model in which each decision maker looks at

the decisions made by previous decision makers in taking her own decision. This is

rational for her because these other decision makers may have some information

that is important for her. We then show that the decision rules that are chosen by

optimizing individuals will be characterized by herd behavior; i.e., people will be

doing what others are doing rather than using their information. We then show that

the resulting equilibrium is inefficient.

I. INTRODUCTION

There are innumerable social and economic situations

in

which we are influenced in our decision making by what others

around us are doing. Perhaps the commonest examples are from

everyday life: we often decide on what stores and restaurants to

patronize or what schools to attend on the basis

of how

popular

they seem to be. But it has been suggested by Keynes [1936], for

example, that this is also

how investors in asset markets often

behave (the famous "beauty contest" example).' In the literature

on fertility choices it has frequently been suggested that various

fertility decisions (how many children to have, whether or not to

use contraception, etc.) are heavily influenced by what other people

in the same area are doing.2 It has also been suggested that the

same kind of factor also influences the decision to adopt new

*I thank Aniruddha

Dasgupta, Mathias

Dewatripont,

Bob

Gibbons,

Tim

Guinnane, Sandy Korenman, Eric Maskin, Andreu Mas-Colell, Barry

Nalebuff,

Klaus Nehring, Avner Shaked, Lin Zhou, and two anonymous referees for helpful

comments and suggestions.

1. See Scharfstein and Stein [1990] for some evidence suggesting that this is

indeed how managers behave.

2. See Cotts Watkins [1990].

?)

1992 by the President and Fellows of Harvard College and the Massachusetts

Institute of

Technology.

The Quarterly Journal of Economics, August 1992

798 QUARTERLY JOURNAL OF ECONOMICS

technologies.3 Voters are known to be influenced by opinion polls to

vote in the direction that the poll predicts will win; this is another

instance of going with the flow.4 The same kind of influence is also

at work when, for example, academic researchers choose to work

on a topic that is currently "hot."

The aim of this paper is to develop a simple model in which we

can study the rationale behind this kind of decision making as well

as its implications. We set up a model in which paying heed to what

everyone else is doing is rational because their decisions may reflect

information that they have and we do not. It then turns out that a

likely consequence of people trying to use this information is what

we call herd behavior-everyone

doing what everyone else is doing,

even when their private information

suggests

doing something

quite different.

But this suggests

that the very act of trying to use the

information contained in the decisions made by others makes each

person's decision less responsive to her own information and hence

less informative to others. Indeed, we find that in equilibrium the

reduction of informativeness

may be so severe that in an ex ante

welfare sense society may actually be better off by constraining

some of the people to use only their own information.

A common real world

example may make our basic argument

clearer.5 Most of us have been in a situation where we have to

choose between

two restaurants that are both more or less

unknown

to us. Consider now a situation

where there is a

population of 100 people who are all facing such a choice.

There are two restaurants A and B that are next to each other,

and it is known that the prior

probabilities are

51

percent

for

restaurant A being the better and 49 percent for restaurant B being

better. People arrive at the restaurants in

sequence, observe the

choices made by the people before

them, and decide on one or the

other of the restaurants. Apart from knowing the prior probabili-

ties, each of these people also got a signal which says either that A

is better or that B is better (of course the signal could be

wrong). It

is also assumed that each person's signal is of the same quality.

Suppose that of the 100 people, 99 have received signals that B

is better but the one person whose signal favors A gets to choose

first. Clearly, the first person will go to A. The second person will

now know that the first person had a signal that favored

A, while

3. See, for example, Kislev and Shchori-Bachrach [1973].

4. See, for example, Cukierman

[1989].

5. I am indebted to an anonymous referee for this very transparent

example.

A SIMPLE MODEL OF HERD BEHAVIOR

799

her own signal favors B. Since the signals are of equal quality, they

effectively cancel out, and the rational choice is to go by the prior

probabilities and go to A.

The second person thus chooses A regardless of her signal.

Her choice therefore provides no new information

to the next

person in line: the third person's situation is thus exactly the same

as that of the second person, and she should make the same choice

and so on. Everyone ends up at restaurant A even

if, given the

aggregate information, it is practically certain that B is better.

To see what went wrong, notice that if instead the second

person had been someone who always followed her own signal, the

third person would have known that the second

person's signal had

favored B. The third person would then have chosen

B,

and so

would have everybody else.

The second person's decision to ignore her own information

and join the herd therefore inflicts a

negative externality on the

rest of the population. If she had used her own information, her

decision would have provided information to the rest of the

population, which would have encouraged them to use their own

information as well. As it is, they all join the herd.

The identification of this externality,

which we call "herd

externality," and the investigation of what it implies, is the main

contribution

of this

paper. The model we present is extremely

simple and does not aspire to capture any specific institutional

detail. There is a set of options represented by a line

segment, and

within this set there is one correct

option. The aim of the game is to

find the correct

option.

All those who find the

correct option get z,

while all others get 0. There is a population of N

people who take

their decisions in a fixed order; each person

moves knowing the

choices made by those before her but not the

information these

choices were based on. Each individual

may either be uninformed,

in which case she has no

signal, or informed, in which case she has

a signal about what the

right option is. This signal, however, may

not be correct. It is correct only with

probability 1; otherwise it is

completely uninformative.

Everybody is rational in the Bayesian

sense,

and the

equilibrium

we look at is a Bayesian-Nash

equilibriums

Because of the extreme simplicity of this model, using quite

6. This formulation is somewhat more

complicated and somewhat less natural

than the two-restaurant

setting discussed above, but it turns out that deriving a

general

result about the

equilibrium decision rule which holds for all parameter

values is actually simpler in this

setting.

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QUARTERLY JOURNAL OF ECONOMICS

elementary arguments we are able to derive a number of rather

striking results that derive directly from the presence of the herd

externality. These results are summarized below.

1. The equilibrium pattern of choices may be

(and for a large

enough population, will be) inefficient in the ex ante welfare sense.

Before people know the order in which they are

choosing, they may

all agree to prevent the first few decision makers from

observing

the choices made by anyone else. This is a direct

consequence of the

herd externality and suggests why herd behavior

may be undesir-

able from the social point of view.

2. The probability that no one in the population

chooses the

correct option is bounded

away

from zero for

any size of the

population. Indeed, by making the

probability

3

small, we can

make this probability as

large as we like. This contrasts with the

case where the decision makers choose without

looking at each

other (i.e., they follow their own

information). Since the informa-

tion they have is

independent, as long as the population is large

enough, someone must choose the right option in this case.

3. Since the herd

externality is of the positive feedback type (if

we join the crowd, we induce others to do the

same), the equilib-

rium pattern of choices will be

very volatile across several plays of

the same game. The

signals (which are partly random and need not

be correct) that the first few decision

makers have will determine

where the first crowd

forms, and from then on, everybody joins the

crowd. This

may shed some light on observations

of "excess

volatility" made in the context of many asset markets7 and the

frequent and apparently unpredictable

changes in fashions.

The emphasis on the herd

externality also distinguishes

our

work from two other

explanations of clustering behavior that have

been

suggested before. One is an explanation based on

strong

complementarities: some things are more worthwhile when

others

are

doing related things.8 Examples of such complementarities are

fashions in

consumption (see, for example, the analysis of fashions

in Karni and Schmeidler

[1989]) and network externalities

in

production (see Arthur [1989], Farrell

and Saloner [1985], and

Katz and

Shapiro [1985]). Whether or not it is as important in

7. The idea that informational externalities may

explain observations of excess

volatility is also discussed in Banerjee [1988].

8. As far as we know, there is actually no formal

model that tries to explain

herd behavior in these

terms;

what exists in the literature is the idea that if

complementarities are sufficiently large, then people will do what

the crowd is doing

even if left to themselves

they would have done something else. Under suitable

conditions this could clearly lead to herd behavior.

A SIMPLE MODEL OF HERD BEHAVIOR

801

other contexts where we observe herd behavior is an open question.

In any case, there is no contradiction between this view and ours;

our point is that many aspects of herd behavior can be explained

quite plausibly

without

invoking

these kinds

of gains

from

association.

A different explanation of herd behavior, which, like the

present work is based on informational

asymmetries,

was sug-

gested in an interesting

recent paper by Scharfstein

and Stein

[1990]. The key difference between their explanation and the one

suggested here is that their explanation is

based on an agency

problem; in their model the agents get rewards for convincing

a

principal that they are right. This distortion

in incentives plays an

important role in generating herd behavior in their model.

By

contrast, in our model agents capture

all of the returns generated

by their choice so that there is no distortion

in incentives.9

In any case, this

approach is not inconsistent

with our

approach. This kind of principal agent problem (trying

to con

someone else into believing that you know something)

seems

common enough, especially in the context of asset markets.

On the

other hand, in many of the other potential instances

of herd

behavior, such as fertility choices, adoption

of innovations, voting

etc., there is no obvious principal agent problem.

It is therefore

useful to establish that inefficient herd behavior

can arise even

when the individuals themselves capture the rewards

from their

decisions.

The strategy of this paper is as follows: the

basic model is

presented in Section

II and analyzed in Section III. The results are

discussed in Section IV. Several extensions

and modifications of the

basic model are presented in Section V. We conclude

in Section VI.

The essential ideas in this paper are generally quite straightfor-

ward. Nonetheless, a fully rigorous treatment

of the results derived

here will certainly be very cumbersome

and repetitive. To prevent

the paper from being too unreadable,

we have omitted some proofs

9. There are two other important differences between the

two models. In the

Scharfstein and Stein model unlike in ours,

all

agents get

a

signal. However, only

some of these signals are potentially informative (though not necessarily correct):

the rest are duds; and the agents cannot distinguish between the

two

types of signals.

In our model this would amount to assuming that agents do not know whether they

have a signal or not. The decision to herd and not use one's signal

is therefore

somewhat less significant (since they actually may not have a signal)

than it is in our

model (where they know they have a signal). On the other hand,

it is possible to

generate instances of herd behavior

in their model even if there are only two agents.

In the model given here the equilibrium is always (second-best) optimal

if there are

two agents.

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and presented others informally. More rigorous proofs

were pre-

sented in an earlier version of this paper

[Banerjee,

1989], which is

available from the author.

II. THE BASIC MODEL

There is a population of agents of size N each of whom

maximizes the identical risk-neutral utility function VNM10

de-

fined on the space of asset returns. For convenience we shall

just

assume that this utility is the same as the monetary

amount

received by the person.

There is a set of assets indexed by numbers in [0,1] . Call the ith

asset a(i). The physical return

to the ith asset to the nth person

investing in that asset is z(i)

E

R. Let us assume that there is a

unique i* such that z(i)

= 0 for all i

?

i* and

z(i*)

=

z, where z

> 0.

This is essentially the assumption that the excess return

on one

asset to the people investing

in it is strictly greater than that on all

other assets.

Of course, everybody, given these payoffs, would want

to invest

in i*. The trouble is no one knows which one it is. We assume

uniform priors so there is not even a likely candidate

for i*.

However, some people have an idea of which one it

might be.

Formally, there is a probability

x

that each person receives a signal

telling her that the true

i* is i'. The signal need not, of course, be

true, and the probability that

it is false is 1 -

P.

If it is false, then

we assume that it is uniformly distributed

on [0,1] and therefore

gives no information about what i* really is.

The decision making in this model is sequential; one person

chosen at random takes her decision first (she

cannot decide to

delay

her decision). The next person, once again chosen at random,

takes her decision next but she is allowed

to observe the choice

made by the previous person and can benefit

from the information

contained in it. However, she is not allowed

to find out whether or

not the person before her actually got a signal.11

The rest of the game proceeds in the same way, with each new

decision maker making her decision

on the basis of the history of

the past decisions and

their own signal if they have one. After

everybody

has made her choice, all the alternatives that have

been

10. It will be evident that for much

of what we

say

it is irrelevant whether we

take our utility function to be

risk neutral or risk averse. The only point where we

need the assumption is when we measure ex

ante welfare.

11. And a fortiori she cannot observe the signal observed by her predecessor.

A SIMPLE MODEL OF HERD BEHAVIOR

803

chosen are tested, and if any of these turn out to work, those who

have chosen it receive their rewards. If no one has chosen an option

that works, the truth remains undiscovered, and no one gets

rewards.

It will be assumed that the structure of the game and Bayesian

rationality are common knowledge. Each person's strategy is a

decision rule that tells us for each possible history what that person

will choose. We are looking for a Bayesian Nash equilibrium in

these strategies. The nature of the equilibrium play, however,

turns out to depend on certain critical tie-breaking assumptions.

Some of these assumptions may be dispensed with by strengthen-

ing the equilibrium concept, but it seems more natural to introduce

these as explicit assumptions. These assumptions are listed below;

the relevance

of these

assumptions

will become clear in the

appropriate context. It should also be possible to see that each of

these assumptions is made to minimize the possibility of herding.

ASSUMPTION A. Whenever a decision maker has no signal and

everyone else has chosen

i =

0,

she always chooses i

= 0.

B. When decision makers are indifferent between

ASSUMPTION

following their own signal and following someone else's choice,

they always follow their own signal.12

ASSUMPTION C. When a decision maker is indifferent between

following more than one of the previous decision makers, she

chooses to follow the one who has the highest value of i.

III.

THE EQUILIBRIuM DECISION RULE

The first decision maker's decision will clearly depend on

whether or not she has a signal. If she has a signal, she will

certainly follow

her

signal. While

if she has no

signal, by

our

Assumption

A she will choose i

=

0. This choice minimizes

12. This assumption is not entirely innocuous.

If the third decision maker

decides to follow the second independently of her informational situation, the fact

that she follows provides no information to the next decision maker. The next

decision maker who has a signal, now faces the same choice; she is indifferent

between following her own signal and joining the second and the third decision

maker. If she too ignores her own signal, the next decision once again has the same

choice and so on. There could, for example, be an equilibrium in which almost

everybody follows either the first decision maker or the first deviant. Since this

involves even more herding than the equilibrium we describe above, its welfare

properties will typically be worse; and in this sense by making the above tie-

breaking assumption, we are choosing to focus

on the best of the set of

possible

equilibria.

804

QUARTERLY JOURNAL OF ECONOMICS

misinformation: the only case where this

will cause confusion is

when i*

= 0, but since this happens with probability 0, we can

ignore this

possibility.13

If the second decision maker has no signal, then she will

of

course imitate the first decision maker and

invest in the same

asset. However, if she has a signal and the

first person has not

chosen i

=

0, she has a problem. She knows that the

first decision

maker had a signal and this signal is as likely to be right

as her own

signal. She is therefore

indifferent between following the first

decision maker's signal and following her own signal.

In this

situation our Assumption

B becomes relevant.'4 By invoking this

assumption, we determine that the second person will,

in this case,

follow her own signal.

The third decision maker can observe four possible

histories:

one or both of her predecessors may have chosen

i = 0, and if

neither of them had chosen i

=

0, they could have still either agreed

or disagreed. If they both chose

i

=

0, the third person should

follow them if she has no signal and follow

her signal otherwise. In

all the other cases, if she does not have a signal, she

should follow

the person who has not chosen i

= 0. If both the others have chosen

i

?

0 but have not agreed with each other,

of course this does not

determine a course of action. Since she is indifferent, however,

we

can invoke our third tie-breaking rule, Assumption C,

which tells

us that she should follow the person

with the highest i.

On the other hand, if the third decision

maker does have a

signal i', she will

follow her own signal, unless both people before

her have chosen the same option and this option

is neither i

=

0 nor

i

=

i'. When both of her predecessors have chosen

i =

0, this is

obvious. When only one of them has

chosen something different

from i

=

0 and

i

=

i' and the other has chosen i

=

0, this is a

consequence of our Assumption

B.

And,

of course, when the third

person's signal matches

the choices made by one or both of

her

predecessors, she must choose to

follow her own signal since this

could not happen unless her signal

was correct.

The last point is much more general than

this specific situa-

tion and deserves to be emphasized.

Whenever some person's

signal matches

the choice made by one of her predecessors,

she

should always follow

her

signal.

This follows from the fact that the

probability that two people should get

the same signal and yet both

13. Of course, there is nothing special about the point

0. The person could just

as well choose some other point i as long as everybody

knows what i is.

14. Once again, this assumption is not entirely innocuous.

A SIMPLE MODEL

OF HERD BEHAVIOR

805

be wrong is zero. To

deal with the

remaining case, we prove the

following simple lemma.

LEMMA

1. If the first and the

second decision

makers have both

chosen the same F

X

0, the third decision

maker should choose

to follow them.

Proof of Lemma 1. Note

that

prob[i*

=

TIH]

=

{a332(1

-

A) + a23(1-

prob[i*

=

i' H]

=

{a2,8(1

-

p)(1-

)(1

-

oa}/prob[H]

o)/prob[H],

where H represents

the event in which

the first two people have

both chosen

I

and the third person

has the signal i'.

Clearly, the first term is greater than the

second. The third

person should therefore choose

i.

Q.E.D.

This has a simple

intuitive

explanation:

the third

person

knows that the first

person must have a signal,

since otherwise she

would have chosen i

=

0. The first person's choice

is therefore at

least as good as the

third person's signal.

Further, the first person

has someone who has

followed her. This is some

extra support for

the first person's choice, since it is

more likely to

happen when the

first person is right than when

she is wrong. It is therefore

always

better to follow the first

person.

The same intuition

tells us what should

happen in any

situation when several

options other than i

=

0 have been chosen

but only one of them

has been chosen by two

people. Assume that

the next person does not have a

signal that matches

any of the

options that have

already been chosen (if she has

a signal that

matches someone else's

choice, she should, of

course, follow her

own signal). In the situation

where this option is not the

one with

the highest i, it is of course

clear (by our

Assumption C) that both

those people must have the

same signal and therefore

they must be

right. In the situation

where it is the one who has the

highest i, the

argument in the previous

paragraph applies, and therefore once

again this option is the best. In

either case,

therefore, the next

person should choose the

option that has been chosen

by

two

people.

Once one option has been chosen

by two people, the next

person

should

always

follow that

option

unless her

signal

matches

one of the options that have

been already

chosen;

in that

case she

806

QUARTERLY JOURNAL OF ECONOMICS

should follow her own signal. A combination of this decision

rule

with Assumption C tells us that the next person will observe

one of

three alternative histories.

i) One option (other than i = 0) has been chosen by

more than

one person, and this is the one that has the

highest i.

ii) One option (other than i

=

0) has been chosen by more than

one person, and this is not the one with the

highest i.

iii) Two options (other than i

=

0) have been chosen by more

than one person, one of which is the one with the highest i.

In the second and the third scenarios, it is clear that the

option

which is not the highest value of i is the correct option, and all

subsequent decision makers should choose it. The argument in the

first case is very similar to the argument for Lemma 1. The

next

decision maker should decide to follow the option that has

already

been chosen by more than one person.

The same argument can now be extended to all

subsequent

decision makers. This yields the following

proposition

which

summarizes all the

arguments

we make

above.15

PROPOSITION

1. Under

Assumptions

A, B, and C, the unique

(Nash) equilibrium decision rule that everyone will adopt is

decision rule D16

given below.

1. The first decision maker follows her

signal if she has one

and chooses i

=

0 otherwise.

2. For k

>

1, if

the kth decision maker has a signal, she will

choose to follow her own signal either if and

only

if

(a) holds or

if (a) does not hold, (b) holds, where (a) and

(b) are given below.

(a) Her signal matches some option that has

already

been

chosen.

(b) No option other than i

=

0 has been chosen

by more than

one person.

3. Assume that the kth decision maker has a

signal. If any

option (among those already chosen) other than the one with

15. A formal proof is available from the author on request.

16. Actually this is not the full decision rule. Technically we have to

specify the

optimal decision in other parts of the decision tree. In

particular, we have

to

specify

the decision rule in the following scenario: one

person finds that his option matches

someone else's and chooses that

option.

In the

equilibrium specified it

is known that

he would not choose that option unless his signal matches that option.

Therefore,

everybody else follows

him

expecting that

no one else will deviate.

However,

now

someone else deviates and chooses another of the

options that

have

already

been

chosen.

It is true that this can happen only with

probability zero, but when it

happens,

we need to say what the subsequent decision makers will choose.

However, it turns

out that agents are

actually indifferent between sticking to the original path and

deviating to the new one, and we can make any assumption we like.

A SIMPLE MODEL OF HERD BEHAVIOR

807

the highest i has been chosen by more than one person, the kth

decision maker will choose this option, unless

her signal

matches one of the other options that has already been chosen.

In this case she chooses the latter option.

4. Assume that the kth decision maker has a signal. If the

option with the highest i (among those already chosen) has

been chosen by more than one person and no other option

(except i

=

0) has been chosen by more than one person, she

will choose this option unless her signal matches one of the

options already chosen. In this case she chooses the latter

option.

5. Assume that the kth decision maker does not have

a signal.

Then she will choose i

=

0 if

and only if that is what everyone

else has chosen. Otherwise, she chooses the option with the

highest value of i that has already been chosen unless one of

the other options (excluding i

=

0) has been chosen by more

than one person. In this case, she chooses the latter

option.

The only part of the statement of this proposition that

is not

explained above is the uniqueness. Because each person's payoff is

completely independent of the choices made by everyone coming

after her in the decision process, there are no

strategic elements

here, and we can solve this game by moving forward in the game

tree. The uniqueness

of the solution is therefore

automatically

guaranteed.

The equilibrium decision rule as stated above is

somewhat

complicated and potentially confusing. In Figure I the decision rule

for decision maker k

(for k

>

2) is presented in a schematic form

that may be easier to follow.

IV. DISCUSSION OF RESULTS

A. Description of the Equilibrium

The equilibrium decision rule in the above model is character-

ized by extensive herding; agents abandon their own signals and

follow others even when

they are not really sure that the other

person is right. The first person always follows her own signal if

she has

one,

and so does the second

person,

but we cannot

guarantee that even the third person follows her own signal. If the

first person chooses i

?

0 and the second person follows her, the

third person will always follow them. All

subsequent

decision

makers will also choose the same

option.

808 QUARTERLY JOURNAL OF ECONOMICS

Has no signal

Has

a signal

ik

signal

Everybody

else has

chosen

i

0

all options

already

chosen

except

i =

have been

chosen by

1 person

only one

option

other than

i = 0 has

been

chosen

by more

than

1 person

two

options

other than

i = 0 have

been chosen

by more than

1 person

some

other

person has

chosen

i =

ik

no one

else

has

chosen

i =

ik

and

no option

other than

i = 0 has

been

chosen

by

more

than

1

person

no other

person

has

chosen

i =

ik

but one

option other

than i = 0

has been

chosen

hy

more

than

1

person

no other

person

has

chosen

=k

but

two

options

other than

0 have

history

heen

chosen

hy

more

than

1

person

choose

i

= 0

choose the

highest

of

the

options

already

chosen

choose

that

option

choose the

lower

of

the

two

options

choose

i =

ik

choose

i =

ik

choose

that

option

choose

the lower

of the

two

options

choice

FIGURE I

The kth decision maker's choice problem (k

>

2)

Herding can also happen when the first and second person,

and for that matter the third and fourth person, choose different

options. After k different options have been chosen,

for

any positive

k,

if the next decision maker does not have a signal, she will choose

the option with the highest value of i (among those already

chosen). Following this, all subsequent decision makers

will choose

the same option unless one of their signals matches one of the

options already chosen. This can happen only if the correct option

has already been chosen. So, there will be herding at an incorrect

option unless the first decision maker to have a signal

or someone

coming after

her but before the first subsequent decision maker

without a signal, made the correct choice.

We can actually calculate the expression for the probability

that no one in the population chooses the right option, however

large the population. A simple calculation establishes

that this

probability is

[1

-

(1

-

)]-1(1

-

&(1

-

A).

This probability is clearly decreasing in both

a

and

3

which

makes intuitive sense. Further, if

3

is sufficiently

small, this

probability will be very close to one.

To see why this is worth a remark, note that if all the decision

makers took their decisions without observing the choices made by

others, some people will always end up choosing the correct option

(in fact, for a large enough population,

the proportion of the

A SIMPLE MODEL OF HERD BEHAVIOR

809

population who will choose the correct option will almost certainly

be close to

(up).

It is also important to note that just the fact that people

observe the decisions made by others does not guarantee that there

will be herd behavior. Consider, for example, the following modi-

fied version of the Normal Learning Model: there is a Normal

(Gaussian) distribution with known variance but unknown mean.

The distribution of the mean is known. In each period a different

agent gets a signal that is a random drawing from this distribution.

The agent then chooses her best guess for the mean of the

distribution (he is minimizing a loss function that is quadratic

in

distance from the mean) given

her signal and the choices made by

her predecessors. It is quite easy to show that this sequence of

choices converges to the mean for almost every sequence of signals.

Therefore, the result we get contrasts sharply with the result from

this superficially quite similar model.

The key reason why we get a different result is that in our

model the choices made by agents

are not always sufficient

statistics for the information they have.

If the choices are always

sufficient statistics, future agents always know what information

their predecessors had acted upon, and therefore there is no herd

externality and no inefficiency. It is when the choices made by some

agents affect the information

that subsequent decision makers

have that there is a potential for herd externality.

The fact that the choices in our model do not have this

sufficient statistic property clearly has to do with how we specify

the payoffs. It is evident that, for a wide range of signals, the agents

in our model will always choose the same option; this lack of

invertibility is what causes the sufficient property to fail. We

conjecture that typically

whenever the space of choices and the

space

of

signals

are of

comparable

dimension and the

payoff

function is continuous, we shall have this kind of invertibility. So,

for example, in the model presented above, we are more likely to

have invertibility if the agents could (and would want to) vary the

size of their investment

in different informational

settings.

On the

other hand, there are many real reasons why it may not be possible

to vary one's choice enough to register all the information one

has-machines,

for example, come in only a small number of sizes.

B. Welfare Properties

The welfare question here is motivated by the herd external-

ity. While joining the herd is optimal for the current agent given

810

QUARTERLY JOURNAL OF ECONOMICS

the play of the game, it reduces the chance that future agents may

discover the truth. So if we consider an ex ante measure of welfare,

imagining that in some primeval state before the game begins all

agents have an equal chance of being at any position in the

sequence of arrivals, we may be able to show that social welfare is

lower in the equilibrium we have described than in other plays of

the game.

This is in fact what we find. Consider another play of the game

in which agents follow the decision rule D* given below.

1. If an agent has a signal, she follows that signal, unless

someone before her has already followed someone else. In that case

she follows suit.

2. If an agent does not have a signal, she picks some option that

has not been picked by anyone else, unless someone before her has

already followed someone else.

In that case she follows suit.

This play of the game is set up to ensure that the right choice

always gets revealed as long as there are enough people

in the

population. With a very large population it is easy to see

that this

would mean that an arbitrarily large fraction of the population will

always make the right choice. Formally, the probability that at

least two people have not received the true signal by the time we get

to the nth person is

1

-

(1

-

ap)n

1

-

(n

-

1)(1

-

z43)n-2o43.

For any

e>

0,

it is easy to see that we can choose an

n(E)

large

enough that this probability is at least

1

-

e.

Now this tells us that

a lower bound for the ex ante expected utility for agents following

this rule is

z[N

-

n(E)](1-

E)-N,

where N is the size of the population. Notice that by making N

large we can make this arbitrarily close to N(1

-

e).

By contrast, the probability that no one will discover the right

choice in the herding equilibrium we described before is

H

[1

-

OL1

-

1)]-1(1

-

&)(1

-

1),

and therefore the ex ante

expected utility is bounded above by

zN[1

-

H].

Since we can choose

e

to be as small as we would like to

by making

A SIMPLE MODEL OF HERD

BEHAVIOR

811

N large, it is easy to see that we can make the

above expression

larger than this expression. There is at least

one decision rule that

for large enough N does better than the

equilibrium we described.

It may be objected here that

we have simply described a

strategy and not explained how it will be

implemented and the

equilibrium may yet be constrained Pareto

optimal. However, this

is not true. This decision rule may be

implemented by using a

number of different incentive schemes.17 One

way that works is to

punish heavily anybody who is a

follower at any option that turns

out to be the incorrect one, while

equally rewarding everybody who

chooses the right option. Given these

rewards, no one will choose to

be a follower unless

they were absolutely sure that the option that

they were choosing was the right one. But this

is, of course, exactly

what we want.

Indeed, even if D* cannot be

implemented,

some of its

advantages can be captured simply by not

allowing the first n

agents to observe anybody else's choice when

they are making their

own choice. The rest of the

population is then allowed to choose

sequentially, with each person observing the choices made

by all

her predecessors.

The first n people will, of course,

choose to follow their own

signals

if

they have one and choose at random if

they have no

signal. The only way that more than one of

these people will choose

the same

option is if they are both right.

Thus, as long as at least

two of the first n

people have chosen the same

option, the rest of

the population will realize that

this must be the correct

option and

choose it.

But an argument exactly

paralleling the one given above in the

case of D* can now be

used to establish that for a large

enough

population, by choosing n suitably, we can

make the fraction of the

population who do not choose the correct

option arbitrarily small.

In other

words, in terms of ex ante welfare, the

economy may be

better off if the early decision

makers are not allowed to observe the

choices made

by the other decision makers than in our

original

equilibrium.

In other

words, destroying information (in this lim-

ited sense), can be

socially beneficial.18

17. This question is

discussed in more detail in an earlier version of

this paper

which is available from the author.

18. This is, of course,

only in terms of ex ante welfare; in terms of

ex post

welfare this is

certainly not true.

812

QUARTERLY JOURNAL OF ECONOMICS

V. EXTENSIONS AND MODIFICATIONS

A. Alternative Payoff Structures

In the model we analyzed in the last section,

everyone who

chooses the right outcome gets the same reward irrespective of how

many others chose this option before and after them. This may be

approximately the correct model of the rewards we get for choosing

the right restaurant, but in many other real world examples the

rewards will depend on the number of people who have chosen this

option and our rank among them. The total amount of the reward

may be fixed, or at least it may not increase as fast as the number of

people who choose the correct option. And in many instances there

are extra rewards for

being first or second to choose the correct

option.

To the extent that we have ignored these

possibilities, our

analysis may overstate the extent to which there will be herding.

Both of these possibilities suggest that it pays more to choose the

correct option when most others have chosen

something else. This

kind of reward for originality clearly discourages herd behavior.

To make this idea precise, consider a model

that is identical to

the previous one except that in this case the first person to choose

the right option gets a bigger output than all the

others, who are all

assumed to get the same amount. A real world institution where

exactly these rewards are not too implausible is academia; the

emphasis is on being first to do something. If you are not first, the

rank does not matter very much.

A little reflection should

persuade the reader that this model is

not really that different from our basic model. It is true that the

relative incentives for being first versus being second are greater

now, which should discourage agents from choosing to be second

but people without signals and people with

signals that exactly

match that of someone else, will still be followers. Consequently, it

is possible that the number of people behind someone becomes so

large and the evidence of their numbers

so convincing,

that

informed agents will decide to ignore their information and the

attraction of the large first prize and join the herd.

Formally, we have Proposition

2.

2. If the return to the first

person choosing

the

right

PROPOSITION

option

is z1 and the return to

everybody

else who chooses that

option

is

Z2,

and 0

<

Z2

/Z1

<

1,

then the

unique equilibrium

A SIMPLE MODEL OF HERD BEHAVIOR

813

decision rule19 under Assumptions A,

B,

and C, is described by

an integer k * > 0 and the following rules.

1. The first decision maker follows her signal if she has one

and chooses i

=

0 otherwise.

2. For k > 1, if the kth decision maker has a signal, she will

choose to follow her own signal either if and only if (a) holds or

if (a) does not hold, (b) holds, or if (b) does not hold, (c) holds,

where (a), (b), and (c) are given below.

(a) Her signal matches some option that has already been

chosen.

(b) No option other than i

=

0 has been chosen by more than

one person.

(c) The option with the highest

i (among those

already

chosen) has not been chosen by more than k* people for some

k*

>

1,

and no other option other than i

=

0 has been chosen

by more than one person.

3. Assume that the kth decision maker has a signal. If any

option (among those already chosen) other than

the one with

the highest i has been chosen by more than one person, the kth

decision maker will choose this option, unless

her

signal

matches one of the other options that have already been

chosen. In this case she chooses the latter option.

4. Assume that the kth decision maker has a signal. If the

option with the highest i (among those already chosen) has

been chosen by more than k* people and no other option

(except i

=

0) has been chosen by more than one person, she

will choose this option unless her signal matches one of the

options already chosen.

In this case she chooses the latter

option.

5. Assume that the kth decision maker does not have a signal.

Then she will choose i

=

0 if and only if that is what everyone

else has chosen. Otherwise she chooses the option with the

highest value of i that has already been chosen unless one of

the other options (excluding

i

=

0) has been chosen by more

than one person. In this case she chooses the latter option.

The proof of this result is omitted since it is a straightforward

extension of arguments used in Section III. In fact, it should be

evident that the result in Section III is a special case of this result

19. Once again we have not actually provided the complete decision rule

here;

for this would need to say what happens at all the unreached information sets.

However, this can be done in a straightforward way.

814

QUARTERLY JOURNAL OF ECONOMICS

for the case where

Z2

=

zj.

What is striking about that special case is

that k*, which we leave as an undetermined integer in this result,

is

one in that special case.

What is more,

if

Z2/Zl

is

not much smaller than one, it is easy to

see that the value of k* will still be one. As

Z2/Zl

goes to zero, k* of

course increases, but at least for values of

Z2/z1

relatively close to

one, there will be

a substantial degree of herding. Therefore, some

of the basic welfare intuitions are going to be quite similar to the

previous case. For N large enough, it will still be true that

the

decision rule

D* will

do better than this rule though the margin of

gain will be less substantial.

While this exercise suggests that the results we got from our

basic model are robust, it also suggests a criticism

of our approach.

What we have done is to take an exogenously given payoff structure

and then argue that as long as the payoff structure falls within

a

certain class we will get socially inefficient herding.

It may be

argued that this is misleading because

if the social costs of herding

were large enough, they would automatically

bring into place

mechanisms

that will modify the payoff structures and reduce

herding. In fact, a message of

the first part of this section is that

one can always reduce and even eliminate herding by having very

high rewards for originality. One might then take

the presence of

the institution of patent laws (which rewards being first)

as proof

that society can always find ways of preventing

inefficient herding.

In our opinion, however, this would be going

too far. We feel

that in many of the cases

we consider there are substantial

informational and transactions costs as

well as institutional

con-

straints which prevent the use

of the

appropriate

incentives to

eliminate herding. In some cases,

like the restaurant

example

we

considered

in the

introduction,

it is difficult to think of how one

could put any incentive scheme

into place. But even in cases where

there is a mechanism for rewarding originality,

as

patent protec-

tion for example, the degree

to which the incentives can be changed

may be quite limited.

Anecdotal evidence suggests,

for

example,

that the amount of protection patents provide

varies a lot from case

to case.

If decreasing returns (average payoffs decline

as the number of

people

who choose it increases) tends to reduce herding,

one would

expect increasing returns,

which rewards

doing

what a lot of others

are doing, to increase the tendency

to herd. This is indeed what we

find. In an earlier version of

this

paper (which

is available from the

author),

we

present

a detailed

analysis

of this case which shows

A SIMPLE MODEL OF HERD BEHAVIOR

815

among other things that if the increasing returns are very strong,

the unique equilibrium decision rule chosen by backward

program-

ming in the game tree involves everyone choosing the same option.

B. Alternative Information Structures

Another possible criticism of our basic model is that it is based

on rather demanding informational assumptions. We assumed that

each person knows the entire history of choices made

by people

before her. This is clearly a strong assumption that

may be valid

in

some cases (like academic research) but not in others.

Examples

where one would like to make a weaker assumption include the

restaurant example suggested in the introduction. In that case we

can usually observe how many people have chosen each restaurant

but not in which order they have made these choices.

However, inspection of the equilibrium decision rules suggests

that in fact they do not make use of

any information about the

order of choice so that the same results

apply as long as the

distribution of choices is observable. It should be noted, however,

that this is possible only under an

assumption like our Assumption

C, which is specified

in terms of a location

("choose the highest

among the options already chosen") rather than in terms of some

order information

("follow

the last decision

maker").

C. Endogenizing the Order of Choice

In this model we have assumed that the order of

choice is

exogenously

fixed: a more natural assumption

(but one which

complicates matters immensely) is to assume that choose when to

move taking into account the fact that waiting is

costly. When this

cost is high, we shall have a game that is similar to the one we

analyzed; when it is low, however, a new and interesting

set of

possibilities arises. A key question here is whether, if the waiting

costs are low enough, all the agents with

signals choose before all

the agents without

signals and whether this results in an efficient

outcome. It turns

out, somewhat surprisingly perhaps, that it is

possible to have situations with low waiting costs where some of

the uninformed move before some of the

informed and the outcome

is inefficient.20 However, the overall analysis is rather complicated,

and we do not

yet have very precise ideas about what happens in

the

general

case.

20. To see the intuition behind this

result, note that it is the marginal and not

the absolute value of information that matters in the decision of

whether or not to

wait.

816

QUARTERLY JOURNAL OF ECONOMICS

VI. CONCLUSIONS

We conclude with remarks on deficiencies of our model and

possible directions of research.

The most serious departure of our model from reality is

probably our assumption that signals to the agents are essentially

free; a more realistic analysis would combine the question

of

incentives for obtaining these signals with the kinds of consider-

ations we discuss. However, it would seem that dropping this

assumption will encourage people to try to "free ride" on other

people's ideas and this would only exacerbate the herding problem.

In this direction, at least, our results seem robust.

Our assumption that there is a continuum

of options and

payoffs exhibit a discontinuity at the true value, is defensible but

somewhat

unorthodox.

Notice, however, that this assumption

would be quite standard if there were only a large but finite

number of options (then there would no discontinuity, for exam-

ple). Preliminary investigations

show that the results we get for

the case where there are a large but finite number of options are

much more complicated but quite similar. We therefore

feel

justified in working with this much more tractable model which we

see as an approximation to the other case.

However, it may still be objected that what is missing from our

model is the fact that options which are in some sense close to the

true option are often better than the other

options. By assuming

that all options other than the right one get the same return, we

have not allowed for this possibility. This is clearly an important

direction for future research.

We have implicitly assumed that the

agents

in this model

cannot actually trade in signals. This

may

be

partly due to the

problem of enforcing contracts describing the exchange of an idea,

and partly due to the presence of transactions costs. Also, if there

are some (perhaps very small) gains to having others choose what

you are choosing, everybody will have the incentive to claim that

they had

a

signal and

that it matched the

option they had chosen.

However, since

the absence of this kind of trade

has serious welfare

consequences,

it is

probably worthwhile to examine this assump-

tion more

closely.

Also, since people gain information by choosing later than

others, there may be strategic aspects of timing that may be worth

investigating. Some current work by the author attempts to extend

this analysis in this direction.

A SIMPLE MODEL OF HERD BEHAVIOR

817

The assumption

that there are only two types of decision

makers-those who have a signal and those who

do not-though

quite strong, can be easily relaxed to allow for

decision makers with

signals of different quality. As long as there are

only a few different

types whose signals differ substantially in

quality, the results are

quite similar.

PRINCETON

UNIVERSITY

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