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HB31
.M415
(9
working paper
department
of economics
SOCIAL LEARNING IN A CHANGING WORLD
Lones Smith
96-34
November, 1996
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
^Y
SOCIAL LEARNING IN A CHANGING WORLD
Lones Smith
96-34
November, 1996
TFR^I
i99
Changing World*
Social Learning in a
Marco
Giuseppe Moscarini^
Lones Sinith§
Ottaviani*
November 1996
(First draft:
September 1994)
Abstract
In the social learning model of B£inerjee
Welch
[2]
[l]
and Bikhchandani,
— can
arise.
— situations where
earlier version of this p>aper is
dissertation at
MIT.
socially valuable information
We
JEL
Classification
is
wasted
contained in the
first
Number: D83.
part of Chapter 3 of Ottaviani's doctoral
thank Massimiliano Amarante, Abhijit Banerjee, Glenn
str6m, Robert Shimer, and Peter S0rensen for helpful suggestions.
seminar participants at Harvard, MIT,
is
the state of the world
Furthermore, no cascade ever arises when the environment changes in
a sufficiently unpredictable way.
The
If
changing stochastically over time during the learning process, only temporeiry
informational cascades
*An
and
individuals take actions sequentially after observing the history of actions
taken by the predecessors and an informative private signal.
is
Hirshleifer
financial support of
SEDC
Banca Nazionale
Ellison,
Bengt Holm-
We also received useful comments from
1995 annual meeting in Barcelona, and University Bocconi.
del Lavoro (Moscarini)
and Mediocredito Centrale (Ottaviani)
gratefully acknowledged.
^Department of Economics, Yale University, 28 Hillhouse Avenue,
New Haven CT
06510,
USA.
E-mail:
gm76@pantheon.yale.edu
^Department of Economics and ELSE, University College London, Cower
UK.
5
St.,
London
WClE
6BT,
E-mail: m.ottaviani@ucl.ac.uk
Department of Economics, Massachusetts Institute of Technology, 50 Memorial
02139,
USA.
E-mail: lones@lones.mit.edu
Dr.,
Cambridge
MA
Introduction
1.
Models of learning and experimentation typically predict that information accumulation
eventually stops.
p>eatedly, if
This prediction contrasts with the observation that information
is
we study the problem
not continuously, collected and used. In this paper
re-
of
information aggregation in a model of observational learning by a society of individuals
acting sequentiedly in a changing environment. In this setting,
new
more than old information and learning
is
after a long
The
is
worth
at least sure to resume
enough period of time.
model of Beinerjee
social learning
(BHW)
[2]
either never stops, or
information
[1]
and Bikhchandani,
Hirshleifer
and Welch
describes the decision problem faced by a sequence of exogenously ordered
individuals each acting under uncertainty about the state of the world. Everyone conditions
his decision in
a Bayes-rationaJ fashion on both a privately observed informative
the ordered history of
all
signal,
and
predecessors' decisions; he observes neither predecessors' realized
private signals, nor their reedized payoffs. Private signals axe
assumed to be drawn from
an identically and independently distributed random variable correlated with the state of
the world. Thus, were
all signals
reveal the state almost siurely,
private,
The
settles
and individueds
by the Strong Law of Large Numbers. Instead,
common
Either outcome
is
is
inefficient action,
that with positive probability everyone eventually
and as
occasioned by the lumpy
BHW
way
in
note,
on a
single action regardless.
which information
is
filtered
Suppose that the evidence from the publicly observed action history
favors one state that
private information
it
is
signals are
only imperfectly infer information from actions.
striking result in this setting
on a
actions.
cein
publicly observable, their aggregation would eventually
swamps
through
sufficiently
the private signals, as can initially happen by chance.
perforce ignored,
and one's action
is
Then
dictated by history. Successors
then learn nothing from the action, and the system has reached a steady-state where
everyone rationally "herds" on the same (either good or bad) action. In this "informational
cascade," socially valuable private signals are lost.^
In this paper
we consider a modified but
natvured
model where
that the state of the world changes stochastically over time.
the above findings to such an evolving environment.
We
We find that
it is
common knowledge
then ask
how
because of the resulting
information depreciation, only temp)oraxy informational cascades can arise.
^
Smith and Sorensen
[7]
show that
this result holds
iff
robust are
the quality of private signals
is
BHW discuss
bounded.
at length the fragility of cascades to the release of small
Here,
we note
come
that any cascade will eventually
to
aji
amount of pubUc information.
end without new informationed
input, but instead simply because of the fading relevance of old information. Ceiscades on
a single action axise only
if
the state of the world
sufficiently persistent.
is
changes are sufficiently unpredictable no cascade ever
When
state
eirises,
because past information
depreciates so fast that the beUef can never be too extreme.
Finally, for completeness
more than
the world
for
is
economic
relevEince,
cascades on alternating actions arise
when the
state of
changing frequently enough, because here too information depreciates slowly.
We conclude that temporary cascades on single actions arise when the environment changes
a temporary phenomenon
slowly. In this sense, herding survives as
many
This model naturally encompasses
making
and consumer
in organizations
interesting economic situations, like decision
choice,
behavior given a discrete action space. This
only.
where
is
social learning induces inertia in the
the natural extension of the never-ending
BHW and Banerjee to a stochastic environment. Provided information does
depreciate too quickly, such inertia
arises. BHW themselves briefly discuss the
herd idea of
not
still
possibility of the state changing in
a nonstationary fashion. They provide a specific numeric
example where the state changes with
reversals are
world
is
more
likely
5%
chance after 100 periods; they find that cascade
than 5%. In this paper, we
feel
that a stochastically evolving
worthy of serious investigation on a par with other recent research on learning.
The problem
of optimal experimentation in a changing environment has already been
addressed by Kiefer
Recently Keller and
[4].
Rady
[3]
have built a tractable continuous-
time version of the model. The complexity of the optimization problem precludes a char-
model of
acterization of the long-run learning outcome. In our simple
optimization problem
Wolinsky
[6]
for
is
straightforward and the aggregation
a manageable frsimework
mentation problem.
is
simple. See Rustichini
for long-run analysis in
and
an individual experi-
Intuitively, information depreciation in these contexts increases the
immediate value of new information but diminishes
its
context, since agents
effect is present,
eire
myopic, only the former
long-term value. In a social learning
unambiguously discourages the noninformative cascade
The paper proceeds
as follows.
changing world. In Section 3
arise,
social learning the
and that cascades
it is
and the depreciation
stage.
Section 2 describes oiu: model of social learning in a
shown that only temporary informationed cascades can
arise only
if
the change in the state of the world
is
sufficiently
predictable. Section 4 discusses the robustness of our results. Section 5 concludes.
2.
A
Model
countable number of individuals take sequentially one of two possible actions, Oq and
ai.
Payoffs to actions are contingent
be the common prior
on an unknown state of the world,
belief that the state
action Oq in state ui, while the opposite
is
Uj
good
is
1 i(
z,"
=
i
j and
if i 7^ j,
is
Action aj
initially Ui.
is
true in state
uio'-
is
more rewarding than
the payoff of action Cj in state
with i,j € {0, 1}. For example, action
and the state of the world
"good
uii:
i \s
ot Ui. Let q^
uiq
may be "buy
Cj
better than the alternative good j
After an individual's decision, the state of the world changes with chance
Pr (w"
E {0,1},
^ j,
i
= Ui
a;"-^
= a;^) =
Pr
= ujj
(cj"
\
a;""^
I
and any
transition matrix at the
n.
= Ui) =
end of next Section.
the public history of action decisions of
see predecessors' signals.
i,j
6
For n
>
2
€
preceding individuals l,2,...,n—
{ctq,
ai }
He cannot
1.
u)j is
a >
1/2
quality of the private signal
is
if z
=
j
and
state.
I
The
—a <
assumed bounded,
i.e.
probability that
1/2
ii i
a <
^
j,
1.
be the public probability
belief that the state
is
Ui
in period
n
conditional
on the publicly observed history of actions chosen by the predecessors of individual
Similarly let r"
= Pr {ui\h", ai)
on both the public action history
n.
be the posterior belief that the state
/i"
and the
„
=
is
n.
ui conditioned
realization a^ of the private signal observed
A simple application of Bayes'
Ti
so that
with
H^ = {oq, aj}"" be the space of all possible period n histories of actions
n — 1 predecessors of individual n. Let /i" denote an element of /f". Let
= Pr {uji\h^)
by individual
and
let
chosen by the
9"
<t"
Private signals are drawn from a state-dependent Bernoulli
realized in state
is
The
{0, 1}.
all
and are independent conditional on the cmrent
the signal ai
e
We generalize our results to a general two-state Markov
Before choosing an action, individual n both observes a private signal
distribution,
assumed
e,
be Maxkovian and independent of the current state of the world:
for simplicity to
ioT i,j
^ z."
rule yields
Pv{uinai\hr)
Pr(a,|/i",c^i)Pr(a;i|/i")
PT{ai\hP)
Pr(<Ti|/i")
These posterior probabilities are used to compute the expected payoffs from
The
the two different actions in the two states.
the action a" which gives her the highest expected payoff.
individual
and only
n
receives the private signal
r"
if
rule can be
1/2. After substituting
summarized
a"
=
if ct"
=
if
>
1
from
to choose
Given our simple payoffs,
optimal to take action a"
it is
(2.2), this
then
a"^
= oq
<^
q^
then
a"
=
<^
g"<l-Q
>
becomes g"
1
=
if
Oj
if
— q. The decision
ao
indifferent
< a and
WLOG
a"
=
and
Ci
=
a"
<=^
g"
>a
ai
<^
q^
>
1
-a
minimizes the possibility of herding.
Informational Cascades with a Changing World
A, Belief Dynamics.
Were
e equal to 0, as in the standard models referenced earlier,
the public prior belief of individual
act according to her signal
=
cr"
n+
at, i.e.
g""*"^
n whenever
the individual's private signal
cr".
>a
(respectively g*
A cascade once steirted
With
£
>
0,
<
1
—
=
r".
There
action Cj
is
an informational cascade
is
a), for then the public belief
by the state switching, since that event occurs only
swamps
belief
either private signal.
would remain unchanged.
after the decision
is
the possibility that the state of the world has changed in the meantime
is
the public prior belief of individual
=
(or
Thus, a cascade on Ci (respectively Oq) arises as soon as
the dynamics change drastically; however, the cascade region
a"^
to
taken by individual n regardless of
would never end, because public
according to her signal
n
equals the posterior belief that leads Mr.
1
cascade) on action Cj at time
q''
ai, then
is
as:
where the action choice when
3.
=
o""
n
decision rule of the agent
talcing
n
+
1,
coming
(1
—
—
+
£ (1
a"
=
On
if a
u
a"
=
-
a,
ai
£)r"
unaffected
made.
When
accounted
n who chose a"
after individual
=
g""*"^
ai, satisfies
is
r"),
for,
=
a,
which can be
rewritten by (2.1) and (2.2) as
ifn
Co")
f lf.n\
JikQ
)
=
=
-
(i-g)(i-°)?"+^"(i-?")
if
(l-e)aq"+e(l-a)(l-q")
q,"+(i-q)(i-5")
Note that anyone can compute these
since the other case can
probabilities.
be treated symmetrically.
regardless of the signal a^.
The next
and computes the public prior
belief
individual k
g*"*"^
=
(1
4
—
+1
e)
We
The
will consider the case g*
action chosen will be a*
knows that a*
g*+e
^
(1
—
=
Oj
is
>
=
'
'
a,
Ci,
uninformative,
g*). In general the following
individual
n
+
>a
as long as q^
1,
or q"
<
—
1
a, will update her prior belief during the
cascade in the same fashion, according to the (uninformative) cascade dynamics below:
=
q-+'
The
q^
<
ct
or g"
<
=
<^(g")
-
(1
1
—a
in
(3.2)
a cascade) and axe deterministic and follow
(3.1) as
long as
when
(3.2)
—a<
1
>q
either g"
(during the cascade). These dynamics are depicted in the figure.
FIGURE
Please insert
Now we
B. Ceiscades are TemporEiry.
Proposition
+ e{l - g")
and determined by
public belief dynamics are stochastic
(when not
£)g"
1.
For any e E
(0, 1), if
here
argue that any cascade
a cascade
exists,
will eventuedly stop.
=
then for some k
k{e)
<
oo, the
cascade must end in k{e) periods.
Proof.
When
in
a cascade, the dynamics of the system axe described by
fixed point of this first order linear difference equation
exactly requires ^^pr\
the fixed point
to q
is
is
=
|1
~
2£|
<
Since
a >
Ci starts
g:
1/2 then
[1
—
k
=
k
1/2. Therefore there exists
One can then compute the maximum
on
<
1, i.e.
<
1.
^
=
For £ €
monotonic, and oscillatory for e £ (1/2,
immediate.
contains q
=
is
1).
1/2) the convergence to
For e
=
1/2 the convergence
a, a] has positive Lebesgue
(e)
such that
length of a cascade.
q'^
E
The
The unique
Global stability of q
1/2.
(0,
(3.2).
[l
—
measure and
ct,a].
longest possible cascade
with a belief /i (a). After h periods in a cascade the belief
is
h-l
^'
ill («))
=
E
e
(1
-
+ (l-2£)Vi(a)
2^)'
.1=0
This cascade terminates as soon as
2a(l
—
a)], so that
K{q,£)
the length of a casceide.
=
Since
log[l
ip'^
—
<
(/i (a))
2q:(1
—
ct,
or equivalently (1
a)]/log(l
dK (a, e) /da >
and
—
2^)
sign
for e 7^ 1/2, the higher the quality of private information
is
(3.1) to
(i.e.
To show the
possibility of
prove that there exists a fixed point of /i
outside
[1
—
a, a]),
and that these
(•)
2e)'*'^^
^
[1
~
a tight upper bound on
dK (a, e) /de =
sign (1
-
2e)
and the more predictable the
evolution in the state of the world, the longer a cascade can possibly
C. Cascades Arise.
—
last.
temporary cascades,
and one of /o
(•)
in
it
suffices to use
the cascade region
fixed points are global attractors.
Consider
world
first
the case of cascades on a single action.
sufficiently persistent.
is
«
middling e
If
2.
(Cascades on a Single Action) For any q^ G
a cascade on some action arises in Bnite time
Proof.
It suffices
is
when
many
some
for
fc
>
The most
1.
f
Note that
equation
"^ 0.
(3.4).
To show
is
any q
=
/i (1)
and fi{q^Y
1
—
>
1)
+ g (1 + £ -
(0, 1)
=
£,
there
=
< £ (a) = a (1 —
on action
Then the dynamics
a).
Oi, requiring
follow
some attractor strictly above q.
a Liapunov function
j^ q;
(in) L{fi{q))
if:
=
=
0.
is
continuous in
It is
left
€
=
sgn
(1
2e)
— e)
for
q; (ii)
any a.
=
[fi{q)
L{f\{q))
~ qHq ~ q)> L{q) >
iff
is
globally stable for e
€
if
(0, 1).
£ satisfies
+ ^{l+e-2af + A{2a-l)a{l-'^)
-{l + e-2a)
^^'^^
2(2a-l)
is it
—
(1/2, 1) these conditions axe satisfied
possible that in finite time the public beliefs surpass a,
neighborhood of this fixed point. Finally,
a), as required.
(3.4)
easy to show that these conditions are
a positive chance of cascade on Oi
For then, and only then,
and reach any
0.
globally stable, let L{q)
is
—1, which always holds; therefore, the fixed point q
=
=
a)
only one positive root q (a, e) of the quadratic
is
Similarly, for e
(0, 1/2).
- £ (1 -
and one can calculate sgn/{(g")
L(.)
(i)
L{q)
2a)
1/2 and q{a,e) E {e,l
that the fixed point q{a,e)
a<q{o^,e)
—
-
Note that q{a, 1/2)
Finally, there is
a(l
if e
favorable case for the occurrence of such a cascade
(g") converges globally to
e ajid f\ (0)
For any e €
always satisfied for £ 6
f'i{q)
only
with probabil-
fixed points of the diff^erence equation (3.3) axe the roots of the quadratic equation
g^ (2a
for
for
(l-£)ag" + £(l-a)(l-g")
.^
,
We claim that for small enough £,
Then L
if and
— a, q),
(1
periods. Consider first the possibility of a ceiscade
n+i
The
i.e.
to show that with positive probability, a c£LScade arises on some action
consecutive signal realizations are a\.
all
the state of the
arise.
ity one,
>a
if
1/2, information depreciates the most. Here, the public behef g" will never
Proposition
that q^
arise only
instead state changes are rather unpredictable,
venture far from 1/2, and so no cascade will ever
in finitely
They
The argument
positive fixed point of /o, then £
for cascades
< £ (q)
is
on Cq
is
(3.5) reduces to £
symmetric:
equivalent to ^ (a, £)
<
1
—
a.
If
< £ (a) =
g(a,£)
is
the
Consider a belief on the border of the non-cascade region,
then
=
fi{cc)
a, so that the posterior belief
the cascade region
less rapidly, so that
If instead
after
<
=
q
=
If e
a.
e (a)
£.{a) the state changes
entered after a signal ai departing from q
the state changes slightly more rapidly, then /i (q)
<
=
a.
q, so that the posterior
a signal Ci would be interior to the non-cascade region.
Temporary cascades
next state
is
when information
arise
most predictable
small, but also
when
it is
near
on current
beised
When
1.
individuals alternate between the
in
is
but for e
fixed,
is
e.g.
two
depreciates the least,
i.e.
when when the
This happens not only when e
beliefs.
the state of the world changes rapidly enough and
may enter an
system
actions, the
alternating cascade
which individuals alternate between the two actions regardless of private
Corollary
1.
is
(Cascades on Alternating Actions) For
q^
6
biEty one, cascades on alternating actions arise in Gnite time if
(1
—
signals.
q,q), with proba-
and only
if
>
e
e (a)
=
I-q(I-q).
The appendicized proof argues
that the public belief oscillates until
it
enters either
cascade region (in finite time). Cascade dynamics (3.2) are nonmonotone given i {a)
From Proposition
environment
Corollary
4.
is
2.
2
and CoroUeiry
1,
and a > 1/2
it
changing in an unpredictable way there
No
follows immediately that
will
>
if
1/2.
the
never be a cascade.
cascade ever arises for e E [1/4, 3/4].
Robustness
Symmetry of the
stochastic process
is
WLOG. If state changes are governed by the Markov
transition matrix
1
then cascades on Oj are temporary for
—Q <
for 1
e
=
Tj
e/ (e
+ 7]),
i.e.
if
and only
a >
a > e/ {e +
while cascades on Oq are temporary
r}),
state
is
in the learning region.
1/2, as required in Proposition
1.
For
Cascades on action
if
V<
and
e
when the Markov steady
these conditions reduce to
Oi arise
—£
similarly for the cascade
conditions are identical to e
=
1-a 1-"
a
a
£
-f-
2q
-
1
on Oq with 9 and £ intercheinged.
rj
< a {1 —
When
e
=
t]
the two
a), the condition given in Proposition 2.
More
generally, our results axe robust to majiy states.
beliefs axe
For during a cascade, public
always drawn back to the fixed point of the cascade dynamics. Such a Markov
steady state exists with any number of states of the world, and Ues outside the (extremal)
cascade
The
[7]);
set.
Thus, our temporaxy cascades Proposition 1
robust to
many
introduction of insurance actions can expeind the cascade set, as
however,
eis
cascades on insiurance actions
the fixed point of the cascade dyneimics
5.
is
may
may
well
is
states.
well-known (see
arise for strictly interior public beliefs,
lie
inside the cascade set.
Conclusion
Social learning decouples the behavior of the agents firom the economy's fundamentals.
During an informational cascade on a
single action, the
same action
while the environment changes with positive probability.
persists predictably
Then the
action eventually
switches and learning resumes. This simple model of observational learning can explain
why common
if
practice can persist
more than
it
should: agents stick to the practice even
they have contrary private information, without possibly knowing in an informational
cascade whether others have similar contrary information.
This paper has dwelt on the short run aspects of leaxning
an analysis of the long-run properties, we
Ottaviani
[5].
As long as there
is
refer to the
in
a changing world. For
companion paper Moscarini and
a positive chance of the state changing,
beliefs
about the
current state of the world axe obviously no longer a martingale. (For instance, E{q"'^^)
(1
— e)q^ + e{l — g")
here.) Since individual actions
depend on such
beliefs,
the martingale
convergence theorem cannot be apphed to study long-run outcomes of learning.
companion paper pursues another approach to
existence, uniqueness,
=
and global
The
stability
of an invaxiant probability measvure over beliefs and states, and thus over actions.
This model
is
amenable to applications
in
macroeconomics and industrial organization.
In a naturzd macroeconomic formulation, information about the state of the
transmitted through the observation of the decisions
economy
made by others. The state would then
be endogenously determined by the aggregate of the individuals. In a microeconomic
ting,
is
set-
where we interpret the action as a pturchase decision by a consumer who observes the
decisions
made by
other consumers, the change in the environment could be endogenized
by allowing producers to innovate on
their products.
8
A. Appendix: Proof of Corollary 1
Proof. The most favorable case
realizations exactly alternate
for
n
for the
up to
n.
occurrence of such a ceiscade
WLOG
supp)ose
first
even. Therefore the public belief dynamics are q^
function /o (/i
(.))
^°^^'
^U be denoted by go
(.).
=
=h
= /i
(9")
[g (1
91
[Q)-
dynamics. Following
(/o (g""'))
by
^1 (9"-^),
/o (/i (g"~^))-
=
ai
The
for ^o
and
<
1
—
>
1/2 the public belief follows
beliefs follow
for the
a)
if
in [0,1],
find that
denoted by qo{a,e) and
qi {a,£].
two dynamics and are
and only
9
if
£
>
monotonic
we
gi the Scime steps as in Proposition 2 for /i,
have exactly one fixed point
ajid 91 (a,£)
where
odd and even subsequences of
Both points are global attractors
>a
ao and a"
- q) (1 - 2£) - £ (q - £)] q + ae{\-e)
-£{2a-l)q + a{l-a-£ + 2a£)
oscillating dynamics, while the
qo (a, e)
=
all signal
Substituting firom (3.1) one gets
For the selected sequence of reaUzations of signals and for £
gi
/o (9""^)
=
when
^ [a{l-a){l-2e)-e{l-a-e)]q + {l-a)e{l-e)
e{2a-\)q^{l-a){a-e + 2ae)
Similarly denote 9"+^
both ^0 and
that a"~^
is
£ (a).
in the cascade region
(i.e.
References
[1]
Banerjee, A. V.:
A
Simple Model of Herd Behavior. Quarterly Journal of Economics
107, 797-817 (1992)
[2]
Bikhchandani,
Cultural
S., Hirshleifer,
Change
D., Welch,
I.:
A Theory of Fads,
Fashion, Custom, and
as Informational Cascades. Journal of Political Exx)nomy 100, 992-
1026 (1992)
[3]
Keller, G.,
versity of
[4]
Rady,
S.:
in
a Changing Environment. The Uni-
Edinburgh Discussion Paper (1995)
Kiefer, N.:
A Dynamic
Cornell University
[5]
Optimal Experimentation
Moscarini,
C,
Model of Optimal Learning with Obsolescence of Information.
mimeo
(1991)
Ottaviani, M.:
Long-Run Outcomes
of Social Learning in a
Changing
World. Yale University and University College London mimeo (1996)
[6]
Rustichini, A.,
WoUnsky,
A.:
Learning about Variable
Demand
in the
Long Run.
Journal of Economic Dynamics and Control 19, 1283-1292 (1995)
[7]
Smith, L., S0rensen,
P.:
Pathological Outcomes of Observational Learning. Nuffield
College Discussion Paper 115 (1996)
10
MIT LIBRARIES
3 9080 01428 2906
Cascade on action a
1/2
Non-cascade region
1-a
<P(.)
Cascade on action a
J
J
i_
Figure A.l:
11
L.
5
232
b
Date Due
Lib-26-67