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DEWEY
HB31
.M415
working paper
department
of economics
PATHOLOGICAL OUTCOMES OF OBSERVATIONAL LEARNING
Lones Smith
Peter Sprensen
June 1996
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
PATHOLOGICAL OUTCOMES OF OBSERVATIONAL LEARNING
Lones Smith
Peter Sprensen
96-19
June 1996
MASSACHUSETTS INSTITUTE
.JUL 18 1996
96-19
Pathological
Outcomes of Observational Learning'
Peter S0rensen
Lones Smith'''
Department of Economics, M.I.T.
June
28,
Nuffield College
: :
'
and M.I.T.
1996
(original version: July, 1994)
Abstract
This paper systematically analyzes and enriches the observational learning
paradigm of Banerjee (1992) and Bikhchandani, Hirshleifer, and Welch (1992).
Our
contributions
First,
fall
into three categories.
we develop what we consider
to be the 'right' analytic framework for
informational herding (convergence of actions and convergence of beliefs, using
a Markov-martingale process).
four major ways: (1)
arise,
We
We
demonstrate
its
power and simplicity
in
decouple herds and cascades: Cascades might never
(2) We show that wrong herds can arise iff the
bounded strength. (3) We determine when the
finite, and show that (absent revealing signals) it is
even though herds must.
private signals have uniformly
mean time to start a herd is
infinite when a correct herd
is
is
inevitable. (4)
We
prove that long-run learning
unaffected by background 'noise' from crazy/trembling decisions.
Second, we explore a
types,
new and economically compelling model with multiple
and discover that a
confounded learning.
Third, we show
may
It
from history even while
'twin' observational
it
how
pathology generically appears:
well be impossible to
draw any further inference
continues to accumulate privately-informed decisions!
the martingale property of likelihood ratios
is
neatly
linked with the stochastic stability of the learning dynamics. This not only al-
lows us to analyze herding with noise, and convergence to our
new confounding
outcome, but also shows promise for optimal experimentation.
'From its initial presentation at the MIT theory lunch, this paper has evolved and been much improved
and polished by feedback from seminars at Yale, IDEI Toulouse, CEPREMAP, IEA Barcelona, Pompeu
Fabra, Copenhagen, IIES Stockholm, Stockholm School of Economics, Nuffield College, LSE, CentER, Free
University of Brussels, Western Ontario, Brown, Pittsburgh, Berkeley, UC San Diego, UCLA, University
College of London, Edinburgh, and Cambridge. We wish also to specifically acknowledge Abhijit Banerjee,
Kong-Pin Chen, Glenn Ellison, John Geanakoplos, David Hirshleifer, David Levine, Preston McAfee,
Thomas Piketty, Ben Polak, and Jorgen Weibull for useful comments, and Chris Avery for a key role at
a very early stage in this project. The usenet group sci. math. research has also proved invaluable. All
errors remain our responsibility. Smith and S0rensen respectively acknowledge financial support for this
work from the National Science Foundation and the Danish Social Sciences Research Council.
t
e-mail address: lones01ones.mit.edu
*e-mail address: SORENSENOcole.nuff.ox.ac.uk
Contents
1
INTRODUCTION
1
2
THE STANDARD MODEL
4
Some Notation
4
2.2
Private Beliefs
5
2.3
Action Choice
6
2.4
Individual Learning
7
2.5
Corporate Learning as a Markov-Martingale Process
8
2.1
3
THE MAIN RESULTS
3.1
Belief Convergence
10
3.2
Action Convergence
12
3.3
Must Cascades Exist with Bounded
Mean Time to Herd
3.4
4
5
Beliefs?
NOISE
14
14
16
4.1
Convergence of Beliefs
16
4.2
Convergence of Actions: The Failure of the Overturning Principle
17
MULTIPLE INDIVIDUAL TYPES
19
5.1
The Model and an Overview
19
5.2
Examples of Confounded Learning
The Basic Theory
20
5.3
6
10
CONCLUSION
22
24
A ON BAYESIAN UPDATING OF DIVERSE SIGNALS
25
B FIXED POINTS OF MARKOV-MARTINGALE SYSTEMS
26
C STABLE STOCHASTIC DIFFERENCE EQUATIONS
28
D MORE STATES AND ACTIONS
31
E SOME ASYMPTOTICS FOR DIFFERENCE EQUATIONS
32
F
OMITTED PROOFS AND EXAMPLES
34
INTRODUCTION
1.
Suppose that a countable number of individuals each must make a once-in-a-lifetime
— encumbered
binary decision
sume that preferences
externalities
all
by uncertainty about the state of the world. As-
solely
are identical, and that there are no congestion effects or network
from acting
alike.
Then
in a world of
complete and symmetric information,
would ideally wish to make the same decision.
But
life is
more complicated than
decide sequentially,
all in
Assume
that.
some preordained
order.
instead that the individuals must
Suppose that each may condition
decision both on his (endowed) private signal about the state of the world
predecessors' decisions, but not their private signals.
and on
his
all his
The above simple framework was
independently introduced in Banerjee (1992) and Bikhchandani, Hirshleifer, and Welch
(1992) (hereafter, simply
BHW).
Their perhaps surprising
with positive probability a 'incorrect herd' would
information, after
In this paper,
as
it
some
Assume
that
we
What
his predecessor
is
identical less profitable decision.
enrich the informational herding paradigm,
is
best motivated by
means of the following
action, but suppose that the very next individual
then could Mr. one million and two conclude? First, he could decide that
had a more powerful
We shall
an accident.
signal than everyone else.
beyond discrete
no uniformly most powerful
irrational or
story
some
generalize the private information
there
Despite the surfeit of available
are in a potential herd in which one million consecutive
individuals have followed suit on
deviates.
conclusion was that
both simple lessons and deeper insights into rational learning.
offer
Our embellishment upon the herding
counterfactual.
make the
we systematically analyze and
happens to
so
point, everyone can
arise:
common
signal.
signals,
To capture
and admit the
no decisive lessons
Here
1.
•
is
shall
possibility that
thus add noise to the herding model. Third, he possibly
model with multiple
only possible 'pathological' outcome:
offers
we
Second, he might opine that the action was
might decide that different preferences provoked the contrary choice.
shall consider the herding
this,
for anyone,
We may
types. Here,
we
On
this score,
find that herding
is
we
not the
well converge to a situation where history
and everyone must forever
rely
on his private
signal!
an overview of the paper, and how we view our contributions.
Proper Analytic Framework; Nonrobustness of Past Results; Extensions.
The
'Right' Theory. Our analysis
of beliefs (learning)
is
focused through the two lenses of convergence
and convergence of actions (herding). As such, two stochastic processes
constitute the building blocks for our theory:
(i)
the public likelihood ratio
given the state, and (u) the vector (action taken, likelihood ratio) a
is
a martingale
Markov
chain.
We
prove the power and simplicity of this informational herding framework in four major ways:
The Markovian aspect
of the dynamics allows us to drastically narrow the range of
possible long run outcomes, as
we need only focus on the ergodic
set.
This set
is
wholly
unrelated to initial conditions, and depends only on the transition dynamics of the model.
By
contrast, the martingale property affords us a different glimpse into the long run
dynamics, tying them down to the
initial
conditions in expectation. As
turns out, this
it
allows us to eliminate from consideration the not inconceivable elements of the ergodic set
where everyone entertains entirely
Bad Herds?
•
false beliefs in the long run.
Rational learning
private signals from a
common
is
fruitful
because Bayes-updating of any space of
prior yields private beliefs that are informative of the true
— a simple corollary of our no introspection property.
state
by the support of the private
have bounded private
itively,
beliefs: Incorrect
beliefs, i.e. there
do not
We
focus on the role played
herds can develop exactly
when
individuals
exist arbitrarily strong private signals. Intu-
whenever individual private signals are uniformly bounded
in strength, history
has
the potential to mislead everyone: Eventually even the most doctrinaire individual dare
not quarrel with the conclusion of histories that aggregate enough information.
With unbounded
private
learning
beliefs,
is
Soon enough, a wise enough
complete.
doctrinaire individual will appear whose contrary action will radically shift public beliefs
and overturn any would-be incorrect herd. Yet by our overturning
cannot be turned on
its
tenacious individuals
is
richer
head to rule out correct herds.
largely a modelling decision.
model than Banerjee (1992) and
in their
framework, but also sheds
BHW,
1
For
and opens
principle, this logic
The admission
it
of arbitrarily
not only provides us with a
lines of inquiry that
critical light into the exact failure of
make no
sense
incomplete learning:
For as we approach this idealized extreme, bad herds become vanishingly implausible.
BHW
Cascades?
•
introduced the colorful terminology of a cascade for an infinite
train of individuals acting irrespective of the content of their signal. Yet
label 'cascades literature'
outside
BHW's
is
malapropos. For cascades are the exception and not the rule
discrete signal world. All but one (rather contrived)
attests to this fact.
With
is
richer
— for
herds must arise. This occurs, we argue, because public
beliefs
and since
time.
1
is
The
find that
link
paper
much
what matters
is
not
between unbounded support
it is
With
these
no longer so clear
must converge
(as per
This simple insight (the
true because contrary actions radically swing beliefs.
Expected Time To Herd. We
We
is
belief convergence implies action convergence.
overturning principle) into herding
•
in this
ever a foregone conclusion (a cascade)!
two notions decoupled, the resulting analysis
usual),
example
generic signal distributions, individuals always eventually settle
on an action (a herd), but no decision
why
we argue that the
characterize
how
beliefs
when herds
arise in finite expected
fast the truth is learned
and complete learning was
but rather how slowly
implicit in
Smith (1991).
error
rooted out: There must be enough contrarians to overturn any temporary herd
is
enough. Our surprising discovery
fast
that while learning
the correct herd always requires infinite
beliefs,
•
is
NOISE. That a
into herding. So
it is
mean time
is
complete with unbounded
some
to start in
single individual can 'overturn the herd'
is
state!
the key analytic insight
only natural to undermine the impact of contrarians by adding noise.
Counterintuitively, even with a constant inflow of crazy/trembling individuals, complete
learning
still
obtains. Everyone (sane) eventually learns the true state of the world.
Besides definitively resolving and further exploring informational herding, the above
results set the context
needed
for
what are our most
New Herding Economics:
2.
We
next relax a critical
namely that
all
(if
and innovative findings below.
Confounding with Multiple Preference Types.
under-appreciated) premise of the original herding results,
individuals have the
same
Surely this
preferences.
is
isolated individuals need not greatly matter
anything but an
why
Multiple types offers a parallel reason
apt description of the world.
fix
striking
the actions of
— but with much richer consequences.
Let's
ideas with a hopefully familiar example. Suppose that on a highway under construction,
depending upon how the detours are arranged, those going to Houston should merge either
right (in state
If
R) or
one knows that
left (in
70%
state L), with the opposite for those
headed toward Dallas.
are headed toward Houston, then absent any strong signal to the
This yields two
contrary, Dallas-bound drivers should take the lane 'less traveled by'.
herding outcomes:
70%
left
or
70%
right.
But another rather subtle
For as the chance q that observed history accords state
r(q) that a
Houston driver merges
for Dallas drivers.
r «(?)
stops!
=
If for
some
q,
R
right gradually increases
cars are equilikely in states
from
R
Of course, whether such a
fixed point exists
is
far
specifications, such a 'confounding'
3.
outcome does
with positive probability
The above convergence
be forever choked
and conversely
and L to merge
result
right, or
learning
all
With
why
if so,
that for non-degenerate
and furthermore, dynamics
will
signals!
Exponentially Fast/Stable Learning.
=*>
and the deduction of
belief convergence.
off
exist,
is
arise.
the chance
— even with arbitrarily strong private
Likelihood Ratios as Martingales
pend on the rate of
1,
1,
from obvious, and even
need we converge there? The surprising content of Theorem 9
it
to
to
no inference can be drawn from additional decisions, and
r L{q), then
converge upon
from
rises
may
possibility
'rational herds' with noise
both de-
noise, the inflow of rational contrarians
without entering a cascade
if
learning
is
exponentially
fast.
may
Such
rapid belief convergence also implies the local stochastic stability of learning needed for
confounded learning to arise with multiple types. The
we have found between the martingale character
stability of learning:
This neat result holds
if
common
ingredient
of the likelihood ratio
is
a simple link
and the exponential
near a fixed point, posterior beliefs aren't
We
(degenerately) equally responsive to prior beliefs for every action taken.
feel this tech-
nique has broad applicability beyond the herding paradigm, into optimal experimentation.
Section 2 outlines the benchmark model and
some preliminary
vides the associated action and belief convergence results; noise
new model with heterogeneous
preferences
is
results. Section 3 pro-
added
is
explored in section
A
5.
and proofs are appendicized, including a convergence
related) results
in section 4.
Our
host of needed (or
criterion for
Markov-
martingale processes, and a new local stability result for stochastic difference equations.
2.
THE STANDARD MODEL
Some Notation
2.1
An
sequence of individuals n
infinite
Everyone observes the ordered actions of
=
1,2,... takes actions in that exogenous order.
A
predecessors.
all
background probability space
random: There
(Q, £, v) captures all uncertainty in our model. First, the action payoffs are
5 =
are
2 possible states of the world (or just states), s
Q
Formally,
partitioned into events
is
=
v{H)
belief be
v(L)
=
significant algebraic cost
2
Our
QH U
f2
L
,
called
H
=
H
and
number
any
finite
and dubious conceptual gain
(see
our 1996 working paper).
finite
action set (a m
might think of investors deciding whether to
,
m£M),
where Ml
'invest' or 'decline'
s
and everyone acts so as to maximize
weakly dominated, and at
least
n
receives a private
Conditional on the state, {an } are
s
)i
in state s
world,
we
\i
L
fi
H And
.
are not the
same measure
Common
priors
3
So
— E
,
.
.
M}. One
all individuals,
assume that
exist.
profile, loosely
an 6 E, about the
WLOG
no
Before selecting
denoted
h.
state of the world.
and drawn according to the probability measure
Radon-Nikodym
to avoid trivialities,
2
.
3
there exists a positive, finite
w.r.t.
i.i.d.,
signal,
{1,
To ensure that no signal will perfectly reveal the state of the
H and [i L be mutually absolutely continuous (a.c.). 4 Thus,
that [i
e {H, L}.
shall insist
random
of states, but at
same for
two undominated actions
an action, an individual observes the entire action history
Individual
We
expected payoff.
his
=
prior
an investment project of
uncertain value. Action a m pays off u (a m ) in state s £ {H, L}, the
is
('low').
common
Let the
L.
=L
s
results extend to
1/2.
Everyone chooses from a
action
and
('high')
—
i.e.
we
derivative g
shall rule out g
some
=
1
=
L
d\x ld\i
H
:
E—
>•
(0,
oo) of
almost surely, 5 so that
fi
L
fi
H and
signals are informative about the state.
WLOG
standard (Harsanyi (1967-68)), and a fiat prior
(see our working paper).
a random variable; /x s = n"n = v s °o~ x where measure u° conditions v on event ft".
"See Rudin (1987). Measure \i L is a.c. w.r.t. fj," if H {S) =
=> fi L (S) =
VS G S, the a-algebra on
H
E. By the Radon-Nikodym Theorem, a unique g € L}{n H ) exists with fi L (S) =
Js gdfi for every S G S.
5
H
L
With fi
mutually a.c, 'almost sure' assertions are well-defined without specifying the measure.
<7
n
:
ft
is
is
,
fJ,
,
/j,
2.2 Private Beliefs
The second source
a € S, an individual
=
p(o)
are
a
F
s
on
(0, 1).
The
F H and
distributions
F
L
each
F
s
has a density.
The next
dF H /dF L
=
a Radon-Nikodym derivative /
Lemma
1
€ {H, L), p
fi
F
H and
L
and ultimately drives
\i
L
are
Thus there
.
f {p)/f
,
L
when
{p)
wisdom that
posterior
learning that occurs.
all
= dF H /dF L
(No Introspection Condition) The derivative f
(a)
distributed with
is
H
distributions F and
which reduces to H
result neatly illustrates the folk
beliefs are sufficient for one's information,
state, private beliefs
are subtly linked. Since
mutually absolutely continuous, so are the associated
exists
shall refer to as his private belief
H. Conditional on the
is
Given signal
private information.
across individuals because signals are. In state s
i.i.d.
c.d.f.
that the state
(0, 1)
is
what we
uses Bayes' rule to arrive at
+ 1)6
\/(g(o)
model
of uncertainty in our
of private
F H F L satisfies {it): f(p) = p/(l—p) almost surely inp € (0, 1). Conversely,
H F L arise from updating a common prior with some signal measures H L
if (-^r) then F
L
H
(b) The difference F (p) — F {p) is nondecreasing or nonincreasing as p ^ 1/2.
L
H
or 1.
(c) (Beliefs are Informative) F (p) < F (p) except when both terms are
belief c.d.f. s
,
p,
,
Proof:
If
the individual further updates his private belief p by asking of
=
the two states of the world, he must learn nothing more. So,
p
= p/{\ — p),
F
Conversely, given f(p)
Part
(a) implies
part (b), and
is
let
distinct,
and
F L (b) =
and
Lemma 1(a)
For
•
clarity,
6
=
a,
=
2a
and thus the
(left
<
=
[l
l
b
<
1,
<
1/2
Let
\x
h
let
n
1).
=
5
<
1
since
L
fj,
no perfectly revealing
will
[b,b]
and
2/(5-2a). Note how p maps
C
[0, 1]
H
p,
signals.
co(supp(F))
if
stochastic
=
are
We
[0, l].
7
be progressively embellished.
So g(a)
=
g
=p <F
2
H be Lebesgue
with Radon-Nikodym derivative g(a)
l/(g(o) + l)
=
have probability density g L (a)
simple formulae F H (p)
Next,
<
and unbounded
panel of figure
Bounded Beliefs Example.
and choose
p{a)
b
6
as there are
1
Unbounded Beliefs Example.
yielding p(a)
<
follows.
common support,
structure of co(supp(F))
we introduce two leading examples which
and n H the density g H {o)
•
<
implies that they have a
The
Note that
if
E {H, L).
in state s, s
when p e supp(F) \ {1/2}. Hence (c)
functions F H and F L can be taken as the
F H (b-) =
the private beliefs bounded
call
s
.
likelihood in
6
say supp(F), equal to the range of p(-) on E.
plays a major role in the paper.
distribution
\i
f(p)/[l+f(p)], as desired.
strict
Thus, the conditional distribution
primitives of the model. Notice that
a have
its
,
H
{cr)/g (a)
L
L
(p)
=
2p
=
=
—p
2
= |^Q =
3/2
[0,1] injectively
onto
on
[b,b]
2a,
(l-a)/a,
2
.
(uniform) measure on
-a
-
[0, 1].
=
[0, 1],
Then
[2/5,2/3],
The support of a measure is the smallest closed set of full measure; co(A) is the convex hull of set A.
To exhaust all possibilities we should also consider supports that are bounded above and not below,
etc., but this exercise in generality yields no new insights.
7
9
Figure
Signal Densities. Graphs
1:
Observe how,
near
for instance, signals
H
{°)
for the
unbounded
and bounded
(left)
(right) beliefs
are very strongly in favor of state L, as in
Lemma
examples.
1.
u L (ai)
u L (a 2 )
u L (a 3 )
f
Figure
=
Payoff Frontier. The diagram
2:
f3 =l
f2
fi
depicts the (expected) payoff of each of three actions as
is H. The individual simply chooses the action yielding
an insurance action, while actions ai and 03 are extreme actions.
a function of the posterior belief r that the state
the highest payoff. Action a?
< p
with p(a)
<£>
2/(5
is
-
2a)
< p
<& (5p
bounded, and the distribution of p €
in state
# and in state
-
>
2)/2p
[2/5, 2/3] is
FH
a.
(p)
=
Here, the support of
H
/i
(5p-2)/2p]
[0,
=
F H FL
,
is
(5p-2)/2p
L:
5p-2
^
2.3
iP
L
(p)
= /
«W da= JOf
Jp(cr)<p
2
\{Zo - a )
Z
l
ru H (a)
Lemma
+
(1
—
belief r
(5p-2)(p
Proof:
2)
Sp^
,
[0, 1]
in state
H, the expected payoff
[0,1] partitions into subintervals,
As noted, the payoff
action a m
[0, 1]
is
strictly best for
where
it
of choosing action a
strictly
of each action
some
dominates
r;
is
is
optimal with posterior belief r 6 Im
a linear function of
therefore, there
all
or action basins, I\,...,Im
must
exist
other actions. That this
r.
We now WLOG order the actions
= f <
G [fm _i,fm = Im where
]
,
so that a m
f\
<
•
<
is
.
But by assumption,
a single open subinterval
is
a partition follows from
the fact that there exists at least one optimal action for each posterior r
r
+
r)u L (a). Figure 2 depicts the content of the next (standard) result.
The interval
2
€
overlapping at endpoints only, such that action a m
of
2"
Action Choice
Given a posterior
is
\-ada=
€
[0, 1].
optimal exactly when the posterior
fM
=
1.
We
employ the tie-breaking
rule
8
that individuals take action a m (versus a m+ i) at r
(resp.
actions 02,
•
optimal when one
is
a,\)
.
.
,
.
=
To
Invest.
posit
u
>
0.
u
both
in
(resp. L), while insurance
H
in state
two actions
and —1
=
when
fu
—
— f),
(1
L
from state
may be
to state
=
a±
=
so f
is
H.
Decline
in state L; to Decline
Since, by assumption, action ai
states.
Indifference prevails
H
is
au
extreme action
.
as confidence shifts
In our running examples, the
Invest yields payoff
investing yields payoff
we
certain that the state
om-i are respectively optimal
Examples Cont'd.
and a 2
is
= fm The
from
undominated,
+ u).
1/(1
2.4 Individual Learning
map from
Individual decision rules
Bayesian equilibrium, everyone knows
compute the ex ante chance
=
likelihood ratio ((h)
q(h) in state
So q(h)
A
is
H,
n
1
s
7r
(h) of
(h) / it
11
own
private beliefs and history to
all
common
decision rules and the
any history h
each state
in
(h) that the state is
L
actions.
prior,
In a
and can
This yields the public
s.
H, and the
versus
9
public belief
i.e.
the posterior ensuing from a neutral private belief and history observation h.
final application of
Bayes rule yields the posterior
terms of the public history
belief r (that the state is
H)
in
— or equivalently the likelihood ratio ((h) — and the private
belief p:
p
ir
H (h)
p
pir H (h) + {l-p)ir L {h)
Lemma
3 (Private Belief Thresholds)
= po(^) < Pi{h) <
pe (pm -i(h),pm (h)],
.
.
.
< pM{h) —
1,
1
p+(l-p)e{h)
Given history
such that a m
is
i
+
h, there are
chosen
iff
(1)
l=t£(h)
M+
1
thresholds
the private belief satisfies
where
Pm{h)
7
RHS of (1) is strictly increasing in p, and
(Hi) the reformulation of (1) as posterior odds (1 — r)/r equal private odds (l—p)/p times
the likelihood ratio (.(h). Note that if pm _i(h) = pm (h) at some h, then a m is never chosen.
This
true given
is
(i)
the tie-break rule,
Since the likelihood ratio
explicit
•
dependence
((h),
is
(ii)
the
informationally sufficient for the history,
and write pm (() instead o(p m (h), where
Examples Cont'd.
In both examples, (2) yields p(()
=
(
h->
(/(u
we suppress the
pm (()
is
+ ()\if=
8
increasing.
1/(1
+ u).
For generic models, this choice does not matter. Even when it does change the probabilistic course of
nongeneric models, the tie-breaking rule does not change the statement of any of our theorems.
9
We
assume common knowledge of
rationality. Section 4 partially
backs away from
this.
£
Optimal action a m
Action a\
:
1
£g(a)elm =
Action a m
is
- r m -i
taken with chance
p(M\LJ)
Action a M
p(m\s,£) in state s G {H, L}
<p(M,£)
£
p(M\H,£)
Figure 3: Individual Black Box. Everyone bases his decision on both the public likelihood ratio
and his private signal <r, resulting in his action choice a m and a likelihood ratio to confront successors.
I
One
takes action a m
iff
one's posterior likelihood
lies in
the interval Im where
,
I\
,
.
.
.
,
Im
partition
[0,
oo]
Corporate Learning as a Markov-Martingale Process
2.5
We
and qn
let t n
chooses action
actions, are
mn
10
,
,
be the public likelihood ratio and belief after Mr. n
respectively,
with
£q
random, and so
=
1
and
(£ n )^=1
and
=F
p(m\s,£)
<p(m, £)
£) is
likelihood
and the true state
we move
Our
to £n+ i
=
s
(<7n)£Li are stochastic processes,
s
(p m (£))
-F
s
and thereby
described by
(p m ^(£))
(3)
= £p(m\L, £)/p{m\H, £)
(4)
e {H, L}. Faced with
£n
,
if
individual n takes action
mn
,
ip(m n £n ). Figure 3 schematically summarizes this transition.
,
insights are best expressed
Mx
(mn +i,(p(m n+ i,£n )) with
is
initial history). Signals,
,
process on the state space
state
1/2 (null
the chance that a (rational) individual takes action o m given the public
Here, p(m\s,
£,
=
go
by considering (m n ,£n ) as a time homogeneous Markov
[0,
Given (m n ,£ n ),
oo).
probability
p(m n+ i\H,
(3)
and
(4)
n ) in state
imply that the next
H.
In principle, such
a two-dimensional process with continuous state variables could be very ill-behaved, and
possibly chaotic.
The next
Lemma
But
result
is
Lemma
5 attests to
how well-behaved
quite standard (but for completeness,
4 (The Unconditional Martingale)
unconditional on the state of the world.
is
is
The public
the likelihood process.
proven in the appendix).
belief (q n )
is
a martingale,
11
This martingale describes the forecast of subsequent public beliefs by individuals in the
model,
who do
not
know
the true state of the world: Prior to receiving his signal, individual
n's expectation of the public belief that will confront his successor
for
our purposes, an unconditional martingale does not
convergence.
tell
us
all
is
the current one. But
we want
to
know about
For that, we will condition on the state of the world, and that will render
m will denote actions, and n individuals.
10
Or
11
We really ought to specify the accompanying sequence of tr-algebras to the stochastic process; however,
equivalently, action a mn
.
Throughout the paper,
these will be suppressed because they are simply the ones generated by the process
itself.
H
the public belief (q n ) a suimartingale in state
E[qn+ i
H,qi,... ,qn ]
|
>
qn
more focused on the true
the modeller, a
—a
state of the world
useful martingale
is
=
—
((1
q n )/q.n)-
The stochastic process of
(b)
The
n
Assume
5 (Likelihood Ratios as a Conditional Martingale)
(a)
=
[0,oo).
So
fully incorrect learning (£ n
—»
H,£ u
...£n
]
= J2 P(m\H,Q
£n
Since the likelihood ratios are non-negative
Martingale Convergence Theorem.
stochastic evolution of (£n )
on state H.
00) almost surely cannot occur.
,
I
(See
=
j^%j\
random
Breiman
H.
= limn-^^,
l^
Proof: Given the value of £n the conditional expectation of £n+ i in state
E[£n+1
state
likelihood ratios (£n ) is a martingale conditional
likelihood ratio process (£ n ) converges almost surely to a r.v.,
with supp(^oo)
For
seek.
one that conditions on knowledge of the
state of the world, namely, the likelihood process (£n )
Lemma
much weaker than we
result
^
4
H
is
= L
P{™\L,£n )
variables, the result follows
(1968),
Theorem
from the
Or, since the
5.14.)
mean-preserving, convergence to any dead wrong belief
is
i.e.
become weakly
Essentially, the public beliefs are expected to
-
much more
(and a supermartingale in state L),
cannot occur: The odds against the truth are not permitted to explode.
a.s.
13
Easley and Kiefer (1988), and others, underscore that unlike statistical learning where
information
may
well accrue at a 'constant rate', complete learning
conclusion in a economic model of costly experimentation.
complete learning
is
generally
When
is
not at
all
a foregone
information has a price,
deemed too expensive. One might expect that the
resulting
pathological outcomes to the learning dynamics persist this observational learning setting.
Guiding Questions:
of the process
1. Is there
(m n ,£n )
settle
down?
Belief convergence obtains
2.
converges
a herd, or action convergence: Does the
If so,
on action o^?
as
— since t^ exists by Lemma
some stage n? Or perhaps only a
n ->
00.
And
is
Otherwise, learning
3.
What
is
learning complete
is
5.
But must a cascade
13
— p(m\H,£
limit cascade
— or do
n)
on a m
beliefs
—
coordinate
expected time?
Markov process (m n ,£n )
arise,
where everyone takes
p(m\L,£ n )
will arise,
i.e.
—
1 for
p(m\H,
converge to the truth,
i.e.
some
£n )
m
—>
1
£ n -> 0?
incomplete (beliefs not eventually focused on state H).
the link between action and belief convergence?
with a discrete signal space,
12
in finite
— the second coordinate of the
action a m irrespective of signal realizations
after
And
first
is:
(i)
cascades must occur, and
See, for instance,
Doob
Another proof of
this fact uses public beliefs (see, for instance,
(1953), section
The rough
(ii)
logic of
BHW,
valid
cascades imply herds, and
II. 7.
Bray and Kreps
(1987)).
The second
thus (Hi) herds occur.
ignores his signal and takes a m
still
p(m\H,
£ n +\)
We
=
shall prove that
we
first
interval
Jm
=
C
{£ supp(F)
|
£
int (Jm ),
With bounded private
(6)
With unbounded private
\p
is
imply that herds eventually
is finite.
beliefs,
£.
So once
all
G
The lemma
1.
[£,
Jm =
oo]
£
is
(0,
=
and Jm
{0}, Ji
=
With bounded
private signals
every public likelihood £
Remarks.
=
beliefs, J\
only boundedly far from
an insurance action,
Im
C
[0, £]
also possible for
Then the insurance
=
the public
if
u L (a 3 )
1,
rearranging an expression
like (2),
-^b + (i-b)£
action.
= u H (a 3 =
)
0,
of private signals.
Jm2 <C Jmi
iff
10
m
2
>
mi.
an action absorbing basin.
beliefs are
and u# (02)
one can show that £
5
oo;
But with unbounded
swamped by some mass
and
<
£
or oo or perhaps even in favor of
are inversely ordered, or
u L (a\)
<
other basins are empty.
all
an 'insurance' action a m when
=
£
private beliefs, the posterior odds are
must lead to the same
oo) will be
<
some
=
bounded. For
is
in
_i
f!
'
+ (i-5)^
m
Jm
=
UL{a 2 )
action a 2 has an action absorbing basin for small enough e
fm
empty
unchanged. Also,
is
for
{oo}, and
sufficiently near
The absorbing basins
instance, take u H (ai)
such that
[0,oo],
asserts that to each 'extreme action' corresponds
is
their signals.
there is a possibly
,
then taken almost surely, and £
intuitive:
By
imply herds?
then the posterior likelihood ratio £g(a) G Im almost surely in
is
3.
limit cascades
For each action a m
m -i(£),pm(£)]} C
The appendicized proof
7^
— and cannot
then prove that this convergence
limit cascades
i.e.
Absorbing Basins)
6 (Action
(a)
But Jm
Do
must understand exactly why individuals might wish to ignore
private signals a. Action a m
2.
So
(2).
Convergence
likelihood ratio £
beliefs,
We
characterize the limit t^.
first
whether the mean time to entry into a herd
Lemma
So the
.
conclude by discussing the speed at which beliefs converge, and the intertwined
3.1 Belief
We
£n
THE MAIN RESULTS
of beliefs implies convergence of actions,
issue of
=
unchanged by
as private belief thresholds are
1,
,
information, and so £n+ i
This leads to a tougher question:
beliefs.
In this section,
We
new
on action a m Mr. n
not robust: Cascades generically needn't arise
(i) is
3.
occur.
irrefutable: In a cascade
too.
1
with unbounded
is
revealing no
,
n+
obtains at stage
cascade
step
precisely
>
1
0.
when
—
e.
J2 2u/Z
2u
Ji
Figure 4: Continuations and Absorbing Basins. Continuation functions for the examples:
unbounded private beliefs (left), and bounded private beliefs (right). By the martingale property, the
expected continuation lies on the diagonal. The stationary points are where both arms hit the diagonal
(impossible here), or where one
arm
is
an action absorbing basin
So,
larger the smaller
is
Unbounded Beliefs Example Cont'd.
F H (p) = p
{(.
=
is
F L (p) =
=
in the left panel; I
2u/2> in the right).
and
the support [b,b],
Only extreme action absorbing basins
the interval [fm _i,fm ].
•
taken with zero chance
exist for large
Private beliefs
p e
(0, 1)
the larger is
enough
[b,b].
are distributed
= [0, 1], and private beliefs are unbounded;
the basins collapse to the extreme points, J\ = {oo}, Jm = {0}. With our two actions,
we have p = £/(u + £), where u > 0. Thus we let p(l\H,£) = £ 2 /{u + £) 2 and p(l\L,t) =
2
We now get <p(lj) = £+2u and ip(2J) = u£/(u + 2£), shown in figure 4.
£{£ + 2u)/(u + £)
H
L
• Bounded Beliefs Example Cont'd. Here, F (p) = (5p - 2)/2p and F (p) =
2
(5p - 2)(p + 2)/8p for p 6 [2/5, 2/3]. With p = £/(u + £), active dynamics occur when
£ € (2u/3, 2u). For £ < 2u/3, we have p(l\H, £) = p(l\L,£) = 0, i.e. action a 2 is taken a.s.,
and thus its absorbing basin is J2 = [0, 2u/3]. For £ > 2u, we similarly find J = [2u, oo].
For£<E (2u/3,2u)wehavep(l\H,£) = (2>£-2u)/2£ and p(l\L, £) = {3£-2u)(3£ + 2u)/8£ 2
By the martingale property, <p(l, £) = u/2 + 3£/4 and ip(2, £) = u/2 + £/4. (See figure 4.)
We now argue that limit cascades must occur: Dynamics must tend to one of the basins.
as
2
and
2p—p2 So supp(F)
.
,
.
x
.
Theorem
1
(Limit Cascades)
=
has support
J
The appendicized proof
is
variable
Any
£<*>
Jy
(we
The
U
•
•
•
U Jm,
=
0) or
feel) intuitive.
it
Theorems B.l-2
H, simply because
Theorem
2 (Long-run Learning:
beliefs,
if£
,
and
(Lemma
process,
=
£).
i.e.
either
If at least
1(c)).
random
,
then with positive chance
11
£00-.
an action
two actions
L
than state
This will violate stationarity.
Complete and Incomplete)
£ JM
the
precisely characterizes
taken with greater chance in state
is
beliefs are informative
With bounded private
Markov
teaches us nothing (<p(m,£)
are taken in the limit, then the least one
£ (X>
the absorbed set.
point £ e supp(^oo) must be stationary for the
doesn't occur (p(m\£)
(a)
likelihood ratio process £ n
Assume
state
H.
i^ G JX U- -UJm-i-
With unbounded private
(b)
For
Proof:
then
(a), if
£& E
beliefs, £ n
—>
with positive chance, we are done. Otherwise,
J\
uniformly bounded above by
(£ n ) is
almost surely.
the infimum of J\, introduced in
£,
Lebesgue's Dominated Convergence Theorem, the
mean
of (£ n )
is
^
if £«,
Ji a.s.,
Lemma
By
6.
preserved in the limit,
i.e.
> £ it cannot be the case that supp(4o) CJM = [0,^].
(b) Theorem 1 asserts £& £ J a.s. By Lemma 6, with unbounded beliefs J — {0} U {+oo},
while Lemma 5 proves supp(£ 00 C [0, oo) if the state is H.
E[£oo]
=
the initial ratio. So,
£o,
if
£
)
To underscore how nonintuitive
where
strict inequality
and so
(£ n )
this conclusion, notice that this
is
Lemma,
holds in Fatou's
or
1
must be unbounded. While the process
=
limn _>0O E[£ n
(£ n )
]
is
one situation
> E^iuvn^^ £ n =
]
0,
occasionally gets arbitrarily large,
corresponding to arbitrarily long trains of individuals choosing the least optimal action,
the longer
all
is
the train, the less likely
such trains must almost surely
(£ n ),
come
Still, this is
Puzzle
#
an arresting
1.
Why
0.
With
On
balance, the
an end. For a
random walk
probability one,
classic
analogy to the behavior of
#
2.
and
eventually hits
is
Why
absorbed.
2,
convergence towards totally incorrect beliefs appears
Why
complete learning?
limit.
aren't correct herds periodically broken
herds are)? Couldn't beliefs be ever cycling betwixt confidence in
(just as incorrect
and L, so that the ergodic distribution assigns weight to non-stationary
establishes that beliefs
BHW
must eventually
settle
H
is
here
fail,
and
beliefs? It
that the martingale convergence theorem succeeds where Markovian arguments
the analysis of
at 1
can't individuals eventually be wholly mistaken about the state of
But the martingale convergence theorem ruled out that
self-enforcing.
2 tells us that
Brownian motion) starting
(or
it
Theorem
on two counts.
result,
the world? For as noted in section
Puzzle
to occur.
to
think of the behavior of a driftless
with an absorbing barrier at
up
it is
down: Limit cycles cannot occur. Note that
— which did not appeal to martingale methods — only succeeded
because their stochastic process necessarily settled down in some (stochastic)
finite time.
Action Convergence
3.2
Suppose a putative herd has begun:
action, but a cascade has not yet begun.
A
string of individuals has taken the identical
If
someone then acts
in a contrary fashion, his
successors have no choice but to concede the strength of his signal, thus sharply revising
the public belief.
action.
his
own
since
More
We
say that the putative herd has been overturned by the unexpected
formally,
if
Mr. n chooses action a m then n
private signal, find
it is
,
it
+
1
should, before he observes
optimal to follow suit because he knows no more than n, and
common knowledge
that
n
rationally chose a m
12
:
So
after n's action, the public
belief
is
G Im
q
=
(r
m _i,r m
The next lemma
].
codifies this logic (proof appendicized), for
proves central to an understanding of the entire observational learning paradigm.
it
Lemma
(The Overturning Principle)
7
action a m
the updated likelihood ratio £
,
<p(l,£)
G
[u,
oo)
and
—
,
(a)
With bounded private
state
(b)
A
H) from
running examples, the principle correctly predicts
G
So
(see figure 4).
[0, it)
Theorems 1-2
beliefs,
on an action other than
(in
with positive probability.
BHW. 14
bounded
Strictly
beliefs
Fix the payoffs and prior
beliefs so
makes
clear.
15
the continuous transition
is
tend from bounded to unbounded.
beliefs.
7/co(supp(F
fc
converges
))
then the chance of an incorrect limit cascade vanishes as k —¥ oo.
Only basins J*
If
Om
(and we think, simplicity) the major
to emphasize the above characterization
Theorem 4 (Continuity)
[0, 1].
om
for their striking result, as the characterization
from incomplete to complete learning as private
Proof:
working paper).
individuals almost surely settle on the optimal action.
beliefs analysis proves in generality
happens to be the mainstay
to [0, l],
(details in
absent a cascade on the most profitable action
beliefs,
pathological herding finding in Banerjee (1992) and
16
convergence implies action
belief
herd on some action will almost surely arise in finite time.
With unbounded private
The major reason
suit.
eg. for the
the outset, a herd arises
The bounded
takes
contrary actions in a limit cascade, for that would
(f(2,£)
3 (Herds)
someone optimally
G Im and an uninformed successor follows
convergence, yielding the action counterparts to
Theorem
history, if
many
This result precludes infinitely
negate belief convergence
For any
fc
7r
is
=
[^*,oo]
and J^ remain once co(supp(F
the chance of a correct limit cascade, then EEqo
by Fatou's Lemma, so that n
k
>
1
—
Eo/fr.
As k —>
Remark: Fragility of Limit Cascades.
00, I*
BHW
—>
this insight.
Even
Our
results
(1
00,
—
close
is
))
fc
7r
enough to
<
)P\ But £"£00
and so
—>
fc
7r
£
1.
devote considerable attention to
the fragility of herds, pointing out that the release of a small
can undo a herd.
>
fc
amount
of public information
on the nonexistence of cascades only serve to strengthen
in the limit,
we have shown that
generically
likelihood ratio lies at the edge of the absorbing basin.
By
public information will break the limit cascade.
bounded away from the edge of the absorbing
(i.e.
without atoms) the
Consequently, arbitrarily
contrast, the limit belief in
basin, thus inoculating the
little
BHW
is
model to the
release of sufficiently small packets of public information.
14
And extended
15
While
BHW
to
M > 2 actions. BHW also handled several states (done in our working paper).
unbounded private signals, their working paper introduces perfectly
informative signals under the rubric of 'pseudo-cascades' (ruled out by our assumption of mutually a.c.
information). Complete learning in that context is clear. For once the public beliefs overwhelm all but the
didn't consider
perfectly revealing signals, the very next contrary action reveals the state of the world,
16
We mean
the HausdorfF topology:
If
co(supp(F*))
13
=
[a k ,b k ],
then a k ->
0,
and
and we're done!
b k -»
1.
.
Must Cascades
3.3
Or
we ask "Can cascades
rather,
Theorem
chain,
£ n -¥ £
Bounded
Exist with
Beliefs?
As (m n ,£n )
exist...?"
not^i finite state Markov
is
—
2 cannot assert finite time convergence (a cascade)
£ J but always
£n
bounded
In the running
BHW,
J. Unlike
$.
beliefs
for
we may have
cascades need not obtain, though herds must.
example, a cascade on action ai obtains
iff
>
£
2u. In a
£„—>£€ [2u, oo), figure 4 clearly shows that (£ n
17
So learning never
its edge. A cascade never starts.
herd (and so limit cascade) on Oi, with
J\, but only
never enters
No matter how many
ceases:
A
approaches
cascade on
With a
<
individuals have followed suit, a contrarian might
why
=
£
BHW
this
is
For
true.
F
inf(Jm ) implies
L
a herd starts on action a m with
if
,
{p m -x{£ n ))
=
FH
=
(p m _i(4))
'jump
into'
Jm
we have
0,
L
))
inf{F L (p)/F H (p)\p G
in
boundedly
and thus are nongeneric.
Mean Time
3.4
We
long
to
illustrate the
is it
int
finite time.
One might prematurely conclude
tributions,
to
-U 1m
To
{£ n )
must
A counterexample to this conjecture is in Appendix F.
if it
(in reverse order). Call
lie
in the
communicating basin C
starts at £$. Inside
C
is
C
[0,
How
oo),
at least one action
1, 2, ...
,
M. Thus,
a string of identical actions that eventually comes
is
8 (Putative Herds)
boundedly positive
There
most n* steps with chance
(£ n )
is e*
is
either a herd or
temporary herd.
This also follows analytically:
possible for (£ n ) to
jump
>
after a fixed
and n*
<
number
of periods.
oo such that a putative herd
at least e*
remaining long
of a putative herd. Intuitively, at least until
It is
and
that cascades can only arise with discrete signal dis-
this end, let £
The appendicized proof rules out
18
A.l,
that temporary herds are of uniformly bounded expected duration, and that the
starts in at
17
Lemma
WLOG the actions that can be taken in 6 as
entry rate into putative herds
Lemma
by
This needn't occur with general signal spaces.
an end a temporary herd; therefore, a putative herd
It suffices
1
power of our framework by answering an important question:
absorbing basin. 18 Relabel
•
>
Herd
until a herd starts?
= I\ U-
()
[b)
X
(supp(F))}
the smallest interval that (£n ) cannot exit
6
0, then
F (Pm (£n))-FHPm-dQ) ... F {Pm {£n))
£)-£
m
tn
n
n)
~
"^
F»(pm (£n - F"(pm - {£n )) ~ F»{p m {£n ))
71+1
>
£ n +i/£ n
^
(Jm )
int
^(
'
So
cp(i, •).
deduced that cascades must occur, and we
L
/
appear.
still
can only arise with a nonmonotonic likelihood transition function
discrete signal distribution,
can easily see
since £ n
a*
)
it
in Uj{Ij\Jj
starts, at least
If i n < 2u, then £ n+1 = <p(l,£ n ) = u/2
over action absorbing basin J{ if tp(i, )
14
—
0}, delaying the start
two actions can occur, and
+
is
ZE/4
< u/2 + 3u/2 = 2u
too.
locally decreasing nearby.
with boundedly positive chance,
Let e n be n's exit chance from a putative herd, and
Mr.
1, 2,
a herd
.
.
,
.
n participate
E^ = F\ =
is
(1
in
— ei)(l — e?)
chance from a herd conditional on
stage n
is
permanent
is
Fn =
En =
— ei)
(1
(since signals are conditionally
it
•
its
•
•
•
— e„)
(1
Then
i.i.d.).
the chance
the chance of
>
exactly
•
•
•,
and
by Bayes
so
— Fn ).
_n — E
=
rule, b n
e n /(l
EE
Since the acid test ^2 n (En
good principle
How
is:
5 (The
With atoms
(a)
Eoc)
<
co can
even when
fail
how
slowly they can radically shift
ofsupp(F),
20
Lemma
by
Until the Herd?)
mean
herds begin infinite
time.
LetF H (p) =c(p-by+0((p-b)i +1 ) andF L (p) =l-d(b-p) 5 +0({b-pY +l
(b)
With bounded
(c)
With unbounded
So
bounded
for
beliefs,
herds begin infinite
beliefs,
beliefs,
mean time
extreme signals
7 <
if
there are
and
2
>
5
2,
many extreme
om
the eventual herd on
starts in finite time in state
time in state L\ So, herds cannot start in
beliefs,
•
finite
om and
is
in
since a
Theorem
If
in its tail.
19
makes
The
For exp(-i)
number
20
5
clear that
result follows
>
1
-x
yields
En =
)
for
from liminfI ^ 1 _(l
exp(-
£
ft
N of terms, followed by N -» 00,
< F H (b) < F H (b~) <
0(l/n
both
6
=
ek )
>
(1
proves (1
With bounded
is (1
some a >
-
S
>
2.
in L, or
herds start in infinite
states.
With unbounded
2,
the
mean time
mean
in infinite
x)
if
—
F
5)
L
, while
•
•
•
>
1
time.
the
tails.
is 1
^J° En 6
n
.
—S n The
.
The proof
has polynomial weight
J2T x "/ na =
- ei)(l - e 2 )
- ei)(l - e 2 )
to herd
of time until a
a herd starts in period n, the discounted fraction of time until then
Theorem
and
our proof explains
(Time Discounting) The expected 5 -discounted fraction
vanishes as 5 —> 1, provided each F s has polynomial weight in
a
2
a t end quickly enough that
H ends
ai in state
expected discounted time to finishing any temporary herd
of
7 <
6
herd starts
Proof:
wrong temporary herd on
andb.
expected (ex ante) to take forever!
Unbounded Beliefs Example Cont'd. With 7 =
is infinite,
iff
beliefs,
H — but then
mean time
complete learning eventually obtains, but
H
H, and conversely few
signals available in state
b
must have an unbounded
With unbounded
then respectively, temporary herds on
near
)
2.
in state
in favor of the true state
density, to ensure a correct herd in finite time.
how
<
5
iff 7,
mean time
herds begin in finite
E.l(b), a
This leads us to
beliefs.
How Long
Speed of Action Convergence:
at the edge
E^ >
—22.
on how quickly individuals become
quickly a herd starts depends not
convinced of a state but on
Theorem
—
0.
when YL e n < °o. 19 Let b n be n's exit
being temporary. The chance that a given herd at
•
— e n )(l - e n+ i)
(1
^
marches monotonically toward some Im with Jm
(£ n )
for all
<
a
<
l.
21
an induction argument on the
e k ) if e 6 [0, 1) for all i.
- (£ k
{
such extreme signals aren't perfectly revealing.
21
Michael Larsen (U. Pennsylvania) gave us a quick proof: Let 77 > 0. As the early terms of S(x) =
2
) = 77.
J2T z"/™" don 't affect our limit, assume l/n° < 77; then (1 - x)S(x) < (1 - 1)77(1 + x + x H
i.e.,
1.
beliefs,
15
4.
The
pivotal role played by the overturning principle
large weight accorded isolated actions
unappealing;
either
NOISE
22
therefore,
by design
that being noisy
we now add
('craziness') or
is
is
somewhat
is
The
both economically implausible and theoretically
system:
'noise' to the
Some percentage
of individuals
We
mistake ('trembling') do not choose wisely.
not public information, and that this trait
across individuals. Since
unsettling.
is
assume
distributed independently
23
actions are expected to occur, none can have drastic effects.
all
Below, we address the simplest case of craziness noise, and appendicize the discussion
of trembling noise. So with chance
We
Km Mr. n chooses action a m
,
,
irrespective of history. 24
= 1 — $3 m =i K m > of rational individuals, each of whom
p(m\s,£) = F (pm (£)) — F (pm -i(£)). So action a m is now
assume a positive fraction k
take action a m with chance
s
s
taken with chance ip(m\s,£) yielding likelihood continuation (p(m,£), where
ip(m\s,£)
<p{m, £)
4.1
= Km +
Kp(m\s,£)
= £i>{m\L, e)/if>(m\H, £)
(7)
Convergence of Beliefs
—
Since £
Ylm=\
sure finite limit 4c-
p(m\H,
£)
=
p(m\L,
xl;
{
rn \H,£)if(m,£), (£n )
The
£)
=
is still
interval structure of
1 for
some
«7i,
.
a martingale in state H, with almost
.
.
be adjudged ex post as noisy, and
Theorems
and 2 are robust
1
,
Jm (Lemma
will
surely.
7 (Long
Run
H, where supp^oo) C
to noise. Statistically
with chance less than
Proof:
As
all ip
or (p(m\H, £)
1 if £
With bounded
beliefs,
£ Jm- With unbounded
are boundedly positive,
Contrary actions
^
m,
will perforce
constant behavior of noisy individuals
Learning with Noise)
[0,oo).
m'
for all
So
simply be ignored. The next result asserts that
doesn't affect long run learning by rational individuals, as
Theorem
6) is also still valid.
m implies p(m'\H, £) = p(m'\L, £) =
and thus rational individuals take action a m almost
it
can be
filtered out.
^
In the noisy model, £n
—>
1^ 6 J almost
and
beliefs, £00
we need only check
=
surely,
in state
4*,
G Jm
almost surely.
stationarity of £
G supp(£00 ),
=I
V>(m\H,t)
22
(6)
+ Kp{mlLj
= £ Km
} =£
K m + Kp(m\H, £)
BHW
Note that the herd fragility discussed in
refers instead to the release of public information.
0ne might think that informational free-riders constitute a different form of noise
i.e. where some
individuals receive no private signal, and simply free-ride off the public information. But they do not
require special treatment: Simply let F H and F L each have an atom at 1/2.
24
Equivalently, every action is commonly misperceived by all successors.
—
23
16
£n+\
.
/45°
•
/45°
2u
2u
\tp{y)s?
\/\/'vlM^
U
/V>^(2,-)
2u/3
2tt/3
-
i
h
Figure 5: Continuations.
Bounded Beliefs Example —
In
>
'
h
J\
2u
2u/3
y
2u
2u/3
J\
Here, we juxtapose the two continuation likelihood functions for the
first for
We
only rational individuals, then with some crazy types.
see in
the right graph that the discontinuity vanishes, corresponding to the failure of the overturning principle.
=
and so p(m\H,£)
p(m\L,£). That £ 6 J
now ensues from
conceptually easier here (one case and not two). Finally,
Theorem
the proof of
but
1,
mimic the proof of Theorem
is
2.
Convergence of Actions: The Failure of the Overturning Principle
4.2
While long-run learning
resilient:
Noisy individuals
rational herds:
Do
unaffected by constant background noise, herding
is
—
— simply do not herd.
like cats
The
the rational agents herd?
It is
not so
is
natural to ask about
failure of the overturning principle offers
a special challenge, as herd violations only minimally impact public beliefs, being deemed
irrational acts. This severs our clean implication: belief convergence => action conformity.
Let us illustrate
running
is
small
By
is
with noise. The
—
the overturning principle, (p(m',£)
amount
left
panel in figure 5 depicts our
can see that for £ near Jm \(p(m,
,
for all
m'
^ m.
£)
—£\
Observe how each
£
w
only for m!
=
m
when
£ near
of noise effects a remarkable sea change in figure
For
in the right panel of figure 5.
ip
fails
only fixed under one continuation, since contrary actions can't occur
Introduction of a small
p and
7
Bounded Beliefs Example, where one
and \ip(m',£) — £\ is bounded away from
stationary point £*
there:
how Lemma
are continuous, ip(m',£)
if all
—
£
5,
Jm
.
as seen
actions occur with boundedly positive chance, and
=
for all
m'
at the fixed point,
by Theorem B.l.
For instance,
<p(m',
£)-£ =
£
p(m',
p(m',£)-£
*[PKIM)-P(m'|ff,l)] =
~1
Kp(m'\H,£)+K m
+ Km>/(K p(m'\H,£))
(8)
,
where P(m',£)
\<p(m', £)
is 0,
and
— £\
for
=
£p(m'\L,
vanishes for
m'
^
m,
£) / p(m'\H, £) is
all
the old noiseless continuation. So, with noise,
m', and any £ G
(p(m', £)
=
£
Jm
:
Indeed, (p(m,
because the denominator
17
is
£)
= £ since
infinite (as
the numerator
p(m'\H,
£)
=0).
That <p(m,
£)
—
=
£
at a fixed point £ allows us to
make simple deductions about
Appendix C develops a theory of
the rate of convergence for this system.
stochastic dynamical systems like this one. Given functions
C
l
and
tp
ip
stability for
that are, respectively,
at the extremes, Corollary C.l asserts that
and continuous
M
=
6
l)^™ *1 ^ =
rate that (£ n ) converges to fixed point I
1
\<Pi(m,
]"J
m=\
mean
the frequency-weighted geometric
i.e.,
The proof
of the continuation derivatives.
of
the next result contains the core essence of our most broadly applicable theoretical finding.
Lemma
L
H
f and f
derivatives
Clearly,
Proof:
£
=
Assume
10 (Rate of Belief Convergence)
Y2 m =i VH
m
If private beliefs are
.
^2^= itp{Tn\H,£) =
l-^> £)
P(
(
m
If
£)'•
y
1,
Y^m'=\ ^i{
than
with equality
(b
>
0),
1:
—
Pe(m,£)
jumps
iff all
1
and informative
Whether a
mean
an
Jm (convergence
<
with
1.
£[f
(p e
=
(m,£)
<
£ for all
m\ and
(ipi(m', £)) is 1. If
rate 0). If
(«;
all
any are
are non-negative,
=
£p(m'\L,
£) / p(m'\H, £)
be
m + Kf3i(m,£))/(K m + re), which is strictly
< 1, given Lemma 1(a), bounded beliefs
So 9
1/2).
Let (3(m',£)
1.
H (b) - L (b)]
f
beliefs (b
=
inequality neatly proves that convergence occurs at a
derivatives are
Then
+
into
<
1.
rational herd eventually starts turns on the speed of convergence of the
public likelihood ratios (£ n ). Suppose
rule out
then 9
25
(9)
in (9) vanishes, since ip(m',£)
So the arithmetic mean of the derivatives
the noiseless continuation.
less
tails,
aU functions are differentiable, we then have
,
m'\Hi £) = 0-
1,
0,
l
while the martingale property yields the identity
G Jm the second sum
the arithmetic mean- geometric
<
C
m=l
negative, then (£n ) eventually
rate 8
have
bounded and f H (b),f L (b) >
m=l
£
,
M
M
- Yl ^H#' £)<Pt{™> t) + Yl Mm\£Mm, £)
1
At a fixed point
F H FL
that
infinite
limit cascade £ n
—>
£
€ Jm
.
When
can we
subsequence of rational 'herd violators', whose private beliefs counteract
the public belief?
In light of the
chance provided ]T^Li
beliefs, this inequality
I
1
~
P(
holds
if
given the positive tail density. So,
25
Namely, each
26
We mean
27
Since not everyone
C
1
in
))
F H and
—>
Borel-Cantelli
(first)
m \H,£n
For then the convergence of £n
is
we have a
0,
<
F
L
°° for almost surely
C
have
and thus
Lemma
l
tails,
this occurs
all (£ n )-
with f H (b) #
- p(m\H, £ n
1
)
->
is
27
with zero
With bounded
and f L (b)
exponentially
^
0.
fast,
10 implies that
some open neighborhood
of b
and
the non-standard conditional version of the
is
Lemma, 26
rational, this inequality in fact
18
b.
Lemma,
e.g.
Corollary 5.29 of Breiman (1968).
assumes a worst
case.
Theorem
and
FL
Lemma
C
have
1
Then
tails.
suffice for the Borel-Cantelli
Lemma
beliefs,
FH
that
arise.
and the slow rate of convergence may not
above. This must remain a (hard) open question.
new
venture into
and investigate a
territory
model with heterogeneous preferences, or
is
must
and
MULTIPLE INDIVIDUAL TYPES
5.
types
that private beliefs are bounded,
rational herds
unbounded
10 doesn't apply to
We now
Assume
Herds)
8 (Rational
A
types.
richer
and often more
realistic
model with multiple but observable
informationally equivalent to the single preference world. So just as with the noise
formulation,
we assume that types
are private information. Still one can learn from history
by comparing the proportions choosing each action with the known type frequencies. This
and
inference intuitively ought to be fruitful, barring nongenericities, like equal frequencies
exactly opposed
learning:
vNM
A
preferences.
Dynamics may
curious
is
then introduced
— confounded
which each action
in
may
it
Given are
by
unbounded private
arise even with
turns out to be an all-round fundamentally different economic beast.
many
finitely
types
—
t
1,
.
.
.
,
T, where
t
determines both
vNM
preferences
over the given action set a\,...,a,M, as well as one's private signal distribution.
denote the known proportion of type
Since an one's type
t,
and assume the types are
private information, this formulation
is
one type has state-independent preferences. In contrast
on (and thus be informative of) private
As
before, {£ n )
is
belief threshold rule
choose action a m
when choosing an
t
of £ yielding p {m\H,£)
=
U
l
U JM
-
p (m\L,£)
=
J{
U
J\
all
£
where each type finds himself
•
•
unbounded private
identical to noise
to noise, all decisions
for each
signals,
of private information,
t,
then J
1
may depend
This radically changes the analysis.
still
employs a posterior
public likelihood
in
J
= J n J2 n
1
•
•
•
D J7
an absorbing basin.
{0, oo}. If £ n
6
is
So
J, every action
l
Jm
will
.
Similarly, let
the overall absorbed
if
is
just one type has
chosen irrespective
and thus provides no information. Conversely,
19
t
on
Next, call the set
£.
the action absorbing basin
so that
=
but
if all
action; however, they will generally disagree
{H,L} and
t
Let A*
across individuals.
i.i.d.
Let p'(m|s,£) be the chance that someone of type
given state s e
,
signals.
is
a convergent martingale in state H. Each type
the desirability of the actions.
set:
is
The Model and an Overview
5.1
J*
taken
is
This twin pathological learning outcome
all states.
one measure more robust than wrong herds, as
beliefs. It also
twist
upon an outcome
well converge
with the same probability in
new
if
£n
£
J,
some
type's action
E J almost
£oo
same
more
surely, as there are potentially
In a limit cascade, a type-specific herd
the
But unlike
not a foregone conclusion.
is
But
action.
if
may
limit points.
Everyone of the same type
arise:
vNM
types differ in their
we cannot conclude that
before,
preferences, unless
and the same action, an overturning principle doesn't work. As with
all
will take
should take one
noise,
we must apply
the speed of convergence reasoning from section 4.2 to conclude that herds must arise.
The dynamics
of the likelihood ratio are described in the usual notation by (7) and
T
i>{m\s, £)
= ^2 AV (mis, £)
(10)
t=i
If -0 is
£.
continuous in
By Theorems
2
£,
and
multiple types, this
outcomes. Such
in the
two
then any fixed points must satisfy (A-4): 4>(m\H,
may
is
solutions to this criterion
7, all
We call its
(p(m, £*) — £* when
no longer
exist since
would ignore
it
the absorbed
lie in
and
is
J confounding
is
most certainly not
their decisions then
(11)
totally informative at £*. For
if it
were, agents
would generally be informative. Rather history has
any additional inferences. The distinction
Both private
both compelling and sweet.
signals
and history
becomes
affect
totally
and private signals wholly inconsequential: Yet both are pathological outcomes.
verify that confounded learning can occur, or £ n —>£*,
inequality (pt(l,£*)
that
=
but with
Vm
decisions in a confounding outcome, whereas in a limit cascade, history
this will
set,
£)
actions are taken with equal chance
= ij;(m\H,e*)
precisely so informative as to choke off
with a cascade
To
or <p(m,
states:
Crucially, history
decisive,
=
solutions £* outside
true.
ip(m\LJ*)
become
£)
imply the
if (£ n )
^
tpg(2,£*), with both
enough to
£*,
suffices to
check the generic
terms positive. For as in the proof of
local stability of the point £*.
starts close
it
—>
then £n
As defined
£*
in
Appendix C,
Lemma
this
10,
means
with positive probability: Since
(£ n )
cannot cycle, limit cascades and confounded learning are the only possible 'pathological'
(incomplete learning) outcomes in finite-action observational learning models.
5.2
Examples of Confounded Learning
We
show that confounding outcomes may
M
and confounded learning can occur.
T = 2 types. Types differ in their
preferences, but not their signal distributions. Type U is 'usual', preferring action a 2 in
state H, ai in state L, and ai for private beliefs below p u (£) = £/{u + £). The preferences
In the examples,
we have
=
exist,
2 actions
20
and
2u/3
U U t2u
Figure
f f t2u
Confounded Learning.
6:
=
Bounded Beliefs Example,
Based on our
with \ u
graph, the curves tp(l\H,£) and ip(l\L,£) cross at the confounding outcome
=
4/5,
v/2. In the
u
where
no additional decisions are informative. At £*, 7/8 choose action oi, and counterintuitively 7/8 lies outside
v and X u
the convex hull of A
eg. in the introductory driving example, more than 70% of cars may
merge right in a confounding outcome. The right graph depicts continuation likelihood dynamics.
left
£*,
—
of type
—1
V
are opposite: action a\ always pays zero, while a 2 yields payoff v in state
He thus chooses
in state L.
WLOG,
v
>
>
u
0,
so
a\ for private beliefs above the threshold
vNM preferences
>
are not exactly opposed if v
p
v
=
(£)
Bounded Beliefs Example Cont'd. The transition probabilities for
nowp u (l\H,£) = (3£-2u)/2£andp u (l\LJ) = {3l-2u)(3£+2u)/8£ 2 where £ 6
v
For type V, p (l\H,£) = (2v - £)/2£ and p v (l\L,£) = (2v + £){2v - £)/8£ 2
,
,
With bounded
beliefs,
7
in the intervals J^
=
dynamics nonessentially
[0,
2u/3) and
differ
confounded learning. The same remark holds
<
v,
no overlap
arises if
mHJ) = -^ + X^
X»
by
(10).
2u/3
<
and
mL
3
2£
U
2£
Figure 6 graphs these functions.
if
J^
=
[2u, oo)
and J^
So consider the dynamics
u.
,l)
We
=
X»
£,
=
where
[0,
for £
^m^+>?
U
€
The
[2u,oo),
in section 3
2w/3] overlap.
G (2u/3,
(2V -
2
for
£
thus precluding
{
can rewrite (11)
are
(2u/3,2u).
J% =
from those
because only one of the types ever makes an informative choice for any
For u
type
the absorbing basins complicate the dynamics.
two types take action 2 with certainty
respectively. If these overlap, then
+ £).
£/(v
u.
•
(2u/3,2v).
H and
2u):
mV+e)
8£ 2
a confounding outcome
as
{2v
(2u
If
u > v then £ maps (2v/3, 2u) onto
X U ,X V
u
(\ /4
.
+
-
(0,
oo),
Next,
Since the functions
ip
2v)
2u)
=
m
and so a confounding outcome
\ v /4)/ij;(l\H,£*)
<p t (l,r) = (3A /4 +
v
3\ /4)/ip(2\H, £*) and both are
t/
£){3£ £)(3£
positive.
exists for
generically differs from
<p t {2,t)
=
So confounded learning can occur.
are increasing, the system either starts in a cascade in
21
any
[0,
2u/3]
and thus
or [2u, oo), or starts
trapped
is
As
in [2u/3,£*] or [£*,2u].
probability mass
all
wrong herd
eventually concentrated at the endpoints, there can only arise only a
is
confounded learning
€
£
[^*,2u],
Just as in the proof of
[2u/3,£*].
E [£oo] =
if
£Q
and a correct herd or confounded learning
Theorem
2,
since (£n )
is
e
£
a bounded martingale and
both possible outcomes have positive probability in each
,
if
or
More
case.
generally,
with only a single confounding outcome, limit cascades must occur with positive chance.
The Basic Theory
5.3
We
• Theoretical Robustness.
comes can
exist,
Theorem
9
(a)
have shown by example that confounding out-
and confounded learning may
(Confounded Learning)
arise.
Assume
We now
there are
If the private signal distribution is atomless,
obtains with positive chance.
No one
T>
2 types.
then for nondegenerate specifications
and type proportions, confounding outcomes
of preference
finish outlining the theory.
exist,
and confounded learning
confounding outcome must occur with bounded
(b)
Generically, at any confounding outcome only two actions are taken.
(c)
Confounded learning
(d)
When confounded
is
is
generically robust to the~addition of craziness noise.
learning does not occur, a limit cascade arises,
almost surely correct with unbounded
(e) If
beliefs.
M > 2 with unbounded
beliefs,
or
£n
i.e.
—»
£
G
J,
and
beliefs.
if
the private signal distributions are discrete, then
generically no confounding outcome exists.
The
Proof:
first
point has been addressed by the example, which
Let us consider the equations that a confounding outcome
(6)
bounded
some actions may not occur
beliefs
are taken with positive probability at
to
M
-
Generically,
sign of
L
jj
—
1
Equality (11)
(c)
d
M
<fii(i,
{i\£)
as
£),
= ^
d£^(i\£)~
when
it
Assume that
1
=
Yl m =i i; ( rni\H^)
still
obtains
if
state independent noise
is
ip(m\£)
Any
=
£
L
i)
{i\£)^{i\£)
_
~
ffi(j|l)
- g(t|g
la
^2i^ii>(
M
with
M actions
<
Walras' Law)
m i\L,£) =
=
satisfy (A-4).
This precludes
all
=
,
The unbounded
beliefs
argument
22
is
a^(i\£)
__
~
1-
2.
sides.
The
-
a^f{i\£)
H
aiP {i\£) + {l-a)Ki
(pt{2,£).
but limit cascades, where
some action a m and confounding outcomes, where
.
M
added to both
H {i\e)
j>
^{i\£f
6 4o must
1 for
two actions a m
—
solve. First,
equals
(11) holds. Finally, only for nongeneric noise will <fe(l,£)
(d)
must
£*
equation in one unknown £ can only be solved when
H {i\£)^{i\£) '
not nongeneric.
Equation (11) reduces in (a
£*.
independent equations, since
1
at all at £*.
is
ip(m\£)
by now standard.
>
for at least
n
Jx
Figure
Continuations and Absorbing Basins.
7:
Example, but
that there
is
no
This
is
interior basin,
but in the second graph there
is
an
graph, preferences are such
first
interior basin.
graph, that with the basin, for any value of I at most two actions are in play. This
one of
M
>
2 types has
Observe
in the
second
the kind of example
is
M > 2 and confounded learning.
which can be used to construct models with
(e) If
Bounded Beliefs
based on the
here with one insurance and two extreme actions. In the
unbounded private
beliefs,
then for no £ G
(0,
oo) are
So confounding outcomes only appear
only two actions taken with positive chance.
in
degenerate models. Next,
v
v
p (l\H,e)-p (l\L,£)
p"(l\L,e)-pV(l\H,e)
A^
\v
is
a reformulation of (11), and so
if
FH
and
FL
are discrete, then the
RHS
will
only assume
a countable number of values, and confounding outcomes will generically not exist.
Remarks.
1.
While only two actions
confounding outcomes, generic models with
outcomes. For with bounded
This
will
2.
I*
beliefs,
M
>
2 actions can
only two actions
happen when there are absorbing basins
Being a fixed point, we can say
occur with positive chance at generic
will
little
may
have confounding
still
well be taken over a range of
for the insurance actions, as in figure 7.
about the uniqueness of confounding outcomes
— except that with discrete distributions, they are not unique when they
interval
3.
only
•
t
will satisfy (11)
FH
because
Our example posited two types with
vNM
•
some
around
preferences differ, then one can
Economic Importance.
light
t.
and
FL
exist (for
an
are locally constant).
different preference orderings over actions. If
show that no confounding outcome can
We now return
to our introductory example,
exist.
and shed
on exactly when one should expect to see our new confounding phenomenon.
The Driving Example Revisited with Unbounded
Houston (type U) drivers should merge
right (action a x ) in state
state L, with the reverse true for Dallas (type
23
V)
drivers.
Going
Beliefs.
H,
left
to the
Posit that
(action a 2 ) in
wrong
city yields
zero always. Getting
home by
the right lane a\
payoff vector of the Houston-bound
drivers,
it is (0, 1)
Claim
and
(u,0).
28
is
preferred, as
is
H
(u, 0) in state
The proportions
are
A^
and
=
it
has fewer potholes. The
(0, 1) in
and X v
.7
With a differentiable signal distribution and unbounded
point exists
if
We
Proof:
preferences are
show that
more disparate than
type frequencies:
ip(l\H, £) lies below -0(1 JZ^, £) near £
=
state L; for Dallas
=
.3.
a confounding
beliefs,
u/v
X u /X v
<
and above
it
<
v/u.
=
near £
and therefore by continuity the transition probability curves must at some
oo,
£ coincide.
Differentiating
V>(l|s, £)
near £
=
>
The
ipe(l\L, 0)
u
F
s
(£/(u
+ £)) +
X
v
[l
-F
merge
nearly
right in state
all
By Lemma
1,
contrarians
when they
there are
(£/{v
H than in state L.
many more
simple,
near p
6.
But otherwise,
beliefs is
(infinitely
are right than
(resp. left) for public beliefs
is
drivers are Houston-bound, then
what matters near extreme
nate, then
s
+ £))]
=
intuition for the existence of confounding points
beliefs. Clearly, if
will
iff
X
v
s
u
L
H
f (0)[X /u - X /v]. Since / (0) > f (0) by Lemma A.l,
X u /X v > u/v. The reverse inequality near £ = oo is similar.
yields ^(l|s,0)
ipe(l \H, 0)
=
=
how many
p
if
b) in
bounded
preference differences domi-
contrarians of either type exist.
are wrong. Thus,
=
to
uniformly true that more
more with unbounded
when they
b (resp.
it is
and common
state
beliefs) doctrinaire
more
merge
will
H than state L
right
(conversely).
CONCLUSION
This paper has explored and expanded upon the so-called herding literature.
We
hope
our analysis underscores a rich theory that ensues from attention to likelihood ratios and
their conditional martingale property in theoretical learning models.
• Related Literature.
theorem
We
for
We think it noteworthy that
Milgrom's (1979) convergence
competitive bidding also turns on the bounded-unbounded signal knife-edge.
must underscore that our complete learning
results are in
no way related to Lee
(1993), where a rich continuous action space effectively allows for a one-to-one
signals
-H-
actions.
same way
the
As roughly
foreseen by Banerjee (1992), this precludes herding in
as statistical decision problems do.
A
key touchstone of herding
inference of predecessors' signals as they pass through a
there
with
28
is
many
no
map
loss of generality in
lumpy
filter (like
is
of
much
the coarse
action choices).
our model with two signal values compared
to a
model
signal values.
By a payoff renormalization, this is equivalent to our standard payoff structure.
To quote from Lee (on page 397): From the standpoint of information revelation,
24
a sparse action
Experimentation Literature.
• Lessons for the
and S0rensen (1996a), or SSI, presents a
allel
different slant
between bad herds and incomplete learning
experimentation, as actions
learning
-f->
makes
clear that even
it
if
on this
field,
drawing a formal par-
optimal experimentation: observational
signals, as belief thresholds
«-»
actions.
SSI also
a patient social planner (unable to observe signals) could control
the action choices, he too would
In light of SSI, that
<->
in
Our companion paper Smith
succumb
to
bad herds, given bounded private
more than two actions
beliefs.
generically cannot be taken at a confounding
outcome speaks to a general property of optimal experimentation models. For instance,
SSI
published examples of incomplete learning, and
cites
in fact, given
beliefs for
Our
exogenous signal functions and just two
which more than two signals optimally
stochastic stability condition also
problems. Since informational herding
is
may be
have binary signals. 30
all
states, only
arise
And
nongeneric models yield
with equal chance in either state.
a useful tool for optimal experimentation
formally equivalent to single person learning, we
suspect that by focusing on the the stochastic dynamic process of likelihood ratios rather
than the intricacies of dynamic optimization, our stability program
offers significant hope.
For example, in the popular two-state, continuous-action learning models,
suffices to
it
verify that near a fixed point, the continuation posterior does not always have the
same
slope as a function of the prior belief after each observed signal. This condition clearly has
nothing to do with the particulars of the actual dynamic optimization. Indeed, this paper
has afforded this simple insight precisely because the optimization problem
•
Where Do We Go Now?
for
why
so trivial.
Smith and Sorensen (1996b) relaxes the key assump-
tion that one can perfectly observe the ordered action history.
(more plausible) reason
is
isolated contrary actions
While
might have
little effect,
motivated by deeper concerns. For absent martingales, the resulting analysis
different that standard rational learning theory,
and
another
this is yet
is
we
are
radically
forces one to think (a la Blackwell)
abstractly about information, and to delve deeply into the theory of urn processes.
A.
ON BAYESIAN UPDATING OF DIVERSE SIGNALS
The set-up (measures
The
s
fi
characterization of the
and distribution functions
Radon-Nikodym
F
derivative
s
,
s
= H,L)
H
of F
,
F
L
is
in
taken from §2.1-2.2.
Lemma
1
implies
set provides little
means
to
signals adds
to the
updating of the posterior distribution and the whole sequence of individuals
little
convey the private information. Consequently the infinite sequence of private
end up choosing the wrong action.
30
An additional one that has come to our attention
25
is
Kihlstrom, Mirman, and Postlewaite (1984).
may
:
Lemma A.l (Extreme Beliefs are Informative)
L
L
H
(a) F (p) < pF (p)/(l - p) holds for all p
(0, 1), and is strict when F (p) >
L
H
is weakly increasing, and strictly so on (b,b].
(b) The ratio F /F
<E
= dF H /dF L
Since /
Proof:
F H (p) = (
is
a strictly increasing function, we have
dF L (r) <
f(r)
f(p)
Jr<p
for
F L (p) >
any p with
F H (p) - F H (q) =
Thus, whenever
0.
f f(r) dF
L
0.
{r)
>
[F
L
(p)
Jf
T <P
dF L (r) = pF L (p)/(l - p)
F L (p) > F L (q) >
- F L (q)]f(q) >
0,
[F
(A-2)
we have
L
(p)
- F L (q))F H (q)/F L (q)
Jq
where we have used (A-2).
It
immediately follows that
FIXED POINTS OF MARKOV-MARTINGALE SYSTEMS
B.
The
general framework that
evolution of the likelihood ratio
Given
ip{-
F H (p)/F L {p) > F H (q)/F L (q).
1
•)
•
ip(-
•
4>
1
is
a
finite set
M x M+
:
£) is also
and
ijj
we introduce here
(£ n )
Ai, and Borel measurable functions
->• [0, 1]
is
<p(-
,
•)
:
a probability measure on
For our application, ip(m\£)
Next, for any
is
and <p(m, £)
£,
to,
the
M. x K+ —¥ K+, and
satisfying:
M
for all £
€ R+, or J2 meM ip{m\£)
jointly satisfy the following 'martingale property' for all £
with likelihood
not confined
31
over time viewed as a stochastic difference equation.
J2i>(rn\£)
P:R+
includes, but
1.
G M+
p(m,£)=£
(A-3)
(
the chance that the next agent takes action a m
is
=
when
faced
the resulting continuation likelihood ratio.
B in the Borel a-algebra B on R+ =
[0,
oc), define a transition probability
xB->[0,l]:
P(£,B)=
^H')
J2
m\<p(m,e)eB
Let
31
(£ n )%Li
be a Markov process 32 with transition from
£n
i->
£ n+ i
governed by P, and
Arthur, Ermoliev, and Kaniovski (1986) consider a stochastic system with a seemingly similar structure
But their approach, differs fundamentally from ours insofar as here
— namely, a 'generalized urn scheme'.
it is
32
of importance not only
Technically, £ n
:
QH
how many times a given action has occurred, but exactly when it occurred.
K+ is a measurable Markov process on (Q H ,£ H ,v H ), where E H and £ L
->
correspond to the restriction of sigma field S to Q H and fi L respectively. Despite a countable state space,
standard convergence results for discrete Markov chains have no bite, as states are in general transitory.
,
26
<
El\
Then
oo.
E[en+1 \e l ,...,£n
By
a martingale, true to the above casual label of (A-3):
(£ n ) is
]
= E[£n+1 \£n ]=
we have
the Martingale Convergence Theorem,
Theorem B.l
(Stationarity)
m G M.,
—»
obtains,
and
£n
£00
If £
]^vH4Mm,4) = 4
f tp(£ n ,dt)=
—>
£n
(p(m,£) and £
>->
almost surely, then for
all £
=
£<*>
i->
lim^oo
a.s.
ip(m\£) are continuous for all
G supp(£ 0O ), stationarity P(£, {£})
=
1
i.e.
=
ip{m\£)
or ip(m,
£)
=
for
£
m
all
Indeed, (£n ) must also converge weakly (in distribution) to
Markov
chain, its limiting distribution
The
Futia (1982).
G
M
(A-4)
Since (£n ,mn )
£00.
convergence then implies that the invariant limit must be pointwise
a.s.
lines,
the continuity assumptions
As we wish
are subtly hard-wired into the final stage of Futia's proof of this fact.
away with
we
continuity,
when
also a
is
intuitively invariant for the transition P, as in
is
While Theorem B.l admits a proof along these
invariant.
exactly
>
£n
That (A-4)
establish an even stronger result.
—
neither ip(m\£) nor p(m,£)
Theorem B.2 (Generalized
is
to
do
violated for
m
£ is zero suggests
Assume
Stationarity)
that the open interval I
C R+
has
the property
W e I 3m G M
>
3e
Then I cannot contain any point from
that there exists £ G /nsupp(4o). Let
m
G
M with
ip(m\£)
>
e
J
=
and <p(m,£)
<£
£
!->
Proof:
is
that £
G supp(^ 0O ).
an
m
£)
—
Theorem B.2
Finally,
£ are both
yields
it is
>
0,
and suppose
(*)
a contradiction
for
£ n +\
J
& J with chance
in the
£e
(•), for all
next period
£,
Then for each
m
G
M,
J
is
J, there
eventually
at least
either £
most
at
That
e.
1
—
e.
h->
tp(m, £) or
or the stationarity condition (A-4) obtains.
such that £ does not satisfy (A-4) and both £
£ 1-4 ip(m\£) are continuous,
and <p(m,
e
eventually in J. Hence, £ cannot exist.
discontinuous at
is
there
If
is
Assume
ip(m\ £)
>
almost surely eventually exits J. This contradicts the claim that with
(£ n )
positive chance (£ n )
Corollary
£\
Since £ G supp(£oo), £n G
J.
the conditional chance that the process stays in
So the process
-
(£-e/2,£+e/2)nl. By
with positive probability. But whenever £n G J,
is,
\<p(m, £)
e,
the support of the limit, £00.
Let / be an open interval satisfying (*) for e
Proof:
exists
>
ip(m\£)
:
then there
is
an open interval / around
bounded away from
0.
This implies that
an immediate contradiction.
obvious that the corollary implies Theorem B.l.
27
i->
£ in
(f(m, £)
and
which ip(m\£)
(•) obtains,
and so
STABLE STOCHASTIC DIFFERENCE EQUATIONS
C.
In this appendix,
We
ence equations.
There
first
then use
develop a global stability criterion for linear stochastic
it
on
to derive a result
local stability of
work
cited as the first
is
Kifer (1986)
in this area, Furstenberg (1963)
a classic article, and
— state-dependent transition chances
many dimensions an added
our needs, and extrapolation to
for
is
Bellman
level.
a modern textbook. Ellison and Fudenberg (1995) have independently ob-
is
tained related 'concrete' results, yet not as flexible
is critical
differ-
a nonlinear systems. 33
a small extant literature that treats models like ours at an abstract
is
(1954)
we
benefit of our
different approach.
we add an
Further,
analysis of rates of convergence. There
Overall, the literature
aimed
is
n
to a fixed point, such that 9~ £n
(£ n )
happy
a small literature which
problems we discuss here, but we have not been able to exactly recognize our
treats the
results.
is
^
C f° r some £
Lemma
dimensional linear case we treat in
discuss later on,
—>
9~ n
upper bound 9 such that
to settle for an
an exact rate 9 of convergence of
at determining
C.l,
we do not determine the exact
£n
We, on the other hand, are
0.
—>
for all 9
>
we do obtain the exact
In the one-
9.
we
rate, but, as
rate in higher dimensions.
• Linear Stochastic Difference Equations.
Fix
G M, p G
a, b
and
[0,1],
let
e R. Let
£
{a£n _i
with probability p
with probability
6£ n _i
1
(A-5)
—p
sequence of stochastic indicator functions defined by yn
=
=
otherwise, where (a n )
random
Markov process
define a
Lemma
Almost
(b)
If 9
£
<
No
1
(a)
Let
(b) If
33
We
Law
—
ab£
n
for all £
i.i.d.
then
—
>•
,
=
then
£)
>
>
all 9
uniform-[0,
1]
aVn b^~ Vn £n -i, where (yn )
1
when an <
is
= (M^I&I 2 ^)"
\£
and yn
a
=
variables.
=
Define 9
—>
then there
0,
\£ n
\
result follows
from
with positive chance jumps to
a.s., all
We
whenever
In particular, £n
p,
is
|a|
almost surely
1-p
p
.
|6|
if
9
<
1.
a positive probability that
€ No-
Numbers, the
(^ n )
9.
around
ball
£Li2/*- Then
are coining terms here.
Pr(lim„_+00 £ n
for
any open
is
of Large
0,
—>
provided £
Yn =
Since £ n
0.
9~ n £
and No
all n,
the Strong
^
surely,
for
Proof:
ab£
a sequence of
is
£n
as:
C.l (Stability of Linear Homogeneous Systems)
(a)
£n
This can be recast
(£ n ).
£
but
call
€
finitely
many terms
lie
Since
\.
|a|"^|&|
s_^~a
at once,
inside
'
Yn /n
—>
9
-> p
by
a.s.
and stays
any open
a.s.
ball
there. Let
No around
a fixed point £ of a stochastic difference equation locally stable
a small enough neighborhood about I. If Pr(lim„_>0o t-n = ^)
J\f(,
£ is globally stable.
28
if
>
some
for
Pr
or
a.s.,
k.
Remark
4
=
4 G A/o}) >
stays inside A/o starting at that 4- WLOG k =
|4 G A/o})
With
invariant.
4=
if
^ ||4||/IKm||A/o
>
So Pr ({w G Q"|Vn
1.
scale invariant dynamics,
time invariance allows us to conclude that
First,
1.
positive probability
then
w
fi
So with positive chance, (4)
dynamics are time
since
e
(\J keN f] n >k{^
any
4
fc,
G A/o
£
G
A/o for all
Second, the equations are linear, so that
£~m-
we have now a smaller
for all n. So,
the system started in the small ball,
would remain
it
ball
around
in the large ball.
will do.
if ||4||
1
D
n with
<
114/11
zero, such that if
But
finally notice
that from any point of the original large ball, one can reach the inner ball in only a finite
number
of steps, which occurs with positive probability too.
Remark
Observe that
2.
but their geometric
p)b,
mean
reformulate the criterion by
first
Remark
taking logarithms, as inplog(|a|) + (l— p)log(|6|)
from the theory of differential equations.
3. It is straightforward to generalize
continuations,
i.e.
where yn has arbitrary
finite
Lemma C.l
support.
The
to the case of
<
It is
—
(1
we
If
then
0,
common
through. Indeed,
4
let
G
n
M
one half of the lemma goes
and assume
A4-i
f
[£4-i
A
and
B
are given real
34
more than two
analysis for multidimensional
also of importance, but unfortunately in that case only
where
+
logarithm to enter when translating from difference to differential equations.
for the
is
of the coefficients pa
that determines the behavior of the linear system.
this is reminiscent of stability results
4
mean
not the arithmetic
it is
n x n
Vn
=
ifyn
=
if
matrices.
Let
1
\\A\\
and
denote the operator
\\B\\
Lemma C.l goes through, with nearly
unchanged proof (using \\AB < \\A\\ \\B\\): If 9 > 9 = \\A\\ p \\B\\ p then 9~ n £n *A 0, i.e.
4 converges a.s. to zero at rate 6. As this part of Lemma C.l is the only result applied in
norms of the matrices.
Then the
following half of
l
\
,
\
the sequel, our local stability assertions will also go through in multidimensional models.
Call 6 a convergence rate of
not very tight, for
9"
G
[9', 1].
->
C.l.
the projection onto
34
That
0~ n
4
->
at the rate
for all 9
9',
then
impractical, for in multidimensional settings
is
Lemma
We
t
(4) converges to
In general,
it is
Consider for instance the case where
time.
£* if
it
>
9.
This definition
also does so at
its
we do not have the converse
is
p||
=
B
the projection onto a linear subspace, and
orthogonal complement, then 9
=
1,
but
is a.s.
reached in
prefer to maintain the possibility of calling 9 a convergence rate, even
is,
rate;
possible to find convergence rates smaller than
A
su P|l|=1 \Ax\.
29
is
any rate
Perhaps we ought to narrow down the convergence rate to the infimum
however, that
part of
if
4
9.
is
finite
if it is
not
the tightest such. See Kifer (1986) for more precise results for linear systems.
• Nonlinear Stochastic Difference Equations.
Consider the revised process
defined by:
{V?(l,4-i)
with probability V(l|4-i)
</?(2,£ n _i)
with probability V(2|4_i)
where transitions are independent:
=
£n
(p(l,£ n -i)
iff
an <
We
ip(l\£ n -i).
care about the
fixed points £ of (A-6):
=
<p(l,i)
Theorem C.l (Local
£ of (A-7), each ip(i\-)
Li
=
<p(2,i)
(A-7)
i
Assume
Stability of Nonlinear Systems)
>
is
continuous and
)
<p(i,
is
Lipschitz,
35
some open
obtains, then for
forever remain inside
Proof:
WLOG
ball
ip(l,
.
•)
with Lipschitz constant
a neighborhood N{£) of
in
£.
£
inside
lies
whenever
around
(A-8)
1
£n
it,
(£ n ) will
—>
£, it
converges at the rate
Lemma C.l
applies to our original non-linear system.
— and thus L\ <
Choose M{£) small enough so that
for
L\l}f p <\, xl)(l\£)>p, and \\<p(i,e)-<p(iJ)\\<Li\\e-i\l
for all £
e M{£). Fix
G M{£). Define a new stochastic process
£
I
where
(y n )
is
our earlier
and ln € M(£)
yielding £n
As
for all
e M{£)
ip(l\£)
for all
ify„
indicator sequence.
Lemma
i.i.d.
n with
£
£n
£ for
—>
The
i
=
'
[0,1],
1,2
=
with £
holds
£
e M{£)
||^n
at x, then
/
—
:
Rm
M{x) such
is
->
Rk
that
is
is 9,
l
C.l then asserts £ n
£\\
>
=
-
\\£ n
since for
=> £ n
1
l\\
£n
—>
=
:
Lipschitz with any constant
£ a.s.,
realization of (an )
(p(l,£n _i),
for all
£
any 9 >
n
\\f(x)
L >
-
f(x)\\
||D/(x)||.
30
<
L\\x
-
and the Lipschitz
(in this realization).
So
with positive probability.
a small enough neighborhood
9,
Lipschitz at the point x with Lipschitz constant
Vx 6 Af(x)
—>
the linear process (£n ) majorizes the non-linear one
any such (an ) realization. In summary,
function /
neighborhood
£,
e N{£), we have yn
Finally, the rate of convergence
35
for
6 N{£). In any
positive chance given £
n and
> p when
=
_'t= (^(In-i-i)
property yields the majorization
—>
(£ n )
some p €
1
and
given,
(£n):
9.
by a linear stochastic difference equation
continuous, (A-8) obtains
is
with positive chance
£
locally
>
£, if
Also,
£.
and then argue that
< L 2 As
L\
—>
while £n
it,
= Li^Ll- mi) <
around
we majorize (A-6)
First,
of the form (A-5),
£n
that at a fixed point
If the stability criterion
.
6
—
and
i
x\\.
If
/
is
L >
if
there exists a
continuously differentiable
Afp (i)
exists for
wherein
it is
p
which L\h\
dominated by
<
Whenever
9.
6=
true with
is
If each
|p/(l,£)|*
when
i.e.
£
and we care about
function,
ip(i,
{1| ' )
•)
its
eventually stays in
p
p
(1_,, (1| ' ))
C.l.
through
In that case <p(m,£)
matrix derivative Di<p(m,
Then
1.
p {£),
1-
carries
£) at
then must assume that the operator norm ||Z?/^(m,£)|| (which
eigenvalue) mis less than
<
'
^(2,*)|
Lemma
by
N
also continuously differentiable,
is
a vector (for several states).
is
(£n )
how Corollary C.l
Finally, let us briefly explain precisely
dimensions,
£,
which converges at rate L 1 Ll~
(£n ),
Corollary C.l (C 1 -Local Stability)
then Theorem C.l
£ n ->
in several
a vector
is
the stationary point.
is
the
same
the proof goes through, largely as before.
We
as the largest
We
must
also
be more careful with the dominance argument. Rather than choosing a constant L\ larger
than
\y>i(l, £)\,
with
all
we have
to choose a matrix
1
rules
finite
s
number S
for each state,
fi
fix
one reference
and use
state,
it
to define
Rather than the simple
open
[0, 1]
36
In
into closed subintervals,
We
convex polytopes.
Theorem B.2 (and
its
proof)
a single action a
it
we need
to refer to the
Rs_1
unit simplex in
to the reader to
it
be impossible to
will
in the limit. So, even with
learning.
But while we do not get
and
Jullien (1991),
correct action
In the
is
same
we
ponder the optimal
open
intervals /
and J
full
statistically distinguish
beliefs,
we cannot
between these
possibly get complete
learning, in the terminology of Aghion, Bolton,
chosen optimally.
vein,
with more than two states, the long-run
state of the world has
The exact formulation
is
optimal in a given state.
ties of
BHW
In that case,
may
of
we
occur,
when the
two or more optimal actions, and there are unbounded
learning will obtain, but
is
unbounded
will arise if there are
get adequate learning: the limit beliefs are such that the
whereby more than one action
(1996b),
leave
optimal in two states of the world, which
is
two states
Harris,
we would now have a
balls.
fewer actions than states,
36
we
would become notationally cumbersome to write down.
notation.
full
Given pairwise mutually absolutely
of states.
But the optimal decision
sliced into closed
If
and
likelihood ratios, each a convergent conditional martingale.
partitioning of
as
£),
MORE STATES AND ACTIONS
can handle any
continuous measures
5—
with the same eigenspaces as D£<p(m,
numerically larger eigenvalues.
D.
We
A
beliefs,
will not necessarily observe that all individuals take
what constitutes full-support
also slightly non-trivial.
31
beliefs,
which
is
true
one
outlined in Smith and S0rensen
particular action. In short, the overturning
But
in the long run.
the learning
The
lemma will
unchanged with a denumerable action space.
Lemma
finite partition of [0, 1] in
thus a countable set of posterior belief thresholds f
threshold functions
p just
The convergence
Theorem
result
.
>
we could
b
we
Lemma
In this way,
we could
minimum
substitute a
was well-defined because there were only
m so that pm
is
unbounded
With
3.
obtain without any qualifications provided a unique action
still
and
teriors sufficiently close to
1,
b
many
such that
beliefs case,
arbitrarily close to 0.
is
comes to Theorem
it
finitely
enough to
close
action threshold pi by one that
Complications are more insidious when
results
model are preserved.
complication arises, as our choice of
the "bounded away" assertions hold. Similarly, in the proof of
all
3 will yield the
depend on the action space being denu-
2 does not
instead just pick
and
get a countable partition,
properties of the
beliefs proof, a technical
such that pm(£)
actions. Otherwise,
2,
37
The martingale
as before.
bounded
In the
m
the actions that are tied
adequate.
is
Rather than a
the least
among
again, since the individuals will eventually get the optimal payoff,
analysis also goes through virtually
merable.
fail
for then the overturning principle
M — oo,
both
optimal for pos-
is
is still
valid near the
extreme actions. But otherwise, we must change our tune. For instance, with unbounded
may
beliefs, there
made such
infinite
[r
strength. Yet this
m _i,fm
is
shrink to a point, which robs the overturning argument of
]
not a serious non-robustness critique, because the payoff functions
of the actions taken by individuals
Under
upon a
Lemma
E.l
(a)
Let Xqq
J2 n
xn = oo
37
is
with the trembling formulation, where we
support of the tremble from any
finite
I.
SOME ASYMPTOTICS FOR DIFFERENCE EQUATIONS
E.
(c)
must then converge!
noise, the only subtlety that arises
shall insist
(b)
sequence of distinct optimal 'insurance' action choices
that the likelihood ratio nonetheless converges. This obviously requires that the
optimality intervals
its
an
exist
Let x, €
=
if
(0, 1)
and x n
<
a
1,
=
£ X
n
for
a/n
n
<
all
+
oo
i,
This
may mean
that
we cannot
define
if
a
>
1,
with
Y>x n
then
,
Xn = (1 — x{)
summable.
b n , with (b n )
LetX^ > 0. Then^iXn-Xoo) <
and x < x < 1 for all i,
// Xoo >
{
and
n
n
if
(1
- x n ).
Xn =
J2
|6„|
0(n~ a ), and
< oo.
so
38
= oo = £»(*»-*«>) =
<oo^> Xoo » 0.
andJ2 n nx n
^ n nx
•
Then
£ X < oo
n
•
oo.
necessarily well order the order the belief thresholds, nor as a result
the actions.
38
Co
Say x 3>
>
c (eg.
c with x(-)
>
x
»
is
x boundedly positive) of a function or random variable x if there exists some
Likewise x < oo (or x boundedly finite) if x(-) < Co for some Co < oo.
Co always.
32
Proof of
(a):
39
The binomial theorem
=
yields d n
- a/n —
log(l
—
bn )
—
log(l
l/n) a
=
— a/n + Cn), for c„ = 0(l/n 2 ). By the mean value theorem,
\d n < \b n + Cn|/min(l — a/n - b n 1 — a/n + c„) < 2\b n + c„| for large enough n (since
a
then l-a/n-bn > 1/2 and (1 - l/n) > 1/2). So £ |6 n < oo iff £ |d„| < oo, and
— a/n —
log(l
—
bn )
log(l
,
\
|
An =
n
°
exp(dj
The boundedly
Proof of
+
•
•
•
+
dn )
= D„n- a
finite parts follow
Dn
<
D
<
<
= 0(l/n a
So A'„
oo.
).
with such an exact bound.
We've shown before that
(b):
D
<
where
,
—
(1
— x2
£i)(l
•
)
>
•
1
—
(xi 4-
+
x-i
•
if all
•)
•
x
{
lie in (0, 1). This means that ^2 k>n x k > 1 — X^/Xn. Summing both sides over n and
appealing to £ n Efc>n x * = E na; „ yields £nx n > EU ~ ^oo/A^) > 52(Xn - *«>)
Conversely, the identity E n (^n-i — -^n) = J2(Xn — -^oo) follows by rearranging terms
of partial sums to n = N, and taking limits as N —> oo. Our sought for result follows from
= X>( A'"-1 -
J2 nx n
Proof of
x E
[0,x].
So
<
Since x
A'oo
=
there exists a(x)
1,
-xi)(l
(1
X>(Xn _l - Xnj/Xoa = Y,( Xn
<
-x
2
>
with
1
X^/X^
~
— x > exp(— a(x)x)
1
£
)---> exp(-a(x)^ k x k ) > exp(-a(x)
for all
D
fcx fc ).
fc
Dynamic Systems) Let g be nondecreasing, with g(0) = 0. 41
= -g{x) and y n+l - y n = -g{yn with x(0) >y >0, then x(n) > yn
< 1 exists, then (l—g'(z))z = —g(z) andy > z(0)—g(z(0)) => yn > z(n)—g(z(n)).
Lemma
Ifx
(a)
40
(c):
Xn )/Xn _i
E.2 (Vanishing
.
)
(b) If g'
Proof of
(a):
By
>
induction, assume x(n)
yn
If
.
assume x(n + l) < yn With x decreasing, there
.
+
x(n
exists
t'
>
1)
€
(n,
>
yn
yn +\, we're done. So
n + 1) with
x(t')
=
yn
Then
.
/•n+1
=
x(n+l)
Proof of
x(t')-
(b): If
yn +i- Since z
for all t
z.
As g
+
z(n
39
40
£"
We
If
<
41
is
1)
also
is
x{t')-g(x(t')){n+l-t')
a least n with z(n)
and z — g(z)
+
1],
monotone,
g(z(n
+
1))
l-x- e~
Since £'(x(a))
<
is
whereupon
=
are very grateful to
ax
=
£(x)
0.
-
not, there
<
6 [n,n
>
g(x(t))dt
,
(1
—
z{n)
increasing in
>
z(t)
yn
g'(z(t)))
-
g(z(n))
x(t')-g{x{t'))
— g(z(n)) <
y n and z(n +
we have
—g(z(t))
z,
z(t)
f
=
n+l
/
—g(z(t))
< —g(yn
0, it follows
=
of Purdue University (Statistics)
and f (1) < 0, and on [0, oo), £(x)
that x'(a)
>
0. Finally,
a(x)
We are very grateful to Anthony Quas of Cambridge University
33
)
is
yn+ i
— g(z(n + l)) >
> yn +i = Vn — gdjn)
l)
for t
is
increasing in
< n+
Then
1.
)
for the proofs of (a)
^
=
y n -g(y n )
—d (z(t) - g{z(t))) dt < yn - g{yn
Herman Rubin
then f(0)
=
— once again because z — g(z)
z(t)
+
>
for
x
^
x(a) e
=
yn+1
and
(b).
(0, 1),
as
the inverse function to x(a).
(Statistics) for result (b)
and
its
proof.
—
OMITTED PROOFS AND EXAMPLES
F.
• Fact: Fair Priors
L
as
not 50/50)
is
WLOG.
is
As alluded
in §2, unfair priors (i.e. states
Lemma
equivalent to a simple payoff renormalization. For
fm u
H (a
m +
(1
)
2
H and
is still
valid,
key defining indifference relation
refers only to the posterior beliefs, while the
it
r
- fm )u L (a m = fm u H (a m+l +
)
)
- fm )u L (a m+l
(1
(A-9)
)
implies that
,,
posterior odds
l-fm
—
=
=
u H (a m )
—
=
—m --
u L {a
rm
Since priors merely multiply the posterior odds by a
f ,...,fM are
all
unchanged
we merely multiply
if
)
u H (am+ i)
z-.
u L (a m+1 )
common
all
constant, the thresholds
H
payoffs in state
constant. Likewise, constants added to payoffs in any state do not affect any
by the same
f{.
The Unconditional Martingale: Proof of Lemma 4.
Given the action
history {mi,
,m n _i}, the conditional expectation of the next public belief is (by the
*
.
.
.
Markovian assumption),
E[q n+ i
qn ]
|
^
= QnYl P(m\H,qn ) + (1 - qn
P(™\L,qn ) p(m|L ^
p{m{Lqn)
i + £"
l + tn
meM
m£M
p(m\H,q n
p(m\H,q n
qn p(m\H,qn + (l-qn )p(m\L,qn
= qn
p{m\H,qn
r—
n p{m\H, q n + (1 - qn )p(m\L, qn
q
meM
)
)
)
E
k
——
Absorbing Basins: Proof of
Lemma
that
)
3,
\p
m -i(£),pm (l)]
is
)
)
)
Lemma
an interval
for all
Since
6.
Then Jm
£.
pm (£)
is
increasing in
m
the closed interval of
is
by
all £
fulfill
Pm-i(£)<b
Then
F
)
interior disjointness
(pm(£))
=
1 for all £
is
obvious.
p m {£)>b
and
Next,
int (Jm )
if
e int(Jm ). The individual
updating occurs; therefore, the continuation value
With bounded
one of the inequalities
beliefs,
simultaneously satisfy both. As
have JM
=
[0, £]
Finally, let
and Jx
m
2
>
=
mu
Pm
2
[£,
Lemma 3
oo],
yields
p
where pM -i{£)
> Pmi {£\)
^
choose action a m
is a.s. £,
=
(£)
=
b
and
a.s.,
and
and so no
as required.
some
and Pm(£)
pi(£)
=
=
.
> Pm
2
-l(£2)
£,
but no
1 for all £,
b define
Jm2 Then
>b>b> pmA^)
34
F H (pm -\(£)) =
then
in (A-10) holds for
with £ x G Jmi and £2 G
-l(£l)
will
(A-10)
<
£
<
£
might
we must
£
<
oo.
and so
£2
<
£\
because pm2 -\
With unbounded
by (A-10). By
=
beliefs, 6
Lemma
strictly increasing in
IS
and
—
at the point £ yields p(m\£)
=
(p(m, £)
equality
£ implies
FH
if
G supp(£oo) with
< F H (pm (£)-) <
£
1,
£
p(m\£)
L
=F
{pm (£))
=F
L
=
£.
so
FH
0,
>
Assume
m satisfying pm (£)
(p m (£))
>
0.
From
>
Since F H y FSD
0.
=
(p m (£))
WLOG
a neighborhood of
in
> F H (pm _
(3), ip(m\£)
=
1
£ for £ near
is
us that
pm
is
F
Indeed,
Case
2.
FH
follows
(p m (£))
£.
on an interval
£
[£,
F
L
=
(p m -i(£))
L
is
=
H
if F
(p m
B.2, £
by
Lemma
Then
0.
1(c), this
is
H. Then
exist a point
some
for
m
we have
some
pm (£) >
and since
There are two
is
po{£)
=
<
6,
the
0.
possibilities:
F H (pm (£)) - F H (pm -i(£)) >
bounded away from
(£))/F H
which
(p m (£)),
is
in this
€ supp(£ 0O )
e for
neighborhood,
bounded away from
also
co(supp(F)), and so
Lemma
1
guarantees
for £ near £ (recall that
therefore cannot occur.
=
has an atom at p m _i(£)
for
k,
{pm _ l {£))-
some
and places no weight on (b,pm (£)].
6,
and pm _ 2 < pm -i, that
Therefore, ip{m
+ T]),
and
this criterion,
,
bounded away from F H (pm (£))
from F H {pm _ l (£)-) =
neighborhood of
ip(m\£)
and
FH
L
0.
B.l, stationarity
So we may assume F H (pm _i(£)-) =
£.
in the interior of
By Theorem
This can only occur
It
pm (£)
(p m (£)) exceeds
continuous).
=
(£)).
p(m\H, £)
= £F
L
£
Thus, £ G Jm as required.
1.
the state
6 is well-defined.
while (4) reduces to (p(m,£)
£.
=F
(p m -i(£))
Here, there will be a neighborhood around £ where
e
meets
£
6 Jm
proceed here under the
first
By Theorem
£.
£
so that individuals will strictly prefer to choose action a m for
Next, F H (pm (£)) >
Case 1. F H
some
£
We
1.
private beliefs and a m+ i for others. Consequently,
least such
and
1
Suppose by way of contradiction that there
Assume
J.
>
(pm (£))
(p m (£))
Next abandon continuity.
£
or (p(m,£)
m such that
only possible
is
m=
are continuous in
ip
= and pm = 1 for
= oo, or m = M and
Hence, p m _i
1.
Proof of Theorem
simplifying assumption that p and
FH
=
only happens for
3, this
• Limit Cascades:
consider the smallest
6
£.
rj
>
l\£)
0.
-
and <p(m
On
1,£)
F H {pm _ 2 {£)) =
-
bounded away from
£ are
the other hand, the choice of
and ip(m, £)-£ are boundedly positive on an
interval {£-rj\
for all £ in a
£],
for
m ensures
some
77'
>
that
So,
0.
once again Theorem B.2 (observe the order of the quantifiers!) proves that £ £ supp(4o).
• Overturning Principle: Proof of
signal
on must
Lemma
7.
If
n optimally chooses a m
,
his
satisfy
1
~ Tm ~ > 'l(h)g(a >
n
X
)
action a m with probability
all
JE(
l
(A-ll)
T
I'm— 1
Let E(/i) denote the set of
-^
m
signals
1
dp*
an that
satisfy (A-ll).
(resp. J^,
h)
35
H
gdp,
)
Then
in state
H
individual
n chooses
(resp. state L).
This
m
yields the continuation
=
£(h, a m )
Now just
£{h)
H
and use inequality (A-ll) to bound the
cross-multiply,
Theorem 3.
and Lemma 7. We
* Herds: Proof of
combine Theorem 2
'
r
then a herd on a m must
With bounded
shall prove that
when the
integral.
we need only
private beliefs,
limit value £
Jm
Notice that £ G
arise in finite time.
hand
right
implies
Jm
in
is
pm (£)
>
,
b.
Consequently, (2) yields
~f
fm
1
>
p
^
-
1
>
~ _m
rm
*
b
— fm -i)/fm _i for all
£ G Jm and so the closed interval Jm lies in the interior of [(1 — fm )/fm (1 — fm _i)/fm -i).
Therefore, whenever £ n —> £ G Jm we have £n G [(1 — fm )/fm (1 — fm -i)/fm _!) for n > N
where the
from
strict inequality follows
b
>
<
Similarly, £
1/2.
(1
,
,
,
N big enough.
and
Now
beliefs.
—>
2 asserts that £ n
Lemma
by
the overturning principle, only action a m
unbounded
posit
Theorem
By
,
Assume
WLOG
and so
a.s.,
whenever any other action than clm
7,
that the state
£ n is eventually in
we
taken,
is
• Calculating the Chance of a Correct Herd.
given £Q
=
1,
let
C
is
H, with a^ optimal.
[0, (1
—
exit that
In the
fM-i)/^M-i)- But
neighborhood of
bounded
{2u,2u/3}.
and by the the reasoning
a.s.,
(Such a tight prediction
in §3
about
1
=
+
ir(2u/3)
(1
— 7r)(2u)
Smooth
* Cascades with
then implicitly defines
Signals.
= £\ =
1
because
-k
For a discrete
jump
<
derivative <^_(l,2u)
odds
F L (p(£))/F H (p(£)),
Since ip{l,£)
0.
which
is
decreasing
\
<
2u.
1
<
2u.
\£n
whenever 2u/3 <
into an absorbing
basin, for instance [2u, oo) in figure 4, simple graphical reasoning tells us
left
example,
this
I
identity
2 tells
clearly impossible with cascades.)
is
Lebesgue's Dominated Convergence assures us that .E^oo H]
The
0.
example,
beliefs
Theorem
a correct herd happen with chance n in state H.
us that a limit cascade arises
supp(£ 00 )
taken after period N.
is
we need the
= £F L (p(£))/F H (p(£)), the private beliefs
by Lemma A.l, must be more than unit-
The trick is to choose smooth but 'nearly' discrete private signal distributions.
Assume that [i H is Lebesgue measure on [0, 1], and that [i L satisfies dfx L /d/j, H = g, where
elastic in
g'
a,
>
0,
£.
g(0)
>
0,
and #(1) <
and has support
+
[1/(1
the belief distributions are
The
oo.
belief p(a)
g(l)), 1/(1
FH
(p)
=
+
f*
g{0))}.
=
1/(1
+ g(cr))
in state
Given the inverse a(p)
.da and
F
L
(p)
= j\
H
=
is
decreasing in
_1
^
g{a)da.
((!
-p)/p),
Now
action a 2
We
thus only
)
requires that £g(a)
need
tp e (l,u/g(0))
<
=
and since g(a)
u,
1
-
#(0)(1
<
g(0),
- #(0)W(0).
36
we have inf^)
It suffices to
=
u/g(0).
choose #'(0) small for fixed
chance
1
O
1
J2 2u/3
2/3
2/5
2u
Ji
H
L
Figure 8: Nonmonotonicity. The left graph exhibits F and F that work in the Cascades
of
their supports, they mimic the
'nearly'
having
atoms
at
the
edge
with Smooth Signals Example,
actions
boundedly
informative,
and so a single decision can
information,
are
effect in BHW: With lumpy
graph
shows
how
the
corresponding
continuation functions
into
The
right
a cascade.
toss all successors
for the likelihood ratio are
£
g(0)
no longer monotonic.
42
Figure 8 gives such an example.
(0, 1).
Proof of
• Putative Herds:
and assume we are not yet
Lemma
Let £n G Im
8.
C
Absent a cascade
in a putative herd.
Q, with
(£ n
WLOG m >
6 Jm
C Im ),
1,
at least
two actions are possible. Then either
a)
Jm y^0 and action a m
b)
an action
c)
actions dj (j
<Zj
(chance ^>
For
if
not
After a m
> m)
occurs with chance 3>
0), £ n +i/^n
(b, b)
taken with chance 3>
are sufficiently
3>
>
and e
0,
so in
1;
then a m
(a) or (b),
G
exists b*
< m)
(j
is
,
a m+1
more
whereupon i n+ G
0,
\
than
likely
boundedly many
is
in
i
>
'-
Now, a putative herd
inductively help establish
= e\
•
Ij
•
•
(some j
M
-e*
a m occurs only
(2a
F L (b*)-F L (b + e)
F H {b*)-F H {b + e)
F L (b*)
>
F H {b*)
< m)
Lemma
»
starts at once with chance
Lemma
8:
There
in k m steps with
is
e*m
>
p <
some
b*,
and a m _i
7;
taken
is
j
or
< m.
So there
iffp
< b + e.
as e I
1
A. 1(a) and
.
(b).
in case (a), while (b)
and k m >
chance e*m
Then
and
(c)
such that £ transits from
set k*
=
k^
+ k*M
+
,
and
(by the conditional independence of the private signals).
r.v.,
the
mean time
By the
9.
alternate formula for the
to exiting from a temporary herd
= (2a + 5)/10 for a e [0,1/4], g(a)
13)/10 for a e [3/4, 1].
Namely, g(a)
+
Lemma
when a m
steps, (£ n ) enters Ij,
for private beliefs
Temporary Herds: Proof of Lemma
of a positive
42
that
by
taken with boundedly positive chance.
where the monotonicity and inequality follow from
£*
< m)
a, (j
Ij,
,
4+
Im into
or
0;
37
=
(18a
+
— that
is,
mean
conditional
1)/10 for a e [1/4,3/4], and g(a)
=
on eventually exiting
is,
the
sum
cc
oo
n
n=l
n=l
fc=l
of chances of exiting after every period
/-
T7>
fc
A
y*
2^11
(1
-
e*)(l
with the
l
71=1
En Fn+ =
because
last equality is true
.
.
.,
or
_ y> g*g ~ gHJ _ y* ^n - fi
Z^
2^ !_^
!__p
fl+i)
i_f"
71=1 fc=l
1, 2,
k
71=1 Jfc=l
~
=
/i\z7>
n
°°
\
n
X
71=1
F\.
i
Mean Time To Herd: Proof of Theorem 5.
Consider a temporary herd on
om in state H: As £n € Im and EyL^H,^ = £n (£n converges Jm with positive chance,
and so E^ > 0. An upper bound to Yl ne n implies a uniform upper bound to the length
•
)
,
of the temporary herds,
•
Part
we wish
(a):
and a uniform lower bound
Extreme Signals are Atoms. We
^2 n ne n
to check that
<
boundedly many, say at most N,
•
Nonatomic
herd on (say) a M
-
As seen
oo.
steps.
Then
Tails: Preliminary.
When
£ +1
-
I
clm
is
have bounded
=
for
(£ n )
beliefs,
and
n > N, and Yl n ne n <
study how fast
taken repeatedly,
E.l(b) and
(c).
as above
a putative herd ends in
in subsection 3.3,
en
We
E^, by Lemma
to
(e n )
N
2
.
vanishes in a putative
obeys the recurrence
- -£ FL ^-M)-F"{p M ^£ n )) =
=
+ £) for some u >
(Lemma 3). If £ solves b = p(£), then Lemma 1(b) implies that t](£) = 0, and that
is increasing near £ (as each of its factors is). Hence, we may apply Lemma E.2 to the
differential equation z = —n(z).
Let preferences result in the private belief threshold Pm-i{£)
£/(u
tj
Fact
The proof will proceed with weaker assumptions than in the theorem, in all
but one case: unbounded beliefs and 6 = 2. So F H (p) = c(p — 6) 7 + o((p — ft) 7 ) and F L (p) =
5
= cy(p - 7 " 1 + o{(p - 7_1 Then with bounded
1 - d(b + o((b - s ). Now H
•
1.
p)
f
p)
6)
(p)
&)
).
7" 1
ft)
L
Lemma 1(a) yields f (p) = cj(l - b)/b(p + o((p - &) 7_1 ), which
'
integrates to F L (p) = c(l - b)/b(p - 6) 7 + o{{p - 6) 7 ); similarly, f L (p) = c^p 1 2 + o(p 7 2
~
and F L (p) = 07/(7 - l)p7_1 + o(p1
with unbounded beliefs. 43
beliefs (b
>
0),
)
l
)
•
Fact
2.
reaches x in finite time. For
we
get
•
43
xt
=x+
Part
= -j(x - x) 1
= x + exp(-jt + d),
Consider the differential equation x
(b):
[(7
-
l)(jt
Bounded
7
+
=
h)}
1
1, we get x
^ 1 -^, h>0
.
For 7
<
1,
the solution
d arbitrary. For 7
>
1,
arbitrary.
Beliefs. One key approximation that we use here and below
Observe that both integrals are valid since necessarily only 7
near b with bounded (resp. unbounded)
beliefs.
38
>
(resp.
7 >
1)
renders f L integrable
is
~k = pM -i{£) - p(£)[u/(u + £) 2 }(£ -£) +
pM -i(£)
and some tedious algebra produces
in (A-12), Fact 1
a=
—
c[u
when
-
\£
(e„)
—
Then
2.
equation of Fact
o((£ n
- £^ <
all cases,
«C 00 when
>
rj(£)
With 7 >
see that £ n
•
2,
,
case, e n
1,
=
e„
and (Hi)
7 e
But outside of N, we
thus reaches
It
is
M
in at
most
(1
see
=
and that
en
=
)
<
^^Coo
_7 ^ 7 ~^),
For the second case,
— F H (pi(£ n ))
L
alone.
A
for
_7/(7_1)
),
en
Lemma
iff
2,
just as with
E.l(a) provides a simple test for
putative herd on a\ in state
(with the derived criterion on
H
so (ne n )
is
,
is
with
some
e
>
0,
E.2(b),
we can
the bounded beliefs
that
want J^ ne n <C 00
7 <
7=1,
(£ n ) is falling
e for
Lemma
= °°-
?7'(0)
=
0.
for a putative
one on a\ (which can't be a herd, and so E^,
and thus Yl ne n -C 00
-
Thus, ]T ne n <C 00.
steps.
Applying
0.
—
£n
1
We
2
£) }(£ n
steps; (ii) for
= 0(n
0{n-'y ^- 1 '>). So J2 n
have two separate cases in state H:
+
c[u/(u
from (A-12) that
— fM )/(efM
differentiable at £ with 77'^)
arbitrarily small
N containing the basin JM
an open neighborhood
is
where
to the above differential
many
(1,2), e„
t) 1 ),
£.
= F H (p M -i{£n)) —
after finitely
for
+ o((£ -
7
K—k
(with
near
f)
Unbounded Beliefs. Mimicking the fist part of
1-7
r](£) = at + o(F), where a — cit
7/(7 — 1). Notice
= 0(n
<2m in state
-
a(£
Recalling our formula
(c):
analysis yields
on clm and
t]
7 for £
).
by boundedly positive decrements, or £ n+ \
= 0{n-^^-^),
Part
We now
e.
7 <
0;
(£ n ) is in J\f.
in our putative herd
then
there
£)
Since e n
E.2(a).
for
(i)
< Ka{£ -
K
<
1
2
£)
bounded above by the solution
(£ n ) is
we have
rj(£)
=
tj(£)
<
exist constants k
converging exponentially to
^ne n
1
There
Lemma
by
2,
£)'1 ),
summable. In
for
.
}
with ka{£
is)
£\
Assume 7 <
tp +
+ 1)
£][u/(u
2 1
-
0({£
=
0).
bounded
^2En
herd
In the
first
beliefs.
<C 00 using e n
=
analytically equivalent to one on
7 then applied
to 6). So,
we
let (£ n )
evolve as
L
M -\(£n))- Now Lemma E.2(a) and (b) yield [(7 — l)(Kan +
< 4 < 7 _ l)(kan + /i)]- ^ 7-1 with k < 1 < K, both arbitrarily close to
one as £ n is close to 0. So, a/[(Kj - l)(an + h)] < e n < a/[(y — l)(kan + h)]. For 7 < 2,
we can find a < 1 with e n < a/n eventually. Then J^ n En = 00. Likewise for 7 > 2, a > 1
before, but consider e n
=F
(p
/>)]-i/(7-i)
1
),
[(
exists with e n
The
case
> a/n
7 = 2
eventually,
F
=
^
n
£n <oo.
requires special attention
assumptions of the theorem
L
and so
in this case.
44
—
Then
it
as already noted,
we go back
to the
can be proved exactly as before that
We have r)(£) = a£ 2 + 0(£ 3 ), and it can be directly verified that
+ 0{p
2
3
T)(£)/(l-T)'(£)) = a£ + 0(£ ). So, we consider dz/dt = -az 2 + bz 3 with a > 0. Integration
yields z(t) = l/[at + c— (b/a) \og(-b + a/z(t))} for z(t) < a/b, i.e. z close enough to zero.
Using our prior knowledge that z(t) = 0(l/t) together with log(l + x) = x + 0(x 2 ), we
get z(t) = \/{at + c- (b/a)[\og(a/z(t)) - b/(at) + 0(\/t 2 )]} = l/(at + d) + 0(\og(t)/t 2 ).
(p)
44
We
If
2cp
2
3
).
we had used the more general assumption we could not conclude below that e„ — 1/n is summable.
we shall refrain from pursuing this.
think a slightly larger error term likely also suffices, but
39
Lemma E.2(6) then yields asymptotically £ n = l/(an + h) + 0(log(n)/n 2 ), and then we
2
get e n = a /(an + h) + 0(\og(n)/n ). Then e„ — 1/n is summable, and we conclude from
Lemma
E.l(a) that
On
•
^
n
En =
Form
the Trembling
instead posits that one
D
oo.
may
A
of Noise.
qualitatively different form of noise
Individuals randomly take a
'tremble', a la Selten (1975):
suboptimal action with some idiosyncratic chance. Someone planning to take action a m
opt for
will instead
simplicity,
we
insist
a,j
with probability K3m (£) for j
that
in state
is
K m (£)
=
[1
,
(7)
is
valid for trembling noise. Indeed, all actions
indeed bounded away from
<p(m, £)
—£ =
and
Let £
2.
is
satisfied
^
than one action
Here, the
We now
invariant
show that Theorem 7
must be taken with positive
by (A-13).
We
.
is
also
probability,
and
wish to argue once more that
under exactly the same conditions as
in the proofs of
Theorems
2
be a stationary point, and assume by way of contradiction that more
taken with positive probability. Then
is
For any such m, we can use
[1
is
Regardless of plans, one accidentally takes action a m with fixed chance K m
£:
is
(A-13)
a special case of trembling, where k™(£)
Proof of Theorem 7 for Trembling Individuals:
so ip(m\£)
and
- Km (£)]p(m\s,£) + Zj#n-*?WPti\s,.Q
Observe crucially that craziness
and
So the chance of deviating
0.
=
H are now described by
iP(m\s,£)
across j
For
t.
]
Y^j^m K m(fy- Even with constant values of K m this
history-dependent, and must be bounded above: K m (£) < 1/2M for all m.
is
The dynamics
m, possibly dependent on
K3m (£) be bounded away from
all
from an optimal action a m
form of noise
^
- K m (£)}
[p(m\L,
sum on
£)
(7) to rewrite (p(m, £)
- p(m\H, £)} =
the right hand side
may
£
=
< F H (pm (£)) <
1 for
some m.
£ as follows:
«?(*) [p(k\H,
£)
-
p(k\L,
(A-14)
£)}
have negative and positive terms, but notice
that
m
Y,[p(k\H,t)-p(k\L,£)]
fc=i
m
=
E
pH
[
^) - FH (P*-i)
- F L (Pk) +
F L (Pk-i)] = F H {pm - F L (p m
)
)
k=l
Recall that the function
pm
is
increasing in
F L -F H
is first
increasing, then decreasing,
m. Thus, the negative terms
40
in the
sum can
at
and that the threshold
most sum to (minus)
number F(£)
the
p(m\H,
£)
= maxm=lt
- p(m\L, £), and
...
M {FL ipm )-FH {pm
k%(£)
Since
)}.
< K k (£) < 1/2M
£
fc#m [p{k\H,£)
by assumption, the
left
-
p(k\L,
hand
£)]
=
side of (A-
obeys the inequality
14)
[1
- K m {£))
[p(m\L,
£)
JL [p(m\L, £)
- p(m\H, £)} <
- p(m\H, £)} + ^F(£)
and so
1As
this holds for all
is
[p(m\L,£)-p(m\HJ)]<^F(£)
m=
m, we may sum over
[F L (pn)
which
M
- F H {Prn)] <
m
1,
.
.
2M-2
impossible by definition of F(£) (and
.
,
m, and discover that
F(£)<
M>
2).
2 obtains once again, while <p{l,£)
Theorem
B.2,
—
£
is
2M-2
is
optimal.
bounded away from
and so Theorem 2 goes through just as before
41
F(£)
Hence, the equations <p(m,£)
could only be solved by an £ for which only one action
rem
M
also.
The proof
=
£
of Theo-
on the interval / of
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43
MIT mimeo.
15-2
15-6
15-7
15-8
15-9
15-10
15-10
15-11
15-12
Booting the System to the Installation Menus
Changing the
Installation
Device
Changing the Network Settings
Changing the Volume Group and Logical Volume Attributes
Starting a
Maintenance Shell
Rebooting the System
Installing the
System
Removing Sysback from a non-licensed machine
Removing the Network Configuration
Chapter
16.
Changing the Volume Group and Logical Volume
16-1
Attributes
16-1
16—5
16—8
Customizing the Volume Group Information
Customizing the Logical Volume Information
Customizing the Filesystem Information
Appendix A. Commands
A— 1
Appendix B. Creating Shell Scripts for Customizing the System
Backup and Install Process
B-l
Creating the Shell Scripts
Example: Appending
Raw Logical Volume Data to System Backup
Appendix C. Device/System -SpeciGc Information
IBM 7208 8mm Tape Drives
IBM 3490 and 3590 9-Track Tape Drives
IBM 7331 8mm Tape Library
IBM 7332 4mm Tape Library
Other Tape Libraries or Autoloaders
IBM Scalable POWERparallel Systems (SP2)
IBM 7133 Serial Storage Architecture (SSA) disk subsystem
IBM 7135 RAIDiant Array
IBM 7137 and Other RAID Devices
Index
B-l
B-2
C-l
C-l
C-l
C-l
C-2
C-2
C-2
C-3
C-4
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AIX System Backup & Recovery/6000
Vll
AIX System Backup & Recovery/6000 (Sysback)
Overview
AIX System Backup & Recovery/6000 (Sysback) is a menu-driven application which
provides system administrators and other AIX users with a simple, efficient way to backup
data and recover from hardware failures. It was designed with the intention of providing AIX
users with the fastest possible method of recovering all or part of the system after a system
or hardware failure. Sysback is flexible enough to allow a system backup created on one
system to be installed or reinstalled onto a machine with either an identical or dissimilar
hardware configuration. Virtual devices provide large storage capacity with little or no
user-interaction.
Sysback provides the following
•
Ability to
specific features:
create various types of backups, including:
-
Full
-
Volume Groups
-
Logical
-
Filesystems (Incremental)
-
Specific Directories or Files
System
Image)
(Installation
Volumes (raw data)
Complete SMIT menus for configuring Sysback options, backing
and restoring files from all backup types.
up, listing,
verifying
Ability to
perform
all
backups
to tape drives or disk
file
on a remote host on
the network. This allows a single system to act as a backup media server for
multiple clients.
Backups may be performed
to multiple sequential devices, automatically
continuing the backup on the next device
Backups may be performed to multiple
backup in a fraction of the time.
A
when
the
first is full.
parallel devices,
completing a single
backup may be written to multiple devices, providing duplicate
same backup in the same time it normally takes to make one.
single
copies of the
Status Indicator displays estimated backup or restore size and time, as well
as performance and percent complete.
User may
All
interactively select files to restore
backups, local and remote,
from a
may be performed
list
of files
on the backup.
to multiple tape
volumes or
multiple devices.
Automated recreation of volume groups, logical volumes and filesystems,
with option to change attributes and physical locations, using the information
stored on the backup media.
Backup compression, reducing backup time and media usage by 25-40%.
exclude specific temporary or non-changeable
backups, saving media space and backup time.
Ability to
from
all
files
Option to create "incremental" backups, specifying that only
a specific time period or since a previous level backup.
V11I AIX System Backup & Recovery/6000
or directories
files
modified
in
•
Simple configuration
of
a network boot server and/or network installation
server used to alleviate the need for local boot or installation media.
•
Ability to "stack" multiple
efficiently
•
and not
backups onto a single tape, using tape media more
media to be changed between backups.
requiring the
User may create custom scripts for performing additional functions before
and after the System Backup, and at completion of the System Install
process.
•
The "System Backup"
-
option provides the following benefits:
Restoration of the original system to
including devices, multiple
its
exact device configuration,
volume groups,
logical
volume placement on
disks, etc.
-
User may customize the installation by changing volume group,
volume and filesystem locations, sizes and other attributes.
-
Installation of a complete system, including multiple volume groups from
a remote tape drive or disk file backups. System to be installed may be
booted from the network server, so no local boot media is required.
-
logical
machines of same or different hardware
custom backup (also referred to as "cloning").
Installation of multiple
configuration from a
-
Preservation of multi-copy (mirrored) logical volumes.
-
Option of retaining exact partition placement of logical volumes or
making contiguous any partitions had become fragmented to improve I/O
performance.
-
Option of either importing, ignoring or recreating and restoring each
individual volume group from a single backup media.
AIX System Backup & Recovery/6000
IX
.
MIT LIBRARIES
3 9080 00978 0252
Chapter
Installing
1.
AIX System Backup & Recovery/6000
AIX System Backup & Recovery/6000 (Sysback), you must have installed the
on your system. Then, follow the installation procedures below for
installation from the Sysback Installation Diskette.
To
install
prerequisite software
Prerequisite Tasks
The
following software
may be
required,
depending on the Sysback functions you
using:
•
AIX Base Operating System (BOS) Runtime Version 3.2 or higher
•
On AIX
in
•
If
Version 4 systems, the "bos.sysmgt.sysbr"
addition to those automatically included with the
you
will
following
•
be using the Remote Backup Services functions
must also be
o
AIX Version
3.2:
bosnet.tcpip.obj
o
AIX Version
4:
bos.rte.net
bos.net.tcp.client
If
you
will
be using the Network Boot function
install
of the
Network
AIX Version
3.2:
bosnet.nfs.obj
o
AIX Version
4:
bos.net.nfs.client
On AIX Version 4 the following are installed automatically with
and may not be removed for Sysback to function properly:
.
o
bos.rte.bosinst
o
bos.rte.archive
o
bos.rte.libnetsvc
(If
Installation
Network
in
Install
Installation functions are to
the system,
be used)
from Diskette
After the prerequisite software
Log
Sysback, the
the following:
o
Procedure for
1
of
installed:
Services, you must also
•
fileset must be installed
base operating system.
is
installed:
as root user. You should see the following:
IBM AIX Operating System
(c) Copyright IBM Corp. 19XX, 19XX
/dev/console)
login: root
(
#
(The default root user prompt
AIX Sysback/6000
2.
Insert the
3.
Enter the following on the AIX
smit install
1—1
Installing
AIX System Backup & Recovery/6000
is
#)
Installation Diskette into the diskette drive.
command
line:
will
be
Date Due
Lib-26-67