Full text of DOI:10.1111/1468-0262.00113

Ž
.
Econometrica, Vol. 68, No. 2 March, 2000 , 371᎐398
PATHOLOGICAL OUTCOMES OF
OBSERVATIONAL LEARNING
1
BY LONES SMITH AND PETER SøRENSEN
This paper explores how Bayes-rational individuals learn sequentially from the discrete
actions of others. Unlike earlier informational herding papers, we admit heterogeneous
preferences. Not only may type-specific ‘‘herds’’ eventually arise, but a new robust
possibility emerges: confounded learning. Beliefs may converge to a limit point where
history offers no decisive lessons for anyone, and each type’s actions forever nontrivially
split between two actions.
To verify that our identified limit outcomes do arise, we exploit the Markov-martingale
character of beliefs. Learning dynamics are stochastically stable near a fixed point in many
Bayesian learning models like this one.
KEYWORDS: Informational herding, cascades, martingale, Markov process, stochastic
stability.
1. INTRODUCTION
SUPPOSE THAT
A
COUNTABLE NUMBER OF INDIVIDUALS each must make a
once-in-a-lifetime binary decisionᎏencumbered solely by uncertainty about the
state of the world. If preferences are identical, there are no congestion effects or
network externalities, and information is complete and symmetric, then all
ideally wish to make the same decision.
But life is more complicated than that. Assume instead that the individuals
must decide sequentially, all in some preordained order. Suppose that each may
condition his decision both on his endowed private signal about the state of the
world and on all his predecessors’ decisions, but not their hidden private signals.
Ž
.
The above framework was independently introduced in Banerjee 1992 and
Ž
. Ž
.
Bikhchandani, Hirshleifer, and Welch 1992 hereafter, simply BHW . Their
common conclusion was that with positive probability an ‘‘incorrect herd’’ would
arise: Despite the wealth of available information, after some point, everyone
might just settle on the identical less profitable decision.
In this paper, we study more generally how Bayes-rational individuals sequen-
tially learn from the actions of others. This leads to a greater understanding of
1 This second revision is based on numerous suggestions from three referees and the Editor. We
have also benefited from seminars at MIT, 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, UCL, Edinburgh, Cambridge, Essex, and IGIER. We wish also to specifically
acknowledge Abhijit Banerjee, Kong-Pin Chen, Glenn Ellison, John Geanakoplos, David Hirsh-
leifer, David Levine, Preston McAfee, Gerhard Orosel, Thomas Piketty, Ben Polak, and Jorgen
¨
Weibull for useful comments, and Chris Avery for early discussions on this project. All errors remain
our responsibility. Smith and Sørensen respectively acknowledge financial support for this work from
the National Science Foundation and the Danish Social Sciences Research Council.
371

372
L. SMITH AND P. SøRENSEN
herding, and why and when it occurs. It also leads to the discovery of a co-equal
robust rival phenomenon to herding that has so far been missed, and that is
economically important.
To motivate our point of departure from Banerjee and BHW, consider the
following counterfactual. Assume that we are in a potential herd in which one
million consecutive individuals have acted alike, but suppose that the next
individual deviates. What then could Mr. one million and two conclude? First,
he could decide that his predecessor had a more powerful signal than everyone
else. To capture this, we generalize the private information beyond discrete
signals, and admit the possibility that there is no uniformly most powerful yet
nonrevealing signal. Second, he might believe that the action was irrational or
an accident. We thus add noise to the herding model. Third, he might decide
that different preferences provoked the contrary choice. On this score, we
consider the model with multiple types, and find that herding is not the only
possible ‘‘pathological’’ longrun outcome: We may converge to an informational
pooling equilibrium where history offers no decisive lessons for anyone, and
everyone must forever rely on his private signal.
Ž .
i What are the robust
The paper is unified by two natural questions:
long-run outcomes of learning in a sequential entry model with observed
Ž .
actions? ii Do we in fact settle on any one? Our inquiry is focused through the
two analytic lenses of convergence of beliefs and convergence of actionsᎏeither
in frequency or, more strongly, with herds. BHW introduced the terminology of
a cascade for an infinite train of individuals acting irrespective of the content of
their signals. With a single rational type and no noiseᎏhenceforth, the herding
modelᎏindividuals always settle on an action, starting a herd. Yet the label
‘‘cascades literature’’ is appropriate outside BHW’s discrete signal world. Among
our simplest findings is that barring discrete signals, cascades need not arise: No
decision need ever be a foregone conclusion even during a herd. With these two
notions decoupled, the analysis is richer, and it suggests why we must admit a
general signal space, and adopt a stochastic process approach. For instance,
Theorem 1 shows that learning is incomplete exactly when private signals are
uniformly bounded in strength. By Theorem 3, then and only then can bad herds
possibly arise in the herding model.
Absent cascades, the explanation we provide for herding is that the standard
belief convergence implies action convergence: The action frequency settles
down, and is consistent with the limit belief. Perfect conformity arises in the
pure herding model because any rational desire to deviate must then be shared
by all successors. Hence, contrary actions radically swing beliefs. But identical
preferences is neither a realistic nor general assumption. Assuming that some
individuals are randomly committed to different actions is a useful interim step.
Yet this form of statistical noise is equivalent to rational agents with different
dominant preference types and strategies, and thus falls far short of our main
contribution. For it washes out in the long run, and does not affect convergence.
In this paper, we more generally assume that individuals entertain possibly
different preferences over actions; further, types are unobserved, so that only

OBSERVATIONAL LEARNING
373
statistical inferences may be drawn about any given individual. Taste diversity
with hidden preferences describes numerous cited motivating examples of herd-
ing in the literature, such as restaurant choice, or investment decisions. This
twist yields our principal new economic findings. The standard herding outcome
Ž
is robust to individuals having identical ordinal but differing cardinal von
.
Neumann Morgenstern preferences. With multiple rational preference types,
not all ordinally alike, an interior rational expectation dynamic steady-state
robustly emerges: It may be impossible to draw any clear inference from history
even while it continues to accumulate privately-informed decisions. Further, this
incomplete learning informational pooling outcome exists even with unbounded
beliefs, when an incorrect herd is impossible.
Let us fix ideas and illustrate this confounded learning outcome with a possibly
familiar example. Suppose that on a highway under construction, depending on
how detours are arranged, those going to Houston should take either the high or
Ž
.
low off-ramps in states H and L , with the opposite for those headed toward
Dallas. If 70% are headed toward Houston, then absent any strong signal to the
contrary, Dallas-bound drivers should take the lane ‘‘less traveled by.’’ This
yields two separating herding outcomes: 70% high or 70% low, as predicted by
armchair application of the herding logic. But another possibility may arise. As
the chance q that observed history accords state H rises from 0 to 1, the
probability that a Houston driver takes the high road gradually rises from 0 to 1,
Ž .
Ž .
and conversely for Dallas drivers. Thus, the fraction ␺ q and ␺ q in the
H
right lane in states H, L each rise from 0.3 to 0.7, perhaps nonmonotLonically so.
If for some q, a random car is equilikely in states H and L to go high, or
Ž .
Ž .
␺
q s␺ q , then no inference can be drawn from additional decisions:
H
Learning tLhen stops. While existence of such a fixed point is not clear, Theorems
1 and 2 prove that confounding outcomes co-exist with the cascade possibilities,
for nondegenerate models.
Our confounded learning outcome is generic when two types have opposed
preferences, assuming uniformly bounded private signals. With unbounded
signals, it emerges for sufficiently strongly opposed von Neumann Morgenstern
Ž .
q
preferences, and not too unequal population frequencies. In either case, ␺
H
Ž .
Ž .
Ž .
)␺ q for small enough q, and ␺ q -␺ q for large enough q.
L
H
L
Two stochastic processes constitute the building blocks for our theory: the
Ž
public likelihood ratio is a conditional martingale, and the vector action taken,
.
likelihood ratio a Markov chain. Martingale and Markovian methods are
standard methods for ruling out potential limit outcomes of learning. But our
major technical contribution concerns their stability: Given multiple limit be-
liefs, must we converge upon any given one? How can we rule in any limit? For
instance, even if our highway driving confounding outcome robustly exists, must
we converge upon it? Theorem 4 gives a simple easily checked condition for the
local stochastic stability of a Markov-martingale process near a fixed point. This
yields a new general property of Bayesian learning dynamics. In our herding
context, assume that near any fixed point, posterior beliefs are not equally
responsive to priors for every action taken, but are monotonely so; that is, a

374
L. SMITH AND P. SøRENSEN
higher prior yields a higher posterior belief. Then the belief process tends with
positive chance and exponentially fast to that fixed point it starts nearby. As an
Ž .
application, Theorem 5 asserts that i an action that can be herded upon, will
Ž .
then be herded upon for nearby beliefs, while ii convergence to the confound-
ing outcome occurs with positive chance, and necessarily rapidly.
Ž
.
The conclusion relates our confounded learning to McLennan 1984 , its first
Ž
.
published instance. More recently, Piketty 1995 has found confounded learning
outcomes in a model where individuals passively learn about social mobility
parameters.
Section 2 gives a common framework for the paper. Section 3 illustrates our
findings in three examples. We then proceed along two technical themes. First,
via Markov-martingale means, Section 4 describes the action and belief limits;
the confounding outcome is the key finding here. Next, Section 5 motivates and
states the stability result, and as an application, shows when a long-run outcome
arises. Extension to finitely many states is addressed in the conclusion; there we
also describe extensions of the paper, as well as some related literature. More
detailed proofs and some essential math results are appendicized.
2. THE COMMON FRAMEWORK
2.1. The Model
States. There are Ss2 payoff-relevant states of the world, the high state
ssH and the low state ssL. As is standard, there is a common prior
Ž
.
Ž .
beliefᎏwithout loss of generality, a flat prior Pr H sPr L s1r2. Our results
extend to any finite number S of states, but at significant algebraic cost, and so
this extension is addressed in the conclusion §6.1 .
Ž
.
Pri¨ate Beliefs. An infinite sequence of individuals ns1, 2, . . . enters in an
exogenous order. Individual n receives a random pri¨ate signal about the state
Ž
.
of the world, and then computes via Bayes’ rule his pri¨ate belief pn g 0, 1 that
ꢀ
4
the state is H. Given the state sg H, L , the private belief stochastic process
pn is i.i.d, with conditional c.d.f. F . These distributions are sufficient for the
s
²
:
state signal distribution, and obey a joint restriction implicit below. The curious
reader may jump immediately to Appendix A, which summarizes this develop-
ment, and explores the results we need.
We assume that no private signal, and thus no private belief, perfectly reveals
the state of the world: This ensures that F H, F L are mutually absolutely
Ž .
continuous, with common support, say supp F . Thus, there exists a positive,
finite Radon-Nikodym derivative fsdF LrdF H: 0, 1 ª 0, ϱ . And to avoid
trivialities, we assume that some signals are informative: This rules out fs1
Ž
.
Ž
.
almost surely, so that F H and F L do not coincide. When F s is differentiable
s
Ž
.
ssL, H , we shall denote its derivative by f .
Ž
Ž ..
w
x
w
x
The convex hull co supp F ' b, b : 0, 1 plays a major role in the paper.
Note that b-1r2-b as some signals are informative. We call the private
Ž
Ž ..
w
x
beliefs bounded if 0-b-b-1, and unbounded if co supp F s 0, 1 .

OBSERVATIONAL LEARNING
375
Indi¨idual Types and Actions. Every individual makes one choice from a finite
ꢀ
4
action menu Ms 1, . . . , M , with MG2 actions. We allow for heterogeneous
preferences of successive individualsᎏthe only other random element. A model
with multiple but observable types is informationally equivalent to a single
preference world. So assume instead that all types are private information.
There are finitely many rational types ts1, . . . , T with different preferences. Let
t
␭ be the known proportion of type t.
We also introduce M crazy types. Crazy type m arrives with chance ␬m G0,
and always chooses action m. One could view these as rational types with state
independent preferences, and unlike everyone else, a single dominant action.
Ž
.
We assume a positive fraction ␬s1y ␬1 q иии q␬ )0 of payoff-motivated
M
rational individuals. Rational and crazy types are spread i.i.d. in sequence, and
²
:
.
independently of the belief sequence pn
ꢀ
4
sŽ
Payoffs. In state sg H, L , each rational type t earns payoff ut m from
.
t
Ž
.
action m for precision, sometimes am , and seeks to maximize his expected
Ž
.
payoff. For each rational type, MG Mt G2 actions are not weakly dominated,
and generically no one action is optimal at just one belief, and no pair of actions
provides identical payoffs in all states. Each type t thus has Ss 2 extreme
actions, each strictly optimal in some state. The other Mt y2 insurance actions
are each taken at distinct intervals of unfocused beliefs.
Ž
.
w
x
Ž
Given a posterior belief rg 0, 1 that the state is H, the expected payoff to
type t of choosing action m is rutH m q 1yr ut m . Figure 1 depicts the next
.
Ž
. .
LŽ
summary result.
LEMMA 1: For each rational type t, 0, 1 partitions into subinter¨als I1t, . . . , IMt
w
x
t
touching at endpoints only, with undominated action mg Mt : M optimal exactly
for beliefs rgImt .
With multiple types, we must introduce T labels for every action. Permuting
Mt, we order rational type t’s actions a1t , . . . , aMt by relative preference in state
H, with atM most preferred. So to be clear, if we torder actions from least to most
t
preferred by type t in state H, then action m has rank ␰s␰ if msat . By
m
␰
FIGURE 1.ᎏExpected payoff frontier. The diagram depicts the expected payoff of each of three
actions as a function of the posterior belief r that the state is H. A rational individual simply
chooses the action yielding the highest payoff. Here 2 is an insurance action, and 1 and 3 are
extreme actions.

376
L. SMITH AND P. SøRENSEN
t
t
Lemma 1, type t’s mth action basin is Im ' rmy1, rmt , with ordered boundaries
w
x
0sr0t -r1t - иии -rMt s1; thus, extreme actions a1t and atM are optimal for type
t in states L and H, tand insurance actions a2t , . . . , atM y1 are each best for some
t
t
interior beliefs. The tie-breaking rule is without loss of generality that type t
chooses atm over amt q1 at belief rmt . Type t has a stronger preference for actions
atm the larger is the basin Imt . Rational types t and tЈ have opposed preferences
t
t
tЈ
tЈ
Ž
.Ž
.
over actions m and mЈ if ␰m y␰
␰m y␰ -0ᎏi.e. their ordinal prefer-
mЈ
mЈ
ences for them in state H, and thus in state L, are reversed. With just a single
rational type, we suppress t superscripts, and likewise strictly order belief
thresholds as 0sr0 -r1 - иии -rM s1.
2.2. The Indi¨idual Bayesian Decision Problem
Before acting, every rational individual observes his type t, his private belief
p, and the entire ordered action history h. His decision rule then maps p and h
into an action. We look for a Bayesian equilibrium, where everyone knows all
sŽ .
decision rules, and can compute the chance ␲ h of any history h in each state
Ž .
H Ž . Ž H Ž .
LŽ ..
s. This yields a public belief q h s␲ h r ␲ h q␲ h that the state is H,
i.e. the posterior given h and a neutral private belief ps1r2. Applying Bayes
rule again yields the posterior belief r in terms of q and p:
H Ž .
p␲
h
pq
Ž .
1
Ž
.
rsr p, q s
s
.
H Ž .
Ž
.
L Ž .
Ž
.Ž .
p␲ h q 1yp ␲
h
pqq 1yp 1yq
As belief q is informationally sufficient for the underlying history data h, we
now suppress h.
Ž .
Since the right hand side of 1 is increasing in p, there are pri¨ate belief
t Ž .
tŽ .
action atm iff his private belief satisfies pg pmy1 q , pm q , given the earlier
Ž .
t
thresholds 0sp0 q Fp1 q F иии FpM q s1, such that type t optimally chooses
t
t
Ž
Ž .
Ž .
Ž .x
tie-break rule. Furthermore, each threshold pmt q is decreasing in q. A type-t
t
t
t
ꢀ
Ž .
w
Ž .
Ž .x
4
cascade set is the set of public beliefs Jm s qNsupp F : pmy1 q , pm
q
. So
t
t
type t a.s. takes action atm for any qgint Jm , since the posterior r p, q gIm
Ž
.
Ž
.
for all p. It follows that any cascade set lies inside the corresponding action
t
t
t
Ž
.
basin, so that Jm :int Im . For if all private beliefs yield action am, then so must
the neutral belief.
Ž
.
As is standard, call a property generic resp. nondegenerate or robust if the
subset of parameters for which it holds is open and dense resp. open and
Ž
.
nonempty .
LEMMA 2: For each action amt and type t, Jmt is a possibly empty inter¨al. Also,
t
t
t
t
Ž .
w
x
w
x
a
with bounded pri¨ate beliefs, J1 s 0, q and JMt s q , 1 for some 0-q -
t
qt -1;
t
t
t
Ž .
ꢀ 4
ꢀ 4
b
with unbounded pri¨ate beliefs, J1 s 0 , JM s 1 , and all other Jm are
t
empty;
Ž .
c
for generic payoffs us, no interior cascade set Jmt is a single point;

OBSERVATIONAL LEARNING
377
Ž .
basin
d
each Jmt is larger the smaller is the support b, b , and the larger is the action
w x
t
t
my1, rmt . Only extreme cascade sets J1 and JMt are nonempty for large
w
x
r
t
w
x
enough b, b .
Ž
.
The appendicized proofs of Lemma 2 a, b are also intuitive: With bounded
private beliefs, posteriors are not far from the public belief q; so for q near
enough 0 or 1, or well inside an insurance action basin, all private beliefs lead to
Ž
.
the same action. With unbounded beliefs, every public belief qg 0, 1 is
swamped by some private beliefs. Next, given the continuous map from payoffs
t
²
sŽ .:
²
:
Ž
.
Ž .
to thresholds ut m ¬ rm , Lemma 2 c, d follows from 1 :
bq
bq
qgJmt mrmt y1
F
and
Frmt .
Ž .
2
Ž
.Ž .
Ž
.Ž .
bqq 1yb 1yq
bqq 1yb 1yq
The paper requires some more notation. Define J t sJ1t jJ2t j иии jJMt . A type t
t
is acti¨e when choosing at least two actions with positive probability. A cascade
arisesᎏeach type’s action choice is independent of private beliefsᎏfor public
beliefs strictly inside JsJ1 lJ 2 l иии lJ T: however, with unbounded beliefs,
H
t
ꢀ 4 ꢀ 4
there are two cascade beliefs: qgJs 0 j 1 . We put J 'F JM , namely
t
where each type t cascades on the action atM , which is optimal in t state H.
t
Similarly, we define J L for state L.
2.3. Learning Dynamics
Let qn be the public belief after Mr. n chooses action mn. It is well-known
ϱ
ns1
²
:
Ž .
<
that qn
obeys an unconditional martingale, E qnq1 qn sqn, and hence
almost surely converges to a limit random variable. While we could in principle
employ this posterior belief process, we care about the conditional stochastic
properties in a given state H. Thus, the public likelihood ratio ln ' 1yqn rqn
that the state is L versus H offers distinct conceptual advantages, and saves
Ž
.
² :
time, as it conditions on the assumed true state. The likelihood process ln
similarly will be a convergent conditional martingale on state H.
We then have likelihood analogues of previous notation, now barred: private
t
t
t
belief thresholds pmt 1yq rq 'pm q ; action basins Im, where rgIm if and
ŽŽ
.
.
Ž .
t
t
t
Ž
.
Ž
.
only if 1yr rrgIm; and cascade sets Jm, where qgJm if and only if 1yq rq
t
H
t
H
t
gJmt ; as well as J , J, J
Ž
w
.
Ž .
for J , J, J . With bounded private beliefs, J1 s
l , ϱ and JM s 0, l for some 0-lt -lt -ϱ. Note: this natural notation
t
t
t
w
x
x
Ž
t
t
t
t
t
t
t
Ž
.
Ž
.
.
implies a reverse correspondence: l s 1yq rq and l s 1yq rq . With
t
t
ꢀ 4
ꢀ 4
unbounded private beliefs, J1 s ϱ , JM s 0 , and all other cascade sets are
t
empty.
² :ϱ
Ž
Likelihood ratios ln
are a stochastic process, described by l0 s1 as
ns1
.
q0 s1r2 and transitions
t
t
t
Ž .
3
t Ž
<
.
s Ž Ž ..
s Ž
Ž ..
l
␳
am s, l sF pm
l
yF pmy1
,

378
L. SMITH AND P. SøRENSEN
FIGURE 2.ᎏIndividual black box. Everyone bases his decision on both the public likelihood ratio
l and his private belief p, resulting in his action choice atm with chance ␺ am L, l , and a likelihood
t
Ž
<
.
t
t
Ž
.
ratio ␸ am, l to confront successors. Type t takes action am if and only if his posterior likelihood
t
t
t
Ž
.
w
x
l 1yp rp lies in the interval Im, where I1, . . . , IM partition 0, ϱ .
t
T
t
Ž .
4
Ž .
<
t Ž
<
.
␺ m s, l s␬ q␬
␭␳ m s, l ,
Ý
m
t
s1
Ž .
5
Ž . Ž . Ž .
<
<
␸ m, l sl␺ m L, l r␺ m H, l .
tŽ
<
.
Here, ␳ m s, l is the chance that a rational type t takes action m, given l, and
t
ꢀ
4
the true state sg H, L . So the cascade set Jm is the interval of likelihoods l
t
t
tŽ
yielding ␳ am H, l s␳ am L, l s1. Faced with ln, Mr. n takes action mn
<
Ž
.
tŽ
<
.
<
.
Ž
.
with chance ␺ mn s, ln in state s, whereupon we move to lnq1 s␸ mn, ln .
Ž . Ž .
Figure 2 summarizes 3 ᎐ 5 .
Our insights are best expressed by considering the pair mn, ln as a discrete-
²
:
w
.
time, time-homogeneous Marko¨ process on the state space M= 0, ϱ . Given
²
:
² Ž Ž < .
.:
mn, ln , the next state is
mnq1, ␸ mnq1, ln with probability ␺ mnq1 s, ln in
² :
state s. Since ln is a martingale, convergence to any dead wrong belief almost
surely cannot occur, since the odds against the truth cannot explode. In
summary we have the following lemma.
Ž .
² :
LEMMA 3: a The likelihood ratio process ln is a martingale conditional on
state H.
Ž .
² :
Assume state H. The process ln con¨erges almost surely to a r.¨., lϱ s
b
Ž .
w
.
Ž
.
limnªϱln, with supp lϱ : 0, ϱ . So fully incorrect learning ln ªϱ almost surely
cannot occur.
Ž
.
² :
PROOF: See Doob 1953 for the martingale character of ln . Convergence
follows from the Martingale Convergence Theorem for nonnegative, perhaps
Ž
Ž
.
.
unbounded, random variables Breiman 1968 , Theorem 5.14 ; Bray and Kreps
Ž
.
1987 prove this with public beliefs.
Q.E.D.
Ž
.
Learning is complete if ‘‘beliefs’’ likelihoods converge to the truth: ln ª0 in
state H. Otherwise, learning is incomplete: Beliefs are not eventually focused on
the true state H.

OBSERVATIONAL LEARNING
379
²
:
Belief convergence then forces action con¨ergence: mn settles down in
Ž
.
Ž
.
ꢀ
frequency, or limnªϱ Nn m rn exists for all m, where Nn m '࠻ mk sm,
4
kFn .
COROLLARY: Action con¨ergence almost surely obtains, if F s has no atoms.
Ž
.
PROOF: Let ␧)0. Given belief convergence Lemma 3 , continuity of
Ž
.
Ž . Ž .
⌿ mNs, l , and the law of motion 4 , the chance ␣m n that action m is chosen
Ž . Ž .
almost surely converges, say ␣m n ª␣m. Thus, ␣m n F␣m q␧ for large n, and
Ž . Ž .
so lim supnªϱ Nm n rnF␣m q␧. Similarly, lim infnªϱ Nm n rnG␣m y␧. Since
Ž .
␧)0 is arbitrary, limnªϱ Nm n rns␣m.
Q.E.D.
The literature has so far focused on two more powerful convergence notions.
As noted, a cascade means ln gJ, or finite time belief convergence. Since every
later rational type’s action is dictated by history, this forces a herd, where
rational individuals of the same type all act alike. By corollary, the weaker
action convergence obtains. We also need the weaker notion of a limit cascade,
or eventual belief convergence to the cascade set: lϱ gJ.
3. EXAMPLES
3.1. Single Rational Type
Consider a simple example, with individuals deciding whether to ‘‘invest’’ in or
Ž
.
‘‘decline’’ an investment project of uncertain value. Investing action ms2 is
Ž
.
risky, paying off u)0 in state H and y1 in state L; declining action ms1 is
a neutral action with zero payoff in both states. Indifference prevails when
Ž
.
Ž
.
.
Ž .
0sruy 1yr , so that rs1r 1qu . Thus, equation 1 defines the private
Ž .
Ž
belief threshold p l slr uql .
A. Unbounded Beliefs Example. Let the private signal ␴g 0, 1 have state-
LŽ
Ž
.
contingent densities g H ␴ s2␴ and g ␴ s2 1y␴ ᎏas in the left panel of
Ž
.
.
Ž
.
Ž
.
Ž
.
Figure 3. With a flat prior, the private belief psp ␴ then satisfies 1yp rps
g L ␴ rg ␴ s 1y␴ r␴ by Bayes’ rule, and has the same conditional densi-
Ž
.
H Ž
Ž .
.
Ž
.
ties f H p '2 p and f p '2 1yp , and c.d.f.’s F p sp and F p s2 p
LŽ .
Ž
.
H Ž .
LŽ .
2
2
Ž .
w
x
yp . So supp F s 0, 1 , and private beliefs are unbounded; the cascade sets
ꢀ 4
ꢀ 4
collapse to the extreme points, J1 s ϱ , J2 s 0 .
Ž
.
Ž
.
FIGURE 3.ᎏSignal densities. Graphs for the unbounded left and bounded right beliefs
examples. Observe how, in the left panel, signals near 0 are very strongly in favor of state L.

380
L. SMITH AND P. SøRENSEN
FIGURE 4.ᎏContinuation and cascade sets. Continuation functions for the examples: unbounded
Ž
.
Ž
.
Ž
.
private beliefs left , and bounded private beliefs right with ␬1 s␬2 s1r10 and without noise
Ž
.
Ž
.
dashed and solid lines . The stationary points are where both arms hit the diagonal as with noise ,
or where one arm is taken with zero chance ls0 or lsϱ in the left panel; lF2ur3 or lGu in the
right panel without noise . With crazy types the discontinuity vanishes, and isolated deviations no
Ž
.
longer have drastic consequences. Graphs here and in Figure 5 were generated analytically with
PostScript.
Given private belief c.d.f.’s F H, F L and threshold p l slr uql , transition
Ž .
Ž
.
2
2
2
Ž <
.
Ž
.
Ž <
.
Ž
. Ž
.
Ž .
by 3 .
chances are ␳ 1 H, l sl r uql and ␳ 1 L, l sl lq2u r uql
Ž
.
Ž
.
Ž
.
Ž . Ž .
Continuations are ␸ 1, l slq2u and ␸ 2, l sulr uq2l by 4 , 5 . As in
Ž
.
Figure 4 left panel , the only stationary finite likelihood ratio in state H is 0;
the limit lϱ of Lemma 3 is focused on the truth. As the suboptimal action 1 lifts
ln G2u, an infinite subsequence of such choices would preclude belief conver-
Ž
.
gence. Hence, there must be action convergence i.e. a herd .
B. Bounded Beliefs Example. Let private signals have density g L ␴ s3r2y␴
on 0, 1 in state L, and uniform on 0, 1 in state H. Given a flat prior, Bayes’
Ž
.
Ž
.
Ž
.
Ž
.
Ž
LŽ
.
.
Ž
.
Ž .
rule yields the private belief p ␴ s1r g ␴ q1 s2r 5y2␴ , i.e. p ␴ Fp
Ž
.
Ž
.
H Ž .
. Ž
Ž
.
.
m2r 5y2␴ Fpm 5py2 r2 pG␴ . Thus, F p s 5py2 r2 p in state H
2
LŽ
.
Ž
.Ž
and F L p sHpŽ␴ .F p g ␴ d␴s 5py2 pq2 r 8 p in state L. Each thus
Ž .
w
x
w
x
has bounded support b, b s 2r5, 2r3 .
Ž .
Ž
.
Ž
.
Since p l slr uql , active dynamics occur when lg 2ur3, 2u , where
H
L
Ž . Ž .
Ž <
.
Ž
.
Ž <
3ly2u 3lq2u r8l , as well as ␸ 1, l sur2q3lr4 and ␸ 2, l sur2qlr4.
The likelihood ratio converges by Lemma 3, so that J1 jJ2 contains all possible
.
equations for F , F , and 3 ᎐ 5 yield ␳ 1 H, l s 3ly2u r2l and ␳ 1 L, l s
2
Ž
.Ž
.
Ž
.
Ž
.
Ž
.
stationary limits, as in Figure 4 right panel . A herd on action 1 or 2 must then
Ž <
.
start eventuallyᎏagain, lest beliefs not converge. If l0 F2ur3, then ␳ 1 H, l s
␳ 1 L, l s0, i.e. action 2 is always taken, and thus ln is absorbed in the set
Ž < ² :
.
w
x
w
x
J2 s 0, 2ur3 . For l0 G2u, we similarly find J1 s 2u, ϱ .
Here, we may strongly conclude that each limit outcome 2ur3 and 2u arises
Ž
.
Ž
.
with positive chance for any l0 g 2ur3, 2u . As in Figure 4 right , dynamics are
Ž
w <
.
< <
forever trapped in 2ur3, 2u . Since ln F2u, the Dominated Convergence
x
Ž
.
Ž
.Ž
.
Theorem yields E lϱ H sl0. Since l0 s␲ 2ur3 q 1y␲ 2u for some 0-␲
-1 whenever 2ur3-l0 -2u, in state H, a herd on action 2 arises with chance
␲, and one on action 1 with chance 1y␲.

OBSERVATIONAL LEARNING
381
In contrast to BHW, if public beliefs are not initially in a cascade set, they
never enter one. This holds even though a herd always eventually starts.
Ž
.
Ž
.
Visually, it is clear that ln g 2ur3, 2u for all n if l0 g 2ur3, 2u . This also
Ž
.
follows analytically: If ln -2u, then lnq1 s␸ 1, ln sur2q3lr4-ur2q3ur2
s2u too. So herds must arise even though a contrarian is never ruled out. This
Ž
.
result obtains whenever continuation functions ␸ i, и are always increasing. For
² :
then ln never jumps into the closed set J1 jJ2 , starting a cascade. Monotonic-
ity is crucially violated in BHW’s discrete signal world.
C. Bounded Beliefs Example with Noise. Suppose that a fraction of individuals
randomly chooses actions. This introduction of a small amount of noise radically
affects dynamics, as seen in the right panel of Figure 4. For since all actions are
Ž
.
expected to occur, none can have drastic effects. Namely, each ␸ i, и is now
continuous near the cascade sets at ls2ur3 and ls2u. Yet, the limit beliefs
Ž
are unaffected by the noise, contrary actions being deemed irrational and
ignored inside the cascade sets.
.
3.2. Multiple Rational Types
With multiple types, one can still learn from history by comparing proportions
choosing each action with the known type frequencies. This inference intuitively
should be fruitful, barring nongenericities. A new twist arises: Dynamics may
lead to each action being taken with the same positive chance in all states,
choking off learning. This incomplete learning outcome is economically different
than herdingᎏinformational pooling, where actions do not reveal types, rather
than the perfect separation that occurs with a type specific informational herd.
Mathematically it is a robust interior sink to the dynamics.
Let us consider the driving example from the introduction. Posit that Houston
Ž
.
Ž
.
type U, our ‘‘usual’’ preferences drivers should go high action 2 in state H,
Ž
.
Ž
.
low action 1 in state L, with the reverse true for Dallas type V drivers. Going
to the wrong city always yields zero, without loss of generality. The payoff vector
Ž
.
Ž
.
of the Houston-bound is 0, u in state H and 1, 0 in state L; for Dallas drivers,
Ž
.
Ž
.
it is 1, 0 and 0, ¨ , where without loss of generality ¨ Gu)0. So the prefer-
ences are opposed, but not exact mirror images if ¨ )u. Types U, V then
respectively choose action 1 for private beliefs below pU l slr uql , and
Ž .
Ž
.
above pV l slr ¨ ql .
Ž .
Ž
.
Assume the same bounded beliefs structure introduced earlier. Assume we
Ž
.
start at l0 g 2¨r3, 2u . The transition probabilities for type U are then just as
UŽ < UŽ <
. Ž .Ž
1 L, l s 3ly2u 3lq
.
.
Ž
.
in Section 3.1.B: ␳ 1 H, l s 3ly2u r2l and
␳
2
.
Ž
.
V Ž <
2u r8l , where lg 2ur3, 2u ; for type V, we likewise have ␳ 1 H, l s 2¨ y
.
Ž
2
.
V Ž <
l r2l and ␳ 1 L, l s 2¨ ql 2¨ yl r8l by applying 3 . The two types take
Ž
.Ž
.
Ž .
U
V
w
x
w
x
action 2 with certainty in the intervals J2 s 0, 2ur3 and J2 s 2¨, ϱ , respec-
U
V
w
x
w
x
tively. If either these sets or J1 s 2u, ϱ and J1 s 0, 2¨r3 overlap, as happens
with 2¨r3-2u or 2ur3-2¨, then only one type ever makes an informative

382
L. SMITH AND P. SøRENSEN
choice for any l, and the resulting analysis for the other type is similar to the
single rational type model: just limit cascades, and therefore herds, arise.
Ž
.
Assume no cascade sets overlap. Consider dynamics for lg 2¨r3, 2u given
Ž .
by 4 :
U 3ly2u
2l
V 2¨ yl
2l
Ž <
.
␺ 1 H, l s␭
q␭
and
Ž
.Ž
8l2
.
Ž
.Ž
8l2
.
3ly2u 3lq2u
2¨ yl 2¨ ql
U
V
Ž <
.
␺ 1 L, l s␭
q␭
.
We are interested in a different fixed point depicted in Figure 5, where neither
rational type takes any action for sure, and decisions always critically depend on
Ž .
Ž <
.
Ž <
.
private signals. The two continuations 5 then coincide: ␺ 1 H, l* s␺ 1 L, l*
Ž
.
g 0, 1 , and actions convey no information:
U
Ž
Ž
.Ž
2¨ yl 3ly2¨
.Ž
2uyl 3ly2u
.
.
␭
␭
Ž .
'h l .
s
V
U
¨
Ž
.
Ž
.
Ž .
If ¨ )u then h maps 2¨r3, 2u onto 0, ϱ , and h l* s␭ r␭ is solvable for
any ␭U, ␭V.
² :
For this example, we can argue that with positive chance, the process ln
w
x
w
x
w
tends to l* if it does not start in a cascade, i.e. in 0, 2ur3 or 2¨, ϱ . Since each
Ž
.
x
likelihood continuation ␸ i, и is increasing, if dynamics start in 2ur3, l* or
w
x
Ž
.
² :
l*, 2¨ , they are forever trapped there. Assume l0 g 2ur3, l* . Then ln is a
bounded martingale that tends to the end-points; therefore, the limit lϱ places
positive probability weight on both a limit cascade on ls2ur3 and convergence
FIGURE 5.ᎏConfounded learning. Based on our Bounded Beliefs Example, with ␭U s4r5,
Ž <
.
Ž <
.
us¨r2. In the left graph, the curves ␺ 1 H, l and ␺ 1 L, l cross at the confounding outcome
l*s8¨r9, where no additional decisions are informative. At l*, 7r8 choose action 1, and strangely
7r8 lies outside the convex hull of ␭ and ␭Uᎏe.g., in the introductory driving example, more than
70% of cars may take the high ramp in a confounding outcome. The right graph depicts continuation
likelihood dynamics.
V

OBSERVATIONAL LEARNING
383
to lsl*. This example verifies that the latter fixed point robustly exists and is
stable, but does not yet explain why.
4. LONG RUN LEARNING
4.1. Belief Con¨ergence: Limit Cascades and Confounded Learning
² :
A. Characterization of Limit Beliefs. That ln is a martingale in state H rules
Ž
.
out nonstationary limit beliefs such as cycles , and convergence to incorrect
point beliefs. Markovian methods then prove that with a single rational type, as
in Section 3.1, limit cascades arise, or ln ªJ. But with multiple rational types,
² :
the example in Section 3.2 exhibits another possibility: ln may converge to
where each action is equilikely in all states. Then define the set K of confound-
ing outcomes as those likelihood ratios l*fJ satisfying
Ž .
6
Ž . Ž .
<
<
␺ m s, l* s␺ m H, l* for all actions m and states s.
Observe that since l*fJ, decisions generically depend on private beliefs. Yet
decisions are not informative of beliefs, given the pooling across types. Also,
history is still of consequence, or otherwise decisions would then generically be
informative. Rather history is precisely so informative as to choke off any new
inferences. The distinction with a cascade is compelling: Decisions reflect own
private beliefs and history at a confounding outcome, whereas in a limit cascade,
history becomes totally decisive, and private beliefs irrelevant.
Markovian and martingale methods together imply that with bounded private
beliefs, fully correct learning is impossible. Only a confounding outcome or limit
cascade on the wrong action are possible incomplete learning outcomes. Theo-
rem 1 summarizes.
THEOREM 1: Suppose without loss of generality that the state is H.
Ž .
Ž .
:
a With a single rational type, a not fully wrong limit cascade occurs: supp lϱ
ꢀ 4
JR ϱ .
Ž .
b With a single rational type, and unbounded pri¨ate beliefs, ln ª0 almost
surely.
Ž .
c With TG2 rational types with different preferences, only a limit cascade that
Ž .
ꢀ 4
is not fully wrong or a confounding outcome may arise: supp lϱ :JjKR ϱ .
H
Ž .
d With bounded pri¨ate beliefs, lϱ gJRJ with positi¨e chance pro¨ided
l0 fJ H. Likewise, if l0 /l*, no single confounding outcome l* arises almost surely.
Ž .
e
Fix payoff functions for all types, and a sequence of pri¨ate belief distributions
w
x Ž
.
Ž
.
with supports bk , bk ks1, 2, . . . . If bk ª1 or bk ª0 in state L , then the
chance of an incorrect limit cascade lϱ gJRJ H ¨anishes as kªϱ.
Ž .
w
.
PROOF: First, Lemma 3 asserts supp lϱ : 0, ϱ in state H, i.e. lϱ -ϱ a.s.

384
L. SMITH AND P. SøRENSEN
ˆ
Ž .
Theorems B.1, 2 tightly prescribe lϱ: Any lgsupp lϱ is stationary for the
Markov process, i.e. either an action doesn’t occur, or it teaches us nothing:
ˆ
.
Ž
<
Absent signal atoms, with continuous transitions, this means that ␺ m s, l s0
ˆ
ˆ
Ž
.
or ␸ m, l sl for all mg M.
Ž .
Proof of a : Assume ␸ continuous in l. A single rational type must take
some action m with positive chance in the limit l, and thus ␸ m, l sl. Since
ˆ
ˆ
.
ˆ
Ž
Ž
<
.
Ž
<
.
␺ m s, l s␬m q␬␳ m s, l ,
ˆ
.
␬m q␬␳ m L, l
Ž
<
ˆ
ˆ
.
ˆ
Ž
ls␸ m, l sl
ˆ
.
␬m q␬␳ m H, l
Ž
<
ˆ
ˆ
.
Ž
<
.
Ž
<
and so ␳ m H, l s␳ m L, l )0. Intuitively, statistically constant noisy behav-
ior does not affect long run learning by rational individuals, as it is filtered out.
Next, pick the least action m taken in the limit by the rational type with positive
Ž
.
chance i.e. for low enough private beliefs . Since private beliefs are informative
Ž ..
Ž
Lemma A.1 c , m is strictly more likely in state L than H, and is thus
ˆ
.
ˆ
.
ˆ
Ž
<
Ž
<
informativeᎏunless ␳ m H, l s␳ m L, l s1. Hence, lgJm. The Appendix
analyzes the case of a discontinuous function ␸.
Ž .
ꢀ
4
Ž .
Proof of b : Js 0, ϱ by Lemma 2 with unbounded beliefs. So supp lϱ :J
and lϱ -ϱ a.s. together jointly imply lϱ s0 a.s.
ˆ
ˆ
.
Ž .
Ž
Ž
<
<
.
Ž
<
Proof of c : With TG2 types, we now have ␺ m s, l s␬m q␬␳ m s, l , and
ˆ
ˆ
.
.
Ž
<
the previous complete learning deduction fails: ␺ m H, l s␺ m L, l for all m
tŽ
need not imply ␳ m H, l s␳ m L, l s1 for all t. Instead, we can only assert
<
.
tŽ
<
ˆ
ˆ
.
ˆ
ˆ
lgJ or lgK.
Proof of d : Recall that J s 0, l where lsmint lt, and l, ϱ :J where
H
Ž .
w
x
w
w
x
t
x
lsmaxt l , across ts1, . . . , T. If lϱ g l, ϱ with positive chance, then we are
done. Otherwise ln Fl-ϱ a.s. for all n. By the Dominated Convergence
² :
Theorem, the mean of the bounded martingale ln is preserved in the limit:
H
w x
Ž .
w
x
E lϱ sl0. So supp lϱ :J ' 0, l fails if l0 )l. Similarly, lϱ sl* with probabil-
ity 1 cannot obtain if l0 /l*.
Ž . x
Ž
.
w
k
Proof of e : By Lemma 2 a, d , only the extreme cascade sets Jk s 0, lk j
w
x
Ž
Ž
.. x Ž
w
.
lk , ϱ exist for co supp Fk close enough to 0, 1 i.e. large k . If ␲ is the
chance of an incorrect limit cascadeᎏnamely, on action 1ᎏthen Elϱ G␲k lk.
Ž .
But Elϱ Fl0 by Fatou’s Lemma, so that ␲k Fl0rlk. By 2 , the knife-edge
cascade public likelihood ratio lkt and the highest private belief bkt yield the
t
t
t
t
t
Ž
.
t Ž
.
posterior r1, by Bayes rule: 1yr1 rr1 slk 1ybk rbk. So lk ªϱ as bk ª1, and
Ž
.
thus ␲k ª0. In state L, ␲k ª0 as bk ª0, and lk ª0.
Q.E.D.
Ž .
Observe how Part e makes sense of the bounded versus unbounded beliefs
knife-edge, since there is a continuous transition from incomplete to complete
learning.2
2
Ž
.
We think it noteworthy that Milgrom’s 1979 auction convergence theorem, which also
concerns information aggregation but in an unrelated context, turns on a bounded-unbounded signal
knife-edge too.

OBSERVATIONAL LEARNING
385
B. Robustness of Confounding Outcomes. Lemma 2 establishes the robustness
of cascade sets. However, unlike cascade sets, the existence of confounding
outcomes is not foretold by the Bayesian decision problem. They are only
inferred ex post, and are nondegenerate phenomena. Fleshing this out, the
²
sŽ
.
model parameters are the preferences and typernoise proportions ut m ,
␬m, ␭ , elements of the Euclidean normed space ޒSM TqMqT. Genericity and
t
:
robustness are defined with respect to this parameter set.
THEOREM 2: Assume there are TG2 rational types.
Ž .
b
a Confounding outcomes robustly exist, and are in¨ariant to noise.
Ž .
At any confounding outcome, at least two rational types are not in a cascade
set.
Ž .
c
For generic parameters, at a confounding outcome, at most two actions are
Ž
.
taken by acti¨e rational types i.e. those who are not in a cascade .
Ž .
d
If belief distributions are discrete, confounded learning is nongeneric.
Ž .
e With M)2 actions and unbounded beliefs, generically no confounding
outcome exists.
Ž .
f
At any confounding outcome, some pair of types has opposed preferences.
Assume Ms2 and some types with opposed preferences. With atomless
Ž .
g
bounded beliefs and Ts2, a confounded learning point exists generically, pro¨ided
both types are acti¨e o¨er some public belief range. With atomless unbounded beliefs
and f H 1 , f 0 )0, a confounding point exists if the opposed types ha¨e suffi-
Ž . LŽ .
ciently different preferences.
Before the proofs, observe that while generically only two actions are active at
Ž
Ž ..
any given confounding outcome Part c , nondegenerate models with M)2
actions still have confounding outcomes. For with bounded beliefs, only two
actions may well be taken over a range of possible likelihoods l. Second, note
that a confounding outcome is in one sense a more robust failure of complete
learning than is an incorrect limit cascade, since it arises even with unbounded
Ž
Ž ..
private beliefs Part g .
PROOF:
Ž .
Proof of a : This has almost been completely addressed by the third example
in Section 3, which is nondegenerate in the specified sense. Invariance to noise
Ž .
follows because shifting ␬ identically affinely transforms both sides of 6 ,
m
Ž .
given 4 .
Ž .
Ž .
Proof of b : By the proof of Theorem 1 a , if all but one rational type is in a
cascade in the limit, then so is that type, given 6 . So at least two rational types
Ž .
are active.
Ž .
Proof of c : Consider the equations that a confounding outcome l* must
solve. First, with bounded beliefs, some actions may never occur at l*. Assume
that M0 FM actions are taken with positive probability at l*. Next, given the
M0
M0
Ž
<
.
Ž
<
.
Ž .
adding-up identity Ýis1␺ mi H, l sÝis1␺ mi L, l s1, 6 reduces to M0 y1
Ž
<
.
Ž
<
.
equations of the form ␺ m H, l s␺ m L, l , in a single unknown l. As the

386
L. SMITH AND P. SøRENSEN
equations generically differ for active rational types, they can only be solved
when M0 s2.
t
t
H
L
Ž .
Ž .
tŽ
<
.
tŽ
<
.
Proof of d : Rewrite
6 as Ýt ␭␳ m H, l sÝt ␭␳ m L, l . For F , F
tŽ
<
.
discrete, each side assumes only countably many values. As ␳ m H, l y
tŽ
<
.
Ž .
␳ m L, l generically varies in t, the solution of 6 generically vanishes as the
t
␭ weights are perturbed.
Proof of e : With unbounded private beliefs, for any lg 0, ϱ , all M actions
are taken with positive chance. By Part c , confounding outcomes generically
Ž .
Ž
.
Ž .
can’t exist.
Ž .
Proof of f : If actions 1 and 2 are taken at l*, and all types prefer ms2 in
state H, then ms2 is good news for state H, whence l* could not be a
confounding outcome.
Ž .
w
x
Proof of g : Consider an interval l, l between any two consecutive portions
Ž <
.
of the cascade set J. The Appendix proves that under our assumptions, ␺ 1 H, l
Ž <
.
Ž <
.
Ž <
.
exceeds ␺ 1 L, l near l iff ␺ 1 L, l exceeds ␺ 1 H, l near l. Without signal
atoms, both are continuous functions, and therefore must cross at some interior
Ž
.
point l*g l, l .
Q.E.D.
Ž
Ž ..
For an intuition of why confounding points exist Theorem 2 g , consider our
binary action Texas driving example depicted in Figure 5. By continuity, it
Ž <
suffices to explain when one should expect the antisymmetric ordering ␺ 1 L, l
c␺ 1 H, l , respectively, for l small near l and large near l . The critical idea
here is that barring a cascade, a partially-informed individual is more likely to
choose a given course of action when he is right than when he is wrong Lemma
A.1 .
Since Houston drivers wish to go high in state H, uniformly across public
.
Ž <
.
Ž
.
Ž
.
Ž
.
beliefs, more Houston drivers will go high in state H than in state L. The
reverse is true for Dallas drivers. The required antisymmetric ordering clearly
occurs if and only if most contrarians are of a different type near l than near l.
With bounded beliefs, as in the example of §3.1B, this is true simply because a
different type is active at each extreme, for generic preferences.
With unbounded beliefs, the above shortcut logic fails, as both types are
active for all unfocused beliefs. The key economic ingredients for the existence
Ž .
of a confounding point are then i not too unequal population type frequencies,
Ž .
Ž .
and ii sufficiently strongly opposed preferences by the rational types. To see i ,
assume the extreme case with rather disparate type frequencies, and nearly
everyone Houston-bound. No antisymmetric ordering can then occur, as
Ž <
.
Ž <
.
Ž
.
Ž .
␺ 1 L, l -␺ 1 H, l for all lg l, l . Next, condition ii ensures that the two
types’ action basins for opposing actions grow, and contrarians of each
typeᎏthose whose private beliefs oppose the public beliefᎏincrease as we
approach one extreme, and decrease as we approach the other, in opposition.
To illustrate the intuition in an unbounded beliefs variant of the driving
Ž
Ž ..
example from the Appendix proof of Theorem 2 g , a confounding point exists

OBSERVATIONAL LEARNING
387
U
V
U
V
Ž
for ur¨ c␭ r␭ c¨ru. In words, type frequencies are not too far apart ␭ r␭
.
not too big or small ᎏand given their disparity, preferences are sufficiently
strongly opposed ur¨ big or small enough .
Ž
.
4.2. The Traditional Case: Herds with a Single Rational Type
and No Noise
We have already argued that actions converge in frequency. Without noise,
this can be easily strengthened. After any deviation from a potential herd, or
finite string of identical actions, an uninformed successor will necessarily follow
suit. In other words, the public belief has radically shifted. As in Section 3, this
logic proves that herds arise.
THEOREM 3: Assume a single rational type and no noise.
Ž .
a
A herd on some action will almost surely arise in finite time.
Ž .
b With unbounded pri¨ate beliefs, indi¨iduals almost surely settle on the optimal
action.
Ž .
c With bounded pri¨ate beliefs, absent a cascade on the most profitable action
from the outset, a herd arises on another action with positi¨e probability.
Ž .
Ž .
PROOF: Part a follows from the convergence result of Theorem 1 a and the
following O¨erturning Principle: If Mr. n chooses any action m, then before
nq1 observes his own private signal, he should find it optimal to follow suit
because he knows no more than n, who rationally chose m. To wit, lnq1 gIm
after n’s action, and a single individual can overturn any given herd. The
Ž
.
Appendix analytically verifies this fact. From Lemma 2, Jm ;int Im , so that
Ž .
Ž
.
when supp lϱ :Jm, eventually ln gint Im , precluding further overturns.
Ž .
Ž .
Ž .
Ž .
Ž .
Finally, Parts b and c are corollaries of Part a and Theorem 1 b and d .
Q.E.D.
This characterization result extends the major herding finding in BHW to
Ž
general signals and noise. BHW also handled several states, addressed here in
.
§6.1 . The analysis of BHWᎏwhich did not appeal to martingale methodsᎏonly
succeeded because their stochastic process necessarily settled down in some
stochastic finite time. Strictly bounded beliefs so happens to be the mainstay for
their bad herds finding. Full learning doesn’t require the perfectly revealing
signals in BHW, ruled out here.
5. STABLE OUTCOMES AND CONVERGENCE
Above, we have identified the candidate limit outcomes. But one question
remains: Are these limits merely possibilities, or probabilities? We address this
with local stability results. Convergence is rapid, and this affords insights into
why herding occurs.

388
L. SMITH AND P. SøRENSEN
5.1. Local Stability of Marko¨-Martingale Processes
²Ž
.:
We state this theoretical finding in some generality. Let mn, xn be a
ꢀ
4
discrete-time Markov process on M=ޒ, for some finite set Ms 1, 2, . . . , M ,
with transitions given by
Ž .
7
Ž
.
Ž
<
.
Ž .
mgM .
xn s␸ m, xny1 with probability ␺ m xny1
M
²
:
Ž
ˆ
< . Ž
.
ˆ
Further assume that xn is a martingale: that is, x'Ýms1␺ m x ␸ m, x . Let
Ž . < .
Ž
.
Ž
x be a fixed point of 7 , so that for all m: either xs␸ m, x or ␺ m x s0.
ˆ
ˆ
Our focus will be on functions ␸ and ␺ that are C1 once continuously
Ž
ˆ
.
ynŽ
differentiable at x. If x ªx, then ␪ is a con¨ergence rate provided ␪
.
ˆ
x yx
ˆ
ˆ
n
n
ˆ
²
:
ª0 at all ␪)␪. Observe that if ␪ is a convergence rate of xn , then so is any
␪Ј)␪. Also, if x sx for some n , then ␪s0.
ˆ
Appendix C denv0elops a local sta0bility theory for such Markov processes. For
²
:
an intuitive overview, recall that near the fixed point x, xn is well approxi-
ˆ
mated by the following linearized stochastic difference equation: starting at
Ž
.
Ž
.
Ž
Ž
.Ž ..
m , x y x , the continuation is
m
nq1, xnq1 y x s m, ␸ m, x x y x
ˆ
ˆ
ˆ
ˆ
n
n
x
n
Ž
< .
ˆ
²
:
with chance ␺ m x . Now, for a linear process yn , where ynq1 sam yn with
chance pm ms1, . . . , M , we have yn sa1 Žn. иии aM Žn. y0, where ␹ n counts
␹
␹
1
m
Ž
.
Ž .
m
Ž .
the m-continuations in the first n steps. Since ␹ n rnªpm almost surely
m
by the Strong Law of Large Numbers, the product a1p1 иии aMpM fixes the long-run
²
:
stability of the stochastic system yn near 0. Accordingly, the product
determines the local stability of the original nonlinear
system 7 near x. Rigorously, we have the following theorem.
M
␺ Žm< x.
ˆ
Ž
.
Łms1␸ m, x
ˆ
x
Ž .
ˆ
THEOREM 4: Assume that at a fixed point x of the Marko¨-martingale process
ˆ
1
1
Ž .
Ž
< .
Ž
.
ˆ
Ž
.
Ž
ˆ
.
7 , ␺ m и and ␸ m, и are C resp. left or right C , for all m. Assume ␸ m, и
Ž
.
Ž
< .
is weakly increasing near x for all m, and ␸ m, x /1 for some m with ␺ m x )0.
ˆ
x
Ž
.
ˆ
Then x is locally stable: with positi¨e chance, x ªx resp. x - x or x x x for x0
ˆ
ˆ
ˆ
ˆ
n
n
n
Ž
.
near x resp. below or abo¨e x . Whene¨er x ªx, con¨ergence is almost surely at
ˆ
␺ Žm< x.
ˆ
ˆ
n
M
Ž
.
ˆ
the rate ␪'Łms1␸ m, x
-1.
x
PROOF: For the in-text proof here, we make the simplifying assumption that
Ž
.
xs␸ m, x for all m. By Corollary C.1, given a frequency-weighted geometric
ˆ
ˆ
mean ␪-1 of the continuation derivatives, x ªx with positive chance, and at
ˆ
n
rate ␪.
M
Ž
< . Ž
.
Differentiate the martingale identity x'Ýms1␺ m x ␸ m, x to get
M
M
Ž .
8
Ž
< .
Ž
.
Ž < . Ž
␺ m x ␸ m, x .
x
.
1s
␺ m x ␸ m, x q
Ý
Ý
x
m
s1
ms1
M
M
Ž
< .
Ž < .
x
Differentiating the probability sum Ýms1␺ m x '1 yields Ýms1␺ m x s0.
Ž
.
ˆ
Ž .
Because ␸ m, x sx for all m at the fixed point x, the second sum in 8
ˆ
ˆ
vanishes at x, and we are left with ÝM ␺ m x ␸ m, x s1. The continuation
< .
Ž
Ž
.
ˆ
ms1
x
Ž
.
Ž
.
ˆ
slopes ␸ m, x G0 are not all equal since we have assumed ␸ m, x /1 for
x
x

OBSERVATIONAL LEARNING
389
Ž
.
some m. Hence, the arithmetic mean᎐geometric mean AM᎐GM inequality
holds strictly. This yields ␪-1.
Ž
.
Ž
.
The proof admitting the event that ␸ m, x /x and ␺ m, x s0 is appendi-
ˆ
ˆ
ˆ
cized.
Q.E.D.
5.2. Cascade Sets Attract Herds, and Confounded Learning Arises
²
:
We now apply Theorem 4 to ln, mn , and prove that both incomplete
Ž
.
learning outcomes do arise. Theorem 5 a, b below shows that a limit cascade
develops with positive chance if beliefs initially lie near a cascade set. Absent
belief atoms, and with monotonic continuation functions ␸ m, и , this follows
Ž
.
Ž
.
Ž
.
from Theorem 4 provided ␸ m1, l* /␸ m2 , l* , some m1 /m2. A positive
l
l
private belief tail density guarantees this inequality. The details, as well as
Ž
.
consideration of nonmonotonic ␸ m, и , are appendicized.
Ž .
² :
Next, Part c proves that ln tends with positive chance to the confounding
outcome l*, even though it is an isolated interior point. The reason for this is
that l* can be locally stochastically stable. This occurs provided ␸ m1, l* /
Ž
.
l
Ž
.
␸ m2 , l* , where actions m1 and m2 are taken with positive probability at l*.
l
Not being an identity, this inequality is generically satisfied. For nondegenerate
parameters, both derivatives are positive, as the example shows. We let con-
founded learning denote convergence to the confounding point, namely the event
where ln ªl*.
THEOREM 5: a Assume bounded pri¨ate beliefs and let F H, F L ha¨e C1 tails,
Ž .
1
Ž . LŽ .
with f H b , f b )0. If l0 is close enough to the cascade set component JsJm
ˆ
1
l иии lJmT ;J, and J is not a single point, then ln ªlgJ with positi¨e chance, and
ˆ
ˆ ˆ
T
t
ˆ
at some rate ␪-1. Whene¨er lϱ gJ, a herd de¨elops: e¨entually type t takes am .
t
Ž .
Ž
.
b
Assume unbounded pri¨ate beliefs. If inf K)0, then Pr lϱ s0 ª1 as
l0 ª0.
Ž .
c Confounded learning occurs: For nondegenerate data, points l*gK are
locally stable.
With atomic tails of the private belief distributionsᎏas in BHW’s analysis
Ž
.
with discrete private signalsᎏeach active continuation ␸ m, и is discontinuous
² :
near its associated cascade set Jm; therefore, ln might well leap over a small
enough cascade set Jm. By graphical reasoning, when F H, F L have C1 tails,
w
.
dynamics can jump into the cascade set 2u, ϱ in Figure 4 with a left derivative
Ž
.
␸
1, 2u -0. More generally, a single action can toss everyone into a cascade
wliyth a nonmonotonic continuation; and a confounding outcome need not be
stable. While we know of no simple sufficient conditions that guarantee mono-
tonic continuation functions, we can show how a nonmonotonicity may arise:
Ž
.
LŽ Ž .. H Ž Ž ..
LŽ Ž .. H Ž Ž ..
Since ␸ 1, l slF p l rF p l , the private belief odds F p l rF p l ,
decreasing by Lemma A.1 e , might be more than unit-elastic in the prior
Ž .

390
L. SMITH AND P. SøRENSEN
likelihood ratio l. This arises with nonatomic private beliefs having spiked
Ž
.
densities see our MIT working paper with the same title .
Ž .
Finally, part a provides a direct explanation as to why herds arise with a
single type. Our current logic is indirect: Martingale methods force belief
convergence, and a failure to herd precludes belief convergence. Yet a herd
arises if and only if eventually the public belief swamps all private beliefs. Here
we see more directly that belief convergence is exponentially fast, and ipso facto
the sequence of contrarian chances is summable. Namely, if Ak denotes the
event that Mr. k deviates once a herd has begun, then on almost surely all
Ž
.
approach paths to the fixed point, Pr Ak vanishes geometrically fast given the
Ž
.
exponential stability, so that Ý Pr Ak -ϱ. By the first Borel-Cantelli Lemma,
Ž
.
ꢀ
4
Pr Ak infinitely often s0ᎏeven though the events Ak are not independent.
In other words, an infinite sequence of contrarians is impossible, and a herd
must eventually start.
6. CONCLUSION AND EXTENSIONS
6.1. Multiple States
Ž .
a The Re¨ised Model. We can admit any finite number S of states. Rather
w
x
than partition 0, 1 into closed subintervals, optimal decision rules instead slice
the unit simplex in ޒSy1 into closed convex polytopes. Belief thresholds become
hyperplanes. Fixing a reference state, the likelihood ratios l1, . . . , lSy1 are each a
convergent conditional martingale.
Ž .
b
Re¨ised Con¨ergence Notions. The extreme interval cascade sets for each
rational type generalize to convex sets around each corner of the belief simplex.
With unbounded beliefs, all cascade sets lie on the boundary of the simplex;
with bounded beliefs, interior public beliefs near the corners lie in cascade sets,
and there can exist insurance action cascade sets away from the boundary. Limit
cascades must arise with a single rational type.
Long-run ties where two or more actions are optimal in a given state may
nongenerically occur, as BHW note. Barring this possibility, a herd occurs
eventually with probability one with a single rational type and no noise. If some
action is optimal in several states of the world, then its cascade set will contain
the simplex face spanned by these states. Even with unbounded beliefs, com-
plete learning need not obtain, as the limit belief may lie on this simplex face
but not at a corner. Yet there is adequate learning, in the terminology of Aghion,
Ž
.
Bolton, Harris, and Jullien 1991 : Eventually an optimal action is chosen.
Ž .
c
Robustness of Confounding Outcomes. At a confounding outcome, for
each ratio l , all M possible continuations must coincide. This produces My
1 Sy1 generically independent equations to be solved in the Sy1 likelihood
ratios: this is possible for nondegenerate data if Ms2, while M)2 yields an
over-identified system. Thus, the existence of confounding outcomes is robust:
Theorem 2 reads exactly as before.
s
Ž
.Ž
.

OBSERVATIONAL LEARNING
391
Ž .
Ž . Ž
Robustness of Local Stability. The extension of Theorem 5 a stable
d
.
interior cascade sets eludes us as it is unclear on which limit point to focus as
we near the hyperplane-frontier. The proof of Theorem 5 b stable extreme
cascade sets is robust to several states, since we may simply consider the
Ž . Ž
.
likelihood ratio of all other states to the reference state.
Ž . Ž
.
For Theorem 5 c stable confounding outcomes , we assume the generic
ˆ
property that in a neighborhood of the confounding outcome l, only two actions
Ž
.
m1, m2 are active taken by rational types with positive chance . For each
Ž
.
Ž
.
inactive action m, we have ␸ m, l sl, and thus the Jacobian Dl ␸ m, l sI.
M
Ž
<
.
Ž
.
Differentiating the martingale property yields Ýms1␺ m S, l Dl ␸ m, l sI.
Since inactive actions leave l unchanged in this neighborhood, we may as well
ˆ
Ž
Ž
<
.
focus entirely on the occurrences of m1 and m2 , now with chances ␺ mi S, l s
ˆ
ˆ
.
Ž
Ž
<
<
. Ž
.
Ž
Ž
<
.
Ž
<
..
Ž
<
.
␺ mi S, l r ␺ m1 S, l q ␺ m2 S, l . Then ␺ m1 S, l Dl ␸ m1, l q
ˆ
ˆ
.
ˆ
␺ m2 S, l Dl ␸ m2 , l sI. By Theorem C.2, l is locally stable under the weak
ˆ
.
Ž
condition that the Jacobians Dl ␸ mi, l both have real, distinct, positive, and
non-unit eigenvalues. Since eigenvalues are zeros of the characteristic polyno-
mial, this requirement is nondegenerate. It is the natural generalization of
nonnegative derivatives of ␸.
6.2. Links to the Experimentation Literature
Ž
.
Our 1998 companion paper draws a formal parallel between bad herds and
the well-known phenomenon of incomplete learning in optimal experimentation.
Ž
It also shows that even a patient social planner unable to observe private
.
signals who controls action choices will with positive probability succumb to bad
herds, assuming bounded private beliefs. The confounded learning that we find
Ž
.
is formally similar to McLennan’s 1984 confounded learning in experimenta-
tion, in the precise sense that probability density functions of the observables
coincide across all states at an interior belief. McLennan’s finding is not robust
to more than two observable signals, nor is ours to more than two active actions.
Thus, if his buyers desire to buy more than one unit of the good, confounded
learning is a nondegenerate outcome. However, as noted, there may be three or
more actions in our informational herding setting, even though only two are
active at some public belief.
Dept. of Economics, Uni¨ersity of Michigan, 611 Tappan Street, Ann Arbor, MI
48109-1220, U.S.A.; econ-theorist@earthling.net; http:rrwww.umich.edur;lones,
and
Institute of Economics, Uni¨ersity of Copenhagen, Studiestræde 6, DK-1455
Copenhagen,Denmark;peter.sorensen@econ.ku.dk;http:rrwww.econ.ku.dkrsorensen.
Manuscript recei¨ed July, 1996; final re¨ision recei¨ed January, 1999.

392
L. SMITH AND P. SøRENSEN
APPENDIX A: ON BAYESIAN UPDATING OF DIVERSE SIGNALS
Ž
.
To justify the private belief structure of §2, let ⌺, ␮ be an arbitrary probability measure space.
L
Partition the signal measure as ␮s ␮H q␮ r2, the average of two state-specific signal measures.
Ž
.
s
ꢀ
4
Conditional on the state, signals ␴ are i.i.d., and drawn according to the probability measure ␮ in
n
H
L
ꢀ
4
state sg H, L . Assume ␮ , ␮ are mutually absolutely continuous, so that there is a Radon-
L
H
Ž
.
Nikodym derivative gsd␮ rd␮ : ⌺ª 0, ϱ . Given signal ␴g⌺, Bayes’ rule yields the private
s
Ž
.
.
Ž Ž
.
.
Ž
.
belief p ␴ s1r g ␴ q1 g 0, 1 that the state is H. The private belief c.d.f. F is the distribution
of p ␴ under the measure ␮ . Then F and F L are mutually absolutely continuous, as ␮ and
s
H
H
Ž
L
␮
are.
LEMMA A.1: Consider the c.d. f.’s F H, F L resulting from ⌺, ␮ and a fair prior on H, L.
Ž
.
The deri¨ati¨e f'dF LrdF H of pri¨ate belief c.d. f.’s F H, F L satisfies f p s 1yp rp almost
Ž . Ž .
Ž .
a
surely in pg 0, 1 . Con¨ersely, if f p s 1yp rp, then F H, F L arise from updating a common prior
Ž
.
Ž
.
Ž
.
with some signal measures ␮H, ␮L.
F L p yF
is nondecreasing or nonincreasing as pb1r2.
.
p except when both terms are 0 or 1.
Ž .
Ž
Ž
Ž
.
HŽ
LŽ
.
p
b
F H p -F
Ž .
.
c
d
F L p rF p G 1yp rp for pg 0, 1 , strictly so if F p )0.
Ž .
.
HŽ
.
Ž
.
Ž
.
HŽ
.
e
The likelihood ratio F HrF L is weakly increasing, and strictly so on supp F .
Ž .
Ž .
PROOF: If the individual further updates his private belief p by asking of its likelihood in the two
.x
w
Ž
4
states of the world, he must learn nothing more. So, ps1r 1qf p , as desired. Conversely, given
s
Ž
.
Ž
.
ꢀ
f p s 1yp rp, let ␴ have distribution F in state s, sg H, L .
Ž .
a
Ž .
Ž
.
ꢀ
Ž .
4
Ž .
c
Part
implies Part b , and is strict when pgsupp F R 1r2 . Hence
follows. Since
fsdF LrdF H is a strictly decreasing function, we can prove Part d :
F L p s
p
dF H r sF p f p sF
Ž .
9
Ž
.
Ž .
f r dF r )f
H Ž .
Ž
.
Ž .
H Ž . Ž
.
H Ž .Ž
.
p 1yp rp
H
H
r
Fp
rFp
for any p with F H p )0. For e , whenever F p )F q )0, 9 implies
Ž
.
Ž .
LŽ
.
LŽ .
Ž .
Ž
.
L Ž
.
F L p yF
q
p
Ž .
H Ž .
w
H Ž
.
H Ž .x Ž
f q - F p yF
.
w
H Ž
.
H Ž .x L Ž
.
H Ž
q rF q .
.
s
f r dF r - F p yF
q
q F
H
q
Expanding the right-hand side, it immediately follows that F H p rF p )F q rF q . Q.E.D.
Ž
.
LŽ
.
HŽ . LŽ .
APPENDIX B: FIXED POINTS OF MARKOV-MARTINGALE SYSTEMS
This appendix establishes results needed to understand limiting behavior of the Markov-martingale
process of beliefs and actions. Despite a countable state space, standard convergence results for
discrete Markov chains have no bite, as states are in general transitory.
Ž
.
Ž < .
Given is a finite set M, and Borel measurable functions ␸ и, и : M=ޒqªޒq, and ␺ и и :
w
x
M=ޒqª 0, 1 satisfying:
ⅷ
Ž <
.
Ž < .
␺ и x is a probability measure on M for all xgޒq, or Ýmg ␺ m x s1.
␾ and ␺ jointly satisfy the following ‘‘martingale property’’ for all xgޒq:
M
ⅷ
Ž
.
Ž . Ž .
<
10
␺ m x ␸ m, x sx.
Ý
m
gM
w
.
For any set B in the Borel ␴-algebra B on ޒqs 0, ϱ , define a transition probability P:
w
x
ޒq=Bª 0, 1 :
Ž
.
Ž
< .
P x, B s
␺ m x .
Ý
.
<
Ž
m ␸ m, x gB

OBSERVATIONAL LEARNING
393
ϱ
²
:
ns1
Let xn
be a Markov process with transition from xn ¬xnq1 governed by P, and Ex1 -ϱ.
²
:
Ž .
Then xn is a martingale, true to the above casual label of 10 :
w
<
x
w
<
x
E xnq1 x1 , . . . , xn sE xnq1 xn
Ž
.
Ž . Ž .
<
s
tP xn , dt s
␺ m xn ␸ m, xn sxn .
H
Ý
ޒq
m
gM
By the Martingale Convergence Theorem, there exists a real, nonnegative stochastic variable xϱ
²
:
such that xn ªxϱ a.s. Since xn is a Markov chain, the distribution of xϱ is intuitively invariant for
Ž
.
the transition P, as in Futia 1982 . The a.s. convergence then suggests that the invariant limit must
be pointwise invariant. While Theorem B.2 below can be proved along these lines, some continuity
assumption will be used. Doing away with continuity, we establish an even stronger result.
THEOREM B.1: Assume that the open inter¨al I:ޒq satisfies
Ž
.
Ž
<
.
Ž .
␸ m, x yx )␧.
11
᭚␧)0 ᭙xgI ᭚mgM: ␺ m x )␧
and
Then I cannot contain any point from the support of the limit xϱ.
Ž
.
PROOF: Let I be an open interval satisfying 11 for ␧)0, and suppose for a contradiction that
Ž
.
Ž
.
Ž .
there exists xgIlsupp xϱ . Let Js xy␧r2, xq␧r2 lI. By 11 , for all xgJ, there exists
Ž
<
.
Ž
.
Ž
.
,
mgM with ␺ m x )␧ and ␸ m, x fJ. Since xgsupp xϱ
xn gJ eventually with positive
probability. But whenever xn gJ, xnq1 fJ with chance at least ␧. That is, the conditional chance
²
:
that the process stays in J in the next period is at most 1y␧. So the process xn almost surely
²
:
eventually exits J. This contradicts the claim that with positive chance xn is eventually in J.
Hence, x cannot exist. Q.E.D.
Ž
.
Ž < .
THEOREM B.2: If x¬␸ m, x and x¬␺ m x are continuous for all mgM, then for all xg
Ž
.
Ž
ꢀ 4.
supp xϱ , stationarity P x, x s1 obtains, i.e.
Ž
.
Ž
<
.
Ž
.
12
Ž < .
PROOF: If there is an m such that x does not satisfy 12 , and both x¬␸ m, x and x¬␺ m x
␺ m x s0 or ␸ m, x sx
for all mgM.
Ž
.
Ž
.
Ž
<
.
Ž
.
are continuous, then there is an open interval I around x in which ␺ m x and ␸ m, x yx are
both bounded away from 0. This implies that 11 obtains, and so Theorem B.1 yields an immediate
contradiction. Q.E.D.
Ž
.
APPENDIX C: STABLE STOCHASTIC DIFFERENCE EQUATIONS
This appendix derives results on the local stability of nonlinear stochastic difference equations.
..
Ž
Ž
There is a very abstract related literature see Kifer 1986 , but Appendix 1 of Ellison and
Ž
.
Fudenberg 1995 treats a model closer to ours. We generalize their stability result to cover
state-dependent transition chances below, ␺ may depend on x , and multi-dimensional states x is
Ž
.
Ž
Ž
.
.
Sy1 -dimensional , and analyze convergence rates.
ꢀ
4
Ž
.
Given is a finite set Ms 1, . . . , M , and Borel measurable functions ␸ и, и : M=ޒqªޒ, and
Ž
<
.
w
x
Ž < .
Ž
.
␺ и и : M=ޒqª 0, 1 satisfying Ýmg ␺ m x s1. Let x0 gޒ. Then 7 defines a Markov process
M
²
²
:
:
²
: Ž .
xn . This can be recast as: Let
␴
be a sequence of i.i.d. uniform- 0, 1 random variables. Let
n
my 1
m
Ž
Ž
.
Ž
.x
yn with values in M be defined by yn sm when ␴n g Ýis1 ␺ i, xny1 , Ýis1␺ i, xny1 . Then
Ž
.
.
xn '␸ yn, xny1
Stability of Linear Equations.
Ž .
Ž
<
.
Ž
.
Consider this special case of 7 : ␺ m xn spm and ␸ m, xn sam xn. Here a1, . . . , aM gޒ and
w
x
p1, . . . , pM g 0, 1 satisfy Ýmg M pm s1.

394
L. SMITH AND P. SøRENSEN
p
M
m
<
<
LEMMA C.1: Define ␪sŁms 1 am
.
Almost surely, ␪yn xn ª0 for all ␪)␪. In particular, xn ª0 almost surely if ␪-1.
Ž .
a
Ž .
b
If ␪-1 and N0 is any open ball around 0, then there is a positi¨e probability that xn gN0 for all
n, pro¨ided x0 gN0.
With ␪)␪ and N0 an open ball around 0, Pr ᭙ngގ: ␪yn xn gN0 x0 gN0 )0.
Ž .
Ž .
c
<
Ž .
a
<
<
Ž
<
<
m.n <
<
Y
M
n
PROOF:
Let Ynm 'Ýnks11ꢀ y sm4, so xn s Łms1 am
x0 . Since Ynmrnªpm a.s. by the
k
m
Y
Strong Law of Large Numbers, the result follows from Łms 1 am
r n ª␪ a.s.
M
n
<
<
Ž .
b
Ž ꢀ 4. Ž .
Since xn ª0 a.s., Pr Dk g ގ FnG k xn gN0 s1. So Pr ᭙nGk, xn gN0 )0 for some k. So
²
:
with positive chance, xn stays inside N0 starting at that xk. Without loss of generality ks0 since
dynamics are time invariant. With linear dynamics, any x0 gN0 will do.
Ž .
c
Ž .
Ž .
Use the result in b on the modification of 7 with constants amr␪.
Q.E.D.
Local Stability of Nonlinear Equations.
Ž .
Ž
.
We care about the fixed points x of 7 : namely, where ␸ m, x sx for all mgM.
ˆ
ˆ
ˆ
Ž .
Ž
< .
Ž
.
THEOREM C.1: At a fixed point x of 7 , assume that each ␺ m и )0 is continuous and ␸ m, и
ˆ
has a Lipschitz constant3 Lm. If the stability criterion ␪'ŁmMs1 Lm
-1 holds, then for all
␺ Žm< x.
ˆ
Ž
.
Ž
.
ˆ
Ž
␪g ␪, 1 there exists an open ball N0 around 0, such that x yxgN «Pr x ªx GPr ᭙ngގ:
ˆ
0
0
n
yn
<
<
.
␪
x yx gN )0. If x ªx, then it con¨erges at rate ␪.
ˆ ˆ
n
0
n
Ž .
PROOF: First, we majorize 7 locally around x by a linear system, and then argue that Lemma
ˆ
C.1’s conclusion applies to our original nonlinear system.
Without loss of generality, assume that 0FL1 F иии FLM . By continuity of ␺ m, и we may
Ž
.
Ž
< .
ˆ
choose N0 small enough and constants pm close enough to ␺ m x so that
M
m
m
L pm -␪ ,
␺ i x G
p , and
i
Ž <
.
Ł
Ý
Ý
m
m
s1
is1
is1
Ž
.
Ž
.
<
<
␸ m, x y␸ m, x FLm xyx ,
᭙mgM
ˆ
ˆ
²
:
with x sx given, and
for all xyxgN . Fix x0 with x yxgN . Define a new process
x
ˆ
ˆ
˜
˜
0
0
0
Ž
0
n
0
x yxsL
x
˜
yx when ␴n g Ýmisy11pi, Ýmis1 pi where
Ž
.
ˆ
x
²
:
␴ is our earlier i.i.d. uniform se-
n
˜
ˆ
n
m
ny1
yn Ž
.
ˆ
quence. Lemma C.1 then asserts ␪
of
the nonlinear one xn : On N0 we have inductively in n,
x yx gN for all n with positive chance. In any realization
x yx gN for all n, the resulting deterministic linear process
˜
n
0
²
:
yn Ž
²
.
ˆ
²
:
x majorizes
˜
n
␴
yielding ␪
˜
n
n
:
0
m
y1
m
<
<
<
<
<
<
<
n
<
␴n g
p ,
p
« x yx sL
x
yx GL xny1 yx G x yx .
˜
n
ˆ
˜
ˆ
ˆ
ˆ
Ý
Ý
i
i
m
ny1
m
ž
i
s1
is1
yn Ž
So ␪
.
ˆ
x yx ª0, and with positive probability.
n
Finally, the rate of convergence is ␪, since for any ␪)␪, a small enough neighborhood exists for
²
:
which the linear system converges at a rate less than ␪. Whenever x ªx, xn eventually stays in
ˆ
n
²
:
.
that neighborhood, wherein it is dominated by
x
˜
Q.E.D.
n
Ž
.
COROLLARY C1: If each ␸ m, и is also continuously differentiable, then Theorem C.1 is true with
␺
Žm< x.
ˆ
Ž
m, x
x
.
␪sŁmg
␸
-1.
ˆ
M
We must generalize Theorem C.1 for xn gޒSy1, S)2. We focus on the Markov-martingale
context relevant for our model, and restrict attention to Ms2. A weaker, more generally applicable,
result could be obtained using the sup-norm of matrices; see our MIT working paper.
3
f: ޒm ªޒk is Lipschitz at x with Lipschitz constant LG0 if ᭙xgN x : f x yf x FL xyx
Ž . 5 Ž .
ˆ
Ž .5
ˆ
5
5
ˆ
ˆ
1
Ž .
ˆ
5
Ž .5
for some neighborhood N x . If f is C at x, then it is Lipschitz with any constant L) Df x .
ˆ
ˆ

OBSERVATIONAL LEARNING
395
Ž .
Ž
THEOREM C.2: Let x be a fixed point of
7
in ޒSy1 with Ms2. Assume that each ␺ m и is
< .
ˆ
continuous at x, with 0-␺ 1 x -1, and that each ␸ m, и is C1 at x. Assume that each D ␸ m, x
Ž <
.
Ž
.
Ž
.
ˆ
ˆ
ˆ
ˆ
x
Ž <
.
ˆ
Ž
.
ˆ
Ž <
.
Ž
.
ˆ
has distinct, real, positi¨e, non-unit eigen¨alues, and that ␺ 1 x D ␸ 1, x q␺ 2 x D ␸ 2, x sI. For
ˆ
x
x
Ž
yn 5
any open ball O2x there exists ␪-1 and an open ball N:O around x, such that x gN«Pr ␪
xn
ˆ
ˆ
0
5
.
Ž
.
yx ª0 GPr ᭙ngގ: x gO )0.
ˆ
n
PROOF: The proof directly extends the methods used in the uni-dimensional case, by considering
Ž <
Ž
.
Ž
x
.
.
ˆ
Ž
.
AsD ␸ 1, x , BsD ␸ 2, x , and ␺s␺ 1 x . Thus, ␺Aq 1y␺ BsI.
ˆ
ˆ
x
First, by basic linear algebra, if A has distinct real eigenvalues, then it can be diagonalized as
JA sQAQy1, where Q is an invertible matrix. Likewise, because JB sQBQy1 sQ Iy␺A Q r 1
y1
Ž
.
Ž
.
Ž
. Ž
.
Ž
.
y␺ s Iy␺ JA r 1y␺ , the matrix Q also diagonalizes B. Rearranging terms, ␺ JA q 1y␺ JB
sI. Since all eigenvalues are positive and not one, JA JB1y␺ has all diagonal entries inside 0, 1 , by
␺
Ž
.
the earlier scalar AM-GM inequality. Let ␪-1 be the maximal element in JA JB1y␺. Put JC s
␺
max JA, JB componentwise , and CsQy1 JC Q.
ꢀ
4 Ž
.
²
:
5
52
5
² :
s x, x . For this yields
On ޒSy1, we use the inner product x, y sxЈQЈQy and the norm
x
X
X
²
:
²
:
5
5
5
Ax, Ax sxЈAЈQЈQAxsxЈQЈJA JAQxFxЈQЈJC JC Qxs Cx, Cx , and so Ax F Cx . By continu-
Ž
.
Ž .
around x such that xgN ␦
1
ous differentiability of ␸, for any ␦)0, there exists an open ball N1
␦
ˆ
Ž
.
Ž
.
ˆ
Ž
.
ˆ
5
Ž .5
5 5
implies ␸ 1, x yxsA xyx q␥ xyx , where ␥ x -␦ x . If the maximal eigenvalue of A is
␭, then we have
ˆ
2
␸ 1, x yx 2 F A xyx
.
Ž
.
Ž
.
ˆ
2 5
q␦ xyx q2 ␦␭ xyx
ˆ
52
5
52
ˆ
ˆ
Ž .
␩ around x so small
ˆ
Since no eigenvalue of A or B is 0, for any ␩)1, there exists an open ball N2
Ž . .5
5
Ž
.
5
5
Ž
5
Ž
.
5
5
Ž
.5
that xgN2
␩
implies ␸ 1, x yx F ␩A xyx and ␸ 2, x yx F ␩B xyx . Clearly then,
ˆ
ˆ
ˆ
ˆ
5
Ž
.
5
5
Ž
.5
Ž .
ˆ
␸ i, x yx F ␩C xyx for both is1, 2, for all xgN2 ␩ .
ˆ
ˆ
Ž
.
Ž < .
Ž
.
For any ␧)0, there exists an open ball N3
␧
around x in which ␺ 1 и g ␺y␧, ␺q␧ . When
␩y1)0 and ␧)0 are both small enough, all diagonal entries of the diagonal matrix
␺
y␧
1y␺y␧
2␧
Ž
.
Ž
.
Ž .
␩JC
Ž
.
Ž .
Ž .
␩JA
␩JB
lie inside 0, 1 . Put N0 sN2 ␩ lN3 ␧ lO.
Consider now
Ž
.
ˆ
␩A x
yx
if ␴n F␺y␧ ,
if ␴n )␺q␧ ,
else.
˜
¡
ny1
~
Ž
.
␩B x
yx
ˆ
x yxs
˜
˜
˜
ˆ
ny1
n
¢
Ž
.
␩C x
yx
ˆ
ny1
y1
Ž
.
ˆ
This linear system is stable. Namely, y sQ x yx Q
follows the stochastic difference equation
˜
n
n
ynq1 s␩JA yn or ynq1 s␩JB yn or ynq1 s␩JC yn with chances ␺y␧, 1y␺y␧, and 2␧. In this
diagonal system, each coordinate follows a scalar equation. By Lemma C.1, each individual
coordinate converges a.s. upon zero at rate ␩␪. The intersection of a finite number of probability one
events has probability one, so a.s. y ª0. Thus, a.s. x ªx at rate ␩␪. Extending the proof of
˜
ˆ
n
n
Ž .
Lemma C.1 b , there exists an open ball N:N0 around zero, such that x gN implies x remains in
˜
n
0
N0 with positive probability.
We have already shown that the linear system x yx dominates in norm the nonlinear one
x yx on N . Hence, x ªx with positive probability, and just as in the proof of Theorem C.1,
²
:
ˆ
˜
n
²
:
ˆ
ˆ
n
0
n
the convergence is at rate ␪.
Q.E.D.
APPENDIX D: OMITTED PROOFS
Cascade Set Characterization: Proof of Lemma 2.
t
t
t
Since pmt
q
is increasing in m by 1 , pmy1 q , pm q is an interval for all q. Then Jm is the
Ž .
Ž . w Ž .x
Ž .
closed interval of all q that fulfill
t
13
pmy 1 q Fb and pmt q Gb.
Ž
.
Ž
.
Ž .
t
t
t
Ž
.
HŽ
Ž ..
HŽ Ž ..
Interior disjointness is immediate. Next, if int Jm /л then F pmy1
q
s0 and F pm q s1
t
Ž
.
for all qgint Jm . The individual will choose action m a.s., and so no updating occurs; therefore,
the continuation belief is a.s. q, as required.

396
L. SMITH AND P. SøRENSEN
Ž
.
With bounded beliefs, one of the inequalities in 13 holds for some q, but no q might
yields p0t q '0 and pMt q '1 for all q, we must have
Ž .
Ž .
Ž .
simultaneously satisfy both. As
1
t
t
t
t
t
t
t
t
t
w
t x
w
x
tŽ
.
Ž
.
q 'b define 0-q -q -1.
J1 s 0, q and JM s q , 1 , where p1
q
'b and pM
t
t
Finally, let m2 )m1, with q1 gJmt and q2 gJmt . Then pmt
q1 Gpm q1 Gb)bGpm
t
t
Ž
.
Ž
.
Ž
y1
.
q2 ;
y1
1
2
2
1
2
and so q2 )q1 because pmt
is strictly decreasing in q.
y1
2
With unbounded beliefs, bs0 and bs1. Hence, pmy 1 s0 and pmt s1 for qgJmt by 13 . By 1 ,
t
Ž
.
Ž .
this only happens for ms1 and qs0, or msMt and qs1.
With bounded beliefs, type t takes only two actions with positive chance in a neighborhood of the
t
t
t
t
Ž
.
Ž .
Ž .
q
nonempty cascade set Jm. This follows from 13 , since all pm are continuous, and pmy1 q -pm
for all m and qg 0, 1 , absent a weakly dominated action for type t.
Ž
.
Q.E.D.
Ž .
Limit Cascades Occur: Proof of Theorem 1 a .
We first proceed here under the simplifying assumption that ␺ and ␸ are continuous in l. By
ˆ
ˆ
ˆ
.
ˆ
ˆ
Ž
Ž
< .
< .
Ž
Theorem B.2, stationarity at the point l yields ␺ m l s0 or ␸ m, l sl. Assume l meets this
HŽ
LŽ
ˆ
Ž ..
l sF pmy1
ˆ
ˆ
Ž ..
l
criterion, and consider the smallest m such that ␸ m l )0, so F pmLy 1
s0.
H
ˆ
.
ˆ
ˆ
l
ˆ
l
Ž
HŽ Ž ..
Then ␸ m, l sl implies F pm
LŽ Ž ..
sF pm
Ž .
l
)0. Since F
s1. Thus, lgJm, as required.
%
FSD F by Lemma A.1 c , this
equality is only possible if F H pm
ˆ
ˆ
Ž
LŽ Ž ..
ˆ
Ž ..
l sF pm
ˆ
Ž
.
Next abandon continuity. Suppose by way of contradiction that there exists a point lgsupp lϱ
HŽ Ž .
with lfJ. Then for some m we have 0-F pm l y -1, so that individuals will strictly prefer
.
ˆ
ˆ
ˆ
Ž .
to choose action m for some private beliefs and mq1 for others. Consequently, pm l )b, and
ˆ
ˆ
Ž .
Ž .
since p0 l s0Fb, the least such m satisfying pm l )b is well-defined. So we may assume
F H pmy 1 l y s0.
Ž .
ˆ
Ž
.
Next, F H pm
)0 in a neighborhood of l. There are two possibilities:
HŽ
Ž .. Ž ..
ˆ
l )F pmy1 . Here, there will be
ˆ
Ž
Ž ..
l
CASE 1: F H pm
l
a
ˆ
ˆ
neighborhood around l where
Ž
F H pm
l
yF pmy1
l )␧ for some ␧)0. From 3 , ␺ m l s␺ m H, l is bounded away from
Ž
Ž ..
HŽ
Ž ..
Ž .
Ž
< .
Ž
<
.
Ž .
Ž
.
LŽ Ž .. HŽ Ž ..
0 in this neighborhood, while 5 reduces to ␥ m, l slF pm rF pm , which is also bounded
l l
ˆ
ˆ
l
ˆ
..
l is in the interior of co supp F , and so Lemma A.1
Ž .
Ž
Ž
l
away from l for l near l. Indeed, pm
Ž ..
guarantees us that F L pm
Ž
HŽ Ž ..
Ž
ˆ
exceeds and is bounded away from F pm for l near l recall that
ˆ
.
Ž .
pm is continuous . By Theorem B.1, lgsupp lϱ therefore cannot occur.
CASE 2: F H pm
places no weight on b, pm l . It follows from F H pmy1 l y s0 and
l
. This can only occur if F has an atom at pmy1 l sb, and
H
HŽ
ˆ
Ž ..
l sF pmy1
Ž ..
Ž .
my2 -pmy1, that
ˆ
ˆ
Ž
ˆ
Ž .x
ˆ
Ž .
Ž
Ž
.
p
F H pmy 2
l s0 for all l in a neighborhood of l. Therefore, ␺ my1 l and ␸ my1, l yl are
ˆ
Ž
Ž ..
Ž
< .
Ž
.
ˆˆ
w
.
bounded away from 0 on an interval l, lq␩ , for some ␩)0. On the other hand, the choice of m
ˆ
ˆ
x
ˆ
Ž
< .
Ž
.
Ž
ensures that ␺ m l and ␸ m, l yl are boundedly positive on an interval ly␩Ј, l , for some
Ž
.
Ž .
.
␩Ј)0. So once again Theorem B.1 observe the order of the quantifiers! proves that lfsupp lϱ
Q.E.D.
Ž .
Confounding Outcomes are Nondegenerate: Rest of Proof of Theorem 2 g .
Ž
.
Ž
Ž
.
Let types U, i prefer action 1 to 2 in state H, and types V, j prefer action 1 to 2 in state L. By
U
U
.
Ž
.
Ž
.
a rescaling, we may assume that the payoff vector of type U, i is bi , ci in state H and 0, 1 in
U
U
V
V
V
V
Ž
.
Ž
.
.
Ž
.
state L, with bi )ci , type V, j respectively earns 0, 1 and bj , cj , with bj )cj . These engender
posterior belief thresholds 1r 1qui and 1r 1q¨j , where ui sbi yci and ¨j sbjV ycVj . As in the
U
U
Ž
.
Ž
example of Section 3.1, we have private belief thresholds piU l slr ui ql and pj l slr ¨j ql .
CASE 1: Bounded Beliefs. Assume Ts2. Generically, u/¨, so assume without loss of generality
Ž .
Ž
.
VŽ .
Ž
.
t
t
w
t x
w
x
w
u-¨. The cascade set for type t is J s 0, l j l , ϱ . With u-¨ and yet a common likelihood
interval of activity, we have lU -lV -lU -l . For lg l , l only type U is active. Thus ␺ 1 L, l
V
U
V
.
Ž < .
V
V
V
Ž <
.
Ž <
.
Ž <
.
rises above ␺ 1 H, l in this interval, so ␺ 1 L, l )␺ 1 H, l . Near l , only type V is active, so
U
U
V
U
Ž
<
.
Ž <
.
Ž <
.
Ž <
.
Ž
.
␺ 1 L, l -␺ 1 H, l . By continuity, ␺ 1 L, l* s␺ 1 H, l* for some l*g l , l .
U
V
CASE 2: Unbounded Beliefs. Let ␭ and ␭ denote the population weights.
i
j
␭UF s lr ui ql q␬
␭
1yF lr ¨j ql
Ž <
.
Ž
Ž
..
V w
sŽ
Ž
..x
␺ 1 s, l s␬ q␬
Ý
Ý
j
1
i
j
i
2
␭ u r u q0 2 y
␭ ¨ r ¨ q0
.
U
V
Ž <
.
sŽ .
Ž
.
Ž
j
.
«␺ 1 s, 0 s␬ f
0
Ý
Ý
j
l
i
i
i
j
j
ž
/
i

OBSERVATIONAL LEARNING
397
Since f L 0 )f 0 by Lemma A.1, ␺ 1 H, 0 -␺ 1 L, 0 when Ýi ␭i rui )Ýj ␭j r¨j. Since f 1 -
U
V
Ž .
Ž <
HŽ .
.
Ž <
.
Ž <
l
.
LŽ .
f H 1 , ␺ 1 H, ϱ )␺ 1 L, ϱ likewise ensues from Ýi ␭ ui -Ýj ␭j ¨j. Finally, both inequalities hold
l
U
V
Ž .
Ž <
.
l
l
i
Ž
.
Ž
.
or both fail for sufficiently different and opposed preferencesᎏnamely ui small enough andror
¨i big enough, in the case above; we get the reverse inequality in each case for ui big enough andror
¨i small enough.
Q.E.D.
Proof of O¨erturning Principle used in Theorem 3:
If n optimally takes m, his belief pn satisfies
1yrmy 1
rmy 1
1ypn
pn
1yrm
rm
Ž
.
Ž .
)l h
14
G
.
Ž .
Ž .
Let ⌺ h denote the set of all beliefs pn that satisfy 14 . Then individual n chooses action m with
probability H⌺Žh. dF H resp. H⌺Žh. f dF H in state H resp. state L . This yields the continuation
⌺Žh. f dF H
Ž
.
Ž
.
H
Ž
.
Ž .
l h, m sl h
.
H⌺Žh. dF H
Ž
.
Ž .
Cross-multiply and use 14 with Lemma A.1 a to bound the right-hand integral.
Q.E.D.
Stability of Marko¨-Martingale Processes: Rest of Proof of Theorem 4.
Ž
.
Ž
.
Next suppose that for m in some subset M0 :M, ␸ m, x /x and thus ␺ m, x s0. Then
␺
Žm< x.
ˆ
Ž .
-1. Choose ␪g ␪, 1 . By
Ž
< .
<
Ž
m, x
x
.<
MRM0 /л since Ým ␺ m и s1. Note that ␪'⌸mf
␸
M
0
ˆyn
Ž
<
<
Corollary C.1, there is an open ball N0 around 0 such that x0 yxgN0 «Pr ᭙ngގ: ␪
xn yx g
.
N0 )0 when only actions in M_M0 are taken.
Define events En s mn fM0
ˆyn
n
ꢀ
4
ꢀ
<
<
<
<4
Ž
.
,
Fn s ␪
xn yx - x0 yx , Gn sFks0 Ek lFk , and Gϱ s
ϱ
Ž
.
Fks0 Ek lFk . Then,
ϱ
ϱ
Ž
.
Ž
.
Ž
.
Ž
.
Ž
<
.
Ž
<
.
Pr Gϱ sPr G0
Pr Gnq1 NGn sPr G0
Pr F
E
nq1 , Gn Pr Enq1 Gn .
Ł
Ł
ns0
nq1
n
s0
1
Corollary C.1 implies 0-Łϱns0 Pr Fnq1
Ž
<
.
Enq1, Gn . When xn ªx exponentially fast and ␺ is C , the
ϱ
Ž
<
.
Ž
<
.
sequence Ýmg ␺ m xn vanishes exponentially fast; therefore 0-Łns1Pr Enq1 Gn . Collecting
M
0
Ž
.
Ž
.
pieces, 0-Pr Gϱ FPr xn converges to x at rate ␪ .
Q.E.D.
Herds and Confounded Learning Arise: Proof of Theorem 5.
ˆ
ˆ
Ž .
Part a : Let l0 be close to, but just below, the fixed point lsinf J. Generically, only one rational
ˆ
type t is active in a neighborhood of l. The other rational types are then equivalent to noise in this
neighborhood, and we continue without loss of generality as if there is only a single rational type.
ˆ
Let m denote the action that this rational type takes with probability one on J. Since private beliefs
1
have C1 tails, ␳ m и , ␺ m и , and ␸ m, и are C near l.
ˆ
Ž
< .
Ž
< .
Ž
.
1
ˆ
ˆ
Ž
.
Ž
<
.
Ž
<
.
Suppose that
␸
m, l -0. As l-l, both C functions ␳ m H, l , ␳ m L, l ª1, and thus
l
ˆ
ˆ
Ž
.
␸ m, l ylª0. Then lnq1 gJ for ln close enough to l. So after finitely many steps, and thus with a
ˆ
²
:
positive probability, ln jumps into J, and convergence is at rate ␪s0.
ˆ
Ž
.
Ž
.
Ž
<
.
Ž
<
.
Next posit ␸ m, l G0. Let ␤ m, l 'l␳ m L, l r␳ m H, l be the continuation without noise.
l
ˆ
.
ˆ
w
Ž
HŽ . LŽ .x
Ž .
Then ␤l m, l s1ql f b yf b -1 by Lemma A.1 a , for bounded and not all uninformative
ˆ
.
ˆ
Ž
.
Ž
Ž
Ž
.. Ž
.
beliefs 0-b-1r2 . So ␸ m, l s ␬m q␬␤l m, l r ␬m q␬ -1, and thus ␪-1 by Theorem 4,
l
ˆ
ˆ
Ž
.
Ž
.
provided ␸ mЈ, l G0 for all actions mЈ. Then ␸ mЈ, l s1 for l near l for all actions except
l
ˆ
.
Ž
mЈsmq1, the only other one rationally taken for l close enough to Jm. But ␤l m, l -1 implies
ˆ
. Ž
mq1, l )1, and thus ␸ mq1, l )1.
l
ˆ
.
Ž
␤
l
ˆ
Finally, assume lϱ gJ, and that the final approach was from the left. Eventually, all but type t
must be inside their cascade sets, and thus herding. The convergence of ln to lϱ is exponentially fast.
With f H b )0, ␳ m H, ln then converges exponentially fast to one. By the first Borel-Cantelli
lemma, eventually type t takes action m.
Ž .
tŽ
<
.
Ž .
w x
Part b : By Fatou’s lemma, E lϱ Fl0. Let l*sinf K)0. Since the limit is concentrated on
ꢀ 4
Ž
.
0 jK, it must be 0 with probability at least l*yl0 rl*.

398
L. SMITH AND P. SøRENSEN
Ž .
Part c : We only need robustness to noise, since the example establishes the noiseless case. Let
˜
˜
Ž
<
.
Ž
<
.
␸, ␺ denote the analogues of ␸, ␺ when noise is added. Then ␺ m H, l* s␬m q␬␺ m H, l* .
˜
˜
˜
Ž
Ž
.
Ž
<
.
<
.
Differentiating ␸ m, l sl␺ m L, l r␺ m H, l then yields
˜
˜
˜
Ž
l
Ž
<
.
<
.
Ž
<
.
Ž
<
.
␺
m L, l* y␺ m H, l*
␺
m L, l* y␺ m H, l*
l
l
l
Ž
.
␸
˜
m, l* s1ql*
s1q␬l*
.
l
Ž .
␬␺ m H, l* q␬
m
<
˜
Ž
<
.
␺ m H, l*
Ž
<
.
Ž
<
.
If ␺ m H, l* )0, then ␸ approaches ␸ as ␬m ª0. If ␺ m H, l* s0, then ␸ does not affect the
˜
l
l
l
˜
Ž
.
Ž
.
Ž
<
.
stability criterion for l*, but does when ␬ )0. For any ␧)0, ␸ m, l - 1q␧ r␺ m H, l* for
˜
m
l
˜
˜
Ž
Ž
<
.
˜
Ž
.
<
.
..
small enough noise, since lim
␺ m H, l* ␸ m, l* s1. As
␬
ŽŽ
m ª0, ␺ m H, l* tends to
˜
l
␬
ª 0
m
˜
␺
Žm< H, l*.
␺ Žm< H, l*.
Ž
<
.
Žl
.
.
Ž
<
␺ m H, l* s0, and so for all ␧)0, eventually ␸ m, l*
-
1q␧ r␺ m H, l*
-
␺
m, l* Žm< H, l*. -1, then l* remains stable for small enough ␬m.
Q.E.D.
Ž
.
1q␧. So if Łm
␸
l
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.
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