Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/dorationaltraderOOsmit
[HB31
.M415
1
\no.%-3L0
working paper
department
of economics
DO RATIONAL TRADERS FRENZY?
Lones Smith
96-20
April
1996
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
DO RATIONAL TRADERS FRENZY?
Lones Smith
96-20
April
1996
reCHNOLOGV
'
FEB
'
:
96-20
Do
Rational Traders Frenzy?*
Lones Smith*
Department of Economics
Massachusetts Institute of Technology
Cambridge,
MA 02139
this version: April 10,
1996
original version: June, 1994
Abstract
I
ing,
develop a simple
generalizing Glosten and
maker
is
new model
faces
n
of strategic trade with endogenous tim-
Milgrom
risk neutral traders
A
(1985):
competitive market
with unit demands or supplies.
private information whether any given trader
is
It
either informed, with
a heterogeneous informative signal about the asset payoff, or a pure noise
trader planning to make a trade at a random time. The market is open
for
an exponential length of time.
This structure
is
recursively soluble into a sequence of 'subgames',
— despite the endogenous timing —
equilibrium.
I
I
find there
prove that necessarily there
is
is
and
a unique separating
incomplete separation, since
only informed traders whose signals are 'good news' ever buy, and only
those with 'bad news' ever
switching sides of the market
if
neutral news signal. Finally,
I
'yes',
that
all
show that traders can only envision
the underlying signal distribution has no
sell!
I
conjecture that the answer to the
title is
trades are self-feeding, and accelerate the time schedule of
any future trades.
My analysis is greatly simplified by focusing on a facetious 'competitive
auction' model, where the market
maker only wishes to
sell units.
*I am grateful to Jennifer Wu for extensive collaboration on §3.1-4, A.2-3, and interaction on
an early version of §7.1-2, while this paper was still a joint effort. I also owe thanks to Preston
McAfee for a key suggestion, to Dimitri Vayanos and Robert Wilson for insightful feedback, and
comments of participants at the Stanford GSB Theory seminar. Daron Acemoglu and Abhijit
Banerjee, commenting on a primordial version of this paper, drew my attention to the option value
of delay. All errors remain my responsibility. Sankar Sunder of MIT has provided the Matlab
the
simulations. Finally, I have benefited from NSF research support while writing this paper.
te-mail address: lonesClones.mit.edu
INTRODUCTION
1.
There has of late been some interest in the question of whether there can be
movements 'without news'. That
nal asset price
this
can in fact arise in equilibrium
1
has in fact already been demonstrated in several papers.
In a world in which in-
may move
dividuals trade on the basis of private information, the price
arrival of
news
if
individuals
may
ratio-
without the
optimally time their trades. This brings
me
to the
question of this paper: In equilibrium, will traders rationally choose to cluster their
buy or
sell
One
transactions?
loose intuition in favor such 'frenzying' seems to be
that the imputed informational content of any given trade acts as a siren
other wavering traders into the fray on the
same
contrarians. This paper explores a simple
model designed
This incentive to herd
is
side of the market,
perhaps most cleanly seen in Bulow and Klemperer (1994)
potentially sparking a buying 'frenzy'.
by an implicit cost of
delay,
stemming from the
(1992) capture a related
with each sale
phenomenon
price.
This behavior
possibility of
in
is
motivated
a stock-out. Gul and
a purely informational context. 2
describe a simple n-player forecasting game, where everyone wishes to accurately
x x +X2
predict
the delay cost
first in
\-x n as
H
is
an
soon as possible, where x*
Crucially observe that there
For instance, the auctioneer in
is
realistic.
financial markets,
1
See
A
Romer
fc's
random
type. Here,
shown that endogenous timing
is
no
I
BK
commits
to maintaining the current price after a
am
for
interested in the
where the price mechanism
(1993), for one,
more applied take on
results
cost to frenzying or clustering in either paper.
many
This casts some doubt as to whether
of behavior in such settings.
2
is
no further buyers wish to purchase. Yet
hardly
type
extreme types able to make more informed decisions. 3
clustered forecasts, with
sale until
is
explicit increasing function of x, so that the highest types forecast
the unique separating equilibrium. It
more
this
first,
Namely, all traders with valuations within
some range immediately purchase at the current
in
and discouraging
in part to test this thesis.
values Dutch auction: In equilibrium, the highest types purchase
They
luring
BK). They study a dynamic multiple-unit independent private
(hereafter, simply
Lundholm
call,
and references
most
situations in real
it is
life,
an apt description
salient
example, namely
explicitly penalizes frenzying.
therein.
found in Caplin and Leahy (1994).
3
Loosely, the frenzying of BK might be seen as an extreme form of clustering, and
forth lump both phenomena together under the rubric of frenzying.
this is
I shall hence-
An
Overview of the Model.
I
have in mind to address the question of
endogenous timing and the issue of financial frenzying in a relatively simple and
yet natural fashion.
Smith (1995) shows that once one ventures outside the world
may
of purely strategic trading, individuals
their trades.
I
thus reach back to one of the earlier models of insider trading, due
to Glosten
and Milgrom (1985)
market
an asset of uncertain value,
for
'market maker' or specialist
informed individuals, the
model
is
have no incentive whatsoever to time
who
(hereafter, simply
denoted GM). They consider a
which prices are set by a competitive
in
faces a sequence of trading orders
'insiders',
from potentially
As
each with unit demands or supplies.
their
primarily concerned with explaining the existence and characteristics of the
bid-ask spread in such a market, they treat the arrivals of informed and uninformed
traders as arising from an exogenous stochastic process
My
adaptation of
known
GM endogenizes the timing of trades.
a known number of traders in the market at time
zero.
to the market maker.
Specifically, I start
An
with
unobservable number
are informed insiders with heterogeneous private information, while the others are
pure
'noise' traders.
I
then allow the informed traders to strategically time their
trades while the noise traders execute their trades according to an exogenously-given
Poisson process.
It is
private information whether any given trader
is
either informed,
having observed a heterogeneous signal about the payoff of the asset, or a pure noise
trader planning a
random trade
at a
random
time.
The market
exponential length of time, after which the payoff of the asset
simplifying innovation,
I largely
closes after a
is
revealed.
random
As a key
focus on a facetious 'competitive auction' model,
where the auctioneer only wishes to
sell units, all
at zero expected profit. Later on,
I
simply expand the strategy space, allowing traders to take both sides of the market.
The Equilibrium.
In any separating equilibrium of this game, informed traders
possessing the most extreme high signal (and also the lowest signal, in the market
maker model) enter
first,
by analogy to Gul and Lundholm (1992). With
tion continuously occurring, the
traders
is
this separa-
updated distribution of possible signals possessed by
constantly being truncated. Barring termination of the game, the solution
of any such truncation equilibrium thus consists of a sequence of
can be recursively solved. That
is,
n subgames
that
each subgame immediately following a trade
simply a rescaled version of the original
game
is
— with fewer players, and a truncated
signal distribution,
and updated
informed traders. Note that this
one trader
left,
separation will
for the
is
still
true even of the one-trader game\ For with only
continue to occur, as the lone individual
who determines when
against nature,
on the value distribution and the number of
beliefs
assumption that trade occurs
the
for
game
will end.
This virtue
is
still
plays
the reason
a random exponential length of time rather
4
than, for instance, on the unit time interval.
This
is
not the
first
instance of a recursive structure in a bid-ask market. Wilson
(1986) studies a private values double auction, and looks for a truncation equilibrium.
In
the ask and bid prices set by the uninformed market maker converge between
it,
and jump up
trades,
(resp.
down)
in response to
my
a few analogies to be drawn between
common
a purchase
equilibrium and
(resp. sale).
his,
There are
but by and large, the
values setting here intuitively makes this an inherently different problem.
With the recursive structure,
it
suffices to
study subgames in isolation by backward
induction, carefully grafting the solution of each into
of 'one-trade-and-out' also allows
me
its
predecessor.
The assumption
to characterize equilibrium strategies as the
solutions to a relatively straightforward continuous time optimal stopping problem.
Trade Dynamics.
model
is
As
have set out expressly to explore trade timing, the
I
set in continuous rather
dynamics
in the ask
in
than discrete time. For the market maker model, the
each subgame can represented by a five-dimensional autonomous system
and bid
prices, as well as the highest
and lowest
signals
x and y that have
not yet purchased, and the remaining mass u of uninformed noise traders.
subgames, the
specialist's zero profit condition implies that the signal truncations
must be continuous, and so the
given.
The bid and
That
is,
and bid
there
is
initial
x and y (and more easily
u) in
any subgame are
the ask, however, must jump, and this indeterminancy potentially
creates a two-dimensional
of initial ask
Across
continuum of
prices
is
equilibria. In fact,
I
argue that only one set
consistent with forward-looking equilibrium behavior.
a unique separating equilibrium.
With a unique
equilibrium, the assumption of unit trades lends itself to a simple
test of the frenzying hypothesis:
Conditional on a trade occurring at time
t,
how do
remaining informed traders subsequently adjust the time that they intend to trade?
I
conjecture that locally at least, rational traders do frenzy.
Also with a
finite horizon,
a nasty singularity ensues from a
last
At
least temporarily,
minute 'mad rush' to trade.
trades are self-feeding. For instance, after a purchase, traders soon planning to
into the fray will purchase sooner
Why
Separation Occurs.
and
sell later
jump
than planned.
That separation even occurs here
is
by no means
transparent, and the argument hinges on the assumption that traders' signals are
conditionally
lier
i.i.d.
The
intuition for
why
traders with stronger signals
revolves around two distinct lines of out-of-equilibrium reasoning.
specifically tied to the
move
The
ear-
first is
dynamic truncation process, while the second stems from the
common-values assumption.
a truncation equilibrium induces heterogeneous beliefs about future price
First,
movements among the informed types (the price
uninformed
specialist, their evolution
between trades and the magnitudes of their
jumps immediately following trades are commonly
however, with conditionally
nals that
may be
i.i.d.
Since prices are set by an
effect).
forecast
signals, insiders disagree
by
all
types of insiders;
on the distribution of sig-
held by other insiders, and thus differentially assess the likelihoods
of future trades; they also have different estimates of the asset value. In particular,
a truncation equilibrium applies more competitive pressure on traders with stronger
The
signals:
highest signal trader
pessimistic of an intervening sale,
ket closure. 5
most confident of an intervening purchase, most
and most
While short of a proof,
nice cross between
I
fearful of the prospect of
believe that the
exogenous mar-
market-maker model
a
offers
an incomplete information game of pre-emption and war of
tion: Intuitively, traders
and just
is
attri-
wish to buy just in advance of competing same side traders
after their oppposite side counterparts.
But while
wait and trade at the higher bid or lower ask price that
all
would
may soon
ideally prefer to
prevail, those with
stronger signals believe an intervening same-side trade and consequent unfavorable
price
jump
to be
more
likely;
they therefore will be more impatient to trade
Observe that an analogous 'price
effect' also
inherently different there, insofar as
(vNM)
it is
6
emerges as the driving force behind
separation in a pure independent private values (IPV) auction, such as
is
earlier.
differences in
BK. But
it
von Neumann-Morgenstern
preferences rather than beliefs that matter: Traders with higher absolute val-
That
is,
except perhaps for some extreme low signal traders,
who might have more
to gain by
shorting the asset.
By corollary, if traders' information signals are merely unconditionally i.i.d., as with Foster and
Viswanathan (1993), a truncation equilibrium doesn't exist: Gul and Lundholm (1992), for instance,
have an explicit cost function to force traders with high signals to move sooner rather than later.
known marginal gains from
uations place a lower value on the
when weighed against the
loss
game, absent further learning,
should an intervening trade occur. In the one-trader
beliefs are fixed,
curring solely from heterogeneous
It
further price reductions
vNM
and one can view separation as oc-
preferences.
should be noted that the insight that competition
among
traders can endoge-
nously generate 'impatience' in the absence of explicit discounting originates with the
But
double auction of Wilson (1986).
common- values
of delay does not emerge: In the
distributions, inframarginal traders
side of the
market to take,
if
any.
more subtle option value
in that paper, the
may be
setting, for
a generic class of signal
uncertain and uncommitted as to which
Having seen the decisions of others, a trader may
be persuaded to change the course of action she had previously planned to take in
the current
subgame
traders, one with
—
for instance, she
may buy
a relatively stronger signal
rather than
will place
sell.
But among such
a lower probability on the
event that other traders will reveal sufficiently strong opposing signals to overturn
her decision, and so her option value of delay
with a weaker signal. As
it
is
lower than that of an informed trader
turns out, the class of signal distributions without any
option value of delay play a key role in this paper.
Is
Separation Complete?
(1982),
it is
equilibrium.
exists
Since the 'no-trade' theorem of Milgrom and Stokey
well-known that trade for purely informational motives cannot occur in
With
noise traders in the picture, such trades
may
occur provided there
a bid-ask spread sufficient to address the remaining adverse selection problem.
GM admitted to one 'failing' of their competitive specialist model, that the required
bid-ask spread effectively screens out marginal but nonetheless potentially valuable
information:
Insiders with only middling information simply
write, "This opens the possibility that another
is
do not
trade.
way of arranging trade may
exist
They
which
Pareto superior to the competitive system.
This paper in part explores whether such an improvement can be effected within
the competitive system simply by endogenizing the timing of trades.
spread
timing,
is
an symptom of an adverse selection problem, and
if
the informed traders enter at a
then the spread can vanish. So could
all
much
in
faster rate
The
bid-ask
a world with endogenous
than the noise traders,
informed traders eventually purchase?
Do
the incentive efficiency characteristics of double auctions that Wilson (1985) finds
IPV
extend beyond the
setting? No.
I
show that
competitive auction model,
in the
equilibrium necessarily entails incomplete separation, simply because no trader having
'bad news' (in the sense of Milgrom (1981)) ever purchases.
model, additionally no one with 'good news' ever
GM,
in
all
good news traders do buy and,
news traders do
sell in
results are true
whose analysis focuses on the
the case of
7
First
many
is
all
bad
the monopolistic model of Kyle
Most
who have extended
closely related to
the Kyle model to
may
heterogeneously informed insiders. Their insiders
trade in any of finitely
show that
market maker model,
'linear' equilibrium.
Foster and Viswanathan (1993),
is
I
There are a handful of informational insider
trading papers with endogenous timing.
work
in the
maker
the unique equilibrium.
Relationship to the Literature.
this
While both
sells.
with endogenous timing, they are both subtle and tight: For
eventually
(1985),
In the market
many
discrete time periods.
8
retained linearity, as have most others.
9
repeatedly
Yet faithful to Kyle, they have
Our dynamic
insider trading paper,
by way
of contrast, requires no such focus on equilibria with special characteristics.
The remainder
of the paper
fictitious zero profit
and because
it
is
organized as follows. In section
auction model. This
saves
much
is
2, I
describe the
introduced both as a conceptual device,
analytic repetition.
Since the model
is
largely new,
I
spend some time deriving the equations of the truncation equilibrium and intuitively
discussing
its
properties in section 3. Section 4
is
a quick march through the analogous
equilibrium laws of motion for the market maker model. In section
theoretical possibility of full separation in either context,
and
the good news and bad news exclusion results.
I
and uniqueness of the separating equilibrium
in section 6.
dynamics of trade
7
5, I
explore the
in particular establish
use this discovery to prove existence
Section 7 explores the
in equilibrium, with attention to the frenzying question.
who studies hedging-motivated trade by insiders.
Foster and Viswanathan underscore the daunting complexity of doing without linearity, noting
7
that they "avoid the usual forecasting the forecasts
problem.
a.k.a. equilibrium theory. In
See also Vayanos (1993),
8
'
. .
.
—
framework, this forecasting problem also rears its head: Individuals must know the equilibria
for all future subgames.
9
For references and a discussion, see Rochet and Vila (1994). On this note, I should comment
this
that the 'justification' in this latter paper for such a restricted focus and that of normality rely on an
assumption that I expressly rule out
that insiders directly observe the amount of noise trading.
—
While an elegant unifying
result, this
seems an awfully powerful presumption.
A ZERO-PROFIT COMMON VALUE AUCTION
2.
Consider an auctioneer
is
selling shares of
Assume that the auction
post value V.
a single asset with uncertain ex
continuously open on a time interval
is
time zero, and no subsequent
arrivals.
all
The number
is
The random auction
closing time
oo) according to a Poisson process with parameter A
chance e~ xt
When
.
T
— that
is
is,
The time-zero prior
beliefs
about
distributed over
T
exceeds
random
the auction closes, the realized value v of the
10
publicly revealed.
V shared
by
all
assume that Go has a continuous density
currently willing to
sell
time-t, given only the information Z*
elapse time
/3t
t,
with
V
participants
I
shall
go.
The Auctioneer. The auctioneer continuously announces a temporary
is
t
variable
are described by the distribution Gq(-) on [Vo,Vi]- For notational simplicity,
at which he
of
trades are always publicly observable.
The Uncertainty.
(0,
T)
Each trader may make at most one purchase
of unit size, while the market remains open ('one-trade-and-out').
remaining traders and
[0,
TV potential buyers (or traders) present in the market at
random duration, with
of
who
unit shares of the asset.
11
That
from the history of purchases in
ask price
is,
at each
(0, t)
and the
the ask price a t earns zero expected profits conditional upon the event
of a purchase.
For instance, the auctioneer
may be an arm
of the government,
with the designated goal of neither losing nor making money in expectation on any
transaction.
in
which case
Later on,
it
will
I
shall allow this auctioneer to take
be the
classic zero-expected-profit
Glosten and Milgrom (1985), that
This
is
The
may be
both sides of a trade,
market maker introduced by
taken as a proxy to Bertrand competition.
the ultimate motivation for this preliminary theoretical construct.
GM
assumption automates the auctioneer's decision
allows one to focus exclusively on insiders' optimization.
price incorporates
Just as in
any information that
GM, I assume
that
if
is
rule,
and thus nicely
In other words, the ask
transmitted by the arrival of the order.
more than one zero-expected-profit ask
price exists,
I
follow the convention of choosing the lowest one (and later on, the highest bid price
too),
10
but unlike
GM I simply assume that the auctioneer has no private information.
Preston McAfee suggested this infinite-horizon reformulation of an earlier finite horizon model,
where the market was open on [0, 1).
11
Throughout the exposition, I shall maintain the convention of referring to the traders as females
and the auctioneer (and later, the market maker) as male.
The
At time
Traders' Private Information.
w€
observes a signal
informed
The
is
of the asset's value.
[0, 1]
signals too are private information,
means that each
signal
w
is
independent across traders.
is
and are conditionally independent. This
an independent draw from the continuous distribution
F(- v), given the true value v.
I
1
assume there
also
is
the strict monotone likelihood ratio property
satisfies
chance q
assume that the event of becoming
I
the trader's private information and
zero, each trader with
a smooth density
/(•
v) that
|
(MLRP), namely,
f{w H v H )/f(w H vL ) > f(w L v H )/f{w L vL )
|
wu > wl
whenever
|
and vh
>
support on
Some
namely,
trivially,
[0,1], so
the strict
that
|
vi. Intuitively, signals
so that traders with higher signals are
Observe that
|
and values are complementary,
more optimistic about the expectation of V.
MLRP
implies that the signal distribution has
can never rule out any signals regardless of
I
of the results shall assume that there exists a signal
it
has the same likelihood for any value
v.
w
that
v.
neutral news:
is
Other results only need the next
weaker notion that necessarily exists with a nonatomic signal space and the
Signal
w
w
equals
as
w^
is
weak neutral news given a prior g
mean.
its prior
Finally,
a signal
w
is
if
the posterior expectation of
The
Traders' Strategies.
I
MLRP:
V
given
(weak) good news or (weak) bad news
w. Appendix A.l summarizes some key implications of the
simple integral inequalities that
full
MLRP
and some
find very useful.
Focusing on the informed traders,
I
assume that
the uninformed (noise) traders have no discretion over the timing of their trades.
Instead, they are
prompted to buy by a positive random
liquidity shock that arrives
according to an independent Poisson process with parameter
12
fi.
In contrast, informed traders purchase for purely speculative motives and fully
optimize over the timing of their trades.
trader with signal
no one
12
An
else
w who
is
open, any informed
has not yet traded chooses an elapse time t(w) to buy
has already purchased. 13
arguably more
While the market
She may well choose never to buy,
in
if
which
(and more complicated) model with explicit stochastic entry of
if the total number of traders were unobserved.
But I feel that the analytic advantage of my approach far outweighs any injustice wrought by my
twisting the standard story. Hopefully the reader will soon agree.
realistic
liquidity traders could also
One
make
informational sense
technical question naturally arises with continuous time:
What happens
if
one trader buys
case t(w)
utility.
14
=
oo.
The
All informed traders are risk neutral
and do not discount future
type-u; trader's expected payoff at each time
she purchases at an ask price of
and
at,
otherwise.
t is
given by E wt
The expectation operator
defined with respect to the public information set Xt (given the
is
and the private signal w.
strategies),
to
t
,
but where the trader
is
I
known
V — at
if
£. wt
commonly known
and 0? be the events analogous
further let 0\
to be informed and uninformed, respectively.
The Equilibrium Concept. As
I
model a dynamic game of incomplete
mation among the informed traders, the appropriate solution concept
is
infor-
sequential
further wish to prune any equilibria in which decisions are predicated
equilibrium.
I
upon payoff
irrelevant information.
For instance,
only wish to consider equilibria
I
that are anonymous, in the sense that each trader's strategy
her signal and the public posterior on the asset value.
I
may depend
only upon
shall address these concerns
simply by restricting focus to the Markov sequential equilibria (MSE) of the model
(due to Maskin and Tirole (1992)).
Among
Separating Equilibria.
that are separating:
buy,
MSE,
the class of
I
am
namely, any traders with different signals
must plan to do so at
each ending in a trade.
Partition the
different times.
The
focus of this paper
is
strictly decreasing truncation
t
1,
such that at time
function
t (i)
xt
£ t-4
is
if
the market
longer
made
is
tion equilibrium
xt
= 0,
,
oo),
subgames,
an especially salient example of
is
a continuous and
differentiable in each
subgame, where
€
[0,x t ].
MSE,
in
which
=
0.
I
am
all
(ii) all
also interested in the notion
informed types eventually buy
Since strategic decisions are no
I
may
>
0.
exploit the recursive structure of a trunca-
and simply analyze a representative n- trader subgame
is
N
the subsequent analysis shall restrict focus to x t
in the introduction,
So a truncation equilibrium
(t
into
plan to
the informed type that buys, and thus
open long enough: lim^oo x t
as soon as
As described
,
w
remaining informed traders have signals
of a completely (or fully) separating
xt
who both
game
a separating equilibrium called a truncation equilibrium: There
<x <
interested in those
fully described
(1
< n < N).
by the monotonic function x
where r marks the start of the current subgame when the
for t
€
last trade occurred.
To save on tedious details, I shall simply ignore this possibility as
be a zero chance event in equilibrium.
"Everyone behaves identically in the models with interest rate r > or, as is the case here, with
no discounting and f exponential with parameter r + A.
'instantaneously after' another?
it will
EQUILIBRIUM ANALYSIS
3.
The Accounting Equations Within a Subgame
3.1
Any delay of trading activity within a subgame simultaneously reveals
the informed status of all remaining traders: remaining traders are less (resp.
about
(i)
more)
likely
informed
if
noise traders entry rate;
their equilibrium entry rate exceeds (resp. is less than) the
and
the underlying asset value: since the informed traders
(ii)
in particular have not purchased, the asset value
I
information
is
condition the probability q that a typical trader
asset value v, then the following
more
likely low. In particular, if
informed upon the underlying
is
two posteriors jointly evolve:
• Qt{v), the chance that any given remaining trader
is
informed, given v and Xt
• g?(v), the public posterior density of v given Xt
Note that qt (-)
is
an interim density
informed trader's private signal.
auctioneer inasmuch as
This, as
it
turns out,
Computing
qt {-)
t
•
u
t
I
= e -M
is easiest.
Given the current truncation
F(x
F(xt v)qQ
t
\v)q
F(x
_
+ e-*(l - go)
F(x t v)q
|
(1) is valid
all
n
traders.
t
x t ],
[0,
\v)q
+u
t
(l
-
U
q
)
t
(since
last purchase,
(So x
left
at time-t
both between and within subgames, and implies that qt (v)
xt
is
To,
immediately after the
endowed with the truncation x^, mass u^, density pTo+ and
= j/o =
,
1,
gTo+
is
continuous in any truncation equilibrium).
Next, consider an n-trader subgame that begins at time
-
and not
1
the (per capita) probability mass of uninformed traders
',
continuous in
9t
—
introduce the notation
Expression
15
and not the
relevant for the informed trader
what the auctioneer cares about.
is
|
where
It is
does not incorporate the
it is
assumes uncertainty only over n
it
\-
a v)
qtK
in the sense that
= gQ
,
and qTo
=q
.)
Before proceeding,
15
it
beliefs
helps to define
Functions are assumed left continuous in time, since a purchase at time t does not affect the
information Xt upon which all variables depend. At a discontinuity of a function h, h(T +) denotes
the right evaluation of h the 'instant after' the trade at time r
i.e., the right hand limit of h(t
—
as
r
4-
tq.
10
• $((•
and
•)
I
<f>t(-
1
•))
an informed trader's time-t conditional signal distribution and
namely
density given the value,
Intuitively, the
MLRP
will
$
imply that
t
(t//|u) is
decreasing in v for fixed
to
< x
t
,
since higher values are associated with higher signals.
I
can now calculate the value density g™ during the current subgame as follows:
nt
v
\
_
[9ro(t;)^T (Xt
V)
I
t
jjjfcd (*)**> fa 2 )
I
/ 9iF(i,|iO
+ (1 - qro{v)){u /u To n gTo+ {v)
+ (1 - 9ro(2))KKo)NT0+ (2)dz
+ (i-a,)«. \*
))
,,
.,,,
oc
for all t
>
the value
r
.
Formula
(3a)
v given that
is
simply an application of Bayes rule for the chance that
is
n
traders remain at time
t.
Here,
I
have
(i)
exploited the
conditional independence of signals held by informed traders, for a fixed value v;
applied the law of total probability in the numerator and denominator to the
number
of those traders
have already bought
oc is
is
with respect to
which
is
who
are informed, and (Hi) used the fact that
already
embedded
v; therefore,
independent of
v.
in the prior
unknown
N — n traders
gTo+ The proportionality sign
.
the simplification (3b) ignores the denominator,
For most uses in this paper, the density g? will appear in
both numerator and denominator of fractions, and thus
this
(ii)
it
will suffice to substitute
reduced form.
Later on,
Lemma
1
I shall
require the following differential laws of motion of g"
(Laws of Motion)
any given remaining trader
qt ( v )
is
_
t
.
Within a subgame, the conditional chance qt (-) that
informed, obeys
= qt
(
while the public posterior density
?(V)
and q
[
v )(l
-
<?"(•)
qt (v))(n
+ <f>(x
t
|
v)x t )
(4)
over the value v evolves according to
[*
qt (v)
qt (z)
1
(5)
9?
11
The proof
—
It
<j> t
is
{x v)x t
|
is
in the appendix.
To
see
how
intuitive
equation
is
(4),
observe that
the flow probability of a purchase by informed traders in equilibrium.
thus simply says that the difference in the rates of change of the proportions of
informed and uninformed traders equals the negative difference in their respective
trading hazard rates:
dlogqt (v)
rflog(l
-q
t
{v))
qt (v)
(1
qt (v)
1
-
qt (v))
_,.,,,
= + <pt\x v)x
x
fi
|
t
The Auctioneer's Problem
3.2
At any moment
in time, the auctioneer's sole task is to control the ask price in
purely automated fashion.
=
• v?
£,
n [V\Xt]
value given
The
=
It is
1
Jy vg?(v)dv, the
n remaining
(auctioneer's) time-t public expectation of the
traders
auctioneer's pricing rule
must
reflect his
informational disadvantage to the inu",
expected valuation of the asset conditional on a purchase at time
the purchase hazard rate for a typical trader at time
being
v.
This
is
the
sum
a
natural to define
formed traders; therefore, he must equate the ask price not to
V
•
qt (v)
t,
but to his updated
t.
Let Pt(v) denote
conditional on the true value of
of the respective entry rates of informed and uninformed
buyers, as follows:
Pt(v)
= q {v)<j> {x v)(-x + (1 - q {v))fi
_ qof(x \v)(-x + (1 - q )utn
qQ F(x \v) + (1 - q )u
t
t
t
t)
|
t
t
t
The
(6)
t
t)
auctioneer only sees that a purchase has occurred at a given ask price, but
does not observe the informed status of the trade. Hence, following a purchase he
updates gt (v) to the new posterior value density gt +(v)
Jvo Pt(z)9t(z)dz
12
= g^' n {v),
where
He
new
carries this
the auctioneer's zero-profit constraint
satisfies
=
at
Observe how
prior g
T2,.
.
.
.
For
I
n
if
Tjv_ n , then
,
£.
n [V\It,
may
=
p
t)
vgt
'
(v)dv
=
an expression
and the
first
The Informed
•
•
-pTN _ n (v)[q
F(x t \v)
(9)
pt {v)g?(v)dv
for
g™ in terms of the original
N — n have
by the conditional independence of
pTl (v)p T2 (v)
price that
———
°
JVq
likewise produce
The ask
therefore
is
Jv
traders remain,
g?(v) oc
3.3
posterior density into the next subgame.
purchased at times
T\,
signals,
+
(1
- qo)u
n
(v)
g
t]
(10)
Trader's Problem
In order to formulate the stopping time problem faced by a typical informed trader
in a truncation equilibrium,
I
describe in turn her flow of payoffs within the subgame,
and then the laws of motion of
I
payoff-relevant variables.
all
remark that any informed trader's time-t interim posterior density given Z* and
the fact that she exists
given that
n—
is
p"
_1
and not
other traders remain at time
1
no more learning occurs by her
particular,
For this
#".
= go(v)
<7 t°(t>)
First, to express the
if
(if
there
she
is
is
only
t.
And
is
the conditional density of v
once only
informed), and so
N=1
n
=
<7°(v) is
trader initially,
1
trader remains,
time invariant. In
who
still
remains.
type-w informed trader's privately estimated valuation
E. wt
V,
introduce the notation
-1
• 7"
tne J
»
sity that
™*
signal-value density:
9t{-
1
•))
mation
n_1
t
(u;,u)
=
l
f(w v)g?~ (v)
\
a given remaining informed trader has signal
given the information It from
•
7
all
w
and the
is
the joint den-
asset value
is v,
previous trades
ner time-t conditional value density given the private signal and the inforIt, defined
W)
9tiV
'
by Bayes' rule as
=
r^w ?fn-u\,
l
Jv f(
\
z )9?
{z)dz
13
«
/(»
I
»)#» = tf-'fo V)
(11)
Then
E wt V
at time
=
t,
an informed trader with signal
E[V\Xt, w]
=
vgt (v w)dV
/
|
=
w
expects the asset
—=
°
is
worth
v
An informed trader with signal w can act as if she receives all payoffs as soon as the
current
subgame
market
closes.
ends. This occurs either because she or another trader buys, or the
If
she buys
then she receives the expected profits
first,
that purchase evaluated using her concurrent beliefs.
nf (w)
from
another purchase triggers
If
the next subgame, she receives her expected continuation value "W(Zt,l3t ,w). If the
market
closes,
she receives nothing. Her instantaneous payoffs are thus:
Uf(w)
n
11? {w)
= EvtV-at
=
J
(
Next,
I
if
another trader buys at time
(12)
t
0, if
the market closes at time
t
describe an informed trader's personal estimate of the instantaneous prob-
abilities of intervening
is
W(Ii, Puw),
<
purchase by any other given trader. For one's private signal
an indication of the underlying value, and by implication of the distribution of
other informed traders' private signals.
p(w)
for
someone with
signal
w
is
The
privately assessed purchase hazard rate
thus obtained simply by integrating pt(v) over the
buyer's private beliefs on the value distribution:
ft (to)
=
p (v)gt (v w)dv
/
t
Jvo
where
I
|
=
=
•
*
l
/^Tf \w,v)dv
fVo
f(w v)g?
|
(13)
Wdv
have substituted from (11).
In light of equations (l)-(3a)
an informed trader at any time
subgame, as well as (ut ,x u at). 16
and (11)-(13),
t is
'
17
variables evolve deterministically,
captured by
all
payoff-relevant information for
initial
conditions for the current
Crucially observe that within a subgame
and
it is
all state
only the concluding time of this subgame
For indeed, (1) and (3a) express qt and$" -1 given these quantities and (u To ,z To ,gro ,g To+ ), while
1
<t>t is a simple function of x t via (2).
All other variables are functions of x t q t 9?'
and t
Observe that the starting time of the current subgame is payoff irrelevant. By way of contrast,
in the natural alternative formulation of the model where the market closes almost surely at time 1
16
,
(my
first choice),
the actual calendar time also matters.
14
,
,
<j>
.
that
(n
—
is
stochastic:
l)p t (w)
+
The type-w informed
X at time
trader thinks
it
ends with Poisson flow rate
Confronting the type-w informed trader
t.
is
the task of
optimally stopping this jump process, given the termination payoff function (12). Her
optimization can clearly be performed at time r
Choose a stopping
T
f
e
rule
as her objective
,
essentially static:
is
r so as to maximize her expected payoff, namely
~ /il(»-i».W+^(n _
i)jS t (
w )W(2t, A, w)dt + e~ f^ n
- 1)Mw)+xVs
Il?(w)
(14)
J To
Recall this optimization
is
performed at time-r
The
.
first
term captures type
assessment of the possibility that the subgame will end before time
r,
tu's
delivering
her the continuation value from the ensuing subgame in the case of an intervening
trade or zero
if
the
possibility that the
game ends
subgame
exogenously.
will persist until her
Incentive Compatibility.
time
I
r,
The second term
describes the remaining
planned purchase time
r.
Differentiating (14) with respect to the stopping
and dividing through by the discount factor exp(—
T
T
[(n
—
l)p r (w)
+ A]dr),
obtain the first-order condition
o
=
(n
-
i)pT (w)[w(ir,
/?T ,
w)
Equilibrium requires that type x t finds
- n? («)] +
it
£n?
H - aii?m
optimal to buy at time
t.
(15)
Thus, (15) yields
the following differential equation:
£n
»
As the 'smooth
= Anf (i,) + (n-
l)p,(i,)
t
-
W(2i, A,i,)]
The type who
is
just willing to
equates the rate of change of her terminal payoff (on the
expected instantaneous change in the value of the
possibility of the
(16)
pasting' condition defining the optimal stopping time, this
equation has a straightforward interpretation.
at time
[nf (i,)
W=Xt
game exogenously ending
15
game
IC
buy
LHS) with the
(on the RHS), due to the
or someone else buying.
Order System
3.4 Analysis of the First
Equation (16) yields an implicit second-order
LHS
function Iff (w) differentiated on the
x embedded
hazard
in the purchase
autonomous
of first-order
From now
rate.
of (16) contains the ask price that itself has
So
,
v) -*
I
I
shall instead derive
an
explicit
system
differential equations in the state variables (u,x, a).
on, I suppress time subscripts,
n
g?(v) -> g 4>t{x t
equation in x: The profit
differential
l
4>x ,
i?- {x t
,
and abbreviate: f{x t \v) -¥ fx qt (v) —>
,
v) -> 7*,
Uf(xt ) -4
II?,
and
W&, ft, x
t)
->
c,
Wf
Proposition 1 (Truncation Equilibria of the Competitive Auction Model)
Within a subgame, the payoff relevant state variable for traders and the auctioneer in
any truncation equilibrium
u
is
(u,x,a), which continuously evolves according to
= — fiu
(17)
ji/fo-q)(l-g)g"
J(v a
= ~A
where
a)q<j>x g
iTX^ "
q,
fx
,
<j)
x,
g
.
.
n
+ (n "
a
1}
n~ l
,
(19)
TJ^
1
J
and g n are
explicit functions via (1)
and
(3a) of (x, a, u)
and
the initial conditions (n,u TO ,g TO ,q To ,x To ).
The
derivation of the equations
The informed
reformulation.
is
in the appendix, but they
traders control their rate of entry
the auctioneer to break even on all trades,
18
have an intuitive
—x
so as to enable
while the auctioneer selects the rate of
decline a of the ask price allowing the traders to behave in an incentive compatible
fashion. Thus, the law of
motion
for
x comes from the ask
price equation (9), while
the law of motion for a comes from the informed traders' IC equation (16):
=
("i)
n
J(v - a)q4>x g + fif(v-
*
v
informed traders'
flow (+) profits
18
More
generally, with
'
o)(l
- q)g n
v
v
'
uninformed traders'
flow
(— )
profits
a profit-maximizing auctioneer, traders would choose
tioneer could satisfy his optimization.
16
—i
so that the auc-
a
=
—AILi
J (V£ -
,
after
continuity of (u,x),
subgame. But the
(n
-
l)jrfc
expected (purchase hazard rate)
x (loss from another purchase)
(expected loss from
exogenous termination)
in every
- a))
/7.
Ax
Remark. By
(»
initial
I
have
initial
ask price
is
conditions
u7rs+
down
not pinned
= u^
and x To+
(intuitively,
= x To
0^+ > Or
a purchase). This indeterminancy potentially creates a multiplicity of equilibria.
Whether
one or more equilibria do in fact
in fact
exist has not yet
been shown.
Are There Other Separating Equilibria?
3.5
With the continuation
subgame can be thought of as a
traders; their
message space
purchase. In this light,
I
down by
the recursive structure, any given
static signalling
game by the remaining informed
payoffs nailed
simply (r
is
now
,
00],
where
t
=
{00}
is
briefly revisit the choice of the truncation equilibrium.
A standard feature of many distributions satisfying the MLRP
of the conditional density,
19
Lemma
and
if
2
the
Any
maximum
most
the n-trader
separating
likely signal
game
too.
is
a single-peakedness
or
• Most Likely Signal Property: For each
unique interior
the message never to
w(v) €
for
v €
(0, 1),
the
map w
»->
f(w\v) has a
(0, 1).
MSE of the
one trader game
property holds, then this
is
is
a truncation equilibrium,
the unique separating
MSE of
20
The (appendicized) proof
is in
three steps. First,
to argue that higher types purchase
first.
I
use the single-crossing property
In a word, while
all
traders prefer the
presumably lower price that comes with time, the higher one's signal the greater the
expected payoff that one places at risk by waiting. But with more than one trader,
this
argument
is
incomplete:
with a lower signal
may
A subgame may end from another purchase,
and a trader
well have a higher estimate of the purchase hazard rate
by
For instance, the normal family f(w\v) = ce _(u,_ "^, but not the Poisson family f(w\v) = ve~ vw
(Of course, these popular examples are ruled out by the bounded signal support assumption.)
20
It is very plausible that no 'pooling equilibria' exist, in the sense of two traders of different types
planning to buy at the same time. In other words, any MSE is a truncation equilibrium.
19
.
17
those other traders
If
if,
perversely, other lower signal traders are expected to enter
the equilibrium in the continuation
more eager
subgame
is
particularly dreaded, they
As
to purchase earlier than the higher signal traders.
it
first.
may be
turns out,
I
show
that this logic cannot be sustained.
The
final
two steps rule out either atoms of informed traders entering (x t drops
discontinuously), or no informed traders entering (x t
rates of informed
=
Essentially, the entry
0).
and uninformed traders always have to be 'mutually absolutely
continuous', neither infinitely larger than the other.
The One-Trader Endgame
3.6
The
differential equations for the n-trader
from the respective (n
by a purchase.
—
l)-trader
subgame contain continuation values
subgames that can be triggered at each point
This significantly complicates the analysis.
I
therefore specialize
the equations to the one-trader endgame, upon whose solution the equilibria of
all
subgames are constructed via backwards induction.
With n
=
l,
equation (19) reduces to
at
=
mB
,
-Allf ,{x t )
J%(v-at)f(xt\v)tf(v)dv
°
„
= -A
(20)
Svl f(xt\v)gHv)dv
Simply put, the lone informed trader buys when her marginal
from waiting
profits
for
a lower price are rising at the rate of time discounting induced by the flow probability A
of exogenous market closing.
When only one trader remains in the market,
neither the
competitive pressures nor the learning opportunities associated with the possibility
of intervening trade in the many-player
trader's only strategic consideration
is
subgame are
present,
and thus the informed
to hide her signal from the auctioneer. Notice
in particular that the fully pooling no-purchase equilibrium
with
a*
=
1 is not
a
solution of (20).
But
as
is
standard with the single-crossing property (from section 3.5),
inframarginal traders must
Corollary
must
strictly
make
strictly positive profits
Uf (x
t)
>
0.
So
all strictly
(20) implies
In any truncation equilibrium of the one trader subgame, the ask price
monotonically decline.
18
I
cannot be as definite about subgames with more than one trader pending a better
In particular, to establish a monotonic
understanding of the continuation values.
decline in the ask price,
engaged
in a
would be
it
sufficient to
argue that the traders are de facto
pre-emption game, with an expected
a slightly higher signal
—
loss
from entry by a trader with
the second term in (19)
i.e.
is
Deducing
negative.
this
remains an open problem.
4.
The Revised Model
4.1
I
of
THE MARKET MAKER MODEL
now complete
GM,
the story of a competitive specialist or market-maker in the spirit
maker continuously announces bid and ask
to
buy and
sell
prices at which he
unit shares of the asset, respectively.
temporary (time-t) prices
profits, conditional
The
The market
rather than the preceding facetious competitive auctioneer.
risk neutral
for
currently willing
is
The market maker chooses
these
each type of transaction so as to earn zero expected
upon the occurrence of the transaction
informed traders' information structure
egy spaces are enlarged. Each of the
N traders
may make
at time
is
at
t
as before, but
all strat-
most one publicly ob-
servable transaction of unit size, either a purchase or a sale of a single share, while
Each noise trader
the market remains open.
probabilities according to
is
prompted to buy or
sale with equal
an independent Poisson process with parameter
The separating truncation
equilibrium
now
consists of continuous
and
creasing (resp. increasing), and hence piecewise differentiable, functions
' *-* yt),
where
< y < x <
t
t
1,
types that respectively buy and
signals
w
lim^oo x t
true that
€
[yt.^t]-
=
lim^oo y t
such that at time
sell,
and thus
Such an equilibrium
.
If
a^ = y^
(i)
strictly de-
M- x t (resp.
x t and y are the informed
t
remaining informed traders have
completely (or fully) separating provided
is
but x t
(ii) all
t,
t
/x.
>
yt for all finite
t,
then
some informed types have not traded while the market
such an equilibrium eventually separating.
19
is
it
may
open.
always be
I
shall call
The Analysis
4.2
Market Maker Model
of the
The Accounting Equations.
in contrast to (1)
and
(4),
Given the current truncation
[y t ,x t ],
the probability q of a typical trader being informed
is
now
and
derivative
its
is
(F(x t \v)-F(yt \v))q
_
,
t
Qt{V)
(F(x t
- F(y
v)
|
= qt(v)(l — q
qt (v)
t
• the public posterior beliefs p"(v) are
.
|
{v))(n
still
+ tfc(l - q
v))q
+ 4>(x
t
\
v)x t
-
K
.
.
$ttK(w\v)=
;
'
F(w
I
v)
„)
- F(yyt
<j){y t
F(x \v)-F(yt
t
Mw\v)=
m
.
,
and
;
\
.
.
,
;
\v)
'
The Market Maker's Problem.
\
v)yt)
given by (3a), but the conditional signal
\v)
t
„)
(
}
)
and density are now conditioned on the truncation
distribution
.,
t
Equations
for
[j/ t
,
x t ], as
flw v)
,\ -=
in
I
]
—,
'
F(a: t
,
jt;)-F(y
t
.
|t;)
7t (iy,v), gt (v\w), and E^V"
remain unchanged. Since noise traders are assumed equally
likely to
buy
or
sell,
the
purchase and sale hazard rates of traders (informed and uninformed alike) are now
Pt(v)
=q
s t (v)
= q (v)<f>t(y
t
{v)<t>t{xt
t
t
v){-x
I
|
v)yt
t)
+
-
(1
+ (1 - q
t
qt (v))(fx/2)
(v))(n/2)
Given a trade that has occurred at a given ask or
the value density gt (v) following a purchase to g t +(v)
gt+(v)
=g
t
'"(u) following
a
bid, the specialist
=
n
updates
g?' (v) as in (8),
and to
a natural (but omitted) analogue to the formula
(10) for
sale,
where
Svo 8t{z)fi(z)dz
With
this equation, there is
the posterior density over valuations v.
Also, the required ask
meet the market maker's zero-profit constraint are given by
h=
Jv
vgt
(v)dv
= In
°
Jvj
20
(9)
vs t {v)g?{v)dv
TT-rr7Z
s t (v)g?(v)dv
and bid
and
prices that
The Informed Trader's Problem. The
at time
t
< T from buying
IlfH =
nfM =
n?H
and delay (D) are as
(B), selling (5),
-E wt V
bt
=
)
Wf(Zt, Pt,w),
if
another trader buys at time
Wf(It,a
if
another trader
t
,w),
the market closes at time
sells at
Vi
,
,
s t (w)
=
f
/
,,
w)dv =
.
,
s t (v)g t (v
,
|
.
$
s t (v)<yt (w,v)dv
-yt (w,
J Vo
=
S%s
'directed stopping rule', or
a time and an action (a
maximize her expected eventual payoff from
f e" />-l)(M"HM"))+A]<*r (n _
1}
J To
+ g" /^[(«-l)(Pr(«)+*r(«))+A]* n «( u
this
t
now
v
buy or
or
=S
=
(n
Mw)WB {w) + §t{w)WS (w)
-
+s
JV H] -
Allf (j/t)
=
(n
sell)
so as to
]
dt
,)
x and y
t
t
find
it
optimal to buy and
1) {ft(x»)
[nf (x t ) -
Wf (*,)]
1U=I(
d
(v)dv
subgame, namely:
respectively, at time t
Xnfixt)
l
entails choosing a
[
incentive compatibility requires that types
-
r {v)dv
v)gt
I
now
To is
=B
also
{v)ttw\v) 9
JVo f(w
v)dv
Note that the informed trader's objective at time
sell,
t
trader's private assessment of the instantaneous probabilities of inter-
Jvo
Thus
time
t
t
vening trade by any other given trader are pt(w) as before, and
.
follows:
E wt V-at
0, if
The informed
type-w trader's instantaneous payoffs
t
-
+s
(x t )[nf(x t )-'Wf( a t )]}
;
1) {pe(2/t)
t
[nf fa)
(y t )[Uf(y t
21
-
Wf (»)]
)-Wf(y )}}
t
The Autonomous
auction model,
I
First
now produce an
explicit
Clearly,
0=
u
(-x)
I\v
In addition,
I
b).
below the intuitive statement of the laws of motion.
have
- a)q^- + (»/2)J(v-a)(l-q)g n
x
'
v
^
'
flow profits of trading
flow profits of trading
yj(b-v)q^-
-All*
Here,
1
+(f,/2)J(b-v)(l-q)g
/ (Wf -
(y
-
a)) (n
(22)
'
uninformed traders'
n
(23)
s
J (Wx -
- lfrg- 1
(v
- a))
(n
-
[
Sir
1
/(ft-^-WfH"- 1 )*^"
hr
I
1)8^ (2i)
Six
expected (purchase hazard rate) x
(A profit from another purchase)
^
differential
Rather than write down an analogue to
informed traders'
d= _ An a
6
shall just include
= — Xu.
<
0=
1, 1
analogy to the solution of the
system of first-order autonomous
equations in the state variables (u, x, y, a,
Proposition
By
Order System.
expected (sale hazard rate) x
(A profit from another sale)
S
f((b-v)-Wy)(n-l)s<yr
1
1
m)
,
'
Itr
l
1
E is the profit loss or gain from nature's exogenous termination of the market.
Remarks:
1. Just as the
competitive auction model had a single degree of freedom in the
choice of o^, in the market
2.
As
maker model there are two degrees of freedom
before, strict individual rationality implies that the ask price
in (a To b To ).
,
rising
is
and
the bid price falling in the one trader game, since there are no continuation values.
To be more
definite
about the n-trader game (n
two terms to be negative in (24) and positive in
a conjecture as
I
>
1), I
(25).
need the sum of the
This
result,
final
however, remains
cannot yet sign these terms.
3. Intuitively (to me), individuals strictly prefer that
an opposite side trade just
precede them, but just wish to pre-empt same-side trades (as with the competitive
auction model),
i.e.
the second term
the reverse true of (25).
I
am
is
negative and the third positive in (24), with
forced to leave open this intriguing question.
22
5.
IS
In this section,
If the
issue:
in the price?
FULL SEPARATION POSSIBLE?
consider the extent of separation. This
I
market
is
open long enough,
As discussed
separation in the
earlier,
is all
is
a very fundamental
information eventually impounded
GM tantalizingly hint at the impossibility of full-
common value setting, and it was
certainly true of their model. This
would provide a sharp contrast to the theory developed
that in standard double auction environments,
full
in
Wilson (1985), who showed
separation
is
incentive compatible.
The Competitive Auction Model
5.1
This paradigm constrains the action set of the traders,
who can only
purchase.
It
so happens that this provides a very natural limit on the extent of entry.
Proposition 2 (Incomplete Separation in the Competitive Auction Model)
In a truncation equilibrium of the n-trader subgame, only those whose signals are good
news given
w
x >
t
Proof:
for
the prior
all t.
g*~ l ever purchase. Moreover,
In either case, complete separation
is
a neutral signal
or, equivalently,
when
o)(l
-
n
q)g
^ <=>
f(v
The appendix shows how
toM
n-i./^r
=
~
s„$
density #"
But
I
will
is
x <
a)q<t> x g
1^
n
exactly
when
£
w
B
ZM]
(26)
f„n-l
n-1
1
<
a t<
Thus, bad and neutral news given the prior p" -1
The
then
this implies
V.
-1
exists,
either of the following inequality pairs obtains:
ej^tf, .
a neutral signal
-
w
thus impossible.
In any n-buyer truncation equilibrium, (18) yields
j{v -
If
if
Jvfxg'
t
is
screened out.
doesn't exist, then incomplete separation
not constant in time, and so u"
_1
might well tend to
with probability one eventually reach the one trader
23
is still
game
in doubt:
over time.
— unless the two
smallest signals are tied, a zero chance event.
traders
who purchase have
that subgame.
5.2
signals above
As soon
as
n
=
1,
the only informed
weak neutral news given the
Thus complete separation cannot occur (almost
prior
0+ for
<?°
surely).
The Market Maker Model
Proposition 2 has the following immediate extension, whose proof
Proposition 3 (On Eventual Separation in the Market
I
omit.
Maker Model)
In a truncation equilibrium of the n-trader subgame, only those with good (resp. bad)
l
news signals given the prior g"~ ever purchase
w
signal
exists,
then x t
>w >
yt for
(resp. sell).
Moreover,
if
a neutral
So with an arbitrary number of
all t.
traders,
complete separation cannot occur in finite time.
In other words, there
is
no option value of changing
sides if the signal distribution
has a neutral signal.
Since (x t ) and (yt ) are each monotonic, they have well-defined limits as
t
—¥
Observe that eventual separation, or
x^ =
informed trade vanishes, since by
an adverse selection problem (namely q^
will arise if the
(1)
oo.
y^, does not imply that the chance of
>
0)
informed traders have entered more slowly than the noise traders.
For eventual separation requires that x t — yt vanish at a faster than the exponential
- 14
rate that e * does. On the other hand, if I do not achieve eventual separation, or
Zoo
>
J/oo?
resolve
then necessarily
q^ =
1
by
(1),
which of these possibilities obtains
and
all
trade
is
choked
in the next section
off.
I
shall try to
by characterizing the
equilibrium.
Corollary (Switching Sides of the Market)
If a neutral
market they
news signal
exists,
then no traders are ever unsure of which side of the
will trade on, if any, in future
signal does not exist, then for
subgames. Conversely,
some intervening
a neutral news
series of transactions, a trader might
well change plans (buy rather than sell, or vice versa).
24
if
6.
I first
EXISTENCE AND UNIQUENESS
must understand how
well
behaved are the laws of motion governing the
model
state variables. It turns out that the
behaved that the slopes
sufficiently well
is
of the equilibrium paths nicely fan out from any given initial point
— at
least in the
one trader game.
Lemma
with
3 (Derivative Monotonicity)
da/dx <
db/db
>
0,
in the one trader
dx/dy >
with
a neutral signal
Remark. The
exists,
>
0,
game. In the market maker model, additionally:
>
and dy/dx
>
then dx/dx
signs of partials of x
fact that only informed traders
In a truncation equilibrium, da/da
0,
ifn
>
dx/da <
1,
0,
and db/dy
dy/dy >
0,
and y with respect to x or y
whose signals are good
<0ifn =
l.
If also
and dy/db
<
0.
crucially rely
on the
bad) news will purchase
(resp.
(resp. sell) in equilibrium.
Absent the encumbering continuation values,
definite with
one than with n
>
1 traders. I
now
results for
both models are more
exploit the above result to provide
a proof of existence and uniqueness of a truncation equilibrium for the n
game.
Essentially, this entails nailing
prices are for each
down
exactly
what the
initial
subgame. For instance, with extremely precise
=
1
trader
ask and bid
signals, the initial
ask price must be set very high, and the bid price very low.
I
argue below that the maximal amount of separation occurs in equilibrium.
Proposition 4 (Existence and Uniqueness)
rium for either model, which
is
There exists a truncation
unique for the one trader subgame. Also, eventually
any informed trader with good news purchases in the competitive auction
in the
Proof:
provide a constructive and visual proof,
model. The idea
by using time
arise
setting,
market maker model, any informed trader with bad news eventually
I
t
is
to revert to the
rather than
u
=
e
nonautonomous
first for
,
is
as in figure
1.
The uniqueness
saddle point stable
structure.
25
and
sells.
the competitive auction
nonautonomous dynamical system
-tlt
because the desired equilibrium
of this given the
equilib-
in (x, a) alone,
essentially will
— although
I
make no
use
Figure 1: Phase Plane Diagram of One Trader Game. For the competitive auction
model with n = 1, 1 project the solutions of (17), (18), and (19) onto x-a space. Notice that a unique
equilibrium arises (namely, a stable manifold starting at aJJ tending to the origin), and it entails the
theoretically maximal amount of separation, with convergence to the neutral signal.
In x-a space, a truncation equilibrium
t
>
0.
Now, x > —oo
21
and x
<
is
a path with xq
=
1
and x <
for all
imply
,^h? <a< hhl
1
The only
9°
(27)
ff*9°
possible destination for the dynamical system
is
the intersection point
7
of
the two boundaries. For as the dynamical system approaches the upper boundary, x
starts to explode while
d vanishes, and so da/dx vanishes; therefore, the paths tend
towards that boundary horizontally and
fast.
22
Conversely, as the system approaches
the lower boundary, x vanishes but d does not, so that da/dx vanishes; therefore,
the paths must transverse the boundary vertically. In either case, the system will hit
either
will
21
boundary
change
in finite time,
sign).
Thus, only
which cannot possibly be an equilibrium
if
the system
(x, a)
approaches
7
can
|±|
(for
+
|d|
then x
->
as
This equivalently comes from individual rationality.
Note that x
reverses course
is
discontinuous across the boundary, and so the dynamical system immediately
upon touching
it,
as seen in the figure.
26
it
—
must
appendix
By
dx/da <
for since
by Proposition
3,
a must also come to a
any such approach must take
in fact verifies that
rest.
The
forever.
continuity of the dynamical system between these two boundaries, such an
approach path (part of the so-called stable manifold) must
exist.
For when oq
below (J vfx g°)/(J fx g°), the system crosses the top boundary, while
(/ V 9°)/(J 9°)
>
it
crosses the lower one.
On
for
is
just
a just above
the other hand, strict monotonicity of the
derivatives of the dynamical system implies uniqueness, for the paths monotonically
'bend down' as ao decreases: Namely,
prices
and
a
k than
for
n
do
> a
>
at any later time t
— so long as (x ,a
t
inequalities apply) for
compare two paths
if I
t)
0,
lies strictly
both paths at time
xt
is
and
tt
strictly higher
starting at ask
and
a* lower for
between the boundaries (and thus the
therefore, there
t;
it
cannot exist two distinct
approach paths to the intersection point.
The market maker problem with n
motion
for (x,y,a,b)
=
1 is
no more
same
fact as
respective boundaries of
above that
x and y are monotonia
Remark. The
if
it
>
1 is as yet missing,
b),
but will
the ask or bid prices start too close to the
x € (— oo, 0) and y €
becomes nonmonotonic. Thus,
both
the law of
decouples into two independent systems in (x, a) and (y,
with analogous analyses. The existence proof for n
exploit the
difficult, since
(0, oo),
then very quickly x or y
should be possible to prove that for some choice,
23
uniqueness for n
>
1
remains an open question, awaiting
my
sharper understanding of the continuation values. For instance, in the competitive
auction model,
I
only lack demonstration that da/dx
appendix proofs of
But
I
all
can say that
<
when n >
1,
since the
other monotonicity results obtain for general n.
if
doubt there does), then
I
there exists a truncation equilibrium for
n >
1
(which no
almost surely eventually reach the one trader game
— at
which point, the above characterization obtains. Thus, in any truncation equilibrium
of the competitive auction or market
maker model with n >
1,
there necessarily obtains
complete separation, with convergence to the neutral signal.
23
Note that the key to this argument will be the
do not hit it at different times.
point, then they
27
fact that
if
x and y take
forever to reach this
PRICE AND TRADE DYNAMICS
7.
7.1 Inter- Subgame Price
The
up
Jumps
GM's exogenous
intuitive result, true in
in response to
any purchase
is
entry model, that prices must
much harder to deduce with endogenous
even with a positive chance that some remaining trader
to conclude that
is difficult
any purchase
pure liquidity reasons, which garbles
its
is
the more likely
it is
trade
may
also
>
decreasing in v by
is
(1):
0), it
come from
FSD
informational content. For by strict
(F(-\v)) in v, the chance of informed trade qt (v)
is v,
A
good news:
timing. For
informed (with x t
is
jump
of
The higher
that there are fewer informed buyers, given that no one has
yet bought in the current subgame.
Fortunately,
Lemma
Let
a neutral signal exists,
and so
all
Proof:
I
whose
fv (x
t
all pt(v) is
purchases are good news and
>
good news
for all
t,
will
Corollary (Bid and
/
x
Pt{
]
Ask
following a purchase, and
yt in the
market maker model.
increasing in v and s t (v) decreasing in v,
bad news about the asset value.
all sales
go/fo
Prices)
is
(1
-
+ (1
-
q )u
F(x t \v)
Thus, a*
>
definition (9) of
for all t
and
n. Next, after
value
the immediately preceding ask price, or
>
u^"
,
price,
>
v?
Lemma
a purchase at time r
and so aT0+
D
a To as claimed.
,
28
>
bt
.
Also, both prices
jump up
sale.
Be
1
go)p
increasing and denominator decreasing.
down following a
an ask
rate
MM) +
Proof:
that a T0+
t
and so the purchase hazard
increasing in v, as the numerator
is
x >
there exists a neutral signal.
purchase in equilibrium. Thus, Fact 1-4 implies that
q
is
Assume
just consider the auction model. Proposition 2 assures us that only traders
signals are
\v)
definite.
in the competitive auction model, or
t
if
can now be more
4 (Transactions are Informative)
x >
Then
I
,
4 and Fact 2.1 then yield at
>
£"
the public expectation of the asset
v^
=
aTo
.
But
I
have just shown
7.2 Transaction Prices
wish to extend the martingale property of
I
To
tion prices are expected to remain constant.
•
the
71*,
fc'th realized
transaction price, k
=
GM, and show
that realized transac-
this end, define
1, 2,
.
.
.
N
,
(if
the market doesn't close
before everyone has a chance to purchase)
Thus,
71-*
is
(Ti,...,Tfc),
a deterministic function of the publicly known
namely
71*
=
SfKIZ*], but 7r*+i
only on the information Z*
known
Tk+i
<
oo,
is
r*
=
Lemma
oo.
5
jump
all
=
£[7rfe+ i|X*]
7r*;
In this notation,
however, in light of
the mere fact that there exists a next transaction, or that
remaining signals are bad news. So
Then
is
£[7r* +1
diately
a random variable conditional
this allows
me
I
=
lim^oo vt
w.r.t.
{2*}
k
:
£[7r*+i \T
\
= 7r*. 24
a standard application of the Law of Iterated Expectations. 25
fc
|X
]
=
fc+1
£[£[t/|r
]|J*]
=
£[E[V\Ik ,Tk+1 }\Ik
shall conclude that following
up,
let's define 71*
to state
Transaction prices {ftk+i} are a martingale
The argument
In section 7,
Ik =
a valuable piece of information in the competitive auction model: This
means that not
when
3,
clearly
k trade times
as of the previous purchase.
the martingale property would assert that
Propositions 2 and
is
first
and then monotonically
}
=
E[V\Ik
]
= nk
any purchase, the ask price must imme-
fall
during the next subgame. Thus, with
the endogenous timing, the martingale property will have a slightly more pointed
implication than
of a
subgame
is
its
precursor in
GM: The
initial
jump
in the ask price at the outset
balanced by an expected subsequent gradual decline during the next
subgame.
Note that
Lemma 5 is
not an arbitrage argument, since any informed trader entails
a risk of the market closing by deferring trade, but no implicit discounting appears in
the martingale expression! Rather,
information structure, and he
is
it is
simply a deduction based on the auctioneer's
not directly affected by the market closing.
24
More generally, this is a martingale relative to the auctioneer's information.
In fact, the martingale result is stronger than that. Namely, I can (and will in the next version)
show that £[7Tjt +1 |Zt] = € t the current expected value of the asset.
25
,
29
Do
7.3
I
Rational Traders Frenzy?
now come
to the motivational question for this paper, which
be tackled given the theory
am
I
hopeful can
have developed to this point. Consider the competitive
I
auction model, and imagine that a purchase occurs at time
traders with signals just below
x To (and thus
How
t.
do informed
On
just about to enter) respond?
the
one hand, favorable information about the asset value has just been released, which
one might presume provides an incentive to jump into the fray sooner. But this urge
is
tempered by the immediate jump
First note that frenzying
deduce
is
Which
in the ask price.
effect
an equilibrium phenomenon, and
dominates?
it is
let
a purchase
So when the ask price jumps up,
this forces
on the basis of one side of the market alone. For instance,
it
occur at time r
.
Recall from (18) that
n
_ tif(v-a)(l-q)g
=
n
x
J{v-
Lemma
3 tells us that
down x
at time r
.
dx/da <
The
0.
or equivalents |x T0+
|
>
|x To
is
By Lemma
x To+ Thus,
so by Fact 2.1, this discretely increases
,
a)q<t> x g
posterior on the valuation
g~'n (v) oc p To (v)g? (v) in the x expression.
x T0+ < x T0
not possible to
.
also
4,
it is
updated from g^
p(v)
is
increasing in v, and
not obvious whether
can show that after a purchase occurs, the
That
is,
there
is
shift
<7"
(u)
•->•
gf
'
n
{v) causes
believe
d to
be used to show that both of the externality terms
diminish, irrespective of whether a war of attrition or a preeemption
I
I
fall.
frenzying on the part of the market maker. For indeed, the basic
integral inequality can
obviously so,
in fact
|.
Let's briefly outline a possible avenue of thinking about this problem.
I
(v) h-»
game
strictly
arises; less
believe (but cannot yet prove) that the profits to the marginal trader
also discretely increase
from the
shift.
Indeed, following a price jump, an informed
trader with a nearby signal places a higher weight on the informed status of the
purchase being than does the market maker, and thus ought to be more encouraged
to enter by the purchase than she
same
token,
someone about to
sell
about to
sell.
But
I
discouraged by the higher ask price.
By
the
places a lower weight on the informed status of the
transaction than the market maker,
for those
is
and thus the bid
price rise should lower profits
need a proof of this inequality.
30
assume that
Let's
in fact
I
have d
<
in equilibrium.
As
it
turns out,
yet been able to demonstrate this intuitive property of the equilibrium.
a/(da/dx), and
RHS
have just argued that the numerator of the
I
have not
I
Now x =
grows in absolute
Unfortunately, with the higher ask price that will obtain in the continuation
size.
subgame, the required slope da/dx also grows. This
and
analysis,
rate, this
relies
is
the equilibrium aspect of the
on simple geometry, and may not admit a
means that
not at
it is
obvious that x
all
falls,
At any
tight proof.
and thus frenzying
is
not
clear.
Remark.
Regardless of whether frenzying occurs or not, by (30), the
terior density g"
density
g^
n
_1
The appendix proves
x t (that
when
is,
subgame
{v)/9-
*
increase
{v)
t
>
lower
-{ qoF(x
>r
To
\v)
by the factor
r
and so by Fact
,
{
2.1,
1
1)
=
I
on the values
2x,
to gradually
26
plan to illustrate the general solution developed up to this point,
of the results depended.
f(x
is
is
thus putting a damper on any frenzy that occurs, or
|x|),
abandon the assumption of an atomless distribution
probabilities
model, this term
the effect of this
turning to a simple example with two states of the world.
porarily
(28)
+ l- qo )uJ
A WORKED EXAMPLE
8.
I
at time t
in fact that in the competitive auction
accelerating any anti-frenzying.
In this section
pos-
immediately but continuously starts to diverge from the prior
for the next
diminishing in v
new
shall let
and
both defined on
1,
g
for
tem-
I
V, upon which none
be a two-point distribution that places equal
and use the two
[0, 1].
In other words,
signal densities
f(x
|
0)
=
1
and
These signal distributions belong to the family
of generalized uniform distributions F(x)
= x*^, a
e
[0, 1).
Observe that in the simulation of this example below, rational traders do frenzy
after a purchase.
(to
26
be continued)
While the proof does not work for the market maker model,
31
I
presume the
result
still
holds.
CI
o
01
\h-^^L
^x^^_^)_
""OS
05
6( 2 )
45
04
•
OJS
OJ
t
<
Frenzying after a Purchase.
Here is graphed the dynamic behavior of the
prices (a, b) and the entering signals (i, y) against time in the model with two traders. A purchase
occurs at time .58, and in response both ask and bid prices jump up, as expected. Also, x and y
accelerate: the entry rate of both buyers and sellers rises, and convergence of i — y is much faster.
Figure
2:
APPENDIX
A.
A.l Useful Mathematical Facts
Fact
1
is
(3)
Let (f(w\v)) be a smooth family of conditional densities. Then
(f(w\v)) has strict
(1)
(2)
(MLRP)
MLRP <&
fv (w\v)/f(w\v)
log-supermodular: f(w\v)fwv (w\v)
f
> fw {w\v)fv {w\v)
MLRP =» f(w\v)/F(w\v) increasing in v, or equivalently the
distribution function F is log-supermodular: F(w\v)Fwv (w\v) > Fw (w\v)Fv (w\v)
(f(w\v)) has strict
(f(w\v)) has strict
MLRP =^
F(- v)
\
is
ordered in v by strict first order stochastic
dominance (FSD), or F(w\v) decreasing in v
(4)
increasing in w, or equivalently
(f(w\v)) has strict
signal
Part
(1) is
follows
x
is
MLRP, and
if
F(w\v)
<
a neutral news signal exists => fv {x\v)
^
as the
good news or bad news
due to Milgrom (1981), and part
(3) is trivial.
Part
by means of an integral inequality. Finally, to see part
Then f(x\v)/f(x\v') >liffv>v', and
Another
1
so
fv (x\v)
(4), let
widely known,
x be good news.
follows.
result is used sufficiently often that it is
32
(2), less
summarized
for easy reference.
Fact 2 (Key Integral Inequalities)
Let
(1)
ip'(v)
>
and N'(v)
>
0.
Then
J* 1,(v)N(v)dv ^> f£ N(v)dv
J* ^(v)D(v)dv J* D(v)dv
if the
both numerators are positive
and both denominators negative and
Let N(v)/D(v) be increasing in v, and N(v),D(v)
(2)
< 0, or
D'(v) > 0.
numerator and denominator in both cases are positive and D'(v)
[* N(v)dv
J
> *
N(Vi)/D<yi)
>
0.
if
Then
> N(V )/D(VQ )
'-
Jv D{v)dv
Lemma
A. 2 Proof of
I
now
1:
Accounting Laws of Motion
describe the evolution of qt
simplest to
first
consider
q. If
and gt as time elapses with no
no trade has occurred between
t
and
t
trade.
+ A,
then
It is
I
may
use Bayes' Rule to update qt pointwise, as follows:
W\ v '
F(xt\v)
%^*W +
F(xt\v)
«p(-/xA)(l - *(«))
~)J\
t+A (t;)\
i-ft(«)
V
After taking logs, dividing by A, and letting
The law
/ Ffa|p) \
A—
>
0, I
of motion for gt (v) within the n-trader
obtain (4) via l'Hopital's rule.
subgame uses
this result
following consequence of Bayes' rule:
[(l-q
9t+M v )
t
(
v ))e-^
= -77,
Svl ((1
+ q (v)^^y 9t (v)
t
7-ha
"
ft
W)e-^ + Qt{z)^g^)
33
gt (z)dz
and the
Taking logs and limits as before,
»log((l-
gt (v)
ft
,
"m
—
TT = a^o
9t(v)
(t,))e-^+
(t,)^Jji)
ft
Aa
log/*
—
obtain
I
((1
-g
t
(z))e-^
+g
t
(z)^ifY 9t (z)d2
lim
A-+0
= n [-/i + q
t
{v)
(fj,
+ <f>{x
|
- n -H+
v)x)]
qt (z)(fi
+ 4>{x\z)x)g
t
(z)dz
JVo
D
A.3 Proof of Proposition
Prom the ask
at
Truncation Equilibria of the Auction Model
1:
price equation (9),
_ f vpt (v)g?(v)dv _ f v[q
'
I
have substituted from
x upon
tion (18) for
+ (1 - g (v))//]ff n (u)<fo
+ (1 - q (v))n}g?(v)dv
(v)<f>(x t \v)(-x t )
/[ft(t;)^(st|t;)(-i t )
/?*(»)$?(*>)*'
where
t
(7) for the
t
t
t
purchase hazard rate
p.
This yields equa-
simplification.
Next, the law of motion for a comes from the informed traders' IC equation (16),
which
I
simplify in parts. Firstly,
dt
W=Xt
(5)
9?
can be rewritten using
»=
(n
-
need to know
J 7x / zjx - / 7Z / z lx
(/%)•
d J Z7t(w,z)dz
^n'ft,™)
Now,
I
n_1
l)^
(w)
J ~f
t
(w,z)dz
W=Xt
(4) as
qt (v)(fi
+ <f>(x
t
\
v)x t )
-
qt {z)(n
/
+ <f>{x
t
\
z)x t )dz
JVo
As the
right
hand
integral
is
constant in
v, it
cancels in
/ "fx J zjx — fix J zjx which
,
therefore expands to
g-i - z
g"-i
J lx f fx
= 1 1x I zfx(n - l)g n - l q(fi + x x) - / z lx / fx (n -
f 7x f zfx
<f>
34
l)g n
-l
q(»
+
4> x x)]
= / 7* / z lx{n -
l)q{n
+
- J z-yx J 7x (n -
x±)
<f>
\)q(n
+
x x)}
<j>
Thus, equation (16) becomes
4
W B_lpi +a
= _AHf + (n - !)Zfatt*)+flzMfc
/7x
J7x
/ 7x / *7x(n -
+
l)g(/x
J)
<ftx
- / £7x / 7x(n -
l)g(/^
+ </>xi)]
2
(/7x)
= -Anf +
(n-l):/
(Wf -
(z
-
a)) («*.(-*)
+ (1 - q)fi) 7x
J 7x
which yields the desired equation
A.4 Completion of Proof of
Step
The
idea
common
Lemma
Higher types purchase
1:
in the spirit of
is
=
x
many
2
first. Start with the one-trader subgame.
related results in auction design, only here
Think of the trader of type
value.
probability
(19).
e
-Ar
w
as sending a message,
of acquiring the good, where r
having to pay a transfer
T = are~ XT
,
is
it is
namely the
the time to purchase, and
and receiving an expected reward
= x£. w [V]-T
u{x,T,w)
For a fixed x and T, higher types place a greater likelihood on the asset value being
high than do lower types. Thus, the Spence-Mirrlees single-crossing property applies:
= — d£, w [V]/dw <
d(ux /u,T)/dw
7.2 in
0.
Standard results
(see, for instance,
Fudenberg and Tirole (1991)) imply that the only implementable (and thus
potentially equilibrium) separating outcomes are weakly monotonic in x, or
Next, with
the
Theorem
first
n >
1 traders,
x <
0.
the informed trader with the lowest signal cannot be
<
to enter, for then E[V\p[,It]
E[V\Pu Xt]
=
a*.
This violates individual
rationality of that informed trader, as the purchase earns her negative profits in
expectation.
enters
first,
Next suppose, by way of contradiction, that some trader
again by the logic that she
informed traders with signals close to
But then
for small
enough e
>
0,
w
is
more confident of an
w €
(0, 1)
early entry by other
(and the putative bad continuation payoff).
a trader with signal
35
w+e
has a
first
order increase
from entry, and only a second order
in expected profits
entry
+ e).
p (w
t
This follows from (13) and the most
earlier single-crossing
argument once more
the private beliefs of
fall in
The
likely signal property.
will tell us that
w
if
has a weak incentive
w + e has a strict incentive to do so.
An atom of informed traders never purchases. With
to enter, then
Step
2:
influx of informed traders, the inelastic noise trade is
swamped
at that instant in time,
and so the standard
'no-trade' result ought to preclude this in equilibrium.
being overly formal:
The ask
Without
price will coincide with the expected value of the asset
more than one type
conditional on which types are buying. If
of such a transaction
a sudden
must be negative
is
buying, the profits
Thus, only one (atomless)
for the lowest type.
type can enter at any 'instant'.
Step
<
xt
3:
always holds.
I
want to
For
rule out 'flats' in x.
if
xt
=
0,
then the noise trade swamps the informed trade, and the ask price coincides with
the expected value of the asset. But then marginally higher types x t
purchased at an ask price that
small enough e
>
is
discontinuously higher
+e
must have
when x Xt+e <
(i.e.
For
0).
could not have been incentive compatible.
0, this
A.5 Proof of Key Inequality for Proposition
First notice that the >,
>
2:
Incomplete Separation
The ask
inequalities in (26) cannot obtain:
price
must
lie
below the expected value given an informed purchase (and thus exceed the expected
value given an uninformed purchase). This follows more formally from
< Jv}x9t~ = / vqt<t>x [qoFXTo +
l
a
'
Jfzg?'
1
J<lt<t> x lqoFXTo
(1
-
q )u To ]g?
+ (l-q
< fvq
"
Jq
)u To }g?
t <f)
x g?
{
t <j>
j
x g?
since (in order)
•
by the informed traders'
• equation (3b)
and then
„
(u)
== f(x
t
a
\v)g
/
n
strict individual rationality, or
(1)
and
(2)
oc
<
Yif (x t )
permit the implication
^(^) + (i- ?oM
-Hv)
at
n-
1(u)
qt (v)<j> t (x t \v)g?(v)[qQ F(x T0
36
\
(30)
v)
+
(1
-
q )u TO
]
• by Fact 1.3,
F(x To
|
v)
is
— and thus equality holds,
Thus, the <,
<
l
-n-i
Vt
inequalities
J v 9t~ <
=
~
-
when x T0 <
decreasing in v
for instance, if
1
=
in v
if
=
x To
1
1
must obtain. Consequently,
l
+ (l-Qo)u]- 9? =
f[qoFx + (l-qQ )u]-ig?
Jv[QoFx
J9?-
n
and constant
1,
fv(l-qt)g?
- ft )rf
J(l
1
Jvf.g?-
f f^'
'
1
(31)
The only
This
is
nontrivial step above
the
first
inequality: It ensues
from (30) and Fact
1.3.
the desired inequality.
A. 6 Proof of
I
is
Lemma
3:
Derivative Monotonicity
prove the inequalities one at a time, and analogize proofs whenever possible
—
although the techniques used for the market maker model are more tricky than for
the auction model.
Claim
1:
and dy/dy >
dx/dx >
model. Equation (3b) allows
me
here
=
[q
Consider the
a)fx [q
Fx + (1 - q
F(x To+ \v) + (l— qo)u] 1
similar to the proof of Proposition 2.
is
first
inequality in the auction
to rewrite (18) as
f(vwhere the density h(v)
0.
~n
l
)u}»-*h
°
;
g TO+ (v), and cancellation of terms
Note that the proportionality
refers to
the variable x, and thus the sign of the partial derivative in x of the two sides must
be the same. Now,
if I
show that whenever vh
g /QcM >
dxf(x\vL )^
then
it
statement of the
dx \q F(x\v L )
[q
F(x\vL )
and thus dx/dx
MLRP,
qo[F(x\v L )f(x\v H )
>
0,
while the second
+ (1 - q
Vl,
)u]q f(x\vH )
-
+
d_ f q F{x\v H )
and
dnU
follows that the negative numerator
are increasing in x,
<=*
Q
U
>
37
qQ )u \
+ \-qQ )u)
2.1.
>
Q
(
But the
first
(32)
both
inequality
is
a
also true:
- [g„F(xK) +
F(x\v H )f(x\vL )]
-
and positive denominator of
by Fact
is
1
(1
+ (1 - q
(1
-
qQ )u]qof(x\v L )
)u[f(x\vH )
-
f(x\vL )}
>
>
(33)
F(x\v L )
> F(x\v H
where f(x\vn)/f(x\vL,)
Now
suppose
am
I
>
1
in the
good news.
is
market maker model. The
-
F(y\v L )]
-
[F(w\vL )
- F(w\v H )] +
>
F(w\vH ) - F(y\v H )
Claim
3:
-
The
1.
now
F(y\vH)]
>
J{x\vl) yields
f(x\vL )
f(x\v H )
)
n >
is
F(y\v H )])
and f(x\vn)
)
IF
-
in (33)
F(y\vL )])
- F(w\v L )
>
F(x\v H - F(w\v H
AND dy/dx >
dx/dy >
term
F(x\v L )
^
1
-
[F(w\v H )
latter difference is positive, because Fact 2.2
F(w\v L ) - F(y\vL )
first
q f(x\vL )[F(x\v H )
- F(w\v L )] +
qQ f(x\v H ) {[F(x\v L )
-qof(x\v L ) ([F(x\v„)
and the
f(x\v L )
obtains because x
Qof(x\vH )[F(x\vL )
=
>
and f(x\v H )
)
proofs are similar to the
latter part of the proof of claim 2.
Claim
4:
and dy/db <
dx/da <
iJ(v-a)(l-q)g
-(
da
f(v o)q<t>x9
^-J(v-
_
J(v
-
a)q<t> x g
-
a)(l
n
\
J
n
q)g
-
which
is
true since <,
Claim
5:
<
6:
0, differentiate (18):
J
(v
-
q)g
a)qcj>x g
a)(l
-
and db/db >
is
n
The
0.
if
first
follows immediately from
n —
1.
Again, the two inequalities are
a and
term of the d expression
in (20) is obviously decreasing in x,
make
it
x and y when n
>
continuation values in (19)
partial derivatives of
<
analogous.
and db/dy <
and what remains
Remark. The
J
q<p x g
n
similar, so let's consider just the first. In this case, the second
vanishes,
n
n
obtains in (26).
da/da >
da/dx <
-
+
Jq4>x9
inspection of (19), and the second
Claim
n
q)g
^ f(v
J(l-q)g n
dx/da <
()
n
(I
see
n
l
\
To
0.
b w.r.t.
short of a proof of existence and uniqueness for
38
n>
1.
by the
strict
MLRP.
impossible (for us) to sign the
1.
This means that
I
just
fall
A.7 Completion of Proof of Proposition 4
Infinite
Time Duration.
I
amount of
point takes the required infinite
= -co
on the upper boundary, x
d
is
wish to show that the approach to the intersection
= 0,
and a
To
time.
see this, consider the fact that
while on the horizontal
line,
x
=
while
bounded. This suggests looking at the product
tif(v-a)(l-q)g l
*J(v-a)g*
Sfx g°
where
a and
oc refers to
But da/dx
t.
= a/x
upper boundary
point, as the slope of the
is
boundedly
is finite.
finite
near the intersection
Thus, d vanishes at most at an
extremely slow exponential rate near the intersection point, and so necessarily a
>
v°
for all finite time.
Equilibrium of
market maker model
the very
for all
t
first
>
0,
The Market Maker Model.
n >
for
1.
The proof is by
and thus
I
shall just address the harder
induction,
=
truncation equilibrium where xo
I
y
1,
=
and
0.
it
suffices to consider
need x
I
<
and y >
must converge to
This yields the extra constaint
with the derivative y exploding as
(y, b)
as (y, b) approaches the upper one.
settle
down
approaches the lower boundary and vanishing
Once more, the dual requirement that x and y
rules out all but the intersection point
destination point
— with convergence by
(x, a)
ft
and
of these boundaries as a possible
(y, b)
from opposite
To
sides.
see
that such an equilibrium must exist, consider the locus of exit points of paths from
the region
ft in (x, y, a,
6)-space defined by (27)
continuous function of (a
,
and
(34).
The
6o)-
Also, just as in the competitive auction proof, for any b
the exit from ft occurs
when
for
(x, a) hits
when
(x, a) hits
the lower horizontal line a
b
is
just above
it,
€ (J vf g / f f
the upper boundary for a just below
=
v°
any oq € (u°,/u/iy ///iy°), the exit from
boundary when
exit point is clearly a
and
hits the
39
when
ft
ao
occurs
is
just above
when
it.
g°,v°),
it,
and
Likewise,
(y,b) hits the lower
upper horizontal
line b
=
v°
when
MIT LIBRARIES
3 9080 01428 3003
below
bo is just
(ao,6o)-space),
occurs
when
it.
it
Since the exit locus
(y, b)
But by the
(x, a) hits 7.
must
a continuous image of a connected
must be connected. Thus,
7
also tend to
some
for
earlier logic
and because the
this takes infinite time,
7,
is
(y, b)
(ao,b
),
this first exit
set (in
from
3£
from the competitive auction proof,
path can only slow down to zero near
along that same path also in infinite time.
A.8 Dousing the Frenzy: Follow-up to Remark in Section 7.3
I
wish to show that (28)
<*
[q
F(x To
(1
-
+ (1 -
,
q )u To ]Fv (x t
v)
- Fv (x
,
Fact 1.2
fv (x\v) >
This implies that the
tells
t
is
term
Fv (x T0
|
v)
«)]
-
[q
v,
and so
F{x t
+ (1 - q
\
v)
)(uT0
is
to establish that
+ (1 - qQ )u ]Fv (x To
t
Ut)Fv {x To
\
\
v)
<
v)
,*„,,*
negative since
is
it suffices
-
Fv <
us that the third term
whenever x
first
|
v)
|
\v)\ „.
the second term
< x To
implies that
|
diminishing in
Fv (x Tn
v (x t \v)
< u T0
because x t
v)
q )u t [Fv (x To
(
Since ut
|
is
is
always (Fact
1.3).
Next,
negative. Finally, Fact 1.4
a good news signal, which
is
true in equilibrium.
negative because
- Fv {x
t
|
v)
40
= rr° fv (w\v)dv
D
References
Bulow, Jeremy and Paul Klemperer,
of Political
Economy,
"Rational Frenzies and Crashes," Journal
1994, 102, 1-23.
Andrew and John
Caplin,
Wisdom
after the Fact,"
Foster, Douglas and
S.
Leahy, "Business as Usual, Market Crashes and
American Economic Review, 1994, 84, 548-565.
Viswanathan,
When
"Strategic Trading
Agents
Forecast the Forecasts of Others," 1993. University of Iowa mimeo.
Fudenberg,
Drew and Jean
Tirole,
Game
Theory, Cambridge,
MA: MIT
Press,
1991.
Glosten, Lawrence R. and Paul R. Milgrom, "Bid, Ask, and Transaction
Prices in a Specialist Market with Heterogeneously Informed Traders," Journal of
Financial Economics, 1985, 14, 71-100.
Gul, Faruk and Russell Lundholm, "On the Clustering of Agents' Decisions:
Herd Behavior versus the Endogenous Timing of Actions," 1992. Stanford GSB
mimeo.
Kyle, Albert S., "Continuous Auctions and Insider Trading," Econometrica, 1985,
53,
1315-1335.
Maskin, Eric and Jean Tirole, "Markov
Perfect Equilibrium," 1992. Harvard
mimeo.
Milgrom, Paul, "Good News and Bad News: Representation Theorems and
Applications," Bell Journal of Economics, 1981, 12, 380-391.
— and Nancy
Stokey, "Information, Trade, and
Common
Knowledge," Journal
of Economic Theory, 1982, 26, 17-27.
Rochet, Jean-Charles and Jean-Luc Vila, "Insider Trading without
Normality," Review of Economic Studies, 1994, 61, 131-152.
Romer, David, "Rational Asset-Price Movements without News," American
Economic Review, 1993, 83, 1112-30.
Smith, Lones, "On the Irrelevance of Trade Timing," 1995. MIT mimeo.
Vayanos, Dimitri, "A Dynamic Model of an Imperfectly Competitive Bid-Ask
Market," 1993.
MIT
mimeo.
Wilson, Robert B., "Incentive
Efficiency of
Double Auctions," Econometrica,
1985, 53, 1101-1115.
— "On
,
Equilibria of Bid- Ask Markets," in George Feiwel, ed.,
Ascent of Economic Theory: Essays in Honor of Kenneth
New York
University Press, 1986.
232
:?
u
(
I
41
J.
Arrow and
Arrow,
the
New
York:
Date Due
Lib-26-67