.M414
.^r
imo, 13198S
K.
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Auctions with Resale Markets:
A Model of Treasury Bill Auctions
by
Sushil Bikhchandani and Chi-fu Huang
WP
#
2
038-88
July 1988
MASSACHUSETTS
INSTITUTE- OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Auctions with Resale Markets;
A Model of Treasury Bill Auctions
by
Sushil Bikhchandani and Chi-fu Huang
WP
//
2038-88
July 1988
AUCTIONS WITH RESALE MARKETS:
A MODEL OF TREASURY BILL AUCTIONS*
Sushil Bikhchandani^
and
Chi-fu Huang*
June 1988
Abstract
This paper develops a model of competitive bidding with a resale market. The
primary market
for the
in a
is
purpose of
modelled as a common-value auction, where bidders participate
resale.
After the auction the winning bidders
the objects
sell
secondary market. Buyers on the secondary market do not have any private
information about the true value of the objects. All their information
is
known and
The
includes information on the bids submitted in the auction.
of this information linkage between the primary auction
on bidding behaviour of the primary auction bidders
is
publicly
effect
and the secondary market
examined. The auctioneer's
expected revenues from organizing the primary market as a discriminatory auction
versus a uniform-price auction are compared, and plausible sufficient conditions
under which the uniform-price au^ tion yields higher expected revenues are obtained.
An example
auction,
is
of our model, with the primary
the U.S. Treasury
bill
market.
*We thank John Cox and Margaret Meyer
author
is
market organized as a discriminatory
for helpful conversations.
The second
grateful for support under the Batterymarch Fellowship Program.
^Anderson Graduate School of Management, University of California, Los Angeles,
CA
90024
^School of
02139.
Management, Massachusetts
Institute of Technology,
Cambridge,
MA
Am
}
8 L
1
Introduction
One
of the
major achievements of economic theory
in the past
decade has been
a deeper understanding of the conduct and design of auctions.
of the literature
and Wilson
rights
on auctions are McAfee and McMillan (1987), Milgrom (1987),
(1987). Auctions account for a large
The
activities.
Recent surveys
volume of economic and
U.S. interior department uses a sealed-bid auction to
financial
sell
mineral
on federally owned properties. Auction houses regularly conduct auctions of
works of
art, antiques, jewellery, etc.
uses a sealed-bid auction to
sell
Every week the U.S. Treasury department
Treasury
bills
worth
economic and financial transactions, although not
Many
billions of dollars.
explicitly
conducted as auctions,
can nevertheless be thought of as implicitly carried out through auctions;
example, an analysis of sales of seasoned new issues
and the market
One
for corporate control in
Tiemann
in
(1986).
is
resale or secondary markets for the objects for sale. This
and
bills
for
the possibility of resale, the auction will be
being the
common
valuation
among
all
that there exist active
is
One may argue
seasoned new issues.
see, for
Parsons and Raviv (1985),
feature often shared by such financial activities
Treasury
other
true, for example, for
when
there exists
common-value with the
resale price
that
the participating bidders.^
Thus
it
might
be argued that the theory of common-value auctions developed by Wilson (1969),
and Milgrom and Weber (1982)
a resale market
that
make
common-value
applicable.
is
bill
The observation
certainly true.
existing theory inapplicable.
In Treasury
who
is
is
A
that an auction with
However, there are situations
case in point
is
a Treasury
bill
auction.
auctions, there are usually about forty bidders or primary dealers
participate in the weekly auction.
These primary dealers are large financial
They submit competitive
sealed bids that are price-quantity pairs.
institutions.
Others, usually individual investors, can submit noncompetitive sealed bids that
specify quantity only.
The noncompetitive
primary dealers compete
the
is,
met
demands
remaining
until all the bills are allocated.
An
auction
is
bills in
if
down, are
The winning competitive bidders pay the
All the noncompetitive bidders
common-value
The
a discriminatory auction. That
of the bidders, starting with the highest price bidder
price they submitted.
^
for the
bids, small in quantity, always win.
unit
pay the quantity-weighted
participating bidders do not value the objects for sale differently.
^
average price of
all
the winning competitive bids. After the auction, the Treasury
department announces some summary
statistics
about the bids submitted. These
include
• total
tender amount received;
• total
tender amount accepted;
• highest
winning bid;
•
lowest winning bid;
•
quantity weighted average of winning bids; and
•
the split between competitive and noncompetitive bids.
The Treasury
bills
are then delivered to the winning bidders
and can be resold
at
an active secondary market.
Since primary bidders are large institutions, they tend to have private information about the
term structure of
interest rates that
possessed by investors in the secondary markets.
in the auction
and
based on both information that
their private information.
The buyers on
is
better than the information
The primary
is
dealers submit bids
publicly available at the time,
the resale market have access only
to public information, including information revealed by the Treasury
bids submitted in the auction.
To the extent that bids submitted
about the
reveal the private
information of the primary dealers, the resale price on the secondary market will be
responsive to the bids. This creates an incentive for the primary dealers to signal
their private information to the secondary
market participants. This information
linkage between the actions taken by the bidders in the auction
is
absent in existing models of common-value auctions and
is
and the
resale price
the primary focus of
this paper.
In Section 2,
we develop a model
Readers can think of the Treasury
bill
dealers or bidders are risk-neutral
value of the objects.
We assume
of competitive bidding with a resale market.
auction as a concrete example.
The primary
and have private information about the true
that the bidders' private signals and the true
^For an analysis of bidding with a resale market when the valuations of the bidders are
knowledge, see Milgrom (1987).
common
value are afGliated
random
variables,
i.e.,
roughly speaking, higher realizations of
a bidder's private signal imply that higher realizations of the true value, and of
the other bidders' private signals, are more
We
likely.
asstime that there are no
noncompetitive bidders. After the auction the auctioneer publicly announces some
information (the prices paid by the winning bidders, for example) about the auction.
The winning
bidders then
the objects on the secondary market, at a price equal
sell
to the expected value of the object conditional
In section 3
we analyze discriminatory
on
all
auctions.
public information.*
It is
assumed that the winning
We
bids and the highest losing bids are revealed at the end of the auction.
sufficient conditions for the existence of a
symmetric Nash equilibrium
crecising strategies in the auction. Unlike the
where bidders participate
market, the
librium.
affiliation
for the
bids
in
in strictly in-
Milgrom and Weber (1982)
purpose of consumption and there
property alone
The equilibrium
model
is
we obtain
provide
no resale
is
not sufficient for the existence of an equiare higher than those derived in
and Weber (1982), because primary bidders have an incentive
Milgrom
to signal. Moreover,
public announcement of the autioneer's private information before the auction
not increase and can decrease the auctioneer's expected revenue even
vate information
is
affiliated
if
may
this pri-
with the true value and the primary bidders' private
information. Sufficient conditions for the public annoxmcement of the auctioneer's
private information to increase his expected revenue are provided.
In Section 4
we
consider a uniform-price auction, that
rule for determining the winning bidders
is
identical to the
is
an auction
one
in the
in
which the
discriminatory
auction, but the winning bidders pay a uniform price equal to the highest losing
bid.
The
existence of an equilibrium depends on the kind of information about the
auction publicly revealed by the auctioneer.
If,
auctions, the winning bids are announced then
exist
an incentive
for the bidders to
the secondary market buyers.
as
we assimied
for discriminatory
we show by example
submit arbitrarily large bids
in
that there
may
order to deceive
Bidders in a discriminatory auction do not have
such an incentive since, upon winning, they must pay what they bid. However, a
symmetric Nash equilibrium always
exists in the uniform-price auction, provided
that only the price paid by the winning bidders
^Riley (1988) investigates a model
depends upon all the bids submitted.
depends on the bids submitted.
in
is
announced.
which conditional upon winning, each bidder's payment
primary bidder
In our model, the expected value for each
Next,
we turn
gained from the theory of auctions without resale markets
tions of buyers are correlated, the greater the
amount
aggressively.
Thus without
that
basic insight
when
the valua-
of information revealed during
about the true value weakens the winners' curse which
more
is
Any reduction
an auction, the greater the expected revenues.
to bid
The
to the question of the auctioneer's revenues.
in
in the
uncertainty
turns causes the bidders'
resale markets, uniform-price auctions yield
higher revenues than discriminatory auctions, since in the former the price
When
to the information of the highest losing bidder.
is
linked
there are resale markets in
which the buyers draw inferences about the true value from the bids, there
additional factor which needs to be considered.
The bidders have an
bid higher in order to signal their information.
If
about the true value, bidders would gain
greater the
amount
there
is
very
little
by submitting higher
little
an
is
incentive to
uncertainty
bids.
Thus the
of information revealed in an auction, the weaker the signalling
motive of the bidders.
Also
in Section 4
tioneer's revenues
we provide
when
plausible sufficient conditions under which the auc-
the primary auction
and only the price paid by winning bidders
discriminatory auction
when
is
organized as a uniform-price auction
is
announced
is
greater than under a
the winning bids are announced.'* Section 5 contains
concluding remarks.
The revenue maximizing mechanism
for selling
Treasury
bills
was a subject of de-
bate in the early 1960s. Friedman (1960) proposed that the Treasury should switch
from a discriminatory auction to a uniform-price auction
bills.
Apart from the
veal their true
fact that uniform-price auctions
demand
curves,
for the sale of
Treasury
would induce bidders to
re-
Friedman asserted that discriminatory auctions en-
couraged collusion and discouraged smaller bidders from participating. Both Goldsein (1960)
and Brimmer (1962) disputed Friedman's contention.
Smith (1966),
on the basis of a mathematical model, concluded that uniform-price auctions yield
greater revenues.
Smith's model, unlike ours,
is
not game-theoretic in that each
bidder's beliefs about the others' bids are not confirmed in an equilibrium.
Also,
the information linkage between the primary auction and the secondary market
is
"When there exists an equilibrium in the uniform-price auction with the winning bids being
announced, the bidders submit higher bids than at the equilibrium when the winning bids are not
announced. Thus, when there exists an equilibrium in the former case, the revenues generated are
even higher.
not modelled. Like Smith, our analysis provides support for Friedman's proposal
that the Treasury
auction should be uniform-price.
bill
The model
2
Consider a common-value auction in which n risk-neutral bidders (the dealers
submit competitive bids
objects, with n
and
>
unknown
is
k.
to
in
The
them
Treasury
bill
who
auctions) bid for k identical, indivisible
true value of the objects
at the time they
is
the
same
for all the bidders,
Each bidder privately
submit bids.
observes a signal about the true value, based on which he submits a bid.
paper we assume that each bidder demands (or
is
In this
allowed) at most one unit of the
object. Also, since noncompetitive bids usually constitute a small percentage of the
total
volume auctioned, we assume that there are no noncompetitive
to incorporate these
We
and other
bids.
We
hope
institutional details in a subsequent paper.
assume that each primary dealer cannot consume the objects being auctioned.
His interest in the objects being auctioned
the secondary market.
If
is
solely for the
it
at the
same
level as the resale buyers,
the private information of the primary bidders
market
will
in
one allows the possibility that the primary bidders can
consume the object and value
auction, the
purpose of resale
is
wt.-
if all
not revealed after the primary
Milgrom and Stokey (1982) no trade theorem
break down. Thus
then
implies that the resale
preclude the possibility that the primary bidders
can consume the object. For similar reasons we assume that primary bidders can
participate in the resale market only as sellers, not as buyers.
After the auction the auctioneer pubhcly announces
auction. For the case of the discriminatory auction
and the highest losing bid submitted
is
in
we assume
may
is
the case.
Thus most
bill
auction. For the case of
not exist an equilibrium
are revealed at the end of the auction.
this
that the winning bids
the auction are publicly announced. This
a simplifying abstraction of the conduct of Treasury
the uniform-price auction, there
some information about the
In Section 4,
if
the winning bids
we provide examples where
of our analysis of the uniform-price auction
assumes
that the winning bids are not announced.
We
allow the possibility that
objects for sale
may become
some additional information about the value
of the
publicly available after the bids are submitted, but
before the opening of the secondary market.
then
The k winners
in the
primary auction
the objects to risk-neutral buyers on the secondary markets.
sell
The buyers on
the secondary markets do not have access to any private information about the true
They
value.
what they can from the information released by the auctioneer
infer
about the primary auction, and any other publicly available information.
regardless of the secondary market
— the resale price
market
—
mechanism
Thus,
an auction or a posted price
be the expected value of the object conditional on
will
publicly available information.^
all
The n
risk-neutral bidders will be indexed
each object being auctioned
prior
V
on
a
is
random
and observes a private
,
t
variable,
signal, X,,
= l,2,...,n. The true value of
V Each bidder i has a common
.
about the true value. Let
any other information that becomes public after the auction
resale
We
market meets.
will
is
denote
over but before the
assume, except when otherwise stated, that given
the bidders' signals are not uninformative about the true value, that
'E[V\Xi,X2,...,Xn,P].
P
If this
condition
violated, for
is
is,
example when
^
EjV'lP]
P=
P
V, our
model reduces to the usual common-value auction without a resale market.
Let /(t;,p,x) denote the joint density function of
X
=
Let
(Xi, Xj, ....,Xn)[t;,t;]
x
product of
the
is
[p,^]
[x,x].
random
X, x'
e
[x,
is
P, and the vector of signals
symmetric
where
n arguments.
in the last
[x,5]" denotes the n-fold
Note that we do not rule out the possibility that the support of
all
the
e
x]", V, v'
[i,
assumed that /
i]" be the support of /,
variables are
assumed that
all
x
It is
V",
unbounded
random
[v,v],
either
from above or from below. Further,
variables in this
and
p, p'
G
f{{v,p,x) V (t;',p',x'))/((r;,p,x) A
model are
affiiiated.
That
is,
it
for
[p,p],
(t;',p',x'))
>
/(t;,p,x)/(t;',p',x'),
where V denotes the componentwise maximum, and A denotes the componentwise
minimum.
Affiliation
implies that
^The
bill
if
H
is
is
said to be strict
if
the above inequality
is
an increasing^ function then 'E[H{V P, Xi, X2,
,
strict. Affiliation
...,
X„)|c,
< Xj <
participants in the secondary market do not have to be risk neutral. In the case of Treasury
auctions, the "true" value of a bill for a primary bidder that pays say $1 in 182 days will be
m
E[m|7|, where
denotes the random marginal rate of substitution between consumption 182 days
from now and that of today and where / denotes the information revealed by all the bids submitted.
This true value will be the secondary market price whether or not the secondary market participants
are risk neutral.
Throughout
this paper,
we
will use
weak
relations.
For example, increasing means nonde-
d,,
I
=
1,
.
n]
,
.
.
is
an increasing function of
and Weber (1982)
plicity that
-^i
<
di,
»
H
if
n]
'^
for other implications of affiliation.
continuously differentiable in
the convention that the derivative at x
left-hand derivative. Moreover,
affiliated so that
then 'ElH {V
di for all c,,
,
P
,
G
(f,
We
referred to
is
Milgrom
further assume for sim-
continuously differentiable then E[jy(V',P,Xi, Xj, ...,X„)|c,
is
is
The reader
d,.
c,,
H
if
is
Xi, X2,
[x,
we
Xn)\ci
<i,,
for all
c,, d,
assume that {V ,P, Xi,
.
.
any of (V, P, Xi,
.
.
strictly increasing in
...,
and
G
< Xi <
di,
i
^
l]
is
.
.
with
[x, x],
the right-hand derivative and at x
is
shall
Ci
,
X„) are
,
Xn), say
<
is
the
strictly
strictly increasing in
in
Xi,
c^
and
x).
Discriminatory auction
3
In a discriminatory auction, the bidders submit sealed bids
win the auction.
we show
that
A
and the k highest bidders
winning bidder pays the price that he or she bids. In
when
this section
the bidders' private signals are information complements, in a
sense to be defined later, there exists a symmetric Nash equilibrium with strictly
increasing strategies in the bidding
game among the primary
auctions examined in Milgrom and
Weber
dealers.
(1982), the affiliation property alone
not sufficient for the existence of a Nash equilibrium. Intuitively,
of the primary bidders
some or
all
is
Unlike the
when
the motive
to resell in a secondary market in which the buyers
of their bids (or a
summary
statistic
is
know
based on their bids), there exists
an incentive for the primary bidders to bid more than they otherwise would and
This
thus signal their private information.
value
is
is
because, by affiliation, the resale
responsive to the bids submitted to the extent the bids reveal the private
information received by the primary bidders.
If
each bidder's incentive to signal
increases with his information realization, then there exists an equilibrium in strictly
increasing strategies.
It is
the information complementarity of the bidders' signals,
with respect to the true value, that ensures that the bidders' incentive to signal
increases with their information realizations
and enables them
to sort themselves
in a separating equilibriiun.
In a
model where bidders participate
creasing, positive
increasing.
means nonnegative,
etc.
If
in
an auction
a relation
is strict,
for final
we
consumption of the
will say, for
example,
strictly
objects,
Milgrom and Weber (1982) show that the auctioneer's expected revenue
he precommits to truthfully reporting his private information
can be increased
if
about the objects
for sale before the auction,
is
announcing
his information, the auctioneer introduces
among the primary
an additional source of
model with a
increased. However, in our
A
resale
market
portion of the bid submitted by a bidder
signal to the resale
affili-
bidders' private information ajid thus weakens the winners'
Hence the bidders compete more aggresively and the expected
curse.
true.
This follows since by publicly
with the bidders' private information.
affiliated
ation
provided that his private information
market participants.
If
this result
selling price
is
not necessarily
is
attributed to his incentive to
is
the auctioneer's private information
is
a
"substitute" for the bidders' information, announcing that information will reduce
This
the responsiveness of the resale price to the bidders' information.
may
reduces the incentive for the bidders to signal and
price to
fall.
On
when
the other hand,
"complement" to that of the bidders',
announce
3.1
By
turn
in
cause the expected selling
the auctioneer's private information
it is
is
a
always beneficial for the auctioneer to
his private information.
Existence of a symmetric Nash equilibrium
the hypothesis that f[v,p,x)
symmetric game. Thus
equilibrium.
We
it is
is
symmetric
in its last
n arguments,
this
is
a
natural to investigate the existence of a symmetric Nash
examine the game from bidder
the other bidders' viewpoint
is
I's
point of view.
The
symmetric.^ Note that at the time
analysis from
when bidder
1
submits his bid, he only observes his private information Xj. Thus a strategy for
bidder
1 is
a function of Xi. Bidder
i's
strategy
denoted
is
6,
:
[x,x]
—
S?.
our analysis by deriving the first-order necessary conditions for an n-tuple
to be a
buyers
Nash equilibrium
in
and
in strictly increasing
the secondary market believe that
(6,
.
.
.
,
We
(6,
differentiable strategies,
6)
begin
.
,
.
.
6)
when
are the strategies followed in
the bidding.
Since buyers in the secondary market do not have access to private information
about the true value, the resale price
is
the expectation of
V
conditional on
all
"To simplify the analysis we arbitrarily assume throughout that in case of a tie the winner is not
chosen randomly. Rather, bidder 1 is declared the winner. This assumption is inconsequential. The
equilibrium strategy will remain unchanged if we assume that in case of a tie, the winner(s) is (are)
chosen from the tied bidders at random.
public information. As mentioned earlier, to simplify the analysis
the auctioneer announces the prices paid by winning bidders
and the highest
bidder
1
losing bid.^
Suppose that bidders
receives information
wins with a bid
where b"^ denotes the
if
=
2,
.
.
.
,
that
the winning bids)
(i.e.,
n adopt the strategy
Xi = x and submits a bid equal
to
6.
Then
if
bidder
b,
1
the resale price will be
b
r^b-'{b),Yr,...,Y,,P)
Note that
t
we assume
P =
V",
=E
x, = b-'{b),b-'{b{Y,)),...,b-'{b{n),p]
= E
X, = b-'{b),Yu...,Y,,p],
inverse^ of b
and
r''{b~'^{b),Yi,.
.
.
Yj
is
the
j'-th
=
,Yk,P)
(1)
order statistic of (X2,
.
.
.
,
X„).
V, and our model reduces to an
ordinary common-value auction without a resale market. Define
v''{x',x,y)=E[r'{x',Yi,...,Y,,P)\Xi
By our hypothesis about
in
strict affiliation,
both
r**
=
and
=
x,Y,
v'^
y].
(2)
are strictly increasing
each of their arguments (provided that given P, the bidders' signals are not
The expected
uninformative about V).
submits a bid equal to
Jl'{b\x)
6
profit for bidder
1
when Xi = x and he
is
^
E[(r^(6-M6),Fi,...,n,P)-6)l^,^j(i>^)j|x,
=
E[B[{r'{b-'{b),Yu
= E
.
.
.
[(v'{b-Hf>),XuY,) -
,Y,,P)
-
b)
=
x]
l^,^^y^^y\Xu^\x, =
b) l{,>S(y,)}|Xi
=
x]
x]
,
r6->(t)
=
/
{v'{b-'{b),x,y)-b)f,{y\x)dy,
JX
where the second equality follows from the law of
iterative expectations,
denotes the conditional density function of Yk given Xy. Taking the
and /i(y|x)
first
derivative
of n''(6|x) with respect to 6 gives
^^
=
[v'^(b-\b),x:b~'{b))
-b) h{b-\b)\x){y[b-\b)))-' - F,{b~\b)\x)
Hy[b-\b)))-'jl~'^'Ki{b-\b),x,y)h{y\x)dy,
(3)
*lf
some
of the other losing bids, or
some function
of them, are also
announced
all
the results
remain unchanged.
^If
fc
<
values of
b
t(i)
then b^(b)
that
lie in
x
and
the range of
if i
6.
>
fc(x)
then b^(b)
x.
Thus we only need
to consider
.
where
the derivative of b{x), Fic{y\x)
b'{x) is
of Yk given Xi,
For (6,...,
is
to be a
6)
for bidder 1
and vf
the partial derivative of
Nash equilibrium,
when bidders
participants believe that
when
=
6
=
t
2,
Rearranging
(4) gives
,
argument.
n adopt strategy
That
b.
and the
b
resale
must be zero
relation (3)
is,
market
Besides
(5),
>
t;''(x,x,x)
fk{y\x) J
d(
u whM^) + y^t;,(x,x,y)^r^dy.
r d(
= (v(x,a:,x)-6(x))^^^^
(
\
\
,
r"^
and the law of
=E
[v\Xr
.
6(x),
Vx G
and the
1
fact that
when Xi =
and
\x,x];
i,
(ii)
6(x)
=
iterative expectations,
= x,n =
(6)
y]
v''(x,i,x).
has to be positive
6(x)
if
<
t;
.
(5)
there are two other necessary conditions that b must satisfy:
expected profit for bidder
e
F,{x\x)
an ordinary differential equation:
t;''(x,x,y)
+
.
its first
= (A^^^^^) - H^)) h{x\x)Cb'{x))-' + {b'{x))-' l^v',{x,x,y)My\x)dy.
Note that, by the definition of
to b{x)
.
with respect to
necessary that 6 be a best response
is
bidders adopt
all
^^U.)
t'l \
6(x)
(i)
.
it
v**
b{x):
=
from
the conditional distribution function
is
Condition
in equilibrium.
(i)
(i)
follows since
Condition
(ii)
follows
then by slightly increasing the bid
(i, i, x),
expected profit can be raised from zero to some strictly
positive amount.
The
solution to (5) with the
b{x)
=
t;<^(x,x,x)
boundary condition
6(i)
- [' L{u\x)dt(u) + f'
Jx
Jx
^
=
i'''(x, i,
;}
.
i) is
dL[u\x),
(7)
}k[u\u)
where
L[u\x)
=
exp|-/J^d.|,
t{u)
—
v'^[u,u,u)^
h{u)
=
/•u
/
vi{u,u,y)fk{y\u)dy.
Jx
Note that L{u\x) and
[i,x].
We
will
show
in
t[u) are increasing functions of
what
follows that (7) also satisfies condition
expected profit under the hypothesis that [Xi,
nients with respect to
V and
,
u and thus are measures on
is
.
.
,
strictly increasing.
10
(i),
maximizes
A'„,P) are information comple-
Definition 1 Random
ments with
variables, [Zi,
respect to another
random
T-^
>
.
.
,Zm), are said to
.
T
variable
0,
Vt
#
=
2i,
OZtOZj
information comple-
be
if
V2i,...,2„,
J,
where
<i)[zi,
.
,
.
.
=E
2m)
Thus, the random variables
respect to
V
if
(A'l,
.
[f|Zi
.
.
,X„,P)
.
.
,
Zm =
2m]
.
complements with
are information
the marginal contribution to the conditional expectation of
higher realization of
A',
For example,
if
4>{zi,.
.
.
.
,
.
Thus
information complements.
tributed, then [Zi,
.
.z^]
is
satisfied
by a large class of
linear in the 2,'s, then [Zi,
[T, Zi,
if
is
.
.
.
.
random
of a
.
.
,
distri-
Z^) are
,Zm) are multivariate normally
Zm) are information complements with respect
give three examples of affiliated
V"
Xj or P.
larger the higher the realization of any other
is
This information complementarity condition
butions.
.
to T.
dis-
We
variables that also satisfy the information
complementarity condition.
Example
1 Let {T,Zi,
function g{t,Zi,
variables
.
.
.
rem
1 of
.
E
E~'
that
be multivariate
,Zm)
.
,Zm). Let
and assume
It is easily verified
.
normally distributed with density
matrix of these random
be the variance-covariance
and has
exists
that d^ \ng/dtdzi
>
strictly negative off-diagonal elements.
and d^lng/dzidzj >
Mil gram and Weber (1982) then implies that [T,Zi,
afffiliated.
Since
ET
Zi.
.
complements with respect
.
.
,
Zm]
is
linear in the Zi
's,
[Zi,
.
.
.
,
for
.
.
.
Zm)
i 7^
,Zm)
j.
Theo-
o.re strictly
are information
to T.
Besides the multivariate normally distributed
random
variables, there
class of distributions with linear conditional expectations.
The
following
is
is
a large
an ex-
ample.
Example
according
2 Let Zi,
gamma
i
=
l,...,m
be
distribution given
g,{zi\t)
=
{
independent conditional on
T=
t:
r(a)^.
^
V
2'
^^'
otherwise,
11
T and
distributed
>
where a
0,
>
t
gamma
according to
and T
0,
gamma
the
is
and a >
>
verifies that [T, Z\,
.
[Zi,.
.
.
.
.
,
Using Theorem 1 of Milgrom and Weber (1982), one
0.
Z^) are
Direct computation yields
strictly affiliated.
,Zm.] ore information
=
glA±^.
ma + 7 —
of "conjugate distributions" of
Other distributions with
1
complements with respect
T in Example
gamma distribution;
Note that the prior distribution of
9).
2
see
to
DeGroot
linear conditional expectations
following example gives
random
T.
an element of the family
is
similarly. Interested readers should consult Ericson (1969)
The
also be distributed
otherwise,
E[f|Z„...,Z^]
Thus
l/T
distribution with a density
I
where 7
Let
function.
(1970, Chapter
can be constructed
and DeGroot
(1970).
variables that are strict information
com-
plements.
Example 3
Let Zi,
i
=
1,2,...,
m,
be
independent conditional on
T
with density
otherwise.
[
The density of
T
is
(r»+l)a
otherwise.
[
It is
easily verified that d^ \ngi{z,\t)/dzidt
and Weber (1982) implies
same theorem
also
that
g,
>
V2,< G
Theorem
1 of
Milgrom
The
satisfies the (strict) affiliation inequality.
shows that
9[t,zuz2,...,zm)~^
is affiliated.
(0, 1).
Otherwise
Direct computation yields
<f>{Zl,Z2...,Z„)
= E[f|Zi =
21,^2
=
Z2,...,Zr„
=
Zm]
=
(nr=i ^.r
(nr=i2.r-i
12
i
ain{nT=iZiy
for Zi,Z2,.
alli^ j
for
The
(0, 1)
Finally, one can also verify that d^(i>{zi, Z2,
.
lemma
is
a direct consequence of Definition
1 Suppose that (Xi,
V Then
.
vf{x,x',y)
derivative of
Proof.
The
.
.
.
is strictly
with respect to
i'"*
.
.
.
,
Zm)/dzidzj
a e (0,1).
if
following
Lemma
to
,Zm G
..
,X„,P) are information complements with
and
increasing in x'
its first
y,
respect
where vf denotes the partial
argument.
joint density of (V',P,Xi,yi,...,y„_i)
(n-
1.
is
l)!/(t;,p,a:,yi,...,y„_i)l{„,>„,>...>„„_,}.
V
As a consequence, the conditional density of
given {P, Xi, Yi,
.
.
.
,
Yn-i)
is
f{v,p,x,yu...,yn-i)
/lp,i,yi,---,yn-ij
Thus [P ,X\,Yi,
of
which
r*^,
is
are information complements. Let rf denote the derivative
....,Yk)
defined in
verified that r5'(a;,yi,
.
.
.
(l),
,yt,p)
with respect to
its first
argument.
It is
a strictly increasing function of
is
p,
then easily
Next
y^, Vj.
note that
vf(x,x',y)
Theorem
is
5 of
=
Elrf(x,Fi,...,n,P)|Xi
Milgrom and Weber (1982) then
a strictly increasing function of
The
following proposition shows that
ments with respect to
VxG
and
x'
V
,
then b of
y].
implies, by affiliation, that vf{x,x' ,y)
I
y.
if
= x',n =
(Xi,
(7) satisfies
.
.
.
,
Xn, P) are information comple-
condition
(i),
that
is,
v'^[x,x,x)
>
b[x)
[1,1].
Proposition 1 Suppose
spect to
where
V
.
Then
that (Xj,
v'^{x,x,x)
>
.
b{x),
.
.
,
X„, P) are information complements with
Vx G
[x,
i]
and
the inequality
is strict
for x
re-
>
x,
b is defined in (7).
Proof.
We
v'[x,x,x)
first
-
write
b[x)
=
fi L{u\x)dt{u)
-
fl
j^^dL{u\x)
(8)
= IxH^l^)
{vi{u,u,u)
+
v^{u,u,u)
13
+
vi{u,u,u)
- p^^)
du.
>
By
to
the hypothesis that (Xi,
V
Lemma
and
1,
.
.
a:)
—
>
h[x)
are information complements with respect
t;i(u,u,t/) for
y
<
u. It follows that
(8) gives
L{u\x) fv2(u,u,u)
/
Note that the above inequality
is
strict for
x G
+
vf{u,u,u)) du
>
{x,x] since Vj
>
0.
and
>
Vj
by
I
strict affiliation.
A
X„,P)
,
>
Vi{u,u,u)
Substituting this relation into
v''(x, X,
.
corollary of Proposition 1
is
that b
strictly increaising.
is
Thus our assumption
that resale market buyers can invert the primary bids to obtain the bidders' signal
realizations
is
justified.
Corollary 1 The
PROOF:
We
v'^{x,x,x)
—
strategy b defined in(7)
show that
will
and noting that
Vi >
>
b{x)
t;f
>
b'{x)
I.
>
is strictly
Vx >
The proof
is
x.
increasing.
From Proposition
completed by inserting
1
we have
this in (5),
I
0.
Before proceeding, we
first
record two
lemmas that
are direct consequences of
the definition of affiliation.
Lemma
2
Lemma
Z Let
is,
(Milgrom and Weber (1982))
x'
>
x
>
y.
Then
Fk{y\x) / fk{y\x)
Fk{y\x') / Fk{x\x')
the distribution function Fic{-\x')/Fic{x'\x')
affiliation
we have
for
>
f,{a\x')f,[0\x)
Thus
for
a, x'
<
decreasing in x.
Fk{y\x) / Fk{x\x)
dominance.
> x
A(a|x)A(/?|x').
X > y
r JXr fk{a\x')fMx)dad0 < Jyr Jxr f,{a\x)f,{/3\x')dadp,
Jy
14
.
That
dominates the distribution function
Fk{-\x) / F)c{x\x) in the sense of first order stochastic
PROOF: By
<
is
.
which
is
equivalent to
F,{y\x'){F,{x\x)
-
F,{y\x))
<
F,{y\x){F,{x\x')
-
F,{y\x')).
Rearranging terms gives
F,{y\x')
^
F,{y\x)
F,{x\x')
-
F,x\x)
•
I
The main
Theorem
result of this section
1 The n-tuple (6,...,
is
6),
with b as defined in (7),
of the discriminatory auction provided that {Xi,
ments with
the strategy
Proof.
V and
resale
x. Recall
from
respect to
.
.
is
a
Nash equilibrium
,X„, P) are information comple-
market buyers believe that
all the
bidders follow
b.
Let
>
x'
0=^^*^
(4)
that
= (6'(x))-F.(x|x)((.^(x,x,x)-6(x))^-S'(x)
<
{b'{x))-'F,{x\x)({v'{x,x',x)
-
6(x))^^
-
b'{x)
<
(6'(x))-^F,(x|x)((t;<'(x,x',x)
-
H^))j^^
"
^'(x)
Fk{x\x) an'^(6(x)|x')
Fk{x\x')
where the
first
inequality follows from Proposition
the second inequality follows from
bids b
=
6(x)
<
6(x'), his
arguments show that
maximized
at 6
'
db
=
Lemma
3.
1,
That
Lemma
is,
1,
and
when Xi —
Lemma
x'
2,
and
and bidder
1
expected profit can be raised by bidding higher. Similar
ah
—
^ ^°^
•'''
^
6(x). Finally, since n'^(6(i)|x)
15
x.
=
As a consequence,
for all x,
we have
n''(6|x)
is
n'^(6(x)|x)
>
for all
I
>
by
I,
strategy for bidder
and when the
b,
When
strict affiliation.
when he
1
resale
We
observes Xi
have thus shown that b{x)
=
x,
when
bidders
market participants believe that
i
=
2,3,
is
.
.
.
the best
,n follow
the bidders follow
all
b.
I
bidders in the discriminatory auction participate only for the purpose of
consumption, Milgrom and Weber (1982) have identified a symmetric Nash equilib-
rium with a bidding strategy
b''[x)
=
v''{x,x,x)
-
f L{u\x)dt{u),
(9)
Jx
where L{u\x] and
Theorem
of resale identified in
amount equal
The bidding
t{u) are as defined in (7).
1 is strictly
higher than
6*^
strategy for the purpose
for every
x G
(i, x]
by an
to
h{u)
r
Ix
Jx
-dL{u\x),
fk{u\u)
the magnitude of which depends on
the submitted bid.^°
It is this
i;^,
the responsiveness of the resale value to
informational link between the resale value and the
bids submitted by bidders in the discriminatory auction that gives the bidders an
Of
incentive to signal.
course, since the b
is
strictly increasing, the resale
buyers
can invert the bids announced by the auctioneer to obtain the private information
of the bidders and, as in Ortega-Reichart (1968)
in equilibrium
and Milgrom and Roberts (1982),
no one gets deceived.
Note that the information complementarity condition
sary for b of
3.2
(7) to
We
variable Xq.
heis
private information about
strategy conditional on
P = V there is no
= V and thus t;f
signalling motive
is
on |— ^, ^1 then as
Xq =
xq.
If
6(-;
bidder
V",
represented by a
impact of announcing Xq before the
will consider the
auction on the expected selling price. Let
)
but not neces-
Public announcement of the auctioneer's information
random
^^li
sufficient
be a symmetric Nash equilibrium.
Suppose that the auctioneer
r'*(
is
zq) be a
1
symmetric equilibrium bidding
bids 6 and wins then the resale price
and the equilibrium strategy is as specified in (9), since
=
and
h{)
0.
Also, when P is "very informative" about V the
(
weak. For example if we let P„ = V + e„ where £„ is, say, uniformly distributed
n increases the resale price becomes less responsive to the players' bids and in
signalling motive
)
=
the limit the incentive to signal disappears.
16
in this case will
be
p''{r\b;xo),Yu...,Y„P;xo)
where
{b;xo);xo)
6(6
w'^{x',x,y;xo)
it is
=
=
[v|Xi
r'(6; lo),^!,
.
.
.
,n,P,Xo =
lo]
,
Putting
b.
=E
=E
[p''(x',Fi,
.
.
.
=
,yit,P;zo)l^i
^ y,Xo =
x,Yi,
Xq]
,
straightforward to show that 6(x; Iq) niust satisfy
6'(x;xo)
=
{'w'^{x,x,x;xo) -b{x;xo)) j^
\ + Jx
/
'
.
,
u;f {i,x,y; Xo)/*(y|a;;xo)<^y.
i'jt(i|i;ioJ
(10)
where /t(y|x;xo) denotes the conditional density of Yk given Xi
In addition, the
boundary condition
6(x; Xq)
=
solution to (10) with this boundary condition
—
b{x;xo)
=
w'^{x,x,x;xo)
=
satisfied.
Xq.
The
is
r^
r*^
-
x and Xq
must be
Xq)
u;''(i, x, x;
=
L{u\x;xo)dt(u;xo)
Jx
+
hftl' Xr\)
rdL(u|x;xo), (11)
Jx /t(u|u;xoj
,'
,
.,
where
L(ulx;xo)
= exp|-y^^r(^j^^rf^|,
f(u;xo)
=
/i(u;xo)
=
iy''(u,u,u;xo),
/u
/
u;f(u,u,y;xo)/t(y|u;xo)rfy.
•'I
Next, we define conditional information complements.
Definition 2 Random
variables, [Zi,
ments conditional on random variable
.
.
Y
.
,Zm), are said to be information complewith respect to
random
variable
T
if
OZiOZj
where
4>{zi,...,Zm,y)
If
Zi
^
Zi,
.
.
.
,
Zm = Zm,Y =
yj
.
{Xi,X2,...,Xn,P) are information complements conditional on Xq with
spect to
is
=E f
V
,
then a proof identical to that of Theorem
a symmetric equilibrium strategy. This
Proposition.
17
is
1
shows that
re-
b defined in (11)
stated without proof in the following
Proposition 2 The n-tuple
(b{-;xo),...,b{-;xo)), with 6(-;xo) as defined in (11),
a Nash equilibrium of the discriminatory auction when the auctioneer announces
is
Xq =
Xq
xq, provided that [Xi,
with respect to
follow strategy
V
and
.
.
X„, P) are information complements conditional on
,
.
the resale market participants believe that all the bidders
xo).
b{-;
Note that the existence of an equilibrium does not depend on whether
{Xo,Xi,
...
,Xn,P) are information complements with respect
a Nash equilibrium as long as (Xi, Xj,
Xo with
conditional on
respect to V.
.
.
.
to V.
There
exists
X„, P) are information complements
,
Our main
result in this subsection will
that the expected selling price under the policy of always reporting
Xq cannot be
lower than that under any other reporting policy provided that [Xq, Xi,
We
are information complements with respect to V.'^
first
show that
.
.
.
,
Xn, P)
6(x; xq)
we record a
increasing function of xq in the next proposition. Before that
be
is
a
technical
lemma.
Lemma
4 (Milgrom and
Weber
(1982,
Lemma
>
and
ferentiable functions for which (i) p[x)
o'{z).
Then p{z) > o{z) for
Proposition 3 Suppose
[Xo, Xi,
is
.
.
.
that
(ii)
p[z)
<
an increasing function of
Xq
>
Xq.
By
i(x;xo)
{V Xq, Xi,
,
we can show
Lemma
Lemmas
1, 2,
and
3,
we have
^^Note that this assumption
for existence of
<
.
.
.
,
Xn, P) are
is
and
affiliated
>
6(x; Xq)
=
to
w"^ [x, x, r, x'q)
6(x;Xo) implies 6(x;xo)
So suppose that
4.
o[z) implies p'{z)
>
V.
that
Thenb[x]Xo)
we know
w"^ [x, X, x; Xo)
that 6(x;xo)
done by
be dif-
Xq.
affiliation
=
and o[z)
x.
Xn, P) are information complements with respect
,
PROOF: Let
If
>
all z
o{x)
2)) Let p[z)
6(x; Xq)
<
that «;i(x,x',y;io)
>
6(x;Xq).
is
.
6(x;Xo), then
we
As generalizations
are
of
increasing in both y and xq,
stronger than the conditional information complementarity required
Nash equilibrium.
18
Fk{y\x; xq) / fk{y\x; Xo)
Fic{-\x; x'q) / Fk{-\x; x'q) in
=
6(x;xo)
[vj
decreasing in Iq, and that Fk{-\x\ Xq) / Fk{-\x; xq) dominates
is
the sense of
[x,x,x\Xq)
first
degree stochastic dominance.
+
-b[x\xo))-—tk\X\X\Xo)
^
>
(
(u;
(x,i,i;Xo) -6(i;2:o))-^r7-r—7T+
l'k(x\x;xo)
=
The main
{Xo,Xi,
.
.
/
J2
fk{y\x;xo)
^ .
^ . dy
_,
,
.
/'jt(x|x;xoj
I
result of this section
is
2 Suppose that {V ,Xo,Xi,
.
\<^y
tk{x\X\XQ)
^(x-.Xo),
which was to be shown.
Theorem
w^[x,x,y\Xo)-=n-\
r ^dx,x,y;Xo)
df
n
is^fkj^^W^^ ^
T,
/^
d,
/
JX
Then
,X„,P)
are information
.
.
.
,X„,P) are
affiliated
and
complements with respect
publicly revealing the seller's information cannot lower,
to
and may
that
V.
A
policy of
raise, the expected
revenue for the seller in a discriminatory auction.
PROOF: Given
Theorem
Proposition
3,
our proof mimics that of Milgrom and Weber (1982,
16). Define
W{x,z) =
which
is
1
<x,Xi =
[b{x;Xo)\Yk
the expected price paid by bidder
Xq, conditional on bidder
By
E
when
1
winning when Xi
Proposition 3 and by the hypothesis that
=
Xq
the auctioneer publicly reveals
and bidder
Xo and Xi
Note that, by symmetry, the expected revenue
publicly reporting
z
z],
1
bids as
if
Xj =
x.
>
0.
are affiliated, W^lx^z)
for the seller
under the policy of
k times
is
E[6(Xi;Xo)|{n<Xi}]
= E[E[6(Xi;Xo)|n<Xi,Xi]|{n<Xi}]
= E[w{XuX,)\{Yk<X,}],
where the
first
equality follows from the law of iterative expectations
from the definition of
revenue for the
seller
W
is
.
On
and the second
the other hand, without reporting Xq, the expected
k times
E[6(Xi)|{n<Xi}].
19
If
we can show
We
W{x,x) >
that
Lemma
will utilize
W
4.
Note
= E
(i, i)
b{x),
where
n<
Xq)
I
=
=
Now we
bidder
to
claim that W{x,x)
some bidding strategy
optimal
when Xq =
Xq.
=
are done.
=
=
x,X^
x]
clW{x,x)/dx
after observing
Xi =
>
x,
b'{x).
Thus W{z,x)
a,t
z
=
x
will
Note
first
that
if
were to commit himself
=
optimal choice will be 2
b[z\-), his
x]
b{x).
b{x) implies
Xo but
prior to learning
1,
<
we
x]
= x,n = i]|n<i,Xi =
Xo,Xi
v'^{x,x,x)
=
Xi
I,
B[w''{x,x,r,Xo)\Yt
= E E
defined in (7), then
is
that by the law of iterative expectations,
first
[b{x;
b{x)
i, since 6(1; zq)
have to satisfy the
first
is
order
condition
Wi{x,x)
=
{v'^{x,x,x)
fk{x\x)
- W{x,x))
+
Fkix\x)
'
fk{y\x)
dy.
vf{x,x,y)
/
Jx
(12)
Fk{x\x)
Then
{v''{x,x,x)-b{x))
b'{x)
fk{x\x)
+
Fk{x\x)
<
h{y\x)
v'l{x,x,y)
/
Jx
fk{x\x)
{v'^{x,x,x)-W{x,x))
< Wi{x,x) +
W:i{x,x)
Jx
W[x,x) <
^2{x,x)
>
0.
Theorem
2
crecLsing in xq.
may
The
dW{x,x)
=
(5),
the
first
inequality follows
and the second inequality follows from
6(x),
assertion then follows from
depends
When
on the
critically
{Xq, Xi,
.
.
.
,
Lemma
fact that
under
incentive to signal. This in turn
.
,
.
.
X„, P) are
4.
its
from the hypothesis
(10)
and the
fact that
I
hypothesis 6(x; xq)
is
in-
X„, P) are not information complements, b{x; xq)
not be an increasing function of xq, and revealing
though {Xq, Xi,
dy
Fk{x\x)
dx
where the equality follows from
that
h{y\x)
v^{x,x,y)
I
Fk{x\x)
dy
Fk{x\x)
may
Xq may reduce the
bidders'
lower the expected revenue for the seller even
affiliated.
20
Uniform-price auction
4
In a uniform-price auction the bidders submit sealed bids and the k highest bidders
win the auction. The price they pay
is
equal to the
(/c
+ l)st
highest bid. Initially
we
obtain a symmetric Nash equilibrium under the assumption that after the auction
the auctioneer reveals the highest losing bid, that
bidders.
(If
is,
the price paid by the winning
any of the lower bids are also revealed, our results remain unchanged.)
The winning
bids are not announced and thus the bidders do not have an incentive
to signal their private information, since
if
they win their bids are not revealed. Even
without the signalling incentive, we are able to show that
in at leeist
the expected revenue generated by this uniform-price auction
is
generated by the discriminatory auction discussed in Section
We
first
scenario
is
a plausible one for the case of the Treasury
In the last part of this section
and
all
we
discuss the case
3.
bill
when
higher than that
believe that the
market.
the highest losing bid
the winning bids are announced. In a uniform-price auction, the price paid
by a bidder conditional upon winning does not increase as his bid
if
two scenarios
the resale price
is
increased.
Thus
very responsive to the bids submitted and the winning bids are
is
announced there may
exist
an incentive
for the bidders to
submit arbitrarily large
We show by example that this is indeed
when = 1, and P is a constant there does
bids and upset any purported equilibrium.
possible,
and more generally show that
A;
not exist any Nash equilibriimi in strictly increasing pure strategies.
4.1
Existence of a symmetric Nash equilibrium
bids are not
As
in the
announced
discriminatory auction, each primary bidder's strategy
[x,x] to the real line.
We show
in strictly increasing
and continuously
buyers
in
when the winning
formation Xi
=
X,
i
=
a function from
below that there exists a symmetric Nash equilibrium
differentiable strategies, (6*, 6', ... ,6*),
the secondary market believe that
Suppose that bidders
is
all
bidders use
2,...,n adopt the strategy
and submits a bid equal
to
6.
If
is
r"(n,p)
= Ef7Zt+i = n,p
21
6*.
6*,
bidder
when
1
bidder
1
receives in-
wins the resale price
.
E[7|xi>n,n,p],
where Zj
is
the j-th order statistic of (ATijXj,
.
.
.
the fact that the signals are identically distributed, r"
its
arguments.
on
Yit
and Xi
If
bidder
bids
1
strictly increasing in
is
both
is
strict affiliation, v"
bidder
equality follows from
wins the auction, the expected resale price conditional
1
=E
t;"(x,y)
By
The
,X„).
his
6,
= 1,^ =
strictly increasing in its
expected profit
= E
-
is
[r"(n,P)|Xi
y]
.
arguments. Thus,
if
A'l
=
x and
is
[e [(r"(n,P)
- 6'(n))
l{*>*.(y,)}|^i,n] |Xi
E[(t;"(Xi,n)-6-(n))lo>fc.(yo}|^i
=
=
x]
^]-
(13)
Define
6'(x)
Note that
6* is strictly
Theorem
increasing.
3 Tht n-tuple
=
t;"{x,x).
We show
(6*, 6* ...,b*)
is
a
(14)
that (6*,6*,
.
.
.
,6*)
Nash equilihrium
auction provided that resale market buyers believe that
is
an equilibrium.
in the uniform-price
all the
bidders follow the
strategy b*
Proof.
Given that bidders 2,3,...,n use
n"(6|x)
where
u"
is
only
/jb(y|x) is
X
>
y.
r
'*'(t;"(x,y)
-
,
we can
rewrite (13) as
v"(y,y))A(y|x)cfy,
(15)
the conditional density of Yk given Xi. Since, by strict affiliation,
strictly increasing in
if
=
b*
both arguments, the integrand
Thus bidder
I's profits
{Xi > Yk}. Therefore bidder
are
I's profits
in (15)
is
positive
maximized when he wins
are maximized^*
if
if
if
and
and only
he uses the strategy
if
6*.
I
^^If
P
independent of Xi, or if there is no post-auction public information, then ti is constant
argument and no bid gives bidder t an expected profit greater than zero. However,
remains an equilibrium.
is
in its first
(b,.
.
.
,b)
22
Milgrom and Weber (1982) have shown that the
a uniform-price auction
is
E[F|Xi
=
yjtjFjt].
when
price paid by winning bidders in
bidders participate for the purposes of consumption
We show
in the
next lermna that the price paid by winning
bidders in the uniform-price auction in our model
is
greater than this. This
is
true
even though the primary bidders do not have a signalling motive.
Lemma
resale
(in
5 The price paid by winning bidders in a uniform-price auction with a
market
is
greater than that in a uniform-price auction without resale markets
which the bidders participate for consumption). That
is
6*(n)>E[i>|xi = n,n].
PROOF: From
the definition of
6*(n)
b*
we have
= E\E[V X,>Y,,Yt,P]Xr =
EEF Xi = yik,n,p]
>
Y,,Y,\
= i*,n]
A'i
= E[K|Xi = n,n]
where the inequality follows from
iterative expectations.
The
if
strict affiliation,
and the equality from the law
I
"true value" of the object for the primary bidders
no additional information becomes available
no signalling motive, one would expect the bids
proved
constant (or
in the next
Lemma
is
is
6 The
when P
is
is
the resale price. Thus,
P
is
Since there
is
after the auction, that
constant, the winners' curse on the primary bidders
when P
of
in the
is
weakened.
is
if
primary auction to increase
independent of {V ,Xi,X2,
.
.
.
,Xn)).
This
is
lemma.
bids in the uniform-price auction increase
when no additional information (other than about
when
P
is
constant, that
the bids submitted in the auc-
tion) becomes public after the auction.
PROOF:
random
Then
Let
6*
be the equilibrium bidding strategy
variable,
and
let 6*
when P
is
a strictly affiliated
be the equilibrium bidding strategy when
since
E[E[v> Xi
> n,n] Xi,n] = E[K Xi > n,n],
23
P
is
constant.
we have
=E[E
-E[K
b'Ax)
=
where the
Xi > x,Yk
'E\V Zk+i
^
last equality follows since
=
=
x
(16)
i],
the signals are identically distributed.
Next
b'{x)
= E[E[K|Xi>n,n,p]x, = x,n =
=
=
x]
< E{E[v\Zk+uP]\Zk>x,Zk+i =
x]
B{-E{v\z,+uP]\Zk
=
x,Z,+i
= B{E[v\z,+uP]\Zk+i =
x]
x]
= E[K|i,+i=x]
where the
first
equality follows since the signals are identically distributed, the
inequality from strict affiliation, the second equality
the last equality from (16).
from the definition of
Z/c,
and
I
Revenue comparison between the discriminatory auction and the
4.2
uniform-price auction
We
obtain two sets of sufficient conditions under which the expected revenues gen-
erated at the symmetric equilibrium of the uniform-price auction obtained in the
previous subsection, are greater than the expected revenues at the symmetric equi-
librium of the discriminatory auction of Section
the case of Treasury
The
bill
3.
The
first set
seems plausible
for
auctions.
following theorem states that
the public information, P,
if
is
not very
in-
formative about the true value of the objects, the uniform-price auction generates
higher expected revenue than the discriminatory auction.
Theorem
4 There
y,P ^
X
[x,x\
[p,p],
exists a scalar
M
>
such that
if
dr"{y,p)/dp <
M for
all
then the uniform price auction generates strictly higher expected
revenue for the auctioneer.
24
Proof.
By symmetry,
the expected revenue for the auctioneer
the unconditional expected payment of bidder
1.
Let
R" and
i?**
is
equal to
n times
denote the expected
revenue of the auctioneer under uniform-price auction and under discriminatory
auction, respectively.
Then
R"
= nxE[6*(n)l(x,>n}]
= kxE[b*{Y,)\x,>Y,],
= kxE[b{X,)\x,>n].
Note that the
total unconditional expected profits for bidders in equilibrium for the
uniform-price auction and for the discriminatory auction are, respectively,
= kxE[v\Xi > n]
-i?",
and
nxE[(r''(Xi,yi,...,n,P)-6-(Xi))l{x.>y,}
= kxE[V Xi>Yk\- R\
Thus, before bidders receive their private information, the two auctions are constant-
sum games between
k xe[f|xi > n
the auctioneer and the bidders, with total payoff equal to
Putting
r"(n)
it is
= E[F|xi>n,n],
easily seen that
E
[v'^iY,,?)
- r"(n)|xi > n,n] =
o,
and hence
E r"(n,p)-r"(n)^i>nl =
o.
Thus
R° =
kxE
r"(n)|Xi
>Y,]>R'^kxE [b{X,)\x,
25
>
Y,]
,
since the (unconditional) expected profit of a bidder in a discriminatory auction
always strictly positive by
{R°
all
-
R'^)l{k[p
y,pe
- p))
X
[x,x]
that the support of
strictly positive.
is
[p,p],
strict affiliation.
we assume
For ease of exposition,
is
then
iE"
>
We
show that
if
M=
p)/dp < M
is finite.
dr''[y,
Then
for
R'^.
First note that by affiliation dr"{y,p)/dp
r"(n,P)
will
P
>
0.
Thus
> r"{n,p)-M{p-P)
> r"(n)-M(p-p).
Hence
r"(n)-r"(n,P)<(i2°-i?')A,
and thus
= -E[r"(n,p)|xi = n,n]
-6*(n)
<
{R°-R'')/k-r''{Y,).
Taking expectation of the above expression conditional on {Xi
i2"
which was to be shown.
In words,
Theorem
=
ik
X
>
Yk} gives
E b*{n)\xi>n] >R'
I
4 says that
if
the ex post public information
P
has
little
impact on the resale price conditional on the information released from the uniformprice auction, then the auctioneer's expected revenue
auction. This
is
true even though there
auction. In the case of the Treasury
every Monday.
resale
The
bill
is
play.
higher in the uniform-price
no signalling aspect
in
the uniform-price
auction, bids are submitted before
results of the auction are
market comes into
is
1:00pm
announced around 4:30pm and the
One would expect
that any public information that
normally arrives between 1:00pm and 4:30pm would not be very informative about
V
conditional on the results of the earlier auction.
The
following theorem gives an alternative scenario under which once again the
uniform-price auction generates higher revenues. Essentially
nalling motive of the bidders
is
it
says that
if
the sig-
not strong, then the uniform-price auction generates
higher expected revenue.
26
Theorem
5 Suppose that {V Xi,
,
M>
positive scalar
[i,
x]*"*"^
X
such that
if
.
.
,
.
X„) are
dr'^[x,yi,
.
.
.
strictly affiliated.
,yic,p)/dx
<
There exists strictly
M for
all
x,yi,.
.
.
,yic,p
&
then the uniform-price auction generates strictly higher expected
[p,p],
revenue for the auctioneer.
Proof.
From Milgrom and Weber
{V ,Xi, Xi,
.
.
.
,
Xn) are
D = kx'E[E
where
6 is
cls
[V\X,
(1982,
strictly affiliated,
= n,n]
\X,
Theorem
we know
15)
and the hypothesis that
that
>n]-kxE [b{X^)
- K{X,)\x, >
Y,]
>
0,
defined in (7) and
K{x) ^
h{u)
r
^fl-dL{u\x),
Jx
/it(u|u)
and where h[u) and L{u\x) are
Let
M
theorem
First
=
>
D/(A:E[A'i|Xi
as defined in (10).^^
We now show
Yk\).
that with
M
as defined, the
true.
is
we
recall
from
Lemma
5 that
6*(n)>E[v>|Xi = n,n].
Therefore,
E[6'(n)|x, > n] > E[E[7|Xi
= n.nii^i > n]
and
A;
X E[6*(n)|A'i
>Y,]-ky. E[6(X0 - K[X,)\X, >
Next the hypothesis that
dr'^{x,yi,.
.
.
,yk,p)/dx
<
M
Y,]
> D.
for all i,yi,
.
.
(17)
.
,yi,p im-
plies that
k X K{x]
<
MXkX
X.
Hence
A;
X
E K{X,)\x,>Yt] <D.
(18)
^^Note that to prove this we also need a technical lemma that is slightly different from Lemma 4
(Milgrom and Weber (1982, Lemma 2)): Let p(z) and cr(z) be differentiable functions for which (i)
/>(?) ^ cr(i) and (ii) p(z) < a(z) imphes p'(z) > <7'(z). Then p(z) > a(z) for all z > x. The reader
should convince herself/himself that this
is
indeed true.
27
,
Substituting (18) into (17) gives
E
X
A;
b*{n)\xi
which was to be shown.
A
5
is
applicable
very informative about the true value
will
be small when bidder
1
wins.
plausible in the case of Treasury
As shown
a uniform-price auction
exists a
V Then
.
when
if
P
the public information
the impact of
Xi on the
is
resale price
This scenario, however, does not seem to be
auctions.
bill
may
not always exist in
the winning bids are announced. However,
symmetric Nash equilibrium
that the symmetric
is
next subsection, a Nash equilibrium
in the
> n]
I
Theorem
scenario where
>n]>kxB [b{x,)\x,
in strictly increasing strategies,
Nash equilibrium strategy
=
6(i)
is
v^{x,x,x)+
it
.
when
there
can be shown
rK
.
It is
then
readily confirmed that the expected revenues of the auctioneer at the sym^metric
equilibrium of the uniform-price auction are greater than his expected revenues
at the
symmetric equilibrimn of the discrminatory auction. However, we are not
aware of an intuitive
equilibrium in the uniform-price auction
4.3
when
the winning bids are announced. ^^
Uniform-price auction when winning bids are announced
we
In this subsection
illustrate the possibilty that strong signalling incentives
the part of the bidders
librium
when winning
may
lead to nonexistence of a pure strategy
,Xn,P)
{Xi,X2,
.
random
variables are only weakly affiliated,
.
when the
.
Example
^^A strong
is
all
x.
We
First
we
a sufficient statistic of
in this
V
is
announced
example the
^
after a
is
that for
f{y\x) f{x'\x')
f(x'\x)
f{y\x')
are unable provide an interpretation for this condition.
28
uniform-
uniform with support
guarantees existence of equilibrium
Mx',x,y) -
>
equi-
illustrates the difficulties that arise
The prior marginal density of
sufficient condition that
ii x'
it
is
Although
V.
the winning bids are
Vi(^',x',y)
and only
^Xn,?}
.
on
very responsive to the winning bids.
4 Suppose that
price auction.
.
for the posterior density of
resale price
Nash
bids are announced in a uniform-price auction.
present an example in which inax{Xi, X2,
if
which guarantee existence of Nash
set of sufficient conditions
all
y
<
[O, l].
x,
The random variables (Xi, X2,.
.
.
,X„,P)
are identically distributed, are indepen-
dent conditional on V, and their conditional density
Z = max{Xi,X2,...,Xn,P}.
= ElV^lZ]. Clearly, the resale
readily confirmed that
It is
E[V\Z =
(61, 62,
.
.
.
,6„) be a candidiate
limit our attention to weakly
Nash
Il"[b\x) be bidder I's expected payoff
verified that if x
<
.
.
,X„,P]
.
and
> Z
=
1]
1.
(19)
equilibrium, with
undominated
strategies
when he
bids b
6,
We
strictly increasing.
<
and thus
=
and Xi
x.
<
6,(X,)
Then
easily
is
it
Let
1.
x then
n"(6i(x)|x)<n"(6i(x)|x),
(20)
bidder 1 bids 61(1), he will always win whenever he would have with a bid of
since
if
61 (x)
and pay
the
same
bid of the others' bid
is
price. In addition he will also
win whenever the k-th highest
greater than 61 (x). Moreover, since 61
(19) implies that the resale price
as large as the resale price
is strict
Let
[0, V"].
E[K|Xi, X2,
price will be equal to ElV'lZ],
E[V\Z]
Let
uniform on
is
as long as there
is
if
if
he bids
61 (x)
is
he wins with a bid of
is strictly
increasing,
equal to one which
61 (x).
at least
is
The inequality in (20)
a nonzero probability that the k-th highest bid
in the
is
interval {bi{x),bi[x)).
Thus the only candidate Nash equilibrium appears
which
to be that in
all the
bidders bid one. At this purported equilibrium the announced bids are uninformative
about the bidders' signals and the resale price will be E[V|P], which has an expected
value of
^.
The price paid by the winning bidders
is 1.
Thus bidding one
is
not an
equilibrium either.
Next we show that
if
A;
=
1
and
if
P
totally uninformative
is
does not exist a symmetric Nash equilibrium
Proposition 4 Suppose
winning
bid,
and
that
A;
=
1
and
that
the highest losing bid are
29
V",
then there
in strictly increasing strategies.
P
is
independent of
announced
equilibrium with strictly increasing strategies.
about
V
.
Then,
there does not exist a
if
the
Nash
Proof.
9f
Let (61,62,
and the
=
1, 2,
.
.
.
resale
show that when Xi
The
=
x,
price that bidder
Bi
Since
We
if
We
1
bidder
faces
1
=
I's profits are
Xi and Bi
6
=
strategy
x.
""^
minimized at a bid of
.
.
we
will
6i(x).
,b„{X„)}.
.
and Bi
atomless.
resale price
b-,'{b),Bi]
Xi = x and he bids
if
is
b is^^
6i(z')
is
.
.
.
,6„(x)},
and
g{-\x)
is
the conditional density of
first-order necessary condition for 6i to be
L,,,
=
(r"(x,fci(x))
-
J5i
an equilibriiun
6i(x))g(6i(x)|x) +|^''^'^7(x,/?)g(/3|x)d^
f"(i,/?) denotes the derivative of f" with respect to its first
Let
6,,
is
^"g^f
where
The
uses strategy
|J'''^(f"(x',/?)-/?)ff(/?|x)rf/?,
min{62(x), 63(x),
—
given Xi
i
bj,
^ E[(f"(x',B0-5l))l{6U^')>B.}|^l=^]
=
where
=
Bi has a density function. The expected
profit for bidder 1
n"(x'|x)
j
uses strategy
are strictly affiliated
r{b,B,)=E[v\Xi =
The expected
j,
>-^
:
is
wins with a bid equal to
1
bidder
if
max{b2{X2),b3{Xs),
assiune, for simplicity, that
bidder
show that
\£,x]
6,
has an incentive to deviate from bi{Xi). In fact
are strictly increasing,
6,
will
Nash equilibrium where
for
market buyers believe that each bidder
n then bidder
,
be a candidate
,6„)
.
are strictly increasing.
2,3, ... ,n
t
..
x'
>
X.
Then by
^
0.
argument.
strict affiliation
=
{r{x\b,{x'))-b,{x'))g{b,{x')\x')
<
9{b,{x')\x')
+
l^'^''^r,{x\P)9{(3\x')df3
((r-(x\6axO) -6i(xO) +_^''''%r(x\/?)^(f|^rf/?)
g(6i(x')|xVn(x'|x)
ff(6i(x')|x)
dx'
^^Here we assume that the identity of the winning bidder
since
we
restricted attention to
symmetric
equilibria, such
30
is
also disclosed. In our earlier analysis,
an assumption was not necessary.
Similarly,
we can show
<
that, for i'
x,
dflix'lx)
Thus
for
any x G
[x,x], n"(i'|i) achieves
a global
minimum
=
at x'
I
xl
Concluding remarks
5
We
consider this paper to be an exploratory study of competitive bidding with a
We
resale market.
have shown that there exists a symmetric Nash equilibrium
discriminatory auctions
when winning
bids and the highest losing bid are annoimced
provided that the relevant variables are
We
also identified conditions
in
affiliated
and are information complements.
under which the auctioneer
will increase his
expected
revenue by precommitting to announce his private information before the auction.
The reader can
verify that a
symmetric Nash equilibrium
exists
of the auctioneer's information no longer holds in this case.
about the general impact of the ex post information
For the case of Treasury
believe that very
little
bill
bill
auctions
is
A
P
markets this
only the
We know
very
little
on the signalling motive
is
not important since
we
additional information becomes publicly available in the
short time period between the closing of the Treasury
of the resale market.
if
However, the result on the pubhc announcement
winning bids are announced.
of bidders.
even
related question
whether there
which
is
bill
auction and the opening
of greater importance for Treasury
exist plausible scenarios in
which the auctioneer can
increase expected revenue by announcing his private information after the auction
(and before the secondary markets convene) rather than before the auction. These
warrant further investigation.
We
showed that there
auction
when only
exists a
symmetric Nash equilibrium
the highest losing bid
is
where the uniform-price auction generates
announced.
Two
strictly higher
auctioneer than the discriminatory auction.
When
existence of a
Whether there
Nash equailibrium
a uniform-price
scenarios are provided
expected revenue for the
winning bids as well as the
highest losing bids are announced, there are situations where a
does not exist.
in
Nash equilibrium
are intuitive sufficient conditions that ensure the
is
an open question.
31
Several extensions of our current model deserve attention. First, in the Treasury
bill
auction, the primary bidders submit price-quantity pairs
than one unit of the Treasury
bill.
This feature
is
missing in our model. Second,
two weeks before the conduct of the weekly Treasury
of the Treasury bills to be auctioned are traded
relationship
among
bill
among
auction, forward contracts
the primary bidders.
The
the forward prices, bids submitted, and the resale price needs to
be investigated. Third, there exists a wide
bills
and demand more
vairiety of close substitutes of
carried as inventories by primary bidders.
Treasury
These close substitutes may have
a significant effect on the interplay between the forward markets and the weekly
auction. These are on-going research topics of the authors.
6
I
1
\
7
u 7 3
22
References
1.
A. Brimmer, 1962, Price determination in the United States Treasury
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/
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Date Due
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