tA
e
">
o/
TEC^.
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
AUCTIONS WITH RESALE MARKETS:
AN EXPLORATORY MODEL OF
TREASURY BILL MARKETS
Sushil Bikhchandani and Chi-fu Huang
June 1988
Revised June 1989
WP
#
3070-89-EFA
MASSACHUSETTS
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
INSTITUTE OF
AUCTIONS WITH RESALE MARKETS:
AN EXPLORATORY MODEL OF
TREASURY BILL MARKETS
Sushil Bikhchandani and Chi-fu Huang
June 1988
Revised June 1989
WP
#
3070-89-EFA
SEP
1
2 1989
RECEIVED
AUCTIONS WITH RESALE MARKETS:
AN EXPLORATORY MODEL OF
TREASURY BILL MARKETS*
Sushil Bikhchandani^and Chi-fu
Huang^
June 1988
Revised June 1989
Abstract
This paper develops a model of competitive bidding with a resale market. The primary
market
is
of resale.
modelled as a common- value auction,
in
which bidders participate
After the auction the winning bidders
sell
for the
purpose
the objects in a secondary market
and the buyers on the secondary market receive information about the bids submitted
in the auction.
The
effect of this
information linkage between the primary auction and
the secondary market on bidding behaviour in the primary auction
is
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 auction yields higher expected revenues are obtained.
example of our model, with the primary market organized
the U.S. Treasury
bill
*We thank John Cox
An
as a discriminatory auction,
is
market.
bill auctions and Margaret Meyer for helpful
Chemical Bank for answering many of our questions
on the organization of Treasury bill auctions. We would also like to acknowledge helpful comments from
seminar participants at City College of New York, Princeton, Stanford and UCLA. The second author is
grateful for support under the Batterymarch Fellowship Program.
^Anderson Graduate School of Management, University of California, Los Angeles, CA 90024
02139
'Sloan School of Management, Massachusetts Institute of Technology, Cambridge,
conversations.
We
for kindling
are grateful to
our interest
Chiang Sung
in
Treasury
of the
MA
Introduction
1
1
Introduction
1
One
of the major achievements of economic theory in the past decade has been a deeper
understanding of the conduct and design of auctions. Recent surveys of the literature on
McAfee and McMillan
auctions are
(1987),
Milgrom (1987), and Wilson
account for a large volume of economic and financial
ment
uses a sealed-bid auction to
sell
activities.
The U.S.
U.S. Treasury Department uses a sealed-bid auction to
Many
Interior Depart-
mineral rights on federally owned properties. Auction
houses regularly conduct auctions of antiques, jewelry, and works of
dollars.
(1988). Auctions
Treasury
sell
Every week the
art.
worth
bills
billions of
other economic and financial transactions, although not explicitly conducted
as auctions, can nevertheless be thought of as implicitly carried out through auctions; see,
example, an analysis of sales of seasoned new issues
for
the market for corporate control in
One
for
seasoned new issues.
resale, the auction will be
among
all
This
for sale.
One may argue
Thus
it
is
that
common-value with the
the participating bidders.'
in
Parsons and Raviv (1985), and
is
that there exist active resale or
(1986).
feature often shared by such financial activities
secondary markets for the objects
and
Tiemann
true, for
when
resale price being the
that an auction with a resale market
is
common
Weber (1982a)
common-value
However, there are situations that make existing theory inapplicable.
Treasury
bills
valuation
might be argued that the theory of common-
value auctions developed by Wilson (1969), and Milgrom and
The observation
example, for Treeisury
there exists the possibility of
is
A
is
applicable.
certainly true.
case in point
is
auctions.
bill
In Treasury bill auctions, there are usually
participate in the weekly auction.
They submit competitive
about forty bidders or primary dealers
who
These primary dealers are large financial institutions.
sealed bids that are price-quantity pairs.
Others, usually indi-
vidual investors, can submit noncompetitive sealed bids that specify quantity (less then a
prespecified
maximum). The noncompetitive
bids, usually small in quantity,
always win.
The primary
dealers compete for the remaining bills in a discriminatory auction.
the
demands
bidder down, are met
the
bills
of the bidders, starting with the highest price
are allocated.
The winning competitive bidders pay the
'An auction
is
After the auction, the Treasury department announces
common-value
if
is,
until all
unit price they submitted.
All the noncompetitive bidders pay the quantity-weighted average price of
competitive bids.
That
all
the winning
some summary
participating bidders do not value the objects for sale differently.
^
Introduction
1
statistics
•
2
about the bids submitted. These include
amount
total tender
received;
amount accepted;
• total tender
-
• 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 thai
in the
is
better than the information possessed by investors
secondary markets. The primary dealers submit bids
information that
is
in
the auction based on both
publicly available at the time, and their private information.
The buyers
on the resale market have access only to public information, including information revealed
by the Treasury about the bids submitted
in the auction.
To the extent that
bids submitted
reveal the private information of the primary dealers, the resale price in 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 and the resale price
in existing
models of common-value auctions and
In Section 2,
we develop
is
is
absent
the primary focus of this paper.
a model of competitive bidding with a resale market.
The
primary dealers or bidders are risk-neutral and have private information about the true value
of the objects.
random
We
variables,
assume that the bidders' private signals and the true value are
i.e.,
affiliated
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
likely.
prices paid
After the auction the auctioneer publicly announces
by the winning bidders,
for
some information
example) about the auction. The winning bidders
^For an analysis of bidding with a resale market when the valuations of the bidders are
see
Milgrom
(1987).
(the
common
knowledge,
Introduction
1
then
3
the objects in the secondary market, at a price equal to the expected value of the
sell
object conditional on
all
public information.^
Although the primary motivation of
many
institutional details,
demand
bidders
And we do
model
which are absent.
most one unit
is
the Treasury
bill
market, there are
For instance, we require that competitive
of the object, instead of being allowed to choose quantity.
not model the effect of any forward contracts, and close substitutes (such as \ast
week's Treasury
we
at
this
bills)
owned by primary bidders on
focus on one aspect of the Treasury
bill
market
their bidding strategies. In this
paper
— the informational linkage of the resale
market and the primary auction.
The
results of this paper
may
also be helpful in analyzing other types of auctions with
resale markets, such as art auctions, in
by Van Gogh
is
auctioned at a price
which bidders have correlated values.
much
In section 3
we analyze discriminatory
auctions.
It is
tions for the existence of a unique symmetric
is
model
in
Nash equilibrium
We
provide sufficient condi-
in
nondecreasing strategies
Milgrom and Weber (1982a) where bidders participate
purpose of consumption and there
for the
in future.
assumed that the winning bids and
the highest losing bids are revealed at the end of the auction.
in the auction. Unlike the
a painting
higher than expected, then one might expect
and other paintings by Van Gogh to be sold at higher prices
this
If
is
no resale market, the
not sufficient for the existence of an equilibrium.
higher than those derived in Milgrom and
Weber
affiliation
The equilibrium
bids
property alone
we obtain
axe
(1982a), because primary bidders have an
incentive to signal.
A
kets
key insight gained from the theory of common-valuation auctions without resale mai-
is
that the greater the
sale) revealed in
duction
in the
amount of information (about the
an auction, the greater the expected revenue
if
Any
for the auctioneer.
uncertainty about the true value weakens the winners' curse which
causes the bidders to bid more aggressively. Thus
affiliated
true value of the objects for
the auctioneer
hats
in
re-
turns
private information
with the bidders' valuations, he can increase his expected revenue by precom-
mitting to announcing his private information before the auction.
market and there
When
there
exists an incentive for the bidders to signal, the auctioneer
the bidders' incentive to signal and thus decrease his expected revenue
if
is
may
a resale
reduce
he announces his
^Riley (1988) investigates a model in which conditional upon winning, each bidder's payment depends
all the bids submitted. In our model, the expected value for each primary bidder depends on the bids
upon
submitted.
1
Introduction
4
private information. Sufficient conditions for the public
announcement of the auctioneer's
private information to increase his expected revenue are provided.
and
In Sections 4
5
we consider a uniform-price auction, that
rule for determining the winning bidders
is
identical to the
one
in
is
an auction
in
which the
the discriminatory auction,
but the winning bidders pay a uniform price equal to the highest losing bid. The existence
of an equilibrium depends in part on the kind of information about the auction publicly
revealed by the auctioneer.
If,
as
we assumed
for discriminatory auctions, the
announced then we show by example that there may
are
exist
an incentive
winning bids
for the bidders
to submit arbitrarily large bids in order to deceive the secondary market buyers. 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
auction, provided that only the price paid by the winning bidders
is
in the
uniform-price
announced (and thus
bidders have no incentive to signal).
Next,
we turn
The key
to the question of the auctioneer's revenues.
the theory of auctions without resale markets mentioned above
is
from
insight gained
Without
also useful here.
resale markets, uniform-price auctions yield higher revenues than discriminatory auctions,
since in the former the price
is
linked to the information of the highest losing bidder.
When
there are resale markets in which the buyers draw inferences about the true value from the
bids and the winning bids are announced, there
same
direction.
Namely that
in a
is
an additional factor which works
uniform-price auction
it is
cheaper to bid high
to signal a high realization of private information, since conditional
does not pay what he bids. Therefore,
in greater
We
when
in
in
in the
order
upon winning a bidder
our model as well, uniform-price auctions result
expected revenues for the auctioneer, when there exists an equilibrium.
also provide plausible sufficient conditions
the primary auction
by winning bidders
is
is
under which the auctioneer's revenues
organized as a uniform-price auction and only the price paid
announced
is
greater than under a discriminatory auction
when the
winning bids are announced. Section 6 contains concluding remarks. All proofs are
in
an
appendix.
The revenue maximizing mechanism
for selling
Treasury
bills
was a subject of debate
in
the early 1960s. Friedman (1960) proposed that the Treasury should switch from a discrim-
inatory auction to a uniform-price auction for the sale of Treasury
bills.
that uniform-price auctions would induce bidders to reveal their true
man
Apart from the
demand
fact
curves, Fried-
asserted that discriminatory auctions encouraged collusion and discouraged smaller
2
The model
5
bidders from participating. Both Goldsein (1960) and
Brimmer (1962) disputed Friedman's
contention. Smith (1966), on the basis of a mathematical model, concluded that uniformprice auctions yield greater revenues.
in that
Unlike Smith's model, our model
each bidder's beliefs about the others' bids are confirmed
we model
in
game-theoretic
is
an equilibrium, and
the information linkage between the primary auction and the secondary market.
Like Smith, our analysis provides support for Friedman's proposal that the Treasury
bill
auction should be uniform-price.
The model
2
Consider a common-value auction
competitive bids
The
in
Treasury
bill
true value of the objects
is
in
which n risk-neutral bidders (the dealers who submit
auctions) bid for k identical, indivisible objects, with n
same
the
for all the bidders,
and
is
unknown
to
them
>
k.
at the
time they submit bids. Each bidder privately observes a signal about the true value, based
on which he submits a
we
indicate
bid.
We
cissume that there are no noncompetitive bids. In Section 6
how noncompetitive
bids can be incorporated in our model.
assume that each bidder demands
The primary
(or
is
allowed) at most one unit of the object.
dealers' interest in the objects being auctioned
of resale in the secondary market.
We
market buyers that they would prefer to
For instance, the primary bidders
week's Treasury
we
bills,
sell
may
may have
sufficiently lower
resell
the Treasury
solely for the
purpose
than the valuations of the
the objects rather then
consume them.
capital constraints so that unless they sell this
not be able to participate in next week's auction. Since, as
primary bidders make positive expected
will establish,
prefer to
they
is
assume that the primary dealers' personal (con-
sumption) valuations of the object are always
resale
Throughout we
bills at their
profits in the auction, they
expected value conditional on
all
publicly
might
known
information.
If
instead one assumes that the primary bidders' personal value of the object
szLme level as that of the resale buyers, then
primary bidders
is
the object
and
their privately
when
at the
the private information of the winning
not revealed after the primary auction, the primary bidders will never
the expected value of the object conditional on
sell
if
if all
is
known information
their expected value
is
is
strictly less
all
public information
greater than the resale price, and will always
than the resale
price.
sell
Therefore the resale market
buyers will make strictly negative expected profits and the resale market will break down.
2
The model
6
Hence we preclude
this possibility. 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 publicly announces some information about the auction.
For simplicity we assume that that the winning bids and the highest losing bid submitted in
the auction are publicly announced. In the case of the uniform-price auction, there
exist an equilibrium
we
when
also analyze uniform-price auctions
the price paid by the winning bidders
We
may become
is
not
the winning bids are not announced and only
announced.
some additional information about the value
allow the possibility that
for sale
may
the winning bids are revealed at the end of the auction. Therefore,
if
of the objects
publicly available after the bids are submitted, but before the opening
of the secondary market.
The
k winners in the primary auction then
risk-neutral buyers on the secondary markets.
The buyers on
the objects to
sell
the secondary markets do
not have access to any private information about the true value. They infer what they can
from the information released by the auctioneer about the primary auction, and any other
publicly available information. Thus, regardless of the secondary market
auction or a posted price market
conditional on
The n
all
— the resale price will be the expected value of the object
is
a
random
variable,
V
i
=
1,2, ...,n.
Each bidder
.
observes a private signal, X,, about the true value. Let
becomes public
except
after the auction
when otherwise
the true value, that
example when
— an
publicly available information.^
risk-neutral bidders will be indexed by
being auctioned
mechanism
BlVlP]
,
^
The
has a
P denote
true value of each object
common
prior on
P
V, and
any other information that
over but before the resale market meets.
stated, that given
is,
P = V
is
i
We will
assume,
the bidders' signals are not uninformative about
E[V\Xi,X2,
,^„,P].
If this
condition
is
violated, for
our model reduces to the usual common-value auction without a
resale market.
Let f{v,p,x) denote the joint density function of
{Xi,X2, .,Xn).
[p,p]
X
[x,
It is
assumed that /
i]" be the support of/, where
V
,
P, and the vector of signals
symmetric
i]" denotes the n-fold product of [x,x].
in the
we do
not rule out the possibility that the support of the
either
from above or from below. Further,
*The participants
n arguments. Let
is
[z,
it
is
Icist
random
assumed that
all
variables
the
is
random
X=
[v,v\
X
Note that
unbounded
variables in
risk neutral. In the case of Treasury bill
primary bidder that pays say $1 in 182 days will be E[m|/], where
denotes the random marginal rate of substitution between consumption 182 days from now and that of
today and where I 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.
in the
secondary market do not have to be
auctions, the "true" value of a
m
bill for
a
Discriminatory auction
3
this
model are
affiliated.
7
That
is,
for ail x, x'
G
[^,^1", v, v'
E
and
{v,v],
e
p, p'
{p,p],
f{iv,p,x) V {v',p',x'))f{{v,p,x) A {v',p'y)) > fiv,p,x]f{v',p',x'),
where V denotes the componentwise maximum, and A denotes the componentwise mini-
mum.
Affiliation
is
said to be strict
if
H
is
an increasing function of
is
if
an increasing^ function then
c,,
d,.
for other implications of affiliation.
the above inequality
E[H{V ,P,Xi,X2,
The reader
We
referred to
is
at X
is
c,
and
for all
c, ,(i,
£
the right-hand derivative and at x
assume that {V ,P,Xi,
a.ny
d,,
.
strictly increasing in
and
c,-
[x,x],
<
in
X,-
<
d,,
H
if
n] is
G
l,...,n]
contin-
is
continuously
that
if
H
is
Moreover, we shall
strictly increasing in
Xi,thenE[H{V,P,Xi,X2,...,Xn)\ci < Xi <
d, for all Ci,di
=
di,
t
7^ 1] is
\x,x).
Discriminatory auction
3
In a discriminatory auction, the bidders
the auction.
that
when
A
bidding game
Weber
winning bidder pays the price that he or she bids. In
symmetric Nash equilibrium with
among
the primary dealers.
in
when the motive
which the buyers know some or
would and thus signal
is
we show
strictly increasing strategies in the
is
of the primary bidders
all
of their bids (or a
their private information.
in
Milgrom and
not sufficient for the existence of a Nash
This
is
to resell in a secondary
summary
their bids), there exists an incentive for the primary bidders to bid
value
this section
Unlike the auctions examined
(1982a), the affiliation property alone
equilibrium. Intuitively,
market
submit sealed bids and the k highest bidders win
the bidders' private signals are infoTmation complements^, in a sense to be defined
later, there exists a
statistic
based on
more than they otherwise
because, by affiliation, the resale
is
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 increeises with his
information realization, then there exists an equilibrium
is
<
i
the left-hand derivative.
is
i
di,
Milgrom and Weber (1982a)
with the convention that the derivative
..,A''„) are strictly affiliated so
o{{V,P,Xi,...,X„),say
Affiliation implies that
< Xi <
further assume for simplicity that
uously differentiable then 'E\H{V ,P,Xi, X^, ..,Xn)\ci
differentiable in
is strict.
...,Xn)\ci
in strictly increasing strategies. It
the information complementarity of the bidders' signals with respect to the true value that
^Throughout this paper, we will use weak relations. For example, increasing means nondecreasing,
means nonnegative, etc. If a relation is strict, we will say, for example, strictly increaaing.
'The reader will see that our notion of information complementarity is different from that in Milgrom
and Weber (1982b)
positive
Discriminatory auction
3
8
ensures that the bidders' incentive to signal increases with their information realizations
and enables them to
secondary markets participant's
exists a unique
beliefs are
monotone,
also
show that
if
a sense to be defined, then there
in
symmetric equilibrium.
model where bidders participate
In a
We
sort themselves in a separating equilibrium.
in
an auction for
final
consumption of the objects,
Milgrom and Weber (1982a) show that the auctioneer's expected revenue can be increased
if
he precommits to truthfully reporting his private information about the objects for sale
before the auction, provided that his private information
is
affiliated
with the bidders'
pri-
vate information. This follows since by publicly announcing his information, the auctioneer
introduces an additional source of affiliation
among
the primary bidders' private informa-
tion and thus weakens the winners' curse. Hence the bidders compete more aggressively
and the expected
result
selling price
not necessarily true.
is
portion of the bid submitted by a bidder
incentive to signal to the resale market participants.
is
model with a
increased. However, in our
is
A
If
market
resale
is
this
attributed to hia
the auctioneer's private information
a "substitute" for the bidders' information, announcing that information will reduce the
This
responsiveness of the resale price to the bidders' information.
incentive for the bidders to signal and
other hand,
bidders',
it is
cause the expected selling price to
the auctioneer's private information
Thus
game.
it
is
is
is
symmetric
symmetric.^
I's
Note that
Bidder
i's
strategy
in its last
is
denoted
6,
:
point of view.
at the time
observes his private information Xi.
[x,x\
Thus
—
9?.
increasing and differentiable strategies,
.
.
,
6)
n arguments,
The
the
this
is
a symmetric
analysis from the other bidders'
when bidder
1
submits his bid, he only
a strategy for bidder
We
order necessary conditions for an n-tuple (6,...,
.
On
natural to investigate the existence of a symmetric Ncish equilibrium.
examine the game from bidder
viewpoint
(6,
fall.
a "complement" to that of the
is
always beneficial for the auctioneer to announce his private information.
the hypothesis that /(v,p,x)
We
may
turn reduces the
Existence of a symmetric Nash equilibrium
3.1
By
when
in
I
is
a function of Xi.
begin our analysis by deriving the
6) to be a
when buyers
in the
Nash equilibrium
first-
in strictly
secondary market believe that
are the strategies followed in the bidding.
^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.
Discriminatory auction
3
9
Since buyers in the secondary market do not have access to private information about
the true value, the resale price
is
earlier, to simplify the analysis
prices paid by
winning bidders
that bidders
=
t
2,
.
.
.
submits a bid equal to
(i.e.,
Then
r''(6-i(6),y'i,...,n,P)
public information.
all
the winning bids) and the highest losing bid.* Suppose
bidder
if
conditional on
we assume that the auctioneer announces the
,n adopt the strategy
6.
ofV
the expectation
As mentioned
=e\v
1
bidder
b,
1
receives information
wins with a bid
b
Xi = x and
the resale price will be
X, = b-'{b),b-HKy^)),---,f>-\biY,),P\
Xi = b-\b),Y,,...,Y,,p],
where b~^ denotes the inverse^ of
Note that
if
P=
common- value
V,
r'^[b~^{b),Yi,
.
and Yj
b
.
.
the j-th order statistic of {X2,
is
=
,Yk,P)
.
.
,Xn).
V, and our model reduces to an ordinary
auction without a resale market. Define
v^[x',x,y)^-E[r'^{x',Yu...,n,P)\Xi
By our hypothesis about
strict affiliation,
both
and v
r
= x,n =
(2)
y]-
are strictly increasing in each of
arguments (provided that given P, the bidders' signals are not uninformative about
their
V). The expected profit for bidder
^
n'{b\x)
1
when Xi = x and he submits
B[{r'{b-'{b)Si,---,y.,P)-b)l^,^i^y^^}\x^
a bid equal to b
=
x]
= E
[e [[r'{b-\b),Yu...,n,P) - b) l{,>i^Y,)y\Xi,n] \x,
= E
[{v'{b-'{b),XuY,) -
=
/
b)
1{,>S(K0}|^1
=
^]
is
=
x]
.
{v'{b-\b),x,y)-b)f,{y\x)dy,
JX
where the second equality follows from the law of
iterative expectations,
the conditional density function of y^ given Xi. Taking the
first
and fk{y\x) denotes
derivative of n'^(6|z) with
respect to b gives
^n^
=
(v^(6-'(6),x,6-'(6))
-
b)
h{b-Hb)\x){b'{b-'{b)))-^ - F,{b-Hb)\x)
+ {b'{b-'{b)))-'fp'Ki{b-\b),x,y)My\x)dy,
where
b'{x)
is
the derivative of 6(1), Ft(t/|i)
given Xi, and vf
'if
some
is
is
the conditional distribution function of Yk
the partial derivative of
v'^
some function
of
of the other losing bids, or
with respect to
its first
them, are also announced
all
argument.
For
the results remain
unchanged.
*If 6
that
<
lie in
b{z)
then
fc~'(fc)
the range of
6.
=
i and
if 6
>
6(r) then 6"'(ii)
=
1.
Thus we only need
to consider values of
b
Discriminatory auction
3
[b,.
.
all
be a Nash equilibrium,
,b) to
.
when
10
bidders
t
=
2,
bidders adopt
.
.
.
,
That
b.
necessary that 6 be a best response for bidder
it is
n adopt strategy
is,
b
and the
must be zero when
relation (3)
= ^^[^j(^) = {A^,x,x) Rearranging
(4) gives
6(x)
=
(v (i,i,x)
-fc(i))-—
v'^
—-+
vi(i,i,y)-—
/
Jx
if
6(x)
profit
[x,x];
and
(ii)
b[x)
has to be positive
<
t;
(i, X, x),
=
—
dy.
(5)
^k[^\^)
= E[y|j^i = x,n-y].
Besides (5), there are two other necessary conditions that
1
- F,{x\x)
and the law of iterative expectations,
v''(x,x,y)
Vi e
b{x):
an ordinary differential equation:
Note that, by the definition of
b{x),
=
6
6(i)) A(i|x)(6'(x))-i
rk{x\i)
bidder
1
market participants believe that
resale
Condition
u'^(x, x, x).
in equilibrium.
Condition
(i)
(ii)
b
(6)
must
satisfy:
(i)
v (x, x, x)
>
follows since expected profit for
follows
then by slightly increasing the bid to 6(i)
from
+
e
(i)
and the
when Xi =
x,
fact that
expected
can be raised from zero to some strictly positive amount.
The
solution to (5) with the boundary condition 6(x)
6(x)
=
v^{x,x,x) -
=
v'^[x,x,x)
is
r L{u\x)dt{u) + Jxr J^dL{u\x),
(7)
hW^)
Jx
where
=exp{-i:^ds},
L(u|x)
=v<^(u,u,u),
t{u)
=
H^)
Note that
will
show
profit
to
V
,
L{ii\x)
in
and
what
is
*^i
(8)
("' "' y) A(y|")'^y-
t(u) are increcising functions of
u and thus are meaisures on
follows that b{x) of (7) also satisfies condition
under the hypothesis that {Xi,
and
Ix
.
,
.
.
(i),
[x, i].
We
maximizes expected
Xn, P) are information complements with respect
strictly increasing.
Definition 1 Random
respect to another
variables, (Zi,
random
T
variable
^\\
aziOZj
.
.
Zm), are said
to be
information complements with
if
""'
'
,
.
>0,
^i^j,Vzu...,Zm,
where
4,{zu...,Z^)=E\f\Zi = Zu---,Zm = Zm\-
3
Discriminatory auction
11
Thus, the random variables
to
V
of Xi
(A'l,
.
.
.
,Xn, P) are information complements with respect
the marginal contribution to the conditional expectation of
if
larger the higher the realization of any other
is
tarity condition
is
.
,
.
or P. This information
.
.
,
.
.
Zm)
random
if
.
(T, Zi,
.
.
.
,
Zm)
.
,
.
is
Zm)
complements with
are information
give three examples of strictly affiliated
complemen-
if <^(2i,
Thus
are information complements.
Zm)
are multivariate normally distributed, then [Zi,
We
of a higher realization
by a large class of distributions. For example,
satisfied
linear in the 2,'s, then (Zi,
respect to T.
Xj
V"
variables that also satisfy
the information complementarity condition.
Example
g{t,zi,
sume
that
.
.
1 Let [T, Zi,
,
.
Zm)- Let
E~^
that
S
exists
d^ln g/dtdzi >
,Zm)
.
.
.
multivariate normally distributed with density function
be
variance-covariance matrix of these random variables and as-
be the
and has
and d^
strictly negative off-diagonal elements.
g/dzidzj >
In
ber (1982a) then implies that {f,Zi,...,
is
linear in the Zi
[Zi,
's,
Zm)
,
.
.
.
for
Zm)
are information
Besides the multivariate normally distributed
gamma
2 Let Zi,
i
=
1,
distribution given
.
,
.
.
T=
(
m
^
The
\A
=
n^
rf^'^r^e-^'/'
)
(
ifZi
«
0,
t
>
0,
and T
is
is
T.
a large class of
an example.
and distributed according
the
>
0,
.
gamma
function. Let
\/T
also be distributed according to
distribution with a density
Ml/0 =
^(')'"^~^^' '^'>0'
I
otherwise,
I
where 7 >
{T,Z\,
T
is
to
otherwise,
I
where a >
Milgrom and We-
variables, there
following
easily verified
Since E[f\Zi, ...,Zm]
complements with respect
independent conditional on
be
1 of
is
t:
gi[Zi\t)
gamma
Theorem
j.
random
distributions with linear conditional expectations.
Example
i
are strictly affiliated.
It
.
.
.
and a >
,
Zm)
0.
Using Theorem
<"£ strictly affiliated.
1 of
-F"ri7
7
t,^!
lZi,...,Zm\
1
Thus {Zi,
.
.
.
Milgrom and Weber (1982), one
Direct computation yields
- £i=i:?Li_^ma + 7 — 1
,Zm) are information complements with
respect to T.
verifies that
3
Discriminatory auction
12
Note that the prior distribution of T
"conjugate distributions" of
gamma
in
Example
distribution; see
2
is
an element of the family of
DeGroot
(1970, Chapter 9).
Other
distributions with linear conditional expectations can be constructed similarly. Interested
readers should consult Ericson (1969)
The
following example gives
Example
3 Let Zi,
i
=
1,2,.
.
.
and DeGroot
random
,m;
independent conditional on
be
with density
is
otherwise.
y
Weber (1982) implies
>
d^ \T\gi{z{\t)/dzidt
It is easily verified that
also
T
otherwise,
[
The density ofT
(1970).
variables that are strict information complements.
Theorem
Vz,t € (0,1).
1
of
Milgrom and
The same theorem
that p, satisfies the strict affiliation inequality.
shows that
,x_
,, ^
,
p(i,zi,Z2,...,Zm)-|
(
is strictly affiliated.
for alii :^j
£
The
ifae
following
otherwise
Direct com.putation yields
=
[,Z2...,Zm) ='E\T\Zi
for zi,Z2,...,2m
t72i,22,...,2,.,<e(o,i)
/»(onr=i 5.(2.10
Zi,Z2
=
=
Z2,...,Zm
Zm\
=
Finally, one can also verify that d^(p[zi, Z2,
(0)1)-
.
,
Zm)/dzidzj >
(0,1).
lemma
is
a direct consequence of the definition of information comple-
mentarity. All proofs are in an appendix.
Lemma
1 Suppose that {Xi,.
Then Vi{x,x',y)
v"*
is strictly
with respect to
The
its first
.
.
,X„, P) are information complements with respect
increasing in
x'
6
to
V.
where vf denotes the partial derivative of
y,
argument.
following proposition shows that
with respect to V, then
and
if
(A'l,
.
of (7) satisfies condition
,
.
.
A'„,
(i),
P) are information complements
that
is,
t;
(i,x,x) > b{x) Vi
6
[x,x].
.
Discriminatory auction
3
Proposition
V.
Then
13
1 Suppose that (A'l,
v'^{x,x,x)
>
b{x),
Vi €
.
,
.
P) are information complements with respect
A'„,
.
and
[z, r]
the inequality is strict for x
>
x,
where
to
b is
defined in (7).
A
resale
is
corollary of Proposition
1
is
that
b is strictly
Thus our assumption that
increasing.
market buyers can invert the primary bids to obtain the bidders' signal realizations
justified.
Corollary 1 The
strategy b defined in (7)
Before proceeding, we
first
is strictly
increasing.
lemmas that
record two
are direct consequences of the defi-
nition of affiliation.
Lemma
2
Lemma
3 Let
(Milgrom and Weber (1982a))
x'
> x >
y.
Then
fjk(y|i)//t(y|i)
Fk{y\x')/Fi,{x\x')
is
decreasing in x.
< Fk{y\x)/ Fk{x\x).
That
is,
the
distribution function Fic{-\x')/ Fk{x'\x') dominates the distribution function Fk{-\x) / Fic{x\x)
in the sense of first order stochastic
The main
Theorem
1
result of this section
The n-tuple
dominance.
is
discriminatory auction provided that [Xi,
spect to
Nash equilibrium of
the
,Xn, P) are information complements with
re-
(6,..., 6), with b as defined in (7), is a
.
.
.
V
When
bidders in the discriminatory auction participate only for the purpose of con-
sumption, Milgrom and Weber (1982a) have identified a symmetric Nash equilibrium with
a bidding strategy
6^(z)
where L{u\x) and
identified in
=
t{u) are as defined in (8).
Theorem
1
is
-
v''(x,x,i)
strictly higher
Jx
r L{u\x)dt{u),
The bidding
than
A uu
b
(9)
strategy for the purpose of resale
for every x
G
{x,x]
by an amount equal to
Discriminatory auction
3
14
the magnitude of which depends on v^
ted bid.^°
It is
,
the responsiveness of the resale value to the submit-
between the resale value and the bids submitted by
this informational link
bidders in the discriminatory auction that gives the bidders an incentive to signal.
since the b
is
strictly increasing, the resale
Of course,
buyers can invert the bids announced by the auc-
tioneer to obtain the private information of the bidders and, as in Ortega-Reichart (1968)
and Milgrom and Roberts (1982),
It is
in
equilibrium no one gets deceived.
worth emphasisng that the primary bidders benefit
better informed about the true value.
relevant information in
Xi,X2,
.
.
bidders and the expected price(s)
P
"informativeness" of
for instance,
the resale market buyers are
if
P =
or
V'
if
F
"centains"
all
the
X„,, then there would be no signalling incentive for the
,
.
If,
in
the auction would be lower.
One might expect
to increase with the length of the time period
the
between the end of
the primary auction and the start of the resale market.
Next we establish that the symmetric equilibrium
metric equilibrium
if
secondary market buyers
identified
above
is
the unique sym-
monotonicity condition. The
beliefs satisfy a
beliefs of the
secondary market buyers about the bidders' private information are said to be
monotone
they are nondecreasing in the bids submitted. For instance,
if
market buyers believe that bidder
when bidder
1
bids
I's private
information
lies in
the secondary
if
the interval [xj(6),i"(6)]
then the monotonicity condition on beliefs would imply that x[{-) and
b,
i"(-) are nondecreasing functions. ^^ Since bidders with higher realizations of their private
information would expect higher resale prices,
it
is
natural to expect
them
to bid more.
Therefore monotonicity of beliefs seems to be a natural restriction to impose. Of course,
when
bids are in the range of the equilibrium bidding strategies, beliefs are obtained by
inverting the bidding strategies.
Since a symmetric equilibrium
and as shown below
when
is
game with symmetric
a natural focal point in a
players,
the only symmetric equilibrium in nondecreasing strategies
(6,..., 6) is
the resale buyers' beliefs are monotone,
we
equilibrium
will use this
when comparing
expected revenues between a discriminatory auction and a uniform-price auction.
Theorem
b is
2 If the secondary market buyers have monotone
as defined in (7),
is the
beliefs
then
(6, 6,..., 6),
where
unique symmetric equilibrium in nondecreasing strategies.
'"If P = V there is no signalling motive and the equilibrium strategy is as specified in (9), since r''() = V
and thus tif (•) =
and h{) = 0. Also, when P is "very informative" about V the signalling motive is weak.
For example if we let P„ = V + €„ where £„ is, say, uniformly distributed on [— ^, ^] then as n increaaes the
resale price becomes less responsive to the players' bids and in the limit the incentive to signal disappears.
"In the symmetric equilibrium
with
z'i(6)
=
x'{b)
=
r'(6). For
of
6
Theorem
>
1,
6(1), x\{b)
the resale buyers beliefs are
=
xt{b)
=
x,
and
for b
<
monotone
6(1), i',(6)
(since
=
itC-)
b is
=
increasing)
Z-
—
3
Discriminatory auction
15
Public announcement of the auctioneer's information
3.2
Suppose that the auctioneer has private information about V, represented by a random
We
variable Xq.
impact of announcing Xq before the auction on the
will consider the
expected selling price. Let b[-;xo) be a symmetric equilibrium bidding strategy conditional
on Xq
b
=
Xq. It
is
assumed that
and wins then the
{b;xo)]Xo)
.
.
.
=
increasing and differentiable in x.
^e[v\x, = r\b-xo),Yu.
,n,P]Xo)
If
bidder
1
bids
.
.
,n,P,Xo =
xo]
,
Putting
b.
u;''(i',i,y;xo)
it is
is
resale price in this case will be
p''Cb~\b;xo),Y\,
where b(b
6(-;xo)
=E
[p^{x' ,Yu
,Yk,
= x,n = y,^o =
P;xo)\Xi
^o]
,
straightforward to show that b{x;xo) must satisfy
V{x;xo)
=
°.
{rv'^{x,x,x;xo) -b{x;xo)) j!
^t(2;|x;xoJ
,
'
,
+
u)f{x,x,y; Xo)fk{y\x-xo)dy.
/
=
where /t(y|x;xo) denotes the conditional density of Y\ given Xi
addition, the
boundary condition b[r,xo)
to (10) with this
=
6(i;io)
boundary condition
it;'^(x,i,x;xo)
-
/
=
(10)
Jx
x and
Xq =
xq.
In
The
solution
-dL{u\x;xo),
/
Jx fk(u\u;xo)
(11)
w'^ [x, x, x; xq)
must be
satisfied.
is
L{u\x;xo)dt[u;xo)
+
Jx
.
.'
where
L{u\x;xo)
=
exp<^-/
t(u;xo)
=
uj
/i(u;xo)
=
—
-ds\,
(u,u,u;xo),
w'}{u,u,y;xo)fk{y\u;xo)dy.
/
Jx
Next,
we
define conditional information complements.
Definition 2 Random
ditional on
random
variables, [Z\,
variable
Y
.
,
.
.
Zm), "re said
with respect to
d^4>{zi,...,zm,y)
to be
information complements con-
random
variable
w-^
W
•
T
if
W
OZiOZj
where
4>{zi,...,Zm,y)
=
'E\f\Zi
=
Zi,...,Zm
= Zm,Y =
y]
.
Discriminatory auction
3
If
V
,
{Xi, A'2,
.
.
.
,
16
Xn, P) are information complements conditional on Xq with respect to
then a proof identical to that of Theorem
equilibrium strategy. This
Proposition 2 The
is
shows that
1
b
defined in (11)
.
the resale
.
.
,Xn,P)
a symmetric
stated without proof in the following proposition.
n-tuple (b{-;xo),
.
.
are information
market participants
Xq =
xq, provided
complements conditional on Xq with respect
complements with respect to
,Xn,P)
as {X\,X2,-
Our main
V
to
and
believe that all the bidders follow strategy 6(-;io).
Note that the existence of an equilibrium does not depend on whether (XcXi,
are information
Nash
,b{-;xo)), withb{-;xo) as defined in (11), is a
.
equilibrium of the discriminatory auction when the auctioneer announces
that [Xi,
is
V
.
.
,X„,P)
.
complements conditional on Xq with respect to V.
are information
be that the expected
result in this subsection will
.
There exists a Nash equilibrium as long
selling price
under the policy of
always reporting A'o cannot be lower than that under any other reporting policy provided
that [Xo,Xi,.
in the
.
.
,Xn,P)
are information
next proposition, that 6(1; zq)
Proposition 3 Suppose
are information
that
complements with respect to
V .^^ We
first
show,
a increasing function of xq.
is
{V ,Xo,Xi,.
complements with respect
..
,Xn,P)
to
are affiliated
V. Then b{x,Xo)
is
and
that {Xq,
Xi,
...
,Xn,P)
an increasing function of
XQ.
The main
Theorem
result of this section
is
3 Suppose that {V ,Xo,Xi,
are information
.
.
.
,Xn, P) are
complements with respect
information cannot lower, and
may
to
V
.
A
affiliated
and
that (A'o,
Ai,
.
,
.
.
X„,P)
policy of publicly revealing the seller's
raise, the expected
revenue for the seller in a discrimi-
natory auction.
Given Proposition
Theorem
Huang
16),
is
the proof of theorem 3 mimics that of
3,
omitted.
The
interested reader
is
Milgrom and Weber (1982a,
referred to Bikhchandani
and
(1988) for details.
Theorem
in iQ.
and
3
When
depends
{Xo,Xi,.
critically
.
.
'^Note that this assumption
existence of
Nash equilibrium.
on the fact that under
its
hypothesis 6(1; xq)
is
increeising
,Xn, P) are not information complements, 6(r;io) may not be
is
stronger than the conditional information complementarity required for
4
17
Uniform-price auction
an increasing function of
This
in
may
turn
xq,
and revealing Xq may reduce the bidders' incentive to
lower the expected revenue for the seller even though (>Yo,Xi,.
.
.
signal.
,Xn,P)
are affiliated.
4
Uniform-price auction
In a uniform-price auction the bidders
The
the auction.
price they
pay
is
submit sealed bids and the k highest bidders win
equal to the {k
l)st highest bid. Initially
-f-
we obtain
a
candidate for a symmetric Nash equilibrium under the assumption that after the auction the
auctioneer reveals the winning bids and the highest losing bid, that
winning bidders.
We
first
If
is,
the price paid by the
any of the lower bids are also revealed, our results remain unchanged.
obtain strategies that satisfy first-order necessary conditions for a symmetric
Ncish equilibrium.
Next we show that when there
exists a
symmetric equilibrium
in the
uniform-price auction, the auctioneer's expected revenues at this equilibrium are greater
than at the symmetric equilibrium of the discriminatory auction.
this result
is
as follows.
From
The
intuition behind
the theory of auctions without resale markets
compared with a discriminatory auction, a greater amount of information
is
we know
that
revealed during
a uniform-price auction. This weakens the winners' curse in the uniform-price auction and
results in greater revenues for the auctioneer.
In addition
when
the resale markets are informationally linked, as in our model,
the primary auction and
cheaper to submit higher
it is
bids in the uniform-price auction in order to signal to the resale market buyers.
This
in
turn further increases the expected revenues from uniform-price auctions.
However, the second of these two factors
price paid by a bidder conditional
can result
in
in
very responsive to the bids submitted there
we show by example
and
P
is
is
uniform-price auction the
cis
a uniform-price auction.
may
exist
an incentive
and upset any purported equilibrium.
that this
in a
upon winning does not increase
nonexistence of equilibrium
arbitrarily large bids
— the fact that
his bid
If
is
increased
the resale price
for the bidders to
—
is
submit
In the last part of this section
indeed possible, and more generally show that
when
)t
=
1,
a constant there does not exist any Nash equilibrium in strictly increasing pure
strategies.
Uniform-price auction
4
Necessary conditions for a symmetric equilibrium
4.1
As
each primary bidder's strategy
in the discriminatory auction,
the real
Suppose that
line.
If all
[bo, bo, ...
when buyers
differentiable strategies,
fco-
18
other bidders use strategy
a bid equal to
b,
then
bidder
if
1
a function from [x,x\ to
in strictly increasing
the secondary market believe that
in
bo,
is
Nash equilibrium
bidder
1
receives information
wins the resale price
=E
r"{6o-i(6),n,...,n,P)
to) is a
,
Xi =
all
and
bidders use
x and submits
is
V\X, = bo'{b),boHbo{Yi)),
.
.
.,boHho{n),P]
= E v\xi = bo\b),Yi,...,n,P].
r"
is
strictly increasing in all its
arguments. Note that r"(-)
auction, the expected resale price conditional on
v"(6o-i(6),i,y)
By
strict affiliation, v"
is
=E
Xi and Yk
=
If
bidder
1
wins the
is
[r"(6o-i(6),y,,...,n,P)|Xi
strictly increasing in its
r'^(-).
= x,n =
(13)
v]
arguments. From the definition of v
it
follows that
v"{x',x,y)
We show
below that
6o
U Xi =
n"(6|x)
is
(14)
=
v^{x,x,x)
+
-^
(15)
as defined in (8).
X and bidder
=
1
bids
b, his
expected profit
is
E[(r"(6o^(6),F,,...,n,P)-6o(n))l^i>,„(y,)}|Xi
=
i]
= E
[e [(r"(6o-^(6),yi,...,n,p) - 6o(n)) i{6>6„(fo}|^i'^*J 1^^
= E
[(v"(6o^(6), A'i,n)
/
Jx
Taking the
Vi',i,y.
v {x',x,y)
must be given by the following equation:
bo{x)
where h{x)
—
first
- 6o(n)) l{6>6o(n)}|^i =
'-^^ =
^1
(v"(6o'(6),x,y)-6o(y))A(y|x)(iy.
derivative of n"(6|i) with respect to
{v"{boHb),^,l>o'{b))
-
b)
b
=^
(16)
gives
hibo\b)\x]{b'o{bo\b)))-'
(17)
+ mbo\b)))-' jf^'\:nboHb),x,y)Uy\x)dy,
4
Uniform-price auction
where 6o(i)
its first
the derivative of bo[x) and n"
is
argument. For
(17) be zero
19
when
b
=
=
6[,(a;)
That
^"'(/
l
the partial derivative of v" with respect to
be a Nash equilibrium,
(60, •••.^o) to
bo[x).
is
necessary that relation
it is
is,
'n
={v"(x,x,x)-bo(x))h{x\x)
^
+ /|t;!j'(i,i,y)A{y|i)(iy.
where we use the eissumption that
60(1)
is
60
is
In a uniform-price auction without resale
markets Milgrom and Weber (1982a) show that
the symmetric Nash equilibrium bidding strategy
auctions, the bidding strategy 60
which depends on
is
strictly higher
is
fc"(i)
than
=
As
v"[x, x, x).
6", for
every x
v", the responsiveness of the resale value to the
G
in discriminatory
(1,1],
by an amount
submitted
bid.
Revenue comparison with the discriminatory auction
4.2
Next we show that when there
tion,
Rearranging terms implies that
strictly increasing.
as defined in (15).
symmetric equilibrium
exists a
in
the uniform-price auc-
generates strictly greater expected revenues than the symmetric equilibrium of the
it
discriminatory auction.^'
Theorem
4 When
(60.
,^0) "
a
symmetric Nash equilibrium
in the uniform-price auction,
the expected revenues generated at this equilibrium are strictly greater than the expected
revenue
4.3
symmetric equilibrium of
at the
PossibiUty of nonexistence of equiUbrium
In this subsection
we
may
of the bidders
illustrate the possibility that strong signalling incentives
lead to nonexistence of a pure strategy
ning bids are announced
max{A'i,X2,
sity of
the discriminatory auction.
V
.
.
.
.
,Xn, P}
Although
in a
is a sufficient statistic of (Ari,vf2,
in this
we
uniform-price auction. First
•
Nash equilibrium when winpresent an example in which
,Xn, P)
for the posterior den-
example the random variables are only weakly
illustrates the difficulties that arise
when
the resale price
is
on the part
affiliated, it
very responsive to the winning
bids.
to that in Theorem 2 establishes that 60 is the only candidate for a symmetric
when the resale market buyers have monotone beliefs. This remark also applies to the symmetric
equilibrium we will obtain in Section 5.
'^An argTiment Bimilar
equilibrium
4
Uniform-price auction
Example
20
4 Suppose that
prior marginal density
ofV
all the
is
uniform on
Let {bi,b2,
.
.
our attention
Let
[0,V^].
=
Y;[V\Xi,X2,...,X„,P]
uniform with support
V
,Xn,P}-
BlVlZ]. Clearly, the resale price
,bn) be a candidate
to
The random variables {Xi, X2,
[0, l].
Z = ma\{Xi,X2,
E[V\Z]
>
Z
E[V\Z=l]
=
1.
Nash
equilibrium, with
Xi =
It
is readily
<
6,(X,)
it is
.
,
.
confirmed that
and
(19)
strictly increasing.
<
.
their conditional den-
will be equal <o EjV'lZ],
6,
Then
x.
and
,
-
weakly undominated strategies and thus
bidder I's expected payoff when he bids b and
The
after a uniform-price auction.
are independent conditional on
are identically distributed,
sity is
announced
bids are
1.
We
limit
Let n"(6|i) be
easily verified that if
x
<x
then
<
n"(6i(x)|x)
n"(6i(i)|i),
(20)
since if bidder 1 bids bi{i), he will always win whenever he would have with a bid 0/61(1)
and pay
same
the
others' bid
is
price.
whenever the k-th highest bid of the
In addition he will also win
in {bi{x),bi{x)).
Moreover, since
bi
is strictly
increasing, (19) implies that
the resale price if he bids 61(1) is equal to one which is at least as large as the resale price
if
he wins with a bid 0/61(1). The inequality in (20)
is strict
as long as there is a nonzero
probability that the k-th highest bid is in the interval (61(1), 61(1)).
Thus
which
the only candidate
at least
Nash equilibrium appears
one of the bidders, say bidder
bid sufficiently low so that they never win,
1,
and
to be a
somewhat degenerate one
always bids one, and at least n
the
n — k lowest bid
bidder 1 always maximizes his profits by bidding one.
But
is
if there is
— k
in
bidders
low enough so that
any
strictly positive
cost of participating in the auction (such as an entry fee or a bid preparation cost) then the
n — k bidders who never win
Next we show that
ii
k
=
at this equilibrium will not participate in the auction.
1
and
if
P
is
totally uninformative
about
V
,
then there does
not exist a symmetric Nash equilibrium in strictly increasing strategies. Essentially
there
is
when
only one object, and no other information becomes public after the auction, there
exists a large incentive for the bidders to
responsive to the winning bid.
submit high bids, since the resale price
is
very
Xn, P)
Uniform-price auction without signalling
5
Proposition 4 Suppose
and
bid
=
that k
1
the highest losing bid are
and
21
P
that
announced
is
V
independent of
.
there does not exist a
Then
if the
winning
Nash equilibrium
in
strictly increasing strategies.
However, there exist other examples
is
uniform on
strategy
and Xi
[0,1]
=
to bid 6o(x)
is
for
uniform on
is
which an equilibrium
then
[V',!],
Whether there
v"(i,i,i).
t;i(-,-,-)
exists.
=
0,
For instance
if V'
and the equilibrium
exist intuitive sufificient conditions
under which an equilibrium exists remains an open question.
Uniform-price auction without signalling
5
Since there exists a possibility of nonexistence of equilibrium in a uniform-price auction
with signalling,
we analyze uniform-price auctions when
in this section
are not announced.
Thus the bidders do not have an incentive to
information, since
they win their bids are not revealed.
incentive,
by
we
if
are able to
show that
this uniform-price auction
discussed in Section
the Treasury
5.1
bill
3.
We
in at least
the winning bids
signal their private
Even without the
signalling
two scenarios the expected revenue generated
higher than that generated by the discriminatory auction
is
believe that the
scenario
first
is
a plausible one for the case of
market.
when the winning
Existence of a symmetric Nash equilibrium
bids are not announced
We show
below that there exists a symmetric Nash equilibrium
difFerentiable strategies, {b' ,b'
all
bidders use
information
6*.
Xi =
,
.
.
.
,b'),
Suppose that bidders
x,
when buyers
i
=
2,
.
.
in strictly increeising
bidder
If
b.
and
secondary market believe that
n adopt the strategy
,
.
and submits a bid equal to
in the
1
6*,
bidder
1
wins the resale price
receives
is
= b[v\xi >n,n,p],
where Zj
is
the j-th order statistic of {Xi, X2,
that the signals are identically distributed, r"
If
bidder
1
.
.
is
.
,Xn)- The equality follows from the fact
strictly increasing in
both
its
arguments.
wins the auction, the expected resale price conditional on V^ and Xi
t}"(i,y)
= E [f"(n,P)LVi =
x,y%
=
yl
.
is
5
Uniform-price auction without signalling
By
strict affiliation, 0"
bids
b,
his
is
22
strictly increasing in its
expected profit
arguments. Thus,
= E
1
is
H E[(f"(n,p)-6-(y',))i{,>,.(^^,j|xi =
n"{6|x)
Xi = x and bidder
\{
z]
[e [(f"(n,p) - 6*(n)) i(.>t.(n)}|^i.n] |^i
= E[(e"(Xx,n)-6'(n))i{,>,.(^^)}|xi =
=
x]
i].
(21)
Define
6*(i)
Note that
Theorem
b' is strictly increasing.
5 The n-tuple {b',b'
=
We show
... ,6')
is a
t)"{i,i).
(22)
that [b',b',
Yk,yk]-
We show
model
an equilibrium.
is
in the uniform-price auction
price paid by winning bidders in a
=
that the price paid by winning bidders in the uniform-
greater than this. This
is
,6*)
participate for the purposes of consumption isEiVjA'i
lemma
the next
in
price auction in our
.
bidders follow the strategy 6*.
all the
Milgrom and Weber (1982a) have shown that the
when bidders
.
Nash equilibrium
provided that resale market buyers believe that
uniform-price auction
.
is
true even though the primary bidders
do not have a signalling motive.
Lemma
4 With probability one, the price paid by winning bidders in
tion with a resale
market
is
a uniform-price auc-
greater than that in a uniform-price auction without resale
markets (in which the bidders participate for consumption). That
is
b'{Y,)>E[v\Xi = Y,,Y,]
with probability one.
The
"true value" of the object for the primary bidders
is
the resale price. Thus,
additional information becomes available after the auction, that
winners' curse on the primary bidders
would expect the bids
Lemma
P
is
5
weakened. Since there
primary auction to increase when
in the
independent o( {V ,X\,X2,.
is
.
,Xn)). This
is
proved
in
is
P
is
constant, that
is
P
is
if
no
constant, the
no signalling motive, one
is
constant (or
when
P
is
the next lemma.
With probability one, the bids in the uniform-price auction
in the auction)
if
strictly increase
when
when no additional information (other than about the bids submitted
becomes public after the auction.
Uniform-price auction without signalling
5
Revenue comparison with the discriminatory auction
5.2
We
23
obtain two sets of sufficient conditions under which the expected revenues generated at
the symmetric equilibrium of the uniform-price auction obtained in the previous subsection,
are greater than the expected revenues at the symmetric equilibrium of the discriminatory
auction of Section
The
The
3.
first set
seems plausible
following theorem states that
if
for the case of Trecisury bill auctions.
the public information, P,
is
not very informative
about the true value of the objects, the uniform-price auction generates higher expected
revenue than the discriminatory auction.
Theorem
[x,x]
X
6 There
[p,p\,
M
exists a scalar
>
such that
if
df"{y,p)/dp <
M for
y,p E
all
then the uniform-price auction without signalling generates strictly higher ex-
pected revenue than the discriminatory auction with signalling, at their respective symmetric
equilibria.
In words.
on the
Theorem 6 says
that
the ex post public information
if
resale price conditional on the information released
then the auctioneer's expected revenue
winning bids are not announced. This
is
is
1:00pm every Monday. The
market comes into
arrives
little
impact
One would expect
is
no signalling
when
eispect in the
auction, bids are submitted before
bill
results of the auction are
play.
has
higher in the uniform-price auction even
true even though there
uniform-price auction. In the case of the Treasury
resale
P
from the uniform-price auction,
announced around 4:30pm and the
that any public information that normally
V
between 1:00pm and 4:30pm would not be very informative about
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
the bidders
Theorem
such that
is
says that
if
the signalling motive of
not strong, then the uniform-price auction generates higher expected revenue.
7 Suppose that [V ,Xi,
if
it
dr'^{x,yi,.
.
.
.
,
.
.
A'„) are strictly affiliated.
,yk,p)/dx <
M for
all
x,yi,
...
,yk,p
There exists a scalar
€
[z,!]*"*"^
x
M>
[£,p], then the
uniform-price auction without signalling generates strictly higher expected revenue than the
discriminatory auction with signalling, at their respective symmetric equilibria.
A
scenario where
Theorem
informative about the true value
7
is
applicable
V Then
.
is
when the
the impact of
public information
Xi on the resale price
P
is
very
will be small
Concluding remarks
6
when bidder
1
Treasury
auctions.
bill
24
wins. This scenario, however, does not seem to be plausible in the case of
Concluding remarks
6
This paper
which
ric
is
an exploratory study of competitive bidding when there exists a resale market
is
informationally linked to the bidding.
Nash equilibrium
in
We
discriminatory auctions
have shown that there exists a symmet-
when winning
bids and the highest losing
bid are announced provided that the relevant variables are affiliated and are information
complements. This
monotone
is
the only symmetric equilibrium
beliefs. If his
will increase his
information
is
when
expected revenue by precommitting to announce his private information
We know
before the auction.
little
about the general impact of the ex post information
the signalling motive of bidders. For the case of Treasury
since
we
believe that very little additional information
short time period between the closing of the Treasury
resale market.
is
the resale market buyers have
complementary to that of the bidders, the auctioneer
A
related question which
whether there
is
bill
markets
this
is
becomes publicly available
bill
in
the
auction and the opening of the
of greater importance for Treasury
exist plausible scenarios in
P on
not important
bill
auctions
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
tion
it
established the possibility of nonexistence of an equilibrium in a uniform-price auc-
when winning
bids are announced. However,
when
there exists a symmetric equilibrium,
generates greater expected revenues than the symmetric equilibrium in a discriminatory
We
auction.
auction
showed that there
also
when only
for the bidders.
exists a
the highest losing bid
Two
is
symmetric Nash equilibrium
announced, so that there
is
in a uniform-price
no signalling incentive
scenarios are provided where the uniform-price auction without sig-
nalling generates strictly higher expected revenue for the auctioneer than the discriminatory
auction.
Some
implications of our model deserve attention. First,
competitive bids
in
common knowledge
bidders and k
price.
+j
it is
eaisy to
incorporate non-
our model, provided the total amount of noncompetitive bids,
before the auction.
objects, j of
The preceding
We
j, is
can model the primary auction as one with n
which are awarded to noncompetitive bidders at the average
analysis remains unchanged.
The expected
profits of the
noncom-
6
Concluding remarks
petitive bidders
25
positive and equal to the ex ante expected profits of the competitive
is
minimum and maximum
bidders.
There
tive bid,
which may be the reason that resale market buyers do not buy Treasury
is
a prespecified
through noncompetitive bids; and
it
rnay also explain
why
quantity for each noncompetibills
competitive bidders do not sub-
mit only noncompetitive bids while avoiding (presumable costly) information collection.^*
Second, the fact that ex ante expected profits of the bidders
when
P
is
average price in the auction
is
strictly less
uniform-price auctions without signalling,
in expectation, the
comparision has been empirically documented by
in section 3.1,
the primary bidders benefit
about the true value, since
it
if
uninformative about
Cammack
This feature
of the weekly Treasury
are traded
among
bill
is
V'')
implies that
than the resale price. This
(1986). Third, as pointed out
decreases the bidders' signalUng incentive.
There are several possible extensions of our model.
bill.
strictly positive (except in
the resale market buyers are better informed
the primary bidders submit price-quantity pairs and
Treasury
is
First, in the Treaisury bill auction,
demand more than one
unit of the
missing in our model. Second, two weeks before the conduct
auction, forward contracts of the Treasury bills to be auctioned
the primary bidders.
The
relationship
among
the forward prices, bids
submitted, and the resale price needs to be investigated. Third, there exists a wide variety
of close substitutes of Treasury bills carried as inventories by primary bidders. These close
substitutes
may have
the weekly auction.
a significant effect on the interplay between the forward markets and
We
hope to investigate some of these
issues in our future research.
'*By renormalizing the units, we can aasume that each competitive bidder demands exactly
This allows uj to keep the prespecified maximum demand of noncompetitive bidders
than that of competitive bidders.
units.
m
>
1
or zero
at a level less
,
7
Appendix
26
Appendix
7
Proof of Lemma
The
l:
-
(n
joint density of (V'.P.A'i,?^!, ...,¥„_!)
is
l)!/(v,p,iiyi,...,yn-i)l{„i>v2>... >„„_,}•
As a consequence, the conditional density
of
given {P,Xi,Yi,.
V'
.
.
,^,,-1)
is
f{v,P,x,yi,...,yn-i)
'^"^'^^•^'"-'>-
f{p,x,yu...,y.-i)
Thus {P,Xi,Yi,
which
fj (i,
j/i
,
information complements. Let rf denote the derivative of
....,Yk) are
defined in (l), with respect to
is
.
.
.
,
j/jt
,
p)
is
=
and
strictly increasing function of i'
PROOF OF Proposition
v'{x, X, x)
By
=
1,
first
=
/| L{u\x) {vfiu, u, u)
rx L[u\x)dt{u)
,
.
.
A'„,
x'
,Y,
=
y]-
implies, by affiliation, that vf{x,x',y)
is
a
- rx j^^dL{u\x)
+
viiu, u, u)
+
viiu, u, u)
-
^)
(23)
dn.
P) are information complements with respect
Vi(u, u,u) > Vi(u,ti,y) for y <
u.
Jx
Fk[n\u)
Next note that
Vj".
=
r'^
then easily verified that
write
b{x)
.
,
It is
I
y.
-
the hypothesis that [Xi,
Lemma
We
i:
j/y
E[Tf{x,Yi,...,n,P)\Xi
Milgrom and Weber (1982a) then
5 of
argument.
a strictly increasing function of p,
vf{x,x',y)
Theorem
its first
It
to
V
and
follows that
Fk{u\vL)
Substituting this relation into (23) gives
v'^{x,x,x)
-
b{x)
Note that the above inequality
affiliation.
r L{u\x) (v^(u,u,ti) + vi{u,u,u)) du >
>
is
strict for
x € {x,x] since Vj
>
and
0.
^3
>
by
strict
I
Proof of Corollary
have v'^{x,x,x) - b{x) >
noting that vf >
0.
I
l:
We
Vx >
will
z.
show that
The proof
b'{x)
is
> O Vi >
z.
From Proposition
completed by inserting this
in (5),
1
we
and
Appendix
7
27
Proof of Lemma
By
3:
affiliation
we have
P >
for
a, x'
>
x
f,{a\x')h{^\x) < A(a|x)A(/3|x')-
Thus
>
for X
y
f j^ h{a\x')h{^\x)dad^ < P jjk{a\x)h[fi\x')dadp,
which
is
equivalent to
Fi(yjx')(n(i|i) - F,(y|x)) < Fk[y\x)[F,{x\x') - F,(y|x')).
Rearranging terms gives
< Fk[y\x)
- Fkx\x)
^*(vl^')
Fk{x\x')
•
I
Proof of Theorem
0-
an'^(6(x')|x')
^j
Let
l:
_
-
<
x'
Recall from (4) that
x.
(^,(,^^-lJ^(,^,^(ld(,,,^
Fk[x\x)\{v (x ,x ,x) v>y^))
^ r'
<
d(
/
AM^
-oyx)
^
h> ( .>\
fk{y\x') , \
^
/
/.r^'U
6(x))
{h\x')r'F,{x'\x')({A-\-,A-K^'))^^^.-y{x')
l'k[x'\x)
\
<
(6'(x'))-^F.(x'|x')
,
r'
d(
UAx\ X, x')
I
-
K^'))Wir\ -
^'(^')
!k{y\x) J \
^
ffc(x'|x')an'^(6(x')|x)
dh
Ft(i'|x)
where the
first
inequality follows from Proposition
second inequality follows from
6(x')
that
<
b[x), his
^^^
'
^
^'
since n''(6(z)ix)
Lemma
3.
That
is,
1,
Lemma
1,
and
Lemma
when X\ = x and bidder
2,
1
and the
bids 6
=
expected profit can be raised by bidding higher. Similar arguments show
<
=
for x'
>
x.
for all x,
have thus shown that 6(x)
is
As
a consequence, n''(6|i)
we have
n'^(6(i)'x)
>
is
maximized
for all x
the best strategy for bidder
1
>
x,
by
at 6
=
h{x). Finally,
strict affiliation.
when he observes X\ —
x,
We
when
Appendix
7
bidders
i
=
28
2,3,
bidders follow
.
,
.
.
n follow
Proof of Theorem
Suppose
2:
ing strategies. Therefore 6,
Weber
(1982a),
and when the resale market participants believe that
6,
the
all
I
6.
Theorem
(6^,6, ,...,6,)
asymmetric equilibrium
is
difFerentiable almost everywhere.
is
proof
14, the
complete once we establish that
is
nondecreas-
in
Thus, as in Milgrom and
b,
must be
strictly
increasing and continuous. Hence b, must satisfy the differential equation (5) and the only
solution to this
First,
6,(i)
=
c,
is 6
of (7).
suppose that
Vi G
not strictly increasing. That
6, is
The
[ia,i(,].
resale price
bi
the
is
random
bidder
=E
r'{b,Bi,...,Bk,P)
where
if
is
affiliated
with
all
the other
random
the conditional expected resale price
=E
v'{b,x, 0)
Affiliation
and monotone
nondecreasing
in all
=
+
Prob{bidder
1
[lo,!*],
both
number
1
£i
the second expression
slightly
>
A'l
market buyers implies that
=
1
r'
and
when Xi = x G
c,bi
=
v' are
[xa,Xi\,
i]
= c}E
(24)
[(t;'(c, Aj, B*) -
c)
left after
those
who
bid
more than
who
c
=
l^^^^jJAi
is
a
x\
,
the
tie is
have been assigned
bid
Since b,[x)
c.
and the probability of the event {Bk
=
c}
=
strictly
is
We must have v'{c,Xa,c) > c else a bid bs{xa) = c
= x^. Therefore, by strict affiliation, for any xq E
(which depends on xq) such that v'{c,Xo,c) > c
xq, there
define
c is
in (24) is strictly positive for
more when Xi =
we
v
wins
of bidders (including bidder 1)
when
nondecreasing, Bi
is
Analogous to
being declared a winner given that there
this probability
results in negative profits
there exists
=
wins|Bt
the /-th order
is
Bu...,Bk,P)\Xi = x,Bk = 0].
b,{x)
positive (and strictly less than one).
(Xfl.ij]
1
E[(v'(c,A'i,B,)-c)l^e^<,j|A'i
objects divided by the
,
,b,{Xn))- Since 6,
.
arguments. The expected profit for bidder
expectation of the number of objects
Vi €
bidder
beliefs of the resale
where the probability of bidder
c,
if
[r'(6,
and he submits an equilibrium bid
n'{c\x)
= b,Bi,...,Bk,P]
variables in the model.
ij such that
b is
variable which denotes bidder I's bid and B/
statistic of the others equilibrium bids, {b,{X2),
<
there exist Xa
wins with a bid
1
\y\^bi
is,
is
x
a discontinuous
=
xq.
jump
If
instead of
c
+
Thus
ci.
bidder
in his probability of
1
bids
winning
Appendix
7
since
Bk
if
v'{c' xq,
=
>
•)
,
29
bidder
c,
v'{c, xq,
now win with
will
1
Vc'
•),
>
Since beliefs are monotone,
probability one.
Therefore, there exists
c.
>
£2
such that a bid of tj(xo)
leads to greater expected profits than 6,(zo)- This contradicts our assumption that 6,
equilibrium strategy. Therefore
Next suppose that
Since 6,
b,{xci).
of v'{b,x,0)
small enough
b,[x')
=
and
b,
and can be inverted we
b,{y)
=
=
B[[v'{x^,xj,n)-b,{x,))l^y^^^^y\Xi
<
E
=
n{b,{xd-€)\xd),
- €,Xd,Yk) -
[(t;'(id
which contradicts our assumption that 6,(id)
We
record a technical
z
>
lemma
6 (Milgrom and
for which
all
(i)
>
p[x)
(t(z)
b,{xd
-
(ii)
x^]
=
^"^J
an equilibrium bid.
is
b,{xd) can be ruled out similarly.
< a{z) implies
p'{z)
I
3.
(1982a)) Let p{z) anda{z)
p[z)
=
l{yj<xj-f}|^i
£)j
before proving Proposition
Weber
and
b{x; Xo)
If
v' implies that for
>
be differentiable functions
(^'{z).
Then p[z) > a{z) for
X.
Proof of Proposition
we can show
follows
>
possibility that limjji^fcj(i)
Lemma
Then, the continuity of
/?.
<
will write v'{x' ,x,y) instead
>
U'{b,{xd)\x^)
The
an
strictly increasing.
has a discontinuity at xj. Consider the case where limiji^ ^ai^)
b,
strictly increasing
is
where
€
must be
6,
+ ^2
is
from
3:
=w
that 6(i;io)
Lemma
6.
we have that
Let xo
<
(z, x,
>
By
x'q.
x]Xo)
>
b{x; x'q)
=w
implies b (i;xo)
^{x-jx'q)
we know
affiliation
>
(i, x, x; x'q)
b {x;Xq),
.
then the proposition
Suppose that b{x;xo) < 6(i;io). As generalizations of Lemmeis
1,
both y and xq, Fk{y\x; io)//it(y|i; iq)
2,
and
is
decreasing in iq, and that Fk{-\x;xo)/Fie[-\x\xo) dominates Fii{-\x;x'q)/ Fk{-\x;x'Q) in the
3,
sense of
first
Vr
6(i;io)\
u;f (i, i', y; iq) is increasing in
degree stochzistic dominance. Then
=
^
>
Id,
(^ {x,x,x;xo) \
f
(u;
r+/f u^jJ,(x, z, y; xq) 77"^
>,^fk{x\x;xo)
b{x;xo))-p-;—^
^iklx|x;ioJ
If
l^
,
b{x;xQ),
^
/
ray
I'k(x\x;xo}
—-—jr+ /' t^i(x,
Jx
I'k(x\x;xQ)
=
Jx
A(x|x;x'o)
^,
u
(x,x,x;xo)-6(x;xo))—
d,
fk{y\x;xQ)
,
»x/*(y|£i^jdy
x, y; Xq)
I'k[x\x,xo)
7
Appendix
30
which was to be shown.
I
Proof of Theorem
that of Milgrom and
Proof.
3:
Weber
(1982a,
Given Proposition
Theorem
3,
the proof of theorem 3 mimics
16).
Define
W{x,z)
which
is
=E
\b{x;Xo)\Yk
the expected price paid by bidder
ditional on bidder
=
winning when X\
1
z
and by the hypothesis that Xq and ^i are
1
when
< i,Xi =
z]
,
the auctioneer publicly reveals Xq, con-
and bidder
affiliated,
1
bids as
Xi =
if
W2{x,z) >
0.
By Proposition
x.
3
Note that, by symmetry,
the expected revenue for the seller under the policy of publicly reporting
Xq
is
k times
E[b{Xi;Xo)\{Yk<Xi}]
= E
< XuX,] \{n < Xi}]
[e [b{XuXo)\n
= E[w{XuXi)\{n<X,}],
where the
equality follows from the law of iterative expectations and the second from
first
the definition of
the seller
is
W
.
On
the other hand, without reporting Xo, the expected revenue for
k times
E
If
we can show
[1982,
Lemma
W{x,x) >
that
Note
2].
W
= E
[b{r,
= E
[w{x,
= E
[e [v\Xo, Xi
=
Now we
we
b[x) then
We
are done.
Xo)\Yk <
strategy b{z]
Thus W{z,x)
•),
Xq but
his
at 2
=
a;
will
Milgrom and Weber
Xi -
x]
= x,n =
will
x]
i]
in <x,Xi =
x]
b{x).
after observing
optimal choice
=
will utilize
Xo)|n = x,Xi =
X, x;
v[x,x,x)
X,
claim that W{x,x) < b{x) implies dW{x,x)/dx
prior to learning
•
that by the law of iterative expectations,
first
{x, i)
[6(Xi)|{n < :^i}]
Xi =
he z
=
have to satisfy the
x,
>
b {x).
Note
first
that
if
bidder
1,
were to commit himself to some bidding
x, since b{x;xo) is
first
order condition
optimal when
Xq =
xq.
7
Appendix
31
Then
h{x)
=
(v
+
i,i,i)-6(z))
<
where the
W{x,x) <
The
first
equality follows from (6), the
b{x),
—-rdy
vi{i,i,y)-—
inequality follows from the hypothesis that
first
Proof of Theorem
4:
Milgrom and Weber
[1982,
Lemma
fact that
W2{x,x) >
when bidders
0.
I
2].
Let P"(z) denote the expected price paid by bidder
uniform-price auction, conditional upon winning,
use strategy 60 and
1
in
Xi =
a
x.
is
P"(z)
^ /^o(y)|
L
If
/
and the second inequality follows from (25) and the
assertion then follows from
That
rk{x\x)
+
{v{x,x,x)-W{x,x))
dy
vx{x,x,y)
/
Jx
i'k[^\x)
dy
'""'--'
'-'
^7^'ili^"when bidders
P'^{x) denotes the corresponding expected price
use strategy 6 in a discrim-
inatory auction then
/^(x)
=
h{x)
(27)
i,
X
(t^'(y,y,y)
My)
+ 77^)d^yix)
,
bo{y)dL{y\x)
where the
(14).
first
(28)
equality follows from using integration by parts on (7)
Note that L{u\x) and ^*rr
increasing, the proof
is
complete
in the sense of first-order, that
are probability distributions on
if
is,
we can show
^*)^l|
<
L[u\x), Vy.
Jx ^k[s\x)
we have
,
.
r
[z, i]-
Then
since 6o(y)
is
that jr^/^LS stochastically dominates L[u\x)
Since
J,
and the second from
n
f k{s\x) ^
Appendix
7
32
Therefore
Fkiy\x)
_
„
Proof of Proposition
where
bj,
t
=
6,
=
j
1,2,
that
>—>
[x,x\
:
.
.
,
.
i,
bidder
price that bidder
Lemma
6,
2.
I
We
show that
will
bidder j, uses strategy
if
market buyers believe that each bidder
t
uses strategy
has an incentive to deviate from bi[Xi). In fact
1
I's profits are
faces
=
Si
Since
fk{s\s)
Let (61,62,. ..,6„) be a candidate for Nash equilibrium
4:
resale
1
r
A(«l^)
'FITm'^^^
Fk[s\x)
Hy\x)
are strictly increasing.
n then bidder
when Xi =
The
5R
and the
2,3, ... ,n
/
jy
,
=
where the inequality follows from
r
,
P{"^^
Fk{x\x)
we
minimized at a bid of 61(1).
max{62(X2), 63(^3),..., 6„(X„)}.
Xi and Bi
are strictly increasing,
and Bi
are strictly affiliated
is
atomless.
if
We
bidder
1
6 is*^
r''{b-\b),Bi)
The expected
6,-,
show
is
assume, for simplicity, that By has a density function. The expected resale price
wins with a bid equal to
will
profit for bidder
n"(x'|z)
1
=
if
=-E[v\Xy =
b-'{b),Bi]
Xi = x and he bids
6i{i')
is
E[(r"(x',5i)-Bi))l^,^(^,)>^,j|Xi
=
x]
^'"^^\r"{x'J)-0)g{^\x)d0,
h
b
where
Xi =
b
X.
=
min{62(i),63(z),
The
,
.
.
6„(z)},
and
g{-\x)
is
the conditional density of Si given
first-order necessary condition for 61 to be
^5^^!
Ox'
,
'x'
where
.
=x
an equilibrium strategy
is
={r-{xM{x))-b,[x))g{h,{x)\x)+ r^'\ux,0)g{0\x)d0 =
O^
Jh
r" (i,^) denotes the derivative of r" with respect to its first
argument.
'^Here we assume that the identity of the winning bidder is also disclosed. In our earlier analysis, since
restricted attention to symmetric equilibria, such an assumption was not necessary.
we
Appendix
7
Let
i'
>
33
X.
Then by
strict affiliation
=
(r"(i',fci{x'))-6i(x'))g(fci(x')Ix')
<
g{h,{x')\J) ^(r"(x',fci(x'))
-
+ |^''*'Vr(^'.^M/3kV/3
6i(x'))
+
1^'''^
^
rr(x',^)
ci^
^^^|^J,^^|^^
g(6i(x')|x') an(x'|x)
ax'
<,(6i(x')ix)
Similarly,
we can show
<
that, for x'
x,
ox'
Thus
for
any x £
[x,
x],
n"(x'|x) achieves a global
Proof of Theorem
Given that bidders
5:
n"(6|x)
where /t(y|x)
is
3,...,
2,
^'\o"(x,y)
Proof of Lemma
4:
From
6-(n)
if
I
x!
we can
rewrite (21) as
if
the definition of h'
is
x >
y.
we
is
positive
and only
he uses the strategy
6*.
have,
(29)
Since, by strict affiliation, v"
in (29)
maximized when he wins
bidder Vs profits are maximized'
6',
=
- t)"(y,y))A(y|x)dy,
both arguments, the integrand
I's profits are
at x'
n use
the conditional density of Yk given Xi.
strictly increasing in
Thus bidder
f
=.
minimum
if
if
if
and only
if
{Xi > Vt}. Therefore
I
V*
7^ i,
= E[E[y|xi>n,n,p]|xi = n,n]
>
E[E[i/|xi
= n,n,p]|A-i = n,n]
= E[^|xi = n,n],
where the inequality follows from
strict affiliation,
and the equality from the law of
expectations. Note that the strict inequality above
zero probability event.
I
Proof of Lemma
Let
affiliated
Then
random
5:
variable,
b'
and
'*If
an equality when Y^
be the equilibrium bidding strategy
let 6*
when
=
x,
iterative
which
is
a
P is a strictly
P is constant.
be the equilibrium bidding strategy when
since
E[E[vlxi > n,n]\xi,n] =
first
is
P
is
independent of Xi, or
if
an equilibrium.
is no post-auction public information, then C" is constant in its
an expected profit greater than zero. However, (b',...,b') remains
there
argument and no bid gives bidder
t
ny\xi > n,n],
7
Appendix
34
we have
=E[E V\Xi>Y,,Y,]\Xy =
= E[y Xi>x,Yk = x\
= E[V^ Zk+i =x],
b;{x)
where the
Next,
x,Yi,
=
x]
(30)
last equality follows since the signals are identically distributed.
when x
y^x,
= E{E[v\Xi>Y,,Yk,P]\Xi =
b'{x)
= E[B[v\z,+ uP]\Zk =
x,Y,
x,Z,+ i
=
=
x]
x]
= B{E{v\Zk+uP]\Zk+i=x]
= E[v\z,^i^x]
=
where the
first
equality follows since the signals are identically distributed, the inequality
from
strict affiliation, the
from
(30).
becomes an
When
x
equality.
=
second equality from the definition of
x,
which
Thus the
Proof of Theorem
to
K{x),
6:
is
Zic,
and the
last equality
a zero probability event, the above strict inequality
assertion of
By symmetry,
lemma
is
proved.
I
the expected revenue for the auctioneer
n times the unconditional expected payment
of bidder
1.
Let
.ff"
and
R"^
is
equal
denote the
expected revenue of the auctioneer under uniform-price auction and under discriminatory
auction, respectively.
Then
R"
=
nxE[6*(n)l{;^.>yJ
= kxE[b-{Y,)\Xi>Yk],
R'
= nxE[6(Xi)l{;e.>yJ
= kxE[b{Xi)\Xi>Yk].
Note that the
total unconditional expected profits for bidders in equilibrium for the uniform-
price auction
and
for the discriminatory auction are, respectively,
nxE[(f"(n,P)-6-(n))l(x,>n}]
=
A:
xE
\v\Xi > Yk] - R",
E
,
Appendix
7
35
and
=
n X
E
[{r'{X,,Yu..
A:x
E
[k|Xi >
,¥,,
n] -
P) - fc*(^i))l{x,>n}]
i?"^.
Thus, before bidders receive their private information, the two auctions are constant-sum
games between the auctioneer and the bidders, with
total payoff equal
tokxY, \v\Xi '>Y]A.
Putting
f"(n) =
EfT/|xi>n,nl,
easily seen that
it is
E
[f''[Yk,p)
- f"(n)|^i >
n,nj =
o,
and hence
Efr"(n,p)-f"(n)|xi>nl=o.
Thus
R°
= kxY.
[f"(n)|xi >
n] >
i?"*
=
A:
[mx^Ixi > n]
X
since the (unconditional) expected profit of a bidder in a discriminatory auction
strictly positive
by
R
[z,i]
X
[p,p],
is
We
strictly positive.
then .R"
>
R*^.
will
show that
P
is
finite.
if
>
f"(n,p)-M{p-P)
>
f«(n)-M(p-p),
from the assumption of
strict inequality follows
Then
df^[y,p)/dp <
First note that by affiliation df''{y,p)/dp
f"(n,P)
where the
always
=
[R" —
for all
y,p G
strict affiliation.
For ease of exposition, we assume that the support of
)/[k[p — p))
is
>
M
0.
strict affiliation.
f"(n)-f"(n,p)<(ir-fl'^)A,
and thus
= -E[f"(n,p)|xi = n,n]
-b'in)
<
[R" -R')/k~f"{Y,).
Taking expectation of the above expression conditional on {Xi > Yt} gives
^" =
it
X
E
f6*(yfc)|A'i
>
Yk]
>
R'^,
M
Thus
Hence
,
7
Appendix
36
which was to be shown.
Proof of Theorem
{V ,Xi,X2,
esis that
D = kxB
I
From Milgrom and Weber
7:
,Xn) are
.
= n,n]
[e [^|i-i
strictly afiRliated,
>
\xi
n]-kxE
it
(1982a,
Theorem
[v\xun =
[e
15)
and the hypoth-
follows that
Xi] - j{Xi)\xi >
n] >
o,
the theorem
is
where
r L{u\x)dt{u),
J{x)=
and where
Let
t{u)
and L{u\x) are as defined
M = D/[kE[Xi\Xi
we
true. First
from
recall
>
Yk]).
Lemma
in (8).^^
We now show
that with
M as defined,
4 that, with probability one,
6*(n)>E[y|Xi = n,n]Therefore,
E[6*(n)i^i > n] > B[E[v\x,
= n,n]i^i > n]
and
k X E[6*(n)|Xi
>n]-kx E[E
[v\XuYk = X,] - J{X,)\Xi >
Y,]
> D.
(31)
Next note that
E [v|Xi,n =
where
A'l]
- J[X,) =
h
j/i,
- K{Xi),
defined in (7) and
6 is
and where h{u) and L{u\x) are
dr'^{x,
fc(Xi)
.
.
.
,
yk,p)/dx <
M
/*(u|u)
Next the hypothesis that
as defined in (8).
for all i, yi,
.
.
.
,
yjk,p implies that
MXkX
k X K{x) <
x.
Hence
Jfc
X
E
> n] < D.
[ii'(A'i)|i'i
(32)
Substituting (32) into (31) gives
fc
X
E
which was to be shown.
'''Note that to prove this
[6*(n)|A-i
> n] >
p{z)
>
a(z) for
all
z
>
X
E
[6(Xi)|xi
> n]
I
we
also need a technical
p{z) and ct{z) be differentiable functions for which
Then
fc
i.
(i)
lemma that is slightly different from Lemma 6: Let
> u(i) and (ii) p{z) < <t(z) implies p'{z) > (7'{z).
p{z)
The reader should convince
herself/himself that this
is
indeed true.
8
References
37
8
1.
S.
Bikhchandani and C. Huang (1988), Auctions with resale markets:
Treasury
2.
bill
markets, working paper,
E.
Commack
Treasury
bill
4.
M. DeGroot
5.
W.
A
model of
UCLA.
A. Brimmer (1962), Price determination
Review of Economics and
3.
References
in the
United States Treasury
bill
market,
Statistics 44, 178-183.
(1986), Evidence on bidding strategies and the information contained in
auctions, working paper. University of Chicago.
Optimal
(1970),
Ericson (1969),
A
Statistical Decisions,
note on the posterior
mean
McGraw
Hill,
New
York.
of a population mean, Journal of
the Roysil StatisticeJ Society 31, pp. 332-334.
6.
M. Friedman
7.
H. Goldstein (1960),
Political
8.
R.
(1960),
Economy
McAfee and
J.
A Program
for
Monetary
The Friedman proposal
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New
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York, 63-65.
Treasury
bills.
Journal of
70, 386-392.
McMillan (1987), Auctions and bidding. Journal of Economic
Lit-
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P.
Milgrom (1987), Auction
theory, in
Advances
of the Fifth World Congress of the Econometric
Cambridge University
10. P.
Milgrom and
J.
in
Economic Theory, Proceedings
Society, edited by
Truman Bewley,
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Roberts (1982), Limit pricing and entry under incomplete
infor-
mation: an equilibrium analysis, Econometrica 50, 443-459.
11. P.
Milgrom and R. Weber (1982a),
A
theory of auctions and competitive bidding,
Econometrica 50, 1089-1122.
12.
P.
Milgrom and R. Weber (1982b), The value of information
in a sealed-bid auction.
Journal of Mathematical Economics 10, 105-114.
13.
A. Ortega-Reichart (1968), Models of Competitive Bidding under Uncertainty, Ph.D.
dissertation, Stanford University.
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References
14. J.
38
Parsons and A. Raviv (1985), Underpricing of seasoned issues, Journal of Financial
Economics 14, 377-397.
15. J. Riley (1988),
Ex post information
in auctions,
forthcoming
in
Review o{ Economic
Studies.
16. J.
Tiemann
(1985), Applications of auction
games
in
mergers and acquisitions: The
white knight takeover defense, mimeo. Graduate School of Business Administration,
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V. Smith (1966), Bidding theory and the Treasury
nation increase
18. R.
bill
prices?
bill
Review of Economics and
auction: Does price discrimi-
Statistics 48, 141-146.
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ence 15, 446-448.
19.
R. Wilson (1988), Strategic analysis of auctions, working paper. Graduate School of
Business, Stanford University.
V(^l
<^^\
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