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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
IDENTIFICATION OF
STANDARD AUCTION MODELS
Susan Athey, MIT
Philip A. Haile, Univ of Wisconsin-Madison
Working Paper 00-18
August 2000
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
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MASSACHUSETTS INSTITUTE
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OCT
2 5 2000
LIBRARIES
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
IDENTIFICATION OF
STANDARD AUCTION MODELS
Susan Athey, MIT
Philip A. Haile, Univ of Wisconsin-Madison
Working Paper 00-18
August 2000
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Network Paper Collection at
http://papers.ssrn. com/paper. taf?abstract_id=XXXXXX
Social Science Research
Identification of
Standard Auction Models
Susan Athey and Philip A. Haile*
This version: August
8,
2000
Abstract
We present new identification resiilts for models of first-price, second-price, ascending (English), and descending (Dutch) auctions. We analyze a general specification
of bidders' preferences and the underlying information structure, nesting as spe-
the pure private values and pure common values models, and allowing
both ex ante symmetric and asymmetric bidders. We address identification of a
series of such models and propose strategies for discriminating between them on
the basis of observed data. In the simplest case, the symmetric independent private values model is nonparametrically identified even if only the transaction price
from each auction is observed. For more complex models, we provide conditions
for identification and testing when additional information of one of the following
types is available: (i) one or more bids in addition to the transaction price; (ii)
exogenous variation in the number of bidders; (iii) bidder-specific covariates that
cial cases
shift
the distribution of valuations;
object sold.
common
results include
(iv)
new
the ex post reahzation of the value of the
tests that distinguish
between private and
values models.
Kejnvords:
common
Our
auctions, nonparametric identification
values,
and
testing, private values,
asymmetric bidders, unobserved bids, order
statistics
M.I.T. and NBER (Athey) and University of Wisconsin-Madison (Haile).
We are grateful to Jerry Hausman, Guido Kuersteiner, Whitney Newey, Harry Paarsch, Rob Porter, Susanne Schennach, Gautam Tripathi, and
seminar participants at the Clarence
Tow Conference on Auctions
(Iowa) for helpful discussions. Astrid Dick, Grig-
ory Kosenok, and Steve Schulenberg provided excellent research assistance.
NSF grants
Web addresses:
Financial support from
SBR-9631760 and SES-9983820 (Athey) and SBR-9809802 (Haile) is gratefully acknowledged.
http://web.mit.edu/athey/www/ and http://www.ssc.wisc.edu/~phaile/.
Introduction
1
This paper derives new results regarding nonparametric identification and testing of models of
price sealed-bid, second-price sealed-bid, ascending (English),
The theory
literature has focused
but not the values of his opponents.
the object
is
spread
among
participation on revenues,
We
may be independent
depend
critically
object,
values models, information about the value of
may be symmetric
classes of models, bidders
across bidders or correlated.
market design, the optimal use of reserve
issues, including
and on the
common
In
in these
knows the value he places on winning the
Within these
bidders.
asymmetric, and information
and descending (Dutch) auctions.
on several alternative models of the economic primitives
auctions. In private values models, each bidder
first-
prices,
and the
A
or
variety of policy
effect of increased
bidder
on which of these models describes the environment
demand and information
specific distributions characterizing the
structure.
consider a general specification of bidders' preferences and information, nesting as special
cases the pure private values
and pure common values models, and allowing correlated private
information and bidder asymmetry.
models and propose strategies
We
address econometric identification of a series of these
for discriminating
between them. Hence, our results address key
demand
empirical challenges facing researchers hoping to evaluate the structure of
at auctions
and
to use this information to guide the design of markets.
The
price
is
types of data available from auctions vary by application. In most cases, the transaction
We
observed.
more bids
numbers
consider four additional types of data that might be available:
in addition to the transaction price;
of bidders;
(iii)
(ii)
auctions
we may
—in
bidder-specific covariates that shift valuations; (iv) the realized value of
In
many
descending auctions losing bids are not even made.
lack confidence in the interpretation of losing bids even
auctions, losing bids
In oral "open outcry"
when they
In contrast, in sealed-bid auctions and ascending "button" auctions (Milgrom and
the interpretation of losing bids
variation in the
number
is
of bidders
clear
may
and records of
arise
when
all
bids
al.
It
may
also arise
by design
may be
are observed.
Weber
available.
(1982)),
Exogenous
auctions are held in diff'erent locations,
internet auctions are conducted over different lengths of time, or
agency) restricts entry.
one or
bids from auctions with exogenously varying
the object for sale or, more generally, auction-specific covariates.
are not recorded
(i)
in field
when the
seller (e.g.,
when
a government
experiments (Engelbrecht-Wiggans et
(1999)). Bidder-specific covariates such as firm size, location, or inventories are often observed,
particularly for government auctions, as are auction-specific covariates such as the appraised value
or other characteristics of an object for sale.
several important
The
realized value of the object sold
is
observed for
government auctions, including those of offshore mineral leases (Hendricks and
Porter (1988)) and timber contracts (Athey and Levin (2000)).
In other cases resale prices can
provide measures of realized values
Most
McAfee, Takacs, and Vincent (1999)).
(e.g.,
prior research on identification of auction models has focused
private values, under the assumption that
mapping between observed bids and the
mapping can be used
and Vuong
to-last bidder
(if
Theory predicts a one-to-one
When
latent private values.
with
all
bids are observed, this
to infer the distribution of values from that of the bids (Guerre, Perrigne
mapping from
the identity function, simplifying the problem. However, while the econometrician
is
may sometimes
bids are observed.
first-price auctions
In second-price and ascending bid auctions the equilibrium
(2000)).
values to bids
all
on
observe
all
bids in second-price auctions, ascending auctions end
when the
next-
drops out. Hence, the winning bidder never reveals her value, implying that at best
exit prices for all losing bidders are observed) the econometrician observes all but the highest
value.
Omitting even one order
distribution of values. Similar but
Our main
results
nonparametric identification of the
statistic creates challenges for
more
difficult issues arise in
common
values models.
demonstrate how different data configurations, characterized by additional
information of types (i)-(iv) described above, can be used for identification and testing of alternative
models.
Due
to the prior attention to first-price auctions (see, e.g., the recent survey
and Vuong (1999)), our primary focus
most common auction forms
(
Lucking- Reiley
(2000)).-^
on recent work on
not
all
is
on second-price and ascending auctions. These are the
in practice, particularly
with the rising popularity of internet auctions
However, the ideas we develop
first-price auctions, yielding identification
and testable
restrictions even
when
bids are observed, including the special case of a descending auction.
dent private values (IPV) model
is
identified
consider a richer data configuration, where
enables testing of the symmetric
IPV model. However, we
trary correlation
among
when only the
IPV model and
if
even one bid
In spite of this lack of identification,
when
shown that
that there
there
is
aids identification
common
is
observed.
is
We
then
observed. This
and testing of the asymmetric
arbi-
unobserved in each auction (always the case
we show that the
is
for
not identified from observed bids.
unrestricted private values model can be
Laff^ont
and Vuong (1996) have
values models are observationally equivalent under the assumptions
no reserve price and the number of bidders
^The prevalence
is
more general model allowing
exogenous variation in the number of bidders.
private and
is
transaction price
also establish a negative result for a
values:
show that the symmetric indepen-
first
more than one bid from each auction
ascending auctions), this unrestricted private values model
of ascending auctions
is
well
known.
resources such as radio spectrum (Crandall (1998)).
who
enable us to build
for these auctions also
Beginning with second-price and ascending auctions, we
tested
by Perrigne
is fixed.
Exogenous variation
in the
number
Second-price auctions have been used to allocate public
In ascending auctions, the use of agents
bid according to pre-specified cutoff prices results in an auction
game
(human
or software)
equivalent to a second-price sealed bid
auction. Bajari and Hortagsu (2000) have argued that due to other features of bidding at online auctions, these are
best viewed as second-price auctions.
of bidders, however, provides a promising avenue for testing because the winner's curse (absent in
private values auctions) causes a bidder's willingness to pay to decrease in the
We
number
of opponents.
use this fact to develop nonparametric tests of private values models that can be used
when
two or more bids are observed from each auction.
Next, we consider a potential remedy for the non-identification result for the unrestricted private
values model, relying on bidder-specific covariates.
(Heckman and Honore
(1990)).
These
(1989),
Han and Hausman
literatures consider situations in
a random sample
is
We
build on the literatures on competing risks
(1990)) and the
Roy model (Heckman and Honore
which the maximal or minimal realization from
observed; they show that covariates with sufficient variation can be used to
These results cannot be applied directly to
recover the underlying joint distribution of interest.
second-price and ascending auctions
when only
the transaction price
is
reveals the second-highest realization rather than the highest. However,
observed because this price
we show that these
results
can be generalized, implying that bidder-specific covariates can give nonparametric identification
in
asymmetric private values models (without restriction on the correlation among bidder values)
even
if
only the transaction price from each auction
Our
fourth approach
is
is
observed.
to use ex post information about the value of the object sold.
This
idea has previously been exploited by Hendricks, Pinkse and Porter (1999) in first-price sealed-bid
auctions.^
We show
can be used
in cases
that ex post value information
for identification
(or,
more
generally, auction-specific covariates)
and testing outside the pure common values model they study, and
where some bids are unobserved.
Finally,
we
consider first-price auctions in which
some bids
and Vuong (1995) have shown that the symmetric IPV model
are unobserved. Guerre, Perrigne,
is
identified
from observation of the
transaction price alone in first-price auctions; however, existing results for more general models
have relied on observation of
is
identified
all
bids.
We
show, for example, that the asymmetric
from the transaction price and identity of the winner. Indeed,
and testing approaches based on observation
of
all
IPV model
our identification
one bid or two bids from each auction extend to
first-price auctions.
Our work
contributes to a growing literature on structural estimation of auction models.^
of this literature has focused
Laffont, Ossard
(1997), Deltas'
Most
on the IPV paradigm. Paarsch (1992b), LafTont and Vuong (1993),
and Vuong (1995), Donald and Paarsch (1996), Baldwin, Marshall, and Richard
and Chakraborty (1997), Bajari (1998), Donald, Paarsch and Robert (1999), Hong
See also Hendricks and Porter (1988) and Athey and Levin (2000),
who
test implications of
common
values
models using ex post values.
'
See Hendricks and Paarsch (1995), Laffont (1997), Perrigne and Vuong (1999) and Hendricks and Porter (2000)
for surveys of the recent empirical literature
on auctions.
and
ric
Shum
(1999),
and Haile (2000) have
all
proposed estimation techniques relying on paramet-
distributional assumptions for identification.
Nonpar am etric approaches have been proposed by
Guerre, Perrigne, and Vuong (2000) for first-price auctions and by Haile and Tamer (2000) for
ascending auctions. Only a few papers, including Paarsch (1992a),
1999), Hendricks, Pinkse
estimation outside the
Vuong
(1998,
and Porter (1999), Bajari and Hortagsu (2000) have considered structural
IPV paradigm. Each
sumptions or addresses only
proposed tests
Perrigne and
Li,
where
first-price auctions
for the winner's curse
(1992a), Haile, Hong, and
of these either relies on parametric distributional as-
Shum
all
bids are observed.
Other authors have
based on variation in the number of bidders
(2000),
(e.g.,
Paarsch
and Bajari and Hortagsu (2000)), although these either
require parametric distributional assumptions or observation of
all
bids.
No
prior
work has consid-
ered nonparametric identification and testing of the standard alternative models of ascending and
second-price sealed-bid auctions.
identification for cases in
To our knowledge, the only
prior
work addressing nonparametric
which some bids are unobserved applies to symmetric IPV
auctions (Guerre, Perrigne, and Vuong, 1995).
Nonparametric
first-price
tests of alternative valuation
and
information structures have been considered only for only a few cases.^
The remainder
of the paper
is
organized as follows.
We
first
describe our general structural
framework and review the equilibrium predictions of the theory of ascending and second-price
auctions.
In Section 3
we consider
then takes up the case of
common
identification
Section 5 extends
values.
auctions in which some bids are unobserved.
bidder uncertainty regarding the
price.
2
We
conclude
number
and testing of private values models. Section 4
many
of our results to first-price
Section 6 discusses the robustness of our results to
of opponents they face
and to the
seller's
use of a reserve
in Section 7.
The Model
Our model
is
based on that of Milgrom and Weber (1982).
indivisible unit, with
n >
common knowledge and
2 risk-neutral bidders.
there
is
no reserve
price.
U^
Other
tests for
common
reserve prices, and Haile and
Below we
We
consider an auction of a single
In the basic model, the
Each bidder
z
=
1,
.
.
.
number
of bidders
n
is
,n would receive utility^
= V + A^
values are considered in Hendricks, Pinkse and Porter (1999), which requires binding
Tamer
IPV assumption for Enghsh auctions.
depend on auction- and/or bidder-specific observables.
(2000), which proposes a test of the
will generalize this structure to allow {/, to
from winning the object, where the random variables
A=
from a joint distribution -Fa,v(0-^ Let Fy
and Faj(-) denote the marginal distributions
A, and Ai,
of V,
convenience we
let
Each bidder
X=
{Xi,
.
.
.
The
respectively.
Fjj
z's
and
(•)
{),
Fa
distribution of
F[/. (•)
{),
(^i,
let
Fx
that the joint distribution of (A, V,
.
denote the induced distributions on
X, that
X)
is
is
is
G
(L'^i,
affiliated
of the
.
M
.
with
z,
etc.
When we
tion Fs{-),
Similarly,
we denote by
Fg^
S^^'^^ the j"^
For
and
C/,.
Ui.
Define
paper we assume
exchangeable with respect to the bidder indices, implying
relax the exchangeability assumption,
as asymmetric. For a sample of generic
drawn
{-{v).
t/„)
,
.
that the joint distributions Fa(-), ^u(-):' arid Fx(-) are exchangeable, in which case F^^
for all
are
Fa
denoted
U=
X. For most
() be the joint distribution of
V
An) G K" and
,
.
A conditional on ^ = f
private information consists of a signal
,X„), and
.
random
variables
S =
we
(•)
= Fa
will explicitly refer to the
{Si,
.
.
.
,
(•)
model
Sn) drawn from a distribu-
order statistic, with ^^""^ the largest value by convention.
{) denotes the marginal distribution of
S^^'""'.
Although bidder-specific random variables enter the bidder's
utility additively in
our model,
the multiplicatively separable specification Ui
= V
and Vuong (1999)) can be incorporated
framework by taking logarithms. Indeed, a natural
interpretation of the
well.
To
scale
and
model
is
that U, V, A, and
we
simplify the exposition,
will allow the
Our model
in this
random
X
Ai
(see, e.g.,
Wilson (1998) or
Li,
Perrigne
are in logarithms, with bids in logarithms as
without reference to the logarithmic
will discuss the variables
variables (as well as bids) to take on negative values.^
nests a wide range of specifications of bidders' preferences and the underlying infor-
mation structure. There are two main categories of models:
Private Values (PV): Xi
Common
=
U, Vi.^
Values (CV): (U,X)
are affihated,
and
for all z,j,
{U^,Xj) are strictly
affiliated
but
not perfectly correlated.
We
will consider
only models that
fall
into one of these classes. In the private values model,
bidder has private information relevant to another's expected
values model, bidder j would update her beliefs about her utility, Uj,
to her
own
signal Xj. Thus, there
is
a "winner's curse" in the
In contrast, in the
utility.^
if
common
no
common
she observed Xi in addition
values model:
upon learning
the distribution of A is unrestricted, the random variable V can be normalized to zero without loss of
we include it in our notation to simplify the exposition of several special Ccises of the model.
some cases, we will assume that random variables are affiliated; recall that this condition is preserved by
When
generality;
In
monotonic transformations such as the logarithm.
V=
with
all
Xi
i.i.d.
is
also a (symmetric) private values model, although here the
two private values models
are observationally equivalent.
Following the literature, in private values models we
interchangeably.
will
use the terms "utility;"
"valuation,"
and "value"
more optimistic than her opponents', and
that she has won, bidder j realizes that her beUefs were
her behefs about her
own
utility
become more
"general affiliated values" model of
to rule out pure private values
model allows
Milgrom and Weber
(1982),
CV
model
Common
is
the
and guarantee that the winner's curse
same
for all bidders. Several of
Values (Pure CV):
In addition to the
a special case of the
is
where we have restricted the model
CV
In general, the
arises.
for utilities to differ across bidders; in contrast, in the "pure
the value of the object
Pure
The
pessimistic.
common
values model,"
our results concern this special case.
CV
assumptions. A,
=
so that
0,
=
f/j
V\/i.
Further refinements of the pure
PV
We
model.
However, we
have not assumed
will
CV
model
be discussed
will
affiliation, since
many
Now
in Section 4.
return to the
of our results hold without this restriction.
pay particular attention to three special cases of an
Conditionally Independent Private Values (CIPV): Xi
=
affiliated private values
= V+A^
Ui
Vi,
model:
with {Ai,. .,An)
.
independent conditional on V.^^
CIPV
with Independent Components (CIPV-I): X^
=
Ur
= V+Ar
=
Ui
=
Vi,
with
(yli,
.
.
.
,/l„,
V)
mutually independent.
Independent Private Values (IPV):
ally
We
V=
so that
0,
Xi
Ai
Vz,
with (Ai
,
.
,
.
.
An) mutu-
independent.
assume that the underlying value and signal
common knowledge among
(English) auctions.-'^
the bidders.
distributions,
and the number of bidders
We initially consider second-price sealed-bid
n, are
and ascending
In a second-price sealed-bid auction bidders submit sealed bids, with the
object going to the high bidder at a price equal to the second- highest bid. For ascending auctions
we assume the standard "button auction" model
of
Milgrom and Weber (1982), where bidders
observably and irreversibly as the price rises exogenously until only one bidder remains.'^
the
'"
random
The
variable S, denote the bid
variable
unobserved
V
made by
can be thought of as the random
player
mean
z,
with Hb,{-)
of the private values.
(to the econometrician) heterogeneity in the objects for sale.
=
V Vi.
Below we generalize the
We
let
distribution.
In practice this can arise from
Equivalently, each U, could be the
common component v about which bidders have
CIPV and CIPV-I models to allow private values
a private component Ai and an independent
E[v]
its
exit
sum
of
identical expectations:
that are independent
conditional on a vector of auction characteristics.
" We
use the terms English auction and ascending auction interchangeably.
forms, see Milgrom and
Weber
McAfee and McMillan
For a review of standard auction
Milgrom (1985), or Wilson (1992).
(1982),
This is a stylized model of an ascending bid auction that may match actual practice better in some applications
than others. Ascending auctions with "activity rules" (e.g., the FCC spectrum auctions discussed in McAfee and
McMillan (1996)), for example, are often designed specifically to replicate the button auction. Many of our results rely
(1987a),
^
only on the interpretation of the transaction price from an ascending auction as the realization of the second-highest
of the bids prescribed by the
Milgrom- Weber equilibrium.
In the second-price auction, each bidder
b^
In the
PV
model
=
=
this reduces to bi
that the bid function 6(-)
is
=
b{x,)
ui
=
z's
equihbrium bid when Xi
E[U^\Xi
x,.
=
ma.xXj
CV
In the
=
model,
=
Xi
is^'^
Xi\.
(1)
strict affiliation of
{Ui,Xj) implies
strictly increasing.
Equilibrium strategies are similar in the English auction, although bidders condition on the
signals of opponents
who have
already dropped out, since these are inferred from their exit prices.
Letting Li denote the set of bidders with signals below
b^
for
each bidder
i.
-
E[U^\X^
= X, =
XiVj
Xi, this implies
^{iULi},Xk=
XfcVfc
In the case of private values, this again reduces to
an equilibrium exit price
G
bi
Li]
=
(2)
Ui
=
Xi.
Note that the
ascending auction ends at the price 6("~^-").
Data Configurations
2.1
We
consider data sets consisting of observations from
the same group of
n bidders
We
will
The
•
independent auctions. In the basic model,
participates in each of the
(A, y, X) (conditional on auction-specific covariates,
represents an independent
T
draw from
if
T
any)
is
auctions.
The
joint distribution of
fixed across auctions.
Each auction
this distribution.^^
assume that the following are observed by the econometrician:
transaction price. This
is
i5("~^-") in a second-price or
ascending auction, but
i?("-") in
a first-price or descending auction.
The number
•
of bidders, n.
In addition, the following data elements
may
or
may
not be observed:
• Other bids (or exit prices), in addition to the transaction price, of the form B^'^'^K
The
•
''
identities of
some
bidders, where /('"") denotes the bidder bidding
fi^'"'").
Throughout the paper we focus on symmetric equilibrium when bidders are ex ante symmetric. When we allow
in the private values framework we will focus on the unique equilibrium in weakly undominated
bidder asymmetry
strategies, in
'^
which each bidder bids her valuation.
Such a dataset might
arise as a result of non-cooperative bidding in auctions for
natural resources, where the underlying competitive environment
the contracts are small from the perspective of the bidders.
is
some applications we might expect dependence of
Examining the empirical implications
Here we follow the vast majority of the theoretical and
In
bidders' information and/or willingness to pay on outcomes of prior auctions.
of such dependence
is
a valuable direction for future research.
procurement contracts or
stationary over the relevant time period, and
empirical literature by focusing on cases in which dependence across auctions
is
absent.
•
K sets
of auctions, each with a different
n/^ finite), where the variation in
•
The
ex post reahzation of
distribution of
V
V + A^+ QiiWi)
arise
is
more
or,
of bidders,
and
possibility that the
random shocks
{ni,
.
.
,
.
n^} [K and each
exogenous.
generally, auction-specific covariates
W
=
(iyi,...,W„), where bidder
Wq
affecting the
i's
number
of bidders
is
a pool of potential bidders
and learn X,. There
no reserve
is
noted already, exogenous variation
in
n can
and Xi.
price, so all bidders
place a bid in the auction. This would lead to exogenous variation in the
number
who
who
learn
Xi
of bidders.
As
from participation restrictions by the
also arise
government auctions or
lengths,
where longer auctions may enable more potential bidders to become aware of the
number
is
common knowledge
we may wish
it is
In contrast, in the
to the participants. Unlike first-price auctions, in private-
not important that the number of bidders be
CV model,
the assumption plays a
focus
is
are taken as the
number
of the
3
identification.
number
of bidders,
n G
.
.
.
In Section 6.1,
how many competitors they
bidders.
we extend
face.
However, when we discuss estimation, the relevant asymptotics
of auctions approaches infinity.
{nj,
role.
of
When we
consider auctions with a varying
,nx}, we consider asymptotics as the number of auctions in each
K sets goes to infinity.
Ascending and Second-Price Auctions with Private Values
If all bids are
values
in
on
number
common knowledge among
more important
the model to the case in which bidders are uncertain
Our
sale.
to revisit the hypothesis that the
value second-price and ascending auctions, bidders' strategies do not depend on the
opponents, so
seller
in field experiments) or as the result of varying online auction
considering variable numbers of bidders,
of bidders
receive
Bidders with
(e.g., in
When
=
might vary exogenously merits comment. This could
to the cost of participating, which are independent of Ui
positive shocks participate
by Ui
utility is given
an unknown function.
g^{-) is
(outside the formal model described above) there
if
n G
U.
• Bidder-specific covariates,
The
n
number
is
observed
in a second-price sealed-bid private values auction,
equal to the distribution of bids. However,
if
any bids are unobserved (always the case
an ascending auction), the problem becomes more complicated.
identification in a series of nested
We
proceed by considering
models of private values auctions, showing how richer models
require richer data configurations for identification
section
the distribution of
and
testing.
As discussed above, we focus
in this
and the next on ascending and second-price auctions, and these auction forms are assumed
unless otherwise stated.
8
Symmetric Independent Private Values
3.1
We
is
begin with the symmetric
IPV
model. In this model, the underlying distribution of valuations
nonparametrically identified even when only one bid per auction
tested
more than one bid
if
Proposition
model
1
(i)
is testable if
In the
(i)
The
i*'*
observed or there
IPV model,
either (a)
at auctions with different
Proof,
is
variation in the
is
F\j{-) is identified
more than one
from
is
observed.
number
The model can be
of bidders.
the transaction price,
(ii)
The
IPV
bid per auction is .observed or (b) transaction prices
numbers of bidders are observed.
order statistic from an
i.i.d.
sample of
size
n from an
arbitrary distribution
F(-) has distribution
^'^"M'
example Arnold
(see for
et.
al
(„-,;(,-i),
(1992, p.
Fjj'^ {)
is.
marginal distribution
(ii)
f/,,
Since the observed transaction price
Fu
(•)
Fy
,
is
identified
and either
[/(""•^•"^
Issues for Estimation
3.1.1
Although a
Fy
is
{•)
or t/^""^'") with
^ j,n ^
i
and testing
is
The
of ways.
(3)
when
n)
must be equal
O
for all u.
FJj
this paper,
we do
offer
Because the right side of
(3) is
theorem and continuous mapping theorem
(z) is substituted for F^^''^\z) is
a consistent
{z).^^
testing principle suggested in part
A
and the
method. The empirical distribution
a pointwise-consistent estimator of F^'"' {).
imply that the value of ^(z) solving
Fu
[/(""-^'"^
distributions of different order
beyond the scope of
differentiable with respect to F{z), the implicit function
estimator of
strictly
F\j{-), the result follows.
several observations. First, note that (3) suggests an estimation
function
is
and Testing
analysis of estimation
full
(3)
whenever any distribution
equal to the order statistic
is
completely determines
()
Under the IPV assumption, values of Fu{u) implied by the
statistics ([/(""'")
(3)
Because the right-hand side of
13)).
increasing in F{z), the marginal distribution of
f'"'-(i -')-<«
direct
Kolmogorov-Smirnov
(ii)
of Proposition 1 can
be implemented
test for equality of distributions
ing bootstrap critical values (see, for example,
Romano
(1988)).
in a
number
can be implemented us-
Alternative possibilities include
standard tests of equality of means or quantiles of the distributions. The same approaches can be
taken for a number of the tests we propose below.
^^
If
more than one bid
observed bids using
(3).
is
observed in each auction, more
efficient
estimates can be obtained by incorporating
all
In real-world ascending auctions, the button auction rules
often free-form. Thus,
some caution
is
may
not be used; instead, bidding
warranted when implementing
from the same auctions. ^^ The IPV button-auction model
may be
is
this test using different bids
rejected because recorded bids
understate the true willingness-to-pay of the bidders. In that case, one could compare estimates
obtained from low-ranked bids to those obtained from high-ranked bids to assess the magnitude of
the bias that arises from bidders' failing to bid at prices as high as their true valuations.
Asymmetric Independent Private Values
3.2
If
the exchangeability assumption
values:
relaxed,
is
we obtain a model
bidders' valuations are independent, but
valuations than others.
While the previous
Specifically,
each Uz
(Berman (1963))
for the closely related
Proposition 2 In
Fui
(•)
is
The
also observed.
is
some bidders may be more
is
Fir^{-).
still
be obtained from observation of
the highest or lowest bid and the identity of the
competing
risks
assume
model with asymmetric independent
that each Fu^
(•)
is
continuous.
risks.
Then each
identified if either of the following hold:
(a) In a second-price auction, the highest hid (B^'^'^' )
(h)
have high
following result follows directly from prior results
IPV model,
the asymm.etric
likely to
drawn from an idiosyncratic distribution
asymmetric private values can
a single bid at each auction, as long as this
corresponding bidder
asymmetric independent private
on valuations' being identically distributed across bidders,
result relied heavily
identification in the case of
is
of
and the identity of the winner are observed,
In either type of auction, the lowest bid and identity of the lowest bidder are observed.
Proof. See Theorem
When
7.3.1
and Remark
7.3.1 in
D
Prakasa-Rao (1982).
there are only two bidders, the lowest bid
is
equal to the transaction price, so part (b)
provides identification from the transaction price and the identity of the loser.
bidders, the transaction price
an auction data
set,
is
an intermediate order
statistic,
and
if
With more than two
any bids are unobserved
in
the lowest bids are most likely to be missing. Unfortunately, the approaches
used to establish identification based on extremal order statistics do not extend immediately to
non-extremal order statistics such as the transaction
bid and two
more observed bidder
Proposition 3 In
the interior of
identified if
its
n >
and next-highest
Haile and
the
Tamer
IPV
support and that
bid
(i.e.,
However, with one additional observed
identities in each auction, identification follows.
asymmetric
3, the identities
price.
model, assume that each
\/i,j,
{suppFir^{-)
D
Fi},{-)
suppFu.{-)}
^
has a positive density on
0.
Then each
Fjj.{-)
is
of the top three bidders are observed, and the transaction price
s("~i'"-)
and
i?("~2-")^ are observed.
(2000) propose an alternative approach to inference and testing motivated by this observation.
10
Proof. By assumption we observe
/X
So (where the derivative
exists)
also observed. Similarly, for
is
/
a2
C?XC?y
The
_
T(n-n)
V
>
y
^
we observe
r(n-2:n)
•
fy(n-2:Ti)
y^
two expressions
ratio of these
x,
_
T{n-l:n)
lj{n-l:n)
\
/"CO
=
\
J.
.,_^
^X
lj^i,j,k
is
{i-FuM)fu,{y)fu,i^)
{l-FuAx))fu,{x)Fu,{x)
Taking the
x approaches y from below
(x)
f
yields ^'^
.
,
which uniquely
,
D
identifies F„^.(-).
Conditionally Independent Private Values
3.3
If
limit as
the
IPV model
rejected,
is
correlated.
We
form of the
CIPV model
is
it
natural to consider the alternative that the private values are
begin with a model where the correlation
or the
more
restrictive
CIPV and CIPV-I models impose significant
private values model. This question
in the auctions literature (see, for
theorem states that any
sequences.^* Hence,
without
infinite
if f/i,
loss of generality.
independence
is
.
.
.
,
is
among
private values takes the special
CIPV-I model. ^'
It is
natural to ask whether the
restrictions relative to the (affiliated
of independent interest since these are
example,
and Vuong
Perrigne,
Li,
and exchangeable)
common
assumptions
The de
(1999)).
Finetti
exchangeable sequence can be represented as a mixture of
C/„ are
exchangeable for
all
However, because we consider a
a restriction on the private values model.
CIPV
n, the
finite
Even
if
assumption can be made
number
U
is
i.i.d.
of bidders, conditional
affiliated
and exchangeable,
conditional independence does not necessarily hold (Shaked (1979)). However, as an approximation
of arbitrary affiliated distributions, even the
it
more
can always match the mean and variance of
restrictive
CIPV-I model may
Ui, as well as the covariances
fit
reasonably well:
among
{Ui,Uj), and
it
allows partial flexibility for higher moments.
" To
understand how CIPV-I
differs
from CIPV, notice that CIPV-I rules out the possibility that the distribution
of Ai depends on V.
'*
all
More
7n>
1,
{X„}~=i are exchangeable random variables, there exists a a-algebra G of events such that
Pr(Xi < xi,
,X„ < x„) = J TYJli^'^i^J < Xj\g)dPi{g). See, e.g., Chow and Teicher (1997).
precisely, if
.
.
.
11
for
Proposition 4 The CIPV-I model (normalized so
that E[At]
of any exchangeable and affiliated distribution -Fu(-)the first through third
Proof.
=
However,
can
0)
it
places a testable restriction on
moments.
First, observe that the
CIPV-I structure induces
same
for all choices of
and Fv{-) such
find distributions Fa(-)
var(f/,)— cov(t/i,
t/j)
By
0.
=
E[V]
E[Ui\, var(y)=cov([/i,
=
=
E[V]E[A'^,]
E[U^]
{var{U^)
if all
-
analogous to an
tification results
and var(ylj)=
This
possible be-
is
cov{U^, Uj))
all
Bids
in
a Second-Price Auction
is
possible (up to a location normaliza-
Vuong
bids from a second-price auction are observed. Li, Perrigne, and
CIPV-I model
identification of the
is
f/j),
D
This section shows that identification of the CIPV-I model
tion)
Then we can
testable.
Observing
3.3.1
that:
0.
cov{Ui,Uj)>
affiliation,
For the third moments, the CIPV-I model implies that
E[UfUj] - E[U,UjUk]
is
and by
j,
distri-
exchangeability, these
(implying that var((7,) =var{Ai)+va.i{V), as required).
cause vai{Ui) '>cov{Ui,Uj)>
This restriction
7^
i
For a given
affihated utilities.
bution Fu(-); suppose that E[Ui\, var(t/j), and cov{Ui,Uj) are given.
values are the
moments
the first two
fit
in first-price sealed-bid auctions. In the
measurement
i.i.d.
(1999) analyze
CIPV-I model, each Ai
error on the variable V, enabling direct application of iden-
from the literature on measurement error (Kotlarski (1966), Prakasa-Rao (1992),
and Vuong (1998)). In second-price and ascending auctions, observed bids correspond to
Li
izations of order statistics [/('"^
=
yl^-'-")
corresponding "measurement errors"
prior results unless
all
Note, however, that
-I-
V, with j from a subset of
{A^^''^^}
{1,
.
-
.
,n}.
real-
Therefore the
are not independent. This precludes application of
order statistics are observed (which
when the asymmetric model
is
is
impossible in an ascending auction).-'^
identified,
it is
possible to test the restriction
that the Ai are identically distributed across bidders.
Proposition 5 In
of the
random
the
CIPV-I model, assume
variables
V
that\fi the characteristic functions '4)y{-) andipj^^{-)
and Ai are nonvanishing.
If all bids are observed in a second-price
auction, then:
(i)
(ii)
'^
The symmetric CIPV-I model
is identified.
The asymmetric CIPV-I model
Although
it
is
identified
seems plausible that since each
distribution, there
may
do not present such a
yl'-'
") is
if,
in addition, all bidder identities are observed.
an order
statistic of
an
i.i.d.
sample from a
common
be sufficient structure to identify the model from only two order statistics f/""',
result here. It
is
interesting to note that the difference
not identify Fa(-) up to location (Arnold et
al.
(1992, p. 143)).
12
U''^''^^
—
f/^*""-'
=
A''^"^''
-
parent
f/'* "',
we
^(''"' does
Proof. Follows
Vuong
An
the ex post realization of
of
V
V
is
CIPV and CIPV-I models
the bidders. Thus,
V
is
V
ex ante
is
When V
observed, the
is
(common knowledge)
CIPV model
from observation of the transaction
identified
available
if
common knowledge
might be some publicly observed auction-specific covariate, such as
the appraised value of an item up for auction or the
for a construction contract.
is
observed. In a second-price or ascending auction with private values,
equilibrium bidding behavior does not depend on whether or not
model)
and
D
alternative approach to identification and testing in the
among
Li, Perrigne,
(2000).
Ex Post Observation
3.3.2
by
as in Kotlarski (1966), or as applied to first-price auctions
cost of
raw materials
(and thus, the nested CIPV-I
price. It is testable
if
more than one bid
is
observed in each auction.
Proposition 6 In
(i) If
(ii)
one bid
is
the
CIPV
model, suppose that the realization of
Given
(i)
is
observed,
observed in each auction, Fv{-) and Fa{-\v) are identified for
CIPV model
If at least two bids are observed in auction, the
Proof,
V
v,
a^^''^'
=
-u^^'"-*
V then follows from Proposition
1.
—
v
=
b^^'^)
—
is testable.
v. Identification of Fji^{-\v)ior
Under the assumptions
(ii)
all v.
of part
(i),
each observed
a consistent estimate
and standard nonparametric smoothing techniques. With
of Fa.{-\v) can be constructed using (3)
observation of two bids in each auction, two estimates of FAi-\v) can be constructed and their
D
asymptotic equality tested.
3.3.3
In
Auction-Specific Covariates
many
private-value auctions, theory points to
more than one source
values; for example, in timber auctions, the species of timber
However,
some applications
in
variation in private values
and CIPV-I models
is
it
may be
and tract
of correlation in bidder
size are typically observed.
the case that conditional on such variables, the remaining
independent across bidders. Formally, we can generalize the
to allow for a vector of auction-specific variables.
Wo, that
shift
CIPV
bidder values
and are observed to the econometrician.^'^ In a generalization of the CIPV model, we assume
A
is
independent conditional on Wq, and analyze identification of Fa{-\wo), the joint distribution
of
A
given
Wq =
wq.
independent, and that
Note that
A more restrictive model, analogous to CIPV-I, assumes that
Wq afi^ects bidder utilities through an unknown function po(')-
since this structure permits
V
G Wo, we can assume
13
V=
without
loss of generahty.
(A, Wq)
is
Proposition 7 Normalize
A
(i) If
is
V=
and suppose
0,
Wq and
that
independent conditional on Wq, then Fa{-\wo)
is
the transaction price are observed.
identified for all wq.
(li)
If (A, Wo) are independent, Fa{-\wq) has density fA{'\u)o)VwQ,
the
unknown function gQ:supp\WQ]
identified
up
^
b
—
(i)
Apply Proposition
Assume
QoiWo)).
the case).
zs
1
m each auction,
for
each wq.
for simplicity that
Then (where
Wq
the models described in
(ii)
Note that
This identifies
(;o(')
up
to a location normalization.
determined through the relationship
Our approaches
(3),
which
=
and
(ii)
=
6|Wo)
the same
are testable.
Pr(^(-''")
when
this
is
<
not
-g'oi'^o)-
Then,
identifies
F^
('i^t'o)
for
FAi-\wo).
,
<
each wq, Pr(A,
b
—
50(^0))
Following the same
(iii)
and equality can be
D
tested.
and testing of the CIPV and CIPV-I models require potentially
to identification
strong independence and/or observability assumptions.
we assume
affiliation
of observed bids can
simply
let
and exchangeability.
It is
from observation of
bids are unobserved?
testing of the
The
And
is
straightforward to see that for fixed n, any set
all
Thus, the
PV
model
bids in a sealed-bid auction, but untestable without further
is
the unrestricted
PV
model
identified
when some
second, what additional information will permit identification and/or
PV
model?
We
PV
Model
Is
We begin by showing that,
address each of these questions below.
Not
Identified
from Incomplete Sets of Bids
without further information, the unrestricted
an ascending auction, and
result
the CIPV-I model
be rationalized in the private values framework (Laffont and Vuong (1996)):
information. This raises two questions. First,
is
if
the distribution of values be equal to the distribution of bids.
identified
3.4.1
Moreover, even
imphes that we cannot rule out the private values model altogether, even
rejected, Proposition 4
This
o-tc
Unrestricted Private Values
3.4
in
^i, where
the derivatives exist)
approach, the additional bids can be used to identify
is
is
(i)
<
Pr(5(-'"")
a scalar (the argument
is
§'^Y>r{A(r-^)<h-go{wo))
if
+
Fx{-\wq) and go{')
difjerentiahle, then for all wq,
^Pr(AO-)<6-(7o(«^o))
is
go{Wo)
to a location normalization.
(Hi) If at least two bids are observed
Proof,
R
=
and Ui
it is
model
not identified in a second-price auction unless
true even under the exchangeability assumption.. This result
from the literature on competing
distribution of competing risks
PV
is
risks.
Cox
not identified
bids are observed.
a generalization of a classic
is
(1959) and Tsiatis (1975)
not identified from the lowest order
14
all
is
statistic.
show that a
We
joint
show that
this
be extended to cases in which any combination of order
result can
the value distribution
method
be exchangeable, so long as not
restricted to
is
statistics
observed, even
is
when
are observed. Since the
all
we proceed by constructing a
of proof of the prior literature cannot be applied directly,
counter-example.-^^
Proposition 8 Consider
from
the vector of bids
unless
[0,4]" is in
=
Then,
[/(*^""-)
is
l{u, G
[3
-l{u^ G
5^=
G
S*
Si
<
and
- £,3 + e],i G
[2
-
e,
2
+ £],i
e
USk-i
<
some k 6
for
l{u, G
Si}
>
=
US'„_fc
l{n, G
0,
/u(-)
-
[1
=
-
[1
e, 1
and similarly
permutations of the vector
f
''T-
"
l{ui
f
l{ui
I
x)
<
<
<
x,i
all x, all
x,i
=
SG
<
x)
5^, and
l}l{ui
is
>
I,
this result
all
x,i
(1,
.
G 5fc_i}
restrict
U
to
be
J ¥" ^}
'
l,|5„_fc|
.
[4
is
-
is
does not impose
affiliated
=n-
k}
.
[4
+ e], Vz G Sn-k}
- e,4 + ^J, V? G S^^k}
e,4
a probability density, with the
to others. For example,
if
A;
=
n
to the neighborhood of (1,
.
—
1,
,
.
.
prob-
1, 3, 4),
pairs.
.
.
.
,m —
liui
\}
>
x,i
=
m+
,'"m-l,2;,1im-Hl,
1,
.
.
.
,n} \
is
•-
du-m,^n)
J
m^k,
=
m+
1,
.
.
.
,n}c((.
is
.
.
,Um-i,x,Um+i,
); S,£)du_rn
affiliation of
U.
Affiliation is
but not independent,
eis
long as
constructing the counter-example.
15
=
0.
replaced with /u(-)' ^-^d the probabihty
D
observationally equivalent to Fjj{-).
potentially disturbed by small perturbations of the distribution.
we
= k-
l{u, G
c{-;S,£)
1, 2, 4)
,
.
unchanged when /u(-)
distribution based on the density /u(-)
'Note that
l{«z G
+ e], V?
•/u(«l,
= l,...,m-
Thus, ^Pr(y(™'")
is
G Sk-i}
I
/
1/2,
positive
is
=
)[
^n-m/J^^_^y
But, for e
/
\
<
l,|5fe-i|
some regions
from the neighborhood of
^ Pr([/('"^")
=
+7Ssg5'=
/u(-)
ability weight shifts
Observe that
and that /x(-)
=
+ e]yi
e, 1
function c shifting probability weight from
for all
X
{!,..., n} a subset of {U^^'"'^
{!,..., n},!^!
1/2, define c(u; 5, e)
G 5i}
For sufficiently small 7
n
not identified
is
unobserved. Define a set of partitions of bidder indices
{{Si,Sk-i,Sn-k)
for
Fu{-)
not identified in a second-price auction
it is
the interior of the support of
Suppose that
throughout this region.
S''
i^u(") is exchangeable.
m an ascending auction; further,
Suppose that
observed but
and assume
are observed.
all bids
Proof.
PV model
the
a delicate condition, and
The
result can
if it
holds weakly,
it
be generalized to the case where
we take a smooth, "small enough" perturbation when
We
PV
Tests of the Symmetric and Asymmetric
3.4.2
PV
have shown that the unrestricted
Models
not identified in general. Despite this,
is
we now show
that the model has testable implications, even in situations where the underlying value distribution
is
Our approach
not identified.
number
requires exogenous variation in the
as the observation of multiple adjacent bids
—
e.g.,
of bidders, as well
the transaction price i5("~^-"^and the second-
highest losing bid i?^""^-") in an ascending auction, or the highest and second-highest bids in the
The
second-price auction.
Lemma
Consider the
1
of the order statistics
following
PV model.
t/(j-"),
Proof.
When
statistics
can be derived
lemma
The distribution ofU^^~^''^^
m—
=
j
to
I
the distribution Fu(-)
(e.g.,
central to our argument.
is
is
n—
with
1,
from
identified
is
the distribution
m < n.
exchangeable, the following relationship between order
David (1981,
105)):
p.
^fI-\u) + '-Fl;^'-\u) = Ft-'\u).
from the distributions of
So,
U^^'''^\
jj(m-\:n-i) ^^ jj{n~2:n-i) ^gj-Qg
^t^q
=
j
m
—
to
1
abovc formula.
n —
If
(4)
we can compute the
1,
m=
Ti
—
this
1,
distributions of
completes the argument.
D
Otherwise, repeating the exercise gives the result.
The
intuition behind (4)
random) from a
of the
bottom
n
set of
is
When
straightforward.
bidders, in each auction, there
r bidders, and a
^^
is
dropping one bidder exogenously
a - chance that the dropped bidder
chance that the bidder
these probabilities as weights, the distribution of ^('""^i)
of
i7('"+i^"^
and
m
>
PV model
bids i?(™--"),
Proof.
^(m-i:m)
^(^-i^")^
2 bidders and bids
.
,
.
5("-")
.
from
L/^"^")
Lemma
and
r bidders.
Using
a weighted average of the distribution
the transaction price
^^("-i:")
.
_
_
1.
[/(""^"'^
^('"-i-'")
m
from auctions with n >
from aucIn
bidders.
also sufficient to observe B^"^''^' at auctions with
strategies, observing the bids
equivalent to observing the order statistics [/("^"i'"),
[/('"-i:'")
The same argument
^('"-i^")^
.
,[/("-^i^")
.
.
.
using only the order statistics
Under the private values assumption,
that observed directly.
to
—
one
m bidders
the n-bidder auctions.
can compute the distribution of
following
it is
Given the equihbrium bidding
jg
we observe
is testable if
the case of a second-price auction,
and
one of the top n
is
f/^'"^").
Proposition 9 The
tions with
is
is
at
(i.e.,
this distribution
.
and
.
^^("-i:") and
t/(™-i:"^).
[/(™~i-")
to
We
jy^""-^'")
must be the same
applies to the case in which the order statistics
as
L'^f'"'")
D
are observed.
16
This result implies that the
PV
model
testable whenever
is
second and third highest) bids from auctions with n and n
symmetry
that Proposition 9 relies critically on
distribution of bidder values
is
—
1
we observe the top two
bidders, holding
of the bidders,
i.e.,
Balakrishnan (1994),
The
variables,
who
m
C N
Vm C Vn
.
.
Vn, and
.
.
let Yi,
.
.
Fjj'
{)
,Ym be a sample
is
some
If for
such that
n
n >
\VTn\
=
m
Tn,
<
then
n,
if
m
>
we observe
of size
Then
If
m=n—
{u)
1, (6)
= Fy
identities in
Yi,
.
m
.
-
drawn without replacement from {Uj,.
.
.
,Un} using
are exchangeable. Define
,^771
'
.
.
,
.
Ym}
+ -Fu
equals
F^'^
= Fu
(•),
must
satisfy (4).
for r
<n
Since the
this gives
(y)-
(5)
^Ft\u) + ^Ft'-^'Hu) = Ft;-'\u).
(6)
(y^
{y)
(u), this simphfies to
can be tested
directly.
For
m < n—
1,
repeated application of
enables testing
(6)
D
^^As with Proposition
and
.
is testable?"^
of (5).
fit'"^'")
of
the distribution of the rth order statistic from {Ui,i G Vm}- Exchangeability
—^^u
'"
.
.
and bidder
asymmetric) private values model
the
,5("~i-")
bids B^'^~^''^\
bidders and the transaction price
distribution of the rth order statistic of {Yi,
F^
is
2, there is positive probability
implies that the distributions of the order statistics of {Yi,...,!^}
Since
Vn
3 such that the probability that
,Un be the random variables corresponding to the valuations of the bidders in
a discrete uniform distribution.
where
=
with \Vn\
bidders, the (unrestricted,
Proof. Let Ui,
identities of
(4).
identities in auctions with
auctions with
we observe the
exchangeable random variables whose distributions must obey
set of participating bidders is positive.
and bidder
on the assumption that the
following result exploits the arguments in Balasubramanian and
set of
Proposition 10 Take any P„
participation by every
Note
note that by taking random draws from samples of arbitrary random
one can obtain a
the recurrence relation
all else fixed.
exchangeable. However, we can extend the result to the case in which
distributions of private values are completely unrestricted, as long as
the participating bidders.
(or the
9, in
at auctions with
the case of a second-price auction,
m bidders
and bids
B'-"''"\
.
.
17
it is
also sufficient to observe
,B<"^"' from the n-bidder auctions.
all
bidder identities
3.4.3
The
and Testing Using Bidder-Specific Covariates
Identification
are available, identification
literatures
When
provided results for testing but not identification.
last section
on competing
may
also
risks (e.g.,
be possible. ^^ To show
Heckman and Honore
the
Roy model (Heckman and Honore
the
Roy model and
this,
(1989),
bidder-specific covariates
we adapt approaches from the
Han and Hausman
(1990)) and
While Heckman and Honore (1990) point out that
(1990)).
the competing risk model share a similar structure, the relationship of these
model to auctions has not been previously exploited.
These prior
is
Observing extreme order
observed.
which either the lowest or highest order
literatures are tailored to cases in
(minima or maxima) creates a
statistics
statistic
relatively simple
inference problem since the distribution of an extreme order statistic provides direct information
about the joint distribution of values. For example,
Pr(i?(n-)
<
6|w)
= Fa(6 -
Inference from non-extremal order statistics
is
if
Ui
=
gi{Wi)
5i(u;i), ... ,6
more
difficult;
-
+ Ai^
then
ff„(«;„)).
however,
when we observe
either the
highest bid in a second-price auction, or the lowest bid in an ascending auction, existing results
can be applied with only minor modification. Analogous to the prior literature, we also need to
observe the identity of the auction winner
Proposition 11 Consider
Qii^yi)
for
+
Alii; (b)
and
all i,j,
identified,
A
is
the
is
and
Proof. The proof
in
Theorem
12 of
identified
model, normalizing
and
{g\{Wi)
up
,
.
gn{Wn))
.
.
,
for the
bid are observed; or
two-bidder case
Heckman and Honore
on the variation
Examples
Assume
0.
(a) Ui
=
is
R".
Then each
g,(-),
i
=
,n, is
1,
to a location normalization if either (i) at a second-price
is
(ii)
at either type of auction, the lowest
identical to that for the two-sector
The proof when n >
(1990).
in the bidder covariates
of such covariates include the distance
backlog of contracts won
in
2
requires a straightforward extension, which
Proposition 11 does not require exchangeability.
ditions
F =
bid are observed.
Roy model. This
multi-sector
^^
continuously distributed with support R"; (c) (Ai,Wj) are independent
auction, the winner's identity
bidder's identity
PV
asymmetric
(d) the support of
and Fa(")
(loser).
It
W.
given
identical to that for a
we
D
omit.
does, however, impose fairly stringent con-
Our next
from the firm
previous auctions, or measures of
is
Roy model
result relies
on similarly strong
to a construction site or tract of timber, a firm's
demand
in the
home market
of bidders at wholesale
used car auctions.
An
alternative identification result can be given
if
bidder identities are unobserved but there
is
sufficient variation
g{Wi) to effectively "pick" the winner (loser) through the choice of lu. More precisely, if the winner's bid
and limu,j__oo 5i(ift) = — oo at a sealed bid auction, or the lowest bid is observed and Vim^^-^oo g{wi)
in
each
gi{-) i
=
1,.
.
.
,n
is
identified,
and Fa(-)
is
identified
up to a location normalization
18
is
=
observed,
oo, then
conditions on
W,
observed,
possible to uncover the underlying joint distribution of values.
it is
but shows that even
Proposition 12 Consider
only the transaction price (the second-highest value)
if
PV
the asymmetric
model, normalizing
y =
0.
is
Suppose hypotheses
(a)-(d) of Proposition 11 hold and that Fx{-) has a differentiate density. Further assume that for
each
i
Then
=
1,
.
.
ind each
-Fa(")
Proof.
,n, gi{-) is differentiahle, g[{-) is absolutely continuous,
.
Let
=
z,
6
gi{-), i
—
=
1,
.
.
.
and
from the transaction
,n, are identified
= — oo.
limu,-_^_oo5i('u^i)
price.
gi{wi). First, observe that (abusing notation for zero-probability events to
simplify exposition)
n
rj
—
<
Pr(i?("-i^")
6
=
|w)
^ 5]
Pr (A,
>
/
- Pr
Aj
=
{Aj
=
Zi,
z„ A^ <
z^ Vfc
^
Ak <
Zk V/c
^
i,j)
n
= I] X] [Pr [Aj =
2„ Ak < Zk
Vfc
z,
J
)
z^,
j
)]
Then
dwn
dwi
<
Pr(i?("-i-)
(^
V db
6
= -(n -
|w ))
1)
fli-g'.iw,))
f^
i=i
j=i
^/a (a)
Observe that
db
\
Thus (using
=
gn{uin))
dwn
dwi
FA{b - gi{wi), ...,b'
=
lim(,^_oo ^^"'"^(6)
F^ib — g\{wi),
1),
.
.
.
,b
gn{iOn))
j
'
F^~'^'''\b)
— Qniuin))
is
=
= W{-g'j{wj))
/a
J
LL
Y^ ^—
Z-^
]
Qd-
and lim^^(^_^^__^-) FA{b -
identified
(a
gi{w-i),
from the observable Pr(5("~-^-"^
.
.
.,b
< 6|W)
by the relation
FA{b-g\{wi),...,b-gn{wn))
1
—
r
I
n-lj
ment
for
.
.
,b
—
each
gn{wn))
i.
=
FA,{b
—
—
and
g\^w\),...,b
gi{-),
.
.
gi{wi)). Varying b
Then, with knowledge of the
Pr(S("-i
")
<
6
\dwi---dwn
Jw={-oo,...-oo)
sider the separate identification of Fa{-)
.
^",
(-
Given that the composite function F^ib
gi{wi),
=
5i(-)'s,
—
gni'Wn))
is
|w)^
)
d-w.
identified,
we now
,5n(0- Note that limu,_,^(_oo,...__oo) •'^a(^
and Wi then
we can
identifies gi{-).
identify Fa{-) at
part of the proof shows
the w^ identify
FA{b —
gi{wi),
.
.
.
,
6
—
how
"~
Repeat the argu-
any point
(ai,
.
.
.
,
a„)
D
through appropriate choices of b and w.
The main
con-
the distribution of the transaction price and variation in
gn{wn)).
The second
19
part of the proof
show how
to identify
the functions
gi{-),
and then varying
.
^gni-)-
-
.
and
b
The
From
lUj.
strategy in this second step entails taking extreme values of w_,,
a practical standpoint, this strategy has disadvantages, as
W.
heavily on observations in the tails of the distribution of
available to identify the functions gi{-)
if
/(j-")
Pr(/("-"^
z,j,
+ gr{wi) >
Pr {A^
We
However, an alternative strategy
We
briefly sketch this approach.
is
^^
denote the identity of the bidder making bid B^^'-^\ From the identity of the
second-highest bidders, for each
relies
the identity of the highest and second-highest bidders are
observed in addition to the transaction price.
Let
it
Aj
=
/("-i-")
i,
+ gj{wj) >
^fc
=
-f-
j|w)
is
gk{wk)
first
and
and equal to
identified
ij)
VA: 7^
can then define the set
5^ =
|w
:
Next, observe that the fohowing
=
Pr (Aj
For Mv G
Sui,
+ gj{wj) <
Pr
=
Vi,i,Pr(/("^")
(/('^'"^
=
/("
-'•"^
j|w)
identified for each
=
/("^")
b
i,
is
=
i,/("-l^"^
=
z,/("-i^")
=
j
|w)
is
J,
=
Pr(/('^^"^
=
i,/("-i^")
= j|w)|
=
z,/("-i^'')
=
.
{i,j)'-
w)
•
Pr
(/("^")
constant. So for any b and any w'
j |w^
7^
w
in Su,, the
b'
that solves
p^^^(n-l:n)
must
up
<
/(«:")
5^
satisfy gj{w'.)
—
^
^j{n-l:n)
gj{wj)
=
b'
—
b.
^j
\y^)
As long
=
Pr(5("-1^")
<
6',
G Sw}
=
M, varying
as {wj
/("^"^
=z,/("-l^") =j|w').
w
and w'
identifies gj{-)
to location.
Assuming that consistent estimators can be constructed using each
PV
model can be implemented by comparing them.
gies in Propositions 11
and
Corollary 1 Suppose
that the hypotheses of Proposition 12 hold.
is testable if
and
^
12,^^ a test of the
each of the following
is
The argument
follows the proof of
statistics in a
Then
the asymmetric
PV model
observed: (a) the transaction price; (b) either the winner's bid
identity at a second-price auction, or the lowest bidder's bid
non-extreme order
of the identification strate-
Theorem
12 in
and
Heckman and Honore
identity.
(1990), but considers the case of
model with more than two bidders (analogous to more than
2 sectors in the
Roy
model).
We
are
unaware of work developing nonparametric estimation approaches
model based on these
identification strategies.
Doing so
is
for
competing
risks
beyond the scope of the present paper.
20
models or the Roy
Ascending and Second-Price Auctions with
4
Testing Based on Multiple Bids and Variable
4.1
In Section 3.4.2,
we
established that the
ber of bidders (Proposition
9).
is
PV
of Bidders
testable using exogenous variation in the
num-
framework. In the second-price auction, the recurrence
CV
replaced with a stochastic dominance relation under the
distribution of transaction prices from auctions with
ticular
Number
Values
Proposition 9 relied on a recurrence relation between distributions
of order statistics of bids within the
relation
PV model is
Common
n—
1
model. For example, the
bidders stochastically dominates a par-
A
convex combination of bid distributions from n-bidder auctions.
similar result holds for
the ascending auction, where a recurrence relation between means of order statistics from samples
of exchangeable
the
CV
random
hypothesis.
variables implies a strict ordering of the
Both
same combination
from the winner's curse, which
results arise
is
of
means under
more severe
in auctions
with larger numbers of bidders.
Proposition 13 The
with
m >2
bidders
Proof. Assume m.
CV model is
and
testable if
bids B^^~^''^\
=n—
I
(the
.
.
argument
If
=
the transaction price B^"^~^''^> in auctions
,ij("~i-") in auctions with
.
auction, where (recalling (1)) each player
bi
we observe
is
m
bidders.
similar for other cases). First consider a second- price
bids
i
= maxXj =
E\U^\X^
n >
xi]
=
b(x,;n).
instead of b{-;n), the bidders used a symmetric monotonic bidding strategy that did not depend
on n,
(4)
would imply
- Pr(5("-2^") <b) +
"^^—-^ Pr(B("-i^")
n
would be
since bids
fixed
n
<
6)
=
Pr(B"-2^"-^
monotonic transformations of exchangeable
signals,
were also exchangeable. However, when the equilibrium strategy b{-;n)
loss of generality
h)
(7)
implying that the bids
used, taking
z
=
1
without
and exploiting exchangeability, we have
6(xi;n)
with the
is
<
= E[U\\Xi= X2-=xi,Xj <xi,
j
=
?,,..., n]
<
E[Ui\Xi=X2 = xuXj<Xi,
j
=
3,...,n-l]
=
b{xi;n
strict inequality following
—
(8)
1)
from the
fact that
E[Ui\Xi,
.
.
.
,X„] strictly increases
in
each
Xj, due to strict affiliation of {Ui,Xi). Hence, 6(xi;n) strictly decreases in n, implying the testable
stochastic
dominance relation
- Pr(B("-2^") <
n
6)
+ ^^—^ p^/^in-l-n) < ^w p
n
21
,^n-2:n-l
<
^x
/g)
instead of (7).
For an ascending auction,
of generality, suppose
it is
fix
a realization of {Xi,
bidder 2
who
—
bidders have these signals in an (n
.
.
who has
1
l)-bidder auction, bidders
1
and
2.
3,
.
.
.
.
.
,n
-
=
is
n
will
E [U2\Xn = Xi = X2 = X,Xj = Xj, j = 3,
x\x_n)E [E
.
.
.
,x„_i)
is
drop out before bidder
2's exit price
having signal
x.
equilibrium)
.
.
,71
.
-
l]
Xn, Xj=Xj,J=3,...,n-l]\
X,
.
.
,Xn-\. To see
revealing x„. If x„
>
y£
\b+ {X^'"^-''~^^
;
X3,
.
.
.
.
,
.
.
>
X^""^'""^))
observe that
all
if
x„
Xn<x]
=
x, bidder
remaining bidders, including n,
=
in
-EfS^^-^^")]
Xn^l)]
<
drop out before bidder n,
,Xn-i gives the testable inequality
L
by exchangeabihty, Pr(JY'„
this,
x, bidder 2 will
based on an expectation that conditions on
J
since,
2,
.
Taking expectations over Xi,
r^(n-2:n-l)l
^
1 will (in
the expected bid of bidder 2 in an auction where an additional bidder
also included, given the realizations of .Yi,
with
= X2 =
[U2 {Xy
When
6+(x2;x3,...,x„_i).
Here b'^{x2;xs,
n
—
loss
1]]
I
+ FT{Xn <
Without
the higher signal.
,n
.
x.
Bidder 2 then drops out at price
_ ^p^ \Xi=X2=X,Xj=XjJ = 3,...,n-l]
= Ex„ [E [U2 Xn, Xi=X2 = X, X, = x„ J = 3,
> Pv{Xn > X |X_„)
=
,X„_i), with x^""^'""^^
has signal x and bidder
drop out and reveal their signals before bidders
^(n-2:n-l)
.
restriction
+ ^^—^^[fi^""!^")]
n
(10)
D
^.
For the ascending auction, a test of the inequality (10) can be implemented using standard onesided hypothesis tests. Since (9) implies (10), the
Of course, other
following
same
test could
be used
for
a sealed-bid auction.
tests of (9) are possible, including direct tests of the stochastic
McFadden
dominance
relation,
(1989).
Note that Propositions 9 and 13 imply that the observational equivalence between the
and
CV
models noted
variation in the
in Li, Perrigne
number
and Vuong (1999)
is
eliminated
when one
CIPV
observes exogenous
of bidders. ^^ In addition, each of these propositions implies that the test
proposed in the other has power against an important alternative. In particular, the
requires that (9) and (10) hold with equality rather than inequahties.
indicate that the inequality in (9) or (10)
is
Note
PV alternative
also that
if
the data
reversed, this will suggest a violation of one of our
more basic maintained assumptions.
One
bidders
limitation of this result
know the number
Other approaches
is
that,
unhke our models of private
values, the assumption that
of competitors they face plays an important role.
for empirically distinguishing these
models based on observation of
auctions are given in Hendricks, Pinkse and Porter (1999) and Haile,
22
While
Hong and Shum
(2000).
all
this
assumption
bids at first-price
does not seem severe in an ascending auction,
may be
it
restrictive in a sealed-bid auction.
We
return to discuss the extent to which this assumption can be relaxed in Section 6.1.
CV model.
identification and testing of the pure CV model.
Without some additional structure, we are unaware
remainder of this section, we consider
Thus,
in the
4.2
Pure
We begin
First,
Common
of an approach to identify the
Values
by highlighting several issues that
arise
when attempting
from an economic perspective, the scaling of the signals
to identify the pure
X
is
monotone transformations. Given
6(xj)
in
= maxXj =
this normalization (recalling (1)), the
=
Xj.
While the same
x,n]
is
preserved under
=
x.
(11)
an ascending auction,
all
much more complex. Because bidder
can be identified
in
change as the
=x
for all bidders
will limit the set of
data configurations under with any
is
However,
equal to the distribution of X, and thus the distribution of
if
some
We
testing approaches developed for private values models,
pursue
to assume that
ditional
we
^*
on V),
not be independent in the
= V + Ei
Xi
for
CV
some £, (where
this additive structure
Further, with pure
=
identified
might hope to apply the identification and
where the same problem
may
formulation. Further, although
E
common
max^^i Bj
values
=
bidders' first-order conditions. This
is
either independent of
not survive the rescaling (11).
consider two (nested) special cases of the pure
estimates of E[V\B,
is
arises,
and indeed,
However, the assumptions required are somewhat more subtle. For example,
this.
X generally will
X
bids are unobserved, the distribution of observed bids provides in-
formation only about certain order statistics of X.
will
model
ascending auctions.
distribution of bids
we
CV
observed in a second-price auction, the normalization (11) implies that the joint
If all bids are
trivially. ^^
This
this reason,
beliefs
auction progresses, no single normahzation can induce the simple strategy b{x)
in the ascending auction.
is
higher bidders
an ascending auction condition their bids on the drop-out points of lower bidders. For
is
bidder
optimal bidding strategy in a second-price auction
logic applies to the lowest bid in
identification in ascending auctions
is,
a particularly salient normalization of signals satisfies
this,
E[V\X^
With
model.
That
arbitrary.
behavior and utility depends on the information content of the signal, and that
CV
CV
may be
natural
or independent con-
To address
this concern,
model:^^
and observability of ex post
values, the
bi,n\ based directly on the observed bids and
is
V
it
model can be tested by comparing
V
to those inferred from bids using
the analog of the approach followed in Hendricks, Pinske, and Porter (1999)
for first-price auctions.
Each
of these
is
receive signals of the
a special case of the mineral rights model described in Milgrom and
common
value v that are independent conditional on v.
23
Weber
(1982),
where bidders
Linear Mineral Rights (LMR):
stants
M
{C,D) €
Ui
—
V. In addition,
R^ and random
X
variables {E\,.
{V
E,)^
ther, conditional
LMR with
.
.
with {E,V,X) non-negative; or
Vz,
on V, the elements of
E
,En) with joint distribution Fe(-)
= maxj^iXj =
such that, with the normalization E'fyjXj
exp(C)
each n there exist two known con-
for
x, either
= C + D{V +
X^
(ii)
=
x, n]
(i)
Xi
=
E^) Vz. Fur-
are mutually independent.
Independent Components (LMR-I):
In the
LMR model,
(V^,E) are mutually in-
dependent.
The
LMR model is analogous to the CIPV model,
is
analogous to CIPV-
However, the functional form assumptions are potentially more demanding
I.^'^
Perrigne, and
Vuong
(1999) show that several plausible models satisfy
LMR definition is satisfied
C+
To
D\n{yi).
bidders see signals (Yi
if
known constants
dent, and there exist
it
LMR-I model
while the
see this, let
X,
=
has the same information content.
C=
In
\
Fr^7iF
E[t^^
\^i
1
Similarly, the following are
and
{V,
We
and
F
— max^^j Ej\
(C, L>)
G
M
,
,
.
.
exp(C) -Yf; since
The
D—
y„), where Yi
x R4. such that
LMR-I. Condition
=V
E^, (^,
In (£'[y|yi
1]
(b) there are
=
(i)
<x
two bidders, and
Li,
in the
E) are indepen-
maxj^j Yj
=
monotone transformation
this is a
following are examples: (a) f{v)
yi])
=
of Yi,
^^ ^ ^R^ whereby
(V,
E) are log-normal.
j
examples of case
(ii);
(a)
the prior /v(-)
(b) there are
is flat;
two bidders,
E) are normally distributed.
LMR-I model. Our main
begin by analyzing identification and testing in the
that our results for the CIPV-I model extend to the
applies only to second-price auctions where
Proposition 14 In
V'a, (")
.
in this case.
of the
auction, the
the
random
LMR-I
variables
LMR-I model
Proof. Consider case
from which we can
(ii)
infer f
e,.
5 then applies directly. Case
(i)
as with Proposition 5, the result
bids are observed.
all
and Ai are nonvanishing.
is identified
-I-
shows
model, assume that for alii the characteristic functions ipy{-) and
V
of the
LMR-I model;
result
and
If all bids are observed in a second-price
testable.
LMR-I model. By
Hence, each bid
follows in the
(1),
ijf-? ")
each bidder
reveals
same manner
V
z's
-\-
bid
bi is
£(•?").
equal to C-\-D{v-\-e^),
The proof
of Proposition
after taking logarithms.
D
Observing information about the ex post value of the object enables us to identify a more general
pure
common
values model.
prove identification of the
fairly
In particular, our results for the
LMR
CIPV model
model. However, in the case of ascending auctions our results are
weak, requiring that the lowest bid be observed. This restriction
'" Identification
of the
See also Paarsch (1992a),
can be generalized to
is
LMR-I model at first-price auctions has been considerd by Li,
who considers several parametric specifications of the mineral
24
required because after
Perrigne and Vuong (1999).
rights model.
the
bidder drops out, the strategies of remaining bidders depend on B^^''^\ implying that no
first
normahzation yields bid functions equal to the identity function.
Proposition 15 Consider
(i)
LMR
the
model. Suppose that the ex post realization
If the transaction price is observed in a second-price auction, F^{-\v)
to the normalization (11). If
(a) If the lowest bid
is
one additional bid
ofV
and Fv{-) are
is
identified
observed in each auction, the model
is
observed,
up
is testable,
observed in either a second-price or ascending auction, Fe{-\v) and Fy{-)
are identified up to the normalization (11).
Proof.
Consider case
C + D(f + Ci),
reveals v
+
(ii)
in the definition of the
from which we can
e^""-^''^^
infer v
-\-
Ci.
LMR
model.
By
each bid
(1),
hi
is
equal to
Hence, the transaction price in a sealed bid auction
while the lowest bid in either type of auction reveals v
+
e^^'").
In either case,
observation of v reveals one order statistic of E, so identification of Fe(-) and testing then follows
directly from the arguments of Proposition
First-Price Auctions
5
Most
we
when Some Bids
are Unobserved
auctions can be extended to first-price auctions, although in
some
require that the second-highest bid be observed in addition to the transaction price.
well
An
PV
of our results for
D
6.
known, the
first-price auction
important difference
winning bid
is
cases
As
is
and descending (Dutch) auction are strategically equivalent.
for the econometrician,
however,
is
that in the descending auction, the
the only bid made. Hence our results below for the first-price auction that require
only the transaction price apply directly to the descending auction. ^^
affiliated values in this section
We
will restrict attention to
because this assumption guarantees monotonicity of the equilibrium
bidding strategies; however, the assumption
is
otherwise unrelated to our identification arguments.
However, here we do make the additional assumptions that Fa{-) and Fx(-) are
strictly increasing
on their respective supports and have associated joint densities /a(-) and /x(0-
A
bidder in a first-price auction views the bids of his opponents as
bidder
i
observes X^
=
max
I
Xi,
random
variables.
When
he solves
E[Ut\Xi
=
Xi,ma:x.Bj
<
hi]
—
5,
|
Pr(maxBj <
bi\Xi
=
Xi).
Define
C(x; n)
^'
Laffont and
Vuong
=
E\Ui\Xi
(1993) and Elyakime et
al.
= x, max JCj =
x]
(1994) address parametric identification and estimation of the
descending auction model.
25
Then
bi H
first-order condition implies that a necessary condition for
i's
—^~
'"^"'^^Jt^'
3
'~_
''
=
By standard arguments,
(^{xi;n).
bidding strategies are strictly increasing.''^ Thus,
an equilibrium, bidder
i's
bid Bi has the
the bids {Bi,
.
.
— —
<
+ ^"B^
R ^
<
-^FT{ma.XJ^^Bj
= h)
=
rv\
= b^)\^^^^^
is
an equilibrium to this model,
in
same information content
PT{maXjJ^^ Bj
.
^'
if
to be an optimal bid
bi
.
,Bn) are generated from such
as Xi, so that
b^\Bi
lo
z\Bi
(12)
Ci^^^^)
This equation, expressing the latent expectation C,{xi;n) in terms of observable bids, has been
widely exploited in the literature; Elyakime et
Vuong
Perrigne, and
for the
asymmetric
(1998) for the affiliated
PV
and Porter (1999) and
models of pure
When
is
all
CV
model
PV
is
identified
In the symmetric
(1995) established that Fu{-)
(6i)
"
)
,
another approach
If bidders are
model
Li,
for a survey).
Hendricks, Pinkse,
approach to identify and estimate
this
is
is
B^;
<
b\Bi
=
z)
from the distribution of bids through (12) (Laffont and Vuong
IPV model
Vuong
of first-price auctions, Guerre, Perrigne, and
also identified
so the left-hand side of (12)
is
from the transaction price alone, since the trans-
=
{HB{b))"'. Thus, Pr(maxj^ji?j
identified.
<
bi\Bi
=
bi)
=
However, when bidders are asymmetric,
required.
Proposition 16 Consider
(i)
IPV model,
model, and Campo, Perrigne and Vuong (2000)
and Vuong (1999) use
action price identifies the distribution i^g " (6)
"
to analyze the
auctions.
observed, so Fu(-)
//g
it
bids are observed (and bidder identities, where relevant), Pr(maxjyj
(1993, 1996)).
(
used
first
and Vuong (1999)
(see Perrigne
Li, Perrigne,
(1994)
al.
the
IPV
model of the
first-price auction.
symmetric and the transaction price
is
observed, then Fu{-)
is
identified
and
the
is testable.
(a) If bidders are asymmetric and the transaction price and the identity of the winner are observed,
then each
Proof,
(12)
Fjj^ (•) is identified
(i)
(ii)
is testable.
Follows from Guerre, Perrigne, and
must be
restriction
and the model
Vuong
(1995), Corollary
strictly increasing in equilibrium (Laffont
on the estimate of
We observe the
distribution of the transaction price, Pr(i?("-")
In asymmetric first-price auction models, existence of equilibrium
A
left-hand side of
this function.
is
is
sufficient condition for
such monotonicity
in
<b),ss
well as the identity of
a strictly increasing function, and
not straightforward. Athey (2000) shows
that as long as each bidder's best response to nondecreasing opponent strategies
exists.
The
and Vuong (1996)), providing a testable
the winning bidder, /(":"). In equihbrium, each bidder's strategy
Nash equihbrium
2.
a
PV
is itself
nondecreasing, a pure strategy
auction
is affiliation.
In
CV
models,
the conditions required for monotone best responses are more stringent when there are more than two asymmetric
bidders, but
Athey (2000) shows that they are
satisfied in the
26
LMR
model.
the bids are independent. Let Hb,{) be the marginal distribution of Bi.
2
imphes that the distributions HBi{-) are
for each bidder
<
Pr(max,^j Bj
b).
Then
From
identified.
(12)
and the
The proof
these distributions
of Proposition
we can compute
joint distribution of (b("^"),/("^")) identify
the joint distribution of ([/("-"^ /("")). Applying Proposition 2 yields identification of each
Testing follows as in
(i).
Note that the assumptions
metric
PV
F[/. (•).
in part
(ii)
contrast with existing results for identification of asym-
models (Laffont and Vuong (1996)), which have
relied
mentioned above, Dutch auctions are strategically equivalent to
feature that only the transaction price
tions as well as first-price auctions
be adapted from those
in, e.g..
first-price auctions,
all
is
bids.
As
and have the
observable; thus, this result will be useful for
where only the winning bid
Dutch auc-
recorded. Estimation strategies can
Guerre, Perrigne, and Vuong (2000). Note that the tests described
may
in the proof of Proposition 16
is
on observation of
not be very powerful
—one can
easily construct non-equilibrium
bidding rules leading to bids that satisfy the required monotonicity. However, additional tests are
possible
when
the top two bids are observed. For example, with symmetric bidders, the approach
taken in Proposition 16 can then be applied, replacing 5("-") with B("~^-"), yielding a second
timate of Fu{-).
equivalent.
This estimate and that based on the transaction price must be asymptotically
^^
Although most
bids, there
may be
of the top
two
auction data sets either contain only the transaction price or else
first-price
example, in procurement auctions, the top bidder
bids. For
is
second-highest bid as "money
left
on the
relevant to strategic bidding behavior.
it is
table,"
approach can be applied when
Proposition 17 Consider
(i)
^^
between the winning bid and the
which has intuitive appeal as information that
The next
result
all
model. Of course, the same testing
bids are observed.
the affiliated
and
PV
PV model
of the first-price auction, and suppose that the
5("~^-"-)j are observed.
If bidders are symmetric, then the joint distribution of ({7("-"), t/^""^'")) is identified,
The
case in which the lowest bid
^(""i-"))
(fit"-"'
top two bids
iH'^~'^'"\h)
+
and
s^H^"^"'(6)
apply (12) to B<"^"' and
be applied to
is
shows that using observations of the top
her value, as in (12). This allows us to test the affiliated
(^i?("-"-)
disquahfied or
compute the equilibrium "markdown," or gap between a bidder's bid and
possible to
highest two bids
may be
awarded, and the second-highest bidder might receive the contract
instead. Further, auction participants often refer to the difference
bids,
all
reasons for a real- world auctioneer (or auction participants) to maintain records
default before the contract
two
es-
and bidder
also allows for
=
t/<"^"';
identity are observed
is
analogous.
the
Note that observing the
an alternative estimation strategies, based on the the relation
;/^"-^="-''(6), which follows from exchangeability.
then, from the observed .ff^"^"^(-)
infer the distribution of Ui
and
from the distribution of
27
we
Given H^""'^""''(6), we can
infer the distribution of f/''"^"^ Finally, (3)
U^"""''
can
model
is testable.
If bidders are
(ii)
asymmetric and the identity of the winner
distribution 0/ ([/("""^ t/^"~^-"\/("'"^)
Proof,
(i)
Let H^{bi,
.
.
.
,bn)
and
is identified,
the
(/("•")) is also observed,
model
be the joint distribution of the
then the joint
is testable.
Taking
bids.
i
=
without loss of
I
generaUty
^ Pr(maxj^i B^
< bi\Bi = 61)
^ Pr(maxj^i Bj < x\Bi = b^)]^^^^
Pr(maxj^i Bj
dxdy
The
first
x,
92
dxdy
x=y=z=bi
Pl.(5(n-l;n)
<X
S(" .n) <
y)
x=y=bi
equality follows by Bayes' rule, canceling terms from the numerator and the denominator.
The second
equality follows by the definitions and exchangeability; the third equality follows from
differentiation
and exchangeability. Since the joint distribution of (^("-i-")^^!"-"))
assumption, identification of the distribution of ((/(""), [/(""' ")) follows
the left-hand side of (12) must be increasing, which
(ii)
Bi<y)
^Pr(i?(- -) < -)x==61
92
&UaI^B(y,z,x,...,a:)
Bi <y)
x=y=bi
y=bi
1)
61
y=bi
Pr(maxj#i Bj <
^HB{y,bi,...,bi]
("~
<
Extending the
from
logic
is
from
<x, Bi<
^Hniy,bi,...,bi]
y)
y=bi
92
y)
Ej#ia^a5--^B(y,22,---,2n]
x=y=bi
£ Pr(B("-")<x,
dxdy
(12). In equilibrium
a testable restriction,
y=bi
a2
observable by
now we have
(i),
^Pr(maxj^aBj<5i, B^ <
dxdy Pr(maxj-^i Bj
is
Pr(B("-i-")
<
/("-")
x, J5("-")
<
=
Z2 =
1)1^^^^
/("")
y,
=
1)
x=y=bi
Identification follows from (12) since the last
Proposition 17 establishes that
we can
term above
is
observable; testing follows as in
identify the joint distribution of the top
(i).
El
two bidder
some important
policy
simulations, including evaluation of alternative reserve prices. This also allows us to apply
many
valuations. Although this
is
not enough to identify F\j{-),
of the results derived in Section
Propositions
8, 6, 7,
and
Corollary 2 Consider
3.
The
it is
sufficient for
following result follows from Propositions 17, together with
11.
the affiliated
PV model
of the first-price auction and supposesuppose that
the highest two bids are observed. If bidders are allowed to be asymmetric,
winner
is
observed as well. Then:
28
assume identity of
the
(i)
Fu(0
not identified
is
(ii)
Under
(Hi)
IfW^
beloiu the highest
observed,
IS
— M
{W\,
>
.
.
.
(A,Wo)
are independent,
=
are observed, and Ui
,Wn)
Proposition 11, then each
i
gi{-),
andUi
then go{-) and Fa{-)
zs differentiable,
=
two are unobserved.^'^
CIPV model, ifV
the additional restrictions of the
go:su-pp\\VQ\
(iv) If
any bids
if
^j(VFi)
l,...,n,
=
is
observed, then -Fa(')
^^ identified.
gQ{WQ) + Ai, where the unknown function
cltc
+ Ai,
identified
up
to a location normalization.
then under the additional restrictions of
is identified,
and Fa{-)
is
up
identified
to a location
normalization.
Further, Proposition 17 implies that the top two bids in a first-price auction give us enough
PV
information to test the
Corollary 3 Consider
model against the
PV model
the affiliated
two bids are observed in an auction with n
—
auction with n
(i) If
(ii)
CV
>
alternative.
of the first-price auction, and suppose that the top
3 bidders, while the highest bid
is
observed in an
bidders.
1
bidders are symmetric, the model can be tested against the
If bidders are permitted to be asymmetric, the
model can
CV alternative.
be tested
if,
in addition,
we observe
the bidder identities corresponding to each of the observed bids.
Proof,
(i)
jj{n-i:n)
^
By
Proposition 17, this information allows us to identify the distributions of [/("-i-"-i),
recurrence relation
Q{x\m)
is
PV
and [/("") under the
(4).
Now
CV
consider the symmetric
strictly increasing in x, so the
random
alternative.
_
^^
with the inequality following from
unobserved
this follows
[j
< n—
1),
distribution of bids Gb(-)-
=
Pr (ciX^^-^^^-^); n)
(8).
from Proposition
and
Now
/3
This inequality
8,
is
suppose that the true distribution of utihties
this,
observe that when player
optimization problem. Since this joint distribution
statistic.
is
Since the jth bid
Proposition 8 does not impose
is
i
<
n-
z)
1)
<
2) (13)
is
F\j{-), the
jth bid
Fu() imply a
Fu(-) and F\jO induce the same
two order
/3
8.
But then, the bidding functions
statistics of the value distribution affect her
and
F\]{-), player i's best
affiliation; as
is
response bid for
together with the value distribution Fu(-) imply a
and Gn{-) induce the same distributions of order
unobserved, Gb(-)
observationally equivalent to
statistics except for the
Gb().
Finally, recall that
discussed in footnote 21, the result can be generalized to that case.
29
/3
computes her best response to opponent bidding
identical for J^u(-)
unchanged. Because the bidding functions
distribution over bids, Gb(-)i where Gb(-)
Jth order
and
the set of equihbrium bidding functions. These together with
is
strategies /3_;, only the joint distribution of the highest
is
fc(X("-i^"-^);
take another value distribution, F\}{-), such that
form an equilibrium. To see
each value of u,
ar-
testable.
distributions of order statistics except for the jth; this exists by Proposition
still
The
(,{X^^''^^;n), (^(X^"~-^'"^;n),
> Pr
''To see how
standard arguments,
are identified. Then, (3) implies
^^—^Pr('c(X("^");n) <2) +ipr('c(X("~^^"^n) <2)
is
By
satisfy the
variables C,{Xi;n) are exchangeable.
guments of Proposition 17 then imply that the distributions of
^(^(n-i:n-i).^
must
hypothesis, in which case these distributions
Using a similar argument,
(ii)
Pr
('[/(«-i^"-i)
<
/("-i^'^-i)
b,
= !?—1 Pr
=i\
+- Pr
The terms
<
('[/("^")
/("^")
b,
ft/("-^^")
in this expression are identified (by Proposition 17), so
<6,/("-i^'^
z— 1;ti)
D
equahty can be tested.
Extensions
6
Bidder Uncertainty Over the
6.1
6.1.1
of
Opponents
Second-Price Auctions
As pointed out above, our
competitors they
analysis of the
CV
models assumed that bidders know the number of
may be
In an ascending auction, this assumption
face.
auction, bidders might not
know the number
as long as bidders observe
CV
Number
an informative signal
model can be extended.
immediately, as long as
(1)
of competitors
ry
Our
rj
when submitting
n
in the
their
bids.'^'^
However,
of the level of competition, our results for the
identification results for the
replaces
natural. In a sealed-bid
LMR definition,
LMR
and
and LMR-I models extend
(2) in
the case of a first-price
auction, the econometrician can distinguish between auctions with different realizations of rj?^
following result shows that the testing approach of Proposition 13 can
Proposition 18 In
a public signal
the
and
CV model
rj
Proof.
Let
argument
is
is strictly affiliated
is testable if
.
7r(r;|7i)
.
be applied.
the second-price sealed-bid auction, suppose bidders do not
that
bids B^'^~^''^\
still
.
we observe
with
n
(with
rj
unobserved
n> m
hi
= En
Given signal
E[Ui\Xi
ry,
each player
= maxXj =
but observe
Then
m
auctions with
m
bidders
bidders.
denote the conditional distribution of the signal
similar for other cases).
known
to the econometrician).
the transaction price B^'^~^'^'
,5("~i-") in auctions with
The
Xi
=
i
rj.
Assume
m
= n—
\
(the
bids
b{xi;T]).
3+i
'^
Matthews (1987) and McAfee and McMillan (1987b) provide
number of bidders.
theoretical analyses of auctions with a stochaistic
Hendricks, Pinkse, and Porter (1999), for example, assume that bidders have the same beliefs about the number
of competitors in
all
auctions at which the
number
of participating bidders falls in a certain range.
30
Taking
z
=
1,
>
the inequality (8) and strict affiliation imply that for ^
=
b{xi;f])
= X2 =
77
Xj <xi,
j
=
3,..., n]\f]]
< En[E[Ui\Xi = X2 = xuX, < xu
j
=
3,...,n]|r?]
En[E[Ui\Xi
xi,
Since
/oo
Pr(6(X("-2^");ry) <6)d7r(r?|n)
-00
strict affiliation implies
/CO
r?)
<
b)dTr{r]\n
-
1).
(14)
Pr(6(X("-i^");77)
<
6) fi7r(7?|n
-
1).
(15)
Pr(6(X("-2^");
-00
Similarly,
/oo
-00
Using
(4)
and the
fact that b{-;ri)
is
strictly increasing,
we obtain the
testable stochastic
dominance
relation
/oo
<
Pr(S(X("-2-"-i);77)
b)dTv{r]\n
-
1)
-00
/oo
r
9
- Pr(6(X("-2^"); 77) <
-00
77—9
-—
^ Pr(6(X("-i^");
t?)
<
"
["
n
6) d7r(77|n
-
1)
J
n
where the inequality follows from
6.1.2
+
6)
(14)
and
D
(15).
First-Price Auctions
Bidder uncertainty over the number of opponents
is
a potentially
case of a first-price auction, since bidding strategies will depend on
private values, after observing signal
max
t]
{ui
each bidder
—
b)
i
more
ij,
difficult
even in a
problem
PV
auction.
in the
With
solves
Pr(max Bj <
b\Ui
=
Ui^rf)
j^i
b
giving first-order condition
PT{ma.Xj^,Bj <b^\B^
"i
+
r)
-^
~^:.
^
Pr(maxj^, Bj <
/
rr;
x\B^
31
= bi,T])
n
=
= bi,r])\^^^^
;
—
""'
(1^)
If
the econometrician observes a set of auctions in which
and vahiations can be used
the symmetric
IPV
in essentially the
same way that
winning bid
case, observation of the
t]
(12)
and
Tj
letting i?^'" denote the
00
00
n=2
n=2
identify Fu{-),
n such
common
7r(n|7/)
Testing of the
different values of
values
>
7r(n|77)
PV
we recover the
in auctions
many
initial
bi\r]),
which
b\T]),
is
rj
is
observed.
equal to
observed directly, Pr{Bi
<
b\r])
is
giving (through (16)) identification
the distribution of
(7^"'"',
(3)
can be used to
D
F\j{-).
models of
PV
first-price auctions
can be extended
hypothesis, these distributions will be identical; with
distribution oi E\Ui\X^^''^'>
where signal
is
<
signal
77
=
772
>
=
maxfc^iXfc
is
observed
77
=
r/j
r/i
is
observed.
=
Xj,//]
rather than that of
will first-order stochastically
auctions the seller announces a reserve price for the auction, which can be viewed as an
bid entered by the
of bidders' valuations.
probability
some
For simplicity we have assumed reserve price
seller.
When
the reserve price tq
is
the
reserve price, ro,
is
If only
IPV model
fixed
and
If either (a)
=
i_jr
that Fu{ro)
°
(^
This creates a discrepancy
results
n.
However,
can be extended. '^^
of first-price, second-price, or ascending auctions, suppose that the
one bid from each auction
denoted F[/(-|ro)
below the support
and the number of participating bidders,
A'',
by conditioning on the participation threshold, many of our
Corollary 4 In
lies
in the interior of the support, with positive
potential bidders will be unwilling to submit a bid.
between the number of potential bidders.
(ii)
n bidders when
sufficient to
is still
77
Reserve Prices
6.2
(i)
with fixed
in
hypothesis can be achieved by comparing distributions of
under the
77:
<
i?^
From
0.
This distribution in auctions where signal
dominate that
In
and
identification results for other private values
fQj.
^(j;n-)_
that
which completely characterizes
in similar fashion.
jjii-n)
b\r])
This enables construction of Pr(maxj^2
of [/("") for each
Our
<
between bids
was used above. For example,
winning bid, we observe Pr(i?™^"
Since (17) strictly increases in Pr(i?i
identified.
fixed, this relation
in auctions
identify Fu{-). Let Tr(n|r/) denote the probability of there being
Fixing
is
,
is
is
E
(0, 1).
observed, the distribution of Ui conditional on Ui
identified
on
>
ro,
[ro,oo].
two bids from each auction are observed or (b) a single bid
is
observed in auctions
For first-price auctions, Bajari and Hortagsu (2000) use a parametric model to simultaneously estimate a model
of entry
and bidding. Guerre, Perrigne, and Vuong (2000) includes a discussion of binding reserve prices
in the
IPV
model. See also the recent paper by Li (2000). Hendricks, Pinkse and Porter (1999) estimate a model of stochastic
participation and reserve prices in
common
value auctions.
32
IPV
with different numbers of participating bidders, the
(Hi) If
N
Proof.
is
fixed
and
number of participating
the
Because each potential bidder
the participating bidders' valuations
1
(i)
is testable.
observed,
is
when
Parts
Fu{-\ro).
is
>
x,
and
made
Similar extensions can be
Corollary 5 In a
fixed
and
PV
and
bids B^^'^'
in all (n
—
Proof.
Participation
_BU+i-"-)
and
then follow from Propositions
Because the
=
Fu{ro), both TV and
D
models considered above. The
models can
also
be applied
in this case.^^
(with 2
<
Both the
j
<
PV model and the CV model are
n) in
all
rg, is
testable
n-bidder auctions and bids B^^''^~^'
l)-bidder auctions.
is
determined as
type to participate in equilibrium.
the recurrence relation (4)
under the
CV
still
Milgrom and Weber
in
Participating bidders
distribution Fx{-) truncated at [xq,
(13) hold
distribution of
or ascending auction, suppose the reserve price,
first-price, second-price,
in the interior of the support of each Ui.
we observe
common
fixed.
A'' is
CV
-Ft/(ro) are identified.
of Proposition 16.
for identification of the other
following result shows that our tests of the
and
binomial with parameter A
is
from the distribution of n when
-F[/(^o) are identified
N
ro, the
(ii)
and 16 and the remarks regarding testing following the proof
participation rule for each potential bidder
if
bidders
participates
i
model
.
,
.
.
xq).
(1982).
draw
their types
Because exchangeability
holds under the
PV
Let xq denote the lowest
is
Xj,
.
.
.
,Xn from
the
preserved by this truncation,
hypothesis, while the inequalities (9), (10), and
D
hypothesis.
Conclusion
7
This paper has established new identification results for standard auction models. For second-price
and ascending auctions, our main
for
results
can be summarized as follows (where the results apply
symmetric bidders unless otherwise noted):
1.
The independent
if
2.
private values
more than one bid
is
model
observed
in
is
identified
from the transaction
price,
and
it is
testable
each auction.
In a second-price auction, observing the winner's identity and bid identifies the asymmetric
independent private values model.
In an ascending auction, this model
is
identified
if
we
observe either the lowest bid and bidder identity or the bids of the top two losing bidders and
the identities of the top three bidders.
With a binding
reserve price, one can also test the
PV
hypothesis (^{xo;m)
CV
=
ro against the alternative (^(xo;
model (Milgrom and Weber (1982)). This testing approach has been proposed
auctions by Hendricks, Pinkse, and Porter (1999).
ro implied by the
33
m) >
for first-price
3.
In second-price auctions, but not ascending auctions, the "conditionally independent private
model
values"
—where bidders' private values are the sum of a common component
idiosyncratic component, and idiosyncratic
common component
—
identified
is
asymmetric bidders
4.
If
if
values
model
is
if all
if
in addition, the
common and
bids are observed in each auction.
idiosyncratic
This extends to
identities are observed as well.
common component
the
components are independent conditional on the
and testable
components are independent, and
identified
of values
is
observed ex post, the conditionally independent private
from the transaction
price,
and
if
two bids are observed,
it is
5.
The
6.
Even with potentially asymmetric bidders, the unrestricted private values model
unrestricted private values model
with
7.
The
if
private values
testable.
not identified from incomplete sets of bids.
is
from the transaction price (and testable when two bids are observed)
specific covariates
and an
if
is
identified
there are bidder-
sufficient variation.
model
is
testable against the
common
values alternative, and vice-versa,
the transaction price and the next-highest bid are observed in two auction data sets with
and n —
1
bidders, respectively (and the variation in participation
is
n
exogenous). This holds
even though these assumptions do not guarantee identification.
8.
In second-price auctions, but not ascending auctions, the "linear mineral rights model"
where bidders' expected values are linear
in
bidder signals, signals are linear in the
common
value and an idiosyncratic component, and idiosyncratic components are independent conditional
is
on the common component
is
identified
and testable
independent of the idiosyncratic component, and
extends to asymmetric bidders
9.
—
In a second-price auction,
model
is
identified
if
if
the
all
if,
in addition, the
common
value
bids are observed in each auction. This
identities are observed as well.
common
from the transaction
is
observed ex post, the linear mineral rights
price. In
an ascending auction, we must observe the
value
lowest bid rather than the transaction price.
Now
consider
was established
how
these results extend to first-price auctions with affiliated signals. Result (1)
for first-price auctions
(8)
were established by
(5)
extend directly to
Li, Perrigne,
by Guerre, Perrigne, and Vuong (1995), and results
and Vuong
(1999). In this paper,
first-price auctions, while results (4), (6), (7),
we show that
and
(9)
extend
(3)
and
results (2)
and
if
the top two
bids in each auction are observed.
Most
of the prior literature on structural estimation in ascending
has focused on parametric identification strategies.
34
and second-price auctions
For first-price auctions, the prior literature
provides nonparametric identification results only
IPV
when
all
bids are observed, or in the symmetric
model. Our results both indicate environments in which weaker identifying assumptions can
be employed and suggest cases
assumptions.
Our
in
which identification can only be obtained via functional form
results further suggest strategies for the design of
new data
researchers are able to collect different elements of auction data (typically at
Increasingly,
sets.
some
cost); this
might
be the case when obtaining data from internet auctions, as the researcher might choose whether
to record the entire history of an auction as opposed to just information about the total
of participants and the transaction price.
Our
results suggest that in
many
number
environments, just the
top two or three bids provide sufficient information for identification and/or testing.
This paper has focused primarily on identification. In general, identification
not sufficient for existence of a consistent estimator.
estimation strategies, and for
estimation approaches that
however, we have
left
many
may be
Many
of the identified models
applicable. In
is
necessary but
of our identification proofs suggest
we have pointed
to nonparametric
most of these cases consistency
is
transparent;
the derivation of the asymptotic properties of these and other potential
estimators for future work, along with their application to bidding data.
35
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