The RAND Corporation
The Winner's Curse, Reserve Prices, and Endogenous Entry: Empirical Insights from eBay
Auctions
Author(s): Patrick Bajari and Ali Hortacsu
Source: The RAND Journal of Economics, Vol. 34, No. 2 (Summer, 2003), pp. 329-355
Published by: Blackwell Publishing on behalf of The RAND Corporation
Stable URL: http://www.jstor.org/stable/1593721
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RANDJournalof Economics
Vol.34, No. 2, Summer2003
pp. 329-355
The
winner's
and
endogenous entry:
auctions
eBay
from
curse,
reserve
prices,
empirical
insights
Patrick Bajari*
and
Ali HortaSsu**
Internetauctions have recentlygained widespreadpopularityand are one of the most successful
forms of electronic commerce.Weexaminea uniquedataset of eBay coin auctions to explorethe
determinantsof bidderand seller behavior Wefirst documenta numberof empiricalregularities.
We then specify and estimate a structuraleconometric model of bidding on eBay. Using our
parameterestimatesfrom this model, we measurethe extent of the winner's curse and simulate
seller revenueunderdifferentreserveprices.
1. Introduction
*
Auctionshavefoundtheirway into millionsof homes with the recentproliferationof auction
sites on the Interet. The web behemotheBay is by far the most popularonline auction site. In
2001, 423 million items in 18,000 unique categories were listed for sale on eBay. Since eBay
archivesdetailedrecordsof completedauctions,it is a sourcefor immenseamountsof high-quality
data and serves as a naturaltesting groundfor existing theoriesof biddingand marketdesign.
In this article we explore the determinantsof bidderand seller behaviorin eBay auctions.1
For our study, we collected a unique datasetof bids at eBay coin auctions from September28
to October2, 1998. Our researchstrategyhas three distinct stages. We first describe the market
and document several empiricalregularitiesin bidding and selling behaviorthat occur in these
auctions.Second,motivatedby theseregularities,we specify andestimatea structuraleconometric
model of bidding on eBay. Finally, we simulateour model at the estimatedparametervalues to
quantifythe extent of the winner'scurse and to characterizeprofit-maximizingseller behavior.
The bidding formatused on eBay is called "proxybidding."Here's how it works. When a
* StanfordUniversityand NBER;[email protected].
**Universityof Chicago;[email protected].
We would like to thankRobertPorterandthreeanonymousrefereesfor theircomments,which helped us to improve
this work considerably.We also would like to thank Susan Athey, Philip Haile, John Krainer,Pueo Keffer,Jon Levin,
David Lucking-Reiley,Steve Tadelis, and participantsat a numberof seminars for their helpful comments. The first
authorwould like to acknowledgethe generousresearchsupportof SIEPR,the National Science Foundation,grantnos.
SES-0112106 and SES-0122747, and the Hoover Institution'sNationalFellows program.
1 For a broad
survey articlecovering 142 online auctionsites, see Lucking-Reiley(2000).
Copyright? 2003, RAND.
329
330 / THERANDJOURNAL
OFECONOMICS
biddersubmitsa proxy bid, she is askedby the eBay computerto enterthe maximumamountshe
is willing to pay for the item. SupposethatbidderA is the firstbidderto submita proxy bid on an
item with a minimumbid of $10 (as set by the seller) and a bid incrementof $.50. Let the amount
of bidderA's proxy bid be $25. eBay automaticallysets the high bid to $10, just enough to make
bidderA the high bidder.Next suppose thatbidderB entersthe auctionwith a proxy bid of $13.
eBay then raises bidderA's bid to $13.50. If anotherbidder submits a proxy bid above $25.50
($25 plus one bid increment),bidderA is no longer the high bidder,and the eBay computerwill
notify her of this via e-mail. If bidderA wishes, she can submit a new proxy bid. This process
continues until the auction ends. The high bidderends up paying the second-highestproxy bid
plus one bid increment.2Once the auctionhas concluded,the winneris notifiedby e-mail. At this
point, eBay's intermediaryrole ends and it is up to the winnerof the auctionto contactthe seller
to arrangeshipmentand paymentdetails.
Thereare a numberof clear empiricalpatternsin our data.First,bidderson eBay frequently
engage in a practicecalled "sniping,"which refers to submittingone's bid as late as possible in
the auction.3The medianwinningbid arrivesafter98.3%of the auctiontime has elapsed. Second,
thereis significantvariationin the numberof biddersin an auction.A low minimumbid anda high
book value are correlatedwith increasedentry into an auction. This is consistent with a model
in which bidding is a costly activity, as in Harstad(1990) and Levin and Smith (1994), where
biddersenter an auctionuntil expected profitsequal the costs of participating.Finally, sellers on
averageset minimumbids at levels considerablyless thanan item's book value, and they tend to
limit the use of secret reserveprices to high-valueobjects.
In Section 4, using a stylized theoreticalmodel, we demonstratethat sniping can be rationalized as equilibriumbehaviorin a common-valueenvironment.If, in equilibrium,bids are an
increasingfunctionof one's privateinformation,then a bidderwould effectively revealherprivate
signal of the common value by bidding early. However,if a bidderbids at the "last second,"she
does not reveal her privateinformationto otherbiddersin the auction.As a result,the symmetric
equilibriuminvolves bidding at the end of the auction.We also demonstratethatthe equilibrium
in the eBay auctionmodel is formally equivalentto the equilibriumin a second-pricesealed-bid
auction.
In Section 5 we specify a structuralmodel of bidding in eBay auctions. Using our results
about last-minutebidding from Section 4, we model eBay auctions as second-pricesealed-bid
auctionswhere there is a common value and the numberof biddersis endogenouslydetermined
by a zero-profitcondition.While the common-valueassumptionmightbe controversial,based on
our discussion in Section 3 of the reduced-formevidence and natureof the good being sold, we
find this specificationpreferableto pureprivatevalues.4
Estimatinga model with a common value is technicallychallenging.The approachwe take
is most closely relatedto the parametric,likelihood-basedmethod of Paarsch(1992). However,
as Donald and Paarsch(1993) point out, the asymptoticsfor maximum-likelihoodestimationof
auction models are not straightforward,because the supportof the likelihood function depends
on the parametervalues. We utilize Bayesian methods used in Bajari (1997) to overcome this
difficulty.
We use our parameterestimates to measure the effect of the winner's curse, to infer the
entrycosts associatedwith bidding,andto characterizeprofit-maximizingreserveprices.Forthe
averageauction, we find that bidderslower theirbids by 3.2% per additionalcompetitor.Based
on our parameterestimates,and a zero-profitconditiongoverningthe entrydecisions of bidders,
we calculate the average implicit cost of submittinga bid in an eBay auction to be $3.20. We
also computea seller's optimalreserve-pricedecision using our parameterestimates.We findthat
2 One
exception to this occurs when the two highest proxybids areidentical.Then the item is awardedto the first
one to submita bid at the price she bid.
3 This fact was also independentlyreportedby Roth and Ockenfels (2000).
4 In
principle,an auctioncould have both a private-and a common-valuecomponent.However,estimatingsuch
a model with endogenousentry would not be computationallyfeasible using currentlyavailableestimationmethods.
? RAND 2003.
BAJARIANDHORTACSU/ 331
the observed practices of setting the minimumbid at significantlyless than the book value and
limiting secretreserveprices to high-valueitems are consistentwith profit-maximizingbehavior.
2. Descriptionof the marketand the data
On any given day, millions of items are listed on eBay, many of which are one-of-a-kind
secondhandgoods or collectibles. The listings are organizedinto thousandsof categories and
subcategories,such as antiques,books, and consumerelectronics.Listings typically containdetailed descriptionsand picturesof the item up for bid. The listing also providesthe seller's name,
the currentbid for the item, the bid increment,the quantitythatis being sold, and the amountof
time left in the auction.Withinany category,buyerscan sortthe listings to firstview the recently
listed items or the auctions that will close soon. eBay also provides a search engine that allows
buyers to searchlistings in each categoryby keywords,price range, or ending time. The search
engine allows users to browse completed auctions,a useful tool for buyersand sellers who wish
to review recent transactions.
In this article we focus on bidding for mint and proof sets of U.S. coins that occurredon
eBay between September28 and October2, 1998.5 Both types of coin sets are prepareddirectly
by the U.S. Treasuryand containuncirculatedspecimens of a given year's coin denominations.6
These sets are representativeof items being tradedon eBay, in that they are mundanelypriced
and are accessible to ordinarycollectors.
For each auctionin our dataset,we collected the "finalbids" as reportedon the "bidhistory
page."These final bids correspondto the maximumwillingness-to-payexpressedby each bidder
throughthe proxy bidding system (except for the winning bidder,whose maximumwillingnessto-pay is not displayed).We found the book values for the auctioneditems in the November 1998
issue of Coins Magazine, which surveys prices from coin dealers and coin auctions aroundthe
United States.7As reportedin Table 1, the averagebook value of the coins is $50, with values
ranging from $3 to $3,700, reflecting the wide dispersion of prices even within this relatively
narrowsection of the collectible coin market.We reviewed item descriptionsto see if the seller
reportedanythingmissing or wrong with the item being sold. This investigationlabelled 9.3% of
the items with self-reportedblemishes.
Sellers on eBay are differentiatedby their feedback rating. For each completed auction,
eBay allows both the seller and the winning bidderto rateone anotherin termsof reliabilityand
timeliness in payment and delivery.The ratingis in the form of a positive, negative, or neutral
response.Next to each buyer'sor seller's ID (which is usually a pseudonymor an e-mail address),
the numberof net positive responses is displayed. By clicking on the ID, bidders can view all
of the seller's feedback,includingall commentsas well as statisticstotallingthe total numberof
positive, neutral,and negativecomments.
In Table 1, we see that the averagebidderhas 41 overall feedback points and the average
seller has 202 overall feedback points. Negative feedback points are very low comparedto the
positive feedback points of the sellers.8 Users of eBay ascribe this to fear of retaliation,since
the identitiesof feedbackgivers are displayed.Therefore,the conventionalwisdom among eBay
users is thatthe negativeratingis a betterindicatorof a seller's reliabilitythanher overallrating.
C
5 Therewere a totalof 516 auctions
completedin themint/proofcategoryduringthe five-daysamplewe considered.
Bid histories for 39 auctionswere lost due to a technicalerrorin datatransfer.Ourdatasetof 407 auctionsrepresentsall
mint/proofset auctions conductedon eBay duringthe dates consideredthat had reliable book values and that were not
Dutch auctions.
6 Mint sets contain uncirculated
specimens of each year's coins for every denominationissued from each mint.
Proof sets also contain a specimen of each year's newly minted coins, but are specially manufacturedto have sharper
details and more-than-ordinary
brilliancecomparedto mint sets.
7 The November issue of the
magazine was bought on October22. The prices quoted were marketprices as of
mid-October.
8 The maximum
negativeratingof 21 correspondsto a seller with 973 overallfeedbackpoints. eBay discontinues
a user's membershipif the net of positive and negative feedbackpoints falls below -4.
? RAND 2003.
332
THE RAND JOURNAL OF ECONOMICS
/
TABLE 1
Summary Statistics
Mean
Standard
Deviation
Minimum
Maximum
$50.1
$212.63
$3
$3700
Highest Bid/Book Value
.81
.43
0
2.08
WinningBid/Book Value(Conditionalon Sale)
.96
.30
.16
2.08
% Sold
.79
$1.02
$1
$5
3.08
2.51
0
14
MinimumBid//Book Value
.63
.41
.0
2.56
% Secret Reserve
.14
201.7
213.18
0
973
Seller's Negative Feedback
.47
1.71
0
21
Bidder'sOverallFeedback
41.2
37.54
Variable
Book Value
Shippingand Handling
% Blemished
Numberof Bidders
Seller's OverallFeedback
$2.15
.09-
-4
199.67
The averagebidder and seller in this markethas a reasonablyhigh numberof overall feedback
points, indicatingthatthey can be classified as a serious collector,or possibly a coin dealer.
3. Some empiricalregularities
*
In this section we reportseveralempiricalregularitieswe have foundaboutbiddingbehavior
on eBay.Wewill use theseregularitiesto motivateourstructuralmodelof biddingin eBay auctions.
a Timing of bids. A strikingfeatureof eBay auctionsis thatbiddingactivityis concentrated
at the end of each auction.Figure 1 is a histogramof final bid submissiontimes. More than 50%
of final bids are submittedafter 90% of the auction durationhas passed, and about 32% of the
bids are submittedafter97% of the auctionhas passed (the last two hoursof a three-dayauction).
Winningbids tend to come even later.The median winning bid arrivesafter 98.3% of the
auction time has elapsed (within the last 73 minutes of a three-dayauction), and 25% of the
winning bids arriveafter99.8% of the auctiontime has elapsed (the last eight minutesof a threeday auction). In our data, we can also observe how many total bids were submittedthroughout
the auction.In the set of auctionswe consider,an averagebiddersubmitsabouttwo proxy bids.
z
Minimum bids and secret reserve prices. A seller on eBay is constrainedto use the proxy
biddingmethodbut has a few options to customize the sale mechanism.9She can set a minimum
bid, a secret reserve price, or both. Sellers frequentlychoose to set the minimumbid at a level
substantiallylower thanthe book value.As can be seen in Table 1, the ratioof the minimumbid to
the book value is on average.63. In a secret reserve-priceauction,biddersare told only whether
the reserveprice has been met or not. The reserveprice is neverannounced,even afterthe auction
ends. As reportedin Table 1, 14%of the auctionswere conductedwith a secret reserveprice.
As we will discuss in Section 6, the use of secret reserveprices is somewhatof a puzzle in
auction theory.Given this, one might wonder whetherthere is a systematic difference between
auctions with secret reserve prices and those without. Table 2 reports the comparisonof the
9 Dutch auctionsfor the sale of multipleobjects are also allowed, but we exclude such auctionsfrom our study.
? RAND 2003.
BAJARIANDHORTACSU/ 333
FIGURE1
OF BIDTIMINGS
DISTRIBUTION
.35
.3
C(
.25
iu .15.1.05
0
.1
.2
.4
.3
.5
.6
.7
.8
Fractionof auctionduration
.9
1
quantilesof book values across both types of auctions.As we see from the table, sellers of items
with high book value aremorelikely to use secretreserveprices comparedto sellers of items with
low book value.
From Table 3, we can see that sellers who decide to use a secret reserve price keep their
minimum bids low to encourage entry.10In secret reserve-priceauctions, the average of the
minimumbid divided by the book value is .28, with severalequal to zero. If we look at auctions
with no secretreserveprice, the averageof this ratiois .69. Since eBay does not divulge the secret
reserveprices at the close of the auction,we cannotcomparethe secretreserveprice to minimum
bid levels in ordinaryauctions;however,the fact thatfarfewer secretreserve-priceauctionsresult
in a sale (49%versus84%)leads one to conjecturethatsecretreserve-pricelevels are,on average,
higherthanthe minimumbids in ordinaryauctions.
]
Determinants of entry. Although the mean numberof bidders in our auctions is 3.08,
there is wide variationin the numberof biddersfor auctions of similar objects. We investigate
the determinantsof entry in Table 4, where we regress the number of bidders in an auction
on various covariates.Our regression specificationinteractsthe minimum bid (normalizedby
dividing throughby the book value) with the presence of a secret reserve price to disentangle
the effects of these seller strategies.After a specification search, we decided to use a Poisson
specificationin the regression.1I
We find that the minimum bid has a significant negative correlationwith the numberof
bidders.The presenceof a secretreservepriceis negativelycorrelatedwith the numberof bidders,
but not in a statisticallysignificantmanner.We findthatitems with higherbook values have more
bidderson averageand thatboth the negative and overallreputationmeasuresof the seller seem
to play an importantrole in determiningentry,with the negative reputationmeasurehaving the
higherelasticity.
D Determinants of auction prices. Perhapsthe most interestingempiricalquestionto ask, at
least from the perspectiveof eBay users, is, "Whatdeterminesprices in eBay auctions?"A seller
might modify this questionto ask, "Whatshould I set as my minimumbid/secretreserveprice to
maximize my revenues?"To answerboth of these questions consistently,one needs to estimate
the primitivesof the demand side of the market,explicitly taking into account the effect of the
supply-sidepolicies chosen by the seller (throughminimumbid and secret reserve-pricelevels)
10Some conversationson
communitychat-boardson eBay suggest this as a strategyfor sellers.
11With the Poisson
specification,the likelihood of observationt, conditional on the regressorsXt, depends on
log(.t) = fi'Xt, where it is the mean of the Poisson process thatwe are tryingto fit to the observednumberof bidders.
? RAND 2003.
334 / THERANDJOURNAL
OFECONOMICS
TABLE 2
Quantiles of Book Value Across Auction
Formats
Variable
Secret Reserve
10%
6.75
6
25%
9
8.5
50%
42.5
13.375
75%
148.5
22.25
90%
620
42.125
TABLE 3
Posted Reserve
Summary Statistics Across Auction Formats
Secret Reserve Mean
Posted Reserve Mean
227.43 (568.36)
21.18 (28.50)
Probabilityof Sale
.491 (.504)
.84 (.3823)
Numberof Bidders
5.0 (2.94)
2.76 (2.285)
.277 (.208)
.686 (.4092)
.911 (.2601)
.961 (.4535)
Variable
Book Value
MinimumBid/ Book Value
WinningBid/Book Value
Numberof observations
57
350
Standarderrorsin parentheses.
on price formation.Before we attemptto performthis exercise using a structuralmodel of price
formation,however,we firstexamine what a straightforwardregressionanalysis yields.
In Table5, we regressthe highest observedbid in the auction(normalizedby the book value
of the item) on auction-specificvariablesfrom Table 1.12 In the first column, we use data from
all auctions, includingthose in which no bids were posted. This regressionis performedusing a
Tobit specification,taking into accountthatthe revenuefrom the auctionis truncatedat zero. In
the second column, we include bids from auctionswith more thantwo bidders.In this regression,
we includeda dummyvariablefor "earlybiddingactivity,"which was coded to be 1 if the bidders
who submittedthe losing bids made their submissions earlier than the last 10 minutes of the
auction.This meansthatthe winnerof the auctionhad time to revise her bid basedon information
revealedbefore the very end of the auction,presumablyresultingin a higherbid. We also include
the averagebidderexperienceas a right-hand-sidevariable.
In both of these specifications,we findthatan increasein the numberof biddersis correlated
with higherrevenue.In the model in Table5, the presenceof an extrabidderresultsin an 11.4%
increase in auction revenue, taking all auctions into account, and a 5.5% increase if we only
considerauctionsin which there are at least two bidders.
Not surprisingly,the coefficientin the secondcolumnof Table5 shows thata higherminimum
bid resultsin higherrevenuesconditionalon entry.However,the firstcolumns show thatonce we
accountfor the truncationcausedby the minimumbid, this positiveeffect diminishesandbecomes
statisticallyinsignificantwith the Tobitcorrection.Based on the estimatesin Table5, one mightbe
temptedto thinkthat an increasein the minimumbid will, on average,always increaserevenues.
However,this reasoningignores the fact thatbidderschoose to enteran auction.It is quite likely
that there will be a thresholdminimum bid level above which expected revenues will start to
12For auctions with a secret reserve
price in which the reserve was not met, we include the highest bid as the
"revenue"of the seller.
? RAND 2003.
BAJARIAND HORTACSU /
Bidder Entry-PoissonRegression
TABLE4
Variable
MinimumBid*(SecretReserve)
MinimumBid*(1-Secret Reserve)
Secret Reserve
ln(Book Value)
ParameterEstimate
-.883* (.329)
-1.571* (.091)
-.162
(.125)
.128* (.025)
Blemish
-.062
ln(NegativeReputation)
-.248* (.072)
ln(OverallReputation)
Constant
Numberof observations
R2
335
(.093)
.108* (.025)
1.102* (.148)
407
.25
The dependentvariableis the numberof bidders.Variablesthataresignificant
at the 5% level are markedwith an asterisk.
decline due to lack of entry. Similarly, using a secret reserve price will also affect the entry process,
as well as interact with the use of a minimum bid. Finally, these regressions treat the reserve-price
policy as exogenous. Therefore, the results in Table 5 should be interpreted with caution from a
policy perspective.
The presence of a secret reserve price is negatively correlated with revenues, both conditional
and unconditional on entry by bidders. The presence of a blemish reduces auction revenues by
15% in the first column and by 11% in the second column. The negative reputation of a seller
seems to reduce revenues, but this reduction is not statistically significant. However, the overall
reputation of the seller seems to increase revenues quite significantly.13 On the other hand, bidder
experience does not seem to play any role in determining auction revenues. We found this result
to be somewhat surprising, since in experimental studies of auctions, bidders frequently adapt
their strategies with successive plays of the game.
Out of the 278 auctions with more than 2 bidders, we found 18 auctions with bids coming
in the last 10 minutes of the auction and 260 with at least one bid coming in early enough that
the winner could revise her bid. However, we see from Table 5 that the presence of early bidding
activity does not appear to change the winning bid in the auction in a statistically significant
fashion. 14
Private values versus common value. As pointed out in the seminal work of Milgrom
o
and Weber (1982), the private-values and common-value assumptions can lead to very different
bidding patterns and policy prescriptions. Although the private-values specification is a useful
benchmark, our belief is that coin auctions on eBay possess a common-value component. Two
types of evidence are consistent with this component. The first is the nature of the object being
sold, and the second is the relationship between the sale price and the number of bidders.
A first piece of evidence for a common value is resale. We have documented that there is a
liquid resale market for coins, from which we obtained our "book values." Given the existence
13
A similar result is also found by Bryan et al. (2000), Houser and Wooders (2000), Melnik and Aim (2002),
Resnick and Zeckhauser(2001), and Resnick et al. (2002).
14
Aside from including a dummy variablefor the presence of early bidding activity,we also tested whetherthe
level of early activitymatteredin the determinationof the finalprice,by interactingan early-bidindicatorwith the secondand third-highestbids in the above regressions.The results indicatedthat the level of early bidding does not affect the
auctionprice.
? RAND 2003.
336 / THERANDJOURNAL
OFECONOMICS
TABLE5
Determinantsof AuctionPrices
Variable
All Auctions (Tobit)
More Than 2 Bidders (OLS)
Numberof Bidders
.1104* (.0114)
.0553* (.0085)
MinimumBid/Book Value
.0896 (.0704)
.4744* (.0579)
Secret Reserve
-.1299* (.0641)
-.0596
Blemish
-.1494* (.0713)
-.1054* (.0495)
In (Negative Feedback)
-.0624
-.0018
In (OverallReputation)
(.0489)
.1033* (.0175)
AverageBidderExperience
Numberof observations
R2
(.0353)
.0383* (.0128)
.0002 (.0004)
-.0777
Early Bidding Activity
Constant
(.0407)
-.00523
(.1028)
(.0604)
.2702* (.0993)
407
278
.2427
.2926
The dependentvariableis the winning bid dividedby the book value.
of this market,it is not surprisingthat a fair numberof coin collectors are drivenby speculative
motives. In this context, learningaboutotherbidders'assessmentsof a coin's resale value might
improvethe forecast of an individualbidder.
A second piece of evidence for a common-valueelement is that bidders on eBay cannot
inspect the coin firsthand.Coins that are scratchedor thatdisplay other signs of wear are valued
less by collectors. Even though we see sellers on eBay expending quite a bit of effort to post
detailed descriptionsand picturesof the object being auctioned,bidderswill probablystill face
some uncertaintyaboutthe conditionof the set being sold.
An empirical test to distinguishbetween the pure private-valuesmodel and the commonvalue model is the "winner'scurse"test suggested by the model of Milgrom and Weber(1982).
In a common-valueauction,the Milgrom and Webermodel predictsthat bidderswill rationally
lower theirbids to preventa winner'scurse fromhappening.The possibility of a winner'scurseis
greaterin an N-person auctionas opposedto an (N - 1)-person auction;therefore,the empirical
predictionis that the averagebid in an N-person second-priceor ascending auction is going to
be lower thanthe averagebid in an (N - 1)-person auction.15But in a private-valueascendingor
second-priceauction setting like eBay, the numberof biddersshould not have an effect on bids
since the dominantstrategyin these auctionsis to bid one's valuation.16
In Table 6, we reportregressionsof the bids normalizedby book value (b) on the realized
numberof bidders(N). We considerfour alternativespecifications.In the firsttwo specifications,
all of the bids are included. As an instrumentfor the numberof bidders, we use the minimum
bid divided by the book value. In Table4, we found that in a Poisson regression,the numberof
bidders was negatively correlatedwith the minimumbid. If the minimumbid is not correlated
with other omitted factors that could raise the bids, it will be a valid instrumentfor the number
of bidders.In the thirdand fourthcolumns of Table6, we limit attentionto bids submittedin the
last minuteof the auction.Since these bids are submittedcloser to the end of the auction,bidders
might have an incentive to bid closer to their true valuationsthan early in the auction. In three
15For a
proof, see Athey and Haile (2002) and Haile, Hong, and Shum (2000).
common value is also exploited by Paarsch(1992) and Haile, Hong, and Shum (2000) in
first-priceauction contexts. A corollaryof Athey and Haile's (2002) Theorem9 is that underthe truncationintroduced
by eBay, the regressioncoefficient on N underthe null of privatevalues is positive.
16 This idea test for a
? RAND 2003.
BAJARIAND HORTA.SU
/
337
Regressionof Bid/BookValueon the Numberof Bidders
TABLE6
Variable
Numberof Bidders
Book Value
All Auctions
All Auctions
Last-Minute Bids
Last-Minute Bids
(OLS)
(IV)
(OLS)
(IV)
-.0204*
(.0035)
-.00007* (.00002)
-.0989*
(.0073)
.0158
(.0119)
-.0610*
(.0277)
.00004 (.00003)
-.00006
(.00007)
.00004 (.00009)
.0798* (.0282)
-.1611'
(.0780)
-.0031
(.1090)
Secret Reserve
-.0436
(.0225)
Blemish
-.1302'
(.0270)
-.0976*
(.0321)
-.2272
(.1176)
-.1669
(.1489)
In (Negative Feedback)
-.0521'
(.0211)
-.1301'
(.0257)
-.0688
(.0844)
-.1755
(.1108)
In (OverallReputation)
.0515* (.0071)
.0557* (.0084)
Bidder'sFeedbackRating
.0003* (.0001)
.0002 (.00016)
Constant
.6831* (.0382)
1.038* (.0527)
Numberof observations
1,248
R2
.1336
1,248
.0571* (.0267)
-.0007
(.0007)
.7256* (.1438)
79
.0586
-.0012
(.0335)
(.0008)
1.1194* (.2164)
79
.2076
In the first two regressions,the dependentvariablesare all of the bids (not just the winning bid), divided by their book
values. In the last two regressions,only bids submittedin the last minuteareincluded.In the IV regressions,the instrument
used is the minimumbid/book value.
of the four specifications,the bids are negatively correlatedwith the numberof bidders in the
auction.These coefficients are significantat conventionallevels.
We take the results of this regression-basedtest as suggestive, but not conclusive, evidence
against a pure private-valuesmodel. Also, we recognize that richerprivate-valuesmodels, such
as those with affiliation,can potentiallydisplay a negative correlationbetween the bids and the
numberof bidders (see Pinske and Tan, 2000). The correctmodel of bidding on eBay auctions
probably involves both a common-value and a private-valueselement. However, the analysis
and structuralestimationof such a model, includingreserveprices and endogenousentry,is not
tractableusing currentlyavailablemethods. Therefore,the rest of the analysis in this article is
based on a purecommon-valuemodel of bidding.
4. Late bids
*
An importantstep in building a structuralmodel of equilibriumbidding in eBay auctions
is to understandthe timing of the bids. In this section we demonstratethat in a common-value
framework,waiting until the end of an auctionto bid is an equilibriumphenomenon.17The eBay
auction formatdoes not fit neatly underthe categoryof "open-exitEnglish auctions"studiedby
Milgrom and Weber (1982).18 In their model, a bidder cannot rejoin the auction once she has
dropped out. Therefore,her dropoutdecision conveys informationto other bidders, who will
update their estimates of the item's true worth and adjust their dropoutpoints. In contrast,on
eBay agents have the opportunityto update their proxy bids anytime before the auction ends.
Withthe option to revise one's proxy bid, biddersmight not be able to infer others' valuationsby
observing theirdropoutdecisions, since dropoutscan be insincere.
Assume that bidders are risk-neutral,expected-utilitymaximizers. We abstractfrom any
17
Roth and Ockenfels (2000) demonstratethat last-minutebidding is (one of the many) equilibriain a stylized
model of eBay auctionswith privatevalues. The crucialfeatureof theirmodel is thatthereis a nonzeroprobabilitythata
last-minutebid will be lost in transmission.
18 Observethatthe
eBay auctionformatis also differentfrom traditionaloral-ascendingauctions,since the proxybidding system precludes"jump"bids-a biddercannot unilaterallyincrease the going price to the amountshe desires.
The auctionprice is always determinedby the proxy bid of the second-highestbidder.
? RAND 2003.
338 /
THE RAND JOURNALOF ECONOMICS
cross-auctionconsiderationsthatbiddersmighthave, andmodel strategicbehaviorin each auction
as being independentfromotherauctions.19Givenourobservationthatthereis not a big difference
in bid levels across bidders with differentfeedback points, we assume that bidders are ex ante
symmetric.
The informationalsetup of the game takes the following familiarform: let vi be the utility
that bidderi gains from winning the auction, and let xi be her privateinformationon the value
of the object. We use the symmetriccommonvalue model of Wilson (1977), in which vi = v is a
randomvariablewhose realizationis not observeduntil after the auction. In this model, private
informationfor the bidderis xi = v + ?i, where Eiare i.i.d. Suppose thatthe minimumbid is zero
and thatthere is no secret reserve.
Let us view the eBay auctionas a two-stage auction.Takingthe total auctiontime to be T,
let the first-stageauction be an open-exit ascending auction played until T - e, where e << T
is the time frame in which bidderson eBay cannotupdatetheir bids in response to others. The
dropoutpoints in this auction are openly observed by all bidders,who will be orderedby their
dropoutpoints in the first-stageauction,01 < 02 < ... < On,with only Onunobservable(bidders
can only infer thatit is higherthanOn- ). The second stage of the auctiontranspiresfrom T - e
to T and is conductedas a sealed-bidsecond-priceauction.In this stage, every bidder,including
those who droppedout in the firststage, is given the optionto submita bid, b. The highest bidder
in the second-stageauctionwins the object.
Given the above setup, we make the following claim:
Proposition1. Biddingzero (or not biddingat all) in the firststage of the auctionandparticipating
only in the second stage of the auctionis a symmetricNash equilibriumof the eBay auction.In
this case the eBay auctionis equivalentto a sealed-bidsecond-priceauction.
This result is based on the following lemma, whose proof is in the Appendix:
Lemma1. The first-stagedropoutpointsOi,i = 1, ..., n cannotbe of the form0(xi), a monotonic
function in bidderi's signal.
This lemma leads to the conclusion thatthe eBay auctionformatgeneratesless information
revelation during the course of the auction than in the Milgrom and Weber (1982) model of
ascendingauctions.
If we takethe extremecase in which nobodybids in the firststageof the auction(oreverybody
bids zero), the second stage becomes a sealed-bid second-price auction, where each bidder
knows only her own signal. In this case, the symmetric equilibriumbid function, as derived
by Milgrom and Weber,will be b(x) = v(x, x), where v(x, y) = E[v xi = x, Yi = y], with
yi = maxjy{1....n}{\i xj}.20
5. A structuralmodel of eBay auctions
*
In this section we specify a structuraleconometricmodel of eBay auctions.The structural
parameterswe estimatearethe distributionof bidder'sprivateinformationanda set of parameters
that govern entry into the auction. Estimates of the structuralparametersallow us to better
understandthe determinantsof bidder behavior.Also, the structuralparameterscan be used
to simulate the magnitude of the winner's curse in this market and the impact of different
19We
recognize that similarobjects are often being sold side by side on eBay and thatbidding strategiescould be
differentin this situation.We believe thatinvestigatingsuch strategiesis an interestingavenueof futureresearch;however,
for the sake of tractability,we have chosen to ignore this aspect of bidding. We recognize, however,that this featureof
eBay auctionsmight play a role in explaininglate bidding.
20 To ruleout
profitabledeviations,observethe following: if bidderj were to unilaterallydeviatefromthe proposed
equilibriumof not biddingat all in the firststage, then the proxy-biddingsystem of eBay would indicatethatthis bidder's
signal xj is greaterthanzero and not muchelse, since bidderj's bid would advancethe "maximumbid"to one increment
above zero;thereforebidderj would not be able to signal or "scare"awaycompetition.Further,since signals areaffiliated,
E[v I xi = x, yi = x, xj > 0] > E[v I xi = x, Yi = x], so bidder j would unilaterallydecrease her probabilityof
winning.
? RAND 2003.
BAJARIAND HORTACSU /
339
reserve prices on seller revenue.Motivatedby the results of the previous sections, we model an
eBay auctionas a second-pricesealed-bidauctionwhere the numberof biddersis endogenously
determinedby a zero-profitcondition in a mannersimilarto Levin and Smith (1994).21Assume
there are N potentialbiddersviewing a particularlisting on eBay. However,not every potential
bidderenters the auction, since each enteringbidderhas to bear a bid-preparationcost, c. In the
context of eBay, c representsthe cost of the effort spent on estimatingthe value of the coin and
the opportunitycost of time spentbidding.Afterpaying c, a bidderreceives her privatesignal,xi,
of the common value of the object, v. Like Levin and Smith (1994), our analysis focuses on the
symmetricequilibriumof the endogenous-entrygame, in which each bidderenters the auction
with an identicalentryprobability,p. In equilibrium,entrywill occur until each bidder'sex ante
expected profitfrom enteringthe auctionis zero.
Let Vnbe the ex ante expectedvalue of the item when it is commonknowledgethatthereare
n biddersin the auction.Let Wnbe the ex ante expected paymentthe winning bidderwill make
to the seller when there are n bidders. Since bidders are ex ante symmetric,a bidder's ex ante
expectedprofit,conditionalon enteringan auctionwithn bidders,will be equalto (Vn- Wn)/n-c.
The seller can set a minimumbid, which we denote by r. Define Tn(r)to be the probabilityof
tradegiven n and a minimumbid r. Given these primitives,the following zero-profitcondition
will hold in equilibrium:
T , (Vn
Wn)(
(1)
pTnin(r)
In
is the probabilitythat there are n biddersin the auction,conditionalon entry
equation
(1),
In equation(1), p/ is the probabilitythatthere are n biddersin the auction,conditionalon entry
by bidderi.
As in LevinandSmith(1994), we assumethatthe unconditionaldistributionof bidderswithin
an auctionis binomial,with each of the N potentialbiddersenteringthe auctionwith probability
p. On eBay, the potential numberof bidders is likely to be quite large comparedto the actual
numberof bidders.Therefore,we will use the Poisson approximationto the binomialdistribution.
Withthe Poisson approximation,the probabilitythattherearen biddersin the auction,conditional
on bidderi's presence,is p = e-~X.n-1/(n- )!,n = 1,... oc, where = E[n] = Np. The mean
of the Poisson entryprocess, ., will be determinedendogenouslyby the zero-profitcondition(1).
In equilibrium,Xis assumedto be common knowledge.
Conditionalon entry, we will assume that the bidders will play the "last-minutebidding"
equilibriumof the game derivedin the previous section. However,biddersdo not observe n, the
actual numberof competitors,when submittingtheir bids. Therefore,by the results in Section
4, the equilibriumbidding strategiesare equivalentto a second-pricesealed-bid auction with a
stochasticnumberof bidders.We also allow the sellers to use a secret reserveprice. To abstract
from any strategicconcernsfrom the seller's side and to make structuralestimationtractable,we
assume that the biddersbelieve that the seller's estimate of the common value comes from the
same distributionas the bidders' estimates.22Given a signal x, we assume that the seller sets a
secretreserveprice using the same bid functionas the biddersin the auction.Therefore,from the
bidders' perspective,the seller is just anotherbidder.With this assumption,the only difference
between the secret reserve-priceauction and a regularauction is that now all potential bidders
know that they face at least one competitor-the seller. Following Milgrom and Weber,define
yn) = maxjy{l ...n}\i {xj } to be the maximumestimate of the n bidders,excluding bidderi. Let
c=
Np
21We
acknowledgethat this equilibriumdoes not entirely fit the observedbiddingbehavioron eBay; as Figure 2
displays, not all of the bids arriveduringthe last 5 to 10 minutes of the auction. However,our finding in Section 3 that
early-biddingactivitydoes not affect the winning bid is consistentwith Lemma 1: early bids cannotbe takenseriouslyin
the "secondstage"if bidderscan jump in at the last second.
22
This strategyis optimalfor the seller if his valuationfor the item is the same as the bidders'and the last-minute
bidding equilibriumis used. We recognize, however,thatit is not clear whetherit is rationalfor the seller to put the item
up for sale in this environment.However,in orderto specify the likelihood function,the model must accountfor bidders'
beliefs aboutthe distributionof the reserveprice.
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340 / THERANDJOURNAL
OFECONOMICS
v(x, y, n) = E[v I xi = x, y(n) = y] denote the expected value of v conditional on bidder i's
signal and the maximumof the other bidders' signals. The propositionbelow characterizesthe
equilibriumbid functions.
Proposition3. Let p/ (X)be the (Poisson)p.d.f. of the numberof biddersin the auctionconditional
upon entryof bidderi into the auction.
(i) The equilibriumbid functionwith a minimumbid, r > 0, and no secret reserveprice is
,Eb(x
A)=
)
b(x,
=
()
I
E00=2 (X, X, n)fy ()(X x)pi
n=2of, ( )nX jPxp.lt )
f
for x > x*
(2)
n
0
forx <x*,
wherex* is the cutoff signal level, above which biddersparticipatein the auction,and is given by
x*(r, )) = inf{EnE[v
I xi = x, Yi < x, n] > r}.
(ii) When there is a minimumbid r and a secret reserveprice, the equilibriumbid function
becomes
n=2 v(X, , n)f(n
bSR(x, ) = .
Yn=2
OI
fn)fyx\(x)p
| x)pX(i
)
i-
0
>
fo
for x > x*
(3)
forX < *,
where the screeninglevel, x*SR(r, X), is in this case the solution to
?
n=2
fvv vfx2(x*Iv)Fxn-(x* I v)fv(v)dv
>3pn1(X
'2(x*
iKvf
=r.
v)Fxn- (*
(4)
v)fv(v)dv
Interestedreaderscan consultthe Appendixfor the derivationof the aboveexpressions.When
the numberof biddersis known to be n, it is well known (see Milgrom and Weber,1982) that a
bidderwith a signal of x should bid her reservationvalue v(x, x, n). In Proposition3(i), we see
that when entry is stochastic and there is no minimumbid, the equilibriumis a straightforward
modificationof the equilibriumto a second-price sealed-bid auction with a known numberof
bidders.The bid functionin this case is a weighted averageof the bids in the deterministiccase,
where the weighting factor is given by
fy(n)(X
I )pI(X)/[Pni =2 fy()(x
x)p '()]. Bidders with
signals falling below the screeninglevel x*(r, X) choose to bid zero. This result is similarto the
analysis in Milgrom and Weber(1982). Proposition3(ii) covers the case when there is a secret
reserve and a minimumbid. The bid function is identicalto 3(i), except for the fact thatbidders
treatthe seller as an additionalbidderwho always enters.
Empirical specification of the information structure and entry process. We assume
that both v and x are normally distributed.The normality assumption is used because it is
computationallyconvenientandthe normaldistributionseems to do a reasonablejob in matching
the observeddistributionof bids.23We specify the priordistributionfv(v) of the common value
]
in auction t to be N(,tt, ct2), where
ltt = ,B1BOOKVALt+ B2BLEMISHt ?BOOKVALt - 2.18
+ /4BLEMISHt?BOOKVALt.
at = B3BOOKVALt
(5)
This specificationis motivatedby our previous findingthat objects with blemishes are sold at a
discountin an auction.Also, since agentsmustpay shippingandhandlingfees, we subtract$2.18
23
fee.
Negative valuationsimplied by a normaldistributionarejustifieddue to the presenceof a shippingandhandling
? RAND 2003.
BAJARIANDHORTACSU/ 341
in computingtt, the averageshippingandhandlingfee for the mint/proofsets.24We assumethat
the distributionof individualsignals fx(x I v) conditionalon the common value v is N(vt, kt2),
where k = P5 is a parameterto be estimated.
The zero-profitcondition (1) suggests that in our estimationprocedure,we should treatthe
entry cost, c, as an exogenous parameterand endogenously derive the equilibriumdistribution
of entrantsas characterizedby Xt. This is a computationallyintractableproblem. Solving for
it requiresthat we find the solution to a nonlinearequation, which can only be evaluatedapproximatelybecausethe expectationtermsinvolve multidimensionalintegrals.Instead,motivated
by the regressionin Table4, we use the following reduced-formspecificationfor it:
log it = P6 + /7SECRETt + f8LN(BOOKt) + B9NEGATIVEt
MINBIDt
MINBIDt
+ fBloSECRETt
BOOKAL + Ill(l - SECRETt)BOOKVAL.
BOOKVALt
BOOKVALt
(6)
In equation(6), the entry probabilitydepends on the book value, the seller's negative feedback,
and the reserveprice. Ourreduced-formspecificationfor it is consistentwith equations(1) and
(5), since these equationsimply thatthe distributionof entrantsshould be a functionof the book
value, the dummyfor a blemish, and the reserveprice used by the seller.
E
Estimation. The model outlinedin the previoussubsectiongeneratesa likelihoodfunction
in a naturalway. Let Q2t= {SECRETt,rt}, t = (1, ..., pfl) and
Zt = (BOOKVALt,
BLEMISHt,NEGATIVEt).
Let b(x I| t, f, Zt) be the equilibriumbid function conditionalon Qt, the structuralparameters
8, and the set of covariates Zt. Since bids are strictly increasingin x, an inverse bid function
exists, which we denote as (p(b Qt, 8, Zt). Let fb(bi I Qt, , Zt, v) denote the p.d.f. for the
distributionof bids conditionalon Qt, B, Zt, and the realizationof the common value v. By a
simple change of variablesargument,fb must satisfy the equationbelow:
fb(bi I Qt, , Zt, v) = fx(((b I Qt, , Zt) I v, k, ot)t'(bi I Qt,,
Zt).
(7)
It is not possible to directly use equation (7) for estimation,since the econometriciandoes not
observe v. In equation(8), we derive the likelihood functionconditionalon the informationthat
is availableto the econometrician.To do this, we must integrateout the latentvariablev. Let nt
be the numberof bidderswho actually submita bid in auction t, and let (b, ..., bn,) be the set
of bids observedby the econometricianin auction t. Let fb,(bl, b2, ..., bn, t,Ot, Q't, ) be the
p.d.f. for the vector of bids observedin auctiont afterwe have integratedout the latentcommon
value. In an auction with no secret reserve,but a minimumbid rt > 0, let x*(rt, At), be defined
as in equation(2). Then
fb, (bl, b2, . .. bbn, I| tt, Ot, Qt)
N
= EP(j
x
I )t)
Fx(x*
v, k, oat)j-n
x (1 - Fx((b,
Qt,
Z)
v, k,
t)l{n>l
J -oo
nt
x
24 Ourdata on
? RAND 2003.
fb(bi I Qt, B
n
i=2
Zt, v)fv(v I Ilt, ort)d,
(8)
shippingand handlinghas a lot of missing values, so we chose not to use the individualvalues.
342 / THERANDJOURNAL
OFECONOMICS
where 1{n > 1} is an indicatorvariablefor the event that there is at least one bidder in the
auction and N is an upper bound on the numberof entrants.25Observe that this specification
allows us to assign a positive likelihood to auctions with no bidders.For auctions with a secret
reserve price, the likelihood function in (8) is modified to reflect that biddersview the seller as
anothercompetitor(who is certain to be there), and as a result the bidders use a differentbid
function.
Whencomputingthese likelihoodfunctions,thefollowing resulthelpsimmenselyin reducing
the computationalcomplexity of the estimationalgorithm.
Proposition4. Let b(x I ut?,a?, k, .) be the bid functionin an auctionwherethe common value,
v, is distributednormallywith mean ut?and standarddeviationa?, generatingsignalx with mean
v and variancek(? )2. If we change the scaling of the common value and signal distributionsuch
thatthe scaled commonvalue, v', is distributedwith meana1l/? + a2 and standarddeviationa la?,
and the scaled signal, x', is distributedwith mean v' and variancek(al a?)2, the equilibriumbid
correspondingto the scaled signal, x', underthe scaled signal and common-valuedistribution,
can be calculatedas aolb([x'- al]/Oa2 I i0, a?, k, X) + a2.
The Appendix contains a proof of Proposition4. This is a key result in our estimation
procedurebecause it allows us to computethe bid functiononly once for a base-case auction.We
can then apply an affinetransformationto this "precomputed"functionto findthe bid functionin
an auction where the distributionof the common value has a differentmean and variance.26
We use recentlydevelopedmethodsin Bayesianeconometricsfor estimation.We firstspecify
a priordistributionfor the parametersand then apply Bayes' theoremto study the propertiesof
the posterior.We use Markov-chainMonte Carlomethodsdescribedin the Appendixto simulate
the posteriordistributionof the model parameters.
There are four advantagesto a Bayesian approachwhen estimating parametricauction
models. First, Bayesian methods are computationallysimple, and we find these methods much
easier to implementthanmaximumlikelihood,which involves numericaloptimizationin several
dimensions. Second, it is not possible to apply standardasymptotictheory in a straightforward
manner in many auction models because the support of the distributionof bids depends on
the parameters.Third, confidence intervals in a classical frameworkare commonly based on
second-orderasymptotic approximations.Our results are correct in finite samples and do not
requireinvoking the assumptionsused in second-orderasymptotics.Fourth,Porterand Hirano
models, Bayesian methodsareasymptotically
(2001) have foundthatin some parametric-auction
efficient while some commonly used classical methodsare often not efficient.The details of how
to computeour estimatescan be found in the Appendix.27
6. Results
*
Table 7 reportsourparameterestimatesfor the structuralmodel.28Since the posteriormean
for P1,the coefficientmultiplyingBOOKVALin equation(5), is almost 1, we infer thatpublished
book values do indeed provide point estimates for how the coins are valued on the market.Our
estimates in Table 7 demonstratethat blemishes reduce the value of the object, but the standard
deviationof the posterioris quitelarge.This could stem fromthe fact thatthese "blemishes"were
25 With a Poisson
process, N = oo. But this makes it impossible to accountfor the truncationin bids due to the
reserve price, hence we take N to be sufficientlylarge so that the differencebetween the truncatedPoisson distribution
and the full distributionis negligible. We use N = 30.
26 Other
researchers,such as Paarsch(1992) and Hong and Shum (2002), have exploitedlinearityof bid functions
in differentmodels.
27 One traditional
objection to Bayesian methods is that the specificationof the prioris typically ad hoc. In our
analysis, thereare only 11 parametersand over 1,000 bids. Ourparameterestimatesare thereforefairly robustto changes
in the priordistribution.
28All resultsarefroma
posteriorsimulationof 30,000 drawswithan initial"bum-in"of 10,000 draws.Convergence
to the posteriorappearedto be rapid.
? RAND 2003.
BAJARIAND HORTAtSU /
343
PosteriorMeans and StandardDeviations:Full Sample
TABLE7
Parameter
Variable
BOOKVAL
-z
BOOKVAL* BLEMISH
a
Standard
Deviation
.9911
.0387
-.1446
.1628
BOOKVAL
.5599
.0260
BOOKVAL* BLEMISH
.0326
.0260
.2545
.0259
1.4069
.0897
k
CONSTANT
SECRET
X
Mean
-.2857
.1080
.0883
LN(BOOK)
.0276
NEGATIVE
-.0414
.0228
SECRET* (MINBID/BOOKVAL)
-.2514
.3268
(1 - SECRET)* (MINBID/BOOKVAL)
-.7893
.0864
coded by "noncollectors"who might have introducednoise into our coding. A $1 increase in
BOOKVALincreases the posteriormean of oa by $.56. Therefore,a large amountof variation
between the values of the differentcoins in our datasetis not capturedby the book value and the
presence of a blemish alone.
Our estimatesimply that the standarddeviationof the bidders' signals, xi, is only k/2a =
28.25% of the book value. Therefore,the bidders' signals of v are much more clustered than
the unconditional distributionof v, suggesting that bidders are quite proficient in acquiring
informationaboutthe items for sale.
The main determinantsof entry are the reserve-pricemechanism,the minimumbid level,
and the book value of the coin. Negative seller reputationwas found to reduce entry,while the
minimumbid level in a secretreserve-priceauctiondoes not affectthe entrydecision significantly.
To gauge the fit of our model, we simulatedthe auctionsin our datasetusing our parameter
estimates. That is, given the auction-specificvariables,we generatednt signals for each auction
t, where nt was generatedby the estimatedentry process. We then computedthe bid functions
correspondingto the signals. The top two plots in Figure 2 comparethe actualand simulatedbid
distributions.
FIGURE2
SIMULATED
VERSUS ACTUALBID DISTRIBUTIONS
.2 -
sdbids/bookvalue
Histogramof simulate
__
0
.5
1
.2 -
2
2.5
Histogiramof bids/bookvalue
f l
.1 I
1.5
I
0
.5
1
.2 i^: 1
.1
0
? RAND 2003.
i
.5
1.5
2
2.5
Histogramof simulatedbids/book
Hg
value,l,
1
1.5
23=.3
2
2.5
344 / THERANDJOURNAL
OFECONOMICS
FIGURE3
SIMULATED
VERSUS ACTUALWITHIN-AUCTION
BID DISPERSION
.6
biddispersion:simulated
Histogramof within-auction
.4 -
.5
.1
.2
.15
.25
.3
.35
.45
.4
.5
.6
biddispersion:actual
Histogramof within-auction
.4
.2
0o
F]r7m
.5
.15
,15
n
.1
.15
I
.25m i,,M.
.3
.2
.25
.3
r
.5
.35
.4
.4
.45
.5
Althoughthe meanvaluesof the bid distributionsappearto match(themeanfor the simulated
distributionis 82.7% of the book value, comparedto 83% for the actual data), Figure 2 tells us
that our estimate of the varianceof cross-auctionheterogeneity(83 in particular)is higher than
what seems apparentin the data.The bottomplot in Figure2 is the simulateddistributionof bids
when we set B3 = .3. This seems to matchthe actualbid distributionbetter.
In interpretingthe above plots, note that Figure 2 alone does not tell us anythingaboutthe
within-auctiondispersion of bids. However, our empirical specification attemptsto match the
within-auctiondispersionof bids and the cross-auctiondispersion.Figure3 is a histogramof the
simulatedandactualdispersionof bids. We findthatthe meandispersion(differencebetweenmax
andmin) of the simulatedbids is 23%of the book value.This is lower thanthe actualdispersionof
bids, which is 32%.In tryingto matchall momentsof the data,our specificationmakes a tradeoff
between overstatingcross-auctionheterogeneityand understatingwithin-auctionheterogeneity.
We believe that adding unobservedheterogeneityinto our econometric specification will help
achieve a tighterfit;however,such an exercise would greatlycomplicatethe estimationprocedure
and is beyond the scope of this article.
To evaluateourparameterestimatesfor the entryprocess, we comparethem with the results
in Table4. The signs of the coefficientson the independentvariablesare in accord.However,the
structuralmodel accountsfor the fact that some bidderswho "enter"the auctionmay not submit
a bid because their signal xi is less thanthe screeninglevel. The structuralmodel predictsthaton
average,3.3 biddersbear the entrycost c. The averagenumberof bids submittedis 3.01.
Controllingfor the minimumbid level, bidder entry is adversely affected when there is a
secretreserve,accordingto both Table4 andTable7. This is plausible,since holdingthe bid level
fixed, the ex ante profitof a bidderenteringa secretreserve-priceauctionis lower, since she will
also have to outbidthe seller.
We also attemptedto explore the robustnessof our results by modifying our likelihood
function in the spiritof Haile and Tamer(2003). The final bids we observe might not reflect the
maximumamounta bidderwas willing to pay. For instance, suppose that bidderA, early in the
auction,submitsa proxybid thatwas muchlower thanher valuation.If a numberof very high bids
arrivein the middle of the auction,bidderA might not findit worthwhileto respondif the current
high bid exceeds her maximumwillingness to pay. Following Haile and Tamer,we interpretthe
final bid of each bidder as a lower bound on v(x, x). Similarly,we interpretthe winning bid as
an upper bound on a bidder's conditional valuation.Haile and Tamerargue that these bounds
hold true for a very rich set of bidding strategies,even including some nonequilibriumbidding
strategies.29
29We
acknowledge that this bounding strategy does not take into account certain complex dynamic elements
like "learningfrom others' bids" that might be underlyingthe data-generationprocess on eBay. The bounds are also
? RAND 2003.
BAJARIANDHORTACSU/ 345
FIGURE4
WINNER'SCURSE EFFECT
90 -Bid functionsforthe
.
80 - representative
auction
70
--
b(x)
- - - b(x)withan extrabidder
- b(x)withsigma =.1
60 ....... b(x)= x
50
.
.
..
.:
-
40 30 20 -
-.5^
10 ,-...
10
20
30
40
50
60
x- estimate($)
70
80
90
The maindifferenceresultingfromthe use of boundsin ourlikelihoodfunctionspecifications
decreasedthe parametercontrollingthe varianceof the signaldistribution(/83)to .28. As discussed
above, this is an improvementover ourpreviousestimatesin termsof matchingthe across-auction
dispersionof bids. However,this estimate predictsa much tighterdistributionof bids within an
auction (10% of book value) than is seen in the data-under the assumptionthat biddersplay
the equilibriumof the second-priceauctiongame when forming the predicteddistribution.This
pointsout the maindifficultyof applyinga "bounds"approachin oursetting:since the econometric
specificationdoes not specify an equilibriummodel, even if structuralparametersare estimated
accurately,it is difficult to make in-sample or out-of-samplepredictions-or to conduct policy
simulationswith the estimatedmodel-unless the boundsused for estimationimply informative
bounds on predictedequilibriumbehavioror policy outcomes.30Therefore,we elect to proceed
with the parameterestimatesin Table7, which are generatedby an empiricalspecificationbased
on a fully specified equilibriummodel of bidding.Nevertheless,we believe thatfurtherresearch
in using a bounds approachis well warranted.
Quantifying the winner's curse. We definethe winner'scursecorrectionas the percentage
o
decline in bids in response to one additionalbidder in the auction. To quantify the magnitude
of the winner's curse correction, we computed the bid function given in equation (3) for a
"representative"auction using the estimated means of the parametersin Table 7 and sample
averages of the independentvariablesenteringequations (5) and (6). The bid function for this
auctionis plottedin Figure4. In this figure,we also plottedthe bid functionin an
"representative"
auctionwhere we change X to add one more bidderin expectation(with all otherparametersand
independentvariableskept the same). We find thataddingan additionalbidderdecreasesbids by
$1.50 in this $47 auction,with a standarderrorof $.20. In otherwords,the winner'scursereduces
bids by 3.2% for every additionalcompetitorin the auction,with a standarderrorof .4%.
The bid function in the representativeauctionis very close to being linear and has a slope
of .9 with an interceptof -.85. Therefore,a bidderwith a signal xi equal to the book value of
$47 should bid $41.5. This is $5.5 less than in a private-valuesauction, which is plotted as the
45-degree line in Figure5. Hence we see thatthe winner'scursecorrectiondoes play an important
role in transactionswithin this market.
To understandhow reducinguncertaintyabout the value of the object affects the bids, we
plot the bid function for the case where the standarddeviation of v is only 10% of book value
problematicwhen participationis costly. For instance, a bidder might not respond to a bid below its valuationif the
chances of eventuallywinning were sufficientlylow.
30 Haile and Tamer
(2003) managed to find informativebounds for the optimal reserve-pricepolicy in their
symmetricindependentprivate-valuesetting. Unfortunately,we have not managedto find an extension of theirresultsto
the common-valuesetting studiedhere.
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346 / THERANDJOURNAL
OFECONOMICS
FIGURE5
EXPECTEDNUMBEROF BIDDERSAS A FUNCTIONOF MINIMUM
BID
4.5
4-o
3.5e
) D
3
-
2.5 -
0
I
I
.2
.4
I
I
.5 -
1
.6
.8
Minimum
bid/book
value
1.2
1.4
(as opposed to 52%). As expected, the bids increase:a bidderwith a signal xi of $47 bids $45.
Clearly, sellers can increase profitsby reducingthe bidders' uncertaintyabout the value of the
coin set.31
m Quantifying the entry cost to an eBay auction. Ourfindingthatthe minimumbid plays an
importantrole in determiningentry into an auctionnaturallyleads us to questionthe magnitude
of the implicit entrycosts faced by the bidders.We can estimateentrycosts using the zero-profit
condition,equation(1).
Using our parameter estimates, assuming that the item characteristics are from the
"representativeauction"used in the winner'scurse calculation,we can calculateall of the terms
on the right-handside of equation(1). This allows us to infer the value of c. Using the simulation
methoddescribedin Bajariand Hortacsu(2000), we findthatthe impliedvalue of c is $3.20, with
a standarderrorof $1.48.
The implied entry cost changes with auction characteristics.However,the entry cost as a
percentageof the book value seems stable. For example, for an auction with book value $1,000
and a minimumbid of $600, the computedentrycost is $66.
D Optimal minimum bids with endogenous entry. Standardresults in auction theory
regardingoptimal minimum bids in private-valueenvironmentstell us that the seller should
set a minimumbid above his reservationvalue. (See, for example, Myerson (1981).) However,
when entry decisions are endogenously determined,the seller has to make a tradeoffbetween
guaranteeinga high sale price conditional on entry versus encouragingsufficientparticipation
to sell the object. In Figure 5, we use the zero-profitcondition in equation (1) to compute the
equilibriumexpected numberof biddersfor differentvalues of the minimumbid. The figurewas
auctionof the previoussection, using an entrycost of $3.20.32
generatedfor the "representative"
When the minimumbid is zero, on average4.5 biddersenterthe auctionas opposed to only 2 if
the minimumbid is set to 80% of the book value.
Suppose that the seller sets the minimumbid in orderto maximize the following objective
function:
}
}
bid I r] + (1 - Pr{Sale r})Residual
E[Revenue] = Pr{Sale r}E[Winning
value,
where the residualvalue is the seller's valuationof the objectin dollarsif he fails to sell the object
to any bidder.In Figure6, we plot differentminimumbid levels on the horizontalaxis, andon the
The full effect would accountfor changes in the entryprocess as well.
Since this amounts to finding the zeros of a nonlinearequation, we used the Newton-Raphsonalgorithmto
numericallyfind the equilibriumexpected numberof bidders.
31
32
? RAND 2003.
BAJARIANDHORTA(SU / 347
FIGURE6
BID
EXPECTEDREVENUEAS A FUNCTIONOF MINIMUM
55
a)
50 -
c
35 -
...................
Residual value = Book value
25
0
.2
1
.4
1
1
1
.6
.8
Minimumbid/book value
1.2
1.4
vertical axis we plot the correspondingexpected revenuesof the seller. We make these plots for
three different residual values. We see that for a $47 book value coin, a seller whose residual value
is 100%of book value maximizes revenueby settinghis minimumbid equal to about90% of the
book value. Similarly, sellers with residual values of 80% or 60% of the book value maximize
expected revenue by setting the minimum bid to 10-20% less than their reservation value. Hence
the optimumminimumbid policy appearsto be for sellers to set the minimumbid approximately
equal to theirreservationvalue. The results in Figure6 also demonstratethat setting a minimum
bid 10%or 20% over the book value will reduce seller revenues.However,setting the minimum
bid at 10-20% less thanbook value is fairlyclose to revenuemaximizing.Note thatin Table3, we
found that sellers tend to set their minimumbid at significantlyless than the book value. Figure
6 suggests thatthis policy is consistentwith seller maximization.
In Section 3 we reportedthat the average minimumbid was set at 70% of book value for
auctions with no secret reserve price. Since we do not know the sellers' residualvalues for the
items on auction, we cannot assess whethereBay sellers are setting an optimal minimumbid.
However, considering that many sellers are coin dealers or coin shop owners who have lower
acquisitioncosts thanretailbuyersand have to balanceinventory,it makes sense thatthey would
have residualvalues less thantt bookhe
valu o e item. At 70%of book value, the residualvalue
of the seller of a $47 coin set would be $33. It is not implausibleto assume that the opportunity
cost of time for the typical seller to launch an auction is at least $5 or $10 less than the book
value. In general, we might expect the seller's residualvalue to depend on privateinformation.
To the extent that this is the case, the minimumbid might signal informationabout the quality
of the object to the buyers. In our analysis, we abstractaway from this problemfor the sake of
tractabilityand because there is a lack of appropriatetheoreticalmodels that we could use as a
basis for structuralestimation.
Secret reserve prices. Why are secretreserveprices used on eBay? The use of these prices
o
as opposed to observableminimum bids has been somewhat of a puzzle in the auction theory
literature.Forexample,Elyakimeet al. (1994) show thatin anindependentprivate-valuefirst-price
auction,a seller is strictlyworse off using a secret reserveprice as opposed to a minimumbid.33
A partialresolution to this puzzle was offered by Vincent (1995). In a common-valuesecondprice sealed-bid auction,keeping a reserve price secret can increase a seller's revenuefor some
particularspecificationsof utilities and information.However,the explanationin Vincent does
not completely characterizethe conditionsunderwhich secret reserveprices might be desirable,
and it also does not accountfor free entryby bidders.
The absence of a clear-cut, analytic answer to the secret reserve price versus observable
33 However, the secret reserve price might be beneficial if bidders were risk averse, as shown by Li and Tan (2000).
? RAND 2003.
OFECONOMICS
348 / THERANDJOURNAL
FIGURE7
EXPECTEDREVENUEDIFFERENCEBETWEENSECRETAND POSTEDMINIMUM
BID
POLICY
1.6
1.4D
1.2-
E
1-
I
.8- -
.4 m
w
.2
20
20
40
40
60
60
120 140
100 120
140 160
160 180
100
180 200
200
Bookvalue ($)
80
80
minimumbid question compels us to provide an answer based on simulationsof our estimated
model of bidding.34We comparetwo differentsellerpolicies in oursimulations.In the firstpolicy,
we assumethatthe seller uses a minimumbid equalto his residualvalue. In the second policy, we
assume thatthe seller sets a secretreserveprice equal to his residualvalue and sets the minimum
bid to zero.35These two policies will lead to differentlevels of bidderentry(due to equation(8))
anddifferentbiddingstrategiesconditionalon entry(dueto equations(3) and(4)). We simulatethe
expectedrevenueof the auctionersby drawingbiddervaluationsfromthe estimateddistributions.
We varythe book value of the objectfrom $5 to $200 in our simulationsand assumethatthe seller
has a residualvalue equal to 80% of the book value of the object.
Figure7 displaysthe resultsof this simulation.We findthatusing a secretreservepricehas a
discerniblerevenueadvantageover using a minimumbid. Fromourparameterestimatesin Table
7, the expected numberof bidderswill be largerunderthe second policy, where a secret reserve
price with no minimumbid is used, than underthe first policy. This "entry"effect of the secret
reserve price is countervailedby a winner's curse effect. Bidders shade their bids more in the
secret reserve-priceauction, since the numberof expected biddersis higher and since the seller
is always present as an additionalbidder.However,Figure 7 seems to suggest that the positive
effect of secret reserveprices on entryoutweighs the bid-shadingeffect.
Given our result above, one might wonder why all auctions on eBay do not have secret
reserveprices. For currentauctions,this is not a mystery.Since August 30, 1999, eBay has been
chargingsellers a $1 fee for using a secretreserve-but refundingthe fee if the auctionends up in
a sale. We would expect few sellers to use a secret reserveprice for items with low book values,
since these auctionsattractfewer biddersand would frequentlyresultin sellers paying the $1 fee.
The explanationgiven above still does not resolve the "toofew secretreserves"puzzle in our
dataset, since there was no $1 fee in the time frame we studied.The events surroundingeBay's
decision to institute the $1 fee are informative.eBay announcedthat its limited supportstaff
could not addressthe many complaints from bidders who were confused by the secret reserve
rule or who claimed that sellers were setting unreasonablyhigh reserves(Wolverton,1999). The
announcementof the reserve-pricefee, however,resultedin a vocal dissentby eBay sellers, some
of whom interpretedthe move as a way for eBay to increase its revenuesratherthan decrease
costs. This is an indication that at least some sellers thought that using a secret reserve was a
revenue-enhancingmechanismfor them. The reluctanceof 84% of sellers to use a secretreserve
can be attributedto the fact that bidderswere frequentlyconfused and frustratedby the reserve.
34Katkarand Lucking-Reiley(2000) approachthe same questionusing a field experimenton eBay.
35We assume thatthe seller sets his secretreserveprice equal to his residualvalue as a simplifying assumption.In
practice,the seller might find it optimalto follow an alternativestrategy.
? RAND 2003.
BAJARIANDHORTA(SU / 349
In fact, sellers using a secretreserveprice had (statisticallysignificant)higher negativefeedback
points than sellers who did not use a secret reserve price. However, secret reserve prices were
more likely to be used for items with higherbook value-items for which therewere highergains
from using the secret reserve. Hence, a potential explanationfor the puzzle of too few secret
reservescan be thatsellers were tradingoff customergoodwill for higherrevenues.In our model,
we do not include bidder"frustration"
with secretreserveprices. However,if we were to modify
the seller's objectivefunctionto include a disutilityfor a higherprobabilityof negativefeedback
from secret reserve prices, it is clear to see that we could rationalizethe increaseduse of secret
reservesfor items with large book values.
Anotherexplanationfor the use of secretreserveprices is that,holding all else fixed, Table7
demonstratesthatthe use of a secretreservediscouragesentryandthereforelowersthe probability
of a sale. It is costly for a seller to list the item twice, in terms of both time and listing fees. For
items with low book value, our results suggest that a low minimumbid and no secret reserve
will increase entry so that the auction will result in a sale. For an item with high book value,
the fixed cost of relisting the object is likely to be less of a concern for the seller. The increased
revenuefromusing a secretreserveprice will probablyoutweighthe expectedcosts of conducting
a second auction.
Our simulations of the effect of a secret reserve will be misleading if the secret reserve
price is not exogenous. The results of Milgrom and Weber (1982) suggest that sellers should
not intentionallyhide informationfrom bidders; otherwise, they will be punished with lower
expected sale prices. However, since bidding on eBay is probablymore complicated than the
MilgromandWebermodel, it is not clear whetherreal-life sellers will follow this policy. In Table
8, we reestimatedthe structuralmodel using only datafrom auctionswith a secret reserveprice.
The main differencewe find is thatthe estimatedmean valuationof the coins, as a percent
of the book value of the item, is lower for items sold with a secretreserve(88%of book value), as
opposed to the entiresample (99% of book value). Observe,however,thatthis estimationis done
using many fewer datapoints, and the posteriorstandarddeviationis 11%,bringingthe posterior
mean of the full sampleestimateswithina standarddeviation.Regardlessof the "significance"of
the differencebetween parameterestimates,this resultis consistentwith the fact that sale prices
of secret reserve items are lower (91% of book value) than items sold without a secret reserve
(96%of book value). Coupledwith the fact that50%of secretreserveitems go unsold, this could
be a sign that sellers are willing to bear the cost of repeatedlyrelisting lower-qualitybut highbook-valueitems in orderto "catch"a bidderwith an unusuallyhigh valuation.Anotherplausible
interpretationis thatthe change in the coefficients is due to fact thatour parametricmodel is not
completely flexible.36
7. Conclusion
*
In this articlewe studythe determinantsof bidderandsellerbehavioron eBay using a unique
datasetof eBay coin auctions.Ourresearchstrategyhas threeparts.First,we describea numberof
empiricalregularitiesin bidderand seller behavior.Second, we develop a structuraleconometric
model of bidding in eBay auctions. Third, we use this model to quantifythe magnitudeof the
winner'scurse and to simulateseller profitunderdifferentreserveprices.
Therearea numberof clearempiricalpatternsin ourdata.First,biddersengage in "sniping,"
submittingtheirbids close to the end of the auction.Second, thereare some surprisingpatternsin
how sellers set reserveprices. Sellers tendto set minimumbids at levels considerablyless thanthe
items' book values. Also, sellers tendto limit the use of secretreservepricesto high-valueobjects.
36
Ouranalysis abstractsaway from the problemof whetherthe seller chooses to drawa signal xi aboutthe object's
worth. We assume that in the model of auctions with no secret reserve, the seller does not draw a signal xi, and in the
model with a secretreserve,the seller does drawa signal. The seller's decision aboutwhetheror not to drawsuch a signal,
and the bidder'sbeliefs about the seller's signal (or lack thereof), can lead to alternativespecificationsof the structural
model.
? RAND 2003.
350 / THERANDJOURNAL
OFECONOMICS
TABLE8
PosteriorMeansandStandardDeviations:
Auctionswith
SecretReservePrices
Mean
Standard
Deviation
BOOKVAL
.8837
.1124
BOOKVAL* BLEMISH
.1249
.2010
BOOKVAL
.5036
.0369
BOOKVAL* BLEMISH
.0515
.0530
.2845
.0483
1.0344
.1301
.1150
.0251
Parameter
,I
a
Variable
k
CONSTANT
A
LN(BOOK)
NEGATIVE
(1 - SECRET)* (MINBID/BOOKVAL)
-.0385
.0282
.0230
.1796
Finally,thereis significantvariationin the numberof biddersin an auction.A low minimumbid,
a high book value, low negativeratings,and high overallratingsall increaseentryto an auction.
We demonstratethat last-minute bidding is an equilibriumin a stylized model of eBay
auctions with a common value, and that, to a first approximation,bidding in eBay auctions
resembles bidding in a second-price sealed-bid auction. Motivatedby our empirical findings,
we specified and estimated a structuralmodel of bidding on eBay with a common-value and
endogenous entry. We then simulate the structuralmodel to study the winner's curse and to
compute seller profitunderdifferentreserveprices. Ourfirstmain findingis thatthe expectation
of one additional bidder decreases bids by 3.2% in a representativeauction. Also, in this
representativeauction, bids are 10% lower than bidders' ex ante estimate of the value of the
coin set. Second, entryplays an importantrole in determiningrevenues.Since biddingis a costly
activity, a high minimumbid can reduce the expected profit of a seller by discouragingentry.
Finally, a seller who uses a secret reserveprice with a low minimumbid can in some cases earn
higher revenue than sellers who only use a minimumbid. However,the use of a secret reserve
discouragesbidderparticipation.
Thereareseveraldirectionsin which ourempiricalspecificationandestimationapproachcan
be potentiallyextended. First, as we mentionedin Section 3, the bidding environmentprobably
has both common-valueand private-valueelements. We focus on the pure common-valuecase
in light of the regression-basedwinner'scurse test and the natureof the collectible coins market.
However,we believe that modifying the empiricalspecificationto allow for both privatevalues
and a common value is an importantdirectionfor futureresearch.
Second, we model the entry decisions of biddersas simultaneous,not sequential.Relaxing
this assumption could lead the way to dynamic bidding strategies that could yield different
empiricalimplications.
natureof eBay andhave
Finally,we have not explicitly takeninto accountthe "multimarket"
treatedeach auction in isolation. Often, identical or very similar goods are sold side by side or
back to back. Since we also observesignificantvariationamongsellers' choices of minimumbids
and reserveprices, it mightbe possible to test theoriesof competitionbetween sellers as in Peters
and Severinov(1997, 2002) and McAfee (1993).
Appendix
*
Proofs of Lemma 1, Propositions3 and 4, descriptionsof the computationof the bid functions,and the estimation
algorithmfollow.
? RAND 2003.
BAJARI AND HORTACSU
/
351
Proof of Lemma1. The first-stagedropoutpoints Oi,i = 1, .., n cannot be of the form 0(xi), a monotonicfunction in
bidderi's signal.
Suppose Oi= 0(xi) is a monotonic functionin xi. Then, since 0(.) is invertible,at the beginningof the second stage
of the auction,the signals of biddersi = 1,..., n - 1 become common knowledge.Then, the second-stagebid functions
will be the maximumof the expected value of the object given the informationavailableto the bidders,and the bid they
committedto at the end of the first stage of the auction:
bin = max{0i, E[v I xl = 0-101) ...,
bn = max{0n, E[v I xl = 0-1(01)
> 0-l(On-1)}
Xn-_ =0-l(0n-1),Xn
=
X_
...,
if>
Xn = Xn}
-l(0n-1),
r
if > r.
Observe that this results in identicalbids for biddersi / n, providedthat their signals are high enough, since they
all form the same expectationof the common value. The expectationformed by the highest bidderin the first stage is a
little different,since she knows her own value with certainty.
In this situation,there is a profitabledeviation.If bidderj decreasesher dropoutpoint in the first stage to 5j < Oj,
then in the second stage,
bi~j(oj) < bij.6j(j),
since the conditionalexpectationsare decreasingin Ojby Milgromand Weber's(1982) Theorem5. But because all other
biddersare biddingsincerely in the firststage, j's bid will not change and she will unilaterallyincreaseher probabilityof
winning the auction.
Proof of Proposition 3. (i) Without loss of generality,focus on the decision of bidder 1. Suppose the bidders j =/ i
adopt strategies bj = b*(xj, X). Then the highest bid among them, conditional on there being n bidders, will be
Wn) = max2</<n bj(xj). Since b* is increasing in x, W) = b*(y(n), ) and the expected payoff of bidder 1 from
biddingb will be
En[E[(v
- b*(y),
I l= x, n > 2]] Pr(n > 2) + E[v I x = x] Pr(n = 1),
))X{b(y(n))b}
where X is the indicatorvariablefor the event thatbidder 1 wins. This is equal to
b*-(b)
En
=I
yI
00
,b*-1(b)
Lp(n, X)v(x, a, n)fy(n)(a Ix) -
x-
,n=2
+ E[v
Pr(n >2) + E[v x = x]Pr(n=1)
)]f(n)(O I x)dca n>2
[v(x, , n)-b*(o,
xl =x]Pr(n=
00
b*(a, A) p(n, A)f (n)((aIx)
n=2
do
),
where
(n - l)
Xi = x)
fyn)(x | x)= Pr(y(n)= x
fx v)F
x
)f(v
I x)dv
and fv(v I x) is the posteriordensity of the common value given the signal, with x, v being the lower supportof the signal
and common-valuedistributions,respectively.
The first-orderconditionwith respectto b yields
N
O=
N
, p(n, X)v(x, x, n)f(n)(X
n=2
x)- b*(x, .) Ep(n,
.)f (n)(x IX);
n=2
V
solving, we get b*(x, X) as in equation(2).
A sufficientcondition for b*(x, .) to be nondecreasingis the log-supermodularitycondition,
a
f(xlv)
av 1 - N(1 - F(x
>
I v))
-
(ii) The "screeninglevel"argumentin the case wherethereis a deterministicnumberof biddersfollows fromMilgrom
and Weber(1982): bidderswith signal level below x*(r, n) = inf{E[v I xi = x, Yi < x, n] > r} do not
participatein
the auction. McAfee, Quan, and Vincent (2002) extend this to the case in which thereis a stochasticnumberof bidders.
In the case analyzed here, the "expectedscreeninglevel" becomes x*(r, X) = inf{EnE[v Ixi = x, Yi < x, n] > r}. A
? RAND 2003.
352
/
THE RAND JOURNAL OF ECONOMICS
sufficientconditionfor b(x, A) to be increasingis given by McAfee, Quan,and Vincent(2002) to be
af -(x Iv)
) >0
av 1 - NX(l - F(x I v)) -
for all x > x*(r, A).
Q.E.D.
Proof of Proposition4. Considerthe von Neumann-Morgenster (vNM) utility of bidderi (the payoff given that other
bidderssubmitbids that are monotonicin their signals, x_i, and conditionalon the numberof competitorsn):
u(bi,b-i,
kz +
j=
+=
0
i,x-i,n)=
ix
- max{b_i(x_)}
ifbi > max{b_i(x_i)}
otherwise.
+
The term(kA, ,j=l
xj)/(n + k) is the posteriormean of the unknowncommon value, given all bidders' signals. Bidder
i maximizes the expectationof the above vNM utility and bids
bi = argmax EnE_i ui(bi, b-i, xi, x-i, n)
= argmax En Exi
bi
i
}
ax{b(
(Al)
-max{b-i(x-)}
The optimalbid in an auctionwherethe signals are scaled linearlyas r1x + r2 andwhereall otherbiddersfollow strategies
rlb-i(x-i) + r2 is
bl = argmax EnErlx_i+r2ui(bl, rlb-i(x-i) + r2, rlxi + r2, rlx-i + r2, n)
x+
-ixi
= argmax EnEr-i_,+r2,b>max{rbi (xi )+r2}
max{b_i(x_i)}J
n+k
= argmax EnE
E
br l2
x_i I
>max{b_i(x_i)}
n+ k
- max{b-i(x_-)}
.
Comparingthe last expressionwith the last line in (Al), we see that - r2 = bi, i.e. b' = rl bi + r2. Hence, the bids follow
the same lineartransformationas the signals.
O Properties and computation of the bid function. Althoughwe have characterizedthe equilibriumbid functions
pertainingto the eBay auctions, to use these findings in an econometricestimationprocedure,we have to evaluatethe
bid functionsexplicitly. For generaldistributionsof x and v, the expectationterms v(x, x, n) and the expressiondefining
the screening level have to be evaluatednumerically.If we assume that v ~ N(,/, oa2)and x I v ~ N(v, ka2), we can
computethese integralswith very high accuracyusing the Gauss-Hermitequadraturemethod(see Judd(1998) andStroud
and Secrest (1966) for details).
The bid functions, b(x, A), in an auction where ,i = 1, a = .6, and k = .3, look linear (with a slope of about .85)
and are indeed increasingin x. More interestingly,bids are also decreasingin A, the expected numberof biddersin the
auction. So the winner'scurse does increasewith the numberof competitors.
The expected screening level, x*(r, A) = inf{ENE[v xi = x, xi < x, N = n] > r}, can be calculated as the
solution of
N
p(l, )E[v
x] +
1(X*
fvf(x v)fv(v)dv2
v)Fx
x
*
rf2(
v)Fn-'(x* I v)fv(v)dv
f
fVU \2(X.
X)
p(n,vf
n=2
(A2)
Once again,normalityassumptionsallow us to computethe integralsin the aboveexpressionswith veryhigh accuracy
using the Gaussian quadraturemethod. However, we also need to solve for x* in expression (11) using a numerical
procedure.We use a simple Newton-Raphsonalgorithmto find this solution. For the auction with it = 1, a = .6, and
k = .3, the screening level appearsto depend on the minimumbid in a linearfashion. The screeninglevel is decreasing
in X and, as shown by Milgrom and Weber(1982), is above the minimumbid.
O Numerical computation of the inverse bid function. Likelihood function evaluationsrequirethat we find the
value of the inversebid function,X, for each bid in auctiont. Since a closed-formexpressionfor the inversebid function
does not exist, we might have to evaluate the bid function b(x I tit, at, k, At) at least a few times for each bid to find
the inverse of b(.) using a numericalalgorithmlike Newton-Raphson.However,as in the closed forms for equilibrium
bid functions, this requiresthe numericalevaluationof integrals.This is a costly step, since we have to evaluatethe full
likelihood about30,000 times in our posteriorsimulation,and with 1,500 bids in the data,the integralswould have to be
evaluatedat least 45 million times.
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353
To reducethe computationalcomplexity,we firstexploitthe linearitypropertyof the bid functions.Recall thatlinearity
implies that if b(x I Ai?,a?, k, A) is a pure-strategyequilibriumbid function to the auction where the common value is
distributednormallywith mean u?oand standarddeviation a?, then rlb(rlx + r2 I ,?, o?, k, A) + r2 is the equilibrium
bid functionin the auctionwhere the common value is distributedwith mean rl L?+ r2 and standarddeviationr a?. This
holds whetherthereis a known or randomnumberof biddersin the auction.Similarly,if thereis a posted reserveprice in
the auction,we can apply the lineartransformationto the screeninglevel x*(R, X, k). Thatis, if x*(R;,z?, oa?,X, k) is the
k, + r2 will be the screeninglevel in the transformed
screeninglevel in the originalauction,then rlx*(rl R + r; I ,0, 0 )k
auction.
How does linearity help us? Suppose at a particularevaluation of the likelihood function, the distributional
parametersare {[dt,rt} for auction t. Then we do not have to evaluate b(x I ,t, cr, k, X). Setting rl, = rt/a? and
r2t = tLt,- rltP?, we interpolatethe value of b((x - r2t)/rlt I I?, c?, k, X) from a set of precomputedvalues and return
rltb((x-r2t)/rlt I ,?, cr0, k, A)+r2t.Assumingthe interpolationis good enough,this scheme lets us avoidcomputingthe
Nash equilibriumin the course of likelihoodfunctionevaluations.The equilibriumis solved beforehandfor a normalized
set of parametervalues and adaptedto the parametervalues generatedby the posteriorsimulationroutine.
When invertingthe bid function,one can use a simple algorithmlike Newton-Raphsonwith relativeease, since the
bid functionis monotonicandb(x) = x is usuallya good startingvalue. Alternatively,one can evaluateb(x I ?o,a?, k, X)
on the grid used for interpolation,and insteadof using polynomialinterpolationto find b as a functionof x, k, X,one can
use polynomialinterpolationto evaluatex as a functionof b, k, X.This allows one to approximatethe inversebid function
directly,instead of having to evaluate the bid function several times until Newton-Raphsonconverges. This method is
much more efficient in termsof calculatingthe derivativeof the inversebid function (the Jacobian),since the derivative
of the polynomialinterpolationcan be writtenout in closed form and, once again, be evaluatedin one step.
O
Posterior simulation.
Let
Xt = (SECRETt,BOOKVALt,BLEMISHt,NEGATIVEt,MINBIDt)
be the auction-specificdatavectorandlet X denotethe vectorX = (X, ..., XT), where T is the totalnumberof auctions.
Let p(fB)denote the priordistributionof the parameters,L(BID I fB,X) denotethe likelihoodfunction,and p(P I BID, X)
denote the posteriordistributionof the vector of parametersfBconditionalupon the observeddata.By Bayes' theorem,
P(f I BID, X,) oc p(f)L(BID I 1, X).
Thatis, the posterioris proportionalto the priortimes the likelihood function.In manyproblems,we are interestedin the
expected value of some known functionof the parameterswhich we representas g(B):
E[g(p) I BIDSt, Xt] =
f g(p)p(p
I BID, X)df.
Examples include the posterior mean and standarddeviation of fB or the expected revenue from an auction with a
particularset of characteristicsX. Recentworkin Bayesianeconometricshas developedmethodsto simulatethe posterior
distribution,so that integrals such as E[g(fB) I BIDSt, Xt] can be evaluatedusing Monte Carlo simulation.Posterior
simulatorsgeneratea sequence of S pseudo-randomvectors 1(1),..,.
(S) from a Markovchain with the propertythat
the invariantdistributionof the chain is the posteriordistributionp(p I BIDSt, Xt). Underregularityconditionsdiscussed
in Geweke (1997) (thatare triviallysatisfiedin our application),it can be shown that
lim
S- oo
Therefore,the sample mean [ys=l g(lS)]/S
s=15s)
sS
S
=
f
g(p)p(p I BID, X)dp.
is a consistentestimatorof E[g(f) I BID, X].
O Priors. We assume thatthe prioris proportionalto a constantif {0 < fB < 3}, {-2 < /2 < 2}, {0 < /3 < 2},
{-2 < 34 < 2}, {0 < P5 < 4}, and {0 < #6,7,8,9,10 < 5} and is equal to zero otherwise.We impose these flat priors
to minimize the effect of priorchoice on the shape of the posteriordistribution.As for the appropriatenessof the cutoff
points, we made sure thatthese containedthe "reasonablebounds"for the possible realizationsof the parameters.
Simulation. We used a variantof the Metropolis-Hastingsalgorithmto simulate the posteriordistributionof the
D
parameters.We startout by findinga roughapproximationto the posteriormode, which we will referto as /. The algorithm
generatesa sequence of S pseudo-randomvectors PB(1), .(S) from a Markovchain. First set B() = , then generate
the other simulationsas follows:
(i) Given 4(n), drawa candidatevalue ff from a normaldistributionwith mean 1(n) and variance-covariancematrix
-H-1.
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THE RAND JOURNAL OF ECONOMICS
(ii) Let
p(P( I BID, X)
a= m
np(B(n)
IBID, X)'
o=mim|
i
(iii) Set B(n+l) = fi with probabilitya.
(iv) Returnto 1.
Geweke(1997) providessufficientconditionsforthe invariantdistributionof this chainto be theposteriordistribution.
It is easy to check thatthese conditions are satisfiedin this application.
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