Information Dispersion and Auction Prices
Pai-Ling Yin, Stanford University
January 9, 2004 (1st draft: November 25, 2002)
Abstract
This paper derives comparative static implications of Nash equilibrium bidding in
a mineral rights model of auctions. It then tests these implications to assess whether
a model of Nash equilibrium behavior in a CV information structure characterizes
eBay online computer auctions. Market data is augmented by survey data in order
to determine the information structure and bidding behavior in these auctions. This
external information permits joint identi…cation of Nash bidding behavior and common
values. By imposing structure on the measurement error in my survey data, I am able
to estimate the extent of the winner’s curse in these markets and the e¤ect of dispersion
and reputation on prices. My estimates show that the strength of sellers incentives to
provide detailed information in their auction descriptions varies. Sellers with good
reputations have powerful incentives to reduce uncertainty and promote e¢ cient trade.
These results are unattainable without employing theory, econometric modeling, and
external survey data.
0
I would like to thank Pat Bajari, Tim Bresnahan, Susan Athey, and Ed Vytlacil for their outstanding
attention and generosity; Paul Hartke for scripting; Jeremy Fox, Ali Hortaçsu, and members of the Stanford
IO seminar and Structural Lunch for feedback; Ti¤any Chow, Yi-Xuan Huynh, Steve Yuan and the many
people who …lled out my survey for their research assistance; and The John M. Olin Foundation through the
Stanford Institute of Economic Policy Research and Stanford School of Humanities and Sciences Graduate
Research Opportunities Grant for their …nancial support. Any mistakes in the paper are completely my own;
comments and suggestions are very welcome.
1
1
Introduction
In the “mineral rights” model of common value (CV) auctions, the item being sold has the
same, unknown value to all bidders. Bidders know the distribution of the common value.
Each bidder also observes a private signal (information) about the common value. Each
bidder’s signal is independently and identically drawn from a commonly known distribution
around the true common value. A bidder who ignores the dispersion of this information may
su¤er the winner’s curse, in which she wins the auction at a price exceeding the common
value.
Milgrom & Weber (1982, henceforth referred to as MW) showed that in equilibrium, if
the seller publicly reveals a signal drawn from the same distribution as those of the bidders’signals, then prices will rise in a second-price sealed-bid common value auction and in
an ascending oral auction.1 The logic behind MW’s result applies to any decrease in the
dispersion of bidders’ information: more certainty about other bidders’ signals diminishes
the winner’s curse. For example, in eBay auctions, a seller provides a product description.
Private information may be dispersed about the product’s idiosyncrasies which a¤ect the
value of the item. The seller can include more details about those idiosyncrasies in the description to raise the price. The act of including more details can be characterized as either
publicly revealing more information about the computer or reducing the uncertainty about
the computer that is being sold. This paper will work with the latter characterization. I
will refer to the level of uncertainty as “information dispersion” and measure information
dispersion through the variance of the information signals.
Note that a positive response in price to a reduction in dispersion is counterintuitive from
a non-strategic perspective. Lowering the variance of the distribution of a signal generates
two e¤ects: 1) the average distance between the highest and second-highest signal decreases,
and 2) both of these signals decline on average. If bidders merely bid their signals, or shade
their bid downward by some …xed percentage or absolute amount dependent on the number
of bidders, then the second e¤ect dominates, and both bids will decline. Prices will rise only
in equilibrium, where bidders account for the more narrow distribution of signals around the
common value by shading less, causing the …rst e¤ect dominate.
This paper derives comparative static implications of Nash equilibrium bidding in a
mineral rights model of auctions. It then tests these implications to assess whether a model of
Nash equilibrium behavior in a CV information structure characterizes eBay online computer
1
In a second-price auction, the person who submits the highest bid wins the item, but only pays the
second-highest bid.
MW assume that the signals and common value are a¢ liated, and that bidders are symmetric and behave rationally. They also assume the existence of some mechanism, such as reputation, which makes the
additional information credible to the bidders.
“Let z and z 0 be points in <m+n . Let z _ z 0 denote the component-wise maximum of z and z 0 , and let
z ^ z 0 denote the component-wise minimum. We say that the variables of the model are a¢ liated if, for all
z and z 0 , f (z _ z 0 )f (z ^ z 0 ) = f (z)f (z 0 ). Roughly, this condition means that large values for some of the
variables make the other variables more likely to be large than small.”(Milgrom and Weber 1982) The MW
public information result will not necessarily hold in …rst price auctions. (Perry & Reny 1999)
2
auctions.
Tests of information structure in an auction (whether auctions are private value or common value) and bidding behavior (whether bidders play Nash equilibrium strategies) are most
often addressed separately. Recent literature on nonparametric identi…cation has shown that
the distribution of private signals is just identi…ed assuming a private values (PV) setting,
and underidenti…ed in a CV setting without further parametric assumptions.2 A joint test
of an auction’s information structure and bidding behavior requires more information.
A novel feature of my approach is that I construct measures of dispersion of information among eBay computer auctions through a survey, allowing joint identi…cation of the
information structure and bidding behavior. In particular, for each auction in my dataset,
an average of 46 survey participants were exposed to information posted on eBay by the
seller about the computer. Each participant then provided an assessment of the value of the
computer.
Numerous empirical studies have examined the e¤ect of eBay seller reputation on price
and bidder entry.3 This paper extends the empirical literature on eBay by examining the
e¤ect of the interaction between reputation and information provided in auction description
on prices, rather than just the e¤ect of reputation on price alone. I …nd that a better
reputation increases the price e¤ect of changes in information dispersion, while a worse
reputation dampens it. The e¤ect of this interaction between reputation and information
dispersion has a more substantial e¤ect on prices than reputation alone. My estimates
indicate that bidders succeed in accounting for a substantial winner’s curse even in the
pedestrian market of online computer auctions.
How have tests of bidding behavior and information structure been addressed in the
empirical literature thus far? Studies of commercial auctions have not yet been able to
directly test implications of information dispersion. Part of the literature has been devoted
to testing between common value and private value settings.4 Authors have explored how
variation in the number of bidders can be used to test between private and common value
settings, since the winner’s curse is more severe with more bidders.5 (Paarsch 1992; Haile,
Hong & Shum 2000, Athey & Haile 2002). The challenge to these approaches (and my
approach as well) is that the true number of bidders may be uncertain in an auction and
endogenously determined. A second avenue for analyzing common value auctions is to exploit
ex post information.6 Ex post values are related to ex ante bids to draw inferences about
2
(La¤ont and Vuong 1996), (Li, Perrigne and Vuong 2000), (Athey and Haile 2002), (Li, Perrigne and
Vuong 2003), (Guerre, Perrigne and Vuong 2000).
3
(Resnick, Zeckhauser, Swanson and Lockwood 2002),(Eaton 2002), (Hauser and Wooders 2000), (Jin and
Kato 2003), (Livingston 2002), (Reiley, Bryan, Prasad and Reeves 2000), (McDonald and Slawson Jr. 2002),
(Melnik and Alm 2002).
4
This empirical literature has been predicated on an extensive theoretical literature identifying empirically
testable conditions for private value and common value settings, e.g. Pinske & Tan (2000); La¤ont, Ossard
& Vuong (1995); Donald & Paarsch (1996); Elyakime, La¤ont, Loisel & Vuong (1994)
5
La¤ont & Vuong (1996) show that bidding data alone with a …xed number of bidders is insu¢ cient to
distinguish common value settings from a¢ liated private value settings.
6
McAfee, Takacs & Vincent (1999) use ex post values to test the information aggregation properties of
3
strategic bidding. In only a few cases have ex post values been available, and all of these
values are typically subject to measurement error.7 A …nal approach imposes the equilibrium
bidding assumption in order to estimate the joint distribution of signals and common value.8
(Hong & Shum 1999, Bajari & Hortaçsu 2002a ) Even with these assumptions and ex post
information on values, the underlying parameters are just identi…ed. By introducing survey
data, the extra information about the distribution of signals allows me to conduct joint tests
of information structure and bidder behavior without such assumptions.
The paper most closely related to my work is Hendricks, Pinkse & Porter (2001). Under the assumption of a common values setting, they test between Nash equilibrium and
alternative, naive bidding strategies where bidders do not take into account the number of
bidders nor the distribution of signals. They conclude that the prices re‡ect the predictions
of common value auction theory in oil tract lease auctions. They then structurally estimate
the information structure.
The experimental literature has tested equilibrium bidding behavior by directly controlling the primitives. Kagel, Levin & Harstad (1994) …nd that while prices rise when information is publicly released in the auctions with fewer bidders, they fall in the larger auctions.9
Although bidders may be attempting to play Nash equilibrium strategies, they may not get
the magnitudes right. When presented with new information, they adjust their bids accordingly, so prices fall instead of rise. However, contrary to strategic bidding, increasing the
number of bidders does not change the bids.10 Most analogous to my work is a study by
Goeree and O¤erman (2002), who test the reaction of prices when the range of signals is
compressed. Prices fall with increased dispersion, but by less than theory predicts.11
auction prices.(McAfee, Takacs and Vincent 1999) In their seminal paper, Hendricks & Porter (1988) showed
that bidders with superior information make a pro…t in auctions, whereas uninformed bidders account for
the winner’s curse and get zero pro…ts. Athey & Levin (2001) show that bidders respond strategically to
private information about the species composition in timber auctions.
7
In fact, di¤erent conclusions regarding whether bidders actually avoided the winner’s curse as evidence
of equilibrium bidding behavior in oil tract lease auctions has been attributed to measurement error. (Capen,
Clapp & Campbell 1971; Mead, Moseidjord & Sorensen 1983; Hendricks, Porter & Boudreau 1987)
8
Li, Perrigne & Vuong (2000) show that in the mineral rights model considered in this paper, the joint
distribution of signals and values is identi…ed under some additional functional form assumptions and if all
bids are observed. Athey & Haile (2002) show identi…cation fails unless all bids are observed, but that ex
post information on the common value combined with partial bid information can identify the primitives in
a common value auction.
9
In experimental auctions with 4-5 and 6-7 bidders, Kagel, Levin & Harstad (1994) provide bidders with
a private signal on a common value item, and then release a public signal after a …rst round of bids on the
item and allow bidders to update their bids.
10
Bidders also fail to account for the winner’s curse in ascending oral auctions. (Kagel & Levin 1992)
Other experimental tests (Kagel & Levin 1991, Lind & Plott 1991, Cox & Smith 1992 ) suggest that this
failure is not the result of strategic considerations with respect to budgets. Even experienced commercial
bidders may fail to shade correctly in experiments, as found in by Dyer, Kagel & Levin (1989) when they
employ construction industry bidders as subjects.
11
A particularly relevant experimental exercise would be to leave the range of signals the same, but change
the density of the distribution to re‡ect a lower variance of signals around the mean. It would also be useful
to then conduct experiments with higher and lower variance for di¤erent numbers of bidders to observe any
interaction e¤ect between variance and number of bidders. Goeree & O¤erman (2002) conduct auctions with
4
This paper builds upon the commercial auction work by employing a measure of the
common value and studying changes in price with respect to the number of bidders, as well
as the experimental work which tests the e¤ects of information dispersion directly.12 By
generating an external measure of dispersion, I do not need to infer that dispersion from
the bids. This permits me to 1) distinguish between common and private value settings
without imposing fully rational bidding behavior, 2) distinguish between Nash and naive
bidding behavior without assuming a particular information structure, 3) employ only price
data from the auctions, as opposed to all bids. Additionally, since I do not need to infer the
distribution of information, I can allow my measures of dispersion and the common value
to be imprecise and estimate any potential bias between those measures and the truth. I
employ parametric functions for modeling prices in order to facilitate the computation of
counterfactuals. These counterfactuals allow me to calculate the potential winner’s curse
in these markets and the impact of changes in information dispersion, reputation, and the
number of bidders on price. The survey design generates a measure of dispersion that is
reputation free, permitting separate identi…cation of reputation e¤ects and dispersion e¤ects
on price.13
The paper is organized as follows: Section 2 derives implications of auction theory regarding dispersion of information. Section 3 reviews the auction and survey data employed from
eBay computer auctions. Section 4 describes the instruments collected. Section 5 presents
the models used for estimation. Section 6 reviews the results of estimation. The last section
summarizes the …ndings of this chapter.
2
Theory and Empirical Implications
This section presents the benchmark theoretical model of Nash CV prices from second-price
sealed-bid auctions. It then presents comparative static implications of the Nash CV model,
the PV and naive CV models, and the PV and naive CV models with risk aversion.
Consider the following model of a pureCV auction. An item is put up for auction. This
< to each of n symmetric bidders; however, none of the
item has a true CV v 2 [v; v]
bidders knows the true CV. Each bidder has a private signal xi 2 X
< and the same a
priori knowledge of the density of v, fv (v), and density of xi conditional on v, fxjv (xi jv). I
assume the form in the mineral rights model, where xi = v + "i , and "i is independently and
identically drawn from an a¢ liated distribution centered around v. The utility of the item
to every bidder i is equal to v, which is unobserved.
In a second-price auction, the person who submits the highest bid wins the auction, and
low and high distributions of signals for 3 bidders, but only conduct high distributions of signals for auctions
with 6 bidders.
12
This work parallels that of David Lucking-Reiley and John Morgan who augment commercial auction
data with experimental data. I augment commercial auction data with survey data. The use of the second
moment as a measure parallels the use of second moments in the …nancial auction literature.
13
The estimation of reputation e¤ects on eBay have been the center of attention for a number of papers.
An excellent overview of the literature is conducted by Resnick, Zeckhauser, Swanson & Lockwood (2002).
5
pays the amount submitted by the second highest bidder. The optimal Nash equilibrium bid
b(xi ) for symmetric bidders in a sealed-price auction is
b(xi )
=
E[vjxi ; max Xj6=i
=
xi ]
where max Xj6=i denotes the maximum of j 6= i signals. (Milgrom and Weber 1982)
The expected winning price p is the expected value of the second order statistic of all
bids. I de…ne the price as a function h of xi ; n; fx (xjv), and fv (v). Letting xn 1:n denote the
2nd highest signal from a set of n signals,
E[p] = E[b(xn
1:n
)]
h(x; n; fxjv ( j ); fv ( )).14
(2)
For distributions which can be characterized by scale and location, such as normal and
lognormal distributions, we can replace fxjv ( j ) and fv ( ) with xjv ; v ; and v ; respectively.
I chose the second-price sealed-bid model as a benchmark for eBay auctions for several
reasons. Since eBay employs an automated bidding mechanism which publishes the current
(and winning) bid as some small increment above the second highest bid, it most closely
resembles a format in between a second-price sealed-bid auction and an oral ascending auction. During the eBay auctions, bidders can see the current second-highest bid plus one
increment. They are free to enter and exit at any time as well as update and resubmit
their bids before the close of the auction. Harstad & Rothkopf (2000) found that English
auctions with re-entry are more closely approximated by second-price sealed bid models.
Empirical observations of the timing of bids on eBay indicate that the majority of auctions
in all categories experience a ‡urry of bidding during the last minutes.15 To the extent that
insu¢ cient time exists to view all the information contained in those bids before the close
of the auction, the auction again tends to favor a second-price sealed-bid model.
2.1
Implications of Nash CV auctions
This section presents the comparative static implications of Nash CV auctions.
1a. The Nash CV price decreases if the dispersion of information signals (denoted by
@p
<0
for normal and lognormal distributions) increases.16
@ xjv
xjv;t
2a. Under normal and uniform distributions, the Nash CV
! price decreases at a decreas2
@ p
ing rate with the dispersion of signals.
> 0 For some parameterizations of
@ 2xjv
14
The analytical derivation is di¢ cult. See Appendix A.
Bajari & Hortaçsu (2002b) review empirical …ndings in online auction settings.
16
This result is translation of Theorems 8 and 12 of MW. McMillan & Kazumori (2002) con…rm this result
for distributions satistfying a¢ liation.
15
6
the lognormal
! distributions, the price decreases at an increasing rate with dispersion.
2
@ p
<0
@ 2xjv
Under symmetric distributions, the distance between the …rst and second order statistics
is monotonically decreasing at a decreasing rate with variance. However, over certain ranges
over the lognormal distribution, this property does not hold. Prices increase as the distance
between the …rst and second highest signals decrease. Simulations run for the normal and
lognormal distribution con…rm these comparative statics of price.
3a. The Nash CV price may be decreasing or increasing with n.
@p
@p
> 0;
<0
@n
@n
Bid shading in response to rising n will not necessarily lower the winning price. As the
number of bidders increases, the probability that a bidder draws a higher signal increases.
Whether the draws from the higher distribution will overcome the amount of bid shading
depends on both the number of bidders and the distribution of signals.17 As a result, the
interaction e¤ect between the number of bidders and dispersion may also have mixed e¤ects
on prices.
4a. The Nash CV price may increase or decrease with an increase in the dispersion of
@2p
@2p
information signals when the number of bidders is higher
> 0;
<0
@n@ xjv
@n@ xjv
2.2
Implications of PV and naive CV auctions
In 2nd price PV auctions, prices equal the second highest signal. In naive CV bidding,
bidders ignore the number of bidders and the level of dispersion, and just bid their signal as
in PV auctions. The resulting comparative static implications are as follows:
1b. Under normal and uniform distributions, the PV and naive CV prices increase with
@p
the dispersion of signals.
> 0 For some parameterizations of the lognormal
@ xjv
@p
distributions, prices may decrease if the dispersion increases.18
<0
@ xjv
2b. Under normal and uniform distributions, the PV and naive
! CV prices increase at a
2
@ p
decreasing rate with the dispersion of signals.
< 0 For some parameteriza@ 2xjv
tions of the lognormal distributions,
the price will decrease at a decreasing rate with
!
2
@ p
dispersion.
>0
@ 2xjv
17
18
In the limit, prices converge to the common value. (Milgrom 1979) (Wilson 1977)
Direct result of expected values of order statistics under uniform, normal, and lognormal distributions.
7
3b. PV and naive CV prices increase with n.19
@p
>0
@n
4b. Under normal and uniform distributions, the PV and naive CV prices increase with
@2p
dispersion at an increasing rate with the number of bidders.
> 0 For some
@n@ xjv
parameterizations of the lognormal distributions, the price may increase or decrease
with an increase in the dispersion of information signals when the number of bidders
@2p
@2p
is higher.
> 0;
<0
@n@ xjv
@n@ xjv
2.3
Implications of PV and naive CV auctions with risk aversion
Consider the PV and naive CV models with risk-averse bidders. If higher dispersion is
interpreted as more risky, then bids will decrease for every bidder. If bidders …nd the presence
of more bidders to be reassuring, then bids increase for every bidder. If bids increase when the
number of bidders is larger, then they should increase more when dispersion is higher. When
there is more uncertainty (risk) about the value of the item, the information contained in
observing more bidders should be weighed more heavily than when there is less uncertainty
about the value of the item. So prices should be increasing with the number of bidders
under risk aversion at a increasing rate with the level of dispersion. Since these behaviors
are symmetric for all bidders, the comparative static implications for prices directly translate
from the comparative static implications for bidders.
1c. Under risk aversion, the PV and naive CV prices may decrease with the dispersion of
@p
signals.
<0
@ xjv
3c. Under risk aversion, the PV and naive CV prices increase with n.
@p
>0
@n
4c. Under risk aversion, the PV and naive CV prices increase with dispersion at an increasing
@2p
rate with the number of bidders.
>0
@n@ xjv
We may not be able to distinguish between risk neutrality and risk aversion combined
with Nash equilibrium common value bidding behavior, since some of the e¤ects are held in
common, while reverse e¤ects may be canceled out.
2.4
Information Asymmetry
This section hypothesizes comparative statics regarding prices and reputation.
Work by Akerlof(1970), Klein & Le- er (1981) and Shapiro (1983) suggest that prices
should rise with better reputation r.
19
Direct result of expected values of order statistics under uniform, normal, and lognormal distributions.
8
5. The expected common value is increasing in reputation (excluding the interaction e¤ects
@p
>0
in Hypothesis 6).
@r
Information asymmetry between the bidders and sellers means that when the seller provides information in an auction, bidders have to determine whether they believe that information. MW note that some mechanism, such as a reputation, is necessary to ensure that the
information provided in the auction is credible. This means that reputation rwill augment
the level of information dispersion provided by the seller. Henceforth, let xjv denote the level
of information dispersion in an auction if all sellers have the same reputation. I hypothesize
that CV Nash prices will fall with dispersion at a faster rate with better reputations.
6. In Nash CV auctions, the perceived level of information dispersion is increasing in the
level of information dispersion provided by the seller at an increasing (decreasing) rate
@2p
with the seller’s (bad) reputation.
<0
@ xjv @r
If a bidder knows that a seller is credible, she will take changes in the level of detail
provided in a seller’s description of an object very seriously. An interesting result of this
implication is that sellers with better reputations will actually su¤er an even larger drop in
price from providing a less informative description than sellers with worse reputations. So
while a good reputation may shift the location of the distribution of the signals upward, it
may also exacerbate the negative e¤ects of high dispersion. As the variance in signals rises,
prices would actually fall more with a better reputation. This interaction e¤ect distinguishes
the role of reputation as a credibility measure of information dispersion from its role as a
reputation premium. The extent to which reputation a¤ects the location and scale of the
distribution of information in the auction is an empirical question.
2.5
Summary of Predictions
Table 7 summarizes the comparative statics predictions for a number of alternative models.
Each row represents a di¤erent model of bidding behavior and information strucutre. Each
column represents a comparative static. Each box in the grid represents the observable sign
for each comparative static under each model. PV and naive CV with risk aversion are ruled
@p
@2p
@p
@2p
out if we observe
> 0,
<
0;
<
0,
or
< 0. Note that if we observe
@ xjv
@ 2xjv
@n
@n@ xjv
@p
> 0, we can also rule out the possibility that the auction being observed is consistent
@ xjv
@p
with Nash CV. If we observe
< 0, we are either in a Nash CV setting.
@n
Table 2 summarizes the comparative statics that distinguish between symmetric distributions and lognormal distributions once the model has been determined. Under the PV or
@p
@2p
@2p
naive CV model, observing
< 0,
>
0,
or
< 0 rules out a symmetric
@ xjv
@ 2xjv
@n@ xjv
9
Table 1: Distinguishing comparative statics, n>3
@2p
@p
@2p
@p
Model
@n@ xjv
@n
@ xjv @ 2xjv
PV/naive CV
Nash CV
PV/naïve CV risk aversion
=+
=+
=+
+
+
=+
+
=+
=+
+
Table 2: Distinguishing comparative statics, n>3
@p
@2p
@2p
Distribution
@ xjv @ 2xjv @n@ xjv
PV/naive CV symmetric
PV/naive CV lognormal
Nash CV symmetric
Nash CV lognormal
+
=+
distribution of signals. Under the CV model, observing
=+
+
=+
+
=+
@2p
< 0 rules out a symmetric
@ 2xjv
distribution of signals.
3
Dataset
The eBay online computer auctions permit me to test the comparative statics implications
of Nash CV. The resale value of the computer or its components induces a common value
element for bidders interested in eventually reselling the computer. Computers vary in the
certainty of their value (e.g., compare a Dell to a no-name PC clone). Di¤erent bidders may
have private information about the availability and price of similar computers at other outlets
or about the reliability of this particular model of computer or its components. The detail in
auction descriptions also varies in this market. Computers most likely have a combination
of private and common values. Some bidders may derive particularly greater bene…t from a
speci…c combination of drive speed, memory, processor, etc. than other bidders. As long as
the common value component dominates, the comparative static implications of Nash CV
should still hold. The sellers of computers on eBay also exhibit relatively large variation in
reputation compared to other markets.
3.1
Survey Data
To obtain a measure of the mean and dispersion of private signals received by bidders in
the 222 auctions, I created a web-based survey. No restrictions were placed on who could
participate in the survey. The survey was distributed to acquantainces by word of mouth.
I refer to the people that responded to all portions of my survey as survey participants or
10
Table 3: Summary statistics for
variable (831 participants) mean
number of responses/auction
46
signals (mean=Vt )
$666.43
dispersion of signals sdt
472.38
survey on 222 auctions
st. dev. min
max
6
25
65
$317.28
$101.48 $1,816.98
153.94
163.57 980.50
survey respondents. I asked people to read computer auction descriptions and then answer
the following question: “If a friend wanted to buy the computer described below, what is the
most she should pay for it?”(full details can be found in Yin, 2003) These descriptions only
contained the information provided by the seller in the “descriptions” section. Information
listed by eBay about the bids, reservation values, number of bidders, and the seller’s identity
and reputation were removed. I also collected background data on survey participants.
On average, I collected 46 valuations per auction. Each auction is indexed by t. I denote
the average of the estimates by Vt . I denote the standard deviation of signals from my survey
by sdt : Data from the survey are summarized in Table 3.
Survey participants may have di¤erent mean valuations and dispersion than bidders in
the auction. This does not mean there is no information in Vt and sdt . I use two strategies to
exploit that information. The …rst only assumes that Vt is positively correlated with vt and
sdt with xjv;t . That assumption lets me test comparative statics implications in a regression
framework. This assumption is not su¢ cient to quantify the relationship between price and
the auction characteristics. So in my second strategy, I add structure to the measurement
error in Vt .
3.1.1
Survey as correlated measure
In Figure 1, a plot of my survey measure of the CV for each auction against the prices
attained in each auction suggests that my survey measure is picking up some information
that is related to the value of the item. This survey measure does not take into account the
dispersion of information or the reputation of the seller, which may be factors that in‡uence
price.
Provision of more or better information should lead bidders to be more certain about the
value of the item, and thus make the variance in their signals lower. One concern may be
that the measure Vt is correlated with sdt . To address this concern, I examine items with
similar Vt to see what my survey respondents considered to be high and low dispersion items.
Figures 2 and 3 show the complete auction descriptions from an item with Vhigh = $313:81
and the auction description excluding picture for an item with Vlow = $290:23, respectively.
The …rst item had SDhigh = 1:61, while the second item had SDlow = 1:05.
Note that the high dispersion item lacks the level of detail in the low dispersion item. Both
of the descriptions show pictures, but the high dispersion picture is of a similar computer,
not of the computer itself. The low dispersion description actually includes a picture of the
computer for sale. The information that is dispersed with respect to the high dispersion
11
Figure 1: Prices vs. survey measure of common value: Auctions have been ordered by
increasing price. Survey measures of the common value follow the price trend, suggesting
that the survey does pick up information correlated with the value of the item.
computer may take the form of di¤erent knowledge in the population of exactly how similar
are the computer for sale and the picture in the ad, or what is the quality of reclaimed
computers generally. Of particular note for the low dispersion item is the detail with which
the seller describes exactly how the computer does not work: although this failing of the
computer may lower the mean valuation, certainty about that valuation also lowered the
variance on the signal. One can imagine that if the bidder merely said “this computer does
not work”or didn’t mention the failure at all, information would be dispersed between those
who were familiar with the types of failures encountered with Hewlett-Packard computers
and those who were not. By revealing exactly what type of problem the computer possesses,
the seller was able to lower the dispersion of that information. If bidders in the actual
auctions respond to the reputation-free measure of dispersion st , then we would expect them
to respond to the correlated measure SDt . Indeed, the high dispersion item was won for
$55.00; the low dispersion item sold in auction for $96.50.
3.1.2
Error structure for survey measure
The error correction structure exploits the background data collected on survey participants.
On average, 20% of the respondents for each auction in my sample said that they had won all
or some of the eBay online computer auctions in which they had participated. I separate out
these "experienced" respondents, and assume that they are identical to the actual bidders
in my sample of auctions. By identical, I mean that the responses from these participants
represent signal draws from the same distribution that the bidders faced. So I consider their
12
Figure 2: High dispersion item with Vhigh = $318:81
13
Figure 3: Low dispersion item with Vlow = $290:23
14
responses as actual realizations of xt =vt + "x;t , where "x;t v (0; xjv ), whereas the responses
from any other respondents I model as x0t = 0 + 1 vt + "0x;t , where "0x;t v (0; 0xjv ). An
unbiased estimate of the true mean can then be written as:
v^t
=
Je;t
ve;t
Jt
+
Ji;t
Jt
vi;t
0
;(3.1.2)
1
where Je;t is the number of experienced survey participants in each auction, Ji;t is the number of inexperienced survey participants in each auction, Jt is the total number of survey
participants in each auction, and ve;t and vi;t are the mean of the survey responses by the
experienced and inexperienced participants in each auction, respectively.
3.2
Auction Data
When an auction opens on eBay, a minimum bid, and possibly a reserve price, is set by
the seller. The seller also determines the length of the auction. Bidders then can submit
bids. Each bid raises the price by one increment above the second highest bid currently
submitted. When the auction has ended, the seller and buyer arrange for shipping and
payment. After the transaction, each person may leave feedback for the other in the form
of a neutral, positive, or negative comment. The comments result in either a +1, 0, or -1
added to the other person’s feedback score.
This feedback mechanism acts as a measure of reputation for the bidders and sellers. It
is a voluntary system. Only those involved in a transaction (the winner and seller) may
participate.20 The feedback score is total positive feedback minus total negative feedback.
So a person with a score of 10 may have a perfect record of 10 sales, or may have sold 100
items, 90 of which received negative feedback. The feedback score is the most prominently
displayed measure of reputation for every user on eBay. For this reason, I record both the
overall feedback score and neagative feedback score of each seller as measures of reputation
r.
I use the winning prices, number of bidders, and reputation of the seller from each of 222
auctions. The auctions were held from June 24, 2002 to July 12, 2002. I selected auctions
from recent Pentium processor subcategories. I excluded auctions with less than two bidders
and auctions for multiple units of computers. I also excluded auctions which were terminated
via "Buy It Now," a feature which allows a bidder to pay a list price for the item and end
the auction. I selected auctions to ensure variation in the sellers’ overall feedback score.
Summary statistics for the auctions are shown in Table 4.
20
Until a few years ago, multiple comments from the same person would not count in the feedback score,
but individuals could leave feedback even if they were not involved in the transaction. Currently, only the
winner and seller in each auction can leave feedback for each other.
15
Table 4: Summary statistics for
variable
mean
price pt
$359.01
overall score (rg;t ) SCOREt
680
negative feedback (rb;t ) N EGt
25.5
number of observed bidders Nt 6.5
4
222 eBay
median
$255.00
27
2
6
computer auctions
st. dev. min
$369.16
$9.51
2601
0
106
0
4
2
max
$2802.00
19456
785
22
Instruments
This section presents the potential endogeneity and measurement error problems in the data
collected, and the instruments collected to address those issues.
4.1
Instruments for the number of bidders
There are various reasons why one would expect the observed number of bidders to be biased
and measured with error. Since there is free entry and exit during the course of the auction,
the number of observed bidders will tend to be less than the actual number of bidders who
received signals.21 If the current price exceeds the expected value of a given bidder, that
bidder will not submit a bid in the auction, even though other bidders would …nd the fact that
the abstaining bidder’s valuation must have been lower than the current price informative
to their estimates of the common value of the true number of bidders. However, since I
impose the structure of the second-price sealed-bid auction on the data, one might expect
the true number of bidders to be less than the observed number of bidders. If the auction is
closer to an oral ascending auction, a given bidder has already conditioned on information
observed about earlier bids, so those earlier bidders won’t enter her conditional expectation.
In addition, “bottom feeders”on eBay may submit extremely low bids on the o¤ chance that
no one else enters the auction. These bidders may not be taken seriously as an entrant who
is drawing a signal about the valuation of the item by the …rst and second highest bidders.
One of the largest advantages of a survey measure of the common value is that survey
readers will tend to pick up the same idiosyncratic aspects of an auctioned items value that
a¤ect a bidders valuation in an auction. Many of the usual selection problems with entry
are thus controlled. However, if the actual bidders in eBay computer auctions are better
equipped than my survey respondents to spot a good deal on eBay, then the number of
bidders may be correlated with unobservable determinants of price.
In selecting excluded regressors to account for this endogeneity, we want variables from
the auction which are correlated with the number of bidders, but, conditional on other
covariates (in particular, the mean of the survey responses), are uncorrelated with anything
unobservable that might also determine prices. The regressors selected should not a¤ect the
21
The number of bidders who draw signals is the important factor for evaluating the winner’s curse, since
the winner cares about how my opponents received signals less than her signal.
16
Table 5: Summary statistics for bidder instruments
variable (222 auctions) mean st. dev. min max
COU N T ER
249.67 167
38
1215
CN T RDU M
0.33
0
1
M IN BID
$58.25 112.95
0.01 650.00
HOU REN D
15
5
1
24
EN DDAY
4
2
1
7
LEN GT H
3.45
1.1
1.5
7
GALLERY
0.33
0
1
F EAT U RE
0.16
0
1
OT HERN
6.56
1.60
3.4
11.6
OT HERN E
6.60
1.85
3
11.8
OT HERN I
6.55
1.58
3.4
11.8
price in the auction except through the number of bidders, nor act as a signal of the value
of an item in the auction.
Several candidates exist as excluded regressors, although they require certain assumptions. Summary statistics are presented in Table 5.
COU N T ER denotes the number of times a site was viewed as described by a website
hits counter. Although the number of times a website is clicked may indicate something
about the desirability of the computer being listed, the counter indiscriminately counts any
access to the site. So there may be noise generated from search engines or repeated access
by the same viewer. Thus COU N T ERt may be less correlated with the value of the item
being auctioned than with the set of people active on eBay during the course of the auction.
Since counters were not present in all auctions, a dummy was included for those auctions
without counters, CN T RDU M . Other regressors which might re‡ect the potential bidders
for the auction but not the value of the item are HOU REN Dt and EN DDAYt , the ending
times of the auction, and LEN GT Ht the length of the auction in days.
The starting bid M IN BID could be an instrument for the number of bidders if we assume
that changing the minimum bid only a¤ects price by changing the number of bidders entering
the auction, but does not give a signal about the value of the item itself. Use of the minimum
bid as a type of reserve price competes as a seller strategy with a "list by lowest price …rst"
feature found on eBay that would increase the eyeballs to an auction site. Anecdotal evidence
suggests that bidders like to generate interest in their auctions by lowering starting bids, so
it is not necessarily a re‡ection of the value of the item. Indeed, this instrument (along with
the rest used in this paper) satisfy overidenti…cation tests. GALLERYt and F EAT U REt
indicate whether an item was included in the photo gallery or listed at the top of the webpage
listings. These characteristics should in‡uence the number of people that enter the auction,
but GALLERYt only costs $0.25 to the seller, so its use is probably not re‡ective of the
value of the computer. F EAT U REt is a more expensive feature at nearly $20.00, but it is
also a variable form of advertising: your item is listed at the top of the page where it would
17
normally appear based on the search criteria that a buyer uses. So the listing may be at the
top of page 1 or the top of page 5. This variability in return on the F EAT U REt investment
suggests that there may be randomness in the type of sellers that would use this feature that
is uncorrelated with the value of the item.
The challenge for all of these instruments is to explain why sellers di¤er in their use of
the instruments just listed. Heterogeneity of the sellers may be one explanation, but we
would also have to justify why this heterogeneity itself is not correlated with the type of
computer being sold. Another possible explanation is that the use of these tactics depends
on the number of auctions the seller is managing. To the extent that the number of auctions
held by the same seller is not correlated with the value of a computer, this may justify the
validity of some of these instruments.
Three instruments that do not involve seller choice are OT HERN , OT HERN E, and
OT HERN I. These variables are the average number of bidders in the ten other auctions
which received the closest common value estimate by the all bidders, experienced computer
bidders, and inexperienced computer bidders, respectively. The variable should give an
indication of the number of bidders in the market for that good, while being uncorrelated
with the particular attributes of particular computer that might drive entry.
4.2
Instruments for dispersion
My survey measure of information dispersion may not match that of the actual bidders, in
particular for those bidders whose background responses indicated that they were inexperienced with respect to the computer market and computer auctions. The excluded regressors
I chose to use for sd were dummies for whether the seller neglected to include information
on various computer components (ram memory RAM M ISS, operating system OSM ISS,
‡oppydrive F LOP P Y M ISS, keyboard KEY BDM ISS, cd/dvd drive CDM ISS, mouse
M OU SEM ISS) and the number of pictures and words included in the auction description
(P ICS, W ORDS). These hedonic measures of the auction descriptions should be correlated
with the amount of information dispersion in the auctions. They should be orthogonal to
the semantics of the descriptions that would lead to di¤erent interpretations between the
bidders and my survey participants and drive measurement error. The summary statistics
of these instruments are presented in Table 10.
4.3
Instruments H for
v
A bidder may be in the market for a certain brand or speed of computer, so she may
search eBay for computers that match those criteria. These criteria thus determine the
expected value of the computer before a bidder receives a signal. I constructed a set of
hedonic characteristics of the computers, collectively denoted H, to be used as regressors for
determining the a priori expected CV. Summary statistics are presented in Table 7.
The dummy variable BRAN DDU M stands for whether the computer had a recognizable
brand name (Toshiba, Dell, Hewlett-Packard, IBM, Compaq) or not. A ranking of the
18
Table 6: Summary statistics for dispersion instruments
variable (222 auctions) mean st. dev. min max
RAM M ISS
0.03
0
1
OSM ISS
0.45
0
1
F LOP P Y M ISS
0.33
0
1
KEY BDM ISS
0.38
0
1
CDM ISS
0.16
0
1
M OU SEM ISS
0.28
0
1
W ORDS
449.4 460.8
23
2727
P ICS
4.05
5.02
0
25
Table 7: Summary statistics for bidder instruments
variable (222 auctions) mean st. dev. min max
BRAN DDU M
0.27
0
1
P ROCESSOR
2.14
1.11
0
3
SP EED
1088
684.88
0
2530
RAM
210.77 196.04
0
1100
HARDDRIV E
27724 27755
0
160000
IN T ERN ET
1.31
0.83
0
2
M ON IT ORDU M
0.21
0
1
CDDU M
0.83
0
1
F LOP P Y DU M
0.66
0
1
19
processor brands in P ROCESSOR ranged from no mention of processor brand (= 0) to
Pentium (= 3). The processor’s speed was denoted as SP EED: The amount of memory
included was characterized by the ram and harddrive capacity (RAM; HARDDRIV E).
I ranked the presence of a communications device in IN T ERN ET (0 for no device, 1 for
modem, 2 for other). Dummies were created for whether a monitor, cd/dvd drive, and ‡oppy
drive was included or not (M ON IT OR; CD; F LOP P Y ). A number of these instruments
have a minimum value of 0, since if the auction did not indicate the value, the value was
coded as 0.
5
The Empirical Model
I test the comparative static implications that distinguish between di¤erent models in a
‡exible reduced-form speci…cation. I then correct for measurement errors and endogeneity
to obtain valid point estimates.
5.1
Comparative Statics
In the …rst set of estimates, I test the comparative statics of CV Nash by regressing price
directly on my survey measures. The price equation that is used in this estimation and
subsequent speci…cations is as follows:
Pt =
0
+
v Vt
+
sdscore sdt
sd sdt
+
2
sdsq sdt
SCOREt +
score SCOREt
+
+
n Nt
+
sdneg sdt
neg N EGt
+ "p;t
nsd Nt sdt +
N EGt +
(5.1)
where t indexes the auctions in my sample and "p;t v(0, p ). The a priori values v and
v are held constant over my sample of auctions, so they are not included in Equation 5.1.
The results are reported in the OLS column of Table 8.
The signs of sd , sdsq , n , and nsd are consistent with Nash CV. The negative sign on
n is inconsistent with PV and naive CV settings, as well as risk aversion. The signs of the
hypotheses on reputation e¤ects correspond with my expectations. score and sdscore are
the coe¢ cients that most closely correspond to Implications 5 and 6. Since reputation is
composed of positive and negative feedback in this case, we would expect the signs on neg
and sdneg to be the reverse of the signs on score and sdscore :
20
Table 8: OLS estimates for prices
222 obs
OLS Std. Error
90.598
117.968
0
z
1.028
0.057
v
z
-1.552
0.449
sd
1.29E-03z
4.30E-04
sdsq
-13.060
11.007
n
0.026
0.022
nsd
z
-1.31E-04
6.68E-05
sdscore
1.43E-03
1.76E-03
sdneg
z
0.079
0.035
score
-0.869
0.817
neg
zsigni…cant at 5%; ysigni…cant at 10% with robust standard errors. R2 = 0:71
5.2
Modeling the Number of Bidders and Dispersion
The following equation is used to model the number of bidders that enter an auction.
Nt = Z + "N;t ;
(5.2)
where Z is a set of regressors, including a constant,
is a vector of parameters to be
estimated, and "N;t v(0, N ).
The set Z includes all of the instruments described in Section 4.1. It also includes all
the regressors included in the price equation, since upon viewing an auction description,
a person on eBay may decide she is not in the market for that computer. In that case,
the person should not be considered as a person who received a signal about the value of
the item; the person’s decision not to bid in this case may be completely independent of the
computer’s value. I also exploit the presence of V in the price equation to employ the hedonic
measures of value H as excluded regressors for the number of bidders. Since V already
contains all the information contained in these measures, plus idiosyncratic information,
these instruments should no longer be correlated with the error on the price equation. A
joint-F test of these hedonic measures in the price equation alongside V shows that they do
not add any explanatory power. However, they would be determinants of whether bidders
enter an auction, since bidders may be in the market for computers of a certain power or
memory size. Bidders may search by these criteria for relevant auctions, thereby limiting the
auctions in which they are potential bidders. The inclusion of these regressors as instruments
for V do not violate overidenti…cation tests. The adjusted R-squared for a least squares
regression of the observed number of bidders on Z is 0.91.
The dispersion equation is denoted as follows
sdt = W + "sd;t ;
(5.2)
where Z is a set of regressors, including a constant,
21
is a vector of parameters to be
estimated, and "sd;t v(0, sd ). The set W include all of the instruments described in Section
4.2 as well as Vt in order to control for the level in sd. The R-squared from least squares
estimation of sd on all instruments is 0.53.
5.3
Exploiting Survey Variance
I replace Vt in Equation 5.1 with v^t from Equation 3.1.2. The new price equation is now:
Pt =
0
+
Ji;t vi;t
Je;t
0
ve;t +
+
Jt
Jt
1
2
sd sdt + sdsq sdt + n Nt + nsd Nt sdt +
v
sdscore sdt
SCOREt +
score SCOREt
+
sdneg sdt
neg N EGt
+ "p;t :
N EGt +
(5.3)
Since I am able to construct two moments from my survey data, a mean and variance,
I can use the de…nition of the variance to generate a moment condition to identify the
parameters that underlie the measurement error in the inexperienced bidders estimates. I
set the standard deviation of the experienced survey responses equal to the de…nition of
the sample standard deviation where I have replaced the sample mean, which is assumed to
be vt , by my constructed estimate of v^t from a larger sample of observations that includes
inexperienced estimates with correction parameters. The following moment condition is then
estimated simultaneously with other equations in the price determination model:
v
u Je;t
uP
t xt v^t
;(5.3)
sde;t
=
Jt 1
where sde;t is the standard deviation of experienced survey participants’responses in each
auction.
5.4
Point Estimates
The results of generalized method of moments estimation of the price, bidder, dispersion,
and second moment restriction equations are presented in Table 9.
From the estimates in 9, we now have signi…cant estimation of every coe¢ cient, and all
the signs correspond to predictions from a common values setting with Nash equilibrium
bidding. In addition, since the negative coe¢ cient on the number of bidders and the positive
coe¢ cient on the interaction between bidders and dispersion are now both signi…cant, we
can rule out risk averse private bidding and risk averse naive bidding behavior in a common
22
Table 9: GMM estimates
Eqns 5.3,5.2,5.2,5.3 Std. Error
1.013z
0.023
1
z
55.358
13.125
0
z
960.284
136.028
0
z
1.150
0.047
v
z
-4.588
0.552
sd
z
3.66E-03
5.22E-04
sdsq
z
-45.844
9.683
n
z
0.087
0.020
nsd
-4.20E-04z
1.18E-04
sdscore
z
8.87E-03
3.30E-03
sdneg
z
0.211
0.058
score
z
-4.138
1.475
neg
zsigni…cant at 1% with robust standard errors.
Reported R2 for Eqn 5.3: 0:64; Eqn 5.2: 0:20, Eqn 5.2: 0:39, Eqn 5.3: 0.85
222 obs
values setting. A check of the partial derivatives of price with respect to dispersion and
the number of bidders con…rms that prices are decreasing in dispersion and the number of
bidders.
The assumptions made regarding the relationship between experienced and inexperienced
responses and the mean value actually will also determine the error correction necessary for
the measure of information dispersion as well, so I could replace instrumenting sdt with
a formula that actually corrects for errors in the inexperienced responses. However, by
continuing to instrument, I impose less assumptions on the nature of measurement error in
dispersion. The signs were all the same and the magnitudes were roughly equivalent in the
more restrictive speci…cation. The largest notable di¤erences were the magnitudes on sd
and n . The more restrictive model had smaller magnitudes for both (-1.973 and -19.074,
respectively).
The next section examines whether the assumption of a common v to all auctions is
restrictive.
5.5
Estimating
v;t
The assumption of a common v over all auctions may be restrictive, so this section proposes
a method for estimating a v;t for each auction.
Throughout the previous speci…cations, I have implicitly assumed that v;t was subsumed
in vt in the following manner:
vt =
v;t
+ "v;t ;
(5.5)
23
where "v v (0; v ). Equation 5.5 re‡ects the initial setup of the mineral rights auction
model that was presented at the beginning of Section 2. If we assume that v;t would only
be determined by hedonics of computers rather than idiosyncratic components, then we can
employ the hedonic regressors H to estimate v;t , and treat the error term as a control for
the unobserved signal xt . This allows a priori beliefs about the expected common value
to enter the price equation di¤erently than a posteriori information based on an observed
signal about the realized common value of the item.Our new price equation then becomes:
Pt =
0
+
v;t
sd sdt
+
+
sdscore sdt
v
v^t
2
sdsq sdt
v;t
+
+
n Nt
SCOREt +
score SCOREt
+
+
nsd Nt sdt +
sdneg sdt
neg N EGt
+ "p;t
N EGt +
(5.5)
To estimate v;t , I assume that v;t is a deterministic function of the hedonic regressors
H. The following equation is simultaneously estimated with Equations 5.5 and 5.2:
v^t = H + "v;t ;
(5.5)
where
is a vector of parameters to be estimated. The results of simultaneous estimation of Equations 5.5, 5.5, ??, 5.2 and 5.2 are presented in Table 10.
The only signi…cant di¤erence between these estimates and the ones in Section 5.3 is the
estimate of the bias shift in the survey measure of the common value. The estimate of the
intercept in Equation 5.5, 0 = 2:919; will force residual to have zero mean. The coe¢ cient
on our estimate of v;t is insigni…cant, and a joint F-test of the signi…cance of separating out
v;t from v fails.
5.6
Summary
The purpose of this section was to test the various predictions of Nash equilibrium bidding
behavior with respect to information dispersion under di¤erent information structures. The
various speci…cations in this section have ruled out PV settings and naive CV settings under
risk aversion and neutrality. Prices fall with dispersion at a decreasing rate. Prices also
fall with the number of bidders over the range of auctions sampled. By exploiting the
second moment from the survey data, we gain more power in identi…cation. The …ndings
are summarized in Table 11. The next section will interpret the coe¢ cients of this section.
24
Table 10: GMM estimates with varying priors
222 obs Eqns 5.5,5.5,??,5.2,5.2 Std. Error
1.107z
0.107
1
z
-5.261
0.192
0
z
1000.176
358.902
0
1.890
1.438
z
1.030
0.091
v
z
-5.026
1.081
sd
-47.439
46.949
n
z
4.04E-03
9.89E-04
sdsq
0.079
0.099
ns
-4.27E-04
2.96E-04
sdscore
8.21E-03
7.60E-03
sdneg
0.228
0.153
score
-4.007
3.476
neg
zsigni…cant at 1% with robust standard errors.
2
Reported R for Eqn 5.5: 0:57; Eqn 5.5: 0.63, Eqn 5.2: 0:27, Eqn ??: 0:85, Eqn 5.2: 0.39:
Model
Table 11: Model predictions revisited
@pt
@ 2 pt
@pt
2
@ xjv;t @ xjv;t @nt
PV/naive CV
Nash CV
PV/naïve CV risk aversion
X
X
X
25
X
X
X
@ 2 pt
@nt @ xjv;t
X
X
Table 12: Corrected
variable
common value v^t
dispersion ^ xjv;t
number of bidders n
^t
6
survey and auction summary
mean
st. dev. min
$604.42 $308.57
$55.26
458.46 95.31
274.94
6.46
2.35
-5.76
statistics
max
$1729.39
804.64
18.20
Analysis
This section examines and quanti…es the presence of the winner’s curse in these auctions,
and quanti…es the e¤ect of reputation on information provision in these markets.
6.1
Comparison to Nash Bidding
In this paper, I have considered Nash equilibrium behavior to be any behavior that accounts
for the number of bidders and dispersion in the correct direction. The magnitudes of the
changes that eBay bidders exhibit does not necessarily re‡ect optimal Nash equilibrium
magnitudes. However, given the corrected regressors, we can generate Nash equilibrium
prices predicted from theory via numerical simulations, and compare the di¤erence in those
prices with the ones observed on eBay. While eBay bidders may behave in a manner that is
consistent with Nash equilibrium behavior in common value settings, they may su¤er from
measurement error themselves in making their bid calculations. Bidders may not be playing
optimal Nash bid strategies.
From the estimates of 0 and 1 , we can apply Equation 5.2 to calculate v^t , which is a
version of Vt that has been corrected for errors. The …tted values from Equations 5.2 and 5.2
provided estimates of the true number of bidders and information dispersion. The resulting
summary statistics for these corrected variables are presented in Table 12.
Compared to the survey data, the true common value is lower on average but with
approximately the same variance. Dispersion is also lower on average than the direct survey
measure with a lower variance comparatively. The survey participants tend to overestimate
items on average (only in one auction did they underestimate the item value). They also
tend to become less informed than the actual bidders after reading the auction, since their
estimated dispersion is quite a bit higher and more variable. In only 30 auctions did the
survey participants underestimate the level of information dispersion. The lower levels of
dispersion perceived by actual bidders is consistent with the possibility that bidders bene…t
from positive externalities of information provided in other auctions on eBay, or information
collected by observed bids during the auction.
The observed number of bidders is lower than the estimated true number of bidders.
The observed number of bidders was higher in only 20 auctions. This …nding supports
the hypothesis that when constructing their bids, eBay bidders are taking into account the
fact that some bidders may not be observed entering the auction. Bidders are shading to
account for those unobserved bidders. In only 2 auctions was the predicted number of bidders
26
negative.
I employ gauss-quadrature simulation methods to numerically calculate the optimal
bids.22 Using ^ xjv;t and v^t , I can generate the the expected second highest signal given
n
^ t in each auction. I examine both normal and lognormal distributions. I then calculate
a common v and v by calculating the mean and standard deviation of the v^t . Plugging
these into the simulation program, I generate the predicted bid for each auction. For each
auction, 1000 draws of the second highest signal were made from the distribution of the
signal conditional on the true value. I then adjust that price for reputation e¤ects based on
the estimates from Table 9 for Equation 5.3. I save the average of the second highest signal
in each simulated auction and interpret it as the predicted private value price if my measures
of the common value and dispersion were actually measures of the average and dispersion of
private values.
Figure 4 shows the comparison of predicted prices and actual prices ordered by predicted
prices for lognormally distributed signals. Predicted private value prices are shown for contrast. Recall that the underlying parameters used to generate the Nash predicted prices were
essentially identi…ed by matching prices from the eBay auctions to survey data. No theory
about how prices were formed was imposed on the estimation. The underlying conditions
that allow us to generate the Nash prices are free from any assumptions about bidding behavior or the information setting. The proximity of predicted and actual prices validates
both the survey data (with corrections) employed and the ability of bidders on eBay to at
least replicate the optimal direction of price responses to changes in the number of bidders
and information dispersion, even if they do not calculate the magnitudes optimally. Bidders
on eBay clearly diverge from behavior expected from a private value auction or a common
value auction with completely naive bidding behavior.
If we then examine how prices as a percentage of the common value correlate to dispersion
as a percentage of common value, we see the pattern predicted by Milgrom and Weber
(1982). In Figure 5, auctions were ordered by increasing corrected information dispersion as
a percentage of the corrected common value. Those auctions were then divided into bins at
every 10% change in normalized dispersion. Prices are falling with increased dispersion.
This section has shown that bidders on eBay generate prices that follow the direction that
Nash equilibrium bidding behavior predicts for prices; however, bidders tend to underbid.
The results are notably similar between Nash predictions and actual prices, especially when
identi…cation comes from survey data and no restrictions imposed by equilibrium theory.
6.2
Winner’s Curse
When we compare the corrected values to the prices paid, we …nd that bidders seem to avoid
overpaying. On average, the bidders paid $194.25 less than the value of the item. There were
only 22 cases where bidders paid more than the estimated true value, of which 5 involved
overpayment under $50.00. However, in the other cases, the average loss was $507.31. The
22
The Gauss-Hermite quadrature method is outlined in Judd (1998). The translation for lognormally
distributed signals is not quite correct, so my algorithm di¤ers slightly from the one implied in his book.
27
Figure 4: Actual and predicted prices: Auctions are ordered by the Nash predicted prices.
Private values prices are derived from simulated second order signals from each auction.
No assumptions were imposed regarding bidding behavior of information setting in order
to derive the initial conditions for generating Nash common value predicted prices. The
similarity between eBay prices and Nash predictions are quite striking, especially in contrast
to predicted private value/naive prices.
Figure 5: Prices vs. information dispersion as % of common value: The auctions were ordered
by increasing disperion as a % of the common value, then divided into bins representing 10%
changes in normalized dispersion. Prices were also normalized by common values. Prices
decline with information dispersion.
28
Table 13: Bidder responses to changes in information dispersion and bidders
Change in price fr. increase xjv;t $1 increase # bidders by 1 increase both
Models
average Pt
average Pt
average Pt
CV, Nash
-$0.80
-$6.13
-$2.93
CV, xjv;t naïve
0
-$45.84
-$45.84
CV, nt naïve
-$1.36
0
-$1.36
CV, all naïve
0
0
0
dispersion of information on these computers tended to be higher than average, as was the
average SCORE of the seller. It is possible that in these cases, the reputation e¤ect was able
to overwhelm the lack of information, or the lack of information was given little credibility
since the seller did not have an established reputation. These results are reported in the last
column of Table 14.
Nash equilibrium behavior is only signi…cant if it actually saves the bidder from a large
loss. Although we have shown that the bidders in these auctions do not overpay on average,
how much is really at risk if they behave naively and do not account for the number of bidders
or dispersion of information? We can answer this question by considering the prices these
bidders might pay if they ignored changes in the number of bidders, dispersion, and their
interaction e¤ects. Again, by employing the corrected common values and the estimates for
Equation 5.3 in Table 9, we can quantify the potential winner’s curse for di¤erent levels of
naive bidding behavior. The results are presented in Table 13.
For each auction, I consider 3 scenarios for a given signal: an increase in information
dispersion by 1%, an increase in the number of bidders by 1 person, and both changes. I
then calculate how much the price would change based on predictions from our estimated
model, Eqn 4a. This model is the benchmark of Nash equilibrium behavior in the common
values setting. I also calculate how much the price would change if bidders were naive with
regard to dispersion, naive with regard to the number of bidders, or both. Those calculations
are made by setting the parameters on dispersion, the number of bidders, and interaction
terms to 0, respectively. The di¤erence in the amounts indicates how much overpricing or
underpricing would take place under alternative models. The problem with underpricing is
that another bidder could have entered and won the auction with a higher price that was
still below the common value. The problem with overpricing is that if you win, you paid too
much.
Under the estimated strategies played by the eBay bidders, prices from their behavior
will lead to a decrease in prices as dispersion falls and as the number of bidders rises. The
benchmarks for how prices should change are listed in the …rst row of numbers in Table 13.
Looking at the second row of numbers, we can see that ignoring the e¤ect of a $1 increase
in dispersion leads to overpricing by $0.80. Prices optimally should drop by $0.80 according
to the …rst row, but prices do not change if all bidders ignore the change in dispersion.
Ignoring the interaction e¤ect of a $1 increase in dispersion on the number of bidders will
generate underpricing of $0.56 (=1.36-0.80). Since prices decrease less with dispersion when
29
Models
CV, Nash
CV, nt & xjv;t
eBay prices
Table 14: Potential winner’s curse
average Pt di¤erence with naive di¤erence with value
$555.27
-$239.84
-$28.31
naïve
$795.11
0
-$211.53
$361.02
-$434.09
-$194.25
the number of bidders is higher, ignoring the number of bidders while taking into account
changes in dispersion will lead to underpricing. When another bidder enters, failure to
account for that bidder leads to overpricing by $6. The failure to account for the interaction
e¤ect of that bidder on dispersion will result in underpricing by $40 on average. If both
changes occur, then ignoring the number of bidders will cause overpricing by a bit over one
dollar, and underpricing by $43.
To get a sense of the potential winner’s curse in these markets, we can again appeal to our
Nash simulations to determine the naive price which would be determined if everyone just
bid their signal, and then compare this to the predicted prices adjusted for reputation, and
the actual prices paid. The summary results are shown in Table 14 A substantial potential
winners curse of $240 exists between what bidders should be paying and what they would
pay if they bid their signals, about 40% of the average value of the computers being sold
in our sample. The last column of Table 14 reports the di¤erence between predicted Nash,
naive, and the actual eBay prices.
This section has shown that bidders not only avoid the winner’s curse by paying less than
the estimated common value of the computers they buy, but they also avoid a potentially
large winner’s curse compared to what they would pay if they were to naively bid their
signals.
6.3
Reputation and Information Provision
What do these estimates mean for seller strategies on eBay? An examination of the implications of Nash bidder behavior on eBay prices from the seller’s perspective will allow us to
speculate on optimal strategies for the seller as well. Table 15 is analogous to Table 13 in
that it examines how prices will change with dispersion and reputation, rather than dispersion and the number of bidders. The results are broken down to examine the partial e¤ects
of dispersion and reputation changes alone on prices, and reputation e¤ects in combination
with dispersion changes.
The bene…t from increasing reputation by one unit is less than the bene…t from decreasing
dispersion by 1 unit when one ignores the interaction e¤ect. The total return from decreasing
dispersion by $1 directly translates into an increase in price on average. However, the bene…ts
from increasing reputation by 1 unit are actually quite small because of the increased penalty
for having high information dispersion. This yields an explanation for why reputation alone
as a price shifter may not show up as a signi…cant driver of price. The value of reputation
is in the interaction with information dispersion: it increases the credibility of information
30
Table 15: Price responses to seller controlled variables
Change in price fr. decrease xjv;t $1 increase reputation by 1
do both
Partial e¤ect
average Pt
average Pt
average Pt
@pt
$0.21
$0.21
*
@rg;t
@pt
*
$0.67
$0.67
@ xjv;t
@ 2 pt
$0.33
-$0.19
$0.14
@ xjv;t @rg;t
Total e¤ect
$1.00
$0.02
$1.02
*These partial e¤ects exclude the interaction e¤ects with respect to reputation.
provided about the auction item. Better information about the auction item seems to have
the larger e¤ect on price. The interaction between information dispersion and reputation
creates an incentive for sellers to maintain a good reputation and provide better information,
which would result in lower uncertainty about the value of items in the eBay markets.
Reducing this uncertainty would help promote e¢ cient trade in this market.
7
Conclusion
This paper has derived comparative static implications of Nash CV models with respect to
information dispersion. By harnessing the dispersion of information in the non-eBay market,
I was able to generate an external measure of information dispersion in eBay auctions. Survey
estimates from an average of 46 responses per auction in my sample generated a mean
valuation and variance of valuations for each auction. I used those means and variances as
measures correlated with the mean and the variance of the signals received by the bidders
in each auction. My use of survey data to augment the data collected from commercial
auctions provided an external measure of dispersion. By testing these implications in eBay
computer auctions, I have concluded that these auctions exhibit Nash bidding behavior and
CV information structures.
I am able to identify two di¤erent e¤ects of reputation: the mean shift that reputation
may have on the expected common value, and the credibility reputation lends to how changes
in the dispersion of information in an auction are perceived by the bidder. The magnitudes
of the estimates indicate that sellers with a good feedback score have an incentive to provide
precise descriptions. They gain more bene…t from the interaction between reputation and
information dispersion than from the e¤ect of reputation directly on price. I am also able
to predict the true common value from my biased survey measure and quantify the winner’s
curse in my sample of auctions. Bidders on eBay seem to do quite well at accounting for the
winner’s curse: they pay less than the common value on average, and overpay in only 9%
of the auctions. Rough calculations of naive bidding models indicate that there is potential
31
for a large winner’s curse, but prices in eBay online computer auctions actually re‡ect Nash
equilibrium common value price behavior in reaction to changes in the dispersion of signals.
Even in the pedestrian market of online computer auctions, prices exhibit the equilibrium
behavior predicted by sophisticated conditioning behavior by strategic bidders.
32
A
1. Analytical derivation of E[p]
The bid function in equation 1 can be written explicitly as
Rv 2
vfx (xi jv)Fxn 2 (xi jv)fv (v)dv
v
b(xi ) = R v
(A1)(Milgrom 1981)
2 (x jv)F n 2 (x jv)f (v)dv
f
i
i
v
x
x
v
where Fx (xi jv) is the cumulative distribution of x. To derive the expected value of the
2nd highest bid, one would have to solve for
Pr[b(xi )
v] = Pr[xi
fx (b 1 (v))
with associated density
b0 (v) =
where
b 1 (v)] = Fx (b 1 (v))
1
b0 (v)
,
@b(v)
;
@v
and 2nd order distribution of
fx(n
1)
(b 1 (v))
1
b0 (v)
= n(n
1)Fx (b 1 (v))n 2 fx (b 1 (v))
and then integrate to get
R1
x (n
1 i
1)Fx (b 1 (v))n 2 fx (b 1 (v))
1
dx.
b0 (v)
33
1
b0 (v)
[1
Fx (b 1 (v))],
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