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Bayesian Inference in Structural Second-Price
Common Value Auctions
Bertil WEGMANN
Department of Statistics, Stockholm University, SE-106 91 Stockholm, Sweden (bertil.wegmann@stat.su.se)
Mattias VILLANI
Research Division, Sveriges Riksbank, SE-103 37 Stockholm, Sweden and Department of Statistics,
Stockholm University, SE-106 91 Stockholm, Sweden (mattias.villani@riksbank.se)
Structural econometric auction models with explicit game-theoretic modeling of bidding strategies have
been quite a challenge from a methodological perspective, especially within the common value frame-
work. We develop a Bayesian analysis of the hierarchical Gaussian common value model with stochastic
entry introduced by Bajari and Hortaçsu. A key component of our approach is an accurate and easily
interpretable analytical approximation of the equilibrium bid function, resulting in a fast and numerically
stable evaluation of the likelihood function. We extend the analysis to situations with positive valuations
using a hierarchical gamma model. We use a Bayesian variable selection algorithm that simultaneously
samples the posterior distribution of the model parameters and does inference on the choice of covariates.
The methodology is applied to simulated data and to a newly collected dataset from eBay with bids and
covariates from 1000 coin auctions. We demonstrate that the Bayesian algorithm is very efficient and that
the approximation error in the bid function has virtually no effect on the model inference. Both models
fit the data well, but the Gaussian model outperforms the gamma model in an out-of-sample forecasting
evaluation of auction prices. This article has supplementary material online.
KEY WORDS: Bid function approximation; eBay; Internet auctions; Likelihood inference; Markov
chain Monte Carlo; Normal valuation; Variable selection.
1. INTRODUCTION
Strategic bidding behavior in auctions has been a widely
studied phenomenon since the pioneering work of Vickrey
(1961), particularly over the last decades (see, e.g., Wolfstet-
ter 1996; Klemperer 1999, 2004; Milgrom 2004 for recent sur-
veys and a general introduction). The advances in auction the-
ory have also found their way into the econometric analysis of
auction data. It seems widely accepted that an explicit model-
ing of bidders’ strategic considerations is a necessary condition
for making economic sense of the observed patterns in the bids.
The availability of high-quality auction data has increased in re-
cent years, especially with the advent of Internet auction sites,
such as eBay (see, e.g., Bajari and Hortaçsu 2004 for a survey).
Paarsch (1992) and Elyakime et al. (1997) have provided ex-
cellent examples of structural econometric analyses of auction
data. Bajari (2005) and Paarsch and Hong (2006) have surveyed
the field.
Analyzing auction data through the lens of a structural game-
theoretic model is not an easy task. It has been quite a chal-
lenge to derive the strategic equilibrium bid function (i.e., the
map from a bidder’s conceived or estimated value of the object
to his optimal bid) under realistic model assumptions. When
such results are available, they typically come in a form una-
mendable to analytical computations, and one needs to resort
to time-consuming and possibly unstable numerical methods,
such as numerical integration. This is an important obstacle to
statistical inference as a single likelihood evaluation typically
requires repeated evaluation of the inverse bid function for all
bids in the dataset.
Common value auctions have been especially difficult to an-
alyze by structural econometric models. In common value auc-
tions, the auctioned object has the same value to every bidder,
but the common value is unknown. The bidders use private in-
formation (their signal) to infer the unknown value. In an influ-
ential work, Bajari and Hortaçsu (2003; henceforth BH) made
a number of important advances that substantially simplify the
analysis of data from common value auctions. BH proved that
it is sufficient to compute the bid function in a selected auc-
tion and then extrapolate linearly to the other auctions in the
dataset. This property considerably speeds up the computation
of the bid function, and thereby likelihood evaluations as well.
BH also show that it is optimal to place a bid in the very last
seconds of commonly used Internet auction formats, such as
eBay’s. This in turn implies that Internet auctions can be mod-
eled as sealed-bid auctions without additional strategic consid-
erations of the timing of the bids. BH also extended the results
of Milgrom and Weber (1982) to the situation with a stochastic
number of bidders, an inherent feature of Internet auctions.
The present work refines and extends the analysis of BH.
Our first contribution is an accurate linear approximation of the
equilibrium bid function for cases with a fixed number and a
stochastic number of bidders. The approximate bid function is
of a particularly simple analytical form with an interesting in-
terpretation. It can be inverted and differentiated analytically,
two extremely valuable properties for fast and numerically sta-
ble evaluations of the likelihood function. The approximation
can be evaluated for all bids in all auctions simultaneously in a
negligible amount of computing time.
An interesting aspect of the BH model is the use of auction-
specific covariates, both in the model for the common value and
© 2011 American Statistical Association
Journal of Business & Economic Statistics
Accepted for publication
DOI: 10.1198/jbes.2011.08289
1

2 Journal of Business & Economic Statistics, ???? 2011
in the stochastic entry process. Our second contribution is the
use of a highly efficient general posterior sampling algorithm
that simultaneously approximates the joint posterior distribu-
tion of the model parameters and does Bayesian variable selec-
tion among the model’s covariates. This allows us to quantify
the importance of the individual covariates in the different parts
of the model, and to correctly account for the uncertainty in the
choice of covariates in, for example, the predictive distribution
of the price. Bayesian variable selection also makes it possible
to use a large number of covariates in the model because it typi-
cally reduces the dimensionality of the parameter space dramat-
ically in every step of the Metropolis–Hastings algorithm (see
Section 3.2).
The model in BH assumes a Gaussian distribution, both for
the common value and for the bidders’ signals around that com-
mon value. The Gaussian distribution is convenient but has the
obvious drawback of allowing negative values and signals. We
also consider an alternative model with nonnegative values and
signals following the gamma distribution. We also derive an an-
alytical approximation to the bid function for this model, par-
alleling the results for the Gaussian model. We propose a full
Bayesian analysis of both the Gaussian and the gamma models.
Finally, we apply the methodology to a newly collected
dataset with bids and auction-specific information from 1000
eBay coin auctions. The dataset was collected by careful human
visual inspection of both photos of the auctioned object and the
seller’s text description. We show that both the Gaussian and the
gamma models fit the data well. The Gaussian model outper-
forms the gamma model in an out-of-sample prediction evalua-
tion on 48 auctions. The variable selection shows that the pub-
licly available book value and the condition of the auctioned ob-
ject are important determinants of bidders’ valuations, whereas
eBay’s detailed seller information, such as bidders’ subjective
ratings of sellers and sellers’ historical selling volumes, is es-
sentially ignored by the bidders. The seller’s posted minimum
bid acts as a safeguard for the seller, to avoid large losses. We
show that it is typically optimal for the seller to post a minimum
bid only slightly below the seller’s valuation of the object, de-
spite the fact that a high minimum bid discourages auction en-
try. The estimation results are shown to be robust to a variety of
modifications of the basic model.
2. TWO MODELS FOR SECOND–PRICE COMMON
VALUE AUCTIONS
2.1 General Setup
Assume that the seller sets a publicly announced minimum
bid (reservation price), r ≥ 0, and risk-neutral bidders compete
for a single object using the same bidding strategy (symmetric
equilibrium). The value of the object, v, is unknown and the
same for each bidder at the time of bidding, but a prior distrib-
ution for v is shared by the bidders. To estimate v, each bidder
relies on his or her own private information of the object mod-
eled as a private signal, x, from a distribution, x|v, that is the
same for all bidders (symmetric bidders). Let fv(v) denote the
probability density function of v, let fx|v(x|v) denote the con-
ditional probability density function of x|v, and let Fx|v(x|v)
denote the conditional cumulative distribution function of x|v.
Because the auction involves symmetric bidders and a symmet-
ric equilibrium, we can focus on a single bidder without loss of
generality. The bid function can be written (BH) as
b(x, λ)
=
∞
∑
n=2
(n − 1) · pn−1(λ) ·
∫
v
v · Fn−2x|v (x|v) · f 2x|v(x|v) · fv(v)dv
/
(
∞
∑
n=2
(n − 1) · pn−1(λ) ·
∫
v
Fn−2x|v (x|v) · f 2x|v(x|v) · fv(v)dv
)
,
if x ≥ x (2.1)
and 0 otherwise, where pn−1(λ) is the Poisson probability of
(n − 1) bidders in the auction with λ as the expected value in
the Poisson entry process. Bidders participate with a positive
bid if their signal, x, is above the cutoff signal level, x. Given
an arbitrary bidder with signal x, let y be the maximum signal
of the other (n−1) bidders. The cutoff signal level is then given
in implicit form as (Milgrom and Weber 1982)
x(r, λ) = inf
x
(EnE[v|X = x,Y < x,n] ≥ r),
which gives the minimum bid, r, as
r(x, λ) =
∞
∑
n=1
pn(λ) ·
∫
v v · F
n−1
x|v (x

|v) · fx|v(x|v) · fv(v)dv
∫
v F
n−1
x|v (x

|v) · fx|v(x|v) · fv(v)dv
.
(2.2)
The minimum bid is exogenously given by the seller and x is
then given as the solution to (2.2).
In both the Gaussian and the gamma models presented
herein, the expected value μ and the variance σ 2 exists in the
distribution of v. Similar to BH, we specify regression models
for (μj, σ 2j , λj) in auction j as
μj = z′μjβμ, σ
2
j = exp(z
′
σ jβσ ), λj = exp(z
′
λjβλ),
(2.3)
where zj = (z′μj, z′σ j, z′λj)′ are auction-specific covariates in auc-
tion j. In addition, let β = (βμ,βσ ,βλ)′. BH modeled σ 2 with
a linear function of covariates and thus needed to restrict the
elements of βσ to ensure that σ 2 is positive, whereas βσ is un-
restricted in our setup.
BH made the assumption that bids in parallel auctions are
independent, and showed that last-minute bidding is a symmet-
ric Nash equilibrium on eBay. This allows us to model eBay
auctions as independent second-price auctions. The likelihood
function of bids is complicated, because some bids are unob-
served. First, some bidders may draw a signal, x < x, in which
case they do not place a bid. Second, the highest bid is usu-
ally not observed because of eBay’s proxy bidding system (see
BH and Section 4.1 for more details). The bid distribution for a
single auction is of the form:
fb(b|β,η, r, z, v) = fx|v[φ(b)|β,η, r, z, v]φ′(b), (2.4)
where φ(b) is the inverse bid function, and η is a vector of
additional parameters in the model. Let ns be the number of
bidders who submit a positive bid in a given auction and let
b = (b2,b3, . . . ,bns) be the vector of observed bids, where b2 >