07350015%2E2013%2E794731

Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

A Point Decision for Partially Identified Auction
Models
Gaurab Aryal & Dong-Hyuk Kim
To cite this article: Gaurab Aryal & Dong-Hyuk Kim (2013) A Point Decision for Partially
Identified Auction Models, Journal of Business & Economic Statistics, 31:4, 384-397, DOI:
10.1080/07350015.2013.794731
To link to this article: http://dx.doi.org/10.1080/07350015.2013.794731

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Date: 11 January 2016, At: 22:16

A Point Decision for Partially Identified
Auction Models
Gaurab ARYAL
Research School of Economics, The Australian National University, Canberra, ACT 2601, Australia
(gaurab.aryal@anu.edu.au)

Dong-Hyuk KIM

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Economic Discipline Group, The University of Technology Sydney, Haymarket, NSW 2000, Australia
(dong-hyuk.kim@vanderbilt.edu)
This article proposes a decision-theoretic method to choose a single reserve price for partially identified
auction models, such as Haile and Tamer (2003), using data on transaction prices from English auctions.

The article employs Gilboa and Schmeidler (1989) for inference that is robust with respect to the prior over
unidentified parameters. It is optimal to interpret the transaction price as the highest value, and maximize
the posterior mean of the seller’s revenue. The Monte Carlo study shows substantial gains relative to the
revenues corresponding to a random point and the midpoint in the Haile and Tamer interval.
KEY WORDS: Maxmin expected utility; Optimal reserve price; Statistical decision theory.

1.

INTRODUCTION

In standard symmetric auctions with independent private values (IPV), the problem of (expected) revenue maximization
boils down to the problem of choosing a reserve price. For
this decision making, it is critical to know the distribution of
values (see Myerson 1981; Harris and Raviv 1981; Riley and
Samuelson 1981). Therefore, a great deal of empirical research
has focused on uncovering the valuation distribution using
bid data and proposing a revenue maximizing reserve price
(RMRP) (see Paarsch 1997; Athey and Haile 2002; Li, Perrigne,
and Vuong 2003; Krasnokutskaya 2011; Kim 2013a,b among
others).

This article proposes a decision theoretical procedure to
choose a single reserve price using a sample of only the transaction prices and the number of bidders from English auctions
where the valuation distribution is partially identified. To choose
a reserve price, Paarsch (1997) used the button auction model
where the price rises continuously and exogenously, each bidder drops out when the price hits his value, and the auction ends
when only one bidder remains at the price equal to the second
highest value. Under the IPV, then, any order statistics of the
bid (drop-out point) identifies the valuation distribution as well
as the RMRP.
Many English auctions, however, differ from the button auction because they employ minimum bid increments and bidders often raise the price by more than these increments (jump
bidding). Moreover, Haile and Tamer (2003) (HT, henceforth)
showed that the button auction hypothesis may lead to incorrect inference, and proposed to employ an incomplete model,
only assuming that the bidders neither overbid their values nor
let the auction terminate at a price they can profitably raise.
Under these weak assumptions, HT showed that the valuation
distribution is partially identified, and proposed a set estimator
of the RMRP (HT interval). Though this approach is robust to
any misspecification of bidding behavior, the HT interval leaves
the seller’s problem unsolved, as the seller can only pick one

reserve price, not an interval. It is, however, unclear how to
choose one point in the HT interval: the Monte Carlo studies in
this article show that a significant fraction of the interval is less
profitable than zero reserve price (see Table 1).
While using the weak behavioral assumptions in HT, the we
propose a procedure to choose a single reserve price by employing the maxmin expected utility theory of Gilboa and Schmeidler
(1989) (GS, henceforth). GS extended the classic expected utility theory to allow an ambiguity averse decision maker to have
many equally reasonable distributions over the random quantity that affects the utility, that is, there is ambiguity about the
distribution. A decision maker is ambiguity averse if he or she
prefers to have fewer such distributions. GS show an ambiguity
averse decision maker behaves as if he or she maximizes the
lower envelope (worst-case) of the equally reasonable expected
utilities.
This article considers the seller as a decision maker with many
equally reasonable priors over the valuation density, which is
the random quantity that affects the revenue (utility). A prior
is said to be reasonable if it conforms that the transaction price
does not exceed the highest value. Thus, the lowest (reasonable)
revenue, at each reserve price, is associated with the valuation
density whose largest order statistics coincides with the transaction price. The optimal decision rule, therefore, regards the

transaction price as the highest value and maximizes the posterior mean of the seller’s revenue. The Monte Carlo study shows
that the method produces substantially higher revenues than the
midpoint as well as a random point in the HT interval, whereas
the loss relative to the RMRP is small.
The article also considers the case when the seller is ambiguous about the number of participating bidders in his auction.
Since an auction with two bidders has the lowest revenue, the

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© 2013 American Statistical Association
Journal of Business & Economic Statistics
October 2013, Vol. 31, No. 4
DOI: 10.1080/07350015.2013.794731

Aryal and Kim: A Point Decision for Partially Identified Auction Models

Table 1. Proportion (%) of ρ in HT with u(0, ·) > u(ρ, ·)
Bidders
n
3

5

Exponential
-like density

Longtail
density

LogNormal
(3,1)

LogNormal
(4,1/2)

54.920
61.425

57.116
58.358

61.757
69.244

56.235
61.048

NOTE: The table shows the proportion of reserve prices in the HT set associated with
revenues that are lower than the revenue at zero reserve price. For example, when values
are distributed as the Longtail density and there are three bidders, 55% of the HT set
produces lower revenues than zero reserve price.

385

the valuation distribution remains unchanged. The future auction
can take any of the “standard” auctions—a standard auction is an
auction where the highest bidder gets the object. Furthermore, in
line with the literature, the article also assumes that if a bidder’s
value is zero, he expects to pay zero. Myerson (1981) and Riley
and Samuelson (1981) showed that when the seller’s values are
zero, the reserve price ρ that maximizes the revenue,

 ∞
max{ρ, ξ }dPvn−1:n (ξ ) − ρPvn:n (ρ)
(1)
u(Pv , ρ) :=
0

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solves the following first-order necessary condition:
seller should behave as if there will be two bidders while treating
the transaction price as the highest value.
Finally, the article considers auctions with correlated values (with a known number of bidders). Then, as long as the
transaction price is equal to the second highest value, an assumption used by Aradillas-Lopez, Gandhi, and Quint (in press), the
main result of the article holds, irrespective of the correlation
structure.
This article is related to some other papers that have used the
maxmin criterion for robust decision in various problems with
partial identification such as Manski (2000), Kitagawa (2011),
Menzel (2011), and Song (2012). This article also has the same
spirit as Handel, Misra, and Roberts (2013) that studied demand

assessment and subsequent pricing problem for a firm using the
minmax-regret criterion.
The next section describes the seller’s problem and Section 3
develops an optimal decision rule. Section 4 shows Monte Carlo
experiments followed by some extensions in Section 5. Section 6
concludes and an appendix collects all computational details.
2.

AUCTION MODELS

A single indivisible object is auctioned among n ≥ 2 risk
neutral bidders in an English auction. Each bidder i observes
only his value vi ≥ 0. Assume v1 , . . . , vn are drawn independently from an identical, absolutely continuous distribution Pv
j :n
j :n
with density pv . Let Pv and pv be the cdf and the pdf of the
jth highest order statistics among the n iid values, respectively.
The auction starts at zero price and at each time bidders raise
the standing price y˜ by at least , the minimum bid increment.
Bidding more than y˜ +  is known as jump bidding. When no

bidder is willing to raise y˜ further, the auction allocates the object to the bidder who offers the final bid at the transaction price
˜ This article makes the assumption:
y : = y.
Assumption 1. (A-HT) The transaction price y ≤ x :=
max{v1 , . . . , vn }.
This assumption is weaker than the assumptions in HT, which
also require that the bidders do not let their opponents win at a
price they can overbid. The data zT available to the seller consist
of iid transaction prices (y1 , . . . , yT ) from past T auctions, each
with n bidders. Only the transaction prices are used because
many public auctions reveal only the transaction prices and
even when all bids are observed, it is not obvious that the losing
bids convey any useful information when  > 0 or when there is
jump bidding (see Lu and Perrigne 2008; Quint 2008; AradillasLopez, Gandhi, and Quint in press).
Now, consider a seller with data zT who wishes to choose a
reserve price to maximize his revenue in a future auction where

ρ=

1 − Pv (ρ)

.
pv (ρ)

(2)

Since the seller does not know Pv , he or she cannot use
Equation (2) to determine ρ. For this problem, Paarsch (1997)
proposed to estimate Pv by treating the observed bids as losers
values in the button auction model, and used the point estimates
in Equation (2) instead of Pv (a.k.a. “plug-in” method). For
nonparametric identification of the button auction model, see
Athey and Haile (2002). When  > 0 or when there is jump
bidding, however, the assumption that bids equal values can
be unreasonable. HT show that this assumption can cause a
significant bias in the estimation of Pv even for a correctly
specified parametric model.
For this reason, HT only assume that bidders do not overbid
their value and do not let the auction terminate at a price that
they can profitably overbid. Then, HT partially identifie Pv and
construct a set estimator for the RMRP. These set estimates are
robust to any misspecification on bidding behavior with jump
bidding and  > 0.
The HT interval itself, however, does not completely solve
the seller’s problem. Moreover, the HT interval includes many
reserve prices with revenue lower than with zero reserve price.
Table 1 shows that such reserve prices are about 55% to 69%
of the HT intervals, each of which is obtained from bid samples
of 200 auctions generated from a fixed data-generating process (DGP), that is, a combination of a valuation distribution in
Figures 1 and 2 and n ∈ {3, 5} bidders (see sec. 4 for detail). In
most cases, zero reserve price produces substantially higher revenues than average revenues of the HT intervals (see Table 2).
This stems from the asymmetric shape of the revenue function.
The revenue gradually increases up to the RMRP, but it drops
sharply thereafter, while the upper limit of the HT interval is
much higher than the RMRP (see Figures 1 and 2).
What should then be the criteria to choose one RMRP from
this interval? The next section proposes a solution to this
question.

Table 2. Percentage revenue gains of ρ = 0
Bidders
n
3
5

Exponential
-like density

Longtail
density

14.487 (7.483)
12.370 (5.734)

13.129 (8.018)
6.210 (1.388)

LogNormal
(3,1)

LogNormal
(4,1/2)

15.850 (11.486) 7.661 (1.370)
13.521 (8.823) 3.473 (0.388)

NOTE: The table compares the average revenue (the revenue at the midpoint) of the HT
set and the revenues at zero reserve price. For example, when values are distributed as the
Longtail density and there are three bidders, using zero reserve price would give revenues
that are 13% (8%) higher than a reserve price randomly drawn from the HT set (the midpoint
of the HT set).

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Journal of Business & Economic Statistics, October 2013

Figure 1. Valuation densities and revenue functions. Panels (a) and (c) plot the exponential-like density function and associated revenue
functions for alternative number of bidders n. Panels (b) and (d) similarly for the longtail values density. On panels (c) and (d), ρ0 indicates the
revenue maximizing reserve price.

3.

Ŵ-MAXMIN SOLUTION

The goal of this section is to develop a statistical decision
method that exploits no information on bidding behavior other
than the no-overbidding assumption (A-HT). That is, the method
models the seller as agnostic across all possible beliefs on bidding behaviors as long as (A-HT) is satisfied. To this end, we
employs the framework of GS because it is a parsimonious
framework to represent the seller’s attitude toward the ambiguous feature of the model. Typically, other decision criteria
require additional assumptions on the seller’s preference and beliefs, which is elaborated at the end of this section. This section
begins by assuming that
Assumption 2. (A-GS) The seller satisfies the axioms (A.1–
A.6) in GS.
(A-GS) coincides with axioms in the classic expected utility
theory, except it allows the seller to weakly prefer any convex
combination of indifferent lotteries to each individual one instead of restricting the combination to be indifferent—ambiguity
aversion, and uses a weaker version of independence. Then, GS
showed that these axioms are necessary and sufficient for a decision maker to have a unique set of distributions and prefer an
action a to b if and only if the minimal expected utility over all
distributions in the set under a is at least as big as under b.

To formulate the seller’s problem within this framework, the
article considers the parameter vector that indexes Pv as the
random vector that affects the revenue. This parameter vector is
divided into two groups: identified and unidentified. The former,
θ ∈ , indexes the distribution of the transaction price y, that
is, Py (y|θ ), and the latter, h ∈ H, captures any discrepancy
between the distributions of y and x = max(v1 , . . . , vn ), so that
Py (y|θ ) = Pvn:n (y|θ, 0)

(3)

as Px (·|θ, h) = Pvn:n (·|θ, h). Therefore, h is inherently infinite
dimensional, and the statistical model Pv (·|θ, h) is nonparametric if θ is also infinite dimensional, or it is semiparametric,
otherwise. (In the statistical decision theory, the term “parameter” refers to the state of world, but does not suggest that the
statistical model is indexed by a finite dimensional parameter.)
The article uses this configuration of parameters, so that h
solely captures the lack of identification because it simplifies
the way the seller’s decision process is framed. Here, h is a
nuisance parameter that is related to the equilibrium bidding
strategy. For example, the button auction model corresponds
to a dogmatic prior on h that puts probability one on h∗ such
that Py (·|θ ) = Pvn−1:n (·|θ, h∗ ). One may also put probability 1
on a different h∗ if he believes that a different bidding model
generates the data. Or, he may have a nondegenerate prior on h

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Aryal and Kim: A Point Decision for Partially Identified Auction Models

387

Figure 2. Valuation densities and revenue functions. Panels (a) and (c) plot the lognormal density with (μ, σ ) = (3, 1) and associated revenue
functions for alternative number of bidders m. Panels (b) and (d) similarly for (μ, σ ) = (4, 1/2). On panels (c) and (d), ρ0 indicates the revenue
maximizing reserve price.

reflecting his (dispersed) beliefs. Any hypothesis on bidding behavior is, however, not verified by the data zT := (y1 , . . . , yT ).
Therefore, the prior on h may have a persistent influence on
decision making.
To see this, suppose that the seller has the sample zT as
well as a prior pθ,h over  × H. For this case, Kim (2013b)
posited choosing a reserve price as the seller’s decision problem
under parameter uncertainty. The article argues that if the seller
behaves rationally in the sense of Savage (1954) and Anscombe
and Aumann (1963), he or she would maximize the expected
revenue given by
E [u(θ, h, ρ)|zT ; ph ]
 
:=
u(θ, h, ρ)pθ,h (θ, h|zT ; ph )dhdθ,


(4)

H

where u(θ, h, ρ) := u[Pv (·|θ, h), ρ] from Equation (1) and the
posterior density can be obtained via Bayes’ theorem,
pθ,h (θ, h|zT ; ph ) := 

T

pθ,h (θ, h) t=1 py (yt |θ )
.

pθ,h (θ, h) Tt=1 py (yt |θ )dhdθ

ever, that since


pθ,h (θ, h|zT ; ph ) = ph (h|θ ) 

Kim (2013b) discussed the optimality of this Bayesian approach
from the frequentist perspective, and shows that it can produce
substantially higher revenues than the plug-in rule. Note, how-

T

t=1 py (yt |θ )

pθ (θ ) Tt=1 py (yt |θ )dθ



= ph (h|θ )pθ (θ |zT ),

the sample zT directly affects θ but not h. Therefore, the impact
of ph (·|θ ) on the solution to maximize Equation (4) would not
disappear even when T is large.
The article takes the framework of GS to handle the ambiguity
about ph (·|·). A conditional prior ph (·|θ ) is said to be reasonable
in this article if it conforms to (A-HT) for every θ ∈ . Let Ŵ be
the set of all priors ph (·|·) such that, for every (θ, h) ∈  × H
pθ (θ )ph (h|θ ) > 0 ⇔ Px (ξ |θ, h) ≤ Py (ξ |θ ),

∀ξ ∈ R. (6)

Under GS, the seller considers every element in Ŵ as equally
reasonable and solves
max min E[u(θ, h, ρ)|zT ; ph ].
ρ∈A ph ∈Ŵ

(5)

pθ (θ )

(7)

That is, the seller should choose a reserve price that maximizes
the revenue in Equation (4) with respect to the most pessimistic
prior in Ŵ.
Definition 1. A decision rule that solves Equation (7) for
every realization of zT is called the Ŵ-maxmin rule.

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Journal of Business & Economic Statistics, October 2013

For an intuitive interpretation of the Ŵ-maxmin rule, suppose
the seller consults with many experts about the bidding behavior,
that is, h, each proposing a different, but reasonable opinion.
Then, the Ŵ-maxmin criterion implies that the seller should
take the most pessimistic prediction about the revenue to secure
against the worst possible outcomes.
The Ŵ-maxmin rule is robust to the choice of prior over the
unidentified parameter. Usually, solving Equation (7) is computationally expensive because the “min” part solves an optimization problem over a space of high-dimensional functions
for every ρ. This issue does not, however, arise for the seller’s
problem as the minimization part has a simple solution, as shown
below.

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Proposition 1. Let the set Ŵ satisfy the property (6). Under
(A-HT) and (A-GS),

Corollary 1. Under (A-HT) and (A-GS), and uncertainty
about nT +1 ∈ N


max min ′ E u(θ, h, ρ; nT +1 )|zT ; ph,nT +1
ρ∈A ph,nT +1 ∈Ŵ

max min E[u(θ, h, ρ)|zT ; ph ] = max E[u(θ, h, ρ)|zT ; δh ],
ρ∈A ph ∈Ŵ

the posterior expectation of the revenue in Equation (4) can
be obtained as before and the worst case for the seller is
that the transaction price is generated from auctions with two
bidders.
¯
Some additional notations are required: let N := {2, . . . , n}
with n¯ < ∞ be the set of all possible number of bidders,
and let Ŵ ′ be the set of all joint priors ph,nT +1 (·, ·|θ ) over
H × N that satisfy (A-HT), that is, pθ (θ )ph,nT +1 (h, n|θ ) > 0 ⇔
Px (ξ |θ, h, n) ≤ Py (ξ |θ, n), where Px (·|θ, h, n) is the cdf of the
highest order statistics of a random sample of values of size n
and Py (·|θ, n) is the cdf of the transaction price of auctions with
n bidders.

ρ∈A

(8)

= max E[u(θ, h, ρ; nT +1 )|zT ; δh,2 ],
ρ∈A

where δh denotes a Dirac measure at h = 0.

where δh,2 is a Dirac measure on h = 0 and n = 2.

Proof. Consider any (θ, h) ∈  × H for which there is some
ph ∈ Ŵ such that pθ (θ )ph (h|θ ) > 0. Then, under the iid assumption, Equation (6) implies that for all ξ ∈ ℜ+ and n ≥ 2,

Proof. Let F(j :n) (·) denote the distribution of the jth highest order statistics among n iid draws from F (·). The fact that
F(n−1:n) (x) ≥ F(n:n+1) (x) for all x ∈ ℜ follows from

F(n−1:n) (x) = n[F (x)]n−1 − (n − 1)[F (x)]n
Px (ξ |θ, h) ≤ Py (ξ |θ ) ⇔ Pvn:n (ξ |θ, h) ≤ Pvn:n (ξ |θ, 0),
⇔ Pvj :n (ξ |θ, h) ≤ Pvj :n (ξ |θ, 0),

+ {F(n:n+1) (x) − F(n:n+1) (x)},
(9)

for any j ≤ n (Lemma 1, HT). Then, using the inequality for
j = n − 1 gives
 ∞
max{ρ, ξ }pvn−1:n (ξ |θ, 0)dξ
0

≤



∞

0

max{ρ, ξ }pvn−1:n (ξ |θ, h)dξ.

= n[F (x)]n−1 (1 − F (x))2 + F(n:n+1) (x)
≥ F(n:n+1) (x).
because F(n−1:n) (·) = n[F (·)]n−1 − (n − 1)[F (x)]n for all n ≥
2. Back to the problem, the seller is uncertain about two parameters n˜ := nT +1 and h. Following the same argument as in
Proposition 1,
˜ ≤ Py (ξ |θ, n)
˜ ⇔ Pvn:˜ n˜ (ξ |θ, h) ≤ Pvn:˜ n˜ (ξ |θ, 0),
Px (ξ |θ, h, n)
˜
˜
n˜
n˜
⇔ Pvn−1:
(ξ |θ, 0)
(ξ |θ, h) ≤ Pvn−1:

These inequalities then imply u(θ, h, ρ) ≥ u(θ, 0, ρ) [see Equation (1)], hence
E[u(θ, h, ρ)|zT , ph ] ≥ E[u(θ, 0, ρ)|zT , ph ]
= E[u(θ, h, ρ)|zT , δh ].
Finally, δh ∈ Ŵ because it implies x ≤ y with equality.



Intuitively, suppose there are two valuation distributions, one
first-order stochastically dominating the other. Then, the revenue
under the latter is smaller for every ρ. The proposition simply
shows that the valuation distribution whose highest order statistics equals y is dominated by all other valuation distributions
that are “reasonable” with respect to (A-HT). It is, therefore,
associated with the smallest revenue. This result implies that
choosing the worst prior amounts to treating y as x.
Until now, the article has assumed that the number of bidders is known. How should a seller choose a reserve price
if it is also ambiguous? This article considers two subcases:
(i) only the number of bidders in future (T + 1)th-auction, nT +1 ,
is ambiguous, and (ii) the number of bidders in the past and future auctions is equal but ambiguous, that is an unknown n
is constant across all t = 1, . . . , T + 1 auctions. Let the utility be indexed by n, that is, u(θ, h, ρ; n). For the case (i),

≤ Pv1:2 (ξ |θ, 0),
where the last inequality is due to inequality shown above. Moreover, for any ρ ∈ A ⊂ ℜ+ , ρPvn:˜ n˜ (ρ|θ, h) ≤ ρPvn:˜ n˜ (ρ|θ, 0) ≤
ρPv2:2 (ρ|θ, 0). Then, the result follows when the seller puts all
mass on h = 0 and n˜ = 2.

Thus, the Ŵ-maxmin rule says when the seller is ambiguous
about nT +1 , he should believe that there will be only two bidders,
that is nT +1 = 2. It is intuitive that since the competition with
two bidders is the lowest, the revenue from two bidder auctions
is the smallest, if all else is equal.
Consider now the case (ii) where the ambiguous n is constant
across auctions. By Proposition 1, for a fixed n ∈ N , the lowest
posterior expectation of the revenue can be written as
En [u(θ, hn , ρ; n)|zT , δhn ]




pθ (θ ) Tt=1 pvn:n (yt |θ, 0)
:=
u(θ, 0, ρ; n) 
dθ,

˜ Tt=1 pvn:n (yt |θ,
˜ 0)d θ˜
pθ (θ)


but since n is ambiguous, the Ŵ-maxmin rule solves
max min En [u(θ, hn , ρ; n)|zT , δhn ].
ρ∈A n∈N

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Aryal and Kim: A Point Decision for Partially Identified Auction Models

Note that n now affects both u(·) and the posterior, and the rule
maximizes the lower envelop of the n¯ posterior expectations of
the revenue, each with a different n ∈ N .
This section concludes with a brief discussion about a few
widely used alternative decision criteria. As noted earlier, the
pure Bayesian decision rule can be sensitive to the choice of
the prior, which motivates the use of the Ŵ-maxmin criteria.
One could employ the Ŵ-maxmin regret rule, instead. It is, however, just the Ŵ-maxmin rule with a modified utility function,
˜ h, ρ) := u(θ, h, ρ) − maxρ˜ u(θ, h, ρ).
˜ But, this modificau(θ,
tion lacks an economic interpretation. Moreover, in general,
such a modification is used when the original (usually, some
maxmin) rule exhibits an odd behavior because the worst scenario is extremely unreasonable (see sec. 5.5; Berger 1985).
But, the Ŵ-maxmin rule does not have this problem because the
worst prior is already reasonable, by construction. The multiplier
preferences of Hansen and Sargent (2001) are also an alternative decision criterion. Its optimal decision rule is equivalent to
a Bayesian decision rule with an initial prior and a modified
utility function (see also Strazalecki 2011). This modification,
however, depends on the decision maker’s confidence about the
initial prior. Thus, the criterion can be used only under an arbitrary assumption about the initial prior and the confidence,
which is against the spirit of the article to not use any information on bidding behavior other than (A-HT). One could also use
the second-order probability model of Klibanoff, Marinacci, and

389

Mukerji (2005), but that too requires unverifiable assumptions
about the second-order priors and the utility function.
4.

COMPARISONS BETWEEN Ŵ-MAXMIN AND HT

This section investigates the performance of the Ŵ-maxmin
rule viz-a-viz the true RMRP’s and the HT interval for four
different valuation densities. The densities and the associated
revenue functions are shown in Figures 1 and 2 for n bidders
with n ∈ {3, 4, 5}. Figure 1 is associated with the density similar
to that of an exponential distribution and a long-tailed density,
and Figure 2 is associated with lognormal densities with different parameters; see Appendix A for construction of these
density functions. (Notice that the revenue structures here are
all asymmetric, so that the cost of choosing ρ > ρ0 is more than
choosing ρ < ρ0 for the same magnitude of bias.)
For each of these valuation densities, the experiments consider (n, T ) ∈ {3, 5} × {100, 200} leading to a total of 16 experiments. Each experiment, that is, each triplet of (Pv , n, T ),
conducts 1000 Monte Carlo replications. For each experiment,
the seller randomly selects a bidder and uses y˜ := 0.05 × y˜
as the minimum bid increment rule. The chosen bidder bids
exactly y˜ + y˜ , as long as it is less than his or her value (no
jump bidding). Each replication uses only transaction prices to
implement alternative decision rules: (i) the rule that randomly
chooses a point from the HT interval, HTRAN , (ii) the rule that

Figure 3. Exponential-like distribution. The revenue function is plotted along with the distributions of the lower and the upper bounds for the
HT set (light lines) and the reserve prices chosen by the Ŵ-maxmin (heavy).

390

Journal of Business & Economic Statistics, October 2013

chooses the midpoint of the HT interval, HTMID , and (iii) the
rule that chooses the true RMRP, Oracle. The comparison with
Oracle provides valuable information about how closely the
Ŵ-maxmin rules approximate the true optimal revenue. Let D
collect these three rules.
For the implementation of the Ŵ-maxmin rules, the experiments specify the distribution of the transaction price using the
Bernstein cdf:
⎤n
⎡
k

θj Beta(y|j, k − j + 1)⎦ ,
Py (y|θ ) = ⎣

(10)

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j =1

where θ ∈ k−1 , the k − 1 dimensional unit
k−1simplex, that
is, θj ≥ 0 for all j = 1, . . . , k − 1 and
j =1 θj ≤ 1, and
Beta(·|a, b) denotes the beta cdf with parameters a and b; see
(Petrone 1999a,b) for a nonparametric Bayesian method that
uses the beta mixture. This article employs the model with
k = 15 with the uniform prior over 14 . Appendix A provides
some information on the flexibility of the model.
Each Monte Carlo replication q = 1, . . . , 1000 simulates a
q
q
sample zT , and implements every decision rule in D. Let ρd be
q
the reserve price chosen by the rule d ∈ D, and let ud be the
q
q
q
true revenue at ρd , that is, ud := u(Pv , ρd ). Then, the percentage
revenue gains of the Ŵ-maxmin approach over alternative rules

are computed by
1000  q
q
1  uŴ − ud
× 100%
q
1000 q=1
ud

(11)

for d ∈ {HTRAN , HTMID , Oracle}.
Table 3 shows the results. For each (Pv , n, T ), columns (I ),
(I I ), and (I I I ) are the percentage revenue gains (11) of the
Ŵ-maxmin rule relative to HTRAN , HTMID , and Oracle, respectively. For example, the first row and column (I ) is associated
with (Pv , n, T ) = (exponential-like, 3, 100). It shows that the
revenue gain of the Ŵ-maxmin approach over the overall HT
interval is around 23.64%. With T = 200, this gain is around
23.62% (second row). The third and fourth rows collect the
results when n = 5. When HTMID is considered, the Ŵ-maxmin
rule still produces substantially larger revenues (7%–15%; see
column I I ). Moreover, the loss of Ŵ-maxmin relative to the true
RMRP is relatively small (see column I I I ).
Figure 3 explains the revenue gains. Each panel depicts
the (simulated) distribution of reserve prices chosen under Ŵmaxmin rule (heavy solid) and the lower and upper bounds for
the HT interval (light solid) along with the revenue function. The
left (right) panels are with T = 100 (T = 200), and the upper
(the lower) panels are with n = 3 (n = 5). The left-upper panel
shows that the upper bound of the HT interval is distributed
around 0.6 with a wide support, while the revenue function

Figure 4. Longtail distribution. The revenue function is plotted along with the distributions of the lower and the upper bounds for the HT set
(light lines) and the reserve prices chosen by the Ŵ-maxmin (heavy).

Aryal and Kim: A Point Decision for Partially Identified Auction Models

Table 3. Percentage revenue gain of the Ŵ-maxmin rule

Pv

n

T

HTRAN
(I )

HTMID
(I I )

Oracle
(I I I )

Exponential

3

100
200
100
200
100
200
100
200
100
200
100
200
100
200
100
200

23.640
23.619
17.079
15.230
19.343
17.767
8.279
6.798
23.785
24.077
17.727
16.549
9.643
8.606
4.380
3.623

15.116
14.912
9.062
7.081
16.152
13.072
3.339
1.575
18.317
18.586
11.911
10.533
2.185
1.641
0.570
0.401

−3.933
−4.618
−0.854
−0.934
−1.265
−1.420
−0.139
−0.149
−2.105
−2.652
−0.497
−0.562
−0.330
−0.373
−0.013
−0.015

5
Longtail

3
5

LogN(3,1)

3
5

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LogN(4,1/2)

3
5

NOTE: This table compares the performance of Ŵ-maxmin rule and the rules using the
HT interval. For example, for (Pv , n, T ) = (exponential, 3, 100), the Ŵ-maxmin rule with
n known produces 23.64% higher revenues than a randomly chosen reserve price from the
HT interval (see column I).

indicates that all the reserve prices larger than approximately
0.3 produces lower revenues than zero reserve price. This implies that a significant portion (54.92%; Table 1) of the HT
interval is less profitable than zero reserve price. On the other

391

hand, the Ŵ-maxmin rule is distributed over the area in which
the revenue is increasing, selecting higher reserve prices than
the lower bound of the HT interval. As a result, the Ŵ-maxmin
rule produces larger revenues than HTRAN and HTMID .
The revenue comparisons for other valuation densities are
similar: see the rest of the table and see also Figures 3–6. In conclusion, the simulation results here suggest that the Ŵ-maxmin
approach can provide a practical policy recommendation.

5.

SOME EXTENSIONS

This section extends the Ŵ-maxmin framework to auctions
with correlated values, auctions with binding reserve price, and
auctions with observed covariates.

5.1

Correlated Private Values

This subsection considers English auctions with correlated
private values as in Aradillas-Lopez, Gandhi, and Quint (in
press). While the environment and the data structure are the
same as before, let the bidders’ value (v1 , . . . , vn ) be jointly
distributed as Pv (·, . . . , ·|θ, h). Since the revenue function is still
given by Equation (1), the choice of a reserve price depends on
the distributions of the second highest value Pvn−1:n (·|θ, h) and
the highest value Pvn:n (·|θ, h). But unlike IPV, when values are

Figure 5. Lognormal (3,1). The revenue function is plotted along with the distributions of the lower and the upper bounds for the HT set
(light lines) and the reserve prices chosen by the Ŵ-maxmin (heavy).

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Journal of Business & Economic Statistics, October 2013

Figure 6. Lognormal (4,1/2). The revenue function is plotted along with the distributions of the lower and the upper bounds for the HT set
(light lines) and the reserve prices chosen by the Ŵ-maxmin (heavy).

correlated, the distributions of order statistics are not necessarily
linked through the identical marginal valuation distribution. In
particular, Equation (9) does not hold anymore. Thus, following
Aradillas-Lopez, Gandhi, and Quint (in press), it is assumed that
Assumption 3. (A-AGQ) The transaction price is equal to the
second highest value.
Under (A-AGQ), zT point identifies Pvn−1:n (·|θ, h), that is,
Py (·|θ ) = Pvn−1:n (·|θ, h). Using Px (·|θ, h) = Pvn:n (·|θ, h), the
revenue function (1) is written as
 ∞
u(θ, h, ρ) =
max{ρ, ξ }dPy (ξ |θ ) − ρPx (ρ|θ, h).
0

Since Px (·|θ, h) = Pvn:n (·|θ, h) ≤ Pvn−1:n (·|θ, h) = Py (·|θ ) and
Py (ξ |θ ) = Pvn:n (·|θ, 0) for all θ , ρ, and h, it follows that the
revenue is bounded below by u(θ, 0, ρ), that is, u(θ, h, ρ) ≥
u(θ, 0, ρ). In other words, the worst case is when h = 0 and
hence the seller should still solve Equation (8).
Proposition 2. Let the set Ŵ satisfy the property (6). When
the values are correlated, under (A-GS) and (A-AGQ), Equation
(8) holds.
Hence, Proposition 2 implies that when the distribution of the
second highest value is identified by the transaction price y, it

is still optimal to treat y as x, the highest value for choosing a
reserve price.

5.2 Binding Reserve Prices
Back to the IPV case. Suppose the data zT contains a binding reserve price ρ1 , that is, Pv (ρ1 |θ, h) > 0. Then, the seller
observes transaction prices only if there is i ∈ {1, . . . , n} such
that vi ≥ ρ1 . Lemma 5 of HT shows that as long as ρ1 < ρ0 :=
arg maxρ u(θ, h, ρ), and u(θ, h, ρ) is strictly pseudo-concave
in ρ, the RMRP with the truncated distribution P˜v (v|θ, h) :=
[Pv (v|θ, h) − Pv (ρ1 |θ, h)]/[1 − Pv (ρ1 |θ, h)] equals the RMRP
associated with the original distribution Pv (v|θ, h). Therefore,
Proposition 1 and Corollary 1 hold with P˜v (v|θ, h) instead of
Pv (v|θ, h).
Whether ρ1 < ρ or not is an empirical question. The empirical
literature has commonly found that the actual reserve prices are
very small for many important public auctions. For example,
see Paarsch (1997) for the British Columbia Timber sales and
Haile and Tamer (2003) and Lu and Perrigne (2008) for the U.S.
timber sales. Those timber auctions set reserve prices following
some simple rule of thumb, which is a function of some observed
characteristics such as the experts’ estimate of the market value
and total volume of timbers. Since the literature testifies that
the actual reserve prices are too small to be optimal, those rules

Aryal and Kim: A Point Decision for Partially Identified Auction Models

may be regarded as some random rule from a formal decision
perspective.

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5.3 Auction Specific Covariates
Many datasets provide the information on auction specific
characteristics, ωt . This subsection discusses the extension of
the Ŵ-maxmin rule to situations where the valuation distribution
depends on ωt . The stochastic dominance relation in Proposition
1 and Corollary 1 still holds for the (conditional) valuation
distribution Pv (·|θ, h, ω) for a given ω, and the lowest revenue is
attained at h = 0. Hence, the Ŵ-maxmin approach is applicable
with the observed characteristics for the future auction, that is,
ωT +1 .
To implement the Ŵ-maxmin rule, therefore, it is sufficient to
model the valuation distribution Pv under the worst prior on h,
that is,
Py (y|θ, ω) = [Pv (y|θ, h = 0, ω)]n ,
where the left-hand side is the cdf of the observed transaction
price, which is equal to the cdf of the highest value on the righthand side with h = 0. One way to specify it would be as follows.
Let Q(·|γ ) be a parametric distribution on the support of the
value that approximates the valuation distribution and (·|ψ)
be a flexible (or nonparametric) distribution defined on [0, 1].
For example,  could be modeled by a series representation
such as the Bernstein polynomial distribution in Equation (10).
Then, it can be specified as
Pv (v|θ, h = 0) := [Q(v|γ )|ψ]
with θ := (γ , ψ). Note that Q approximates the valuation distribution first and H improves the approximation (under the worst
prior on h).
If each auction is characterized by ωt , one may model the
parameter γ as a function of ωt ∈ ℜd . That is, the distribution of the transaction price can be expressed as Py (y|θ, ωt ) =
{ (Q[y|γ (ωt )]|ψ)}n . For example, if Q(·|μ, σ 2 ) is lognormal
distribution with parameter γ := (μ, σ 2 ), then μ(ωt ) = ωt′ a and
σ 2 (ωt ) = exp(ωt′ b) with a, b ∈ ℜd can be used.
6.

CONCLUSION

The problem of choosing a reserve price using a bid sample has been widely studied in the auction literature, since
Paarsch (1997). This article proposes to use decision making
under ambiguity to determine a reserve price when a sample of
transaction prices from English auctions does not point-identify
the valuation distribution and the seller is unwilling to make
strong (unverifiable) bidding assumptions other than bidders
never overbidding.
This article extends Kim (2013b) to the partially identified
auction models of HT, and proposes to use the maxmin decision
criteria axiomatized by Gilboa and Schmeidler (1989), where
the seller in the face of ambiguity chooses a reserve price that
maximizes the worst case revenue. Under the only behavioral
assumption of no-overbidding, this article shows that it is optimal to employ the Bayesian decision rule of Kim (2013b)
interpreting the transaction prices as the highest value. A Monte
Carlo exercise shows that this procedure performs well. Exten-

393

sions to auctions with uncertain number of bidders, correlated
values, binding reserve prices, and observed characteristics are
also considered.
For the IPV case, this article abstracts away the choice of
auction format on account of the revenue equivalence theorem,
and for the correlated values, it implicitly assumes the seller uses
the English auction as it gives higher revenues than the first price
auction (see Milgrom and Weber 1982). However, if bidders
are asymmetric, neither the revenue equivalence theorem nor
the revenue ranking holds even when values are independent
(see Marshall et al. 1994; Maskin and Riley 2000; Cantillon
2008). Hence, the choice of auction format is also an important
issue. For this reason, this article does not pursue the case of
asymmetric bidders, but a few words on it are in order.
If the format is fixed to be the English auction, the Ŵ-maxmin
procedure can be applied to auctions with the asymmetric
bidders under some assumptions. For example, if bidders are
grouped by different types and the probability for each type is
known, then it is still optimal for the seller to treat the transaction prices as the valuation and compute the optimal reserve
price with respect to the mixture distribution, where the mixing weights can be identified from the data. In the U.S. timber
auctions, for example, information on the fractions for small
mills or huge mills is available. Similarly, the bidding history
of highway procurements reveals the types of large companies
and small local companies.
To sum up, this article takes a first step for empirical auction
design where the structural parameter is partially identified, by
proposing a method that is decision theoretically well-founded,
simple, and profitable. A further investigation for more general
situations including auctions with asymmetric bidders would be
important future research.
APPENDIX A: VALUATION DENSITIES AND
FLEXIBILITY OF THE BERNSTEIN DENSITY
A.1 Construction of True Valuation Densities
Figure 1 is associated with the density similar to that of an exponential distribution and a long-tailed density. These densities
have the form of Equation (10) with k = 15. For the exponentiallike density, the parameter values are θ :=(0.3548, 0.2350,
0.1486, 0.0946, 0.0466, 0.0440, 0.0217, 0.0119, 0.0089, 0.0080,
0.0084, 0.0081, 0.0049, 0.0028, 0.0017) and for the long-tailed
density, θ :=(0.0748, 0.1403, 0.1871, 0.5145, 0.0009, 0.0009,
0.0009, 0.0009, 0.0009, 0.0009, 0.0009, 0.0750, 0.0009, 0.0009,
0.0002). Similarly, Figure 2 is associated with lognormal densities with different parameters. The lognormal distributions with
(μ, σ ) = (3, 1) and (4,1/2) are truncated at the 99th percentile
and rescaled, so that their supports are the unit interval. HT
employed the lognormal densities that appear in Figure 2 for
Monte Carlo studies.
A.2. Flexibility of the Bernstein Density
Figure 7 gives some information on the flexibility of the BPD
in Equation (10) with k = 15. Panel (a) plots 15 beta density
functions (basis functions for the model) and panel (b) plots five
valuation densities that are drawn from the uniform prior over

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394

Journal of Business & Economic Statistics, October 2013

Figure 7. Bernstein density with k = 15. This figure gives some information on how flexible the Bernstein density model with k = 1 would
be. Panel (a) plots 15 basis functions and panel (b) plots five densities drawn from a uniform prior over 14 .

14 , that is, each of the five curves on panel (b) is a mixture of
the 15 beta densities on panel (a).

for s < 50,
0

1

(θ , θ , . . . , θ
APPENDIX B: COMPUTATION
As mentioned in Section 4, there are 16 experiments. Each
experiment employs 1000 Monte Carlo replication to make a
long-run comparison between the Ŵ-maxmin approach and the
average performance of the HT interval. The first replication
employs the Adaptive Metropolis (AM) algorithm of Haario,
Saksman, and Tamminen (2001) to draw {θ s }Ss=1 from the posterior distribution, and all other replications q = 2, . . . , 1000
employ the importance sampling using the {θ s }Ss=1 from the first
q
replication and the new data zT .
B.1 Sampling from the Posterior with a Flat Prior:
1st Replication
Let zT1 := (y11 , . . . , yT1 ) denote the simulated auction data of
size T in the first Monte Carlo replication; q = 1. Let also θ 0 :=
( k1 , . . . , k1 )(k−1)×1 be the initial vector for the AM algorithm. At
each s = 1, . . . , S, a candidate parameter vector is drawn from
θ˜ |θ 0 , θ 1 , . . . , θ s−1 ∼ N(θ s−1 , (θ 0 , θ 1 , . . . , θ s−1 )),
where N(a, b) is the k − 1 dimensional multivariate normal
distribution with mean vector a and covariance matrix b, and

s−1

) :=




2.382
Ik−1
k−1

with Ik−1 being the k − 1 dimensional identity matrix, but for
s ≥ 50,
(θ 0 , θ 1 , . . . , θ s−1 )
⎧
2
⎪
⎨ 2.38 I
with probability 0.05
k−1
k−1
:=
⎪
⎩ cov(θ 0 , θ 1 , . . . , θ s−1 ) with probability 0.95,

where cov(a1 , . . . , an ) denotes the sample covariance matrix of
n vectors a1 , . . . , an . Then, let θ s := θ˜ with probability


T

˜
py (yt1 |θ)
0
s−1
˜
˜
,
α(θ , θ , . . . , θ ) = min 1, Ik−1 (θ ) ·
p (y 1 |θ s−1 )
t=1 y t
(B.1)
with probability 1 − α(θ˜ , θ , . . . , θ ). Note
or let θ := θ
that since the uniform prior is employed, the prior ratio in α(·)
reduces to the indicator for the k − 1 dimensional simplex,
Ik−1 (·).
The likelihood can be evaluated as follows: let x be a column vector of N numbers in [0, 1], and let Bk (x) be a N × k
matrix with the (i, j )th element being beta(xi |j, k − j + 1), for
s

s−1

0

s−1

Aryal and Kim: A Point Decision for Partially Identified Auction Models
(i,j )

example, if k = 15,

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B15 (x)
⎡

395

⎤

beta(x1 |1, 15)

beta(x1 |2, 14)

···

beta(x1 |15, 1)

⎢
⎢ beta(x2 |1, 15)
⎢
=⎢
⎢
⎣
beta(xN |1, 15)

beta(x2 |2, 14)

···

⎥
beta(x2 |15, 1) ⎥
⎥
⎥,
⎥
⎦
beta(xN |15, 1)

...
beta(xN |2, 14)

···

(i,j )

and let fk (x, θ ) and Fk (x, θ ) be the (i, j )th element of
fk (x, θ ) and Fk (x, θ ), respectively. Then, the likelihood ratio in
Equation (B.1) is given by
 1    1 m−1
k


T
T
˜ Fk yt , θ˜


py yt1 |θ˜
j =1 fk yt , θ
=
 1

  1
m−1 .

k
1
s−1
p y |θ s−1
Fk yt , θ s−1
t=1 y t
t=1
j =1 fk yt , θ

Note that Bk (zT1 ) and Ck (zT1 ) are computed only once.
Each experiment iterates the AM algorithm 20,000,000 times,
where beta(c|a, b) is the beta density at c with parameters a and
b. Similarly, define Ck (x) with the cdf’s instead of pdf’s, for recording every 1000th iteration. Figure 8 shows the mixing quality of the MCMC traces for each parameters for the
example, if k = 15,
experiment (Pv , T , m) = (exponential-like, 100, 3). The first
C15 (x)
⎤ half of the outcomes is discarded as a burn-in phase. Thus,
⎡
the experiment employs S = 10,000 draws from the posterior
Beta(x1 |1, 15) Beta(x1 |2, 14) · · · Beta(x1 |15, 1)
⎥ to implement the Ŵ-maxmin rule.
⎢
⎥,
...
= ⎢
⎦
⎣
In particular, the Ŵ-maximin rule solves
Beta(xN |1, 15) Beta(xN |2, 14) · · · Beta(xN |15, 1)
S
1
where Beta(c|a, b) is the beta cdf at c with parameters a and b.
u(θ s , 0, ρ; m),
(B.4)
max
ρ∈A S
In addition, let ιT be the T-dimensional column vector of
s=1
¯ denote the element-wise multiplication operator,
ones. Let ×
¯ 1 , b2 ) = (a1 b1 , a2 b2 ). Define
for example, (a1 , a2 )×(b
where the experiments employ A := {xj }1000
j =0 with xj =
j/1000.
The
revenue
u
is
computed
via
the
trapezoid
rule over
¯ T · θ ′ ),
fk (x, θ ) := Bk (x)×(ι
(B.2)
the discretized unit interval A. The next subsection describes
¯ T · θ ′ ),
Fk (x, θ ) := Ck (x)×(ι
(B.3) how to evaluate the revenue functions.

Figure 8. MCMC outcomes: exponential-like, (T , m) = (100, 3). The figure shows the MCMC outcomes for each parameter θj with
j = 1, . . . , 14 for the experiment with the exponential-like valuation density and (T , m) = (100, 3). Each panel records every 1000th iteration
from 20,000,000 AM outcomes. The MCMC outcomes for other experiments are similar to the ones here.

396

Journal of Business & Economic Statistics, October 2013

B.2. Evaluation of the Revenue Functions
The seller’s revenue in Equation (1) can be written as
u(θ, 0, ρ; m)
(B.5)
:= mρ(1 − Pv (ρ|θ ))Pv (ρ|θ )m−1
 1
y(1 − Pv (y|θ ))Pv (y|θ )m−2 pv (y|θ )dy.
+ m(m − 1)

for each s = 1, . . . , S. Then,


S

ωq (θ s )
u(θ s , 0, ρ; m)
T
˜
s
s˜ =1 ωq (θ )
s=1

 q
a.s
u(θ s , 0, ρ; m)p θ |zT dθ.
→


ρ

The trapezoid rule approximates all the integrals of Equation
(B.5) using the J = 1001 equidistant reference points on the
unit interval, A.
For xi ∈ A, compute

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pv (xi |θ ) =

k


fk

k


Fk

(i,j )

(A, θ ),

j =1

Pv (xi |θ ) =

(i,j )

(A, θ ),

ACKNOWLEDGMENTS
The authors thank Keisuke Hirano, Aleksey Tetenov, Amit
Gandhi, Laurent Lamy, Myoung-jae Lee, and seminar audience
at Universities of Auckland, Chicago, and Queensland, and at
the 4th World Congress of the Game Theory Society. The authors thank the referee and the associate editor for useful and
insightful suggestions, which improved the quality and the scope
of the article. Kim acknowledges the financial support from the
Faculty of Business Research Grant, UTS.

j =1

(i,j )

where fk

(i,j )

(A|θ ) and Fk

(A|θ ) are the (i, j )th elements of

[Received June 2012. Revised January 2013.]

¯ T · θ ′ ),
fk (A, θ ) = Bk (A)×(ι
¯ T · θ ′ ),
Fk (A, θ ) = Ck (A)×(ι
respectively. They are the 1001 dimensional column vectors of
pdf’s and cdf’s evaluated at each x ∈ A. It is straightforward to
compute the first term of Equation (B.5). Finally, let
D1 :=

1
x1 (1 − Pv (x1 |θ ))Pv (x1 |θ )m−1 pv (x1 |θ ),
2
1

Di := Di−1 +

1 
xi−j (1 − Pv (xi−j |θ ))
2000 j =0

× Pv (xi−j |θ )m−1 pv (xi−j |θ ),
for all i = 2, . . . , 1000. Then, for any j > 1,
1000


Di −

j


Di ≈

i=1

i=1



1

x(1 − Pv (x|θ ))Pv (x|θ )m−1 pv (x|θ )dx.
xj

B.3. Replications 2 to 1000: Importance Sampling
q

Let zT := (y11 , . . . , yT1 ) denote the simulated auction data of
size T in the qth Monte Carlo replication with q = 2, . . . , 100