Int. J. Production Economics 127 (2010) 147–157
Contents lists available at ScienceDirect
Int. J. Production Economics
journal homepage: www.elsevier.com/locate/ijpe
The optimal format to sell a product through the internet: Posted price,
auction, and buy-price auction
Daewon Sun a,1, Erick Li b,2, Jack C. Hayya c,
a
Department of Management, University of Notre Dame, Room 366, Mendoza CoB, Notre Dame, IN 46556, USA
Department of Operations Management and Econometrics, The University of Sydney, Room 490, H04 - Merewether Building, NSW 2006, Australia
c
Department of Supply Chain & Information Systems, The Pennsylvania State University, 362 School of Business, University Park, PA 16802, USA
b
a r t i c l e in fo
abstract
Article history:
Received 9 November 2009
Accepted 9 May 2010
Available online 1 June 2010
Motivated by the proliferation of online selling, we study a seller’s decision problem: The seller has one
unit of product and needs to choose a selling format among three different ones (posted price, auction,
and buy-price auction), as well as to decide upon the corresponding decision variables. By incorporating
the auction participation cost of the bidders and the operational cost of the seller, we demonstrate that
these two costs play an important role in the choice of a selling format. We first characterize and
identify the customers’ decision rule for buy-price auction and then present the seller’s optimal choices
by comparing the performance of the three formats. Through a comprehensive numerical study, we
investigate the performance of the three selling formats by changing the experimental factors.
& 2010 Elsevier B.V. All rights reserved.
Keywords:
Online auction
E-commerce
Decision making optimization
Optimal decision rule
Game theory
Strategic behavior
Pricing
1. Introduction
The development of the Internet has led to increased use of
electronic commerce in business to business, business to
consumers, and consumers to consumers. One of the most
noticeable changes is the widespread use of online auctions to
sell products through the Internet. This increased interest and
demand for online auctions have resulted in the proliferation of
online auction sites (e.g., eBay, Yahoo Auctions, UBid, Sam’s
Auction, etc.) that allow a seller to establish an online auction for
products. Away from the traditional simple auction format or
posting, some of the major auction sites have been offering
various features and different selling formats. For example, eBay
offers various features (adding subtitles, more or larger pictures,
bold, highlight, etc.) at additional cost.
More interestingly, we note that a seller can even choose a
selling format, e.g., eBay offers auction, fixed price, and Buy-ItNow auctions. Yahoo Auction provides similar choices, but Yahoo
offers a permanent buy price, whereas eBay’s Buy-It-Now is
temporary (i.e., eBay’s Buy-It-Now option disappears immediately
after receiving the first bid above a reservation price, but Yahoo’s
permanent buy price remains until the end of the auction or a
Corresponding author. Tel.: +1 814 862 2219; fax: +1 814 863 7067.
E-mail addresses: [email protected] (D. Sun), [email protected] (E. Li),
[email protected] (J.C. Hayya).
1
Tel.: + 1 574 631 0982; fax: + 1 574 631 5127.
2
Tel.: + 61 2 9114 0751; fax: + 61 2 9351 6409.
0925-5273/$ - see front matter & 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.ijpe.2010.05.006
customer exercises the buy option). Based on these observations,
we study the performance of buy-price auctions (i.e., auctions
with a permanent buy price) as compared to pure selling formats.
Specifically, we consider a seller who has one unit of product. The
seller wants to sell the product through one of the major online
stores, but faces a decision problem: to choose a selling format
from three choices (posted price, auction, or buy-price auction)
and then decide upon the corresponding decision variables, e.g.,
fixed price or buy price.
In this paper, we present analytical models to examine the
impact of auction participation and operational costs on the choice
of the three selling formats. First, Bajari and Hortacsu (2003, 2004)
show empirical evidence of the existence of auction entry costs,
stating that ‘‘The analysis of online auctions suggests that entry
costs, and hence the endogenous entry decisions of bidders, are
indeed quite important, and should be taken into account when
modeling the performance of alternative trading rulesy’’ (Bajari
and Hortacsu, 2004, p. 483). In incorporating this comment in our
present paper, we are assuming that there is disutility to
participating in an auction, and, hence, some customers would
not bid although they know of the existence of an auction.
Second, when a seller uses posted price, it is possible that the
valuations of all potential buyers would be lower than the posted
price, and hence no sale would take place. When the seller uses
auction formats, in the presence of auction participation costs, it
is possible that there would be no bidding even though customers
observe the auction at a web store. In either circumstance, the
seller incurs a non-negative operational cost, which can be
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D. Sun et al. / Int. J. Production Economics 127 (2010) 147–157
understood in two different ways. One can consider it as an
additional holding cost. When the seller is unable to sell the
product, she has to keep it until she can find a buyer. Another way
to understand the cost is the additional fee to post the product in
the next period. For both interpretations, an operational cost
would be incurred if the product is not sold.
The rest of this paper is organized as follows. Section 2
discusses the existing literature on comparing selling formats and
studying auctions with a buy option. Section 3 describes the
problem setting and presents the analysis of buy-price auctions,
focusing on characterizing the customers’ decision rule and
formulating the seller’s profit function. We demonstrate that
there exists a symmetric equilibrium (i.e., a unique threshold for
exercising the buy option) and present our formulation of the
seller’s expected profit under the buy-price auction. A comparison
of the three choices is presented in Section 4. We provide an
analytical analysis for the comparison of the pure formats and
discuss the performance of the buy-price auctions, as compared to
the pure formats, through a comprehensive numerical study.
Section 5 discusses our extensions and Section 6 concludes our
discussion with future research directions. All proofs are provided
in the Appendix.
2. Literature review
Since auctions are widely studied, numerous researchers have
examined fundamental aspects and invented applications of
auctions. For a broad overview of the auction literature, we refer
the readers to Wolfstetter (1994), Matthews (1995), and Klemperer (1999) that summarize and explain the overview of
auctions. Further, we note that there are a few interesting
empirical studies that report on the proliferation and various
characteristics of online auctions (see, e.g., Lucking-Reiley, 2000;
Roth and Ockenfels, 2002; Ockenfels and Roth, 2002; Bajari and
Hortacsu, 2003, 2004).
The comparison between auctions and posted price is first
considered by Wang (1993). In Wang’s model, a retailer has only
one unit and customers arrive stochastically according to a
Poisson distribution. He assumes that the retailer incurs different
stocking fees for the unit depending on the choice of a selling
format: display cost for posted price and storage cost for auctions.
He also assumes that storage cost is cheaper than display cost and
demonstrates that auctions are preferable when there is no
auctioning cost. Further, he shows that auctions with positive
auctioning costs can be superior to posted price, when customer
valuations are widely dispersed (i.e., the marginal revenue curve
is sufficiently stiff). Wang (1998) extends Wang (1993) by
considering correlated private valuations. He shows that the
main results of the previous study still hold for this relaxed model
setting. Kultti (1999) studies the comparison of the two selling
formats under a game-theoretic setting where there are multiple
sellers and buyers. He demonstrates that the two formats in his
model setting are practically equivalent by showing the existence
of a Nash equilibrium with co-existence of the two formats.
Inventory decisions under auctions have been considered in a
few papers. Pinker et al. (2005) consider sequential and multi-unit
auctions to dispose of a given number of units. They show that the
lot size of sequential auctions decreases from period to period
when there exist positive holding costs. Vulcano et al. (2002)
consider an auction where a seller has a constant number of units
and a random number of buyers arrive during certain periods.
They compare the optimal auctions to the optimal list prices and
demonstrate that the optimal auctions outperform the optimal
list prices under certain conditions (i.e., moderate number of
selling periods, constrained capacity, and not too large total
volume of sales). They also point out that a wide dispersion of
buyers’ valuations and large number of buyers per period make
auction favorable. van Ryzin and Vulcano (2004) study a joint
inventory-pricing decision under auctions over an infinite
horizon. They compare the optimal design of auctions to that of
list pricing (Federgruen and Heching, 1999) and demonstrate that
auctions outperform list pricing significantly when the number of
buyers per period is moderate, holding cost is relatively high, or
there exists great uncertainty in demand.
The Buy-It-Now option available at eBay has been one of the
widely studied features offered by many online auction sites.
Budish and Takeyama (2001) demonstrate that the buy price
enables a seller to extract a risk premium and hence improve
profits when bidders are risk-averse. Their paper sheds light on
this interesting topic, but the study is limited in the sense that the
model considered has only two bidders and discrete valuation for
the product. Matthews (2004) studies an auction with a buy
option similar to eBay’s Buy-It-Now (a temporary option). He
demonstrates that the seller would set the buy option high
enough so that it would not be exercised by bidders, if bidders are
indifferent in transaction times (i.e., bidders are time-insensitive).
However, he shows that when bidders discount their future
utility, the buy price can be set optimally and could lead to higher
profits. Wang et al. (2004) discuss the impact of eBay’s Buy-ItNow when there exists auction participation cost and demonstrate that this feature of premature ending with Buy-It-Now
price can outperform regular auctions or posted price under
certain conditions. Hidvegi et al. (2006) extend Budish and
Takeyama (2001)’s work by considering n bidders with continuously distributed valuations. They find that an optimally chosen
buy price does not reduce the expected payoff of any agent (the
seller or bidders) and increases expected social welfare, when
either the seller or bidders are risk-averse. Reynolds and Wooder
(2006) also consider the case where bidders are risk-averse, and
demonstrate that auctions with buy price are preferable in that
the seller can increase expected revenue. Gallien and Gupta
(2007) analyze and compare the benefits of temporary and
permanent buy price. They consider a model setting where the
bidders’ arrival process follows a Poisson distribution and bidding
times are endogenized. They find that a permanent buy price
makes auction participants bid at the last minute to eliminate
other bidders’ opportunity to respond. In contrast, they demonstrate that the first auction participant who sees a temporary buy
price should bid immediately to remove the buy option. In terms
of seller’s profit, they find that a permanent buy price is superior
to a temporary option.
There are a few studies that discuss a similar problem: ‘‘Dual
Channel’’, where a retailer offers two selling formats (auction and
posted price) simultaneously (see, e.g., Etzion et al., 2006;
Caldentey and Vulcano, 2007; Sun, 2008). Under ‘‘Dual Channel’’,
the retailer optimally allocates multiple units for the two
channels. Hence, it is still possible that a customer can buy a
product after some other customers already bought some units of
the product at a fixed price or made bids for auctions. However,
under the buy price option, customers need to take into
consideration the fact that the product would not be available
after another customer exercised the buy option.
Our work is inspired by Wang et al. (2004), who study eBay’s
temporary buy option. Although we consider Yahoo Auction’s
permanent buy option, our basic model formulation follows
theirs. Interestingly, we find that in our model setting the
structure of the customer optimal bidding strategy is identical,
whether the seller offers a temporary or a permanent buy option.
However, we find that the seller’s expected profits are different,
which is consistent with Gallien and Gupta (2007)’s finding. Since
Wang et al. (2004) present customer bidding strategy under the
D. Sun et al. / Int. J. Production Economics 127 (2010) 147–157
uniform distribution, we extend their results by considering a
general distribution of customer valuation. In addition, in Section
5 we relax two of major assumptions of the basic model setting
and present our findings.
To summarize, our focus is to investigate the performance of
the permanent buy option and then compare the three selling
formats: (1) whether one selling format always dominates the
other two or at least one and (2) under what circumstance a
particular selling format would be preferable.
3. Model setting and buy-price auctions
We first describe the model setting considered in this paper
and present our analysis on buy-price auctions. We demonstrate
that there exists a unique participation and buy threshold when
customer valuations are generally distributed and present the
seller’s profit function under the customers’ symmetric equilibrium strategy.
149
Table 1
Notation.
h
n
FðÞ
f ðÞ
o
i
vi
v
P
pa
pb
X(n,i)
G(n,i)(v)
g(n,i)(v)
Unit operational cost
Number of customers
cdf of customer valuation
pdf of customer valuation
Auction participation cost
Index for customer
Customer i’s valuation
Participation threshold
Probability
Auction winning price
Buy price
ith highest valuation among n customers
Probability of Xðn,iÞ r v, where
P
j
nj
, i ¼ 1,2, . . . ,n
Gðn,iÞ ðvÞ ¼ ji1
¼ 0 ðn!=ðnjÞ!j!Þ½1FðvÞ ½FðvÞ
pdf of G(n,i), where
gðn,iÞ ðvÞ ¼ ði=1FðvÞÞðn!=ðniÞ!i!Þ½1FðvÞi ½FðvÞni f ðvÞ,i ¼ 1,2, . . . ,n
v
buy threshold
H(i)(j)(x,y) Joint density function of X(n,i) and X(n,j), where io j
k
Index for selling format, k¼ p (posted price), a (auction), b (buyprice auction)
pk
Expected profit with selling format k
r
Announced price for posted price
3.1. Model and assumptions
We consider a seller who wants to sell a single unit of product
through an online auction site. Without loss of generality, the
procurement cost of the product is normalized at zero. To reflect
current practice of the major Internet auction sites, we assume
that the seller can choose a selling format among posted price,
auction, and buy-price auction. We adopt the second-price
auction format to determine the winning price so that our model
reflects online bidders’ behavior in major auction sites (last
minute bidding and/or sniping, see, e.g., Roth and Ockenfels,
2002; Ockenfels and Roth, 2002). Taking into account the
customers’ strategic behavior, the seller has to choose a selling
format and set related decision variables, if any, to maximize her
profit. If no one buys the product at the end of the period, then the
seller incurs a non-negative operational cost, h.
We assume that the number of (potential) customers, n, is
public information (i.e., the customers and seller know how many
customers are interested in the product). Customer valuations are
identically and independently distributed. The cumulative distribution function is represented by FðÞ and the density function
is represented by f ðÞ. It is also assumed that the customers and
seller know only the aggregate information of customers (i.e., they
do not know the specific valuations of customers, but know the
number of customers and the probability distribution of customer
valuations, FðÞ and f ðÞ). We assume that upon a customer’s
arrival, the customer does not know how many other customers
have already visited the store. Following the findings of existing
literature (Bajari and Hortacsu, 2003, 2004), we assume that a
customer incurs an auction participation cost ðoÞ if that customer
participates in an auction. Table 1 summarizes the notation used
in the paper.
3.2. Buy-price auctions
Following the existing literature (e.g., Wang et al., 2004;
Caldentey and Vulcano, 2007; Gallien and Gupta, 2007), we limit
our discussion to the case where the customers use a symmetric
equilibrium strategy. Also to reflect the current format of the
option (Yahoo Auction, Amazon.com), we assume that the buy
option is available until the end of the auction if no customer
exercises the buy option. However, note that the product would
not be available after a customer exercised the buy option. Facing
this type of auction, each customer can therefore take one action
among the three choices: (1) buying immediately, (2) bidding, and
(3) leaving the store.
When there exists non-negative auction participation cost, o,
customer i will join the auction if her expected surplus is not
negative. Let v be the participation threshold. Then, for a given
auction participation cost, o, the expected utility of the customer
with valuation vi is
E½Utility ¼ E½Surplus by winningo
Z vi
¼ ðvi 0ÞP½pa ¼ 0 þ
ðvi uÞP½pa ¼ u duo
¼ vi ½Gðn1,1Þ ðvÞ þ
Z
v
vi
v
ðvi uÞ½gðn1,1Þ ðuÞ duo,
ð1Þ
where GðÞ and gðÞ represent the cdf and pdf of order statistics,
respectively. The expected utility of bidding for an auction with
buy price is identical to that of bidding for a pure auction. This can
be understood by considering all the possible cases for customer
i’s winning. It is easy to see that the customer gets the product
when the rest of customer valuations are below the participation
threshold, and she pays pa ¼0 and keeps surplus vi. Next, she can
get the product if the rest of customer valuations are below her
valuation, but the second highest valuation is between the
participation threshold and her valuation. In this case, she gets
the product by paying the second-highest customer valuation.
Except for these two cases, customer i cannot win the product by
bidding.
Since the expected utility of a customer with valuation v is
indifferent between joining or staying out, we replace vi with v in
Eq. (1) and have the condition v ½Gðn1,1Þ ðvÞo ¼ 0. Therefore, to
derive the threshold, we need to solve one of the following
equations:
Gðn1,1Þ ðvÞ ¼
o
v
or
FðvÞn1 ¼
o
v:
ð2Þ
Note that the LHS of Eq. (2) is strictly increasing in v and that the
RHS of Eq. (2) is a strictly decreasing convex function in v. This
implies that when customer valuations are generally distributed, the
participation threshold v specified in Eq. (2) exists and is unique.
Now consider the buy option. For a given buy price, pb, and
participation cost, o, customer i with valuation vi gets (vi pb)
utility if the customer chooses to buy immediately. Note that a
150
D. Sun et al. / Int. J. Production Economics 127 (2010) 147–157
customer with valuation v should be indifferent to buying
immediately or bidding. Therefore, we have
Z v
vpb ¼ v ½Gðn1,1Þ ðvÞ þ
ðvuÞ½gðn1,1Þ ðuÞ duo:
ð3Þ
Next, we derive the joint density function H(1)(2)(x,y). We
observe that for x 4 y,
n!
PðX1 A dx,X2 A dy,all otherso yÞ
ðn2Þ!
n!
f ðxÞ dx f ðyÞ dy FðyÞn2 :
¼
ðn2Þ!
PðXðn,1Þ A dx,Xðn,2Þ A dyÞ ¼
v
Since Eq. (3) does not yield a closed-form solution for a general
distribution, we need to investigate whether there exist multiple
buy thresholds. Now we demonstrate the characteristics of the
buy threshold.
Proposition 1. There exists a unique buy threshold, v, for any given
pb. Specifically,
If the buy price is set higher than the participation threshold
ðpb ZvÞ, then there exists a unique buy threshold, v ðv ZvÞ, that
solves
Z v
Gðn1,1Þ ðuÞ du:
ð4Þ
pb ¼ v
v
Hence, the joint density function satisfies
(
nðn1Þf ðxÞf ðyÞFðyÞn2 if x 4 y4 0,
Hð1Þð2Þ ðx,yÞ ¼
0
otherwise:
It should be mentioned that
Z vZ x
Z vZ
yHð1Þð2Þ ðx,yÞ dy dx ¼
v
v
Z
v
¼
To summarize, a customer’s (with valuation vi) symmetric
equilibrium strategy for buy-price auctions is as follows:
If v o pb , then
leave the store, if vi o v,
J bid vi, if v r vi ov,
J buy immediately at the buy price, if vi Z v.
If v Z pb , then
J leave the store, if vi o pb ,
J buy immediately at the buy price, if vi Z pb .
J
Now we consider the seller’s expected profit for buy-price
auctions and limit the discussion to the case where v opb , since
the other case is straightforward. We note that there could be
three outcomes from the auction. First, the seller is unable to sell
the product, because all customer valuations are less than the
participation threshold, v, and an operational cost occurs. Second,
a customer buys immediately at the buy price when at least one
customer’s valuation is higher than the buy threshold v. Third,
there will be an auction winner. Therefore, the seller’s expected
profit can be expressed as
Z vZ x
pb ¼ h ½FðvÞn þð1½FðvÞn Þ pb þ
yHð1Þð2Þ ðx,yÞ dy dx,
v
v
where the first term captures the expected operational cost due to
a no bid; the second captures the expected profit when at least
one customer valuation is higher than v; and the third captures
the expected winning price conditioned on both the first-highest
and second-highest valuations lying between v and v. Additionally, when the second highest valuation is less than v and the
highest valuation is between v and v, the winning price is zero but
the seller does not incur an operational cost.
v
y
Z
yHð1Þð2Þ ðx,yÞ dx dy
v
f ðxÞ dx dy
y
v
¼
ðv ¼ pb Þ.
This result can be understood as follows. When the buy price,
pb, is set above the participation threshold, v, the buy threshold, v
is greater than the buy price. Therefore, some customers whose
valuation is between pb and v participate in the auction by
expecting to win the auction at a lower price (hence, higher
expected utility compared to buying immediately). However, it
may happen that the winning price of the auction is higher than
the buy price, which implies that the winner is paying more than
the buy price. Now consider the other case when the buy price is
set too low ðpb o vÞ. In this case, all customers whose valuation is
higher than the buy price prefer buying immediately. Therefore,
the winning price of the auction cannot exceed the buy price.
ynðn1Þf ðyÞFðyÞn2
v
Z
ynðn1Þf ðyÞFðyÞn2 ðFðvÞFðyÞÞ dy
v
Otherwise ðpb o vÞ, the buy threshold is equal to the buy price
v
ð5Þ
r
Z
v
ynðn1Þf ðyÞFðyÞn2 ð1FðyÞÞ dy ¼
v
Z
v
v
yg ðn,2Þ ðyÞ dy:
Further derivation of analytical results concerning the profit
function pb is intractable.
4. Analysis of the seller’s choices
This section discusses the performance of the three selling
formats. Our analysis seeks both to derive analytical results and to
develop managerial insights through comprehensive numerical
experiments, when the analytical analysis is intractable.
4.1. Pure formats
Since it is difficult to obtain closed-form solutions for the two
selling formats under a general distribution, we consider a special
case to gain managerial insights. Suppose that customer valuations are uniformly distributed on support [0,1]. Then, the seller’s
profit under posted price (Eq. (9) in Appendix A.1) becomes
pp ðrÞ ¼ ½1ðrÞn rh ðrÞn :
Next, consider the case where the seller opens an auction for the
product. The seller’s expected profit from the auction (Eq. (12) in
Appendix A.2) becomes
Z 1
pa ¼
u ½nðn1Þð1uÞun2 duh o:
o1=n
We are ready to compare the expected profit between posted
price and pure auction.
4.1.1. The impact of participation and holding costs
As a benchmark, we consider a case where auction participation and operational cost are ignored (i.e., o ¼ 0 and h¼0). Since
the expected profits are
1
1 1=n
n1
,
pp ¼ 1
and pa ¼
n þ1 1 þ n
n þ1
we observe
Lemma 1. When o ¼ 0 and h¼0, pure auction dominates posted
price if there are more than two potential buyers (i.e., n 4 2).
Otherwise (i.e., n r 2), posted price is strictly better than pure
auction.
D. Sun et al. / Int. J. Production Economics 127 (2010) 147–157
Since it is more likely that there will be more than two customers
for a product, this result reinforces the conventional belief that
auction is preferable (see, e.g., Wang, 1993, 1998; Kultti, 1999).
Next, we investigate the impact of auction participation cost.
Lemma 2. When o 4 0 & h¼0, for any given nð Z3Þ, there exists a
unique threshold, o , such that if o r o , then pure auction dominates
posted price; otherwise ðo 4 o Þ, posted price dominates pure
auction.
Fig. 1 demonstrates the impact of the two costs on the cutoff
point where the seller is indifferent to choosing between posted
price and auction. Finally, we find
Proposition 2. For any given nð Z 3Þ and h 40,
1. there is only one intersection, o , of the two profit functions,
2. if o r o , then pure auction dominates posted price. Otherwise
ðo 4 o Þ, posted price dominates pure auction, and
3. if the optimal price of posted price is greater than the participation
threshold ðr 4vÞ, then the intersection is strictly greater than the
intersection when there is no operational cost ðo 4 o Þ.
The first result is illustrated in Fig. 2(a), and the impact of the
two costs is illustrated in Fig. 2(b). As can be seen, if the
participation threshold is relatively low, taking into account
operational cost makes auctions preferable. However, if there is
relatively higher participation cost, the consideration of
operational cost results in posted price being a more attractive
selling format.
profit to the seller (e.g., the upper limit of customer value). In fact,
this belief is half-true.
Lemma 3. When o 4 0, it holds that (1) limd-1 pp ¼ 1; and (2)
limn-1 pa ¼ 1hooð1lnoÞ o 1.
This result demonstrates that the expected profit of posted
price does converge to 1, but that of auctions does not when
auction participation cost is positive.
Observation 1. For any given o,h 4 0, there exists n1 r n2 such that
when n rn1 , posted price dominates auction; when n1 o n o n2 ,
auction dominates posted-price; when n Zn2 , posted price dominates auction.
We provide a numerical example to illustrate Observation 1
when h ¼ o ¼ 0:02. As can be seen in Table 2, posted price
dominates when n ¼2 or n Z 23.
Table 2
Optimal solutions and profits ðh ¼ o ¼ 0:02Þ.
4.1.2. The impact of the number of customers
A common belief is that when the number of customers is
infinite, both posted price or auction should give the maximum
0.06
n
pp
rn
pa
pb
pb
2
3
4
22
23
24
25
50
0.3783
0.4625
0.5310
0.8286
0.8338
0.8387
0.8433
0.9059
0.5707
0.6250
0.6648
0.8663
0.8701
0.8737
0.8770
0.9240
0.3148
0.4677
0.5577
0.8289
0.8320
0.8348
0.8374
0.8689
0.3394
0.4922
0.5811
0.8524
0.8557
0.8588
0.8617
0.9059
0.4556
0.6204
0.7062
0.9247
0.9265
0.9281
0.9297
0.9240
1
Posted Price dominates.
0.055
Holding Cost Cutoff
Participation Cost Cutoff
151
0.05
0.045
0.04
Auction dominates.
0.035
0.03
0.8
Auction dominates.
0.6
0.4
0.2
Posted Price dominates.
0
3
4
5
6
7
3
8
4
5
6
7
Number of Customers
Number of Customers
Fig. 1. Changes of cutoff point by varying costs: (a) impact of participating cost, h ¼0.15 and (b) impact of operational cost, o ¼ 0:05.
0.59
0.07
Participation Cost
Posted Price
Auction
0.58
Profit
0.57
0.56
0.55
0.54
0.53
Posted Price dominates.
0.06
0.05
0.04
Auction dominates.
0.03
0.02
0
0.1
0.2
Operational Cost
0.3
0.4
0
0.1
0.2
0.3
0.4
0.5
Operational Cost
Fig. 2. Graphical illustration of Proposition 2: (a) profit comparison (n¼ 5, o ¼ 0:05) and (b) impact of h and o.
0.6
0.7
152
D. Sun et al. / Int. J. Production Economics 127 (2010) 147–157
4.2. Performance of buy-price auctions
Since analytical analysis of buy-price auctions is not tractable, we
conduct a comprehensive computational experiment to derive
managerial insights. In the experiment, we vary the three parameters
(i.e., n A f5,10,20,40,80,160g, hA f0:05,0:1,0:15,0:2,0:25,0:3g, and
o A f0:01,0:02,0:03,0:04,0:05,0:06g). The selection of these parameter values was made by surveying some existing literature, e.g.,
Bajari and Hortacsu (2003, 2004), van Ryzin and Vulcano (2004), and
Bhargava et al. (2006). To summarize, we have a total of 216
(¼6 6 6) cases and 648 (¼216 3) optimal solution sets for the
three selling formats. For the sake of clarity, we write down the profit
function for buy-price auction as follows:
8
h o þ ð1v n Þ pb
>
>
>
< n1
ðv n þ 1 oðn þ 1Þ=n Þðn1Þoðvo1=n Þ if pb Z o1=n ;
pb ðpb Þ ¼ þ
nþ1
>
>
>
: ½1ðp Þn p h ðp Þn
otherwise:
b
b
b
1=n
In particular, when pb Z o
(i.e., when the buy price is set higher
than the participation threshold), v is determined by
8
1o
>
>
>
,
if pb 41
<1
n
v¼
1 n
>
>
otherwise:
> arg pb ¼ v ðv oÞ
:
n
Since buy-price auctions are identical to posted price when pb o o1=n
(i.e., when the buy price is set lower than the participation threshold),
buy-price auctions weakly dominate posted price. To understand this
result, recall that when the buy option is set lower than the
participation threshold, Buy-Price Auctions and Posted Price are
identical (i.e., they have the same expected profit). However, when
the seller is able to make more profit by setting the buy option higher
than the participation threshold, Buy-Price Auctions outperform
Posted Price. Therefore, Buy-Price Auctions weakly dominate Posted
Price. For comparison, we assume that the seller prefers posted price
to buy-price auctions when the performances are identical. We
Table 3
Summary of sensitivity analysis by varying parameters.
Change
r
pp
v
pa
pb
v
pb
nm
hm
om
m
k
–
m
k
–
m
–
m
m
k
k
m
–
k
m
–
k
m
k
k
conduct a sensitivity analysis to see the general behavior of the
optimal solutions, thresholds, and profits by varying the key
parameters.
Observation 2. The impacts of the three parameters ðn,h, oÞ are as
follows:
1. The increase of the number of customers, n, results in an increase
in profit, selling price, auction winning price, optimal buy price,
and the two thresholds.
2. As the operational cost, h, increases, the optimal posted selling
price and the profit of the three selling formats decrease.
3. The increase of participation cost, o, causes a reduction of profit
from pure auctions and buy-price auctions. As the participation
cost increases, the participation threshold increases, but the buy
threshold and optimal buy price decrease.
The above observations are summarized in Table 3. First, the
impact of the parameters on posted price is intuitive. Specifically,
an increase in the number of customers will result in a higher
optimal price and profit, but operational cost plays a negative role
in both of them. As expected, the increase of operational and
participation cost reduces the seller’s profit from auctions (pure
and buy-price auctions); however, an increase in the number of
customers will result in higher profit. Second, we observe that the
participation threshold increases as participation cost and the
number of customers increase (refer to Fig. 3(a)). The impact of
the participation cost is not surprising, and that of the number of
customers can be easily understood when we consider the
stochasticity of auctions. Specifically, when there are many
customers, uncertainty is reduced, and hence the participation
threshold will be increased as compared to one with fewer
customers.
The behavior of the optimal buy price is identical to that of the
buy threshold. The impact of the number of customers can be
understood by considering the stochastic nature of auctions, and
the seller sets a higher buy price to maximize the profit as
the participation threshold increases (refer to Fig. 3(b)). Finally,
the buy threshold increases as the number of customers increases,
because there is a higher probability that at least one customer’s
valuation is higher than the threshold when there are many
customers. We observe that the increase in participation cost
results in a decrease of the buy threshold.
The most interesting question concerning the performance
of buy-price auctions would be whether this selling format
Fig. 3. Impact of participation cost ðoÞ on threshold ðvÞ and buy price (pb): (a) participation threshold ðvÞ and (b) optimal buy price (pb).
D. Sun et al. / Int. J. Production Economics 127 (2010) 147–157
153
Fig. 4. Performance of buy-price auction: (a) percentage of profit increase compared to auctions (i.e., ðpb pa Þ=pa ) and (b) percentage of profit increase compared to posted
price (i.e., ðpb pp Þ=pp ).
dominates the pure formats or at least one of them. Surprisingly,
we observe.
Observation 3. Buy-price auctions outperform pure auctions;
however, posted price can outperform buy-price auctions.
This is interesting in the sense that one would think that the
hybrid format might increase the seller’s profit, but our observation suggests that the seller needs to choose the selling format by
carefully considering the three parameters (n, h, and o). Now we
shed light on the favorable conditions for a selling format. We
compare two auction formats. It is easy to see that operational
cost does not play any role in choosing an auction format, since
the change in operational cost will produce an identical impact on
both auction formats. By investigating the results of our
experiment, we find that the buy option is preferable when there
is a large number of customers and participation cost (refer to
Fig. 4(a)), which can be easily understood when we consider the
stochasticity and the impact of the participation cost on pure
auctions. Specifically, when the seller has a large number of
customers, it is more likely that the product will be sold through
buy price, and hence the buy option is preferable. Further,
participation cost plays a negative role on the performance of
pure auctions. Therefore, the hybrid selling format is desirable
when there exists high a participation cost.
Next, we consider the performance of posted price and
auctions with a buy option. Although the impact of the three
parameters for this comparison is not straightforward, we observe
the following:
5. Extensions and discussion
In this paper, we use a stylized model to ensure the tractability
of the analysis. Now we discuss the implications of relaxing some
of our assumptions. First, we present our findings on the
relaxation of the deterministic number of customers. Second, we
discuss the role of reservation price on the expected profit and the
customers’ optimal strategy.
5.1. Random number of customer
To incorporate the stochasticity of potential customers, we use
l to denote the random number of customers, where l Z 1. First,
we consider the posted price format. When the posted price is set
to be r, a customer i with valuation vi greater than r will buy the
product. Hence, the expected profit for the seller is
pp ðrÞ ¼ PrðSellingÞ rPrðNotSellingÞ h
1
X
¼
Prðl ¼ nÞ
Z
1
r
n Z
f ðuÞ du
r
0
n¼1
¼ rðr þhÞ
1
X
r
n f ðuÞ du h
0
n
Prðl ¼ dÞFðrÞ :
ð6Þ
n¼1
We find
Lemma 4. When customer valuations are uniformly distributed
between 0 and 1, there exists a unique solution that maximizes the
seller’s profit specified in Eq. (6).
Observation 4.
1. As operational cost increases, buy-price auctions perform better.
2. When the operational cost is high, a small number of customers
and low participation cost make auctions with buy option
preferable (refer to Fig. 4(b)).
Our observations demonstrate that the seller should consider
using buy-price auctions when there is a small number of
customers, high operational cost, and low participation cost. In
addition, we find that the hybrid selling format is not always
preferable. Therefore, the seller needs to carefully consider the
three important factors in choosing the best selling format for a
product.
Next, consider the pure auction. Consistent with our early
analysis, we assume that each customer uses a threshold policy
with the same participation threshold v. Conditioning on n
customer, customer i’s expected surplus for winning the auction
is given in Eq. (11) in Appendix A.2. Hence, the customer’s
expected surplus with a random number of customers is
(
)
Z vi
1
X
EðUÞ ¼ o þ
Prðl ¼ nÞ vi ½Gðn1,1Þ ðvÞþ
ðvi uÞ½gðn1,1Þ ðuÞ du :
n¼1
v
Now note that the participation threshold v must satisfy
o¼
1
X
n¼1
Prðl ¼ nÞv ½Gðn1,1Þ ðvÞ:
ð7Þ
154
D. Sun et al. / Int. J. Production Economics 127 (2010) 147–157
When Gðn1,1Þ ðvÞ is strictly increasing (e.g., the value density
function f ðxÞ 4 0 for any x), the RHS is strictly increasing, which
implies
Lemma 5. When the probability density function of customer
valuations satisfies f ðxÞ 4 0 for any x, then there exists a unique
participation threshold that solves Eq. (7).
Finally, consider a buy-price auction. If customer i chooses to
bid, her expected surplus with a random number of customers is
(
)
Z vi
1
X
EðUÞ ¼ o þ
Prðl ¼ nÞ vi ½Gðn1,1Þ ðvÞþ
ðvi uÞ½gðn1,1Þ ðuÞ du :
n¼1
o¼
Reservation price
Participation threshold
Valuation
Action
0oRrv
v o pb
vi o v
v r vi o v
vi Z v
vi o p b
vi Z p b
Leave the store
Bid vi
Buy immediately
Leave the store
Buy immediately
vi o R
R r vi o v
vi Z v
vi o p b
vi Z p b
Leave the store
Bid vi
Buy immediately
Leave the store
Buy immediately
v Z pb
v o Ro v
v o pb
v
v Z pb
It is easy to see that the participation threshold v satisfies
1
X
Table 4
Customers’ optimal strategic choice with minimum bid amount.
Prðl ¼ nÞv ½Gðn1,1Þ ðvÞ,
n¼1
which is the same as the auction threshold. Now we see that the
buyout threshold satisfies
(
)
Z v
1
X
vpb ¼ o þ
Prðl ¼ nÞ v ½Gðn1,1Þ ðvÞ þ
ðvuÞ½gðn1,1Þ ðuÞ du :
n¼1
v
amount is more effective, the secret reservation price is widely
used. In addition, the choice between secret and public reservation might depend on many other factors not considered in our
paper, e.g., different item prices, risk aversion, etc. Hence,
studying the two reservation price options with a richer model
calls for future research.
ð8Þ
We find
6. Conclusion and future directions
Lemma 6. When there exists a solution that solves Eq. (8), the
buyout threshold is unique.
We note that it is possible that Eq. (8) has no solution. For
instance, when the participation cost is too high, it is always
optimal to buy the product if v 4pb . Our result indicates that
except for this extreme case, there exists a unique buyout
threshold.
5.2. Reservation price
In most online auction sites, a seller can use a reservation price
option to ensure a minimum auction winning price. For example,
in eBay’s auctions there are two types of this feature: secret
reservation price and minimum bid amount (public reservation
price). Since Katkar and Reiley (2006) found that public reservation price outperforms the secret one, we discuss the implications
of our results when the seller incorporates the minimum bid
amount feature. First, note that when a minimum bid amount is
added to our model setting, there are three possible cases:
0 o R rv, v oR o v, and R Zv, where R represents the minimum
bid amount. Next, it is easy to see that the third case (i.e., R Zv)
cannot happen unless the seller sets the minimum bid amount
above the buy price. In other words, when the seller sets the
minimum bid amount below the buy price, then the buy
threshold is always greater than the reservation price (i.e.,
R opb ov). Therefore, we only need to consider the first two
cases and find
Lemma 7. When there exists a minimum bid amount, the customers’
optimal strategy is as summarized in Table 4:
Given this customers’ optimal strategy, the analysis of the seller’s
profit compared to no minimum bid amount is rather straightforward. First, the seller’s expected profit is unchanged when 0 oR rv.
Next, we see that the seller’s profit decreases when v o R ov. This
result can be easily understood by considering the fact that R serves
as the participation threshold. As we discussed earlier, as the
participation threshold increases, the seller’s profit decreases.
Finally, we emphasize that the analysis in incorporating the
secret reservation price would be a good extension of our study.
Although Katkar and Reiley (2006) found that the minimum bid
The widespread use of the Internet has caused many changes
in the way business transactions are made, and correspondingly
new creative selling formats and technologies are being developed. Some examples are online auctions, reverse online auctions,
eBay’s reputation mechanism, eBay’s Buy-it-Now (temporary
option), Yahoo’s Buy price (permanent option), price comparison
engines, etc. In this paper, we investigate choosing the best selling
format for a product.
We characterize the customers’ strategic behavior and then
derive the seller’s profit function for each selling format by
incorporating customers’ strategic decision making. We compare
pure selling formats and demonstrate that one does not dominate
the other, and that there exists a unique cutoff that determines
the preferred selling format. In general, the existence of auction
participation cost reduces the efficiency of auctions, but the
consideration of operational cost induces the seller to prefer
auctions. When we include buy-price auctions in the choice set,
we find that posted price can outperform buy-price auctions.
Therefore, a careful consideration of the important factors
(number of customers, auction participation cost, and operational
cost) can guide the seller in choosing the best selling format.
In this paper, we formulate the problem in a stylized manner.
Of course, a more realistic and richer model can be used to
analyze the same problem; however, the model we present is
sufficient to capture the main managerial insights without losing
tractability. We believe that our analytical analysis and observations from the computational experiment provide another
explanation why online auctions do not dominate e-commerce
transactions. Further, our study explains why we can observe the
three selling formats on the major auction sites. Our future
studies include a generalization of the results and investigation of
the problem in the environment of a more general model setting.
Acknowledgements
The authors are grateful for valuable comments on earlier
drafts of this work from Hemant K. Bhargava, Alan L. Montgomery, Xin Wang, Susan H. Xu, and conference participants at 37th
Annual Meeting of the Decision Science Institute (DSI2006) and
D. Sun et al. / Int. J. Production Economics 127 (2010) 147–157
18th Annual Conference of POMS (2007). This article was
improved with the exhaustive and instructive comments of two
anonymous reviewers.
Appendix A. Pure posted price and auctions
For completeness, this appendix discusses pure posted price
and auctions. For pure auctions, we demonstrate that there exists
a unique participation threshold when customer valuations are
generally distributed and show that both the seller’s expected
profit and the customer’s expected surplus are decreasing in
auction participation cost.
A.1. Posted price
When the seller chooses the posted price selling format, her
decision is to announce a fixed price, r. After announcing the fixed
price, we assume that customers are risk-neutral in that if
customer i with valuation vi finds that vi Zr, that customer will
buy the product. Since operational cost occurs only when all the
customer valuations are less than the fixed price, the expected
profit for the seller with posted price, r, is
pp ðrÞ ¼ PðSellingÞ rPðNotSellingÞ h
Z
¼ 1
r
n Z
f ðuÞ du
r
0
r
n
f ðuÞ du h
0
¼ rðr þ hÞ½FðrÞn :
ð9Þ
155
highest bidder’s valuation is less than v and the highest bidder’s
valuation is higher than v, the winning price is zero, but the seller
does not incur an operational cost.
Although deriving closed-form solutions of Eq. (12) is
analytically intractable, we see
Lemma 9. Under the auction format, the seller’s expected profit and
the bidder’s expected utility are both decreasing in o.
This result is intuitive in that as the auction participation cost,
o, increases, the customer’s expected utility decreases (refer to
Eq. (1)), and the seller’s expected profit decreases due to the fact
that fewer customers will participate in the auction. This result
implies one important managerial insight. It is in the seller’s
interest to lower the auction participation cost, if possible,
because a lower o leads to a higher expected profit. Therefore,
this result can explain why many bidding assistance features (e.g.,
automatic bidding) are widely offered and used in Internet
auctions and partly why auctions have proliferated after the
development of the Internet.
Appendix B. Technical details and proofs
Proof of Proposition 1. Using integration by parts, Eq. (3)
becomes
Z v
vpb ¼ v ½Gðn1,1Þ ðvÞ þ
ðvuÞ½gðn1,1Þ ðuÞ duo
v
The first-order condition of Eq. (9) yields
¼ v ½Gðn1,1Þ ðvÞ þ ½ðvuÞGðn1,1Þ ðuÞv v þ
@pp ðrÞ
¼ 1½FðrÞn nðr þhÞ½FðrÞn1 f ðrÞ ¼ 0,
@r
ð10Þ
¼ v ½Gðn1,1Þ ðvÞ þ
which determines the optimal posted price. We see
Z
Z
v
½Gðn1,1Þ ðuÞ duo
v
v
½Gðn1,1Þ ðuÞ duo ¼
v
Z
v
½Gðn1,1Þ ðuÞ du,
v
n
Lemma 8. There exists a unique price, r , that maximizes pp ðrÞ,
when the distribution of customer valuations has an increasing
failure rate (i.e., f ðrÞ=ð1FðrÞÞ is increasing in r).
ð13Þ
Most of the commonly used distributions satisfy the condition
of increasing failure rate. Finally, from Eq. (9), it is easy to see that
the seller’s expected profit, pp ðrÞ, is decreasing in operational cost,
h.
where the last equation is derived by using Eq. (2) (i.e.,
v ½Gðn1,1Þ ðvÞ ¼ o). Let
8
Ry
if y Z v,
< y ½Gðn1,1Þ ðuÞ du
v
LðyÞ ¼
:
0
otherwise:
A.2. Pure auction format
For the case with pb Z v, we have two scenarios.
Now consider pure auctions when customers use the symmetric participation strategy. If a customer with valuation vi
chooses to bid, the expected utility is
(1) xi is unbounded. We see that @LðyÞ=@y ¼ 1Gðn1,1Þ ðyÞ 4 0,
which implies that LðÞ is strictly increasing and unbounded.
Because Lð1Þ ¼ 1 and LðvÞ ¼ v rpb , there exists a unique
solution v ð Zpb Þ such that LðvÞ ¼ pb .
(2) xi is bounded, i.e., xi rM. If LðMÞ 4 pb , then L(y)¼pb has a
unique solution v ðM 4 v Zpb Þ; if LðMÞ o pb , then v ¼ M, i.e., no
customer will buy immediately.
E½Utility ¼ E½Surplus by winningo
Z vi
¼ vi ½Gðn1,1Þ ðvÞ þ
ðvi uÞ½gðn1,1Þ ðuÞ duo:
ð11Þ
v
Since Eq. (11) is identical to Eq. (1), it is easy to see that the
participation threshold with buy price is identical to that with
pure auction. Then, the seller’s expected profit can be expressed
as
Z 1
pa ¼
yg ðn,2Þ ðyÞ dyh Gðn,1Þ ðvÞ:
ð12Þ
v
This can be understood as follows. When the second highest
bidder has valuation Xðn,2Þ Z v, there will be an auction winner and
the winning price is Xðn,2Þ , captured by the first term in Eq. (12).
When the highest bidder’s valuation is lower than v, no bidding
occurs and the seller incurs the operational cost, captured by the
second term in the RHS of Eq. (12). Note that when the second
On the other hand, when pb ov, it holds that LðvÞ ¼ v 4 pb ,
implying that all customers whose valuation is higher than the
buy price pb will choose to buy immediately. As such, v ¼ pb . &
Proof of Lemma 1. By taking the difference of two profit
functions, we have
"
1=n #
1
1
nn E½pa E½pp ¼
1 :
n þ1
1þ n
It is easy to see that
ðE½pa E½pp Þjn ¼ 1 o 0,
ðE½pa E½pp Þjn ¼ 2 o0
and
ðE½pa E½pp Þjn ¼ 3 40:
Now we show that if n Z4, E½pa 4E½pp . To this end, we rewrite
156
D. Sun et al. / Int. J. Production Economics 127 (2010) 147–157
the difference of the profit functions as
"
"
1=n #
1=n #
1
1
1
1
E½pa E½pp ¼
4
n1n n1n nþ1
1þn
nþ1
n
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi
1
n
n
ðn1Þn nn1 :
¼
nþ1
We note that both (n 1)n and nn 1 are increasing functions in n
and that the tangent of (n 1)n is greater than that of nn 1.
Therefore, there must be only one intersection or no intersection
of two functions on n. It can be easily shown that there is only one
intersection, which occurs between 3 and 4. This implies that
ðn1Þn 4 nn1 (and hence E½pa 4 E½pp ) if n Z 4. &
Proof of Lemma 2. Note that E½pa is strictly decreasing in o
(Lemma 9) and that E½pp is a constant. By Lemma 1,
E½pa jo ¼ 0 4 E½pp jo ¼ 0 when n Z 3. Combining these implies that
there is only one intersection (i.e., o ) and that E½pa 4 E½pp ðE½pa oE½pp Þ, if o o o (o 4 o , respectively). &
Proof of Proposition 2. The first and second part of Proposition 2
can be proved by following the same procedures in Lemmas 9 and
2. To prove the third part, note that o is the intersection of
ðn1Þ½1ðn þ 1Þ o þ n oðn þ 1Þ=n ,
nþ1
ð1pn Þ p ¼
ð14Þ
and that o is the intersection of
ð1pn Þ ph pn ¼
ðn1Þ½1ðn þ 1Þ o þ n oðn þ 1Þ=n h o:
n þ1
ð15Þ
ðn1Þ½1ðn þ 1Þ o þ n oðn þ 1Þ=n þ h ðpn oÞ:
nþ1
ð16Þ
The LHS of Eq. (16) is identical to that of Eq. (14) and both are
constant in o. Since the RHS of Eq. (16) is larger than that of
Eq. (14) when pn 4 o, the intersection of Eq. (16) must be greater
than that of Eq. (14) when the precondition of Proposition 2 is
satisfied. &
Proof of Lemma 4. The first derivative of the seller’s expected
profit (i.e., Eq. (6)) with respect to r is
1
1
X
X
dpp ðrÞ
¼ 1ðr þ hÞ
Prðl ¼ nÞnFðrÞn1 f ðrÞ
Prðl ¼ nÞFðrÞn ,
dr
n¼1
n¼1
and the second derivative is
d2 pp ðrÞ
dr
2
¼ ðr þ hÞ
1
X
df ðrÞ
Prðl ¼ nÞn ðn1ÞFðrÞn2 f ðrÞ þ FðrÞn1
dr
n¼1
1
X
Prðl ¼ nÞnFðrÞn1 f ðrÞ
n¼1
1
X
Prðl ¼ nÞnFðrÞn1 f ðrÞ:
n¼1
Now note that f(r)¼1 and df ðrÞ=dr ¼ 0 when customer valuations
are uniformly distributed between 0 and 1. Then, pp ðrÞ is
guaranteed to be concave. This completes the proof. &
Proof of Lemma 6. Suppose that v is given. The first derivative of
the LHS of Eq. (8) with respect to v is 1 and that of the RHS is
(
)
Z v
1
X
@RHS
¼
Prðl ¼ nÞ ½Gðn1,1Þ ðvÞ þ
gðn1,1Þ ðuÞ du
@v
v
n¼1
¼
1
X
1¼
nðr þ hÞðFðrÞÞn1 f ðrÞ
nðr þ hÞðFðrÞÞn1
f ðrÞ
¼ jðrÞ:
¼
n
1ðFðrÞÞ
1 þFðrÞ þ þ ðFðrÞÞn1 1FðrÞ
If f1 ðrÞ and f2 ðrÞ are two non-negative increasing functions of r,
then the product of f1 ðrÞ and f2 ðrÞ is also non-negative and
increasing in r. By the definition of increasing failure rate,
f ðrÞ=ð1FðrÞÞ is non-negative and increasing in r. Also, r + h and
ðFðrÞÞn1 =ð1þ FðrÞ þ þ ðFðrÞÞn1 Þ are non-negative and increasing
in r. Therefore, jðrÞ is non-negative and increasing in r.
Furthermore, it can be verified that jð0Þ ¼ 0 and jð1Þ ¼ 1.
Therefore, there exists a unique solution, rn, such that jðr Þ ¼ 1.
In other words, the first-order condition yields a unique
solution. &
Proof of Lemma 9. According to the equation Gðn1,1Þ ðvÞ ¼ o=v,
we apply the implicit function theorem and obtain
@v
v
¼
@o v 2 gðn1,1Þ ðvÞ þ 14 0,
which implies that when o increases, v increases. When v
R1
v y gðn,2Þ ðyÞ dy reduces (e.g., the integral
increases, the integral
interval ðv,1Þ shrinks) and so does the second term in Eq. (12).
Therefore, pa is decreasing in o.
On the other hand, the bidder’s expected utility is given by
Z vi
ðvi uÞ½gðn1,1Þ ðuÞ duo:
Ua ¼ vi ½Gðn1,1Þ ðvÞ þ
v
Taking the first derivative with respect to o, we obtain
We rewrite the latter equation as
ð1pn Þ p ¼
Proof of Lemma 8. Rearranging Eq. (10), we have
Prðl ¼ nÞ½Gðn1,1Þ ðvÞ,
n¼1
which can be obtained from the Leibniz Rule. Since Gðn1,1Þ ðvÞ o1
for any v o M, where M is the upper bound of the customer
valuation, we see that @RHS=@v o1. Hence, if Eq. (8) has a solution
that must be unique. &
v
v
@Ua
¼ vi gðn1,1Þ ðvÞ@
ðvi vÞgðn1,1Þ ðvÞ@
1
@o
@o
@o
v
1
¼ vgðn1,1Þ ðvÞ@
1 ¼ 2
@o
v gðn1,1Þ ðvÞ þ 1 o0:
This completes the proof.
&
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