An Experimental Study of Auctions with A Buy Price Under Private and Common Values Summary 1. The summaries must at least be one full page (no more than two pages)2. Summarize the major points of the reading.3. Provide some of your opinion about the paper and any questions you might have. Games and Economic Behavior 72 (2011) 558–573
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Games and Economic Behavior
www.elsevier.com/locate/geb
An experimental study of auctions with a buy price under private and
common values ✩
Quazi Shahriar a , John Wooders b,∗
a
b
Department of Economics, San Diego State University, San Diego, CA 92182, United States
Department of Economics, Eller College of Management, University of Arizona, Tucson, AZ 85721, United States
a r t i c l e
i n f o
a b s t r a c t
Article history:
Received 14 August 2008
Available online 12 November 2010
JEL classification:
C72
C91
D03
D44
Keywords:
Auction
Buy price
eBay
Laboratory experiments
Risk aversion
Private values
Common values
eBay’s Buy It Now format allows a seller to list an auction with a “buy price” at which
a bidder may purchase the item immediately and end the auction. When bidders are risk
averse, then theoretically a buy price can raise seller revenue when values are private (but
not when values are common). We report the results of laboratory experiments designed
to determine whether in practice a buy price is advantageous to the seller. We find that a
suitably chosen buy price yields a substantial increase in seller revenue when values are
private, and a small (but statistically insignificant) increase in revenue when values are
common. In both cases a buy price reduces the variance of seller revenue. A behavioral
model which incorporates the winner’s curse and the overweighting by bidders of their
own signal explains the common value auction data better than the rational model.
© 2010 Elsevier Inc. All rights reserved.
1. Introduction
In a “buy it now” auction the seller sets a fixed price, termed a “buy price,” at which a bidder may purchase the item,
thereby ending the auction immediately. If no bidder accepts the buy price, then in the ascending bid auction that follows,
the bidder with the highest bid wins and pays the second highest bid. The buy-now auction format has proven to be
extremely popular. Yahoo introduced the buy-now auction format in 1999. eBay followed with its own buy-now auction
format in 2000 and by the end of 2001 about 40% of all eBay auctions were buy-now auctions (see Hof, 2001).
Several theoretical explanations for the popularity of buy prices have been proposed for private value auctions. Reynolds
and Wooders (2003, 2009) show for both eBay and Yahoo auctions that when bidders are risk averse then a suitably
chosen buy price raises seller revenue; in this case the buy price extracts a risk premium from bidders who wish to avoid
uncertainty over whether they win and the price that they pay. For eBay auctions, Mathews (2003) establishes that a seller
✩
We are grateful to two referees and an associate editor for thoughtful comments and suggestions. Part of this work was completed while Wooders was
a visitor at the National University of Singapore, and he is grateful for their hospitality. We are grateful to seminar participants at CIDE, George Mason
University, HKUST, ITAM, NUS, Ohio State University SMU, University of Maryland, University Carlos III of Madrid, and Vanderbilt University for comments
and suggestions. We thank Dan Ackerberg for helpful comments. We are especially grateful to Stan Reynolds whose input has significantly shaped our
thinking about this project.
Corresponding author.
E-mail addresses: qshahria@mail.sdsu.edu (Q. Shahriar), jwooders@eller.arizona.edu (J. Wooders).
*
0899-8256/$ – see front matter
doi:10.1016/j.geb.2010.10.009
© 2010
Elsevier Inc. All rights reserved.
Q. Shahriar, J. Wooders / Games and Economic Behavior 72 (2011) 558–573
559
can increase his revenue by setting a buy price when bidders are impatient; in this case the buy price extracts a premium
from bidders who end the auction early. Mathews and Katzman (2006) establish that a buy price may be advantageous for
a risk-averse seller as it reduces the variance of seller revenue.
In the present paper we investigate experimentally the properties of eBay buy-now auctions in both pure private value
and pure common value settings. For private value auctions our objective is to determine whether a buy price raises seller
revenue and whether it reduces the variance of revenue, as theory suggests it can. We find that in private value auctions the
use of a buy price has a positive and statistically significant effect on seller revenue. We also find that a buy price lowers
the standard deviation of revenue. In fact, the empirical c.d.f. of revenue in buy price auctions second order stochastically
dominates revenue in auctions without a buy price, suggesting that a risk averse seller is better off with a buy price. These
experimental results provide support to the main theoretical explanations offered for the use of buy prices in private value
auctions.1
In private value auctions a bidder who accepts a buy price eliminates uncertainty about his payment (and hence his
payoff). In common value auctions, in contrast, a bidder who accepts a buy price eliminates uncertainty about his payment
but not about his payoff, since he is also uncertain about the item’s value. Common value buy price auctions have, as a
result, quite different theoretical properties than private value auctions, as shown in Shahriar (2008). In the common value
setting we study, (i) any buy price that in equilibrium is accepted with positive probability reduces seller revenue when
bidders are either risk neutral or risk averse, and (ii) seller revenue and the probability the buy price is accepted are both
decreasing in the degree of bidder risk aversion. These different theoretical properties suggest that buy prices are worth
studying experimentally in common value settings. Common value auctions are of practical interest as well since in some
eBay auctions a common value model may be more appropriate than a private value model. Bajari and Hortaçsu (2003), for
example, argue that a common value model is correct for eBay coin auctions.2
Prior experimental studies of common value auctions, beginning with the seminal work of Kagel and Levin (1986), have
found that bidders are naive, conditioning only on their own signal (and failing to condition on their rivals’ signals) when
forming their bid.3 Buy-now auctions are of special interest since they provide a new mechanism to distinguish naive
from rational bidding when values are common. To see this, suppose that bidders follow “cutoff” strategies, with a bidder
accepting the buy price if his signal exceeds some cutoff “c,” and rejecting it otherwise. When all other bidders reject the
buy price, this is informative to a naive bidder since he then infers that all his rivals have signals less than the cutoff c.
A naive bidder, consequently, drops out earlier when the buy price is rejected than he would in an ascending bid auction
where no buy price is offered. In contrast, a rational bidder forms his bid in an ascending bid auction by conditioning on
both his own signal and on the highest signal of a rival bidder being equal to his own signal (see Milgrom and Weber,
1982). Hence a rational bidder who rejects the buy price (since his signal is less than c) obtains no information relevant to
his bidding when all other bidders reject the buy price. He, therefore, chooses the same dropout price in an ascending bid
auction, whether or not the auction has a buy price.
We find, in fact, that bidders drop out earlier when the buy price is rejected than they do in identical auctions where no
buy price was offered. This finding provides additional support for the naive bidding model. We find that in the ascending
bid phase of the auction, bidders tend to overbid relative to equilibrium when they have low signals and underbid when
they have high signals. This last finding is consistent with prior studies of ascending bid common value auctions without
a buy price. In contrast to the predictions of the rational model, (i) the use of a buy price in common value auctions had
a small positive (although statistically insignificant) effect on seller revenue, and (ii) the buy price is accepted with high
frequency, although theoretically it is always rejected if bidders are risk neutral or risk averse. We also find the variance of
seller revenue is significantly lower in the buy-now auctions.
In order to better explain the common value data, we build on the naive bidding model of Kagel and Levin (1986) to
develop and estimate a behavioral model of common value buy-now auctions. In our model a bidder fails to condition his
value for the item on winning, both when deciding whether to accept the buy price and when deciding whether to drop
out in the ascending bid phase of the auction (i.e., a bidder suffers from the “winner’s curse” when making either type of
decision). However, a bidder does update regarding his rivals’ signals if none takes the buy price. The model also allows
bidders to overweight their private information, and we find that overweighting of own signal is important in explaining
the high frequency with which the buy price is accepted. The behavioral model explains (i) the high rate at which the buy
price is accepted, (ii) the higher than expected revenue realized in the buy-now auction, and (iii) that bidding behavior is
less aggressive following the rejection of the buy price.
1.1. Results from field data
There are several important empirical studies of buy-now auctions based on field data from eBay. We focus here on
their findings regarding which sellers tend to use a buy price and the revenue effects of a buy price. In an extensive study
1
Since there is no meaningful delay in laboratory experiments, our results do not provide insight into whether a buy price can be used to exploit bidder
impatience.
2
They find the sale price is decreasing in the number of bidders, which is consistent with common values.
3
Kagel and Levin’s (1986) paper and their other foundational contributions on common value auctions are conveniently assembled in Kagel and Levin
(2002).
560
Q. Shahriar, J. Wooders / Games and Economic Behavior 72 (2011) 558–573
of Palm Vx auctions, Anderson et al. (2008) find that sellers are more likely to use the buy-now auction format as they
are more experienced. However, since their sample consists entirely of auctions which end with a sale, the effect on seller
revenue of employing a buy price is not clear. Durham et al. (2004) study a sample of 138 auctions of American silver dollars
ending with a sale. They find that the 41 auctions listed with a buy price had an average selling price of $10.27, while the
remaining auctions had an average selling price of $9.56, a statistically significant difference. They also find that a buy price
tended to be offered by the more experienced sellers. Durham, Roelofs, and Standifird report, in addition, the results of a
field experiment in which they conducted 84 eBay auctions of 2001 American Eagle Silver dollars, with varying buy prices
but the same low reserve price of $1.00. All of these auctions ended with a sale. The average sale price in auctions without a
buy price was $8.82. Setting a buy price equal to $8.80 raised average revenue to $9.83, a difference which was statistically
significant.4
Laboratory experiments complement the analysis of field data obtained from either naturally occurring or experimental
auctions for several reasons. First, the theory doesn’t make unambiguous predictions regarding the revenue effect of buy
prices when sellers are risk averse since, as shown in Mathews and Katzman (2006), a risk averse seller has an incentive
to set a buy price even if it reduces expected revenue. Second, as noted earlier, the theoretical properties of a buy price
differ depending on whether values are private or common. In the field it may be difficult to determine whether either
pure private values or pure common values is appropriate, whereas in the lab the experimenter controls the structure of
values. Third, in the field bidders may be both risk averse and impatient. The short duration of experiments in the lab
allows the experimenter to focus on the effects of bidder risk aversion in isolation. Fourth, when analyzing field data one
must control for the seller’s reputation, which is known to have an effect on price (see, e.g., Houser and Wooders, 2006),
whereas reputation is not a factor in lab experiments. Finally, in a laboratory experiment one can focus on the effect of a
buy price by varying whether or not the auction has a buy price, while holding all the other aspects of the auction – e.g.,
the number of bidders, the reserve price – fixed.
The present paper complements a number of other experimental studies of Internet auction formats. Ariely et al. (2005),
Ockenfels and Roth (2006), and Houser and Wooders (2005) study the effects of the auction closing rule, “hard” or “soft,”
on bidding behavior. Ely and Hossain (2009) and Grey and Reiley (2007) use field experiments to investigate the reasons
for, and consequences of, late bidding (or “sniping”) in hard close auctions. Salmon and Wilson (2008) is an experimental
study of eBay’s “second chance offer” auction format.
The present paper is organized as follows. In Section 2 we present the experimental design. The theoretical background
on buy-now auctions is presented in Section 3. Section 4 presents our experimental results for both the private value and
common value auctions. In Section 5 we estimate a behavioral model of common value buy-now auctions. Section 6 provides
concluding remarks.
2. Experimental design
We conducted experiments for the four treatments shown in Table 1. In the private value (henceforth PV) treatments,
each bidder’s value for the item was an independent draw from the U [$0, $10] distribution. In the common value (henceforth CV) treatments each bidder’s signal was an independent draw from the U [$0, $10] distribution; the value of the item
was the same for each bidder and equal to the average of the signals.5 For expositional convenience we used the same
distribution for value/signal draws in both the private and common value treatments. This implies, however, no theoretical
connection between the two treatments.
We conducted both ascending bid auctions and buy-now auctions. In the ascending bid auctions the price increased
by $0.05 every 0.2 seconds (i.e., $0.25 per second) so long as at least one bidder remained active. At any point a bidder
could exit the auction by clicking on a “Drop Out” button. Bidders did not observe the number of bidders remaining in the
auction, i.e., they did not observe when a rival bidder dropped out. The auction ended when only one bidder remained; the
remaining bidder won the auction and paid a price equal to the amount at which the last bidder dropped out.6 At the end
of each round, each bidder observed the sale price. In the common value auctions bidders observed, in addition, the signals
of all four bidders. There was no reserve price, and the clock began ascending from a bid of $0.
The buy-now auctions had two stages. At the first stage the four bidders simultaneously decided whether to accept or
reject the buy price. The buy price was $8.10 in the private value auctions and was $5.60 in the common value auctions. (We
motivate these choices in the next section.) If a bidder accepted the buy price, then he won the item at the buy price and
the auction ended.7 If all the bidders rejected the buy price, then at the second stage the item was sold via the ascending
bid auction described above.
4
Lee and Malmendier (forthcoming) report the provocative finding that when an item is simultaneously available on eBay at a posted price and at an
auction, the closing auction price frequently exceeds the posted price. They attribute this finding to limited attention on the part of bidders.
5
Experimental papers in which the common value is the average of the bidders’ signals include Avery and Kagel (1997), Holt and Sherman (2000),
Goeree and Offerman (2002).
6
This auction format is sometimes referred to as a Japanese, or button auction.
7
If more than one bidder accepted the buy price, then the item was randomly allocated to one of the accepting bidders.
Q. Shahriar, J. Wooders / Games and Economic Behavior 72 (2011) 558–573
561
Table 1
Experimental design.
Buy
price
Number
sessions
Periods/
session
Bidders/
session
Dist. of
values/signals
PV
Ascending
Buy-now
na
$8.10
6
6
30
30
4
4
U [$0, $10]
U [$0, $10]
CV
Ascending
Buy-now
na
$5.60
6
6
30
30
4
4
U [$0, $10]
U [$0, $10]
As eBay implements buy-now auctions, the buy price disappears as soon as the first bid is placed and hence it is only
available to the first bidder in the auction.8 Our experimental design, in contrast, follows the developed theory (Mathews and Katzman, 2006; Reynolds and Wooders, 2009; Shahriar, 2008) which models bidders as simultaneously deciding
whether to accept or reject the buy price. This modeling choice, however, is not significant: If only the first bidder to arrive
can accept the buy price, the main theoretical results are qualitatively the same, e.g., when bidders are risk averse a buy
price may raise seller revenue when values are private but it cannot raise revenue when values are common.
The experiments were conducted at the University of Arizona where subjects were recruited in groups of eight. Each
group of 8 subjects was split into two groups of four bidders, and each group of four bidders participated in 30 periods of
an auction.9 We refer to a single group of four bidders participating in 30 rounds of a given auction format as a “session.”
We conducted six sessions for each of the four treatments, and hence a total of 96 subjects participated in the experiments.
The bidders’ values/signals were determined randomly once, i.e., the same set of 120 values/signals (4 bidders per auction
and 30 auctions) was used in all 24 sessions.10 Table 1 summarizes our experimental design.
In common value auctions it has been observed in prior experiments (e.g., Kagel and Levin, 1986; and Kagel et al., 1989)
that subjects sometimes go “bankrupt,” with their accumulated earnings becoming negative. Bankruptcy affects a bidder’s
incentives as he is no longer liable for his losses. A low but positive balance also affects a bidder’s incentives as he can lose
at most his current balance if he wins an auction and the price exceeds the item’s value. In our common value experiments,
a bidder began with an initial balance of $25 (experimental dollars) and was declared bankrupt if his current balance fell
below $5.11 In the private value sessions, each subject began with an initial balance of $5 and was declared bankrupt if
his balance fell below $0. A bankrupt bidder would exit the experiment and be replaced by an additional subject who was
standing by. In fact, no subject went bankrupt over the course of the experiment.12
3. Theoretical background
The theoretical foundation of our experimental design is Reynolds and Wooders (2003) and Shahriar (2008) for, respectively, private and common value buy-now auctions. In each case, there are n bidders who are assumed to have constant
absolute risk aversion (CARA) with utility function u ( w ) = (1 − e −α w )/α , where α 0 is the index of risk aversion. Since
limα →0 u ( w ) = w, then α = 0 corresponds to risk neutrality. Let B denote the buy price in a buy-now auction. Denote
by F ( v ) the cumulative distribution function of values/signals with support [ v , v̄ ]. Let G ( v ) = F ( v )n−1 be the c.d.f. of the
highest of n − 1 values/signals. The densities of F and G are denoted by f and g, respectively. For our experimental design
v
for v ∈ [0, 10]. To simplify, we assume the seller sets no reserve price.
we have n = 4 and F ( v ) …
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