1. No category

How to Make an Offer? A Theoretical Model and Empirical
Evidence from eBay Motors∗
Ching-I Huang†
Jong-Rong Chen‡
Chiu-Yu Lee§
Abstract
The popular Internet marketplace, eBay, introduced a Name-Your-Own-Price mechanism based on the Best Offer format in 2005. This study focuses on the buyer’s behavior
under such a mechanism. We characterize this mechanism as a sequential-move game
between the seller of an item and its buyers. Our model suggests that a rational buyer’s
offer price increases in relations to the number of buyers who have previously made an
offer on the item and the Buy-It-Now price chosen by the seller. On the other hand, the
offer price decreases for items which have been listed on eBay for a longer period of time.
We empirically test our theoretical predictions using data on the sale of Toyota Camry
cars on eBay. The empirical evidence is found to be consistent with our model.
Keywords: Name-Your-Own-Price mechanism, eBay, Best Offer, sequential equilibrium, ecommerce
JEL Classification: C72, D49, L81
∗
We thank seminar participants at National Tsing Hua University, Academia Sinica, National Taiwan
University, National Taipei University, 2010 EARIE conference, and the Workshop on Econometric Theory
and Application for beneficial suggestions. Financial support from the Center of Institution and Behavior
Studies at Academia Sinica are gratefully acknowledged. All remaining errors are ours.
†
Department of Economics, National Taiwan University, Taipei, Taiwan.
‡
Graduate Institute of Industrial Economics, National Central University, Jhongli, Taiwan.
§
Graduate Institute of Industrial Economics, National Central University, Jhongli, Taiwan.
1
1
Introduction
With the vigorous development of the Internet, several innovative retail formats have evolved
over the past decade. Among them, the so-called Name-Your-Own-Price (NYOP) is a popular
interactive pricing mechanism that lets both buyer and seller suggest the price for a product.
This mechanism is known as the “Best Offer” in eBay. In this study, we analyze the buyer’s
haggling behavior for car transactions under the eBay Best Offer format.
NYOP is not really a new transaction mechanism. Historically, haggling over prices has
been a common practice when doing business in most countries. Using fixed prices is a
relatively new retail practice. Fixed price transactions involve a lower transaction cost than
haggling over prices when the size of the retailer becomes larger. (Terwiesch, Savin, and
Hann, 2005) Although haggling over prices is still used in many transactions nowadays, it
is normally difficult to empirically analyze such behavior due to a lack of observable data.
In this study, we collect data on the haggling behavior from public information on the eBay
website. We utilize the game-theoretical model to predict a buyer’s decision and then use the
data to empirically verify our predictions.
Using the NYOP mechanism for Internet transactions was pioneered by Priceline.com,
a website which specializes in selling travel tickets (such as flights and hotel rooms). To
purchase a ticket (say, at a 3-star hotel room in the Hyde Park area of Chicago on April 1) in
Priceline, a buyer offers a price and enters his or her credit card information on the website.
Once the offer is accepted, the credit card is charged immediately and the purchase is not
refundable. In 2005, eBay.com introduced a new selling format called “Best offer”, which is
very similar to the NYOP mechanism used by Priceline. The Best Offer mechanism on eBay
works as follows. A seller can choose to use the Best Offer format when listing a product on
eBay. Choosing this format incurs no additional listing fee from eBay. When initiating the
listing, the seller needs to announce a Buy-It-Now (BIN) price for the product. If a buyer
is willing to pay the BIN price, he/she can buy the product immediately at the BIN price.
Otherwise, a buyer can make up to ten offers for each product listed under the Best Offer
2
format.1 Once a buyer makes an offer, the seller can either accept the offer, reject the offer,
make a counteroffer, or simply ignore the offer. An offer or a counteroffer is valid only if it is
responded to within 48 hours. When the seller accepts a buyer’s offer or a buyer accepts a
seller’s counteroffer, the transaction is completed. The transaction price is the accepted offer
price or the accepted counteroffer price.
Although the two mechanisms are similar, there are several important differences between
the Priceline NYOP mechanism and the eBay Best Offer mechanism. First, since eBay is
basically a consumer-to-consumer platform, there is usually only one unit of a given item
for sale in each listing. This is particularly true in our study of car sales. In contrast,
Priceline is a business-to-consumer platform specializing in selling travel tickets and there
are usually multiple units of a particular travel ticket available for sale. Consequently, an
item is usually sold to one buyer exclusively under the eBay Best Offer mechanism, but not
under the Priceline NYOP mechanism. It is important to account for the supply constraints
when analyzing the eBay Best Offer mechanism. Secondly, Priceline does not disclose the
transaction history of one buyer to another buyer, but eBay does display certain information
of previous offers on the website. Specifically, the disclosed information includes how many
buyers have made an offer on the item up to now, when these previous offers were made,
and the status of these previous offers. This information can have an important impact on
a buyer’s behavior. In fact, our empirical analysis shows that buyers do use the available
information to make their offer decisions.
The main focus in the literature on the NYOP mechanism has been analyzing the rules
used by Priceline or other similar retailers. For example, there have been numerous theoretical
works attempting to analyze the specific design of the mechanism, such as whether multiple
offers are allowed and the time interval between two offers (Fay, 2009; Chen, 2009). Other
attempts involved comparing the tradeoff between the NYOP mechanism and the fixed-price
mechanism (Wang, Gal-Or, and Chatterjee, 2009; Fay and Laran, 2009). Some researchers
have used laboratory experiments to test the theoretical predictions regarding the NYOP
1
A maximum of ten offers is allowed for vehicles and three for other products.
3
mechanism (Shapiro and Zillante, 2009; Amaldoss and Jain, 2008; Fay and Laran, 2009).
Nonetheless, due to a lack of available data, there have been very few empirical studies using
real world data. Hann and Terwiesch (2003) use data from a German retailer to estimate the
cost of making an offer. Using the same data, Terwiesch, Savin, and Hann (2005) evaluate the
retailer’s potential profit gain from setting an optimal rejection threshold. In addition, Spann
and Tellis (2006) propose a model which implies that a buyer’s offer prices monotonically
increase over time. However, their empirical study finds that buyers are irrational because the
offer prices in the data reject the monotonicity hypothesis. Our current study is a departure
from previous ones in that we use data on eBay’s Best Offer format. As explained above,
competition between buyers is important in the eBay platform. While previous studies on
the Priceline-style NYOP mechanism typically ignore this competition, we formally include
this factor in both the theoretical model and the empirical estimation. To the best of our
knowledge there has still been no other academic research focusing on eBay’s Best Offer
mechanism.2
We focus on a buyer’s first offer for the listing of a Toyota Camry car on eBay Motors.
There are two primary reasons for focusing only on a buyer’s first offer during the listing
period. First, our data reveal that most of the accepted offers (89%) are actually the buyer’s
first offer. Secondly, a buyer’s second or later offers depend on the seller’s responses to earlier
offers. However, our data constraint does not allow us to know details of these responses.
Nonetheless, by restricting our attention to first offers we can safely exclude the effects of
these later responses.
The rest of the paper is organized as follows. In the next section, we briefly summarize the
features of the Best Offer mechanism used on eBay. In Section 3, we describe our unique data
collected from eBay Motors on sales of Toyota Camry cars and show the stylized facts of the
Best Offer mechanism. These stylized facts motivate our theoretical model in Section 4. The
model characterizes the interactions between a seller and a series of buyers in a sequential2
We descriptively summarize a buyer’s first offers under the eBay Best Offer mechanism in an accompanying
paper (Chen, Huang, and Lee, 2009).
4
Table 1: Transaction Results for different selling formats of Toyota Camry on eBay Motors
Format
No. of listings
Percentage of successful listings
Avg. BIN price among all listings
Avg. BIN price of successful listings
Avg. transaction prices
Auction
w/o BIN
Auction
w/ BIN
Fixed Price
w/o Best Offer
Fixed Price
w/ Best Offer
2,065
32.93%
1,462
4.86%
13,478
9,747
9,558
170
4.71%
16,475
6,003
6,003
650
18.31%
13,876
10,825
10,182
5,100
Notes: The data include all sales of Toyota Camry cars listed on eBay Motors between June 18, 2008 and
March 6, 2009. A successful listing means a listing which results in a transaction.
move game. Each buyer chooses his/her offer price based on all the disclosed information. In
particular, the offer is affected by other buyers’ behavior as revealed in the “offer history”.
Our empirical study in Section 5 tests our theoretical hypotheses. The estimation results
are consistent with our theoretical predictions, implying that buyers are rational in using
available information to make an offer. Concluding remarks are in the final section.
2
The Best Offer Mechanism on eBay
There are two primary categories of selling formats on eBay: an auction-style format and a
fixed-price format. For an auction-style listing, a seller can add a BIN price. For a fixed-price
format listing, a seller can optionally add the Best Offer feature. We collect all transactions
of Toyota Camry cars listed on eBay during a nine-month period from June 2008 to March
2009. Table 1 summarizes the transaction results for each type of the four selling formats on
eBay. The first format, auction without BIN, is a modified ascending auction with a secret
reserve price. It allows a seller to list a product for a specific duration. During the listing
period, a buyer can submit one or more bids for the product. At the end of the listing period,
the product is sold to the buyer with the highest bid at a price equal to the second highest
bid as long as the bid is higher than the secret reserve price.3 When a seller sets a BIN price
3
To be more accurate, it is a proxy-bidding mechanism. A buyer enters the maximum amount of his/her
willingness to bid on the eBay website, and the eBay computer will bid for him/her by adding a small increment
5
in an auction-style listing, a buyer can purchase the product at the BIN price immediately
without submitting any bid. As for a listing under the fixed-price format without using Best
Offer, a seller posts a BIN price for the product. During the listing period, any buyer who
agrees to pay the BIN price can get the product immediately at that price. The last format,
fixed-price with Best Offer, was introduced in 2005. When a seller chooses this format, a
buyer can still purchase the product immediately at the BIN price. Alternatively, the buyer
can negotiate with the seller for a price lower than the listed BIN price during the listing
period. A transaction occurs when the buyer and the seller mutually agree on the offer price.
The focus here is on the last format and we use the term “Best Offer” to refer to this format.
Several features in Table 1 motivate the importance of studying the Best Offer mechanism.
First, the Best Offer mechanism makes up 15.0% of all listings. This is not a trivial proportion.
Second, 18.31% of listings under the Best Offer format result in a successful transaction. This
percentage is the second highest among the four formats. Third, the Best Offer format has
the highest average transaction price among the successful transactions.4
At first glance, one may think our research is related to Internet auctions, especially
because eBay is well known for its auction mechanism and there is a rich literature on
Internet auctions.5 However, the Best Offer mechanism is quite different from the auction
mechanism. In particular, while a seller is obliged to accept the highest bid under the auction
mechanism, he/she can reject any offer under the Best Offer mechanism. Moreover, a buyer
can observe previous bids in the eBay auction mechanism, but no information about other
buyer’s offer price is revealed during the listing period with the Best Offer mechanism.6 As a
result, under the Best Offer mechanism, a buyer cannot learn any information about others’
offer prices on which to decide his/her own offer price.
to the current standing price until the standing price reaches the maximum amount entered by the buyer.
4
Note that Table 1 simply shows the summary statistics in the data, but does not account for the heterogeneity of format selection. A formal analysis to compare these formats needs to consider the heterogeneity,
but it is beyond the scope of the current paper.
5
See Ockenfels, Reiley, and Sadrieh (2007) and Hasker and Sickles (2010) for surveys of recent works.
6
Offer prices are disclosed at the end of a listing. We use the disclosed prices in our empirical analysis, but
buyers cannot observe the prices when they are making their offering decisions.
6
Table 2: Transaction Results of Toyota Camry sales listed under the Best Offer Format
Result
Number of Listings
Avg. BIN Price
Avg. Transaction Prices
3
Ended by an
Accepted Offer
Ended by
BIN Option
Ended without
Transaction
Total
68
10,795
9,670
51
10,864
10,864
531
14,560
na
650
13,876
10,182
Data on Best Offer Sales
We collected data on the sale of all Toyota Camry cars listed on eBay Motors during a ninemonth period from June 18, 2008 to March 6, 2009. The original dataset included 4,347
listings. As shown in Table 2, the Best Offer format is used in 650 of the listings. Table
2 categorizes these 650 listings according to the transaction results. Roughly 10.5% of the
listings ended with an accepted offer while 7.8% ended with a BIN purchase. Even though
we do not control for heterogeneities among listings, successful listings tend to have a lower
BIN price than unsuccessful ones. In addition, the average transaction price is slightly lower
for listing ended by an accepted offer than by a BIN purchase.
The “offer history” on the eBay website discloses information regarding each buyer’s
offers. In general, the disclosed information includes the buyer’s username, the offer price,
the offering time, and the results of the offer. Although most of the information is publicly
observed by everyone during the listing period, the offer price is observed only by the seller
during the listing period. Since offer prices become publicly observable only after the end of
the listing period, we can use them in our empirical analysis. Nonetheless, a seller can choose
to list an item on eBay privately. For a private listing, the buyer’s username and/or the offer
price is hidden in the offer history even after the end of the listing period. When the buyer’s
username is hidden, we cannot tell whether two offers are made by the same buyer using the
disclosed information. As a result, we cannot study buyer behavior in these private listings.
Similarly, when the offer price is hidden in a private listing, we cannot analyze the choice of
offer prices.
7
Table 3: Frequency of the Number of Offers Made by a Buyer in a Listing
Buyer does not use BIN.
Buyer chooses BIN.
Number of Offers
Made by a Buyer
Total
Frequency
Frequency
of Accepted
Frequency
0
1
2
3
4
5
6
7
8
9
759
135
53
17
9
4
0
0
1
34
3
0
0
2
0
0
0
0
10
0
0
0
0
0
0
0
0
0
Total
978
39
10
Among the 650 Best Offer listings, buyer’s IDs are observed in 326 listings. We can know
the number of offers made by each buyer in these listings. Table 3 shows the frequency of
offers made by each buyer for a particular listing. The second column shows the distribution
of buyers who do not use the BIN option in the listing. The third columns indicate whether
the last offer of an buyer is accepted by the seller. The last column shows the number of
offers by a buyer who ends up using the BIN option to purchase. There are 1,334 offers from
978 buyers in these listings.7 In addition, ten of the listings ended by BIN without any offer
coming from these buyers. Although eBay allows a buyer to make up to ten offers in a listing,
most buyers do not make many offers. Roughly 78% (759/978) of the buyers make only one
offer in a listing. Among those who use the BIN option, none of them have made an offer
before choosing the BIN. Besides, 87% (34/39) of accept offers occurs in a buyer’s first offer.
Table 3 indicates that the first offers are the most important phenomenon in the market.
Our remaining analysis focuses on a buyer’s first decision in a listing. Consequently, the
observations consist of 978 offers and 10 BIN choices.8 We will use these 988 observations in
7
The same ID makes offers in different listings are treated as separate buyers. We do not consider the
interaction among listings which have overlapping durations in this paper.
8
When a buyer chooses the BIN option before making any offer, we treat the BIN decision as an observation,
8
Nature
A buyer
arrives.
T
not BIN
Buyer
T
No T
T
buyer
T
arrives.T
J
Buyer
offer
price
J
J
J
Nature
J
J
accept
Seller
J
J
JJ
game ends
game ends
@
@
reject@
@ period t ends
no responseJ
BIN J
T
response
JJ
period t ends
period t ends
Figure 1: Game Structure during Period t
our empirical analysis.
4
Model
In this section we set up a sequential-move game to investigate how a buyer’s first offer is
determined under the eBay Best Offer mechanism. We restrict our attention to a sequential
equilibrium in which the seller’s outside choice value θ is fully revealed after Period Zero.
We will explain the effects on the offer price resulting from the BIN price, the number of
buyers who have submitted an offer beforehand, and the timing of the offer. The theoretical
predictions will be verified in later sections.
4.1
The Game Structure
The listing duration of a product lasts for T periods. In period zero, the seller sets a BIN price
p¯. Figure 1 shows the timeline of events during each subsequent period t = 1, 2, . . . , T . At
the beginning of a period t, one buyer may arrive. When no buyer arrives, nothing happens
during that period. When a buyer arrives, he/she can purchase the product either by using
the BIN option or by making an offer. At the end of a period, the buyer exits the game and
never returns.
The probability of buyer arrival in each period is λ. The arrival probability λ reflects the
demand for the item. To model demand uncertainty, assume the value of λ is constant over
and the BIN price is treated as the “offer price”.
9
time but unobserved by everyone. Both the seller and the buyers have a common belief on
its distribution,9


 λH , with probability 0.5;
λ=

 λL , with probability 0.5,
for some 0 < λL < λH < 1. Both the seller and buyers are uncertain about the arrival
probability λ. Participants in later periods observe the offer history (except the offer prices)
of earlier periods and use that information to form a posterior belief of λ, which in turn
affects their decisions.
When a buyer arrives, he/she can decide whether to select the BIN option or make an
offer. If the buyer chooses the BIN option, the item is sold to the buyer at the BIN price p¯
and the game ends. On the other hand, if the buyer chooses to make an offer, nature draws
an exogenous shock on whether the seller can respond to the offer. With probability ρ, the
seller can decide whether to accept or reject the offer. When an offer is accepted, the item is
sold to the buyer at the offer price, and the game ends. When an offer is rejected, the buyer
gets nothing and leaves, and the period ends. With probability (1 − ρ), the seller does not
have a chance to respond to the offer, and the period ends. When an offer is not responded to
by the seller, the offer is recorded as “expired” in the offer history. We can think of random
shocks which prevent the seller from responding as some unavoidable events, such as Internet
connection problems.
Since we assume that a buyer stays in the game for only one period, our model rules out
the possibility of multiple offers from the same buyer. This simplification seems reasonable
for our empirical study since more than three quarters of buyers make only one offer and
87% of the accepted offers are a buyer’s first offer (Table 3). Our model also rules out the
possibility of choosing the BIN option after a rejected offer. This is consistent with the data
shown in Table 3.
When a buyer submits an offer, all future participants can observe the outcome of the
9
In theory we can generalize the distribution of λ to any distribution, but the model analysis would be
more complicated without qualitatively changing the results.
10
offer. If the item has not been sold, a previous offer is either not responded to or rejected by
the seller. However, the offer price is only observed by the seller, but not by other buyers.
The commonly observed offer history indicates whether a buyer submits a bid in each of
the previous periods. We denote the offer history of the first t periods by Ωt ∈ {0, 1}t .
For instance, if an offer is submitted in all previous periods, denote the history as Ωt =
(1, 1, 1, . . . , 1). In addition, we use Ω+1
t to denote the history of observing an offer in period
t + 1 after a history of Ωt in period t. Similarly, denote the history of no offer in period t + 1
after an history Ωt as Ω+0
t . The null history in period zero is denoted as ∅. If history Ωt2
occurs in period t2 after the history Ωt1 in period t1 where t2 > t1 ,10 we denote Ωt1 ⊂ Ωt2 .
When the seller sells an item at the BIN price p¯, the payoff is p¯. When the seller accepts
an offer price x, the payoff is x. Additionally, the seller has a value on the outside option. For
example, the seller can sell the product on another platform or simply retain the product for
his/her own use. Denoted the seller’s outside choice value as θ, which is the seller’s private
information with a publicly known distribution.11 When the item is unsold at the end of the
game, the payoff is θ. We also assume that in each period t, the seller gets a random shock, ηt ,
which has zero mean and a commonly known symmetric atomless distribution Fη .12 Denote
its density function as fη . We assume that the distribution of ηt has an increasing hazard
rate. A convenient example is a normal distribution, ηt ∼ N (0, σ 2 ). We can think of the
shock as the uncertainty regarding the handling and shipping cost of sending the item to the
buyer who arrives in period t. We are interested in the case with small variance of shock.13
The seller is willing to accept an offer x if and only if the expected surplus from accepting
the offer relative to that from rejecting it is greater than ηt .
Each buyer independently draws a private value v of the item from a commonly known
10
To be more precise, it means the first t1 elements in the sequence Ωt2 are identical to the entire sequence
Ωt1 .
11
We do not specify the distribution of θ in the model because buyers would infer the value of θ in equilibrium
and the distribution is irrelevant for the analysis.
12
We normalize the mean of the shock to zero. If its mean is nonzero, our result will be qualitatively the
same.
13
There are two technical reasons for including such a shock in the model. First, rejection of an offer would
not occur in equilibrium without such a shock. Second, a buyer’s offer price is not a smooth function of his/her
valuation of the product.
11
distribution Fv . The value is private information. A buyer’s payoff is v − p when buying
the item at a transaction price p (which is either the BIN price or the buyer’s accepted offer
price), and zero otherwise. For simplicity, assume that both making an offer and choosing
the BIN option incur no cost.14
When a buyer chooses the BIN option, the game terminates. Alternatively, when an offer
is accepted by the seller, the game also terminates. We are going to analyze each participant’s
optimal behavior in a non-terminal history.
4.2
Equilibrium after Period Zero
In this subsection, we assume that the seller’s choice of the BIN price in period zero is a
monotonic function of his/her outside option value θ to find the equilibrium behavior during
subsequent periods. Because of the monotonicity, θ is fully revealed after observing the BIN
ˆ In any sequential equilibrium, all buyers believe θ = θˆ
price p¯. Denote the inferred value as θ.
with probability one.
In each period t = 1, 2, . . . , T , a buyer may arrive. The buyer has an independently drawn
value of the product, v. A buyer needs to decide whether to use the BIN option or decide on
the offer price if he/she does not choose BIN. Whether an offer is accepted depends on the
ˆ p¯, Ωt ) denote the seller’s expected payoff
seller’s expected surplus of rejecting it. Let πt (θ, θ,
at period t conditional on not selling the item until that period, where Ωt ∈ {0, 1}t is the
offer history.
4.2.1
Buyer Decisions
First, consider the situation where a buyer does not choose to make a BIN purchase. Given
the seller’s expected payoff π, the choice of the optimal offer price for a buyer with value v is
to solve the maximization problem
max ρEηt 1{x − π > ηt }(v − x) + (1 − ρ)0,
x
14
(1)
Allowing positive cost of making an offer or choosing the BIN option does not qualitatively change the
model result although computing the equilibrium will be more complicated.
12
where the first term is the expected surplus when the offer price x is accepted, and the
second term is surplus when the seller does not respond. Here we use the fact that the
seller is willing to accept an offer if and only if the expected surplus from accepting the offer
relative to waiting for the next period is greater than the seller’s random shock ηt . Denote
the optimal offer price as
x(v, π).
The following proposition shows that x(v, π) is a well-defined function with positive partial
derivatives in both arguments:
Proposition 1. There is a unique solution in a buyer’s choice of the offer price. Moreover,
0<
∂x(v, π)
<1
∂v
0<
and
∂x(v, π)
< 1.
∂π
Proof. The maximization problem in (1) can be equivalently written as
max ρFη (x − π)(v − x).
x
The first order condition implies
fη (x − π)(v − x) − Fη (x − π) = 0.
To show the existence and uniqueness of the maximum, let z = x − π. The first order
condition can be rewritten as
v−π = z+
Fη (z)
.
fη (z)
(2)
Because the shock η has an increasing hazard rate, fη (z)/(1 − Fη (z)) increases in z. Since
η is symmetric with zero mean, fη (z)/(1 − Fη (z)) = f (−z)/F (−z). This implies that z +
Fη (z)/fη (z) is an increasing function of z. Therefore, there exists a unique z which satisfies
the first order condition (2) for any given value of v and π. This implies that a unique solution
exists for the maximization problem (1).
13
Next, consider the partial derivative of x(v, π) with respect to v. Note that when v
increases by an amount δ while holding π fixed, the right hand side of Equation (2) must also
increase by δ. Since the slope of the right hand size as a function of z is greater than one, z
has to increase by an amount less than δ. This implies that x is an increasing function of v
with a slope less than one.
Finally, consider an increase of π by an amount δ while holding v fixed. The right hand
side of Equation (2) has to decrease by δ. Consequently, z must go down by an amount less
than δ. Since x = z + π, x increase by an amount less than δ.
ˆ the optimal offer price of the
Given the inferred value of the seller’s outside option θ,
buyer who arrives in period t is
ˆ θ,
ˆ p¯, Ωt ))
x(v, πt (θ,
in a sequential equilibrium.
Next, to decide whether to use the BIN option, the buyer compares the expected payoff
with and without using this option. Conditional on the seller’s expected payoff π, a buyer’s
expected payoff from making the optimal offer x(v, π) is
ρFη (x(v, π) − π)[v − x(v, π)].
(3)
Lemma 1. The expected payoff from making the best offer (3) is an increasing function of
v, but the slope with respect to v is less than one.
Proof. By the envelope theorem,
d
ρFη x(v, π) − π [v − x(v, π)] = ρFη (x(v, π) − π).
dv
Since Fη is a distribution function and 0 < ρ < 1, the above derivative must lie between zero
and one.
14
On the other hand, the payoff from making the BIN purchase is
v − p¯.
(4)
Its slope with respect to v is one. Comparison of the payoffs in (3) and (4) makes it clear
ˆ p¯, Ωt ), a
that the choice of the BIN option follows a cutoff rule. Given the information (θ,
ˆ p¯, Ωt ), where the cutoff point
buyer with valuation v chooses BIN if and only if v ≥ Bt (θ,
ˆ p¯, Ωt ) is the solution of v to the following equation:
Bt (θ,
h
i
ˆ θ,
ˆ p¯, Ωt ) = v − p¯.
ˆ
ˆ
ˆ
ˆ
ρFη x v, πt (θ, θ, p¯, Ωt ) − πt (θ, θ, p¯, Ωt ) v − x v, πt (θ,
(5)
When a buyer arrives in period t, he/she will either choose the BIN option or make
an offer regardless of his/her valuation on the item. Consequently, in subsequent periods,
participants will always have full information on whether a buyer has arrived in period t. “No
action” would not occur in equilibrium. If we drop the assumption of zero cost in making an
offer, a buyer with low valuation may choose no action in equilibrium. Such extension will
not qualitatively change our model predictions.
4.2.2
Bayesian Updating of the Arrival Probability
The belief of the arrival probability λ is updated according to the Bayes’ rule. The probability
of λ = λ0 after observing an offer in period t is
ˆ p¯, Ωt−1 ]λ0 Pr[v ≤ B(θ,
ˆ p¯, Ω+1 )]
Pr[λ = λ0 |θ,
t−1
ˆ
ˆ
¯, Ωt−1 ]λ Pr[v ≤ B(θ, p¯, Ω+1
t−1 )]
λ=λH ,λL Pr[λ = λ|θ, p
ˆ p¯, Ωt−1 ]λ0
P r[λ = λ0 |θ,
=P
ˆ ¯, Ωt−1 ]λ
λ=λH ,λL P r[λ = λ|θ, p
ˆ p¯, Ω+1 ] = P
Pr[λ = λ0 |θ,
t−1
(6)
for λ0 = λH , λL . Similarly, the probability of λ = λ0 after observing no offer in period t is
ˆ p¯, Ω+0 ] = P
Pr[λ = λ0 |θ,
t−1
ˆ p¯, Ωt−1 ](1 − λ0 )
Pr[λ = λ0 |θ,
.
ˆ p¯, Ωt−1 ](1 − λ)
Pr[λ = λ0 |θ,
λ=λH ,λL
15
(7)
Note that the independence between the arrival process and a buyer’s valuation v allows us
to simplify (6).
ˆ p¯, ∅] = Pr[λ = λL |θ,
ˆ p¯, ∅] = 1/2, by induction, it is easy to see that if
Since Pr[λ = λH |θ,
there are n offers in the history Ωt ,
ˆ p¯, Ωt ] =
Pr[λ = λ0 |θ,
λn0 (1 − λ0 )t−n
.
λnH (1 − λH )t−n + λnL (1 − λL )t−n
for λ0 = λH , λL . The posterior belief does not depend on the inferred value of the seller’s
outside option θˆ and the BIN price p¯. As a consequence, we will express it simply as Pr(λ =
λ0 |Ωt ) henceforth.
4.2.3
Seller’s Expected Surplus of Continuation
The next step is to compute the seller’s expected surplus in each history. Obviously, the
expected surplus of not selling the product until the final period equals the outside choice
value,
ˆ p¯, ΩT ) = θ for any ΩT ∈ {0, 1}T .
πT (θ, θ,
For period t < T , the expected surplus can be computed recursively from the possible histories
ˆ p¯, Ωt ) and the offer
in the next period. Given the buyer’s BIN decision function Bt+1 (θ,
price function xt+1 (v, π), the expected surplus of not selling the product until period t after
16
observing the history Ωt is
ˆ p¯, Ωt ) =
πt (θ, θ,
X
Pr(λ|Ωt )×
λ=λH ,λL
i
n
h
ˆ p¯, Ω+1 ) p¯
λ Pr v ≥ Bt+1 (θ,
t
i
h
ˆ p¯, Ω+1 ) + ηt+1 ×
ˆ θ,
ˆ p¯, Ω+1 )) ≥ πt+1 (θ, θ,
ˆ p¯, Ω+1 ), x(v, π(θ,
+λρ Pr v < Bt+1 (θ,
t
t
t
i
h
ˆ p¯, Ω+1 )+ηt+1
ˆ p¯, Ωt ), x(v, πt+1 (θ,
ˆ θ,
ˆ p¯, Ω+1 )) ≥ πt+1 (θ, θ,
ˆ θ,
ˆ p¯, Ω+1 )) v < Bt+1 (θ,
E x(v, π(θ,
t
t
t
i
h
ˆ p¯, Ω+1 )
ˆ θ,
ˆ p¯, Ω+1 )) < πt+1 (θ; p¯, Ω+1 ) + ηt+1 πt+1 (θ, θ,
ˆ p¯, Ω+1 ), x(v, π(θ,
+λρ Pr v < Bt+1 (θ,
t
t
t
t
h
i
ˆ p¯, Ω+1 ) πt+1 (θ, θ,
ˆ p¯, Ω+1 )
+λ(1 − ρ) Pr v < Bt+1 (θ,
t
t
o
ˆ p¯, Ω+0 ) ,
+(1 − λ)πt+1 (θ, θ,
(8)
t
where the first term inside the bracket is the surplus for selling by BIN during the next period;
the second term is the surplus for accepting an offer during the next period; the third term
is the surplus for rejecting an offer during the next period; the fourth term is the surplus for
not responding to an offer during the next period; the last term is the surplus for no offer
arriving during the next period.
We can use backward induction to compute πt (θ, θ, p¯, Ωt ) for all possible histories.
4.2.4
Equilibrium Properties
The following propositions show how the seller’s expected surplus changes with the offer
history and over time:
ˆ p¯, Ωt ) increases
Proposition 2. For any given period t, the seller’s expected payoff πt (θ, θ,
in the number of offers made in the history Ωt .
Proof. Because λ/(1 − λ) is an increasing function of λ, λH /(1 − λH ) > λL /(1 − λL ). Suppose
17
that history Ωt consists of n offers, and Ω′t consists of n′ offers where n > n′ ,
ˆ p¯, Ωt ] =
Pr[λ = λH |θ,
λnH (1 − λH )t−n
λaH (1 − λH )t−n + λnL (1 − λL )t−n
′
′
λnH (1 − λH )t−n
ˆ p¯, Ω′ ].
> n′
= Pr[λ = λH |θ,
′
t
λH (1 − λH )t−n′ + λnL (1 − λL )t−n′
ˆ p¯, ΩT ) = θ for any
As a result, E[λ|Ωt ] > E[λ|Ω′t ]. We know that in the final period, πT (θ, θ,
ˆ p¯, Ωt ) ≥ πt (θ, θ,
ˆ p¯, Ω′ ) for any given
history ΩT . By applying induction to (8), we have πt (θ, θ,
t
ˆ p¯).
(θ, θ,
Proposition 3. Conditional on the information of any given history Ωt , the seller’s expected
ˆ p¯, Ωt ) decreases over time.
surplus πt (θ, θ,
ˆ p¯, Ωt )|Ωt ] ≥ E[πt (θ, θ,
ˆ p¯, Ωt )|Ωt ]
E[πt1 (θ, θ,
1
2
2
(9)
for any Ωt ⊂ Ωt1 ⊂ Ωt2 with t ≤ t1 ≤ t2 .
Proof. By equation (8), the seller’s expected payoff of not selling the product until period t
after observing the history Ωt is
ˆ p¯, Ωt ) ≥E[λ|Ωt ]πt+1 (θ, θ,
ˆ p¯, Ω+1 ) + (1 − E[λ|Ωt ])πt+1 (θ, θ,
ˆ p¯, Ω+0 )
πt (θ, θ,
t
t
ˆ p¯, Ωt+1 )|Ωt ].
=E[πt+1 (θ, θ,
Consequently, the inequality (9) holds for any Ωt ⊂ Ωt1 ⊂ Ωt2 by applying induction and the
law of iterated expectations to the above inequality.
Proposition 1 shows that a buyer’s offer price increases in the seller’s expected surplus.
Therefore, we have the following two corollaries.
Corollary 1. For any given period t, a buyer’s offer price increases in the number of offers
made in the history Ωt .
18
Corollary 2. Conditional on the information of any given history Ωt , a buyer’s offer price
decreases over time.
4.3
Choice of the Buy-It-Now Price in Period Zero
In period zero, the seller needs to decide upon the BIN price p¯. While slightly abusing the
notation, let p¯(θ) denote the optimal BIN price when the outside option value is θ and denote
ˆ p). In a sequential equilibrium, θ(¯
ˆ p) = θ.
its inverse function as θ(¯
The choice of the BIN price has both a direct effect and a ratchet effect on the seller’s
expected surplus. The direct effect of a higher BIN price is twofold. On the one hand, it
reduces the probability of selling by the BIN option. On the other hand, it raises the revenue
from selling by BIN. There is a tradeoff between these two impacts. Moreover, since buyers
will infer the seller’s outside choice value θ from the observed BIN price p¯ in later periods,
the buyer’s choice of the BIN price is affected by the ratchet effect. The ratchet effect leads
buyers to infer a higher outside choice value θˆ from observing a higher BIN price, which
in turn increases their offer prices and increases the probability of choosing the BIN price.
Hence, the seller has an incentive to choose a higher BIN price. The optimal BIN price for
any given θ is the solution to the constrained maximization problem:
ˆ p), p¯, ∅)
max π0 (θ, θ(¯
p¯
ˆ p) = θ.
s.t. θ(¯
ˆ p) satisfies the differentiation equation15
Hence, the function θ(¯
ˆ p)
∂π0 (θ, θ, p¯, ∅) ∂π0 (θ, θ, p¯, ∅) dθ(¯
+
= 0.
∂ p¯
d¯
p
∂ θˆ
(10)
ˆ p).
The function for the optimal BIN price, p¯(θ), is the inverse function of θ(¯
We use the following numerical example to demonstrate the existence of a monotonic
The partial derivative ∂π0 /∂ θˆ denotes the partial derivative with respect to the second argument in
ˆ p¯, ∅).
π0 (θ, θ,
15
19
10000
9000
Buy−It−Now Price
8000
7000
6000
5000
4000
3000
1500
2000
2500
3000 3500 4000 4500 5000
Seller’s Outside Choice Value
5500
6000
Figure 2: The Seller’s Optimal Choice of the BIN Price in Equilibrium
function p¯(θ). In addition, we show that the seller’s expected surplus increases in the outside
option value θ for any given history in any period.
Example 1. Assume that the distribution of the buyer’s valuation, Fv , follows a log-normal
distribution. The mean of log(v) is 8 and variance is 0.25. Then the distribution of v has
a mean of 3,378, median of 2,980, and standard deviation of 1,800. In addition, assume
λH = 0.3, λL = 0.1, ρ = 0.25, T = 3, and the distribution of ηt degenerates at zero.
Figure 2 shows the optimal BIN price as a function of the seller’s outside choice value θ in
a sequential equilibrium. The graph shows that p¯(θ) is an increasing function. Furthermore,
we compute the expected surplus for each possible history in the game. Each curve in Figure 3
20
6500
6000
Seller’s Expected Surplus
5500
5000
4500
4000
φ
{1}
{1,1}
{1,1,1}
3500
3000
2500
2000
1500
1500
2000
2500
3000 3500 4000 4500 5000
Seller’s Outside Choice Value
5500
6000
Figure 3: Seller’s Expected Payoff for a Given History
corresponds to a history with a buyer entering in all previous periods, i.e. Ωt = (1, 1, . . . , 1).16
We find that the expected surplus increases in θ for any given history.
As the above example suggests, the seller’s expected surplus increases in θ. Since the
seller’s optimal BIN also increase in θ, Proposition 1 implies a buyer’s offer price increases in
the BIN price.
16
We compute the expected payoff for all 24 − 1 = 15 possible histories. To ease the demonstration in the
figure, we only draw those histories in which there is a buyer entering in all previous periods. Nonetheless, all
the curves are similar in shape.
21
5
Estimation
In this section we propose an empirical model to verify our theoretical predictions of a buyer’s
offer prices. Specifically, we predict the offer price to increase in the number of buyers who
have made on offer on the listing, but to decrease in the remaining time of the listing.
Moreover, the BIN price is positively correlated with the offer price. Since the seller’s choice
of the BIN price is likely to be an endogenous variable, we use exclusive instruments to form
moment conditions and estimate the coefficients by the Generalized Method of Moments
(GMM).
5.1
Observed Variables
Table 4 presents definitions and descriptive statistics for the variables used in the study. The
upper panel of the table show listing-specific variables, including product characteristics and
seller characteristics. The variables in the lower panel are offer-specific, such as the offer
price, offering time, and buyer characteristics.
The value of a car varies with many observable characteristics, such as age, mileage, and
mechanical condition. The Kelly Blue Book (more commonly known as the “blue book”) is a
commonly used reference for estimating the value of a used car. Since our focus is the impact
of the trading system on the first offer, not the impact of the observed characteristics, the
blue book price is used to control for heterogeneities among cars. Specifically, we obtain the
blue book price for each car according to its age, mileage, feature, and the zip code of the
seller’s location. Then, we normalize a price (seller’s listed BIN price or buyer’s offer price)
by first dividing the price by the blue book price for the listed car and then take a logarithm
of the ratio. Therefore, the normalized price can be interpreted as the percentage of deviation
from the blue book price. Table 4 indicates the median BIN price to be approximately 14.6%
(exp(0.136) = 1.146) higher than the blue book price while the median offer price is 22.3%
(exp(−0.252) = 0.777) lower than the blue book price.
Even though we use the blue book price to control for several aspects of quality hetero-
22
Table 4: Descriptive Statistics
Variable
Definition
Listing characteristics
BIN
Buy-It-Now price (in $)
BLU E
Kelly Blue Book price (in $)
ln(BIN/BLU E)
Log of (BIN price/ Kelly Blue Book price)
CLEAR
Whether the title is clear
W ARRAN T Y
Whether the vehicle is still under warranty
SCORES
Seller’s feedback score on eBay
DEALER
Whether the seller is a dealer
DU RAT ION
Duration of the listing (in days)
OT HER
Whether the seller lists other vehicle concurrently
ln(BIN/BLU E)other Log of (BIN price/ Kelly Blue Book price) of
other vehicles listed by the seller concurrently
Offer characteristics
OF F ER
Buyer’s first offer price (in $)
ln(OF F ER/BLU E) Log of (first offer/ Kelly Blue Book price)
P RIOR
Number of buyers having made an offer
F IRST
Whether the offer is the first one of the listing
T IM E
Length of time since the start of the listing (days)
SCOREB
Buyer’s feedback score on eBay
W EEKEN D
Whether the offer is made on weekend
M ORN IN G
Whether the offer is made between 6–10 am
23
Mean
Median
Std. Dev.
11746
10578
0.094
0.779
0.305
347
0.493
11.379
0.564
0.032
9844
9130
0.135
1
0
137
0
10.000
1
0.057
7968
6746
0.323
0.415
0.460
1051
0.500
6.121
0.496
0.332
7996
-0.450
2.022
0.332
6.022
75
0.299
0.229
6000
-0.251
1
0
4.991
11
0
0
6571
0.847
2.488
0.471
4.844
203
0.458
0.420
geneities, car quality may still differ in other aspects. We include two variables to account for
additional quality difference. The dummy variable CLEAR takes a value of one if the vehicle
title is clear and zero otherwise. A clear title basically means that there are no outstanding
payments to be made on the car. It also means that the car has never been reported as being
excessively damaged or stolen. The other dummy variable W ARRAN T Y takes a value of
one if vehicle is still under warranty and zero if its warranty has expired.
Literature on Internet transactions indicates that sellers and buyers can use feedback
ratings on the platform to build reputations and overcome problems caused by information
asymmetry. (Livingston, 2005; Houser and Wooders, 2006) We include the seller’s feedback
score on eBay, SCORES, as an explanatory variable to determine a buyer’s willingness to
pay. The higher the feedback score a seller has, the higher the probability that the seller
will deliver the subjects as described. Besides, we also include a buyer’s feedback score
(SCOREB) to control for the effects due to a buyer’s reputation.
In collecting data on a seller’s listing for a Toyota Camry, we use the hyperlinks to follow
all of his/her listings on eBay at the same time. The dummy variable OT HER indicates
whether the seller has listed any other vehicle on eBay at the same time. When OT HER = 1,
we compute the normalized BIN prices for other vehicles listed by the seller at the same time
and denote it by ln(BIN/BLU E)other .17 We use these two variables as exclusive instruments
in our estimation to control for the endogeneity problem.
17
We set the value of this variable to zero when the seller does not list any other vehicle on eBay at the
same time. When the seller has listed more than one other vehicle at the same time, we randomly choose one
of the other listed vehicles to compute the value of this variable.
24
5.2
Empirical Specification
Our empirical model is specified as
ln(OF F ER/BLU E) = β0 + β1 ln(BIN/BLU E) + β2 F IRST + β3 P RIOR + β4 T IM E
+ β5 CLEAR + β6 W ARRAN T Y + β7 ln(SCORES) + β8 DEALER
+ β9 ln(SCOREB) + β10 W EEKEN D + β11 M ORN IN G + ε, (11)
where ε represents product characteristics observed by the seller and buyers but unobserved
to econometricians. For instance, ε may capture quality information presented by the photos
posted by the seller on the webpage. Buyers can use the information to determine their
offer prices, but that information is not observed by econometricians.18 We assume that ε
is independent across listing and has a mean of zero. We allow ε to be correlated for offers
within the same listing.
Our theoretical model suggests that a buyer tends to offer a higher price when observing
a higher BIN price, because the BIN price is positively correlated with the seller’s outside
choice value, which determines the seller’s willingness to accept an offer. Therefore we expect
β1 > 0. A buyer’s offer price will also increase when the demand for the product is believed
to be stronger. Corollary 1 shows that the inferred demand is stronger when more buyers
have made an offer. We use F IRST and P RIOR to capture this effect and expect β2 < 0
and β3 > 0. By using these two variables as covariates, we allow the marginal effect of the
first buyer in a listing to be different from that of subsequent buyers. According to Corollary
2, a buyer will decrease the offer price toward the end of the listing period because the seller
will have less chance of selling the product to other buyers. The variable T IM E captures
this effect, and we expect β4 < 0.
When a car has a better unobserved quality ε, the seller will post a higher BIN price and
the buyer will be willing to offer a higher price. Therefore, the endogeneity of the BIN price
may be a concern for establishing a causal relationship from its estimated coefficient. We
18
Although we collect these photo files for our sample, it is difficult to quantify the photo information.
25
overcome the endogeneity problem by using exclusive instruments to form moment conditions.
We estimate the regression equation (11) based on the GMM. Although the equation can
be also estimated by performing the two-stage least squares (2SLS) method, the unobserved
characteristics ε are assumed to be homoscedastic across listings in the 2SLS. Instead, in our
GMM estimation, the standard errors are robust to the presence of arbitrary heteroskedasticity.19
5.3
Exclusive Instruments
We propose two variables to control the endogeneity of the BIN price in the regression equation: OT HER and ln(BIN/BLU E)other . The primary intuition for choosing these instruments is that these variables may correlate with a seller’s outside option value, but they are
unlikely to affect a consumer’s demand directly (after controlling for the seller’s reputation
and observed characteristics).
When a seller has a better outside choice for one car, he/she is likely to have a better
outside choice for another one as well. For instance, if local demand for used cars is high
in the seller’s location, the seller has a higher outside choice value for any of the cars listed
on eBay. Hence, we expect the BIN price to be correlated with the average BIN price of
other cars listed by the same seller at the same time. Here, we normalize the price by taking
logarithm over the ratio of the BIN price and the blue book price and denote the normalized
price as ln(BIN/BLU E)other . When a seller does not concurrently list other vehicle on eBay,
this variable is missing. In such cases, we set its value to zero. The binary variable OT HER
indicates whether a buyer lists any other vehicle on eBay at the same time.20 As long as the
unobserved characteristic ε of a particular car is uncorrelated with the demand for used cars
19
We use the user-written command in Stata ‘ivreg2’ to estimate the regression equation. We estimate the
model under the option ‘gmm’ so that the parameters are estimated using the optimal weighting matrix in
the GMM.
20
If we simply drop the observations without listing other vehicles on eBay concurrently, we would
lose 44% of the observations in the estimation, causing much larger standard errors. Instead, by setting
ln(BIN/BLU E)other = 0 for these observations and adding the dummy variable OT HER to the regression
equation, we can use all observations in the estimation and the dummy variable OT HER will control for the
selection bias resulting from missing values.
26
at the seller’s location, these two variables are reasonable choices for exclusive instruments.
We assume that the unobserved characteristic ε is uncorrelated with all the explanatory variables in the regression equation (11) except for ln(BIN/BLU E). We estimate the
coefficients using GMM with the following orthogonality condition:
E[εz] = 0
(12)
where the vector z includes the two exclusive instruments, OT HER and ln(BIN/BLU E)other ,
and all the explanatory variables in the regression equation except the endogenous variable,
ln(BIN/BLU E).
Before moving on to discuss our estimation results, we show the validity of our exclusive
instruments. The first column in Table 5 reports the first stage of the two stage least squares
(2SLS) estimation. The variables in the upper panel are the exclusive instruments. The F
statistic is over ten and the χ2 statistic for the Anderson-Rubin test is 14.22, which has a
p-value of 0.0008. These statistics indicate that these two instruments are relevant to the
endogenous variable, ln(BIN/BLU E).
5.4
Estimation Results
Table 6 shows a comparison our preferred specification in the last column with other choices of
explanatory variables. After deleting incomplete samples, we are left with 848 observations.21
The t-values in Table 6 are computed based on robust standard errors. As has been pointed
out in Subsection 5.2, we allow for correlated unobserved characteristics ε for offers within
the same listing.
We now discuss the estimation results for the preferred specification, as shown in Table
6. All the estimated coefficients have the expected signs. The results are robust across most
of the specifications we tried, including unreported specifications. Our empirical results are
21
Some sellers refused to disclose the buyer’s offer in the offer history. We thus lost 86 observations. Also,
we could not obtain a blue book price for some cars, mainly due to unavailability of the owner’s ZIP code or
vehicle’s age, eliminating another 9 observations.
27
Table 5: First Stage of the 2SLS for the Instruments
Dependent variable
ln(BIN/BLU E)
0.406∗∗∗
(0.090)
-0.049
(0.045)
ln(BIN/BLU E)other
OT HER
Listing characteristics
ln(BIN/BLU E)
0.239∗∗∗
(0.069)
-0.052
(0.040)
-0.004
(0.013)
0.138∗∗∗
(0.044)
CLEAR
W ARRAN T Y
ln SCORES
DEALER
Offer characteristics
P RIOR
-0.013∗∗
(0.006)
0.009
(0.016)
0.007∗∗
(0.004)
0.003
(0.005)
0.001
(0.022)
0.010
(0.020)
-0.136∗
(0.081)
F IRST
T IM E
ln SCOREB
W EEKEN D
M ORN IN G
constant
Number of offers
R-squared
F-statistic
850
0.347
8.10
Notes: Robust standard errors are given in parentheses. Superscripts ∗∗∗ , ∗∗ , and ∗ represent significance at 1%, 5%, and
10%, respectively.
28
Table 6: Estimated Coefficients with Different Explanatory Variables
Dependent variable: ln(OF F ER/BLU E)
Constant
Model 1
Model 2
Model 3
Preferred Model
-0.842∗∗∗
-0.628∗∗∗
-0.817∗∗∗
(0.155)
-0.842∗∗∗
(0.152)
1.130∗∗∗
(0.171)
1.140∗∗∗
(0.199)
0.043
(0.104)
0.146∗∗∗
(0.056)
0.033∗∗
(0.018)
0.065
(0.054)
1.147∗∗∗
(0.198)
0.041
(0.104)
0.142∗∗∗
(0.055)
0.034∗
(0.018)
0.054
(0.054)
0.023∗∗
(0.011)
-0.140∗∗
(0.072)
-0.010
(0.010)
0.043∗∗∗
(0.013)
-0.034
(0.057)
0.161∗∗∗
(0.059)
0.022∗∗
(0.011)
-0.116∗
(0.069)
-0.013
(0.010)
0.041∗∗∗
(0.013)
-0.031
(0.054)
0.133∗∗
(0.057)
-0.004
(0.005)
0.022∗∗
(0.011)
-0.120∗
(0.069)
-0.015∗
(0.009)
0.041∗∗∗
(0.013)
-0.030
(0.054)
0.135∗∗
(0.058)
880
0.178
0.247
0.619
850
0.256
0.007
0.934
850
0.255
0.001
0.975
(0.123)
Listing characteristics
ln(BIN/BLU E)
1.102∗∗∗
(0.211)
CLEAR
0.091
(0.102)
W ARRAN T Y
0.135∗∗
(0.057)
ln SCORES
0.038∗∗
(0.019)
DEALER
0.019
(0.057)
Offer characteristics
P RIOR
F IRST
T IM E
ln SCOREB
W EEKEN D
M ORN IN G
(0.073)
DU RAT ION
Number of offers
R-squared
Hansen J-statistic
p-value
862
0.226
0.402
0.526
Notes: Robust standard errors are given in parentheses. Superscripts
∗
represent significance at 1%, 5%, and 10%, respectively.
29
∗∗∗
,
∗∗
, and
consistent with our theoretical predictions and suggest that buyers are rational in the sense
of using the available information to determine the offer price.
Our theoretical model suggests that a higher BIN price means that the seller has a higher
outside choice value and is less likely to accept an offer. As a result, we expect to see a
positive relationship between the offer price and the BIN price. Our estimation indicate that
a 1% increase in ln(BIN/BLU E) is associated with approximately 1.15% increase in the
ln(OF F ER/BLU E), which confirms our theoretical prediction.
On average, buyers begin to make offers roughly 6.1 days after a car is listed. The length
of time since the start of the listing has a significantly negative effect on the offer price. Other
things being equal, the price that a buyer is willing to offer is reduced by 1.5% with every
passing day. Our model predicts that a buyer is likely to offer a lower price toward the end
of a listing period because the seller is willing to accept a relatively lower offer to avoid no
transaction.22
Although the offer price is not disclosed on the eBay website during the listing period, it
reveals the buyer ID for each offer having made for the product. Hence, a potential buyer can
infer the demand for the product in a particular listing by observing the number of buyers
who have already made an offer for it beforehand. Our estimation finds that the first buyer
to make an offer tends to offer a significantly lower price. The first buyer’s offer price is
13.3% (exp(−0.120 − 0.023) = 0.867) lower than the second buyer. Moreover, when more
buyers have made an offer, the offer price becomes significantly higher. Each additional buyer
increases the offer price by 2.3%. Both indicate that the inferred demand positively affects a
buyer’s offer price.
In addition to comparing the estimated coefficients with our theoretical model, we can only
compare them with the literature on Internet transactions. A seller’s reputation, as measured
by the feedback score, is recognized as an important factor in determining a buyer’s willingness
to pay in Internet auctions (Melnik and Alm, 2002; Livingston, 2005; Houser and Wooders,
22
An alternative explanation for the negative sign is that an individual with a stronger desire to buy tends
to make an offer earlier and offer a higher price.
30
2006). Our analysis of the offer prices shows similar results. The seller’s feedback score on
eBay has a significantly positive impact on offer prices. Besides, a buyer’s feedback score
also has a significantly positive effect on the offer price. There are different explanations for
the effect of a buyer’s feedback score. On one hand, the score is representative of the buyer’s
reputation. A buyer with a good reputation can expect the seller to be willing to accept a
relatively lower price since he is more likely to honestly complete the transaction process. On
the other hand, a higher score indicates the buyer has completed more transaction on eBay
in the past. A buyer who is willing to pay more for products listed on eBay can have more
successful transactions in the past and accumulate a higher score. Our regression suggests the
latter explanation is more important than the former one. These findings are different from
those for Internet auctions. For instance, Houser and Wooders (2006) show that a buyer’s
reputation has no effect on price.
Following the finding in Lucking-Reiley, Bryan, Prasad, and Reeves (2007), we also consider the timing effect. Offers made between 6 am to 10 am Pacific Time tend to be a
significantly higher than those made at other times. Offer prices are also higher on weekdays,
although the difference is insignificant. The reason could be that buyers making an offer on
weekday mornings have a stronger desire to buy a car.
The J-statistic of the Hansen overidentification test is 0.001 with a p-value equal to 0.975.
Therefore, we cannot reject the hypothesis that the instruments and ε are jointly uncorrelated
at any conventional significance level.
5.5
Robustness
Finally, we use several alternative estimation approaches to consider the robustness of the
GMM estimation. The first column in Table 7 is our preferred specification using the GMM.
In the second column, we ignore the endogeneity problem and estimate the regression equation
(11) by the Ordinary Least Squares (OLS) method. Although most estimated coefficients are
very similar to our GMM estimation result, the coefficient of ln(BIN/BLU E) is 8% smaller
than that in the GMM estimation. The last column consider the censoring problem in our
31
Table 7: Robustness Checking with Different Estimation Methods
Dependent variable: ln(OF F ER/BLU E)
GMM
-0.842∗∗∗
(0.152)
Listing characteristics
ln(BIN/BLU E)
1.147∗∗∗
(0.198)
CLEAR
0.041
(0.104)
W ARRAN T Y
0.142∗∗∗
(0.055)
ln SCORES
0.034∗
(0.018)
DEALER
0.054
(0.054)
Offer characteristics
P RIOR
0.022∗∗
(0.011)
F IRST
-0.120∗
(0.069)
T IM E
-0.015∗
(0.009)
ln SCOREB
0.041∗∗∗
(0.013)
W EEKEN D
-0.030
(0.054)
M ORN IN G
0.135∗∗
(0.058)
Constant
Number of offers
R-squared
Hansen J-statistic
p-value
OLS
Tobit
-0.861∗∗∗
(0.163)
-0.856∗∗∗
(0.163)
1.051∗∗∗
(0.099)
0.070
(0.083)
0.135∗∗
(0.054)
0.033∗
(0.018)
0.070
(0.052)
1.040∗∗∗
(0.100)
0.073
(0.084)
0.129∗∗
(0.055)
0.032∗
(0.018)
0.068
(0.052)
0.021∗∗
(0.010)
-0.118∗
(0.071)
-0.014
(0.009)
0.042∗∗∗
(0.013)
-0.031
(0.054)
0.135∗∗
(0.058)
0.021∗∗
(0.010)
-0.115∗∗
(0.070)
-0.014
(0.009)
0.042∗∗∗
(0.013)
-0.034
(0.054)
0.137∗∗∗
(0.059)
850
0.256
850
850
0.255
0.001
0.975
Notes: Robust standard errors are given in parentheses. Superscripts
, ∗∗ , and ∗ represent significance at 1%, 5%, and 10%, respectively.
∗∗∗
32
data. In the Best Offer mechanism, a buyer can always obtain the product by paying the
BIN price. Therefore, a buyer’s offer is bounded from above by the BIN price. Our GMM
estimation ignores the censoring problem, but there are 7 offers equal to the BIN price in our
sample. We use the standard Tobit model to account for censoring of the data but ignore
the endogeneity of the BIN price. Since less than 1% of the observations are censored, it
is unsurprising to find the estimated coefficients do not change much after accounting for
censoring.
6
Conclusion
Our study contributes to the understanding of the buyer’s behavior using the Best Offer
selling format on eBay. Contrary to the finding of Spann and Tellis (2006) regarding the
NYOP mechanism used by a German retailer, we find that buyers are rational in making
their decisions. Consistent with our sequential-game model, a buyer’s offer price crucially
depends on the information available to them. The length of time when the offer is made
after the start of listing is negatively related to the offer price. Buyers are able to infer
the seller’s outside choice value from the BIN price, and hence make a higher offer when
observing a higher BIN price. Although the values of other buyers’ offer prices are unknown,
the number of buyers who have made offers beforehand is an indicator for demand for the
product and affects each potential buyer’s offer price. We hope to extend our work to study
the interactions between the buyer and the seller after the first offer in future work.
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