1  

Higher Revenue with the Same Demand: Distributing Bidders Evenly Across Multi-Object Sequential

Auctions

By: Eric Overby, Georgia Institute of Technology and Karthik Kannan, Purdue University

1.0 Introduction

Suppose you are a consultant advising a seller on how to use auctions to sell products. The seller has

multiple products to auction, and he has retained you to help him achieve higher prices. You might advise the

seller to implement strategies to attract more bidders to his auctions, as more bidders will translate into higher

prices, ceteris paribus. What you might not realize is that the seller can achieve higher prices without

increasing the number of bidders; instead he can achieve them by distributing existing bidders more evenly

across auctions.

In this study, we show both analytically and empirically how a more even distribution of bidders across a

series of auctions increases average prices. We also show how lower transaction costs online facilitate this

distribution of demand. The study contributes to the broader auction literature by investigating how the

distribution of demand influences outcomes in sequential auctions. The study also contributes to the research

literature related to reduced transactions costs in electronic channels (e.g., Clemons 1991; Malone et al. 1987.)

Much of this literature has focused on how electronic channels reduce buyer and seller search costs and the

corresponding effects on market outcomes (e.g., Bakos, 1997; Kuruzovich, Viswanathan, & Agarwal, 2010).

We extend this literature into a sequential auctions context by investigating how electronic channels reduce the

cost that buyers incur when waiting for a product to be auctioned, which we refer to as the “cost of waiting.” In

addition to research implications, the study has clear practical implications for auction sellers and for market-

makers whose transaction fees are based on a percentage of the selling price.

Section 2 presents the motivation and analytical model. Section 3 discusses the empirical context and the

data we use to test the insights from the model. Section 4 presents the empirical analysis and results. We

conclude in Section 5.

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2.0 Motivation and Analytical Models

In many traditional auctions, products are auctioned sequentially. In other words, the first product is

auctioned, followed by the second product, followed by the third product, and so on. Collectibles, antiques,

wine, livestock, used vehicles, and agricultural commodities such as coffee beans are auctioned in this fashion.

Sequential auctions in which successive products are identical are referred to as “multi-unit” sequential

auctions. Those in which successive products are different are referred to as “multi-object” sequential auctions.

We focus on multi-object sequential auctions, specifically those for used vehicles, although our results can be

generalized to other types of multi-object sequential auctions.1

We investigate how bidder participation across the auctions in the sequence affects outcomes for the seller.

A straightforward approach for improving seller outcomes is to increase the number of bidders who

participate, as more bidders will lead to higher prices. We take a different view and investigate whether and

how the seller can achieve better outcomes without attracting more bidders. Instead, we examine how changing

the distribution of bidders across the auctions in the sequence can yield improved outcomes for the seller.

To illustrate, consider the following simple example. Assume that a seller is auctioning 5 products

sequentially and that she is able to attract a total of 25 bidders, each of whom participates in the bidding for

one of the products. The distribution of bidders across the products in the sequence can take multiple forms,

examples of which are shown in Figure 1. For example, if there is bidder attrition across the sequence, then the

bidder distribution will show a declining pattern. If bidder participation remains steady throughout the

sequence, then the bidder distribution will appear flat. If bidder participation increases over the sequence, then

the distribution will show an increasing pattern. In each case, it is critical to note that the total number of

bidders (n=25) is unchanged; the only difference is how these bidders are distributed across the products in the

sequence.

                                                         1 Modeling multi-unit sequential auctions can be challenging because bidders learn each other’s valuations as the auctions progress. This affects each bidder’s valuation, thereby complicating bidders’ strategies. This is not an issue with multi-object sequential auctions.

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Figure 1: Examples of declining, flat, and increasing bidder distribution patterns.

Ceteris paribus, products with high bidder participation will fetch higher prices than products with low

bidder participation (Brannman, Klein, & Weiss, 1987). Thus, in the example above, products 1 and 2 will

fetch higher prices in the auction sequence with the “Declining” distribution pattern than in the auction

sequence with the “Flat” distribution pattern, while the converse will be true for products 4 and 5. At first

glance, it appears that these price differences will cancel each other out, such that total revenues will be the

same regardless of the bidder distribution. Interestingly, this is not the case, which we show analytically in the

proofs below.

2.1 Analytical Models

We present two proofs. The first is based in a simple setting and is designed to demonstrate the intuition

behind our analytical result. The second relaxes the assumptions of the first setting to demonstrate the

robustness of the result.

2.1.1 Simple Setting: Two Products, n Bidders, Bidder Valuations Distributed Uniformly

Consider two scenarios in which two products are auctioned sequentially. Suppose in scenario 1 that the

number of bidders for each product are n and n with n < n , and that in scenario 2 the number of bidders for

each product is the same n. Next suppose that the total number of bidders for the two products is the same in

each scenario. Hence, 2 * n = n + n . Now, suppose bidders in both scenarios draw their valuations from the

same uniform [a,b] distribution. Then, the following proposition holds:

Proposition 1: The total expected seller revenue is higher in scenario 2 than in scenario 1. In other words,

a more even distribution of bidders across the auctions yields higher revenue than a less even distribution of

the same total number of bidders.

0

5

10

1 2 3 4 5

Declining

0

5

10

1 2 3 4 5

Flat

0

5

10

1 2 3 4 5

Increasing

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Proof: Following theory pertaining to ascending auctions, the high bid is equal to the second-highest

valuation for each product. For n bids from a uniform [0,1] distribution, this will equal 11

+−

nn

in expectation.

For n bids from a uniform [a,b] distribution, this expression becomes )(*1)1( ab

nna −+−

+ . Thus, in the

second scenario, the total expected revenue from the two auctions is ⎟⎠⎞

⎜⎝⎛ −

+−

+ )(*1)1(*2 ab

nna . In the first

scenario, the total expected revenue from the two auctions is )(*1)1( ab

nna −+−

+ + )(*1)1( ab

nna −+

−+ =

⎟⎟⎠

⎞⎜⎜⎝

⎛−

++−

+ )(*)1)(1(

)1(*2 abnn

nna . The proof for the proposition is accomplished by showing that the first term

is greater than the second, i.e., that (after simplifying and canceling the a and b parameters) 11

+−

nn

>

)1)(1(1++

−nn

nn.

Consider the difference 11

+−

nn

−)1)(1(

1++

−nn

nn=

)1)(1)(1(22+++

−++−−nnn

nnnnnnnnn. As long as the numerator is

positive, we have proved the result. Substituting n = 2n − n and simplifying the terms, the condition

022 >−++−− nnnnnnnnn simplifies to nn > . QED.

Figure 2 provides a graphical illustration of the intuition behind the proof. If bidders draw their valuations

from the same uniform [0,1] distribution, then the expected price given the number of bidders is a concave

function. Thus, the price increase from adding an additional bidder to an auction with low participation

outweighs the price decrease from taking that bidder away from another auction with high participation. Figure

2 illustrates this by showing that the average expected price per product is higher when 10 bidders are

distributed across two auctions as 5 and 5 than when they are distributed as 3 and 7.

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Figure 2: Graphical illustration of the concavity of the expected price given n bidders drawing valuations from a uniform [0,1] distribution.

2.1.2 Generalized Setting: m Products, n Bidders, Bidder Valuations Distributed Log-Concave

Consider again the two scenarios, 1 and 2, but each involving products that are auctioned sequentially.

Let us represent the number of bidders in scenario 1 by set and in scenario 2 by . Assume that, consistent

with the above, the total number of bidders in each scenario is the same, i.e., the sum of the elements in sets

and are equal. Suppose the distribution of bidders across the products is more even in scenario 2 than in

scenario 1. We capture this by making assumptions regarding the majorization of vectors (e.g., see Arnold

1987). In particular, we assume that set A majorizes B, i.e., A Bf . Suppose the bidder valuations are drawn

from a log-concave distribution with cdf . A wide range of commonly-used distributions such as uniform

and normal satisfy the corresponding properties. Then, the following proposition holds:

Proposition 2: The total expected seller revenue is higher in scenario 2 than in scenario 1.

Proof: We compare the sum of the winning prices across the two scenarios. Let and

be the sum of the winning prices in scenarios 1 and 2, respectively. Note that, for any set

, , is a function separable with respect to each product. Thus,

∑ , where is the expected winning price when the number of bidders is .

Following auction theory about ascending auctions, is the 1 -th order statistic from

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Expe

cted

Rev

enue

Number of Bidders

Avg. revenu e given 5 and 5 bidders for products 1 and 2, respectively

Avg. revenu e given 3 and 7 bidders for products 1

and 2, respectively

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draws. Given the log-concavity of the valuation distribution, is a concave function (see appendix

for this proof.) Then, according to Theorem 2.9 in Arnold (1987), ∑ ∑ .

This is the same as the condition . QED

3.0 Empirical Context and Data

3.1 Empirical Context

We test the insights from the analytical model in the context of the wholesale automotive market. The

buyers in this market are used car dealers, who use the market to source vehicles to sell to the retail public. The

sellers are other dealers or institutional sellers such as rental car companies and banks. Sellers use the market

to dispose of large numbers of vehicles at a predictable price. There are multiple intermediaries who broker the

transactions, typically referred to as automotive auction companies.

Sellers transport their vehicles to auction facilities operated by the automotive auction companies. There

are multiple facilities throughout the United States. Vehicles are auctioned sequentially in what is referred to as

a “sales event.” Each sales event consists of multiple vehicles, typically grouped based on seller. For example,

a sales event might consist of 200 vehicles being sold by Avis Rent A Car. Each vehicle in a sales event is

driven -- one at a time -- into a warehouse-type building, where a human auctioneer solicits bids on them. The

bidding for each vehicle takes approximately 45 seconds, after which the next vehicle is auctioned. An entire

sales event can last several hours. For each vehicle, the seller has the option to accept or reject the high bid;

there is no binding reserve price to which the seller must commit prior to the bidding. The position in the

sequence in which each vehicle is auctioned is referred to as the “run number.”

Bidders can bid on each vehicle in one of two ways. First, they can travel to the facilities and place bids in

person. We refer to these as physical bidders and physical bids. Second, they can use a web-based application

to place bids. This application simulcasts live audio and video of the sales event; bidders using this application

place bids in competition with bidders who are physically present at the facility. We refer to these as electronic

bidders and electronic bids. Electronic bidders place bids by clicking a Bid button that continuously refreshes

to show the current asking bid.

3.2 Data and Variables

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Data were provided by an automotive auction company and consist of 19,297,068 vehicles auctioned in

the United States from January 11, 2005 to May 31, 2007 and January 1, 2009 to March 31, 2010. Complete

data for the June 1, 2007 to December 31, 2008 period were not available. For each auctioned vehicle i, the

data consist of the sales event j in which the vehicle was auctioned, the bid log (described below), the vehicle’s

run number (RunNumberi,j), the day the vehicle was auctioned (Dayi,j), which ranges from 1 (Jan. 11, 2005) to

1,906 (March 31, 2010), the vehicle’s mileage (Mileagei,j), and whether the auction resulted in a sale (Soldi,j).

For sold vehicles, the data include the sales price (SalesPricei,j) and a valuation estimate for the vehicle

calculated by the auction company based on prices of similar vehicles sold over the prior 30 days (Valuationi,j).

The vehicles were auctioned in 163,766 sales events, for an average of approximately 118 vehicles per sales

event.

Some institutional detail is necessary to understand the data contained within the bid log. Prior to the

introduction of the webcast channel, individual bids were not recorded; i.e., there was no bid log. Only the

winning bid for each vehicle was recorded, along with the identity of the winning bidder. This changed when

the electronic “webcast” channel was introduced. Each time a physical bidder placed a bid, an auction clerk

would manually register the bid amount in a database. This generated a running bid log for each vehicle as it

was being auctioned. The bid log is displayed in the webcast application to assist electronic bidders with

tracking the bidding activity. Importantly, the auction clerk does not record the identity of physical bidders,

only the bid amounts, unless the bidder was the winning bidder. Thus, it is impossible to tell from the bid log

which bidder placed a physical bid if that bid was not a winning bid. By contrast, when an electronic bidder

places a bid via the webcast channel, this bid is automatically registered into the bid log along with the identity

of the bidder based on the credentials s/he used to log into the webcast channel. Thus, each row of a vehicle’s

bid log contains the bid amount (BidAmount), whether the bid was placed by a physical or electronic bidder

(BidType), and the ID number of the bidder for electronic bids only (BidderID). Table 1 provides a sample bid

log.

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BidAmount BidType BidderID 10,000 P (for physical) 10,100 E (for electronic) 111 10,200 E 222 10,300 P 10,400 P 10,500 P 10,600 E 111

Table 1: Example of the bid log for a vehicle.

3.2.1 Bid Log Variables: For each vehicle, we recorded the sequence of physical and electronic bids as a

string variable (BidPatterni,j.) For example, for the bid log shown in Table 1, BidPatterni,j = “PEEPPPE”. We

recorded the starting bid (StartingBidi,j) and high bid (HighBidi,j) , the difference between Valuationi,j and

StartingBidi,j (ValuationMinusStartingBidi,j), and the bid increment (BidIncrementi,j). We also counted the

number of physical bids (PBidsi,j), the number of electronic bids (EBidsi,j), and the number of electronic

bidders (EBiddersi,j.) We imputed the number of physical bidders (PBiddersi,j) by using the information

contained within the bid pattern and the relationship between the number of electronic bids and electronic

bidders. Imputation was trivial for three cases. If a vehicle had no physical bids (n=49,303), we set PBiddersi,j

= 0. If a vehicle had only one physical bid (n=1,313,520), we set PBiddersi,j = 1. If a vehicle had only two

physical bids that were placed in succession (n=1,145,492; this would include bid patterns such as “PP,”

“PPEE,” and “EEPPEE”), then those bids had to have been placed by two physical bidders, because a single

bidder would not immediately outbid himself. We set PBiddersi,j = 2 for these vehicles. We used the

relationship between electronic bids and electronic bidders to impute the other cases. We estimated this

relationship as follows. First, we constructed TwoConsecutiveEBidsi,j, which is a dummy variable set to 1 for

vehicles in which at least two electronic bids were placed in succession (0 otherwise.) Examples of bid patterns

for vehicles with TwoConsecutiveEBidsi,j = 1 include “PEE” and “PEPEEEEE.” This represents vehicles for

which at least two electronic bidders placed bids. We then regressed EBiddersi,j on EBidsi,j and

TwoConsecutiveEBidsi,j, after removing observations in which: a) EBidsi,j = 0, b) EBidsi,j = 1, and c) EBidsi,j =

2 and TwoConsecutiveEBidsi,j = 1.2 This yielded a sample of n=2,422,176 and ensured that the sample

                                                         2 We used different explanatory variables in the regression, including: a) dummy variables for each level of EBidsi,j, thereby permitting its relationship to EBiddersi,j to be estimated as a fully flexible spline, and b)

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contained the appropriate data for imputing the value of PBiddersi,j for those observations not covered by the

trivial cases. Results using both an OLS and a Poisson specification appear in Table 2. EBidsi,j and

TwoConsecutiveEBidsi,j are both positively and significantly correlated with EBiddersi,j. Because

TwoConsecutiveEBidsi,j is a dummy variable, it effectively shifts the intercept when TwoConsecutiveEBidsi,j =

1. Thus, if two of the electronic bids placed on a vehicle are consecutive, the intercept in the OLS specification

becomes 2.115 (i.e., 1.117 + 0.938 = 2.115). This makes sense given that a vehicle with two consecutive

electronic bids must also have at least two electronic bidders. If none of the electronic bids are consecutive, the

intercept is 0.938. We used the OLS coefficients to impute PBiddersi,j. Specifically, PBiddersi,j = 0.938 +

(1.177 * TwoConsecutivePBidsi,j) + (0.049 * PBidsi,j), where TwoConsecutivePBidsi,j is defined analogously to

TwoConsecutiveEBidsi,j.

A: OLS Specification B: Poisson Specification Coefficient Coefficient

EBidsi,j 0.049 (0.000) *** 0.015 (0.000) *** TwoConsecutiveEBidsi,j 1.177 (0.001) *** 0.739 (0.001) *** Intercept 0.938 (0.001) *** 0.053 (0.000) *** n 2,422,176 2,422,176 R2 0.63 n/a Log Likelihood n/a -2960840.7 Robust standard errors in parentheses. * p < 0.01, ** p < 0.05, * p < 0.10.

Table 2: Results of OLS and Poisson regression of EBiddersi,j on EBidsi,j and TwoConsecutiveEBidsi,j.

4.0 Empirical Analysis and Results

4.1 Vehicle-Level Analysis

Figure 3 shows the average number of physical and electronics bids (first row) and bidders (second row)

by run number for vehicles auctioned in the first quarters of 2005 and 2010. Figure 3 also displays linear

regression trend lines and R2 statistics.

                                                                                                                                                                                       dummy variables corresponding to the number of consecutive electronic bids in each vehicle’s bid pattern. None of these variables improved model fit, so we used the more parsimonious expression.

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Q1 2005 Q1 2010

Figure 3: Average number of total, physical, and electronic bids by run number for the first quarters of 2005 and 2010.

As shown in Figure 3, the average number of physical bids and bidders declines over the sequence, with

the slope of this decline similar in 2010 (regression slope = -0.0078 for bids and -0.0007 for bidders) and in

2005 (regression slope = -0.0079 for bids and -0.0006 for bidders.) The average number of electronic bids

increases over the sequence, with the slope of the increase larger in 2010 (regression slope = 0.0031 for bids

and 0.0012 for bidders) than in 2005 (regression slope = 0.0014 for bids and 0.0007 for bidders.)

This indicates attrition among physical bidders and the converse for electronic bidders. One potential

explanation relates to the transaction cost of waiting for subsequent vehicles in the sequence to be auctioned.

This cost of waiting consists of the opportunity cost of not performing other activities while waiting for

vehicles to be auctioned. This cost is high for many bidders, as time spent away from the dealership may result

Avg. num

ber of bids

Run Number

y = ‐0.0079x + 10.455R² = 0.9146

y = 0.0014x + 0.4682R² = 0.7727

0123456789101112

0 25 50 75 100

125

150

175

200

225

250

275

300

325

350

375

400

425

450

475

500

y = ‐0.0078x + 9.3928R² = 0.8891

y = 0.0031x + 0.6915R² = 0.7961

0123456789101112

0 25 50 75 100

125

150

175

200

225

250

275

300

325

350

375

400

425

450

475

500

Physical BidsElectronic  Bids

Run Number

Avg. num

ber of bidde

rs

Run Number

y = ‐0.0006x + 2.5533R² = 0.9234

y = 0.0007x + 0.1589R² = 0.8813

0

0.5

1

1.5

2

2.5

3

3.5

4

0 25 50 75 100

125

150

175

200

225

250

275

300

325

350

375

400

425

450

475

500

y = ‐0.0007x + 2.5028R² = 0.8939

y = 0.0012x + 0.2269R² = 0.8888

0

0.5

1

1.5

2

2.5

3

3.5

4

0 25 50 75 100

125

150

175

200

225

250

275

300

325

350

375

400

425

450

475

500

PhysicalBiddersElectronic  Bidders

Run Number

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in missed sales opportunities. The cost of waiting is lower for electronic bidders than for physical bidders,

because electronic bidders can perform other tasks at their dealerships during the sales event. Not only are

electronic bidders better able to wait until later in the sequence to bid, but they also have an incentive to do so

to avoid competing with physical bidders who are active at the beginning of the sequence. This could explain

the patterns shown in Figure 3.

There are alternative explanations, including the possibilities that the patterns are idiosyncratic to the first

quarters of 2005 and 2010 and that vehicles that appeal to electronic (physical) bidders are auctioned later

(earlier) in the sequence. The latter possibility is important to consider because prior research has shown that

the electronic channel is often used to purchase low mileage, relatively high-value vehicles whose quality is

predictable and can be adequately represented online (Overby and Jap 2009). We used the following regression

specifications to examine these possibilities.

PBiddersi,j = α + β1*RunNumberi,j + β2*Dayi,j + β3*(RunNumberi,j * Dayi,j) + β4*Mileagei,j + β5*Mileage2

i,j + β6*Valuationi,j + β7*Valuation2i,j + β8* ValuationMinusStartingBidi,j +

β9*BidIncrementi,j + ε.

(1a)

EBiddersi,j = α + β1*RunNumberi,j + β2*Dayi,j + β3*(RunNumberi,j * Dayi,j) + β4*Mileagei,j + β5*Mileage2

i,j + β6*Valuationi,j + β7*Valuation2i,j + β8* ValuationMinusStartingBidi,j +

β9*BidIncrementi,j + ε.

(1b)

Specifications 1a and 1b are identical except for the dependent variables PBiddersi,j and EBiddersi,j. The

interaction between RunNumberi,j and Dayi,j allows the relationship of RunNumberi,j to the dependent variables

to change continuously over time. Mileagei,j and Valuationi,j (and their squares) control for the possibility that

physical or electronic bidders are attracted to vehicles of certain quality and/or value.

ValuationMinusStartingBidi,j controls for the possibility that bidders are attracted to vehicles for which the

auctioneer starts the bidding substantially below market value. BidIncrementi,j controls for the possibility that

smaller increments allow a higher number of bidders to bid. We estimated specifications 1a and 1b

simultaneously using seemingly unrelated regression (“SUR”) and individually using Poisson regression.3 We

scaled Dayi,j, ValuationMinusStartingBidi,j, Mileagei,j, and Valuationi,j by dividing by 1000, 1000, 10000, and

10000, respectively. Results appear in Table 3. Including the variables containing Valuationi,j reduces the

                                                         3 EBiddersi,j is a count variable, but PBiddersi,j is not as a result of the imputation. We rounded PBiddersi,j to the nearest integer when using Poisson regression.

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sample size because Valuationi,j is only recorded for vehicles that sold. For robustness, we also estimated the

models without these variables. Results are qualitatively unchanged and are not reported.

PBiddersi,j EBiddersi,j SUR Poisson SUR Poisson

Coefficient Coefficient Coefficient Coefficient

RunNumberi,j -0.00023 (0.0000)

*** -0.00011 (0.0000)

*** 0.00029 (0.0000)

*** 0.00076 (0.0000)

***

Dayi,j -0.05248 (0.0000)

*** -0.02125 (0.0000)

*** 0.17020 (0.0005)

*** 0.19860 (0.0037)

***

RunNumberi,j * Dayi,j 0.00004 (0.0000)

*** 0.00005 (0.0000)

*** 0.00010 (0.0000)

*** 0.00016 (0.0000)

***

Mileagei,j 0.01997 (0.0000)

*** 0.00986 (0.0000)

*** -0.02890 (0.0001)

*** -0.15006 (0.0003)

***

Mileage2i,j

-0.00029 (0.0000)

*** -0.00015 (0.0000)

*** 0.00039 (0.0000)

*** 0.00148 (0.0000)

***

Valuationi,j 0.00248 (0.0003)

*** 0.01643 (0.0006)

*** 0.19360 (0.0006)

*** 0.57278 (0.0025)

***

Valuation2i,j

-0.00442 (0.0001)

*** -0.00284 (0.0001)

*** -0.01371 (0.0001)

*** -0.07307 (0.0005)

***

ValuationMinusStartingBidi,j 0.05620 (0.0000)

*** 0.02593 (0.0000)

*** 0.03031 (0.0000)

*** 0.06233 (0.0000)

***

BidIncrementi,j -0.00013 (0.0000)

*** -0.00015 (0.0000)

*** -0.00007 (0.0000)

*** -0.00091 (0.0000)

***

Intercept 2.49706 (0.0006)

*** 0.87043 (0.0011)

*** 0.07887 (0.0010)

*** -1.48459 (0.0037)

***

na 11,484,013 11,484,013 11,484,013 11,484,013 R2 (or Pseudo-R2) 0.07 0.07 0.10 0.10 Standard errors in parentheses. * p < 0.01, ** p < 0.05, * p < 0.10. a Sample size is reduced due to the inclusion of variables including Valuationi,j, as discussed in the text.

Table 3: Results of specifications 1a and 1b.

Results indicate that the negative relationship between RunNumberi,j and PBiddersi,j shown in Figure 3

holds after controlling for vehicle quality, the starting bid, and the bid increment. The same is true for the

positive relationship between RunNumberi,j and EBiddersi,j. Both relationships become more positive over

time, although the magnitude of this interaction effect is greater for EBiddersi,j. This increases our confidence

that the physical and electronic bid patterns can be explained by differences in the transaction costs associated

with waiting rather than by other factors.

4.2 Sales Event Level Analysis

The above results show that physical bidders participate more heavily early in the sequence while

electronic bidders participate more heavily later in the sequence. This has the effect of distributing the bidders

more evenly across the sequence. Building on the analytical results presented in Section 2, we next examine

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whether the evenness of the bidder distribution positively impacts average revenues in sales events in our

empirical context.

The unit of analysis here is the sales event. For each sales event j, we constructed several variables.

AvgHighBidj is the mean of the high bids for the vehicles in sales event j. VehiclesAuctionedj is the number of

vehicles auctioned in sales event j. AvgValuationj is the mean of the valuation of the vehicles sold in sales

event j. As above, Dayj is the day on which sales event j occur, scaled by dividing by 1000. TotalBiddersj is the

number of bidders who participated in sales event j. GiniBiddersj is the gini coefficient of the bidder

distribution for sales event j. This measure the (un)evenness of the bidder distribution across the vehicles in

sales event j. Smaller values represent a more even distribution. For example, assume that sales event A

consists of 5 vehicles on which 4, 2, 3, 5, and 2 bidders bid, and that sales event B consists of 5 vehicles on

which received 3, 3, 3, 3, and 4 bidders bid. For sales event A, GiniBidsj = 0.20. For sales event B, GiniBidsj =

0.05. We regressed AvgHighBidj on GiniBiddersj, TotalBiddersj, AvgValuationj, Dayj, VehiclesAuctionedj, and

VehiclesAuctionedj2. Including TotalBiddersj allowed us to determine whether GiniBiddersj had an effect while

holding constant the total number of bidders participating in the sequence. Results appear in Table 4.

Coefficient GiniBiddersj -6289.43 (102.96) *** TotalBiddersj 0.32 (0.09) *** AvgValuationj 0.99 (0.00) *** Dayj 181.09 (7.59) *** VehiclesAuctionedj 6.63 (0.27) *** VehiclesAuctionedj

2 -0.01 (0.00) *** Intercept 180.28 (17.20) *** N 154,742 R2 0.91 Robust standard errors in parentheses. * p < 0.01, ** p < 0.05, * p < 0.10.

Table 4: Results of regressing AvgHighBidj on GiniBiddersj, TotalBiddersj, AvgValuationj, Dayj,

VehiclesAuctionedj, and VehiclesAuctionedj2.

The coefficient for GiniBiddersj is negative and significant, indicating that a more even bid distribution

(i.e., a smaller value for GiniBiddersj) is associated with higher bids. The standard deviation of GiniBiddersj is

0.052; thus, a one standard deviation decrease in GiniBiddersj is associated with a $326 increase in the average

high bid across the sequence.

14  

5.0 Conclusion

Intuition suggests that the key to achieving higher prices in a sequential auction setting is to increase

demand by attracting more bidders. However, this is not strictly required. Our analytical and empirical results

show that redistributing existing bidders more evenly also yields higher prices. We provide evidence in our

empirical context that electronic bidding has facilitated a more even distribution of bidders across the

sequence, theoretically because the electronic channel’s low transaction costs help bidders shift their bids from

early to late in the sequence where competition is lower. Our analysis contributes to the theoretical

understanding of multi-object sequential auctions and transaction costs in online environments and also yields

an alternative strategy for sellers seeking to improve outcomes in sequential auctions without necessarily

having to attract new bidders.

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