9
The Effects of Proxy Bidding and Minimum
Bid Increments within eBay Auctions
ALEX ROGERS, ESTHER DAVID, and NICHOLAS R. JENNINGS
University of Southampton
and
JEREMY SCHIFF
Bar-Ilan University
We present a mathematical model of the eBay auction protocol and perform a detailed analysis of
the effects that the eBay proxy bidding system and the minimum bid increment have on the auction
properties. We first consider the revenue of the auction, and we show analytically that when two
bidders with independent private valuations use the eBay proxy bidding system there exists an
optimal value for the minimum bid increment at which the auctioneer’s revenue is maximized. We
then consider the sequential way in which bids are placed within the auction, and we show ana-
lytically that independent of assumptions regarding the bidders’ valuation distribution or bidding
strategy the number of visible bids placed is related to the logarithm of the number of potential
bidders. Thus, in many cases, it is only a minority of the potential bidders that are able to submit
bids and are visible in the auction bid history (despite the fact that the other hidden bidders are
still effectively competing for the item). Furthermore, we show through simulation that the min-
imum bid increment also introduces an inefficiency to the auction, whereby a bidder who enters
the auction late may find that its valuation is insufficient to allow them to advance the current bid
by the minimum bid increment despite them actually having the highest valuation for the item.
Finally, we use these results to consider appropriate strategies for bidders within real world eBay
auctions. We show that while last-minute bidding (sniping) is an effective strategy against bidders
engaging in incremental bidding (and against those with common values), in general, delaying
bidding is disadvantageous even if delayed bids are sure to be received before the auction closes.
Thus, when several bidders submit last-minute bids, we show that rather than seeking to bid as
late as possible, a bidder should try to be the first sniper to bid (i.e., it should “snipe before the
snipers”).
This research was funded by the ARGUS II DARP (Defence and Aerospace Research Partnership)
and the DIF-DTC project (8.6) on Agent-Based Control. The ARGUS II DARP is a collaborative
project involving BAE SYSTEMS, QinetiQ, Rolls-Royce, Oxford University, and Southampton Uni-
versity, and is funded by the industrial partners together with the EPSRC, Ministry of Defence
(MoD), and Department of Trade and Industry (DTI).
Authors’ addresses: A. Rogers, E. David, and N. R. Jennings, Department of Electronics and
Computer Science, Southampton University, Southampton, SO17 1BJ, U.K; email: {acr, ed,
nrj}@soton.ac.uk; J. Schiff, Department of Mathematics, Bar-Ilan University, Ramat Gan, 5290
Israel; email: Schiff@math.biu.ac.il.
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C
2007 ACM 1559-1131/2007/08-ART9 $5.00 DOI 10.1145/1255438.1255441 http://doi.acm.org/
10.1145/1255438.1255441
ACM Transactions on The Web, Vol. 1, No. 2, Article 9, Publication date: August 2007.
Article 9 / 2
•
A. Rogers et al.
Categories and Subject Descriptors: J.4 [Social and Behavioral Sciences]:—Economics
General Terms: Design, Economics, Theory
Additional Key Words and Phrases: Online auctions, electronic commerce, sniping, proxy bidding,
bid increment
ACM Reference Format:
Rogers, A., David, E., Jennings, N. R., and Schiff, J. 2007. The effects of proxy bidding and minimum
bid increments within eBay auctions. ACM Trans. Web. 1, 2, Article 9 (August 2007), 28 pages. DOI
= 10.1145/1255438.1255441 http://doi.acm.org/10.1145/1255438.1255441
1. INTRODUCTION
The growth of the Web has ensured that electronic commerce, once the preserve
of large corporations, is now an everyday activity for millions of private indi-
viduals and small businesses. eBay, the world’s most popular online auction
house, represents a canonical example of this phenomenon. Through a simple
and intuitive Web interface, it allows buyers and sellers to come together within
a worldwide virtual market, and its popularity is evidence of both its successful
design and the public demand for these services
1
.
Described briefly, the eBay auction protocol is a variation of an ascending
price auction with a minimum bid increment and a fixed closing time. Buyers
interested in bidding within an auction do not submit bids directly. Rather, they
must use a proxy bidding system that requires that they specify the maximum
amount that they are willing bid (with the constraint that this amount must
exceed the current auction price plus a minimum bid increment, d ). The proxy
bidding system then automatically submits bids on their behalf, and the eBay
protocol guarantees that the bidder who has entered the highest amount wins
the item, but pays no more than the amount entered by the second highest
bidder plus the minimum bid increment. Indeed, eBay recommends that bidders
simply enter the maximum amount that they are willing to pay for the item, and
leave the proxy bidding system to perform all the resulting bidding (see eBay
help: Bid increments), available online at http://pages.ebay.com/help/buy/
bid-increments.html.
While eBay has been the subject of significant academic research, the major-
ity of this work has been limited to phenomenology (observing what happens in
actual auctions and supplying qualitative explanations of the observed effects).
As such, it has often been stated that the eBay auction protocol behaves as a
second price auction whereby the expected auction revenue is equal to the sec-
ond highest bidder’s valuation plus the minimum bid increment [Bapna 2003;
Ockenfels and Roth 2006]. However, this assumption apparently contradicts
previous results (including our recent work) that shows that, whenever the
bids that may be submitted to the auctioneer are restricted to certain discrete
levels, then the auction generates less revenue than the second highest bid-
ders’ valuation [Rothkopf and Harstad 1994; David et al. 2007]. To resolve this
contradiction, in this article we construct a mathematical model of the eBay
1
In 2006, eBay listed more than 2.2 billion items and generated gross merchandise sales of $52.2
billion [eBay 2006].
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auction protocol. Using this model, we carry out, for the first time, a detailed
analysis of how the minimum bid increment and the eBay proxy bidding system
affect the properties of the eBay auction.
First, we consider the revenue of an eBay auction, and we calculate analyti-
cally the expected revenue in the case that two bidders with private valuation
use the eBay proxy bidding system (and, as suggested by eBay, these bidders en-
ter their valuations as the maximum amount that they are willing to pay for the
item as soon as they become aware of the auction). We show that, in this case, the
revenue of the auction is dependent on the value of the minimum bid increment,
and that there exists an optimal value at which this revenue is maximized. At
this optimal value, the revenue does indeed exceed the second highest bidder’s
valuation (but by an amount that is less than the minimum bid increment). To
fully understand this effect, we compare it to the case where the two bidders
employ a pedestrian bidding strategy (i.e., each time they are outbid, they in-
crease the current bid by the minimum bid increment). We again analytically
calculate the expected revenue of the auction, and we show that, in this case,
the auction generates a revenue that is less than the valuation of the second
highest bidder. Comparing these two results indicates that it is the interaction
of the eBay proxy bidding system and the minimum bid increment, and not the
minimum bid increment alone, that is responsible for the increased auction rev-
enue. We confirm this result for larger numbers of bidders through simulation.
Second, we consider the fact that, unlike conventional auctions, an eBay
auction does not commence with all the bidders being present. Instead, the
bidders arrive in a random order that is dependent on the time at which they
first become aware of the auction’s existence (typically through searching the
eBay Web site). We show through simulation that the proxy bids of the early
bidders cause the current price of the auction to increase rapidly, and hence
bidders who enter the auction late may find that their valuation is insufficient
to allow them to advance the current bid by the minimum bid increment. Thus
the number of bids that are observed in an eBay auction (and recorded in the
auction history) can be substantially smaller than the number of bidders who
would have liked to place a bid. We calculate an analytic expression for this
relationship in the limiting case that the minimum bid increment d = 0 and
we show that the number of bids received is approximately proportional to the
logarithm of the number of bidders who would have liked to place a bid. This
analysis is attractive as it is independent of any assumptions regarding the dis-
tribution of bids placed within the auction (or the valuations of the bidders). It
also represents the maximum number of bids that may be observed; increasing
either the minimum bid increment or the starting bid only reduces the chance
that a bidder will have a valuation that is sufficient to allow them to place a
bid. This result suggests that care must be taken when attempting to infer the
behavior of bidders through observation of the bid history, and thus empirical
studies that neglect this effect may be incorporating systematic errors. In addi-
tion, it shows that the minimum bid increment introduces an inefficiency in to
the auction, in that the bidder with the highest valuation does not always win.
We show through simulation that, the earlier bidders submit their valuation to
the eBay proxy bidding system, the greater their chance of winning the auction.
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A. Rogers et al.
This last observation appears to contradict the extensive literature that sug-
gests that last minute bidding or sniping is an effective strategy within an eBay
auction [Roth and Ockenfels 2003]. Thus we consider how these observations
determine the bidding strategy that a bidder should adopt within a real eBay
auction. Our results show that, while late bidding is effective in the case of com-
mon values or when other bidders engage in incremental bidding (i.e., rather
than using the proxy bidding system as eBay intended, they repeatedly increase
the amount that they are willing to pay whenever they are outbid
2
), in general,
due to the inefficiency discussed above, bidding late is disadvantageous. Thus,
when several bidders snipe, we show that, rather than seeking to place the last
bid, bidders should try to be the first sniper to submit their bid (i.e., they should
“snipe before the snipers”).
These results significantly advance the understanding of the eBay auction
protocol. They can be used by institutions attempting to construct similar on-
line auctions, since our analysis indicates how the value of the minimum bid
increment should be chosen to maximize the revenue that the auction gen-
erates. In addition, our observation that the number of bids observed within
an eBay auction is significantly smaller than the number of bidders who at-
tempted to place a bid (and the related analytical calculation) is extremely
important for researchers performing empirical studies of eBay auctions using
the auction bid history. Finally, our insights into the timing of snipe bids can be
used by bidders in real eBay auctions to maximize their chance of winning the
item.
The remainder of the article is organized as follows. Section 2 presents re-
lated work. Section 3 describes the eBay protocol and the proxy bidding system
in more detail and Section 4 presents our analysis of the revenue of the eBay
auction. Section 5 considers how the sequential update of the current bid affects
the auction properties and Section 6 considers the resulting bidding behavior.
Finally, we conclude and discuss future work in Section 7.
2. RELATED WORK
The growth of Web-based electronic commerce has initiated much research
into the design of novel mechanisms for online auctions [Fontoura et al. 2002],
and also effective bidding strategies for automated bidding agents [Guo 2002;
Dumas et al. 2002; Anthony and Jennings 2003]. However, much less research
has addressed the specific implementation details of auction mechanisms that
have proved to be popular and effective in real online settings (such as eBay).
What work does exist in this area has tended to focus on observing bidding
within eBay auctions (typically through an analysis of the bid history that eBay
makes available on its Web site) and then attempting to form qualitative expla-
nations of the observed effects (see Bajari and Hortacsu (2003) and Ockenfells
et al. (2007) for reviews of this work). Typical work within this area has con-
sidered how the closing price of the auction is affected by the various auction
settings, such as the starting bid, the reserve price, the shipping costs, and the
2
Pedestrian bidding, where the bid is raised by the minimum bid increment each time, is an extreme
form of incremental bidding.
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Proxy Bidding and Minimum Bid Increments within eBay Auctions
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auction duration [Lucking-Reiley 2000; Bajari and Hortacsu 2003; Hossain and
Morgan 2006; Gerding et al. 2007].
In addition, other research has considered the behavior of bidders within
these auctions [Shah et al. 2003; Bapna 2003; Roth and Ockenfels 2003]. A
key observation in this respect has been the fact that in many eBay auctions
a significant number of bids are placed in the last few minutes before the auc-
tion closes. Many explanations have been proposed for this last-minute bidding
and it is commonly believed that sniping is the best response to na
¨
ive bidders
engaging in incremental bidding [Roth and Ockenfels 2003]. It has also been
suggested that, in common-value auctions, such last-minute bidding prevents
the disclosure of information, and thus prevents other bidders from responding
with an updated bid [Bajari and Hortacsu 2003].
3
Furthermore, a number of
researchers have suggested that last-minute bidding may also protect a bid-
der from a form of shill bidding, commonly known as maximum-bid fishing,
whereby the seller submits small incremental shill bids to expose the value of
the highest bid, and thus forces the highest bidder to pay their full bid [Barbaro
and Bracht 2004; Engelberg and Williams 2005].
However, despite this extensive body of research, little work has considered
the specific details of the eBay auction protocol, such as the operation of the eBay
proxy bidding system, the use of a minimum bid increment,
4
and the fact that
bidders generally arrive within the eBay auction after bidding has commenced.
Instead, the eBay auction protocol is commonly assumed to behave as a second-
price auction, and it is often stated that the expected auction revenue is equal to
the second highest bidders’ valuation plus the minimum bid increment [Bapna
2003; Ockenfels and Roth 2006]. In the work that we present here, we show that
these assumptions are too simplistic. By constructing a mathematical model
of the eBay auction protocol and then performing a detailed analysis of its
properties, we are able to gain significant and novel insights into the operation
of the eBay auction protocol that can be used by designers of online auctions,
by other researchers performing empirical studies of real eBay auctions and
finally, by bidders within these auctions.
3. THE EBAY AUCTION
We consider the single-item eBay auction protocol
5
and, to explain its operation,
we describe the sequence of events as the bidding proceeds. The auction process
3
Last-minute bidding has also been shown to be an equilibrium strategy in private-valuation auc-
tions where there is some small probability that last-minute bids are dropped [Ockenfels and Roth
2006].
4
Other work has considered the effect of the minimum bid increment within ascending-price
discrete-bid English auctions [Rothkopf and Harstad 1994; David et al 2005, 2007]. However, the
additional details of the eBay auction make it difficult to apply these general results in this specific
case, and this motivates the more detailed study that we present here.
5
eBay also offers a multiple-item auction format that they often refer to as a Dutch auction
(see eBay help: Multiple item auction, avalaible online at http://pages.ebay.com/help/buy/
buyer-multiple.html). This auction format is actually quite different from the descending price
auctions used in the Dutch flower markets that are commonly referred to as Dutch auctions within
the academic literature of auctions. In the multiple-item eBay auction, bidders specify the quantity
of items that they wish to purchase and the maximum price that they are willing to pay per item.
ACM Transactions on The Web, Vol. 1, No. 2, Article 9, Publication date: August 2007.
![Fig. 4. Simulation results showing the dependence of the expected auction revenue on the minimum bid increment for both the eBay proxy bidding system and pedestrian bidding. Results are for 20 bidders with valuations drawn uniformly on [0,1] and s = 0. Results are averaged over 500,000 auctions.](/figures/fig-4-simulation-results-showing-the-dependence-of-the-10wbjhkt.webp)
![Fig. 8. Simulation results showing the probability of each bidder winning the auction depending on the order in which they bid. There are 20 bidders with valuations drawn uniformly on [0,1] and s = 0. Results are averaged over 107 auctions.](/figures/fig-8-simulation-results-showing-the-probability-of-each-24hl2t62.webp)
![Fig. 9. Simulation results showing (a) the probability of winning, and (b) the expected profit, when the bidders with the highest and second highest valuations bid first, at a random time, and last. There are 20 bidders with valuations drawn uniformly on [0,1] and s = 0. Results are averaged over 107 auctions.](/figures/fig-9-simulation-results-showing-a-the-probability-of-3v3jt7gk.webp)
