This note concerns bidding in a hybrid first-price and second-price auction. The winning bidder sometimes pays his bid and sometimes pays an amount determined by the next highest bid. In internet auctions where bidders wait until the end of the auction to bid the auction reduces to a sealed-bid auction and the bid function we derive may be relevant in such cases.sniper bidding, bid increments

Richard Cox

English

Research Papers in Economics

Bid Increments in Second-Price Sealed Bid
Auctions1
Abstract: This note concerns bidding in a hybrid ﬁrst-price and second-
price auction. The winning bidder sometimes pays his bid and sometimes
pays an amount determined by the next highest bid. In internet auctions
where bidders wait until the end of the auction to bid the auction reduces to a
sealed-bid auction and the bid function we derive may be relevant in such cases.
JEL classiﬁcation code: D44
This version: June 23, 2005
1Richard Guy Cox, University of Arkansas at Little Rock, Department of Economics and
Finance, Little Rock, AR 72204, USA; email: rgcox@ualr.edu; ph: (501) 569-8875.
11. Motivation
Recently some economists have focused their attention on explaining the
phenomenon of sniper bidding on eBay (see for example Ockenfels and Roth
2003, Roth and Ockenfels 2002, Wang 2003, and Bajari and Horta¸ csu 2004).
While I do not address the issue of sniper bidding here I note that when
bidders wait until the ﬁnal moments of an auction to bid, the ascending auction
becomes a sealed-bid auction. Simply, when bidders snipe there are no further
opportunities to respond to previous bids. This suggests that sniper bids
may deviate from the bid-your-valuation bids suggested by eBay as the ideal
strategy. However, to see this we must ﬁrst establish that the equilibrium
bids in a sealed-bid auction with eBay’s price determination rules are below
the bidders’ valuations. I demonstrate this for an independent private values
model.
In the pure second-price sealed-bid auction the bid-your-valuation strategy
is an equilibrium. Under the rules at eBay sometimes the winner pays his
bid and sometimes he pays an amount determined by the second highest bid.
The actual amount depends on the speciﬁc bidding increment and how close
the winning bid and next highest bid are to each other. Accordingly, the bid
increment plays an important role in the result. In fact, I provide an example
in which the equilibrium bids are below the bidders’ corresponding valuations
and that this distance is between the bids and valuations is greater than the
bid increment. Accordingly, if bidders submit comparable early bids then there
is the possibility that they may submit new bids which implies that waiting
until the end of the auction to bid could be preferable for bidders when all bid-
ders wait. Chwe (1989) addresses the question of bid increments in ﬁrst-price
auctions where bids are drawn from a set. I am not aware of results for the
ﬁrst- second-price hybrid case where bids are drawn from a continuum. The
model and results follow.
2. Model
There is an auction for a single item. There are n ≥ 2 risk neutral bidders.
Let vi be the valuation of bidder i ∈ {1,...,n}. The valuations are indepen-
dently and identically distributed and represent draws from the cumulative
distribution function F with support [v,v]. The cdf F is twice diﬀerentiable
and strictly increasing over its support without any atoms. The realized val-
uations are private knowledge while F is common knowledge.
2The auctioneer establishes a reservation price or screening level r. Bidder
i’s bid, Bi must exceed r; otherwise, the bidder does not participate. All bid-
ders simultaneously submit their bids. Without loss of generality we order the
bids Bn > Bn−1 > ... > B1. If only one bidder participates then Bn−1 = r.
If we can ﬁnd a monotonic bid function, B : supp F → supp F, then the
probability that any 2 bids are equal when F has no atoms is 0 and I ignore
the possibility of a tie. The high bidder, i = n, receives the object after the
bids are submitted. Let ∆ ∈ R+ be the bidding increment. The payoﬀ to
bidder i is vi − min{Bn,Bn−1 + ∆} if he wins, i = n, and 0 otherwise. In
words, the high bidder will sometimes pay his own bid and will sometimes pay
the second highest bid plus ∆. The transaction price depends on the bidder’s
bid with some positive probability. In this way the standard argument that
in a second-price auction a bidder gains nothing by lowering his bid does not
apply here.
3. Results
Building on the results of Riley and Samuelson (1981) we can derive a
strictly increasing equilibrium bid function which is applicable to both ﬁrst-
and second-price auctions. For r < v we specify that a bidder with valuation
v bids his valuation.
Riley and Samuelson (1981) show that in equilibrium the expected payment
E(v) must satisfy
E
0(vi) = vi(n − 1)F(vi)
n−1F
0(vi).
The solution to the diﬀerential equation is
E(vi) = viF(vi)
n−1 −
Z vi
r
F(x)
n−1dx,
where vi ≥ r.
In this auction we must calculate the expected payment in two separate
cases. Let ˆ v satisfy B(ˆ v) = max{r + ∆,v + ∆}. In the ﬁrst case vi < ˆ v. This
auction is a ﬁrst-price auction from this bidder’s point of view. The expected
payment is
BiG(B)
n−1,
where, G is the distribution of equilibrium bids. Note that a strictly increasing
bid function implies G(B(v)) = F(v). In this case the bid function is just
B(v ≤ ˆ v) = vi −
1
F(vi)n−1
Z vi
r
F(x)
n−1dx.
3The second case to consider is that of high signal bidders, where vi > ˆ v. In
this case the expected payment can be written by recognizing what happens
on two intervals. Let Bn−1:n denote the the second highest bid when there
are n bids. On the interval Bn−1:n ∈ [r,B − ∆] the bidder pays the expected
value of the second highest bidder’s bid plus the bid increment ∆, while on
the interval Bn−1:n ∈ (B − ∆,B) the winner pays his bid. Thus we have
E(v) =BG(B)
n−1

1 −
G(B − ∆)n−1
G(B)n−1

+
Z B−∆
r
(x + ∆)(n − 1)G(x)
n−2G
0(x)dx
=BG(B)
n−1 − BG(B − ∆)
n−1
+ (B − ∆)G(B − ∆)
n−1 −
Z B−∆
r
G(x)
n−1dx + ∆G(B − ∆)
n−1
=BG(B)
n−1 −
Z B−∆
r
G(x)
n−1dx.
Notice that the expectation using the pdf for the truncated order statistic
where i = n − 1 times the probability of winning gives the same expression
as the expectation using the pdf for the order statistic where i = n which
we use above. (The distribution of the order statistic is found in Arnold,
Balakrishnan, and Nagaraja 1992).
Now we can solve for the bid function for the high valuation bidders and
obtain
B(v > ˆ v) =v −
1
F(v)n−1
Z v
r
F(x)
n−1dx −
Z B−∆
r
G(x)
n−1dx

, (1)
where we make use of the fact that G(B) = F(v),∀v ≥ r.
Because B is an element of a compact convex subset of R and the bid
function is continuous then it has a ﬁxed point. Actual solutions could be
obtained by using a Newton-Raphson method and piece-wise evaluation of G
beginning at the distribution for low signal bidders. However, in general G is
unknown and we must obtain solutions by construction.
The bid function itself is a straightforward extension of the results of Riley
and Samuelson (1981). However, we can now specify how B changes as ∆
changes and provide an example of bids that are “far” from bidders’ true
valuations as a relevant argument for sniper bidding.
By the mean value theorem we can rewrite (1) as
4B = v −
1
F(v)n−1
Z v
r
F(x)
n−1dx + G(c)
n−1(B − ∆ − r),
where c ∈ [r,B − ∆] is the solution to the mean value theorem. Then, rear-
ranging we have,
B =
1
1 − G(c)n−1

v −
1
F(v)n−1
Z v
r
F(x)
n−1dx + G(c)
n−1(−∆ − r)

,
and the derivative is,
∂B
∂∆
= −
G(c)n−1
1 − G(c)n−1.
Now consider a simple example where n = 2 and v is uniform on [0,1].
Table 1 provides bids for several bidder valuations and bid increments. We
observe that some bids are less than the bidder’s valuation by an amount
greater than ∆. This suggests that if bidders do not wait until the end of
the auction to bid, and bid according to (1), then they face the possibility of
submitting larger bids later in the auction. If everyone waits until the end of
the auction to bid, then equilibrium bids will be lower than otherwise.
Table 1. Bids
∆ v = .2 v = .25 v = .3
0 .2000 .2500 .3000
.01 .1900 .2400 .2900
.05 .1355 .2078 .2566
.1 .1000 .1282 .1634
.15 .1000 .1250 .1500
Then in this simple setting the ﬁrst and second order statistics for the bids
when ∆ = .1 are .5690 and .1921 respectively. The expected revenue is .2921
while the sealed bid auction with ∆ = 0 has expected revenue of .3333. If an
English auction proceeds with .1 unit jumps and bidders alternate then half
the time the high valuation bidder starts the auction and half the time the low
valuation bidder begins the auction and the expected revenue is .35. Because
the distribution G is unknown, I cannot make more general observations on
revenue. The result suggests that converting the ascending auction with a ﬁxed
closing time to a sealed-bid auction by waiting until the last moment to bid
implies that equilibrium bids will fall below the bidders’ true valuations with-
out allowing the opportunity for new feasible bids. As such, equilibrium bids
would be lower than when bidders apply eBay’s suggested bid-your-valuation
5strategy. However, this does not imply that waiting until the last moment to
bid is an equilibrium.
References
Arnold, Barry C., N. Balakrishnan, and H. N. Nagaraja, 1992, A First
Course in Order Statistics. John Wiley & Sons, New York.
Bajari, Patrick and Ali Horta¸ csu, 2004, Economic Insights from Internet
Auctions. Journal of Economic Literature 42:457-486.
Chwe, Michael Suk-Young, 1989, The Discrete Bid First Auction. Eco-
nomics Letters 31:303-306.
Ockenfels, Axel and Alvin E. Roth, 2003, Late and Multiple Bidding in
Second Price Internet Auctions: Theory and Evidence Concerning Diﬀerent
Rules for Ending an Auction. CESifo working paper no. 992 (to appear in
Games and Economic Behavior).
Riley, John G. and William F. Samuelson, 1981, Optimal Auctions. Amer-
ican Economic Review 71:381-392.
Roth, Alvin E. and Axel Ockenfels, 2002, Last-Minute Bidding and the
Rules for Ending Second-Price Auctions: Evidence from eBay and Amazon
Auctions on the Internet. American Economic Review 92:1093-1103.
Wang, Joseph Tao-yi, 2003, Is Last Minute Bidding Bad? UCLA working
paper.
6

A First Course in Order Statistics.

Economic Insights from Internet Auctions.

Is Last Minute Bidding Bad? UCLA working paper.

Last-Minute Bidding and the Rules for Ending Second-Price Auctions: Evidence from eBay and Amazon Auctions on the Internet.

Late and Multiple Bidding in Second Price Internet Auctions: Theory and Evidence Concerning Diﬀerent Rules for Ending an Auction. CESifo working paper no. 992 (to appear in Games and Economic Behavior).

The Discrete Bid First Auction.

Bid Increments in Second-Price Sealed Bid Auctions

http://128.118.178.162/eps/mic/papers/0506/0506008.pdf

Bid Increments in Second-Price Sealed Bid Auctions

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