Quantifying Bid Shading in Online Auctions via A Functional Approach
Galit Shmueli, Wolfgang Jank
University of Maryland
Ravi Bapna
University of Connecticut
1. Bid shading and the winner’s curse
Winner’s curse is a phenomenon that occurs in auctions for common-value items. These are auctions where
the item being auctioned is valued the same by each participant, although the participants may not know
the precise value1 . Since each bid is assumed to be a realization from the value distribution, the average
bid is assumed to be the estimate of the common value. For this reason, if bidders bid their estimate then
the highest bid is an over-estimate of the common value, leading to the “Winner’s Curse” phenomenon.
To avoid this curse, bidders in such auctions must “shade” their bid, i.e. shrink their estimate. According
to auction theory, the amount of shrinkage depends on the number of bidders in the auction: the more
bidders, the more shading is required. Using this reasoning, if we see bid shading taking place, this can be
indicative of a common-value component. If there is no bid shading, this can indicate either a private-value
item or else a common-value auction that experiences the winner’s curse. But more importantly, from the
degree of bid shading we can learn about how bidders are behaving regardless of our knowledge of whether
the item is common or private value.
2. The winner’s curse test and its limitations
The winner’s curse phenomenon has been investigated in the context of real online auction data by several
authors. They look for the presence of bid shading which would be indicative of a common-value auction.
This is done by determining whether the change in bid magnitude is affected by the number of bidders.
Had the series of bids during an auction consisted of independent events, then we could have treated the
pairs {bids,# Bidders} as a random sample and examine the change in bid magnitude as a function of
the number of bidders. A linear relationship would indicate that the bid increase is equally affected by an
additional bidder irrespective of the number of bidders (i.e., whether the increase is from 2 to 3 bidders
or from 10 to 11 bidders, the bid increase is similar). This is what we would expect to see in a common
value auction with no bid shading. Alternatively, in the presence of bid shading we would expect to see a
nonlinear relationship, where the rate of bid increase per additional bidder decreases as a function of the
number of bidders.
This is essentially the setting that Bajari & Hortacsu (2003)2 use, relying on Milgrom & Weber’s (1982)3
“winner’s curse test” which compares the average bid in auctions with different numbers of bidders. Like
the above setting, they treat the set of all bids as an iid sample. They consider the bid magnitude as their
response variable and then make the restricting assumption that the relationship of bid magnitude and
number of bidders is linear, whether or not there is bid shading. They then fit a linear regression of bids
to the number of bidders, and find a negative correlation between the two. They take this as suggestive
evidence for the presence of bid shading, and thus a common-value auction.
However, a negative correlation between bid magnitudes and the number of bidders can be explained by
other factors: First, there are usually less auctions with many bidders, which means fewer data points with
many bidders. The regression estimates are therefore susceptible to influential points at the high number of
bidders range. Even a single auction with many bidders and a relatively low last bid can lead to a negative
coefficient. Second, if the underlying relationship between the rate of bid change and the number of bidders
is linear (y 0 = β0 + β1 x) when there is no bid shading, then the relationship between the bid magnitude and
number of bidders is expected to be quadratic: y ∝ β0 x + β1 x2 . In fact, a scatterplot of bids vs. number of
bidders for a sample of 97 Cartier watch auctions does show some curvature (Figure 1, top left). Finding
a negative slope coefficient in a linear regression of bids vs. number of bidders is therefore not necessarily
1 from
“Common value auction” in Wikipedia.
P and Hortacsu, A (2003) “The Winner’s Curse, Reserve Prices and Endogenous Entry: Empirical Insights from
eBay Auctions”, The Rand Journal of Economics 3(2), 329-355.
3 Milgrom, P R and Weber, R J (1982), “A Theory of Auctions and Competitive Bidding”, Econometrica, 50, 10891122.
2 Bajari,
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Figure 1: Top: Bids vs number of bidders for 97 Cartier wristwatch auctions (left) and auction curves
(right). Bottom: bid velocity per number of bidders before (left) and after (right) factoring out bid timing.
indicative of no bid shading. Instead, we should look at the rate of bid change, or bid velocity.And third,
the assumption that the price at time t is not related to the price at time t0 , (t0 < t) is not reasonable.
In fact, the underlying assumption in common-value item auctions is that bidders derive information from
the other signals, and in our case this information is manifested by the observable previous bids.
3. Quantifying the curse: a functional approach
We introduce a new method that directly measures the degree of bid shading and overcomes the shortcomings above. The method is based on a functional data analytic approach, which is capable of capturing
the dynamics that take place during an auction. The functional approach uses the entire set of bids, not
just those in the last minutes of the auction; it maintains the relationship between the bid amount and bid
timing; and it captures the temporal dependencies between bids in the same auction.
Using a functional approach, we treat the set of {bid,# bidders} pairs from an auction as a time series
and use a curve to represent it (Figure 1, top right). We then treat the set of auction curves as our sample.
The first derivative of these curves tells us about the price increase per additional bidder (Figure 1, bottom
left). This is almost a reasonable measure of bid shading, except that it is biased by the bid timing: in
closed-ended online auctions there tends to be a surge of bidding towards the auction end, and sometimes
some early bidding as well. This means that the bid-increase-per-bidder is likely to vary during different
times of the auction. Bajari & Hortacsu (2003) addressed this by using data only from the last few minutes
of the auction. However, with our method we can use the entire bid data. Our solution is to first factor
out the bid time and then fit curves and estimate derivatives. The factoring out of bid timing is done by
fitting a (linear or nonlinear) regression between the bid amounts and bid times. The residuals from this
regression are then free of bid timing effects, and are used to obtain curves and curve derivatives (Figure
1, bottom right).
Our Cartier watch auction sample shows that before removing the bid timing the average bid velocity
is slightly positive and constant in the number of bidders, indicative of no shading (Figure 1, bottom left).
However, the average velocity changes, once bid timing is factored out, to a constant zero bid velocity
(Figure 1, bottom right) again indicative of no shading (and therefore presence of winner’s curse if these
are common value items). Thus, the initial positive velocity was reflecting the heat of the auction rather
than the effect of the number of bidders, which in other instances might conceal the presence of bid shading.
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