The Effect of Corruption on Bidding Behavior in First-Price Auctions

The E?ect of Corruption on Bidding Behavior in
First-Price Auctions
Leandro Arozamena? and Federico Weinschelbaum†
April 22, 2004.
Very preliminary version
Abstract
Most of the literature on auctions assumes that the auctioneer owns the object on
sale. However most auctions are organized and run by an agent of the owner. This
separation generates the possibility of corruption. We analyze the e?ect of a particular
form of corruption on bidding behavior in a single-object, private-value auction with
risk-neutral bidders. Bidders believe that, with a certain probability, the auctioneer
has reached an agreement with one of the bidders by which, after receiving all bids, (i)
she will reveal to that bidder all of her rivals’ bids, and (ii) she will allow that bidder to
change her original bid upwards or downwards. We study how an honest bidder would
adjust her bidding behavior when facing this type of collusion between a dishonest
rival and the auctioneer. In a ?rst price auction, an honest bidder can become more
or less aggressive than she would be without corruption, or her behavior can remain
unchanged. We identify su?cient conditions for each of the three possibilities. We also
examine the extent to which the most commonly used distributions satisfy each of the
three conditions, and we provide a preliminary assessment of the e?ect of corruption on
e?ciency and revenue. In fact, we provide an example where the existence of corruption
causes the seller’s revenue to grow.
?Universidad Torcuato Di Tella, Business School and Economics Dept. E-mail: larozamena@utdt.edu .
†Universidad de San Andrés, Economics Department. E-mail: fweinsch@udesa.edu.ar .
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Keywords: Auctions; Corruption.
JEL classi?cation: C72, D44
1
Introduction
Most of the literature on auctions assumes that the auctioneer owns the object on sale.
However, most auctions are organized and run by an agent of the owner. This separation
between the owner and the auctioneer creates the scope for corruption to appear. The
auctioneer may be tempted to enter into corrupt agreements with one of the bidders to tilt
the auction in her favor.
In this paper, we examine a particular form of corruption in a single-object, private-value
auction with risk-neutral bidders. We focus on the case where corrupt dealings between the
auctioneer and any bidder consist of revealing information on how much other participants
in the auction have bid. Speci?cally, we assume that bidders believe that the auctioneer may
have reached an agreement with one of them by which, after receiving all bids, (i) she will
reveal to that bidder all of her rivals’ bids, and (ii) she will allow that bidder to change her
original bid upwards or downwards if she wishes to do so. We are particularly interested in
the e?ect of possible existence of this form of corruption on how honest bidders behave in
the auction. That is, we want to ascertain whether the fact that her bid will, with a certain
probability, be revealed to a rival makes a bidder more or less aggressive as compared to the
case where corruption is absent.
In a second-price auction, since bidding her own valuation is a weakly dominant strategy
for every bidder, this form of corruption has no e?ect on bidding behavior or on the auction’s
result. In a ?rst-price auction, however, the situation is more complex. We show below that
an honest bidder can become more or less aggressive than she would be without corruption,
or her behavior can remain unchanged. We provide su?cient conditions for each of the three
possibilities. Namely, if F is the cumulative distribution function of a bidder’s valuation for
the object being auctioned and f is the corresponding density, if the ratio F/f is strictly convex
(respectively, strictly concave, linear) in the valuation, the honest bidder will become more
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aggressive (respectively, less aggressive, equally aggressive) with corruption. Furthermore,
we establish the extent to which the most commonly used distributions satisfy one of those
conditions.
Other papers have dealt with similar forms of corruption. In particular, Jones and
Menezes (1995), Lengwiler and Wolfstetter (2000), Burguet and Perry (2002) and Menezes
and Monteiro (2003) consider cases where a corrupt arrangement, just as in this paper,
implies revealing to one of the bidders what her rivals have bid. However, their analyses di?er
from the one we present in several respects. In Jones and Menezes (1995), bidders are not
aware of the possibility of corruption when choosing their bids, so bidding behavior remains
unaltered by assumption.1 Lengwiler and Wolfstetter (2000) and Menezes and Monteiro
(2003) consider situations where the auctioneer approaches the winning bidder o?ering the
chance to lower her bid (while still winning the auction) in exchange for a bribe. In this
paper, on the contrary, who the auctioneer may conspire with is independent of the auction’s
result. That is, their agreement is reached before the auction takes place. In addition, the
favored bidder is allowed to raise or lower her bid according to her interests. Burguet and
Perry (2002) is closest to our analysis. They study several variants as to the exact form
corruption may take. They are particularly interested in the e?ect of the bargaining game
between the auctioneer and the dishonest bidder on the auction’s result. Only one of the
variants they deal with involves allowing the favored bidder to freely revise her bid upwards
or downwards, but, as in all the remaining cases, they only focus on the two-bidder case
when corruption is certain to all parties. Furthermore, they do not concentrate on the
e?ect corruption has on equilibrium bids. Here, we allow for the existence of corruption to
be uncertain, extend the analysis to the general, N -bidder case and are able to provide a
fuller characterization of the e?ect of corruption on bidding behavior. Finally, Compte et
al. (forthcoming), consider the situation where the auctioneer reveals the winning bid to all
participants and allows them to compete for the chance to resubmit their bids. Their focus,
1 In addition, they consider a setting where bidders draw their valuations from uniform distributions. We
prove below that, for such distributions, even if bidders were aware of the possibility of corruption their
behavior would remain unaltered.
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then, is on bribing competition and its e?ects.
While our interest here limits to the case where a single-dimensional object is auctioned,
another related strand in the literature focuses on the possibility of corruption in multi-
dimensional procurement auctions. Speci?cally, in addition to the price, the object being
procured has a quality dimension that a?ects the procurer’s welfare. The procurer delegates
the assessment of quality on an agent, and the scope for corruption is thereby created. Celen-
tani and Ganuza (2002) and Burguet and Che (2004) examine di?erent forms of corruption
in that procurement environment.
In Section 2 below, we present the auctioning context and provide su?cient conditions to
characterize the e?ect of corruption on an honest bidder’s behavior. First, we examine the
case where the presence of corruption is certain to all parties and prove the su?ciency of the
proposed conditions. Then, we show that su?ciency extends to the more general case with
uncertain corruption. Finally, we try to ascertain the extent to which standard distributions
satisfy each of the su?cient conditions. In Section 3, we provide a preliminary assessment
of the e?ect of corruption on e?ciency and revenue. In fact, we provide an example where
the existence of corruption causes the seller’s revenue to grow.
2
The Model
The owner of a single, indivisible object is selling it through an auction organized and run by
an agent of hers. We assume that there are two bidders2 in the auction whose valuations vi
(i = 1, 2) for the object are distributed identically and independently according to the c.d.f.
F with support on the interval [0, 1]3 and a density f that is positive on the whole support.
The context is, then, one of independent private values. Both bidders are risk neutral. We
will focus below on sealed-bid auctions with no reserve prices, and assume that the c.d.f.
2 See the Appendix for the n-bidder case (to be written).
3 This is done purely for ease of exposition and without loss of generality. We could use any interval on
the real line instead.
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F is logconcave.4 For future use, let ?(vi) = F (vi)/f (vi). Note that the logconcavity of F
means that ? is increasing.
Given the fact that the owner of the object being sold and the auctioneer are not the
same, there is scope for corruption. The auctioneer may tilt the auction in favor of one of the
bidders in exchange for a compensation. The exact form this collusion between the auctioneer
and one of the bidders5 may take is open to multiple possibilities. Here, we concentrate on
one particular case. We assume that before the auction takes place the auctioneer may
approach one of the bidders and o?er to provide her with information during the auction,
and that she chooses to approach each of the bidders with equal probability. Speci?cally,
the auctioneer will reveal to the favored bidder her rival’s bid, and then allow her to modify
her bid upwards or downwards if she wishes to do so.
In what follows, any given bidder may be in one of two situations. If she is colluding with
the auctioneer, she will be sure that her rival is not colluding at the same time –we rule out
the possibility that the auctioneer colludes with both bidders. If she is not colluding with
the auctioneer, she believes that her rival is with probability p. Our main object of analysis
will be how an “honest” bidder who believes that her rival colludes with the auctioneer with
probability p > 0 will bid, as compared with how she would bid in the absence of corruption
(i.e. with p = 0).
There are at least two possible interpretations of this setup. Our analysis may be viewed
as part of a more complete and speci?c study of how corruption a?ects the result of an
auction. We will not model how the auctioneer and the bidder she approaches will bargain
when deciding if they will collude. In addition, we will assume that both bidders will stay
in the auction independently of the value of p. One of the most relevant considerations in
any such bargaining game, and when a bidder decides whether or not to take part in the
auction, will be what would happen if both bidders stayed and an honest bidder attached
4 Logconcavity of the c.d.f. function holds for most well known distributions, such as the uniform, nor-
mal, logistic, extreme value, chi-squared, chi, exponential, Laplace, Pareto and any truncation of these
distributions. For details see Bagnoli and Bergstrom (1989).
5 We are not allowing for the possibility of collusion between the bidders (see Hendricks and Porter, 1989,
for a general analysis of this phenomenon), but only between one bidder and the auctioneer.
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a given probability to the fact that her rival will be favored. So our results below will be
crucial in any study of this form of corruption. Furthermore, examining how the auctioneer
and a bidder bargain is open to many modelling alternatives. Di?erent assumptions could
be made, for instance, on what knowledge the auctioneer has of the bidder’s valuation, or
on how much bargaining power each party has. Our own results will be relevant to any such
speci?cation. We will assume, though, that the auctioneer and the colluding bidder bargain
e?ciently: they will reach an agreement whenever they can both gain by doing so.
Our setup could be understood in a more speci?c way as well. We may assume that,
before the auctioneer may approach any of them, both bidders believe that the auctioneer
is corrupt –and then expect her to make an approach– with probability q. Then, if a
bidder has not been approached, this may mean that the auctioneer is not corrupt or that
the auctioneer has approached the rival. The updated probability that an honest bidder
attaches to the fact that the auctioneer be corrupt is then p = q .
2?q
All that will matter below, though, is that an honest bidder believes that her rival is
colluding with the auctioneer with probability p. When deciding how much to bid, any
bidder will ?nd herself in one of two situations. If she is not colluding with the auctioneer,
then with probability (1 ? p) she will be competing against another honest bidder in her
exact same situation (i.e. the rival will herself believe that the original bidder is colluding
with the auctioneer with probability p). With probability p, she will face a rival that will
be informed of her bid and be allowed to rebid accordingly. She will have to choose her bid
weighing both possibilities. If she is colluding with the auctioneer, her bid will be irrelevant,
since she will later be allowed to change it in any way she may wish after learning her rival’s
bid.
The e?ect of corruption on bidding behavior will clearly di?er according to the sealed-bid
auction format. In a second price auction, bidding her own valuation is a weakly dominant
strategy for every bidder without corruption, and, of course, remains so once the possibility
of corruption appears. Thus, bidding behavior remains unaltered. Furthermore, the result
of the auction will not change. Knowing the rival’s bid, no bidder will have any incentive
to modify her original bid. It remains true that the bidder with the highest valuation wins
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in equilibrium, and she pays her rival’s valuation, exactly as occurs when the possibility of
corruption is absent.
In a ?rst-price auction, however, the possibility of corruption has a signi?cant e?ect. If
a bidder colludes with the auctioneer and learns her rival’s bid, he may have an incentive
to change her original bid. If, according to her original bid, she is winning the auction,
then she will revise her bid down to her rival’s.6 If she is losing the auction, there are two
possibilities. If her rival’s bid lies below the colluding bidder’s valuation, then she will raise
her bid up to her rival’s and win the auction. If her rival’s bid lies above her valuation, then
her original bid will remain unchanged, since she would have to bid above her valuation to
win the auction. Viewing the auction from the standpoint of an honest bidder that faces a
colluding rival, this means that the former will have to bid above the latter’s valuation to
win. In other words, she will be competing against the rival’s valuation instead of competing
against her bid.
Let bpi : [0, 1] ?? IR be bidder i’s bidding function in this setup. For convenience, we will
use the inverse of her bidding function, ?pi(.). An honest bidder with valuation vi who faces
a rival j that, if honest, has a bidding function with inverse ?p(.) that is strictly increasing
j
will choose her bid by solving the following expected utility maximization problem
max (vi
(b)) + pF (b)i
b
? b)h(1?p)F(?pj
with ?rst-order condition
(1
(b)) + pF (b)
v
? p)F(?pj
i ? b = (1 ? p)f(?p(b))?p0(b) + pf(b)
j
j
In a symmetric equilibrium –i.e. one where both bidders exhibit the same behavior when
they are honest– we will have ?p = ?p = ?p, and the ?rst-order condition becomes
i
j
(1
?p(b) ? b =
? p)F(?p(b)) + pF(b)
(1)
(1 ? p)f(?p(b))?p0(b) + pf(b)
In the absence of corruption, we would have a standard symmetric ?rst-price auction
with independent private values and risk-neutral bidders. Let ? be bidder i’s inverse bidding
i
6 We assume that, in the event of a tie, the auctioneer chooses the winner. Therefore, she will always
chose the bidder she is trying to favor.
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function in this case. Thus, when bidder i has valuation vi and faces a rival that behaves
according to the inverse bidding function ? she would solve the problem
j
max (vi
(b))
b
? b)F(?j
At a symmetric equilibrium, ? = ? = ?, and the ?rst-order condition is
i
j
F (?(b)
?(?(b))
?(b) ? b =
=
(2)
f (?(b))?0(b)
?0(b)
Our main objective is to compare ?p with ?. That is, we want to establish whether the
fact that her rival will learn her bid with probability p makes a bidder more aggressive
(?p(b) < ?(b) for all b ? (0,1]) or less aggressive (?p(b) > ?(b) for all b ? (0,1]) as compared
to how she would behave if corruption were absent. We view the case where p < 1 as more
relevant and realistic in the analysis of corruption. With p = 1, every party to the auction
except the owner of the object on sale is aware of the existence of corruption and of who will
be favored by the auctioneer. With p < 1, the existence of corruption is uncertain and so is
the identity of the favored bidder.
For expositional ease, we make this comparison in two steps. The next subsection deals
with the case where p = 1. Using the results obtained for this case, subsection 2.2 derives
analogous results for any p > 0.
2.1
First-Price Auctions with Certain Corruption
We assume here that the auctioneer has agreed to reveal to one of the bidders the value of
her rival’s bid and then allow her to modify her own bid if necessary. This can be viewed as a
sequential game were the honest agent bids ?rst and her rival bids after observing the honest
bid. We compare bidding behavior in this environment with the standard, non-corruption
case.
Let ?c be the honest bidder’s inverse bidding function when p = 1. The ?rst-order
condition (1), which characterizes the honest bidder’s behavior, becomes
F (b)
?c(b) ? b =
= ?(b)
(3)
f (b)
The comparison between (2) and (3) yields Proposition 1.
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Proposition 1 If ? (v) is strictly convex (respectively, strictly concave, linear), the honest
bidder will become more aggressive (respectively, less aggressive, equally aggressive) with
corruption. That is, for all b ? (0,1], ?c(b) < ?(b) (respectively, ?c(b) > ?(b), ?c(b) = ?(b)).
Proof. We provide a proof only for the case where ?(v) is strictly convex. In the remaining
two cases, the proof is analogous.
We proceed in two steps:
(a) First, we show that when ? (v) is convex, for all b ? (0,1], ?(b) = ?c(b) ? ?0(b) >
?c0(b). That means that if there is a value b > 0 at which the bidding functions cross, to the
right of this value we have ?(b) > ?c(b).
Let ?(b) = ?(b) ? ?c(b). From (2) and (3),
?(?(b))
?(b) = ?(b) ? b ? (?c(b) ? b) = ?0(b) ? ?(b)
Using (3) again,
?0(b) = ?0(b) ? ?c0(b) = ?0(b) ? 1 ? (?c(b) ? 1) = ?0(b) ? 1 ? ?0(b)
When ?(b) = 0, it must be the case that ?(?(b)) = ?(b). Consequently,
?0(b)
?(?(b))
?(?(b))
?(?)
?0(b) =
? ?(b)
? ?(b)
?(b)
? 1 ? ?0(b) =
?(b)
? ?0(b) =
? ? b
? ?0(b)
where the last equality, once more, follows from (3).
Given that ?(.) is convex, and that b < ?(b), we have
?(?(b)) ? ?(b) > ?0(b) .
?(b) ? b
Hence ?0(b) > 0.
(b) It can be easily checked that ?c(0) = ?(0) = 0. Our second step is to show that,
arbitrarily to zero, ?c(b) < ?(b). If this is the case, then, by step (a) above we will know that
both bidding functions cannot cross for any b ? (0,1], and the proof will be complete.
Let ? > 0 be an arbitrarily small number. We want to show that for all b ? (0,?),
?(b) > 0.
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Assume this is not true. Then, there is a b0 ? (0,?) such that, ?(b0) ? 0. By step (a)
above, if ?(b0) = 0, ?(b) < 0 for b lower than, but close to, b0. Then, without loss of generality
we can concentrate on the case where ?(b0) < 0.
Since ?(0) = 0, there has to exist a b1 < b0 such that ?(b1) < 0 and ?0(b1) < 0 (b1 > 0,
because -as can be easily shown- ?0(0) = 0).
For any b > 0, if ?0(b) < 0 it follows that ?0(b) ? 1 < ?c0(b) ? 1, or
?(?(b))
?0(b) ? 1 < ?0(b) <
? ?(b)
?(b) ? b
where the second inequality derives form the strict convexity of ?(v). Therefore,
?(?(b))
?0(b) ? 1 <
? ?(b)
?(b) ? b
or
?0(b)(?(b) ? b) ? (?(b) ? b) < ?(?(b)) ? ?(b)
Using (2),
?(?(b)) ? (?(b) ? b) < ?(?(b)) ? ?(b)
which yields
?(b) < ?(b) ? b
From (3), then,
?c(b) ? b < ?(b) ? b
Hence, ?0(b) < 0 implies that ?c(b) < ?(b), or ?(b) > 0..We conclude that it is not possible
that there exist a b1 ful?lling the conditions mentioned above.
2.2
The General Case: p<1
We return now to the most interesting and realistic case. We model corruption as an uncer-
tain phenomenon. We will show that the conditions mentioned in the previous subsection
are also su?cient to characterize the e?ect of corruption on bidding behavior in the general
case. Furthermore, as Proposition 2 asserts, the bidding function of an honest bidder who
is facing a bidder who is corrupt with probability p lies between the two bidding functions
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