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Who Pays When Auction Rules Are Bent?
David McAdams and Michael Schwarz ∗
Abstract
In many negotiations, rules are soft in the sense that the seller and/or buyers may
break them at some cost. When buyers have private values, we show that the cost
of such opportunistic behavior (whether by the buyers or the seller) is borne entirely
by the seller in equilibrium, in the form of lower revenues. Consequently, the seller is
willing to pay an auctioneer to credibly commit to a mechanism in which no one has the
ability or the incentive to break the rules. Examples of “costly rule bending” considered
here include hiring shill bidders and trying to learn others’ bids before making one’s
own.
1 Introduction
Breaking or bending rules is not uncommon in auctions or negotiations. This paper shows
that costly rule-breaking hurts the seller ex ante even if it benefits the seller ex post. Indeed,
∗David McAdams at MIT Sloan School of Management, [email protected]. Michael Schwarz at Yahoo!
Research and NBER, [email protected]. We thank seminar audiences at UC Berkeley, University
of Pennsylvania, UC Riverside, Stanford, USC, and Yahoo! Research for helpful comments, as well as the
editor and two referees.
1
the seller bears all costs associated with rule-breaking in equilibrium, no matter whether
it is the seller or the buyers who cheat. Thus, the seller is willing to pay an intermediary
(“auctioneer”) to credibly commit to a sales mechanism in which no one – including the
seller – has the ability and incentive to break the rules.
The recent high-profile sale of the General Motors Building in Manhattan provides an
example of alleged seller misbehavior. The owners of the building advertised that they
would sell it using a first-price sealed-bid auction.1 According to the New York Times, this
auction “drew the highest bids ever for a skyscraper in the United States”, up to $1.4 billion.
However, the owners allegedly revealed others’ bids to one of the bidders and allowed that
bidder to submit a late, winning offer. (The bidder who had submitted the highest bid in
the auction round unsuccessfully sued to block the deal.)
The Netherlands 3G auction provides a recent example of alleged buyer misbehavior.
The bidder Telfort allegedly induced another bidder Versatel to stop bidding by threatening
to sue Versatel for driving up the price by its participation in the auction; see Klemperer
(2002) for details. More broadly, Klemperer argues that “Ascending auctions are particularly
vulnerable to rule-breaking by the bidders since they necessarily pass through a stage where
there is just one (or a few) excess bidders”. During this final stage of the auction, each bidder
has a strong incentive to drive others from active bidding. Such anti-competitive behavior
is clearly harmful to the seller.
Less obvious is whether a seller, or an auctioneer acting on the seller’s behalf, would prefer
to stop buyers from making more offers than allowed by the auction rules. As Klemperer
(2002) argues, “Sealed-bid auctions [may be] vulnerable to rule-changing by the auctioneer.
1Solow Building Corporation vs 767 Fifth Avenue LLC, et al (Delaware Court of Chancery, CA#20542,
2003). See “Lawsuit Seeks to Block Sale of G.M. Building” by Charles Bagli, New York Times, September
20, 2003. We thank Robert Gertner for sharing this example.
2
For example, excuses for not accepting a winning bid can often be found if losing bidders
are willing to bid higher. The famous RJR-Nabisco sale went through several supposedly
final sealed-bid auctions (Burrough and Helyar, 1990)”. It may seem good for the seller to
be able to solicit additional rounds of offers, but McAdams and Schwarz (2006) show that
the seller is worse off when unable to commit to a single round of offers. The reason is that
buyers anticipate that they will have another chance in the future to make a serious bid, and
hence bid less aggressively than if there were a single round.
Fraudulent bidding by the seller or her accomplices (“shill bidding”) is another sort of
rule-breaking by sellers. eBay threatens shill bidders with a variety of internal and external
punishments, such as loss of “eBay Powerseller” status and reporting to law enforcement.
In November 2004, Eliot Spitzer successfully prosecuted several cases arising from an inves-
tigation of shill bidding in online auctions.2 Despite the risks, a seller may find it difficult
to commit not to hire a shill if she interacts infrequently with buyers and/or if buyers find
it difficult to detect shills.
The possibility of cheating in negotiations can have implications for industry structure.
An upstream firm whose negotiations with suppliers are prone to costly cheating will earn
less profit and have a greater incentive to merge with its suppliers than otherwise. Indeed,
by eliminating the efficiency losses due to cheating, such a merger can constitute a Pareto
improvement. Thus, this paper provides a new sort of efficiency rationale for vertical integra-
tion. Indeed, the efficiency losses due to cheating in negotiations between unintegrated firms
can be substantial, in some cases as large as the extra revenue associated with committing
to an optimal reserve price in an auction. In this sense, our paper relates to the growing
2“Shill bidding’ exposed in online auctions”, Press Release, Office of New York State Attorney General
Eliot Spitzer, November 8, 2004.
3
literature on the interaction between negotiation institutions and the incentives for vertical
integration. See e.g. Hart and Tirole (1990), Chemla (2003), Inderst and Wey (2003), and
de Fontenay and Gans (2005).
When mergers are not possible, cheating creates an opportunity for an auctioneer to arise
to help conduct an orderly sales process. Since buyers and sellers may need to interact in
the future with the auctioneer, even if they never expect to trade with each other again,
the auctioneer may be in a better position to sanction opportunistic behavior as well as to
tie its own hands. Since an auctioneer can be involved in many more transactions than any
individual buyer or seller, it may also enjoy economies of scale in preventing, detecting, and
punishing opportunistic behavior by the other players (as well as in marketing, conducting
sales, and so on).3
Sales intermediaries are ubiquitous in markets ranging from corporate acquisitions to col-
lectible art, and prominent intermediaries such as WorldWide Retail Exchange have recently
emerged to facilitate business-to-business trading and intermediate input markets. The share
of asset value captured by such intermediaries ranges from about 1% in the market for ac-
quisitions4 to about 6% for real estate transactions and routinely exceeds 20% in the art
world. An obvious source of value creation by intermediaries stems from their marketing
3 Of course, it can still be difficult for an auctioneer to deter such behavior. For instance, despite eBay’s
efforts to deter shill bidding, the Internet Crime Complaint Center (IC3) continues to identify shill bidding
as one of the most common types of e-auction fraud. (In 2004, 71.2% of all internet fraud complaints were
of e-auction fraud, and about 74,000 e-auction fraud complaints were referred to law enforcement. In 2001,
42.8% of all internet fraud complaints were of e-auction fraud, and about 7,200 e-auction fraud complaints
were referred to law enforcement.) Separately, an analysis of rare coin auctions on eBay indicated that about
5% of all such auctions have at least one buyer who acts as if to “run up the bid” rather than to win the
auction (Kauffman and Wood (2000)).4 McLaughlin (1990) reports average investment banking fees of 1.29%.
4
and negotiation skills as well as their ability to verify quality and create a marketplace.5
This paper argues that reputable auctioneers may also create value by enforcing the rules of
a sales mechanism and by creating a marketplace in which neither buyers nor sellers have
the incentive to behave opportunistically.
The idea of using third-parties to commit to a negotiation strategy was noted by Schelling
(1956) and has been explored in the extensive literature on commitment through delegation.
Delegation in this literature allows the seller to strengthen her bargaining position by incen-
tivizing the delegate to be a tough bargainer. For example, a seller may benefit by sending a
representative who can commit to a reserve price (Bester (1994)), who has a different utility
function than the seller (Crawford and Varian (1979) and Sobel (1981)), or who is more
patient than the seller. By contrast, here the seller only delegates the process of conducting
the sale. The auctioneer need not negotiate on the seller’s behalf, per se, but helps the seller
indirectly by committing to a transparent sales mechanism.
A related literature on corruption in auctions explores a variety of settings in which
a corrupt auctioneer manipulates the outcome of an auction on behalf of an unscrupulous
buyer. For example, Menezes and Monteiro (2005) consider a first-price auction in which the
auctioneer approaches the winning bidder and offers to sell the object for the second-highest
price in exchange for a bribe.6 Cheating arises in this context because of an agency problem
between the seller and the auctioneer. This paper complements the corruption literature
by considering possibilities for cheating when there is no agency problem. Here, auctioneers
arise as professionals with sufficiently valuable at-risk reputation to be trusted. For example,
5 See Ellison, Fudenberg, and Mobius (2004) on network effects associated with creating a marketplace.6See also Burguet and Perry (1999), Burguet and Che (2004), Ingraham (2005), Arozamena and Wein-
schelbaum (2004) and Lengwiler and Wolfstetter (2002). We thank an anonymous referee for pointing out
the connection with the corruption literature.
5
the slogan for uBid.com, a relatively recent entrant in the online auctioneering market, is
“The marketplace you can trust” (emphasis in original).7
Our work also complements the existing literature on auction theory and mechanism
design. Relative to the benchmark performance of a standard efficient auction,8 this litera-
ture shows how much extra expected revenue a seller can raise given complete commitment
power. See e.g. Myerson (1981). Our work shows how a seller who lacks the power to
credibly commit to hard rules may achieve significantly less than this benchmark.
There is a vast literature on bargaining with private information in the context of a single
seller and a single buyer.9 In such settings, typically, the seller is worse off and the buyer
is better off when the seller lacks the power to commit to a negotiation strategy. See e.g.
Bulow (1982). By contrast, in our model the seller’s inability to commit to rules hurts the
seller but buyers are neither better nor worse off. The reason is that each buyer’s expected
surplus is determined by his “information rent”, which does not depend on how much players
spend to break the rules.
The rest of the paper is organized as follows. Section 2 presents the main result on the
revenue effects of cheating, with applications to “spying” in Section 2.1 and to “shill bidding”
in Section 2.2. Section 3 considers how an auctioneer may be able to raise expected revenue
by preventing cheating. Concluding remarks are in Section 4, with most proofs relegated to
an Appendix.
7 uBid.com accessed August 22, 2006.8A standard efficient auction is one in which the buyer with the highest value wins, and any buyer who
never wins pays nothing, e.g. an English auction or (in a symmetric setting) a first-price auction with zero
reserve price. By the Revenue Equivalence Theorem, all such auctions generate the same expected revenue.9 See Ausubel, Cramton, and Deneckere (2002) for a survey.
6
2 Who bears the cost of broken commitments?
A (female) risk-neutral seller has one object to sell to N ≥ 2 risk-neutral (male) buyers who
have i.i.d. private values drawn from interval support in [0,∞) and density f(·). The seller
announces a procedure to sell the object, but the seller and/or the buyers have the ability
to violate that procedure (“cheat”) at some cost.
For example, a seller holding a “second-price auction” or an “English auction” could
employ shill bidders. Or a seller holding a “first-price auction” could reveal the sealed bids
and delay the sale in hopes of negotiating an even higher price. Employing shill bidders,
lying to buyers, credibly revealing sealed bids, and delay may all be costly to the seller. Even
so, if buyers are naive and do not anticipate such behavior, the seller may gain by her ability
to cheat them. If buyers are sophisticated, however, they will compete less aggressively in
anticipation that the seller may cheat.
In other contexts it may be the buyers who incur costs as they try to gain an unfair
advantage. For instance, in a first-price auction, a bidder who learns that he has not won
might try to improve his offer after the fact. It may be difficult for the seller to commit to
refuse such revised offers, or otherwise stop the buyers from cheating in ways that increase
the price. Our analysis also applies to sales that are conducted by a corruptible auctioneer
since, from the buyers’ point of view, bribing the auctioneer is just another cheating cost.
Thus, an auction with a corrupt auctioneer can be viewed as an auction with costly buyer
cheating.
In the larger game played by a possibly cheating seller and possibly cheating buyers, any
perfect Bayesian equilibrium induces an incentive-compatible mechanism. For each buyer i,
let pi(vi) denote buyer i’s probability of winning, zi(vi) his expected payment, and ci(vi)
his expected spending on cheating conditional on having value vi. Let cs denote the seller’s
7
(ex ante) expected spending on cheating. Define shorthand p(·) = (p1(·), ..., pN(·)), z(·) =
(z1(·), ..., zN(·)), and c(·) = (c1(·), ..., cN(·)). Finally, let
Π(p(·), z(·), c(·), cs) =N∑
i=1
E[zi(vi)]− cs
denote the seller’s expected profit in mechanism (p(·), z(·), c(·), cs).
Theorem 1. Suppose that (p(·), z(·), c(·), cs) is an incentive-compatible mechanism. Then
(p(·), z(·),0, 0) is also incentive-compatible, where zi(vi) = zi(vi) + ci(vi). Further,
Π(p(·), z(·), c(·), cs) = Π(p(·), z(·),0, 0)− cs −N∑
i=1
E[ci(vi)] (1)
Theorem 1 shows that, holding the allocation probabilities p(·) fixed, the seller is always
better off using a mechanism with hard rules, in which neither she nor the buyers have the
ability and incentive to cheat. Indeed, again holding p(·) fixed, (1) implies that the seller
pays for all cheating costs, including those incurred by the buyers.10 As cheating changes the
way that the object is allocated, however, it is conceivable that the seller might prefer to allow
cheating rather than commit to a non-optimal auction with hard rules. On the other hand,
every incentive-compatible mechanism with costly cheating leads to strictly lower expected
revenue for the seller than an optimal auction.
The following remarks are straightforward corollaries of Theorem 1.
Remark 1. In any incentive-compatible mechanism, the seller’s expected revenue is less
than or equal to the seller’s expected revenue in an optimal auction minus the sum of all
cheating costs incurred by all players.
10In the case of a corruptible auctioneer, all auctioneer profit translates into lost seller profit. This is
consistent with previous findings in the corruption literature, e.g. Menezes and Monteiro (2005) and Burguet
and Che (2004).
8
Remark 2. An incentive-compatible mechanism that allocates the object to the highest
value buyer always results in a seller’s payoff equal to the revenues from an efficient auction
minus the sum of cheating costs incurred by all players.
The rest of this section illustrates this basic result through two applications. Section 2.1
considers a seller who announces a first-price auction with zero reserve price, but is unable to
keep bids secret and unable to refuse to consider late offers. This gives buyers the incentive
to “spy” on one another. Section 2.2 considers a seller who announces a second-price auction
with zero reserve price, but is unable to commit not to hire a shill bidder.
In each of these applications, the seller’s expected equilibrium profit can be substantially
lower than if she could have committed to the announced rules, with losses on the same order
of magnitude as the gain from committing to an optimal reserve price. For instance, suppose
that there are two bidders, each having values uniformly distributed on [0, 1]. An auction
with zero reserve price raises expected revenue .333. An optimal auction raises expected
revenue .417, a 20% increase. In the spying example, expected revenue can be as low as .259,
a 22% decrease. In the shill bidding example, expected revenue can be as low as .250, a 25%
decrease.
2.1 Spying in the first-price auction
In this section, we consider a seller who announces a first-price auction with zero reserve
price, but can not stop buyers from “spying”.11
Model. In period 0, each bidder learns his private value vi, where vi ∼ U [0, 1] iid. In period
11Our “first-price auction with spying” model differs from the “first-price auction with corruption” models
studied by Lengwiler and Wolfstetter (2002), Menezes and Monteiro (2005), and others. One difference is
that, in our model, bidders’ costs to learn others’ values are socially wasteful rather than a transfer.
9
1, each bidder simultaneously decides whether to submit his final bid bi,1 or pay c ≥ 0 to
spy. In period 2, all those who decided to spy learn all bids made by those who did not spy.
(Spying can be interpreted as attempting to discover others’ bids as well as making sure that
one’s own bid can not be discovered.) In period 3, all those who decided to spy then submit
their final offers bi,3. (Each bidder submits a bid in either period 1 or period 3.) The highest
bidder pays his bid and receives the object. Any bid submitted in period 3 wins in a tie over
any bid submitted in period 1, whereas ties between bids submitted in the same period are
broken by coin-flip.
We shall restrict attention to perfect Bayesian equilibria in which each bidder chooses
to spy iff his private value vi > v∗ for some threshold v∗ ∈ [0, 1] (“threshold equilibria”).
Theorem 2 establishes that there is a unique threshold equilibrium.12
Theorem 2. The first-price auction with spying has a unique threshold equilibrium. In this
equilibrium, the threshold v∗(N, c) = min{
N
√c N2
(N−1), 1}
and the seller’s expected revenue is
Nc(1− v∗(N, c)) less than in a first-price auction without spying.
Intuition for the threshold v∗(N, c). In any threshold equilibrium, a bidder having the
threshold value must be indifferent between spying and not spying. Those with lower val-
ues vi < v∗(N, c) have no chance of winning except in the first period, and hence bid
as if in a first-price auction: bi,1(vi) = N−1N
vi. If a bidder having the threshold value
were also to bid as if in a first-price auction, he would win with probability Pr(maxj 6=i ≤
v∗(N, c)) = (v∗(N, c))N−1 and pay v∗(N, c)N−1N
. If this bidder were to spy, however, he
could pay maxj 6=i bj,1 < v∗(N, c)N−1N
instead when he wins.13 Bidder i’s expected payment
in this case conditional on spying and winning is E[maxj 6=i bj,1|maxj 6=i vj < v∗(N, c)] =
12We conjecture but have not proven that this is the unique perfect Bayesian equilibrium.13A bidder having the threshold value v∗(N, c) wins in equilibrium iff he has the highest value, whether
he chooses to spy or not to spy. (See the proof in the Appendix.)
10
N−1N
E[maxj 6=i vj|maxj 6=i vj < v∗(N, c)] =(
N−1N
)2v∗(N, c). A threshold bidder’s expected
savings from spying due to paying a lower price is therefore
(v∗(N, c))N−1
(v∗(N, c)
N − 1
N− v∗(N, c)
(N − 1
N
)2)
= (v∗(N, c))N N − 1
N2= c
So, bidder i is indifferent between spying and not spying given value v∗(N, c).
If the cost of spying is zero, the first-price auction with spying becomes equivalent to
a true first-price auction (with zero reserve price). Each bidder waits until period 3 before
making his bid, but such delay is costless. At the same time, if the cost of spying is sufficiently
large (c ≥ N−1N2 ), every bidder submits his bid in period 1 and the game is again equivalent
to a true first-price auction. Spying hurts the seller in our model the most when the cost
of spying is small enough that bidders sometimes choose to spy but large enough that the
expected losses due to spying are not inconsequential. (By assumption, each bidder has only
one opportunity to spy, so the total expected losses due to spying are bounded by Nc.)
Special case. Suppose that N = 2 and c = 1/9. v∗(N, c) = 2/3 in the unique threshold
equilibrium. Each bidder spies with probability 1/3, so total expected spying costs are 2/27.
In this setting, a first-price auction with zero reserve price raises expected revenue 1/3. Since
the first-price auction with spying also allocates the object to the buyer with the highest
value in equilibrium, Theorem 1 (Remark 2) implies that the seller’s expected revenue must
be 1/3 − 2/27 ≈ .259, about 22% less than the expected revenue in a standard efficient
auction.
In related work, McAdams and Schwarz (2006) studies costly cheating in the first-price
auction, but considers the possibility that the seller may cheat, soliciting additional rounds
of offers although this leads to costly delay. As long as the per-round cost of delay is large
enough, the seller can credibly commit to a first-price auction. However, unlike in the present
study of spying, the total cost of cheating does not disappear as the cost of each cheating
11
episode goes to zero. Indeed, as the delay between rounds of offers falls, the seller asks for
so many more rounds that total delay may actually increase.
Also related is Edelman, Ostrovsky, and Schwarz (2006)’s analysis of the “generalized
first-price auction” that was used in the early days of Internet advertisement.14 In such
auctions a bidder can gain a significant advantage by investing in robots that are faster than
others’ robots; this in turn can be detrimental for auctions’ revenues. Intuitively, each bidder
who invests in a fast robot is able to choose his bid after observing the bids made by those
with a slow robot. This is similar to spying in our model, in which each bidder who spies is
able to choose his bid after observing the bids made by those who did not spy.
2.2 Second-price auction with shill bidding
Second-price auctions are vulnerable to different sorts of cheating than first-price auctions.
In this section, we consider an example in which the seller announces a second-price auction
with zero reserve price, but may hire a shill bidder. In the unique equilibrium, buyers shade
their bids below their values in anticipation that the seller will sometimes hire a shill, and
expected (gross) revenue is the same as if shills were impossible to hire. Thus, all resources
spent hiring shills translate into lost seller profit.
Model. In period 0, each bidder learns his private value vi, where vi ∼ U [0, 1] iid, and the
seller decides whether to (unobservably) hire a shill at cost w > 0.15 The seller knows w,
14 In the generalized first-price auction, advertisers submit bids per click and their sponsored links are
arranged on the web page in order of their bids. Thus, higher bidders get more visible positions on the page
that yield more clicks, but each advertiser pays his bid per click regardless of the position that he gets.15A very similar analysis applies to an alternative game in which the seller decides whether to pay the
cost w > 0 after observing the bids. In equilibrium in this case, buyers correctly anticipate that the winner
will pay his own bid whenever the (ex post) difference between the highest and second-highest bids exceeds
12
while the buyers only know its atomless c.d.f. Fw(·). in period 1, each bidder submits a
bid and the winner is the highest bidder, with ties broken by coin-flip. If the seller has
not hired a shill, the winner pays the second-highest bid. If the seller has hired a shill, on
the other hand, the winner pays his own bid. (Without shill bidding, the winner pays the
second-highest bid which is less than or equal to than the highest bid. With shill bidding,
the second-highest bid is always equal to the highest bid.)
Theorem 3. The second-price auction with shill bidding has a unique perfect Bayesian
equilibrium. The seller hires a shill whenever w < 1
(N+1)(1+ pN−1)
, where p∗ is the unique
solution of (3) below. Buyers then follow the symmetric bidding strategy b(vi; p∗) = vi
1+ p∗N−1
.
Proof. Consider first the bidding subgame. Buyers care about the probability p that the
seller employs a shill. Taking this probability as given, the resulting game amongst the
buyers is strategically equivalent to one of the “exotic auctions” considered in Lizzeri and
Persico (2000) (LP): the winner is the highest bidder and pays a convex combination of
the highest and second-highest bids. In our context, LP’s analysis can be easily adapted to
prove that all such auctions have a unique bidding equilibrium. Given the symmetry of the
environment, the unique bidding equilibrium must be symmetric, with each buyer bidding
b(vi; p) given private value vi. The Revenue Equivalence Theorem allows us to compute
b(vi; p), since each buyer’s expected payment conditional on winning must be the same as in
a second-price auction: b(·; p) solves
pb(vi; p) + (1− p)E
(maxj 6=i
b(vj; p)|maxj 6=i
vj < vi
)= E
(maxj 6=i
vj|maxj 6=i
vj < vi
)=
N − 1
Nvi (2)
The unique solution to (2) is b(vi; p) = vi
1+ pN−1
. Next, consider the seller’s initial decision
regarding whether to hire a shill. The seller cares about how much shill bidding will increase
w, and shade their bids accordingly.
13
her expected revenues. If bidders believe that the seller will hire a shill with probability p,
this increased revenue equals
E(maxi
b(vi; p)− 2nd maxi
b(vi; p)) =1
(N + 1)(1 + p
N−1
)where 2nd maxi b(vi; p) denotes the second-highest bid. Thus, the seller will hire a shill
whenever w < 1
(N+1)(1+ pN−1)
. In equilibrium, buyers correctly anticipate the use of shills, so
the probability p = p∗ where p∗ is defined as follows:16
p∗ = 0 if Fw
(1
N + 1
)= 0; otherwise
p∗ = 1 if Fw
(N − 1
N(N + 1)
)= 1; otherwise
p∗ ∈ (0, 1) solves Fw
(1
(N + 1)(1 + p∗
N−1
)) = p∗ (3)
This completes the proof of Theorem 3.
Corollary. The seller’s expected profit in the second-price auction with shill bidding is
Pr
(w < 1
(N+1)(1+ p∗N−1
)
)E
(w|w < 1
(N+1)(1+ p∗N−1
)
)less than in a second-price auction with-
out shill bidding.
Numerical example. Suppose that N = 2 and that the cost of hiring a shill w ∼
U [1/9, 1/3]. It is easy to check that p∗ = 1/2 and that the seller hires a shill whenever
w < 13(1+1/2)
= 2/9. The seller spends cs = 1/12 on shills on average, whereas the winner’s
expected payment is 1/3. Since the second-price auction with shill bidding allocates the ob-
ject to the buyer with the highest value in equilibrium, Theorem 1 (Remark 2) implies that
the seller’s expected revenue is only 1/4, compared to expected revenue 1/3 in a second-price
auction with zero reserve price and no shill bidding. Note that, relative to a second-price
16The left-hand-side of (3) is decreasing in p∗ (equal to 1N+1 when p∗ = 0 and equal to N−1
N(N+1) when
p∗ = 1) while the right-hand-side is strictly increasing in p∗. Thus, p∗ is uniquely defined.
14
auction with zero reserve price, the seller’s expected loss due to shill bidding here is as large
as the expected gain from committing to an optimal reserve price. (Expected revenue in a
second-price auction with optimal reserve price is 5/12 = 1/3 + 1/12.)
Now suppose that the seller can take some (observable) action that increases the cost of
hiring a shill by 1/6. Given shill cost w ∼ U [1/3, 5/9], it is easy to check that the seller
never hires a shill bidder (p∗ = 0) and her expected profit increases to 1/3. Thus, the seller
is willing to pay up to 1/12 (25% of expected revenue) to increase her cost of hiring a shill.
However, raising the cost of using shills may be difficult.17 For example, a seller may not
be able to credibly impose a fine for withdrawing a winning bid (a standard ploy of shill
bidders), since for a shill bid this fine would simply “travel from the seller’s left pocket to
her right pocket”.
On the other hand, a sufficiently reputable third-party may be able to credibly deter the
seller from employing shills. By investigating buyers’ credentials and/or forcing buyers to
post a bond, such an intermediary can raise the cost of employing a shill. By not allowing
bid withdrawal and by charging a commission, similarly, the intermediary can make it more
costly for the seller to have a shill “win”.
Interestingly, not all intermediaries have taken such steps. In contrast to traditional
auction houses such as Sotheby’s, online auctioneer eBay charges low commissions and makes
it relatively easy to withdraw bids. According to eBay rules, a buyer can withdraw a bid if
there is a “clear typo” such as bidding $1200 instead of $120. An unscrupulous seller can
therefore use a shill who purposefully submits a high bid with a typo and then withdraws
17Committing always to use shills is easy, since this is equivalent to making the winner pay his own bid.
However, as we have seen, the first-price auction may be vulnerable to other sorts of cheating. Also, beyond
the scope of our model in asymmetric environments, the seller’s expected revenue from a first-price auction
may be less than in a second-price auction (Maskin and Riley (2000)).
15
it. eBay rules are explicit on this matter, suggesting that it is more than a theoretical
possibility: “Manipulating bids to discover the maximum bid of the current high bidder, and
then retracting the bid, is prohibited”.18 Yet, detecting that a seller has violated this eBay
rule may be difficult.
Rothkopf, Teisberg, and Kahn (1990) discuss why buyers will be reluctant to bid their true
values in a Vickrey auction, if they fear that the seller may insert a shill bid. Chakraborty and
Kosmopoulou (2004) examine a common-value setting with costless shill bidding, and show
that the seller is worse off having the option to sometimes use a shill bidder. The underlying
logic of their result is very different from ours. In Chakraborty and Kosmopoulou (2004), an
effect related to the “winner’s curse” depresses expected revenue. By contrast, we consider
a private-value setting with costly shill bidding, where there is no winner’s curse.
3 Can an auctioneer change the game?
An incorruptible auctioneer who is able to both detect and punish cheating can help the
seller to increase her expected revenue by conducting a clean auction. In what settings can
we expect such intermediaries to arise?
Incorruptibility is necessary so that the auctioneer does not cheat on behalf of any of the
other players. The auctioneer may not be able sign formal contracts promising not to cheat,
especially if cheating is difficult to verify in court. If so, to be incorruptible, the auctioneer
must have a profitable repeat business or be able to link his reputation in other markets to
his performance as an auctioneer. Network effects, economies of scale, and switching costs
may all make it difficult to enter the auctioneering market, and thereby allow auctioneers to
sustain positive economic profit. There must also be some chance that auctioneer cheating
18http://pages.ebay.com/help/rulesandsafety/43015002.html accessed October 18, 2005.
16
will be detected by the buyers and sellers with whom the auctioneer will interact in the
future. Once auctioneer cheating is detected, sellers can credibly commit to punish the
auctioneer by refusing to pay as much for its services, since they expect lower revenue when
hiring a corrupt auctioneer.
At the same time, the auctioneer must be able to deter buyers and sellers from cheating
on their own. Such deterrence requires the threat of sanctions in future interactions, which in
turn requires that cheating players not be able to switch too easily in the future to another
auctioneer who is not aware of the cheater’s past. For this reason, deterrence requires
either that switching costs be relatively high, or that the auctioneering market be relatively
concentrated, or that auctioneers be able to share information about cheaters.
Numerical example reprise: spying. Consider again our spying example, in the special
case in which two bidders have i.i.d. values uniform on [0, 1] and the cost of spying is 1/9.
(A similar analysis of the shill bidding example is omitted to save space.) Spying may or
may not require the auctioneer’s cooperation but, again to save space, we will focus on
the case in which it does. The auctioneer can eliminate spying by resisting the temptation
to accept a bribe from any buyer. Each bidder i’s willingness to bribe the auctioneer is
greatest when he has value vi = 1. Assuming that the auctioneer can not be bribed, so that
the other bidder acts as if in a first-price auction, a bidder with value vi = 1 would pay
an average price of 1/4 after spying, rather than his equilibrium bid of 1/2. Suppose that
the auctioneer makes profit πA per sale and has discount factor δA between sales, and that
there is a probability pA ∈ [0, 1] that accepting a bribe today will lead it to lose all future
business. In present value terms, the auctioneer’s at-risk reputation is pAπAδA
1−δA. Since the
cost of spying is 1/9, a buyer with value vi = 1 will be able to bribe the auctioneer unless
1/4− 1/9 = 5/36 ≤ pAπAδA
1−δA.
17
4 Concluding Remarks
Committing to a mechanism is difficult for a seller, especially when she has an ex post
incentive to change the rules of the game or allow buyers to break the rules of the game. As
long as buyers are sophisticated, however, the seller is better off committing to a mechanism
with hard rules in which neither the seller nor any buyer has the ability and incentive to
break the rules. Furthermore, the lost revenue due to the seller’s inability to enforce rules
can be substantial, on the same order of magnitude as the extra revenue from an optimal
reserve price.
When a seller would otherwise lose substantial expected revenue due to cheating, insti-
tutions may arise to help sellers enforce mechanism rules and hence avoid wasteful spending
on cheating. Professional auctioneers who help conduct orderly sales are an example. Such
intermediaries can more easily commit to hard rules than a seller if they interact repeatedly
with buyers and sellers and stand to lose a valuable reputation by not punishing rule-breakers.
5 Appendix
Proof of Theorem 1. By the Revenue Equivalence Theorem, each bidder’s interim ex-
pected surplus given value vi must be∫ v
0pi(x)dx given any incentive-compatible mechanism
with allocation p(·). Since buyer surplus depends only on the probability of winning pi(vi)
and the net payment zi(vi)+ ci(vi), mechanism (p(·), z(·),0, 0) is incentive-compatible given
our presumption that (p(·), z(·), c(·), cS) is incentive-compatible.
Total expected surplus gross of cheating costs in either mechanism is∑N
i=1
∫∞0
vf(v)pi(v)dv.
18
Since seller profit equal total gross surplus minus buyer surplus,
Π(p(·), z(·),0, 0) =N∑
i=1
∫ ∞
0
vf(v)
(pi(v)−
∫ v
0pi(x)dx
v
)dx
Π(p(·), z(·), c(·), cS) = Π(p(·), z(·),0, 0)− cs −N∑
i=1
∫ ∞
0
ci(vi)dvi
This completes the proof.
Proof of Theorem 2. The proof is in four steps. First, in any threshold equilibrium,
those who bid in the first period and those who spy / bid in the third period must bid
as if in a first-price auction. Second, there is only one threshold level v∗(N, c) consistent
with (perfect Bayesian) equilibrium. Third, the strategies corresponding to this threshold
constitute an equilibrium. Fourth, on the seller’s expected revenue in this equilibrium.
Step 1. Suppose that a threshold equilibrium exists with threshold v∗. Let I = {i : vi ≥
v∗} denote the set of bidders who choose to spy. Consider first any bidder i 6∈ I. Let bi,1(vi)
denote the support of bidder i’s (possibly random) bid in period 1. (By convention, let bj,1 =
0 for all j ∈ I.) By standard dominance arguments, note first that bi,1(vi) ⊂ [0, vi] ⊂ [0, v∗].
Thus, bidder i is certain to lose if any other bidder has value greater than v∗, since any such
bidder will spy and then outbid bidder i. We may therefore express bidder i’s optimization
problem as
bi,1(vi) ⊂ arg maxb
Pr(maxj 6=i
vj < v∗) Pr(b > maxj 6=i
bj,1(vj)|maxj 6=i
vj < v∗))
= arg maxb
Pr(b > maxj 6=i
bj,1(vj)|maxj 6=i
vj < v∗)) (4)
(4) is identical to each bidder’s optimization problem in a first-price with zero reserve price
and N bidders having values iid uniformly distributed on [0, v∗]. As is well-known, the unique
equilibrium in this setting is for each bidder to adopt a pure strategy bi,1(vi) = viN−1
N.
Next, consider the set of bidders I who spied. In equilibrium, every bidder who spied
believes that all others who spied have values in [v∗, 1]. Repeating the logic surrounding (4),
19
every bidder in I must bid as if in a first-price auction in which (i) there are #(I) bidders
whose values are iid uniform on [v∗, 1] and (ii) the “reserve price” r = maxi bi,1. In particular,
on the equilibrium path when maxi bi,1 < v∗, bi,3(vi; I) = v∗ + #(I)−1#(I)
(vi − v∗) for all vi > v∗
when #(I) > 1 and bi,3(vi; 1) = maxj 6=i bj,1 when I = {i}.19
Step 2. Any bidder having the threshold type vi = v∗ must be indifferent between bidding
in period 1 and spying. As argued above, if a bidder with value v∗ were to bid in period
1, his best bid would be bi,1 = v∗N−1N
. If he spies, on the other hand, a bidder with the
threshold type only pays E[maxj 6=i bj,1(vj)|maxj 6=i vj < v∗] = N−1N
E[maxj 6=i vj|maxj 6=i vj <
v∗] = v∗(
N−1N
)2when he wins.20 Thus, the expected gain from spying is
Pr
(maxj 6=i
vj < v∗)
(N − 1)v∗
N2= (v∗)N (N − 1)
N2(5)
for the threshold type. Since the cost of spying is c ≥ 0, the threshold type must be
v∗(N, c) ≡ N
√c
N2
(N − 1)∈ [0, 1) when c <
N − 1
N2(6)
≡ 1 when c ≥ N − 1
N2
Step 3. Do the threshold strategies corresponding to threshold v∗(N, c) constitute an
equilibrium? In Step 1, we established the (unique) equilibrium bidding strategies for those
who do not spy, in period 1, and for those who do spy, in period 3. Now we need to check
that bidder i prefers to spy iff vi ≥ v∗(N, c). If vi < v∗(N, c), the gain from spying is strictly
less than (5). By definition of v∗(N, c), bidder i therefore prefers not to spy. Finally, suppose
that vi ≥ v∗(N, c). We prove that bidder i prefers to spy in this case, in a few smaller steps.
19Off the equilibrium path, if any bidder were to have value vi ≤ v∗ but choose to spy, all other bidders
would incorrectly believe that vi > v∗ and bid as described in the text. Thus, we may assume without loss
that bi,3(vi; I) = vi for all vi ≤ v∗.20In either case, a bidder with value v∗ wins iff maxj 6=i vj < v∗. (If any other bidder has value vj > v∗,
that bidder will spy and then outbid bidder i in period 3.)
20
First, observe that bidder i gets the same expected equilibrium surplus as if in a first-
price auction with N bidders having values iid U [0, 1] and zero reserve price (“basic first-
price auction”). In a basic first-price auction, bidder i wins iff vi ≥ maxj 6=i vj and pays
E[maxj 6=i vj|vi ≥ maxj 6=i vj] on average when he wins. Now, suppose that bidders I ⊂
{1, ..., N} spy, where i ∈( I. Recall from Step 1 that all bidders in I bid in period 3 as if in
a first-price auction with #(I) bidders having values iid U [v∗(N, c), 1] (and with irrelevant
“reserve price” maxj bj,1 < v∗(N, c)) in equilibrium. Thus, conditional on any given I ) i,
bidder i wins iff vi ≥ maxj 6=i vj and pays E[maxj 6=i vj|vi ≥ maxj 6=i vj and vj ≥ v∗(N, c)∀j ∈ I]
when he wins. Next, conditional on I = {i}, bidder i always wins and pays
E[maxj 6=i
bj,1(vj)|v∗(N, c) ≥ maxj 6=i
vj] = v∗(N, c)
(N − 1
N
)2
= E[maxj 6=i
vj|v∗(N, c) ≥ maxj 6=i
vj]− v∗(N, c)N − 1
N2
on average after spying. Since v∗(N, c) ≥ maxj 6=i vj occurs with probability (v∗(N, c))N−1,
the ex ante expected savings due to paying less after spying is (v∗(N, c))N N−1N2 = c by the
definition of v∗(N, c). Thus, spying has zero net effect on bidder i’s expected surplus.
To show that bidder i prefers to spy, it suffices to show that deviating by bidding bi,1 in the
first period gives less expected payoff than in a basic first-price auction. First, any deviation
bi,1 ≤ N−1N
v∗(N, c) wins with the same probability Pr(maxj 6=i vj ≤ NN−1
bi,1) and hence gives
the same payoff as in the basic first-price auction. Since bidding bi,1 is an unprofitable
deviation in the basic first-price auction, and it is also less profitable than spying in our
game. Second, any deviation bi,1 > N−1N
v∗(N, c) wins with strictly lower probability than in
the basic first-price auction.21 Thus, every such bid gives less than bidder i’s equilibrium
surplus in the basic first-price auction, and hence is less profitable than spying in our game.
21When bi,1 ∈ (N−1N v∗(N, c), v∗(N, c)), bidder i wins with probability Pr(maxj 6=i vj ≤ v∗(N, c)) <
Pr(maxj 6=i vj ≤ NN−1bi,1). When bi,1 > v∗(N, c), Pr(maxj 6=i vj ≤ bi,1) < Pr(maxj 6=i vj ≤ N
N−1bi,1).
21
Step 4: seller expected revenue. Since each bidder spies with probability Pr(vi >
v∗(N, c)) = 1− v∗(N, c), the total expected cost of spying is Nc (1− v∗(N, c)). By Theorem
1, all of these costs are borne by the seller in the form of lost equilibrium revenues. This
completes the proof.
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