Who Pays When Auction Rules Are Bent?

Home , Revenue equivalence, Shill

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 integration. 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

N X Π(p(·), z(·), c(·), cs) = E[zi(vi)] − cs i=1

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(·), ˆz(·), 0, 0) is also incentive-compatible, where zˆi(vi) = zi(vi) + ci(vi). Further,

N X Π(p(·), z(·), c(·), cs) = Π(p(·), ˆz(·), 0, 0) − cs − E[ci(vi)] (1) i=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

n q 2 o ∗ N N equilibrium, the threshold v (N, c) = min c (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

N−1 as if in a ﬁrst-price auction: bi,1(vi) = N 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(maxj6=i ≤

∗ ∗ N−1 ∗ N−1 v (N, c)) = (v (N, c)) and pay v (N, c) N . If this bidder were to spy, however, he

∗ N−1 13 could pay maxj6=i bj,1 < v (N, c) N instead when he wins. Bidder i’s expected payment

∗ in this case conditional on spying and winning is E[maxj6=i bj,1| maxj6=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−1 ∗ N−1 2 ∗ N E[maxj6=i vj| maxj6=i vj < v (N, c)] = N v (N, c). A threshold bidder’s expected savings from spying due to paying a lower price is therefore ! N − 1 N − 12 N − 1 (v∗(N, c))N−1 v∗(N, c) − v∗(N, c) = (v∗(N, c))N = c N N N 2

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

N−1 large (c ≥ N 2 ), 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

1 ∗ equilibrium. The seller hires a shill whenever w < p , where p is the unique (N+1)(1+ N−1 )

∗ vi solution of (3) below. Buyers then follow the symmetric bidding strategy b(vi; p ) = p∗ . 1+ 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

N − 1 pb(vi; p) + (1 − p)E max b(vj; p)| max vj < vi = E max vj| max vj < vi = vi (2) j6=i j6=i j6=i j6=i N

vi The unique solution to (2) is b(vi; p) = p . Next, consider the seller’s initial decision 1+ N−1 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

1 E(max b(vi; p) − 2nd max b(vi; p)) = i i p (N + 1) 1 + N−1

where 2nd maxi b(vi; p) denotes the second-highest bid. Thus, the seller will hire a shill

1 whenever w < p . In equilibrium, buyers correctly anticipate the use of shills, so (N+1)(1+ N−1 ) the probability p = p∗ where p∗ is deﬁned as follows:16

1 p∗ = 0 if F = 0; otherwise w N + 1 N − 1 p∗ = 1 if F = 1; otherwise w N(N + 1) ! ∗ 1 ∗ p ∈ (0, 1) solves Fw p∗ = p (3) (N + 1) 1 + N−1 This completes the proof of Theorem 3.

Corollary. The seller’s expected proﬁt in the second-price auction with shill bidding is 1 1 Pr w < p∗ E w|w < p∗ less than in a second-price auction with- (N+1)(1+ N−1 ) (N+1)(1+ N−1 ) 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

1 w < 3(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

16 ∗ 1 ∗ N−1 The left-hand-side of (3) is decreasing in p (equal to N+1 when p = 0 and equal to 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

δA business. In present value terms, the auctioneer’s at-risk reputation is pAπA . Since the 1−δA

cost of spying is 1/9, a buyer with value vi = 1 will be able to bribe the auctioneer unless

δA 1/4 − 1/9 = 5/36 ≤ pAπA . 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, institutions 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- R v pected surplus given value vi must be 0 pi(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(·), ˆz(·), 0, 0) is incentive-compatible given our presumption that (p(·), z(·), c(·), cS) is incentive-compatible. PN R ∞ Total expected surplus gross of cheating costs in either mechanism is i=1 0 vf(v)pi(v)dv.

18 Since seller proﬁt equal total gross surplus minus buyer surplus, N Z ∞ R v X pi(x)dx Π(p(·), ˆz(·), 0, 0) = vf(v) p (v) − 0 dx i v i=1 0 N X Z ∞ Π(p(·), z(·), c(·), cS) = Π(p(·), ˆz(·), 0, 0) − cs − ci(vi)dvi i=1 0 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 max Pr(max vj < v ) Pr(b > max bj,1(vj)| max vj < v )) b j6=i j6=i j6=i

∗ = arg max Pr(b > max bj,1(vj)| max vj < v )) (4) b j6=i j6=i

(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

N−1 equilibrium in this setting is for each bidder to adopt a pure strategy bi,1(vi) = vi 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,

∗ ∗ #(I)−1 ∗ ∗ on the equilibrium path when maxi bi,1 < v , bi,3(vi; I) = v + #(I) (vi − v ) for all vi > v

19 when #(I) > 1 and bi,3(vi; 1) = maxj6=i bj,1 when I = {i}.

∗ 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

∗ N−1 1, his best bid would be bi,1 = v N . If he spies, on the other hand, a bidder with the

∗ N−1 threshold type only pays E[maxj6=i bj,1(vj)| maxj6=i vj < v ] = N E[maxj6=i vj| maxj6=i vj <

∗ ∗ N−1 2 20 v ] = v N when he wins. Thus, the expected gain from spying is

∗ ∗ (N − 1)v ∗ N (N − 1) Pr max vj < v = (v ) (5) j6=i N 2 N 2

for the threshold type. Since the cost of spying is c ≥ 0, the threshold type must be s N 2 N − 1 v∗(N, c) ≡ N c ∈ [0, 1) when c < (6) (N − 1) N 2 N − 1 ≡ 1 when c ≥ N 2

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.

19 ∗ Oﬀ 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 . 20 ∗ ∗ ∗ In either case, a bidder with value v wins iﬀ maxj6=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 ≥ maxj6=i vj and pays

E[maxj6=i vj|vi ≥ maxj6=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 ≥ maxj6=i vj and pays E[maxj6=i vj|vi ≥ maxj6=i vj and vj ≥ v (N, c)∀j ∈ I]

when he wins. Next, conditional on I = {i}, bidder i always wins and pays

2 ∗ ∗ N − 1 E[max bj,1(vj)|v (N, c) ≥ max vj] = v (N, c) j6=i j6=i N

∗ ∗ N − 1 = E[max vj|v (N, c) ≥ max vj] − v (N, c) j6=i j6=i N 2

∗ ∗ N−1 on average after spying. Since v (N, c) ≥ maxj6=i vj occurs with probability (v (N, c)) ,

∗ N N−1 the ex ante expected savings due to paying less after spying is (v (N, c)) N 2 = 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

N−1 ∗ N bi,1 ≤ N v (N, c) wins with the same probability Pr(maxj6=i vj ≤ N−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

N−1 ∗ game. Second, any deviation bi,1 > N 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.

21 N−1 ∗ ∗ ∗ When bi,1 ∈ ( N v (N, c), v (N, c)), bidder i wins with probability Pr(maxj6=i vj ≤ v (N, c)) < N ∗ N Pr(maxj6=i vj ≤ N−1 bi,1). When bi,1 > v (N, c), Pr(maxj6=i vj ≤ bi,1) < Pr(maxj6=i vj ≤ N−1 bi,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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