1. shopping
2. auctions

‘Upping the Ante’: how to design efficient auctions with entry?∗
Laurent Lamy†
Abstract
Of primary importance in auction design is the set of strategies available to the seller at
the auction stage, in particular whether she is able to submit shill bids or to cancel the sale.
The literature in law emphasizes the hold-up problem regarding entry costs that preys on
auctions when those instruments are tolerated, if not allowed. However, the lack of formalization does not help to disentangle the differences between those various instruments that
may also give some valuable flexibility if the seller is ex ante uncertain about her reservation
value. We first formalize hold-up in second-price auctions with entry, derive the optimal
reserve and show how shill bidding can make posted-prices outperforming auctions. Second,
we advocate for a new regulation where shill bidding would be banned but with the possibility to cancel sales ex post: the English auction with jump bids implements then the first
best in large environments.
Keywords: Auctions, Auctions with entry, Shill bidding, Commitment failure, Hold-up,
Posted-price, Cancelation rights, Jump bids, Bilateral asymmetric information
JEL classification: D44, D82, K12
The exercise of rights to bid by or on behalf of the seller presents difficult problems. [...]
So well established are these practices that their existence and effects appear to be accepted
without demur, not only by the auctions world but also by judges. It is suggested, therefore,
that specific external regulation is required if the practice of puffing is to be outlawed as has
been recommended. (Frank Meisel, The Modern Law Review, 1996)
1
Introduction
A major discrepancy between auction theory in economics1 and auction practice is the tactical
use of seller’s bids, labeled henceforth as shill bidding, a pervasive phenomenon whose real-life
∗
I would like to thank Pierre-André Chiappori, Philippe Jehiel, Mike Riordan, Bernard Salanié, Robert Wilson
and seminar participants at Columbia University, Robert Wilson’s seminar (Stanford), at the EEA meeting 2009
(Barcelona) and at the University of Bristol for discussions at various stages of this research which started while
the author was visiting Stanford University. All errors are mine.
†
Paris School of Economics, 48 Bd Jourdan 75014 Paris. e-mail: lamy@pse.ens.fr
1
See Krishna (2002) and Milgrom (2004) for recent textbooks.
1
importance is first suggested by the linguistic profusion to describe more and less the same
activity: among them lift-lining, by-bidding, trotting, running, puffing, phantom bids, dummy
bids, phony bids, fake bidders, sham bidders, cappers, decoy ducks, white bonnets, barkers,
fictitious bids. In Cassady’s (1967) book that describes how real-life auctions work and that had
a major influence among the pioneering auction theorists, the topic appears in various chapters.
In auctions law, the topic is also of central importance and is sometimes a source of litigations
between sellers and bidders.2 If the seller advertises the sale as being ‘without reserve’, the
seller is supposed to propose a contract whose acceptance is by bidding: she commits to sell
the object to the highest bona fide bidder whatever the price can be, i.e. for any price above
the opening bid.3 The seller is thus neither allowed to withdraw the good from sale once the
auction has started nor can she submit a bid as any other bidder. On the whole, shill bidding
and sale cancelations have bad reputations among jurists where they are perceived as breaches
of contract that are commonly observed and then tolerated as an usage in the auctions world.
More precisely, shill bidding is mainly viewed either as an hold-up on the time and expenses in
preparing and attending the auction or as a simulation fostering a false appearance of genuine
competition that distorts bidders’ representation of the value of the good. The latter deceitful
aspect has been formalized in Lamy (2009) in a model with interdependent valuations while the
former hold-up issue is the central feature of the present model with purely private valuations.
While there is a large literature on hold-up problems (Che and Sakovics, 2008), the present paper
is, to the best of our knowledge, the first in the economic literature to analyze this hold-up issue
that preys on auctions.
Shill bidding continuously attracts the attention of the media all over the world. In 2007,
Eftis Paraskevaides, one of Britain’s top eBay traders who used to sell more than £1.4m worth
of antiquities a year on eBay, got caught by a journalist of the Sunday Times.4 He has then been
banned for life by eBay once the investigation found that his ex-wife was bidding up the price of
goods he was selling without a reserve price. The aim of this strategy was to boost participation
according to his own allegations. Snapnames is the largest reseller of Web site names that held
above 1,000,000 domain name auctions over the years. The Washington Post5 reports that the
2
See Harvey and Meisel (1995) for U.K. and Mauger-Vielpeau (2002) for France for general textbooks on
auctions law and Yale-Law-Journal (1922) and Meisel (1996 & 2001) for specific papers on seller’s bids and
cancelation rights. We emphasize that this is a controversial topic in law such that no consensus has been reached
so far, as testified by the quotation above from the conclusion in Meisel (1996).
3
The terminology ‘auction without reserve’ is sometimes used only for the case where the opening bid is also
set to zero.
4
See the series of articles: Make me an offer: the eBay bid scam - dealers fix online auctions with a little help
from their friends. The Sunday Times, January 28 2007; Revealed: how eBay sellers fix auctions. The Sunday
Times, January 28 2007; Top trader banned for life over fixed eBay auctions. The Sunday Times, February 4
2007.
5
B. Krebs, Snapnames: Former exec. bid up domain prices, The Washington Post, November 4 2009.
2
company admits the allegation that a former top executive regularly boosted the price on some
auctions since 2005 through a bid account that has been explicitly created for this fraudulent
usage. The recent attention of the media mainly covers online auctions partly because it affects
consumers that were unaware of such unethical if not forbidden manipulations, contrary to
audiences with professional traders where everybody knows that shill bidding is an integral part
of the game such that nobody is actually deceived as argued by Meisel (1996), but also because
shill bidding can take “industrial” proportions on the net with the use of dozen of shill bidders
for a single auction as in the manipulation reported in Lamy (2009) or with the creation of a
commercial company providing a service that automates the process of shill bidding (Ockenfels
et al., 2006). We emphasize that shill bidding is not confined to online auctions. E.g., shill
bidding is pervasive in real estate auctions in Australia and New Zealand, two countries where
private houses are commonly sold through auctions. In 2004, a new legislation has been brought
in to change the way auctions were conducted in New South Wales and Victoria. However, it
is not clear whether shill bidding prohibition can actually be enforced if prosecutions are never
engaged if suspected shill bidders deny the allegations.6 The Australian experience leads to a
new profession: buyers’ advocates. Pioneering in this activity is the former real-estate agent
David Morrell who does not solely explain how to spot shill bidding in Morrell (2004) but whose
company offers to bid for a 1% fee in case of success. In other words, the client finds the property
he is interested to bid on and then go to a buyer’s advocate to which he reports his valuation and
who will bid in the auction on the client’s behalf. If real-life English auctions were fitting with
their theoretical counterparts, then there will be few room for such an activity. David Morrell
argues that a key element of their technical expertise is their ability to unmask the shill bidders
throughout the auction process such that they switch to a bargaining phase with the seller once
there is no real bidder to bid against.
We consider second-price and English auctions in the symmetric independent private value
model and in an environment with entry à la Levin and Smith (1994) (henceforth L&S), i.e.
where potential buyers have to incur a sunk cost in order to be able to learn their valuation and
participate to the auction. L&S has shown that the second-price auction where the reserve price
is set at the seller’s reservation value is an optimal mechanism: the direct gain from raising the
reserve ceteris paribus would be counterbalanced by the loss resulting from the induced reduction
in buyers’ participation. Furthermore, in an ex ante perspective, buyers have no informational
rents such that the seller’s expected payoff and the welfare coincide. The optimal mechanism
implements thus the first best: it allocates the good in an ex post efficient way while implementing
6
T. Giles and G. Mitchell. Dodgy bids clamp fails. Herald Sun, July 30 2005
3
the efficient level of entry. This normative analysis relies on two crucial assumptions: first, the
seller is able to commit fully to the auction mechanism before potential buyers make their entry
decisions and more precisely she is able to commit not to hire a shill bidder; second, the seller does
not revise her reservation value in the course of the game, either from exogenous information or
from endogenous information that comes from the auction itself. We revisit auctions with entry
à la Levin and Smith while relaxing successively those two assumptions.
In a first step, we derive the optimal reserve price in second-price auctions with a strategic
seller deciding whether to incur a cost to hire secretly a shill bidder being able to bid in the same
way as any real bidder. If she hires a shill bidder, she will instruct him to bid up to the optimal
reserve price under exogenous entry, i.e. the optimal reserve à la Myerson (1981). If the cost to
hire a shill bidder is small enough, then the seller would strictly benefit from shill bidding once
the announced reserve is low enough. If the shill bidding constraint is binding insofar as the seller
can not credibly commit not to shill bid if she sets the reserve price at her reservation value,
we show more generally that the optimal reserve lies strictly above her reservation value and
that the corresponding equilibrium involves no shill bidding activity. Under the threat of shill
bidding, the costs from imperfect commitment are then reflected by higher reserves. Those costs
can be so important, in particular if entry costs are high enough ceteris paribus, that postedprices may outperform auctions since the former do not suffer from any commitment failure.
Furthermore, the seller’s expected payoff may not increase in her reservation value such that
she may strictly benefit from ‘burning her ships’. The literature in law notes that surprisingly
it is not uncommon that courts show clemency toward shill bidding: “The consideration which
moves the few dissenters from the rule against puffing appears to be a desire to shield the seller
of property from an unjust price” (Yale-Law-Journal 1922, p.432). Bennett (1883) and Meisel
(1996) also emphasize how courts of equity recognized the right of the seller to participate in the
auction to prevent a sale at a sacrifice. However, this literature has been powerless to formalize
the possible benefits from shill bidding: if it allows to prevent a sale at an undervalue, then it is
argued that it could have been substituted by a public reserve price. The argument is actually
true in the stylized models that are typically considered in auction theory where it is assumed
that the seller knows her reservation value ex ante, but becomes fallacious when this reservation
value is only partially known, e.g. because the seller refines her valuation in the course of the
auction itself, such that shill bidding may offer some valuable flexibility.
In a second step, we consider thus a larger framework where the seller may receive additional
information about her reservation value at the interim stage, i.e. after the choice of the selling
mechanism and potential buyers’ entry decisions. In such a setup, the design of an efficient auc-
4
tion with entry is no longer straightforward, even if shill bidding could be banned. In particular,
auctions where the seller is inactive at the auction stage yield ex post inefficiencies. By avoiding
sales when the seller has the highest valuation, shill bidding provides some valuable flexibility
but at some costs since shill bidders bid strictly above the seller’s valuation which opens the door
for allocative inefficiencies where the seller keeps the good though the highest valuation among
the entrants is above hers. Flexibility can be introduced by an alternative instrument: the right
to cancel the sale after the price has been fixed, which has far less bad reputation among jurists
compared to shill bidding. On the one hand, it prevents the seller to sell at undervalue and
thus seems a perfect substitute to the legitimate motive for shill bidding. On the other hand,
cancelation rights do not entail the undesirable effects associated with shill bidding, i.e. the
possible beliefs’ manipulation effect (absent in the present paper with private values)7 and the
“unfair” rent extraction that consists in making the winner pay a strictly higher price than the
maximum of the seller’s reservation value and the highest valuation among his opponents. With
cancelation rights, the winning price is always equal to the second highest valuation among the
entrants which should be larger than the seller’s reservation value since the sale would have been
canceled otherwise. The second-price auction without reserve but with cancelation rights seems
thus a good candidate for ex post efficiency. Nevertheless, a new source of inefficiency arises: it
may occur that the seller cancels the sale because the winning price is lower than her reservation
value while ex post efficiency would dictate to allocate the good to the highest bidder. In other
words, inefficiency occurs when the seller’s reservation value lies between the first and the second
highest bids. If the seller’s reservation value is common knowledge at the interim stage, then a
simple mechanism can avoid the aforementioned various pitfalls. It is the English auction with
jump bids and with cancelation rights: if the second highest valuation lies below the seller’s
reservation value, then the remaining bidder has the opportunity to make a jump bid up to the
seller’s reservation value. Our analysis provides thus a completely new argument in favor of this
popular auction format which is shown to implement the first best. On the whole, its practical
implementation calls for a regulation that makes a clear distinction between shill bidding and
cancelation rights, while those latter are still associated with a form of hold-up such that allowing
them is often viewed as allowing also shill bidding.
This paper is organized as follows. Section 2 introduces the basic auction model with the
opportunity to hire a shill bidder and lays the foundation of our equilibrium analysis when the
seller’s reservation value is common knowledge ex ante. Section 3 moves to the characterization
7
This yields jurists to make a distinction between anonymous and non-anonymous shill bidding activities.
With the latter, English auctions are transparent such that bidders would not feel ex post regret as in Lamy
(2009).
5
of the optimal reserve price and sheds some light on the influence of the seller’s reservation
value on her expected payoff. Section 4 is devoted to the analysis of posted-prices and their
relative performance with respect to auctions. Section 5 introduces uncertainty about the seller’s
reservation value, the second main ingredient of the paper. We consider then the problem of
designing efficient/optimal auctions which thus asks for flexible mechanisms that leave some
active role to the seller at the auction stage. The proofs are all relegated in Appendices A-H.
2
The Model
Consider a setting with N ≥ 2 symmetric risk-neutral potential buyers and a risk-neutral
seller who wants to sell an indivisible good for which her valuation (or reservation value) is
Xs ≥ 0. Apart from Section 5, XS is assumed to be common knowledge at the beginning
of the game. If they incur a fixed sunk cost ce > 0, then potential buyers are learning their
valuations and become eligible to participate in the subsequent mechanism. Buyers’ valuations
are private information and are independently distributed with a common CDF F (.) that has a
continuously strictly positive density f (.) on its support [x, x] where x > Xs . The hazard rate
function x →
1−F (x)
f (x)
is assumed to be strictly decreasing on [x, x]. For k ≤ n, let F (k:n) denote
the CDF of the k th order statistic among n i.i.d variables distributed according to the CDF F (.).
The timing of the game is as follows. First, the seller announces a reserve price (or opening
bid) r ∈ R+ . Second, each potential buyer decides whether or not to incur the sunk cost ce in
order to enter the auction game and the seller decides simultaneously whether to enroll a shill
bidder. Hiring a shill bidder costs cshill ≥ 0.8 Third, entrants are privately informed about their
valuations and the number of entrants is observed by the seller who gives instructions to the shill
bidder on how to bid in the following auction. Fourth, the entrants and the shill bidder, if any,
are playing a second-price auction. Fifth, if the seller has ex post rights to cancel the sale, then
she is able to decide whether to trade at the winning price or to withdraw the good from sale.
We assume furthermore that buyers are bidding their valuation, i.e. are playing their standard
weakly dominant strategy in second-price auctions:
Assumption A 1 Entrants that have a valuation greater than r are bidding their value.
By default, we consider next that the auction game does not involve cancelation rights. This
last instrument is not considered until Section 5. Without those rights, note that second-price
auctions are then equivalent to English auctions. Our analysis where shill bidding involves some
8
Assuming that the shill bidding cost is constant may abstract from some interesting aspects: in some environments as eBay, a part of the shill bidding cost results from transaction fees paid to the auctioneer in the event
where the shill bidder buys the good. Those fees are typically a function of the final price. Such developments
are left for further research.
6
cost covers thus the two extreme cases where shill bidding is banned (cshill = +∞) and where shill
bidding is allowed for free (cshill = 0), two cases which are respectively referred to as ‘auctions
without shill bidding’ and ‘auctions with shill bidding’. Variations on the selling mechanism at
the fourth stage of the game, in particular posted-prices, are introduced later in the course of
the paper. We do not allow for entry fees/subsidies: we have shown in the working paper version
that our insights extend if such instruments were available.
As in L&S, our analysis is restricted to so-called bidder-symmetric equilibria where potential buyers are using the same strategy at the entry stage. Let q ∈ [0, 1] denote the corresponding
probability to enter the auction. If shill bidding is banned or equivalently if the cost to hire a shill
bidder is prohibitively costly, then our auction model corresponds exactly to the private value
model in L&S and the optimal reserve price policy is to set r = Xs . Once the entry decisions
have been taken, the seller would like to update the reserve price up to the optimal reserve under
exogenous participation, denoted by rM , which is uniquely characterized by the implicit equation
X s = rM −
1−F (rM )
f (rM )
if x −
1
f (x)
< 0 and rM = x otherwise. In our private value model, having
a shill bidder is strategically equivalent to the opportunity to update the previously advertised
reserve price. As a corollary, we obtain in equilibrium that if the seller enrolls a shill bidder, then
she will instruct him to bid rM . Note that the instruction to the shill bidder depends neither
on the number of entrants in the auction nor on the bidding history in an English auction, a
well-known insight that comes from the i.i.d. assumption of the model.9
The rest of this section is devoted to the equilibrium analysis of the auction game when the
seller has announced a reserve r ≤ rM at the first stage.10 An equilibrium is fully characterized
by two variables: potential buyers’ probability to enter the auction q ∈ [0, 1] and the probability,
denoted by p ∈ [0, 1], that the seller enrolls a shill bidder.
From the mechanism design approach (Myerson, 1981), buyers’ expected gross payoffs (i.e.
after the entry costs have been sunk) can be computed from the way the good is assigned to the
buyers according to their valuation profile. More precisely, the expected gross payoff of a buyer
with valuation x and facing n competing entrants is given by:
Z
ϑn (x) +
x
P robwin
n (s)ds,
(1)
x
9
In models with exogenous entry, Graham et al. (1990) and Lopomo (2000) show how departures from the
symmetric independent private value model make shill bidding benefic in English auctions insofar as it allows the
seller to adapt her reserve price policy as a function of respectively the identity of the remaining active bidder
and of the bidding amounts of early dropouts.
10
For r > rM , the analysis is straightforward: there is a unique equilibrium which coincides with the one in
L&S. We obtain from L&S that those reserves are suboptimal since they are outperformed by the reserve price
rM .
7
where P robwin
n (s) corresponds to the probability of winning the good of a buyer with valuation
s and ϑn (x) to the expected gross payoff of a buyer having the lowest valuation x. This formula
holds for any selling mechanism (in particular also for posted-prices). In auctions without shill
n
bidding, we have P robwin
n (s) = F (s) if s ≥ r and 0 otherwise. If a given potential buyer
faces n competing entrants and if there is no shill bidding activity, then the integration of
Eq. (1) over the buyer’s valuation CDF leads to the following ex ante expected gross payoff
Rx Ru
in the auction: r ( r F n (s)ds)dF (u). If the seller enrolls a shill bidder with probability p,
then the way the good is assigned corresponds to the following one: with probability (1 − p),
the good is assigned exactly in the same way as in the auction without shill bidding and with
probability p in the same way as in an auction without shill bidding where the reserve price
would have been set to rM instead of r. The ex ante expected gross payoff is now given by
Rx Ru
Rx Ru
(1 − p) · r ( r F n (s)ds)dF (u) + p · rM ( rM F n (s)ds)dF (u).
Let B(q, r, p) denote the expected gross payoff (i.e. after the entry cost have been sunk) of
a given entrant in the auction if the reserve price is r and the strategies of his opponents are
characterized by q and p. A straightforward calculation leads to:
Z
x
B(q, r, p) = (1 − p) ·
r
Z
(
u
N −1
[qF (s) + (1 − q)]
Z
x
ds)dF (u) + p ·
r
rM
Z
(
u
[qF (s) + (1 − q)]N −1 ds)dF (u).
rM
(2)
Note that B(q, r, p) is thus strictly decreasing in q on [0, 1] and strictly decreasing in p if r < rM .
We make the following two assumptions in order to guarantee that any equilibrium of the game
involves mixed entry, i.e. q ∈ (0, 1).11
Assumption A 2 The entry cost is large enough such that full entry is never an equilibrium:
Z
x
ce >
Z
(
0
u
F (N −1) (s)ds)dF (u) ≡ B(1, 0, 0).
(3)
0
Assumption A 3 The entry cost is small enough such that no entry is not an equilibrium with
the reserve price rM :
Z
x
(u − rM )dF (u) ≡ B(0, rM , 0).
ce <
(4)
rM
Inequality (3) means that entry is not individually rational if all potential buyers decide to enter
the auction even if the reserve price is null and without any shill bidding activity. A fortiori,
entry can not be individually rational if all potential buyers decide to enter the auction for
general reserve prices and with possibly some shill bidding activity, i.e. B(1, r, p) < ce , since
such strategy profiles would leave less rents to the buyers (B(1, r, p) ≤ B(1, 0, 0)). Inequality
Rx Ru
Rx
If N is large enough, then 0 ( 0 F (N −1) (s)ds)dF (u) < r (u − rM )dF (u) such that there exists some entry
M
costs such that both conditions (3) and (4) are satisfied.
11
8
(4) means that entry is individually rational if all the remaining potential buyers decide not to
enter the auction if the reserve price is rM and thus a fortiori for any reserve price below rM and
with possibly some shill bidding activity. Only A2 is actually crucial to our analysis by avoiding
the exogenous entry of all potential buyers. If A3 fails note that there will be no participation
in auctions with shill bidding, which is another illustration of how shill bidding can be harmful.
The aim of this assumption is only to alleviate the analysis by avoiding such degenerate cases.
Since entry is mixed in equilibrium, the potential buyers’ equilibrium equation is given by:
B(q, r, p) = ce .
(5)
For a given p and r, we have B(0, r, p) > ce while B(1, r, p) < ce such that Eq. (5) has a unique
solution on (0, 1), denoted by q ∗ (r, p). For p = 0, let q ∗ (r) denote the solution. By integrating
by parts Eq. (2), we obtain equivalently:
∗
Z
B(q (r), r, 0) =
x
(1 − F (u))[q ∗ (r)F (u) + (1 − q ∗ (r))]N −1 du = ce .
(6)
r
The differentiation of (6) leads to:
(1 − F (r))[q ∗ (r)F (r) + (1 − q ∗ (r))]N −1
∂q ∗
=−
.
Rx
∂r
(N − 1) r (1 − F (u))2 [q ∗ (r)F (u) + (1 − q ∗ (r))]N −2 du
(7)
We obtain thus that q ∗ (.) is a strictly decreasing function for any r ≤ rM (< x). If p = 1, the
solution of the equilibrium equation (5) equals q ∗ (rM ). Furthermore, since B(q, r, p) is strictly
decreasing in q on [0, 1] and decreasing in p on [0, 1], then for any p ∈ [0, 1], any probability of
entry q that satisfies Eq. (5) should satisfy q ∈ [q ∗ (rM ), q ∗ (r)].
We now move to the seller’s side. Let Πn (r) denote the expected payoff of the seller with a
reserve price r when there are n entrants and when she does not hire a shill bidder. We have
Π0 (r) = Xs , Π1 (r) = F (r) · Xs + [1 − F (r)] · r and Πn (r) = F (1:n) (r) · Xs + [F (2:n) (r) − F (1:n) (r)] ·
Rx
r + r udF (2:n) (u) for n ≥ 2. For n ≥ 1, a straightforward calculation leads to:12
Z
Πn (r) − Πn (0) =
r
[Xs +
0
1 − F (u)
− u]d[F (1:n) (u)].
f (u)
(8)
Let RA (XS , r, q) and WA (Xs , r, q) denote respectively the seller’s expected payoff and the expected welfare at the ex ante stage when the reserve price equals r, the probability of entry equals
q and if she does not hire a shill bidder. We have RA (XS , r, q) =
12
PN
n=0
N
n
q n (1 − q)N −n Πn (r). Since
The expression (8) is slightly awkward outside the range [x, x] where f (u) = 0. All over the paper, we should
(u)
read [Xs + 1−F
− u] · f (u) as (1 − F (u)) when f (u) = 0.
f (u)
9
Πn (r) is strictly increasing in the number of entrants n if r ≥ XS , we obtain that RA (XS , r, q)
is strictly increasing in q if r ≥ XS . A straightforward calculation leads also to:
RA (XS , r, q) = [
N X
N
n=0
n
N −n
n
q (1 − q)
r
Z
Πn (0)] +
[Xs +
0
1 − F (u)
− u]d[(qF (u) + (1 − q))N ],
f (u)
(9)
which implies that r → RA (XS , r, q) is strictly increasing on [0, rM ] for any q > 0. A similar
calculation as for RA (., ., .) leads to:
Z
N
WA (Xs , r, q) = [qF (r) + (1 − q)] · XS +
x
ud[(qF (u) + (1 − q))N ] − qN · ce .
(10)
r
As it is already known from L&S, the seller’s expected payoff coincides with the expected welfare
in an equilibrium where potential buyers strictly mix, i.e. WA (Xs , r, q ∗ (r)) = RA (Xs , r, q ∗ (r)).
Let H(q, r) denote the expected benefit of the shill bidding activity before the number of
entrants has been revealed and after the cost to enroll a shill bidder has been sunk, i.e. the
difference between the expected payoff in the auction when the seller has hired a shill bidder
(that bids then rM ) and the expected payoff in the auction when the seller has not hired a shill
bidder: H(q, r) := RA (XS , rM , q) − RA (XS , r, q).13 From Eq. (9), we obtain:
Z
rM
[Xs +
H(q, r) =
r
1 − F (u)
− u]d[(qF (u) + (1 − q))N ].
f (u)
(11)
For any equilibrium where the seller strictly mix, i.e. if p ∈ (0, 1), the seller’s equilibrium
equation is characterized by the indifference between enrolling or not a shill bidder:
H(q, r) = cshill .
(12)
For an equilibrium where the seller never enrolls a shill bidder, we have q = q ∗ (r) and the
following inequality needs to be satisfied:
H(q ∗ (r), r) ≤ cshill .
(13)
For an equilibrium where the seller always enrolls a shill bidder, we have q = q ∗ (rM ) and the
following inequality needs to be satisfied:
H(q ∗ (rM ), r) ≥ cshill .
13
To alleviate the notation, we have dropped the dependence on the variable XS .
10
(14)
Under assumptions A2-A3, we have thus three kinds of candidates to be an equilibrium:
• Type 1: a unique equilibrium without any shill bidding activity and where the entry
probability equals q ∗ (r),
• Type 2: a unique equilibrium where the seller always enrolls a shill bidder and where the
entry probability equals q ∗ (rM ),
• Type 3: equilibria where the seller strictly mix and that are characterized as the solutions
of the equations (5) and (12) and where the entry probability necessarily belongs to the
interval (q ∗ (rM ), q ∗ (r)) for r < rM .
Remark 2.1 If maxq∈[q∗ (rM ),q∗ (r)] H(q, r) < cshill , then the seller would strictly prefer not to
hire a shill bidder for any equilibrium candidate: there is then a unique equilibrium and it is of
type 1. Since H(q, r) is strictly decreasing in r on the range [0, rM ] for any q > 0 and since the
interval [q ∗ (rM ), q ∗ (r)] shrinks when r increases, then the function r → maxq∈[q∗ (rM ),q∗ (r)] H(q, r)
is strictly decreasing on [0, rM ] while the expression equals zero at rM . If cshill > 0, there is
then a threshold rb < rM such that if the reserve price r is (strictly) larger than rb then there is
a unique equilibrium and it involves no shill bidding. On the contrary, if we take re such that
minq∈[q∗ (rM ),q∗ (er)] H(q, re) > cshill (which exists if cshill is small enough), then there is a unique
equilibrium and it is of type 2. When we move from the reserve price re to a reserve r ∈ (b
r, rM ),
then the participation in equilibrium rises from q ∗ (rM ) to q ∗ (r). On the whole, we obtain thus
the surprising insight that participation may be enhanced by higher reserve prices. This results
from the fact that the incentives to hire a shill bidder shrink with the reserve price.
Proposition 2.1 There exists a bidder-symmetric equilibrium in second-price auctions. Biddersymmetric equilibria are of type 1,2 or 3. For a given reserve price r such that H(q ∗ (rM ), r) >
cshill > H(q ∗ (r), r), then equilibria of the three kinds of type coexist.14
The intuition for the multiplicity of equilibria for intermediate shill bidding costs is a strategic
complementarity between the seller’s shill bidding activity and buyers’ participation decisions: if
the seller shill bids more, then buyers participate less which enhances the incentives to shill bid.
For our subsequent normative analysis, we assume that the seller can select her most preferred
equilibrium, which is actually (weakly) Pareto dominant in the set of equilibria since potential
buyers are indifferent between all equilibria. With this selection criterium, we obtain thus a
lower bound on how shill bidding can be harmful. For any r ≥ XS , note that if a type 1
14
In the working paper version, we give a mild sufficient condition such that the inequality H(q ∗ (rM ), r) >
H(q ∗ (r), r) holds for any r < rM .
11
equilibrium exists, then it is the seller’s preferred equilibrium: the seller’s expected payoff equals
RA (XS , r, q ∗ (r)) which is larger than both RA (XS , rM , q ∗ (rM )) − cshill the payoff in a type 2
equilibrium and RA (XS , r, q ∗ (r, p)) the payoff in a type 3 equilibrium.15
3
Optimal reserve price policy
We now consider the strategic choice of the announced reserve price by the seller: we charac-
terize the optimal reserve price that maximizes her expected payoff. The following lemma shows
that for any equilibrium which involves some shill bidding activity then we can pick a reserve
price and an equilibrium that raises a strictly higher expected payoff than the former one.
Lemma 3.1 Suppose that cshill > 0. If there is a reserve price r such that the probability of entry
q and the shill bidding probability p > 0 is an equilibrium profile, then there is a reserve price
r0 ∈ [r, rM ] such that the same probability of entry q and no shill bidding activity is an equilibrium
profile if the announced reserve price is r0 . Furthermore, this latter equilibrium raises a strictly
higher expected payoff than the former one.
As a corollary to Lemma 3.1, there is no loss of generality to restrict the analysis to type 1
equilibria when we are looking for an optimal reserve price, or equivalently to limit ourselves to
reserve prices such that H(q ∗ (r), r) ≤ cshill . Such reserve prices are now called “implementable
reserve prices” according to the following definition.
Definition 1 A reserve price is implementable if there exists an equilibrium without any shill
bidding activity when this reserve price is announced.
Reserve prices above rM are obviously implementable since the incentives to hire a shill bidder
are then null. Below rM , lowering the reserve price has two impacts on the incentives to hire a
shill bidder: on the one hand, the standard channel through the variation of the reserve price
ceteris paribus makes shill bidding more profitable; on the other hand, lowering the reserve price
raises the probability of entry which may negatively impact the incentives to shill bid. Next
lemma shows that the first channel always dominates the second one under a mild additional
assumption.
f (x)
16 then the gain from hiring a shill
Lemma 3.2 If x → (XS − x) 1−F
(x) is decreasing on [0, x),
15
Formally, the arguments result respectively from Appendix B and since RA (XS , r, q ∗ (r)) = WA (XS , r, q ∗ (r))
for any r ≤ rM (under assumptions A2-A3) and from the fact that RA (XS , r, q) increases with q for r ≥ XS .
f (x)
16
Note that the assumption on the hazard rate guarantees that x → (XS − x) 1−F
is decreasing on [XS , x).
(x)
The monotonicity holds also on the range [0, x) since the function is null and thus constant on this range. On the
whole potential failures of this additional assumption may arise only on the range [x, XS ] if XS > x.
12
bidder decreases with the announced reserve price if potential buyers participate as if there were
no shill bidding activity: r → H(q ∗ (r), r) is decreasing in r.
We obtain thus that the set of implementable reserve prices is an interval, denoted by [r∗ , ∞).
If H(q ∗ (0), 0) < cshill , then any reserve price is implementable, i.e. r∗ = 0. Otherwise, r∗ is the
lowest solution in R+ of the equation
H(q ∗ (r), r) = cshill .
(15)
For the rest of the paper, we only need to assume that the set of implementable reserve prices
is an interval, denoted by [r∗ , ∞), as guaranteed by the monotonicity assumption in Lemma 3.2.17
Note that this is always true if cshill = 0 where H(q ∗ (r), r) > 0 if and only if r < rM such that
r∗ = rM . As a corollary to Lemma 3.1, the seller’s maximization program reduces then to:
max∗ WA (Xs , r, q ∗ (r)).
r≥r
(16)
The constraint r ≥ r∗ is called the ‘shill bidding constraint’.
Proposition 3.3 The seller maximizes her expected payoff by setting the lowest implementable
reserve price above her reservation value, i.e. her optimal reserve price ropt equals max {Xs , r∗ },
while the equilibrium without shill bidding is played.
We obtain in particular that the optimal reserve induces no shill bidding activity. This
contrasts with Lamy (2009) where shill bidding always arises in a model with interdependent
valuations and where the bidding strategy of the shill bidder is mixed in equilibrium. This contrasts also with the evidence we present in the introduction about the practical importance of
shill bidding. However, we emphasize that we could easily generate some shill bidding activity on the equilibrium path while maintaining the private value framework once we introduce
uncertainty about the seller’s valuation as it is done in Section 5: if the support of the seller’s
valuation is large enough, then it would be too costly to announce a reserve that makes shill
bidding non profitable for each possible realization of the seller’s valuation. The insight that
should be retained from Proposition 3.3 is rather that even if shill bidders are rarely used, this
threat can have a huge impact on the seller’s optimal reserve price policy. It gives incentives to
set higher reserves in order to credibly commit not to shill bid. In Proposition 3.3, when r∗ > Xs ,
which is the case if cshill = 0 and more generally if the shill bidding cost is small enough, then
the seller’s expected payoff and the welfare are strictly below the ones that would prevail in an
17
This assumption is far from being necessary and has been chosen mainly for simplicity.
13
environment where shill bidding is banned. In other words, from an empirical perspective, we
emphasize that shill bidding could have a first-order detrimental impact on the welfare/revenue
even if the percentage of auctions with shill bidders is rather limited. On the contrary, if cshill is
large enough, then r∗ = 0 such that hiring a shill bidder is never profitable if potential entrants
base their entry decisions on a no-shill bidding scenario and we are back to models without shill
bidding as in L&S. Next we say that shill bidding is binding (resp. not binding) if r∗ > Xs (resp.
r∗ ≤ XS ).
Some rationale for ‘burning your ships’
In the auction literature, the seller’s reservation value is typically normalized to zero since
it plays no salient role for auction design. Furthermore, under either exogenous or endogenous
entry, if buyers’ valuations do not depend on the seller’s reservation value, then the seller’s
expected payoff is non-decreasing in her reservation value: by choosing the mechanism that was
optimal with a lower reservation value, she is guaranteed to obtain at least the same payoff while
strictly benefiting from her higher valuation in the case she keeps the good. Most of the auction
design literature considers that the seller is able to commit to be inactive in the announced
mechanism and there is thus no rationale for a seller to ‘burn her ships’. This general insight
is no longer valid under imperfect commitment if the set of implementable mechanisms, i.e. the
set of announced mechanisms she can credibly commit to, may depend on the seller’s (known)
reservation value and more precisely if this set may shrink with the seller’s reservation value.
More formally, when shill bidding is binding, we have:18
dWA
∂WA
dWA
∂r∗
(XS , q ∗ (ropt ), ropt ) =
(XS , q ∗ (ropt ), ropt ) +
(XS , q ∗ (ropt ), ropt ) ·
.
dXS
∂XS
dr
∂XS
|
{z
}
|
{z
}
| {z }
>0
≤0
from Eq. (10)
from Lemma B.1
(17)
≥0
The first term is the standard term which is driven by the event where the seller keeps the good
and which is always positive while the second, which is the one resulting from the shill bidding
constraint, is negative since the set of implementable reserve prices shrinks with the seller’s
reservation value. Next proposition gives a simple sufficient condition that guarantees that the
seller’s expected payoff fails to be increasing in her reservation value when cshill = 0.
Proposition 3.4 In auctions with shill bidding, if
1
f (x)
< x and ce ≤ e−1 ·
Rx
x
(1 − F (u))du, then
the seller’s expected payoff is non-monotonic with respect to her reservation value.
18
We have
dr ∗
dXS
=−
∗
dH (q ∗ (r ∗ ),r ∗ )
dXS
dH (q ∗ (r ∗ ),r ∗ )
dr
which is positive since
dH
(q ∗ (r∗ ), r∗ )
dr
≤ 0 [Lemma 3.2] and
dH(q ∗ (r),r)
dXs
≥0
[Eq. (11) and since q (r) does not depend on XS ]. See the working paper version for more comparative statics.
14
The condition
1
f (x)
< x guarantees the existence of a reservation value such that rM = x
and could be dropped if we allow the seller’s reservation value to be negative. When rM = x,
the direct impact of increasing XS ,
∂WA
∂XS ,
is given by the event where there is no entrants while
the impact through the shill bidding constraint, i.e. the last term in (17), is driven by the event
with a unique entrant, which is the unique case where the reserve influences the final allocation.
If ce is small, then the level of entry is high which makes this later event much more probable
than the former. It is thus intuitive that if ce is small enough then the term resulting from the
imperfect commitment channel may overwhelm the standard force in favor of a larger reservation
value, and so that the seller may finally benefit from ‘burning her ships’.
Arozamena and Cantillon (2004) analyzed a related issue: the benefits for a bidder to invest
in order to upgrade the CDF from which his valuation is picked. In first price auctions, it could
happen that a bidder losses from an upgrade or equivalently gains if his valuation is picked from
a less favorable distribution. Here we have considered the incentives for the seller to invest in
order to upgrade her valuation and we have obtained a similar counterintuitive loss.19
4
Auctions with shill bidding versus posted-price selling
The current evolution of eBay’s business and the success of Amazon is a puzzle from a
standard auction design perspective: 56% of worldwide sales on eBay are now concluded at a
price predetermined by the seller (e.g. by means of the “Buy it Now” option in auctions) while
eBay’s new CIO made a U-turn to the company’s motto by emphasizing that eBay’s future lies
in fixed-price sales. In early 2008, eBay has also modified her fee structure in favor of fixed-price
selling mechanisms.20 This section is devoted to posted-prices and how they may outperform
auctions though they do suffer intrinsically from ex post inefficiencies.21
The posted-price game we analyze corresponds to the variation of the previous auction game
where the seller announces a posted-price, also denoted by r ∈ R+ , instead of a reserve price
at the first stage, while the entrants report simultaneously whether they are willing to buy the
19
The simulations reported in the working paper version show that the incentives to ‘burn its ships’ arise under
natural parametrizations and that their extent can be important.
20
See C. Holahan, Auctions on eBay: a dying breed, BusinessWeek, June 3 2008 and A. Jamieson, eBay auctions
overtaken by fixed price sales, The Telegraph, December 10 2009.
21
There are several papers advocating for such an insight but with completely orthogonal arguments. Campbell
and Levin (2006) adopt a standard mechanism design perspective à la Myerson (1981), but consider buyers with
interdependent valuations. Wang (1993) consider different transaction costs between fixed price and auction
mechanisms. In a setting with competing mechanisms that involve the same transaction costs, Eeckhout and
Kircher (2010) consider different search technologies. Our model with entry costs can also be viewed as a reduced
form of a model with competing mechanisms: ce would capture the payoff from taking an alternative mechanism
as an outside option. However, with respect to Eeckhout and Kircher’s (2010) terminology, our analysis is limited
to a search technology where ‘meeting are non-rival’ such that their channel in favor posted-prices would never
arise.
15
good at the price r in the fourth stage. If several entrants are demanding the good at that price
then the winner is selected at random with equal probability. Obviously, only posted-prices with
r ≥ Xs make sense and there is then no room for any shill bidding activity since it would only
reduce the probability to sell the good while it can not manipulate the final price. Contrary to the
literature on optimal search procedures as in Weitzman (1979) or Crémer et al. (2007), the way
we model posted-price selling does not involve a sequential structure that would allow to save up
entry costs. Our model abstracts thus from the gains that result from a coordination of the entry
process. Indeed the optimal mechanisms in such environments would involve both posted-price
and auction-like ingredients. Furthermore, auctions would also gain from coordinated entry, e.g.
by means of a sequential procedure as in Engelbrecht-Wiggans (1993). Our insights are robust
to coordinated entry.
If a given potential buyer faces n entrants and if the posted-price equals r, then his probability
to obtain the good if his valuation is above r is given by:
P (r, n) :=
n X
n
j=0
j
F (r)j (1 − F (r))n−j ·
1
1 (1 − F n+1 (r))
=
.
n−j+1
n + 1 (1 − F (r))
His expected gross payoff if he incurs the entry cost is thus given by
Rx
r
(u − r) · P (r, n)dF (u).
Let BP (q, r) denote the ex ante gross payoff of a given entrant under the posted-price r and if
the probability of entry of his opponents equals q. A straightforward calculation leads to:
BP (q, r) =
[1 − (qF (r) + (1 − q))N ]
·
N q(1 − F (r))
Z
x
(u − r)dF (u).
(18)
r
Note that BP (0, r) = B(0, r, 0) such that Assumption A3 implies that participation is strictly
positive for any posted-price below rM , which guarantees that optimal posted-prices involve
strictly positive entry. Suppose now that an optimal posted-price rP∗ exists where entrants make
profit, i.e. BP (1, rP∗ ) > ce , then there exists r > rP∗ such that BP (1, r) = ce (by continuity and
since BP (1, x) = 0), which raises a contradiction with the optimality of rP∗ . On the whole, we
have shown that the equilibrium level of entry q satisfies BP (q, rP∗ ) = ce at an optimal postedprice rP∗ .
Let WP (Xs , r, q) denote the expected welfare if the posted-price equals r and the probability
of entry equals q. A straightforward calculation leads to:
N
N
Z
WP (Xs , r, q) = (qF (r) + (1 − q)) · XS + (1 − (qF (r) + (1 − q)) ) ·
u
r
16
x
dF (u)
− qN · ce . (19)
(1 − F (r))
The seller’s maximization program with posted-prices is then given by:
max
q,r:r≥XS
BP (q,r)=ce
WP (Xs , r, q).
(20)
For any r < x and q > 0, we have WP (Xs , r, q) < WA (Xs , r, q): by assigning the good
randomly if there are multiple entrants, posted-prices are less efficient than auctions since when
the good is sold, it is not necessarily assigned to the entrant with the highest valuation. When
shill bidding is not binding, the optimal auction corresponds to the solution of the maximization
program maxr,q WA (Xs , r, q) (Appendix E) and the corresponding welfare and thus also the
seller’s expected payoff are then greater than in any posted-price. The picture is completely
different when shill bidding is binding: posted-prices may then outperform the optimal auction.
The intuition is that the seller can credibly commit to any possible posted-price above her
reservation value and in particular the optimal one, while in auctions she can credibly commit
only to a limited set of reserve prices.
Next proposition formalizes how the level of the entry costs influences the relative performance
of posted-prices and auctions with shill bidding. We obtain in particular that posted-prices
become benefic in environments with high entry costs. Nevertheless, we emphasize that the
result does not come from the standard channel where posted-prices are a way to save on entry
costs due to miscoordination as in the aforementioned literature. The point here is that when
the entry cost raises, then there is a threshold from which participation becomes null in auctions
with shill bidding. On the contrary, at this threshold, there is still some participation for postedprices that are strictly lower than rM . When ce lies above this threshold, then the welfare is
stuck to XS in auctions with shill bidding while it is strictly above XS under the optimal postedprice as long as ce lies below the entry cost threshold from which participation becomes null in
an auction without shill bidding and with the reserve price XS , which corresponds also to the
threshold from which participation becomes null with the posted-price XS . Below this threshold,
it is thus possible to raise some participation and then to make some benefit from trade with a
posted-price that lies strictly above XS .
Proposition 4.1 Ceteris paribus, there is a threshold c > B(1, rM , 0) such that auctions with
shill bidding strictly outperform optimal posted-prices if ce < c and a threshold c < B(0, rM , 0)
such that optimal posted-prices strictly outperform auctions with shill bidding if ce > c.22
22
If B(1, 0, 0) ≥ c > B(1, rM , 0), then the condition ce < c is vacuous since it stands in conflict with Assumption
A2. In auctions with shill bidding, all our analysis extends straightforwardly if we replace (3) in A2 by the
inequality ce > B(1, rM , 0) such that there is always a non-vanishing range of entry costs such that the proposition
applies. On the contrary, the condition ce > c does not stand in conflict with Assumption A3.
17
Proposition 4.1 emphasizes the role of entry costs in the comparison between auctions and
posted-prices. Another crucial ingredient is the relative heterogeneity between buyers. When
the difference (x − x) goes to zero and with XS < x, then the ex post inefficiencies inherent to
posted-price selling vanish such that the performance of the optimal posted-price converges to
the one of the optimal auction without shill bidding.23,24 On the contrary, the gain from trade
in auctions with shill bidding goes to zero (it is actually equal to zero once x − x < ce ): we have
rM ≥ x such that entrants have to pay at least x, which discourages entry and makes the welfare
converging to XS .
Remark 4.1 On the one hand, online markets seem to favor auctions against posted-prices
since those markets are reducing entry costs compared to traditional brick and mortar markets.
On the other hand, online markets homogenize buyers’ valuations since opportunity costs are
inherent to valuations and that online markets homogenize those costs, at least for goods that
are frequently sold.
5
Efficient auctions with uncertain reservation values?
Our analysis so far has formalized the rigorous perspective in the literature in law. Commit-
ting to low reserve prices encourages participation. To be effective, the reserve price commitment
should also include a banning of any shill bidding activity or of withdrawing the good from sale,
two kinds of instruments that would make an hold-up on the entry costs that have been sunk
by the entrants. The literature in law puts forward two kinds of motives for judges’ clemency
toward shill bidding. First, it seems legitimate in presence of a ring: Marshall et al’s (1990)
early work on shill bidding in the economic literature actually adopts this view, shill bidding
allows the seller to adjust the reserve depending on the strength of the highest bidder, in particular whether she is facing a ring or not, and would then reduce buyers’ incentives to collude.
Second, shill bidding may be legitimate if it is used to prevent a sacrifice, i.e. a sale at a price
below the seller’s reservation value, as courts of equity recognized at some points. The literature
in law is actually embarrassed with both arguments. On the one hand, they argue that rings
should not be fought with a fraudulent activity but with legal means. On the other hand, they
∗
N q (XS )
Consider the limiting case x := x = x. The welfare is maximized with the posted-price rP∗ = x− 1−(1−q
∗ (X ))N ·
S
∗
∗
∗
ce . From Eq. (18), this implies that BP (q (XS ), rP ) = ce , i.e. the level of participation equals q (XS ), the one
prevailing in the auction without shill bidding when the reserve equals XS . The difference between those two
optimal mechanisms lies in the price paid when there is at least one entrant: under the posted-price rP∗ , it is always
equal to rP∗ while in the optimal auction without shill bidding the price is equal either to XS or x, respectively
in the case where there is a unique entrant or several entrants. However, the expected price paid by the winner
is the same in both mechanisms.
24
Wang (1993) also obtains that less dispersed valuations among buyers is a driving force in favor of postedprices.
23
18
emphasize that if the seller wants to secure herself against a sacrifice, the use of an appropriate
reserve price is sufficient. This second argument does actually hold in the limited theoretical
model considered so far where we have assumed that the seller knows her reservation value ex
ante: second-price/English auctions with a reserve price equal to the seller’s reservation value
and where the seller would be able to commit not to shill bid are efficient/optimal mechanisms.25
This argument becomes fallacious in a larger framework when the seller is uncertain about her
reservation value ex ante. It could seem awkward that the seller significantly revises her reservation value in the short period of time between the advertisement and the course of the auction.
Nevertheless, a large part of the seller’s reservation value comes from the good’s “market value”
(e.g. due to future re-auctioning opportunities) which may be largely unknown to the seller ex
ante and where the auction itself -through the number of participants and the bid history- could
matter a lot.
From now on, we consider more general environments where the seller’s reservation value is
eS may depend on an interim public signal SeP , on
a random variable whose realization interim X
an ex ante private signal VeA and on an interim private signal VeI . The interim stage refers to the
stage after potential buyers’ entry decisions and before the selling mechanism starts. We assume
that those signals are distributed independently of the number of entrants and their valuations.
eS = U (SeP , VeA , VeI ).
X
(21)
Remark 5.1 To simplify our analysis, we have implicitly considered that those additional signals
are exogenous and disclosed before the auction starts. While it seems to contradict our initial
motivations where we emphasize that the seller may refine her valuation through the course of
the auction itself, most of our insights could be also obtained with models with endogenous
information that hinges on the bids themselves, e.g. from the prices at which the losing buyers
have exited in the English button auction.
We use then the terminology ‘complete information on the reservation value’ for the cases
where U depends solely on SeP while ‘incomplete information on the reservation value’ refers to
more general environments where U may depend also on additional information that is private
information of the seller. Among the latter environments, ‘ex ante private information’ refers to
the environments where U does not depend on the interim private signal VeI .
Definition 2 For a given environment, we say that a mechanism implements the first best if,
in equilibrium, it maximizes the ex ante expected welfare.
25
As shown by L&S, this optimality result holds more generally in the symmetric conditionally independent
private value model. All the results of this section extend to such a framework.
19
The first best criterium in definition 2 does not take into account the incentive compatibility
and individual rationality constraints that may limit what can be implemented in equilibrium
with a (feasible) mechanism. Furthermore, we also implicitly assume that entry is uncoordinated
as in L&S:26 the welfare is then characterized by an entry probability (that can only depend on
the information known at the ex ante stage) and an assignment rule among the entrants.
Lemma 5.1 (Characterization of first best mechanisms) A mechanism implements the first
best if and only if it is almost surely ex post efficient, i.e. it allocates the good to the agent (including the seller) with the highest valuation with probability one, and the probability of entry is
the same as the one that would prevail in a second-price auction where the reserve price would
eS and if potential entrants knew the
be set at the interim stage at the seller’s reservation value X
signal VeA .
If entry is mixed in equilibrium, then the ex ante welfare coincides with the seller’s expected
payoff such that any mechanism that implements the first best is also optimal for the seller.
5.1
Complete information on the reservation value
In the following, we consider that the seller is uncertain about her reservation value ex ante
eS takes multiple values in (x, x) with positive probability. The second-price/English
insofar as X
auctions analyzed so far (i.e. without cancelation rights) are then powerless to implement the
first best. Auctions where shill bidding does not occur in equilibrium, i.e. with a unique reserve
price r announced ex ante, are ex post inefficient with positive probability: for some realizations
of buyers’ valuations, the good will stay in the seller’s hand while it would be optimal to sell (if
eS ) or the good will be sold while it would be optimal that the seller keeps it (if r < X
eS ).
r>X
eS ) that are strictly above X
eS , we know that shill bidding
Since shill bidders submit bids rM (X
is not an appropriate solution insofar as it yields some ex post inefficiencies.
A milder instrument that introduces flexibility and that receives attention in the literature
in law is the possibility to cancel the auction: the seller’s optimal strategy is then to cancel
the auction if and only if the auction price lies below her valuation.27 We enter slightly into
the details of the analysis of the second-price auction with cancelation rights. To simplify the
presentation we assume that there is no reserve price while it will become straightforward that
positive reserves will not restore the first best. The probability to win the good for a bidder
26
Our first best corresponds thus to some constrained efficiency according, e.g., to Eeckhout and Kircher’s
(2010) terminology.
27
We emphasize that what we call cancelation rights here is a right to withdraw the good from sale after the
price has been fixed, a much stronger device than the right to withdraw the good from sale before the auction
has begun which is, on the contrary, never precluded in auctions law.
20
eS is given by
with valuation u, with n opponents and the seller’s valuation being equal to X
eS )] for u > X
eS and 0 otherwise: winning occurs if he has the highest valuation
[F n (u) − F n (X
among the entrants while the second highest valuation is above the seller’s valuation such that the
seller does not cancel the sale. From a straightforward calculation, we obtain that the expected
welfare is given by:
Wcsp (q) =
Z x
eS )(qF (X
eS ) + (1 − q))N −1 d[F (u)]], (22)
(u − X
− EXeS [
e
XS
{z
}
|
welfare under ex post efficiency
eS , X
eS , q)]
EXeS [WA (X
|
{z
}
inefficiency term
if the probability of entry equals q. In second-price auctions, cancelation rights are opening
the door for a new source of ex post inefficiency: if only one bidder has a valuation above the
seller’s reservation value, then the final price would lie strictly below the seller’s reservation
value such that the seller will keep the good while it would be ex post efficient to allocate it to
the provisionally winning bidder.28 Note that the latter would be prepared to pay the seller’s
reservation value. Nevertheless, the second-price auction does not leave him the opportunity
to make such an offer to the seller contrary to the English auction where he could top his own
winning bid by submitting a jump bid up to the seller’s reservation value since it is common
knowledge at the auction stage under complete information. From Lemma 5.1, we obtain thus
that, with the opportunity to submit jump bids, the English auction implements the first best.
Our findings are summarized in the following proposition.
Proposition 5.2 Second-price auctions (with or without cancelation rights) and English auctions without cancelation rights do not implement the first best if the seller is uncertain about
her reservation value ex ante.
The English auction with cancelation rights, no reserve price and the possibility to submit jump
bids always implements the first best under complete information on the seller’s reservation value.
In pure private value environments, second-price and English auctions are typically viewed
as strategically equivalent: Proposition 5.2 provides a new argument in favor of the latter in
presence of cancelation rights and the opportunity to use jump bids. Jump bids have received
some attention in the literature where they have been perceived as offering buyers an opportunity
to signal their strength in the auction process.29 Here the motives for submitting a jump bid are
radically different: while they typically occur at the beginning of the bidding process in previous
works, as in Fishman (1988) where a firm aims to discourage information acquisition from her
28
The kinds of ex post inefficiency are different in a second-price auction without shill bidding, when the seller
has hired a shill bidder or with cancelation rights such that there is no general ranking between those standard
formats as illustrated by the simulations in the working paper version.
29
See Hörner and Sahuguet (2007) for a recent contribution.
21
opponents, here a given bidder may benefit from topping his own bid at the end of the sale when
he is the unique remaining active bidder. The motives are thus seller-oriented: it allows to take
into account the seller’s preferences without giving her to much bargaining power at the auction
stage.
Proposition 5.2 pleads in favor of a ban of shill bidding, if such a prohibition could be enforced
which does not seem an easy task. This relies also on a regulation where a clear distinction has
been made between shill bidding and cancelation rights, which is far to be always the case in
auctions law, and where the latter are allowed contrary to shill bids. Nevertheless, we would
like to emphasize that allowing cancelation rights and raising only mildly the shill bidding costs
could be rather counterproductive in practice: under the perspective that an important part of
the shill bidding costs comes from the auction fees that the seller has to pay to the auctioneer in
the case where the shill bidder wins the auction (see footnote 8), then cancelation rights are a
licence to shill since it allows the seller to cancel the auction once his shill is the winner in order
to avoid to pay the final fees.30
Remark 5.2 If entrants can submit multiple bids (e.g. by using shill bidders themselves), then
second-price auctions can implement the first best: each participant would submit two bids,
one up to his valuation and the other up to the seller’s valuation. However, with regards to
remark 5.1, such a solution is awkward: it would not be robust to learning from the course of
the auction, i.e. to the motives to cancel the sale that come from the bidding record itself. On
the contrary, in the English auction, once all buyers except one have exited the auction, then the
price below which the seller would cancel the sale would become common knowledge between the
seller and the buyers. In other words, when only one bidder remains in the English auction, then
the bulk of the uncertainty about the seller’s reservation value is cleared up. General models
where the seller would possibly refine her knowledge about her reservation value and about
buyers’ valuation distributions from the bidding history raise additional issues: buyers may have
incentives to submit multiple bids to influence the seller. In this vein, David Morrell’s company
reports that one of their tricks is to raise some fictitious bids on their own, a strategy whose aim
is precisely to influence the seller’s beliefs which matters at the bargaining stage when there is
only one remaining active bidder.31 A similar roundabout motive for buyers’ shill bidding occurs
30
Sellers withdrawing goods from sale or withdrawing bids (which is allowed under drastic conditions) are
perceived as suspect on forum devoted to online auctions where it is interpreted as a (prohibited but pervasive)
strategy to avoid the shill bidder being the winner. On the contrary, the top eBay trader mentioned in the
introduction confessed that he thought that eBay knew that he was using shills regularly but that it was tolerated
because he was paying regularly the fees when his shill was winning the auction.
31
Consistent with this practitioner’s perspective, Levin and Smith (1996) show that under affiliated private
values the optimal reserve price ex post is decreasing in the number of participants such that buyers would benefit
from hiring shill bidders in order to convince the seller that there are more participants.
22
in a case before the Court reported by Bennett (1883) (p.16) as an example of bidder’s fraud
where a bidder employs a shill bidder known to have had connections with the seller in the past:
the purpose of buyers’ shill bidding was to influence the strategy of other potential real buyers
and more precisely to discourage them from entering the auction because they thought that the
seller will surely shill bid.
Remark 5.3 When the seller’s reservation value is known and if shill bidding could not be
banned, then the first best can be implemented by an alternative standard auction format:
the first price auction with the reserve price set at the seller’s reservation value. However,
with uncertain reservation values, it is straightforward that first price auctions (possibly with
cancelation rights) can not implement the first best.
5.2
Incomplete information on the reservation value
Consider first environments with ex ante private information. It can be easily checked that
if the seller can communicate about her ex ante private information, but only on a pure cheap
talk basis and at the ex-ante stage, then truthful revelation is typically not an equilibrium in the
English auction with cancelation rights since the seller would always prefer to make believe to
potential entrants that she has a lower valuation in order to boost participation.32 Nevertheless,
the first best can be implemented if the seller delegates the exercise of her cancelation rights to
a third party.
Definition 3 A delegated cancelation rights policy is a function D : SP → R+ that maps to any
interim public signal SeP a threshold D(SeP ) such that the third party cancels the auction if and
only if the final price is strictly below D(SeP ).
The truthful delegated cancelation rights policy corresponds to the one where D(.) := U (., VeA ) if
her private signal is VeA .
Proposition 5.3 The English auction with the truthful delegated cancelation rights policy, no
reserve price and the possibility to submit jump bids implements the first best if the seller is
privately informed ex ante.
We are not aware of any formal use of a delegated cancelation rights policy in real-world auctions. Nevertheless, the expertise role of traditional auction houses33 could be viewed as playing
32
Making believe that she has a “lower” signal that she actually has has also an additional effect: in the event
where only one bidder has a valuation above the seller’s reservation value, then the seller will cancel the sale,
while she would have sold the good if buyers knew her true ex ante private signal. However, this side effect is
neutral from the perspective of the seller’s payoff since this is an event where the seller’s payoff would have been
eS if buyers knew her true signal, i.e. exactly the same payoff when she keeps the good.
X
33
See Ashenfelter (1989) and Ashenfelter and Graddy (2003).
23
a related role: the information they disclose reduces the informational asymmetry between the
seller and the potential buyers, it can be partially viewed as an agent certifying the signal VeA .
In any cases, the solution in Proposition 5.3 is powerless to implement the first best in more
general environments with private information at the interim stage. With bilateral asymmetric
information at the interim stage, then we know from Myerson and Satterthwaite (1983) that no
budget-balanced mechanism is ex post efficient. As a corollary of Lemma 5.1, no budget-balanced
mechanism can implement the first best.34
Corollary 5.4 (General impossibility with bilateral asymmetric information) No budgetbalanced mechanism can implement the first best if the seller is privately informed at the interim
stage.
References
[1] L. Arozamena and E. Cantillon. Investment incentives in procurement auctions. Rev. Econ.
Stud., 71(1):1–18, 2004.
[2] O. Ashenfelter. How auctions work for wine and art. J. Econ. Perspect., 3(3):23–36, 1989.
[3] O. Ashenfelter and K. Graddy. Auctions and the price of art. J. Econ. Lit., 41(3):763–787,
2003.
[4] E. Bennett. Auctions sales. The American Law Register, 31(1):1–21, 1883.
[5] C. Campbell and D. Levin. When and why not to auction. Econ. Theory, 27(3):583–596,
2006.
[6] R. Cassady. Auctions and Auctioneering. University of California Press, 1967.
[7] Y.-K. Che and J. Sákovics. Hold-up problem. In S. N. Durlauf and L. E. Blume, editors,
The New Palgrave Dictionary of Economics. Palgrave Macmillan, 2008.
[8] J. Crémer, Y. Spiegel, and C. Z. Zheng. Optimal search auctions. J. Econ. Theory,
134(1):226–248, 2007.
[9] J. Eeckhout and P. Kircher. Sorting versus screening: Search frictions and competing
mechanisms. J. Econ. Theory, 145(4):1354–1385, 2010.
[10] R. Engelbrecht-Wiggans. Optimal auctions revisited. Games Econ. Behav., 5(2):227–239,
1993.
[11] M. J. Fishman. A theory of preemptive takeover bidding. RAND J. Econ., 19(1):88–101,
1988.
[12] D. A. Graham, R. C. Marshall, and J.-F. Richard. Phantom bidding against heterogenous
bidders. Econ. Letters, 32:13–17, 1990.
34
We do not enter into the technical details. In particular, Myerson and Satterthwaite’s impossibility result
eS may lie strictly above x. Furthermore, this general impossibility result relies on the
relies on the point that X
statistical independence that has been assumed both between buyers and also between the seller and the buyers.
As in McAfee and Reny (1992), correlated information would enable the seller to implement the first best.
24
[13] B. Harvey and F. Meisel. Auctions Law and Practice. Oxford University Press, 1995.
[14] J. Hörner and N. Sahuguet. Costly signalling in auctions. Rev. Econ. Stud., 74(1):173–206,
2007.
[15] V. Krishna. Auction Theory. Academic Press, 2002.
[16] L. Lamy. The shill bidding effect versus the linkage principle. J. Econ. Theory, 144:390–413,
2009.
[17] L. Lamy. ‘Upping the ante’: how to design efficient auctions with entry? miméo, 2011.
[18] D. Levin and J. Smith. Equilibrium in auctions with entry. Amer. Econ. Rev., 84(3):585–599,
1994.
[19] D. Levin and J. Smith. Optimal reservation prices in auctions. Econ. J., 106(438):1271–1283,
1996.
[20] G. Lopomo. Optimality and robustness of the english auction. Games Econ. Behav., 36:219–
240, 2000.
[21] L. Mauger-Vielpeau. Les ventes aux enchères publiques. Economica, 2002.
[22] R. P. McAfee and P. J. Reny. Correlated information and mechanism design. Econometrica,
60(2):395–421, 1992.
[23] F. Meisel. ‘Upping the ante’: Market distortion in auction sales. Mod. Law Rev., 59(3):398–
413, 1996.
[24] F. Meisel. What price auctions without reserve. Mod. Law Rev., 64(3):468–472, 2001.
[25] P. Milgrom. Putting Auction Theory to Work. Cambridge Univ. Press, Cambridge, 2004.
[26] D. Morrell. Real Estate Myths exploded. Lothian Books, 2004.
[27] R. Myerson. Optimal auction design. Mathematics of Operation Research, 6(1):58–73, 1981.
[28] R. Myerson and M. Satterthwaite. Efficient mechanisms for bilateral trading. J. Econ.
Theory, 29(2):265–281, 1983.
[29] A. Ockenfels, D. Reiley, and A. Sadrieh. Online Auctions, chapter 12, pages 571–628.
Handbook of Economics and Information Systems. Elsevier Science, 2006.
[30] R. Wang. Auctions versus posted-price selling. Amer. Econ. Rev., 83(4):838–51, 1993.
[31] M. Weitzman. Optimal search for the best alternative. Econometrica, 47(3):641–54, 1979.
[32] Yale-Law-Journal. Agreements for fictitious bids at auctions. Yale Law Journal, 31(4):431–
436, 1922.
25
Appendix
A
Proof of Proposition 2.1
If there is no equilibria of type 1 or 2, then both inequalities (13) and (14) fail to hold: we
have thus H(q ∗ (rM ), r) < cshill < H(q ∗ (r), r). Note that H(q ∗ (r), r) > 0 implies that r < rM .
From the intermediate value theorem, there exists qb ∈ (q ∗ (rM ), q ∗ (r)) such that H(b
q , r) =
cshill . Since B(q, r, p) is strictly decreasing in q for any p, we obtain from B(q ∗ (r), r, 0) = ce
and B(q ∗ (rM ), r, 1) = ce that B(b
q , r, 0) < ce and B(b
q , r, 1) > ce . Another application of the
intermediate value theorem guarantees the existence of pb ∈ (0, 1) such that B(b
q , r, pb) = ce . On
the whole, the pair (b
q , pb) defines an equilibrium of type 3.
If H(q ∗ (rM ), r) > cshill > H(q ∗ (r), r), then equilibria of type 1 and 2 coexist. Note also that
H(q ∗ (rM ), r) > 0 implies that r < rM . Furthermore, from the intermediate value theorem, there
exists qb ∈ (q ∗ (rM ), q ∗ (r)) such that H(b
q , r) = cshill . The existence of an equilibrium of type 3 is
obtained exactly as above.
B
Some properties of q → WA (Xs , r, q)
Lemma B.1
•
• If x > r ≥ XS , then q → WA (Xs , r, q) is strictly concave on [0, 1].
∂WA (XS ,XS ,q)
|q=q∗ (r)
∂q
= 0.
• If x > r > XS and q ≤ q ∗ (r), then
∂WA (Xs ,r,q)
∂q
> 0.
Proof Using the decomposition WA (Xs , r, q) = [WA (Xs , r, q) − WA (r, r, q)] + WA (r, r, q) and
an integration by parts of Eq. (10) for WA (r, r, q) leads to:
Z
N
WA (Xs , r, q) = −[r − XS ] · (qF (r) + (1 − q)) + x −
x
(qF (u) + (1 − q))N du − qN · ce .
r
Since
d2 (q·F (u)+(1−q))N
d2 q
> 0 for q ∈ (0, 1) and u < x, we obtain that q → WA (Xs , r, q) is strictly
concave on [0, 1] if x > r ≥ XS . The second bullet has been shown previously by L&S (Proof of
their Proposition 6). Note that
∂WA (r,r,q)
.
∂q
= [r − XS ] · N (1 − F (r))(q · F (r) + (1 − q))N −1 +
The first term is strictly positive for x > r > XS . Since WA is concave with respect
to q, we obtain finally that
C
∂WA (Xs ,r,q)
∂q
∂WA (XS ,r,q)
∂q
>
∂WA (r,r,q)
∂q
≥
∂WA (r,r,q ∗ (r))
∂q
= 0 for q ≤ q ∗ (r). Q.E.D.
Proof of Lemma 3.1
Note first that if the reserve price r implements an equilibrium profile involving some shill
bidding activity, then we must have r < rM : otherwise the seller would never hire a costly shill
26
bidder that could not raise her expected payoff. If p = 1, then the result is straightforward: take
r0 = rM , the seller raises a higher payoff than in the former equilibrium since she saves up the shill
bidding costs. Consider now p ∈ (0, 1). From (2) and since r < rM , we obtain that
∂B
0
∂p (q, r, p )
<0
for any p0 ∈ [0, 1]. Consequently, B(q, r, p) = ce implies that B(q, r, 0) > ce and B(q, r, 1) < ce .
Finally we obtain B(q, r, 0) > ce and B(q, rM , 0) = B(q, r, 1) < ce . Since
∂B
00
∂r (q, r , p)
< 0
for r00 ∈ [r, rM ] if p < 1, there exists a (unique) r0 ∈ (r, rM ) such that B(q, r0 , 0) = ce which
guarantees that the buyers’ equilibrium equation is satisfied. Furthermore, under the probability
of entry q, not hiring a shill bidder is a seller’s best response if the reserve price is r0 since the gain
from shill bidding r00 → H(q, r00 ) is decreasing in r00 while H(q, r) = cshill . Since r0 ∈ (r, rM ),
then RA (XS , r0 , q) > RA (XS , r, q) and the seller is thus strictly better off with the reserve r0 .
D
Proof of Lemma 3.2
For r ≥ rM , we have H(q ∗ (r), r) = 0. Below we consider in our calculations that r ≤ rM .
dH(q ∗ (r), r)
∂H(q ∗ (r), r) ∂q ∗ (r) ∂H(q ∗ (r), r)
=
+
·
dr
∂r
∂r
∂q
After differentiating Eq. (11) with respect to r and q, we obtain:
∂H(q ∗ (r),r)
(r)
= −q ∗ (r)N [q ∗ (r)F (r) + (1 − q ∗ (r))]N −1 [Xs + 1−F
∂r
f (r) − r]f (r)
Eq.(7)
Rx
z}|{ ∂q∗ (r)
2 ∗
∗
N −2 du[X + 1−F (r) − r] f (r)
∗
=
s
∂r N (N − 1)q (r) r (1 − F (u)) [q (r)F (u) + (1 − q (r))]
f (r)
(1−F (r))
R
r
1−F (r)
∂q ∗ (r)
f (r)
M
2
∗
∗
N
−2
∗
du[Xs + f (r) − r] (1−F
≤ ∂r N (N − 1)q (r) r (1 − F (u)) [q (r)F (u) + (1 − q (r))]
(r))
where the last inequality comes from
∂H(q ∗ (r),r)
∂q
=N
R rM
r
−N (N − 1)q ∗ (r)
< 0 and Xs +
1−F (r)
f (r)
[q ∗ (r)F (u) + (1 − q ∗ (r))]N −1 [Xs +
R rM
r
≥ −N (N − 1)q ∗ (r)
∂q ∗ (r)
∂r
1−F (u)
f (u)
− u]dF (u)
1−F (u)
f (u) −
(u)
q ∗ (r))]N −2 [Xs + 1−F
f (u)
(1 − F (u))[q ∗ (r)F (u) + (1 − q ∗ (r))]N −2 [Xs +
R rM
r
− r ≥ 0, and
(1 − F (u))[q ∗ (r)F (u) + (1 −
u]dF (u)
.
− u]dF (u)
We obtain then:
dH(q ∗ (r),r)
≤
dr
∂q ∗ (r)
+ ∂r N (N
Rr
∗
(u)
− ∂q∂r(r) N (N − 1)q ∗ (r) r M (1 − F (u))[q ∗ (r)F (u) + (1 − q ∗ (r))]N −2 [Xs + 1−F
f (u) − u]dF (u)
.
R
r
(r)
f (r)
− 1)q ∗ (r) r M (1 − F (u))2 [q ∗ (r)F (u) + (1 − q ∗ (r))]N −2 du[Xs + 1−F
f (r) − r] (1−F (r))
In order to obtain
R rM
r
≤
≤ 0, it is thus sufficient to show that:
1−F (u)
f (u) − u]f (u)du
(r)
f (r)
q ∗ (r))]N −2 [Xs + 1−F
f (r) − r] (1−F (r)) du
(1 − F (u))[q ∗ (r)F (u) + (1 − q ∗ (r))]N −2 [Xs +
R rM
r
dH(q ∗ (r),r)
dr
(1 − F (u))2 [q ∗ (r)F (u) + (1 −
27
,
(u)
f (u)
1−F (r)
f (r)
This last inequality holds since [Xs + 1−F
f (u) − u] (1−F (u)) ≤ [Xs + f (r) − r] (1−F (r)) for any
f (u)
f (r)
u ≥ r, an inequality which is equivalent to [Xs − u] (1−F
(u)) ≤ [Xs − r] (1−F (r)) which holds since
f (x)
x → (XS − x) 1−F
(x) is assumed to be decreasing on [0, x).
E
Proof of Proposition 3.3
Without the shill bidding constraint, L&S have shown that the optimal reserve price equals
XS . To be more precise, they have shown only that that q → WA (XS , XS , q) is locally optimal
at q ∗ (XS ) while global optimality is needed to obtain the result formally (see their footnote 17).
From Lemma B.1, we have that q → WA (XS , XS , q) is concave such that global optimality is
guaranteed. In our framework and if r∗ ≤ XS , we obtain then that the seller still maximizes her
expected payoff by setting the implementable reserve price XS .
∗
∗ (r)) = ∂WA + ∂q (r) · ∂WA . At q = q ∗ (r), we have
A
For rM ≥ r > XS , we have dW
(X
,
r,
q
s
dr
∂q
| ∂r
{z } | ∂r
{z }
≤0
from Lemma B.1 that
∂WA
∂q
> 0. If
r∗
> XS , we have thus
<0
dWA
∗
dr (Xs , r, q (r))
< 0 on the interval
[r∗ , rM ]. More generally, for r > XS , the seller’s expected payoff is first strictly decreasing (once
entry is mixed) and then constant (once the reserve price is to high such that the equilibrium
level of entry is null). On the whole, the optimal reserve price equals r∗ .
F
Proof of Proposition 3.4
35
bS := x −
We consider auctions with shill bidding: we have thus ropt = r∗ = rM . Let X
1
f (x) .
From the assumption
1
f (x)
bS > 0. Note first that r∗ = x for
< x, we obtain that X
bS ]. From Eq. (17), we obtain that the seller’s expected payoff is strictly increasing
XS ∈ [0, X
bS ]. To conclude the proof, we show
with respect to her reservation value on the interval [0, X
Rx
below that the assumption ce ≤ e−1 x (1 − F (u))du guarantees that the function WA∗ (Xs ) :=
bS . Now we consider
WA (Xs , rM , q ∗ (rM )) is strictly decreasing in the right neighborhood of X
bS .
XS ≥ X
dWA∗ (Xs )
∂WA
∂rM ∂WA
∂q ∗ (rM ) ∂WA
=
+
·[
+
·
].
dXs
∂Xs
∂Xs
∂r
∂r
∂q
From the definition of rM , we have
hazard rate function guarantees that
∂rM
∂Xs
∂rM
∂Xs
= [2 +
f 0 (rM )(1−F (rM )) −1
] .
(f (rM ))2
The assumption on the
≥ 1. In the proof of Proposition 3.3, we have shown
35
The assumption A2 does not depend on XS . However, A3 depends on XS through rM . Assuming that A3
holds for any realization of XS < x would not make sense for Proposition 3.4. For the precise statement, we
assume rather that B(0, x, 0) > ce , i.e. (4) holds only for rM = x.
28
that the term
∂q ∗ (rM )
∂r
·
∂WA
∂q
≤ 0 while
∂WA
∂r
≤ 0 for r ≥ XS . We obtain thus:
∂WA ∂WA
dWA∗ (Xs )
≤
+
= (q ∗ (rM )F (rM )+(1−q ∗ (rM )))N −(rM −XS )N f (rM )(q ∗ (rM )F (rM )+(1−q ∗ (rM )))N −1 .
dXs
∂Xs
∂r
bS ) = x such that F (rM (X
bS )) = 0, we obtain:
After the substitution rM (X
dWA∗ (Xs )
|Xs =Xb + ≤ (1 − q ∗ (x))N − N q ∗ (x)(1 − q ∗ (x))(N −1) = (1 − q ∗ (x))(N −1) [1 − (N + 1)q ∗ (x)].
S
dXs
To conclude the proof, we show below that q ∗ (x) > N 1+1 for any N . From (2), an integration
Rx
by parts leads to B(q, x, 0) = x (1 − F (u)) · [qF (u) + (1 − q)](N −1) du. We obtain then:
1
B(
, x, 0) ≥
N +1
Z
x
x
1 (N −1)
1 (N −1)
·
(1 − F (u)) · [1 −
]
du = [1 −
]
N +1
N +1
{z
}
|
Z
x
(1 − F (u))du.
x
>e−1
The last inequality can be viewed as a corollary of the well-known inequality ln(1 − x) >
Rx
x for any x ∈ (0, 1). From the assumption ce ≤ e−1 x (1 − F (u))du, we obtain thus that
B( N 1+1 , x, 0) > ce and finally that q ∗ (x) >
G
1
N +1
which concludes the proof.
Proof of Proposition 4.1
If entry costs are small enough, then there is full participation in auctions with shill bidding
and the model converges thus to the one with exogenous entry such that the expected payoff of
the seller converges to the optimal auction. On the contrary, the allocative inefficiency associated
to posted-price selling does not vanish which shows the first part of the result. Any choice larger
and sufficiently close to B(1, rM , 0) for c will work.
Rx
Let e
c := rM (u − rM )dF (u) = B(0, rM , 0) > 0: e
c corresponds to the smallest participation cost
such that the equilibrium level of entry is null and the expected welfare in equilibrium is thus
equal to XS . By continuity, if ce is close enough to e
c then the expected welfare in auctions
with shill bidding is close to XS . Consider now posted-prices with ce = e
c. Note first that
Rx
BP (0, r) = r (u − r)dF (u) = B(0, r, 0). If r ≤ r0 < rM , then BP (0, r) > e
c uniformly or
equivalently the equilibrium level of entry (i.e. q that satisfies BP (q, r) = e
c) is bounded away
from zero. There is thus a reserve price r ∈ (XS , rM ) such that BP (0, r) > ce or equivalently such
that there is some participation with the posted-price r and so such that the seller’s expected
payoff is strictly greater than XS . By continuity, if ce is close enough to e
c then the expected
welfare in the optimal posted-price is bounded away from XS , which concludes the proof.
29
H
Proof of Lemma 5.1
The ex ante welfare depends solely on the probability of entry and the allocation rule among
the seller and the entrants. For a given probability of entry, the ex ante welfare is maximized
if and only if the allocation rule gives the good to the agent with the highest valuation with
probability one. In other words, the ex ante welfare is maximized if a second-price auction
eS is used. At best, the probability of entry can
(without shill bidding) where the reserve price X
be fine-tuned as a function of VeA . For a given probability of entry function q(VeA ), the ex ante
eS , X
eS , q(VeA ))]]. For any VeA , the function
expected welfare is then given by EVeA [EXeS |VeA [WA (X
eS , X
eS , q)] is strictly concave on [0, 1] since q → WA (X
eS , X
eS , q) is strictly
q → EXeS |VeA [WA (X
eS (Lemma B.1). There is thus a unique optimal solution
concave on [0, 1] for any realization X
q opt (VeA ) for any VeA . It remains to show that this solution coincides with the one that would
prevail in a second-price auction where the reserve price would be equal to the seller’s reservation
value and if potential entrants knew the signal VeA , a result that is a straightforward extension
of L&S to the case where there is some uncertainty on the seller’s valuation. The right intuition
for this result is exposed in Milgrom (2004) (Theorem 6.1): in a second-price auction where
the reserve price is set at the seller’s reservation value, the net profit of an entrant coincides
with his incremental contribution to the welfare. This congruence holds for any realization of
eS and remains thus true on expectation which
the entrants’ valuations and any realization of X
guarantees that decentralized participation decisions from potential buyers lead to q opt (VeA ).
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