GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.215

Endogenous Competition Alters the Structure of Optimal
Auctions

Ronald M. Harstad

September 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Endogenous Competition Alters the Structure of Optimal
Auctions
Ronald M. Harstad
harstadr@missouri.edu
Economics, University of Missouri, andInstitute for Social and Economic Research, Osaka University
Revised version, c September, 2011

Abstract
Potential bidders respond to a seller’s choice of auction mechanism for a common-value ora¢ liated-values asset by endogenous decisions whether to incur an information-acquisition cost(and observe a private estimate), or forgo competing. Privately informed participants decide
whether to incur a bid-preparation cost and pay an entry fee, or cease competing. Auction
rules and information ‡ows are quite general; participation decisions may be simultaneous orsequential. The resulting revenue identity for any auction mechanism implies that optimal
auctions are allocatively e¢ cient; a nontrivial reserve price is revenue-inferior. Optimal auctions
are otherwise contentless: any auction that sells without reserve becomes optimal by adjusting
any one of the continuous, spanning parameters, e.g., the entry fee. Seller’s surplus-extracting
tools are now substitutes, not complements. Many econometric studies of auction markets areseen to be ‡awed in their identi…cation of the number of bidders.
D44; D82; C72; Keywords: optimal auctions, endegenous bidder participation, a¢ liatedvalues, common-value auctions, surplus-extracting devices

I thank, without implicating, Masaki Aoyagi, Johannes Horner and Jeroen Swinkels for helpful comments, andthe ISER at Osaka University for faciliating extensive rewriting. A less thorough version of this paper was circulated
under the title “Selling without Reserve as the Content of Optimal Auctions.”

1

Introduction

How should an owner or auctioneer select a selling procedure when bidders’value estimates for an
asset are private information? That fundamental question has for centuries received a variety ofanswers from experienced auctioneers, who in di¤erent markets persist in conducting their businessin quite di¤erent ways. In contrast, theoretical models of “optimal auctions” with rational riskneutral bidders have tended to provide a unique answer.1
While the particular answer provided depends fragilely on the model assumed, optimal auctions in the literature share two common characteristics. First, the optimal auction is ine¢ cient(unless surplus can be fully extracted), primarily due to a nontrivial reserve price.2 Second, it is
a complicated mechanism. Depending on the particular assumptions, it has involved: distributionand bidder-speci…c reserve prices, disjoint sets of prices at which seller refuses to sell, requiring
payments from losing bidders that vary with their bids and rivals’, requiring bidders to accept
lotteries with unboundedly large losses, or to accept lotteries before their terms are speci…ed.3
When the assumption of a …xed number of bidders (often simply treated as notation) is realistically opened to having the number of bidders respond endogenously to the expected pro…tabilityof bidding,4 the strikingly di¤erent characterizations that arise are shown here also to be strikingly
robust. The optimal auction is e¢ cient, and the argument for a nontrivial reserve price absent.
The characterization of equilibrium expected revenue becomes more general, and indeed can be
expressed as a function solely of the expected number of privately informed participants.
This paper’s second major purpose, less well known but no less important: to demonstrate1

In essence, the models cited in the following footnote each de…ne a very narrow equivalence class of auctions, andshow that optimal auctions all fall in a single equivalence class, which serves to characterize nearly all auction forms
as necessarily suboptimal, even with adjustments in parameters of that auction form.
2
Myerson [1981], Harris and Raviv [1981] and Riley and Samuelson [1981] derive optimal auctions when bidders’private information (their types) are independent. Of these models, Myerson’s is most general. All revolve around a
nontrivial reseve price (below which the seller prevents the asset from ever being sold); so do more recent papers (seesurveys in Klemperer [2000] and Krishna [2002]). The only optimal auctions attaining e¢ ciency are in models thatuse strong informational assumptions and correlated types to extract full surplus: Crémer and McLean [1985], [1988],
McAfee, McMillan and Reny [1989], McAfee and Reny [1992] and Crémer, Spiegel and Zheng [2009]. The criticism
of these models in Robert [1991], that the weakest form of limited liability or in…nitesimal risk aversion renders themdiscontinuously suboptimal, is similar in spirit to the present e¤ort. Mares and Harstad [2007] provide an accessible
treatment of necessary and su¢ cient conditions for full surplus extraction.
3
Examples of these complications, in order: Harris and Raviv [1981], Myerson [1981], Crémer and McLean [1985],
McAfee, McMillan and Reny [1989], McAfee and Reny [1992].
4
Headline-grabbing auctions–airwaves licenses, privatization of governmental enterprises, o¤shore oil leases,
investment-quality art, initial public o¤erings, acquisitions of new, established, and distressed corporations–all …t
the mold of endogenous decisons about whether to compete, rather than exogenous number of bidders. So do suchmundane markets as used-car auctions, timber sales and routine art auctions. (Buying at auction and selling at retailis a sensible stylization of the art-gallery business.)1that when bidder participation is modeled as a rational decision, characterization of an auction
as optimal does precious little to constrain the structure of the auction. Di¤erent auctioneers canemploy di¤erent mechanisms without necessarily implying that any are behaving suboptimally. The
Content Theorem below yields a most useful interpretation, given that all auction theory models
abstract from reality: an unmodeled aspect of a particular market–prior practice, the need forthe speed of a Dutch auction, unwarranted costliness of congregating bidders in time or space,bidders’ preference for dynamic ascending prices, or for a mechanism protecting bidders’ private
information from a bidtaker or rival or observer who lacks knowledge of the bidder’s beliefs or of the
distribution of private information but can observe his bids–each can be accommodated without
sacri…ce of e¢ ciency or revenue. Indeed, adjusting any spanning variable su¢ ces.
These characterizations stem from two overriding assumptions: symmetry and single-dimensional
types (though quite strong assumptions, these are ubiquitous in auction theory), and two sensible
constraints on auction forms: anonymity and uniqueness. If potential bidders are asymmetric in
their behavior, in their beliefs or in the distribution generating their private information (types),
multiple equilibria are unavoidable, and participation decisions either cannot be analyzed or cannot
be separated from an ad hoc equilibrium selection (which will bury the interesting economics). The
same problem arises if auction rules are not anonymous, and arises unless the seller is constrained
to auction rules which admit a unique symmetric equilibrium. Very little is known about auction
theory when a bidder’s private information cannot be summarized by a single random variable;
the impossibility theorem for equilibrium existence in Jackson [2009] suggests broad, fundamental
problems.
With these two assumptions and two constraints, a model quite general by the usual standardsof auction theory can be analyzed. A¢ liated-values auctions are considered, an extension of the“General Symmetric Model” in Milgrom and Weber [1982] to allow for an endogenous number ofbidders. Thus, both private-values and common-value motivations can be present. N “potential
bidders”, not yet privately informed, make decisions whether to obtain costly private information;
the decisions may be simultaneous or sequential (the sequential model is treated in Appendix B).
The (endogenous) n “participants” then acquire private information (an estimate of asset value,equivalently, their type, in the usual terminology of games of incomplete information) and may–or
may not–learn the realization n of the number of privately informed participants who are competing.
Next a (possibly costly) decision as to whether to prepare for and conduct bidding is made; thea “actual bidders” who do so may–or may not–learn the value of a before bids are submitted.
2

Seller’s information disclosure decisions include options as to how accurately to disclose, when todisclose, and whether to disclose publicly or privately.5 It adds both generality and realism toincorporate resource costs both before and after participants become privately informed; limiting
revenue generation to arise after participants become privately informed avoids incorporating what
is in essence a lump-sum tax.
Hence, it is determination of the level of competition by expected pro…tability decisions thatalters the structure of optimal auctions, not details of how this determination is modeled.
I emphasize at the outset that the model avoids any assumption of monotone equilibrium, inparticular avoiding the assumption of a screening level (a threshold level of private information
above which a participant chooses to pay the entry fee). An impossibility theorem by Landsberger
and Tsirelson [2000] makes this complication critical.6 This aspect, together with allowing for possible sequential participation decisions (Appendix B), for general a¢ liated valuations, for resource
costs facing potential bidders both before (an information-acquisition cost) and after becoming privately informed (a bid-preparation cost), for seller to wield the widest variety of surplus-extracting
tools, as well as allowing for players either to observe or not observe the number of players stillcompeting at each stage, greatly distinguish the generality of this model from prior auction models
which endogenize the number of bidders.7

2

A Small Example

Auctions are where market transactions usefully re‡ect private information because the mechanism
can be explicit about the rules underlying transaction determination. This advantage of auctions
forces robust models to deal with a variety of potentially cumbersome rule choices. So I begin witha simple illustration of the di¤erence endogenous participation makes.
The example asset has a common value T

U [0; 10]. First assume each of an exogenously

5

The model treats situations where a seller o¤ers an asset to potential bidders who decide whether to compete tobuy. A corresponding model where a buyer details a contractual obligation, and potential bidders decide whether tocompete to supply, has completely corresponding results.
6
A more accessible discussion of this theroem is in Landsberger [2007].
7
Harstad [1990] introduces the notion that the number of bidders ought to be considered an endogenous variable,
in a simpler model employing monotone equilibrium and with a smaller set of surplus-extracting tools. Levin andSmith [1994] also depend on monotone equilibrium, and critically on assumptions that the seller [i] cannot disclose
an appraisal, and [ii] cannot impose an entry fee after bidders have private information, for their result supporting a
positive reserve price. Chakraborty and Kosmopoulou [2001] partially specify a simpler model employing monotone
equilibrium and with a smaller set of surplus-extracting tools, and argue for a zero entry fee when negative entry
fees are precluded. McAfee [1993] describes a steady state in a private-values model where an exogenous number ofbidders is endogenously partitioned across sellers; the relation to his paper is reconsidered below in footnote 33.

3speci…ed 3 bidders observes an estimate Xj

U [t; t + 1jT = t]. Suppose seller held auction form

m2 , a second-price auction with no reserve price, no entry fee and no information about asset
value disclosed by seller. In m2 ’s symmetric equilibrium, expected revenue would be 4:833, thewinner’s expected pro…t would be 0:167, and an individual bidder’s ex ante expected pro…t onethird of that. If seller switched to auction form mE , an otherwise identical auction conducted underEnglish auction rules, and still drew 3 bidders, expected revenue would rise to 4:875, the winner’s
expected pro…t falling to 0:125.
Now adjust, instead to have N = 5 potential bidders decide whether to compete. If a potential
bidder wishes to acquire a privately-observed estimate of asset value, he must incur an informationacquisition cost c = 0:177; each participant j who does so draws his estimate from the same
distribution as in the prior paragraph, Xj

U [t; t + 1jT = t]. For simplicity, this example sets thebid-preparation cost to 0, and assumes that participants learn the number of participants in theauction before selecting bidding strategies. Calculations are simpli…ed for both second-price andEnglish auctions by a pathology of the double-uniform distribution: for a participant observing an
estimate Xj 2 [0:5; 10:5], expected pro…tability of continuing to compete is constant (it is lower on[10:5; 11] and 0 on [0; 0:5]).
Suppose each of the 5 potential bidders participates (that is, pays c to acquire an estimate,necessary to compete) with probability= 0:6. Then the probability

a of a bids submitted islisted in the middle row of Table 1, with the expected number of bids submitted, 3, shown as ain the right-hand column. However, any given potential bidder, if he chooses to compete, instead
calculates the probabilities of n 1 rival participants when 4 rival potential bidders each participate
with probability 0:6; this yields as expected number of bids submitted thea calculations in thebottom row. Thus, he rationally expects 3.4 bidders on average.
# bids:Table 1
0
1

2

3

4

5

a

a

0.01

0.08

0.23

0.35

0.21

0.05

3.0

a

0

0.03

0.15

0.35

0.35

0.13

3.4

With endogenous participation, the auction rules need further speci…cation; assume both m2
and mE inform participants of the number n of participants before bidding strategies are selected.
Then for the second-price auction m2 , if a potential bidder’s 4 rivals all take part with probability 0:6(and each then submits the equilibrium bid for the estimate he observed), his expected pro…t should
he compete will be 0:177, making him just indi¤erent over whether he competes. The resulting4

equilibrium attains expected revenue 4:42 (including allowance for 0 revenue in the 0:11-probability[rounded] event that less than 2 bids are submitted).
Should the same 5 potential bidders participate with the same probability= 0:6 in the English

auction mE , expected revenue would be 4:46. However, this is not an equilibrium: assuming a
potential bidder’s 4 rivals all take part with probability 0:6, his expected pro…t should he compete
will be less, only 0.166, so he would prefer not to pay the 0:177 information-acquisition cost. Rather,in equilibrium for auction mE , the 5 potential bidders participate with probability 0:59, yielding
the 0:177 expected pro…t that makes each indi¤erent over competing. The equilibrium revenue formE exceeds that for m2 by less than 0:002.
As potential bidders are in equilibrium indi¤erent over participating, Theorem 2 below showsthat equilibrium expected revenue equals the expected excess of gains from trade over resource
costs incurred (here, those costs simplify to c a); Corollary 3 shows that revenue can be expressedsolely as a function of the expected number of bidders. For these parameters, any auction thatattains 3:29 bidders on average (equivalently, a participation probability of approximately 0.574)
will reach the maximum attainable level of revenue, 4:4224. One way of attaining this is to keep
the second-price auction m2 , but impose an entry fee of ' = 0:038 (which is just enough to force
down the equilibrium participation probability to the desired level). Among many other ways toreach maximum attainable revenue is to use the English auction mE , but impose an entry fee of
0:028; the desired entry fee is lower because an English auciton with an exogenous n bidders reduces
the winner’s expected pro…t relative to a second-price auction. If some unmodeled aspect of themarketplace yielded a reason to conduct a …rst-price auction, there would be some entry fee, larger
than the 0:038 for the second-price auction, that would allow a …rst-price auction to be optimal.

3

A General Model with Endogenous Participation

Begin with the notion that potential bidders choose among a variety of auctions, and other uncertain
economic opportunities, in which to invest their attention, time and money. Only a segment of theextensive form of such a game, that relating to a particular auction, appears explicitly here. One
indivisible asset is sold in the explicit model. A subset of the (exogenously determined) N potential
bidders will participate, and a subset of the n participants will become the a actual bidders. It will
be convenient to describe seller’s strategies both in terms of recognized aspects of auctions, such as
pricing rules and information disclosure policies, and also as elements of an abstract set.

5

The game segment unfolds as follows, cf. Figure 1. First, seller announces an auction mechanism M := (m; '; r) 2 M

(M

<

<+ )

<d+1 , where m is an auction form, M the set of

auction forms, ' an entry fee, and r a reserve price. An auction form m speci…es not just pricing
rules, but the entire ‡ow of information and hence the nature of the extensive form continuation.
For example, m = m0 might specify a second-price auction with seller releasing an uncensoredindependent appraisal to all participants, as well as specifying that neither participants nor active
bidders learn their number before bidding. Or m = m1 might specify an English (oral ascending)auction without any seller-released public information, but with an appraisal privately revealed to
one actual bidder chosen at random, in which the number of participants is not learned but the
number of actual bidders is, and alternating recognition rules determine the probabilistic revelationof bidders’exit prices to remaining bidders. A particular seller in a particular situation may face
additional constraints: he may, for example, …nd credibly imposing a nontrivial reserve price impossible, or may be unable to inform participants of the number of competitors who acquired private
information, or may not have a reputation that would allow using a second-price auction withoutbidders assuming he could well insert a fake bid just below the highest bid;8 all such constraints
are treated via making M the feasible set of auction mechanisms for a particular auction. The set
M consists of discrete choices in some variables (e.g., English or second-price or …rst-price auction)and continuous choices in others (e.g., the signal-to-noise ratio of a public announcement of an
appraisal possessed by seller); without loss of generality, it can be embedded in a real space. For
later convenience, the dimensionality of this space is d + 1. Note that when a seller has the optionof credibly announcing how many participants are still competing (or how many actual bidders)before continuing, or of preventing the participants (or actual bidders) from knowing n (or a),seller’s choice is simply modeled as a choice between two (otherwise identical) auction forms, justas if it were a choice between …rst- and second-price auction rules.
Second, a pool of potential bidders N := f1; : : : ; N g simultaneously select probabilities

i ofbecoming a participant in this auction, basing those decisions on M (or select these probabilities
sequentially, considered in Appendix B).9 Participation has two consequences: each participant j
obtains some private information Xj 2 X about the asset’s value to him (call this j’s estimate), and
8

The impact of privately revealed information is considered in Mares and Harstad [2003]; alternating recognition
rules for English auctions are analyzed in Harstad and Rothkopf [2000]. Impacts of bidtaker cheating are consideredin Rothkopf and Harstad [1995].
9
Similar results to those in the main text when participation decisions are sequential depend on an equilibrium
selection favorable to seller.

6Seller
sets rulesn participants
observe signals, maylearn value of n.

N potential bidders decideon participating; if so,
incur cost c.

a actual bidders
paid j + b, maylearn value of a.

Privately informed
participants
decide on paying j,
Incurring b.

Winner obtains asset,
pays p; losers pay 0.

Auction conductedwith a actual bidders.

Value V learned.

Figure 1: Time Lineeach incurs an information-acquisition cost, c > 0. The information-acquisition cost is exogenously
speci…ed, and does not generate revenue for seller. It may represent resource costs of acquiring
information about the quality of the asset being auctioned and/or represent foregone pro…table
opportunities (e.g., passing on the potentially pro…table option to participate in another auction
occurring elsewhere).10 These two possibilities alter the interpretation of results, an issue consideredin Concluding Remarks. This cost is likely to vary across auctions, but c is the same for all potential
bidders in a given auction, and invariant to the mechanism by which the auction is run. The payo¤of a potential bidder who does not participate is normalized to 0.
Third, each participant j = 1; : : : ; n decides whether or not to incur a bid-preparation costb0 (which does not accrue to seller),11 plus pay the entry fee ' to seller and thereby become an

actual bidder, based on information available at the time. This information includes the auction
mechanism M , and participant j’s private estimate Xj . If the component m of M characterizing
the auction form speci…es that participants are informed of the number n of participants, then n
is taken into account. If n is not known, then the vector ( 1 ; : : : ;N ) of rational participation

probabilities of potential bidders is taken into account. A participant who chooses not to continue
attains a payo¤ ofc.

Fourth, each actual bidder k = 1; : : : ; a selects a bidding strategy for the auction form mwith reserve price r. In addition to M and Xk , n if known, and ( 1 ; : : : ;N ) if n is not known,

this decision takes into account the number a of actual bidders if the auction form m releases this10
This consideration is missed if a view of substitute auctions is not at least implicitly present. Unwillingness ofan additional potential bidder to participate need not imply zero (gross) expected pro…t.
11
The bid-preparation cost is treated as the same no matter what auction mechanism is employed. This assumption
is not innocuous; I return to it in Concluding Remarks.

7

information. If not, the bidding decision takes into account the functional structure of participants’decisions on whether to pay the entry fee, and includes strategizing to learn about a (and perhaps
useful inferences about rivals’private information) as soon as information ‡ows permit.
The winning bidder pays a price p for the asset, if this price is no less than the reserve pricer; otherwise the asset goes unsold, which implies that all actual bidders would then be losers.12
Losing actual bidders attain a payo¤ of

'

b

c.

Asset value to a particular participant observing estimate Xi a continuous function v (T; Xi ),increasing in both variables (common across participants, in that v does not have a subscript).
Here T is a common trend to asset values, or perhaps the common-value component of asset value.
A sale yields the winning bidder a payo¤ of v (T; Xi )

uS =

3.1

8
< 0,

p

'

b

if a = 0 or p < r,

: p + a'; if a

1 and pc. Seller’s payo¤ is

13r.

Assumptions: Auction Environment

A.1. The in…nite sequence fX1 ; X2 ; : : :g from which participants will observe estimates is a sequenceof exchangeable, positively a¢ liated, real-valued random variables with nonatomic measure , andmarginal

1 onto support X<.

A¢ liation is de…ned and characterized in Milgrom and Weber [1982], pp. 1098-1100 and 11181121; it is referred to as the MLRP (monotone likelihood ratio property) in several auction models.
Roughly, a¢ liation means that higher realizations for any subset of the variables fX1 ; X2 ; : : :gmake higher realizations for any disjoint subset more likely. Exchangeability means that the joint
distribution is una¤ected by any …nite permutation of the indices. Let Xz = fX1 ; : : : ; Xz g, andPz
Tz = z1
i=1 Xi .

A.2. The common trend T = limz!1 Tz ; c + b < inf v (T; X) < sup v (T; X) < 1.
A variant of DiFinetti’s Theorem justi…es the use of a limit in A.2:
12

Notation that is already more cumbersome than might be hoped for would …nd signi…cant additional complication
if the seller were allowed to announce a vector of reserve prices ra , with the ra that corresponded to the number aof bids actually submitted enforced after bids were submitted. That complication would not a¤ect any of the results
below. In particular, the result in Levin and Smith [1994] that a nontrivial reserve price would be employed at leastin the case where only 1 actual bidder showed up is still seen to depend on their assumptions that the seller cannot
utilize an entry fee, and cannot publicly reveal such information as an appraisal.
13
Considering the value to seller of an unsold asset to be 0 is, as usual, a harmless normalization. It bears emphasis,
however, that failing to meet the reserve price implies that the seller is irrevocably constrained from ever o¤ering thisasset to this set of potential bidders in the future (this assumption is nearly ubiquitous in auction theory, though
seldom mentioned). I return to this consideration in Concluding Remarks.

8

Theorem 1 (Kingman [1980]) Given A.1, the sequence fT1 ; T2 ; : : :g almost surely converges pointwise. Moreover, conditional on T , the fXi g are mutually independent.
Letting the common-value component equal the asymptotic mean is without loss of generality(Milgrom and Weber [1986]).14

3.2

Assumptions: Auction Rules

A.3. The price paid is an anonymous, nondecreasing, continuous function of the pro…le of actual
bids submitted.15
A.4. Each auction form m 2 M provides all bidders with the same strategy set, determines awinning bidder anonymously,16 and attains a unique symmetric equilibrium continuation for any
exogenously speci…ed binomial distribution of the number of actual bidders (including degenerate).
Allowing for payments to or from losing bidders greatly complicates the notation, but would
not change any results below. Uniqueness of the symmetric equilibrium continuation (a constraint
on the set of mechanisms available to seller) is critical to being able to predict the pro…tability ofparticipating and actually bidding; it is satis…ed for a wide variety of auction forms.17

3.3

Assumptions: Behavior

A.5. All N

2 potential bidders are risk-neutral.18

A.6. Symmetric behavior: each potential bidder selects the same probabilityof participating,each participant selects the same function of known and inferred information to determine whether
to actually bid, and each actual bidder selects the same bid function. These selections constitute a
Bayesian equilibrium continuation.
14
I chose these assumptions about informational and valuational variables to be as natural as possible for the
purpose of modeling endogenous participation. They are also mild in comparison to the bulk of auction theorymodels. However, the results below are quite general, and do not depend on the speci…c nature of A.1 and A.2. For
example, all results extend, under virtually unchanged proofs, if the corresponding assumptions in Pesendorfer andSwinkels [1997] are substituted.
15
The sort of revenue-maximizing, non-capricious discriminaton across bidders in Myerson [1981] has already been
ruled out by exchangeability (in A.1). The sort of capricious discriminaton contemplated in McAfee, McMillan
and Reny [1989] is ruled out here, solely for notational ease. Footnote 29 below explains how their mechanism,
which extracts full surplus whenever the exogenous number of bidders is at least two, becomes revenue-inferior withendogenous bidder participation.
16
Though it may be natural, nothing depends on this being the highest bidder. For example, the lottery-quali…cation
auction (Harstad and Bordley [1986]) meets these assumptions.
17
Cf. Levin and Harstad [1986], Bikhchandani and Riley [1991], Pesendorfer and Swinkels [1997], Harstad andRothkopf [2000], and Maskin and Riley [2000].
18
All results below readily extend to the case where all N potential bidders have the same concave utility function,with much more cumbersome notation and no further insight.

9If, in addition, seller selects the mechanism M to maximize expected revenue given the assumed
behavior of bidders, a full Bayesian equilibrium is attained. As our focus is on the behavior thatvarious announcements of M will induce, and thus upon the expected revenue attained, equilibrium
continuation is the key assumption.19
Anonymity appears in A.3 and A.4, distributional symmetry of private information in A.1,
and symmetry of behavior in A.6 (symmetry of beliefs is suppressed, as it would add notation
without altering content). The last part of A.1, that X<, imposes single-dimensionality ofprivate information. Seller is constrained in A.4 to announce auction rules satisfying uniqueness.20

4

The Participation Decision

In this model, the equilibrium expected number of participants is not invariant to seller’s choice ofauction mechanism. Rather, it adjusts to the auction mechanism M so that expected pro…t equals
information-acquisition cost. The straightforward logic is, ultimately, independent of many detailsof the mechanism.
In the symmetric equilibrium, each of the N potential bidders will participate with the sameprobability(M ); the expected number of participants is then n (M ) = N (M ). This section

exploits symmetry to consider the participation decision of potential bidder 1 when N

1 rivalpotential bidders all participate with probability ; the main text merely summarizes the intuition,with the horrendously cumbersome details diverted to Appendix A (and the case of sequential
participation decisions to Appendix B). Appendix A develops the set of estimates bidder 1 could
observe that would imply a su¢ cient expected pro…tability of competing to justify paying theentry fee. This development di¤ers somewhat according to whether or not the n participantslearn the number of participants; both cases are complicated by the need to avoid assuming athreshold determines such a set. The set characterized is used to identify, in Conclusion 1, the exante probabilities that [i] a particular potential bidder, if he chooses to become a participant, willdraw a signal justifying continued competition, [ii] he will face any particular amount of bidding
competition, and [iii] only the …rst

participants will become actual bidders (which is di¤erent

19
Note that there exist Nash equilibria in which seller selects an otherwise inferior M 0 because, for example, all N
potential bidders respond to any M 6= M 0 by not participating, or otherwise punishing seller. These equilibria fail tobe subgame-perfect; in ignoring them, I follow a standard but usually implicit practice.
20
Which auction forms meet that constraint may depend on particular details of parameters; for example, only fornarrow classes of has a unique equilibrium been found when seller’s information is privately disclosed to a randomly
selected bidder (Mares and Harstad [2003]).

10

from [ii]).Probability [iii] is denoted

a (M;; n), though degenerate inif participants learn their

number, and in n if they don’t. Next, it develops the probability sr that the reserve price r is met.
Finally, using a function p ( ) that expresses the expected auction price conditioned on various
arguments, expected pro…tability is characterized, …rst if the bidding stage is reached, then moving
backward to the decision to become an actual bidder and then to the decision to participate. The“expected pro…t equals information-acquisition cost” characterization is Conclusion 2.
For all the gory details, the underlying intuition models the only possibilities: a privately
informed participant continues competing to become an actual bidder if and only if the expected
pro…tability of doing so exceeds the costs not yet sunk (entry fee ' and bid-preparation cost b);a not-yet-privately-informed potential bidder competes (participates) if and only if the expected
pro…tability of doing so (net of ' and b, should they be incurred) at least covers the informationacquisition cost c. In equilibrium, expected pro…tability exactly equals c.
Appendix A details de…nitions for the following notation:
Table 2
V (M; ; n; '; t) Expected asset value for mechanism M , participation probability ,n participants, entry fee ', common trend t (ex ante, before learning Xj ),p (M; a; ; n; t)

Expected price, given M , a actual bidders, , n, t,s (M; a; ; n)(Abbreviated sr ), sale probability given M , which speci…es the reserve price r , for a; ; n,

a (M;

; n)

(z; Z; )Probability that only the …rst a participants will become actual bidders, for M; ; n,Binomial probability of z successes in Z trials, given success probability :Depending on the information ‡ows speci…ed by a mechanism M , each of the …rst four functions
above will typically be degenerate in one of its variables.21

5

General Revenue Formulation

Begin, naturally, at the end: with a speci…cation of seller’s expected revenue, conditional on assuming a

1 actual bidders and na participants:
sr E [p (M; a; ; n; )] + a';

21

For example, p ( ) is degenerate in a for a second-price auction,bidders of n, and degenerate in n if bidders are not so informed.

11

a is degenerate inif the auction form informswhich simply sums the price paid by the winner (multiplied by the probability of a sale), andentry fees paid by all actual bidders. It is harmless to condition on the common-value component’s
expected value, and simpli…es the characterization to come.
Stepping back by replacing an assumed number of actual bidders and then of participants withequilibrium continuation probabilities gives our speci…cation of expected revenue:
R (M; n) =R (M ) ==Xa(sr E fE [p (M; a; ; n; ) jT ]g + a')n

R (M; n) [n; N; (M )]

X

(
X X

n
(sr E fE [p ( ) jT ]g + a')
a
a

n

n
a

a (M;

a (M;

; n) :

(1)

)

(2)

; n)

n:

Equation (2) is still a simple sum of the price paid and entry fees, itself summed over the objective
probabilities of a actual bidders and n participants (there arenaways thata (M;; n) might

correctly predict the number of actual bidders). (The summation harmlessly ignores the events of
0 participants and 0 actual bidders, which contribute 0 revenue.) To interpret expected revenue,
natural de…nitions of the expected value transferred and expected number of actual bidders, areinvoked:V (M ) =XXna (M ) =asr E fV [M; (M ) ; n; '; T ]g

(
X X
n

n
a
a
a

)

a (M;

; n)

n
a

a (M;

; n)

n;

n:

(These summations also harmlessly ignore the cases n = 0, a = 0.) Note that these de…nitions
depend on the mechanism; in particular, V (M ) treats as a zero transfer an asset that does not sell.
Theorem 2 (The Fundamental Revenue Identity): In symmetric equilibrium continuation withendogenous bidder participation, for any M 2 M,
R (M ) = V (M )ba (M )cn (M ) :(3)

Theorem 2 is proven simply by separating out terms in (2) that are zero by equilibrium participation (eq. (13) in Appendix A). The notationally cumbersome details are diverted to Appendix
C.

12In simple language, the Identity says that revenue in symmetric equilibrium continuation isequal in expectation to the expected value transferred (or, if you wish, expected gains from trade)less aggregate participation and bid-preparation costs.22 It is particularly important that thisidentity provides a simple formula for revenue for all M ; there is no need for separate formulas for…rst-price, second-price, and English auctions, or for di¤erent information-revealing policies (except
to determine[M ]), and the entry fee does not directly enter the calculation. The reserve priceenters only through the probability of a sale.
Viewing e¢ ciency as the sum of expected surplus of seller and all N potential bidders, Theorem
2 yields a general and striking contrast to prior optimal auctions models (in which revenue ismaximized by enforcing allocative ine¢ ciencies):
Corollary 1 The Bayesian equilibrium in which seller maximizes expected revenue is allocativelye¢ cient. Indeed, seller’s preferences over any set of auction mechanisms match those of an e¢ cientsocial planner.
Proof. The right-hand-side of (3) is an e¢ ciency measure, and in equilibrium continuation is
also seller’s objective.
Due to mixed-strategy participation decisions, when(M ) < 1, there is a probability (1)Nof the event that the asset goes unsold because all N potential bidders happen not to participate.
This is not an ine¢ ciency: as it can only be avoided by making the auction so attractive that all
N potential bidders participate, and the added resource costs accrue to seller, seller and a central
planner identically accept this event. A more subtle issue: since the set of estimates justifying
continued competition are not in general upper contours, in a revenue-maximal auction, there
can be a positive probability that the participant observing the highest estimate chooses to cease
competing rather than pay the entry fee, with the asset then sold to a participant who values it less
22
No result approaching comparable generality is in the literature, but this Theorem has many antecedents. Relative
to Levin and Smith [1994], for example, it is original in extension to a¢ liated-values environments, in its allowancefor costs incurred both before and after bidders observe private information, in allowance for numbers of participants
and actual bidders to be either learned or inferred, in the number and variety of surplus-extracting devices allowed
for, and in dealing with the impossibility of a screening level.
In special cases, a corresponding result is found by Samuelson [1985] and Hausch and Li [1990], can be calculated in
the example of Theorem 5.2 in Milgrom [1981], and found as an asymptotic approximation in Matthews [1984] (where
the number of bidders is not necessarily an equilibrium level, but the information-acquisition costs are). Theorem 2
veri…es shortcuts taken, but not justi…ed explicitly, in equations (2) and (3) in Harstad [1990]. French and McCormick[1984] discuss a similar heuristic feature of …rst-price, common-value auctions, but do not provide a complete modelor equilibrium characterization. McAfee and McMillan [1987b] assert the corresponding equation for a nonstochastic
but supposedly endogenous n, without justi…cation either for the equation or the source of n, and proceed incorrectly
to dismiss the possibility that seller could enhance expected revenue via a positive entry fee.

13

highly than he does. While this reduces the expected value transferred, seller and a social plannerboth accept it. A su¢ cient reduction in ' will eliminate the possibility of selling to an ine¢ cientacquirer, but the lower ' will increase n, and with it increase the resource costs borne by seller.23

6

Inferiority of a Nontrivial Reserve Price

A seller can attain the entire interval of equilibrium participation probabilities, 0 through 1; thisresult is shown next for a second-price auction, chosen purely for convenience. The range ofequilibrium values ofis attained by varying only the entry fee ' (including possibly ' < 0,reimbursing a fraction of participation and bid-preparation costs), while keeping the reserve price…xed at r = 0. Let M ' = (m; '; 0), where m is a “vanilla” second-price auction with no disclosureof seller’s information, and with n and a revealed to bidders; M ' “sells without reserve.”

Theorem 3 For any

0 2 [0; 1], there exists an entry fee '0 such thatM '0 =

0.

The proof (in Appendix C) simply sets up the Intermediate Value Theorem.
A reserve price r is nontrivial if at least one actual bidder does not guarantee a sale, that is, if
there is an a > 0 such that fs (M; a; ; n) < 1;a (M;; n) > 0g :Corollary 2 Any auction mechanism M with a nontrivial reserve price, yielding(M ) 2 (0; 1),

is an expected-revenue-inferior mechanism for seller to adopt.
Though the proof (in Appendix C) conveys little insight, Corollary 2 is quite intuitive. Thereis an unavoidable detriment in this model, whenever< 1. That is, with probability (1)N ,the independent mixed-strategy decisions lead to no potential bidder participating, which is a costthat both seller and a social planner would take into account. (Appendix B considers an alternativemodel avoiding this detriment.)
23

Indeed, with some additional notation, one can readily build an extension of this model featuring a number NLof potential bidders who face information-acquisition costs cL and NH who face costs cH > cL . In such a model,
compare any auction mechanism M1 for which the high-cost potential bidders participate with positive probability(in which case, all low-cost potential bidders strictly prefer to participate) with any mechanism M2 for which thehigh-cost potential bidders do not participate. It is straightforward to show that a seller will prefer M1 to M2 if andonly if an e¢ cient social planner does. Cox, Dinkins and Swarthout [2001] outline a model in which each potential
bidder draws his information-acquisition costs from a smooth distribution.

14

Should a nontrivial reserve price be used, a further detriment that is an ine¢ ciency is introduced:
not only is there no sale with probability (1
n
XX

[1

)N , there is also a probabilitys (M; a; (M ) ; n)]

n a=0

n
a

a (M;

(M ) ; n)n(4)that one or more potential bidders participate, but none of them are willing to pay the reserve
price.
When the number of bidders responds endogenously to the pro…tability of competing, there isno counterbalance to make up for the loss of a sale due to a nontrivial reserve price. Occasionally,a reserve price would have, for example, fallen between the highest and second-highest bids in asecond-price auction, or prevented a single participant from obtaining the asset for merely the entry
fee, but the increased revenue such events create will have been taken into account in bidders’calculations of the probability with which to participate. Levin and Smith [1994] …nd that a
nontrivial reserve price enhances revenue in common-value auctions with entry; their result is
entirely due to disallowing entry fees, disclosure of seller’s information, and other surplus-extracting
devices that shed the revenue losses in (4).24
Via Corollary 2, endogenizing bidder participation turns much of standard auction theory on
its head. A reserve price (or bidder-speci…c reserve prices if bidders draw types asymmetrically) is
the focus of Myerson’s [1981] original “Optimal Auction Design”paper, and of much of the optimal
auctions and mechanism design literature since (see, for examples, surveys in Klemperer [2000] andKrishna [2002]). Indeed, auction policy papers also focus on the reserve price as if it were a key
variable (Klemperer [2002]). Yet when the number of bidders becomes an endogenous variable, the24Levin and Smith [1994] have a more primitive device in their model that they call an entry fee, but it is an
information fee, in that it must be paid before bidders learn their signals; it in essence allows seller to employ a lumpsum tax on participants before they become privately informed. They obtain the result that a nontrivial reserve
price is called for when they assume this information fee has to be set to zero. This paper follows a tradition in theliterature, led by Cassady [1967], Milgrom and Weber [1982] and Samuelson [1985] in the normal usage of the term
entry fee (as a fee incurred after participants become privately informed).
It also follows the tradition in the optimal auctions literature, and indeed in auction theory more generally (theonly other exceptions I know of are McAfee and Reny [1992] and Crémer, Spiegel and Zheng [2009]), of assumingthat any surplus-extracting device is potentially distortive, and thus ruling out devices that are in essence lump-sum
taxes. I thank Jeroen Swinkels for emphasizing this issue, and for pointing out that a limit to the generality of thispaper’s results is that they apply only after a seller has exhausted usage of devices that are essentially lump-sum
taxes.
Levin and Smith criticize Samuelson [1985] for considering the impact of entry fees in a model where the aggregate
expenditures on becoming privately informed are exogenous. The current model withstands that criticism.
Chakraborty and Kosmopoulou [2001] report a similar characterization to Corollary 2, that with entry, an auction
with a nontrivial reserve price is revenue-inferior to some auction with a lower reserve and an entry fee. It is not clear
what model of entry yields the nonstochastic number of participants in their paper, which depends on a screening
level.

15

reserve price becomes a uniquely inferior tool for extracting surplus from bidders; a rational seller
does not use it, and an e¢ cient social planner is glad he doesn’t.

7

Revenue and Participation

In view of Corollary 2, the remainder of the text limits mechanisms to M 2 MZ = fM 2 M j r = 0g,
zero-reserve-price auctions. This section shows that seller’s announcement of M a¤ects expected
revenue solely through its e¤ects on the participation probability . To develop and understandthis result, I begin with some natural comparative statics: two auction mechanisms with the same
have the same expected revenue, and a change in mechanism which would lead to a higher expected
revenue for any exogenously given number of bidders will lead to a lower . Formally,
Proposition 1 For any fM; M 0 g

MZ ,[i]: f (M ) =(M 0 )g ) fR (M ) = R (M 0 )g;[ii]: f (M ) =(M 0 )g ) V (M )[iii]:ba (M ) = V (M 0 )

R (M; n) R R (M 0 ; n) 8n 2 N )(M ) Qba (M 0 ) ;(M 0 ) .

The proof is in Appendix C.25
Proposition 1 implies that revenue consequences of an increased number of bidders are necessarily less rosy when the extra bidders arrive via a rational participation calculation:26
Remark 1 Suppose a seller can switch to an auction mechanism that increases equilibrium participation. Then each participant has a lower chance of winning, and so in equilibrium requires
a higher expected pro…t in the event of winning. The winner’s higher expected pro…t means an
expected revenue further below expected value transferred.
A host of econometric studies of auction markets are not sensible in this context, cases whererevenue, the high bid, or some similar variable is estimated using the number of bidders as an
exogenous explanatory variable. If ten potential bidders decided to participate expecting about
three participants, but mixed strategy participation decisions happened to lead to six showing up,
no wonder the extra bidders led to higher revenue: the auction rules were su¢ ciently extractive
25

Proposition 1 applies only when (M ) is a function. The only auction mechanism this appears to rule out is the
McAfee, McMillan and Reny [1989] mechanism that extracts full surplus if n
2 bidders are exogenously given. I
cover the details of comparison with that mechanism in footnote 29.
26
This is the intuition behind the negative answer in Harstad [2008].

16

of bidders’surplus that no one wanted to be a fourth bidder. It would be interesting to discover
the circumstances under which a higher expected number of bidders was associated with a higher
expected price, but the historical record of auctions (outside carefully designed laboratory experiments, cf., e.g., Levin and Smith [2002]) does not include data on the expected number of bidders.
The actual number of bidders is no substitute.27 That revenue is higher when the realized numberof bidders is higher does not imply that a seller prefers to take steps to increase the expected numberof bidders.
For many empirical studies, especially merger-and-acquisition studies, it remains a problem thatthe record does not indicate the number of participants, but at most the number of actual bidders.
Hence, the size of a winner’s curse adjustment a bidder ought rationally to make depends on avariable (or inferences about that variable) unavailable to the empirical analyst.
Proposition 1.[ii ] …nds the di¤erence V (M ) ba (M ) takes on the same value for any mechanisms
M; M 0 that attain the same ; let W ( ) denote this di¤erence. Then let a simple function R on
the entire unit interval be de…ned byR( ) = W ( )

c N:

(5)

Corollary 3 R (M ) = R [ (M )] for all M 2 M Z ; that is, R (M ) can be projected onto [0; 1] toyield R ( ).
Proof. Proposition 1 implies that M in‡uences R (M ) only through its in‡uence on(M ).

Theorem 3 shows that the entire interval [0; 1] can be reached.
Remark 2 Corollary 3 …nally changes auction theory’s view of the comparative roles of the various
surplus-extracting devices available to a seller, from complements (their role in exogenous-numberof-bidders models in the tradition of Milgrom and Weber [1982]) into their common-sense roleof substitutes. With exogenous n, a seller who had introduced some subset of: switching to an
English auction, releasing public information, setting an entry fee, and adding a nontrival reserve
price, would still gain by incorporating the remaining surplus-extracting devices. With endogenous
27
If a historical series of auctions arguably results from the same equilibrium for each auction, then the binomial
distribution that is the number of participants can be estimated from the series. Many empirical auction databases,
however, arise from situations far enough from ex-ante symmetry to warn against direct application of this model.
Nonetheless, observed data on the number of bidders may incorporate entry decisions based on which rivals were
expected to take part with what probabilities. If the identities of participants are recorded in the database, a separate
binomial distribution representing participation of each potential bidder could be estimated.

17

participation, it is (solely) the equilibrium probability

that interests seller, and alternative methodsof accomplishing an improvement in this variable are substitutes for each other.

8

The Content of Optimal Auctions

Theorem 3 and Corollary 3 imply that R ( ) in (5) is continuous. As its range is obviously
bounded, it attains a maximum.

Let R> 0 be the maximum attainable level of revenue.

Since a screening level is impossible, it is unsurprising that I have not been able to demonstrate strict concavity of R ( ). Accordingly, there may be multiple values ofattaining R ;let A = f 2 [0; 1] j R ( ) = R g. By continuity, A contains a minimal and a maximal element,and

1 (not necessarily di¤erent). As R (0) = 0,

Proposition 2 If N > 2,0

0 > 0.

1 < 1:

Proof. Evaluation of (5) as

decreases from 1 to 1

4 shows a revenue gain from decreased

information-acquisition costs that is linear in 4 , and a revenue loss from a probability of 0
participants that is of the order (4 )N .

Proposition 2 applies to N = 2 potential bidders in a common-value auction.28 In that case,
it is revenue-inferior to adopt an auction that leads to both bidders participating with probability
one. The seller will have some revenue-superior alternative which will lead potential bidder 2 to
be indi¤erent over participating even when he infers that there will be at least a (11) > 0probability of facing no competition.
A little structure enables characterizing the size of the set of optimal auctions. As mentioned
in section 3, without loss of generality, the set M of seller’s feasible auction mechanisms can beembedded in a real space, M<d+1 . Then the set of zero-reserve-price auctions can be viewed as

one dimension smaller (by omitting the r = 0 component): MZ<d , let D = f1; : : : ; dg. D canbe called the set of components the seller chooses.29
De…ne MC to be the set of auction forms m for which some component im 2 D continuously
altersand allows spanning [0; 1]. Theorem 3 above shows that MC 6= ?; the continuity used

28

Harstad [2008] proves and interprets this result in a somewhat simpler model.
The exact dimensionality depends on modeling choices (as to what constitute the components) that otherwise
distract from the paper. The entry fee is one component. Whether n and a are revealed generates two more. At least
one component could be generated by whether the auction is dynamic (if the degree of information dispersal during
the course of the auction is an issue, more than one component), and still …rst-price and second-price auctions havenot been distinguished. Mares and Harstad [2003] show that seller’s information disclosure options cannot be fully
speci…ed via a single component.
29

18

in that proof is known to hold for the English and …rst-price auctions (and for public information
disclosure).

Next, de…ne a set of auction mechanisms M = M 2 MZ jm 2 MC

<d ; this is

the set of mechanisms selling without reserve for which the auction form exhibits continuity andspanning via some component of D.
Consider M the domain of choice for seller, in light of Corollary 2. Without loss of generality,
the d components of D can be ordered so that component i 2= f1; : : : ; d g (0 < d < d) denotes

a spanning, continuous component for all m 2 MC (' is one such component, guaranteeing 0 < d ;other examples are provided below). For arbitrary auction mechanism M 2 M and arbitrarycomponent

2, let M jM

value of the component

M denote the set of mechanisms which di¤er from M only in the(thus, for example, if

connotes the entry fee, all mechanisms in M jM

di¤er only in the entry fee, so if M incorporates the auction form m for which only the number ofactual bidders is learned before bidding under …rst-price rules, then this holds for all auctions in
M jM ). Let M = fM

M j (M ) 2 Ag, a collection of optimal auctions.30 Then,

Theorem 4 (The “Content” Theorem): For arbitrary auction mechanism M
component

2, M jM \ M

6= ?. That is, any auction in M can be converted into an optimal

auction in M merely by adjusting any one component in
Proof. Select an arbitrary M0 2 (MnM ). If
assumec(M0 ) 2
= A. Select an arbitrary componentb, for example), else if(M0 ) >.

(M0 ) 2 A, nothing remains to be proven, so

solely in component , as follows. Take an arbitrary'=

M and arbitrary

2c 2 M jM by changing M0. Construct M

2 A; if(M0 ) <

, setc) = 0 (by ' = E [T ]);, set (M2

is possible. As in Theorem 3, continuity implies the existence of a value for thisyielding M 2 M , with M di¤ering from M0 only in this30thc) = 1 (by(M
insures thisthcomponent

component.31

Some readers may question how these optimal auction mechanisms compare to the mechanism which extracts
full surplus in McAfee, McMillan and Reny [1989]. For their mechanism, call it M M M R , the unique equilibrium
continuation is
M M M R = 0, hence R M M M R = 0. To arrive at a sensible comparison, consider mechanisms
MMR
M'
with negative entry fees appended. Setting ' < c b necessarily generates a revenue-inferior auction; thereverse inequality su¤ers the same R M'M M R = 0 problem as M M M R . So consider ' = c b: for M McMbR , every
2 [0; 1] is an equilibrium continuation. Selecting 2 ( 1 ; 1] yields excessive incurrence of information-acquisition
costs with no compensation; selecting 2 [0; 1 ] runs into the same problem as a reserve price: there is an excessprobability of no sale (happening anytime a < 2), with no compensation. So any equilibrium selection is revenueinferior to the second-price auction M ' of Theorem 3 that attains the same , hence suboptimal. If a mechanism
similar to Crémer and McLean [1985], [1988] were to apply to a common-value auction, it would su¤er the sameproblems. So would the mechanism of McAfee and Reny [1992], which also depends on using information fees, ruledout here (cf. footnote 23).
31
A geometric interpretation: construct the projection mapping proj : <d ! <d 1 by deleting the component
corresponding to an arbitrary 2
from any d-vector M 2 M. Then the proof of Theorem 4 has shown for every
M 2 M an element MM 2 M such that proj (M ) = proj (MM ). Note that M is typically a strict superset of theset thus obtained.

19

Figure 2: The Set of Optimal Auctions

Thus, prior optimal-auction characterizations depend critically on the implicit assumption thata seller has a captive audience: there will be exactly n bidders no matter how the seller changes
auction rules to extract more surplus. When the number of bidders responds endogenously to the
pro…tability of competing, the content of optimal common-value auctions is merely this: choose
any auction form, commit to sell without reserve, and adjust any continuous parameter to avoid
overly encouraging or overly discouraging bidder participation.
Figure 2 may help to visualize Theorem 4. It simpli…es by imagining that M has three dimensions: ' on the vertical axis, plus one dimension in which M can take on one of three discrete values(e.g., English, second-price, or …rst-price auction form, for a seller we imagine to be constrained
to those three choices), and one dimension in which a variable can be chosen over an interval, but
may not necessarily span the range of . The set M is then the union of three rectangles in parallel
vertical planes, outlined in Figure 2 by dashed lines. The set of optimal auctions M is represented
by the union of a collection of curves shown lying in the three rectangles. The optimal auctions
M span M in that, from any point in one of the three rectangles, it is possible to reach one of
the curves by moving only vertically, that is, by adjusting only the entry fee. (To be consistent
with the Theorem, the curves must, collectively, contain continuous paths from the left to the rightedges of all three rectangles.)A variety of surplus-extracting devices might exhibit su¢ cient continuity to apply this logic.
For example, suppose seller observes XN +1 (which is a¢ liated with asset value T ), and consider

20

mechanisms My1 , all …rst-price auctions with ' = '0 (arbitrary), r = 0, and with seller making apublic announcement of Zy = XN +1 +y , whereis an independent standard normal (white noise),and y a scalar parameter of the noisiness of this public announcement. Then (by Theorem 17 in
Milgrom and Weber [1982] and Proposition 1.[iii ] above),(naturally) that it is continuous. Suppose

M01 <

0 <

My1 is nondecreasing in y; assume
MY1 for su¢ ciently large Y . That is,full and honest public announcement of seller’s information is overly extractive of surplus, but nota public announcement where the signal-to-noise ratio is very small (this is, in essence, a spanningsupposition). Then the argument of Theorem 4 can be applied to derive the existence of a y suchthat My1 is an optimal auction.32
c denote an auction (with seller’s information-disclosure policy
For another example, let M

and ' = '0 arbitrary) in which the highest sealed bid win at a price p =b(1) + (1convex combination of the two highest bids. Naturally assuming continuity in , if

) b(2) , a(M2 ) <

1 <(M1 ) (a spanning assumption that the …rst-price but not the second-price auction is insu¢ ciently
extractive of surplus given the arbitrary entry fee and information policy), then only the weighton the highest bid needs to be adjusted to obtain an optimal auction.33
In a world where a wide variety of auction mechanisms are employed by experienced and apparently successful auctioneers and frequent auction sellers or bid-taking procurers, Theorem 4 has
the comforting conclusion that this variety is not per se unambiguous evidence that some of thesesellers and auctioneers must be choosing suboptimally. If some aspect of a particular situation
falling outside the model creates a preference for one auction form over another, nothing in themodel surmounts that preference, so long as some variable remains su¢ ciently adjustable.
The sharpness of Theorem 4 stems partly from the exactness attained via a mixed-strategy participation decision arrived at simultaneously by ex-ante symmetric potential bidders (thus not yet
privately informed). Appendix B …nds the bulk of these results attainable, if granted a su¢ ciently
useful equilibrium selection, when potential bidders sequentially decide whether to participate. The
intuition is this: if behavior once participating is symmetric, and one potential bidder who is indi¤erent over whether to participate does take part, then that one participant’s indi¤erence drives
32

A similar example: some sellers categorize assets being sold, so each asset in a category has an appraisal in agiven range (e.g., $20K-$40K for one range, $40K-$80K for the next). For any auction form with the property thata broad enough range is insu¢ ciently extractive and an exact announcement of the appraisal overly extractive, there
exists a range width yielding an optimal auction. Thus, from an arbitrary auction form, only the range width of thiscategorization need be altered to obtain optimality.
33
The same analysis would apply if were the probability that a …rst-price auction would be run, if seller had
precommitted to a rule wherein each bidder submitted both a bid for a …rst-price and another bid (presumably
distinct) for a second-price auction, before a credible randomization determined which set of bids would be used.

21

revenue whether his participation decision was made simultaneously or sequentially.

9

Concluding Remarks on Generality

The contrast is striking: Many papers calculate an “optimal auction,” having innocuously (or atleast without comment) assumed there are n bidders. When this number remains …xed as therole of being a bidder is made far less pro…table, these authors are in essence assuming irrational
behavior, for most situations where they would have us apply their results. Those results typically…nd a particular auction form to be optimal, and it typically revolves around strategically setting
a reserve price which has a signi…cant chance of preventing a sale.
When bidders are also rational in deciding whether to bid, and the number of bidders is explicitly
recognized as an endogenous variable, these results are completely overturned: seller optimallyconducts an allocatively e¢ cient auction (even in some situations where this means sale to a bidder
other than the highest-valuing participant–because in equilibrium that participant does not pay
the entry fee); the only aspect of an auction design that, per se, characterizes it as suboptimal
is a nontrivial reserve price. Selling without reserve is the full content of optimal auctions whenparticipation is endogenous.
As contemplated in auction theory, a nontrivial reserve price is almost never seen in practice(Cassady [1967]). The contemplated reserve price is a credible binding commitment that, if no bid
exceeds it, the asset will not now and never in the future be available to the potential bidders.
In some situations, such a commitment may stretch credibility, but in many, I suspect a tool so
impacting yet so blunt is not used because it would pointlessly introduce ine¢ ciency. What iscommon, and in the industry usually called a reserve price, is a price below which the current
auction will end without a sale, but the same asset will be put up for sale again later34 (often,
34

Though he does not comment on the di¤erent usage of the term reserve price, this is the sense in which McAfee[1993] …nds a nontrivial reserve price in the steady state of a model where sellers repeatedly hold auctions, withprivate-values buyers each time deciding which auction to attend. When a reserve price is not met, the seller simply
holds another auction, possibly with a di¤erent reserve price, in the next period. Successful transactors, buyer andseller, exit the market. Each period, new buyers and sellers are exogenously placed in the market (though all buyers,
new or having bid before, choose among sellers’auctions each period).
Though quite di¤erent models address quite di¤erent issues, this model and McAfee’s are strikingly congruent. In
particular, his sellers’ steady state choices, essentially just in-period reserve prices, are those that a social planner
would choose. In his model, a buyer’s cost of attending an auction is foregoing opportunities to compete in otherauctions, which is an acceptable interpretation of the information-acquisition cost c here. His model explicitly solves
for sellers’steady state behavior; here other sellers are implicitly represented via potential bidders only participating
if expected pro…tability is high enough. His bidders decide which auction to attend, but the number of bidders in themarketplace is exogenous; thus, most of the questions addressed here cannot be addressed in his model.

22there may be negotiations between the seller and the high bidder to buy between the …nal bid andthe reserve price). Such a policy, of course, limits potential ine¢ ciencies to a wholly lower orderof magnitude; introducing it would bring complications of dynamic negotiations into the model,
which I have avoided.
This paper has a¢ rmed how natural and robust are the two principal result reversals whenthe expected number of bidders is endogenous: optimal auctions do not generate ine¢ ciencies, anddo not set a nontrivial reserve price. The principal result is that while selling without reserve,
the set of optimal auctions is large, consisting of single-parameter adjustments of all auctions.
While the demonstration has hopefully been as straightforward as possible, it is clear that thecharacterizations allow several natural extensions:
The oft-seen formulation of an auction problem as an abstract mechanism design problem
contemplates payments to or from losing bidders. Such payments are easily incorporated
here, although Theorem 3 and Proposition 1 render them pointless.
The bid-preparation cost above was exogenous, and independent of the form of the auction.
If strategic issues were to make it more costly to prepare a bid in a …rst-price auction thanin a second-price or an English auction (due to some variant on incentive compatibility), a
more complicated twist on the tools provided here would be needed.35
The assumption of a single asset for sale does not seem critical to the qualitative results.
However, the ease with which extension to the modal multiple-unit auction model (where
each bidder can acquire but one asset) arises in Milgrom [1981] and Pesendorfer and Swinkels[1997] would be somewhat misleading. If k identical assets are sold, the probabilities of0; : : : ; k

1 actual bidders would signi…cantly clutter up the expected revenue formula.

35

Engelbrecht-Wiggans [2001] argues that the strategic simplicity of English auctions, in some simple settings, yields
lower bid-preparation costs than …rst-price sealed-bid auctions, and so English auctions might attract more bidders
and attain higher expected revenue. He demonstrates the possibility in a simple example with independent, uniformly
distributed private values and in…nitely many potential bidders. Unfortunately, the notion of strategic simplicity does
not admit nearly as facile a sensible de…nition when going beyond such a simple setting. In particular, it may be that an
English auction becomes far less simple strategically when a signi…cant entry fee is prepended. (Engelbrecht-Wiggans
does not write as if he is comfortable with the notion that an English auction necessarily remains strategically simpler
once common-value elements enter the model.) Since in the equilibrium above, aggregate expected bid-preparation
costs fall on the seller (Theorem 2), this by itself gives an incentive to favor devices which extract surplus while facingbidders with lower bid-preparation costs. For a given , the mechanisms with lower bid-preparation costs yield higher
revenue. For there to be a su¢ ciently large set of such lower-bid-preparation-cost mechanisms to span continuously
the range of participation probabilities, and thus render all higher-bid-preparation-cost mechanisms inferior, is likely
to depend on some controversial assumptions about strategic simplicity.

23

The model has been designed so that adding other sellers who are auctioning related assets isvirtually automatic. (No problems are created if one seller’s auction exhibits an informationacquisition cost of c, while a possibly more distant seller has a cost c0 > c.)A nontrivial dynamic structure would, however, introduce concerns not yet addressed. Amongthem, both sellers and potential bidders may have incentives to invest in building reputations.
Nonetheless, a stride in this direction is made here: if an analysis of such reputational issues is tobe applicable to markets where a subset of …rms in an industry appear as bidders, reputational
investments need to be viewed in terms of their discounted expected pro…tability when responsesof other players include an endogenous decision as to whether and when to play.
This model assumes rational behavior consistent with a symmetric equilibrium. Asymmetricequilibria at the bidding stage are certainly not going to be unique, so it is unclear how to prepend an
entry stage, without a unique expected pro…tability calculation. Asymmetric participation decisions
are presumably rife for signaling a preferred asymmetric equilibrium. A “symmetric sequential”
entry model which then assumed symmetric behavior following sequential entry decisions can bebuilt; it yields similar but less sharp results. An outline is provided in Appendix B.
Laboratory evidence suggests the winner’s curse is not easily overcome in common-value auctions(Kagel, Levin and Harstad [1995]); however, it is far from clear how to model participation decisionsof potential bidders who will not follow up by bidding rationally. Nor can I envision how to model
usefully the participation decision of a potential bidder who will himself behave rationally, but who
cannot predict even the number of irrationally-behaving rivals who will participate.

10

Appendix A: Mathematics of the Participation Decision

Recall that N is the exogenous number of potential bidders,(M ) the symmetric equilibrium

probability of participating given mechanism M , and n and a numbers of participants and of
actual bidders. Throughout, the usual binomial formula for the probability of z successes in Z
trials, each with independent success probability & is denoted(z; Z; &) =

Z z
& (1
z

&)Z z :

Thus, if N potential bidders each participate with probability , the probability of n participantsis(n; N; ). A potential bidder analyzing the consequences of proceeding to the next step of the24

game (participating or actually bidding) rationally evaluates the likelihood of di¤erent numbers of
rival competitors according to

(n

1; N

1; ), which accounts for the presumption that he (theanalyzing potential bidder) proceeds–even if this is not a certainty, all behavior is otherwise payo¤irrelevant.36 When context makes clear, I will shorten
to

n 1.

(n; N; ) to

n and

(n

1; N

1; )

When an arbitrary potential bidder i becomes a participant, I will harmlessly treat therenumbering function ren(i; n; N ) that would provide his position in the numerically ordered set ofparticipants as if it were the identity function, and refer to the continuing roles of the player who
begins as potential bidder i as if he becomes participant i if he participates, and actual bidder iif he pays the entry fee. Symmetry attained through A.1 and A.6 allows a focus throughout on
potential bidder 1, participant 1, and actual bidder 1.
At the point that the decision to become an actual bidder (to incur the bid-preparation cost b
and pay the entry fee ') is made, participant 1 has observed estimate X1 = x. Two cases must
be developed.
Case 1: M 2 MK , where MK is the subset of mechanisms for which auction form m speci…esthat participants know (perhaps because the seller informed them, perhaps because the seller could
not prevent their knowing) the number of participants, n, before deciding whether to pay the entryfee.
Let
Let ( ) :denote the set ofx

1 -measurable subsets of X , with element

X , andx =f2

jx 2 g.

X ! [0; 1] be de…ned by ( ; x) = Pr [X2 2 jX1 = x], the probability that a given

rival participant observed an estimate in the set , conditional on participant 1 observing estimate
x. Next de…ne the binomial distribution B ( ; x; n) attaching density
values j = 1; : : : ; n. Let

K

[j

1; n

1; ( ; x)] to(M; a; n; ; x) be the expected pro…tability (gross of bid-preparation

cost and entry fee, but net of information-acquisition cost) of actually bidding in auction M , when
there are a actual bidders, n participants, the a actual bidders all observed estimates in
actual bidder 1 observes X1 = x. De…ne forK(M; n; ; x) =

n
X

K

2

, and

x

(M; j; n; ; x) [j1; n1; ( ; x)] ;j=1

which is expected pro…tability when the number of actual bidders is determined by the number ofrival participants who observe estimates in.

36

This insight is originally due to Matthews [1987] (who is credited in McAfee and McMillan [1987a]), and is
employed in Harstad, Kagel and Levin [1990]. In all three papers, the uncertain number of bidders follows anexogenous distribution.

25

Next letdence

K (M; n; ';M;n;';x ( ) :x !

n; x) = & 2(M; n; ; x)x can be de…ned byK
M;n;';x (WhatK

x jx 2 & ))= &2

x j& 2

K does is take an arbitrary subset

Ko' + b , so that the correspon-(M; n; '; ; x) :of the space of estimates, a subset consistent withparticipant 1 observing X1 = x, use it to …nd out expected pro…tability if rival participants pay
the entry fee ' i¤ they observe an estimate in
the subsets of, and then map that expected pro…tability into

x for which paying the entry fee is rational (i.e., yields an expected pro…tability

preferable to ceasing further competition). Finally,K(M; n; ') =
K

Here,

(M; n) =Z

x 2 X j9 2
K (M;n;')K
M;n;';x () j f9 ( 0; z) 2 (x

X1 ) j0; z 2 int ( 0n )g ) 0 2=

d 1 (x) :K (M; n; ') is the maximal subset of X

consisting of those estimates with a su¢ ciently

high interim expected pro…tability to justify continuing to compete in auction M with nparticipants, and1 rival

K (M; n) is the ex-ante probability that a potential bidder whose action is toparticipate will end up becoming an actual bidder.
Continuing with case 1, M 2 MK , de…ne, for n = 1; : : : ; N; i = 1; : : : ; n, the event that participants 1; : : : ; i observe estimates leading them to continue, while participants i + 1; : : : ; n observe
estimates leading them to cease competing:K(M; i; n) =

Xj 2

K

(M; n; ') , fj

ig ; j = 1; : : : ; n :LetK(M; i; n) = Pr Xn 2K

(M; i; n) , andK(M; i; n) = Pr Xn 2K

(M; i; n) jX1 2

K(M; n; ')denote the probability of this event, and its probability conditional on participant 1 observing an
estimate leading to continued competition. By Bayes’Formula, for n = 1; : : : ; N; i = 1; : : : ; n,K

(M; i; n) =K

(M; i; n)

26

K(M; n) :(7)

M;n;';x (0)(

Case 2: M 2 MU = MnMK , when the auction form m speci…es that the number of participants
is unknown when deciding whether to pay the entry fee.
LetU

(M; a; ; ; x) be the expected pro…tability (gross of bid-preparation cost and entry fee,but net of information-acquisition cost) of actually bidding in auction M , when there are a actual
bidders, N

1 rival potential bidders participated with probability

observed estimates inU, the a actual bidders all, and actual bidder 1 observes X1 = x.37 De…ne for(M; ; ; x) =

N X
i
X

U

(M; j; i; ; x) (i1; N1; ) [j

2

1; i

x

1; ( ; x)] :i=1 j=1

The next four steps of case 2 correspond exactly to those of case 1, substituting
for

K

de…ne

to de…ne
U (M;

U (M;

; '; ; x),U

to de…neU
M; ;';x (

),

U

U (M;

to de…ne

U

; '),

above
U

to

). In case 2, the event that participants 1; : : : ; i actually bid, and i + 1; : : : ; n cease

competing:U(M; ; i; n) =

Xj 2

U

(M; ; ') , fj

ig ; j = 1; : : : ; n ;which is well-de…ned although the participants do not know that they number n. The probabilityof the …rst i participants becoming the only actual bidders, given that N potential bidders eachbecome a participant with probability , must take the probabilities of events

U (M;; i; n) andweight them according to their likelihood:
UU(M; ; i) =(M; ; i) =

N
X

n=1
N
X

n=1

(n; N; ) Pr Xn 2

U

(M; ; i; n) , and(n; N; ) Pr Xn 2

U

(M; ; i; n) jX1 2

U(M; ; ')

is the conditional probability given that participant 1 observes a estimate leading to continued
competition. As before,U(M; ; i) =U(M; ; i)37U(M; ) :

(8)

If the auction form m speci…es that information disclosed by seller is disclosed before the entry fee is paid, K andare speci…ed with respect to the equilibrium in the bidding subgame corresponding to the information actually
disclosed. If m speci…es that information disclosed by seller is disclosed after the entry fee is paid, K and U are
speci…ed with respect to the distribution of equilibria expected given the prior distribution of seller’s information.
U

27

Conclusion 1 The relevant ex-ante probabilities combine the two cases :(M; ; n) =i (M; ; n) =i (M; ; n) =with, for any M , each of
Thus,

;

i;

8
<

:
8
<

:
8
<
:

K (M; n) ;M 2 MK ;U (M;M 2 MU ;);;K (M; i; n) ;M 2 MK ;U (M;M 2 MU ;K
U; i) ;(M; i; n) ; M 2 MK ;(M; ; i) ;

; and

;

M 2 MU ;i degenerate in one of its last two variables.

(M; ; n) takes an ex-ante view, from the viewpoint of a potential bidder:

it is theprobability, should he participate, that he will go on to become an actual bidder, evaluated before
the estimate x is observed. Similarly,a (M;; n) is the ex-ante probability, should he become

an actual bidder, that a potential bidder will …nd himself to be one of the set f1; : : : ; ag actual
bidders, anda (M;; n) is the (unconditional) ex-ante probability of f1; : : : ; ag being the set of

actual bidders.38
Note that prior models of endogenous participation have, explicitly or implicitly, assumed ascreening level : some xe (M; n; ') 2 X such that x 2

K (M; n; '), fxxe (M; n; ')g. Lands-

berger and Tsirelson [2000] demonstrate that this is impossible in a common-value auction, forlarge numbers of potential bidders, under mild assumptions, satis…ed by this and most prior models. Whether a corresponding impossibility theorem extends to a¢ liated-values auctions is unclear.
This paper is careful to allow for the fact that

K (M; n; ') andU (M;; ') may not be upper

contours of X ; much of the complication in cases 1 and 2 above is due to that allowance.39
Two cases are also distinguished with respect to actual bidders.

An auction form m 2 MK0

if the number a of actual bidders becomes known before bidding strategies are selected; let the38
What has been shown is that the two extreme possibilities, that participants are informed of their number andthat they are not, can be combined into a sensible description of the relevant ex-ante probabilities. This sameprocedure could readily be followed (with yet more notation) should seller also have some third or fourth options,such as informing participants with some interior probability, or informing them whenever n > 3 but not informing
them when n 3. In other words, the procedures just shown are illustrative, rather than limiting. As stated in the
introduction, it is the endogeneity of the degree of competition that changes the picture, not the details of how thatis modeled.
39
A second-price auction with ' = r = 0 is one special case where K and U are upper contours of X . In general,
the sets K ('; ) of estimates implying at least ' expected pro…tability are unions of nondegenerate intervals (“ins”)
separated by nondegenerate intervals (“outs”). An adjustment improving pro…tability–decreasing or '–continuously
expands the “in” intervals. Harmlessly de…ning the “ins” as closed sets then justi…es the reference to maximality ofK
and U .

28

0

probability of a sale be sK (M; a), which is the probability that at least one of a actual bidders
is willing to pay the reserve price r.
000

For m 2 MU = MnMK , the number of actual bidders

is unknown when bidding; let sU (M; ; n) be the probability that at least one actual bidder iswilling to pay the reserve price r when either [a] each of n participants becomes an actual bidderi¤ Xj 2

K (M; n; '), if m

2 MK (degenerate in), or [b] if m 2 MU , each of N potential

bidders becomes a participant with probability , and if a participant, becomes an actual bidderi¤ Xj 2

U (M;; ') (degenerate in n). Again, combine these cases via

s (M; a; ; n) =8< sK 0 (M; a) ;0

M 2 MK ;(9): sU 0 (M; ; n) ; M 2 MU 0 :Notation will be slightly abused when context makes clear by representing this probability as sr(the reserve price r is the principal component of M a¤ecting this probability).
Getting closer to a characterization: Relying on A.4, let p (M; a; ; n; t) be a function indicating
the expected price paid by the winning bidder, given auction M , a actual bidders,probability ofparticipating, n participants, and conditional on a realization t of common trend T . Depending on
which cases above apply, p ( ) will typically be degenerate in at least one variable. It bears emphasisthat p ( ) is an ex-ante calculation, and thus is symmetric across potential bidders. De…ne
8 R
K
K
< K
(M;n;') G (M; n; '; x; t) v (t; x) d 1 (xjt) ; M 2 M ;V (M; ; n; '; t) =R:U
M 2 MU ;U (M; ;') G (M; ; '; x; t) v (t; x) d 1 (xjt) ;(10)

where GK (M; n; '; x; t) [resp., GU (M; ; '; x; t)] is the probability of becoming the winning bidder
for a potential bidder who will participate in auction M , when there are n[when N

1 other potential bidders participate with probability

1 other participants], the entry fee is ', he willobserve estimate x, and underlying asset value is t. Then V (M; ; n; '; t) is the expected asset
value to a potential bidder who will participate, conditional on his winning in the circumstances
speci…ed by its arguments.
Momentarily assume a potential bidder is one of n participants and one of an actual bidders;

his ex-ante expected payo¤ iss (M; a; ; n)
E fV (M; ; n; '; T )a

E [p (M; a; ; n; ) jT = t]g

29

'

b

c:

(11)

In essence, conditioning the price on the common-value component (the inner expectation) makes
the outer expectation simply the expected di¤erence between what the winner gets and what hepays for it. The probability that the winner obtains this di¤erence is simply the probability ofa sale (s). Ex ante, given a winner, the probability that any one of the a actual bidders is thewinner is 1/a, by A.1 and A.6. For an actual bidder, the bid-preparation cost b, entry fee ' andinformation-acquisition cost c are subtracted with certainty. (Note that this calculation need not
require that the actual bidder know the value of n or a.)Continuing to assume n participants, the ex-ante probability of being an actual bidder is(M; ; n), and of any particular formula (11) being the relevant calculation for an assumed actual
bidder is

n 1
a 1

a (M;

; n), since there are

n 1
a 1

ways in which actual bidder 1 could face a1remaining rivals. Now to step back, assume only that a potential bidder is one of n participants.
His expected pro…t, for n = 1; : : : ; N; isw (M; n) =X h sr(M; ; n)aa

Throughout,

P

a and

E fV ( )E [p (M; ) jT ]g'P

n are to be taken as abbreviated forms of

i n
c
a

b
Pn

a=1 and(12), for di¤erent n, is relevant (assuming participation) with probability

1
1

a (M;

!

; n) :(12)PN

n=1 . Each formula

n 1 =

[n

1; N

1; (M )].

Thus,
Conclusion 2 Equilibrium participation is that0 =X

n 1 w (M; n)

n

=

X
n

2 (0; 1) characterized byn 1((M; ; n)X h sraa

E fV ( )E [p (M; ) jT ]g

'

b

i n
c
a

1
1

a (M;

)

; n)
(13);

equating the payo¤ from nonparticipation to the net expected bene…ts.
The right-hand side of (13) can be lowered by increasing . If

= 1 is allowed as an equilib-

rium possibility (Proposition 2 …nds this a revenue-inferior option for seller), 0 = r:h:s:(13) must
be replaced by [r:h:s:(13) ]0=(

1) [r:h:s:(13)]. Equation (13), by implicitly de…ning the sym-

metric participation probability function(M ), together with equilibrium continuation, provides

a complete characterization of potential bidders’behavior.40
40

A corresponding equation is asserted by French and McCormick [1984], and found in simpler models by Harstad

30

11

Appendix B: An Alternative Sequential-Entry Model

Consider the following “symmetric sequential” participation model. First, the seller announces anauction mechanism M , as above. An exogenous randomization assigns to the N potential bidders arelabeling of their indices, with a potential bidder’s realization that he is number i in this relabelinghis own private information, and all reorderings equally likely. Then potential bidders are in order
given the opportunity to participate (at cost c, as above). As soon as a potential bidder declines
to participate (an action that may be the result of a mixed strategy), seller is committed to giving
no other potential bidder the opportunity.
As a potential bidder knows the step in this order in which he makes his decision, he knows how
many potential bidders have already chosen to participate. As a participant’s stage in the orderdoes not get revealed, an opportunity to signal a favorite asymmetric equilibrium via becoming
participant 1, for example, is unavailable. After participation decisions have been made, one of theparticipants may know privately that he is the marginal participant, but none knows the order inwhich rivals became participants.41 For symmetric behavior to be possible, the private informationof the last participant, as to the equilibrium number of participants, must become public; denote
this number ne (M ). Hence only mechanisms where participants learn the number of participants
can be considered (in Appendix A, this is the set MK ).
The ex-ante probabilities of a participant becoming an actual bidder, and of participants
1; : : : ; a becoming the actual bidders (unconditional and conditional), are exactly the same as
K [M; ne (M )],

K [M; a; ne (M )], andK[M; a; ne (M )], as these terms are deined in Appendix

A, and s ( ) is unchanged from (9) except that it no longer can depend on . Lack of dependenceonis also the only change in p ( ) above, so the expected pro…tability of being the ne (M )th par-ticipant is still (12) above. Hence, ne (M ) is determined by the equilibrium participation constraint
w [M; ne (M )]0 > w [M; ne (M ) + 1] :For equality in (14), participation by ne (M )the ne (M )th potential bidder with probabilityof the(14)

1 potential bidders with probability 1 and by

2 [0; 1] are equilibria for all values of . Selection= 1 equilibrium (revenue-maximal in this set of equilibria) can be based on it being the[1990] and Levin and Smith [1994]. The current development is original in extending to a¢ liated-values settings,
avoiding a monotonicity assumption and allowing for the full variety of information ‡ows.
41
It is solely for this reason that the model builds a counterfactual where a potential bidder’s sequence order is his
own private infomation.

31

unique element of this set which is the limit of equilibria for mechanisms di¤ering from M byhaving in…nitesimally smaller entry fees. Of course, virtually as strong a selection argument canbe made for the= 0 equilibrium, as the unique limit of equilibria for mechanisms di¤ering fromM by having in…nitesimally larger entry fees. However, usual problems with limits of open sets
prevent existence of optimal auctions if theself-servingness of the= 0 equilibrium is selected. I will just consider the= 1 selection be a weakness of the alternative model, and proceed with it.

Revenue is now R (M ) = R [M; ne (M )], from (1). De…ne8< 1; n = ne (M ) ;# (M; n) =: 0; otherwise:This substitutes for the binomial coe¢ cients( ) in the formulas for expected value transferred

and the expected number of actual bidders:V (M ) =XXna (M ) =s (M; a; n) E [T ]

a

(
X X
n

a

a

n
a

K

n K
(M; a; n) # (M; n) ;a)

(M; a; n) # (M; n) :Then expected revenue satis…es
R (M )V (M )ba (M )cne (M ) ;(15)with equality for and only for the selected equilibria attaining equality in (14); let M= be the subset
of MK consisting of those mechanisms M for which equality in (14) and (15) can be attained. Notethat MK nM= contains a dense subset of MK (open in the dimensions of MK with an interior).
De…ne Mknk = M 2 MK j ne (M ) = n , for n = 1; : : : ; N , and MZ = M 2 MK jr = 0 .
The following results can be obtained for such a model. [i ]. fM0 ; M1 g

M= and ne (M0 ) =ne (M1 ) implies R (M0 ) = R (M1 ). This corresponds to a comparative static of the simultaneousentry model.
[ii ]. Suppose a mechanism M0 2 M= with ne (M0 ) = n0 participants in equilibrium. Then
there exists M1 2 Mkn0 k \ M= \ MZ (i.e., M1 does not use a positive reserve price). This M1
is revenue-maximal in the set Mkn0 k ; revenue comparisons across auction forms for an exogenous
number of bidders apply within Mkn0 k , and surplus-extracting devices are substitutes within Mkn0 k ,32with the exception that nontrivial reserve prices are revenue-inferior.
[iii ]. Suppose there exist Msm 2 M= , MLg 2 MK such that 1

ne (Msm ) < ne (MLg )N , and

R (MLg ) > R (Msm ). Then there exists n > ne (Msm ) such that [a] M = Mkn k \ M= \ MZ 6= ;(these are all zero-reserve-price auctions attaining equality in (14) for n participants), and [b] every
auction in M is an optimal auction. Moreover, for an arbitrary auction form m0 for which expected
pro…tability is continuous in the entry fee ', if there exists M 0 = (m0 ; '0 ; 0) such that ne (M 0 ) < n ,then there exists ' such that (m0 ; ' ; 0) 2 M . In this sense, an arbitrary auction can be madeoptimal by the change of a single parameter, attaining a quite similar characterization to theprincipal result of the simultaneous entry model.
[iv ]. Let Msm ; MLg 2 MK be such that ne (Msm ) < ne (MLg ). SupposeXs [Msm ; a; ne (Msm )]ne (Msm )aa [M;; ne (Msm )]s [MLg ; a; ne (MLg )]ne (MLg )aa [M;; ne (MLg )] ;aXathat is, suppose a sale is at least as likely ex ante under Msm as under MLg . Then R (Msm ) >R (MLg ), a sense in which the bidder-discouragement ‡avor of the simultaneous entry model extends
to this model. Note that the sale-probability supposition is critical to result [iv ]. (Proofs of theseresults correspond closely to methods used in the main text and Appendix C.)
Figure 3 illustrates these results, for auctions that use a 0 reserve price.

The entry fee ' isshown horizontally, expected revenue R vertically. The solid curve illustrates one type of auction,
the dotted curve a second type which extracts less surplus for a given number of bidders.

For

concreteness, we may call the solid curve English auction revenue, and the dotted curve …rstprice auction revenue.

The vertical line segments on each correspond to values of ' for which

the speci…ed auction mechanism lies in M= . In particular, each point in a vertical line segment
is revenue associated with one of the multiple equilibria: the lower endpoint is associated withthe marginal participant selecting to enter with probability 0, the upper endpoint associated withprobability 1.
The rightmost vertical segments are where one participant in an English (solid vertical segment)
and in a …rst-price (dotted) auction is enough to make a second potential bidder indi¤erent overparticipating. Along the sloped segment of each curve to the left of its rightmost vertical segment,
the second potential bidder strictly prefers to participate, while staying out is the third potential

33

Revenue

Entry Fee

Figure 3: The Sequential Participation Modelbidder’s strict preference.

Then each curve reaches another vertical segment where the third

potential bidder’s indi¤erence yields multiple equilibria, followed further left by a sloped segment
along which there are three participants.
Each pair of vertical segments corresponding to the multiple equilibria where the ith potential
bidder is indi¤erent over participating peak at exactly the same height. This is a result of equalityin (14) and (15). A curve like those shown could be drawn for any auction form; for example, thecurve for a second-price auction would have vertical segments that lie between the paired verticalsegments shown. The vertical segments shown for the English auction would be shifted to the left
if seller’s information were publicly disclosed.

All such curves for auction forms with 0 reserve

prices would reach identical heights at the peaks of vertical segments.
The case illustrated, which would …t a pure common-value environment, will have as an optimal
auction (given the self-serving equilibrium selection mentioned above) any auction without a reserve
price where the entry fee is set so that the second potential bidder is indi¤erent over participating.
Any auction form which is su¢ ciently extractive to strictly discourage the second potential bidder
via a high enough entry fee will have an entry fee which makes that auction form (with r = 0)
optimal.
In the general a¢ liated-values environment, it is possible that the rightmost pair of vertical line
segments in Figure 3 do not attain the height of the pair to the left of them. If so, then optimal
auctions are those where potential bidder 3 is indi¤erent over participating but does take part (cf.
34

Harstad [2008]).
Consider, for an arbitrary auction form m, beginning with n0 participants in equilibrium, impacts of increasing '. Increasing from small enough ', revenue is monotonically increasing, andw, the expected pro…tability of participating (the l.h.s. of (14)), is monotonically decreasing, whiles, the probability of a sale (here, with r = 0, the probability that a > 0), holds nearly constant.
As ' continues to increase, past some level s starts to decrease nonnegligibly. There will be somethreshold 'b at which revenue from n0 participants hits a local maximum and starts to decrease.

Figure 3 is drawn assuming potential bidder n0 is driven down to indi¤erence over participating

before ' reaches 'b.

I know of no assumption on the primitives of the model guaranteeing this will always be the case(this is why the results above in this appendix are stated with such speci…c conditions). In general,
little is known about the behavior of auction mechanisms above 'b . Revenue need not be monotonic

in ' above 'b , nor need w be monotonic. It is the case that, for (m; '0 ; 0) 2 M= , revenue approaches

R (m; '0 ; 0) from below as ' approaches '0 from below. Also, (m; '; 0) 2 Mkn0 k ) 9'0 j (m; '0 ; 0) 2
M= \ Mkn0 k . However, '0 > 'b will mean multiple local maxima of revenue in ' for given m, acrossthe set of ' for which an equilibrium with n0 participants is selectable. In the presence of such
multiple local maxima, I know of no argument from primitives that implies the global maximum
revenue must lie in M= . Should it not, in essence the theory of auctions with an exogenous numberof bidders applies.
Several seminar attendees have insistently pursued the following assertion: a seller who (somehow) had a choice between selling via an auction following the “sequential symmetric” entry ofthis Appendix and via an auction following the “simultaneous symmetric” entry in the main text
above would always prefer the former. I …rst provide a counterexample, and then discuss why theassertion appears to be so appealing.
Example:Let there be N = 2 potential bidders for a common-value asset: v (T; Xi ) = T .

Denote M = (m; 0; 0), a second-price auction. Fix ; then E [T ] and R M ; 2 are …xed.
choosing information-acquisition cost c = E [T ]R M; 2Then=2 and bid preparation cost b = 0

yields an environment for which M is an optimal auction in the sequential entry model, if theequilibrium is selected in which the second potential bidder is indi¤erent over participating but
participates with probability 1. The expected revenue attained is R M ; 2 . In the simultaneousentry model,

M = 1 and R M = R M ; 2 . However, by Proposition 2, an increase in ' from0 to d' increases revenue, to a level unattainable in the sequential entry model.
35

The assertion pays attention to an obvious detriment in the main model, the probability(1)N that no potential bidder participates, and thus no gains from trade occur. It neglects amore subtle advantage: for optimal mechanisms, the probability that a participant faces a smaller)N . In the example, a potential

than-average number of rival participants is far larger than (1bidder making a sequential participation decision knows for sure that he faces one rival bidder, andis indi¤erent over participating when ' = 0; an entry fee of d' > 0 will lead to his nonparticipation
and a plunge in revenue (to d'). However, a potential bidder making a simultaneous participation
decision will face one rival bidder with probability
bidder, his net expected pro…tability isslightly less than 1.

If he faces one rival

d'. Countering this loss is the (1) probability that

he faces no opposition and obtains the asset for a price of d'. The seller gains because the resource
costs have been reduced from 2c to 2c, and in each case there is a participant who is indi¤erent.

For those auction forms where the relationship between a bidder’s expected pro…tability and an
exogenously speci…ed number of bidders is known, this relationship is strictly convex. Hence, a seller
can sometimes attain a sizable , even though the mechanism is strongly surplus-extractive, because
a bidder is weighing in the chances of being the only participant or one of very few participants.
With “sequential symmetric” entry, an optimal auction never faces a participant with fewer thanne (M )

12

1 rival participants. On average, the seller may be able to gain from this di¤erence.

Appendix C: Proofs

Proof of Theorem 2: For the proof, shortena (M;; n) toa,(n; N; ) ton , and(n(M ) to1; N,(M; ; n) to1; ) to

n 1.

,

a (M;

; n) toa,

Begin by harmlessly con-ditioning the price on the common trend, and then adding 0 in useful forms at two locations in

36

(2):
R (M ) = (sr E fE [p (M; a; ; n; ) jT ]g + a')
(
)
X X
n
=
(sr E fE [p ( ) V ( ) ab + V ( ) + abjT ]g + a')
cn + cn
a a
n
a
(
)
X n
X
sr E [V ( )] bacn=a (M; ; n)
n
a
a
n
(
)
X X
n
(sr E fE [p ( ) V ( ) jT ]g + a [' + b])
+ cn
+
n
a a
n
a
)
(
X n
X
(M; ; n) cn
asr E [V ( )] b
=
n
a a
a
n
(
)
X X
n n 1
+
(sr E fE [p ( ) jT ] V ( )g + a [' + b])
a + cn
n;
a a 1
n
a

n

where the last equality uses the Bayes’formulas [(7) or (8)].
R (M )(
X=
sr E [V ( )]
n

+

(
X X
n

a

X

n
b
a
a
a

a (M;

(sr E fE [p ( ) jT ]awhere the …rst equality sorts n out of

cn

n

n
V ( )g + [' + b])a= T (M ) ba (M ) cn (M )
"
(
X
X
+
(sr E fE [p ( ) jT ]
n

; n)

)

1
1

a

n
V ( )g + [' + b])
a

1
1

)

P
i (i; N; )+ c n n Pi
i i (i; N; )
)

a+c

n 1

#

N ;

P

a and multiplies by 1 in a useful form, and the …nal equality

simpli…es the numerator and combines the denominator with n n . The term in large [ ] is 0 by
(13).
Proof of Theorem 3: Setting ' = sup v (T; X) generates
Interim expected pro…tability= 0; ' =cb generates

M' ; n; ; x is degenerate in ', so the mapping ' 7!

[(6)] is continuous. Smoothness of B1 (from A.1) implies thatK

M ' ; n and

K

K

= 1.

M ' ; n; 'M ' ; a; n arecontinuous in '. Since, for m, the price function p ( ) is degenerate in ', it follows that theM'function implicitly de…ned in (13) is continuous in '. The Intermediate Value Theorem yields theconclusion.
Proof of Corollary 2: Consider any M = (m; '; r) such that37

(M ) = b 2 (0; 1), with r

nontrivial. By Theorem 3, there exists M 'b = (m; 'b ; 0) so thatM 'b = b. Expected revenue,

R (M ), with r nontrivial, is
"
X X
n

<

a

"
X X
n

a

n
ab) s (M; a; b; n)a(E fV [M; (M ) ; n; '; T ]gab) s M 'b ; a; b; n

(E fV [M; (M ) ; n; '; T ]g

a (M;

n
a

; n)

#

M 'b ; ; n

a

cbN

n

#

n

cb(16)
N,

which is R M 'b . Naturally, the probability that a given participant pays the reserve price is strictly

less than the probability that he wins, while the probability that he pays the entry fee is strictly
greater than the probability that he wins. Hence, with both mechanisms attaining participation
probability b, 'b < ' + r. The inequality then results from the terms in f g on the left-hand side of(16) summing to less than the corresponding terms on the right-hand side (as terms corresponding

to the cases where no participant is willing to pay the reserve price are zero on the left-hand side,
but positive on the right, while the aggregate expected pro…t of participants is the same).
Proof of Proposition 1: [i ]: The same

implies that (r.h.s.) of (13), which is monotone,

attains the same value. Now reversing the substitutions used in the proof of Theorem 2 demonstrates revenue equality. [ii ]: f (M ) =(M 0 )g ) fn (M ) = n (M 0 )g, so [ii ] follows from [i ]. [iii ]:The proof for the equality has already been shown. Suppose R (M; n) > R (M 0 ; n) 8n 2 N. From(2), 8n 2 N,
X
a

X

>

(E fE [p (M; ) jT ]g + a [' + b])
EE

p M 0;

a

jT

n
a

a (M;

n
a

+ a '0 + ba(M ) ; n)M 0;

M0 ; nimplies, using Bayes’formula as in the previous proof:
X

n 1

n

implying(

0M;(M ) <

0

X 1
;n
E
a
a

T

E

0

p M;

jT

'

0

(M 0 ). The reverse inequality is identical.

38

b

n
a

1
1

a

0

M;

0

M ;n

)

> c,

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42