Tsukuba Economics Working Papers
No. 2010-002	
 Equilibria in Asymmetric Auctions with Entry
by
Jun Nakabayashi	 	 January 2010

UNIVERSITY OF TSUKUBA
Department of Economics
1-1-1 Tennodai
Tsukuba, Ibaraki 305-8571
JAPANEquilibria in Asymmetric Auctions with Entry
Jun Nakabayashi∗
January 26, 2010

Abstract
Regarding optimal design in the private value environment, there is an unsolved
discrepancy in the literature regarding asymmetric auctions and auctions with endogenous participation; Literature on the former suggests that well-designed distortive
mechanisms are optimal (revenue maximizing) assuming the bidding costs are negligible, while that on the latter insists that the mechanisms with free entry and no
distortion are optimal provided that the potential bidders are ex ante symmetric.
This paper is the first attempt to reconcile the two views by establishing a model
for asymmetric auctions with costly participation. The main findings are threefold;First, an optimal outcome is possible if and only if the mechanism is ex post efficient.
Second, without any participation control, a coordination problem is likely in which
only the weak bidders participate and the strong bidders stay out. Finally, there is anentry fee/subsidization scheme which, together with an ex post efficient mechanism,
induces the optimal outcome as a unique equilibrium.

Key words: auctions, endogenous participation, asymmetric bidders
JEL classification: C72, D44, L22

1

Introduction

In high-valued asset or procurement auctions, the costs for preparing bids are typically
non-trivial. The costs each bidder incurs prior to bidding range from information acquisition costs to transportation costs and even opportunity costs of awarding. Potential
bidders who anticipate that bidding is unprofitable may hesitate to do so; therefore, designing a mechanism that accounts for the bidder’s endogenous participation is crucial forthe auction to be successful. The model of auction with endogenous entry has received∗
Doctoral Program in Economics, Graduate School of Humanity and Social Sciences, University of Tsukuba Tenodai 1-1-1, Tsukuba, Ibaraki 305-8571.
Tel.: +81(29)853-7432, e-mail:nakabayashi@dpipe.tsukuba.ac.jp

1

comprehensive analysis motivated by such practical situations faced by the auctioneer(e.g., McAfee and McMillan (1987), Engelbrecht-Wiggans (1993), Levin and Smith (1994),
Kjerstad and Vagstad (2000), Ye (2004)). Furthermore, empirical studies of auctions withendogenous participation are growing following the development in the theoretical endogenous participation models.1
The notable insights presented in the existing literature, however, crucially depend on
the assumption that all the potential bidders are ex ante identical. This strong assumption
may result in analyses being restrictive. Consider the case, for example, where a limited
number of bidders participate frequently but there are many other potential bidders whorarely enter the auction. The question of whether revenue-maximizing auctioneers should
precommit to running a distortive auction to encourage entry by one-shot customers orgive up promoting competition and simply set an entry fee to extract more surplus from thefrequenters must be asked. The possibility that one-shot customers have a higher valuation
for an item must also be examined. Furthermore, it must be determined whether more
entries and stronger competition create higher revenue for the seller. The existing studies
provide an ambiguous prediction for auctions with asymmetric potential bidders.2
In this research, we provide the first theoretical analysis for asymmetric auctions withendogenous participation. The model we establish is an extension of the model of auctions
with costly participation, in which risk-neutral potential bidders randomize their participation in the auction. The bidders who actually enter incur a fixed participation cost andacquire their private information. In this formulation, we relax the symmetric assumption
in the following manner. First, we suppose that potential bidders consist of two groups andthat the bidders in each group participate in the auction with probability p1 for one groupand p2 for the other.3 The equilibrium we focus on in this paper is the type-symmetric
mixed-strategy entry equilibrium which is constituted by a pair of probabilities (p1 , p2 ).
Second, we consider that the potential bidders in one group may be stronger than those in
the other group, i.e., the value distribution of a group of bidders stochastically dominatesthat of the remaining bidders.4
It is shown that there is at least one and typically multiple type-symmetric equilibria.
1

See e.g. Athey and Haile (2006).
The theoretical model provided by Pevnitskaya (2003a) characterizes an equilibrium with asymmetricbidders that assumes that different potential bidders have different risk attitudes. Pevnitskaya finds thatsome bidders who are less risk-averse tend to participate in auctions more frequently than others in equilibrium provided that each bidder’s risk attitude is common knowledge. However, the equilibrium bidding
function analyzed in her study is essentially symmetric.
3
Although the model considers only two groups, it can be extended to a case involving three or moregroups.
4
There can be many forms of asymmetry. Our model does not restrict the form of asymmetry tostochastic dominance.
22

In some cases, a group of weak potential bidders enters with positive probability, and all
the strong bidders stay out in equilibrium. We then demonstrate that, if the mechanism
is ex post efficient, participation is always efficient in the sense that the expected marginalcontribution of an additional participant to social surplus equals the marginal costs forparticipation. However, we also show that the “efficient entry” is not always optimal forthe seller and society. Due to the stochastic entry process, some equilibria are more likely
to end up with “too many” or “too few” bidders. These coordination costs are anothersource of efficiency loss, which makes all the efficient entry equilibria sub-optimal, except
one. In the case of symmetric potential bidders, for example, a symmetric equilibrium
creates the lowest revenue because of the highest coordination costs among equilibria.
It must be asked, therefore, how, when facing such a multiplicity problem, the auctioneer induces optimal participation. Our model suggests that, by introducing a participation
fee/subsidization contingent on the realization of participation, the desired entry is always
induced as a unique equilibrium regardless of the form of asymmetry. Moreover, thetransfer scheme enables auctioneers to extract the entire surplus generated by the auction. Hence, sellers can implement the optimal entry by using a simple auction (English
or second-price sealed-bid auction) with a well-chosen monetary transfer scheme.
These findings shed new light on the literature of auctions with asymmetric bidders aswell as auctions with endogenous participation; First, our results contradict the theorems
for an optimal design problem with a fixed number of bidders, which assert that a positive
reservation price or some distortive allocation favoring a group of bidders improves revenue
for the seller (e.g., Riley and Samuelson (1981), Myerson (1981), Bulow and Roberts(1989), McAfee and McMillan (1989)). However, this argument ignores the point that therent extraction from a group of potential bidders could depress their participation. Our
results illustrate that ex post efficiency is essential for efficient entry and, thus, for optimal
outcome, taking endogenous participation into account. It follows that any distortive
mechanism entails the sub-optimal outcome in asymmetric auctions with potential bidders.
Second, the optimality of ex post efficient mechanisms challenges the ranking theorems.
In asymmetric auctions, revenue ranking between first- and second-price mechanisms is
generally ambiguous. This ambiguity, however, disappears, considering entry. We show
that, by asymmetry, first-price auctions may attract more or fewer bidders, but the resulting excessive or deficient entry inflicts a greater burden on the seller, who bears all theparticipation costs. This story can easily be extended to the case of asymmetric auctions
with affiliated private value (APV). Since affiliation gives greater advantage to secondprice mechanisms, first-price mechanisms are still dominated by second-price mechanisms
under the APV environment.

3

Third, our model extends the theoretical analysis of auctions with endogenous participation in several aspects. There are two groups of literature for auctions with costly
participation, investigating either an asymmetric equilibrium (e.g., McAfee and McMillan(1987)) or a symmetric equilibrium (e.g., Levin and Smith (1994)). We provide a general
theory that analyzes both simultaneously. This enables us to obtain a ranking method forsocial surplus and revenue across equilibria. Furthermore, the theorem is robust in casesof heterogeneous potential bidders. The symmetry assumption is somewhat disturbing
since it is violated if it is common knowledge that some potential bidders are even slightly
likely to have higher valuation for the item.
Finally, our results provide a theoretical background for the experimental analysis
for auctions with endogenous participation. Pevnitskaya (2003b) observed evidence inlaboratory experiments that some subjects are more likely to participate than the others.
Since we show that the participation game is similar to a coordination game in which there
are multiple equilibria, such an outcome is possible as an equilibrium.
The reminder of this paper is organized as follows; In Section 2 we describe the model.
The Discussion is provided in Section 6. Section 7 is the conclusion. Proofs are presented
in an appendix.

2

Model

Consider a risk-neutral seller auctions a single indivisible item to two groups of risk-neutral
potential bidders with unit demand. Suppose that there are Nτ potential bidders in groupτ ∈ {1, 2}, and let Nτ ≡ {1, · · · , Nτ } denote the index set of group τ potential bidders.
The transaction is described as a two-stage game. In the second stage, an auction
takes place with n = (n1 , n2 ) actual bidders to allocate the item subject to an allocation
rule set by the auctioneer. In the first stage, each potential bidder simultaneously makes
his or her decision to incur a fixed participation cost cτ and enter the auction game.
Throughout this paper, we suppose private values, i.e., that one buyer’s signal does
not affect the other’s preferences. Each bidder draws his or her own signal denoted by σ,which is, without loss of generality, uniformly distributed between 0 and 1. The value ofthe item for a bidder in group τ is captured by the valuation function vτ : [0, 1] → <+ ,which is strictly increasing and continuously differentiable. Finally, the seller’s value forthe item is normalized to equal zero.
The auction game consists of a set Bτi ∈ B of bids for each bidder, an allocation rulex(·|n) : B → [0, 1]n , and a payment rule k(·|n) : B → T n . If the ith bidder in groupτ chooses a bid biτ ∈ Bτi , then, given a bid profile b = (b11 , . . . bn1 1 , b12 , . . . bn2 2 ), the bidder4

obtains the item with probability equal xiτ (b|n) ∈ [0, 1] and makes the expected paymentkτi (b|n) to the seller.
The entry game begins with the seller’s announcement on the assignment rule ξ =(ξ1 , ξ2 ) with ξτ : ×n∈N ξτ (b|n) and ξτ (b|n) = (ξτ1 (b|n), . . . , ξτnτ (b|n)), the payment rulek = (k1 , k2 ) with kτ : ×n∈N kτ (b|n) and kτ (b|n) = (kτ1 (b|n), . . . , kτnτ (b|n)), and the transfer
schedule y = (y1 , y2 ) from the seller to the participants with yτ : ×n∈N yτ (n) for τ = {1, 2}.
Given {ξ, k, y}, each of N1 +N2 potential bidders simultaneously makes their entry decisionby assigning a probability piτ on his or her entry. Those who actually participate in theauction observe n, incur a participation cost cτ , obtain a monetary transfer yτ (n) and bid
following a Nash bidding strategy βτi (σ|n).
Now, let πτi (n|ξ, k) denote the expected payoff of a bidder in group τ from the auction
prior to drawing his or her private information. Then, the net gain of the bidder fromparticipating in the auction, uiτ (n|·), is given byuiτ (n|ξ, k, y) = πτi (n|ξ, k) + yτ (n) − cτ .

(1)

Throughout the paper, we focus on the class of transfer schemes Y such that yτ (n) is
decreasing in n. We also suppose that πτ (n|xτ ) is decreasing in n so that uτ (n|xτ , yτ ) is
decreasing in n. Therefore, there are a set of pairs of numbers n∗ = (n∗1 , n∗2 ) such thatuτ (n∗τ , n∗−τ |·) ≥ 0 > uτ (n∗τ + 1, n∗−τ |·) for some τ = {1, 2}.5 To keep the model general, we
do not assume that the participation costs c must be moderate. Therefore, for some n∗−τ ,we could have n∗τ = 0 or n∗τ > Nτ .
Hereafter, our analysis will proceed backward, beginning with the analysis of thesecond-stage auction game. After we obtain the equilibrium bidding strategy and theassociated ex ante expected payoffs from the auction, we will investigate the entry decision in the first stage.

3

The Nash bidding strategy in asymmetric auctions

By the Revelation Principle, the asymmetric auction analyzed here can be described as the
incentive-compatible (IC) direct-selling mechanism. Let στ = (στ1 , . . . , στnτ ). Then, given areport profile on signals σ = (στ , σ−τ ), a direct mechanism is characterized as an allocation
rule x(σ|n) = {xiτ (σ|n)}i∈nτ ,τ ∈{1,2} and a payment rule λ(σ) = {λiτ (σ)}i∈nτ ,τ ∈{1,2} , wherexiτ (t) ∈ [0, 1] is the probability with which the ith bidder in group τ obtains the item andλiτ (t) is the expected payment the bidder makes to the auctioneer when σ is reported.
5

Throughout, we assume that, for any function η, the first and second arguments for n1 and n2 areexchangeable, i.e., η(n1 , n2 |·) ≡ η(n2 , n1 |·)5In our analysis, we focus on the class of the mechanism in which the assignment rulex and the payment λ for bidders in the same group are identical, respectively. In otherwords, we have xiτ ≡ xτ , and λiτ ≡ λτ holds for all i. Hence, without using the superscript
i, let wτ be the conditional expected payoff a group τ bidder provided that his or her
signal equals στi . If the remaining n − 1 participants report (στ−i , σ−τ ) truthfully, then theenvelope integral formula makes w satisfywτ (στi |n, xτ ) = wτ (0|n, xτ ) +

Z στi
0

d
vτ (σ̂)xτ (σ̂|n)dσ̂.
dσ̂

Next, we derive the bidder’s ex ante expected payoff from the asymmetric auction prior
to drawing his or her signal στi . Since σ is uniformly distributed between 0 and 1, theexpected payoff πτ (n|xτ ) ≡ E[wτ (σ|·)] is given by

Z 1 Z στidvτ (σ̂)xτ (σ̂|n)dσ̂dσ
dσ̂
0
0
Z 1
d
=
(1− σ̂) vτ (σ̂)xτ (σ̂|n)dσ̂,dσ̂0πτ (n|xτ ) =(2)

where we normalize wτ (0|n, xτ ) = 0. If there is a unique Nash bidding strategy denotedby β(σ|n), then we have x(σ|n) ≡ ξ(β(σ)) for all σ ∈ [0, 1]n . Thus πτ (n|ξτ ) ≡ πτ (n|xτ )holds.

4

Type-symmetric entry equilibria

A type-symmetric equilibrium is an equilibrium in which all the bidders in the samegroup assign an identical probability on their participation. Suppose that bidders ingroup τ except i enter the auction with probability pτ and each bidder in group −τenters the auction with probability p−τ . In general, if Nt − k potential bidders in group tenter the auction with probability pt and N−t − ` potential bidders in group −t enter itwith probability p−t , then the probability that the number of actual entrants is equal ton̂ = (n̂t , n̂−t ) is given by
Nt −k,N−t −`

Pn̂,p


≡

Nt−k



n̂t

N−t−`
n̂−t


[pt ]n̂t [1−pt ]Nt −k−n̂t [p−t ]n̂−t [1−p−t ]N−t −`−n̂−t .

Therefore, provided that all the remaining potential bidders follow p ∈ (p1 , p2 ), the conditional expected gain of the ith bidder in group τ from participating in the auctionUτi (p1 , p2 |xτ , yτ ) is written as

6


NX
N2
1 −1 X
N −1,N2u1 (n̂1 + 1, n̂2 |x1 , y1 )
U
(p
,
p
|x
,
y
)
≡
Pn̂,p1
1 1 2 1 1







 U2 (p1 , p2 |x2 , y2 ) ≡n̂1 =0 n̂2 =0
N1 NX
2 −1
X

N1 ,N2 −1

Pn̂,p

(3)

u2 (n̂1 , n̂2 + 1|x2 , y2 ),n̂1 =0 n̂2 =0

where we omit N and c = (c1 , c2 ) on Uτi since these are exogenous throughout our research.
Given {x, y}, the ith bidder in group τ will randomize his participation if and only ifUτ (p1 , p2 |xτ , yτ ) = 0; otherwise, he will choose to enter or stay out as a pure strategy. To
clarify, let hτ : [0, 1]Nτ −1 × [0, 1]N−τ → [0, 1] be the best response entry decision of the ith
potential bidder in group τ . Then, this best response function can be described ashτ (p1 , p2 |·)




= 1

if {p1 , p2 |Uτ (p1 , p2 |·) > 0}=0 ∈ [0, 1]

if {p1 , p2 |Uτ (p1 , p2 |·) < 0}
if {p1 , p2 |Uτ (p1 , p2 |·) = 0}.

A type-symmetric mixed strategy entry equilibrium is characterized by a pair of probabilities (p∗1 , p∗2 ). Hence, the best response participation decision of the ith bidder in groupτ , hτ , must be equal to pτ in equilibrium, implying that (p∗1 , p∗2 ) satisfieshτ (p∗1 , p∗2 |·) = p∗τ ,(4)for all τ ) ∈ {1, 2}. In other words, if we define Aτ (p1 , p2 |·) ≡ hτ (p1 , p2 |·) − pτ , thenp = (p1 , p2 ) is a type-symmetric equilibrium if and only if Aτ (p|·) = 0 for all τ = {1, 2}.
Using this result, we have the following proposition which verifies the existence of such
mixed-strategy equilibria in the asymmetric entry game.
Proposition 1. There exists at least one mixed-strategy type-symmetric entry equilibriumin the participation game.
The proof given in the Appendix is absolutely in line with a regular proof of theexistence of a mixed-strategy Nash equilibrium.

5

The property of the entry equilibria

The following figures depict Aτ as well as the mixed strategy entry equilibria as theintersections of Aτ on (p1 , p2 ) space.

7

p1

p1
1

1

E∗

s

A1 (p|·)A1 (p|·)
p¯1 s 
p̃1A2 (p|·)cc

A2 (p|·)


45◦

s
p¯2 1

p˜2

O

p2

p2
1

O

Figure(a)Figure(b)We define p̃τ such that U2 (p̃1 , 0|x2 , y2 ) = 0 and U1 (0, p˜2 |x1 , y1 ) = 0. In addition, we definep̄τ such that U1 (p̄1 , 0|x1 , y1 ) = 0 and U2 (0, p̄2 |x2 , y2 ) = 0.
There are two types of mixed-strategy equilibria. To clarify, let Gτ (p|xτ , yτ ) ≡ ∂p1 /∂p2 |Aτ =0
denote the absolute value of the slope of (10) on the p1 -p2 square. Upon taking the totalτ
derivative of Aτ (·) with respect to p1 and p2 , one obtains dAτ (p|xτ , yτ ) = ( ∂A
∂p1 )dp1 +τ( ∂A∂p2 )dp2 .

Therefore, we have 6
Gτ (p|xτ , yτ ) =∂Aτ /∂p2∂Aτ /∂p1(5)

if pτ ∈ (0, 1).
Let an “odd” mixed-strategy type-symmetric equilibrium be the equilibrium such thatG1 (p∗ |x, y) − G2 (p∗ |x, y) ≥ 0, and let an “even” mixed-strategy type-symmetric equilibrium
be the equilibrium such that G1 (p∗ |x, y) − G2 (p∗ |x, y) < 0.
Since Aτ (·) is continuous and non-increasing, the number of even equilibria is always
one fewer than the number of odd equilibria. Hence, by Proposition 1, there exists at
6

We can compute G by
N1 −1N2 −1∂A1 /∂p2 = N2 ·X X

N −1,N2 −1Pn̂,p1[u1 (n̂1 +1, n̂2 +1|·) − u1 (n̂1 +1, n̂2 |·)] ,n̂1 =0 n̂2 =0
N1 −2 N2

∂A1 /∂p1 = (N1 −1)·

X X N −2,N2
Pn̂,p1[u1 (n̂1 +2, n̂2 |·) − u1 (n̂1 +1, n̂2 |·)] ,n̂1 =0 n̂2 =0
N1 N2 −2

∂A2 /∂p2 = (N2 −1)·

X X

N ,N2 −2Pn̂,p1[u2 (n̂1 , n̂2 +2|·) − u2 (n̂1 , n̂2 +1|·)] ,n̂1 =0 n̂2 =0
N1 −1N2 −1∂A2 /∂p1 = N1 ·X X

N −1,N2 −1Pn̂,p1[u2 (n̂1 +1, n̂2 +1|·) − u2 (n̂1 +1, n̂2 |·)] .

n̂1 =0 n̂2 =08

least one odd equilibrium and an even equilibrium exists if and only if there are multiple
equilibria. Furthermore, if all potential bidders are identical and y = 0, the symmetricequilibrium is always even since G1 (ρ, ρ) = N2 /(N1 − 1) and G2 (ρ, ρ) = (N2 − 1)/N1 forany ρ ∈ (0, 1).
To seek strategic interaction in the participation equilibrium, it is convenient to formulate the relative strength between the two groups by the difference in their expected
gain from participation as follows:
Definition 1. The group 1 potential bidders are more profitable than the group 2 potential bidders if and only if i) u1 (n1 + 1, n2 |x1 , y1 ) ≥ u2 (n1 , n2 + 1|x2 , y2 ) holds for somex and y and ii) u1 (n1 , n1 |x1 , y1 ) ≥ u2 (n1 , n2 |x2 , y2 ) holds for such x and y.
Condition i) implies that the ex ante payoff of a particular bidder is monotonicallyincreased by the change of his or her type from 2 to 1, whereas ii) implies that the exante payoff of a type 1 potential bidder is always greater than that of a type 2 bidder in
the auction. If the mechanisms are ex post efficient, a more profitable bidder is equivalent
to a strong bidder in the sense that his or her value distribution stochastically dominates
the value distribution of a weak bidder.7 Then, one obtains the following lemma about anequilibrium participation decision.
Proposition 2. Suppose that a potential bidder in one group is more profitable thana potential bidder in the other group for some mechanism and transfer scheme. Then,the probability with which a less profitable bidder enters the auction is greater than theprobability with which a more profitable bidder enters the auction if each potential bidder
is indifferent between participating and staying out in equilibrium.
See the Appendix for proof.
A symmetric equilibrium with ex ante identical bidders corresponds to the special casein which the equations in condition i) and ii) hold with equality, as shown in Figure (a). On
the other hand, if asymmetry between two groups is sufficiently large, no even equilibrium
7

If a mechanism is ex post efficient, we haveR
π1 (n1 + 1, n2 |x) = Rv (1−F1 (v))(F1 (v))n1 (F2 (v))n2 dv,
π2 (n1 , n2 + 1|x) = v (1−F2 (v))(F1 (v))n1 (F2 (v))n2 dv,

for any n. Hence, F1 (v) ≤ F2 (v) for any v implies π1 (n1 + 1, n2 |x) ≥ π2 (n1 , n2 + 1|x). In addition, underthe ex post efficient mechanism, we haveR
π1 (n1 , n2 |x) = vR(1−F1 (v))(F1 (v))n1 −1 (F2 (v))n2 dv=R v [F2 (v)−F1 (v)F2 (v)](F1 (v))n1 −1 (F2 (v))n2 −1 dv
π2 (n1 , n2 |x) = vR(1−F2 (v))(F1 (v))n1 (F2 (v))n2 −1 dv= v [F1 (v)−F1 (v)F2 (v)](F1 (v))n1 −1 (F2 (v))n2 −1 dv
for any n. Thus, if F1 (v) ≤ F2 (v) for any v, then π1 (n1 , n2 |x) ≥ π2 (n1 , n2 |x).

9

is likely. The marginal case is shown in Figure (b), in which an even equilibrium will
disappear if group 1 bidders become more profitable.8
If potential bidders across groups are identical and c is moderate, there exist two
additional asymmetric equilibria. The asymmetric equilibrium analyzed by McAfee andMcMillan (1987), Engelbrecht-Wiggans (1993) is the case in which p∗τ = 1 and p∗−τ = 0
for some τ . Depending on N1 and N2 , there could be many other asymmetric equilibria
e.g., p∗1 = 1 and p∗2 ∈ (0, 1), as shown in Figure (b), where group 1 bidders obtain positive
expected rents.
The auctioneer has no reason to keep the bidders obtaining strictly positive rents. The
following lemma shows that full extraction of rents is trivially possible by an entry fee.
00
Lemma 1. For any p, there exists a constant participation fee schedule yp,x,y= (y1,p,x,1 ,y10y2,p,x) ∈ <2 such that full extraction of rents is possible for some x ∈ X and y ∈ Y .
2 ,y20Moreover, yp,x,y
implements the rent extraction by holding p constant.
00
Proof. Set yτ,p,x
such that 0 = yτ,p,x+ Uτ (p|xτ , yτ ) for some p, x and y. Then, by
τ ,yτ
τ ,yτ

(1) and (3),00 = yτ,p,x
+
τ ,yτ

NX
−τ
τ −1 NX

N −1,N−τPn̂,pτ[πτ (n̂τ + 1, n̂−τ |xτ ) + yτ − cτ ]n̂τ =0 n̂−τ =0

=

NX
−τ
τ −1 NX

N −1,N−τPn̂,pτ0[πτ (n̂τ + 1, n̂−τ |xτ ) + yτ + yτ,p,x− cτ ]τ ,yτn̂τ =0 n̂−τ =0
0= Uτ (p|xτ , yτ + yτ,p,x
),
τ ,yτ

for any p and x. x and p remain unchanged throughout.
A simple entry fee schedule, although it might be discriminatory, allows the seller toextract full rents without disturbing equilibrium p. Therefore, the optimal design problem
in auctions with asymmetric potential bidders is equivalent to the maximization problemof the social surplus.
The social surplus associated with the transaction here is defined as the winning bidder’s valuation for the item minus the sum of participation costs incurred by participants.
Let β(n|·) be the incentive-compatible expected payment from the winning bidder to the8

Sufficiency for multiple equilibria is p̄τ ≥ p̃τ for all τ . Intuitively, this is the case in which, if eachparty commits to assigning its maximum probability p̄τ , there is no room for the other group to participate
profitably.

10

seller in the auction taking n as given. Then the auction revenue R(p|·) is written as
N1 X
N2
X
N
R(p|x, y) =Pn̂,p[β(n̂1 , n̂2 |x)−y(n̂1 , n̂2 )].
n̂1 =0n̂2 =0

From (1), the sum of the bidder’s ex ante expected payoffs U (p|x, y) = N1 p1 U1 (p|x1 , y1 ) +N2 p2 U2 (p|x2 , y2 ) is
N1 X
N2
X
N
U (p|x, y) =Pn̂,p[V(n̂1 , n̂2 |x)−β(n̂1 , n̂2 |x)+y(n̂1 , n̂2 )]−N1 p1 c1 −N2 p2 c2 ,n̂1 =0 n̂2 =0

where V (·) is the expected valuation of the winning bidder given n̂1 , n̂2 and x. On theother hand, the total surplus S(·) is given by
S(p|x) = R(p|x, y) + U (p|x, y)
=

N1 X
N2
X
N
V (n̂1 , n̂2 |x) − N1 p1 c1 − N2 p2 c2 .
Pn̂,p

(6)

n̂1 =0 n̂2 =0
0
By lemma 1, yτ,p,x
enables the auctioneer to extract the bidder’s rent at all for any p,τ ,yτ0i.e., U (p|x, yp,x,y) = 0. Hence, we obtain
0
S(p|x) = R(p|x, yp,x,y).

This implies that, if the seller sets a monetary transfer scheme such that bidder’s rent iszero, then the seller’s revenue is identical to the social surplus for any mechanism.
In addition, (6) may remind us of the revenue equivalence theorem for asymmetricbidders with entry, as described in the following statement:
Theorem 1. Suppose that the bidder’s rent is fully extracted. If any two mechanisms havethe same probability assignment functions and induce equal entry, then the two mechanisms
generate the same revenue for the seller.
This determines that the pure transfer y is redundant for the expected revenue as longas rents are fully extracted and the equilibrium entry p is unchanged. It follows that
auctioneers have many alternative transfer schedules y that do not influence p.
In the reminder of this section, we explore the maximization problem over S(p|x) tofind the upper bound of social surplus Ŝ. Among the arguments on S, we first investigate
x and then control p to seek Ŝ. It is trivially true that any ex post inefficiency stemmingfrom distortive allocation or a positive reservation price decreases the social surplus.

11

Proposition 3. Let x∗ represent the ex post efficient mechanisms. For any p ∈ [0, 1] ×
[0, 1], the social surplus is maximized if and only if the mechanism is ex post efficient,
namely for any p S(p|x∗ ) > S(p|x) ∀ x ⊂ X\x∗ .
Proof. V (n|x∗ ) ≥ V (n|x) holds for any x ⊂ X. Hence, by (6), S(p|x∗ ) ≥ S(p|x) holds forany p.
After focusing on ex post efficient mechanisms, we can discuss a useful theorem forequilibrium analysis as follows. This theorem provides a relationship between the ex ante
payoff for each potential bidder and the expected marginal contribution to the social
surplus of the bidder.
(1)

Theorem 2. Let vτ

be the highest valuation among group τ bidders. Let φτ (·|x) be a
(1)

matching function such that the bidder with vτ
(1)

(1)

and the bidder with v−τ tie under some(1)mechanism x if and only if v−τ = φτ (vτ |x). Suppose that φτ (vτ ) ≥ vτ for some τ . Then,for any nτ ,(π1 (n1 +1, n2 |x) = V (n1 +1, n2 |x) − V (n1 , n2 |x),π2 (n1 , n2 +1|x) = V (n1 , n2 +1|x) − V (n1 , n2 |x),(7)

if and only if φτ (vτ ) = vτ .
See the Appendix for proof. This property is first introduced by Engelbrecht-Wiggans(1993) in an IPV setting. We extend the results to the asymmetric private value environment and give a sufficient condition for this property to hold. Most standard auctions satisfy the condition φτ (vτ ) ≥ vτ . For example, in asymmetric first-price auctions with a strong bidder and a weak bidder, we have φτ (vτ ) > vτ , which suggeststhat (7) does not hold.9 On the other hand, second-price auctions with a positive andconstant reservation price satisfy φτ (vτ ) = vτ . Such examples assure that ex post efficiency in allocation is not necessary for the marginal contribution theorem to hold asshown in Engelbrecht-Wiggans (1993). Obviously, the amount of contribution by one
more additional bidder under the ex post inefficient mechanism is strictly smaller sinceV (nτ +1, n−τ |x∗ ) − V (nτ , n−τ |x∗ ) > V (nτ +1, n−τ |x) − V (nτ , n−τ |x) for any x ∈ X\{x∗ }.
Now we return to the maximization problem of S. Taking a partial derivative of S
9

Suppose that there are two bidders j = {1, 2} whose value distribution is Fj (v) on [v̄, v] and fj (v)denotes the corresponding density. Assume that lnFj (v) is supermodular. Let βj be the equilibrium bidding
functions. The range of the equilibrium bidding functions should be identical so that β1 (v̄) = β2 (v̄) holds.

12characterized in (6) with respect to p gives the first-order condition as follows
N2
NX
1 −1 X∂S
N −1,N2[V (n1 +1, n2 |x) − V (n1 , n2 |x)] − N1 c1
Pn̂,p1
=
N
·
1

 ∂p1
n1 =0 n2 =0

N1 NX
2 −1

X
∂S

N ,N −1Pn̂,p1 2 [V (n1 , n2 +1|x) − V (n1 , n2 |x)] − N2 c2 .
= N2 ·

 ∂p
2

(8)

n1 =0 n2 =0

Note that the second derivative is negative under the ex post efficient mechanism.10 Then,by Theorem 2, (8) can be rewritten as
NXN2
1 −1 X∂S
N −1,N2[π1 (n1 +1, n2 |x) − c1 ]Pn̂,p1 ∂p1 = N1 ·n1 =0 n2 =0

N1 NX
2 −1

X
∂S

N ,N −1= N2 ·
Pn̂,p1 2 [π2 (n1 , n2 +1|x) − c2 ].

 ∂p2

(9)

n1 =0 n2 =0

Recall that (9) coincides with Aτ (·|x∗τ , 0). Therefore, (9) vanishes if (p∗1 , p∗2 ) ∈ (0, 1)×(0, 1)
and ∂S(p|x∗ , ·)/∂pτ ≤ 0 or ∂S(p|x∗ , ·)/∂pτ ≥ 0 if p∗τ = 0 or 1. This implies that the social
gain from an incremental change in pτ equals the social cost of an incremental change inthe probability for each τ .11 We thus establish the following proposition about efficient
entry.
Proposition 4. Suppose that a mechanism is ex post efficient and involves no transfer.
Then, the bidder’s profit-maximizing entry decision is the necessary and sufficient condition for efficient entry in the sense that the social gain from entry equals the cost of theentry.
The result is quite reasonable, as we already know that the bidder’s expected payoff
is identical to the marginal contribution to the social surplus under x∗ . Since the agent’s
problem is the same as the society’s problem, the bidder’s rational participation decisionleads to an efficient outcome in the entry.
The sufficient condition for our maximization problem is a little more involved thanthat in the symmetric case. The following lemma demonstrates that, as a result of Hessian
analysis over S(p|x), any local maximum (or saddle) point on S(p|x) is formed at theShow ∂ 2 S(p|x∗ , ·)/∂p21 < 0. Under the ex post efficient mechanism, V (n1 + 2, n2 |x∗ , ·) − 2V (n1 +1, n2 |x∗ , ·) + VR (n1 , n2 |x∗ , ·) = π1 (n1 +1, n2 |x∗ , ·) − π1 (n1 , n2 |x∗ , ·) holds. Furthermore, we obtain π1 (n1 +1−1
d s
1, n2 |x∗ , ·) = 0 (1−σ) dσv (σ)σ n1 ·(φ1 (σ))n2 dσ, where φτ (·) ≡ v−τ(vτ (σ)) is a matching function. Then,
R1∗dfor any n1 and n2 , π1 (n1 + 1, n2 |·) − π1 (n1 , n2 |x , ·) = − 0 (1 − σ)2 dσ
v1 (σ)σ n1−1 · (φ1 (σ))n2 dσ < 0. The
2∗
2
same is true for ∂ S(p1 , p2 |x , ·)/∂p2 < 0.
11
In addition, if the gain is greater (or less) than the costs, deterministic participation (or staying out)
occurs.
1013intersection between Aτ (·) if the mechanism is ex post efficient without monetary transfer.
Lemma 2. If x = x∗ and y = 0, then odd equilibria form a local maximum and evenequilibria form a saddle point on the social welfare function S(p|x∗ ).
See the Appendix for proof. Again, the linkage between A(·) and S(p|x∗ ) is shown here
under the ex post efficient mechanisms with no transfer. The proof demonstrates that thegradient of S(p|x∗ ) is determined by the positional relationship between A1 (·) and A2 (·).
If an analysis focuses on a symmetric equilibrium with ex ante symmetric bidders,
efficiency and optimality in an entry are equivalent. Taking into account multiple equilibria, however, an efficient entry may not be the optimal entry. Different equilibria create
different level of social surplus due to the difference in coordination costs associated withthe randomness of the actual number of participants. Lemma 2 implies that even equilibria entail more waste of social welfare than odd equilibria and, hence, lower revenue. For
instance, the symmetric equilibrium in a case of symmetric potential bidders suffers thehighest coordination costs among three equilibria.
There is at least one odd equilibrium in asymmetric auctions with endogenous participation. In addition, in the case of multiple equilibria, one of the odd equilibria yields thehighest social welfare. Hence, Ŝ is achieved at an odd equilibrium under ex post efficientmechanisms. The following proposition addresses the question of whether this occurs onlyin the case of ex post efficient mechanisms.
Proposition 5. The social surplus created in an odd equilibrium attains Ŝ if the mechanism is ex post efficient and no transfer is used. Efficient entry is necessary for socially
optimal entry, but not sufficient.
Proof. Proof for the “if ” part is obvious by proposition 3. The proof for the “only if ” partis shown as follows. Let p̂x denote the maximizer for S(p|x) for any x. By construction,∗S(p̂x |x∗ ) ≥ S(p|x∗ ) for any p. By proposition 3, S(p̂x |x∗ ) > S(p̂x |x) for any x 6= x∗ .
∗Hence, S(p̂x |x∗ ) > S(p̂x |x) for any x 6= x∗ .
We conclude this section by considering implementation in the entry game with multiple equilibria. The auctioneer has two devices to influence entry i.e., x and y. For
example, setting a discriminatory reservation price which favors a group of bidders affectsAτ (·) and, hence, the equilibrium formulation. In addition, bidding credits may help aparticular group of potential bidders to participate more frequently and discourage theremaining bidders’ participation. Furthermore, monetary transfer y enables the seller tocontrol Aτ (·) or extract more surplus from a group of bidders. The question of whether
the auctioneer should choose any x 6= x∗ to reach the best equilibrium must then be asked.

14

The following lemma illustrates that the seller can implement any p as a unique equilibrium by controlling y ∈ Y . 12
Lemma 3. Given x ∈ X, there exists a set of transfer schedule Ŷp,x,y with which any p
will be induced as a unique mixed-strategy entry equilibrium.
See the Appendix for proof. Recall lemma 1 insisting that full rent extraction is00possible at any p by any yp,x,y∈ Yp,x,y. Furthermore, we already know that Ŝ is neverachieved unless the mechanism is ex post efficient in proposition 5. Hence, the optimalmechanism in auctions with asymmetric potential bidders involves a participation control,
as reported in the following proposition.
Proposition 6. The participation game has a unique mixed-strategy type-symmetric equilibrium in which the auctioneer’s revenue is maximized if and only if the auctioneer employs the ex post efficient mechanism with an appropriately chosen transfer scheme.
Proof. By lemma 3, there is a set of ŷ that induces p̂ as a unique odd equilibrium. Furthermore, if type τ potential bidders have positive expected rents, the auctioneer can extract0 ). Then, by lemma 3, ŷ and ŷ 0 induce thethem by setting ŷτ0 = ŷτ + yτ0 . Let ŷ 0 = (ŷτ0 , ŷ−τsame entry if and only if it is unique.

5.1

Affiliated Private Value

Suppose that there exists bτ (v|x) for some x such that the bidder who has the highest b−1wins the item. Define φτ (v|x) = b−τ(bτ (v|x)|x) and Fτ (v|x) = Fτ (φ−τ (v|x)). Then, (2)becomesZ Z
π1 (n1 , n2 |x) =
Zz Zv
=
z

(1−F1 (v|z))(F1 (v|z))n1 −1 (F2 (v|z))n2 dvg(z)dz,
Z Z
n1
n2 0
v[(F1 (v|z)) (F2 (v|z)) ] dvdz −v[(F1 (v|z))n1 −1 (F2 (v|z))n2 ]0 dvg(z)dz

v

z

v

= V (n1 , n2 |x∗ )−V (n1 −1, n2 |x∗ ),which extends theorem 2 to an affiliated private-value (APV) paradigm. Now, we haveextended the results to the asymmetric APV environment.

6

Discussion

We show that, if endogenous participation is accounted for, ex post efficiency is necessary
for an optimal mechanism. The results contradict the existing theorems for optimal design
12This is primarily because y is a function of n1 and n2 rather than a negative constant variable i.e., afixed entry fee. However, nothing would be gained in the symmetric model if y is a function of n.

15

with a fixed set of asymmetric bidders, which insist that ex post efficient mechanisms are
not optimal (See Myerson (1981), McAfee and McMillan (1989), and Bulow and Roberts(1989)). Our model illustrates that the costs of introducing a distortive assignment rule forrent extraction always outweigh the benefits due to a serious inefficiency in participation.
Proposition 6 also impacts the ranking theorems with IPV asymmetric auctions. Vickrey (1961) showed that there is no general ranking between first and second-price asymmetric auctions, if the number of bidders is exogenously determined.13 Considering endogenous participation, we obtain a clear ranking. Second-price mechanisms always dominate
first-price mechanisms. In addition, Milgrom and Weber (1982) suggests that second-pricemechanisms yield higher revenue than first-price ones if signals are affiliated. Therefore,
our proposition about the superiority of second-price mechanisms still holds in the case ofasymmetric APV environments.
The proposition in which efficient entry is a necessary condition for optimal outcome
is also referred to in Levin and Smith (1994).14 They address the issue that when morepotential bidders exceed the point of transition between pure- and mixed-entry strategies,
the result is a waste of social welfare.15 Our approach is a non-trivial extension from
Levin and Smith (1994), since we provide a generalized scheme that makes it possible toevaluate revenue across equilibria with asymmetric potential bidders.
For ways of promoting competition in asymmetric auctions with participation, ouranalysis relates to Ayres and Cramton (1996) and Gilbert and Klemperer (2000). Motivated by auctioneers’ concern about insufficient competition among well-qualified bidders,
their studies explore whether subsidizing weak buyers through a distortive allocation ruleincreases revenue. Both conclude that promoting entry by weak buyers will enhance revenue. Since an even equilibrium is a saddle point on S(·|x∗ ), the change of an allocation
13
The existence of an equilibrium bidding function in asymmetric first-price auctions with a fixed numberof bidders is shown by Lebrun (1999). In general, weak bidders would bid more aggressively than strong
bidders in the first-price auction, resulting in ex post inefficient allocation. See Maskin and Riley (2000).
14
Ye (2004) also shows the advantage of asymmetric equilibrium by using Jensen’s inequality. However,
the results are crucially dependent upon the assumption that potential bidders are ex ante identical.
15
This argument is true withoutPa symmetry
assumption if the mechanisms are ex post effiN1 −1,N2
N1−1 PN2
cient. Since S(p1 , p2 |x, N1 , N2 ) =P[p1 · V (n1 + 1, n2 |x) + (1 − p1 ) · V (n1 , n2 |x)] −=0
n2 =0 n̂,pPN1 PN2−1 N1 ,Nn21−1
N1 p1 c1 − N2 p2 c2 =
P
[p
·
V
(n
,
n
+
1|x)
+ (1 − p2 ) · V (n1 , n2 |x)] − N1 p1 c1 − N2 p2 c2 ,
2
1
2
n1 =0
=0 n̂,pPN1−n12P
N1 ,N2 −1
N2
S(p1 , p2 |x, N1 −1, N2 ) = n1 =0 n2 =0Pn̂,pV (n1 , n2 |x)−(N1 −1)p1 c1 −N2 p2 c2 andS(p1 , p2 |x, N1 , N2 1) =PN1 PN2−1 N1 ,N2 −1
P
V
(n
,
n
|x)−N
p
c
−(N
1
2
1
1
1
2 −1)p2 c2 for any p1 , p2 , and x, we have
n1 =0
n2 =0 n̂,p(P −1 PN2
N1 −1,N2
S(p|x∗, N1, N2 )−S(p|x∗, N1 −1, N2 ) = p1 · nN11=0[V(n1+1, n2 |·)−V(n1 , n2 |·)]−p1 ·c1
n2 =0 Pn̂,pP
P−1 N1 ,N2 −1
S(p|x∗, N1, N2 )−S(p|x∗, N1, N2 −1) = p2 · nN11=0 nN22=0
Pn̂,p[V(n1 , n2+1|·)−V(n1 , n2 |·)]−p2 ·c2 .

If x = x∗ , then, by theorem 2, both are equal to zero. This indicates that, if a potential bidder, regardlessof its type, is eliminated and the remaining potential bidders still use p, then social welfare is unchanged.
If the remaining potential bidders choose p0 according to their new best response, which accounts one fewer
potential bidders, then, social surplus increases by construction.

16

rule from x∗ to some x0 may induce another p, which creates greater S despite some efficiency loss in allocation. However, our study suggests that the outcome is sub-optimal.
The first best outcome is achieved only through an ex post efficient allocation with a puretransfer.
A transfer may often be seen in the real-world procurement auction as the requirement
for a higher financial guarantee for bidders. It is costly for bidders but beneficial to theauctioneer by reducing the risk of facing default. Government spending for improving
small business access can also be considered to be a transfer. An important aspect ofparticipation control, which includes an implicit one, is that any change in participationfrom the optimal equilibrium results in both efficiency loss due to coordination costs andfluctuation in participation.

7

Conclusion

Over the past decade, the model of auctions with endogenous participation has provided
a striking result, namely, that both efficiency and revenue maximization can be achieved
simultaneously. Despite the contribution of endogenous participation models, they depend
heavily on the assumption that the potential bidders are ex ante the same. Little progresshas been made in the theory of auction with asymmetric endogenous entry.
The relaxation of symmetric assumption is not a trivial extension from the existing
symmetric model. First, with symmetric bidders, optimal design problems boil down tothe optimal choice of a reservation price, as investigated in Riley and Samuelson (1981).
Introducing asymmetry, the optimal design problem becomes more complicated. As discovered in Myerson (1981), the appropriately chosen distortive mechanisms enhance revenue. This proposition, however, absolutely ignores the effects of a potential bidder’sparticipation. Accounting for the rational decision of a potential bidder’s participation,
any rent extraction by distortive allocation causes inefficient entry, and, hence, simple
auctions are optimal. The results provide a new interpretation for the widespread use of
simple auctions.
Second, the endogenous entry model becomes applicable to a more general environment. In procurement auctions for international projects, for example, a number of contractors randomly participate. Based on the fact that traveling costs as well as currency
values differ from country to country, it is impossible to suppose that all potential bidders are ex ante the same. Almost all empirical models for auctions with endogenous
participation, so far, have been based on the symmetric assumption (e.g. Li and Zheng(2006)). We hope that our model contributes to the enrichment of the empirical analysis

17for auctions with endogenous participation.

Appendix
Proof for Proposition 1
Define Hτ (p−τ |·) : [0, 1] → [0, 1] as the solution of (4) for pτ . Since Uτi is decreasing in p, we
0
i00
ihave p0−τ ≤ p00−τ ≤ p000
−τ , where pτ ∈ {p−τ |Uτ (1, p−τ |·) > 0}, p−τ ∈ [0, 1]\{p−τ |Uτ (1, p−τ |·) >i0} ∪ {p−τ |Uτi (0, p−τ |·) < 0}, and p000
−τ ∈ {p−τ |Uτ (0, p−τ |·) < 0}. Therefore, Hτ can be de-scribed asHτ (p−τ |·) =






if {p−τ |Uτi (1, p−τ |·) > 0}

1

0


 {p |U i (p , p |·) = 0}
τ

τ

1

2

if {p−τ |Uτi (0, p−τ |·) < 0}(10)otherwise.

Since (p∗1 , p∗2 ) must satisfy (10), we have(

H1 (p∗2 |x1 , y1 ) = p∗1 ,H2 (p∗1 |x2 , y2 ) = p∗2 .

In other words, if we define H(p|·) = ×τ ∈{1,2} Hτ (p−τ |·), a type-symmetric equilibrium∗p = (p∗1 , p∗2 ) is a fixed point of H i.e., p∗ ∈ H(p∗ |x, y).
uτ (n|·) is decreasing in n. Since n follows a binomial distribution, Uτ is continuous.
Hence, by (10), Hτ (p−τ |·) is continuous. For each τ the set Hτ (p−τ ) is nonempty and has
a closed graph since Hτ (·) is continuous. Thus, by the fixed-point theorem, there exists
at least one fixed point on H. A mixed-strategy equilibrium is a fixed point.

18

Proof for Proposition 2
By the characteristic of binomial distribution, the following identities hold:
NXN2
1 −1 X
N −1,N2u1(n̂1 +1, n̂2 |·)Pn̂,p1




n̂
=0
n̂
=0
1
2



NX
1 −1N
2 −1

X

N −1,N2 −1[p2 ·u1(n̂1 +1, n̂2 +1|·)+(1−p2 )·u1(n̂1 +1, n̂2 |·)]≡
Pn̂,p1n̂ =0 n̂ =0
1

2

N1 NX

2 −1
X


N ,N −1Pn̂,p1 2 u2(n̂1 , n̂2 +1|·) n̂1 =0n̂2 =0


NX

1 −1N
2 −1
X


N −1,N2 −1[p1 ·u2(n̂1 +1, n̂2 +1|·)+(1−p1 )·u2(n̂1 , n̂2 +1|·)].
≡
Pn̂,p1(11)n̂1 =0 n̂2 =0

If U1 = U2 , then by monotonicity,p2

XXPn̂ u1(n̂1 + 1, n̂2 + 1) + (1 − p2 )XXPn̂ u1(n̂1 + 1, n̂2 )n̂1 n̂2n̂1 n̂2≤ p1

XXPn̂ u1(n̂1 + 1, n̂2 + 1) + (1 − p1 )n̂1 n̂2

Hence, (p2 − p1 )for any p1 and p2 ,n̂2 Pn̂ u1(n̂1 + 1, n̂2 + 1) ≤ (p2 − p1 )

P P
n̂1

Pn̂ u1(n̂1 + 1, n̂2 ).

n̂1 n̂2

P P
n̂1

XX

n̂2 Pn̂ u1(n̂1 + 1, n̂2 + 1) −

P P
n̂

P P1
n̂1

n̂2 Pn̂ u1(n̂1 + 1, n̂2 ). Since,n̂2 Pn̂ u1(n̂1 + 1, n̂2 )< 0, onemust obtain p2 ≥ p1 . Equality holds if both u1(n1 + 1, n2 + 1) = u2(n1 + 1, n2 + 1) andu1(n1 + 1, n2 ) = u2(n1 , n2 + 1) hold for all n.

Proof for Lemma 3
Suppose, by contradiction, that the set Ŷ is empty. Set y1 = ỹ1 such that u1 (n1 + 1, n2 +1|x1 , ỹ1 ) − u1 (n1 + 1, n2 |x1 , ỹ1 ) = 0 and u1 (n1 + 2, n2 |x1 , ỹ1 ) − u1 (n1 + 1, n2 |x1 , ỹ1 ) > 0.
Then, G1 (p|x1 , ỹ1 ) = 0 for any p. Moreover, set y2 = ỹ2 such that u2 (n1 , n2 + 2|x2 , ỹ2 ) −u2 (n1 , n2 + 1|x2 , ỹ2 ) > 0 and u2 (n1 + 1, n2 + 1|x2 , ỹ2 ) − u2 (n1 + 1, n2 |x2 , ỹ2 ) = 0. Then,
G2 (p|x2 , ỹ2 ) → ∞ for any p. Since G1 < G2 , ỹ = (ỹ1 , ỹ2 ) ∈ Ŷ , we have reached acontradiction. Thus Ŷ is nonempty.
Let ŷ ∈ Ŷ . Set yτ0 (p) = −Uτ (p|xτ , ŷτ ) for some p. Then, Uτ (p|xτ , ŷτ + yτ0 (p)) = 0 for
any p. Since yτ0 (p) is constant for all n, G1 (p|x1 , ŷ1 + y10 ) < G2 (p|x2 , ŷ2 + y20 ) for any x.
Hence, ŷ(p) + y 0 (p) ∈ Ŷ .

19

Proof for Theorem 2
Proof. Suppose that there exists bτ (v|x) for some x such that the bidder who has the−1highest b wins the item. Define φτ (v|x) = b−τ(bτ (v|x)|x) and F̂τ (v|x) = Fτ (φ−τ (v|x)).

Then, (2) becomesZ(1−F1 (v))(F1 (v))n1 −1 (F̂2 (v|x))n2 dv,
Zv
Z= (F1 (v))n1 −1 (F̂2 (v|x))n2 dv − (F1 (v))n1 (F̂2 (v|x))n2 dv,
v
Z
Zv
n1
n2 0
= v[(F1 (v)) (F̂2 (v|x)) ] dv − v[(F1 (v))n1 −1 (F̂2 (v|x))n2 ]0 dv.

π1 (n1 , n2 |x) =v

v

where Fτ (·) = vτ−1 (·) and fτ (·) = Fτ0 (·). Integral by parts yields the last equality. SinceF̂1 (v|x) = F1 (φ2 (v)), F̂1 (φ1 (v)) = F1 (φ2 (φ1 (v))) = F1 (v) and φ2 (φ1 (v)) = v1 (v), letting
φ1 (v) = v̂, it is possible to obtainZv(F1 (v))n1 [(F̂2 (v|x))n2 ]0 dv =Z

φ2 (φ1 (v)) · [F̂1 (φ1 (v))]n1 n2 f2 (φ1 (v))(F2 (φ1 (v)))n2 −1 φ01 (v)dv

Z

φ2 (v̂) · [F̂1 (v̂)]n1 n2 f2 (φ1 (v))(F2 (φ1 (v)))n2 −1 dv̂.

=v̂RecallZZ
n1−2n2
V(n1 −1, n2 |x) = v(n1 −1)f1 (v)(F1 (v))(F̂2 (v|x)) dv+ v(F̂1 (v|x))n1−1 n2 f2 (v)(F2 (v))n2−1 dv
Z
Z
V(n1 , n2 |x) = vn1 f1 (v)(F1 (v))n1−1 (F̂2 (v|x))n2 dv + v(F̂1 (v|x))n1 n2 f2 (v)(F2 (v))n2−1 dv.

Therefore,Z
V (n1 , n2 |x)−V (n1 −1, n2 |x)+0[φ2 (v)−v][1− F̂1 (v|x)](F̂1 (v|x))n1−1 [(F2 (v))n2 ] dv = π1 (n1 , n2 |x).

v

φτ (v) = v for any v under the ex post efficient mechanism. Therefore, the third termon the left-hand side vanishes if x = x∗ . Clearly, the term is typically non-zero for any
x ⊂ X \{x∗ }.

Proof for Lemma 2
Since ∂ 2 S(p|x∗ , ·)/∂p2 < 0, the first-order principle minor of the Hessian on S(p|·) is
negative.

20

Take the second-order derivative on S.
N −2 N

1
2X
X∂ 2 S(p|·)
N1 −2,N2
=
N
(N
−1)·
Pn̂,p
[V (n1 +2, n2 |·) − 2V (n1 +1, n2 |·) + V (n1 , n2 |·)]
1
1
∂p21

n̂1 =0 n̂2 =0∂ 2 S(p|·)= N1 N2 ·∂p1 ∂p2

NX
1 −1 N
2 −1
X

N1 −1,N2 −1
Pn̂,p[V(n1 +1, n2 +1|·)−V(n1 , n2 +1|·)n̂1 =0 n̂2 =0−V(n1 +1, n2 |·)+V(n1 , n2 |·)]
2

∂ S(p|·)= N2 (N2 −1)·
∂p22

N1 NX
2 −2
X

N1 ,N2 −2Pn̂,p[V (n1 , n2 +2|·) − 2V (n1 , n2 +1|·) + V (n1 , n2 |·)]n̂1 =0 n̂2 =0

The second-order principal minor of the Hessian on S(p1 , p2 |·), which is given as (N1 −P −2 N1 ,N2 −2P
P −2PN2
N1 −2,N2[π2 (n1 +Pn̂,p[π1 (n1 , n2 +2|x)−π1 (n1 , n2 +1|x)]· n̂N11=0 n̂N22=0
1)(N2 −1)· n̂N11=0n̂2 =0Pn̂,p
PN1 −1 PN2 −1 N1 −1,N2 −1[π1 (n1 , n2 +1|x)−π1 (n1 , n2 |x)]·2, n2 |x)−π2 (n1 +1, n2 |x)]−N1 N2 · n̂1 =0 n̂2 =0 Pn̂,p
PN1 −1 PN2 −1 N1 −1,N2 −1[π2 (n1 +1, n2 |x)−π2 (n1 , n2 |x)] by Lemma 2, is positive if and onlyn̂1 =0n̂2 =0 Pn̂,pif G1 (p|·)−G2 (p|·) > 0. Hence, the even equilibrium is a saddle point. Finally, we show thatS(p1 , p2 |x∗ , U1 (p1 , p2 ) = 0) is increasing in p1 if and only if U1 (p1 , p2 |x∗ ) > U2 (p1 , p2 |x∗ ).
Suppose, by contradiction, that there exist p01 and p02 such that S(p01 , p02 |x∗ )|A1 =0 is decreasing in p1 . Since U2 (·) is continuous in p2 , there exists some p002 < p02 such that U2 (p01 , p002 ) = 0.
By (8), S(p01 , p002 |x∗ ) > S(p01 , p2 |x∗ ) for any p2 . Therefore, S(p01 , p002 |x∗ ) > S(p01 , p02 |x∗ ).
Since U1 (·) is continuous in p1 , there exists p00 > p0 such that U1 (p001 , p002 ) = 0. By (8),S(p001 , p002 |x∗ ) > S(p1 , p002 |x∗ ) for any p1 . Therefore, S(p001 , p002 |x∗ ) > S(p01 , p002 |x∗ ). Hence,
S(p001 , p002 |x∗ ) > S(p01 , p02 |x∗ ), which leads to contradiction.

References
Athey, Susan and Philip A. Haile, “Empirical Models of Auctions,” Levine’s Bibliography 122247000000001045, UCLA Department of Economics March 2006.
Ayres, Ian and Peter Cramton, “Deficit Reduction Through Diversity: How Affirmative Action at the FCC Increased Auction Competition,” Technical Report 1996.
Bulow, Jeremy and John Roberts, “The Simple Economics of Optimal Auctions,”
Journal of Political Economy, October 1989, 97 (5), 1060–90.
Engelbrecht-Wiggans, Richard, “Optimal Auctions Revisited,” Games and Economic
Behavior, April 1993, 5 (2), 227–239.
Gilbert, Richard J. and Paul Klemperer, “An Equilibrium Theory of Rationing,”
RAND Journal of Economics, Spring 2000, 31 (1), 1–21.

21

Kjerstad, Egil and Steinar Vagstad, “Procurement auctions with entry of bidders,”
International Journal of Industrial Organization, December 2000, 18 (8), 1243–1257.
Lebrun, Bernard, “First Price Auctions in the Asymmetric N Bidder Case,” International Economic Review, February 1999, 40 (1), 125–42.
Levin, Dan and James L Smith, “Equilibrium in Auctions with Entry,” American
Economic Review, June 1994, 84 (3), 585–99.
Li, Tong and Xiaoyong Zheng, “Entry and competition effects in first-price auctions: theory and evidence from procurement auctions,” CeMMAP working papers
CWP13/06, Centre for Microdata Methods and Practice, Institute for Fiscal Studies
August 2006.
Maskin, Eric and John Riley, “Asymmetric Auctions,” Review of Economic Studies,
July 2000, 67 (3), 413–38.
McAfee, R. Preston and John McMillan, “Auctions with entry,” Economics Letters,
1987, 23 (4), 343–347.
and, “Government procurement and international trade,” Journal of International

Economics, May 1989, 26 (3-4), 291–308.
Milgrom, Paul R and Robert J Weber, “A Theory of Auctions and Competitive
Bidding,” Econometrica, September 1982, 50 (5), 1089–1122.
Myerson, Roger B, “Optimal auction design,” Mathematics of Operations Research,
1981, 6 (1), 58–73.
Pevnitskaya, Svetlana, “Endogenous Entry and Self-selection in Private Value Auctions: An Experimental Study,” 2003. working paper,California Institute of Technology,
Division of the Humanities and Social Sciences.
, “Endogenous Entry in First-Price Private Value Auctions: the Self-Selection Effect,”
2003. working paper,Ohio State University.
Riley, John G and William F Samuelson, “Optimal Auctions,” American Economic
Review, June 1981, 71 (3), 381–92.
Vickrey, William, “Counterspeculation, Auctions, and Competitive Sealed Tenders,”
Journal of Finance, Mar 1961, 16 (1), 8–37.

22

Ye, Lixin, “Optimal Auctions with Endogenous Entry,” Contributions to Theoretical
Economics, 2004, 4 (1), 1158–1158.

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