Optimal Entry in Auctions with Value Discovery
Costs∗
Jingfeng Lu
National University of Singapore
March 2007
∗
I benefited from discussions with Indranil Chakraborty, Daniel Friedman, Guofu Tan, Ruqu
Wang and Julian Wright. I thank the associated editor and two anonymous referees for insightful comments and suggestions. Previous versions have been presented at the 17th International
Game Theory Conference at Stony Brook and the 2006 Hong Kong Economic Association Meeting. All errors are mine. Financial support from National University of Singapore (R-122-000088-112) is gratefully acknowledged.
Address: Department of Economics, National University of Singapore, Singapore 117570. Tel :
(65) 6516-6026. Fax : (65) 6775-2646. Email: [email protected].
1
Abstract This paper studies auction design in an independent private value setting where
bidders have value discovery costs. First, when these entry costs are fixed, there is no loss
of generality in considering entry patterns where every bidder participates with probability of either 0 or 1, for the revenue-maximizing (also total-expected-surplus-maximizing)
auction. Second, there is also no loss of generality if we consider nondiscriminatory auctions for revenue maximization when bidders are symmetric. With asymmetric bidders,
considering only nondiscriminatory auctions are restrictive and the revenue-maximizing
nondiscriminatory auction may have to induce mixed-strategy entry. Third, when the
entry costs are private information of bidders, the revenue-maximizing entry generally
differs from the ex ante efficient one and both can be asymmetric across bidders. While a
second-price auction with no entry fee and a reserve price equal to seller’s value is ex ante
efficient, the revenue-maximizing auction employs ex ante entry fees that equal the hazard
rates of the bidders’ entry cost distribution, evaluated at the desired entry-thresholds. We
find that large dispersion in entry costs and increasing hazard rate of the cost distribution ensure the symmetry in the efficient/revenue-maximizing entry and guarantee the
uniqueness of entry equilibrium for the proposed auctions.
Keywords: Auction Design, Endogenous Participation, Value Discovery Cost.
JEL classifications: D44, D82.
2
1
Introduction
The impact of value discovery costs on bidders’ entry decision and auction design, has
been extensively studied in the literature. Here, value discovery costs refer to the costs
that bidders must incur to discover their values of the auctioned object. According to
the characteristics of these costs, the literature can be summarized into three branches.
The first branch includes Milgrom (1981), French and McCormick (1984), McAfee and
McMillan (1987), Engelbrecht-Wiggans (1987, 1993), Harstad (1990), Levin and Smith
(1994) and Ye (2004) among others.1 All these studies assume that the costs are fixed.
The second branch instead assumes that the quality of the acquired information about
the bidders’ values depends on the costs incurred by them. This branch of literature
is composed of Matthews (1984), Tan (1992), Persico (2000), Bergemann and Välimäki
(2002). The third branch assumes that the information acquisition costs are bidders’
private information. Rezende (2005) belongs to this branch. Bergemann and Välimäki
(2005) present a thorough review of the literature.
In this paper, we begin with a basic independent private value (IPV) setting where
symmetric bidders have fixed value discovery costs, and study the optimal entry in terms
of the seller’s expected revenue and the expected total surplus of seller and bidders.
McAfee and McMillan (1987) study an IPV framework with infinite number of potential
bidders. They show that the optimal reserve price for seller is his own value.2 By allowing
the optimal number of entrants to be a non-integer, they show that a first-price auction
with a reserve price equal to seller’s value and an entry fee is revenue-maximizing. Since
a non-integer number of entrants is not implementable, they further establish that a
first-price auction with a reserve price equal to seller’s value and zero entry fee leads to
the closest discrete number of participants to the unrestricted optimum. Engelbrecht1
There are many studies that focus on other entry costs that are incurred even when bidders know their
values. These studies include Green and Laffont (1984), Samuelson (1985), Stegeman (1996), Menezes
and Monteiro (2000), Lu (2004) and Tan and Yilankaya (forthcoming).
2
Engelbrecht-Wiggans (1987) independently finds a similar result.
3
Wiggans (1993) further examines the revenue-maximizing auction in a setting allowing
for affiliation among bidders’ private values. In all these works, it is assumed that bidders
participate in either probability of 1 or 0. Levin and Smith (1994) and Ye (2004) look
at the symmetric mixed strategy entry equilibrium in a symmetric setting, i.e., every
potential bidder participates in a common probability that falls in the interval (0, 1).
Levin and Smith (1994) show that the revenue-maximizing auction must also induce
socially optimal entry. Particularly, for the private value case, the optimal entry occurs
when seller charges no entry fee and a reserve price equal to seller’s value. Ye (2004)
shows a similar result in a private value framework, which differs from Levin and Smith
(1994) in assuming that after incurring an entry cost, each entrant not only observes his
own value, but also observes information that updates his belief on the distributions of
other bidders’ values.
The feature of our analysis lies in that we allow each bidder to participate in any probability in interval [0, 1]. This results in new findings. First, there is no loss of generality
in considering the entry patterns where every bidder participates with probability of 0 or
1 for the revenue-maximizing (meanwhile ex ante efficient) auction. This result eliminates
the necessity of considering noninteger number of entrants for the analysis on optimal entry. The significance of this result is two-fold. On one hand, this finding greatly simplifies
the auction design problems. On the other hand, the actual number of participants is
always discrete in reality. Our result means that this causes no loss to optimal auction
design. Second, our result means that the revenue-maximizing entry pattern is generally
an asymmetric corner solution rather than the symmetric inner solution in terms of the
participation probabilities. This explains why seller’s expected revenue may drop with
the number of potential bidders when the entry pattern are restricted to be symmetric
in Levin and Smith (1994). Third, a second-price auction with no entry fee and a reserve price equal to the seller’s value remains to be ex ante efficient though the optimal
entry pattern is asymmetric across bidders. Fourth, individual entry fees for each bidder
generally are essential to extract all the expected surplus of the participants for revenue
maximization. While the revenue-maximizing entry is asymmetric across bidders, there is
4
no loss of generality to solely consider nondiscriminatory auctions for its implementation.
We further study two generalizations of the basic setting. We first allow the bidders to
be asymmetric in terms of their value distributions and entry costs. All the findings from
the symmetric setting still hold except that considering only nondiscriminatory auctions
becomes restrictive for revenue maximization and the revenue-maximizing nondiscriminatory auction may have to induce mixed-strategy entry.
The second generalization allows entry costs to be bidders’ private information. In this
case, the seller cannot extract all the surplus of the bidders. As a result, unlike the case of
fixed entry costs, the revenue-maximizing entry diverges from the ex ante efficient entry.
While a second-price auction with no entry fee and a reserve price equal to seller’s value is
still ex ante efficient, the revenue-maximizing auction employs ex ante entry fees that equal
the hazard rates of the bidders’ entry cost distribution, evaluated at the desired entrythresholds. Like the case of fixed entry costs, the efficient and/or revenue-maximizing
entry can be asymmetric across symmetric bidders. When the cumulative distribution
function of the entry costs changes rather slowly with respect to its argument, the efficient
entry must be symmetric across bidders and it is the unique entry equilibrium of the
proposed efficient auction. If the hazard rate of the entry cost distribution is additionally
increasing, then the revenue-maximizing entry must also be symmetric and it is the unique
entry equilibrium of the revenue-maximizing auction. Thus, large dispersion in the entry
costs and increasing hazard rate of the cost distribution ensure the symmetry of the
efficient/revenue-maximizing entry.
Stegeman (1996) shows that a second-price auction with no entry fee and a reserve
price equal to the seller’s value is also ex ante efficient in a setting where fixed entry costs
are incurred by participating bidders who know their values. Bergemann and Välimäki
(2002) also show that in a private value environment where quality of the acquired information by the bidders depends on the amount of their investment, the Vickrey-ClarkGroves mechanism renders both ex ante and ex-post efficiency. Our results regarding the
efficiency of the second price auction obtained from different settings extend the scope of
applicability of their findings.
5
This paper is organized as follows. In Section 2, we study a symmetric IPV setting
where potential bidders have fixed value discovery costs, and derive the ex ante efficient
auction and revenue-maximizing auction. The optimality of discrete participations is
established. The issues of nondiscriminatory auction, symmetric entry and zero entry
fee are addressed. Section 3 considers the first generalized setting where bidders can be
heterogeneous in terms of value distributions and value discovery costs. Section 4 considers
the second generalized setting where bidders’ entry costs are their private information.
Section 5 concludes.
2
Symmetric IPV Setting
We begin with a basic symmetric IPV setting. There are N potential bidders who are
interested in a single item auctioned, where N is public information. Denote this group
of potential bidders by N = {1, 2, · · · , N}. The seller’s value is v0 , which is public
information. Every bidder has to incur a fixed entry cost of c(> 0) in order to enter the
auction.3 After incurring an entry cost c, he observes his private value vi , where vi is
the private information of bidder i. The cumulative distribution function of vi , ∀i ∈ N
is F (·) with density function of f (·) defined on [v, v]. The distribution of vi , i ∈ N are
common knowledge. We assume vi , ∀i ∈ N are mutually independent. The seller and
bidders are risk neutral.
The timing of the auction is as follows.
Time 0: The group of potential bidders N , the bidders’ entry costs c, the seller’s value
v0 and the distribution of the bidders’ values are revealed by Nature as public information.
Time 1: The seller announces the rule of the auction.
Time 2: The bidders simultaneously and confidentially make their entry decisions.4
3
This assumption is adopted in the literature. However, this assumption precludes the possibility that
a bidder may simply submit a bid equal to the unconditional expected value.
4
In Section 2.2, we will show that the optimal simultaneous entry can be implemented through an
sequential-move game.
6
If they do not enter, they take the outside option which gives them zero payoff. If they
enter, they incur their entry costs and observe their private values.
Time 3: Every participant bids.5
Time 4: The payoffs of the seller and all the participating bidders are determined
according to the announced rule at time 1.
We study the auction rules announced at time 1, which maximize the expected total
surplus and/or the seller’s expected revenue. Hereafter, the auction maximizing the expected total surplus of seller and bidders is called the ex ante efficient auction; the
auction maximizing the expected revenue of the seller is called the revenue-maximizing
auction. We use A0 to denote the second-price auction with no entry fee and a reserve
price equal to the seller’s value v0 .
2.1
Auction Design
Our analysis allows every possible entry pattern which can be described through a vector of entry probabilities P = (p1 , p2 , · · · , pN ) where pi ∈ [0, 1], ∀i ∈ N is bidder i’s
entry probability. We first establish the following results regarding the restricted efficient auction and revenue-maximizing auction, which implement a given entry pattern
P = (p1 , p2 , · · · , pN ).
Lemma 1: Among all auctions implementing any given entry pattern P = (p1 , p2 , · · · , pN ),
a second-price auction with a reserve price equal to seller’s value and appropriately set
entry fee (or subsidy) for each bidder provides the highest seller’s expected revenue as well
as the highest expected total surplus. The entry fees (or subsidies) are charged upon entry
at time 2 before the values are learned by the entrants, and are set at levels such that each
entrant gets zero expected payoff at the equilibrium.
Proof: See Appendix.
The implications of Lemma 1 are two fold. First, for a given entry pattern, the auction
5
Every participant may or may not observe the other participants. The auctions designed later work
in both cases.
7
that maximizes seller’s expected revenue also maximizes expected total surplus. Note
that Proposition 1 in Levin and Smith (1994) has shown this result for symmetric entry.
Second, this auction can take the form of a second-price auction with appropriate reserve
price and entry fees. Every entrant pays the personalized entry fees and then learns their
values. As pointed out by McAfee and McMillan (1987), the seller can demand these fees
before potential bidders inspect the item for sale. All entrants bid, the highest bidder
wins given his bid is higher than the reserve price. The winner pays the second highest
bid or the reserve price, whichever is higher. If no bidder enters, the seller keeps the
object.
According to Lemma 1, the expected surplus of bidders must be zero at the optimum.
Thus the seller’s optimal expected revenue equals the optimal expected total surplus.
Denote this optimal expected total surplus by ES(P). We next introduce a way of
writing ES(P) that facilitates our analysis and the interpretation of the results.
We define set K = {(k1 , k2 , · · · , kN )|ki ∈ {0, 1}, i ∈ N }, where ki denotes bidder
i’s ex post entry status. Specifically, ki = 1 stands for the participation of bidder i,
while ki = 0 represents the non-participation of bidder i. In addition, k0 ≡ 1 symbolizes
the participation of the seller. For any k = (k1 , k2 , · · · , kN ) ∈ K, we use vk to denote the
highest value of all ex post participants including the seller, i.e., vk = max{kj =1,0≤j≤N } {vj }.
We use fk (vk ) and Fk (vk ) to denote the density and cumulative distribution function of
vk , respectively. Furthermore, we use Vk to denote the expectation of vk . ES(P) can
then be written as
ES(P) =
X Vk
{k∈K}
Y
pki i (1 − pi )1−ki −
i∈N
X
pi c.
(1)
i∈N
The first summation term in (1) is the contribution of the values of all participants
including the seller to the expected total surplus if potential bidders participate according
to probabilities P. The second term corresponds to the contribution (negative) of their
entry costs.
We next characterize the ex ante efficient auction and the revenue-maximizing auction.
Clearly, ES(·) is a continuous function defined on a compact set S0 = [0, 1]N . Thus,
8
according to the Weierstrass extreme value theorem, there must exist P∗ = (p∗i ) ∈ S0
that maximizes ES(·) within S0 , where P∗ is allowed to be corner solution.
Define K−i = {(k1 , · · · , ki−1 , ki+1 , · · · , kN )|kj ∈ {0, 1}, j 6= i.}, ∀i ∈ N . Let us
consider bidder i’s optimal entry probability p∗i , ∀i ∈ N . For any k−i = (k1 , · · · , ki−1 ,
ki+1 , · · · , kN ) ∈ K−i , we use k1 (k−i ) to denote the N -element vector where the i-th element
is 1 and other elements are same with k−i , while we use k0 (k−i ) to denote the N -element
vector where the i-th element is 0 and other elements are same with k−i . We then have
X
Y k
∂ES(P)
=
{(Vk1 (k−i ) − Vk0 (k−i ) − c) pj j (1 − pj )1−kj },
∂pi
j6=i
{k−i ∈K−i }
as
P
{k−i ∈K−i } (
Q
j6=i
(2)
k
pj j (1 − pj )1−kj ) = 1. As P∗ maximizes ES(P), we must have the
following first order conditions for P∗ . For all i ∈ N ,
∗






if p∗i ∈ (0, 1),
0,
∂ES(P )
= ≥ 0, if p∗i = 1,

∂pi



 ≤ 0, if p∗ = 0.
i
Define MSi (P) =
P
{k−i ∈K−i } {(Vk1 (k−i )
− Vk0 (k−i ) −c)
(3)
Q
k
j6=i
pj j (1 − pj )1−kj }. This term
in (2) is the marginal contribution of bidder i to the expected total surplus in any second
price auction with a reserve price equal to the seller’s value such as A0 , given that other
bidders participate according to P.
Alternatively, MSi (P) can be interpreted as the expected payoff of bidder i from
participating in auction A0 , provided that other bidders participate according to P. This
can be seen from the following arguments. Note that the economic meaning of Vk1 (k−i ) −
Vk0 (k−i ) − c is the marginal contribution of bidder i to the expected total surplus if he
participates in auction A0 , and all other participants are those bidders with kj = 1, j 6= i
in vector k−i . Hence, Vk1 (k−i ) − Vk0 (k−i ) − c can be alternatively written as
Z
v
v0
n
(vi − v0 )Fk0 (k−i ) (v0 ) +
Z
o
vi
v0
(vi − v)fk0 (k−i ) (v)dv f (vi )dvi − c,
(4)
where Fk0 (k−i ) (·) and fk0 (k−i ) (·) are the cumulative distribution function and density function of vk0 (k−i ) , respectively. (4) can also be interpreted as the expected payoff of bidder
9
i when he participates in auction A0 , if all other participants are those bidders with
kj = 1, j 6= i in vector k−i . It then follows that MSi (P) is the expected payoff of bidder i
when he participates in auction A0 , if all other potential bidders participate according to
P. This insight together with (2) and (3) lead to the following proposition that addresses
the ex ante efficient auction and revenue-maximizing auction.
Proposition 1: (i) A second-price auction with a reserve price equal to seller’s value
and no entry fee maximizes the expected total surplus. (ii) A second-price auction with a
reserve price equal to seller’s value v0 and time-2 entry fee ei for bidder i defined below
maximizes the seller’s expected revenue. The time-2 entry fees ei , ∀i ∈ N are defined as
following,
ei =











0,
if p∗i ∈ (0, 1),
MSi (P∗ )(≥ 0),
if p∗i = 1,
(5)
any number ≥ MSi (P∗ )(≤ 0), if p∗i = 0,
where P∗ = argmaxP∈S0 ES(P). (iii) The efficient and revenue-maximizing reserve price
must equal the seller’s value v0 .
Proof: See Appendix.
As Proposition 1 points out, the revenue-maximizing reserve price must equal the
seller’s value v0 . This result holds as the revenue-maximizing auction extracts the entire
expected surplus of all entrants and thus the seller captures the entire social surplus.
Clearly, the entire social surplus cannot be maximized unless the reserve price is v0 . If
the reserve price deviates from v0 , there must exist efficiency loss as the item is not always
allocated to the player with the highest value.
A feature of the revenue-maximizing auction lies in the time-2 entry fees that are
essential to extract the expected surplus of the entrants. In Section 2.5, we will further
study auction design if seller cannot charge entry fees.
2.2
Optimality of {0, 1} Entry
We first establish the following result regarding the optimal entry.
10
Proposition 2: There must exist an entry probability vector P∗ where p∗i ∈ {0, 1}, ∀i ∈
N , which maximizes ES(P). In other words, there is no loss of generality to consider
only pure-strategy entry patterns for efficient and/or revenue-maximizing auctions.
Proof: See Appendix.
The significance of this result is as follows. First, this finding greatly simplifies the auction design problems, as the necessity of considering mixed-strategy entry is eliminated.
Second, the actual number of participants is always discrete in reality. Proposition 2
shows that discrete entry does not cause loss to auction design.
The following examples illustrate the insights of Proposition 2. Assume v0 = 0, c = 0.1
and N = 3. Assume F (vi ) = viα on [0, 1], i ∈ N where α > 0. Based on (1), direct
calculation gives the following results. If α = 0.3, P∗ = (1, 1, 0) maximizes ES(P) and
ES(P∗ ) = 0.175. If α = 0.5, P∗ = (1, 1, p) where p is any number in [0, 1] maximizes
ES(P) and ES(P∗ ) = 0.30. If α = 1.0, P∗ = (1, 1, 0) maximizes ES(P) and ES(P∗ ) =
0.4667.
For the example of α = 1.0, the symmetric mixed-strategy entry equilibrium of the
efficient auction A0 is P† = (.91, .91, .91). This symmetric entry leads to a total expected
surplus of 0.4524. The loss in total surplus due to this inefficient entry is thus 0.0143.
This suggests that the asymmetric pure-strategy entry P∗ = (1, 1, 0) Pareto dominates
the symmetric P† of mixed-strategy when corresponding revenue-maximizing auctions
of Proposition 1(ii) are adopted, since the revenue-maximizing auctions extract all the
expected surplus of the entrants. In other words, the seller is strictly better off at entry
(1, 1, 0) while the bidders are indifferent. We next show that this result holds in general.
First, we have the following result, which will be used in the proof of Corollary 1 and
Section 4. Define Zl as the expectation of the highest value of the seller and any l(≥ 0)
bidders. The following Lemma provides some properties of the series Zl , l ≥ 0.
Lemma 2: Both Zl+1 − Zl and (Zl+1 − Zl ) − (Zl+2 − Zl+1 ) decrease with l(≥ 0).
Proof: See Appendix.
We next show the Pareto dominance of the asymmetric pure-strategy entry.
Corollary 1: If a symmetric mixed-strategy (strictly) entry exists for auction A0 , this
11
symmetric entry is Pareto dominated (strictly) by the asymmetric pure-strategy entry of
A0 when corresponding revenue-maximizing auctions of Proposition 1(ii) are adopted.
Proof: See Appendix.
Define MS(n) = MSi (P∗ ) where P∗ 1×n = (1, 1, · · · , 1). MS(n) is then the marginal
contribution of any participant to the expected total surplus in auction A0 if n potential
bidders participate. Clearly, MS(n) decreases with n. If MS(1) < 0, define N ∗ = 0,
otherwise define N ∗ = max{M S(n)≥0} {n}. Based on Proposition 2, integer N ∗ ≥ 0 must
satisfy the following property. If the number of potential bidders N ≤ N ∗ , it must be
optimal that all bidders participate with probability of 1. On the other hand, if the
number of potential bidders N ≥ N ∗ , it must be optimal that any N ∗ bidders participate
with probability of 1, and other bidders participate with probability of zero.
Levin and Smith (1994) show that expected total surplus may decrease with the number of potential bidders when the entry patterns are restricted to be symmetric across
bidders. If no restriction is imposed on the entry pattern, N ∗ is the optimal number of
bidders if the number of potential bidders N > N ∗ . Thus, additional potential bidders
beyond N ∗ neither increases nor decreases the seller’s optimal expected revenue. Consider
an example where c = 0.1, F (v) = v 2 on [0, 1]. Calculation shows that when N = 2,
P∗ = (1, 1) is the optimal entry, which gives ES(P∗ ) = 0.60. Note that it is a symmetric
solution. When N = 3, P∗ = (1, 1, 0) maximizes ES(P) and ES(P∗ ) = 0.60. In addition,
P† = (.83, .83, .83) is the optimal symmetric participation, which gives ES(P† ) = 0.57017.
As pointed out by Vagstad (2006), coordination device is necessary to ensure purestrategy entry equilibria. When entry costs are intermediate in the sense that there is room
for at least one bidder but not for all potential bidders in auction A0 , there is a symmetric
mixed strategy entry equilibrium besides the pure strategy entry equilibrium. As argued
by Vagstad (2006), “While mixed-strategy entry is a plausible assumption if there are
no coordination devices, pure-strategy entry is plausible if entry can be coordinated,
e.g. if prospective bidders make their entry decisions sequentially and the decisions are
observable or can be communicated to other prospective bidders before they decide”.
We assume the existence of these coordination devices proposed by Vagstad (2006) when
12
pure-strategy entry equilibria are desired and thus to be implemented.
2.3
The Issue of Nondiscriminatory Auctions
Suppose P∗ = (p∗1 , p∗2 , · · · , p∗N ) maximizes expected total surplus ES(P), where p∗i , i ∈ N
is either 0 or 1. We use M to denote the number of elements in P∗ , which are equal to
1. Without loss of generality, we can assume p∗i = 1, i ≤ M and p∗i = 0, i > M. From
Proposition 1, we have the following result.
Proposition 3: A second-price auction with a reserve price equal to seller’s value and a
uniform time-2 entry fee e = MS1 (P ∗ ) is revenue-maximizing.
From the revenue equivalence theorem and Propositions 1 and 3, we have the following
alternative efficient and revenue-maximizing auctions.
Corollary 2: (i) A first-price auction with a reserve price equal to seller’s value and no
entry fee maximizes the expected total surplus. (ii) A first-price auction with a reserve
price equal to seller’s value and a uniform time-2 entry fee e = MS1 (P ∗ ) is revenuemaximizing.
Proposition 3 and Corollary 2(ii) show that we always have nondiscriminatory auctions
that are revenue-maximizing in symmetric setting. In addition, Corollary 2 shows that
modified first-price auctions can also be efficient and/or revenue-maximizing in symmetric
setting.
2.4
The Issue of Symmetric Entry
As pointed at the end of Section 2.2, symmetric entry is more plausible if there are no
coordination devices. We now study the optimal entry and auction design when entry is
restricted to be symmetric. Specifically, we restrict the entry probabilities to be symmetric across bidders, i.e., pi = p ∈ [0, 1]. For notational simplicity, define ES(p) = ES(Ps),
where vector Ps = (p, p, · · · , p). Under the symmetry restriction, we have
P
i∈N
∂ES(Ps )
∂pi
= N
∂ES(Ps )
∂pi
, ∀i ∈ N . This leads to
∂ES(Ps )
∂pi
=
dES(p)
/N, ∀i
dp
s∗
pose p∗ maximizes ES(p), where p∗ could be corner solutions. Use P
13
dES(p)
dp
=
∈ N . Sup-
to denote the
entry probability vector in which every element equals p∗ . Thus, we have the following
characterization for p∗ . For all i ∈ N ,
s∗
MSi (Ps∗ ) =
∗






0,
if p∗ ∈ (0, 1),
∂ES(P )
dES(p )
=
/N = ≥ 0, if p∗ = 1,

∂pi
dp



 ≤ 0, if p∗ = 0.
Note that MSi (Ps∗ ) is the expected payoff of bidder i when he participates in auction A0 ,
if all other potential bidders participate according to Ps∗ . Thus, we have the following
proposition.6
Proposition 4: Consider the class of auctions implementing symmetric entry across
bidders. Let p∗ = argmaxp∈[0,1] ES(p) and Ps∗ = (p∗ , · · · , p∗ ). (i) Auction A0 is ex ante
efficient. (ii) Auction A0 is also revenue-maximizing if p∗ ∈ [0, 1). If p∗ = 1, a secondprice auction with reserve price equal to the seller’s value v0 and a time-2 entry fee of
e = M S1 (Ps∗ )(≥ 0) is revenue-maximizing. (iii) The efficient and revenue-maximizing
reserve price must equal the seller’s value v0 .
Clearly, Proposition 4 still holds if the “second price auctions” were replaced by “first
price auctions” in it.
The result for p∗ falling in (0, 1) corresponds to that of Levin and Smith (1994). At
a mixed-strategy entry equilibrium, the bidders’ expected net profits must be zero, so
the seller captures the entire social surplus. At a reserve price of v0 , a bidder enters if
and only if his expected profit, which is the same as his expected contribution to total
surplus, exceeds his entry costs. Thus, the probability of entry that maximizes expected
total surplus is an equilibrium entry probability.
On the other hand, proposition 4 shows that when p∗ = 1, generally A0 fails to
be revenue-maximizing as the entrants may enjoy positive expected surplus. In this
case, entry fees need to be applied to extract the surplus of the entrants for revenue
maximization.
6
The proof is similar to that of Proposition 1.
14
2.5
The Case of Zero Entry fees
In previous sections, we assume that the seller can use positive entry fees to extract the
surplus of the entrants for revenue maximization. Here, we instead assume that entry
fees or subsidies are not available to the seller as instruments. We thus study how the
revenue-maximizing auction should set the reserve price when entry is endogenous and
entry fees must be zero.7
Here, we focus on auctions inducing symmetric entry.8 Define S(r) as the expected
payoff of every entrant in a second price auction with reserve price r and no entry fees
when all N bidders enter, if their entry costs are zero. When c ≥ S(v0 ), Proposition 4
implies that A0 is revenue-maximizing. Thus, the issue of entry fees does not exist in this
case.
Assume the usual regularity condition of increasing marginal revenue function J(v) =
v−
1−F (v)
f (v)
that is defined in Bulow and Roberts (1989). Let r∗ = J −1 (v0 ). We have
J −1 (v0 ) > v0 and S(r∗ ) < S(v0 ) as S(·) is a decreasing function. According to Myerson
(1981) and Bulow and Roberts (1989), a second price auction with reserve price r∗ is
the revenue-maximizing auction when there is no entry problem. When c ≤ S(r∗ ), the
expected payoff of every entrant is nonnegative when all bidders participate in this auction. This auction is then clearly the revenue-maximizing auction when no entry fees are
allowed.
We now turn to the case where c ∈ (S(r∗ ), S(v0 )). Define r̂ = S −1 (c) ∈ (v0 , r∗ ). In this
case, the optimal reserve price must be r̂, and the expected payoff of every bidder must
be zero. Following Myerson (1981) and Bulow and Roberts (1989), clearly the seller’s
expected revenue increases with reserve price r when r ≤ r̂. In addition, r̂ dominates
any r(> r̂) in terms of the total expected surplus and seller’s revenue. When r ≥ r̂, the
expected surplus of bidders is zero. Thus the seller’s revenue equals to the total expected
7
Note that the issue of entry fees is not relevant to the efficient auction according to Propositions 1
and 4.
8
Engelbrecht-Wiggans (1993) has studied how to set the reserve price for the case of pure-strategy
entry.
15
surplus. For any r(> r̂), it induces an equilibrium entry probability of p(r) ∈ [0, 1).
The total surplus in a second price auction with reserve price r is not higher than that
in a second price auction with reserve price r̂ given v0 < r̂ < r, if every bidder enters
with probability of p(r). The second price auction with reserve price r̂ actually induces
equilibrium entry probability of 1 and every bidder’s expected payoff is zero. Thus,
the marginal contribution of every bidder to the total expected surplus is zero if the
seller’s value were r̂. However, since the seller’s value is actually v0 (< r̂), the marginal
contribution of every bidder to the total expected surplus must be positive, even when
all other bidders enter with probability 1.9 Therefore, the total expected surplus from a
second price auction with reserve price r̂ increases when every bidder’s entry probability
increases from p(r) to 1. Thus, reserve price r̂ dominates any r(> r̂) in terms of the total
surplus and seller’s revenue.
We next show that mixed-strategy entry can dominate pure-strategy entry if entry fee
is not available as an instrument. This result contrasts with the optimality of pure-strategy
entry when entry fees are allowed. Consider an example where N = 2, v0 = 0, F (v) = v
on [0, 1], and c =
1
6
+ ε where ε is a small positive number. Note that even if we set the
reserve price at zero in a second price auction, the auction has no room for two entrants.10
Thus, we only need to consider the optimal reserve price when there is only one entrant
for the case of pure-strategy entry. According to Myerson (1981) and Bulow and Roberts
(1989), the optimal reserve price is J −1 (0) =
1
2
even if we can ignore the entry problem.
Thus, the seller’s expected revenue must be smaller than 14 . If we allow mixed-strategy
entry, auction A0 implements an equilibrium entry probability that is very close to 1 as ε is
very small. The seller’s expected revenue can thus be very close to 13 . This example shows
that mixed-strategy entry can dominate pure-strategy entry if entry fee is not allowed.
9
10
One additional bidder decreases the probability that the item is not sold out.
When ε = 0, the auction can accommodate exactly two entrants.
16
3
Asymmetric IPV Setting
In this generalization of the basic setting of Section 2, the bidders’ value distributions and
entry costs are allowed to be asymmetric across potential bidders. After incurring entry
costs ci (≥ 0), bidder i observes his private value vi , i ∈ N . The cumulative distribution
function of vi is Fi (·) with density function fi (·) defined on [vi , vi ]. In this section, we
investigate the implications of asymmetry across bidders to the optimal entry and auction
design.
Following the same procedure as in Section 2, the optimal expected total surplus for
given entry probabilities P is
ES(P) =
X {k∈K}
Define MSi (P) =
Y
Vk
P
pki i (1 − pi )1−ki −
i∈N
{k−i ∈K−i } [(Vk1 (k−i )
− Vk0 (k−i ) −ci )
X
pi ci .
(6)
i∈N
Q
j6=i
k
pj j (1 − pj )1−kj ], which is the
expected payoff of bidder i from participating in auction A0 , provided that other bidders
participate according to P. With these redefined MSi (·) functions, the ex ante efficient
auction and revenue-maximizing auction are still defined as the second-price auctions in
Proposition 1. In addition, the optimality of {0, 1} entry also holds.11
Unlike the symmetric setting, it no longer holds that first-price auctions can also
be efficient or revenue-maximizing in asymmetric setting. The reason is that first-price
auctions generally do not always allocate the item to the bidder with the highest value in
an asymmetric setting.12
While discriminatory auctions do exist in reality (McAfee and J. McMillan, 1989),
they might not always be allowed. Clearly, considering only nondiscriminatory auctions
becomes restrictive for revenue maximization in asymmetric settings due to the necessity
of the personalized entry fees. Next, we show that when bidders are asymmetric, the
11
12
The proofs are similar to those of Propositions 1 and 2.
For example, consider a setting with two asymmetric bidders whose values follow different density
functions defined on the same support. Generally, the equilibrium bidding functions of the two bidders
have to be asymmetric in a first-price auction.
17
revenue-maximizing nondiscriminatory auction may have to induce mixed-strategy entry.
This result contrasts with the optimality of pure-strategy entry when personalized entry
fees are allowed.
Since solving for the revenue-maximizing nondiscriminatory auctions turns out to
be much more difficult than the unrestricted optimum, we adopt the following special
case to illustrate our points. In this setting, bidders differ only in entry costs, i.e.,
Fi (·) = F (·), ∀i ∈ N . Without loss of generality, we assume that c1 ≤ c2 ≤ · · · ≤ cN .
Suppose it is optimal for n(≥ 2) bidders to enter when discriminatory auctions are allowed. These entrants must be the first n bidders who have smaller costs. Recall that Zl
is the expectation of the highest value of the seller and any l(≥ 0) bidders. Then, n must
satisfy the following conditions:
Zn − Zl −
n
X
ci ≥ 0, ∀l < n, and Zl − Zn −
l
X
ci ≤ 0, ∀l > n.
(7)
i=n+1
i=l+1
We first consider discrete entry decisions. Clearly, a revenue-maximizing nondiscriminatory auction that induces l ∈ N bidders must induce the first l bidders. Since the
auction is nondiscriminatory, the l-th bidder must enjoy zero surplus and bidder i(< l)
must enjoy a surplus of cl − ci . Thus, the highest possible expected revenue from these l
bidders is
ER(l) = Zl −
l
X
i=1
ci −
l
X
(cl − ci ), l ∈ N ,
(8)
i=1
which is strictly lower than the unrestricted maximal expected revenue due to the heterogeneity in entry costs cj , j ≤ l. Note that revenue ER(l) is achievable through a nondiscriminatory second price/first price auction with a reserve price equal to seller’s value and
an entry fee el = Zl − Zl−1 − cl . Let l∗ = argmaxl∈N ER(l) and el∗ = Zl∗ − Zl∗ −1 − cl∗ .
We have the following result.
Proposition 5: Suppose the bidders are symmetric in their value distribution and the
entry decisions are discrete. (i) The revenue-maximizing nondiscriminatory auction induces the first l∗ (≤ n) bidders with smaller entry costs to enter, where n is the optimal
18
number of entrants at the unrestricted maximum. (ii) This auction can take the form of a
second price or first price auction with a reserve price equal to seller’s value and an entry
fee el∗ .
Proof: See Appendix.
It deserves to be pointed out that l∗ can be any positive integer smaller than n. From
(8), the necessary and sufficient conditions for l∗ ∈ {1, 2, · · · , n − 1} to be optimal are
ER(l) − ER(l∗ ) ≤ 0, ∀l 6= l∗ , i.e.,
[Zl − Zl∗ −
l
X
ci ] − [
i=l∗ +1
[Z − Zl −
l∗
l∗
X
i=l+1
l
X
(cl − ci )] − l∗ (cl − cl∗ ) ≤ 0, ∀l ∈ {l∗ + 1, · · · , n}, and
i=l∗ +1
∗
ci ] − [
l
X
(cl∗ − ci )] − l(cl∗ − cl ) ≥ 0, ∀l ∈ {1, · · · , l∗ − 1}.
(9)
i=l+1
Let ci = ∞, ∀i > n and choose any cn such that 0 < Zn − Zl∗ − (n − l∗ )cn < l∗ cn . Let
ci = 0, ∀i ≤ l∗ , and ci = cn , ∀i ∈ {l∗ + 1, · · · , n}. In this setting, all conditions in (7) and
(9) are satisfied.
We next show that the revenue-maximizing nondiscriminatory auction may have to
induce mixed-strategy entry. The following example shows this point. Assume N =
2, F (v) = v on [0, 1], v0 = 0, c2 =
1
6
and c1 = 16 −δ where δ ∈ [0, 16 ). Since Z2 = 23 , Z1 = 12 ,
we have Z2 − Z1 − c2 = 0 and l∗ = 2 for all δ ∈ [0, 16 ). We consider a second-price or
first-price auction with zero reserve price and a uniform entry fee e. Suppose we want to
induce entry probabilities p1 ∈ (0, 1) and p2 = 1, and use entry fee e to extract all the
expected surplus of the participants, then
MS1 (p1 , p2 ) = Z2 − Z1 − c1 − e = 0, and
(10)
MS2 (p1 , p2 ) = p1 (Z2 − Z1 − c2 ) + (1 − p1 )(Z1 − c2 ) − e = 0.
(11)
(10) and (11) give that p1 = 1 − 3δ and e = δ. For these auctions, seller’s expected
revenue ER equals expected total surplus. From (6), we thus have ER =
Thus
dER
dδ
1
3
+ δ − 3δ 2 .
= 1 − 6δ, which is positive when δ ∈ (0, 16 ).
Note that the above identified auctions corresponding to δ = 0 coincide with the
Proposition 5(ii) nondiscriminatory auctions, which are not contingent on δ ∈ [0, 16 ).
19
Therefore, for all δ ∈ (0, 16 ), we have identified a nondiscriminatory auction that renders
higher expected revenue than that from the nondiscriminatory revenue-maximizing auction inducing discrete entry. Thus, the revenue-maximizing nondiscriminatory auction
must not induce pure-strategy entry for all δ ∈ (0, 16 ).
4
Private-Information Entry Costs
In this second generalization of the basic setting of Section 2, we allow the entry costs to
be the bidders’ private information. The cumulative distribution function of entry costs
is G(·) with density function of g(·)(> 0) on interval [c, c].13 This distribution is public
information. The entry cost of bidder i is denoted by ci , i ∈ N . We assume ci and vj ,
∀i, j ∈ N are mutually independent. All bidders observe their private entry costs before
their simultaneous entry decisions. If they enter, they incur their private entry costs and
observe their private values.
First, all the feasible equilibrium entry patterns can be characterized as in the following
Lemma.
Lemma 3: Any equilibrium entry pattern can be described through a vector of entry
thresholds Ce = (ce1 , · · · , ceN ) satisfying the following properties: (i) cei ∈ [c, c], ∀i ∈ N ;
(ii) if ci < cei , bidder i participates with probability 1; if ci > cei , bidder i participates with
probability 0.
Proof: See Appendix.
Next, we establish the following results regarding the restricted ex ante efficient
auction/revenue-maximizing auction that implements given entry thresholds Ce = (ce1 , · · · ,
ceN ), where cei ∈ [c, c], ∀i ∈ N .
Lemma 4: (i) Among all auctions implementing any given entry thresholds Ce =
(ce1 , · · · , ceN ), a second-price auction with a reserve price equal to seller’s value and appropriate ex ante entry fee (or subsidy) for every bidder provides the highest seller’s
13
We focus on a symmetric setting here. The results of this section can be easily generalized to an
asymmetric setting following the same methodology.
20
expected revenue as well as the highest expected total surplus. (ii) The ex ante entry fees
(or subsidies) are charged upon entry before the values are learned by the entrants, and
are set at levels such that the threshold-type entrants get zero expected payoff. (iii) In the
above auction, the expected surplus of bidder i with entry cost ci (< cei ) is cei − ci .
Proof: See Appendix.
For given entry thresholds Ce where cei ∈ [ci , ci ], ∀i ∈ N , we denote the highest
expected total surplus and the highest seller’s expected revenue attainable through the
Lemma 4 auction by S(Ce ) and R(Ce ), respectively. According to Lemma 4, S(Ce ) and
R(Ce ) can be written as the following
S(Ce ) =
X
Vk P r(k) −
{k∈K}
R(Ce ) =
X
Vk P r(k) −
Q
e ki
i∈N (G(ci )) (1
c
i∈N
X
cei
ci g(ci )dci ,
(12)
cei G(cei ),
(13)
i∈N
{k∈K}
where P r(k) =
XZ
− G(cei ))1−ki is the ex ante probability that the ex post
participation status denoted by k happens. The term
P
{k∈K}
Vk P r(k) is the ex ante
expected contribution of the values of all players including the seller to the expected
total surplus if potential bidders participate according to threshold-vector Ce . The term
P
i∈N
R ce
c
i
ci g(ci )dci is the expected entry costs of entrants. The difference between these
two terms is then the expected total surplus S(Ce ). Following Lemma 4(iii), we know
that the ex ante expected information rent of bidder i is
R ce
c
i
(cei − ci )g(ci )dci . This leads to
the seller’s expected revenue R(Ce ) in (13), which is the difference between the expected
total surplus S(Ce ) and all the bidders’ ex ante expected information rents
P
i∈N
R ce
c
i
(cei −
ci )g(ci )dci .
4.1
Ex Ante Efficient Auction
We first characterize the first order conditions for the efficient threshold-vector Ce∗ =
e∗
e
e∗
(ce∗
1 , · · · , cN ) = argmax{Ce ∈[c,c]N } S(C ). Let us consider the entry threshold ci for bidder
21
i, ∀i ∈ N . Clearly, we have
X
∂S(Ce )
= g(cei )
{(Vk1 (k−i ) − Vk0 (k−i ) −cei )P r(k−i )},
e
∂ci
{k−i ∈K−i }
as
P
{k−i ∈K−i }
(14)
P r(k−i ) = 1. We then have the following characterization for Ce∗ . For all
i ∈ N,






0,
if ce∗
i ∈ (c, c),
∂S(Ce∗ )
= ≥ 0, if ce∗
i = c,

∂cei



 ≤ 0, if ce∗ = c.
i
Define Si (Ce ) =
P
e
{k−i ∈K−i } {(Vk1 (k−i ) −Vk0 (k−i )−ci )P r(k−i )},
(15)
which is the expected payoff
of bidder i with cost cei when he participates in auction A0 , when all other potential
bidders participate according to Ce . This insight, together with (14) and (15), leads to
the following proposition that addresses the ex ante efficient auction.
Proposition 6: The second-price auction A0 is ex ante efficient.
Proof: See Appendix.
4.2
Revenue-Maximizing Auction
We now turn to the revenue-maximizing auction. From (12), (13) and (14), we have
G(cei )
∂R(Ce )
e
e
}, ∀i ∈ N .
=
g(c
){S
(C
)
−
i
i
∂cei
g(cei )
(16)
e†
e
Suppose that Ce† = (ce†
1 , · · · , cN ) = argmax{Ce ∈[c,c]N } R(C ), then we have the follow-
ing characterization for Ce† . For all i ∈ N ,
e†






0,
if ce†
i ∈ (c, c),
∂R(C )
= ≥ 0, if ce†
i = c,

∂cei



 ≤ 0, if ce† = c.
i
Define Ri (Ce ) = Si (Ce ) −
G(cei )
.
g(cei )
(17)
Clearly, Ri (Ce† ) is the expected payoff of bidder i with
cost cei , if he participates in a second-price auction with a reserve price equal to v0 and an
22
ex ante entry fee of
G(cei )
g(cei )
for bidder i, provided that all other potential bidders participate
according to Ce† . We thus obtain from (16) and (17) the following proposition that
addresses the revenue-maximizing auction.
Proposition 7: A second-price auction with reserve price equal to seller’s value and ex
ante entry fees ei for bidder i defined below leads to the seller the highest expected revenue.
The entrants pay their ex ante entry fees before their values are learned. The ex ante entry
fees ei , i ∈ N are defined as






Si (Ce† ) =
G(ce†
i )
g(ce†
i )
if cie† ∈ (c, c),
,
ei =  Si (Ce† ) ≥ 1 ,
if cie† = c,
g(c)



 any number ≥ S (Ce† )(≤ 0), if ce† = c,
i
i
(18)
where Ce† = argmax{Ce ∈[c,c]N } R(Ce ).
Proof: See Appendix.
4.3
Asymmetry in the Efficient/Revenue-Maximizing Entry
Consider a setting where v0 = 0, N = 2, F (v) = v, v ∈ [0, 1], and G(c) = 10(c − 0.4), c ∈
[0.4, 0.5]. Direct calculations using (12) and (13) give the following results. S(Ce ) takes
the maximum of 0.05 when ce1 = 0.5 and ce2 = 0.4, and R(Ce ) takes the maximum of
0.025 when ce1 = 0.45 and ce2 = 0.4. If we restrict ce1 = ce2 , then we have that S(Ce ) takes
the maximum of 0.023 when ce1 = ce2 = 0.4231, and R(Ce ) takes the maximum of 0.01875
when ce1 = ce2 = 0.4187. In other words, the ex ante efficient/revenue-maximizing entry
is asymmetric. This example indicates that “symmetric” entry is generally restrictive for
auction design.
The intuitions behind the asymmetry of the efficient/revenue-maximizing entry are as
follows. Let us consider the case with 2 potential bidders (N=2). From (12) and (13), the
first common component of S(ce1 , ce2 ) and R(ce1 , ce2 ) can be written as
Z2
2
Y
G(cei ) + Z1 [G(ce1 )(1 − G(ce2 )) + G(ce2 )(1 − G(ce1 ))] + Z0
i=1
2
Y
i=1
23
(1 − G(cei ))
= (Z1 − Z0 )
2
X
G(cei ) +
i=1
2
(Z2 − Z1 ) − (Z1 − Z0 ) X
[( G(cei ))2 − (G(ce1 ) − G(ce2 ))2 ].
4
i=1
From Lemma 2, Z2 − Z1 < Z1 − Z0 as Zl+1 − Zl decreases with l. Thus for given
P2
i=1
G(cei ), we want to maximize or minimize G(ce1 )−G(ce2 ) in order to maximize the above
common component of S(ce1 , ce2 ) and R(ce1 , ce2 ). Therefore, if the symmetric entry threshold
maximizing the above common component is an inner solution, we must have that the
unrestricted solution must be asymmetric. The above arguments can be generalized to
the case where N > 2 by focusing on the entry of any two bidders while assuming the
entry thresholds of all other bidders are fixed.
It is clear that if c and c are close enough, increasing the difference between the entry
thresholds of any two bidders while keeping the sum of their ex ante entry probabilities
unchanged will lead to higher expected total surplus/the seller’s expected revenue because
the second terms in (12) and (13) do not change much.14
4.4
Optimality and Uniqueness of Symmetric Entry
For the efficient auction A0 , there may exist multiple entry equilibria. Recall the Section
4.3 example, the unrestricted efficient entry thresholds are ce1 = 0.5 and ce2 = 0.4, while
the symmetric efficient thresholds are ce1 = ce2 = 0.4231. Both of these two entry patterns
are implemented through the same auction A0 . Similarly, for the proposed revenuemaximizing auction (Proposition 7), there may exist multiple entry equilibria. To address
the multiplicity of entry equilibria and a closely related issue of optimality of symmetric
entry, we present the following results.
Lemma 5: If
G(c1 )−G(c2 )
c1 −c2
<
1
,
(Z1 −Z0 )−(Z2 −Z1 )
∀c1 , c2 ∈ [c, c], then there exists a unique
entry equilibrium for the efficient auction A0 . The entry equilibrium is symmetric.
Proof: See Appendix.
Since G(·) belongs to [0, 1], the condition
G(c1 )−G(c2 )
c1 −c2
<
1
(Z1 −Z0 )−(Z2 −Z1 )
can easily be
satisfied if the distribution of the entry costs is well dispersed. A sufficient condition for
14
In particular, this second term does not change when the entry costs are fixed.
24
G(c1 )−G(c2 )
c1 −c2
<
1
(Z1 −Z0 )−(Z2 −Z1 )
is g(·) <
1
.
(Z1 −Z0 )−(Z2 −Z1 )
Therefore, if the entry costs follow
a uniform distribution, then the bigger the range of the entry costs, the less likely there
exist asymmetric equilibria. In this sense, greater dispersion of entry costs decreases the
likelihood of asymmetric entry equilibria. In the Section 4.3 example where asymmetric
entry emerges for auction A0 , the range of the entry costs is rather small. As a result,
the condition in Lemma 5 is violated.
The efficient entry is always implemented through A0 according to Proposition 6.
Lemma 5 further provides sufficient condition for A0 to implement a unique entry equilibrium, which is symmetric. We thus have the following results.
Proposition 8: If
G(c1 )−G(c2 )
c1 −c2
<
1
,
(Z1 −Z0 )−(Z2 −Z1 )
∀c1 , c2 ∈ [c, c], then the following results
hold. (i) The ex ante efficient entry must be symmetric. (ii) This ex ante efficient entry
is the unique simultaneous-move entry equilibrium of the efficient auction A0 .
The following proposition presents further complementary results for the revenuemaximizing entry.
Proposition 9: If
G(c1 )−G(c2 )
c1 −c2
<
1
,
(Z1 −Z0 )−(Z2 −Z1 )
∀c1 , c2 ∈ [c, c] and the hazard rate
G(·)
g(·)
increases on [c, c], then the following results hold. (i) The revenue-maximizing entry must
be symmetric. (ii) This entry is the unique simultaneous-move entry equilibrium of the
corresponding revenue-maximizing auction of Proposition 7.
Proof: See Appendix.
Proposition 9 provides sufficient conditions for the symmetry of revenue-maximizing
entry and the uniqueness of the entry equilibrium for the revenue-maximizing auction. For
this symmetric entry, the revenue-maximizing auction defined in Proposition 7 becomes
a second price auction with a uniform reserve price and a uniform entry fee. Therefore,
there is no loss to consider only nondiscriminatory auctions for revenue maximization
when the conditions of Proposition 9 hold. Moreover, the uniform entry fee is actually
zero whenever the optimal entry threshold is less than c. Thus, the issue of entry fees
does not exist in this case.
25
5
Conclusion
This paper studies optimal entry and auction design in an independent private value
(IPV) framework, where bidders have to incur costs to discover their values. When the
entry costs are fixed, there is no loss of generality in considering entry patterns where
every bidder participates with probability of either 0 or 1, for the revenue-maximizing
and/or efficient auction. This result not only eliminates the necessity of considering
noninteger number of entrants for the analysis on optimal entry, but also explains why
seller’s expected revenue may drop with the number of potential bidders when the entry
pattern are restricted to be symmetric. While the second price auction with no entry fee
and a reserve price equal to the seller’s value is ex ante efficient, the revenue-maximizing
auction generally employs personalized entry fees to extract the surplus of the participants.
Although the desired entry is asymmetric across bidders, there is no loss of generality
to consider only nondiscriminatory auctions for revenue maximization when bidders are
symmetric. However, with asymmetric bidders, considering only nondiscriminatory auctions are restrictive and the revenue-maximizing nondiscriminatory auction may have to
induce mixed-strategy entry. Moreover, we find that mixed-strategy entry can dominate
pure-strategy entry if entry fee is not available as an instrument.
When entry costs are bidders’ private information, the seller cannot extract all the
surplus of the bidders. Unlike the case with fixed entry costs, the revenue-maximizing
entry diverges from the ex ante efficient entry. While zero entry fee remains ex ante efficient, the revenue-maximizing entry fees must equal the hazard rates of the bidders’ entry
cost distribution, evaluated at the desired entry thresholds. While the efficient and/or
revenue-maximizing entry can be asymmetric across symmetric bidders as in the case of
fixed entry cost, we find that large dispersion in the entry costs and increasing hazard rate
of the cost distribution ensure the symmetry in the efficient/revenue-maximizing entry
and guarantee the uniqueness of entry equilibrium for the proposed auctions.
The second-price auction with a reserve price equal to the seller’s value and zero entry fee is shown to be ex ante efficient in more general settings than those considered in
26
the existing literature. The efficiency of this simple auction is based on key connections
between the first order conditions characterizing the optimal entry patterns and the expected payoffs of the entrants. When the entry costs are fixed, the first order derivative
of the expected total surplus attainable with respect to a bidder’s entry probability coincides with his expected payoffs if he enters. Similarly when the entry costs are private
information of the bidders, the first order conditions are also solely determined by the
expected payoffs of the threshold types.
In this paper, we focused on settings with independent private values. The robustness of the main results should be considered when values are affiliated. Note that the
methodology we developed does not clinch on independent values. Thus, my conjecture
is that our main results and the economic intuitions should still be valid in setting with
affiliated values, no matter the entry costs are fixed or private information of the bidders.
For a given entry pattern, clearly a second price auction with reserve price equal to the
seller’s value and appropriate entry fees is still ex ante efficient and revenue-maximizing.
The established connections between the first order conditions characterizing the efficient
and/or revenue-maximizing entry and the expected payoffs of the entrants also remain to
be valid.
27
Appendix
Proof of Lemma 1: Let us first consider auction A0 . Suppose all bidders other than i
participate in auction A0 according to their entry probabilities specified by P = (p1 , p2 , · · · , pN ).
Denote bidder i’s expected surplus by Si (P) if he participates in A0 . Set a time-2 entry fee (or
subsidy) for bidder i as ei = Si (P), ∀i ∈ N . Clearly, for a second-price auction with entry fee
(or subsidy) ei for bidder i and a reserve price equal to seller’s value, bidder i’s expected payoff
is 0 if he participates with probability pi . Hence, the above auction with entry fee ei for bidder
i implements participation pattern P.
Note that for any auction implementing P, the total expected entry costs are the same.
Thus the above designed auction achieves the highest possible expected total surplus among the
class of auctions implementing P, since the auction always awards the item to the participant
(including the seller) with the highest value.
Moreover, for any auction implementing entry pattern P = (p1 , p2 , · · · , pN ), the expected
payoff of bidder i cannot be smaller than 0 due to the participation constraint. As a result,
the proposed auction achieves the highest possible seller’s expected revenue among all auctions
implementing any given entry pattern P. 2
Proof of Proposition 1: Clearly, auction A0 implements entry pattern P∗ that maximizes
ES(P) on S0 . For A0 , every participant bids his true value, thus the participant (including the
seller) with the highest value wins. As a result, A0 renders the highest possible expected total
surplus. In other words, A0 is ex ante efficient.
The second price auction defined in Proposition 1 (ii) also allocates the item to the participant (including the seller) with the highest value, while implementing entry pattern P∗ . In the
same time, all the surplus of the participating bidders are extracted. Therefore, the auction is
revenue-maximizing for the seller.
If the reserve price deviates from v0 , there must exist efficiency loss as the item is not always
allocated to the player with the highest value. The efficiency loss is also reflected in the revenue
as the seller extracts all the expected surplus from the bidders.2
∗(0)
Proof of Proposition 2: Suppose P∗(0) = {p1
[0, 1], ∀i ∈ N . If
∗(0)
pi
∗(0)
∗(0)
, · · · , pN } maximizes ES(P), where pi
∈
∈ {0, 1}, ∀i ∈ N , then the proof is completed. Otherwise, there must
28
∗(0)
exist i0 ∈ N satisfying pi0
∗(0)
∈ (0, 1). Since pi0
M Si0 (P∗(0) ) = 0. Note that M Si0 (P∗(0) ) actually does not depend on the value
∗(0)
∗(0)
ES(P∗(0) ) does not depend on the value of pi0 . Therefore, setting pi0
will not change the expected total surplus. This change in
vector
P∗(1)
∂ES(P∗(0) )
=
∂pi0
∗(0)
of pi0 . Thus
is an inner solution, we must have
∗(0)
pi 0
to be either 0 or 1
leads to a new entry probability
which also maximizes ES(P). The only difference between P∗(0) and P∗(1) lies in
∗(1)
the values that the i0 th elements of the two vectors take. If now pi
∈ {0, 1}, ∀i ∈ N , then the
proof is completed. Otherwise, the above process can continue until every element in the entry
probability vector becomes either 0 or 1. 2
Proof of Lemma 2: We use Hl (·) to denote the cumulative distribution function of the highest
value of the seller and l(≥ 0) symmetric bidders. Without loss of generality, we assume v0 ∈
(v, v). Thus, Hl (·) = F l (·) on its support [v0 , v], ∀l ≥ 1. Hl (·) has a mass point at v0 . It follows
that Zl = v0 F l (v0 ) +
Rv
v0 (1
Rv
v0
xdF l (x) = v −
Rv
v0
F l (x)dx, ∀l ≥ 0. This leads to that Zl − Zl−1 =
− F (x))F l−1 (x)dx and (Zl − Zl−1 ) − (Zl+1 − Zl ) =
Rv
v0 (1
− F (x))2 F l−1 (x)dx, ∀l ≥ 1.
Therefore, we have both Zl+1 − Zl and (Zl+1 − Zl ) − (Zl+2 − Zl+1 ) decrease with l(≥ 0). 2
Proof of Corollary 1: We only need to show that the symmetric entry P = (p, p, · · · , p), p ∈
(0, 1) cannot deliver the highest expected total surplus, since the revenue-maximizing auction
always extracts all the expected surplus. To this end, we will show that when all bidders i, i ≥ 3
participate with probability p, the total expected surplus cannot be maximized if p1 = p2 = p.
We use P r(l) to denote the probability that l of the N − 2 bidders (3 ≤ i ≤ N ) participate, then
ES(p1 , p2 , p, · · · , p)
=
N
−2
X
P r(l) p1 p2 Zl+2 + [p1 (1 − p2 ) + (1 − p1 )p2 ]Zl+1 + (1 − p1 )(1 − p2 )Zl
k=0
−[p1 + p2 + (N − 2)p]c
=
N
−2
X
P r(l) (Zl+1 − Zl )(p1 + p2 ) +
k=0
(Zl+2 − Zl+1 ) − (Zl+1 − Zl )
[(p1 + p2 )2 − (p1 − p2 )2 ]
4
−(p1 + p2 + (N − 2)p)c.
From Lemma 2, (Zl+2 − Zl+1 ) − (Zl+1 − Zl ) < 0, ∀0 ≤ l ≤ N − 2. Thus, given p1 + p2 = 2p, the
total expected surplus is maximized when p1 − p2 is maximized or minimized. This means that
29
P = (p, p, · · · , p) cannot deliver the highest expected total surplus. It follows that the seller is
strictly better off at the asymmetric pure-strategy entry of A0 while the bidders are indifferent.2
Proof of Proposition 5: We only need to show that ER(l) ≤ ER(n), ∀l > n. From (8),
ER(l) − ER(n) = Zl −
l
X
ci −
i=1
= [Zl − Zn −
l
X
(cl − ci ) − [Zn −
i=1
l
X
i=1
ci ] − [
i=n+1
From (7), Zl − Zn −
Pl
i=n+1 ci
n
X
l
X
ci −
n
X
(cn − ci )]
i=1
(cl − ci )] − n(cl − cn ), ∀l > n.
i=n+1
≤ 0, ∀l > n. We thus have ER(l) − ER(n) ≤ 0, ∀l > n. 2
Proof of Lemma 3: Let us consider any entry equilibrium E implemented by an auction. If
all bidders other than i adopt their equilibrium entry strategy described by E, the bidder i’s
equilibrium entry strategy in E must be his best entry strategy. Given that all bidders other
than i adopt the equilibrium entry strategy in E, there must exist an entry threshold cei ∈ [c, c]
such that bidder i’s best entry strategy is described by property (ii) in Lemma 3. This is true
because the expected payoff of bidder i from participating in any given auction decreases strictly
and continuously with his entry cost, given that all bidders other than i adopt their equilibrium
entry strategy in E. 2
Proof of Lemma 4: Let us first consider auction A0 . Suppose all bidders other than i
participate in auction A0 according to thresholds Ce = (ce1 , · · · , ceN ). Denote bidder i’s expected
surplus by Si (ci ; Ce ) if he participates in A0 while his entry cost is ci . Then Si (ci ; Ce ) decreases
strictly and continuously with ci . Set an ex ante entry fee (or subsidy) for bidder i as Ei =
Si (cei ; Ce ), ∀i ∈ N . Clearly, for a second-price auction with ex ante entry fee (or subsidy) Ei
for bidder i and a reserve price equal to seller’s value, bidder i’s expected payoff is cei − ci if
he participates and his entry cost is ci . Hence, the above auction with ex ante entry fee Ei
for bidder i implements entry thresholds Ce . Note that for any auction implementing entry
thresholds Ce , the total expected entry costs are the same. Thus the above designed auction
achieves the highest possible expected total surplus among the class of auctions implementing
Ce , as the auction always awards the item to the participant (including the seller) with the
highest value.
30
Moreover, for any auction implementing entry thresholds Ce , the expected surplus of bidder
i with entry cost ci ≤ cei cannot be smaller than cei − ci , if he participates. This is due to the fact
that a type ci can always mimic a type cei , and by doing so he gets at least a payoff of cei − ci .
Recall that in a second-price auction with ex ante entry fee Ei for bidder i and a reserve price
equal to seller’s value, bidder i’s expected surplus is exactly cei − ci if he participates and his
entry cost is ci . As a result, this auction achieves the highest possible seller’s expected revenue
among all auctions implementing any given entry threshold-vector Ce . 2
Proof of Proposition 6: Since g(·) > 0, it is clearly a Nash equilibrium that every bidder
participates in auction A0 according to Ce∗ , as (15) is satisfied for Ce∗ . In addition, A0 clearly
renders an expected total surplus of S(Ce∗ ). 2
Proof of Proposition 7: Since g(·) > 0, it is clearly a Nash equilibrium that every bidder
participates in this auction according to Ce† , while (16) and (17) hold.
From Lemma 4(ii), if ce†
i > c, the entry fees should be set at the levels such that the threshold
types get zero expected payoff. Therefore, ei should be Si (Ce† ) if ce†
i > c. From (16) and (17),
Si (Ce† ) =
G(ce†
i )
g(ce†
i )
e†
if ce†
i ∈ (c, c), and Si (C ) ≥
1
g(c)
e†
if ce†
i = c. If ci = c, any entry fee which is
bigger than Si (Ce† )(≤ 0) implements the threshold participation. 2
Proof of Lemma 5: We prove the lemma using contradiction. Note that a symmetric entry
equilibrium always exists. Suppose that there is another asymmetric entry equilibrium Ce .
Then, we can find i1 , i2 ∈ N such that cei1 > cei2 . We use P r(n), n = 0, 1, · · · , N − 2 to denote
the probabilities with which there are n bidders from the other N − 2 bidders participating in
the auction. Note that Zn+1 −Zn is the expected payoff of a bidder from participating in auction
A0 , if his entry cost is zero and there are other n participants.
Since entry thresholds cei1 , cei2 can be corner solutions, we must have that
N
−2
X
P r(n){(1 − G(cei2 ))(Zn+1 − Zn ) + G(cei2 )(Zn+2 − Zn+1 )} ≥ cei1 ,
(A.1)
n=0
N
−2
X
P r(n){(1 − G(cei1 ))(Zn+1 − Zn ) + G(cei1 )(Zn+2 − Zn+1 )} ≤ cei2 .
(A.2)
n=0
31
(A.1) and (A.2) lead to
(G(cei1 )
−
G(cei2 ))
N
−2
X
P r(n)[(Zn+1 − Zn ) − (Zn+2 − Zn+1 )] ≥ cei1 − cei2 .
(A.3)
n=0
From Lemma 2, we have
PN −2
n=0 P r(n)[(Zn+1 − Zn ) − (Zn+2 − Zn+1 )] ≤ (Z1 − Z0 ) − (Z2 − Z1 ). As
G(c1 )−G(c2 )
1
< (Z1 −Z0 )−(Z2 −Z1 ) , ∀c1 , c2 , we have the left hand side of (A.3) must be smaller than
c1 −c2
cei1 − cei2 . This contradicts with (A.3). Therefore, there exists no asymmetric entry equilibrium
for the efficient auction A0 . 2
Proof of Proposition 9: We prove the proposition using contradiction. Suppose that the
revenue-maximizing entry Ce is asymmetric. Then, we can find i1 , i2 ∈ N such that cei1 > cei2 .
We use P r(n), n = 0, 1, · · · , N − 2 to denote the probabilities with which there are n bidders
from the other N − 2 bidders participating in the auction.
Since entry thresholds cei1 , cei2 can be corner solutions, condition (17) gives
N
−2
X
P r(n){(1 −
n=0
N
−2
X
P r(n){(1 − G(cei2 ))(Zn+1 − Zn ) + G(cei2 )(Zn+2 − Zn+1 )} ≤ cei2 +
G(cei2 ))(Zn+1
− Zn ) +
G(cei2 )(Zn+2
− Zn+1 )} ≥
n=0
cei1
G(cei1 )
+
,
g(cei1 )
(A.4)
G(cei2 )
.
g(cei2 )
(A.5)
(A.4) and (A.5) lead to
(G(cei1 ) − G(cei2 ))
≥ (cei1 − cei2 ) +
as hazard rate
Zn+1 )] ≤
hand side
G(·)
g(·)
N
−2
X
P r(n)[(Zn+1 − Zn ) − (Zn+2 − Zn+1 )]
n=0
G(cei1 )
(
g(cei1 )
−
G(cei2 )
) ≥ cei1 − cei2 ,
g(cei2 )
increases. From Lemma 2, we have
(A.6)
PN −2
n=0 P r(n)[(Zn+1 − Zn ) − (Zn+2 −
G(c1 )−G(c2 )
1
(Z1 − Z0 ) − (Z2 − Z1 ). As
< (Z1 −Z0 )−(Z
, ∀c1 , c2 , we have the left
c1 −c2
2 −Z1 )
of (A.6) must be smaller than cei1 − cei2 . This contradicts with (A.6). Therefore, the
revenue-maximizing entry must be symmetric.
For this symmetric entry, the revenue-maximizing auction defined in Proposition 7 becomes
a second price auction with a reserve price equal to the seller’s value and a symmetric entry fee.
Similar to Lemma 5, we can show that this auction induces a unique entry equilibrium when
the condition
G(c1 )−G(c2 )
c1 −c2
<
1
(Z1 −Z0 )−(Z2 −Z1 ) ,
∀c1 , c2 holds. 2
32
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