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\newtheorem{proposition}{Proposition}
\newtheorem{example}{Example}
\author{Flavio M. Menezes \\
%EndAName
Department of Economics\\
Faculty of Economics and Commerce\\
Australian National University\\
Canberra, ACT 0200, Australia\\
Email: Flavio.Menezes@anu.edu.au \and Paulo K. Monteiro \\
%EndAName
IMPA\ - Instituto de Matem\'atica Pura e Aplicada\\
Estrada Dona Castorina 110, Jardim Bot\^anico\\
Rio de Janeiro, RJ, CEP\ 22460, Brazil\\
Email: pklm@impa.br}
\title{A Note on Auctions with Endogenous Participation\thanks{
\qquad Flavio Menezes acknowledges the financial assistance from the
Australian National University and from IMPA/CNPq (Brazil). }}
\date{20 September 1996 }
\maketitle
\begin{abstract}
In this paper, we study an auction where bidders only know the number of
potential applicants. After seeing their values for the object, bidders
decide whether or not to enter the auction. Players may not want to enter
the auction since they have to pay participation costs.
We characterize the optimal bidding strategies for both first- and
second-price sealed-bid auction when participation is endogenous. We show
that only bidders with values greater than a certain cut-off point will bid
in these auctions. In this context, both auctions generate the same expected
revenue.
We also show that, contrarily to the predictions of the fixed-$n$
literature, the seller's expected revenue may decrease when the number of
potential participants increases. In addition, we show that it is optimal
for the seller to charge an entry fee, which contrasts greatly with results
from the existing literature on auctions with entry.
\end{abstract}
\section{Introduction}
In the standard Independent Private Values (IPV) \ auction model it is
assumed that bidders submit bids knowing how many opponents they will face.
This assumption is clearly not appropriated for sealed-bid auctions, where
participants may know the pool of potential applicants, but by definition
cannot know how many bidders have submitted bids (unless this information is
released by the auctioneer). In oral auctions, even though bidders have a
more precise idea about the number of competitors, they only know who has
bid before them and not who will bid in the future.
We departure from this hypothesis and examine an auction where bidders only
know the number of potential applicants. After seeing their values for the
object, bidders decide whether or not to enter the auction. Players may not
want to enter the auction since they have to pay participation costs. (In
the context of procurement auctions, participation costs can be interpreted
as the costs of preparing a bid.)
We characterize the optimal bidding strategies for both first- and
second-price sealed-bid auctions when participation is endogenous. We show
that only bidders with values greater than a certain cut-off point will bid
in these auctions. In this framework, both auctions generate the same
expected revenue.
Contrarily to the prediction of the fixed-$n$ literature, the seller's
expected revenue may decrease when competition increases: when the number of
potential participants increases so does the cut-off value. Thus, for some
distributions of values, it may be optimal for the seller to engage in
policies to reduce the number of potential participants. In addition, we
show that it is optimal for the seller to charge an entry fee. This
contrasts greatly with results from the existing literature on auctions with
entry.
This paper is organized as follows. In the next section we provide a brief
survey of auction models which consider entry. In section 3, we specify the
model and the notation, derive the optimal bidding strategies and compute
the expected revenue for both first- and second-price auctions. In section
4, we compare the expected revenue generated by the two auction formats and
provide some new insights on the effects of increased competition and entry
fees on expected revenue. In section 5, we examine the case when bidders
learn the number of opponents before submitting their bids. Section 6
concludes.
\section{Auctions with Entry:\ A\ Brief Survey}
The question of entry in auctions has been addressed before. For instance,
McAfee and McMillan (1987-a) consider a model where participants have to
incur a cost before learning their values for the object. These authors use
the revelation principle to show that a first-price auction induces the
optimal number of bidders to enter. Engelbrecht-Wiggans (1987, 1993) also
examines auctions with entry, assuming that potential bidders use pure entry
strategies. That is, if only $n<N$ bidders enter the auction, then an
asymmetric equilibrium is considered in which $N-n$ bidders stayed out. Note
that neither McAfee and McMillan nor Engelbrecht-Wiggans identify the
process by which identical bidders are divided into those who participate
and those who do not participate in the auction.
In contrast, Levin and Smith (1994) explicitly consider a mechanism by which
individuals decide whether or not to participate. Their model is different
from ours in at least two dimensions. First, bidders incur a fixed cost of
entry before seeing their values for the object. Second, a bidder who enters
the auction knows how many opponents she faces. In their model, if there are 
$N$ potential bidders, but only $n<N$ could enter the auction and make
nonnegative profits, then each potential bidder enters with probability $q$
and stays out with probability $1-q$. In equilibrium, $n$ varies
stochastically between $0$ and $N$ with probabilities exogenously determined
by the auction format. For the IPV model they provide a revenue-equivalence
theorem. They also show that in the IPV auctions with entry, the seller
should not set a reservation prices or charge entry fees, since the seller
has no reasons to discourage entry.
As Levin and Smith, we also explicitly consider a mechanism by which
individuals decide whether or not to participate. We characterize the
optimal bidding strategies when agents learn their values before deciding
whether or not to incur a bid preparation cost - a very reasonable
assumption in procurement auctions for example. We consider two cases,
namely, when bidders know how many opponents they face and when they do not
know. As Levin and Smith, we find that, for risk-neutral bidders, the
expected revenue does not depend on whether or not the seller reveals the
actual number of bidders.
Moreover, since in our model bid preparation costs screen low valuation
bidders, entry fees are optimal and increased competition may actually
reduce expected revenue. This should be contrasted to the unrestricted-entry
result in Levin and Smith. The difference between the two models lies on the
fact that, in our model, the participation decision is such that only
induces high types to participate.
Our results should be seen as complementary to those of Levin and Smith.
While we study the effects of entry when agents learn their values prior to
incurring bid preparation costs, they examine the case where agents have to
incur the cost prior to learning their values. One can think of examples of
auctions where our model is more appropriate and vice-versa.
A different class of model is examined by Harstad et al. (1990), Matthews
(1987) and McAfee and McMillan (1987-b) who have made the number of bidders
random, with a known distribution. The last two papers adopt the IPV model,
and analyze effects of bidder risk aversion across auction institutions. The
institutional choice is whether the seller reveals the number of bidders,
known by him in advance. Their analysis is indirect in the sense that
equilibrium bid functions are not derived. For a first-price auction with
bidders having constant absolute risk aversion, McAfee and McMillan show
that the expected selling price is higher when bidders do not know how many
other bidders there are than when they do know. Thus, the seller should
conceal the number of bidders. Matthews has derived a similar conclusion
when bidders have decreasing absolute risk aversion. Harstad et al. retain
the hypothesis of risk-neutrality and investigate a general independent
model which allows for asset value uncertainty. Assuming that the
probability of participation is fixed, they show that the following five
auctions are revenue-equivalent: FPSB and SPSB auctions, each with the
number of bidders known or uncertain, and oral auctions. We replicate their
results for the case of endogenous participation.
Finally, our model is very similar to that of Samuelson (1985), who
considers an indirect approach for a generic mechanism satisfying some
properties and finds that equilibrium entry achieves a welfare optimum but
that increased competition may cause the expected revenue to fall. In
contrast, we characterize equilibrium bidding strategies, including the
cut-off value, for both first-price and second-price sealed-bid auctions
when agents do not know the number of opponents they face and when they do.
We compute the expected revenue generated by the two auctions and show that
they are equal. We obtain a similar result to that of Samuelson with respect
to the effects of increased potential competition and provide a new result
with respect to entry fees. The advantage of our direct approach of
characterizing bidding strategies is to allow empirical tests of auction
theory with entry and to provide proper theoretical foundations for
laboratory experiments.
\section{The Model}
We consider the sale of a single indivisible good through a sealed-bid
auction. We assume that the reserve price is equal to zero. We denote by $I$
the finite set of potential risk neutral participants, with $\#I=n\in {\em N}
$, where $\#$ denotes the cardinality of the set. Accordingly to the IPV
assumption, Bidder $i\in I$ knows her own value $(v_i)$ for the object but
only knows the distribution $F(v_j),$ $\forall j\neq i,$ of other bidders'
values. It is assumed that values are independently drawn from the
continuous distribution $F\,$with support $[0,\overline{v}]$.
Bidders face participation costs $c,$which might be interpreted as the costs
of preparing a bid. Given their values, each bidder decides whether or not
to submit a bid (and pay $c$) without knowing how many bidders will submit
bids. In what follows, we characterize the individual participation decision
and derive the optimal bidding strategy for both first- and second-price
sealed-bid auctions.
\subsection{First-Price Sealed-Bid Auctions}
To derive the optimal bidding strategy for Bidder $i$, let us define first $%
i $ 's expected profits from participation: \\(1) $\pi
_i(v_i,b_i,b_{j,\text{ }j\neq i})=$
$\dsum\limits_{H\subset I\setminus \{i\}}(v_i-b_i)E\left[ \chi _{_{_{b_i>%
\stackunder{j\in H}{\max \text{ }}b_j(v_j)}}}\text{ }\chi _{_{v_j>v_{\rho
_{_F}},\text{ }\forall j\in H}}\chi _{_{v_j<v_{\rho _F},\text{ }\forall j\in
(I\setminus \{i\}-H)}}\right] -c$ \\Where $b_i$ denotes $i$'s bid,
and $b_j$, Player $j$'s bidding strategy, for $j\neq i.$ $H$ denotes the set
of participants and $v_\rho $ is such that $\pi _i(v_\rho ,b^{*})=0$ in a
first-price sealed-bid auction$,$ i.e., the cut-off value when all bidders
use the equilibrium strategy $b^{*}$.\footnote{%
In the appendix we show that such strategy indeed exists. Note that Levin
and Smith (1994), for example, assume the existence of a unique increasing
symmetric Nash equilibrium bidding function.} Equation (1) states that
Player $i$'s expected profits is equal to the difference between her value
and her bid times the probability that she wins with a bid equal to $b_i(.)$%
. For every possible set of additional participants $H$, this probability is
simply the probability that $i$'s bid is greater than the maximum bid among
the $\#H$ other participants. In addition, Player $i$ must take into account
the fact that, in equilibrium, other bidders will participate if and only if
their value are greater than the cut-off point $v_\rho $. This cut-off point
is such that if a participant has a value $v_\rho ,$ then she is indifferent
between entering and not entering the auction. Thus, $v_\rho $ solves\bigskip%
\ \\(2) $v_\rho F(v_\rho )^{n-1}-c=0$ \\where $v_\rho $ denotes her
profits conditional on winning and given that her bid is equal to zero and $%
F(v_\rho )^{n-1}$ the probability that she wins with a zero bid. That is, a
participant with value $v_\rho $ wins only if she is the sole participant.
If we assume that everyone else except Player $i$ uses the same strategy $b$%
, we can rewrite (1) as follows: \\(3) $\pi
_i(v_i,b_i,b)=(v_i-b_i)\dsum\limits_{k=0}^{n-1}\left[ \left( F(v_\rho
)\right) ^{n-1-k}C_{n-1}^k\left( F(b^{-1}(b_i))-F(v_\rho )\right) ^k\right]
-c$ \\From the binomial expansion formula we have: \\(4) $%
\pi _i(v_i,b_i,b)=(v_i-b_i)\left[ F(v_\rho )+F(b^{-1}(b_i))-F(v_\rho
)\right] ^{n-1}-c$ \\Thus, maximizing $\pi _i$ with respect to $b_i 
$ yields:\\(5) $b^{*}(v)=\left\{ 
\begin{array}{l}
\dfrac{\dint\limits_{v_\rho }^v(n-1)xF(x)^{n-2}f(x)dx}{F(v)^{n-1}},\text{ }%
v\geq v_\rho \\ 
0,\text{ }v<v_\rho
\end{array}
\right. $\\According to our definition of $v_\rho ,$ we choose the
normalization $b(v_\rho )=0.$ \footnote{%
It is straightforward to check that $b(v_\rho )=0$ is the correct
normalization. Consider the expected profits of a player who has a value $%
v\leq v_\rho .$ If this player enters the auction and bids zero, her
expected profits are equal to $\pi (v,0)=vF(v_\rho )-c\leq v_\rho F(v_\rho
)-c=0,$ where the non-participation profits are equal to 0. On the other
hand, if a player has value $v\geq v_\rho ,$ then we can prove that her
expected profits from participation are greater than zero. In this case,\\$%
\pi (v,b(v))=\left( v-b(v)\right)
F(v)^{n-1}-c=vF(v)^{n-1}-c-(n-1)\int\limits_{v_\rho }^vxF(x)^{n-2}f(x)dx\geq 
$\\$vF(v)^{n-1}-c-(n-1)v\int\limits_{v_\rho
}^vF(x)^{n-2}f(x)dx=vF(v)^{n-1}-c-(n-1) v \left( \dfrac{F(v)^{n-1}-F(v_\rho
)^{n-1}}{n-1}\right)\\ = vF(v_\rho )^{n-1}-c = vF(v_\rho )^{n-1}-v_\rho
F(v_\rho )^{n-1}\geq 0.$}
Next we compute the expected revenue generated by the first-price sealed-bid
auction when bidders face participation costs. The expected revenue, $R^1$,
is simply the expected value of the highest bid among those players who
decide to participate:\\(6) $R^1=\dsum\limits_{k=1}^nC_n^k\left(
F(v_\rho )\right) ^{n-k}E\left[ b^{*}(v_1\vee ...\vee v_k)\chi _{_{v_j\geq
v_\rho ,\text{ }1\leq j\leq k}}\right] $\\Since this is a symmetric
problem, we can rewrite (6) as follows:\\(7)$R^1=\dsum%
\limits_{k=1}^nC_n^k\left( F(v_\rho )\right) ^{n-k}kE\left[
b^{*}(v_1)(F(v_1)-F(v_\rho ))^{k-1}\chi _{_{v_1\geq v_\rho }}\right] $%
\\Taking the expected value we obtain:\\(8) $%
R^1=\dint\limits_{v_\rho }^{\overline{v}}b^{*}(x)\dsum\limits_{k=1}^nC_n^k%
\left( F(v_\rho )\right) ^{n-k}k(F(x)-F(v_\rho ))^{k-1}f(x)dx$
\\Since $\dsum\limits_{k=1}^nkC_n^k\left( F(v_\rho )\right)
^{n-k}\left( F(x)-F(v_\rho )\right) ^{k-1}=n(F(v_\rho )+F(x)-F(v_\rho
))^{n-1},$ we can write (8) as:\\(9) $R^1=\dint\limits_{v_\rho }^{%
\overline{v}}b^{*}(x)nF^{n-1}(x)f(x)dx$\\From the previous
expression, we can conclude that allowing participation to be endogenous
reduces the auctioneer's revenue given a fixed population of potential
bidders. This is, per se, quite straightforward. The revenue comparison and
the other results of the next section are not. We now solve the individual
problem and compute the expected revenue when a second-price sealed-bid
auction is used.
\subsection{Second-Price Sealed-Bid Auctions}
It is not difficult to show that the optimal strategy in a second-price
sealed-bid auction with endogenous participation is to bid the true value.
(The proof is identical to the case of a fixed number of participants.) In
the appendix we show that such a strategy is an equilibrium. Thus, we can
write Bidder $i$ 's expected profits in equilibrium as a function of her own
value and the value of the other opponents who participate:\\(10) $%
\pi _i(v_i,v_{j,j\neq i})=$
$\dsum\limits_{H\subset I\setminus \{i\}}E\left[ \chi _{_{v_j>v_{\rho _{_S}},%
\text{ }\forall j\in H}}\chi _{_{v_j<v_{\rho _S},\text{ }\forall j\in
(I\setminus \{i\}-H)}}(v_i-\stackunder{j\in H}{\max }v_j)^{+}\right] -c$%
\\Where $v_{\rho _s}$ is the cut-off value for the second-price
auction$.$ Since this is a symmetric problem, we have:\\(11) $\pi
_i(v_i,v_{j,j\neq i})=v_i\left( F(v_{\rho _s})\right) ^{n-1}-c+$
$\dsum\limits_{k=1}^{n-1}C_{n-1}^k\left( F(v_{\rho _s})\right)
^{n-k-1}kE_{v_i}[(v_i-v_{_2})\chi _{_{v_i\geq v_2\geq v_{\rho _{_S}}}}\left(
F(x)-F(v_{\rho _s})\right) ^{k-1}]$\\Taking the expected value we
obtain:\\(12) $\pi _i(v_i,v_{j,j\neq i})=v_i\left( F(v_{\rho
_s})\right) ^{n-1}-c+$
$(n-1)\dint\limits_{v_{\rho _{_S}}}^{v_i}(v_i-v_2)(F(v_2))^{n-2}f(v_2)dv_2$%
\\Integrating by parts yields:\\(13) $\pi _i(v_i,v_{j,j\neq
i})=v_{\rho _{_S}}\left( F(v_{\rho _s})\right)
^{n-1}-c+\dint\limits_{v_{\rho _{_S}}}^{v_i}(F(x))^{n-1}dx$\\The
cut-off point in the case of second-price auctions is obtained by solving
the following equation:\\(14) $v_{\rho _{_S}}\left( F(v_{\rho
_s})\right) ^{n-1}=c$\bigskip 
\\Therefore, $v_{\rho _{_S}}=v_\rho .$ We now compute the auctioneer's
expected revenue, $R^2$. It is simply the expected value of the second
highest valuation among those who participate $(Y_H^{(2)})$:\\(15) $%
R^2=E\left[ \sum_{H\subset I,\text{ }\#H\geq 2}Y_H^{(2)}\chi _{\stackunder{%
v_i<v_\rho \text{ },\text{ }i\notin H}{_{v_i\geq v_\rho \text{ },\text{ }%
i\in H}\text{ }}}\right] $\\By symmetry we can write:\\(16) $%
R^2=\dsum\limits_{k=2}^nk(k-1)C_n^k\left( F(v_\rho )\right) ^{n-k}E\left[
(1-F(v_2))v_2\left( F(v_2)-F(v_\rho )\right) ^{k-2}\text{ }\chi
_{_{v_2>v_\rho }}\right] $\\Taking the expected value, writing the
sum inside the brackets and using a binomial expansion yields:\\(17) 
$R^2=n(n-1)\dint\limits_{v_\rho }^{\overline{v}}(1-F(x))x(F(x))^{n-2}f(x)dx$%
\\The expected revenue generated by a second-price auction is
trivially smaller than in the case of a fixed number of bidders equal to $n$%
. In the next section we compare the expected revenue from the two auctions
when participation is endogenous and obtain some insights into the effects
of increasing the number of potential participants and charging entry frees.
\section{Revenue Comparison, Effects of increased Competition and Entry Fees}
We are now in a position to provide a direct proof of the revenue
equivalence between the two auction formats.
\begin{proposition}
In an IPV model with a fixed number of potential players where participation
is endogenous, first-price and second-price sealed-bid auctions generate the
same revenue.
\end{proposition}
\begin{description}
\item[Proof:]  Define $h(v_\rho )=R_1-R_2=\dint\limits_{v_\rho }^{\overline{v%
}}b^{*}(x)nF^{n-1}(x)f(x)dx-n(n-1)\dint\limits_{v_\rho }^{\overline{v}%
}(1-F(x))x(F(x))^{n-2}f(x)dx.$\\Notice that $h(0)=0.$ Moreover,%
\\$h^{\prime }(v_\rho ))=\dint\limits_{v_\rho }^{\overline{v}}\dfrac{%
(n-1)}{\left( F(x)\right) ^{n-1}}nv_\rho \left( F(v_\rho )\right)
^{n-2}f(v_\rho )\left( F(x)\right) ^{n-1}f(x)dx-$\\$nb(v_\rho
)\left( F(v_\rho )\right) ^{n-2}f(v_\rho )+n(n-1)\left( 1-F(v_\rho )\right)
v_\rho \left( F(v_\rho )\right) ^{n-2}f(v_\rho )=0.\Box $
\end{description}
The above result can be explained in the context of the Revelation
principle: we know that if the two mechanisms yield the same allocation of
the object - in this case the high type wins, and if this type at least
equals a reservation type, then the two mechanisms should generate the same
revenue. As in the case we examine above the reservation type is the same
for both auctions, the revelation principle can be applied to prove revenue
equivalence.
The advantage of our direct approach of characterizing bidding strategies is
to allow empirical tests of auction theory with entry and to provide proper
theoretical foundations for laboratory experiments. Moreover, this direct
approach allows us to obtain some new insights on the effects of increased
potential competition and entry fees on IPV auctions.
The effect of increased competition in a standard fixed-$n$ IPV auction is
straightforward: expected revenue increases as $n$ increases. Thus, the
seller has the incentive to use policies - e.g., advertisement, to boost the
number of bidders attending the auction. When bidders face bid preparation
costs, however, there is a potential trade-off between the number of
potential participants and expected revenue since the cut-off value
increases as $n$ increases. The intuition for the possible existence of a
trade-off is as follows. Consider the effect of raising the number of
potential players from $n$ to $n+1$; we are comparing the second-order
statistics of $n+1$ draws from a fixed distribution truncated at some point $%
\bar{x}$ with the second-order statistics of $n$ draws from the same
distribution truncated at some point $\bar{y}<\bar{x}$. The next examples
will clarify this point.
\begin{example}
Let's assume there are $n$ potential players with values uniformly
distributed on the interval [0,1]. In this case, $v_\rho =c^{\frac 1n}$ and
the expected revenue given by equation (17) can be rewritten as: \\$\Psi (n)=%
\dfrac{n-1}{n+1}\left( 1-(n+1)c+nc^{\frac{n+1}n}\right) $\\Thus, \\$\Psi
^{\prime }(n)=\dfrac 2{(n+1)^2}\left( 1-(n+1)c+nc^{\frac{n+1}n}\right) +%
\dfrac{n-1}{n+1}\left( -c+c^{\frac{n+1}n}-\frac 1nc^{\frac{n+1}n}\log
c\right) $\\Let $g(c)=\Psi ^{\prime }(n).$ Note that $g(1)=0$. The first and
second derivative of $g$ are given by:\\$g^{\prime }(c)=-1+c^{\frac
1n}\left( 1-\dfrac{n-1}{n^2}\log c\right) $\\$g^{\prime \prime }(c)=c^{\frac{%
1-n}n}(\frac 1{n^2}-\frac{n-1}{n^3}\log c)\geq 0$\\We can then conclude that 
$g^{\prime }(c)\leq g^{\prime }(1)=0$ and $g(c)\geq g(1)=0.$ That is, we
have shown that, when values are uniformly distributed on [0,1], the
expected revenue increases in $n$ for any $c$ between 0 and 1.
\end{example}
\begin{example}
Suppose that the $n$ players are represented by random draws from the
distribution $F(x)=x^a,a>0,$ $x\in [0,1]$. The cutoff $v_\rho $ for this
distribution when there are $n$ potential participants is $v_\rho =c^{\frac
1{1+a(n-1)}}.$ The expected revenue is (after simplification): 
\[
R_n=n(n-1)\dint\limits_{c^{\frac 1{1+a(n-1)}}}^1a(1-x^a)x^{a(n-1)}dx 
\]
Letting $y=x^a$ yields: 
\[
R_n=\dint\limits_{c^{\frac a{1+a(n-1)}}}^1n(n-1)y^{\frac 1a+n-2}(1-y)dy. 
\]
The limit of $R_n$ as $a\rightarrow \infty $ is: 
\[
r_n=\dint\limits_{c^{\frac 1{n-1}}}^1n(n-1)y^{n-2}(1-y)dy=1-nc+(n-1)c^{\frac
n{n-1}}. 
\]
Thus 
\[
r_n-r_{n+1}=c+(n-1)c^{\frac n{n-1}}-nc^{\frac{n+1}n}=c\left( 1+(n-1)c^{\frac
1n}-nc^{\frac 1n}\right) . 
\]
Therefore $r_n>r_{n+1}$ if and only if $g(c)=\left( 1+(n-1)c^{\frac
1n}-nc^{\frac 1n}\right) >0.$ Since $g\prime (c)=c^{\frac 1n-1}-c^{\frac
1n-1}=c^{\frac 1n-1}(c^{\frac 1{n-1}-\frac 1n}-1)<0$ we have that $%
g(c)>g(1)=0$ if $c<1$. Hence we conclude that for sufficiently large $a$ and
cost $c<1$ the expected revenue decreases when the number of potential
participants increases. For example, if $a=7$ and $c=0.1,$ $r_n$ is
maximized when $n=4$.
\end{example}
The first example conforms with the fixed-$n$ literature, where increased
competition increases revenue. (e.g., Holt,1979.) In the second example,
however, increased potential competition decreases expected revenue because
its effect on raising the cut-off value dominates the effect of having a
larger pool of participants. In this case, it may be optimal for the seller
to engage in policies that limit the number of potential participants. This
is summarized by the following proposition.
\begin{proposition}
In a IPV auction model with endogenous participation, the expected revenue
may decrease when the number of potential participants increases.
\end{proposition}
Next, we consider a fixed number of potential participants and ask whether
is optimal to charge an entry fee or to establish a participation subsidy.
Formally, the problem facing the seller is to choose an entry fee/subsidy, $%
\delta ,$ so as to maximize total revenue:\\(18) $\varphi (\delta )=%
\stackunder{-c<\delta <1-c}{Max}\left( n(n-1)\dint\limits_{v_\rho (\delta
)}^{\overline{v}}(1-F(x))x(F(x))^{n-2}f(x)dx+n\delta \left( 1-F(v_\rho
(\delta ))\right) \right) $
\\Where $v_\rho (\delta )$ is defined so that $v_\rho (\delta
)\left( F(v_\rho (\delta ))\right) ^{n-1}=c+\delta $ and the last term in
the brackets denotes the expected revenue from the entry fees, i.e., the
entry fee times the expected number of actual bidders.
As one may recall from section 2, when entry is modelled as in Levin and
Smith (1994), the seller should never charge entry fees or use a reservation
price since unrestricted entry is optimal (Proposition 6 in Levin and Smith)
. In our model, however, since the bid preparation costs screen low
valuation bidders, it may be optimal to set either an entry fee or a
subsidy. We show next that charging an entry fee maximizes expected revenue.
\begin{description}
\item[Theorem.]  {\it In a IPV auction model with endogenous participation
it is optimal to charge an entry fee.}
\item[Proof:]  It suffices to show that $\delta ^{*}$ that maximizes the
problem defined in (18) is positive. Taking the derivative of $\varphi
(\delta )$ with respect to $\delta $ yields:\\$\varphi ^{\prime
}(\delta )=\dfrac{n\left( 1-F(v_\rho (\delta ))\right) f(v_\rho (\delta
))F(v_\rho (\delta ))^{n-1}-\delta f(v_\rho (\delta ))}{F(v_\rho (\delta
))^{n-1}+(n-1)f(v_\rho (\delta ))F(v_\rho (\delta ))^{n-2}}$\\This
expression is positive whenever $\delta $ $\leq 0.$ Thus, we conclude that $%
\delta ^{*}>0.$ $\Box $
\end{description}
The intuition again is in the nature of the trade-off between the effects of
an entry fee on raising the cut-off value and expected revenue as determined
by the underlying distribution. The next two examples help to illustrate
this point.
\begin{example}
As in example 1, let's assume there are $n$ potential players with values
uniformly distributed on the interval [0,1]. Expected revenue is given by:%
\\$\dfrac{n-1}{n+1}\left( 1-(n+1)(c+\delta )+n(c+\delta )^{\frac{n+1}%
n}\right) +n\delta \left( 1-(c+\delta )^{\frac 1n}\right) $\\When $%
n=10$ and $c=.2$, the optimal entry fee is $\delta ^{*}=.0367.$
\end{example}
\begin{example}
Suppose that the $n$ players are represented by random draws from a
distribution $F(x)=x^4,0<x<1.$expected revenue given by\\$%
4n(n-1)\left( \dfrac{1-c-\delta }{4n-3}-\dfrac{1-(c+\delta )^{\frac{4n-1}{%
4n+3}}}{4n+1}\right) +n\delta \left( 1-(c+\delta )^{\frac{4n-1}{4n+3}%
}\right) $\\If $n=10$ and $c=.4$, the optimal entry fee is $\delta
^{*}=.2271.$\ 
\end{example}
\section{Is it profitable for the seller to reveal the number of actual
bidders?}
We now modify the benchmark model by examining individual behavior when
bidders learn the number of opponents before submitting their bids. The
timing now is as follows. Individuals learn their values for the object;
based on this information and on how other players behave in equilibrium
they decide whether or not to enter the auction and pay the bid preparation
cost; bidders observe the number of opponents and then submit a bid. As in
Levin and Smith (1994), we find first- and second-price auctions to be
revenue-equivalent. Our proof, however, is direct in that we derive
equilibrium bidding strategies for both auctions. Moreover, as in Harstad,
Kagel and Levin (1990), who consider an auction where the number of bidders
is a random variable with a known distribution, we prove that the following
five auctions are revenue-equivalent: first-price and second-price auctions,
each with the number of bidders known or unknown, and English auctions.
Note that the individual behavior in a second-price auction does not change
when bidders learn how many opponents there are. Once they have paid their
participation costs, bidding the true values is still a dominant strategy.
Thus, the cut-off point is the same as in the previous section as it is the
expected revenue. On the other hand, individual behavior in a first-price
auction may be quite different, since the amount they shade depends on the
number of participants. Next, we derive the optimal strategy and the
expected revenue for FPSB auctions under this new informational assumption.
Bidder $i$ 's expected profits from participation is given by: \\%
(19) $\pi _i(v_i,b_i,b_{j,\text{ }j\neq i})=v_iF(\hat{v})^{n-1}+$
$\dsum\limits_{\emptyset \neq H\subset I\backslash
\{i\}}(v_i-b_i^H(v_i))\left( F(v_i)-F(\hat{v})\right) ^{\#H}F(\hat{v}%
)^{\#(I\backslash \{\{i\}\cup H\})}-c$ \\Where $\hat{v}$ denotes
the value that sets expected profits from participation equal to zero when
the equilibrium number of bidders enters the auction and $H$ denotes the set
of participants excluding player $i$. Note that the cardinality of $H$ is at
least equal to one. When Player $i$ faces no competition he bids zero. For a
given set $H$, Player $i$ chooses $b_i^H$ in order to maximize (19) yielding%
 \\(20) $b_i^H=v_i-\dfrac{\dint\limits_{\hat{v}}^{v_i}\left( F(y)-F(%
\hat{v})\right) ^{\#H}dy}{\left( F(v_i)-F(\hat{v})\right) ^{\#H}}$ %
\\When players bid knowing the set of participants, the amount of shading
changes accordingly. If we replace (20) into the individual decision (19)
and set it equal to zero to find $\hat{v},$ we obtain: \\(21) $\hat{%
v}F(\hat{v})^{n-1}=c.$ \\That is, the cut-off point now is
identical to the cut-off point for the second-price auction. Next we compute
the expected revenue, which can be written as: \\(22) $\overline{R}%
^1=$ $\dsum\limits_{k=2}^nC_n^kF(\hat{v})^{n-k}kE\left[ b_i^H\left( F(y)-F(%
\hat{v})\right) ^{k-1}\right] $ \\Replacing (22) into (20) yields:%
 \\(23) $\overline{R}^1=E\left[ v_i\dsum\limits_{k=2}^nC_n^kF(\hat{v%
})^{n-k}k\left( F(v_i)-F(\hat{v})\right) ^{k-1}- \\\dint\limits_{%
\overline{v}}^{v_i}\dsum\limits_{k=2}^nC_n^kF(\overline{v})^{n-k}k\left(
F(v_i)-F(\overline{v})\right) ^{k-1}dy\right] $ \\After writing
explicitly the expected value, integrating by parts and simplifying, we
obtain: \\(24) $\overline{R}^1=n(n-1)\dint\limits_{\hat{v}}^{%
\overline{v}}(1-F(x))x(F(x))^{n-2}f(x)dx$
This expression is identical to the expression obtained for the second-price
sealed-bid auction. As a corollary of proposition 1, this expression is also
identical to the expected revenue generated by a first-price sealed-bid
auction where bidders do not know how many opponents they actually face.
This can be summarized in the following proposition:
\begin{description}
\item[Proposition 3.]  {\it When participation is endogenous, the four
auctions are revenue-equivalent: first- and second-price sealed-bid
auctions, each with the number of bidders known and unknown.}
\end{description}
\subsection{Oral Auctions}
In oral auctions players typically observe the number of opponents. In our
formulation, if we allow bidders to submit the number of bids they want and
observe the number of opponents at each time (natural assumptions in the
case of oral auctions), then we claim that the oral auction will generate
the same expected revenue as the first-price sealed-bid auction.\footnote{%
Note that this characterization of oral auctions is different from the one
introduced by Milgrom and Weber (1982), who consider a button auction:\ the
last player to remove her finger from the button wins. Once a bidder removes
her finger from the button, she cannot go back to the auction.} We will not
present a formal proof, but the argument is quite straightforward. A bidder
in a oral auction submits, for every subset $H$, a bid that is at most equal
to the last bid plus a small epsilon. (The epsilon is of course determined
by the auctioneer - who calls the bids in discrete intervals.) But this is
how individuals bid in first-price auctions. As a consequence, the Revenue
Equivalence theorem holds when agents are allowed to observe the number of
participants before submitting their bids and the last proposition can be
extended to include oral auctions.
\section{Conclusion}
We examined individual behavior and the expected revenue in first and
second-price sealed-bid auctions when participation is endogenous. We found
that in a IPV model, when agents decide whether or not to enter the auction
after seeing their values and without knowing the exact number of opponents
they will face - a natural assumption in procurement auctions, the Revenue
Equivalence Theorem holds.
This result can be explained in the context of the Revelation principle: We
know that if the two mechanisms yield the same allocation of the object - in
this case the high type wins, and if this type at least equals a reservation
type, then the two mechanisms should generate the same revenue. The
revelation principle can be applied to prove revenue equivalence in the case
we examine since the reservation type is the same for both auctions. The
advantage of our direct approach of characterizing bidding strategies is to
allow empirical tests of auction theory with entry and to provide proper
theoretical foundations for laboratory experiments. Moreover, this direct
approach allows us to obtain some new insights on the effects of potential
competition and entry fees on IPV auctions.
Analogously to Samuelson (1985), and contrarily to the fixed-$n$ literature,
we found that the expected revenue may or may not increase with increased
potential competition. The nature of the possible trade-off between expected
revenue and the number of potential participants can be understood in the
following way. When examining the effects of increasing the number of
potential bidders from $n$ to $n+1$, we are comparing the second-order
statistics of $n$ independent draws from a fixed distribution truncated at a
point $\bar{y}$ with the second-order statistics of $n+1$ independent draws
from the same distribution truncated at a point $\bar{x}>\bar{y}$. Thus, the
existence of the trade-off depends on the distribution of bidders' values.
In contrast to existing entry models, we show that charging an entry fee is
optimal from the seller's perspective. When entry decisions occur prior to
bidders knowing their values, as in Levin and Smith, entry should not be
discouraged in a IPV model, since the private gains from further entry
corresponds exactly to that of the seller. In our model, since low
valuations bidders are screened, an entry fee increases total revenue.
We also prove that revealing the number of bidders does not raise the
seller's revenue. If participation is endogenous but agents know the number
of opponents when submitting their bids, then the expected revenue generated
by both sealed-bid auctions are equivalent. More specifically, the following
auctions are revenue-equivalent: first- and second-price sealed-bid
auctions, each with the number of bidders known and unknown, and oral
auctions.
Finally, the analysis for the case of correlated values remained to be
examined; it is of interest to determine whether the ranking of auction
formats by the amount of revenue they generate, as it was obtained by
Milgrom and Weber (1982), holds when participation is endogenous.
\begin{thebibliography}{99}
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in Auctions,'' {\bf Management Science 33(6)}, 763-770.
\bibitem{}  Engelbrecht-Wiggans, R., 1993, ``Optimal Auctions Revisited, '' 
{\bf Games and Economic Behavior 5}, 227-239.
\bibitem{}  Harstad, R. M., J. H. Kagel and D. Levin, 1990, ``Equilibrium
Bid Functions for Auctions with an Uncertain Number of Bidders,'' {\bf %
Economics Letters 33}, 35-40.
\bibitem{}  Holt, C. A., Jr., 1979, ``Uncertainty and the Bidding for
Incentive Contracts,'' {\bf American Economic Review 69(4)}, 697-705.
\bibitem{}  Levin, D. and J. L. Smith, 1994, ``Equilibrium in Auctions with
Entry, '' {\bf American Economic Review 84(3)}, 585-599.
\bibitem{}  Matthews, S., 1987, ``On Comparing Auctions for Risk-Averse
Buyers:\ A Buyer's Point of View,'' {\bf Econometrica 55}, 633-646.
\bibitem{}  McAfee, R. P. and J. McMillan, 1987-a, ``Auction with Entry,'' 
{\bf Economics Letters 23}, 343-347.
\bibitem{}  McAfee, R. P. and J. McMillan, 1987-b, ``Auctions with a
Stochastic Number of Bidders,'' {\bf Journal of Economic Theory 43}, 1-19.
\bibitem{}  Milgrom, P. R. and R. J. Weber, 1982, ``A Theory of Auctions and
Competitive Bidding,'' {\bf Econometrica 50}, 1089-1122.
\bibitem{}  Samuelson,\ W. F., 1985, ``Competitive Bidding with Entry
Costs,'' {\bf Economics Letters 17}, 53-57. \newpage\ 
\end{thebibliography}
\begin{center}
{\bf Appendix 1} 
\end{center}
(1) {\bf Proof of Existence of Optimal Bidding Strategy for First-Price
Auctions}
We will show that if players participate whenever their values are greater
than a certain cut-off point $\tilde{v}$, then there is a unique optimal
bidding strategy defined by: \\$b(v)=\left\{ 
\begin{array}{l}
\dfrac{\dint\limits_{\tilde{v}}^v(n-1)xF(x)^{n-1}dx}{F(v)^{n-1}},v\geq 
\tilde{v} \\ 
0,v<\tilde{v}
\end{array}
\right. .$  \\Let $b^{*}$ be a bid, set $v^i=v$ and define $%
I^i=I\backslash \{i\}.$ Agent $i$'s profits from participation are given by: %
 \\$\pi _i=(v-b^{*})\dsum\limits_{H\subset I^i}\Pr \left( v_j\geq 
\tilde{v},\text{ }\forall j\in H;v_j<\tilde{v},\text{ }\forall j\in
I^i\backslash H;b^{*}>\stackunder{j\in H}{\max }b(v_j)\right) -c$ \\%
The above expression can be reduced even further given the independence
assumption and the individual problem can be written as: \\$%
\stackunder{b^{*}}{max}$ $(v-b^{*})\left( F(b^{-1}(b^{*}))\right) ^{n-1}-c$%
 \\Since $g(z)=(v-z)(F(z))^{n-1}$ is such that $g(0)=-c,$ $g(\tilde{%
v})\leq -c,$ there is an interior maximum. The first order condition is
given by: \\$-\left( F(b^{-1}(b^{*}))\right)
^{n-1}+(x-b^{*})(n-1)\left( F(b^{-1}(b^{*}))\right) ^{n-2}F^{\prime
}(b^{-1}(b^{*}))(b^{-1})^{\prime }(b^{*})=0$ \\Let $b(y)=b^{*}$,
then the first order condition can be rewritten as: \\$-\left(
F(y)\right) ^{n-1}+$
$(v-b(y))(n-1)\dfrac{\left( F(y)\right) ^{n-2}F^{\prime }(y)}{b^{\prime }(y)}%
=0$ \\Replacing the value of $b$ in the above expression and
rearranging we obtain that $y=v$ is indeed the maximum. $\Box $\newpage\ 
(2) {\bf Proof of Existence of Optimal Bidding Strategy for Second-Price
Auctions}
We will show that there exists a cut-off value $\overline{x}\in [0,\overline{%
v}]$ such that, for all $i\in I,$ the following strategy is a Nash
equilibrium: ``agent $i$ participates and bids his/her true value in the
auction if and only if $v_i\geq \overline{v}."$ We assume that $F(v)$ is
continuous and strictly increasing. To simplify the notation, define $%
I^i=I\backslash \{i\}$ as in the previous proof. Let's define the variable $%
p $ $\in [0,1].$ For a given realization $\varpi \in \Omega $ of the state
of nature, Player $i$'s expected profits are given by: \\$\pi
_i=E\left[ \dsum\limits_{H\subset I^i}\left( v_i(\varpi )-\stackunder{j\in H%
}{\max }v_j(\varpi )\right) ^{+}\dprod\limits_{j\in H}\chi _{F(v_j(\varpi
))\geq p}\dprod\limits_{j\in I^i\backslash H}\chi _{F(v_j(\varpi
))<p}\right] -c$ \\Thus, for $v_i(\varpi )=v$, define \\$%
\varphi _i(v,p)=$
$E\left[ \dsum\limits_{H\subset I^i}\left( v-\stackunder{j\in H}{\max }%
v_j(\varpi )\right) ^{+}\dprod\limits_{j\in H}\chi _{F(v_j(\varpi ))\geq
p}\dprod\limits_{j\in I^i\backslash H}\chi _{F(v_j(\varpi ))<p}\right] -c$%
 \\Then, we define the function $g:[0,\overline{v}]$ $\rightarrow
[0,1]$ as follows: \\$g(p)=\left( \Pr (\varphi _i(v,p)\leq
0)\right) _{i\in I}$ \\Since $g$ is continuous and increasing, the
Intermediate Value Theorem guarantees that there exists $\overline{p}$ where 
$g(\overline{p})=\overline{p}$ such that $\varphi _i(v,p)=0.$ But $\overline{%
p}=\Pr (\varphi _i(v,p)\leq 0)=\Pr (v\leq \overline{v})=F(\overline{v}).\Box 
$
\end{document}