From:	SMTP%"Flavio.Menezes@anu.edu.au" 19-NOV-1996 03:23:58.22
To:	pklm@feunl.fe.unl.pt
CC:	
Subj:	pooled.tex

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Paulo,

Estou mandando a versao revisada do pooled.tex. Voce pode colocar no netec.
Eu deixei a referencia ao revelation principle (so tirei a palavra
straigtforward) pois esta ja em forma de uma conjectura...

Um abraco,

Flavio.





11:52 AM 11/18/96 +0100, you wrote:
>Alo Flavio. O arquivo que enviei ontem nao entrou na conta corretamente.
>Agora espero que sim. 
>Veja como esta a nossa profissao. Recebi uma copia do artigo de Che e I. GAle
>que trata do mesmo tipo de problema que eu trato com Page. E' claro que
>o artigo deles e' muito inferior servindo so para um monopolista com custo
>identicamente nulo e utilidades lineares e um unico bem. Pois bem esse 
>artigo ja esta aceito com agradecimentos ao editor Guy Laroque que era o
>editor da Econometrica quando submetemos. O nosso artigo nem foi aceito
>para apresentacao no congresso da sociedade econometrica nos EUA!
>Um abraco,
>Paulo.
>
>

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\begin{document}

\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\\
Phone: 61-6-2492651\\
Fax: 61-6-2495124 \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{Simultaneous Pooled Auctions\thanks{
\quad Flavio Menezes acknowledges the financial assistance from the
Australian National University and from IMPA/CNPq (Brazil). We thank G.
Mailath and seminar participants at ANU for helpful comments. The usual
disclaimer applies.}}
\date{05 November 1996}
\maketitle

\newpage\ 

\begin{center}
{\bf Abstract}
\end{center}

Suppose a seller wants to sell $k$ similar or identical objects and there
are $n>k$ potential buyers. Suppose that buyers want only one object. (This
is a reasonable assumption in the sale of condominiums or in the sale of
government-owned residential units to low-income families). In this case, we
suggest the use of a simultaneous auction that would work as follows.\
Players are asked to submit sealed bids for one object. The individual with
the highest bid chooses an object first; the individual with the second
highest bid chooses the next object; and this process continues until the
individual with the $k^{th}$ highest bid receives the last object. Each
individual pays the equivalent to his/her bid.\ \\When objects are
identical, we show that the proposed auction generates the same revenue as a
first-price sealed-bid sequential auction. When objects are perfectly
correlated, there is no known solution for sequential auctions, whereas we
can characterize bidding strategies for the proposed pooled auctions.
Moreover, the pooled auction is optimal since it satisfies a straightforward
generalization of the revelation principle (Myerson, 1981) to $k$ perfectly
correlated objects. Thus, if the first-price sequential auction is optimal
then it generates the same revenue as the pooled auctions. Otherwise, it
generates less revenue. Therefore, the first-price sequential auction
generates at most as much revenue than the pooled auction for identical and
perfectly correlated objects. In addition, the pooled auction may be easier
and cheaper to run, and bidders' strategies are simpler to compute since
there are no interdependencies between sales as in the case of sequential
auctions, i.e., the strategy space is smaller.\ \\{\bf Keywords:}
Simultaneous auctions; Revenue-equivalence; Condominium auctions.\ \\{\bf %
JEL Classification:}\ D44

\newpage\ 

\section{Introduction}

In some areas of the US and in most parts of Australia auctions are a
primary method for selling real estate. According to Kravets (1993), the
number of properties sold by auction ``is growing at a geometric rate.''
While auctions were previously viewed as a way to dispose of distressed
properties, the current view now is that real estate auctions are ``an
acceptable and profitable way for the real estate person to do business.''
(Sherman and Bussio (1994).) Although precise numbers are not available, it
has been said that real estate sold at auction in 1988 reached more than
\$2.5 billion. (Chicago Tribune, November 13, 1988.)

This expansion in the use of auctions to sell real estate has been
accompanied by a diversification of the auction terms being used. Kravets
(1993) distinguishes between pooled unit bidding\footnote{%
Similar properties form a lot and the highest bidder chooses any one
property in the pool.} and sequential bidding\footnote{%
Objects are auctioneed off one at at time in a pre-arrenged order.}. For
example, Vanderporten (1992) reports the use of particular pooled auction in
the sale of similar condominium units and of like-sized adjoining tracts of
land. In this auction, a lot is formed with all items for sale. Oral bids
are submitted with the highest bidder winning the right to choose one item
from among the objects in the lot. A second auction follows for the right to
choose another object from the remaining unclaimed objects. This procedure
is repeated until all objects are disposed.

Auctions can also be described by their rules of bidding. Sherman and Bussio
(1994), for example, list four types of rules; (i) public or oral auctions -
buyers attend the auction site and bid against each other until someone wins
(as in the auction reported by Vanderporten). The most popular of such
auctions is known as the English auction where the auctioneer requests bids
in an increasing order; (ii) sealed bid auctions - bids are mailed in, and
each buyer is notified by mail of the highest bid; (iii) spot bid auction -
buyers bring sealed bids to the auction site where the auctioneer announces
the winning bid; and (iv) negotiated sales - written and telephone offers
are taken by the auctioneer before the auction date. The highest offer is
accepted on the auction date.

Thus, one can think of various combinations between auction terms and rules
of bidding. However, not all combinations are used in practice.\footnote{%
For a survey of the theory and practice of real estate auctions, see Quan
(1994).} In this paper we explore the properties of a simultaneous pooled
auction that would work as follows. Suppose a seller wants to sell $k$
similar or identical objects and there are $n>k$ potential buyers. Suppose
buyers want only one object.\footnote{%
For condominium auctions of similar units, for example, this assumption is
realistic. (See Vanderporten (1992).) This assumption is also particularly
realistic in the sale of government-owned residential units to low-income
families.} Players are asked to submit sealed bids for the right to choose
an object. The individual with the highest bid chooses an object first; the
individual with the second highest bid chooses the next object; and this
process continues until all objects are sold.

When objects are identical, we show that the proposed auction generates the
same revenue as a first-price sealed-bid sequential auction.\footnote{%
For an analysis of sequential auctions of identical objects, see Weber
(1983).} When objects are perfectly correlated, however, there is no known
solution for sequential auctions,\footnote{%
In the next section we survey the existing papers on sequential pooled
auctions. There are no general results for these auctions either. The
reasons are analogous to the inexistence of results for sequential auctions,
namely, the interdependencies between sales. See the discussion in Menezes
and Monteiro (1995).} whereas we can characterize bidding strategies for the
simultaneous pooled auctions. Moreover, the pooled auction is optimal since
it satisfies a straightforward generalization of the revelation principle
(Myerson, 1981) to $k$ perfectly correlated objects. Thus, if the
first-price sequential auction is optimal then it generates the same revenue
as the pooled auction. Otherwise, it generates less revenue. Therefore, the
first-price sequential auction generates at most as much revenue as the
proposed pooled auction for identical and perfectly correlated objects. In
addition, the pooled auction may be easier and cheaper to run, and bidders'
strategies are simpler to compute since there are no interdependencies
between sales as in the case of sequential auctions, i.e., the strategy
space is smaller. Ultimately, the choice of the auction mechanism will be
determined by the comparison of expected revenue and expected costs. One of
the advantages of our direct approach of characterizing bidding strategies
is to allow empirical tests of the theory and to provide proper theoretical
foundations for laboratory experiments.

Notice that we are not arguing that auctions must be used instead of
traditional broker channels.\footnote{%
This issue is addressed, for example, by Mayer (1995) and Vandell and
Riddiough (1992). These authors emphasize a trade-off between the speed of
the sale and expected revenue. The underlying idea is that auctions are
faster than a traditional sale by a broker but generate less revenue because
a quick sale implies a poorer ``match'', on average, between property and
buyers.}Instead, we are pointing out that whenever a decision has been made
to use an auction, the proposed simultaneous pooled auction is an attractive
option. We have to add the following caveats. This method is particularly
appropriate when potential buyers want to buy only one unit, and are
reasonably indifferent between the units they buy, but expect to pay a lower
price for a less valuable unit. Moreover, units must be either identical or
individuals must agree on how to rank them. For example, in the sale of
apartments to first-time home-buyers, our model would be appropriate if
buyers agree that, {\it ceteris paribus}, first-floor apartments should be
cheaper than second-floor apartments, and so on. Thus, this method may be a
viable alternative to be used by the Resolution Trust Corporation or by the
various local housing trust funds in Australia in the disposition of
properties with the characteristics above.

This paper is organized as follows. In the next section we review the
existing research on (sequential) pooled auctions. Section 3 formalizes our
model of simultaneous pooled auctions, while in Section 4 we derive the
equilibrium bidding strategies. Section 5 contains the revenue comparison
for the case of identical objects and the expected revenue in the case of
perfectly correlated objects. We also develop in this section a comparative
statics exercise examining what happens to the expected revenue when the
degree of correlation between objects changes. Our conclusions are
summarized in Section 6.

\section{Literature Review}

Vanderporten (1992) was, to the best of our knowledge, the first author to
specifically analyse pooled or right-to-choose auctions. He examined
auctions of the type used to sell condominiums, where oral bids are
solicited by the auctioneer with the highest bidder winning the right to
choose one condominium among the ones being offered for sale. After the
choice is made, a second round follows for the right to choose another
condominium from the remaining units. The auction continues until all units
are sold.

Vanderporten considers an auction with two homes for sale and two potential
bidders. Each bidder wants to purchase only one of the two homes and will
drop out of the auction if that home is sold to the other bidder. A simple
discrete probability distribution (a binomial distribution) for the
bidders's values is considered. For a numerical example, it is shown that
the expected revenue to the seller from the pooled auction can never be
greater than that from a sequential English auction and may be as much as
14\% less. Moreover, the variance of the expected revenue is lower in the
sequential pooled auctions than in the sequential English auctions. The
author argues that this lower variance reduces the risk to the seller of
default by the buyer.

Vanderporten's model provides an insightful approach to sequential pooled
auctions. However, his conclusions may not hold under alternative assumption
(e.g., when the two objects are stochastically identical, i.e., when
valuations for the two objects are drawn from the same distribution; or when
there are more bidders than objects).

Gale and Hausch (1994) consider the sale of two stochastically identical
objects but allow a bidder's valuation to be dependent across objects.%
\footnote{%
That is, Buyer $i$ has valuations $x_i$ and $y_i$, which are privately known
and drawn independently from a strictly positive density function; $x_i$ and 
$y_i$ may be correlated, and $x_j$ and $y_j$ may be correlated, $i\neq j,$
but all other pairs such as $x_i$ and $x_j$ are independent.} They examine
two second-price sealed-bid auction formats: standard sequential auctions
and sequential pooled auctions. In their model, the two buyers decide
whether to enter prior to each auction. For the standard sequential auction,
if one of the buyers bids in the first auction and loses, then she wins the
second auction and pays zero. If none of the buyers enters the first
auction, then they bid their true valuations for the second object. There
are three possible equilibrium bids in the first auction, depending on the
relationship between the values of the two objects for a particular bidder:
(1) she does not enter the first auction; (2) she enters the auction and
submits a bid of zero ; or (3) she bids the difference between the first and
second object. A bidder may want to bid zero in order to guarantee the first
object (her least preferred object) for a price of zero in the case her
opponent does not participate in the first auction. Gale and Hausch refer to
this property of the equilibrium bid as bottom-fishing.

The analysis of sequential pooled auctions is more straightforward since the
loser of the first auction is guaranteed to win the second object for a
price equal to zero. Gale and Hausch show that both bidders submit a bid
equal to half of the difference between their values for the two objects in
the first auction. (The intuition is that a bidder will shade her bid in the
first auction by the expected profits from participating in the last
auction. In this context, If bidder $i$ loses the first auction, her
opponent is equally likely, by assumption, to choose either object. Then
Bidder $i$ wins the second object and pays zero. Therefore, the net surplus
from losing the first auction is equal to one half of the sum of the value
of the two objects and her bid in the first auction is equal to the
difference between her value and this net surplus.) As a result, the
sequential pooled auctions generate more revenue than the standard
sequential auctions whenever the latter exhibits declining expected revenue.

The above result depends crucially on the assumption of two bidders and two
objects. For example, if we have three bidders and two objects, the expected
revenue from the second pooled auction is different from zero. Moreover, by
bidding zero in the first round of the sequential auction, a bidder does not
hedge against not receiving her most-preferred object. In this case, bidders
may bid more aggressively in the first auction depending on the degree of
correlation between the two objects for each bidder. There are no general
results for the case of $k$ objects and $n$ bidders ($n>k$) and the
difficulties arise from the interdependencies between the results of the
auction. (Since first-round bids may provide information regarding bidders'
values, bidders' behavior may be very complex). In contrast, these
interdependencies between sales are - by definition - inexistent in the
simultaneous pooled auction that we examine in the next section.

\section{The Model.}

Suppose a seller wants to sell $k$ units to $n$ potential buyers ($n>k$).
Suppose, further, that each buyer only wants one object. For simplicity, we
consider the case where the seller's reserve price is zero. We assume that
objects are correlated as follows. If the value of the first object for
buyer $i$ is equal to $x\in [0,1]$ \footnote{%
This restriction is just for normalization purposes. Our results can be
generalized for any distribution with bounded support.}, then the value of
the second object is $V_2(x)$, the third $V_3(x),$... , the $k^{th}$ $%
V_k(x). $ This function is assumed to be nonincreasing in the following
sense: $x=V_1(x)\geq V_2(x)\geq ...\geq V_k(x)$. The analysis of this paper
can be generalized to the case of nondecreasing values and to the case of
players with distinct functions $V$. In our framework, each player knows his
own vector of values, but only knows the distribution of his opponents'
values. The value $x^i$ is draw from the distribution $F$ with continuous
density $f(x)>0$ and support $[0,1]$.

Each bidder submits only one bid and is allocated one object if the bid is
among the $k$ highest bids. The object he receives depends on the ranking of
his bid. The bidder with the highest bid receives his most-valued object and
so on. Thus, when submitting their bids, players must take into account the
possibility that they may receive any of the objects (or none if the bids
falls below the $k^{th}$ highest bid). Let's examine the game from the
perspective of an arbitrary player, say Player 1. Player $1$'s expected
profit given that his value is equal to $x$, he submits a bid equal to $l$,
and everyone else submits a bid $b(x_i)$, $\forall i\neq 1$, is equal to:\ \\%
\ (1) $\pi _1\left( x,l\right) =(V_1(x)-l)\Pr \left( l>\left( b(x_i)\right)
_{\forall i\neq 1}\right) +(V_2(x)-l)\Pr ($ largest bid among all $i\neq
1>l\geq $ second largest bid among all $i\neq 1)+...+(V_k(x)-l)\Pr (k-1^{th}$
largest bid among all $i\neq 1>l\geq $ $k^{th}$ largest bid among all $i\neq
1).$\ \\Let's set $g=b^{-1}$ and assume that $b$ is increasing and
differentiable (we will show that in the next section). Thus, we can rewrite
(1) as follows:\ \ \\(2) $\pi _1\left( x,l\right) =(V_1(x)-l)\Pr \left(
g(l)>\left( x_i\right) _{\forall i\neq 1}\right) +(V_2(x)-l)\Pr ($ largest
value among all $i\neq 1>g(l)>$ second largest value among all $i\neq
1)+...+(V_k(x)-l)\Pr (k-1^{th}$ largest value among all $i\neq 1>g(l)>$ $%
k^{th}$ largest value among all $i\neq 1).$\ \\Using symmetry the above
expression can be written as:\\(3) $\pi _1\left( x,l\right)
=(V_1(x)-l)\left( F(g(l))\right) ^{n-1}+(V_2(x)-l)(n-1)\left(
1-F(g(l)\right) \left( F(g(l)\right) ^{n-2}+$\ \\$%
...+(V_k(x)-l)C_{n-1}^{k-1}\left( 1-F(g(l)\right) ^{k-1}\left( F(g(l)\right)
^{n-k}$\ \ \\Expression (3) can be written more economically as:\ \ \ \\(4) $%
\pi _1\left( x,l\right) =\dsum\limits_{t=1}^k(V_t(x)-l)C_{n-1}^{t-1}\left(
1-F(g(l)\right) ^{t-1}\left( F(g(l)\right) ^{n-t}$\ \ \\The problem that our
arbitrary player faces is to choose $l$ (given $x$) to maximize the above
expression. This problem will be dealt with in the next section.

\section{Equilibrium Bidding Strategies.}

In the next proposition we characterize the equilibrium bidding strategy in
a symmetric equilibrium. First, however, we need the following lemma. Define 
$\Psi _t(x)=\left( 1-F(x)\right) ^{t-1}\left( F(x)\right) ^{n-t}.$%
\TeXButton{lemma}{\begin{lemma}}\ For any $l,1\leq l\leq n,$ and every $a\in
R^{l+1},a_{l+1}=0$: 
\[
\dsum\limits_{t=1}^la_tC_{n-1}^{t-1}\Psi _t^{\prime
}(x)=f(x)\sum_{t=1}^l(a_t-a_{t+1})(n-t)C_{n-1}^{t-1}\left( 1-F(x)\right)
^{t-1}\left( F(x)\right) ^{n-t-1}. 
\]
\TeXButton{endlemma}{\end{lemma}}\TeXButton{Proof}{\proof}The proof is easy
and will be omitted.\ \\{\bf Proposition 1}: {\rm In a symmetric
equilibrium, bidding strategies are given by}{\bf :} \ \\(5) $b(x)=\dfrac{%
\dint\limits_0^x\dsum\limits_{t=1}^kV_t(y)C_{n-1}^{t-1}\left( \left(
1-F(y)\right) ^{t-1}\left( F(y)\right) ^{n-t}\right) ^{\prime }dy}{%
\dsum\limits_{s=1}^kC_{n-1}^{s-1}(1-F(x))^{s-1}F(x)^{n-s}}$\ \ \ 

\TeXButton{Proof}{\proof} In the symmetric equilibrium, it must be the case
that for any $x$, $l$ is such that maximizes profits. Thus, we must have $%
\dfrac{\partial \pi _1}{\partial l}=0,$ i.e.(recall $\Psi _t$ definition and
(4)):\ \ \\(6)$\;\dsum\limits_{t=1}^k(V_t(x)-l)C_{n-1}^{t-1}\Psi _t^{\prime
}(g(l))g^{\prime }(l)-\dsum\limits_{t=1}^kC_{n-1}^{t-1}\Psi _t(g(l))=0.$\ \
\ \\We can now use the fact that in a symmetric equilibrium $l$ must be set
equal to $b$, and we can replace $g^{\prime }(l)$ in expression (6) by $%
\dfrac 1{b^{\prime }(x)}.$ Moreover, since by definition $g(l)=b^{-1}(l)$ ,
we can replace $g(l)$ by $x$. Condition (6) then becomes:\ \ \ \\(7) $%
\dsum\limits_{t=1}^k(V_t(x)-b(x))C_{n-1}^{t-1}\Psi _t^{\prime }(x)\frac
1{b^{\prime }(x)}=\dsum\limits_{t=1}^kC_{n-1}^{t-1}\Psi _t(x).$\ \ \ \\%
Notice that we can write this expression as a first-order differential
equation as follows: \ \ \\(8) $b^{\prime }(x)$ $\dsum%
\limits_{t=1}^kC_{n-1}^{t-1}\Psi
_t(x)+b(x)\dsum\limits_{t=1}^kC_{n-1}^{t-1}\Psi _t^{\prime }(x)=$\ $%
\dsum\limits_{t=1}^kV_t(x)C_{n-1}^{t-1}\Psi _t^{\prime }(x)$.\ \ \ \\We
have: 
\[
\begin{array}{c}
\left( b(x)\ \dsum\limits_{t=1}^kC_{n-1}^{t-1}\Psi _t(x)\right) ^{\prime
}=b^{\prime }(x)\dsum\limits_{t=1}^kC_{n-1}^{t-1}\Psi _t(x)+b(x)\
\dsum\limits_{t=1}^kC_{n-1}^{t-1}\Psi _t^{\prime }(x)= \\ 
\dsum\limits_{t=1}^kV_t(x)C_{n-1}^{t-1}\Psi _t^{\prime }(x)
\end{array}
\]
Integrating and choosing $b(0)=0$ we obtain $b(x)$. $\Box $

From inspection of (5), equilibrium bids are an average over the set of all
possible values so that bidders behave as if they were bidding for an
``average '' object. Of course, this auction is very different from a
standard single-object independent private value auction, since a bidder's
value for this average object depends on how other bidders behave. However,
this analogy will be useful later when we compare the expected revenue
resulting from this auction with the expected revenue from a standard
sequential auction when objects are identical.

Contrarily to standard auction theory models, the equilibrium bidding
function given by (5) may not be increasing in $x$. For example, when
objects are not identical, a player faces a potential trade-off: increasing
her bid increases the chances of receiving her most-valued object but it may
decrease her profits if she wins. The existence of the trade-off depends on
the difference between the values of the objects. \footnote{%
For example, consider the following example. Let $k=2$ and $n=3$. Let the
function $V(x)$ be given by: 
\[
\begin{array}{l}
V_1(x)=x \\ 
V_2(x)=\left\{ 
\begin{array}{ll}
x & 0\leq x\leq a \\ 
(2a-x)^{+} & a\leq x
\end{array}
\right. 
\end{array}
\]
If we set $a=1/10$ and use our equilibrium bidding strategy we obtain: 
\[
b(x)=\left\{ 
\begin{array}{lll}
\frac{x(3-2x)}{3(2-x)} &  & 0\leq x\leq a \\ 
\frac{-7/325+2x/5-7x^2/5+2x^3}{2x-x^2} &  & a\leq x\leq 2a \\ 
\frac{\frac 2{125}+2x^3/3}{2x-x^2} &  & 2a\leq x
\end{array}
\right. 
\]
Note that $b^{\prime }(2a)=-0.04.$ Therefore, (5) cannot be an equilibrium
bidding strategy for this example.} Thus, we need an additional assumption
to guarantee that any convex combination of the values for the objects is
increasing in $x$. In particular, we assume that $\dfrac{\partial V_t(x)}{%
\partial x}\geq 0.$ This condition simply says that the $t^{th}$ highest
value increases as $x$ increases. For example, in the case of a constant
degree of correlation (e.g., $t^{th}$ object is worth $\lambda $ times the
value of the $t-1^{th}$ object, $0<\lambda <1$, for any $1\leq t<k$) this
assumption is trivially satisfied. We then have the following proposition.\ %
\\ {\bf Proposition 2}: {\rm If\ }$\dfrac{\partial V_t(x)}{\partial x}\geq
0,t\geq 1${\rm \ then\ }$b^{\prime }(x)>0${\rm \ }$\forall x,x\in (0,1).$\ %
\\ {\bf Proof.} We can write $b(x)$ as:\ \ \\ (9) $b(x)=\left(
\dsum\limits_{t=1}^kC_{n-1}^{t-1}\Psi _t(x)\right) ^{-1}\left(
\dsum\limits_{s=1}^kV_sC_{n-1}^{s-1}\Psi
_s(x)-\dint\limits_0^x\dsum\limits_{s=1}^k\dfrac{\partial V_s(y)}{\partial y}%
C_{n-1}^{s-1}\Psi _s(y)dy\right) $.\ \\ Taking the derivative yields:\\ $%
b^{\prime }(x)=\left( \dsum\limits_{s=1}^kV_sC_{n-1}^{s-1}\Psi
_s(x)-\dint\limits_0^x\dsum\limits_{s=1}^k\dfrac{\partial V_s(y)}{\partial y}%
C_{n-1}^{s-1}\Psi _s(y)dy\right) \left( -\dfrac{\dsum%
\limits_{t=1}^kC_{n-1}^{t-1}\Psi _t^{\prime }(x)}{\left(
\dsum\limits_{t=1}^kC_{n-1}^{t-1}\Psi _t(x)\right) ^2}\right) +$

\ \\$\left( \dsum\limits_{t=1}^kC_{n-1}^{t-1}\Psi _t(x)\right) ^{-1}\left(
\dsum\limits_{s=1}^kC_{n-1}^{s-1}V_s(x)\Psi _s^{\prime }(x)\right) $\ \\By
the lemma, $\dsum\limits_{t=1}^kC_{n-1}^{t-1}\Psi _t^{\prime
}(x)=f(x)C_{n-1}^{k-1}(n-k)(1-F(x))^{k-1}F(x)^{n-k-1}>0$ if $x\in (0,1).$ By
our assumption,\ \ \\$\dint\limits_0^x\dsum\limits_{s=1}^k\dfrac{\partial
V_s(y)}{\partial y}C_{n-1}^{s-1}\Psi _s(y)dy\geq 0.$\ \ \\Thus, we can
write\ \\(10) $b^{\prime }(x)\left( \dsum\limits_{t=1}^kC_{n-1}^{t-1}\Psi
_t(x)\right) ^2>$\\$-\dsum\limits_{t=1}^k\dsum%
\limits_{s=1}^kC_{n-1}^{s-1}C_{n-1}^{t-1}V_s(x)\Psi _t^{\prime }(x)\Psi
_s(x)+\dsum\limits_{t=1}^k\dsum%
\limits_{s=1}^kC_{n-1}^{s-1}C_{n-1}^{t-1}V_s(x)\Psi _s^{\prime }(x)\Psi
_t(x) $\ \\If we define $w_k(t)=\dsum\limits_{t=1}^k\dsum%
\limits_{s=1}^kC_{n-1}^{s-1}C_{n-1}^{t-1}V_s(x)\left( \Psi _t\Psi _s^{\prime
}-\Psi _s\Psi _t^{\prime }\right) ,$ then it suffices to show that $%
w_k(t)\geq 0.$ We will use induction to prove that $w_k(t)\geq 0$. For $k=1$
we have:\ \ \\$w_1(x)=V_1(x)\left( \Psi _1\Psi _1^{\prime }-\Psi _1\Psi
_1^{\prime }\right) =0.$\ \ \\Before we proceed with the induction, we need
to compute $\left( \Psi _t\Psi _s^{\prime }-\Psi _s\Psi _t^{\prime }\right) $
for arbitrary $t$ and $s$: From\ \\(11)\ $\Psi _s^{\prime
}=(1-F)^{s-2}F^{n-s-1}f\left( (n-s)(1-F)-(s-1)F\right) $\ \\we obtain that:\
\ \ \\(12) $\Psi _t\Psi _s^{\prime }$ $=(1-F)^{t+s-3}F^{2n-s-t-1}f\left(
(n-s)(1-F)-(s-1)F\right) $\ \ \\(13) $\Psi _s\Psi _t^{\prime
}=(1-F)^{t+s-3}F^{2n-s-t-1}f\left( (n-t)(1-F)-(t-1)F\right) $\ \ \\Using
(11), (12) and (13) we obtain: \ \ \\(14) $\left( \Psi _t\Psi _s^{\prime
}-\Psi _s\Psi _t^{\prime }\right) =$ $f(x)\left( t-s\right)
(1-F)^{t+s-3}F^{2n-s-t-1}$\ \bigskip\ \\Thus, we can rewrite $w_k(t)$:\ \ \\%
(15) $w_k(t)=\dsum\limits_{t=1}^k\dsum%
\limits_{s=1}^kC_{n-1}^{s-1}C_{n-1}^{t-1}V_s(1-F)^{t+s-3}F^{2n-s-t-1}f\cdot
\left( t-s\right) $.\bigskip\ \ \\The induction step is to assume that $%
w_k(t)$ as given in (15) is greater or equal to zero for $k<n$ and check
whether $w_{k+1}(t)$ is also greater or equal to zero, where $k+1\leq n.$
Using (15) we can write:\ \ \\(16) $w_{k+1}(t)=\dsum\limits_{t=1}^{k+1}\dsum%
\limits_{s=1}^{k+1}C_{n-1}^{s-1}C_{n-1}^{t-1}V_s(1-F)^{t+s-3}F^{2n-s-t-1}f%
\cdot \left( t-s\right) $\ \ \\But this is equivalent to write:\ \ \\(17)\ $%
w_{k+1}(t)=\dsum\limits_{t=1}^k\dsum%
\limits_{s=1}^kC_{n-1}^{s-1}C_{n-1}^{t-1}V_s(1-F)^{t+s-3}F^{2n-s-t-1}f\cdot
\left( t-s\right) +$\ \\$\dsum%
\limits_{t=1}^kC_{n-1}^kC_{n-1}^{t-1}V_{k+1}(1-F)^{t+k-2}F^{2n-k-t-2}f\cdot
\left( t-k-1\right) +$\ \\$\dsum%
\limits_{s=1}^kC_{n-1}^kC_{n-1}^{s-1}V_s(1-F)^{k+s-2}F^{2n-k-t-2}f\cdot
(k+1-s)$\ \ \\That is,\ \bigskip\ \\$w_{k+1}(t)=w_k(t)+f(x)\dsum%
\limits_{t=1}^kC_{n-1}^kC_{n-1}^{t-1}(1-F)^{t+k-2}F^{2n-t-k-2}\left(
(t-(k+1))V_{k+1}+(k+1-t)V_t\right) $\ \ \\The above expression can be
reduced further to:\bigskip\ \\$w_{k+1}(t)=w_k(t)+f\dsum%
\limits_{t=1}^kC_{n-1}^kC_{n-1}^{t-1}(1-F)^{t+k-2}F^{2n-t-k-2}(k+1-t)(V_t-V_{k+1}) 
$\\The induction is completed since $(V_t-V_{k+1})$ is greater than zero by
assumption and $w_k(t)$ is greater than zero by the induction step. $\Box $

\section{The Revenue Comparison}

In this section we first show that when objects are identical, the proposed
auction generates the same revenue as a first-price sealed-bid sequential
auction where $k$ objects are sold one at a time. As in our model, bidders
only want one object. Thus, in each round, the remaining bidders (i.e.,
those who have not received an object) submit sealed bids. The highest
bidder is awarded an object and pays his bid. Each remaining bidder knows
the sale price. We use Weber's (1983) Theorem 2 that states that the
expected revenue in this sequential auction is given by $k$ $\cdot
E[x_{(k+1)}]$ , i.e., $k$ times the $(k+1)^{th}$ order statistics. The
expected revenue from a simultaneous pooled auction when objects are
identical is simply the expected value of the sum of the $k$ highest bids
(setting $V_t(x)=x,t=1,...,n,$ in $(5)$).

Let's denote by ${\bf R}_S$ the expected revenue in the sequential auction
and by ${\bf R}_P$ the expected revenue in the simultaneous pooled auction.
We then have the following proposition:\ \\{\bf Proposition 3}: {\rm If the
objects are identical, then }$R_S=R_P.$\ \\{\bf Proof:} \ The density of the 
$l^{\text{th}}$ greatest value is\ 
\[
f^l(x)=\sum_{u=0}^{l-1}C_n^u[(1-F(x))^u(F(x))^{n-u}]^{\prime },1\le l\leq n. 
\]
Therefore 
\begin{equation}
R_S=k\int x\sum_{u=0}^kC_n^u[(1-F(x))^u(F(x))^{n-u}]^{\prime }dx
\label{dois}
\end{equation}
and 
\begin{equation}
R_P=\int
b(x)\sum_{l=1}^k\sum_{u=0}^{l-1}C_n^u[(1-F(x))^u(F(x))^{n-u}]^{\prime }dx
\label{tres}
\end{equation}
\ Let us first consider (18). By the lemma with $n+1$ in place of $n$ and $%
l=k+1$ we have 
\begin{equation}
\begin{array}{c}
\sum_{u=0}^kC_n^u[(1-F(x))^u(F(x))^{n-u}]^{\prime
}=\sum_{u=0}^{k+1}C_n^{u-1}[(1-F(x))^{u-1}(F(x))^{n-u+1}]^{\prime }= \\ 
f(x)C_n^k(n-k)(1-F(x))^kF(x)^{n-k-1}
\end{array}
\label{qqq}
\end{equation}
Substituting this expression in (1) we obtain 
\[
R_S=k\int xf(x)C_n^k(n-k)(1-F(x))^kF(x)^{n-k-1}dx 
\]
Let us consider now the integrand of ${\bf R}_P.$ From (3) with $l-1$ in the
place of $k,$%
\[
\begin{array}{c}
\sum_{u=0}^{l-1}C_n^u[(1-F(x))^u(F(x))^{n-u}]^{\prime
}=f(x)C_n^{l-1}(n-l+1)(1-F(x))^{l-1}F(x)^{n-l}= \\ 
nf(x)C_{n-1}^{l-1}(1-F(x))^{l-1}F(x)^{n-l}.
\end{array}
\]
Therefore substituting $b(x)$ in ( 2) we have that 
\[
\begin{array}{c}
R_P=n\int f(x)\left( \int_0^xy\sum_{t=1}^kC_{n-1}^{t-1}\Psi _t^{\prime
}(y)dy\right) dx=n\int y(1-F(y))\sum_{t=1}^kC_{n-1}^{t-1}\Psi _t^{\prime
}(y)dy= \\ 
n\int yf(y)C_{n-1}^{k-1}(n-k)(1-F(y))^kF(y)^{n-k-1}dy.
\end{array}
\]
Since $kC_n^k(n-k)=nC_{n-1}^{k-1}(n-k)$ we finish the proof.\ \ \ $\Box $

The above result seems less surprising if we use the interpretation that
bidders behave in a simultaneous pooled auction as if they were bidding for
an average object. When objects are identical, the average is equal to the
value of the object so that effectively bidders bid as to beat the
individual with the marginal valuation. But this is how individuals bid at
the last round of the sequential auction which determines the expected price
at all rounds.

The above result can also be explained in the context of the revelation
principle. For the case of one object, Myerson (1981) shows, as a corollary
to the revelation principle, that an auction is optimal if it satisfies two
properties, namely, (1) the object always goes to the bidder with the
highest value, and (2) bidders with the lowest possible valuation should
expect zero profits. A generalization of Myerson's result for the case of $k$
objects (identical or perfectly correlated) would require for an auction to
be optimal, in addition to (2), that the highest-valued object should go to
the bidder with the highest value, the second-highest-valued object should
go the bidder with the second highest value, and so on; when the $k$ objects
are identical, the bidders with the $k^{th}$ highest values should each
receive an object.

It is easy to verify that when the objects are identical, both the pooled
and the sequential auctions are optimal and, therefore, should generate the
same revenue. When the objects are perfectly correlated it is still the case
that the pooled auction is optimal; we can guarantee that the bidder with
the $t^{th}$ highest value receives the $t^{th}$ highest-valued object since
strategies are increasing functions. For the case of sequential auctions of
perfectly correlated objects, however, there is no known solution. If
strategies in each auction are increasing in values, then the revelation
principle will guarantee its optimality and the revenue it generates will be
equal to the revenue generated by the pooled auction. Otherwise, the
sequential auction will generate less revenue than the pooled auction.

Therefore, a sequential auction will generate at most as much revenue as the
proposed pooled auction for identical or perfectly correlated objects. In
addition, we argue that the total social cost of running pooled auctions is
smaller than the total social cost of running sequential auctions. From the
auctioneer's perspective, running a simultaneous auction is cheaper since he
only collects bids once; whereas the auctioneer has to collect bids $k$
times in a sequential auction. From the bidders' point of view, pooled
auctions are faster and they have to spend less time at the auction site.
Moreover, bidding strategies are easier to compute for pooled auctions than
for sequential auctions when there are independencies between bidding
behavior in the various rounds. That is, the strategy space for the pooled
auction is smaller than the strategy space for the sequential auction.
Nevertheless, further evidence by means of either empirical tests or
laboratory experiments is needed and one of the advantages of our direct
approach is exactly to allow for that.

Finally, we can compute what happens to the expected revenue when the degree
of correlation changes. Consider the sale of two objects and when the value
of the first object for Player $i$ is denoted by $x_i$ and the value of the
second object is denoted by $\lambda x_i,$ with $x_i\geq \lambda x_i,\forall
i,i=1,...,n.$ The expected revenue can be written as:\ \bigskip\ \\${\bf R}%
_P=n(n-1)\dint\limits_0^1f(x)[\dint\limits_0^xf(y)y\left( F(y)\right)
^{n-2}dy+$\ \\$\lambda \dint\limits_0^x\left( (n-2)\left( F(y)\right)
^{n-3}-(n-1)\left( F(y)\right) ^{n-2}\right) dy]dx$\ \bigskip\ \\Thus, we
can compute the derivative of ${\bf R}_P$ with respect to $\lambda $ as
follows:\ \bigskip\ \\$\dfrac{\partial {\bf R}_P}{\partial \lambda }%
=n(n-1)\dint\limits_0^1f(x)\left( \dint\limits_0^x\left( (n-2)\left(
F(y)\right) ^{n-3}-(n-1)\left( F(y)\right) ^{n-2}\right) dy\right) dx$\ %
\bigskip\\This can be simplified to:\bigskip\ \\(21) $\dfrac{\partial {\bf R}%
_P}{\partial \lambda }=n(n-1)\dint\limits_0^1\left( 1-F(x)\right) \left(
(n-2)\left( F(x)\right) ^{n-3}-(n-1)\left( F(x)\right) ^{n-2}\right) dx$\ %
\bigskip\\From inspection of (21), we can conclude that $\dfrac{\partial 
{\bf R}_P}{\partial \lambda }>0.$ For any particular distribution $F$, it is
possible to compute the precise value of the derivative. For the case of the
uniform distribution, for example, $\dfrac{\partial {\bf R}_P}{\partial
\lambda }=1,.i.e.,$ if the degree of correlation doubles, the expected
revenue also doubles.

\section{Conclusion}

In this paper we examine a simultaneous pooled auction. When objects are
identical, we show that the proposed auction generates the same revenue as a
first-price sealed-bid sequential auction. When objects are perfectly
correlated, we are able to characterize bidding strategies for the
simultaneous pooled auction - which is optimal, whereas there is no known
solution for sequential auctions. These strategies are not very different
from the case of identical objects. Moreover, we can determine precisely the
expected revenue and how it changes as the degree of correlation between
objects changes.

Simultaneous pooled auctions may be easier and cheaper to run than
sequential auctions since players submit only one bid. Bidders' strategies
are simpler to compute given that there are no interdependencies between
sales as in the case of sequential auctions. Moreover, our direct approach
of characterizing bidding strategies allows empirical tests and provides
proper theoretical foundations for laboratory experiments who could offer
further evidence of the appropriateness of polled auctions. Our results do
provide, however, a {\it prima facie} case for the use of simultaneous
pooled auctions to sell government-owned residential units to low-income
home buyers.

\begin{thebibliography}{99}
\bibitem{}  Gale, I. L. and D. B. Hausch (1994), ``Bottom-Fishing and
Declining Prices in Sequential Auctions,'' {\bf Games and Economic Behavior 7%
}, 318-331.

\bibitem{}  Kravets, A. R. (May/June1993), ``Going, going, gone! Real Estate
Auctions in the 90s,'' {\bf Probate and Property}, 39-42.

\bibitem{}  Mayer, C. J. (1995), ``A Model of Negotiated Sales Applied to
Real Estate Auctions,'' J{\bf ournal of Urban Economics 38}, 1-22.

\bibitem{}  Menezes, F. M. and P. K. Monteiro (1995), ``Sequential
Asymmetric Auctions with Endogenous Participation, '' Forthcoming in {\bf %
Theory and Decision}.

\bibitem{}  Myerson, R. B., 1981, ``Optimal Auction Design,'' {\bf %
Mathematics of Operations Research 6(1)}, 58-73.

\bibitem{}  Quan, D.C. (1994), ``Real Estate Auctions: A Survey of Theory
and Practice,'' {\bf Journal of Real Estate Finance and Economics 9}, 23-49.

\bibitem{}  Sherman, L. F. and J. Bussio (April 1994), ``Real Estate
Auctions:\ The New Method to Sell Real Estate,'' {\bf Real Estate Issues},
36-40.

\bibitem{}  Vandell, K. and T. Riddiough (1992), ``On the Use of Auctions as
a Disposition Strategy for RTC Real Estate Assets:\ \ A Policy
Perspective,'' {\bf Housing Policy Debate 3}, 117-141.

\bibitem{}  Vanderporten, B. (1992), ``Strategic Behavior in Pooled
Condominium Auctions,'' {\bf Journal of Urban Economics 31}, 123-137.

\bibitem{}  {Weber, R.J., 1983, ``Multi-Object Auctions'', in }{\bf Auction,
Bidding, and Contracting: Uses and Theory}\/{, edited by R.
Engelbrecht-Wiggans, M. Shubik, and R.M. Stark, New York University Press,
165--194.}
\end{thebibliography}

\end{document}

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Dr. Flavio Menezes
Department of Economics
Faculty of Economics and Commerce
Australian National University
Canberra, ACT 0200, Australia
Phone 61-6-2492651
Fax   61-6-2495124
Email: Flavio.Menezes@anu.edu.au

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