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\begin{document}
\title{Sequential English auctions:\ A theory of opening-bid fishing\thanks{%
Antitrust Division, U.S. Department of Justice, USA.
joseph.podwol@usdoj.gov. The views expressed herein are the author's and do
not necessarily reflect those of the U.S. Department of Justice. This paper
is adapted from a chapter of my Ph.D. dissertation at Cornell University.
Thanks to Talia Bar, Justin Johnson, Deborah Minehart, Henry Schneider,
Michael Waldman and various seminar participants for valuable suggestions.
All errors are my own. }}
\author{Joseph Uri Podwol}
\maketitle
\begin{abstract}
\cite{Cassady1967} describes an auction in which the auctioneer
\textquotedblleft fishes\textquotedblright\ for an opening bid, calling out
lower and lower amounts until an opening bid is eventually placed. Once a
bid is placed, it is not uncommon for the bidding to escalate above the
initial starting price. The current study explains this puzzle in a model in
which an auctioneer sells an indivisible good via English ascending-price
auction and cannot commit to keeping the item off the market should the
initial starting price fail to elicit any bids. A key insight of the paper
is that the well-known strategy equivalence between the English auction and
the second-price auction fails to extend to the sequential setting. This
difference has important implications for the equilibrium starting-price
path, giving rise to a \textit{Coase conjecture} in the English auction but
not in the second-price auction.
\end{abstract}
\section{Introduction}
In his oft-cited survey of all things auction, Ralph Cassady (1967, pages
57, 105, 113) describes auctions for a range of products from livestock to
antiques in which the auctioneer \textit{fishes }for an opening bid. The
auctioneer begins by announcing an opening bid and goes on to solicit bids
from the set of assembled buyers. If a bid is placed, the auction proceeds
in typical English ascending-price fashion -- soliciting higher and higher
bids until the final amount is hammered down with a gavel. But should the
auctioneer fail to find a bidder, the opening bid is reduced and a second
attempt is made. This process continues until an opening bid ultimately
elicits a response from a buyer. Cassady points out an astonishing outcome
of this process: once a bid is placed, it is not uncommon for the bidding to
progress beyond the amount of the initially proposed opening bid.
More recent evidence suggests that this outcome extends to ascending-price
auctions conducted online. In a quote culled from an eBay message board,
this seller of homemade jewelry makes a similar discovery:
\begin{quotation}
\textit{I have been in the practice of listing items for the least amount of
money I would be willing to sell them for. I discovered that another seller
was selling items similar to mine and starting them at 99 cents [...] I
noticed that many of her bids went way above what my similar items started
at, where many of my items started at the higher price just stagnated
without bids. -- Treasures\_by\_Cynthia}
\end{quotation}
It is surprising, given the breadth of the literature on optimal auctions,
that the tactic of opening bid fishing has not received more attention. That
the high bid is negatively correlated with the starting price (or opening
bid) may well be explained by \textit{affiliated values} or by \textit{%
auction fever}. In the affiliated values model of \cite{MW1982}, buyers
shade their bids in order to avoid the winner's curse. A lower starting
price allows for more bids, which reveal information to buyers and leads to
less bid shading. \cite{OckenfelsReileySadrieh2007} define auction fever as
\textquotedblleft an excited and competitive state-of-mind, in wich the
thrill of competing against other bidders increases a bidder's willingness
to pay in an auction.\textquotedblright\ This definition encompasses a
variety of cognitive biases which offer a common prediction that a lower
starting price leads to increased competition, which serves as a trigger for
excitement and consequently higher bids.\footnote{\cite%
{KuGalinskyMurninghan2006} study a similar mechanism in which auction prices
are negatively correlated with starting prices, which they attribute to the
tendency for: i) bidders who enter the bidding early to justify the sunk
cost of participation and ii) bidders who arrive late to infer greater value
to auctions with more bids.} While models of affiliated values and auction
fever offer compelling insights, neither explains why the seller wouldn't
simply set a low initial opening bid, as opposed to fishing for one.
To rationalize opening bid fishing and the concurrent \textit{start low end
high }phenomenon, we analyze a dynamic model in which each period offers the
seller an opportunity to sell a single item via English auction with an
announced starting price. Buyers' valuations are independently and
identically distributed and the lowest possible bidder type exceeds the
seller's valuation. The seller cannot commit in any period to a
take-it-or-leave-it offer. Therefore, if the auction fails to elicit a bid,
she relists in the following period and continues to conduct auctions
successively as long as it is sequentially rational to do so. Anticipating
this behavior, buyers may wish to hold off bidding in the current auction
and instead wait for a subsequent auction where the item may be obtained at
a lower price. For instance, a buyer with a valuation of $S$, say, gains
nothing by winning an auction started at $S$. But should the item go unsold
in the intial auction, his expected surplus in the following period is
positive since the starting price will be reduced. In equilibrium, a buyer
whose valuation lies between $S$ and some threshold value holds off bidding
in the auction with a starting price of $S$, but bids up to his valuation
when the item is relisted with a lower starting price.
The preceding argument explains the outcome described by Cassady that after
the initial starting price fails to induce any bids, the item may later be
hammered down at a price greater than the initial starting price. This would
be the case if the two highest valuations among the assembled buyers were
both greater than the initial starting price but below the threshold value.
This is actually the most likely outcome when the time between auctions is
sufficiently short. In such circumstances, waiting for the subsequent
auction is virtually costless to buyers. Consequently, the seller must set
the initial starting price low in order to induce buyers with high
valuations (should there be any present) to bid. Despite the low initial
starting price, there will still be buyers that prefer to wait for the
subsequent auction--where the starting price will be even lower--before
bidding. Eventually, the seller reduces the starting price low enough to
induce all buyers to bid. The high bid (i.e. the amount paid by the high
bidder) in this auction will be equal to the second-highest bidder's
valuation, which is greater than the low initial starting price.
This research contributes to the budding literature on credible sales
mechanisms. \cite{McAdamsSchwartz2007} and \cite{Vartiainen2013} are recent
studies which model environments in which the seller, having announced her
intention to sell an item using a particular auction mechanism, cannot keep
from deviating when information revealed during the auction makes it
profitable to do so.\footnote{%
Along these lines, \cite{BesterStrausz2001} model a principle-agent
relationship where the principle cannot commit to following the rules of the
mechanism.} The most closely related research is \cite{MV1997}, who model a
seller that cannot commit to a reserve-price policy when offering an item
for sale via first-price or second-price sealed-bid auction, respectively.%
\footnote{%
A \textquotedblleft reserve price\textquotedblright\ in a sealed-bid auction
indicates the lowest allowable bid. This acts similar to a \textquotedblleft
starting price\textquotedblright\ in an open-outry or English auction.} They
show that under both mechanisms, the reserve-price path is declining similar
to the declining starting-price path in the current study. The current study
shows, however, that Cassady's start low end high phenomenon arises when the
item is offered via English auction but not when offered via first- or
second-price auction. A seller that offers the item via English auction
follows a starting price path that is lower than the corresponding reserve
price path of a seller that uses only first- or second-price auctions,
respectively. In the limit, as the time between successive English auctions
becomes short, the initial starting price convergest to the lowest possible
buyer valuation. A related result in the sequential bargaining literature is
the \textquotedblleft Coase conjecture,\textquotedblright\ which says that
when a seller cannot commit to a take-it-or-leave-it offer, her initial
offer converges to the lowest buyer valuation type as the time between
successive offers becomes short.\footnote{\cite{Coase} proposed that a
durable-good monopolist that cannot commit to not make additional sales in
subsequent periods must reduce the price in the initial period to marginal
cost. \cite{GSW1986} formalized the argument and pointed out that the model
is mathematically equivalent to one in which a monopoly seller bargains with
a buyer whose valuation is known only to himself and the the seller is
unable to commit to a take-it-or-leave-it offer.} The current study is the
first to extend this result to a setting with multiple buyers.
\cite{Skreta2013} studies a similar model to \cite{MV1997} but where the
seller can commit, after a fixed number of periods, to keeping the item off
the market.\footnote{%
The primary contribution of \cite{Skreta2013} is endogenizing the seller's
choice of mechanism.} \cite{Burguet&Sakovics1996} consider a sequential
first-price auction model where the number of bidders is endogenous. In both
instances, the starting price path is quite different from the one studied
here.
This research speaks to the literature on auction theory and mechanism
design. \cite{Vickrey} first demonstrated that the English and second-price
auctions are strategically equivalent and that they yield the same expected
revenue to the seller.\footnote{%
They are also revenue equivalent to other common mechanisms such as the
first-price and Dutch auctions. See \cite{Meyerson} and \cite{RS1981}.}
Subsequent research showed that in more complex informational environments,
they are not strategically equivalent nor are they revenue equivalent.%
\footnote{\cite{MW1982} and \cite{Caillaud&Mezzetti2004} are two examples.}
In the setup considered here, the English and second-price auctions are not
strategically equivalent yet they are revenue equivalent. The English
auction conveys information to buyers that once a bid is placed, the current
auction will be the last. This information is not available to buyers in a
second-price auction where bids are sealed until the winner is announced.
The additional information revealed on in the English auction induces buyers
to bid in the current period when they otherwise would have waited for a
later period. Anticipating this response from competing bidders, each bidder
becomes more passive in deciding whether to be the first to bid. This in
turn forces the seller to reduce the starting price relative to the reserve
price in a second-price auction in order to induce buyers to be the first to
bid. These effects turn out to be offsetting so that the expected revenue is
the same whether the seller uses English or second-price auctions.
The paper is organized as follows. Section 2 describes the model. Section 3
characterizes the equilibrium of the sequential English auction game and
proves the Coase conjecture. Section 4 demonstrates how the start low end
high phenomenon arises from equilibrium behavior. Section 5 reconciles our
results with those of \cite{MV1997} and Section 6 concludes.\emph{\ }
\section{The model}
Consider an auction marketplace consisting of a single seller and $n\geq 1$
potential buyers indexed $i=1,2,...,n$. The seller has one unit of a
particular item to sell. Buyers are risk neutral, have unit demands, and
differ only in their valuation of the item. Each buyer's valuation, denoted $%
v$, is known only to himself. Valuations are assumed to be independent and
identically distributed according to $F$, a continuous distribution with
density $f,$ bounded between zero and infinity, over the support $\left[
\underline{v},\bar{v}\right] $, where \underline{$v$}$>0$. The seller's
valuation is normalized to zero and the seller is risk neutral.\footnote{%
The seller's valuation need not be zero, but it is important for the
analysis that it be strictly less than $\underline{v}.$ This is known as the
\textquotedblleft gap\textquotedblright\ case in the literature. The
significance of this assumption is discussed later in this section.} Assume
the seller's valuation, the distribution of buyer valuations $F$, and $n$
are common knowledge.
The time horizon is infinite with periods indexed $t=1,2,...$. The pool of
buyers is fixed over time as are their valuations. In period 1, the seller
conducts an English auction and announces an opening bid or
\textquotedblleft starting price,\textquotedblright\ $s_{1}$. The English
auction proceeds as follows:\ Beginning with $s_{1}$, the auctioneer calls
for bids. If a bid is placed, the seller pauses to invite additional bidders
to enter. Each buyer indicates his active status by pressing a button. Once
all the buyers that wish to enter have done so, the price increases
continuously. Bidders may drop out at any point by releasing their button.
The price continues to increase until all but one bidder has dropped out. At
that point, the one remaining bidder is awarded the item and pays the
\textquotedblleft high bid,\textquotedblright\ equal to the price at which
the last bidder dropped out, or $s_{1}$ if no other bidders have entered. If
no bids are placed at $s_{1}$, the seller may conduct an English auction in
period 2. If the auction item fails to elicit a bid in period 2, the game
continues on to period 3 and so on. As the game advances from one period to
the next, all buyers and the seller discount returns accrued in the
following period by factor $\delta \in \left( 0,1\right) $, which is common
knowledge.
This setup models open-bid fishing as a repeated game, where the sales
mechanism in each period is an English auction. The English auction modeled
here differs from the typical \textquotedblleft button\textquotedblright\
auction introduced by \cite{MW1982} as a stylized approximation to the
English auction. In the button auction, the seller announces a starting
price and buyers decide simultaneously whether to enter, which they indicate
by pressing a button. The price then increases continuously and buyers may
exit by releasing their button. The decision to exit is irrevocable. The
English auction modeled here adds a bit of realism in that in the English
open-outcry auctions often seen in practice, buyers who were not active at
the beginning can submit bids later on. In fact, the auctioneer generally
cannot discern which bidders are active and which are not until all but one
have dropped out. Here, we allow the entry decision to take place over
possibly two stages. In the first stage, buyers simultaneously decide
whether to press their buttons. If at least one buyer presses his button in
stage 1, then a stage 2 is entered, whereby all buyers are made aware that a
bid has been placed in stage 1, and all those who did not bid in stage 1,
may do so in stage 2. Following stage 2, the price rises continuously as in
\cite{MW1982} until all but one bidder has dropped out. If no bids are
placed in stage 1, then stage 2 is foregone and the game moves immediately
to the next period, where the seller may announce a new starting price.
The sequential nature of the game arises because the seller cannot -- as is
typically assumed in the literature on optimal auctions -- commit to keeping
the item off the market should the initial auction fail to elicit any bids.
Absent commitment power, the seller continues to relist in subsequent
periods since selling the item to even the lowest valuation buyer is always
preferred to keeping it. It should be noted that due to the assumption that
\underline{$v$}$>0$, all equilibria of the game are necessarily stationary.
This precludes the existence of \textit{reputational equilibria }(%
\citeNP{Ausubel&Deneckere1989}),\textit{\ }in which the seller reduces the
starting price at an arbitrarily slow rate.
Though the seller's objective function need not be concave to insure a
stationary equilibrium,\footnote{%
See \cite{MV1997}.} such considerations complicate the analysis without
adding to the economic substance. Therefore, we\ impose the following
regularity condition.
\begin{condition}
For any\emph{\ }$u,v\in \left[ \underline{v},\bar{v}\right] $\emph{, }$v-%
\frac{F\left( u\right) -F\left( v\right) }{f\left( v\right) }$ is strictly
increasing in $v$\emph{.}
\end{condition}
The sequential English auction game is shown to have multiple equilibria.
All of the equilibria have a declining starting-price path and have the same
expected revenue and the same probability of sale in each period. The
equilibria differ only in the starting price used in each period. We employ
a selection criteron to settle on an equilibrium of the game in which
rationalizes the start low end high phenomenon described by \cite%
{Cassady1967}.
\section{Equilibrium characterization}
The equilibrium concept is \textit{perfect-Bayesian equilibrium }(PBE). Any
equilibrium is a history-contingent sequence of the seller's starting
prices, $s_{t}$, buyers' bidding decisions, and updated beliefs about the
valuations of existing buyers satisfying the typical consistency conditions.
Formally, let $H_{\tau }=\left\{ s_{1},s_{2},...s_{\tau }\right\} $ denote
the history through period $\tau $ of a game that has not ended prior to
period $\tau $. Since a bid placed in any period $t<\tau $ necessarily
results in the game ending in that period, $H_{\tau }$ consists only of the
seller's starting prices with the implicit assertion that no bids have been
placed to that point. A strategy for the seller in period $t$ is a starting
price which maximizes expected discounted revenue given her beliefs over
buyer valuations and given equilibrium behavior in what follows. A strategy
for each buyer in period $t$ involves two decisions -- whether to bid in the
period-$t$ auction and if so, what price to drop out at -- which jointly
maximize expected discounted surplus given his beliefs over valuations and
given equilibrium behavior. We restrict attention to monotonic bidding
strategies, which is necessary for an equilibrium in symmetric strategies.
Whether or not the game proceeds to the next period relies on a buyer's
decision of whether to bid at the starting price, that is when it is not
known if other buyers intend to bid in that period. When a buyer does
ultimately place a bid, it insures that the item will sell and the game will
end at the conclusion of the bidding. An important distinction then is
between \textit{initial bidders}$,$ those willing to bid at the starting
price, and \textit{interim bidders}, those who bid only after the bidding
has begun. Note that there can be more than one initial bidder since the
distinction is a counter-factual, determined by what the buyer \textit{would
do} if no other buyers had submitted a bid.
\begin{lemma}
\label{initial}The following characterize the equilibrium in any PBE:
\begin{enumerate}
\item It is weakly dominant for an initial bidder to drop out when the price
equals his valuation.
\item In any period $t$, there exists a marginal type $\beta _{t}$, such
that every buyer whose valuation exceeds $\beta _{t},$ bids at the starting
price.
\item Regardless of the history, all buyer types bid at the starting price
when the starting price is at or below \underline{$v$}.
\item There exists a period $T<\infty $ , endogenously determined, such that
the game ends in at most $T$ periods.
\end{enumerate}
\end{lemma}
All proofs are in the Appendix. Result 1 of the lemma is a standard result
in auction theory but applied to the sequential setting. Dropping out at
one's valuation is unique among all symmetric decision rules and so we
assume initial bidders follow that in what follows. The bidding of interim
bidders is as yet undetermined; this will be pinned down by the equilibrium
refinement.
Result 2 of the lemma is the \textit{successive skimming} property, which is
common to the dual literatures on sequential bargaining and durable good
monopoly. Analogous to those models, a buyer bids at a given starting price
only if the payoff from bidding in the current period is sufficiently large
to have him forgoe future opportunities. Since any buyer whose valuation
exceeds $\beta _{t}$ necessarily bids in period $t$, it follows that if the
item remains unsold after period $t$, it must be that all valuations are
below $\beta _{t}$. This makes for a simple updating rule in which $\beta
_{t}$ becomes the highest type in period $t+1$, denoted $u_{t+1}$. In what
follows, we refer to $u_{\tau }$ as the \textit{state }in period $\tau $ and
$\beta _{\tau }$ as the \textit{screening level} in period $\tau $, both of
which are standard usage in the aforementioned literature.
A second implication of Result 2 is that initial bidders have higher
valuations than interim bidders. Since $\beta _{t}$ determines a minimum
valuation type, any interim bidder in period $t$ must not have had a
valuation above $\beta _{t}$, otherwise he would have been an initial
bidder. Thus the allocation of the item amongst buyers will be unaffected by
the strategies of interim bidders since the item can only be obtained by the
buyer with the highest valuation. However, the bidding of interim bidders
raises the price above what it would have been absent such bidding, taking
everything else as given. To see this, consider the bidding decision of an
interim bidder after an initial bid has been placed. The fact that he was
not an initial bidder indicates to him that he will ultimately lose the
bidding and so has no reason to bid. On the other hand, he has no reason not
to bid as doing so is costless. Therefore, any strategy that assigns
positive probability to any amount up to and including the buyer's own
valuation is individually rational. Only bids above the buyer's own
valuation are ruled out in equilibrium.
To understand how interim bidders affect the payoffs of initial bidders,
consider some period $t$ and suppose that there exists only one initial
bidder. If interim bidders were precluded from bidding, the lone initial
bidder would be the only buyer to submit a bid and he would pay a price
equal to the starting price, $s_{t}$. This is the outcome in the analogous
second-price auction, where there are no interim bidders. If instead,
interim bidders are allowed to bid, the price paid by the initial bidder
would be determined by the highest bid placed among all interim bidders and
would only equal $s_{t}$ if all initial bidders declined to bid.
Let $\rho _{t}$ denote the expected price paid by a lone initial bidder,
taking into account the (possibly mixed) strategy followed by interim
bidders. Without applying an equilibrium refinement criterion, $\rho _{t}$
is indeterminate. We assume in what follows that $\rho _{t}\left(
s_{t},\beta _{t}\right) $ is increasing in both arguments and strictly so
for the first argument.\footnote{%
This assumption is satisfied trivially when interim bidders opt not to bid,
in which case $\rho _{t}\left( s_{t},\beta _{t}\right) =s_{t},$ and will be
shown to be true when they bid up to their valuations as in the equilibrium
selected under our refinement.}
\subsection{Characteristics of all equilibria}
That the screening level eventually reaches \underline{$v$} (Result 4 of the
Lemma), whereby the game necessarily ends (Result 3), implies that the
equilibrium can be derived via backward induction. A complication arises
because the number of periods required for the starting price to reach
\underline{$v$} is determined endogenously, so the number of periods to be
inducted upon must also be solved for. The details are presented in Appendix
A.2.
\begin{proposition}
\label{PBE}A PBE consists of a sequence of screening levels $\left\{ \beta
_{t}\right\} _{t=1}^{T}$ and corresponding starting prices $\left\{
s_{t}\right\} _{t=1}^{T}$ such that:
\begin{enumerate}
\item In any period $t$, the seller chooses screening level $\beta
_{t}=\beta \left( u_{t}\right) $ to satisfy
\begin{equation}
\beta _{t}f\left( \beta _{t}\right) +F\left( \beta _{t}\right) -F\left(
u_{t}\right) \leq 0, \label{FOC}
\end{equation}%
with a strict equality when $\beta _{t}>\underline{v}$. The choice of $\beta
_{t}$ depends only on $u_{t}$ and the density $f$, and is independent of $%
\delta $, $n$, and $\rho _{t}$.
\item To induce a screening level of $\beta _{t}$, given $\rho _{t}$, the
period-$t$ starting price $s_{t}=\sigma \left( \beta _{t}\right) $ satisfies
the following sequential rationality condition,
\begin{equation}
\left[ \beta _{t}-\rho _{t}\left( \sigma \left( \beta _{t}\right) ,\beta
_{t}\right) \right] F_{Y_{1}}\left( \beta _{t}\right) =\delta \left( \left[
\beta _{t+1}-\rho _{t+1}\right] F_{Y_{1}}\left( \beta _{t+1}\right)
+\int_{\beta _{t+1}}^{x}F_{Y_{1}}\left( Y_{1}\right) dY_{1}\right) ,
\label{sequential rationality}
\end{equation}%
where $\beta _{t+1}=\beta \left( \beta _{t}\right) $ and $\rho _{t+1}=\rho
_{t+1}\left( \sigma \left( \beta _{t+1}\right) ,\beta _{t+1}\right) $.
\item The allocation of the item, the seller's revenue, and the probability
of sale in a given period are independent of $\rho _{t}$.
\end{enumerate}
\end{proposition}
Equation $\left( \ref{FOC}\right) $ is the solution to the seller's problem,
taking into account the equilibrium strategies of buyers and the optimal
screening levels in subsequent periods. In a static version of the game,
this same condition solves the seller's problem in selecting the optimal
starting price, where $\overline{v}$ would take the place of $u_{t}$.
However, in a static setting, the screening level and starting price are one
and the same. Within the sequential setting, sequential rationality requires
that the seller set the starting price below the screening level in order to
induce bids from the desired set of buyer types.
Equation $\left( \ref{sequential rationality}\right) $ indicates that the
seller's choice of starting price makes a type $\beta _{t}$ buyer
indifferent between bidding at the starting price in period $t$ and waiting
one more period, taking into account equilibrium behavior. The left-hand
side of the expression indicates a type $\beta _{t}$ buyer's expected
surplus from bidding at the starting price, $\sigma \left( \beta _{t}\right)
$, when he is the lowest type to do so. His expected payment would then be
given by $\rho _{t}\left( \sigma \left( \beta _{t}\right) ,\beta _{t}\right)
$. $F_{Y_{1}}\left( \beta _{t}\right) $ denotes the probability that type $%
\beta _{t}$ is the lone initial bidder.\footnote{%
Using conventional notation, let $Y_{1}$ denote the maximum of the $n-1$
competing buyers' valuations. As the $v_{i}$ are independent, $%
F_{Y_{1}}\left( \beta _{t}\right) \equiv F\left( \beta _{t}\right) ^{n-1}$.}
The right-hand side of equation $\left( \ref{sequential rationality}\right) $
indicates a type $\beta _{t}$ buyer's expected continuation surplus were he
not to bid, given an equilibrium starting price of $\sigma _{t+1}$ and
screening level of $\beta _{t+1}$ in the period to follow. In the following
period, since $\beta _{t}>\beta _{t+1}$, he receives an amount given by the
first term in the event he is the lone initial bidder, and an amount given
by the second term when he is bidding against at least one other initial
bidder.\footnote{%
All expectations are taken conditional on state $u_{t}$ being reached.
Expression $\left( \ref{sequential rationality}\right) $ does not explicitly
include $u_{t}$ due to a canceling of terms.}
Result 3 of the proposition, that the seller's revenue is independent of $%
\rho _{t},$ would seem counter-intuitive since a higher value of $\rho _{t}$
translates to a larger expected payment for the winning bidder, holding
fixed the seller's starting price. However, the seller takes $\rho _{t}$
into account when selecting the starting price. The higher is $\rho _{t}$,
the lower the starting price must be in order to induce bids from buyers
whose valuations are at least $\beta _{t}$, the desired screening level.%
\footnote{%
This follows from equation $\left( \ref{sequential rationality}\right) $
along with the assumption that $\rho _{t}$ is strictly increasing in $s_{t}$.%
} In this way, the equilibrium starting prices corresponding to every
possible value of $\rho _{t}$ lie along a continuum. The equilibrium we will
select for using our refinement criterion yields the highest possible $\rho
_{t}$ and hence lowest possible starting price. The equilibrium yielding the
highest possible starting price is the case in which $\rho _{t}=s_{t}$,
which is the payment the lone bidder would make in the analogous
second-price auction. One implication is that the starting price path in any
equilibrium of the sequential English auction game is lower period by period
(strictly so for $\rho _{t}>s_{t}$) than the equilibrium starting price path
in the sequential second-price auction game. A second implication is that
all equilibria of the English auction game are revenue-equivalent to the
equilibrium of the second-price auction game.
\subsection{Equilibrium refinement}
Having characterized equilibrium behavior of sellers and initial bidders
under any equilibrium strategy played by interim bidders, we now apply a
refinement to select the equilibrium in which interim bidders are most
aggressive. This equilibrium is important as it gives rise to the Coase
conjecture.
Consider a perturbed version of the game in which, in a given auction, every
possible move is played by each buyer with positive probability.\footnote{%
Trembles by the seller are uninteresting, since the starting price is known
to all buyers before the auction begins.} Each buyer bids at the starting
price, $s_{t}$, with some probability strictly between zero and one and each
buyer drops out at any price in $\left( s_{t},\bar{v}\right] $ with some
probability strictly between zero and one. The limits of equilibria of such
perturbed games as the tremble probabilities go to zero are \textit{%
extensive-form trembling-hand-perfect equilibria }(ETE).\footnote{%
The ETE is an extension, due to \cite{Selten1983}, of the \textit{%
trembling-hand-perfect} equilibrium concept of \cite{Selten1975}. A
trembling-hand perfect equilibrium is one that takes the possibility of
off-the-equilibrium play into account by assuming that the players, through
a tremble, may choose unintended strategies, albeit with negligible
probability. When extending this concept to extensive-form games, the
modeler may choose to interpret a tremble as a mistake in a player's choice
of action at a particular information set or as a mistake in a player's
entire strategy choice. The ETE concept employs the former interpretation.}
\begin{proposition}
\label{THP}There exists a unique ETE in which each interim bidder bids up to
his valuation with probability 1$.$
\end{proposition}
The logic behind the proposition is that trembles make it so that an interim
bidder may with positive probability win the auction at a price below his
valuation. This is because, in contrast to equilibrium behavior, a buyer who
bids at the starting price before it is known whether others will bid may
drop out at a price less than the interim bidder's valuation. It is then in
the interim bidder's interest to remain active until his valuation is
reached. The same conclusion could also be reached less formally by
appealing to the seller's incentive to entice interim bidders to remain
active. It is costless for interim bidders to continue bidding up to their
valuations, while the seller strictly prefers that they do so. Thus the
equililbria in which interim bidders do not stay active up to their
valuations are not robust to the possibility that the seller can pay buyers
to remain active. We assume in all further discussion of the English auction
that the selected equilibrium is the one that is played. The expected
payment made by a lone initial bidder is then,
\begin{eqnarray}
\rho \left( s_{t},\beta _{t}\right) &=&E\left[ \max \left\{
s_{t},Y_{1}\right\} |Y_{1}<\beta _{t}\right] \notag \\
&=&\beta _{t}-\frac{\int_{s_{t}}^{\beta _{t}}F_{Y_{1}}(Y_{1})dY_{1}}{%
F_{Y_{1}}\left( \beta _{t}\right) }. \label{rho}
\end{eqnarray}
The aggressive bidding by interim bidders in the ETE puts downward pressure
on the seller's starting price. In the limit as $\delta $ goes to unity,
this gives rise to the following \textit{Coase conjecture. }
\begin{proposition}
\label{Coase}In the ETE, for every $\varepsilon >0$, there exists a $\bar{%
\delta}<1$ such that for all $\delta \geq \bar{\delta}$, and for any initial
screening level, $\beta _{1}=\beta \left( \bar{v}\right) \in \left[
\underline{v},\bar{v}\right] $, the seller's initial starting price, $\sigma
\left( \beta _{1}\right) $, is less than \underline{$v$}$+\varepsilon $.
\end{proposition}
The intuition for this result is fairly straightforward. If all buyers
believe that in the event they are the lone initial bidder, the price they
will pay will be driven up by interim bidders, they will be more hesitant to
place a bid at the starting price when it is not known if any other buyers
will do so. It may make more sense to wait for a subsequent period, where
the starting price will be reduced, before bidding. When the time between
periods is very short (i.e. $\delta $ is close to 1), there is virtually no
cost to waiting until the terminal period where starting price is reduced to
the lowest valuation type. For the seller to induce the marginal type to bid
in period 1, she must guarantee him the surplus he would have obtained in
the terminal period. The only way to do this is to drop the starting price
in period 1 to the lowest valuation type. For reasons that I\ discuss in
Section 5, this result does not extend to values of $\rho _{t}$ less than
the value given by expression $\left( \ref{rho}\right) $.
\section{Bidding dynamics}
The Introduction described a sequence of English auctions which give rise to
a counter-intuitive result. \cite{Cassady1967} explains that after failing
to elicit any bids in the initial auction, the seller lowers the starting
price whereupon the bidding escalates beyond the amount of the initial
starting price. The current section seeks to rationalize this \textit{start
low end high phenomenon}. The first result -- the \textquotedblleft weak
gap\textquotedblright\ property -- demonstrates that this is a possible
outcome. The second result-- the \textquotedblleft strong
gap\textquotedblright\ property -- demonstrates that when the time between
auctions is sufficiently short, the subsequent auction is actually more
likely to end with a price at least as high as the initial starting price
than is the initial auction.
We begin by motivating the general results with a simplifying example. Let $%
n=2$ and suppose that buyer valuations are drawn from a Uniform $\left[ a,1+a%
\right] $ distribution, $a\in \left( 0,1\right) $. Within the PBE, the
subgame beginning at some arbitrary period $t$ is characterized by a state
variable, $u$, which denotes the highest possible valuation among the
contingent of buyers, given that the item is still available to that point.
This value is obviously equal to $1+a$ in period 1 and decreases from there.
The PBE is stationary so $u$ does not depend explicitly on $t$. The seller's
period-$t$ problem reduces to one of choosing the optimal screening level in
the current period, $\beta \left( u\right) $, subject to the constraint that
the type-$\beta $ buyer is indifferent between bidding in period $t$ and
bidding in period $t+1$ and given sequential rationality in what follows.
Inserting $F\left( x\right) =x-a$ into $\left( \ref{FOC}\right) $, the
solution to the seller's problem is given by,
\begin{equation}
\beta \left( u\right) =\left\{
\begin{array}{cc}
u/2 & \text{if }u\geq 2a \\
a & \text{otherwise}%
\end{array}%
\right. . \label{stationary}
\end{equation}%
At any state $u$, the seller cuts the demand curve in half, serving the top
half, until $u$ becomes sufficiently small so that $u/2$ falls below $a$,
whereby the seller reduces the starting price to $a$ which induces all buyer
types to bid.
Examination of $\left( \ref{stationary}\right) $ shows that the length of
the game (i.e. the maximum number of periods until a sale is made) is
determined by the value of $a$. For any integer, $k$, the game will last at
most $k$ periods for $a\in \left[ \frac{1}{2^{k}-1},\frac{1}{2^{k-1}-1}%
\right) .$ In what follows, we\ solve a two-period game and then show how
those results extent to a game of any finite length.
\subsection{A linear two-period game}
Let $a\in \lbrack 1/3,1)$ so that the game ends in at most two periods. The
sequence of screening levels is $\left\{ \beta _{1},\beta _{2}\right\}
=\left\{ \frac{1+a}{2},a\right\} $ following from $\left( \ref{stationary}%
\right) $. To induce all buyer types to bid in period 2, the seller must set
a non-binding starting price of $s_{2}=a$.\footnote{%
Any starting price strictly less than $a$ would also work except in the case
of $n=1$.} All buyers bid because there is no chance that the starting price
will be reduced in subsequent periods.
To induce buyers whose valuations are at least $\beta _{1}$ to bid in period
1, the seller must set $s_{1}$ low enough that these buyer types strictly
prefer bidding in period 1. Solving for $s_{1}$ requires making the type-$%
\beta _{1}$buyer indifferent between bidding in period 1 when he does not
know if other buyers will bid in period 1 and waiting for period 2 as shown
in $\left( \ref{sequential rationality}\right) $. Since all buyer types bid
in period 2, $\beta _{2}=\rho _{2}=a.$ Subsituting $\beta _{2}=\rho _{2}=a$,
$F_{Y_{1}}\left( y\right) =y-a$ and $\beta _{1}=\left( 1+a\right) /2$ into $%
\left( \ref{sequential rationality}\right) $ and solving for $s_{1}$ yields,
\begin{equation}
s_{1}=a+\frac{1-a}{2}\sqrt{1-\delta }. \label{s1}
\end{equation}%
Notice that as $\delta $ goes to 1, $s_{1}$ converges to $a$ as guaranteed
by the Coase conjecture.
The \textit{start low end high} phenomenon is explained within the model as
follows. Conditional on period-2 being reached, the high bid in period 2
exceeds $s_{1}$ if there are at least two buyers whose valuations exceed $%
s_{1}$. The fact that the period-1 auction failed to elicit any bids implies
that there are no buyers whose valuation is as high as $\beta _{1}$.
However, since $\beta _{1}>s_{1}$, the high bid in period-2 exceeds $s_{1}$
with positive probability after conditioning on the fact that the period-1
auction failed to receive any bids. We refer to this as the
\textquotedblleft weak gap\textquotedblright\ property, as it is the gap
between $\beta _{1}$ and $s_{1}$ that makes this result possible.
A somewhat surprising stronger result is that conditional on period 2 being
reached, the period-2 is actually \textit{more likely} to have a high bid
exceeding $s_{1}$ than the period-1 auction when the time between relistings
is sufficiently short. Let $p_{t}$ denote the high bid in period $t$. This
is equal to: the price at which all but one bidder has dropped out if two or
more bidders entered the bidding in period $t$; or the starting price
otherwise.\footnote{%
The thought experiments that follow are concerned with whether $p_{t}$ is
greater than $s_{t}$ or not. Therefore, it is immaterial whether $p_{t}$ is
assigned a value of $s_{t}$ or zero if no bidders entered in period $t$.}
Let $\eta _{t}$ be an indicator of whether the period-$t$ auction elicits a
bid (equal to unity if it does, zero otherwise). Further, let $X_{1}$ and $%
X_{2}$ denote the realizations of the highest and second-highest buyer
valuations, respectively. The probability that the period-2 high bid exceeds
$s_{1}$ is given by,
\begin{eqnarray}
\Pr \left\{ p_{2}>s_{1}|\eta _{1}=0\right\} &=&\Pr \left\{
X_{2}>s_{1}|X_{1}<\beta _{1}\right\} \notag \\
&=&\frac{\beta _{1}-s_{1}}{\beta _{1}-a}. \label{p2-twoperiod}
\end{eqnarray}%
Recall that as $\delta $ goes to 1, $s_{1}$ goes to $a$, so that the
right-hand side of $\left( \ref{p2-twoperiod}\right) $ goes to 1.
In period 1, the high bid exceeds $s_{1}$ if there is at least one initial
bidder, whose valuation is at least $\beta _{1}$, and a second bidder who
could be an initial bidder or an interim bidder but whose valuation exceeds $%
s_{1}$. The probability of this occuring is,
\begin{eqnarray}
\Pr \left\{ p_{1}>s_{1}\right\} &=&\Pr \left\{ X_{1}\geq \beta
_{1},X_{2}>s_{1}\right\} \notag \\
&<&\text{Pr}\left\{ X_{2}>s_{1}\right\} \notag \\
&=&\left[ 1-(s_{1}-a)\right] ^{2} \label{p1-twoperiod}
\end{eqnarray}%
where the inequality in the second line follows from the fact that $\beta
_{1}$ is bounded above $s_{1}$. As $\delta $ goes to $1$, the right-hand
side of $\left( \ref{p1-twoperiod}\right) $ converges to $1$ from below. It
follows that for $\delta $ sufficiently close to 1, $G\equiv \Pr \left\{
p_{2}>s_{1}|\eta _{1}=0\right\} -\Pr \left\{ p_{1}>s_{1}\right\} $ is
positive. We refer to this result as the \textquotedblleft strong
gap\textquotedblright\ property as it requires $\beta _{1}$ to be
sufficiently larger than $s_{1}$.\footnote{%
Since $\beta _{1}$ is independent of $\delta $ and $s_{1}=\sigma \left(
\beta _{1}\right) $ is decreasing in $\delta $, $\beta _{1}-s_{1}$ is
increasing in $\delta $. } The strong gap property is illustrated in Figure
1.1, which plots $G$ over $\delta $. The lower curves corresond to higher
values of $a$. It is evident that each curve crosses the horizontal axis for
$\delta $ sufficiently close to 1.
\FRAME{dtbpFU}{3.0441in}{1.881in}{0pt}{\Qcb{Figure 1.1: $G$ as a function of
$\protect\delta $ for $a\in \left\{ 1/3,1/2,2/3,3/4\right\} .$}}{\Qlb{Figure
1.1}}{Figure 1.1}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width
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\subsection{A linear k-period game}
\bigskip The above result can be extended to a game of any finite length. As
in the two-period case, the screening level in the last period of the game
is $\beta _{k}^{\left( k\right) }=a$, so that all buyers are induced to bid.
From $\left( \ref{stationary}\right) $, the state variable in the last
period of a $k-$period game is
\begin{equation*}
\beta _{k-1}^{\left( k\right) }=\frac{1+a}{2^{k-1}},
\end{equation*}%
so that conditional on period $k$ being reached, all buyers' valuations will
lie in the interval $\left[ a,\beta _{k-1}^{\left( k\right) }\right] .$%
\footnote{%
That $\beta _{k-1}^{\left( k\right) }>a$ follows from the fact that for the
game to end in at most $k$ periods, it must be that $a\in \left[ \frac{1}{%
2^{k}-1},\frac{1}{2^{k-1}-1}\right) .$} The Coase conjecture guarantees that
the initial starting price converges to the minimum valuation type, $a,$ as
the time to relisting becomes sufficiently short. Therefore, conditional on
period $k$ being reached, the high bid in period $k$ will necessarily exceed
the initial starting price. In contrast, the high bid in the intial period
exceeds the initial starting price only if there is one buyer whose
valuation exceeds the initial screening level, $\beta _{1}^{\left( k\right)
}=\frac{1+a}{2}.$ Since $\beta _{1}^{\left( k\right) }>a$, there may be no
buyers whose valuations are high enough to justify bidding in the initial
period. Therefore, the probability that a high bid of at least $s_{1}$ is
obtained in the period-$k$ auction, conditional on period $k$ being reached,
is greater than the probability that a high bid of at least $s_{1}$ is
obtained in period 1.
\subsection{General results}
\bigskip I\ now establish in more general terms the weak and strong gap
properties for the general model. The weak gap property shows that if the
auction in some arbitrary period $t$ fails to receive any bids, then some
subsequent auction may end with a high bid strictly greater than the period-$%
t$ starting price with positive probability.
\begin{proposition}
\label{weak gap}Whenever $T>1$, there exists a $\bar{\tau}>1$ such that for
any $\tau \leq \bar{\tau}$,%
\begin{equation*}
\Pr \left\{ p_{t+\tau }>s_{t}|\eta _{t+\tau -1}=0\right\} >0.
\end{equation*}
\end{proposition}
This says that if we consider any period $ts_{t}.$ We know that this is necessarily true for $\tau =1$ since the
marginal type in a given period will always strictly exceed the starting
price whenever there is at least one more period remaining in which the item
could be obtained at a lower price. $\bar{\tau}$ is the highest integer
value for which this continues to be true.
The strong gap property shows that as the time between auctions becomes
sufficiently short, the auction in the terminal period is more likely to end
at a price strictly greater than the period-$t$ starting price than is the
period-$t$ auction itself.
\begin{proposition}
\label{strong gap}Whenever $T>1$, there exists some $\tilde{\delta}<1$ such
that for any $\delta \geq \tilde{\delta}$, and any $ts_{t}|\eta _{T-1}=0\right\} >\Pr \left\{
p_{t}>s_{t}\right\}
\end{equation*}%
regardless of $T-t$, the number of periods required for the starting price
to reach $\underline{v}$.
\end{proposition}
Proposition \ref{strong gap} says that the start low end high phenomenon is
the most likely outcome when the auctioneer lowers the start price in quick
succession (i.e. when $\delta $ is close to 1). This is because when doing
so, the initial starting price (and all subsequent starting prices) become
quite small. As such, upon failing to sell in the initial auction, and the $%
T-2$ subsequent auctions, the auction in period $T$ will necessarily have a
high bid of at least $s_{1}$. This logic continues to hold when the period-1
starting price is replaced with period$-t$ starting price for any $t$\underline{$v$}.%
\footnote{%
The right-hand side of the equality is the area under the $F_{Y_{1}}$ curve
over the domain $\left[ \underline{v},\beta _{1}\right] $. The left-hand
side of the equality is a rectangle with height $F_{Y_{1}}\left( \beta
_{1}\right) $ and base $\beta _{1}-\sigma \left( \beta _{1}\right) $. Since $%
F_{Y_{1}}$ is weakly increasing, if $\sigma \left( \beta _{1}\right) $ were
to be equal to $\underline{v}$, then the area under the curve would lie
entirely inside the rectangle so the equality fails. Thus, the value of $%
\sigma \left( \beta _{1}\right) $ must be sufficiently above $\underline{v}$
to bring about an equality.} The idea is that in the second-price auction,
the seller screens out types in the interval $[\sigma \left( \beta
_{1}\right) ,\beta _{1})$ with a binding reserve price of $\sigma \left(
\beta _{1}\right) $. Thus the marginal buyer pays $\sigma \left( \beta
_{1}\right) $ for realizations of $Y_{1}$ in $[\sigma \left( \beta
_{1}\right) ,\beta _{1})$. By choosing $\sigma \left( \beta _{1}\right) $
small enough (but still greater than $\underline{v}$), the seller can make
bidding in the initial period just profitable for marginal buyer.
Contrast this to the ETE of the English auction. Substituting $\left( \ref%
{rho}\right) $ into $\left( \ref{delta equal 1}\right) $ yields,
\begin{equation*}
\underset{\delta \rightarrow 1}{\lim }\int_{\sigma \left( \beta _{1}\right)
}^{\beta _{1}}F_{Y_{1}}(Y_{1})dY_{1}=\int_{\underline{v}}^{\beta
_{1}}F_{Y_{1}}\left( Y_{1}\right) dY_{1}.
\end{equation*}%
For this equality to hold, $\sigma \left( \beta _{1}\right) $ must be equal
to \underline{$v$} in the limit. The distinction is that unlike in the
second-price auction, the seller cannot screen out buyers whose valuations
exceed the starting price since interim bidders are free to enter the
bidding after the opening bid. Thus the marginal bidder pays $Y_{1}>\sigma
\left( \beta _{1}\right) $ for realizations of $Y_{1}$ in $[\sigma \left(
\beta _{1}\right) ,\beta _{1})$. Under those circumstances, the surplus
received by the marginal bidder is equal to that of a one-shot auction with
starting price $\sigma \left( \beta _{1}\right) $. A buyer comparing the
surplus from a one-shot auction with starting price $\sigma \left( \beta
_{1}\right) $ to that of a one-shot auction with starting price \underline{$%
v $} will choose the former only if $\sigma \left( \beta _{1}\right) =%
\underline{v}.$ Thus the only way the seller can guarantee the type-$\beta
_{1}$ buyer his reservation surplus is by running an auction with a starting
price close to \underline{$v$}.
Another important difference between the two auction formats is that the
second-price auction does not have the strong gap property. The strong gap
property in the sequential English auction follows from the fact that the
initial starting price converges to zero as the time between auctions
becomes short. Conditional upon reaching the terminal period in the English
auction, the second-highest buyer's valuation exceeds the initial starting
price with probability approaching one. In the second-price auction, the
initial starting price is bounded above $\underline{v}$ and as a result may
exceed the second-highest buyer's valuation.
\section{Conclusion}
This paper began with the task of understanding the bidding dynamics that
result when an auctioneer fishes for an opening bid. We have studied a model
of an English auction in which the seller goes fishing for an opening bid
when she cannot commit to a predetermined starting-price path. The outcome
observed by \cite{Cassady1967}, in which the price in a subsequent auction
exceeds the amount of the starting price in the initial auction, is shown to
be a natural consequence of the model. Further, it is shown to be the most
likely outcome when the time between successive auctions is sufficiently
short.
An implication of the model is that when auctions are conducted
sequentially, the English auction is no longer strategically equivalent to
the second-price auction. The English auction is shown to induce bids from a
larger set of buyer types than does the second-price auction. The difference
is in the participation of \textit{interim bidders}, who enter the bidding
only after first observing a bid from a competing buyer\textit{.} Because of
the incentive for buyers to take a wait-and-see approach, the seller in the
sequential English auction must lower the sequence of starting prices from
what they would be were the auction mechanism a second-price auction. The
sequence of starting prices is set in such a way as to equalize revenues
across the two formats as well as to make the probability of sale and the
allocation of the item identical. Since the participation of interim bidders
in the English auction allows the seller to lower her starting price while
keeping revenues unchanged, one could say that the interim bidders are
\textquotedblleft doing the seller's bidding.\textquotedblright\
A key finding of the paper is that the seller's initial starting price
converges to the lowest buyer valuation when the time between auctions is
sufficiently short. This suggests that the act of fishing has very little
value to the seller. The result is analogous to the Coase conjecture in the
sequential bargaining or durable goods monopoly settings. It is interesting
to note that while the inability to commit to a starting/reserve price
policy has the same impact on revenue when the seller conducts second-price
or first-price auctions, it does not force the seller's initial reserve
price to its lowest possible level. As a consequence, the \textit{start low
end high }phenomenon reported by Cassady could not be explained by treating
the English auction as equivalent to the second-price auction.
Of course the seller's revenues would be greater could she commit to not
relist the item if the initial auction fails to induce any bids. It is then
natural to ask why institutions have not arisen to solve the seller's
commitment problem. After all, \cite{McAdamsSchwartz2007} argue that a role
for the auction house is to eliminate commitment problems that prevent
mutually beneficial transactions from taking place. But in the current
setting, a seller's inability to commit to a starting-price path is
efficiency enhancing as it forces the seller to offer the item at a low
initial starting price, thereby insuring the item ends up in the hands of
whomever values it the most. The auction house would have an interest in the
efficient outcome since in many instances, the auction house is responsible
for assembling the group of bidders. It should therefore not be surprising
that opening bid fishing persists.
An interesting empirical question is whether dynamic considerations are
responsible for the \textit{start low end high }phenomenon described by
Cassady and not other explanations such as affiliated values or auction
fever. The model provides a testable prediction between the time to
relisting and the probability of having a subsequent auction end in a price
of at least $S$ after the initial auction started at $S$ fails to sell:\ the
shorter is the time to relisting, the more likely is a subsequent auction to
reach a price of $S$. In contrast, considerations of affiliated values or
auction fever would reasonably be unaffected by the time to relisting. The
implementation of such a test is left to future research.
\appendix
\begin{center}
\pagebreak
\bigskip {\LARGE APPENDIX}
\end{center}
\section{Proofs}
\subsection{Proof of Lemma \protect\ref{initial}}
1. Fix a starting price $s_{t}$ and for bidder 1, say, let $dB_{1}$ denote
the density of the highest of maximum bid prices of all other $n-1$ buyers
should buyer 1 submit a bid. Upon bidding at the starting price, the auction
will necessarily result in a sale. The expected return to bidder 1 of
playing a strategy of bidding up to some amount $b$ is
\begin{equation*}
\left( v-s_{t}\right) \int_{0}^{s_{t}}dB_{1}+\int_{s_{t}}^{b}\left(
v-B_{1}\right) dB_{1},
\end{equation*}%
where we let $v$ denote buyer 1's valuation. This expression is maximized at
$b=v$ for any set of strategies giving rise to the arbitrary density $dB_{1}$%
.
2. The proof proceeds to show that if some type $v>s_{t}$ is an initial
bidder, then so too is any type $v^{\prime }>v$. Assuming that buyer 1, upon
bidding at the starting price, bids up to $v$ and let $dB_{1}$ denote the
highest of maximum bid prices of all other $n-1$ bidders in the current
period. Let $V_{B}\left( z;v,H_{t}\right) $ denote the continuation payoff
of a type $v$ buyer from the following period on, given history $H_{t}$,
playing the strategy of a type-$z$ buyer in what follows. Further, let $%
dA_{1}$ denote the density of the highest of maximum bid prices of all other
buyers should buyer 1 abstain from bidding. If a type-$v$ buyer is an
initial bidder in period $t$, then%
\begin{equation}
\left( v-s_{t}\right) \int_{0}^{s_{t}}dB_{1}+\int_{s_{t}}^{v}\left(
v-B_{1}\right) dB_{1}\geq \delta V_{B}\left( v;v,H_{t}\right)
\int_{0}^{s_{t}}dA_{1}. \label{v bids}
\end{equation}%
Expression $\left( \ref{v bids}\right) $ states that the surplus from
bidding up to the buyer's valuation must exceed the expected value of not
bidding.
Now suppose, by way of contradiction, that some type $v^{\prime }>v$ finds
it unprofitable to bid at the start price in period $t$. This implies that
\begin{equation}
\left( v^{\prime }-s_{t}\right)
\int_{0}^{s_{t}}dB_{1}+\int_{s_{t}}^{v}\left( v^{\prime }-B_{1}\right)
dB_{1}<\delta V_{B}\left( v^{\prime };v^{\prime },H_{t}\right)
\int_{0}^{s_{t}}dA_{1}\text{.} \label{v' no bid}
\end{equation}%
Since a type $v$ buyer can always adopt the strategy of type $v^{\prime }$,
incentive compatibility implies
\begin{eqnarray*}
V_{B}\left( v;v,H_{t}\right) &\geq &V_{B}\left( v^{\prime };v,H_{t}\right) \\
&=&\sum_{j=0}^{\infty }\delta ^{j}\alpha _{t+1+j}\left( v^{\prime }\right)
\left[ v-m_{t+1+j}\left( v^{\prime }\right) \right] ,
\end{eqnarray*}%
where $\alpha _{t+1+j}\left( v^{\prime };H_{t}\right) $ denotes the
probability, conditional on $H_{t+j}$, that the item is obtained in period $%
t+1+j$ playing the strategy of type $v^{\prime }$ and $m_{t+1+j}\left(
v^{\prime };H_{t}\right) $ is the analogous expected payment. It follows
that
\begin{equation}
V_{B}\left( v^{\prime };v^{\prime },H_{t}\right) -V_{B}\left(
v;v,H_{t}\right) \leq \left( v^{\prime }-v\right) \sum_{j=0}^{\infty }\delta
^{j}\alpha _{t+1+j}\left( v^{\prime };H_{t}\right) . \label{squinching}
\end{equation}
From $\left( \ref{v bids}\right) $ and $\left( \ref{v' no bid}\right) ,$ we
have that%
\begin{eqnarray}
\left( v^{\prime }-v\right) \int_{0}^{v}dB_{1} &<&\delta \left[ V_{B}\left(
v^{\prime };v^{\prime },H_{t}\right) -V_{B}\left( v;v,H_{t}\right) \right]
\int_{0}^{s_{t}}dA_{1} \notag \\
&<&\left( v^{\prime }-v\right) \delta \sum_{j=0}^{\infty }\delta ^{j}\alpha
_{t+1+j}\left( v^{\prime };H_{t}\right) \int_{0}^{s_{t}}dA_{1},
\label{contradiction}
\end{eqnarray}%
where the second inequality follows from $\left( \ref{squinching}\right) .$
Equation $\left( \ref{contradiction}\right) $ necessarily leads to a
contradiction as long as $\int_{0}^{v}dB_{1}\geq \int_{0}^{s_{t}}dA_{1}$
since $\sum_{j=0}^{\infty }\delta ^{j}\alpha _{t+1+j}\left( v^{\prime
};H_{t}\right) $ can be no greater than 1$.$
Now, $\int_{0}^{v}dB_{1}$is the probability of obtaining the item for the
type$-v$ buyer and $\int_{0}^{s_{t}}dA_{1}$ is the probability that the item
goes unsold when the buyer in question abstains from bidding. So too, $%
\int_{0}^{s_{t}}dB_{1}$ is the probability of obtaining the item for a type$%
-s_{t}$ buyer. Since a type $s_{t}$ buyer wins only when he is the lone
bidder, we have that
\begin{equation}
\int_{0}^{s_{t}}dB_{1}=\int_{0}^{s_{t}}dA_{1}. \label{A=B}
\end{equation}%
It follows from $\left( \ref{A=B}\right) ,$ that if we choose $v$ to be some
increment greater than $s_{t}$ and increase the upper integrand on the
left-hand side of $\left( \ref{A=B}\right) $ by that increment, we have that
$\int_{0}^{v}dB_{1}\geq \int_{0}^{s_{t}}dA_{1}.$
Since $v^{\prime }$ was chosen arbitrarily, it must be the case that if some
type $v$ submits a bid in period $t$, then so does every buyer whose
valuation exceeds $v.$
3. We begin by asserting that there exists a minimum starting price such
that all bidder types bid whenever the starting price is less than or equal
to the minimum, regardless of the history. The claim is that \underline{$v$}-%
$\bar{v}$ is one such starting price. We know that in equilibrium, the
seller's expected receipts must be nonnegative--since she can always opt not
to sell--and that a buyer's expected surplus cannot exceed $\bar{v}$ by the
same token. Therefore, the expected surplus for a buyer with valuation
\underline{$v$} is \textit{at most }\underline{$v$} minus the starting
price. This is less than $\bar{v}$ as long as the starting price is less
than \underline{$v$}-$\bar{v}$. Thus, all types bid when the starting price
is less than or equal to \underline{$v$}-$\bar{v}.$
We now calculate a buyer's expected surplus at the minimum starting price.
When all buyer types bid and the starting price is less than \underline{$v$}%
, a given buyer's expected surplus is $\int_{\underline{v}%
}^{v}F_{Y_{1}}\left( y\right) dy\geq 0.$ Notice that a buyer's expected
surplus is independent of the actual starting price as the price will
necessarily be determined by the bid of the second-highest valuation buyer.
This is crucial in what follows.
We work recursively to show that the minimum starting price is in fact
\underline{$v$}. Consider a starting price, $s_{\varepsilon _{1}}=$%
\underline{$v$}$-\bar{v}+\varepsilon ,$ just slightly greater than
\underline{$v$}-$\bar{v},$ such that if the auction started at $%
s_{\varepsilon _{1}}$ fails to sell, the starting price is reduced to
\underline{$v$}-$\bar{v}$ in the following period. When the starting price
is $s_{\varepsilon _{1}}$, a given buyer bids at the start as long as the
surplus gained in the current period exceeds the surplus gained in the
following period should the item go unsold. Assume by way of contradiction
that there exists some $\beta _{\varepsilon _{1}}>$\underline{$v$} that is
the lowest type to bid at the starting price. Consider then a buyer with
valuation $v<\beta _{\varepsilon _{1}}$. That the valuation$-v$ buyer does
not bid in the current period, it must be that,
\begin{equation}
\left[ v-\rho \left( s_{\varepsilon _{1}},v\right) \right] F_{Y_{1}}\left(
\beta _{\varepsilon _{1}}\right) <\delta \int_{\underline{v}%
}^{v}F_{Y_{1}}\left( Y_{1}\right) dY_{1}. \label{epsilon}
\end{equation}%
The left-hand side of $\left( \ref{epsilon}\right) $ indicates the type$-v$
buyer's surplus in the current period, where $\rho \left( s_{\varepsilon
_{1}},v\right) $ denotes his expected payment, taking into account that
interim bidders' highest bid price. Interim bidders' strategies are as yet
undetermined, but we assume that they do not play any dominated strategies
such as bidding above their valuation. The right-hand side of $\left( \ref%
{epsilon}\right) $ indicates the type$-v$ buyer's surplus in the following
period, where the starting price is reduced to \underline{$v$}$-\bar{v}$ and
all remaining buyer types bid.
As the mixed strategy followed by interim bidders is yet undetermined, so is
$\rho \left( s_{\varepsilon _{1}},v\right) $. However, the highest payment
made by a type-$v$ buyer is when all interim bidders bid up to their
valuations, which implies,
\begin{equation*}
\rho \left( s_{\varepsilon _{1}},v\right) =E\left[ Y_{1}|Y_{1}\int_{\underline{v}}^{v}F_{Y_{1}}\left(
Y_{1}\right) dY_{1}. \label{sub-epsilon}
\end{equation}%
Combining $\left( \ref{epsilon}\right) $ and $\left( \ref{sub-epsilon}%
\right) $, we have that,
\begin{equation*}
\int_{\underline{v}}^{v}F_{Y_{1}}\left( Y_{1}\right) dY_{1}<\delta \int_{%
\underline{v}}^{v}F_{Y_{1}}\left( Y_{1}\right) dY_{1},
\end{equation*}%
which is a contradiction. Therefore, it cannot be the case that $\beta
_{\varepsilon _{1}}>$\underline{$v$}$.$We conclude that all types bid when
the starting price is $s_{\varepsilon }$ or less.
For the inductive step, consider a starting price $s_{\varepsilon
_{k}}=s_{\varepsilon _{k-1}}+\varepsilon <$\underline{$v$}, such that if the
item fails to sell at $s_{\varepsilon _{k}}$ the seller reduces the starting
price to $s_{\varepsilon _{k-1}}$in the following period, wherein all
remaining buyers will bid. As before, assume by way of contradiction that
there exists some $\beta _{\varepsilon _{k}}>$\underline{$v$} that is the
lowest type to bid. Consider then a buyer with valuation $v<\beta
_{\varepsilon _{k}}$. That the valuation$-v$ buyer does not bid in the
current period, it must be that,
\begin{equation}
\left[ v-\rho \left( s_{\varepsilon _{k}},v\right) \right] F_{Y_{1}}\left(
\beta _{\varepsilon _{1}}\right) <\delta \int_{\underline{v}%
}^{v}F_{Y_{1}}\left( Y_{1}\right) dY_{1}. \label{epsilon-k}
\end{equation}%
As before, the left-hand side can be minimized by substituting
\begin{equation*}
\rho \left( s_{\varepsilon _{k}},v\right) =E\left[ Y_{1}|Y_{1}$ \underline{$v$} since such starting prices may
actually determine the price, meaning that a buyer's participation decision
does not give rise to equation $\left( \ref{sub-epsilon}\right) $.
4. Let $g\left( u_{t},\beta _{t},\beta _{t+1}\right) $ denote the seller's
expected payoff in state $u_{t}$, when choosing a starting price in the
current period that induces a screening level of $\beta _{t}$, which
subsequently induces a screening level of $\beta _{t+1}$ in the following
period. For ease of notation let $Y_{1}$ denote the highest valuation among
all buyers other than buyer 1 (the reference buyer) and let $F_{Y_{1}}\equiv
F^{n-1}$ denote the distribution of $Y_{1}.$Note that from part 2 of the
lemma, $\beta _{t}$ exceeds $\beta _{t+1}$ and from part 3, $\beta _{\tau }$
is equal to \underline{$v$} if $s_{\tau }\leq $\underline{$v$}. We have that,%
\footnote{%
For a more thorough explanation, see the discussion following expression $%
\left( \ref{g-j}\right) $.}%
\begin{eqnarray*}
{\small g}\left( u_{t},\beta _{t},\beta _{t+1}\right) &{\small =}&{\small n}%
\rho _{t}\left( s_{t},\beta _{t}\right) \left[ F\left( u_{t}\right) -F\left(
\beta _{t}\right) \right] F_{Y_{1}}\left( \beta _{t}\right) \\
&&+n\int_{\beta _{t}}^{u_{t}}\int_{\beta _{t}}^{X_{1}}Y_{1}dF_{Y_{1}}f\left(
X_{1}\right) dX_{1} \\
&&{\small +\delta }\Gamma _{t+1}\left( \beta _{t}\right) {\small ,}
\end{eqnarray*}%
where $\Gamma _{t+1}\left( \beta _{t}\right) $ is the seller's optimal
payoff from period $t+1$ on, beginning at state $\beta _{t}$. The
first-order condition for the seller's optimal choice of $\beta _{t}$
reduces to
\begin{equation*}
\left( 1-\delta \right) \left[ F\left( u_{t}\right) -F\left( \beta
_{t}\right) -\beta _{t}f\left( \beta _{t}\right) \right] F_{Y_{1}}\left(
\beta _{t}\right) \leq 0.
\end{equation*}%
Since $f\left( \cdot \right) $ is positive, there exists some $u^{\ast }>$%
\underline{$v$} $\ $such that for $u_{t}__u^{\ast }$, the
screening level jumps down in discrete steps so that some $u^{\ast }>$%
\underline{$v$} is eventually reached. If the optimum is not an interior
solution, then by definition, $u_{t}____1$, the sequences $\left\{ \beta
_{j}\right\} _{j=0}^{k}$, $\left\{ \rho _{j}\right\} _{j=0}^{k}$, $\left\{
\Gamma _{j}\right\} _{j=0}^{k}$ are such that:
\begin{enumerate}
\item The $\sigma _{j}\underline{v}$.
\item The $\Gamma _{j}\left( x\right) $ satisfying $\left( \ref{gamma-j}%
\right) $ are increasing and continuous.
\item The $\beta _{j}\left( u\right) $ satisfying $\left( \ref{beta-j}%
\right) $ are strictly less than $u$ and increasing.
\end{enumerate}
\end{lemma}
\begin{proof}
Property 1 is proven directly from $\left( \ref{sigma-j}\right) .$
Uniqueness follows from the fact that the left-hand side of $\left( \ref%
{sigma-j}\right) $ is strictly decreasing in $\sigma $ while the right-hand
side is constant in $\sigma $ for $x>\underline{v}$ thus implying a single
point of intersection. Differentiating both sides of $\left( \ref{sigma-j}%
\right) $ with respect to $x$ and rearranging, we have
\begin{equation*}
\frac{d\sigma _{j}}{dx}=\frac{\left( 1-\delta \right) F_{Y_{1}}\left(
x\right) +\left[ x-\rho _{j}\left( \sigma _{j},x\right) \right]
f_{Y_{1}}\left( x\right) }{\frac{\partial \rho _{j}\left( \sigma
_{j},x\right) }{\partial \sigma _{j}}F_{Y_{1}}\left( x\right) }.
\end{equation*}%
The numerator of this expression is positive by the fact that $\rho _{j}\leq
x.$ The denominator is positive under the assumption that $\rho _{j}\left(
\sigma _{j},x\right) $ is increasing in $\sigma _{j}$.
Properties 2 and 3 are proven by induction. It is straightforward to show
that both of properties 2 and 3 are satisfied for $j=2$. Now assume, by way
of induction, that each of properties 2 and 3 are satisfied for $j=k-1.$
Since $\rho _{j}\left( \sigma _{k}\left( x\right) ,x\right) $ is continuous
in both arguments and $\sigma _{k}\left( x\right) $ is continuous in $x$,
then $g_{k}\left( u,x\right) $ is continuous in both arguments. It follows
using standard arguments that $\Gamma _{k}$ is continuous and increasing.
Consider now $u____0$ such that for all $\rho $, $\delta $, and $%
n $, $z_{1}\geq \underline{v}+\varepsilon $. Further, there exists a $%
T<\infty $ such that $z_{T}=\bar{v}.$
\end{lemma}
The proof is identical to that of Lemma 2 in \cite{MV1997}, only with $\rho $
taking the place of $\sigma $, so there is no need to repeat it here.
With the $z_{j}$ so defined, we can define the seller's problem uniquely by $%
u$, independent of $j$. In this way, if $u\in (z_{j-1},z_{j}]$, the seller
chooses the optimal screening level independent of $j$; it just so happens
that such a screening level will lead, assuming optimal behavior in what
follows, to the game ending in $j-1$ more periods should the item fail to
sell. In what follows, we change our notational convention so that a
subscript $t$ denotes $\left( t-1\right) $ periods \textit{after} the
initial period as opposed to $t$ periods \textit{before} the terminal
period. In this way, given $u_{1}=\bar{v}$, we have that $u_{2}=\beta \left(
\bar{v}\right) $, $u_{t}=\beta \left( u_{t-1}\right) $ and $s_{t}=\sigma
\left( \beta _{t}\right) $ for any $t>1.$
The following addresses the three individual components of Proposition \ref%
{PBE}.
Taking as given the sequence of screening levels, $\left\{ \rho _{t}\right\}
$ is the sequence of the expected payment made by the marginal bidder type
in each period. From $\left( \ref{sigma-j}\right) $, $\rho _{T-1}$ satisfies
\begin{equation*}
\left( \beta _{T-1}-\rho _{T-1}\right) F_{Y_{1}}\left( \beta _{T-1}\right)
=\delta \int_{\underline{v}}^{\beta _{T-1}}F_{Y_{1}}dY_{1}.
\end{equation*}%
Working backward, we that in any period $t$\underline{$v$}$.$ To induce $\beta _{1}$, the
seller chooses a reserve $\sigma \left( \beta _{1}\right) $ giving rise to $%
\rho _{1}\equiv \rho \left( \sigma \left( \beta _{1}\right) ,\beta
_{1}\right) $, solving
\begin{equation}
\left( \beta _{1}-\rho _{1}\right) F_{Y_{1}}\left( \beta _{t}\right) =\delta
\left[ \int_{\beta _{2}}^{\beta _{1}}F_{Y_{1}}dY_{1}+\left( \beta _{2}-\rho
_{2}\right) F_{Y_{1}}\left( \beta _{2}\right) \right] , \label{sigma-1}
\end{equation}%
where $\beta _{2}=\beta \left( \beta _{1}\right) $ and $\rho _{2}=\rho
\left( \sigma \left( \beta _{2}\right) ,\beta _{2}\right) $. By the same
logic, the second term on the right-hand side of $\left( \ref{sigma-1}%
\right) $, assuming $\beta _{2}>$\underline{$v$} satisfies
\begin{equation*}
\left( \beta _{2}-\rho _{2}\right) F_{Y_{1}}\left( \beta _{2}\right) =\delta
\left[ \int_{\beta _{3}}^{\beta _{2}}F_{Y_{1}}dY_{1}+\left( \beta _{3}-\rho
_{3}\right) F_{Y_{1}}\left( \beta _{3}\right) \right] .
\end{equation*}%
Following this logic recursively, and noting that
\begin{equation*}
\left( \beta _{T-1}-\rho _{T-1}\right) F_{Y_{1}}\left( \beta _{T-1}\right)
=\delta \int_{\underline{v}}^{\beta _{T-1}}F_{Y_{1}}dY_{1},
\end{equation*}%
since $\beta _{T}=\underline{v},\left( \ref{sigma-1}\right) $ becomes
\begin{equation}
\left( \beta _{1}-\rho _{1}\right) F_{Y_{1}}\left( \beta _{t}\right) =\delta
\tsum\limits_{j=0}^{T-\left( t+1\right) }\delta ^{j}\int_{\beta
_{t+j+1}}^{\beta _{t+j}}F_{Y_{1}}dY_{1}. \label{summation}
\end{equation}%
Using $\left( \ref{rho}\right) $ on the left-hand side of $\left( \ref%
{summation}\right) $,
\begin{equation}
\left( \beta _{1}-\rho _{1}\right) F_{Y_{1}}\left( \beta _{t}\right)
=\int_{\sigma \left( \beta _{1}\right) }^{\beta _{1}}F_{Y_{1}}dY_{1}.
\label{rho-1}
\end{equation}
We are interested in the value of $\sigma \left( \beta _{1}\right) $ as $%
\delta $ gets arbitrarily close to unity. Therefore, in $\left( \ref%
{summation}\right) ,$
\begin{equation}
\underset{\delta \rightarrow 1}{\lim }\delta \tsum\limits_{j=0}^{T-\left(
t+1\right) }\delta ^{j}\int_{\beta _{t+j+1}}^{\beta
_{t+j}}F_{Y_{1}}dY_{1}=\int_{\underline{v}}^{\beta _{1}}F_{Y_{1}}dY_{1}.
\label{limit}
\end{equation}%
Putting $\left( \ref{rho-1}\right) $ together with $\left( \ref{limit}%
\right) $, $\left( \ref{summation}\right) $ implies
\begin{equation*}
\underset{\delta \rightarrow 1}{\lim }\int_{\sigma \left( \beta _{1}\right)
}^{\beta _{1}}F_{Y_{1}}dY_{1}=\int_{\underline{v}}^{\beta
_{1}}F_{Y_{1}}dY_{1}.
\end{equation*}%
The only way this can hold is if $\sigma \left( \beta _{1}\right)
\rightarrow $\underline{$v$}.
\subsection{Proof of Proposition \protect\ref{weak gap}}
Consider the auction in period $t+1$ assuming all prior auctions failed to
induce any bids. From equation $\left( \ref{sequential rationality}\right) $%
, we have that $\beta _{t}>\rho _{t}>s_{t}$. Therefore, the fact that all
auctions conducted through period $t$ failed to induce any bids implies that
the highest possible valuation as of period $t+1$ is $\beta _{t}$. It
follows that
\begin{eqnarray*}
\Pr \left\{ p_{t+1}>s_{t}|\eta _{t}=0\right\} &=&\Pr \left\{ X_{1}\geq \beta
_{t+1},X_{2}>s_{t}|X_{1}<\beta _{t}\right\} \\
&>&\Pr \left\{ X_{1}\geq \beta _{t},X_{2}>s_{t}|X_{1}<\beta _{t}\right\} \\
&=&0
\end{eqnarray*}%
where the inequality in the second line follows from the fact that $\beta
_{t}>\beta _{t+1}$ (which was established in Lemma \ref{initial}. This
establishes that $\bar{\tau}$ is at least unity.
\subsection{Proof of Proposition \protect\ref{strong gap}}
Consider the equilibrium of a game with an arbitrary number of periods,
denoted $T$. Conditional upon period $T$ being reached, the highest possible
valuation is $\beta _{T-1}$. In period $T$, the starting price is reduced to
$\underline{v}$, so that all buyers bid up to their valuations. It follows
that the probability that the period$-T$ auction ends with a price exceeding
the period-$t$ starting price, $s_{t}\equiv \sigma \left( \beta _{t}\right) $%
, is the probability that the second-highest buyer valuation is at least $%
s_{t}$. $\ $Formally,
\begin{equation}
\Pr \left\{ p_{T}>s_{t}|\eta _{T-1}=0\right\} =\Pr \left\{
X_{2}>s_{t}|X_{1}<\beta _{T-1}\right\} . \label{after}
\end{equation}%
In the limit as $\delta \rightarrow 1$, we have from Proposition \ref{Coase}
that $\sigma \left( \beta _{t}\right) \rightarrow \underline{v}$. Combining
this with $\left( \ref{after}\right) $, we have that,
\begin{eqnarray*}
\underset{\delta \rightarrow 1}{\lim }\Pr \left\{ p_{T}>s_{t}|\eta
_{T-1}=0\right\} &=&\Pr \left\{ X_{2}>\underline{v}|X_{1}<\beta
_{T-1}\right\} \\
&=&1.
\end{eqnarray*}
In period $t$, buyers realize that if no bids are received in that period's
auction, they may obtain the item at a lower price in a subsequent auction.
Therefore, a bid is placed at the starting price only if there is at least
one bidder whose valuation exceeds $\beta _{t}$. Therefore, the probability
that the price exceeds $s_{t}$ is,
\begin{equation}
\Pr \left\{ p_{t}>s_{t}|\eta _{t-1}=0\right\} =\Pr \left\{ X_{1}\geq \beta
_{t},X_{2}>s_{t}|X_{1}<\beta _{t-1}\right\} . \label{before}
\end{equation}%
As $\delta \rightarrow 1$, this becomes,
\begin{eqnarray*}
\underset{\delta \rightarrow 1}{\lim }\Pr \left\{ p_{t}>s_{t}|\eta
_{t-1}=0\right\} &=&\Pr \left\{ X_{1}\geq \beta _{t},X_{2}>\underline{v}%
|X_{1}<\beta _{t-1}\right\} \\
&=&\Pr \left\{ X_{1}\geq \beta _{t}|X_{1}<\beta _{t-1}\right\} \\
&<&1
\end{eqnarray*}%
where the inequality in the third line follows from Result 1 of Proposition %
\ref{PBE} which says that the sequence $\left\{ \beta _{t}\right\} $ is
independent of $\delta $, and that $\beta _{t}<\beta _{t-1}$. It follows
that for $\delta $ sufficiently close to 1, the quantity in $\left( \ref%
{after}\right) $ exceeds the quantity in $\left( \ref{before}\right) .$
\pagebreak
\begin{center}
\bibliographystyle{chicago}
\bibliography{acompat,auction}
\end{center}
\end{document}
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