ECONOMIC ANALYSIS GROUP 

DISCUSSION PAPER 

SEQUENTIAL ENGLISH AUCTIONS: A THEORY OF OPENING-BID FISHING 

by 

Joseph Uri Podwol 

EAG 15-1 March 2015 

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U.S. Department of Justice. 
Sequential English auctions: A theory of

opening-bid Öshing

Joseph Uri Podwol

March 4, 2015

Abstract

Cassady (1967) describes an auction in which the auctioneer ìÖshesî 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 o§ 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 di§erence

has important implications for the equilibrium starting-price path, giving rise

to a Coase conjecture in the English auction but not in the second-price auction.

1 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 Öshes 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 ñsoliciting

higher and higher bids until the Önal amount is hammered down with a gavel. But

should the auctioneer fail to Önd 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.

Antitrust Division, U.S. Department of Justice, USA. joseph.podwol@usdoj.gov. The views

expressed herein are the authorís and do not necessarily reáect 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.

1
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:

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. ñTreasures_by_Cynthia

It is surprising, given the breadth of the literature on optimal auctions, that the

tactic of opening bid Öshing has not received more attention. That the high bid is

negatively correlated with the starting price (or opening bid) may well be explained

by a¢ liated values or by auction fever. In the a¢ liated values model of Milgrom

and Weber (1982), buyers shade their bids in order to avoid the winnerís curse. A

lower starting price allows for more bids, which reveal information to buyers and

leads to less bid shading. Ockenfels, Reiley, and Sadrieh (2007) deÖne auction fever

as ìan excited and competitive state-of-mind, in wich the thrill of competing against

other bidders increases a bidderís willingness to pay in an auction.î This deÖnition

encompasses a variety of cognitive biases which o§er a common prediction that a lower

starting price leads to increased competition, which serves as a trigger for excitement

and consequently higher bids.1 While models of a¢ liated values and auction fever

o§er compelling insights, neither explains why the seller wouldnít simply set a low

initial opening bid, as opposed to Öshing for one.

To rationalize opening bid Öshing and the concurrent start low end high phenomenon, we analyze a dynamic model in which each period o§ers the seller an opportunity to sell a single item via English auction with an announced starting price. Buyersí

valuations are independently and identically distributed and the lowest possible bidder type exceeds the sellerís valuation. The seller cannot commit in any period to

a take-it-or-leave-it o§er. 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

o§ 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 o§ 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.

1Ku, Galinsky, and Murninghan (2006) 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.

2
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 su¢ ciently 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 auctionñwhere the starting price

will be even lowerñbefore 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 bidderís valuation,

which is greater than the low initial starting price.

This research contributes to the budding literature on credible sales mechanisms.

McAdams and Schwartz (2007) and Vartiainen (2013) 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 proÖtable to do so.2 The most closely related

research is McAfee and Vincent (1997), who model a seller that cannot commit to a

reserve-price policy when o§ering an item for sale via Örst-price or second-price sealedbid auction, respectively.3 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 Cassadyís start low end high phenomenon arises

when the item is o§ered via English auction but not when o§ered via Örst- or secondprice auction. A seller that o§ers 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 Örst- 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 ìCoase conjecture,îwhich says that when a seller cannot commit to

a take-it-or-leave-it o§er, her initial o§er converges to the lowest buyer valuation type

as the time between successive o§ers becomes short.4 The current study is the Örst

to extend this result to a setting with multiple buyers.

Skreta (2013) studies a similar model to McAfee and Vincent (1997) but where

2Along these lines, Bester and Strausz (2001) model a principle-agent relationship where the

principle cannot commit to following the rules of the mechanism.

3A ìreserve priceîin a sealed-bid auction indicates the lowest allowable bid. This acts similar to

a ìstarting priceîin an open-outry or English auction.

4Coase (1972) 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.

Gul, Sonnenschein, and Wilson (1986) 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 o§er.

3
the seller can commit, after a Öxed number of periods, to keeping the item o§ the

market.5 Burguet and Sakovics (1996) consider a sequential Örst-price auction model

where the number of bidders is endogenous. In both instances, the starting price path

is quite di§erent from the one studied here.

This research speaks to the literature on auction theory and mechanism design.

Vickrey (1961) Örst demonstrated that the English and second-price auctions are

strategically equivalent and that they yield the same expected revenue to the seller.6

Subsequent research showed that in more complex informational environments, they

are not strategically equivalent nor are they revenue equivalent.7In 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 Örst 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 Örst to bid. These e§ects turn out to be

o§setting 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 McAfee

and Vincent (1997) and Section 6 concludes.

2 The model

Consider an auction marketplace consisting of a single seller and n  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 di§er only in their valuation of the

item. Each buyerís 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 inÖnity, over the support

[v; v], where v> 0. The sellerís valuation is normalized to zero and the seller is risk

neutral.8 Assume the sellerís valuation, the distribution of buyer valuations F, and

n are common knowledge.

5The primary contribution of Skreta (2013) is endogenizing the sellerís choice of mechanism.

6They are also revenue equivalent to other common mechanisms such as the Örst-price and Dutch

auctions. See Meyerson (1981) and Riley and Samuelson (1981).

7Milgrom and Weber (1982) and Caillaud and Mezzetti (2004) are two examples.

8The sellerís valuation need not be zero, but it is important for the analysis that it be strictly

less than v: This is known as the ìgapîcase in the literature. The signiÖcance of this assumption is

4
The time horizon is inÖnite with periods indexed t = 1; 2; :::. The pool of buyers

is Öxed over time as are their valuations. In period 1, the seller conducts an English

auction and announces an opening bid or ìstarting price,îs1. The English auction

proceeds as follows: Beginning with s1, 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 ìhigh bid,î

equal to the price at which the last bidder dropped out, or s1 if no other bidders have

entered. If no bids are placed at s1, 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  2 (0; 1),

which is common knowledge.

This setup models open-bid Öshing as a repeated game, where the sales mechanism

in each period is an English auction. The English auction modeled here di§ers from

the typical ìbuttonî auction introduced by Milgrom and Weber (1982) 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 Örst 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 Milgrom and Weber (1982) 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 ñas is typically

assumed in the literature on optimal auctions ñcommit to keeping the item o§ 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 v> 0, all equilibria of the game are necessarily stationary.

This precludes the existence of reputational equilibria (Ausubel and Deneckere 1989),

in which the seller reduces the starting price at an arbitrarily slow rate.

discussed later in this section.

5
Though the sellerís objective function need not be concave to insure a stationary equilibrium,9

such considerations complicate the analysis without adding to the

economic substance. Therefore, we impose the following regularity condition.

Assumption 1 For any u; v 2 [v; v], v 

F(u)F(v)

f(v)

is strictly increasing in v.

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 di§er 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 Cassady (1967).

3 Equilibrium characterization

The equilibrium concept is perfect-Bayesian equilibrium (PBE). Any equilibrium is a

history-contingent sequence of the sellerís starting prices, st, buyersíbidding decisions,

and updated beliefs about the valuations of existing buyers satisfying the typical

consistency conditions. Formally, let H = fs1; s2; :::sg denote the history through

period  of a game that has not ended prior to period  . Since a bid placed in

any period t <  necessarily results in the game ending in that period, H consists

only of the sellerís 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 ñwhether to bid in the period-t auction and if so, what price

to drop out at ñwhich 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 buyerís 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 initial bidders; those willing to bid at the starting price,

and 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 would do if no other buyers had submitted a bid.

Lemma 1 The following characterize the equilibrium in any PBE:

1. It is weakly dominant for an initial bidder to drop out when the price equals his

valuation.

9See McAfee and Vincent (1997).

6
2. In any period t, there exists a marginal type t

, such that every buyer whose

valuation exceeds t; bids at the starting price.

3. Regardless of the history, all buyer types bid at the starting price when the

starting price is at or below v.

4. There exists a period T < 1 , endogenously determined, such that the game

ends in at most T periods.

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 oneís 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 reÖnement.

Result 2 of the lemma is the 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 payo§ from bidding

in the current period is su¢ ciently large to have him forgoe future opportunities.

Since any buyer whose valuation exceeds 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

t. This makes for a simple updating rule in which t becomes the highest type in

period t + 1, denoted ut+1. In what follows, we refer to u as the state in period 

and  as the screening level in period  , 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 t determines a minimum valuation type, any interim bidder

in period t must not have had a valuation above t, otherwise he would have been

an initial bidder. Thus the allocation of the item amongst buyers will be una§ected

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 buyerís own valuation

is individually rational. Only bids above the buyerís own valuation are ruled out in

equilibrium.

To understand how interim bidders a§ect the payo§s 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, st. 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

7
determined by the highest bid placed among all interim bidders and would only equal

stif all initial bidders declined to bid.

Let 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 reÖnement criterion, tis indeterminate. We assume in what follows that

t(st; t) is increasing in both arguments and strictly so for the Örst argument.10

3.1 Characteristics of all equilibria

That the screening level eventually reaches 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 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.

Proposition 1 A PBE consists of a sequence of screening levels ftg

T

t=1 and corresponding starting prices fstg

T

t=1 such that:

1. In any period t, the seller chooses screening level t =  (ut) to satisfy

tf (t) + F (t)  F (ut)  0; (1)

with a strict equality when t > v. The choice of t depends only on ut and the

density f, and is independent of , n, and t.

2. To induce a screening level of t, given t, the period-t starting price st =  (t)

satisÖes the following sequential rationality condition,

[t  t( (t); t)] FY1(t) = 

 



t+1  t+1FY1



t+1+

Z x

t+1

FY1(Y1) dY1

!

;

(2)

where t+1 =  (t) and t+1 = t+1 





t+1; t+1.

3. The allocation of the item, the sellerís revenue, and the probability of sale in a

given period are independent of t.

Equation (1) is the solution to the sellerís 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 sellerís problem in

selecting the optimal starting price, where v would take the place of ut

. However, in

a static setting, the screening level and starting price are one and the same. Within

10This assumption is satisÖed trivially when interim bidders opt not to bid, in which case

t(st; t) = st; and will be shown to be true when they bid up to their valuations as in the equilibrium selected under our reÖnement.

8
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 (2) indicates that the sellerís choice of starting price makes a type t

buyer indi§erent 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 t buyerís expected surplus from bidding at the starting

price,  (t), when he is the lowest type to do so. His expected payment would then be

given by t( (t); t). FY1(t) denotes the probability that type tis the lone initial

bidder.11 The right-hand side of equation (2) indicates a type t buyerís expected

continuation surplus were he not to bid, given an equilibrium starting price of t+1

and screening level of t+1 in the period to follow. In the following period, since

t > t+1, he receives an amount given by the Örst 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.12

Result 3 of the proposition, that the sellerís revenue is independent of t; would

seem counter-intuitive since a higher value of ttranslates to a larger expected payment for the winning bidder, holding Öxed the sellerís starting price. However, the

seller takes tinto account when selecting the starting price. The higher is t, the

lower the starting price must be in order to induce bids from buyers whose valuations

are at least t, the desired screening level.13 In this way, the equilibrium starting

prices corresponding to every possible value of tlie along a continuum. The equilibrium we will select for using our reÖnement criterion yields the highest possible

t and hence lowest possible starting price. The equilibrium yielding the highest

possible starting price is the case in which t = st, 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 t > st) 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.

3.2 Equilibrium reÖnement

Having characterized equilibrium behavior of sellers and initial bidders under any

equilibrium strategy played by interim bidders, we now apply a reÖnement 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

11Using conventional notation, let Y1 denote the maximum of the n  1 competing buyersívaluations. As the vi are independent, FY1

(t)  F (t)

n1

.

12All expectations are taken conditional on state ut being reached. Expression (2) does not

explicitly include ut due to a canceling of terms.

13This follows from equation (2) along with the assumption that t

is strictly increasing in st.

9
possible move is played by each buyer with positive probability.14 Each buyer bids at

the starting price, st, with some probability strictly between zero and one and each

buyer drops out at any price in (st; v] with some probability strictly between zero and

one. The limits of equilibria of such perturbed games as the tremble probabilities go

to zero are extensive-form trembling-hand-perfect equilibria (ETE).15

Proposition 2 There exists a unique ETE in which each interim bidder bids up to

his valuation with probability 1:

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 bidderís valuation. It is then in the interim bidderís interest to remain active

until his valuation is reached. The same conclusion could also be reached less formally

by appealing to the sellerís 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,

 (st; t) = E [max fst; Y1g jY1 < t]

= t 

R t

st

FY1(Y1)dY1

FY1(t)

: (3)

The aggressive bidding by interim bidders in the ETE puts downward pressure

on the sellerís starting price. In the limit as  goes to unity, this gives rise to the

following Coase conjecture.

Proposition 3 In the ETE, for every " > 0, there exists a  < 1 such that for all

  , and for any initial screening level, 1 =  (v) 2 [v; v], the sellerís initial

starting price,  (1), is less than v+".

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

14Trembles by the seller are uninteresting, since the starting price is known to all buyers before

the auction begins.

15The ETE is an extension, due to Selten (1983), of the trembling-hand-perfect equilibrium concept

of Selten (1975). A trembling-hand perfect equilibrium is one that takes the possibility of o§-

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 extensiveform games, the modeler may choose to interpret a tremble as a mistake in a playerís choice of action

at a particular information set or as a mistake in a playerís entire strategy choice. The ETE concept

employs the former interpretation.

10
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.  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 tless

than the value given by expression (3).

4 Bidding dynamics

The Introduction described a sequence of English auctions which give rise to a counterintuitive result. Cassady (1967) 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 start low end high phenomenon. The Örst result ñthe ìweak gapî property

ñdemonstrates that this is a possible outcome. The second resultñthe ìstrong gapî

property ñ demonstrates that when the time between auctions is su¢ ciently 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 [a; 1 + a] distribution,

a 2 (0; 1). 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 sellerís period-t problem reduces

to one of choosing the optimal screening level in the current period,  (u), subject to

the constraint that the type- buyer is indi§erent between bidding in period t and

bidding in period t + 1 and given sequential rationality in what follows. Inserting

F (x) = x  a into (1), the solution to the sellerís problem is given by,

 (u) = 

u=2 if u  2a

a otherwise : (4)

At any state u, the seller cuts the demand curve in half, serving the top half, until

u becomes su¢ ciently 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 (4) 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 2

h

1

2

k1

;

1

2

k11



: In what follows, we solve

11
a two-period game and then show how those results extent to a game of any Önite

length.

4.1 A linear two-period game

Let a 2 [1=3; 1) so that the game ends in at most two periods. The sequence of

screening levels is f1; 2g =

1+a

2

; a	following from (4). To induce all buyer types

to bid in period 2, the seller must set a non-binding starting price of s2 = a.

16

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 1to bid in period 1, the seller

must set s1 low enough that these buyer types strictly prefer bidding in period 1.

Solving for s1 requires making the type-1buyer indi§erent 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 (2). Since all buyer types bid in period 2, 2 = 2 = a: Subsituting

2 = 2 = a, FY1(y) = y  a and 1 = (1 + a) =2 into (2) and solving for s1 yields,

s1 = a +

1  a

2

p

1  : (5)

Notice that as  goes to 1, s1 converges to a as guaranteed by the Coase conjecture.

The 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 s1 if there

are at least two buyers whose valuations exceed s1. The fact that the period-1 auction

failed to elicit any bids implies that there are no buyers whose valuation is as high

as 1. However, since 1 > s1, the high bid in period-2 exceeds s1 with positive

probability after conditioning on the fact that the period-1 auction failed to receive

any bids. We refer to this as the ìweak gapî property, as it is the gap between 1

and s1 that makes this result possible.

A somewhat surprising stronger result is that conditional on period 2 being reached,

the period-2 is actually more likely to have a high bid exceeding s1 than the period-1

auction when the time between relistings is su¢ ciently short. Let pt 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.17 Let t be an indicator of whether the period-t auction elicits a bid (equal

to unity if it does, zero otherwise). Further, let X1 and X2 denote the realizations of

the highest and second-highest buyer valuations, respectively. The probability that

the period-2 high bid exceeds s1 is given by,

Pr fp2 > s1j1 = 0g = Pr fX2 > s1jX1 < 1g

=

1  s1

1  a

: (6)

16Any starting price strictly less than a would also work except in the case of n = 1.

17The thought experiments that follow are concerned with whether pt is greater than st or not.

Therefore, it is immaterial whether pt is assigned a value of st or zero if no bidders entered in period

t.

12
Recall that as  goes to 1, s1 goes to a, so that the right-hand side of (6) goes to 1.

In period 1, the high bid exceeds s1 if there is at least one initial bidder, whose

valuation is at least 1, and a second bidder who could be an initial bidder or an

interim bidder but whose valuation exceeds s1. The probability of this occuring is,

Pr fp1 > s1g = Pr fX1  1; X2 > s1g

< Pr fX2 > s1g

= [1  (s1  a)]2(7)

where the inequality in the second line follows from the fact that 1is bounded above

s1. As  goes to 1, the right-hand side of (7) converges to 1 from below. It follows

that for  su¢ ciently close to 1, G  Pr fp2 > s1j1 = 0g  Pr fp1 > s1g is positive.

We refer to this result as the ìstrong gapîproperty as it requires 1to be su¢ ciently

larger than s1.

18 The strong gap property is illustrated in Figure 1.1, which plots G

over . The lower curves corresond to higher values of a. It is evident that each curve

crosses the horizontal axis for  su¢ ciently close to 1.

0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

Figure 1.1: G as a function of  for a 2 f1=3; 1=2; 2=3; 3=4g :

4.2 A linear k-period game

The above result can be extended to a game of any Önite length. As in the two-period

case, the screening level in the last period of the game is 

(k)

k = a, so that all buyers

are induced to bid. From (4), the state variable in the last period of a kperiod game

is



(k)

k1 =

1 + a

2

k1

;

so that conditional on period k being reached, all buyersí valuations will lie in the

interval ha; (k)

k1

i

:

19 The Coase conjecture guarantees that the initial starting price

18Since 1

is independent of  and s1 =  (1) is decreasing in , 1  s1 is increasing in .

19That 

(k)

k1 > a follows from the fact that for the game to end in at most k periods, it must be

that a 2

h

1

2

k1

;

1

2

k11



:

13
converges to the minimum valuation type, a; as the time to relisting becomes sufÖciently 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, 

(k)

1 =

1+a

2

: Since 

(k)

1 > 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 s1 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 s1 is obtained in period 1.

4.3 General results

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.

Proposition 4 Whenever T > 1, there exists a  > 1 such that for any   ,

Pr pt+ > stjt+1 = 0	> 0:

This says that if we consider any period t < T in which the period-t auction fails

to elicit a bid, any subsequent periodís auction, say period t +  , may have a high

bid of at least st as long as t+1 > st: We know that this is necessarily true for

 = 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.  is the highest integer value for which this continues

to be true.

The strong gap property shows that as the time between auctions becomes sufÖciently 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.

Proposition 5 Whenever T > 1, there exists some ~ < 1 such that for any   ~,

and any t < T,

Pr pT > stjT 1 = 0	> Pr fpt > stg

regardless of T  t, the number of periods required for the starting price to reach v.

Proposition 5 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  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 s1. This logic continues to hold when the period-1 starting price

is replaced with periodt starting price for any t < T.

14
5 English versus second-price auctions

We now return to the case where tcan take on di§erent values and discuss the

di§erences in outcomes between English and second-price auctions when the seller

lacks the ability to commit. Recall that the equilibrium in which t = st generates the

same outcomes (e.g., the allocation of the item, the probability of sale and the sellerís

revenue) as the second-price auction. This is the equilibrium in which interim bidders

do not bid. In contrast, within the ETE of the English auction, interim bidders bid up

to their valuations so that tis given by (3). We now consider di§erences in outcomes

when between equilibria the equilibrium when tis given by (3) (the English auction)

and when t = st (the second-price auction).

The Örst di§erence to notice is that the Coase conjecture (Proposition 3) does

not extend to the second-price auction.20 Theorem 3 in McAfee and Vincent (1997)

shows that the sellerís proÖts converge to that of a one-shot auction with no reserve

price in the limit as  ! 1, but even in the limit, the sellerís initial starting price

remains bounded above the minimum valuation type. Conditional on the sequence

of screening levels, the sellerís initial starting price (or reserve price in the case of

the second-price auction) must be such that it makes the marginal bidder in period

1 indi§erent between bidding in the current auction and waiting for the following

period. When the time between auctions is very short, the seller has to guarantee the

marginal bidder the same expected surplus that he would get in the terminal period,

when the starting price is reduced to v: Formally, the sequential rationality constraint

reduces to:

lim

!1

(1  1) FY1(1) = Z 1

v

FY1(Y1) dY1; (8)

where 1is still fully general as in equation (2) and Y1 denotes the highest valuation of

all buyers conditional on the marginal bidder being the highest valuation type among

the n bidders. If 1 =  (1) as in a second-price auction, then (8) becomes,

lim

!1

[1   (1)] FY1(1) = Z 1

v

FY1(Y1) dY1:

For this equality to hold in the limit,  (1) must strictly exceed v for any 1 >v.

21

The idea is that in the second-price auction, the seller screens out types in the interval

[ (1); 1) with a binding reserve price of  (1). Thus the marginal buyer pays  (1)

for realizations of Y1 in [ (1); 1). By choosing  (1) small enough (but still greater

than v), the seller can make bidding in the initial period just proÖtable for marginal

buyer.

20Further, it does not extend to any equilibrium other than the ETE.

21The right-hand side of the equality is the area under the FY1

curve over the domain [v; 1

]. The

left-hand side of the equality is a rectangle with height FY1

(1) and base 1   (1). Since FY1

is

weakly increasing, if  (1) were to be equal to v, then the area under the curve would lie entirely

inside the rectangle so the equality fails. Thus, the value of  (1) must be su¢ ciently above v to

bring about an equality.

15
Contrast this to the ETE of the English auction. Substituting (3) into (8) yields,

lim

!1

Z 1

(1)

FY1(Y1)dY1 =

Z 1

v

FY1(Y1) dY1:

For this equality to hold,  (1) must be equal to 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 Y1 >  (1) for realizations of

Y1 in [ (1); 1). Under those circumstances, the surplus received by the marginal

bidder is equal to that of a one-shot auction with starting price  (1). A buyer

comparing the surplus from a one-shot auction with starting price  (1) to that of a

one-shot auction with starting price v will choose the former only if  (1

) = v: Thus

the only way the seller can guarantee the type-1 buyer his reservation surplus is by

running an auction with a starting price close to v.

Another important di§erence between the two auction formats is that the secondprice 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 buyerís valuation exceeds the initial starting price with probability approaching one. In the second-price

auction, the initial starting price is bounded above v and as a result may exceed the

second-highest buyerís valuation.

6 Conclusion

This paper began with the task of understanding the bidding dynamics that result

when an auctioneer Öshes for an opening bid. We have studied a model of an English

auction in which the seller goes Öshing for an opening bid when she cannot commit

to a predetermined starting-price path. The outcome observed by Cassady (1967),

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

su¢ ciently 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 di§erence is in the participation of interim bidders, who

enter the bidding only after Örst observing a bid from a competing buyer. 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

16
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 ìdoing

the sellerís bidding.î

A key Önding of the paper is that the sellerís initial starting price converges to the

lowest buyer valuation when the time between auctions is su¢ ciently short. This suggests that the act of Öshing 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 Örst-price auctions, it does not force the sellerís initial reserve price to its lowest

possible level. As a consequence, the 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 sellerís 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 sellerís commitment problem. After

all, McAdams and Schwartz (2007) argue that a role for the auction house is to

eliminate commitment problems that prevent mutually beneÖcial transactions from

taking place. But in the current setting, a sellerís inability to commit to a startingprice path is e¢ ciency enhancing as it forces the seller to o§er 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 e¢ cient 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 Öshing persists.

An interesting empirical question is whether dynamic considerations are responsible for the start low end high phenomenon described by Cassady and not other

explanations such as a¢ liated 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 a¢ liated values or auction fever would

reasonably be una§ected by the time to relisting. The implementation of such a test

is left to future research.

17
APPENDIX

A Proofs

A.1 Proof of Lemma 1

1. Fix a starting price st and for bidder 1, say, let dB1 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

(v  st)

Z st

0

dB1 +

Z b

st

(v  B1) dB1;

where we let v denote buyer 1ís valuation. This expression is maximized at b = v for

any set of strategies giving rise to the arbitrary density dB1.

2. The proof proceeds to show that if some type v > stis an initial bidder, then

so too is any type v

0 > v. Assuming that buyer 1, upon bidding at the starting price,

bids up to v and let dB1 denote the highest of maximum bid prices of all other n  1

bidders in the current period. Let VB (z; v; Ht) denote the continuation payo§ of a

type v buyer from the following period on, given history Ht, playing the strategy of

a type-z buyer in what follows. Further, let dA1 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

(v  st)

Z st

0

dB1 +

Z v

st

(v  B1) dB1  VB (v; v; Ht)

Z st

0

dA1: (9)

Expression (9) states that the surplus from bidding up to the buyerís valuation must

exceed the expected value of not bidding.

Now suppose, by way of contradiction, that some type v

0 > v Önds it unproÖtable

to bid at the start price in period t. This implies that

(v

0  st)

Z st

0

dB1 +

Z v

st

(v

0  B1) dB1 < VB (v0

; v

0

; Ht)

Z st

0

dA1. (10)

Since a type v buyer can always adopt the strategy of type v

0

, incentive compatibility

implies

VB (v; v; Ht)  VB (v

0

; v; Ht)

=

X1

j=0



jt+1+j (v

0

) [v  mt+1+j (v

0

)] ;

where t+1+j (v

0

; Ht) denotes the probability, conditional on Ht+j, that the item is

obtained in period t + 1 + j playing the strategy of type v

0 and mt+1+j (v0

; Ht) is the

18
analogous expected payment. It follows that

VB (v

0

; v

0

; Ht)  VB (v; v; Ht)  (v

0  v)

X1

j=0



jt+1+j (v

0

; Ht): (11)

From (9) and (10); we have that

(v

0  v)

Z v

0

dB1 <  [VB (v

0

; v

0

; Ht)  VB (v; v; Ht)] Z st

0

dA1

< (v

0  v) 

X1

j=0



jt+1+j (v

0

; Ht)

Z st

0

dA1; (12)

where the second inequality follows from (11): Equation (12) necessarily leads to

a contradiction as long as R v

0

dB1 

R st

0

dA1 since P1

j=0 

jt+1+j (v

0

; Ht) can be no

greater than 1:

Now, R v

0

dB1is the probability of obtaining the item for the typev buyer and

R st

0

dA1 is the probability that the item goes unsold when the buyer in question

abstains from bidding. So too, R st

0

dB1 is the probability of obtaining the item for a

typest buyer. Since a type st buyer wins only when he is the lone bidder, we have

that

Z st

0

dB1 =

Z st

0

dA1: (13)

It follows from (13); that if we choose v to be some increment greater than st and

increase the upper integrand on the left-hand side of (13) by that increment, we have

that R v

0

dB1 

R st

0

dA1:

Since v

0 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 v-v is one such starting price. We know

that in equilibrium, the sellerís expected receipts must be nonnegativeñsince she can

always opt not to sellñand that a buyerís expected surplus cannot exceed v by the

same token. Therefore, the expected surplus for a buyer with valuation v is at most

v minus the starting price. This is less than v as long as the starting price is less than

v-v. Thus, all types bid when the starting price is less than or equal to v-v: 

We now calculate a buyerís expected surplus at the minimum starting price. When

all buyer types bid and the starting price is less than v, a given buyerís expected

surplus is R v

v

FY1(y) dy  0: Notice that a buyerís 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 v. Consider

a starting price, s"1 =vv + "; just slightly greater than v-v;  such that if the auction

started at s"1fails to sell, the starting price is reduced to v-v in the following period.

When the starting price is s"1, a given buyer bids at the start as long as the surplus

19
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 "1 >v

that is the lowest type to bid at the starting price. Consider then a buyer with

valuation v < "1. That the valuationv buyer does not bid in the current period, it

must be that,

[v   (s"1; v)] FY1



"1



<  Z v

v

FY1(Y1) dY1: (14)

The left-hand side of (14) indicates the typev buyerís surplus in the current period,

where  (s"1; v) denotes his expected payment, taking into account that interim biddersíhighest bid price. Interim biddersístrategies 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 (14) indicates the typev buyerís surplus in the

following period, where the starting price is reduced to vv and all remaining buyer

types bid.

As the mixed strategy followed by interim bidders is yet undetermined, so is

 (s"1; v). However, the highest payment made by a type-v buyer is when all interim

bidders bid up to their valuations, which implies,

 (s"1; v) = E [Y1jY1 < v] :

Combining this with the left-hand side of (14), we have that

[v   (s"1; v)] FY1



"1



>

Z v

v

FY1(Y1) dY1: (15)

Combining (14) and (15), we have that,

Z v

v

FY1(Y1) dY1 <  Z v

v

FY1(Y1) dY1;

which is a contradiction. Therefore, it cannot be the case that "1 >v:We conclude

that all types bid when the starting price is s" or less.

For the inductive step, consider a starting price s"k = s"k1 + " <v, such that

if the item fails to sell at s"kthe seller reduces the starting price to s"k1in the

following period, wherein all remaining buyers will bid. As before, assume by way of

contradiction that there exists some "k >v that is the lowest type to bid. Consider

then a buyer with valuation v < "k. That the valuationv buyer does not bid in

the current period, it must be that,

[v   (s"k; v)] FY1



"1



<  Z v

v

FY1(Y1) dY1: (16)

As before, the left-hand side can be minimized by substituting

 (s"k; v) = E [Y1jY1 < v]

20
into (16). This leads to condition (15) only with "k

in place of "1, which contradicts

the assumption that "k v. We then conclude that all types bid when the starting

price is s"k

or less. This establishes the result. Note that the recursion does not extend

to s"k+1 > v since such starting prices may actually determine the price, meaning that

a buyerís participation decision does not give rise to equation (15).

4. Let g



ut; t; t+1

denote the sellerís expected payo§ in state ut, when choosing

a starting price in the current period that induces a screening level of t, which

subsequently induces a screening level of t+1 in the following period. For ease of

notation let Y1 denote the highest valuation among all buyers other than buyer 1 (the

reference buyer) and let FY1  F

n1 denote the distribution of Y1:Note that from part

2 of the lemma, texceeds t+1 and from part 3, is equal to v if s v. We have

that,22

g



ut; t; t+1= nt(st; t) [F (ut)  F (t)] FY1(t)

+n

Z ut

t

Z X1

t

Y1dFY1f (X1) dX1

+t+1 (t);

where t+1 (t) is the sellerís optimal payo§ from period t + 1 on, beginning at state

t. The Örst-order condition for the sellerís optimal choice of treduces to

(1  ) [F (ut)  F (t)  tf (t)] FY1(t)  0:

Since f () is positive, there exists some u

 >v such that for ut < u

, F (ut)F (t)

tf (t) is strictly negative. Thus for u < u, it is optimal for the seller to induce

bids from all types thus ending the game.

Next we show that the sellerís beliefs fall below u



in Önite time. For this, we again

examine the sellerís Örst-order condition for the optimal screening level. Solving for

an interior optimum and rearranging terms yields

F (ut)  F (t) = tf (t):

Since f is bounded away from zero, so too is the distance between ut and t. So in an

interior optimum, implying ut > u, the screening level jumps down in discrete steps

so that some u

 >v is eventually reached. If the optimum is not an interior solution,

then by deÖnition, ut < u:

A.2 Proof of Proposition 1

We begin by deÖning a kperiod game in which the seller and each buyer behaves

optimally given the constraint that should the item fail to sell, the game necessarily

ends after k1 periods. For this constrained game, denote the screening level, sellerís

starting price, and expected revenue in the terminal period:

0  v, s0  v, 0 (u)  n

Z u

v

Z X1

v

Y1dFY1f (X1) dX1.

22For a more thorough explanation, see the discussion following expression (18).

21
Expected revenue is calculated by considering Örst the expected payment of some

buyer, say buyer 1, with valuation X1. Since the auction in the Önal period has a

starting price of v; buyer 1ís price will be determined by the maximum of n  1 valuations of other buyers, denoted Y1: Buyer 1ís expected payment is the expectation of

Y1 over [v; X1], where FY1  F

n1 denotes the distribution of Y1: The sellerís expected

revenue is simply n-times the expectation of a given buyerís expected payment.

DeÖne the sequences



j

	k

j=0 , fjg

k

j=0 ,



j

	k

j=0 , fjg

k

j=0 , fgjg

k

j=0

iteratively as follows.

Let j (x) denote the starting price that induces a screening level of x in the jthto-last period of the k period game. Since the type-x buyer wins the auction only

upon being the lone initial bidder, j (x) and j(j (x); x) satisfy



x  j(j (x); x)



FY1(x) = 

 



j1  j1



FY1



j1



+

Z x

j1

FY1(Y1) dY1

!

:

(17)

The left-hand side of the expression indicates a type-x buyerís expected surplus from

bidding at starting price j (x), when he is the lowest type to be an initial bidder.

His expected payment under the circumstance is j(j (x); x). The right-hand side of

the expression gives a type-x buyerís expected continuation surplus, given a starting

price of j1 and a screening level of j1in the period to follow. In the following

period, since j  j1, he receives an amount given by the Örst term, where j1 

j



j1; j1



; in the event that he is the lone initial buyer and an amount given by

the second term when he is bidding against at least one other initial bidder.

Let gj (u; x) denote the sellerís revenue in the jth-to-last period, at state u; when

choosing a starting price that induces a screening level of x. We have that

gj (u; x) = nj(j (x); x) FY1(x) [F (u)  F (x)] + n

Z u

x

Z X1

x

Y1dFY1f (X1) dX1

+j1 (x): (18)

The Örst term represents the sellerís revenue from having only one initial buyer and the

second from having at least two. The third term represents her maximum discounted

return from the following period on, at state x, in the event the current auction fails

to produce a sale. The seller solves the problem,

j (u) = max

xu

gj (u; x); (19)

of which jis the solution, satisfying

j = arg max

xu

fgj (u; x)g : (20)

22
Lemma 2 For a given k > 1, the sequences 

j

	k

j=0,



j

	k

j=0, fjg

k

j=0 are such that:

1. The j < x satisfying (17) are unique and increasing in x for x > v.

2. The j (x) satisfying (19) are increasing and continuous.

3. The j(u) satisfying (20) are strictly less than u and increasing.

Proof. Property 1 is proven directly from (17): Uniqueness follows from the fact

that the left-hand side of (17) is strictly decreasing in  while the right-hand side is

constant in  for x > v thus implying a single point of intersection. Di§erentiating

both sides of (17) with respect to x and rearranging, we have

dj

dx =

(1  ) FY1(x) + x  j(j; x)



fY1(x)

@j(j ;x)

@j

FY1(x)

:

The numerator of this expression is positive by the fact that j  x: The denominator

is positive under the assumption that j(j; x) is increasing in j.

Properties 2 and 3 are proven by induction. It is straightforward to show that both

of properties 2 and 3 are satisÖed for j = 2. Now assume, by way of induction, that

each of properties 2 and 3 are satisÖed for j = k 1: Since j(k (x); x) is continuous

in both arguments and k (x) is continuous in x, then gk (u; x) is continuous in both

arguments. It follows using standard arguments that k is continuous and increasing.

Consider now u < u0 and let x 2 arg max

y

fgk (u; y)gand let x

0 2 arg max

y

fgk (u

0

; y)g :

Suppose, by way of contradiction, that x

0 < x. We have that

gk (u

0

; x) = gk (u; x) + nk(k (x); x) FY1(x) [F (u

0

)  F (u)]

+n

Z u

0

u

Z X1

x

Y1dFY1(Y1) f (X1) dX1 (21)

and

gk (u

0

; x0) = gk (u; x0) + nk(k (x

0

); x0) FY1(x) [F (u

0

)  F (u)]

+n

Z u

0

u

Z X1

x0

Y1dFY1(Y1) f (X1) dX1: (22)

Subtracting (22) from (21), we have

gk (u

0

; x)  gk (u

0

; x0)  [gk (u; x)  gk (u; x0)] =

n



k(k (x); x) FY1(x)  k(k (x

0

); x0) FY1(x

0

) 

Z x

x0

Y1dFY1(Y1)



[F (u

0

)  F (u)] :

(23)

The left-hand side of (23) is non-positive since x is a maximizer of gk (u; ) and x

0

is a maximizer of gk (u

0

; ): We now want to show the right-hand side of (23) to be

positive, so resulting in a contradiction.

23
For compactness, let   k

(k (x); x) FY1(x) and 0  k(k (x

0

); x0) FY1(x

0

).

Using this notation, the right-hand side of (23) is positive if

  

0 

Z x

x0

Y1dFY1(Y1)  0: (24)

From the periodk analogue of (18); we have

 = (1  ) xFY1(x) + k1 + 

Z x

k1

Y1dFY1(Y1),

where k1  k1



k1



k1



; k1



FY1



k1



and k1  k1(x). Using the

fact that, from (18), d=dk1  0,

  (1  ) xFY1(x) + 

0

k1 + 

Z x



0

k1

Y1dFY1(Y1)

= 0 + (1  ) [xFY1(x)  x

0FY1

(x

0

)] + 

Z x

x0

Y1dFY1,

where 

0

k1 < k1 and so too 0

k1 < k1 by the assumption that x

0 < x: It follows

that (24) holds if

(1  )



[xFY1(x)  x

0FY1

(x

0

)] 

Z x

x0

Y1dFY1



 0.

The above is true under the assumption that x

0 < x, thus yielding the desired contradiction.

Having characterized the equilibrium to the arbitrarily constrained k period game,

we can extend the results of Lemma 2 to the unconstrained game. We show that at

any stage of the game, the number of remaining periods is determined solely by u,

the highest potential buyer valuation. This is done by constructing a sequence of

numbers fzjg

T

j=0 iteratively as follows. Let

z1 = sup fuj1(u) = vg

denote the largest value of u such that the seller chooses to end the game immediately

and

zj = min sup ujj(u)  zj1

	

; v

	

denote the largest value of u such that the seller chooses a screening level in the

current period such that the optimal policy from the following period onward has her

end the game in j  1 periods. The following lemma shows that there exists some

T such that when u = v, the seller chooses a screening level such that the optimal

policy from the following period onward has her end the game in T  1 periods.

Lemma 3 There exists an " > 0 such that for all , , and n, z1  v + ". Further,

there exists a T < 1 such that zT = v:

24
The proof is identical to that of Lemma 2 in McAfee and Vincent (1997), only

with  taking the place of , so there is no need to repeat it here.

With the zj so deÖned, we can deÖne the sellerís problem uniquely by u, independent of j. In this way, if u 2 (zj1; zj

], 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 (t  1) periods after the initial period as opposed to t periods

before the terminal period. In this way, given u1 = v, we have that u2 =  (v),

ut =  (ut1) and st =  (t) for any t > 1:

The following addresses the three individual components of Proposition 1.

Taking as given the sequence of screening levels, ftg is the sequence of the expected payment made by the marginal bidder type in each period. From (17), T 1

satisÖes



T 1  T 1



FY1



T 1



= 

Z T 1

v

FY1 dY1:

Working backward, we that in any period t < T  1, tsatisÖes

(t  t) FY1(t) = 

 



t+1  t+1FY1



t+1+

Z t

t+1

FY1(Y1) dY1

!

; (25)

so that the sequence ftg is unique to a given sequence of ftg. The payment by the

marginal type is a known function, t(; ) of the starting price and screening level.

Therefore, for the optimal sequence of screening levels ftg and the corresponding

sequence of payments ftg, an equilibrium starting price  (t) is some price such that

t( (t); t) = t. Under the assumption that t be increasing in both arguments,

 (t) is unique and increasing in t: This establishes part 2 of the proposition.

The sellerís revenue in period T is

g (uT ; v) = n

Z uT

v

Z X1

v

Y1dFY1f (X1) dX1:

Working backward, the solution to the sellers problem gives rise to the Örst-order

condition

tf (t) + F (t)  F (ut) = 0

which characterizes a solution in u, independent of n, ; and . This establishes part

1 of the proposition.

Since the seller continues to relist until a sale is transacted, the item is allocated

to the buyer with the highest valuation, regardless of t. The probability of sale in

some period t is F



t1

n

 F (t)

n

. Since the sequence of screening levels is the

same for each value of t, so is the probability of sale in each period. The sellerís

revenue in a given period is given by the Örst two terms in g (ut; t). Using the fact

25
that ut = t1

, after some manipulation, we have the sellerís period-t revenue is

R



t1; t



= n

Z t1

t



vf (v) + F (v)  F



t1

 FY1

(v) dv

n



F



t1



 F (t)



(t  t) FY1(t):

Substituting in from (25) recursively (T  t) times, we have

R



t1; t



= n

Z t1

t



vf (v) + F (v)  F



t1

 FY1

(v) dv

n



F



t1



 F (t)



T

Pt

j=0



j+1 Z t+j

t+j+1

FY1(Y1) dY1;

which is independent of t.

A.3 Proof of Proposition 2

The e§ect of the trembles on an interim bidders is to create the possibility of winning

the auction when bidding no more than his valuation. This is because 1. the buyer

who bid at the starting price may simply be an interim bidder who bid by mistake;

or 2. all true initial bidders may mistakenly drop out of the bidding before their

valuations are reached. It is su¢ cient to show that given the possibility of obtaining the item, all interim biddersístrategies other than the one proposed are weakly

dominated.

Consider bidder 1, an interim bidder with valuation v  st, and let dB" denote

the density of the highest maximum bid price of all n  1 other buyers, given that

a bid was placed at the starting price. Note that if the probability of all trembles

were zero to be zero, dB" would be equal to dFY1. Suppose buyer 1 chooses some

maximum price b at which to drop out. His expected payo§ from that strategy is

Z b

st

(v  B") dB":

This expression is maximized at b = v for any proÖle of behavioral strategies played

by the other bidders which gives rise to dB":

A.4 Proof of Proposition 3

Suppose the seller wishes to induce a screening level of 1in the initial period. If

1 =v, then she simply chooses a starting price no greater than v and the result is

shown. Assume then that 1 >v: To induce 1

, the seller chooses a reserve  (1)

giving rise to 1   ( (1); 1), solving

(1  1) FY1(t) = 

Z 1

2

FY1 dY1 + (2  2) FY1(2)



; (26)

26
where 2 =  (1

) and 2 =  ( (2); 2). By the same logic, the second term on the

right-hand side of (26), assuming 2 >v satisÖes

(2  2) FY1(2) = 

Z 2

3

FY1 dY1 + (3  3) FY1(3)



:

Following this logic recursively, and noting that



T 1  T 1



FY1



T 1



= 

Z T 1

v

FY1 dY1;

since T = v;(26) becomes

(1  1) FY1(t) = 

T P(t+1)

j=0



j

Z t+j

t+j+1

FY1 dY1: (27)

Using (3) on the left-hand side of (27),

(1  1) FY1(t) = Z 1

(1)

FY1 dY1: (28)

We are interested in the value of  (1) as  gets arbitrarily close to unity. Therefore, in (27);

lim

!1



T P(t+1)

j=0



j

Z t+j

t+j+1

FY1 dY1 =

Z 1

v

FY1 dY1: (29)

Putting (28) together with (29), (27) implies

lim

!1

Z 1

(1)

FY1 dY1 =

Z 1

v

FY1 dY1:

The only way this can hold is if  (1) !v.

A.5 Proof of Proposition 4

Consider the auction in period t + 1 assuming all prior auctions failed to induce

any bids. From equation (2), we have that t > t > st. 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 t. It follows that

Pr fpt+1 > stjt = 0g = Pr X1  t+1; X2 > stjX1 < t

	

> Pr fX1  t; X2 > stjX1 < tg

= 0

where the inequality in the second line follows from the fact that t > t+1 (which

was established in Lemma 1. This establishes that  is at least unity.

27
A.6 Proof of Proposition 5

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 T 1.

In period T, the starting price is reduced to v, so that all buyers bid up to their

valuations. It follows that the probability that the periodT auction ends with a

price exceeding the period-t starting price, st   (t), is the probability that the

second-highest buyer valuation is at least st. Formally,

Pr pT > stjT 1 = 0	= Pr X2 > stjX1 < T 1

	

: (30)

In the limit as  ! 1, we have from Proposition 3 that  (t) ! v. Combining this

with (30), we have that,

lim

!1

Pr pT > stjT 1 = 0	= Pr X2 > vjX1 < T 1

	

= 1:

In period t, buyers realize that if no bids are received in that periodís 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 t. Therefore, the probability that the price exceeds stis,

Pr pt > stjt1 = 0	= Pr X1  t; X2 > stjX1 < t1

	

: (31)

As  ! 1, this becomes,

lim

!1

Pr pt > stjt1 = 0	= Pr X1  t; X2 > vjX1 < t1

	

= Pr X1  tjX1 < t1

	

< 1

where the inequality in the third line follows from Result 1 of Proposition 1 which

says that the sequence ftg is independent of , and that t < t1. It follows that

for  su¢ ciently close to 1, the quantity in (30) exceeds the quantity in (31):

28
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