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FACULTY WORKING
PAPER NO. 1486
CO
H
sample of Auction Design: A Theoretical
Basis for 19th Century Modifications to the Port
of New York Imported Goods Market
Richard Engelbrecht-Wiggans
College of Commerce and Business Administration
Bureau of Economic and Business Research
University of Illinois. Urbana-Champaign
BEBR
FACULTY WORKING PAPER NO. 1486
College of Commerce and Business Administration
University of Illinois at Urbana- Champaign
August 1988
An Example of Auction Design:
A Theoretical Basis for 19th Century Modifications
to the Port of New York Imported Goods Market
Richard Engelbrecht-Wiggans , Associate Professor
Department of Business Administration
My thanks to Michael H. Rothkopf for bringing the historical
example that motivated this paper to my attention. Parts of
this paper draw heavily on, and supersede, an unpublished
working paper by Richard Engelbrecht-Wiggans (1987a).
Abstract
Over 150 years ago, with the express purpose of assuring the
future prosperity of the Port, a New York, auctioneer persuaded the
State to lower the tax rate on goods imported through dockside auc-
tions, and simultaneously, to extend the tax to goods offered for sale
but not actually sold. He suggested that this would encourage the
absolute sale of all goods offered in auctions, that the absolute sale
of offered goods — possibly at bargain prices — would attract more
buyers, and that, ultimately, the State would benefit. Neither the
historical record nor the current theory of auctions provides much
insight into these suggestions. Still, after the change, the Port
prospered as never before.
This paper defines a model of auctions in which potential bidders
join the auction so long as it is in their own best interest to do so,
and the potential bidders presume that the seller will act in his own
best interests — independent of any promises — in the auction itself.
In an analytic example, taxing — at an appropriately lower rate — all
goods offered for sale reduces the sellers' benefits from retaining
the goods, lowers the anticipated reservation price, attracts more
bidders, drives up the expected price, and ultimately benefits both
the sellers and the tax collector. An examination of the example's
underlying structure reveals the, rather general, factors driving this
sequence of results. Finally, a more general argument suggests simi-
lar results for a wider class of models.
1. Introduction
Thousands of years ago, the Babylonians gathered annually to auc-
tion marriable maidens. The auctioneer awarded each damsel to the
prospective bridegroom offering to pay the most to wed her, or, in the
case of less attractive damsels, to the prospective bridegroom asking
to be paid the least to wed her. (See Shubik (1983) for more details,
and for a history of auctions in general.) Since then, an individual
offering to pay the most, or asking to be paid the least, has been
awarded just about any good or service imaginable. By now, the com-
bined value of all the automobiles, houses, horses, farm machinery,
farms, tobacco, antiques, paintings, financial instruments, miscella-
neous junk, contracts of many forms, and — well you name it! — auctioned
in a single day runs to many billions of dollars. Few, if any, market
mechanisms rival the illustrious history and current prominence of
auctions and related forms of competitive bidding in pricing goods and
services .
In particular, auctions play an important role in the history of
the Port of New York. Albion (1961, pp. 276-279, 410) reports that
the War of 1812 cut off outside markets and created high surpluses of
British goods. With the end of the war, markets reopened, and a flood
of imported goods hit American ports. Importers anxious to quickly
dispose of their goods turned to auctions. Soon, New York's dockside
auctions alone handled one-fifth of the whole nation's imports.
Perhaps surprisingly for a field of such immense practical impor-
tance, the theory of auctions and competitive bidding in general, and
the design of such mechanisms in particular, blossomed only very
-2-
recently. As indicated by the survey of the field by Engelbrecht-
Wiggans (1980), very little of the currently available theory predates
1960. Within two decades, the Stark and Rothkopf (1979) bibliography
listed almost 500 works studying various aspects of auctions and com-
petitive bidding; much of the current, formal mathematical theory
postdates this bibliography. Still, even now, the theory still leaves
many questions unanswered.
Without the current theory to guide design decisions, how did so
many different forms of auctions and competitive bidding come about —
progressive oral auctions of farm machinery, descending silent auctions
of Dutch cut flowers, sealed bid auctions on Federal mineral rights,
multi-stage auctions of defense system contracts, as well as many
variations within each of these basic forms. Presumably, evolution
played some role; an auctioneer might make a haphazard choice, but
only the fittest mechanism for each type of situation would tend to
survive. In addition, we suggest that conscious design, experimenta-
tion, and evaluation also played a role.
In fact, a timely, well conceived proposal appears to have helped
New York establish itself as the chief American seaport and metropolis.
Specifically, Albion reports that by 1817 the volume of imports auc-
tioned in New York had grown to the point that the city's auctioneers
feared the British might soon divert imports to less glutted markets.
To forestall this possibility, a New York auctioneer by the name of
Abraham Thompson proposed legislation that would, as he later boasted,
"cause all of the Atlantic cities to become tributary to New York."
The proposed legislation became law and reduced the tax rate on goods
sold at dockside auctions by one-third to two-thirds.
-3-
At first glance, Thompson's proposal suggests a very simple argu-
ment. Reducing the tax rate makes New York's auctions relatively more
attractive than before. By attracting more sellers — or by retaining a
larger fraction of current sellers — the State could make up in volume
for any taxes lost on individual goods.
Thompson, however, clearly had a much more profound argument in
mind. In fact he later explained in Hunt's Merchant Magazine (as
quoted by Albion):
Every piece of goods offered at auction should posi-
tively be sold, and to encourage a sale, the duty
should always be paid upon every article offered at
auction... The truth was, that both in Boston and
Philadelphia, the free and absolute sale of goods
by auction was not encouraged. (Lt did not appear
to be understood.) In Philadelphia, goods were
allowed to be offered, and withdrawn, free from
state duty, and the purchaser went to the auction
rooms of that city with no certainty of making his
purchases. He was not certain that the goods would
be sold to the highest bidder.
Not only did Boston and Philadelphia apparently not understand the
benefits to be gained from encouraging the absolute sale of all goods
offered for sale, even the current theory of auctions and competitive
bidding hardly addresses the subject! We shall take steps to develop
the missing theory.
Clearly, Thompson makes two points. One, the proposed change in
taxes would encourage the absolute sale of all goods offered.
Although the record fails to record Thompson's thoughts on why the
proposed changes should result in lower reservation prices, there can
be little doubt that he felt that they would do so. Two, the State —
and presumably also the auctioneers — would benefit from the absolute
-4-
sale of all goods offered. Again, the record provides few details.
However, it does reveal that the price at which goods sold varied con-
siderably from sale to sale. Thus, a wiley seller might speculate; if
the bidding on a particular good failed to reach some critical
"reservation price," the seller could withhold the merchandise from
sale in the hope of obtaining a better price for it later, possibly
elsewhere. In fact, Myerson (1981) suggests that a strategic seller
should set the reservation price strictly greater than his residual
expected value for the goods if they fail to sell in this particular
auction. So, at the time that a merchant was trying to decide whether
to incur the cost of travelling to New York to bid in a particular
good of interest to him, he was, according to Thompson, not even
"certain that the goods would be sold." Apparently Thompson felt that
this discouraged merchants from even attending the auctions, to the
ultimate disadvantage of both the State and the auctioneers.
In short, Thompson felt that lowering the tax rate, but extending
the tax to goods offered for sale even if not sold, would encourage
lower reservation prices. Lower reservation prices — more specifically,
the increased probability of advertised goods actually being sold,
possibly at bargain prices — would attract more merchants to the auc-
tion. This increased number of bidders would firm up prices depressed
by the glutted nature of the market, and ultimately the State would
benefit, or so Thompson seems to have argued.
It worked! At the time of the change, Boston, New York, and
Philadelphia handled roughly equal volumes of imports. By the comple-
tion of the Erie Canal in 1825, however, New York's volume of imports
-5-
had grown to three times that of the other two ports. Perhaps
Thompson deserves at least some of the credit so often accorded to the
Erie Canal in establishing New York as the nation's chief seaport and
metropolis.
This paper develops a theoretical basis for Thompson's sug-
gestions. Section 2 presents an example in which things do work, out
as Thompson seems to have expected. Subsequent sections examine what
underlies the example, and argue that the results illustrated by the
example occur much more generally than just under the specific con-
ditions of the example.
Specifically, two phenomena underlie the example. One, ex-ante
(before potential bidders decide whether or not to attend and partici-
pate in the auction) the seller prefers a lower reservation price than
ex-post (after bidders have committed themselves to participating);
that means, as Thompson suspected, ex-ante, sellers would benefit from
changes which ex-post encourage the absolute sale of goods. Two,
Thompson's proposal reduces the ex-post optimal reservation price,
thereby in effect assuring potential bidders of a reservation price
closer to the (lower) reservation price that the seller would have
like to have been able to commit to ex-ante; by setting the tax rates
appropriately, both the seller and tax collector share in the
resulting gain, a gain that comes not at the expense of the bidders,
but rather as a result of a more efficient market.
Our investigation of these phenomona comprises three sections.
Specifically, Section 3 defines a general model of oral auctions with
bidders who privately know their respective values for the object
-6-
being offered, and establishes that a reservation price equal to the
seller's reservation value results in the socially efficient number of
participants. An investigation of reservation prices and subsidies
from the seller's perspective; reveals that the seller prefers a res-
ervation price affecting the socially optimal number of participants,
or a subsidy that in effect rounds the number of participants up to
the next integer. Thus, roughly speaking, except for the discreteness
in the number of bidders, in the oral auction model, the seller would
prefer to commit ex-ante a reservation price equal to the reservation
value; Engelbrecht-Wiggans (1987b) illustrated this for specific
examples. We then argue more generally that the seller may prefer an
ex-ante reservation price strictly less than the ex-post optimal
reservation price. Next, Section 4 argues that Thompson's proposed
change reduces the ex-post optimal reservation price, thereby if
effect allowing the seller to commit to a lower — and closer to the
ex-ante opt imal--reservat ion price than previously possible. Finally,
Section 5 summarizes the paper.
2 . The Example
This example demonstrates that things can work out as Thompson
suggested. In the example, taxing all goods offered for sale, but
taxing them at an appropriately lower rate, drives down the seller's
optimal reservation price; as Thompson suggested, the change in tax
structure encourages the absolute sale of all goods offered for sale.
Specifically, regardless of what promises may have been made, little
prevents the seller from arbitrarily resetting the reservation price
at any time throughout the auction. The potential bidders know this
-7-
and correctly anticipate (on average) what reservation price the
seller will ultimately settle on. This affects how many potential
bidders decide to attend (and bid in) the auction; with everything
else equal, lower reservation prices attract more bidders. The addi-
tional bidders, one of whom might value the goods offered more highly
than any other bidder, push the expected price up more than enough to
offset any losses attributable to the lowering of the reservation
price. In the end, the final tax structure happens to split the
increased revenues between the seller and the tax collector so that
both parties benefit from the change, again just as Thompson seems to
have expected.
In some ways, the example — and the subsequent models — resembles
previously studied models. Most notably, the seller (in setting the
reservation price) and the potential bidders (in deciding whether or
not to attend and bid in the auction, and in deciding how they bid)
act to maximize their own expected net profit conditional on what they
know or presume about the state of Nature and subject to any restric-
tions placed on them by the model. Specifically, not only are the
individual decision makers risk neutral, but nothing other than mone-
tary profit enters into their utility functions. While this assump-
tion simplifies the analysis significantly, it is not critical to the
basic nature of the results.
Two aspects of the example — and of the subsequent models —
distinguish it from most previously studied models. First, the tax
structure appears explicitly in the model; this allows us to study how
the outcome of the auction varies with the tax structure. Second, the
expected number of bidders varies with the potential bidders' percep-
tions of how profitable it would be to attend and bid in the auction;
so endogenizing the number of bidders recognizes that the auction's
characteristics — say, for example, the specific reservation price that
the seller ultimately settles on — may affect who attends and bids in
the auction. Each of these two aspects will be discussed in some
detail.
To parameterize the tax structure, imagine the tax to consist of
two components. One, the seller pays a fraction a (0Ka<l) of the
winning bid to the tax collector if someone other than an agent of the
seller (or the seller himself) ends up with the goods. Second, the
seller also pays a fraction 6 (CK8, a+8<l) of the winning bid regard-
less of who wins. In the example, imagine that the seller implements
a reservation price of r by having an agent submit an opening bid of
r; thus the "winning price" equals the reservation price when the
seller retains the goods. In the subsequent models, however, other
definitions of the winning price for the case when the seller retains
the goods work just as well. Thus, in terras of this notation,
Thompson proposed going from a system with a positive and 8 zero to
one with a zero and 8 positive, but less than the original a.
The tax rates end up affecting the seller's choice of reservation
price. In particular, the seller sets a reservation price r below
which he will not sell a particular good; this reservation price may
exceed the reservation value the seller derives from the good when
he retains it. To avoid confounding the analysis with issues extra-
neous to our primary investigation, ignore the principal-agent problem
-9-
by imagining that the seller acts as his own auctioneer; the seller
directly sets the reservation price to maximize his expected net sales
revenue. As the tax structure changes, the tradeoff between selling
and retaining the goods changes, and therefore the seller's choice of
reservation price changes.
The model's assumptions as to when the seller sets the reservation
price critically affect the outcome. A seller might advertise that
all goods will be sold without reserve, or otherwise try to commit to
some specific reservation price before potential bidders decide
whether or not to attend and bid in the auction. In practice though,
little prevents the seller from implementing some higher reservation
price in the auction itself. For example, the seller might have a
"shill" bid on his behalf, or have the auctioneer pretend to observe
bids when none occur if so doing is in the best interests of the
seller. Given the difficulty of enforcing a pre-announced reservation
price or of detecting deviations from it — especially in the case of
many, many different individual foreign ship owners each importing
goods to the Port of New York only every once in a while — we model the
seller as setting the reservation price only after potential, bidders
have decided whether or not to attend and bid in the auction; as we
will see, this results in higher reservation prices than when sellers
can convincingly commit themselves to whatever reservation price they
want before potential bidders decide whether or not to attend and bid
in the auction.
Goods retained by the seller in one auction might be offered for
sale again in another, later auction. Let v denote the expected
o
-10-
value that the seller derives from the goods conditional in retaining
thera in some specific auction; in some cases, this reservation value
v might be a constant, while in other cases it might depend on the
reservation price and the number of bidders in the auction in which
the goods failed to sell. To keep things as simple as possible, in
the example the reservation value (as opposed to reservation price)
equals zero; for instance, in the Dutch cut flower auctions, any goods
not sold are destroyed and therefore of no value to the seller. (This
assumption that v equals zero will be relaxed later.) In general,
the seller need never consider reservation prices less than his reser-
vation value, but, as Myerson (1981) suggests, may profit from setting
the reservation price strictly greater than the reservation value.
Presumably, potential bidders have some idea as to what the reser-
vation price will actually be. In practice, potential bidders may
base their ideas on past experience. In our analysis, to keep things
simple, we presume that potential bidders perfectly anticipate what
reservation price the seller ultimately chooses to implement; perhaps
our potential bidders are examples of the proverbial "perfectly
rational player" and correctly analyze the seller's problem Co deduce
the "optimal" reservation price, or perhaps they have enough prior
experience with similar auctions.
Given the potential bidder's ability to anticipate the actual
reservation price, any changes in tax structure that affect the choice
of reservation price also indirectly affect the number of bidders. In
fact, for any fixed collection of bidders, lowering the reservation
price both increases the probability that some bidder (as opposed to
-11-
an agent of the seller) wins the good being auctioned, as well as
decreases the expected price paid by a winning bidder; in short,
decreasing the reservation price makes the auction more profitable to
any fixed collection of bidders. This increased profitability makes
the auction more attractive to those who would otherwise decide that
the auction was not attractive enough to attend and bid in it. In
fact, in our models, lowering the reservation price results in an
increased expected number of bidders.
By endogenizing the setting of the number of bidders, we obtain
results not possible from the more traditional models that exogenously
specify who bids. In fact, this endogenizat ion results in the tax
structure affecting the expected number of bidders, and therefore also
affecting the outcome of the auction. This seems to lie at the core
of Thompson's arguments.
For the purpose of the example, consider a specific mechanism for
determining the expected number of bidders. As already suggested, in
the long terra, the number of individuals in the business of retailing
goods bought wholesale at Port of New York dockside auctions — the
number of "merchants" in the greater New York area — varies with the
expected profitability of the auction to the bidders. Specifically,
if another individual could profitably enter the business, we presume
that such an individual would have already entered the business. Con-
versely, if a merchant would be better off leaving the business, that
individual would have already left. Thus, at least in the symmetric
case, for any given reservation price, the number of merchants will be
as large as possible without driving individual merchant's expected
profits negative.
-12-
Not every merchant necessarily attends every auction. For in-
stance, a merchant may travel to New York from time to time depending
on his current inventory and on what the auction's advance notices
list as to be offered for sale. Therefore, we view the number of
merchants attending any particular auction and bidding in it — the
"number of bidders" — as the outcome of some stochastic, possibly
degenerate, process. Not only does this add realism to our model, but
it also reduces the effect of any one additional merchant on the
expected number of bidders; this allows small changes in the reser-
vation price to be large enough to affect a discrete change in the
number of merchants.
To define the distribution of the number of bidders, let P denote
K.
the probability that exactly k merchants decide to go attend the auc-
tion and bid in it. In the example, P equals the binomial probability
K.
( )p (l-p) , where n denotes the number of merchants and p denotes
the probability of any particular merchant participating in the auc-
tion; this distribution serves the purposes of the example par-
ticularly well in its being mathematically very tractable while still
allowing us to vary the expected number of bidders in sufficiently
small increments. Other distributions, including the degenerate case
of a' constant collection of bidders, produce results similar to those
illustrated by the example.
The example focuses on the auctioning of a single object. On the
one hand, this only crudely approximates the actual situation with its
many, some simultaneous, auctions on any given day. On the other
hand, this focusing on a single object allows us to isolate the effect
-13-
that the change of tax structure has on the expected revenue from each
unit of good offered from any more direct effect the lowered tax rates
might have in attracting additional goods to the market. In fact,
extending the tax — at an appropriately lower rate — to all goods
offered for sale increases the seller's (as well as the tax collec-
tor's) expected revenue from each item offered, and this increases the
attractiveness of the auction to sellers beyond the more direct effects
of a lowered tax rate (and also assures that the tax collector bene-
fits regardless of how many — or few — additional sellers the new rules
attract) .
Individuals' decisions whether to attend the auction and how to
bid if they do attend depend on what they know about the goods to be
offered. For our purposes, imagine that before bidders arrive at the
auction site, they all have exactly the same information about the
goods; perhaps this common public knowledge coraes from the auction
advertisement. Thus, the decision whether to attend the auction must
be made before merchants have any unshared individual-specific infor-
mation about the goods. This not only simplifies the analysis, but
avoids confounding the results with any selection effects, and does so
without changing the basic nature of the results.
Once at the auction site, however, a merchant inspects the mer-
chandise and thereby gains some private insights about his own value
for the goods, or possibly about how others might value the goods.
The example presumes that each bidder inspects the merchandise care-
fully enough to remove any uncertainty about his own value (gross of
any costs already incurred) for the merchandise. Moreover, the
-14-
exaraple models these values as being dependent draws from a random
variable with cumulative distribution function F(») and independent of
the number k of attendees; specifically, the example models the values
as being uniformly distributed on the unit interval. Of course, both
the distribution of the number of attendees and of their values might
vary with what is being offered for sale, or more generally, with the
actual pre-sale common public information; we avoid this complication
by simply- holding what is offered — and the pre-sale information — fixed.
To attend the auction, a merchant incurs a known fixed cost of c.
Perhaps this represents the cost of travelling between the merchant's
retail location and the Port of New York. More generally, this may be
viewed as the cost of private information, for example, the cost of
collecting seismic data about an offshore tract (possibly) containing
oil and other minerals. Alternatively, one might view this cost as
some amor it izat ion of the merchant's fixed cost of being in the busi-
ness of reselling imported goods obtained at Port of New York, auctions.
In any case, merchants incur a fixed cost before obtaining any private
information about the merchandise. However, once merchants incur this
cost, only the winner of an item incurs any additional cost; thus we
presume that all of the merchants attending an auction actually bid,
and therefore refer to them simply as "bidders" throughout the paper.
Given the independent privately-known values nature of the
example, the revenue equivalence results of Engelbrecht-Wiggans (1988)
and Myerson (1981) assure quite generally that any pricing rule for an
auction with a continuum of allowable bids — whether it be first price,
second price, the outcome of an oral auction, or some other function
-15-
of the equilibrium bids — generates the same expected selling price at
equilibrium (for that pricing rule) for a fixed number of k bidders so
long as the bidder with the highest value for the good wins it if and
only if his value exceeds a fixed screening level. Clearly, this
invariance of the expected selling price also assures that the bid-
ders' expected profits, the seller's expected net revenue, and the
tax collector's expected receipts be independent of the pricing rule.
So, with little loss of generality, within the example, derive the
expected profit and revenue expressions as if we conducted a sealed-
bid second-price auction; specifically, the seller implements a res-
ervation price of r by submitting a sealed bid of r himself, the high
bidder wins, and the winning bidder (who might be the seller) pays the
seller an amount equal to the winning bid; the seller must then pay
the appropriate taxes. The equilibrium in this unrealistic auction
mechanism generates the same expected revenues and profits in the
example as would any equilibrium of any more realistic or commonly
used high valuer wins auction mechanism with the same screening level
r .
For this second price auction mechanism, Vickery (1961) established
that each bidder has the dominant optimal bidding strategy of bidding
equal to his own value. In this case, with everyone bidding equal to
their value at equilibrium, the screening level coincides with the
seller's reservation price. (If the pricing rule changes, the seller
may have to change his reservation price in order to still effect the
same screening level.) All this makes for relatively simple deriva-
t ions .
-16-
In particular, start by looking the expected value of goods trans-
ferred by the auction to merchants. In independent private values
auctions, this value equals
k=°° x=°°
Z P J xkFk_1(x)dF(x)
k=0 x=r
and, in our example with its Binomial distribution for the number of
bidders and standard uniform distribution for each bidder's value,
evaluates to
k=0°
Z C)pkU-p)n~kh~T- U-rk+1)].
k=0 k K L
Now turn to the seller's and tax collector's revenues. Start by
defining G, (r) — to be interpreted as the expected payments to the
seller by his agents when the seller retains the goods — to be
oo x=r
Z P J xkFk_1(x)dF(x),
k=0 x=0
which, in our example, evaluates to
„ ,nN k, , xn-kr k k+1
Z ( )p (1-p) [— r
k=0 K K l
Then, define G„(r) — to be interpreted as the expected payments by
merchants to the seller — to be
-17-
W= oo x= °°
l P [rk(l-F(r))Fk"1(r) + J xk(k-l) ( l-F(x) )Fk~2(x)dF(x) ,
k=0 x=r
which, in our example, evaluates to
\n ,^ '<,, ,n-krk-l k 2k k+1
k=0
Now, the expected net revenue to the seller after taxes may be written
as
-8G1(r) + (l-(a+B))G2(r)
while the tax collector's expected receipts may be written as
8GL(r) + (a+6)G2(r)
Thus, the expected profit to the bidders collectively (net of payments
and of participation costs) equals
E P [ J xkFk 1(x)dF(x)-kc] - G (r)
k=0 x=r
which, our our example, evaluates to
.. , nN k, . ,n-k, 1 k k k+1 ,
,sn (k)p (1_p) [k+T" r + k+Tr <c
k=0
To characterize the reservation price, differentiate the seller's
expected net revenue with respect to r, set the resulting expression
equal to zero, simplify, and remember the correspondence between
-18-
reservation price and screening level to obtain the following neces-
sary condition for a nontrivial optimal reservation price r* :
(l-(ct+6))(l-F(r*)) = (l-a)r*dF(r*)
For the standard uniform distribution, this necessary condition
simplifies to
r
* =
l-(a+B)
(l-(a+8)) + (1-a)
and happens to be a sufficient condition.
Note that this condition is independent of the P, 's. In par-
ticular, for a binoraially distributed number of actual bidders with
independent private values, the optimal reservation price does not
depend on the parameters p and n. Also note that for 8=0, this con-
dition is independent of a. However, for a=0, the condition does
involve 6. While this dependence of the optimal reservation price on
the tax rate might at first appear to be a drawback of Thompson's pro-
posed change, we suggest that in practice sellers discover the optimal
reservation price by some iterated trial and error process rather than
by solving the above stated necessary condition; in practice, the
optimal reservation price should be no more difficult to discover
after the change in tax structure than before.
Given these expressions, consider what happens for specific choices
of the parameters. In particular, start with 8=0, p=0.1, c in the
interval (0.0897, 0.0926], and a < 0.1449; this illustrates the pre-
Thorapson situation. The first column of Table 1 summarizes the
results .
-19-
Then consider the post-Thompson cases illustrated by 6=0.1, a=0,
p=0.1, and c either in the interval (0*0882, 0.0912] or the interval
(0.0912, 0.0944]; these two intervals for c result in different
numbers of merchants n, but the two intervals together cover the
interval for c in the pre-Thompson case. The second and third columns
of Table 1 summarize the results for these cases. Notice that just as
Thompson anticipated, changing from taxing only those goods sold to
taxing — at an appropriately lower rate — all goods offered for sale
increases the expected number of bidders; n increased from 10 to 12 or
13 depending on the exact value of c, and so the expected number of
bidders increased from 1.0 to 1.2 or 1.3. The change in tax structure
also resulted in the seller adopting a lower reservation price; again
just as Thompson anticipated. In the end, the expected total revenue
increased, and we restricted a so that both the seller and tax collec-
tor benefit from the change.
3. Ex-Post vs. Ex-Ante Optimal Reservation Prices
This section examines the basic structure of the previous example
in attempt to understand what affects the amount of money available to
be split between the seller and the top collector. In particular, we
define "oral auctions with privately known (but not necessarily inde-
pendent) values." In such auctions, the winner pays a price closely
related Co the nearest competitors estimated value for the object, and
therefore the winner has an expected profit closely related to the
increase in social value generated by the auction as a result of his
participation. If the number of bidders increases continuously until
-20-
Table 1: Summary of Example Parameters and Numerical Results
Case:
Parameter values:
Pre-Thorapson
I
< 0.1449
0
0.1
(0.0897,0.0926;
Post-Thompson
I la lib
0
0.1
0.1
0
0.1
0.1
(0.0882,0.0912] (0.0912,0
Consequences
r*
n
1/2 (=0.5)
10
9/19(s0.4737)
13
9/19(50
12
Expected Revenues
total
seller
taxes
0.2160
0.2160 (1-a)
(< 0.2160 for all a 2 0
0.2160a
(50.0313 for a = 0.1449)
0.2676
0.2346
0.0330
0.250
0.219
0.031
(Mote: All numbers rounded to four decimal places)
-21-
none else could profitably enter, then bidders make zero profit, the
seller and tax collector together capture the full net social value
generated by the auction for any fixed number of bidders, and bidders
enter until the social value is maximized as a function of the number
of bidders. Therefore, as Engelbrecht-Wiggans (1987b) previously
established for a specific example, if the seller could commit to a
reservation price ex-ante, setting it equal to the seller's reser-
vation value and letting the bidders then in effect set the number of
bidders maximizes the expected total revenue. Given the integrality
of bidders, total revenue may benefit from setting a higher reser-
vation price, changing an entry fee, or providing a subsidy, but in
any case only to the extent of in effect rounding the number of bid-
ders up or down to the next integer.
We start by defining a model of oral auctions. In particular, the
auctioneer starts by asking a price low enough so that at least two
bidders (one of which may be a shill acting on behalf of the seller)
would be willing to pay that price if offered the object on a take-it-
or-leave-it basis. If someone "bids" — indicates a willingness to take
the object at the current asking price — the auctioneer increases the
asking price by some pre-specif ied increment. If no one bids, then
the last bidder wins and pays the amount he last bid.
Look, at the action from the viewpoint of the next to last bidder;
we call this bidder the "price setter." Let p„ denote the amount bid
by the price setter, and let p. denote the amount paid by the winner
(of course, p. > p~). We presume that by bidding p„ , the price setter
indicates that his expected value for the object conditional on
-22-
everything he curreatly knows and conditional on the presumption (in-
correct, as it turns out) that no one will outbid him equals at least
p?; in particular, we rule out the possibility that the price setter
had such accurate information about what other bidders would do so
that he bid up the price beyond his own value certain that someone
else would eventually save him from winning the object. Furthermore,
in not being willing to outbid the winner, we presume that each bidder
other than the winner indicates that for each allowable bid level p >
p., his expected value for the object conditional on what he now knows
and conditional on no one else outbidding him if he were to bid p is
strictly less than p.
Our example illustrates a special case of the oral auction model
defined so far. In particular, the expected values just mentioned
equal the expected value of the object to a bidder conditional on what
he knew at the beginning of the auction and conditional on no one else
bidding higher than the current asking price; that is, the expected
value is functionally independent of anything a bidder learns about
his own value for the object through the actions of other bidders.
Since the bidder "knows" his (expected) value independent of what
others reveal, we call this the case of "privately known values."
Also, in the example, the asking price in effect rises con-
tinuously. As a result, the bidder with the highest (privately known)
value wins the object and pays an amount equal to the second highest
(privately known) value. Raising the price continuously guarantees
that the highest valuer wins, and that the second highest valuer
becomes the price setter. In addition, raising the price continuously
-23-
guarantees that the winner's price is exactly equal to the price set-
ters' value, rather than the price setter's value rounded up the next
allowable bid level. Taken together, this results in the winner
paying an amount equal to the second highest (privately known) value.
For the remainder of this section, consider only oral auctions
with privately known values in which the asking price rises con-
tinuously. This simplifies the statement of the results. The example
satisfies these restrictions. And, as we relax these restrictions
slightly, we expect the results to change only incrementally. Thus,
adopting these restrictions simplifies the analysis without unduly
limiting our insights into how the phenomona underlying the example
affect the results of the example more generally.
To proceed, start by defining some notation. Let v. denote the
privately known value to bidder i of the object net of any amounts
paid to individuals other than the auctioneer (e.g., unlike the uni-
formly distributed value in the example, v. is now net of travel
costs paid to attend the auction); v denotes the seller's privately
known reservation value; assume v > 0. Then, as a function of the
o —
reservation price r below which the object will not be sold, and of
the set of bidders N, let V(N,r) denote the social value
E[max (v.-v )] generated by the auction. (For the moment,
i : ieN & v^ _> r
think of N as deterministic — in our example, this would correspond to
p= 1 ; most of the results would (more or less obviously) carry over to
the random case, but at a great cost in the complexity of the exposi-
tion.) Note that for fixed N, V(N,r) is concave in r and is maximized
at r = v . If $.(N,r,d) denotes the expected profit to bidder i from
o 1 r r
-24-
attending the auction as a function of the set N of bidders, the
reservation price r that the seller ends up irapliraenting, and any
entry fee d that each bidder must pay to the seller on entering the
auction, then the total expected revenue R(N,r,d,v ) equals V(N,r) -
Ii£N$.(N,r,d) + vo.
In the symmetric case, drop the subscript "i" and replace the set
N by the number of its elements n. Assume that $(n,r,d) is continuous
in r and a decreasing function of n; quite plausible, bidders' profits
suffer as the number of competitors, or the competitiveness of the
shill, increases. Finally, define n*(r,d) as the integer n such that
$(n,r,d) > 0, but $(n+l,r,d) < 0. Note that n*(r,d) is a non-
increasing function of r and d.
Theorem 1, In symmetric oral auctions with privately known values and
continuously increasing asking prices, for any fixed n, r, and d,
$(n,r,d) = (V(n,r) - V(n-l,r))/n - d.
Proof : For any fixed N, r, and d, the fact that the winner pays an
amount equal to the second highest privately known value — the highest
value if the winner weren't present — implies that $.(N,r,d) ■ V(N,r) -
V(N\i,r) - d. In the symmetric case, i wins with probability 1/n, and
conditional on i winning, V(N,r) - V(N\i,r) = V(n,r) - V(n-l,r); the
remaining (n-l)/n of the time, i loses and conditional on i losing,
V(N,r) - V(N\i,r) = 0. Thus, $(n,r,d) = (l/n)(V(n,r) - V(n-l,r)) - d
as claimed. Q.E.D.
This relationship between bidder profit and contribution to social
value may be the extreme case of practical interest. In particular,
-25-
in the contrasting "common values" case in which each bidder has the
same, unknown value for the object, the social value is independent of
n so long as n > 1, and, since bidders have a strictly positive
expected profit in typical common values models, $(n,r,d) > (V(n,r) -
V(n-l,r))/n - d. Many, if not most, practical situations fall
somewhere in between the common values and the privately known values
extremes: this author knows of no practical auction model in which
$(n,r,d) < (V(n,r) - V(n-l,r))/n - d.
Theorem 2. If for each fixed n,r, and d bidders bid so that $(n,r,d)
> (O (V(n,r) - V(n-l,r))/n - d, then V(n,r) - V(n-l,r) < (>) 0 for
all n > (O n*(r,0).
Proof: By hypothesis, V(n,r) - V(n-l,r) < (>) n$(n,r,d) + nd for all
n,r,d. Since the left hand side of this inequality is independent of
d, the right hand side must also be independent of d, and so must
equal n$(n,r,0) — the value obtained when d=0 — for all n,r,d. But, by
the definition of n*(r,d) and the monotonicity of $(n,r,d) in n,
$(n,r,d) is < (2) 0 for all n > (O n*(r,0), as claimed. Q.E.D.
Corollary. In the symmetric oral auctions with privately known values
and continuously increasing asking prices, for each fixed r and d,
V(n,r) is maximized at n = n*(r,0).
Roughly speaking, this corollary states that if bidders enter and
leave the auction in their own best interests, then the socially opti-
mal number of bidders results. This result plays a crucial role
throughout this section. In particular, the social value is an upper
-26-
bound on the total revenue; bidders must make a non-negative profit,
and that profit must come out of the social value generated by the
auction. As subsequent theorems establish, deviations from the
socially optimal number of bidders typically hurts the total revenue
(presumably because of its relationship to the social value generated
by the auction) more than any gains achieved by deviating from an ex-
ante reservation price equal to the reservation value and/or deviating
from an entry fee (subsidy) of zero. In fact, the seller should
deviate from r = v and d = 0 only to the extent that it has no effect
o
on the number of bidders other than rounding the number to an integer
if n*(r,d) would have been non-integer had we allowed non-integer num-
bers of bidders.
Theorem 3. In symmetric oral auctions with privately known values and
continuously increasing asking prices, if bidders enter and leave
until n = n*(r,d), then for any fixed v , r = v and d = (V(n,r) -
o o
V(n-l,r))/n maximizes the expected revenue R(n,r,d,v ) with respect to
r and d.
Proof: By definition of d, $(n,r,d) = 0 and n*(r,d) = n*(r,0). But,
$(n,r,d) = 0 implies that R(n,r,d,v ) = V(n,r) + v , and thus setting
r and d as specified yields an expected revenue of V(n*(v ,0),v ) +
r J r o o
v . Since r = v maximizes V(n,r), V(n,r) + v < V(n,v ) + v . By
o o o—oo
the corollary to Theorem 2, V(n,v ) + v < V(n*(v ,0),v ) + v . To
o o — oo o
summarize, R(n,r,d,v ) < V(n*(v ,0) ,v ) + v . Thus, for all n, r and
o — o o o
d, R(n,r,d,v ) is at most the expected revenue actually achieved by
setting r and d as specified in the hypothesis. Q.E.D
-27-
In effect, an appropriate entry fee adjusts for the integrality in
the number of bidders, thereby allowing the seller and tax collector
to capture any profit that the bidders would have otherwise obtained
simply because our model did not allow a fractional number of bidders;
in short, with an appropriate entry fee, this auction generates as
much total revenue as any mechanism can. However, barring a positive
entry fee, the revenue suffers, and adjusting for the integrality in
the number of bidders calls for setting an appropriate reservation
price strictly greater than v , or appropriately subsidizing bidders
by setting d < 0, The next two theorems — and a subsequent example —
examine these options.
To characterize the optimal ex-ante reservation price when d = 0,
define r as the largest reservation price r such that n*(r,0) =
n*(v ,0). Assume that n*(r,0) = n*(v ,0) for all r between v and r .
o o o o
Then, define r* as the r that maximizes R(n*( r ,0) , r ,0,v ) subject to
v < r < r .
o — — o
Theorem 4. If 1) d restricted to be zero and r restricted to be no
less than v ; 2) all bidders bid such that $(n,r,0) _< (V(n,r) -
V(n-l,r))/n; and 3) bidders enter/leave until n = n*(r,0), then r* is
an ex-ante optimal reservation price, and at any ex-ante optimal r,
n*(r,0) = n*(v ,0).
o
Proof: Consider two cases; v < r < r and r > r . First, if v < r
o — — o o o —
< r , then by the definition of r* , R(n*(r ,0) , r ,0 , v ) <
— o o —
R(n*(r*,0) , r*,0,v ). Second, if r > r , then the definitions of
o o
R(n,r,d,v ) <_ V(n*(r,0),r) + v . Since V(n,r) is decreasing in r for
-28-
r > v , for r > r , V(n*(r,0),r) + v < V(n*(r,0),r ) + v . Since r >
— O O O 0 0
r implies that n*(r,0) _< n*(r ,0), Theorem 2 establishes that
V(n*(r,0),r ) + v < V(n*(r ,0),r ) + v . By the definitions of
o o — 0 0 o
r and R(n,r,d,v ), V(n*(r ,0),r ) + v = R(n*(r ,0),r ,0,v ). But by
O O 0 0 O 0 0 0
the definition of r*, R(n*(r ,0),r ,0,v ) _< R(n*(r*,0) ,r* ,0, v ).
Summarizing, for r > r , R(n*(r ,0) , r ,0 , v ) < R(n* (r*,0) , r*,0 , v ).
This together with the first case gives the desired results. Q.E.D.
Theorem 4 says that in symmetric oral auctions with privately
known values and continuously increasing asking prices, barring entry
fees paid by the bidders to the auctioneer (or subsidies paid the
other direction), the seller benefits from a reservation price in
excess of v only because it in effect rounds off any fractional
o J
bidder that would occur if r = v and fractional bidders were allowed.
o
Roughly speaking, except for the discreteness of the number of bid-
ders, an ex-ante reservation price equal to the seller's reservation
value, maximizes expected revenue. This contrasts to the situation
ex-post to bidders having committed themselves to attend — a situation
similar to the model of Myerson (1981) with its exogenously fixed
number of bidders — in which a reservation price strictly larger than
the seller's reservation value maximizes revenue. Thus, for d = 0,
the seller and tax collector together would benefit from committing to
a lower reservation price ex-ante to bidders committing themselves
than would be chosen ex-post.
The result that at the optimal reservation price r, n*(r,0) =
n*(v ,0) requires some restriction on the assymetry of the model. To
-29-
illustrate, consider a second-price sealed-bid auction with statisti-
cally independent privately known values; for each bidder, the value
gross of the participation cost is either zero or one, each with pro-
bability one half, independent of the other bidders' values. The
seller has a reservation value of zero, and pays no taxes. To intro-
duce assymetry, let bidder i = 1 have a participation cost of c, = e, ,
and each bidder i > 1 have a participation cost c. = 1/4 - e~, where we
think of e. and e„ as sufficiently small, but still strictly positive,
quantities .
What happens at equilibrium? For a reservation price set to zero
ex-ante, and appropriately small e and e„, the number of entrants n*
equals two. In particular, at the dominant strategy Nash equilibrium
each bidder bids his privately known value; the (or "any," in the case
of ties) high bidder wins and pays an amount equal to the highest
amount bid by any non-winning bidder. Thus, each bidder has an
expected profit gross of the participation cost equal to the probabi-
lity that his value for the object equals one times the probability
that the value to all other bidders is zero. For two bidders, each
has an expected gross profit of one-fourth; for e. < 1/4 and e~ > 0,
this leaves both bidders with a strictly positive expected profit net
of the participated costs. For three or more bidders, the gross
expected profit per bidder equals one-eighth, which is less than all
but the first bidder's participation costs so long as e9 < 1/8, and
some bidder (other than the first bidder) should leave if e9 < 1/8.
Thus for an ex-ante reservation price of zero, 0 < e. < 1/4 and 0 <
-30-
e~ < 1/8 results in two bidders. Note that for two bidders, the auc-
tioneer has an expected revenue equal to one times the probability
that at least two bidders have a value of one for the object; this
equals one-fourth for the case of two bidders.
Now consider a reservation price equal to 1 - 3e.. Instead of the
winner sometimes getting an object of value one for free, the winner
must pay the reservation price. Thus, the gross expected profit has
been reduced from one-fourth to one-fourth of 1 - r, that is, to ■
(3/4)e.. No longer will the market support two bidders. However, any
one bidder alone would have a gross expected profit of (1-r) times the
probability that he has a value of one for the object; in other words,
the expected gross profit equals (3/2)e.. Thus, the first bidder by
himself would have a strictly positive expected net profit (as would
any other bidder by theraself under appropriate choices for e, and e~ ) .
So, for r = 1 - 3e., only one bidder participates. But now, the
expected revenue equals r times the probability that the lone bidder
has a value of one for the object. That is, the expected revenue
equals (l-3e.)/2 which exceeds one-fourth for e. < 1/6.
Now pull the two cases together. For 0 < e. < 1/6 and 0 < e~ <
1/8, and ex-ante r of zero results in two bidders and an expected
revenue of one-fourth, while an ex-ante r of 1 - 3e . results in one
bidder and a (strictly greater) expected revenue of (l-3e,)/2. In
short, for this assymetric example, the seller benefits from using a
reservation price enough larger than his reservation value of zero to
drive away a bidder. Thus, we cannot hope to significantly weaken the
symmetry presumed by Theorem 4 without affecting the results.
-31-
The restriction that d = 0 rather than d \ 0 also affects the
results of Theorem 4. The case of negative entry fees d corresponds
to the auctioneer subsidizing bidders, something which seems to happen
in some real world auctions. The next theorem examines optimal sub-
sidies .
Theorem 5. If 1) the reservation price r is restricted to equal
v and the entry fee d is restricted to be non-positive (that is, to
o
be, in effect, a subsidy), 2) all bidders bid such that $(n,v ,d) _>
(V(n,v ) - V(n-l,v ))/n - d, and 3) bidders enter/leave until n =
n*(r,d), then for the optimal d < 0, n*(v ,0) _< n*(v ,d) _< n*(v ,0) +
1. (In words, don't subsidize more than necessary to round up the
number of bidders.)
Proof: For k > 1, define d, = $(n*(v ,0) + k, v ,d); in words, Id, I
— k o o ' k1
is the subsidy needed to just attract k more bidders than when d = 0.
Since $(n,r,d) is non-increasing in n, d, will be non-positive for all
k >_ 1 , and non-increasing in k. Also, as k goes to infinity, d, must
go to negative infinity.
Since for d = d, , n*(v ,d) = n*(v ,0) + 1, and for d, < d < 0,
loo 1 —
n*(v ,d) = n*(v ,0), it suffices to prove that any optimal d satisfies
o o
the condition d, '. d _< 0. We will do so in two steps, first showing
that R(n*(vo,dk),vQ,dk,vo) < R(n*(v .d^.v .d^v ) for all k < 1, and
then showing that for each k > 1 R(n*(v ,d),v ,d,v ) <
— 0 0 0
R(n*(v ,d, ),v ,d, ,v ) for all d such that d. ^ . < d < d, .
okoko k+1 k
To show the first part, note that by the definition of d ,
K.
R(n*(v ,d, ), v ,d, ,v ) = R(n*(v ,0)+k,v ,d ,v ), which in turn equals
-32-
V(n*(v ,0)+k,v ) + (n*(v ,0)+k)d + v by the definition of
o o o o
R(n,r,d,v ). For all k > 1 and d, < d, < 0, V(n*(v ,0)+k,v ) +
o k. — 1 — o o
(n*(v ,0)+k)d. + v < V(n*(v ,0)+k,v ) + (n*(v ,0)+l)d, + v . By
o ko — o o o 1 o
Theorem 2, the right hand side of this last inequality must be less
than V(n*(v ,0)+l,v ) + (n*(v ,0)+l)d, + v , which by the definition
o o o 1 o
of d, equals V(n*(v ,d,),v ) + n*(v ,d. )d, + v , and which is simply
1 olo olio
R(n*(v ,d.),v ,d. ,v ). In short, R(n*(v ,d.),v >d ,v ) <
\j L \j 1. \) 0-LOK.O
R(n*(v ,d,),v ,d, ,v ) for all k > 1.
O 1 O 1 0
To show the second part, for d, . . < d < d, , R(n*(v ,d),v ,d,v ) =
K.+ 1 k 000
R(n*(vo,dk),vo,d,vo) = V(nMvo,dk),vo) - n*( vq ^ $ (n* ( Vq , d^) , vq) +
nA(v ,d. )d + v , which in turn equals V(n*(v ,d, ),v ) -
ok o o k o
n*(v ,d, )(d -d) + v , which since d < d, , must be strictly less than
O K. K, O K.
V(n*(v ,d. ),v ) + v , which is simply R(n*(v ,d, ),v ,d, ,v ). In
ok o o okoko
short, for d, , , < d < d, , R(n*(v ,d),v ,d,v ) < R(n*(v ,d. ),v ,d, ,v ),
k+1 k oo o okoko
Q.E.D
Theorem 5 says that the auctioneer should not subsidize more than
needed to simply round up the number of bidders. In fact, given the
corollary to Theorem 2, this seems intuitive; a subsidy of |d. | just
attracts an additional bidder, thus reducing each bidders' expected
net profit to zero, and giving a total revenue equal to the full
social value generated by this slightly inefficient auction. Any
larger subsidy either increases the bidders' profits, or attracts
additional bidders thereby decreasing the social value generated in
addition to decreasing the revenues by the amount of the subsidies.
(This theorem also raises the question of just why do bid takers sub-
sidize bidders to the extent that they appear to in certain actual
-33-
auctions, auctions which though perhaps not symmetric oral auctions
with privately known values, nonetheless seem to satisfy the
conditions — condition 2 in particular — of the theorem.)
Depending on how close n*(r,d) would be to the next smaller or
next larger integer if it were allowed to take non-integer values
determines whether d = 0 and some appropriate r > v out performs r =
v and some appropriate subsidy -d, or the other way around.
Intuitively, the optimal subsidy (when r=v ) and the optimal reser-
vaton price (when d=0) in effect round the number of bidders. But any
such rounding comes at a cost; increasing the reservation price
decreases the social value, as does subsidizing to the point of
increasing the number of bidders. Thus, as an example illustrates, we
might expect the choice between subsidizing versus increased reser-
vation price to depend, roughly speaking, on which direction requires
less rounding.
To ilLustrate optimal reservation prices and subsidies in general,
and more specifically, to ilLustrate that sometimes a reservation
price should be preferred to a subsidy, and sometimes vice versa,
again consider the case of independent private values (gross of par-
ticipation costs) distributed uniformly on the unit interval. Assume
that all potential participants have the same entry cost c, and that
the reservation value is zero. Then with a reservation price and sub-
sidy both equal to zero, the n bidders would have an expected
equilibrium profit of l/(n+l) gross of participation costs. Thus, for
1/12 < c < 1/6, the equilibrium number of bidders will be two.
-34-
The bidding equilibrium to this example with a reservation price
of r generates an expected price of r + ((n+l)-2nr )/(n+l), and an
expected total profit the the n bidders (gross of participation costs)
of -r + (nr +l)/(n+l). We consider two different levels for the
entry cost c. First, for a c just a hair above 1/12, on the one hand,
a very small subsidy (and zero reservation price) would result in an
equilibrium with three bidders, an expected price of 2/4 = 1/2, and an
expected net revenue to the seller of just under 1/2. On the other
hand, for zero subsidy and a reservation price of just under 1/2, two
bidders would have a combined expected profit (gross of participation
costs) of just over 1/6 — just enough to cover the participation costs
of two bidders. Thus, as c drops to 1/12, the optimal reservation
price rises to 1/2. But even for reservation price of 1/2, the
expected price from two bidders would be only 5/12 — strictly less than
the just under 1/2 that can be obtained from an appropriate subsidy.
In short, as c drops to 1/12, an optimal subsidy together with a
reservation price of zero results in a greater expected equilibrium
revenue to the seller than that possible from an optimal reservation
price and no subsidy.
Second, consider the case of c = 5/36. Now a reservation p price
of 1/4 (and no subsidy) would give two bidders a combined expected
profit 54/192 = 162/576 — more than enough to cover their participation
costs of 2(5/36) = 160/576. Thus, a reservation price of 1/4 (and no
subsidy) would result in an equilibrium with two bidders and an
expected price of 3/8. But, to get three bidders would require a sub-
sidy of 3(5/36) - 1/4 (the bidders' expected profit when n=3) = 1/6.
Three bidders would give rise to an expected equilibrium price of 1/2.
-35-
Net of the subsidy, the seller could expect a revenue of 1/2 - 1/6 =
1/3 — strictly less than the 3/8 possible with a reservation price of
1/4 and no subsidy. Here, when c = 5/36, even a suboptimal reser-
vation price and no subsidy does better for the seller than optimal-
subsidy and zero reservation price; in fact this will be the case for
.1160256 < c _< 1/6, while the reverse is true for 1/12 < c < .1160255.
Thus a small change in one parameter of the model may swing the seller
from preferring a subsidy over a reservation price to the other way
around.
Despite this inconclusiveness , we can conclude something of
interest from these last three theorems. In particular, even if the
seller uses an entry fee, a reservation price, or a subsidy to custom
tailor the basic oral auction to a specific situation, the resulting
number of potential buyers need never be less than the original
equilibrium number, nor need it ever exceed the original equilibrium
by more than one. Thus, roughly speaking, in our oral auction model,
the seller should set the reservation price equal to his reservation
value; if the seller has a reservation value of zero, then as we pre-
viously quoted Thompson, "Every piece of goods offered at auction
should be positively sold."
In fact, Engelbrecht-Wiggans (1987b) suggests an argument that
quite generally, the ex-post optimal reservation price exceeds the ex-
ante reservation price even if we can't show that the ex-ante optimal
reservation price is essentially equal to the seller's reservation
value. In particular, imagine that u parameterizes the distribution
of the set of actual bidders; the example suggests thinking of u as
-36-
the mean number of bidders — which, indeed, is how we will refer to
it — even though it could be some other parameterization. Let
R (u,r,d,v ,a,6) denote the seller's expected net revenue as a func-
tion of the parameter u, the reservation price r, the entry fee d, the
reservation value v , and the tax rates a and 3; this revenue may be
o J
from a single auction, or from several auctions (each with the same
reservation price r >_ v ) of the same object if it was won back by the
seller in all but at most one of them. Assume that the derivative of
R with respect to u will be positive; plausibly, as u increases, so
too does the probability of a bonifide bidder winning (as well as the
expected price conditional on a bonifide bidder winning) and since
bonifide bidders pay at least r(r > v ) , an increase in their
— o
probability of winning increases the seller's expected revenue.
Ex-post to the auctioneer seeing u, for fixed d and v , the opti-
mal reservaton price still depends on a and 8 as well as on u. Assume
that the indicated derivatives exist and are well enough behaved so
" d
that for some function r(u,a,8), - — R (,r,d,v ,a,8)| „ =0 for
i dr s o ' , oN
d2 r=r(u,a,6)
all u, ot and 8, and — r- R (u,r,d,v ,a,8) | A < 0 for all u, a
dr""^ ' r=r(u,a, 8)
and 8. Interpret this r as the ex-post optimal reservation price.
Now, ex-ante, u depends on r; therefore write u(r) when the depen-
dence matters. Assume that the derivative of u( r) with respect to r
exists and is negative. That is, as the reservation price increases,
the mean number of bidders decreases. To characterize the ex-ante
optimal reservation price, look at the derivative of R (u(r),r,d,v ,ot,8)
with respect to r when r = r (u( r ) , ot, 8) . (This is an implicit equation
for r.) In particular,
-37-
~ R (u(r),r,d,v ,a,6)
dr s o
= 4~ R (u,r,d,v ,a,B) | + T~ R (u,r,d,v ,a,6) -j~ u(r) |
dr s o . x du s o dr , .
u=u(r) u=uCr)
When r = r(u( r) ,a, B) , the first right hand side terra is zero by the
definition of r. The second right hand side terra is the product of
two quantities, both negative by assumption. Therefore, ex-ante, the
seller's revenue decreases with r at the ex-post optimal r, and so the
seller would prefer a smaller r ex-ante. While this argument presumes
more continuity than is present in the original example, it does illu-
minate why we might reasonably expect that the ex-post optimal reser-
vation price exceeds the ex-ante optimal reservation price even more
generally than the symmetric oral auctions with privately known
values. Thus, the seller and tax collector benefit quite generally
from at least moving in the direction of encouraging the absolute sale
of all goods offered.
4. Lowering the Expected Reservation Price
In practice, sellers do try to commit to a lower reservation price
than might be e:<-post optimal. Some auction notices advertise that
all goods will be sold without reserve. Certain laws and auctioneers'
codes of ethics prohibit shills. But does it work?
On several occasions, this author observed what appeared to be
"cheating" by auctioneers who had promised to sell everything without
reserve. On one occasion, an object which sold in one auction resur-
faced a couple of weeks later in another auction by the same auc-
tioneer. On another occasion, an auctioneer indicated receiving a bid
-38-
frora a part of the audience unlikely to have bid (this author knew
that the individuals in question had never registered for the "bid
numbers" required in order to bid). On yet other occasions, an auc-
tioneer lost track of who made the current — and apparently final — bid
and then backed up the bidding to a previous, lower level, in order to
sell the object.
Whether or not auctioneers actually cheat is not the question.
Rather, what matters is how much potential bidders expect an auc-
tioneer to cheat, and what they expect the reservation price to be.
As long as this author — and presumably other potential bidders as
well — suspect certain auctioneers of implementing higher reservation
prices than others, the number of actual bidders attending these auc-
tioneers' auctions will be affected.
Thompson suggested that changing the tax rates would encourage the
absolute sale of goods offered. In fact, the change does not directly
provide the sellers with a mean for committing ex-ante to lower reser-
vation prices. Rather, as will be shown in this section, the changes
reduce the ex-post optimal reservation price. This reduces potential
bidders' ex-ante expectations of the reservation price that a per-
fectly honest auctioneer will end up using, and reduces the incentive
for — and therefore, possibly, the degree of — cheating by an auctioneer
who promised to sell goods without reserve.
As before, let R (u,r,d,v ,a,0) denote the seller's expected net
s o
revenue. Since attention focuses on how the tax rates a and 3 affect
the ex-post optimal reservation price r(u,a,6) (as defined before)
think of u as being fixed. As before, restrict a > 0, 8 > 0, and a +
-39-
8 < 1. Since non-zero entry fees or subsidies d seemingly arose only
in response to the original example's integrality of bidders, hold d
fixed at zero throughout this section; in fact this appears to be the
appropriate choice of d in modelling the Port of New York auctions.
Again, the seller acts as his own auctioneer and sells a single
object. As before, the seller's utility depends only on money, and
the seller is risk neutral. Also as before, the seller implements the
reservation price through a (real or imagined) shill who bids so as to
assure that no bonifide bidder wins the object at a price less than r;
other than this effect on the selling price to bonifide bidders, we
need not specify how the shill bids. Unlike before, the seller may
re-auction (always with the same reservation price r) the object until
a bonifide bidder wins it.
Now that the seller may repeatedly offer a single object for sale,
the reservation value v must be defined more carefully. To do so,
o
split the world into two markets — the market affected by changes in ct
and 8, and the market not affected by such changes; in terms of the
original example, the world consists of the Port of New York, versus
everything else. Then let v denote the expected net value of the
o
object to the seller conditional on not selling it in the market
affected by a and 3, and let F(u,r) denote the probability of the
object not selling in the market affected by ot and 3; making F depend
on only u and r implicitly assumes restrictions such as that the tax
rates affect the probability only through their effects — directly or
indirectly — on u and r.
The total tax paid breaks into two components. If G.(u,r) deno-
tes the expected payments by the shills (who may win more than once)
-40-
in the market affected by a and 8, and G^(u,r) denotes the expected
payments by bonifide bidders in the market affected by a and 8, then
the total tax
T(u,r,a,8) = gG1(u,r) + (a+8)G2(u,r) .
Using the same notation,
R (u,r,0,v ,a,8) = -BG^u.r) + ( l-a-8)G2(u,r) + v F(a,r).
As with F, making G. and G~ depend only on r and u places implicit
restrictions on these functions.
Now make five assumptions, each holding over the (unspecified)
range of allowable u and r. One, the derivative of G. with respect to
r exists and is positive. This seems plausible because as the reser-
vation price increases, so does the likelihood of the shill winning,
and so does the expected amount paid by the shill conditional on
winning. Two, the derivative of G. with respect to u exists and is
negative. This occurs if when the mean number of bidders increases,
the shill's probability of winning drops rapidly enough to more than
offset any increase (toward r) in the shill's expected payment con-
ditional on winning. Three, the derivative of G,? with respect to u
exists and is positive. This seems plausible because as the number of
bidders increases, so too does the probability of a bonifide bidder
winnirttf, and so too does the expected amount paid by the bonifide bid-
ders conditional on winning. Four, both G, and G.? are positive. This
occurs if both the shill and the bonifide bidders have positive proba-
bility of winning, and each pays a positive expected amount con-
ditional on winning. Finally, five, the derivative of F with respect
-41-
to r exists and is non-negative; as the reservation price increases,
the probability of selling the object (to a bonifide bidder) in the
market affected by a and 8 doesn't increase.
(In the example, u = np, and as the integer n varies for fixed p,
u varies discontinuously. Thus, the current section does not include
the example as a special case; we feel that the previous section ade-
quately deals with the effects of discrete changes in the (mean)
number of bidders. Still, this section does include the example
modified so that p varies with n fixed, and includes the example in
the limiting case of u being the mean of a Poisson distribution.)
To see how a and 8 affect the ex-post optimal reservation price,
examine the first order condition for r(u,a,B). In particular,
0 - [-3 f- G.(u,r) + (1-a-B) ~ G.(u,r) + v ~ F(u,r)] | A
dr 1 dr 2 o dr ' * , „.
r=r(u,a, 8)
Notice, for use later, that by the assumptions on a, 8, v , and F, the
derivative of G» with respect to r evaluated at r = r(u,ct,B) must be
non-positive. Differentiating the first order condition with respect
to a and rearranging yields
dr(u,a,8) .. d7G2(u'r) ,
da d2
— x- R (u,r,0,v ,a,8) ' *, „ aN
,2 s o r=r(a,a,8)
dr
The second order condition defining r(u,a,8), together with the above
note on the derivative of G.. at the ex-post optimal reservation price
when 8 equals zero, implies that the derivative of r will be non-
negative; when 8 equals zero, as a decreases, the ex-post optima]
reservation price stays the same or decreases.
-42-
Similarly, differentiating the first order condition with respect to
and rearranging yields
dr(u,3, 8) dr 1 dr 2
d8
dr
2- Rs(u,r,0,vo>a,8)
r=r(u,a,8)
To sign this second derivative requires establishing the relative
sizes of the two derivatives in the numerator. Since G. depends on
how the shill bids — does the shill initially bid r, does the shill bid
only as needed until the price reaches r or he wins, or does the shill
bid differently still--we have been unable to establish a general
relationship. (Still, for the first two choices of shill behavior
just mentioned, the numerator can be shown to be strictly positive in
independently distributed, private values models.)
As an alternative approach to looking at specific models, hold the
tax T(u,r,a,8) constant; that is, as 3 varies, vary a so that the tax
remains the same. Thus, we now think, of a as some function a( 8) of 8.
Then, see how varying 8 affects the ex-post optima reservation price,
by evaluating
ib ^.*<«.e> ■ & r"(u>*>6) + h ka-*-s) ^fr1
oFa(e)
Differentiating T(u , r ,a( 8) , 3) implicitly with respect to 8 and
rearranging yields
da(6) . "Gl(u'r) + G2(u»r)
d8
G2(u,r)
-43-
Substituting this together with the two previously obtained
expressions for the derivatives of r(u,a,8) yields
jg r(u,a(8),8) = —
fcGjCa.r) ^G2(u,r)
dr
j Rs(u,r,0,vo>a,S)
r=r(u,a(3),8)
— G2(u,r)
G2(u,r) - G^u^)
dr
- R (u,r,0,v ,a,8)
I s o
G2(u,r)
r=v(u,a(3),3)
which simplifies to
A G9(u,r) — G (u,r) - G (u,r) — G (u,r)
^ r(u,a(8),3) = — dr A ^^-^
G7(u,r) — r R (u,r,0,v ,a,8)
I ,2s o
dr
r=r(a,a(3),3
By assumption, G. , -r— G., and G„ are strictly positive. By the defi-
d^
nition of r, — =■ R is strictly negative when evaluated at r = r(u,ct,8).
dr d
And, by a previous note, -r~ G„ is non-negative when 3=0. Therefore,
at 3 = 0, j£ r(u,a(3),B) < 0.
In short, if we hold the total tax receipts constant, then ini-
tially as we raise 3 above zero (and correspondingly drop a toward
zero), the ex-post optimal reservation price drops. That is, quite
generally, moving at least somewhat in the direction of Thompson's
proposal (to raise 3 while dropping a to zero) has the desired effect
of lowering the reservation price that potential bidders can reason-
ably anticipate will be used in the auction. While this analysis says
nothing once 3 > 0, the original example establishes that increasing 3
— AA —
until a drops all the way to zero decreases the ex-ante optimal reser-
vation price.
5. Summary
In the early nineteenth century, an astute auctioneer proposed
that rather than just taxing imported goods actually sold, the tax
should be reduced, but at the same time extended to goods offered for
sale but not sold. He felt it important to encourage the absolute
sale of all goods offered, and claimed that his proposal would provide
such an encouragement. Presumably, in making the proposal, the auc-
tioneer felt that the auction houses — and therefore presumably also
the sellers — would benefit; by adopting the proposal, the New York
State Legislature presumably indicated it too expected to benefit.
This paper develops a theory to explain what may have happened.
First, an analytically tractable example establishes that in at least
certain specific models the proposal has the anticipated effects;
lowering the tax rate while extending it to goods offered but not sold
reduces the reservation price a wiley seller would wish to use,
thereby making the auction more attractive to bidders and attracting
more of them, thus raising the expected revenue, and in the end
increasing both the seller and tax collector's receipts.
Subsequent sections attempt to understand what might drive the
results more generally. The analysis breaks into two pieces. The
first argues that for a variety of models, if the seller could convin-
cingly commit to a reservation price ex-ante to potential bidders
deciding whether or not to attend the auction, the seller would prefer
-45-
a lower reservation price ex-ante than he would chose ex-post to bid-
ders having committed themselves to attend. The second argues that
moving in the direction of the proposed change at least initially
decreases the ex-post optimal reservation price, thereby in effect
enabling the seller to make the potential bidders expect that a lower-
and more attractive to bidders — reservation price will be used than
before, and as a result, increasing the revenue to be split between
the seller and tax collector.
-46-
6. References
Albion, R. G. , The Rise and Fall of New York Port: 1815-1360, Hamden,
CT, Anchor Books, 1961.
Engelbrecht-Wiggans, R. , "Auctions and Bidding Models: A Survey,"
Management Science, Vol. 26, No. 2, February 1980, pp. 119-142.
, "Optimal Auctions: The Efficiency of Oral Auctions
without Reserve for Risk Neutral Bidders with Private Values and
Costly Information," BEBR Faculty Working Paper No. 1316,
University of Illinois at Urbana-Charapaign, February 1987a.
, "Optimal Reservation Prices in Auctions," Management
Science, Vol. 33, No. 6, June 1987b, pp. 763-770.
, "Revenue Equivalence in Multi-Object Auctions," Economics
Letters, Vol. 26, 1988, pp. 15-19.
Myerson, R. B., "Optimal Auction Design," Mathematics of Operations
Research, Vol. 6, No. 1, February 1981, pp. 58-73.
Shubik, M., "On Auctions, Bidding and Contracting," in Auct ions ,
Bidding, and Contracting: Uses and Theory, R. Engelbrecht-
Wiggans, M. Shubik, and R. M. Stark (eds.), New York University
Press, New York, 1983, pp. 3-32.
Stark, R. M., and M. H. Rothkopf, "Competitive Bidding: A
Comprehensive Bibliography," Operations Research, Vol. 27, No. 2,
1979, pp. 364-390.
Vickrey, W. , "Counterspeculat ion, Auctions and Competitive Sealed
Tenders," Journal of Finance, Vol. 41, No. 1, 1961, pp. 8-37.
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