The Simple Economics of Optimal Auctions
Jeremy Bulow; John Roberts
The Journal of Political Economy, Vol. 97, No. 5. (Oct., 1989), pp. 1060-1090.
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The Simple Economics of Optimal Auctions
Jeremy Bulow and John Roberts
Stanford C'nzz~rrs~t)
We show that the seller's problem in devising an optimal auction is
virtually identical to the n~onopolist's problem in third-degree price
discrimination. More generally, many of the important results and
elegant techniques developed in the field of mechanism design can
be reinterpreted in the language of standard micro theory. We illus-
trate this by considering the problem of bilateral exchange with pri-
vatelv known values.
I.
Introduction
An "optimal auction" is a bidding mechanism designed to maximize
a
seller's expected profit. The literature on optimal auctions has mush-
roomed in recent years (see, e.g., Harris and Raviv 1981a, 19816;
Myerson 198 1; Riley and Samuelson 198 1; Milgrom and Weber 1982;
Matthews 1983; Maskin and Riley 1984a, 19848) and has provided the
basis for the more general study of efficient mechanism design.'
Unfortunately, this field has been a difficult one for most econo-
mists, seemingly bearing little relation to traditional price theory. This
paper greatly simplifies the analysis of optimal auctions by showing
hluch of the work reported here was done while Bulow was tisiting the Graduate
School of Business at the University of Chicago. We wish to thank Milton Harris, Paul
Milgrom, Kevin
M.
Murphy, Roger hlyerson, Hugo Sonnenschein, Sherwin Rosen,
Bob M'ilson, and an anonynlous referee for valuable discussions and comments. The
financial support of the Sloan Foundation and the National Science Foundation is
gratefully acknowledged.
'
Examples of research employing mechanism design techniques include studies of
bilateral monopoly and bargaining (Myerson and Satterthwaite 1983), multilateral par-
tial equilibrium exchange (LVilson 1985), regulation of
a
monopolist without knowing
its costs (Baron and Myerson 1982), and dissolution of a partnership (Cramton, Gib-
bons, and Klemperer 1987).
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OPTIMAL
AUCTIONS
1061
that
it
is essentially equivalent to the analysis of standard monopoly
third-degree price discrimination. The auctions problem can there-
fore be solved by applying the usual logic of marginal revenue versus
marginal cost. The same logic also clarifies the celebrated results on
the revenue equivalence of various auctions and many of the results
concerning bilateral monopoly.
Our primary purpose is not to obtain new results, though we do
somewhat extend earlier work, but rather to
connect existing results
to familiar ideas. We hope that this reinterpretation of optimal auc-
tion theory will give readers a better understanding of a wide range of
private information problems and, thus, perhaps lead to new insights.
11.
The Optimal Auction Problem
The literature on optimal auctions begins with the work of L'ickrey
(1961). He considered the following situation.
A
seller values an item
at zero. She has the option to sell
it
to one of
n
risk-neutral bidders.
Each bidder alone knows the value, 7~, that he places on the good.
Because the seller and the other buyers are uncertain about this value,
it
appears to them to be a random variable. It is assumed to be com-
rllon knowledge among all the buyers and the seller that everyone
views the buyers' values as independent draws from a common distri-
bution F(v), with
F(u)
=
0 and F(z)
=
1. That is, everyone (including
i)
agrees that the probability that
71i
is less than
71
is given by F(71),
everyone knows that everyone knows this, everyone knows that every-
one knows that everyone knows this, ad infinitum. The seller is con-
sidering two alternative ways to sell the item: a sealed-bid first-price
auction, in which the high bidder wins and pays the amount he bid,
and a sealed-bid second-price auction, in which the high bidder wins
but pays only the amount bid by the second-highest bidder. The issue
is, Which of these methods should the seller use to maximize her
expected profit?
Either choice induces a game among the potential buyers, with each
having to decide how much to bid as a function of his valuation for
the good. Vickrey noted that when the second-price auction is used,
each bidder has a dominant strategy of bidding his true valuation.
This is because altering his bid from his value changes the outcome of
the auction in only two cases: when underbidding causes him not to
be the high bidder although his value is highest (so he gets a zero
surplus instead of a positive one), and when bidding more than his
value causes him to get the good but pay more than
it
is worth to him
(because the next-highest bid exceeds his actual valuation).
Given that each bidder will announce his true value in the second-
price auction.
it
is clear that an individual with value
-0
will
win with
1062
JOURNAL
OF
POLITICAL
ECONOMY
probability
F('"
=
[F(zI)]("
'I,
which is just the probability that all
')((71)
others will have values (and bids) below
v.
It is also clear that his
expected payment will be
( I )
I
-
(I)
=
I
(I)
-
6
F'"
-
"(2Ll)dul
and that his expected payment, contingent on winning, will be
which is the expected value of the second-highest valuation, given
that
-0
is the highest.
With a first-price auction, the strategic interaction among the bid-
ders is not trivial; there is an incentive for each bidder to shade his bid
below his value, trading off a reduced probability of winning against a
lower payment if he wins. Vickrey showed that, in fact,
it
is equilib-
rium behavior for a bidder with value to bid the amount B(n)
11
defined above.
(A
largely graphical treatment of these results is pro-
vided in the Appendix.) Vickrey's celebrated revenue equivalence
result follows immediately from
B(71) being the optimal bid. Given
symmetric, risk-neutral bidders with independent valuations, the ex-
pected revenue from the first- and second-price auctions is the same
because the bidder with the highest value always wins under either
auction (note that B(71) is increasing) and the winner's expected pay-
ment conditional on winning is the same amount, B(~I), under either
auction.
Although there was a steady stream of work on auctions over the
two decades following Vickrey's paper, the literature expanded dra-
matically during the 1980s. A particularly seminal piece-both for
the results obtained and for the methods introduced-is that of
My-
erson (1981). Myerson extended earlier work in two important ways.
The first was to consider the case of asymmetric bidders, that is,
bidders whose (privately known) valuations of the object are drawn
from independent, but not necessarily identical, probability distribu-
tions. These distributions are assumed to be common knowledge, so
that all bidders and the seller know the distribution from which each
bidder's value is drawn. With resale excluded, this allows the seller to
discriminate among bidders. The second extension was to consider
ull
possible ways of selling the good rather than just a prespecified set of
alternative auction forms as Vickrey had done. In this context, Myer-
son formulated and solved the "optimal auction design problem":
among
all
possible ways of selling the good, which one should the
seller use if she wants to maximize her expected net revenues?
OPTIMAL
AUCTIONS
1063
Myerson's solution gives an explicit formula for the optimal auction
with asymmetric bidders. He also extended Vickrey's revenue equiva-
lence result to show that any two mechanisms that always lead
;o the
same allocation of the good (and meet one further trivial condition)
would yield the same expected revenue. Finally, the methods he in-
troduced have since been widely applied to other problems involving
private information.
Note that the optimal auction design problem is extremely com-
plex. There are a myriad of possible ways of allocating and charging
for the good: a simple posted price; the various common forms of
auctions, including first- and second-price sealed-bid auctions, and
ascending and descending oral auctions (each of which could be
modified by using reservation prices and fees charged for the
privilege of bidding); less common auction forms such as the "all-pay"
auction, in which the highest bidder wins but everyone pays the
amount he bid2-the list seems limited only by the imagination.
The crucial breakthrough in this research effort was to identify a
method of formulating the optimal auction problem that immensely
simplified its solution. The key insight is to use the "revelation princi-
ple" (see Myerson
[1981] for an exposition). This states that in search-
ing for an optimal method of selling the good
it
is sufficient to con-
sider only "direct revelation mechanisms." In these, bidders are asked
to announce their valuations directly, and the seller commits herself
to using rules for allocating the good and for charging the buyers that
ensure both that the buyers will be willing to participate and that each
will find
it
in his interest to announce his true valuation. Given this
result, the problem of selecting an optimal way to sell the good re-
duces to a relatively simple constrained maximization problem: max-
imize the seller's expected revenues by the choice of functions giving
the probability of allocating the good to each buyer and the payment
to be made by each (both as functions of the announced values),
subject to the
"participation constraints" that each bidder receive non-
negative expected surplus and the "incentive constraints" that
it
be
equilibrium behavior for bidders to reveal their true valuations.
While this is an immense simplification of the problem, its solution
is still complex, and the literature on optimal auctions has remained
difficult despite such excellent surveys as those of Milgrom (1985,
1987, 1989) and
McAfee and McMillan (1987).
In the next section we present a simple and easily interpreted rec-
ipe for constructing the optimal auction. This approach connects di-
rectly to the standard monopoly problem of third-degree price dis-
'
Lobbying and other rent-seeking activities and wars of attrition are examples of
economic situations relevantly modeled as all-pay auctions.
