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## Dec04_article 5

dec04_Article 5 12/14/04 2:27 PM Page 1102

*Journal of Economic Literature Vol. XVII (December 2004) pp. 1102–1115*
seriously on auctions in the late 1970s and
early 1980s,6 just when the right game-the-
after 25 years of intensive work, the literature
oretic methods for studying this subject—
continues to grow at a prodigious, even accel-
erating rate;3 it has spawned much empirical
Harsanyi 1967–68) and perfect equilibrium
and experimental research;4 its tentacles
have spread into other disciplines; 5 and auc-
tion theorists have been influential in the
fields, e.g., industrial organization (I.O.),
design of mechanisms for the privatization of
benefited from the same symbiosis of tech-
public assets (such as spectrum bands) and
nique and application; collectively, they
for the allocation of electricity and other
resulted in the game theory revolution. But
goods (they have also often served as con-
the study of auctions has had more staying
sultants to the bidders in those mechanisms).

power than many other applications ofgame theory. Whereas enthusiasm for theo-
1 Paul Milgrom,

*Putting Auction Theory to Work*.

retical industrial organization has cooled
Cambridge, Churchill Lectures in Economics: Cambridge
University Press, 2004. NY, Melbourne, Pp.

*xxi*, 368. ISBN
years ago, research on auctions, as I have
2 Institute for Advanced Study and Princeton
University. I thank the NSF (SES-0318103) for research
believe, several reasons why auction theory
3 To give just one indication: at the August 2004 joint
meeting of the Econometric Society and European
First, theorists of I.O. and other applied
Economic Association, there were seven separate sessions
fields labor under the constraint that they do
on auction theory, a figure well beyond that for any other
4 Again, to cite only one, conference-related datum, the
organizers of the 2005 Econometric Society World
6 Auction theory actually began well before then.

Congress, who attempt to invite special talks on the most
Indeed, the seminal contribution was William Vickrey
lively and interesting developments in recent economics,
(1961). But until game theory came into its own fifteen
are planning a set of talks on empirical auctions work.

years later, Vickrey’s work—as well as that of other early
5 So, for example, there is now a sizeable computer-sci-
pioneers such as James Griesmer, Richard Levitan, and
ence literature on auction theory, often focusing on com-
Martin Shubik (1967), Armando Ortega Reichert (1968),
and Robert Wilson (1969)—remained largely ignored.

dec04_Article 5 12/14/04 2:27 PM Page 1103

*Maskin: The Unity of Auction Theory*
study (e.g., firms or consumers) are

*actually*
playing; models are at best approximations
(indeed, one implication of the papers of
of reality. By contrast, auction theorists typi-
footnote 8 is that, as numbers grow, most
cally know the rules that

*their *players follow
reasonable sorts of auctions converge in per-

*precisely*. If, for example, a high-bid auction
formance; only in the small-numbers case do
is the object of study, the theorist

*knows *that
the differences between auction forms come
(i) the bidders submit nonnegative real num-
bers as sealed bids; (ii) the winner is the bid-
der submitting the highest bid; and (iii) the
beautiful edifice:9 many of its major proposi-
winner pays his bid (of course, there may
tions deliver remarkably powerful conclu-
still be uncertainty about how the buyers
sions from apparently modest hypotheses.

*behave *under these rules). This precision
helps put the auction theorist’s findings on a
relatively strong footing; it also simplifies the
job of the experimentalist or empiricist.

worldwide impulse toward privatization that
mists’ “social engineering” instincts. Many
people go into economics at least in part
brought on by the fall of communism in the
because they want to improve the world. The
East—and the consequent need to sell off
mechanism-design 7 aspect of auction theo-
state assets—and the disenchantment with
ry—tinkering with the rules of the game in
order to achieve a better outcome—helps
bureaucracy’s loss proved to be auction the-
ory’s gain, as auction mechanisms, to great
Vickrey’s work is so celebrated is that his
public acclaim, were increasingly invoked to
famous creation, the Vickrey auction, provides
implement the transfer of resources. More
an attractive solution to an important social
recently, online auction enterprises such as
problem: designing an efficient allocation
eBay have provided further impetus for the
tion—the transfer of a good from seller to
Paul Milgrom has played a starring role in
ment—is fundamental to all of economics,
auction theory’s success story. Not only has
and so auction theory has been nourished by
he been a seminal contributor to the theo-
its connection with other theoretical areas.

retical literature (e.g., Milgrom 1981; and
For example, it has sometimes been used as
a foundation for understanding the workings
together with Robert Wilson, he had a major
of competitive markets.8 Of course, there
hand in designing the simultaneous ascend-
are important differences: competitive theo-
numbers of buyers and sellers, whereas in
much of the radio spectrum in the UnitedStates. Thus, his book

*Putting Auction*
7 Of course, auction theory is only a small part of a vast
mechanism design/implementation theory literature. For

*Theory to Work *has been eagerly awaited
recent surveys of the literature from a general perspectivesee Thomas Palfrey (2002); Roberto Serrano (2004); and
9 Economists, being a hard-boiled lot, sometimes deny
Eric Maskin and Tomas Sjöström (2002).

that esthetics have anything to do with what they are up to.

8 See for example Milgrom (1981), Wolfgang
But this sentiment belies the fact that the most important
Pesendorfer and Jeroen Swinkels (1997); Mark
economic ideas, e.g., the first welfare theorem of compet-
Satterthwaite and Steven Williams (1989); and Wilson
itive theory or the principle of comparative advantage, are
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*Journal of Economic Literature, Vol. XLII (December 2004)*
since his 1995 Churchill lectures, on which it
judgment (albeit very well-informed judg-
ment) as logic, they occasionally contrast jar-
book covers a great deal of theoretical mate-
ringly with the authority and precision of the
rial and does so with extraordinary economy
theory. For example, in Milgrom’s opinion,
(without sacrificing rigor). This economy
derives from Milgrom’s conception of auc-
tion theory as a subspecies of demand theo-
ry, in which a few key tools—the envelope
“unsuitable for most applications”—a con-
theorem in particular—do most of the work.

clusion that is far from being a theorem and
Indeed, once these tools are in place, he
that I will come back to in section 7.

establishes most theorems with just a few
But putting such quibbles aside, I should
lines of proof. As the title suggests, he also
emphasize that Milgrom is completely per-
discusses the extent to which the results bear
suasive on the general point that auction the-
ory matters in practice. In chapter one, he
Admittedly, the monograph is not the only
current volume of reflections by a leading
auction’s failure to raise the revenue antici-
auction expert on theory and practice.10 Nor,
pated can be traced to its seriously flawed
despite its unfailing clarity, is it the most
likely candidate for a graduate text on the
auctions for each license. Specifically, he
subject.11 Rather, its signal contribution is to
lay out Milgrom’s unified view of the theory.

This vision is notably distinct from that of
other major auction scholars. For example,
responds to those who argue that how gov-
ernment assets are sold off is irrelevant for
avoiding the revelation principle as an auc-
efficiency (because, in their view, the “mar-
ket” will correct any misallocation after-
learn to enjoy seeing things his way. In this
wards) with the theoretical riposte that,
sense, the book is more a “master class” (to
under incomplete information, there exists
quote Al Roth’s blurb on the back cover)

*no *nonconfiscatory mechanism (market-
than a text. And, of course, a master class is
based or otherwise) capable of attaining effi-
ciency, once the assets are in private hands
As for his ideas on how to apply (or not to
apply) the theory to actual auctions, these are
The heart of chapters 2–8 consists of a suc-
certainly most welcome and enlightening.

cession of formal results, almost all proved indetail. I will try to reinforce the book’s
10 Coincidentally, Paul Klemperer—like Milgrom, a
important lesson that auction theorems are
theorist of the first rank and also a principal architect ofthe United Kingdom’s 3G mobile-phone auction—has
easy to prove by stating and proving some of
almost simultaneously produced his own take on the sub-
them below (although I will not attempt to
ject (Klemperer 2004). The two books differ markedly in
replicate Milgrom’s rigor or generality).

style and substance. Milgrom’s is primarily a compendiumof theorems and proofs, together with less-formal observa-tions about their application to actual auctions. Except forthe appendices of chapters 1 and 2, Klemperer’s is almost
wholly nontechnical and consists largely of his views on thedesign of large-scale auctions in practice (although these
In chapter two, Milgrom turns to the most
11 Indeed, Vijay Krishna, yet another prominent auc-
the Vickrey (or “second-price”) auction (and
tion theorist, has recently produced a beautifully lucidtreatment (Krishna 2002), that in its organization and cov-
its Groves–Clarke extension). Suppose that
erage may be more suitable as an introductory textbook.

there is one unit of an indivisible good for
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*Maskin: The Unity of Auction Theory*
sale. There are

*n *potential buyers, indexed
buyers to bid truthfully and the high bidder
by

*i *= 1,…,

*n*, and each buyer

*i *has a valua-
wins, the second-price auction is efficient.
tion

*v *for the good (the most he is willing to
pay for it). Thus if he pays

*p*, his net payoff
dominant-strategy property is the fact that a
An

*auction *is a game in which (i) buyers
winning buyer’s payment does not depend
make “bids” for the good (for now we will be
on his bid. Next, we show that, under mild
permissive about what a bid can be), on the
hypotheses, the second-price auction is the
basis of which (ii) the good is allocated to (at

*only *efficient auction with this property
most) one of the buyers, and (iii) buyers
(modulo adding a term not depending on

*bi*
negative) to the seller. An auction is

*efficient*
*Proposition 2 *(Jerry Green and Jean-Jacques
if, in equilibrium (we need not worry about
the precise concept of equilibrium at this
point), the winner is the buyer

*i *with the
Theorem 2.3): Suppose that, for all

*i*,

*v *can
assume any value in [0, 1]. Then an auction is
efficient and bidding truthfully is weakly dom-
attained by a

*second-price auction*: an auc-
inant if and only if (a) the high bidder wins
and (b) for all

*i*, buyer

*i*’s payment

*p *satisfies
numbers as bids, the winner is the high bid-
der (ties can be broken randomly), and the
winner pays the second-highest bid (nobody
for some function

*t *, where

*b *is the vector

*Proposition 1 *(Vickrey 1961; Theorem 2.1 in
Milgrom 2004): In a second-price auction, it
Proof

*: *Consider an efficient auction in which
is (weakly) dominant for each buyer

*i *to bid
truthful bidding is dominant. Then, the high
his valuation

*v *(i.e., regardless of how other
bidder must win (property (a)). As for (b), let
buyers bid, it is optimal for buyer

*i *to set a

*tL*(

*b *,

*b *) be buyer

*i*’s payment if he loses and
bid of

*b *ϭ

*v *). Furthermore, the auction is
Proof: Suppose that buyer

*i *bids

*b *Ͻ

*v *. The

*t *(

*b*Ј,

*b *)Ͼ

*t *(

*b*Љ,

*b *)
only circumstance in which the outcome for

*i *is changed by his bidding

*b *rather than

*v *is
for bids

*b*′

*,b*′ ≤ max

*b*
when the highest bid

*b *by other bidders sat-
off bidding

*b*Љwhen

*v *ϭ

*b*Ј, contradicting the
isfies

*v *Ͼ

*b*Ͼ

*b *. In that event, buyer

*i *loses
dominant-strategy property. Hence, we can
by bidding

*b *(for which his net payoff is 0)
but wins by bidding

*v *(for which his net pay-
off is

*v *Ϫ

*b*). Thus, he is

*worse off *bidding

*t *(

*b *,

*b*- )ϭ

*t *(

*b*- ).

*b *Ͻ

*v *. By symmetric argument, he can only
Similarly, we can write buyer

*i’s *payment
be worse off bidding

*b *Ͼ

*v *. We conclude

*tW*(

*b *,

*b *) if he wins as
that bidding his valuation (truthful bidding)
is weakly dominant. Because it is optimal for

*tW*(

*b *,

*b *)ϭˆ

*t *(

*b *).

*j *, buyer

*i*’s winning or los-
Efficiency is often an important criterion in auction
design, particularly in the case of privatization. Indeed, the
ing are both efficient, and so for truthful
U.S. Congress directed the FCC to choose an auction
bidding to be dominant, buyer

*i *must be
design for allocating spectrum licenses that (to quote AlGore) puts “licenses into the hands of those who value
indifferent between them. From (1) and (2),
them the most” (see Milgrom 2004, p. 4).

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*Journal of Economic Literature, Vol. XLII (December 2004)*
max

*b *−

*tˆ *(

*b *) = −

*t *(

*b *)
with c.d.f.

*F *and support [0, 1]. Then there
exists an efficient and payment-balanced auc-tion in which bidding truthfully constitutes a

*ˆt *(

*b *) = max

*b *+

*t *(

*b *)
Proof

*: *For convenience, assume

*n*ϭ2. In an
i.e., (b) holds. Conversely, if (a) and (b) hold,
it is immediate that the auction is efficient
and, from Proposition 1, that truthful bidding
1, buyer 1 will choose

*b*1 to maximize
Call the auctions of Proposition 2 “gener-
1

*v dF *(

*x*) −

*P *(

*b *)
alized Vickrey” auctions. It is easy to see that
there is no generalized Vickrey auction in
if buyer 2 bids truthfully. The first-order
which payments “balance,” i.e., sum to zero.

*Proposition 3 *(Green and Laffont 1977;

*b F*′(

*b *) =

*P*′(

*b *)
Laffont and Maskin 1980; Milgrom’s Theorem2.2): Under the hypotheses of Proposition 2
there exists no generalized Vickrey auction in
buyer 1’s best reply to 2 is to bid truthfully(because the first-order condition holds at
Proof

*: *For convenience assume

*n*ϭ2.

Consider a generalized Vickrey auction.

2 (

*v*1)Ն0). Similarly, truthfulness is a best
reply for buyer 2 if his payment function is

*p*2

*v*2

*t*1(

*v*2)ϩ

*t*2(

*v*1).

If the right-hand side of (3) equals zero for

*p *(

*b ,b *) =

*P *(

*b *) −

*P *(

*b *)

*p *(

*b ,P *) =

*P *(

*b *) −

*P *(

*b *)
1 is a constant. Similarly, for

*v*2
Then it is evident that the players’ paymentssum to zero and that truthfulness remains an
equilibrium (the latter follows because sub-
for constant

*k*2. From (4) and (5), we can
tracting

*P*2(

*b*2) from buyer 1’s payment does
not affect his incentives and similarly forbuyer 2).
which clearly cannot equal zero for all

*v*1.

tent with efficiency once we relax the solu-
Hence, balanced payments are impossible.

*individual rationality *if one of the players
already owns the good. More specifically,
Gérard-Varet (1979) show, the failure of bal-
suppose that player 1 owns the good and that

*n*ϭ2. An efficient mechanism will transfer
relaxing the solution concept from dominant-
the good to player 2 if and only if

*v *Ͼ
Thus, in a payment-balanced and efficient

*Proposition 4 *(d’Aspremont and Gérard-
mechanism,

*individual rationality *for player
Varet 1979): Suppose that, for all

*i*,

*v *is
1 (the “seller”) is the requirement that
dec04_Article 5 12/14/04 2:27 PM Page 1107

*Maskin: The Unity of Auction Theory*
∫ (

*b *(

*v *)

*,b *(

*v *))

*dF *(

*v *)−
whereas individual rationality for player 2
where

*k*1 and

*k*2 are constants of integration.

*v dF *(

*v *) −

*p *(

*b *(

*v *)

*,b *(

*v *))

*dF *(

*v *)
1(

*v*1) and

*b*2(

*v*2) are the (Bayesian)
their valuations are

*v*1 and

*v*2 respectively.

*Proposition 5 *(Laffont and Maskin 1979;
1983; Milgrom’s Theorem 3.6 ): Let

*n*ϭ2.

there exists no efficient and payment-bal-
equilibrium is the solution concept.

Proof

*: *The proof is considerably simplified
Propositions 1 and 4 on the one hand (which
by supposing that

*F*1 and

*F*2 are uniform dis-
exhibit efficient auctions) and Proposition 5
tributions on [0, 1]. Consider a balanced-
on the other (which denies the existence of
payment and efficient mechanism for which
such a mechanism). The reason for the dif-
ference lies in the issue of ownership. In the
former two propositions, no player yet owns
∫ (

*b *(

*v *)

*,b *(

*v *))

*dv*
the good. We can interpret the latter propo-
sition, however, as applying to the circum-
stance in which there has already been an
∫ (

*b *(

*v *)

*,b *(

*v *))

*dv*
auction that player 1 won—so that he now
where

*p *(

*b *(

*v *)

*,b *(

*v *))
has the opportunity to resell. Together, these
two sets of propositions validate Milgrom’s
um payment when valuations are (

*v*1,

*v*2).

refutation of the claim that auctions are
Hence, in equilibrium, player 1’s and player
unnecessary for efficiency, that

*ex post *free
trade among the players will ensure the right
allocation. According to this claim we might
just as well assign assets like spectrum licens-
them later to correct misallocations. But
licenses have been distributed, efficiency

*v *−

*Pˆ*′(

*v *) = 0
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*Journal of Economic Literature, Vol. XLII (December 2004)*
very different auction forms. Milgrom pres-
Clearly,

*G *(

*v *) must be lie between 0 and
ents his view of this material in chapters 3
1, but there are other restrictions on it as
and 4. The central result is what he calls the
well. In particular, it must be nondecreasing.

*payoff-equivalence theorem *(which implies

*Proposition 7 *(Myerson 1981, Riley and
the considerably weaker but more familiar
Samuelson 1981, Milgrom’s Theorem 4.1):
In any Bayesian equilibrium a buyer’s proba-

*Proposition 6 *(Vickrey 1961; Myerson 1981;
bility of winning is a nondecreasing function
Proof

*: *From (12) and (13), the derivative of
hypotheses of Proposition 4, if there are two
buyer

*i*’s equilibrium expected payoff if his val-
auctions such that, in Bayesian equilibrium,
uation is

*v *but he bids as though it were ˆ

*v *is
(a) for all

*i *and

*v *, the probability of winning
for a buyer

*i *with valuation

*v *is the same in

*G*Ј(ˆ

*v *)(

*v *Ϫ ˆ

*v *).

both auctions, and (b) for all

*i*, the amount
But if

*G*Ј(

*v *)Ͻ0 for some

*v *then from
that buyer

*i *with valuation 0 pays is the same
in both auctions, then, for all

*i *and

*v *, the
maximum (

*G*Ј(

*v *)Ն0) is violated at ˆ

*v *ϭ

*v *, a

*equilibrium expected payoff *for buyer

*i *with
valuation

*v *is the

*same *in both auctions.

Proof

*: *Choose one of the two auctions andlet (

*b*
1(

*v*1),…,

*b *(

*v *)) be Bayesian equilibri-
um bids by the buyers when valuations are(

*v*
in equilibrium of the second-price auction
1,…,

*v *). Because buyer

*i *has the option
of behaving as though his valuation is ˆ

*v*
when in fact it is

*v *, he, in effect, faces the
truthfully, and so a buyer’s probability of
winning is simply the probability that theother buyers all have lower valuations (the
where

*G *( ˆ

*v *)ϭbuyer

*i*’s probability of win-
ning and

*P *( ˆ

*v *)ϭbuyer

*i*’s expected payment
if he bids

*b *( ˆ

*v *) and each of the other buyers
in that auction. But (16) and (17) also hold for

*j *bids according to the equilibrium bid
equilibrium of the English auction, the mech-
function

*b *(·). By definition of equilibrium,
anism in which buyers call out bids openly,
the maximizing choice of ˆ

*v *in (12) is ˆ

*v *ϭ

*v *,
each successive bid must be higher than the
previous one, and the winner is the last buyer

*P*′(

*v *) =

*G*′(

*v *)

*v*
to bid (and pays his bid). To see this, notice
that a buyer will continue to bid higher in the
English auction until the current price reach-

*P *(

*v *) =

*v G *(

*v *) −
es his valuation, and so the high-valuation
buyer will win, i.e., (16) holds. Thus we have:
where

*k *is a constant of integration. Notice

*Proposition 8 *(Vickrey 1961): The second-
from (14) that buyer

*i*’s expected payment if

*v *ϭ0 is

*k *. By hypothesis (b), this is true of
the other auction as well. Furthermore, by
hypothesis (a),

*i*’s probability of winning in
the other auction is

*G *(

*v *) for all

*v *. Hence,
“standard” auctions are equivalent.

from (14), buyer

*i*’s expected payment is

*Proposition 9 *(Vickrey 1961; Riley and

*P *(

*v *) and his equilibrium expected payoff is
Samuelson 1981; Myerson 1981; Milgrom’s

*G *(

*v *)

*v *Ϫ

*P *(

*v *) in both auctions.

Theorems 4.6 and 4.9): When each

*v *is
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*Maskin: The Unity of Auction Theory*
with c.d.f.

*F *and support [0, 1], then the
Proposition 9 can be modified to maximize
high-bid, second-price, English, Dutch, and
all-pay auctions are payoff-equivalent.

*Proposition 10 *(Riley and Samuelson 1981;
Proof

*: *We have already described the rules
of all but the Dutch and all-pay auctions. In
Assume the hypotheses of Proposition 9 and
the

*Dutch auction*, the auctioneer continu-
ously lowers the price, starting from some

*J*Ј(

*v*)Ͼ0 for all

*v*,
high level, until some buyer (the winner)agrees to buy at the current price. Notice
that this is formally the same as the high-bid
where

*J*(

*v*) =

*v *− −
auction, since the price at which a buyer
auctions of Proposition 9 maximizes the sell-
agrees to buy in the Dutch auction is the
er’s expected revenue provided that the sell-
same as the bid he would make in the high-
er sets a reserve price

*v*∗ (i.e., he refuses to
bid auction.13 In the

*all-pay auction*, buyers
sell for less than

*v*∗), where

*J*(

*v*∗)ϭ0.

Proof

*: *Given buyers’ ex ante symmetry, (14)
high bidder, but all buyers pay their bids.

implies that, for any symmetric auction (we
Consider a symmetric equilibrium

*b*(·) of
restrict to symmetric auctions without loss of
the high-bid auction; i.e.,

*b*(

*v*) is the bid any
generality), the seller’s expected revenue is
buyer with valuation

*v *makes (a symmetric
equilibrium exists because of the ex ante

*vG*(

*v*) −

*G*(

*x*)

*dx *+

*k dF*(

*v*),
symmetry of the buyers). From Proposition
7,

*b*(·) must be nondecreasing. Suppose it is
not strictly increasing, i.e., suppose

*b*(

*v*)ϭ

*b*∗
for all

*v*ʦ[

*v*∗,

*v*∗∗]. We have

*v*∗Ϫ

*b*∗Ն0
) (

*v*)

*dF*(

*v*)+

*n *.

because, thanks to the atom at

*b*∗, a bid of

*b*∗
We have already noted that

*G*(

*v*) must satisfy
wins with positive probability (and thus if

*v*∗Ϫ

*b*∗Ͻ0, the buyer would have a negativepayoff). Hence we obtain
As Steven Matthews (1984) shows, it mustalso satisfy
But if a buyer with reservation price

*v*∗∗ bids

*b*∗, he ties for the high bid with positive prob-
−

*G*(

*x*)

*dF*(

*x*) 0 for all

*v*. (22)
ability. Thus if he slightly increases his bid,
he discontinuously raises his chances of win-
subject to (21) and (22). Note from (12) and
ning (since a tie then has zero probability),
(14) that

*k*Յ0, otherwise a buyer with zero
which is worthwhile in view of (18). We con-
valuation has a negative expected payoff (an
clude that

*b*(·) must be

*strictly *increasing,
impossibility, since he always has the option
which means that the high-valuation buyer
of not participating). Hence the maximizing
always wins. Thus, Proposition 6 implies
that the high-bid auction is equivalent to thesecond-price auction. This same argument

*P*(0)ϭpayment by a 0-valuatioin buyerϭ0.

applies also to the all-pay auction.

From (19)–(22), the optimal choice of

*G*(

*v*) is
This equivalence relies on the assumption that buy-

*F*(

*v*)

*, if v v*∗
ers obey the usual axioms of expected utility; see Daisuke
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*Journal of Economic Literature, Vol. XLII (December 2004)*
where

*J*(

*v*∗)ϭ0. But all of the auctions of
4.12): Assume that buyers are

*risk averse*;
Proposition 9, modified by a reserve price of
i.e., buyer

*i*’s utility from winning is

*u *(

*v *–

*p *),

*v*∗, satisfy (23) and (24), and so they are solu-
where

*u *is strictly concave. Suppose that
buyers are ex ante symmetric, i.e., the

*v *’s are
The fact that the Dutch and high-bid auc-
drawn (independently) from the same distri-
bution with c.d.f.

*F *and support [0, 1] and
forms. Nor is the equivalence (Proposition
generates higher expected revenue than the
8) between the English auction and the sec-
ond-price auction very deep. But the sense
Proof

*: *First, observe that risk aversion does
in which all four auctions are equivalent
not affect behavior in the second-price auc-
(Proposition 9) is more interesting, as is the
tion; it is still optimal to bid truthfully. If

*b*(·)
idea that any of them—modified by setting a
is a symmetric-equilibrium bid function in
reserve price15—can be used to maximize
the high-bid auction, a buyer with valuation
the seller’s revenue (Proposition 10).

5.

*Departures from the Benchmark Model*
on some restrictive hypotheses, viz., (i) buy-
−

*F u*′

*b*′ + (

*n *− 1

*F*
ers’ risk neutrality, (ii) private values (to be
defined below), (iii) independent valuations,
) ′(

*v*)

*u*(

*v *−

*b*(

*v *))
(iv) ex ante symmetry, and (v) financially
unconstrained buyers.16 We will now relax

*F*(

*v*)

*u*′(

*v *−

*b*(

*v *))
each of (i)–(iii) in turn (for relaxation of (iv),
Now, if buyers were risk neutral, (25) would
see Milgrom’s Theorems 4.24–4.27, and
Maskin and Riley 2000; for relaxation of (v),
) ′(

*v*)(

*v *−

*b *(

*v*)
see Milgrom’s Theorem 4.17, and Che and
Note first that in Propositions 6–10, we
suppose that buyer

*i*’s objective function is
second-price auctions are payoff-equivalent
given by (12), i.e., that he is

*risk neutral*. If
we replace risk neutrality with risk aversion,
symmetric. Hence,

*b *(

*v*) is also the expect-
then in particular Proposition 9 no longer
ed payment by a winning

*v*-valuation buyer
in the second-price auction (whether he is

*Proposition 11 *(Holt 1980; Maskin and Riley
risk averse or not). Because

*u*ЉϽ0,

*u*(

*v *−

*b*(

*v *))

*v b*(

*v*),
We have ignored the constraint

*G*ЈՆ0 because it is sat-
isfied by the solution to the program in which it is omitted.

15 The reason a reserve price helps the seller is that it
puts a lower bound on what buyers pay. Of course, by set-

*b*′(

*v*) >

*b*′ (

*v*)

*whenever b*(

*v*) =

*b *(

*v*)
ting a positive reserve, the seller runs the risk of not sellingat all, but this effect is outweighed by the lower-bound
Because

*b*(0)ϭ

*b *(0)ϭ0, we conclude that
consideration. To see this, imagine that there is just one
buyer. Then in a high-bid auction, that buyer would bid

*b*(

*v*) >

*b *(

*v*)
zero; the only way to get him to pay

*anything *is to makethe reserve positive.

which implies that, for every

*v*Ͼ0, a buyer
16 The efficiency of the second-price auction (Proposition
1) invokes (ii), as we will see in Proposition 12, and (v) (see
pays more in the high-bid than in the second-
Maskin 2000), although it does not demand (i), (iii), or (iv).

dec04_Article 5 12/14/04 2:27 PM Page 1111

*Maskin: The Unity of Auction Theory*
2 and 3 is that buyers actually

*know *their
valuations (more to the point, that their val-

*j*≠

*i *∂

*si*1 ≠ ∂

*si*1
uations do not depend on the private infor-
mation of

*other *buyers). This is called the

*private values *assumption. Let us now relax
Then, if Bayesian equilibrium is the solution
it to accommodate

*interdependent values*
concept, there exists no efficient auction.

(sometimes called generalized

*common val-*
Proof (sketch): Choose parameter values

*s*Ј

*ues*). Specifically, suppose that each buyer

*i*
and

*s*Љ such that (

*s*Ј)ϭ (

*s*Љ). From (26),
receives a private signal

*s *and that his valu-
buyer

*i*’s preferences are

*identical *for

*s *ϭ

*s*Ј
ation is a function of

*all *the signals: i.e., his
and

*s *ϭ

*s*Љ and so, in equilibrium, he must be
valuation is

*v *(

*s *,

*s *), where

*s *is the vector of
indifferent between the outcomes that result
other buyers’ signals. In such a setting, the
from the two cases. But, from (27), which of
second-price auction will no longer be effi-

*s *ϭ

*s*Ј or

*s *ϭ

*s*Љholds will in general lead to dif-
ferent efficient allocations—e.g., perhaps
because buyers no longer know their valua-
buyer

*i *wins when

*s *ϭ

*s*Ј and loses when
tions, their bids (reflecting their

*expected*
*s *ϭ

*s*Љ. Thus buyer

*i *cannot be indifferent
valuations) do not guarantee that the high
bidder actually has the highest valuation.

Despite this negative result, matters are
Nevertheless, if the signals are one-dimen-
not as bleak as they may seem, at least in the
case of

*single-good *auctions. Specifically,
introduce a one-dimensional “reduced” signal

*r *for buyer

*i *and, for all

*j*≠

*i*, define
nisms that ensure efficiency (provided that
“single crossing” holds, i.e., buyer

*i*’s signal
(

*,s *) =

*E v *(

*s ,s *) ϕ (

*s *) =

*r*
has a greater marginal effect on his own val-
i.e., ˆ

*v *(

*r *,

*s *) is

*v *(

*s *,

*s *) expected over all those
uation than on other buyers’ valuations:
values

*s *such that (

*s *)ϭ

*r *. Because we are

*j *for

*j i*). More concretely, Partha
back to one-dimensional signals, the extend-
“reduced” valuations

*v *(

*r *,

*s *) and { ˆ

*v *}
(assuming that the single-crossing property
above holds). That is, the auction is efficient
accommodate

*contingent *bids so that effi-
subject to the constraint that buyer

*i *behaves
ciency is restored (in a more limited class of
the same way for any signal values

*s *for
which (

*s *)ϭ

*r *, i.e., it is

*incentive efficient*
that the English auction is efficient with
(see Dasgupta and Maskin 2000). Jehiel and
Moldovanu (2001) show, however, that this
signals are multidimensional, then efficien-
“reduction” technique does not generalize to

*Proposition 12 *(Maskin 1992; Philippe Jehiel
Finally, let us explore what happens when
we drop the assumption of independence in
Theorem 3.8): Suppose that, for some buyer

*v *(

*s ,s *) = ϕ (

*s *) + ψ (

*s *)
that

*n *ϭ 2,18 and that

*v*1 and

*v*2 are jointly
17 For convenience, suppose that all the other signals
18 Milgrom and Weber (1982) generalize this result to

*n*Ն3.

dec04_Article 5 12/14/04 2:27 PM Page 1112

*Journal of Economic Literature, Vol. XLII (December 2004)*
symmetrically distributed with support [0, 1]
and

*affiliated *(positively correlated) in the

*v*1

*v dF*(

*v | v *)

*F*2(

*v | v *)
But (34) follows from affiliation, i.e., from
where

*F*(

*v*2|

*v*1) is the c.d.f. for

*v*2 conditional
on

*v*1. Then, revenue from the second-price
Notice that Propositions 11 and 13 pull in
opposite directions: the former favors the
high-bid auction, the latter the second-price
Proof: Let

*P*1(

*v*1) be buyer 1’s equilibrium
expected payment when his valuation is

*v*1.

Milgrom’s introductory points: that the kind
of auction a seller will want to use depends

*PH*(

*v *) =

*F*(

*v | v *)

*b*(

*v * ),
Milgrom’s observations about auctions in
where the superscripts

*S *and

*H *denote sec-
practice are a good deal less compelling than
ond-price and high-bid auctions, respective-
the book’s theoretical results. But this con-
ly, and

*b*(·) is the symmetric equilibrium bid
trast is not primarily his fault. In spite of all
function in the high-bid auction. Clearly,
that it has accomplished, auction theory still
1 (0) ϭ

*P*1 (0). We wish to show that
has not developed far enough to be directly
1 (

*v*1)Ͼ

*P*1 (

*v*1) for all

*v*1
applicable to situations as complex as, say,

*dPS *(

*v *)

*dPH *(

*v *)
1 (

*v*1)ϭ

*P*1 (

*v*1).

To begin with, those auctions involve mul-
tiple goods. Observe, however, that all the
results presented above are for single-good
auctions. This is no coincidence; the litera-
ture on auction theory has overwhelmingly
focused on the single-good case. Apart fromthe efficiency of the multigood second-price
′(

*v *) +

*b*(

*v *)

*F*1
auction (the Vickrey-Clarke-Groves mecha-
nism; see section 7) with private values,
+

*b*(

*v *)

*F*2(

*v | v *),
there are very few general results for more
where superscripts of

*F *denote partial deri-
attempt to extrapolate from well-analyzed
vations. The second equation can be rewrit-
ten, using buyer 1’s first-order condition, as
stances (multiple goods). But doing so is
Another difficulty for theory is that real
auctions impose constraints that are difficult
to formalize. Milgrom notes that prospective
bidders and sellers are typically nervous
about participating in auction mechanisms
From (31)–(33), it remains to show that
dec04_Article 5 12/14/04 2:27 PM Page 1113

*Maskin: The Unity of Auction Theory*
giving a precise meaning to “unfamiliar” or
“complicated” is forbiddingly difficult.

lowing sort of problem. Assume that there
The upshot is that giving advice on real auc-
tion design is, at this stage, far less a science
than an art. And the essence of an art is far
wants these goods only as a

*package*, i.e., his
harder than a science to convey convincingly
valuation for A or B alone is zero. Suppose
that buyer 1 has a valuation of $100 for Aand B together, but that buyer 2 has a pack-
7.

*The Vickrey-Clarke-Groves Mechanism*
age valuation of $200. If the buyers bidtruthfully in the VCG mechanism, then
buyer 2 will win both A and B (and pay $100,
the winning bid were 2 not present). Buyer
(VCG) mechanism, the generalization of the
second-price auction to multiple goods.

Indeed, his unhappiness with it has led him
bids through two

*different *proxy buyers, 1

*x*
and 1

*y*. As buyer 1

*x *he enters a bid of $201
(Ausubel and Milgrom 2002) on an interest-
package AB). As 1

*y*, he enters a bid of $201
for good B (and zero for both A and AB).

Then 1

*x *and 1

*y *will be the winners of A and
(i) each buyer makes a bid not just for each
B respectively, and so 1 will obtain both
good but for each combination (or “pack-
goods. Furthermore, notice that had 1

*x *not
age”) of goods; (ii) goods are allocated to
bid at all, 1

*y *would still be the winner of
buyers in the way that maximizes the sum of
good B (good A would just be thrown away),
the winning bids (a bid for a package is “win-
and so the sum of the other buyers’ winning
ning” if the buyer making that bid is allocat-
bids is the same (namely, $201) whether 1

*x*
ed the package); (iii) each winning buyer

*i*
participates or not. Thus, by VCG rules, 1

*x*
pays nothing at all (and, similarly, neither
between (a) the sum of the bids that would
does 1

*y*), which means that the ploy of pass-
win if

*i *were not a participant in the auction
ing himself off as multiple buyers is worth-
and (b) the sum of the other buyers’ (actual)
winning bids. Following the line of argu-
revenue for the seller and leads to an ineffi-
ment in the proof of Proposition 1, one can
cient allocation (1 wins the goods, rather
show that truthful bidding (reporting one’s
than 2), which is why Milgrom is led to reject
true valuation for each package) is dominant.

Thus the auction results in an efficient allo-
But notice that having 1

*x *and 1

*y *enter
cation (an allocation that maximizes the sum
these bids makes sense for 1 only if he is
quite sure that buyer 2 does not value A and
B as single goods. As soon as there is a seri-
ous risk that 2 will make single-good bids
both buyers (placing bids on each package
that add up to $101 or more, buyer 1 will
can be an onerous task) and the auctioneer
come out behind with this strategy (relative
to truthful bidding). If, for example, buyer 2
potentially difficult maximization problem).

bid $51 for each of A and B alone (as well as
This, however, is not the shortcoming that
$200 for the package), 1

*x *and 1

*y *would still
be awarded A and B with their $201 bids but
Milgrom paper is subject to the same sort of
dec04_Article 5 12/14/04 2:27 PM Page 1114

*Journal of Economic Literature, Vol. XLII (December 2004)*
Indeed, with sufficient uncertainty about
Restricted Domains,”

*Econometrica *47, pp. 1137–44.

how other buyers will bid, it is not hard to see
Holt, Charles. 1980. “Competitive Bidding for
Contracts under Alternative Auction Procedures,”

*J.*
*Polit. Econ. *88:3, pp. 433–45.

makes sense for a buyer in a VCG auction.

Jehiel, Philippe and Benny Moldovanu. 2001.

And since I would venture to say that consid-
“Efficient Design with Interdependent Valuations,”

*Econometrica *69:5, pp. 1237–59.

erable uncertainty is quite common in real
Klemperer, Paul. 2004.

*Auctions: Theory and Practice*.

auction settings, I believe that Milgrom is too
harsh when he deems VCG “unsuitable” for
Krishna, Vijay. 2002.

*Auction Theory*. San Diego:
———. 2003. “Asymmetric English Auctions,”

*J.*
*Econ. Theory *112:2, pp. 261–68.

Laffont, Jean-Jacques and Eric Maskin. 1979. “A
Differential Approach to Expected UtilityMaximizing Mechanisms,” in

*Aggregation and*
This is a minor reservation about a volume

*Revelation of Preferences*. Jean-Jacques Laffont, ed.

that covers a cornucopia of material in mag-
———. 1980. “A Differentiable Approach to
isterial fashion and gives us deep insight into
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*Econometrica *48,
the thinking of an outstanding theorist. The
book is not for everybody; one needs at least
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*Privatization: Symposium in Honor of Herbert*
enough technique to be able to follow the

*Giersch*. H. Siebert, ed. Tubingen: Mohr (Siebek),
proofs of the propositions above. But, with
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———. 2000. “Auctions, Development, and
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*European Econ. Rev*. 44, pp.

Maskin, Eric and John Riley. 1984. “Optimal Auctions
with Risk Averse Buyers,”

*Econometrica *52:6, pp.

Ausubel, Lawrence and Paul Milgrom. 2002.

Maskin, Eric and Tomas Sjöström. 2002.

“Ascending Auctions with Package Bidding,”
“Implementation Theory,” in

*Handbook of Social*
*Frontiers Theoret. Econ. *1:1.

*Choice and Welfare*, vol. I. Kenneth Arrow, Amartya
Che, Yeon-Koo and Ian Gale. 1998. “Standard Auctions
Sen and Kataro Suzumara, eds. NY: Elsevier
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*Rev. Econ.*
Matthews, Steven. 1983. “Selling to Risk Averse Buyers
Clarke, Edward. 1971. “Multipart Pricing of Public
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*J. Econ. Theory *30, pp.

Goods,”

*Public Choice *11, pp. 17–33.

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Selling Strategies under Uncertainly for a
Form Auctions,”

*Econometrica *52, pp. 1519–22.

Discriminating Monopolist When Demands Are
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*,” J.*
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*Quart. J. Econ. *115:2, pp. 341–88.

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Source: http://luciemenager.free.fr/agreg/JEL/auction.pdf

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The Third International Conference on International Business in Transition Economies September 9-11, 2004, The Stockhlm School of Economics in Riga, Latvia Doctoral Seminar on The Theory and Methodology in International Entrepreneurship, Innovation and Competitiveness Research in the CEE Context September 7-8, 2004, Stockholm School of Economics in Riga, Latvia