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Games and Economic Behavior 45 (2003) 488–497
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Auction Theory
By Vijay Krishna. Academic Press, 2002.
1. Introduction
Writing a book about auctions requires making a number of difficult decisions. How
much of the rich history of auctions will be presented? Should current practice and the
ongoing debate over alternative procedures be covered? Should empirical and experimental
evidence be discussed? What, for that matter, is an auction?
Vijay Krishna makes his choices clear from the start: This is a book about auction
theory. Furthermore, for purposes of the book, an “auction” is an explicit economic
mechanism by which a single seller permits bidders, through their actions, to determine
both payments to the seller and an allocation of the item (or items) being sold. He
has chosen to “concentrate on selected themes. . . central to the theory,” via “detailed
consideration of a few basic models.”
One cannot fault an author for choosing to write a book, rather than an encyclopedia.
And indeed, this is quite a book, and it fills a major gap in the economic literature: An
economist (or game-theorist) looking to come up to research speed, or an educator looking
to inject either a few lectures on auctions into an advanced course, or an entire new course
into a curriculum, will find Krishna’s Auction Theory to be a well-written, logicallyorganized,
self-contained, and very complete introduction both to the foundations of the
subject, and to the issues drawing current research attention.
Most of the results presented in this book can be found in their original form in Paul
Klemperer’s (2000) superb two-volume collection of papers, The Economic Theory of
Auctions.With the advantage of historical perspective, though, Krishna brings these results
together under a uniform notation, and chooses to present those examples, derivations, and
proofs that have been found to best illustrate the underlying intuitions.
2. Approaching auction theory
It has become traditional to take a bottom-up approach to auction theory, and Krishna
follows that tradition. Rather than start with a broad overview of the many economic
0899-8256/2003 Published by Elsevier Inc.
doi:10.1016/j.geb.2003.08.001
Book review / Games and Economic Behavior 45 (2003) 488–497 489
environments in which auctions may be conducted, or the many varieties of auction
procedures that might be implemented, he begins with a comprehensive treatment of the
four best-known single-itemauction procedures, in the simplest of economic environments.
A number of risk-neutral bidders are independently drawn from a general population.
The distribution of “tastes” (i.e., private valuations) across the population is commonly
known by all of the bidders, and each bidder knows perfectly what the item being sold is
worth to him.
In this “symmetric independent private values” environment, an auction is conducted.
The seller might choose to accept sealed bids, and award the item to the highest bidder.
But even after the allocation rule is specified, the seller must still announce a pricing rule.
For example, he might declare that the highest bid establishes the price to be paid: This
is the “first-price” auction procedure. But he could just as easily use the second-highest
submitted bid to set the price: This is the “second-price” auction procedure first studied by
Vickrey in the late 1950s.
Alternatively, the seller could employ a price “clock,” which gradually rises, and permit
bidders to permanently (and publicly) withdraw from the auction as the price increases.
When the next-to-last bidder withdraws, the clock is stopped, and the remaining bidder
pays the displayed price for the item: this is commonly called an “English auction.” Of
course, the seller could instead start the clock at a price above the most any bidder could
ever be expected to value the item, and let the price descend until some bidder signals a
willingness to pay the displayed price: This is a “Dutch auction” (so named due to its use
in wholesale cut-flower markets in the Netherlands).
Why would a seller choose one type of auction procedure over another? A financially
self-interested seller would likely wish to use a procedure which is expected to generate
a high price. On the other hand, a public agency might be motivated by the desire for an
efficient outcome, expecting that such an allocation will ultimately generate the greatest
social good. Throughout his book, Krishna balances his discussion between these two
objectives, highlighting in particular those cases where the two are incompatible.
No matter which objective is selected, the seller faces one final challenge: How to
compare the predicted results from the use of various auction procedures? Predictions can’t
be made until some assumption is made concerning how bidders will behave.
This is where game theory quite naturally comes into play. A rationality-based
benchmark for comparisons is set by assuming that each bidder will correctly form
expectations as to the way the other bidders will behave, and will act optimally in light
of those expectations, i.e., that the bidders’ strategies (viewed as functions of their initial
information, and any additional information revealed as the auction progresses) together
constitute a (Bayesian Nash) equilibrium of the auction (viewed as a game of incomplete
information). Krishna provides a four-page appendix defining the game-theoretic concepts
required for a treatment of auction theory, but appropriately recommends that any reader
have a basic understanding of game-theoretic principles, or acquire such, before trying to
dig deeply into the material presented.
Even this, though, is not quite enough to permit a direct comparison of predicted
results across different auction procedures, since some auction “games” have multiple
equilibria. (For example, in a sealed-bid second-price auction, any one bidder could ignore
all the information he holds, and simply submit a ridiculously-high bid, while the other
490 Book review / Games and Economic Behavior 45 (2003) 488–497
bidders all bid 0. The first bidder gets the item for free, yet none of the others could gain
from a change in strategy.) Where appropriate throughout his book, Krishna presents four
ways of dealing with this issue: In some settings, for some auction procedures, equilibria
are unique. In cases where both the economic environment and the auction rules are
symmetric across all bidders, symmetric equilibria (which are frequently unique) can be
compared. In other cases, specific equilibria can be selected for their special properties
(such as involving only dominant strategies, or, at least, undominated strategies). Finally,
mechanism-based analysis (exploiting the Revelation Principle) provides a method for
exploring, in various contexts, the full range of outcomes achievable at any equilibrium
of any auction procedure.
3. Overview of the book
There are three fundamental principles that form a foundation for most of the auction
theory literature. Each of the three is (very) roughly sketched below.
The “Revenue Equivalence Principle” arises in the independent private values setting
(with risk-neutral bidders). It asserts that, at (an unsubsidized) equilibrium, the seller’s
expected revenue from an auction depends only on the way the final allocation of the
item being sold relates to the actual private valuations of the bidders (and not on the
details of the pricing rule). For example, if at equilibrium two different auction procedures
always end up awarding the item to the bidder who values it most highly, then the
seller’s expected revenue will be the same (even though his actual revenues might differ in
specific instances) no matter which procedure is used. This principle facilitates the study of
revenue-maximizing auction procedures. The first five chapters of Auction Theory explore
this and related topics.
When the private signals received by the bidders (which might, for example, be
estimates of the quality of the item being sold, rather than direct private valuations) are
not independent, the “Linkage Principle” comes into play. The payments made by a bidder
can, through the choice of auction procedure, be made to depend not only on that bidder’s
actions, but also (indirectly) on the bidder’s private signal, since the signals of the others
bidders, and therefore their observed actions, will be statistically linked to that private
signal. A seller can exploit this additional linkage, and revenue equivalence no longer
holds across the most-commonly-used auction procedures. Chapters 6 through 8 focus on
auctions held in settings where the bidders’ signals are dependent, and Chapters 9 and 10
continue to explore settings where the bidders’ valuations are dependent.
Finally, when multiple items are to be sold, and when each bidder privately knows his
valuation of each subset of items, there is an auction method—the Vickrey–Clarke–Groves
procedure—under which the bidders are simply asked to report their valuations, truthful
reporting by all constitutes an equilibrium, and an efficient allocation of the items results at
that equilibrium. The final six chapters of Auction Theory contrast this and other methods
for organizing the sale of multiple items in a variety of economic settings.
Chapter 11, which appears between the single-item and multiple-item treatments, stands
somewhat outside the scope of these three areas. Entitled “Bidding Rings,” it examines a
particular formal type of collusive behavior amongst bidders. Since issues of collusion
Book review / Games and Economic Behavior 45 (2003) 488–497 491
also arise in multiple-item auctions, where bidders might find opportunities for mutuallyadvantageous
“strategic demand reduction,” this chapter might have been better placed
(and expanded upon) at the end of the book.
4. Independent private values
In Chapter 1, Krishna presents the four “standard” single-item auction procedures. The
strategic equivalence of first-price and Dutch auctions in general is noted, as is the practical
equivalence of second-price and English auctions in the private values setting.
Chapter 2 begins by showing that, in a second-price auction, each bidder has a dominant
strategy of simply bidding his own valuation. For the case of symmetrically-distributed
bidder valuations, the strategy used by all bidders at the symmetric equilibrium of the
first-price auction (which is known to be the unique equilibrium) is derived. This permits
direct computation of the seller’s expected revenue in each case, yielding Vickrey’s striking
result: While the actual outcomes of the two auctions might differ from one instance to
the next, the average price generated by the two auction formats is the same! (Indeed, it
is the expected value of the second-highest valuation amongst the bidders.) The chapter
concludes with a discussion of reserve prices and entry fees as revenue-enhancement
devices, and shows that some threat to withhold the item from sale is always better for the
seller than no threat at all: Efficiency and revenue-maximization are incompatible goals.
The heart of Chapter 3 is that this “revenue equivalence” holds, in the symmetric
independent private values setting, across all auction formats which, at equilibrium,
(i) deliver the item to the bidder with the highest valuation, and
(ii) leave a bidder with the lowest of all possible valuations with an expected profit of 0.
Remarkably, the classical proof, presented in this chapter, is nonconstructive: The actual
derivation of equilibrium strategies is not required. All-pay auctions (in which each bidder
pays his submitted bid) and third-price auctions are presented as further examples.
The revenue equivalence result depends on the bidders being risk-neutral, facing
no budgetary constraints, and having their valuations drawn independently from a
common distribution. Chapter 4 examines the consequences of relaxing these assumptions,
considering first risk-averse bidders, then bidders with limited budgets, and finally bidders
with valuations drawn independently from distinct distributions. Under the first two
relaxations, first-price auctions will yield expected prices at least as high as (and typically
higher than) the expected prices resulting from second-price auctions. Two examples show
that asymmetrically-drawnbidder valuationsmay result in either of the two auction formats
being the higher revenue-generator.Moreover, asymmetrymay lead to inefficient outcomes
at (the unique) equilibrium of a first-price auction. The final part of this chapter considers
whether opportunities for post-auction resale might restore efficiency, and shows that such
opportunities cannot, in general, do so.
One of the most important ideas in modern economic theory is the “Revelation
Principle.” Imagine throwing a group of strategic competitors into an arena. A set of
rules is announced, and they must follow these rules as they compete. At the end, an
492 Book review / Games and Economic Behavior 45 (2003) 488–497
allocation of ownership rights, and payments to be made by (or to) each competitor, are
the results. (Auctions, negotiations over price, and partnership dissolution all lay within
this framework.) Assuming that the competitors’ strategies constitute an equilibrium of
the rules-based “game” they are playing, what possible results could be expected at
equilibrium?
This seems, at first glance, to be an intractable question. It is not even apparent how
one might categorize all the different sets of rules that might be imposed. However, for
any rule-set and corresponding collection of strategies constituting an equilibrium, one
can construct a new game: An automaton is built to carry out each player’s strategy in
the original game. Each player’s sole strategic action in the new game is to “tell” the
automaton that player’s private information; the automaton then acts accordingly. Clearly,
in this new “information revelation” game, if each other player follows the strategy of
truthfully revealing his private information, the remaining player cannot do better than
to whisper his own private information accurately to his automaton, i.e., “truth-telling”
strategies constitute an equilibrium in the new game.
And this means that any outcome achievable via any equilibrium, under any set of rules,
must be attainable as the outcome of a “revelation game” in which each player finds it
optimal to truthfully reveal his information, given the assumption that all other players will
truthfully reveal their information as well.
Chapter 5 presents the Revelation Principle, and uses it to prove that the Revenue
Equivalence Principle—that the seller’s expected revenue, using any auction procedure,
at any equilibrium in which a buyer with the lowest of all possible valuations has an
expected profit of 0 (this simply rules out seller giveaways), is fully determined by the
manner in which the auction allocates ownership given the bidders’ actual valuations—
holds across all independent private value environments, symmetric or not. By applying
this result to the question of expected seller revenue, “optimal” auctionmechanisms (which
typically depend on the distributions from which the bidders’ valuations are drawn) can be
characterized. The Vickrey–Clarke–Groves (henceforth VCG) mechanism is presented in
its single-item form, and is shown to be outcome-equivalent to a second-price auction
in which all bidders follow their dominant bid-equals-valuation strategies, yielding an
efficient allocation of ownership. In a slight digression from the auction track, the results
derived in this chapter are invoked to yield a simple demonstration of a striking fact
concerning two-party bargaining: In a buyer-seller negotiation over price, if the two parties
have private valuations drawn from distributions with (non-trivial) overlapping supports,
then at no equilibrium of any negotiating “game” will trade occur in all cases where
mutually-advantageous trades exist, i.e., in a rational world, inefficiency is sometimes
inevitable.
5. Dependent signals
Chapter 6 presents the general symmetricmodel, in which the expected value of the item
to a bidder might depend on both his own private information, and the information held by
the other bidders. The bidders’ signals are assumed to be observations of affiliated random
variables. (Affiliation means, roughly, that—both absolutely, and when conditioned on a
Book review / Games and Economic Behavior 45 (2003) 488–497 493
natural class of events—when some of the signals are large, the others are also likely
to be large. Krishna provides an appendix discussing affiliation and some of its useful
consequences.)
The symmetric equilibria of the English, second-price, and first-price auctions are
derived, and it is shown that the Revenue Equivalence Principle no longer holds: The
English auction will generate the highest expected price, the first-price auction the lowest,
and all three procedureswill typically (i.e., except in boundary cases, such as the symmetric
independent private values model, which is a special case of the general symmetric
model) generate different expected prices. All three procedures are shown to yield efficient
outcomes (at their symmetric equilibria), as long as a natural “single-crossing” condition
holds: Each bidder’s expected valuation is at least as responsive to his own signal as are
the others’ valuations to his signal (or, equivalently in this symmetric environment, as is
his expected valuation to the signals of others).
The Linkage Principle—that the seller can benefit by “linking” a bidder’s payments
to information held by others (even indirectly, by linking his payments to the bids of
others)—is presented in Chapter 7. Applications of this principle yield a generalization
of the revenue-ordering results of the previous chapter, as well as showing that a seller’s
revenue-maximizing long-term policy is to fully reveal any relevant information he holds
prior to a sale.
In Chapter 8, a series of examples show that both the revenue-ordering and informationrelease
results of the previous chapter can fail if the bidders’ valuations are not
symmetrically distributed. As well, an example shows that—unlike in the independent
private values case, where some non-trivial reserve price will always enhance revenues—
in a symmetric private value setting with affiliated valuations, a seller might be best off
setting no reserve price.
In a private values environment (symmetric or not), when bidders follow their dominant
“keep bidding until the price passes your valuation” strategies, an English auction always
sells the item to the bidder who values it most highly, i.e., it always yields an efficient
outcome. In Chapter 6, it was shown that, in the general symmetric environment, efficiency
still holds if a single-crossing condition is satisfied. Chapter 9, drawing heavily from
the work of Krishna and his coauthors, presents generalizations of the single-crossing
condition—all focused on a bidder’s signal playing a larger role in determining his own
valuation than it does in determining the valuations of the other bidders—which suffice to
guarantee that English auctions will have efficient equilibria in settings where the bidders
have asymmetrically-determined interdependent valuations.
The first section of Chapter 10 could have just as well been put at the end of Chapter 9.
It continues the discussion of the interdependent-values case, and begins by showing that
no auction mechanism can have an efficient equilibrium if the economic environment fails
to satisfy the single-crossing condition. However, as long as the single-crossing condition
holds, a generalization of the VCG mechanism will always have a truth-telling equilibrium,
and at that equilibrium, the item will always be sold to the bidderwith the highest valuation.
(Unfortunately, a seller would, in general, need to know the bidders’ specific valuation
functions in order to implement this mechanism.)
The remainder of Chapter 10 returns to a focus on seller revenues in the correlatedsignals
case. Exploiting the correlations, the seller can create side-bets (with payoffs
494 Book review / Games and Economic Behavior 45 (2003) 488–497
determined by the bids of others) which the bidders will be willing to accept as a condition
to auction participation, but which transfer, in expectation, essentially all of the bidders’
gains to the seller. While of theoretical interest, this result lacks practical application since
it not only requires that the seller know the valuation functions and the signal distributions,
but also often requires the use of lotteries that assign a positive probability to extremely
large bidder-to-seller payments.
6. Collusion
Chapter 11 returns to the independent private values environment, and examines bidder
rings. A “ring” is composed of bidders who play a two-stage game: They participate in an
internal “knockout” auction that determines a ring representative. Then the representative
participates in an external auction against non-ring members for the item up for sale.
The representative—if he wins the external auction—compensates the other ring members
according to how they bid in the internal auction, and how much he paid in the external
one.
When both stages are conducted using second-price auctions, issues arise concerning
the internal compensation rule, and whether ring members have an incentive to accurately
report their valuations through the internal auction. By construction, it is shown that the use
of an internal “bank” (which sometimes subsidizes the ring members, at other times nets a
profit, and which balances its books over the long haul) can solve the reporting problem.
If, on the other hand, the ring members’ accountsmust balance at the end of each sale, they
will have an incentive to distort their reports.
The external auction—even if those outside the ring know of the ring’s existence—is
typically asymmetric, since the ring representative’s valuation is actually the maximum
of several separate valuations. This makes a general analysis of first-price auctions with
bidder rings difficult. An example shows the equilibrium in the case in which the ring
consists of all potential bidders (so the representative must pay the seller only the reserve
price), and the non-representatives each receive their internal bids in compensation.
7. Multiple items
When the seller seeks to sell multiple items, many generalizations of both the singleitem
economic environment and single-itemauction procedures are available. Are the items
identical, or not? Are the items complements, substitutes, or something in between? Do the
bidders each receive a single valuation (or signal), or several? Are the items to be sold in
sequence, or simultaneously?
The received literature contains fewer positive results here than in the single-item case.
This serves to limit the issues Krishna can discuss.
Chapter 12 deals with the sale of multiple identical items, and presents three sealed-bid
procedures. Under each, each bidder submits a list of bids, all of the bids are ordered, and
the highest bids are the ones which obtain items. A “discriminatory auction” charges each
winning bidder the sum of his winning bids. A “uniform-price auction” charges winning
Book review / Games and Economic Behavior 45 (2003) 488–497 495
bidders the same price for all won items, and that price is the highest of the losing bids. The
“Vickrey procedure” charges a winning bidder the sum of the losing bids that would have
won had that bidder not been present at the auction. An example is presented to illustrate
the differences in pricing rules, and dynamic (i.e., ascending- or descending-bid) versions
of the three are described.
These three procedures are explored further in Chapter 13, in an independent (across
bidders) private values setting within which additional items offer decreasing marginal
value to the buyers. Under the Vickrey procedure, there is an equilibrium at which each
bidder bids precisely his valuation for the first item, and his successive marginal valuations
for the remaining items. This equilibriumclearly yields an efficient allocation. Equilibrium
behavior is not described (and, in general, has not yet been characterized) in the other cases.
However, for symmetric settings, it is shown that, under uniform pricing and at a symmetric
equilibrium, a bidder will bid his first-item value, and his second-highest bid will typically
be less then the marginal value of a second item. (Downward pressure on the second bid
is exerted by the possibility that his second-highest bid might determine the price he must
pay if he wins a single item.) Under discriminatory pricing, it is shown that a symmetric
equilibrium typically involves cases in which a bidder submits equal highest and secondhighest
bids, even when the marginal value of the second item is less than the value of the
first. These observations are sufficient to show that both uniform-price and discriminatory
procedures will sometimes yield inefficient allocations.
Chapter 14 presents a generalization of the Revenue Equivalence Principle in the setting
of Chapter 13: Once again, it is the allocation of items at equilibrium that determines
the seller’s expected revenue. An enlightening example uses this principle to explicitly
determine symmetric equilibria under all three procedures. The example involves two
bidders, who each seek to acquire at most two of three items. Since each must get at
least one item, and it can be shown under each procedure that the final item must go to
the bidder who values a second item most highly, all three procedures will be efficient, and
hence must generate the same expected revenue for the seller.
Sequential sales are examined in Chapter 15, in the symmetric independent private
values setting with the further assumption that no bidder seeks more than a single item. The
symmetric equilibria of both sequential first-price and sequential second-price auctions
are determined. Both sequential auctions are efficient, and therefore generate equal seller
revenues. In both cases, the time-series of prices at equilibrium is a martingale, i.e., it
displays neither systematic upward nor downward drift.
The sale of non-identical items is considered in Chapter 16. Here, a simultaneous
auction of all items should permit the bidders to submit separate bids for every subset. Only
the VCG mechanism is discussed. This mechanism allocates the items in a manner which
maximizes the sum of the winning bids, charges each bidder his winning bid, but then
rebates to him the amount by which his presence at the auction increased the maximum bid
total. (The second-price auction, and the Vickrey identical-items auction, are both special
cases of this procedure.) It is shown that this is a direct-revelation mechanism, i.e., that
if the other bidders all simply bid their valuations for the various subsets, the remaining
bidder can do no better than to bid his valuations as well. An immediate consequence
is that the VCG mechanism will always yield an efficient allocation of the items at this
“truth-telling” equilibrium.
496 Book review / Games and Economic Behavior 45 (2003) 488–497
Might the seller gain by bundling some of the items together? Indeed, when there are
only two bidders, the seller is always at least as well off (and typically, better off) selling the
items as a single bundle (and using a second-price auction, the single-item specialization of
the VCG procedure) as he is using the VCG procedure on the unbundled set. An example
shows, however, that even in the three-bidder, two-item case, an unbundled sale might
generate higher revenues.
The chapter closes with a clever example involving the sequential sale of two items
using English auctions, where, at equilibrium, budget-constrained bidders will generate
higher revenue for the seller than would unconstrained bidders. (The intuition behind the
example is that one bidder might be willing to bid more than his valuation for the first
item, in order to drain the resources of the other bidder before the sale of the second item
begins.)
Chapter 17 returns to the topic of auctions where the bidders have interdependent
valuations, considering first multiple-item auctions in which the bidders’ signals are onedimensional,
and then single-item auctions where the signals are multi-dimensional. In the
former case, some of the efficiency results of Chapter 9 are shown to generalize as long as
there are only two bidders:With three or more bidders, the generalization fails. In the latter
case, efficiency is generically unattainable.
Krishna’s introduction to this final chapter is quite appropriate: “In some sense we have
reached the limits of what auction-like mechanisms can accomplish in terms of allocating
efficiently. . . It is perhaps fitting, therefore, that this is the last chapter.”
8. Appendices
Auction Theory concludes with seven appendices. The first five briefly review fundamentals
from probability and linear algebra that are referenced on occasion in the text. The
sixth introduces the fundamental game-theoretic notions used throughout. The final appendix
outlines a proof that, in the independent private values setting, first-price auctions
always (i.e., even in asymmetric settings) have equilibria.
9. Final comments
The book offers no direct in-line references. Instead, the chapter summaries provide
references to the original sources of presented results, as well as to papers that reach beyond
the included material. On the one hand, this makes the text easier to read, and frees the
author to somewhat restate results from the literature in a manner that fits his presentation.
On the other hand, it makes it a bit more difficult for the reader to search out the original
presentations in the literature. The writing style alternates nicely between theorem-proof
presentations, and embedded-proof discussions with summarizing theorems.
Some connections to the broader economic literature might have been included. For
example, it would seem appropriate to connect the “Winner’s Curse” to the more general
phenomenon of adverse selection, or to connect internal bidder-ring auctions to the
dissolution (via auction) of partnerships. The complete eschewal of applications leads to
Book review / Games and Economic Behavior 45 (2003) 488–497 497
the omission of interesting issues surrounding new auction formats. Ascending-bid secondprice
auctions (used in many online settings) might have been mentioned. Certainly, one of
the most dramatic applications of multiple-item auctions in recent years, to the allocation
of telecommunication spectrum rights, has prompted a number of intriguing theoretical
treatments.
The book appears to have been carefully edited and proofread. Other than a single
misplaced superscript, and a few instances where some features of the assumed economic
environment or details of an auction procedure weren’t immediately clear (but could be
inferred), all the details seem to be properly in place. In reading the summary to Chapter
12, one might argue that the US Treasury Department’s shift from discriminatory (debtplacement)
auctions to uniform-price auctionswas motivated more by the Salomon scandal
(an attempt to control the market for certain debt issues by creating fictitious bidders) than
by revenue concerns. But these are minor criticisms.
In the course of reading Krishna’s Auction Theory, I was pleased to find several
examples and perspectives that were new to me. Revisiting familiar territory, and seeing
it through Krishna’s eyes, was invigorating. For a first-time visitor to the field of auction
theory, this is a superb guidebook that I am happy to recommend.
References
Klemperer, P. (Ed.), 2000. The Economic Theory of Auctions. Edward Elgar. 2 Vols.
Robert J. Weber
J.L. Kellogg School of Management, Northwestern University,
Evanston, IL 60208, USA
E-mail address: rjweber@northwestern.edu
Received 25 August 2003
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