Combinatorial Auction Design
Aleksandar Pekeč • Michael H. Rothkopf
Decision Sciences, The Fuqua School of Business, Duke University, Durham, North Carolina 27708-0120
RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, New Jersey 08854-8003
[email protected] • [email protected]
C
ombinatorial auctions have two features that greatly affect their design: computational
complexity of winner determination and opportunities for cooperation among competitors. Dealing with these forces trade-offs between desirable auction properties such as
allocative efficiency, revenue maximization, low transaction costs, fairness, failure freeness,
and scalability. Computational complexity can be dealt with algorithmically by relegating
the computational burden to bidders, by maintaining fairness in the face of computational
limitations, by limiting biddable combinations, and by limiting the use of combinatorial
bids. Combinatorial auction designs include single-round, first-price sealed bidding, VickreyClarke-Groves (VCG) mechanisms, uniform and market-clearing price auctions, and iterative
combinatorial auctions. Combinatorial auction designs must deal with exposure problems,
threshold problems, ways to keep the bidding moving at a reasonable pace, avoiding and
resolving ties, and controlling complexity.
(Auction Design; Combinatorial Bidding; Bidding with Synergies)
1.
Introduction
Auctions have recently come into the spotlight
because of their use in deregulation and their explosive propagation on the Internet. Sales of the rights
to the use of radio spectrum by the U.S. Federal
Communications Commission (FCC) and daily electricity supply auctions are examples of recent innovations in the use of auctions in deregulation. While
millions participate in Internet auctions such as those
on Ebay.com, the main impact of auctions has been
in business-to-business (B2B) applications. For example, in its remarkable makeover that led to the
title “E-Business of the Year 2000” by InternetWeek
magazine, General Electric has pushed toward online
auctions for most of its procurement operations, conducting more than $6 billion in online auctions in
2000 (General Electric Corp. 2001). It is likely that
online auctions will continue to play an important role in procurement and, perhaps, in other
aspects of business operations of most successful large
organizations.
Auctions have a particularly convenient property of
aggregating information. Even if no information on
0025-1909/03/4911/1485
1526-5501 electronic ISSN
the bidders’ valuations of an item is available, if there
is sufficient demand, the bidtaker can arrange the sale
of an item to the bidder who values it the most at
a “fair” price. However, when multiple items are for
sale, potential auction mechanisms may be impractical due to inherent complexities. There may be further complexities in analyzing equilibrium bidding
strategies. While some theoretical limits still constrain
what can realistically be done in practice, what is
practical has changed considerably with advances in
both communications and computational technology.
Combinatorial auctions—simultaneous multiple-item
auctions that allow submission of “all or nothing
bids” for combinations of the items being sold—are
an important example.
Bids on combinations of items are important to bidders whose value for combinations is greater than
the sum of their values for the individual items
in the combination. Such “complementarities” commonly arise from cost savings in procurement (such
as back hauls in trucking) and synergies between
assets (such as spectrum licenses for adjacent areas).
Complementarities appear to be the main practical
Management Science © 2003 INFORMS
Vol. 49, No. 11, November 2003, pp. 1485–1503
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
motivation for interest in combinatorial auctions. On
the other hand, sometimes bidders will view items
offered in an auction as substitutes. Such situations
can arise when bidders have resource and capacity
constraints or when winning alternative items offered
can meet the same need, e.g., two different broadband
licenses covering the same region. In fact, it is hard to
think of a realistic auction of multiple items where all
bidders would value all combinations as the sum of
values of individual items.
The last few years have witnessed considerable
interest in combinatorial auctions. In the early to
mid-1990s, during the debate on the design of
the FCC’s frequency spectrum auctions, combinatorial auctions were branded as inapplicable in
practice due to the computational complexity of
their implementation (e.g., McMillan 1994). However, in recent years, the computational ease of
implementation of combinatorial auctions in a variety of practical situations made combinatorial auctions a hot research topic, as well as a lively topic
of interest in the public sector and in the B2B
community. In fact, the first Internet B2B exchange,
Automated Credit Exchange (www.acemarket.com)
that trades air emission credits, allows combinatorial
bidding. Combinatorial auction models are becoming increasingly popular among the next generation
of B2B marketplaces. One of the first such auctions
was conducted by Net Exchange (www.nex.com) in
1993/1994, procuring transportation services for Sears
(Ledyard et al. 2002). Today, a range of companies
use combinatorial auctions: from combinatorial auction “specialists” in the B2B arena (e.g., Combine
Net, www.combinenet.com; Trade Extensions, www.
tradeextensions.com; etc.), to industry-specific B2B
procurement specialists who conduct combinatorial
auctions (e.g., Logsitics.com in the transportation
and shipping arena), to generalist e-business solution companies that conduct combinatorial auctions
when appropriate (from recently established companies such as Ariba, www.ariba.com, to information
technology giants such as IBM, www.ibm.com). The
short time between the first scientific and academic
discussions to actual business applications exhibits
an amazing speed of adoption of combinatorial auctions as a practical e-business tool. Thus, it is likely
1486
that combinatorial auctions will continue to play a
role in auction and e-market design, to be of interest to researchers in several fields, and to be implemented and used in a variety of contexts. Business
Week Online (2001) estimated that the FCC’s planned,
but later postponed, first combinatorial auction (Auction No. 31) would raise tens of billions of dollars.
There is a disparity between the current state of
many academic discussions of combinatorial auctions
and the current e-business implementations of combinatorial auctions. On one side, most academic papers
on combinatorial auctions focus on narrow technical issues. Computer scientists have concentrated
on developing fast heuristics and further analyzing
the complexity of winner determination for various
possible combinatorial auction models; operations
researchers have focused on the integer programming
(IP) formulation of the problem and tried to apply IP’s
heavy machinery to it (de Vries and Vohra 2003 is an
excellent survey); while game theorists have focused
mainly on certain theoretically desirable properties
within simplified models (Krishna and Rosenthal
1996, Rothkopf et al. 1998, de Vries and Vohra 2003
provide some relevant references). Economists have
also considered lab experiments (Banks et al. 2001
gives an overview) and the evaluation of noncombinatorial simultaneous ascending auctions (e.g., Cramton 1995, 1997; Cramton and Schwartz 2000). On the
other side, in spite of the theoretical difficulties, combinatorial auctions are being implemented successfully. However, possibly because of efforts to protect
proprietary information and competitive advantages,
there is little documentation and public information
on details of combinatorial auction design and implementations in the e-business arena. Apart from scattered corporate promotional “success” stories,1 only a
1
For example, Net Exchange white papers (www.nex.com/nex_
files/WhitePapers.htm) describe the use of combinatorial auctions
in logistics and bond trading; Combine Net describe a wide variety of applicable markets (www.combinenet.com/applicable.html)
and claim that it “ has tackled and solved a variety of complex allocation problems resulting in client savings of more
than $100 million dollars in the first part of 2002 alone.”
(www.combinenet.com/Press7.html); Trade Extensions provide
basic details of one of the combinatorial auctions they have conducted (www.tradeextensions.com/press/volvoPackCase.html).
Management Science/Vol. 49, No. 11, November 2003
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
handful of papers describe implementations of combinatorial auctions such as Net Exchange’s combinatorial auction for Sears Logistics (Ledyard et al.
2002), a combinatorial auction at The Home Depot
(Elmaghraby and Keskinocak 2003), a combinatorial auction for providing school meals in Chile
(Epstein et al. 2002), and IBM’s procurement combinatorial auction for Mars Incorporated (Hohner et al.
2003).2 So far, there is limited empirical evidence on
what the central combinatorial auction implementation issues are and on how theoretically challenging
issues should be resolved in practice.
Given the potential practical value of combinatorial
auctions, it is important to discuss the central issues in
designing such auctions. Rather than giving a detailed
review of technical developments (many of limited
practical value) in what has become a lively interdisciplinary research area, we present a critical assessment of the current state of the art and discuss some
important issues for designing combinatorial auctions
that will work well in practice. These issues should
interest not only practitioners, but also researchers in
the emerging field of combinatorial auctions. They
involve interdisciplinary problems that combine auction and mechanism design, game theory, operations
research, and computer science.
2.
Distinguishing Features of
Combinatorial Auctions
The term “combinatorial auction”3 is used to describe
any auction mechanism that (1) simultaneously sells
2
Cantillon and Pesendorfer (2002) analyze bidding data from a
combinatorial auction of bus routes in the United Kingdom. Some
electricity supply auctions and a gas pipeline capacity auction, both
discussed briefly below, involve combinatorial bids. Combinatorial
auctions can be found in real estate markets: Bayers (2000) reports
their use in sales of apartment complexes; for some farm land sales,
see www.landandfarm.com. Also, de Vries and Vohra (2003) report
examples of ideas for the use of combinatorial auctions.
3
“Combinational auction” might be more appropriate because bids
on combinations of items are allowed. The term “combinatorial”
could, somewhat inappropriately, be suggestive of combinatorics—
an area of mathematics that encompasses deep results and theories,
just a few of which are all that are needed to analyze the mathematical issues in “combinatorial” auctions. However, we bow to
the widespread use of the term “combinatorial.”
Management Science/Vol. 49, No. 11, November 2003
multiple items, and (2) allows “all-or-nothing” bids
on combinations of these items. For example, if items
a, b, and c are auctioned, an “all-or-nothing” bid on
combination {a b} will either win both a and b, or
neither. No partial allocation is allowed.
Even in noncombinatorial auctions, difficulties arise
in the theoretical analysis of equilibrium behavior,
of efficiency, and of revenue expectations. Such
theoretical analysis faces considerable difficulties even
on issues that are utterly trivial in noncombinatorial mechanisms. In particular, combinatorial auctions
have two features that distinguish them from other
auction models: complexity of winner determination
and a cooperative aspect.
Complexity of Winner Determination. The problem of determining auction winners is normally a
trivial exercise in noncombinatorial auctions because
all that has to be done is to identify the bidder who
placed the highest bid. However, in a combinatorial
auction, the highest bid on a combination of items
is not guaranteed to win. For example, suppose that
three items are for sale: a, b, and c and the highest bids
on combinations {a}, {b}, {c}, {a b}, {a c}, {b c}, {a b c}
are 1, 3, 2, 5, 5, 4, 6 (dollars), respectively. The auction
revenue is maximized if {a c} is sold for $5 and item b
is sold for $3. Hence, the high bids on {a}, {c}, {a b},
{b c}, and {a b c} do not win. Thus, determining
auction winners is not completely straightforward.
More formally, let items = a b c denote the set
of n items being auctioned and let bidders = i j k be identities of participating bidders. Suppose biddable (allowable) combinations are nonempty subsets
of items, A B C (If bidding on all nonempty
combinations is allowed, there will be 2n − 1 such
combinations. A noncombinatorial auction is one in
which biddable combinations consist only of n singleton sets, i.e., a b c ) Let bidcomb denote the
set of all biddable combinations. Finally, let allocs
denote the set of all possible allocations, that is the
set of all ⊆ bidcomb such that no two elements of intersect, i.e., such that for any two distinct biddable
combinations C D ∈ , C ∩D = , and such that there
is at least one bid submitted for every C ∈ .4 In other
4
These conditions ensure that each item is assigned to at most one
bidder and that only items for which bids are received (either as
1487
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
words, any defines a possible auction outcome: For
any C ∈ , all items defining biddable combination C
are allocated to the bidder who submitted the highest
bid on C
Winner determination is the problem of finding an
allocation ∈ allocs that is optimal with respect to
some preannounced objective. The normal objective
in both theory and practice is the sum of high bids
on all C in . More precisely, let rev = C∈ bC,
where bC is the highest submitted bid on C (assuming bC = 0 if there are no bids submitted for C, and
assuming no bid contingencies such as budget constraints or XOR bids, which are discussed in §6.1). In
this notation, the winner determination problem is
max rev
∈ allocs
This problem is equivalent to the set-packing problem
on hypergraphs, a prototypical NP-complete problem
(see Rothkopf et al. 1998),5 that has a straightforward
IP formulation6
max
bCxC
C∈bidcomb
Subject to
for every a ∈ items
xC ≤ 1 xC ∈ 0 1
C∈ bidcomb a∈C
Thus, the task of determining auction winners, trivial
in noncombinatorial auctions, becomes a potentially
bids on that individual item or as bids on a combination containing
that item) get allocated.
5
The set-packing problem was shown to be NP-complete by Karp
(1972). It belongs to a class of “harder” NP-hard problems because,
for any , there is no polynomial time algorithm (unless ZPP = NP,
which is an open question but believed to be unlikely) that would
guarantee an approximate solution within a factor of n1− from an
optimal solution, where n is the number of submitted bids (Håstad
1999).
6
This formulation can be modified to accommodate some auctionspecific requirements, e.g., changing all inequality constraints to
equalities, thus, requiring that every item be allocated. de Vries
and Vohra (2003) discuss several modifications of the IP formulation, e.g., a requirement that each bidder have at most one winning bid. Also, note that it is cast in the context of high-bid-wins
auctions. As in noncombinatorial auctions, low-bid-wins auctions,
e.g., procurement auctions (most of the auctions that are mentioned
in this paper are of this type), are logically isomorphic.
1488
computationally intractable combinatorial problem in
a combinatorial auction. Without mastery of the determination of auction winners, any serious strategic
analysis such as game-theoretic analysis of optimal
bidding strategies is impossible.
Cooperative Flavor of Bidding in Combinatorial
Auctions. Combinatorial auctions have a distinct
cooperative flavor. Because there are different ways
of partitioning the set of items for sale into a feasible allocation, a bidder for any combination of items
(except the combination of all items) benefits from
bids on complementary items, i.e., ones not in that
combination. In the previous example, a bid of $5
on combination a b cannot become a winning bid
without a high enough bid on c. This cooperative
feature stems from the way auction winners are determined and contrasts starkly to noncombinatorial auctions where, absent collusion, all bidders are direct
competitors or, for complementary items, gain nothing from cooperation.
These two fundamental features play a prominent
role in our discussion. They require auction designers
to compromise between different, potentially desirable auction properties. We will address this below.
First, we discuss some potential goals of auction
designs.
3.
Desirable Properties of
Auction Mechanisms
Allocative efficiency is a desirable property of an auction. It is achieved when one maximizes the total
value to the winners of the items being auctioned.
In a combinatorial auction, achieving allocative efficiency, even in a theoretical model, is demanding.
The most notable way to attempt to achieve it is the
Vickrey-Clarke-Groves (VCG) mechanism, which is
not practical. It is discussed below. Related to, but
not necessarily the same as allocative efficiency, is
overall economic efficiency. In addition to the efficiency
of the allocation, this concept takes into account the
effect of auction revenue on economic efficiency. In
particular, if an auction reduces revenue that must be
replaced with money from inefficient taxes, the inefficiency of the taxes affects overall economic efficiency
(see Rothkopf and Harstad 1990).
Management Science/Vol. 49, No. 11, November 2003
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
Revenue maximization (or cost minimization) is often
another goal for auctions. Companies running procurement auctions often have cost minimization as
a primary goal. In government auctions, revenue
maximization is controversial.7 Given the difficulty
in finding equilibrium bidding strategies, designing revenue-optimizing combinatorial auctions is not
easy. In addition to the difficulty of predicting bidding strategies, a combinatorial auction designer faces
a more basic obstacle: determining the revenuemaximizing (or cost-minimizing) allocation for a
given set of bids.
Low transaction costs are another potential goal for
auction designers. Both the bidders and the bidtaker
care about their costs of participating in the auction.
Obviously, auctions with lower participation costs are
desirable. Delay in concluding the auction is also a
transaction cost. Thus, high auction speed is desirable.
Fairness, while sometimes difficult to define, is often
a vital goal of auctions. Concern about equal treatment of competitors (and the appearance of it) is often
a key reason for government use of auctions. In private auctions, fairness influences bidders’ willingness
to participate.
Failure freeness is a design goal closely related to
fairness. Auction designs should work as intended
under all but the most extreme conditions. If failures
cannot be avoided completely, they should be minimized and their impact mitigated. The probability
of failure in determining auction winners according
to preannounced auction rules should be carefully
assessed. This issue is application dependent. In some
applications, it may be acceptable to fail occasionally to allocate the items to the bidders who offer
to pay the most. Tolerating rare failures of optimal
winner determination can ease construction of workable combinatorial auction mechanisms. For example,
7
Even though allocative efficiency is often a proclaimed primary
goal in government-run auctions (e.g., sales of frequency spectrum
licenses), efficient allocations that would bring virtually no revenue
are politically risky. However, the nongovernmental participants in
government auction sales usually have a lot at stake and are well
informed about the auction design. They prefer that less revenue
be collected, and the ultimate beneficiaries of the extra revenue, the
taxpayers, are relatively ill-informed and individually have little at
stake. Hence, there is substantial political pressure not to maximize
revenue.
Management Science/Vol. 49, No. 11, November 2003
auction winners in corporate procurement auctions
could be chosen using an algorithm not guaranteed
to maximize revenue when the bidtaker’s only loss
comes from the revenue difference between the optimal and the announced allocations. This could be
a wise choice if committing to always find the
revenue maximizing allocation is lengthy, costly,
and yields only marginal revenue improvements.
However, in government auctions, especially “big
stakes,” one-time auctions such as the FCC spectrum
auctions, the perception of fairness is critical, and the
possibility of failing to determine winners correctly
may be intolerable. Bidders who lose even though
their bids would have won with an exact calculation
might cause lengthy and costly litigation. Note that
the optimal and a marginally suboptimal allocation
could consist of completely different sets of bidders.
As discussed below, it may be possible to head off
such contests by giving the bidders themselves a fair
chance to provide better solutions to the winner determination problem.
In combinatorial auctions, the winner determination problem may not only be computationally unmanageable, but also opaque. Transparency is
important in auctions for two reasons: (1) it simplifies
bidders’ understanding of the situation, thus, easing
their decision making, and (2) it increases their trust
in the auction process by improving their ability to
verify that the auction rules have, in fact, been followed. This issue needs special attention in the design
of combinatorial auctions for practical use. Note that
polynomial time algorithms do not equate to transparency. Some such algorithms mentioned in the context of combinatorial auctions are far from transparent.
The issue of computational complexity can be
ignored when only a few items are being sold. However, if combinatorial bidding is to become a standard
practice, not just an isolated occurrence, the auction
design has to have scalability, i.e., be workable for
sales of many items. This may be particularly important in the design of B2B marketplaces based on combinatorial auctions. Note that the very definition of
the items to be sold may be affected by the size of
the combinatorial auction the seller can handle. For
example, in selling rights to 30 MHz of spectrum, the
1489
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
FCC could sell one 30 MHz license, or divide it into
several licenses for less bandwidth. It can also define
the geographic areas that licenses cover broadly or
narrowly. The larger the auction it and the bidders
can handle, the more freedom it has to offer many
smaller licenses if it prefers.
4.
Coping with Computational
Complexity
The computational complexity of winner determination in combinatorial auctions is not an incidental topic. Rather, complexity is a central reason for
difficulties in designing such auctions. This section
reviews strategies for coping with it.
Algorithmic Approach. When winner determination is NP-complete, a guaranteed efficient algorithm is unlikely to be found. However, a variety of
tools and techniques perform well overall in practice even though they lack theoretical performance
guarantees. IP can easily handle winner determination in combinatorial auctions with a small number
of items, and the number that is “small” becomes
larger with improvements in computational power
and IP codes. However, general IP techniques are far
from transparent and not amenable to scaling. On
the other hand, various heuristics and approximation
algorithms are likely to produce solutions which, in
most cases, are optimal and which are rarely far from
optimal. However, closeness to optimality, as already
mentioned, may not be adequate in some contexts.
Thus, unless suboptimality cannot be ruled out, failure freeness can be a serious concern. The survey by
de Vries and Vohra (2003) addresses the huge literature of the last few years on algorithmic approaches.
However, algorithms are not the only way to cope
with computational complexity.
Relegating Computational Complexity. One of the
earliest combinatorial auction mechanisms is the
“AUSM” mechanism (Banks et al. 1989). In it, computational complexity is relegated to the bidders. They
must show with which other valid bids a new bid
has to be combined to make it part of the optimal set.
This approach hides rather than solves computational
problems. It shifts them from the bidtaker’s winner
1490
determination problem to the bidder’s bid preparation problem, where they are hidden and may need
to be solved under extreme time pressure. This favors
bidders with superior computational abilities. This is
problematic; the allocation might better depend on the
bidders’ values than on their computational prowess.
The possibility of separating computation from the
allocation decision merits further investigation. An
auction mechanism could treat computation as a service that the auctioneer can outsource. Perhaps the
auctioneer could pay a fee to whoever finds the best
solution to the winner determination problem. Note
that this approach might require the public release of
bids (but not necessarily bidder identification). If protecting bidders’ private information is important, this
might be a disadvantage.
Maintaining Fairness in the Face of Computational Limits. Rothkopf et al. (1998) pointed out that
even if the winner selection problem could not be
solved optimally, fairness could be maintained by
giving bidders an opportunity to improve on proposed solutions not guaranteed to be optimal. This
approach may work when the optimal solution to
the winner determination problem, while desirable,
is not required. Such an approach ought to guarantee that the best solution found will be perceived as
fair, and blunt any court challenge. In particular, if
after the award, a bidder does find a solution to the
winner determination problem with a higher value,
that bidder cannot fairly complain since she had a fair
chance to find that solution before the award.
Limiting Biddable Combinations. Rothkopf et al.
(1995, 1998) discuss limiting biddable combinations.
Winner determination can become computationally
manageable—possibly transparent—with some restrictions on biddable combinations. They demonstrate several such restriction strategies that allow
bids on what may be economically sensible combinations. Park and Rothkopf (2001) take a different approach to limiting biddable combinations. They
have each bidder prioritize combinations of importance to it. They then use as many priorities of all bidders as computation will allow. When the worst-case
bounds of computational complexity are loose, as
they often are in practice, this allows all combinations
Management Science/Vol. 49, No. 11, November 2003
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
of interest to the bidders. When computation is limiting, it allows the bidders’ most important combinations. The auctioneer need not prespecify allowable
combinations, something that could be perceived as
unfair.
Prescribing biddable combinations may limit bidders’ ability to express perfectly their synergetic values. The view that the auctioneer should not limit
biddable combinations in any way overlooks the fact,
discussed above, that the very definition of individual items limits what is biddable. That is, limiting
biddable combinations is a natural part of auction
design. This has two counterintuitive consequences.
First, no auction design can be completely neutral.
Second, a finer and less restrictive feasible auction
might result from dividing what is to be sold into
more pieces and restricting sensibly the combinations
that can be bid upon than from allowing bids on all
possible combinations of less finely divided assets.
Thus, an auction designer needs to consider computational complexity while defining the items and the
biddable combinations.
consider this possibility even though we have little
general theory.
In overall summary, there is no one clear-cut way
to deal with the complexity of winner determination.
Luckily, in some applications this might not be an
issue. In general, however, it is, and any choice an
auction designer makes may involve some unwanted
consequences.
5.
Combinatorial Auction
Mechanisms
8
Revenue comparisons between auction mechanisms
and the analysis of equilibrium bidding strategies
are the essence of economic auction theory (e.g.,
Milgrom and Weber 1982, McAfee and McMillan
1987). However, for several reasons, there are few
such results for general combinatorial auctions.
First, combinatorial auctions emerged in the academic literature only two decades ago (Rassenti et al.
1982 and Banks et al. 1989 are often considered the
earliest work on the topic) and have received considerable attention only in the last few years—mainly
due to the design of the FCC’s spectrum auctions and
an explosion of interest in e-business auctions. The
electronic context of e-business granted plausibility
and sometimes applicability to auction mechanisms
previously thought impractically complex. Second,
combinatorial auctions are much harder to study and
analyze. As discussed, even winner determination,
a nonissue in single-item auctions, is an important
obstacle to analysis. In general, equilibrium bidding
strategies are not known except for special, but not
necessarily useful, cases. Third, even mere attempts
to define natural extensions of standard noncombinatorial auction models open interesting questions.
9
5.1.
Limiting the Use of Combinatorial Bids. The difficulty of winner determination arises primarily from
the intersecting structure of all biddable combinations rather than the combination size8 or the number
of combinations. However, restricting the quantity of
combinatorial bids might ease somewhat the computational burden and yield more tractable combinatorial auction mechanisms. The FCC took this approach
(FCC 2000b). Sometimes, limiting the use of combinatorial bids can interact favorably with concerns
about bidder incentives.9 Auction designers need to
Even with bids restricted to arbitrary sets at most three items,
winner determination is NP-complete (Rothkopf et al. 1998).
This occurred in the design of a single-round sealed bid auction for pipeline capacity by Natural Gas Company of America.
Bidders wanted to submit minimum quantity restrictions on their
bids. This adds integer constraints to the winner selection problem.
The pipeline company suggested that minimum quantity restrictions
be allowed, but if such a constraint on a bid was binding, the
bid be rejected even though including it would increase revenue.
While this eased the computational problem, the primary reason
for this choice was the pipeline company’s view that it fixed an
incentive problem with the bids. Otherwise, a bidder who is flexible (i.e., has no minimum quantity restriction on its bid) and offers
Management Science/Vol. 49, No. 11, November 2003
Single-Round (Sealed Bid) First-Price
Combinatorial Auctions
This basic model can be defined clearly in combinatorial auctions. All bids are submitted before a
a high price may lose capacity to a bidder who is inflexible and
offers a lower price. For example, suppose there is a fixed capacity
of 100 units and three bids: one for up to 70 units at 10 per unit,
one for exactly 40 units at 8 per unit, and one for up to 30 units at
6 per unit. Revenue maximization gives the second bid 10 units at
a price lower than the first bid just because it is inflexible.
1491
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
deadline, then the winner determination problem is
solved. Items go to the revenue-maximizing collection of bids, and winners pay the amounts of their
bids. The main obstacle to implementing such auctions is the complexity of the winner determination.
For reasonably sized and structured auctions, stateof-the-art IP can overcome this obstacle for practical
purposes. In fact, previously mentioned combinatorial auctions were designed this way (Epstein et al.
2002, Elmaghraby and Keskinocak 2003, Cantillon and
Pesendorfer 2002).
A number of important properties of single-round
sealed bid auctions carry over to combinatorial sealed
bid auctions. Such auctions are resistant to collusive conspiracies by bidders. Unlike progressive and
Vickrey auctions discussed below, agreements by bidders to collude in single, isolated, sealed bid auctions
are unstable because violations of the collusive agreement cannot be detected by coconspirators until it is
too late to react (Robinson 1985). Also, bid signaling to
arrange tacit collusion (“Don’t bid on my item, and I’ll
stop bidding on yours”) is impossible. Of course, collusion in richer environments such as repeated auctions is always a risk. Sealed bid auctions, including
combinatorial ones, encourage participation. In a progressive auction, a bidder who knows that another
bidder has a higher value for an item or combination
has no incentive to bid on it, because the other bidder would surely outbid it. However, in a sealed bid
auction, the bidder with the known higher value is
uncertain about the best competitive bid. Sometimes,
the bidder will bid too “greedily” and lose. Hence, the
disadvantaged bidder has a chance to win and, therefore, an incentive to participate.10 Single-item sealed
bid auctions have strategic complexity as compared
to progressive or Vickrey auctions. In particular, bidders in first-price auctions, including combinatorial
ones, must worry about competitive information and
strategies and cannot just bid up to their own values.
In them, information about competitors is valuable.
10
Much academic literature assumes bidder participation and a priori symmetry or uses weak dominance or weak equilibria to analyze
auctions. See, for example, McAfee and McMillan’s (1987) survey.
Conclusions in this literature about progressive or Vickrey auctions
are unlikely to apply when there are asymmetries in values known
to bidders.
1492
However, the corresponding progressive auctions also
have strategic complexity. Bidders in them have
incentive to try to signal and coordinate bids. Similarly, another advantage of single-item progressive
auctions over first price—information sharing to mitigate the winner’s curse—is greatly reduced in the
combinatorial context because bids are being used to
signal and, thus, do not clearly reveal values.11
The “pay-your-bid” property has a nice transparency. This is especially important in e-business
applications where participants have little, if any,
chance to verify the validity of other bids. This may be
critical when bids other than the winning ones affect
auction prices. On the other hand, sometimes winning
bidders care about visibly paying much more than the
highest losing bids. This can be embarrassing organizationally or politically. For that reason, they may
prefer uniform pricing or progressive auctions.
5.2. Vickrey-Clarke-Groves Mechanisms
VCG mechanisms (Vickrey 1961, Clarke 1971, Groves
1973) generalize Vickrey’s classical proposal for a
single-item auction design and can be applied in the
context of combinatorial bidding. Like the single-item
Vickrey auction, a VCG combinatorial auction mechanism is superficially attractive. In particular, in an
isolated combinatorial auction, the VCG mechanism
makes it a dominant strategy for bidders to bid their
true values for every possible combination of items. It
does this by refunding to the bidders the increase in
value caused by their bids. These refunds are sometimes called Vickrey payments. For example, suppose
that two items, a and b, are for sale and there are two
bidders, one offering 10 for a, 5 for b, and 15 for
a b; the other offering 1 for a, 6 for b, and 12 for
a b. Apparent value is maximized at 16 by selling
a to the first bidder and b to the second. The first
bidder pays her bid for a of 10 but gets a refund of
4, because without her bids, the total value would be
12, not 16. The second bidder pays her bid for b of
6, but gets a refund of 1, because without her bids,
the total value would be only 15.
11
Even in single-item auctions, this advantage is often not as great
as was believed (Harstad and Rothkopf 2000).
Management Science/Vol. 49, No. 11, November 2003
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
With the bidders’ truthful evaluations, the bidtaker can achieve allocative efficiency. However, VCG
mechanisms are impractical and are rarely, if ever,
used. Because they pay bidders, sometimes handsomely, to give them the incentive to bid their true
values, they are revenue deficient by the amount of
the Vickrey payments. (Who funds these payments
and what incentive effects do they have on those
who pay them?) They are also subject to several
kinds of cheating and are unsustainable in realistic
dynamic environments in which the revelation of the
bidders’ values has consequences beyond the auction.
Rothkopf et al. (1990) discuss the information revelation problem in single-item Vickrey auctions. This
problem carries over to combinatorial auctions, but in
the context of e-auctions, cryptography, as suggested
by Nurmi and Salomaa (1993), may help. Rothkopf
et al. (1990) and Rothkopf and Harstad (1995) discuss
cheating by the bidtaker in single-item Vickrey auctions. (The bidtaker can reduce the Vickrey payments
by procuring insincere bids.) Hobbs et al. (2000) point
out that the cheating problem carries over to VCG
mechanisms and that, in addition, if bids are made
by both suppliers and purchasers, then it is possible
for a seller and buyer to collude and increase their
Vickrey payments. Finally, Sakurai et al. (1999) point
out that bidders can cheat by submitting “false name”
bids. To see this, consider this example: Four bidders
bid their valuations in a VCG of three items, a, b,
and c. One bidder values a b c at $2. Each of the
remaining three bidders values only a single item (a
for the second bidder, b for the third, and c for the
fourth) at $1. The VCG allocation has value $3 and
allocates a single item to each of the last three bidders. The winners will get the items for free because
their Vickrey payments, $3 − $2 = $1, equal their bids.
Suppose, however, that in this example, the “three
remaining bidders” were really a single bidder who
valued a b c at $3. A truthful single bid would win
a b c at a net cost of $2, not for free. Thus, this bidder has an incentive to submit three “false name” bids
instead of bidding truthfully.12
12
This example assumes that the first bidder does not distinguish
between the allocation that assigns it the set a b c and the
allocation that assigns it each of the sets a, b, and c. In practice,
Management Science/Vol. 49, No. 11, November 2003
In addition to the appeal of the theoretical allocative
efficiency of the VCG mechanism, it has the mathematical appeal that the bidders’ equilibrium strategies
are easily calculated.
In a combinatorial VCG auction, every bidder is
supposed to submit its valuations for all possible allocations. This is at least 2n − 1 different valuations,
where n is the number of items auctioned.13 If bidders
are free to omit submission of bids they believe will
lose, even if the bids would lose, this can affect the
payments by the other bidders.14
Discussion of VCG mechanisms in combinatorial auctions are in Ausubel and Milgrom (2002),
Bikhchandani et al. (2002), de Vries and Vohra (2003),
and Parkes (2001). While VCG mechanisms may continue to be a topic of theoretical combinatorial auction
research, the potential implementation disasters make
us believe that e-business practice is likely to continue
to ignore them, except in special cases.
5.3. Uniform Price Mechanisms
When multiple identical items are being sold, marketclearing price mechanisms have appeal. In them, bids
there could be a difference: in one case, the bidder gets all items
in a single contract and might be subject to constraints not present
with three separate contracts. For example, the items might correspond to contracts for services covered by state laws only, while the
single contract for all three would involve services crossing state
boundaries and come under federal law. Thus, one may have to
distinguish between allocations defined as (1) partitions of sets of
winning combinations that belong to the same bidder, from those
defined as (2) partitions of winning combinations (two or more can
belong to the same bidder). Interestingly, all papers so far on VCG
mechanisms in combinatorial auctions assume (1) without explicitly stating this assumption (e.g., Bikhchandani et al. 2002, de Vries
and Vohra 2003, Parkes 2001, Ausubel and Milgrom 2002).
13
Even more valuations are needed with the distinction discussed
in footnote 12.
14
Here is an example. Suppose that there are two items, a and b
and two bidders with the following values: for bidder 1, va = 5,
vb = 15, va b = 20. For bidder 2, va = 10, vb = 2, va b =
12. The VCG result is bidder 2 wins a and pays 5; bidder 1 wins b
and pays 2. Suppose, however, bidder 1 knows that its bid for a
is hopeless, bidder 2 knows that its bid for b is hopeless, and both
know that their bids for a b will lose. If bidders do not make
hopeless bids, the result is the same award, but neither bidder pays
anything.
1493
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
are ranked by price, and the highest bids are allocated
items until the supply is exhausted. The last item supplied or the highest bid not honored sets the price
for all items sold. A version of this mechanism can
be used when some bids are offers to buy and others
are offers to sell. In this case, the highest offer to buy
a unit is matched with the lowest offer to sell, then
the next highest with the next lowest, and so forth
until the price of the sell offer exceeds the price of the
buy offer. All transactions are priced by a prespecified rule between the lowest accepted offer to buy and
the highest accepted offer to sell.15 When all offers are
from one side of the market, when bidders want no
more than one unit each, when the price is set by the
best losing bid, and when the bidder who will make
that bid does not know the bid will lose, this kind of
auction, considered in isolation, has some nice theoretical properties.16
When there are clear ways of accounting for the
differences, the idea of a market-clearing price auction may be used to sell nonidentical items.17 However, the way to define a uniform price for all
15
Occasionally, there are similar auctions in which the price is not
uniform. In the EPA’s emission rights auction, all sellers receive the
bid price of a specific buyer. Those with the lowest asking prices
receive the highest bids. There is, apparently, little to recommend
this method (Carson 1993, Carson and Plott 1996).
16
As in single-item Vickrey auctions, bidders have no reason not
to make bids reflecting their true values. This is simple and makes
the auction perfectly efficient. However, when some bidders want
more than one item, this is no longer true. Bidders wanting more
than one item have incentive to reduce below their value any bid
(other than their highest one) that has a chance of being the pricesetting bid. However, even when some bidders do want somewhat
more than one item, such auctions may still be competitive with
alternative mechanisms.
17
An example is electricity supply auctions in which electricity is
offered at different locations into a congested transmission grid. As
long as the transmission constraints are convex and a mathematical program that the cheapest feasible way of meeting the demand
with the bids can be formulated and solved, shadow prices from
the constraints will give the appropriate adjustments to the uniform price for electricity offered at different locations. (This is called
“locational marginal pricing.”) The electricity supply auctions in
some systems go even further. Not only do bidders offer prices for
electricity supply for each of the 24 hours of the coming day, their
bids also include startup costs and constraints, e.g., minimum run
levels and ramp rates. (These are reported periodically, not varied
from day to day.) The system operator solves the IP problem for
1494
winning bidders’ bids is not obvious in general
sales of heterogeneous items or in sales where bidders’ valuations of subsets are not additive. Consider an example: three items are for sale, a, b,
and c, five bids are submitted, each by a different
bidder, ba = bb = 3, bc = 1, ba b c = 6, and
ba b c = 5. Assigning the items to three different
bidders maximizes revenue at 7, but how could prices
be defined to implement uniform pricing? If the items
were homogenous, the lowest winning bid per unit
would be bc = 1, thus, all winners would pay this
price bringing the auction revenue to 3. In this example, it is not even clear what is the highest losing
bid. Would it be 0 because there are no bids other
than winning ones on any of the winning items?18 Or
should the highest losing bid be a value derived from
other losing bids, e.g., ba b c = 5 and, if so, how
should it be derived?
Bikhchandani and Ostroy (2002) show that determining market-clearing prices (so that each bidder
is maximizing its profits in the revenue-maximizing
allocation) is not trivial. Such prices might not be
additive (i.e., pA ∪ B might differ from pA + pB
nor anonymous (i.e., the price quote for a set
might have to differ between bidders). Nonanonymity
might affect bidders’ perception of the fairness of
the auction. Somewhat amenable and potentially
applicable market-clearing prices exist under certain
conditions; Parkes (2001) has a systematic survey.
These difficulties with market-clearing prices make
uniform price combinatorial auctions unattractive for
practical use except, perhaps, when bidders’ valuations have special structure. However, as discussed
later, these prices are central to setting minimum bid
increments in iterative combinatorial auctions.
the cheapest supply plan for the 24 hours and calculates a marketclearing electricity price for each hour, ignoring startup costs. If
a generator’s bid is accepted, it is paid the market-clearing price
(adjusted for location) in each hour. In addition, if the difference
between its offers and the market-clearing prices will not cover its
startup costs, it is also paid the difference. Hence, it will not lose
money if its bids reflect its true costs.
18
If so, the auctioneer would be better off selling all items to the
bidder who bid ba b c = 6 at the price, in this case clearly
defined, of the highest losing bid of ba b c = 5.
Management Science/Vol. 49, No. 11, November 2003
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
6.
Iterative Combinatorial
Auctions
Iterative auctions are predominant in e-business. Use
of the Internet enhances communication capabilities,
and iterative auctions are easier to implement with
use of computers. (Lack of communication has effects
even in simple, single-item, English auctions. For
example, Harstad and Rothkopf 2000 show that the
apparent revenue advantage of English auctions is
reduced in an auction in which the willingness to
pay the current price of every bidder is not revealed.)
Conducting auctions electronically anonymizes and
widens the prospective participant pool. An iterative format, electronic or not, allows bidders to learn
about their rivals’ valuations through the bidding process, which might lead them to adjust their own valuations. On the other hand, iterative procedures open
up space for strategizing and collusive behavior.19
Another important feature of iterative combinatorial
auctions is that, unlike single-round combinatorial
auctions that require bidders to submit many bids
(potentially 2n −1 of them), it allows bidders to submit
only a small number of bids in each round, thereby
reducing the bid submission burden. However, the
total bid preparation process could be as burdensome
as submitting a large number of bids in a single-round
auction. Also, all of the pathologies from single-round
designs get magnified in iterative combinatorial auctions. Furthermore, iterative auctions have additional
difficult design issues discussed below. Nevertheless,
this clearly is the most popular combinatorial auction
format in practice: it is almost standard in corporate
procurement (all companies mentioned in the introduction tend to use it), and the FCC has not even seriously considered anything but multiround formats
for its combinatorial auction design.
Several iterative combinatorial auction designs
come from academic research. Some were primarily
19
For example, in an early FCC auction, one bidder started bidding
on licenses of interest to a competing bidder using insignificant
digits (most bids were rounded to the nearest thousand dollars)
that were the identification number of the license the bidder was
interested in. This sent a clear message to its rival to stop bidding
on the license it wanted.
Management Science/Vol. 49, No. 11, November 2003
motivated by potential uses in practice (e.g., Brewer
and Plott 1996), some were motivated by theoretical questions or developed as tools for theoretical
analysis (e.g., Parkes 1999, Wurman and Wellman
2000, Bikhchandani et al. 2002, Ausubel and Milgrom
2002), and some were motivated by improving known
designs to get a generic implementable off-the-shelf
design (e.g., DeMartini et al. 1999, Porter et al. 2003).
We have already mentioned one iterative mechanism:
the AUSM procedure (Banks et al. 1989). It is a continuous time auction (new bids are accepted at any
time), which, as discussed, transfers all of the difficulties of combinatorial auctions to bidders.20 Continuous time combinatorial auctions may be reasonable
when the auction is small enough that computation is
not a concern; their use in real estate auctions involving only a few parcels of land appears to predate
academic analysis.21 However, more complicated situations may be handled better with discrete bidding
rounds. In any case, a design for a substantial combinatorial auction that is not going to be burdensome for
participants has to address successfully several important issues.
6.1.
Dealing with the Exposure Problem and the
Expressiveness of the Bidding Language
The exposure problem in auctions of multiple items
involves the risk of bidders winning unwanted items,
i.e., winning items at prices above bidders’ values for
them. For example, in an auction of two items a and b,
a bidder who wants to win both items, but not just one
of them, faces an exposure problem if bids on a b
are not allowed because it risks winning only one
item. Combinatorial bidding eliminates this type of
the exposure problem. But a related type of exposure
problem can remain an issue in combinatorial bidding.
Suppose a bidder wants to win a or b, but not both.
In that case, the bidder might be reluctant to submit
competitive bids for both a and b if by doing so
20
The PAUSE mechanism (Kelly and Steinberg 2000) is a polished
multiround version of AUSM.
21
Englebrecht-Wiggans (1995) describes a progressive farmland
auction held in Illinois on December 12, 1989, involving bids on
seven different combinations of four different parcels. The sale
method was not novel in the area.
1495
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
it risked winning both. Thus, to avoid exposure problems completely, a mechanism has to allow bidders to
express valuations for quite a few and potentially all
possible allocations. Allowing bids on combinations
only partially addresses this need. Computer scientists
approached this problem by designing and analyzing “bidding languages,” i.e., different forms of bids
aimed at being (1) as succinct as possible, and (2) as
expressive as possible. The goal is to allow bidders
to express as fully as practically possible their valuations of all possible allocations naturally, simply, and
briefly. An example of a fully expressive22 bidding
language is allowing bidders to submit “OR of XOR
ti
bids,” ORsi=1 XOR j=1 bCij , where OR and XOR represent the logical “OR” and the exclusive “OR” operation, respectively. These are composite bids, where s
and all ti are any positive integers with bids b(Cij ) on
combinations Cij , i = 1 s, j = 1 ti, with the
restriction that, for every i, the bidder is allocated at
most one of the combinations Ci1 , Ci2 Cit i. Nisan
(2000) discusses various bidding languages. In short,
the only completely general way to deal with the exposure problem is to allow bidders to submit bids in a
lengthy and complex form, thereby further plaguing
the winner determination computation.23
One compromise approach is to allow bidders a limited amount of flexibility. For example, each bidder
might be allowed to have a limited number of “pseudoitems” to include in its bid. By making some of its
bids include one of its pseudoitems, these bids are
made mutually exclusive.24 In addition, bidders might
22
We assume the bidder is indifferent to identities of other winners.
If this assumption is not met, serious problems can arise.
23
Note that a bid on combination, bC could be viewed as an
“AND” bid: ANDa∈C ba, where bC = a∈C ba. In other
words, building blocks of any bidding language are individual
items. One could relate the most common logical clauses to the
nature of bids as follows: AND bids correspond to combinatorial bidding, OR bids correspond to allowing multiple bids from
the same bidder, and XOR bids correspond to allowing contingent
bids.
24
This simple idea is discussed in Fujishima et al. (1999). It is
equivalent to having a limited number of nontrivial XOR clauses
in the “OR of XOR” bidding language.
1496
want to be able to include a budget constraint with
their bids.25
An iterative mechanism that allows bidders to
reveal their valuations dynamically is another or, perhaps, a complimentary approach. However, combinatorial bidding in any iterative auction is plagued
by its cooperative flavor. When submitting bids for a
new round, bidders would like to rely on the validity
of current high bids on complementary combinations.
However, a current high bidder on a combination that
is not part of a currently optimal allocation might
want to submit a competitive bid on a different combination without being exposed to the possibility of
its current high, but losing, bid becoming a winning
one. In other words, the bidder may not want to win
both the combination on which she is currently a
high but losing bidder and a combination on which
she plans to bid. Proposals considered during the
FCC combinatorial auction design debate attempted
to find a middle ground. One suggested allowing
bid withdrawals with sufficient notice when compensated by competitive bidding on other combinations
(Pekeč and Rothkopf 2000). Another suggested preceding every bidding round by an opportunity for
bidder communication before firm bid commitment
(Vohra and Weber 2000). Another suggestion was time
limits on the validity of nonwinning high bids on
combinations, e.g., by automatically removing a nonwinning high bid on a combination C from the system
unless it is renewed or becomes part of a winning
combination within some number of rounds (Harstad
2000).
Clearly, there is no way to reconcile completely both
concerns. Possible solutions could range from focusing exclusively on the former concern to focusing
exclusively on the latter. Interestingly, the evolution
of FCC’s combinatorial auction went from focusing
on the former issue by requiring that all high bids
on all combinations remain valid until topped by a
new high bid, thereby leaving bidders who are high
bidders on nonwinning combinations with an exposure problem, to retaining only bids from the last
25
Again, any such bid can be represented in “OR of XOR” language. However, statements that are simple, e.g., using budget
constraints, could be rather clumsy and lengthy in the “OR of
XOR” bidding language.
Management Science/Vol. 49, No. 11, November 2003
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
two rounds in which a bidder submitted any bids
(FCC 2000a, b). Any iterative auction mechanism will
either endure an exposure problem or restrict severely
the cooperative nature of combinatorial bidding, thus
limiting the key benefit of iterative combinatorial auctions. The auction designer should choose a mechanism that best compromises these concerns for the
application in question.
The existing practice in combinatorial corporate
procurement auctions favors mitigating the exposure
problem by allowing (or imposing) budget and capacity constraints (e.g., Hohner et al. 2003) or XOR bids
or both (e.g., Elmaghraby and Keskinocak 2003) even
in the multiround format. This helps bidders but adds
complicating computations during the auction. Thus,
if computational complexity is not an issue due to
the small number of items for sale or the structure
of bidder values, this seems to be a sensible strategy.
Even when XOR bids are not allowed, bidders find
inexpensive items and use them as pseudoitems as
described above (Ledyard et al. 2002). At the same
time, keeping bidders accountable for their bids by
retaining current best bids on all items and combinations (not just winning ones) seems common.
6.2.
The Threshold Problem and Procedures for
Keeping the Bidding Moving
The threshold problem in combinatorial auctions
refers to the difficulty that multiple bidders desiring
combinations that constitute a larger one may have
in outbidding a single bidder bidding for that larger
combination. Suppose four items are being auctioned,
a, b, c, and d, and that bids submitted by five different bidders are ba = bb = bc = bd = 4 and
ba b c d = 18. Suppose further that each bidder’s
valuation of the combination she bid on is one higher
than the bid value, i.e., va = vb = vc = vd = 5
and va b c d = 19. None of the bidders bidding
on single items can singlehandedly overcome the difference (the “threshold”) between the currently winning allocation of 18 and 16, the sum of the current
bids on the individual items. However, coordinated
bid increases would allow the individual bidders to
bid 4.51 each and win. Thus, a threshold problem cannot be overcome without allowing for bid increases
that are insufficient to win on their own (i.e., absent
Management Science/Vol. 49, No. 11, November 2003
bid increases on complementary combinations). However, allowing such individually noncompetitive bids
might prevent an auction from moving at a reasonable pace and also open the door to collusive bidding
and signaling. Note that reaching the best solution
by distributing the burden to overcome the threshold among bidders in this case requires cooperative
behavior. Because each bidder for an individual item
would prefer a “free ride” by letting other bidders
pay more of the cost of beating the currently winning
combination bid, this cooperative behavior involves
coordinating on a choice among many different (equilibrium) possibilities. Nonetheless, such cooperation
is quite plausible. Thus, an auction designer is forced
to trade-off overcoming potential threshold problems
with ensuring a reasonable pace26 and limiting anticompetitive strategizing. Depending on which is more
critical, an auction designer could focus on either of
these.
Note that (non)anonymous minimum bid increases
for items or combinations in a given round correspond to announcing (non)anonymous prices for the
items or combination. Thus, determining minimum
bid increases is equivalent to determining marketclearing prices given the current set of valid bids discussed in §5.3. As noted, anonymous prices might not
exist and prices for combinations might not be calculated easily. If a bidtaker has to resort to nonanonymous prices, she will be facing a fairness problem
or the problem of keeping price information private
to herself and each individual bidder. If the prices
are nonlinear, she may be facing computational problems (that have to be dealt with not once, but in
every auction round). An alternative approach that
avoids these difficulties is to approximate marketclearing prices. The most popular method seems to be
using the set of additive anonymous prices (i.e., prices
for the individual items and combinations priced as
the sum of its item prices) that are calculated from
the shadow prices of the linear relaxation of the
winner determination problem (e.g., Hohner et al.
2003). While these shadow prices are easy to compute
and report, they are market-clearing prices only in
26
Experimental evidence suggests that the pace can be unacceptably slow (Cybernomics Inc. 2000, Banks et al. 2001).
1497
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
special circumstances (e.g., see Bikhchandani and
Ostroy 2002, Parkes 2001, Ausubel and Milgrom 2002)
and could be overestimating and underestimating at
the same time. Still, in the absence of better, easily
computable feedback and guidance, such prices can
help keep the auction moving.
A somewhat different approach toward calculating
minimum bid increments and mitigating the threshold problem has been taken in the FCC’s combinatorial auction design. In such big stakes, one-time
government sales, the perception of fairness and the
public release of all information are crucial. Also, any
ad hoc methods for keeping the auction moving at
the expense of not properly addressing the threshold
problem, could result in a perception of discriminating against small bidders (who are most likely to face
the threshold problem) and also in leaving money on
the table. (This might not be a major concern in corporate procurement because bidders’ valuations could
be well understood by everyone, possibly from performance in previous auctions, and also because of
the relative value of auction speed.) Initial FCC minimum bid increment rules were aimed at finding the
minimum amount a nonwinning but high bid bC
on combination C would have to be increased, keeping all other bids unchanged, for C to enter a winning allocation. Call this value GapC. (As noted by
Rothkopf et al. 1998, the computational complexity
of determining GapC is the same as the complexity of winner determination.) To mitigate the threshold problem, the FCC wanted to set a minimum bid
increment for C at some proportion of GapC, taking
into account that bidders on complementary combinations might contribute by increasing their bids. This
approach required the auctioneer to calculate GapC
for all biddable combinations C after every round of
an auction. If bids on all combinations are allowed,
as many as 2n − 1 different NP-hard problems have
to be solved between two rounds of an auction.27 If
an auctioneer desires to provide such minimum bid
27
Although these NP-hard problems are similar, substantial
“economies of scale” are unlikely. This calculation, not winner determination, took essentially all of the computational
time between rounds in a simulated FCC combinatorial auction
(Hoffman 2001).
1498
increments, he should consider making the winner
determination problem computationally tractable by
restricting biddable combinations. After lively public
discussion, the FCC finally abandoned this approach
and settled for a somewhat complicated, but computationally easier, bid increment rule based on shadow
prices of the LP relaxation of the winner determination problem (FCC 2002).
Auction stopping rules should also be chosen with
threshold problems in mind. In fact, one approach to
dealing with minimum bid increments is to ignore
them completely and use an auction stopping rule
as the only guidance. For example, a bidtaker can
simply announce that the auction will terminate if
her total revenue does not improve by at least x%
from the revenue in the previous round (or, alternatively, revenue can be compared to the revenue some
k rounds before the current round). This approach has
been taken in Ledyard et al. (2002). (Interestingly, this
paper mentions calculation of shadow prices as a possible improvement.) In general, the rules should allow
enough flexibility to bidders facing a threshold problem so that they can submit improved but possibly
noncompetitive bids in the hope that other bidders
interested in complementary combinations will help
overcome the threshold problem by increasing their
bids. In other words, the bidtaker has to allow some
signaling to address possible threshold problems. For
example, the FCC’s combinatorial auction ends after
two rounds in which there is no improvement in total
revenue.
In summary, an auction designer may face a difficult choice in defining minimum bid increments
for iterative combinatorial auctions. These increments
have to be large enough to keep the auction moving, but also must create opportunities for overcoming threshold problems. There are several possible
approaches, but some lead to computations that may
be too hard in practice.28
6.3. Avoiding and Resolving Ties
Ties in an auction are undesirable; they lead to arbitrary rather than value-based allocations. In addition,
28
When minimum bid increments are not a central issue in combinatorial auction design, it may not pay to invest too much time
and computational power in precise calculations of them.
Management Science/Vol. 49, No. 11, November 2003
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
it is usually critical that ties that do occur be resolved
in a way perceived to be fair. In noncombinatorial auctions, ties in determining winners occur only
if there are two or more bidders who placed the
same bid for the same item. This makes tie breaking
both rare and straightforward. For example, the earliest of identical bids could be proclaimed a winner.
When fairness and transparency are not of critical
importance, like some corporate procurements, ties
can be broken arbitrarily. From this perspective, tie
breaking seems unimportant. However, when fairness
and transparent auction rules matter, resolving ties
must concern auction designers.
Ties in combinatorial auctions can occur in a
variety of ways. Consider a sale of three items a,
b, and c, with the following seven bids submitted: bb =bc =2, ba = 3, bb c = ba c =
4 ba b = 5 ba b c = 7. Note that there are no
identical bids for any combination. However, there is
a four-way tie: each of the allocations a b c,
a b c, a b c, and a b c has a revenue
of 7. Care in designing combinatorial auctions is
needed to minimize the probability of ties occurring
and in designing good ways to break ties. The winners in a combinatorial auction are not determined
based on a single bid, nor does a collection of winning
bids have to consist of the same number of bids. (In
the above example, it ranged from three different bids
for single items to a single bid for all three items.) In
general, the probability of ties occurring is related to
the strategies of bidders and bidding agents and to
the choice of acceptable bid increments. As in many
e-auctions, if bidding is constrained to prespecified
bid increments (resulting in a discrete set of possible
new bids), these could be interrelated in a way that
makes ties more likely. For example, suppose that in
the example, new bids have to be 10% larger than
current high bids. Then, if in the next round, six new
bidders chose to outbid the current winners on a, b,
c, a b, b c, and a b c by submitting the lowest allowed bids (10% above the current bids), there
would again be a four-way tie. More generally, if initial bids define many ties (e.g., if the initial bid on a
combination is set to be the sum of the initial bids
on single items that define the combination), and if
bid increments are constrained to a uniform rule (like
Management Science/Vol. 49, No. 11, November 2003
a fixed percentage or a fixed amount), ties become
more likely. Similarly, in iterative auctions, one has to
be careful in defining suggested bid improvements to
avoid unnecessary ties. For example, if an increment
is calculated according to additive prices from the
dual of the linear programming relaxation of the winner determination problem as described in the previous section, ties become much more likely. (This
seemed to cause a large number of ties in the combinatorial auction at Mars Incorporated, see Hohner
et al. 2003.)
From these examples, some suggestions for avoiding ties are obvious. Do not pick minimum bids for
combinations as the sum of the minimums for their
components. Instead, use a combination differential,
perhaps a small one. In addition, ties are less likely
with a wide range of allowable bid advances.29
What rules can be used to break ties? On an abstract
level, tie breaking corresponds to choosing among different optimal allocations, so as long as a preference
ordering on all allocations is defined, it could break
ties. Such an ordering could even change from round
to round. For example, one could use a generalized
time stamp method (Pekeč and Rothkopf 2000) in
which every allocation gets a sequence of time stamps
corresponding to the time of the most recent bid submission that is part of it. Then, preference can be
given to the allocation that was completed first, i.e.,
sequences of time stamps can be set in a decreasing
order (assuming bids cannot be submitted simultaneously) and the allocation whose sorted sequence is the
lowest in the lexicographic ordering can be declared
the winner.30 For example, suppose that in the example above, the times into the auction of bid submis29
However, this facilitates signaling. If signaling is being combated
by limiting allowable bid increments, bidtakers can allow bidders
to increase their bids by a private arbitrary subincrement that
would be binding, used to break ties, and kept secret until the end
of the auction, except perhaps when used to break a tie.
30
This rule mimics the behavior of auctioneers in simple progressive auctions who accept only bids that improve the previous bid. It has desirable incentive properties. However, a different rule is used by some e-auctioneers. uBid.com (see http://www.
ubid.com/help/topic8.asp) breaks ties according to the earliest bid
placed by the bidder involved in a tie, not the time of placing the
tied bid. This rule allows bidders to submit early low bids that
stand no chance of being competitive but that will be useful if a
1499
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
sion for a, b, c, ab, ac, bc, and abc are 17, 10,
18, 16, 8, 20, and 19 seconds, respectively. Then, the
sequence corresponding to the allocation a b c
is 18, 17, 10, i.e., this allocation was completed only
after the bid for c was submitted 18 seconds into the
bidding. The allocation to be selected according to the
lexicographic rule is a b c whose sequence is 18,
16.31 Other orderings derived from time stamp information (or any other labeling of combinations) could
also be used to define a tie-breaking rule. In general, any one-to-one function that gives a real value
to any collection of labels (corresponding to combinations in the allocation) could be used to break ties.
The average value of the time stamp was used by
Hohner et al. (2003). In our example, this rule picks
a b c because its average time stamp, 15, is the
lowest among all revenue-maximizing allocations.
Another way to break ties is to select a winning
allocation at random among all optimal allocations.
In general, however, finding all optimal solutions to
the winner determination problem is much more time
consuming than finding one such solution. One way
to get around this issue is to assign a random number
rC to each combination and then find max rCxC
subject to the same constraints as in the winner determination problem, plus the following constraint that
ensures selecting an optimal collection: bCxC = M,
where M is the optimal value of the winner determination problem. This tie-breaking IP problem has
the same complexity as does winner determination.
Extensive simulations by the FCC show that it should
pose no serious additional computational difficulties
in practice (Hoffman 2001).
6.4. Complexity of Communication
Compared to simple auctions, combinatorial auctions
generally place much higher information and communication burdens on both bidders and the bidtaker.
In general, bidders should be able to evaluate and
tie occurs. Such bidders can start bidding competitively at the end
of an auction with the advantage of needing only to tie the current
high bid rather than top it.
31
Note that the bid for c is part of both allocations and happened
to be the last combination bid on in both. Hence, the lexicographic
rule used the time of the bid on the next to last combination in
each allocation.
1500
possibly bid on all biddable combinations, and the
bidtaker should process those bids to determine winning allocation and prices. In an iterative auction, the
bidtaker has to provide feedback after each bidding
round and the bidders have to process that information and consider new bids. The complexity stems
from the potentially exponentially many (in the number of items) biddable combinations and the cooperative aspect of combinatorial bidding. For example,
consider a spectrum auction of 12 frequency licenses
in which bids on any of 212 − 1 = 4095 combinations
are allowed. How should 4,095 different valuations be
elicited from a telecom executive? Equivalently, given
the 4,095 prices for bundles, which bundles should
an executive consider bidding on? Thus, complexities
exist even at the level of valuation elicitation, and it is
not clear whether the multiround format helps overcoming some of these issues. The prices on bundles
could fluctuate wildly (and in both directions unless
high but nonwinning bids are always retained). Efficient methods and strategies for elicitation of combination valuations are needed, and neither theory nor
practice provides any firm guidelines at this point.32
For another example, consider a bidder who placed
a high but not winning bid on combination C. What
information would such bidder require to verify why
it did not win? One way to handle this is by having a
bidtaker announce market-clearing prices. As already
noted in §5.3, such prices might have to be nonanonymous and are hard to compute.
In fact, Nisan and Segal (2002) analyze the communication complexity of eliciting subset valuations, and
solving (and even approximating) the winner determination problem. They show that, in the model in
which the bidtaker communicates prices on single
items (one such example is the already mentioned
shadow prices of the linear programming relaxation
of the winner determination problem) and bidders
reply with most-preferred combinations given these
prices, the lower bound on the number of queries
32
In general, one cannot avoid facing all 2n − 1 questions, but if
valuations have specific structure, e.g., if all synergies are due to
few factors such as volume, adjacency, and so on, it is plausible that
valuations have succinct representation and that efficient elicitation
strategies exist. Among the rare papers dealing with such issues
are Hudson and Sandholm (2002) and Fishburn et al. (2002).
Management Science/Vol. 49, No. 11, November 2003
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
that need to be communicated in the worst-case scenario is exponential in the number of items. One way
to mitigate communication burdens is to limit the
structure of biddable combinations, e.g., as proposed
in Rothkopf et al. (1998). However, if this is not an
option, bidtakers should take particular care in providing tools that help bidders in bid preparation. In
particular, bid submission should not be burdensome,
and feedback has to be easily understandable, possibly providing sufficient or suggested bid improvements. Furthermore, and this is a common practice even in noncombinatorial auctions on the Internet, electronic communication allows for automating
some of these decisions for bidders via proxy-bidding
tools.33 Proxy bidding enables participation of bidders
who are not capable of dealing with the communication and computation demands of bidding in combinatorial auctions by adopting a proxy-bidding strategy and relegating these complexities to the proxy
agent.34
7.
Discussion
Combinatorial auctions are largely a new phenomenon. They have already been applied in a variety of contexts. One is scheduled for a multibilliondollar spectrum auction. They may play a prominent
role in the next generation of e-business auctions.
Given the computation and communication complexities involved, combinatorial auctions for more than
a handful of objects require computers in the auction
process. The future of combinatorial auctions is tied
to the future of e-business and of the Internet as a
medium for business communication.
In a nutshell, combinatorial auctions allow bidders
to express interdependencies in their valuations of
33
Most popular Internet auction sites offer a proxy-bidding option.
For one example, see http://pages.ebay.com/help/buyerguide/
bidding-prxy.html.
34
Parkes and Ungar (2000) were among the first to propose
proxy-bidding agents in combinatorial auctions, noting that use
of proxy bidding prevents some forms of strategic manipulation
(e.g., jump bidding). Some even propose combinatorial auction
designs that mandate use of proxy-bidding agents (Ausubel and
Milgrom 2002).
Management Science/Vol. 49, No. 11, November 2003
combinations of items for sale at the cost of creating considerable implementation complexities. Auction designers must decide whether the costs of
implementing a combinatorial auction outweigh the
benefits of allowing combinatorial bidding. Unlike
simple auctions, combinatorial auctions require, at the
very minimum, the ability to solve winner determination problems, often by using combinatorial optimization techniques. Section 4 discusses ways to ease the
computational burden, and these sometimes involve
avoiding rather than solving a mathematical problem.
However, the computational issue cannot be completely brushed away, and auction designers should
have linear and integer programming expertise.
One of the most important decisions about combinatorial auctions is the decision whether to hold one
or not. In particular, auction designers must define
what is to be sold. This task involves significant and
sometimes subtle choices. If definitions of the items
to be sold can be found that make it appropriate to
sell them independently, the complications of combinatorial auctions need not be faced. However, when
the definition process yields a set of items with interrelated values, allowing combinatorial bidding is an
important option.
The simplest combinatorial auction to implement is
a sealed bid first-price auction as discussed in §5.1.
However, for various reasons, iterative formats are
of interest. Iterative combinatorial auctions were discussed in §6 and involve several issues specific to
combinatorial bidding. In particular, iterative combinatorial auction designs that fail to address the
exposure and threshold problem facing bidders could
easily wipe out any potential benefits of combinatorial
bidding and yield inefficient and revenue-deficient
outcomes.
Theoretical research will continue to define and
affect what is desirable, possible, and implementable
in combinatorial auctions, but currently there are
many more open questions than definitive answers.
Research directions we find most interesting and
promising include further analysis of the usefulness
of proxy bidding, development of efficient methods
of representation and elicitation of interdependent
valuations, and development of classical gametheoretic equilibrium bidding strategies for practical
1501
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
combinatorial auction designs. While general theorems can be useful, practical auction design has much
to gain from taking advantage of particulars. What is
best for government sales of billion-dollar spectrum
rights will not be best for purchasing a firm’s raw
materials or tomorrow’s electricity. While the results
will be less sweeping, research on dealing with the
realities of particular auctions should prove useful.
Allowing bidders to express their synergetic values is a valuable option that auction designers should
not ignore. However, even though there is no longer
need to reject complex mechanisms out of hand, auction designers need to weigh carefully the important
trade-offs that combinatorial bidding brings. Caution
is needed due to the many potentially damaging
implications of design decisions that have to be made
and that may lack theoretical or empirical support.
However, carefully exploring combinatorial auction
design possibilities, keeping in mind potential pitfalls and that the devil is in the details, could be
worthwhile. Given the potential importance of combinatorial bidding, both the theory and practice of combinatorial auctions should quickly move out of their
current infancy.
Acknowledgments
The authors wish to acknowledge helpful and extensive comments from the special issue editors, an associated editor, and three
referees.
References
Ausubel, L., P. Milgrom. 2002. Ascending auctions with package
bidding. Frontiers Theoret. Econom. 1(1).
Banks, J. S., J. O. Ledyard, D. Porter. 1989. Allocating uncertain and
unresponsive resources: An experimental approach. RAND
J. Econom. 20(1) 1–25.
, M. Olson, D. Porter, S. Rassenti, V. Smith. 2001. Theory,
experiment and the Federal Communications Commission
spectrum auctions. Working paper. http://wireless.fcc.gov/
auctions/conferences/combin2001/papers/vsmith.pdf.
Bayers, C. 2000. Capitalist econstruction. Wired Magazine 8(3) 210–
212.
Bikhchandani, S., J. Ostroy 2002. The package assignment model.
J. Econom. Theory 107 377–406.
, S. de Vries, J. Schummer, R. V. Vohra. 2002. Linear programming and Vickrey auctions. B. Dietrich, R. V. Vohra, eds.
Mathematics of the Internet: E-Auctions and Markets. Springer,
New York, 75–116.
1502
Brewer, P. J., C. R. Plott. 1996. A binary conflict ascending price
(BICAP) mechanism for the decentralized allocation of the
right to use railroad tracks. Internat. J. Indust. Organ. 14(6)
857–886.
Business Week Online. 2001. A way out of the HDTV mess and
Wireless-spectrum bidders beware (March 1).
Cantillon, E., M. Pesendorfer. 2002. Combination bidding in multiunit auctions. Working paper, Harvard University, Boston, MA
and Yale University, New Haven, CT.
Carson, T. N. 1993. Seller incentive properties of EPA’s emission
trading auction. J. Environ. Econom. Management 25 177–195.
, C. R. Plott. 1996. An experimental investigation of the seller
incentives in EPA’s emission trading mechanism: A laboratory
evaluation. J. Environ. Econom. Management 30 133–160.
Clarke, E. H. 1971. Multipart pricing of public goods. Public Choice
11 17–33.
Cramton, P. 1995. Money out of thin air: The nationwide narrowband PCS auction. J. Econom. Management Strategy 4 431–495.
. 1997. The FCC spectrum auctions: An early assessment.
J. Econom. Management Strategy 6 267–343.
, J. A. Schwartz. 2000. Collusive bidding: Lessons from the FCC
spectrum auctions. J. Regulatory Econom. 17(3) 229–252.
Cybernomics Inc. 2000. An experimental comparison of the simultaneous multi-round auction and the CRA combinatorial auction
(March 15). http://wireless.fcc.gov/auctions/conferences/
combin2000/releases/98540191.pdf.
DeMartini, C., A. Kwasnica, J. O. Ledyard, D. Porter. 1999. A new
and improved design for multi-object iterative auctions. Social
science working paper 1054, California Institute of Technology,
Pasadena, CA.
de Vries, S., R. V. Vohra. 2003. Combinatorial auctions: A survey.
INFORMS J. Comput. 15(3) 284–309.
Elmaghraby, W., P. Keskinocak. 2003. Technology for transportation
bidding at the Home Depot. C. Billington, T. Harrison, H. Lee,
J. Neale, eds. The Practice of Supply Chain Management: Where
Theory and Applications Converge. Kluwer.
Englebrecht-Wiggans, R. 1995. Private Communication.
Epstein, R., L. Henriquez, J. Catalan, G. Y. Weintraub, C. Martinez.
2002. A combinational auction improves school meals in Chile.
Interfaces 32(6) 1–14.
FCC. 2000a. The Federal Communications Commission public
notice
DA00-1075.
http://wireless.fcc.gov/auctions/31/
releases/da001075.pdf.
. 2000b. The Federal Communications Commission public notice DA00-1486. http://www.fcc.gov/wireless/auctions/
31/releases/da001486.pdf.
. 2002. The Federal Communications Commission public notice DA02-260. http://wireless.fcc.gov/auctions/31/
releases.html#da020260.
Fishburn, P. C., A. Pekeč, J. A. Reeds. 2002. Subset comparisons for
additive linear orders. Math. Oper. Res. 27 227–243.
Fujishima, Y., K. Leyton-Brown, Y. Shoham. 1999. Taming the computational complexity of combinatorial auctions: Optimal and
approximate approaches. Proc. Internat. Joint Conf. Artificial
Intelligence (IJCAI). Stockholm, Sweden, 548–553.
Management Science/Vol. 49, No. 11, November 2003
PEKEČ AND ROTHKOPF
Combinatorial Auction Design
General Electric Corp. 2001. Letter to share owners. GE Annual
Report 2000.
Groves, T. 1973. Incentives in teams. Econometrica 41 617–631.
Harstad, R. M. 2000. A blueprint for a multi-round auction with
package bidding (June 14). http://wireless.fcc.gov/auctions/
31/releases/harstad.pdf.
, M. H. Rothkopf. 2000. An “alternating recognition” model of
English auctions. Management Sci. 46 1–12.
Håstad, J. 1999. Clique is hard to approximate within n1− . Acta
Math. 182 105–142.
Hobbs, B. F., M. H. Rothkopf, L. C. Hyde, R. P. O’Neill. 2000. Evaluation of a truthful revelation auction for energy markets with
nonconcave benefits. J. Regulatory Econom. 18(1) 5–32.
Hoffman, K. 2001. Issues in scaling up the 700 MHz auction
design. Presented at 2001 Combin. Bidding Conf., Queenstown,
MD.
Hohner, G., J. Rich, E. Ng, G. Reid, A. J. Davenport, J. R.
Kalagnanam, H. S. Lee, C. An. 2003. Combinatorial and qualitydiscount procurement auctions with mutual benefits at Mars,
Incorporated. Interfaces 33(1) 23–35.
Hudson, B., T. Sandholm. 2002. Effectiveness of preference
elicitation in combinatorial auctions. Technical report CMUCS-02-124, Computer Science Department, Carnegie Mellon
University, Pittsburgh, PA.
Karp, R. M. 1972. Reducibility among combinatorial problems.
R. E. Miller, J. W. Thatcher, eds. Complexity of Computer
Computations. Plenum Press, NY, 85–103.
Kelly, F., R. Steinberg. 2000. A combinatorial auction with multiple
winners for universal service. Management Sci. 46 586–596.
Krishna, V., R. W. Rosenthal. 1996. Simultaneous auctions with synergies. Games Econom. Behavior 17 1–31.
Ledyard, J. O., M. Olson, D. Porter, J. A. Swanson, D. P. Torma.
2002. The first use of a combined value auction for transportation services. Interfaces 32(5) 4–12.
McAfee, R. P., J. McMillan. 1987. Auctions and bidding. J. Econom.
Literature 25 699–738.
. 1994. Selling sectrum rights. J. Econom. Perspectives 8
145–162.
Milgrom, P., R. Weber. 1982. A theory of auctions and competitive
bidding. Econometrica 50 1089–1122.
Nisan, N. 2000. Bidding and allocation in combinatorial auctions.
Proc. Second ACM Conf. Electronic Commerce (EC00). ACM Press,
New York, 1–12.
, I. Segal. 2002. The communication complexity of efficient allocation problems. Working paper, Hebrew University,
Jerusalem, Israel, and Stanford University, Stanford, CA.
Nurmi, H., A. Salomaa. 1993. Cryptographic protocols for Vickrey
auctions. Group Decision Negotiation 4 363–373.
Park, S., M. H. Rothkopf. 2001. Auctions with endogenously
determined allowable combinations. RUTCOR research report
3-2001, Rutgers University, New Brunswick, NJ.
Parkes, D. C. 1999. iBundle: An efficient ascending price bundle
auction. Proc. Second ACM Conf. Electronic Commerce (EC00).
ACM Press, New York, 148–157.
. 2001. Iterative combinatorial auctions: Achieving economic
and computational efficiency. Doctoral dissertation, Computer
and Information Science, University of Pennsylvania, Philadelphia, PA.
, L. H. Ungar 2000. Preventing strategic manipulation in iterative auctions: Proxy-agents and price-adjustment. Proc. 17th
National Conf. Artificial Intelligence (AAAI-00), AAAI Press,
Menlo Park, CA, 82–89.
Pekeč, A., M. H. Rothkopf. 2000. Making the FCC’s first combinatorial auction work well. An official filing with the Federal
Communications Commission, June 7. http://wireless.fcc.gov/
auctions/31/releases/workwell.pdf.
Porter, D., S. Rassenti, A. Roopnarine, V. Smith. 2003. Combinatorial
auction design. Proc. National Acad. Sci. 100 11153–11157.
Rassenti, S. J., V. L. Smith, R. L. Bulfin. 1982. A combinatorial auction mechanism for airport time slot allocation. Bell J. Econom.
13 402–417.
Robinson, M. S. 1985. Collusion and the choice of auction. RAND
J. Econom. 16 141–145.
Rothkopf, M. H., R. M. Harstad. 1990. Reconciling efficiency arguments in taxation and public sector resource leasing. RUTCOR
research report #66-90, Rutgers University, New Brunswick, NJ.
,
. 1995. Two models of bid-taker cheating in Vickrey auctions. J. Bus. 68 257–267.
, A. Pekeč, R. M. Harstad. 1995. Computationally manageable combinatorial auctions. RUTCOR research report #1395 and DIMACS technical report 95-09, Rutgers University,
New Brunswick, NJ.
,
,
. 1998. Computationally manageable combinational auctions. Management Sci. 44 1131–1147.
, T. J. Teisberg, E. P. Kahn. 1990. Why are Vickrey auctions
rare? J. Political Econom. 98 94–109.
Sakurai, Y., M. Yokoo, S. Matsubara. 1999. An efficient approximate algorithm for winner determination in combinatorial
auctions. Proc. Second ACM Conf. Electronic Commerce (EC-00).
ACM Press, New York, 30–37.
Vickrey, W. 1961. Counterspeculation, auctions, and competitive
sealed tenders. J. Finance 16 8–37.
Vohra, R. V., R. J. Weber. 2000. A “bids and offers” approach
to package bidding. An official filing with the Federal Communications Commission, May 16. http://wireless.fcc.gov/
auctions/31/releases/bid_offr.pdf.
Wurman, P. R., M. P. Wellman. 2000. AkBA: A progressive,
anonymous-price combinatorial auction. Proc. Second ACM
Conf. Electronic Commerce (EC00). ACM Press, New York,
21–29.
Accepted by Arthur Geoffrion and Ramayya Krishnan; received December 2001. This paper was with the authors 15 months for 4 revisions.
Management Science/Vol. 49, No. 11, November 2003
1503