Combinatorial Auctions: A Survey - Department of … Auctions: A Survey Sven de Vries • Rakesh V. Vohra Zentrum Mathematik, TU München, D-80290 München, Germany Department of Managerial

Combinatorial Auctions: A Survey

Sven de Vries • Rakesh V. VohraZentrum Mathematik, TU München, D-80290 München, Germany

Department of Managerial Economics and Decision Sciences, Kellogg School of Management,Northwestern University, Evanston, Illinois 60208, USA

[email protected] • [email protected]

Many auctions involve the sale of a variety of distinct assets. Examples are airport timeslots, delivery routes, network routing, and furniture. Because of complementarities

or substitution effects between the different assets, bidders have preferences not just forparticular items but for sets of items. For this reason, economic efficiency is enhanced ifbidders are allowed to bid on bundles or combinations of different assets. This paper sur-veys the state of knowledge about the design of combinatorial auctions and presents somenew insights. Periodic updates of portions of this survey will be posted to this journal’sOnline Supplements web page at http://joc.pubs.informs.org/OnlineSupplements.html.(Auctions; Combinatorial Optimization)

1. IntroductionMany auctions involve the sale of a variety ofdistinct assets. Examples are the FCC spectrum auc-tions (http://www.fcc.gov/wtb/auctions/) and auc-tions for airport time slots (Rassenti et al. 1982),railroad segments (Brewer 1999), delivery routes(Caplice 1996) and network routing (Hershberger andSuri 2001). Because of complementarities or substi-tution effects between different assets, bidders havepreferences not just for particular items but for sets ofitems, sometimes called bundles.

To illustrate, suppose you must auction off a diningroom set consisting of four chairs and a table. Wouldyou wish to auction off the entire set or run five sep-arate auctions for each piece? The answer depends,of course, on what bidders care about. If every bid-der is interested in the dining room set and nothingless, the first option is preferable. If some bidders areinterested in the set but others are interested onlyin a chair or two it is not obvious what to do. Ifyou believe that you can raise more by selling offthe chairs separately than the set, the second optionis preferable. Notice, deciding requires a knowledgeof just how much bidders value different parts ofthe ensemble. For this reason, economic efficiency

is enhanced if bidders are allowed to bid directlyon combinations of different assets instead of biddingonly on individual items. Auctions where bidders areallowed to submit bids on combinations of items areusually called combinatorial auctions. “Combinationalauctions” is more accurate, but in this survey we willcomply with convention.

Auctions where bidders submit bids on combi-nations have recently received much attention. Seefor example Caplice (1996), Rothkopf et al. (1998),Fujishima et al. (1999), and Sandholm (1999). How-ever, such auctions were proposed as early as 1976(Jackson 1976) for radio spectrum rights. Rassentiet al. (1982), a little later, propose such auctions toallocate airport time slots. Srinivasan et al. (1998) haveproposed a mechanism for trading financial securi-ties that allows buyers and sellers to offer bundles offinancial instruments; their mechanism treats financialsecurities as divisible. Increases in computing powerhave made combinatorial auctions more attractive toimplement.

Perhaps the best known auction of heterogenousobjects has been the 1994 FCC’s “Nationwide Narrow-band Auction” of spectrum rights. Here bidders wereinterested in different collections of spectrum licences.

INFORMS Journal on Computing © 2003 INFORMSVol. 15, No. 3, Summer 2003, pp. 284–309

0899-1499/03/1503/0284$05.001526-5528 electronic ISSN

DE VRIES AND VOHRACombinatorial Auctions: A Survey

The FCC decided against an auction in which bidderswould bid on subsets of licences as it was thought,at the time, that such an auction would be cumber-some to run. Instead, the FCC used a separate auctionfor each licence. However, the auctions were run inparallel and bidders were allowed to participate in asmany of them as possible. A more detailed descrip-tion of these FCC auctions can be found in Cramton(2002), whereas the European spectrum auctions of2001 are covered by Binmore and Klemperer (2002),Grimm et al. (2001), Jehiel and Moldovanu (2001b),Klemperer 2002, and de Vries and Vohra (2001a). Inthe Spring of 2003 the FCC plans to run its first auc-tion (Auction #31) in which bidders will be allowedto bid on combinations of spectrum licences.

In contrast to the FCC, a number of large firmshave actively embraced combinatorial auctions to pro-cure logistics services. Ledyard et al. (2000) describethe design and use of a combinatorial auction thatwas employed by Sears in 1993 to select carriers.Here the objects bid upon were delivery routes (calledlanes). Since a truck must return to its depot, itwas more profitable for bidders to have their trucksfull on the return journey. Being allowed to bid onbundles gave bidders the opportunity to constructroutes that utilized their trucks as efficiently as pos-sible. In fact, a number of logistics consulting firmstout software to implement combinatorial auctions.SAITECH-INC, for example, offers a software productcalled SBIDS that allows trucking companies to bidon bundles of lanes. Logistics.com’s system is calledOptiBidTM. Logistics.com claims that more than $5billion in transportation contracts have been bid byJanuary 2000 using OptiBidTM by Ford Motor Com-pany, Wal-Mart, and K-Mart. Two more companieshave been formed to provide software for combi-natorial auctions. They are CombineNet and TradeExtensions.

Since about 1995, London Transport has been auc-tioning off bus routes using a combinatorial auction.About once a month, existing contracts to servicesome routes expire and these are offered for auc-tion. Bidders can submit bids on subsets of routes,winning bidders are awarded a five-year contractto service the routes they win. In this way, about20 percent of London Transport’s 800 bus routes are

auctioned off every year. Details of the auction can befound at http://www.londontransport.co.uk/buses/images/tend_rpt.pdf.

Graves et al. (1993) describes the auction of seats ina course that is executed regularly at the University ofChicago’s Business School. Strevell and Chong (1985)describe the use of an auction to allocate vacationtime slots. Banks et al. (1989) propose a combinato-rial auction for selecting projects on the space shuttle.It was tested experimentally but never implemented,for political reasons.

Procurement auctions where bidders are asked tosubmit a collection of price-quantity pairs, for exam-ple, $4 a unit for 100 units; $3.95 a unit for 200 units,etc., are also examples of combinatorial auctions.Here each price-quantity pair corresponds to a bun-dle of homogenous goods and a bid. If the goods areendowed with attributes like payment terms, delivery,and quality guarantees, they become bundles of het-erogenous objects. Davenport and Kalagnanam (2002)describe a combinatorial auction for such a contextthat is used by a large food manufacturer. Ausubeland Cramton (1998) and Bikhchandani and Huang(1993) describe the auction for Treasury Securities thatis actually used by the U.S. Department of Treasuryand compare it with other mechanisms.

In 1998 OptiMark Technologies (http://www.optimark.com/markets.html) offered an automatedtrading system that allows bidders to submit price-quantity-stock triples (along with a priority list). TheSecurities and Exchange Commission (SEC) approveda proposal by the Pacific Stock Exchange to imple-ment this electronic trading system. That same yearthe NASDAQ market announced plans to introducethis technology to its dealers and investors tradingstocks listed on it. The system was adopted by theOsaka Securities exchange, but suspended in June of2001.

The designer of a combinatorial auction faces a sur-feit of choices, some of which we list below:

1. Should the collection of bundles on which bidsare allowed be restricted? If so, to what?

2. Should the auction involve a single round of bid-ding? If so, how should the bundles be allocated asa function of the bids and what should the paymentrules be?

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3. If the auction is to involve multiple rounds (callsuch auctions iterative), what information should berevealed to bidders from one round to the next?The choice depends on the objectives of the auction-eer. For example, is it to maximize revenue or eco-nomic efficiency? Other considerations also matter:Speed, practicality, bidders preferences, and the needto discourage collusion and encourage competitionamong the bidders.

Nevertheless, no matter how one chooses thereare three problems that every auction designer mustresolve. The first has to do with bid expression. Thesecond is how to allocate bundles among bidders soas to optimize some criterion. Third, what are theincentive implications of the solutions offered to thefirst two. We discuss the first two issues in Sections 2and 3. In Section 4 we will discuss iterative auctionsand conclude with a discussion of the third majorproblem—incentive issues—in Section 5.

In the interests of space, a number of issues relevantto auction design in general are omitted. These areinterdependent values (Jehiel and Moldovanu 2001),privacy in bidding (Naor et al. 1999) and “false name”bidding (Yokoo et al. 2001).

2. Bid ExpressionThe first and most obvious difficulty faced by an auc-tion that allows bidders to bid on combinations is thateach bidder may have to determine a bid for everybundle he is interested in. The second problem is howto transmit this bidding function in a succinct way tothe auctioneer.

In theory, a bidder could be interested in everycombination of items possible. In practice resourceconstraints on the part of bidders will limit the num-ber of combinations on which they will submit bids.For example, in the auction of spectra, estimating thevalue of a bundle of spectra requires putting togethera business plan. Having decided on which combina-tions to place a bid, the next step is to communicatethis to the auctioneer.

The difficulty now is to communicate this list, if itis particularly large, in a way that will be computa-tionally useful to the auctioneer. One approach, notmuch explored, is to rely on an “oracle.” An oracle

is a program (black box) that, for example, given abidder and a subset computes the bid for it. Thus bid-ders submit oracles rather than bids. The auctioneercan simply invoke the relevant oracle at any stage todetermine the bid for a particular subset. (Sandholm1999 points out that another advantage of oracles isthat bidders need not be present. Their applicationdoes rely on the probity of the auctioneer.) Effective-ness of this approach depends on the computationalefficiency of the oracle.

Alternatively, the auctioneer may specify a biddinglanguage that all bidders must use to encode theirpreferences. A discussion of various ways in whichbids can be restricted and their consequences can befound in Nisan (2000). In that paper Nisan asks, givena language for expressing bids, what preferences oversubsets of objects can be correctly represented by thelanguage. What seems clear is that a computationallyefficient oracle or language relies on restricting thepreferences of bidders, or combinations on which bid-ders can bid.

Another way to overcome the complexity of com-municating bids and determining the winning bid-ders is to restrict the collection of bundles on whichbidders might bid. Different scenarios along this ideaare developed by Rothkopf et al. (1998); see alsoSubsection 3.4.

Even if this problem is resolved (in a non-trivialway) to the satisfaction of the parties involved, it stillleaves open the problem of deciding which collectionof bids to accept.

3. Winner DeterminationThe problem of identifying which set of bids to accepthas usually been dubbed the winner-determinationproblem. The precise formulation will depend on theobjectives of the auctioneer. Here we focus on the for-mulation described in Rothkopf et al. (1998) and bySandholm (1999). To distinguish it from other possibleformulations we call it the combinatorial auction prob-lem (CAP). (We assume that the auctioneer is a sellerand bidders are buyers.) CAP can be formulated asan integer program. We will survey what is knownabout the CAP. It assumes a knowledge of linear pro-gramming and familiarity with basic graph-theoreticterminology.

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3.1. The CAPTo formulate CAP as an integer program, let N bethe set of bidders and M the set of m distinct objects.For every subset S of M let bj�S� be the bid thatagent j ∈ N has announced he is willing to pay for S.(Implicit is the assumption that bidders care onlyabout the combinations they receive and not on whatother bidders receive.) From the formulation it will beclear that bids with bj�S� < 0 will never be selected.So, without loss of generality, we can assume thatbj�S�≥ 0. Let y�S� j� = 1 if the bundle S ⊆M is allo-cated to j ∈ N and zero otherwise.

max∑j∈N

∑S⊆M

bj�S�y�S� j�

s.t.∑S�i

∑j∈Ny�S� j�≤ 1 ∀ i ∈M

∑S⊆M

y�S� j�≤ 1 ∀ j ∈ N

y�S� j�= 0�1 ∀S ⊆M� j ∈ N

The first constraint ensures that overlapping sets ofgoods are never assigned. The second ensures that nobidder receives more than one subset. Call this formu-lation CAP1. Problem CAP as formulated here is aninstance of what is known as the set-packing problem(SPP), which is described below.

When bid functions are superadditive, a more suc-cinct formulation is possible. Let b�S� = maxj∈N bj�S�and set xS = 1 if the highest bid on the set S is to beaccepted and zero otherwise. Then CAP can be for-mulated as:

max∑S⊆M

b�S�xS

s.t.∑S�ixS ≤ 1 ∀ i ∈M

xS = 0�1 ∀S ⊆M

Here the constraint∑

S�i xS ≤ 1∀ i ∈ M ensures thatno object in M is assigned to more than one bidder.Call this formulation CAP2. It is also an instance ofthe SPP. (In the absence of superadditivity, one mustimpose the additional constraints of CAP1 that pre-vent any bidder from receiving more than one bundlein an optimal solution.)

There is another interpretation of the CAP possible.If the bids submitted are the actual values that bid-ders have for various combinations, then the solutionto the CAP is the economically efficient allocation ofindivisible objects in an exchange economy.

We have formulated CAP1 under the assumptionthat there is at most one copy of each object. It isan easy matter to extend the formulation to the casewhen there are multiple copies of the same objectand bidders may want more than one copy of thesame unit. Such extensions, called multi-unit combina-torial auctions, are investigated by Leyton-Brown et al.(2000a) as well as by Gonen and Lehmann (2000). Ifthe number of units of each type is large, then onecould approximate the problem of selecting the win-ning set of bids using a linear program. The relevantdecision variables would be the percentage of eachtype of good allocated to a bidder.

The formulation for winner determination justgiven is not flexible enough to encompass some of thevariations that have been considered in the literature.Here is a more comprehensive formulation:

max∑j∈N

∑q∈�j

bj�q�y�q� j�

s.t.∑j∈N

∑q∈�j

y�q� j�qi ≤mi ∀ i ∈M (GCAP1)

yj ∈ PAj ∀ j ∈ N (GCAP2)

y ∈ PA (GCAP3)

yj ∈ PBj ∀ j ∈ N (GCAP4)

y�q� j�= 0�1 ∀ q ∈�j ∀ j ∈ N (GCAP5)

Here mi is the number of units of object i availableand q is an integral vector whose ith component rep-resents the number of units of object i demanded. Ify�q� j� = 1 this means agent j is allocated the bundlerepresented by the vector q.

The sets �j ⊆ �M ∩ �0�m1�× �0�m2�× · · · × �0�mm�model restrictions on what bidders can bid on. Theycan be fixed by the auctioneer or she might permitbidders to specify them (subject to some constraintsas the FCC-restrictions for auction #31 on the numberof packages on which a bidder might bid).

The constraints (GCAP1) ensure that no more of anitem is allocated than the available supply. The con-straints (GCAP2) are imposed by the auctioneer



and enforce capacity constraints on the bidders; forexample no bidder is supposed to win more than twoitems, no bidder is supposed to win more than 40%of the total business, etc. Here PAj denotes the poly-hedron of feasible solutions to these constraints. (Theauctioneer could choose PAj that in effect restrict �j�

To avoid this one could require that PAj ∩��j be fulldimensional.)

Constraints (GCAP3) permit the auctioneer torestrict the overall allocation. For example, the alloca-tion must be edges that form a path or a tree. Here PAdenotes the polyhedron of feasible solutions to theserestrictions. (We assume that PA cannot be describedas the cartesian product of an interval on the coordi-nate axis and a lower-dimensional polytope.)

Constraints (GCAP4) allow each bidder to restrictthe allocations he might win. The feasible solutionssatisfying these bidder-imposed restrictions are rep-resented by the polyhedron PBi . For example, if hehas a subadditive valuation, he might put PBi = �y ∈��j �∑S∈�j y�S� j� ≤ 1� to ensure that he does not paymore than what he bid.

Finally, (GCAP5) ensures that we end with an inte-gral allocation.

3.2. The Set-Packing ProblemThe SPP is a well-studied integer program. Givena ground set M of elements and a collection V ofsubsets with non-negative weights, find the largest-weight collection of subsets that are pairwise disjoint.To formulate this problem as an integer program, letxj = 1 if the jth set in V with weight cj is selectedand xj = 0 otherwise. Define aij to be 1 if the jth setin V contains element i ∈M . Given this, the SPP canbe formulated as:

max∑j∈Vcjxj

s.t.∑j∈Vaijxj ≤ 1 ∀ i ∈M

xj = 0�1 ∀ j ∈ V

As noted by Rothkopf et al. (1998) and Sandholm(1999) the CAP is an instance of the SPP. Just take Mto be the set of objects and V the set of all subsetsof M .

Before continuing with a discussion of the SPP wemention two of its close relatives. The first is calledthe set-partitioning problem (SPA) and the second iscalled the set-covering problem (SCP). Both would berelevant had we cast the auction problem in procure-ment rather than selling terms. The auctions used inthe transport industry are of this set-covering type. Inthat setting, objects are origin-destination pairs, calledlanes. Bidders submit bids on bundles of lanes thatrepresent how much they must be offered to under-take the deliveries on the specified lanes. The auction-eer wishes to choose a collection of bids of lowest costsuch that all lanes are served. (In fact, one must spec-ify not only lanes but volume as well, so this problemconstitutes an instance of a multi-unit combinatorialauction.)

While SPA and SCP are cosmetically similar to theSPP they have different computational and structuralproperties. The survey by Balas and Padberg (1976)contains a bibliography on applications of the SPP,SCP, and SPA.

3.3. Complexity of the SPPHow hard is the SPP to solve? By enumerating all pos-sible 0-1 solutions we can find an optimal solution ina finite number of steps. If �V � is the number of vari-ables, then the number of solutions to check would be2�V �, clearly impractical for all but small values of �V �.For the instances of SPP that arise in the CAP, the car-dinality of V is the number of bids; possibly a largenumber.

No polynomial-time algorithm for the SPP isknown and is unlikely to exist because SPP isNP-hard. (More precisely, the recognition version ofSPP is NP-complete.)

For the CAP, this discussion of complexity mayhave little relevance. To see why, suppose one takesthe number of bids as a measure of the size of theinput and this number is exponential in �M �. Anyalgorithm for CAP that is polynomial in the numberof bids but exponential in the number of items would,formally, be polynomial in the input size but imprac-tical for �M � large. Thus, effective solution proceduresfor the CAP must rely on two things. The first is thatthe number of distinct bids is not large and is struc-tured in computationally useful ways. The second is



that the underlying SPP can be solved reasonablyquickly.

3.4. Solvable Instances of the SPPThe usual way in which instances of the SPPcan be solved by a polynomial algorithm is whenthe extreme points of the polyhedron P�A� = �x �∑

j∈V aijxj ≤ 1 ∀ i ∈M ; xj ≥ 0 ∀ j ∈V� are all integral, i.e.0-1. In these cases we can simply drop the integralityrequirement from the SPP and solve it as a linear pro-gram. Linear programs can be solved in polynomialtime. It turns out that in most of these cases, becauseof the special structure of these problems, algorithmsmore efficient than linear-programming ones exist.Nevertheless, the connection to linear programmingis important because it allows one to interpret dualvariables as prices for the objects being auctioned. Wewill say more about this later in the paper.

A polyhedron with all integral extreme points iscalled integral. Identifying sufficient conditions forwhen a polyhedron is integral has been a cot-tage industry in integer programming. These suffi-cient conditions involve restrictions on the constraintmatrix, which in this case amount to restrictions onthe kinds of subsets for which bids are submitted. Welist the most important ones here.

Rothkopf et al. (1998) cover the same ground butorganizes the solvable instances differently as wellas suggesting auction contexts in which they maybe salient. An example of one such context is givenbelow.

3.4.1. Total Unimodularity. The most well knownof these sufficient conditions is total unimodularity,sometimes abbreviated to “TU.” A matrix is said tobe TU if the determinant of every square submatrixis 0, 1 or −1. If the matrix A= �aij�i∈M�j∈V is TU thenall extreme points of the polyhedron P�A� are integral(see Nemhauser and Wolsey 1988).

A special case of TU matrices are those with theconsecutive-ones property (Nemhauser and Wolsey1988). A 0-1 matrix has the consecutive-ones propertyif the non-zero entries in each column occur con-secutively. Rothkopf et al. (1998) offer the follow-ing to motivate the consecutive-ones property in theauction context. Suppose the objects to be auctioned

are parcels of land along a shore line. The shoreline is important as it imposes a linear order on theparcels. In this case it is easy to imagine that themost interesting combinations (in the bidders’ eyes)would be contiguous. If this were true it would havetwo computational consequences. The first is that thenumber of distinct bids would be limited (to inter-vals of various length) by a polynomial in the num-ber of objects. Second, the constraint matrix A ofCAP2 would have the consecutive-ones property inthe columns. If the valuation of each bidder is addi-tive over sets of nonadjacent intervals and superaddi-tive over sets of adjacent intervals, then CAP2 modelsthe situation correctly and the problem is polynomi-ally solvable. Otherwise one has to use CAP1, whichadds constraints that violate the consecutive-onesproperty. Müller (personal communication) pointedout that it is a consequence of Keil (1992) that thisproblem becomes NP-hard in general. For more effi-ciently solvable subcases see van Hoesel and Müller(2001).

3.4.2. Balanced Matrices. A 0-1 matrix B is bal-anced if it has no square submatrix of odd order withexactly two 1s in each row and column. If the matrixB is balanced then (see Schrijver 1986) the linear pro-gram

max

{∑j

cjxj �∑j

bijxj ≤ 1 ∀ i� xj ≥ 0 ∀ j}

has an integral optimal solution whenever the cj ’s areintegral.

For one instance of balancedness that may be rel-evant to the CAP, consider a tree T with a distancefunction d. For each vertex v in T let N�v� r� denotethe set of all vertices in T that are within distance rof v. If you like, the vertices represent parcels of landconnected by a road network with no cycles. Bidderscan bid for subsets of parcels but the subsets are con-strained to be of the form N�v� r� for some vertex vand some number r . Now the constraint matrix of thecorresponding SPP will have one column for each setof the form N�v� r� and one row for each vertex of T .This constraint matrix is balanced. See Nemhauserand Wolsey (1988) for a proof as well as efficient algo-rithms. In the case when the underlying tree T is



a path the constraint matrix reduces to having theconsecutive-ones property. If the underlying networkwere not a tree then the corresponding version of SPPbecomes NP-hard.

3.4.3. Perfect Matrices. More generally, if the con-straint matrix A can be identified with the vertex-clique adjacency matrix of what is known as a perfectgraph, then SPP can be solved in polynomial time.The interested reader should consult Chapter 9 of thebook by Grötschel et al. (1988) for more details. Thealgorithm, while polynomial, is impractical.

We now describe one instance of perfection thatmay be relevant to the CAP. It is related to the exam-ple on balancedness. Consider a tree T . As beforeimagine the vertices represent parcels of land con-nected by a road network with no cycles. Bidders canbid for any connected subset of parcels. Now the con-straint matrix of the corresponding SPP will have onecolumn for each connected subset of T and one rowfor each vertex of T . This constraint matrix is perfect(Nemhauser and Wolsey 1988).

3.4.4. Graph-Theoretic Methods. There are situ-ations where P�A� is not integral yet the SPP canbe solved in polynomial time because the constraintmatrix of A admits a graph-theoretic interpretation interms of an easy problem. The best-known instanceof this is when each column of the matrix A con-tains at most two 1s. In this case the SPP becomesan instance of the maximum-weight matching prob-lem in a graph, which can be solved in polynomialtime.

Each row (object) corresponds to a vertex in agraph. Each column (bid) corresponds to an edge.The identification of columns of A with edges comesfrom the fact that each column contains two non-zeroentries. It is well known that P�A� contains fractionalextreme points. Consider for example a graph that isa cycle on three vertices. A comprehensive discussionof the matching problem can be found in the bookby Lovász and Plummer (1986). The subclass of SPPwhere each column has at most K ≥ 3 non-zero entriesis NP-hard.

It is natural to ask what happens if one restrictsthe number of 1s in each row rather than column.The subclass of SPP with at most two non-zero entries

per row of A is NP-hard. These instances correspondto what is called the stable-set problem in graphs,a notoriously difficult problem. (The instance of CAPproduced by the radio spectrum auction in Jackson1976 reduces to just such a problem.)

Another case is when the matrix A has the cir-cular ones property. A 0-1 matrix has the circular-ones property if the non-zero entries in each column(row) are consecutive; first and last entries in eachcolumn (row) are treated consecutively. Notice theresemblance to the consecutive-ones property. In thiscase the constraint matrix can be identified with whatis known as the vertex-clique adjacency matrix of a cir-cular arc graph. (Take a circle and a collection ofarcs of the circle. To each arc associate a vertex. Twovertices will be adjacent if the corresponding arcsoverlap. The consecutive-ones property also bears agraph-theoretic interpretation. Take intervals of thereal line and associate them with vertices. Two ver-tices are adjacent if the corresponding intervals over-lap. Such graphs are called interval graphs.) The SPPthen becomes the maximum-weight independent setproblem for a circular arc graph. This problem canalso be solved in polynomial time; see Golumbic andHammer (1988). Following the parcels of land on theseashore example, the circular-ones structure makessense when the land parcels lie on the shores of anisland or lake.

3.4.5. Using Preferences. The solvable instancesabove work by restricting the sets of objectsover which preferences can be expressed. Anotherapproach would be to study the implications ofrestrictions in the preference orderings of the biddersthemselves.

One common restriction that is placed on bj�·�is that it be non-decreasing (that is, bj�S� ≤ bj�T �

for S ⊆ T ) and supermodular (that is, bj�S�+ bj�T � ≤bj�S ∪ T� + bj�S ∩ T�). Suppose now that bidderscome in two types. The type-one bidders havebj�·� = g1�·� and those of type two have bj�·� =g2�·�� where gr�·� are non-decreasing, integer-valuedsupermodular functions. Let Nr be the set of type-rbidders.



Consider now the dual to the linear programmingrelaxation of CAP1:

min∑i∈M

pi+∑j∈Nqj

s.t.∑i∈Spi+ qj ≥ g1�S� ∀S ⊆M� j ∈ N 1

∑i∈Spi+ qj ≥ g2�S� ∀S ⊆M� j ∈ N 2

pi� qj ≥ 0 ∀ i ∈M� j ∈ N

This problem is an instance of the polymatroid-intersection problem and is polynomially solvable;see Theorem 10.1.13 in the book by Grötschel et al.(1988). More importantly it has the property of beingtotally dual integral, which means that its linear-programming dual, the linear relaxation of the origi-nal primal problem, has an integer optimal solution.This last observation is used in Bikhchandani andMamer (1997) to establish the existence of competitiveequilibria in exchange economies with indivisibilities.Utilizing the method to solve problems with three ormore types of bidders is not possible because it isknown in those cases that the dual problem aboveadmits fractional extreme points. In fact the problemof finding an integer optimal solution for the inter-section of three or more polymatroids is NP-hard;see Section 12.6.3 of the book by Papadimitriou andSteiglitz (1982).

Another restriction on bids/preferences that hasbeen studied is the gross substitutes property (Kelsoand Crawford 1982). To describe it let the value thatbidder j assigns to the set S ⊆M of objects be vj�S�.Given a vector of prices p on objects, let the collectionof subsets of objects that maximize bidder j’s utilitybe denoted Dj�p�, and defined as

Dj�p�={S⊆M �vj�S�−

∑i∈Spi≥vj�T �−

∑i∈Tpi ∀T ⊆M

}�

The gross-substitutes condition requires that for allprice vectors p�p′ such that p′ ≥p, and all A∈Dj�p�,there exists B∈Dj�p

′� such that �i∈A�pi=p′i�⊆B. Aspecial case of the gross-substitutes condition is whenbidders are interested in multiple units of the sameitem and have diminishing marginal utilities.

In the case when each of the bj�·� have the gross-substitutes property, the linear-programming relax-ation of CAP1 and CAP2 have an optimal integersolution. This is proved in Kelso and Crawford (1982)as well as Gul and Stacchetti (2000). In both casesa primal-dual algorithm for the linear relaxation ofCAP1 is offered and interpreted as an auction. Murotaand Tamura (2000) point out the connection betweengross substitutes and M$-concavity. From this connec-tion it follows from results about M$-concave func-tions that CAP1 can be solved in time polynomial inthe number of objects under the assumption of grosssubstitutes by using a proper oracle.

3.5. Exact MethodsAn exact method for solving the SPP and the CAPis one that is guaranteed to return a solution that isboth feasible and optimal. They come in three vari-eties: branch and bound, cutting planes, and a hybridcalled branch and cut. Fast exact approaches to solvingthe SPP require algorithms that generate both goodlower and upper bounds on the maximum objective-function value of the instance. In general, the upperbound on the optimal solution value is obtained bysolving a relaxation of the optimization problem. Thereare two standard relaxations for SPP: Lagrangeanrelaxation (where the feasible set is usually requiredto maintain 0-1 feasibility, but many if not all of theconstraints are moved to the objective function with apenalty term), and the linear-programming relaxation(where only the integrality constraints are relaxed—the objective function remains the original function).Lagrangean relaxation will be discussed in greaterdetail in Section 5 on iterative auctions.

Because even small instances of the CAP1 mayinvolve a huge number of columns (bids) the tech-niques described above need to be augmented withanother method known as column generation. Intro-duced by Gilmore and Gomory (1961), it works bygenerating columns when needed rather than all atonce. An overview of such methods can be found inBarnhart et al. (1998). Later in this paper we illustratehow this idea could be implemented in an auction.

One sign of how successful exact approaches arecan be found in Hoffman and Padberg (1993). Theyreport being able to find an optimal solution to an



instance of SPA with 1,053,137 variables and 145 con-straints in under 25 minutes. In auction terms this cor-responds to a problem with 145 items and 1,053,137bids. A major impetus behind the desire to solvelarge instances of SPA (and SCP) quickly has beenthe airline industry. The problem of assigning crewsto routes can be formulated as an SPA. The rows ofthe SPA correspond to flight legs and the columnsto a sequence of flight legs that would be assignedto a crew. Like the CAP, in this problem the num-ber of columns grows exponentially with the numberof rows. (However, these crew-scheduling problemsgive rise to instances of SPA that have a large num-ber of duplicate columns in the constraint matrix, insome cases as many as 60% of them.) For the SPP, thelarge instances that have been studied have usuallyarisen from relaxations of SPAs. Given the above webelieve that established integer-programming meth-ods will prove quite successful when applied to thesolution of CAP.

Logistics.com’s OptiBidTM software has been usedin situations where the number of bidders is between12 to 350 with the average being around 120. Thenumber of lanes (objects) has ranged between 500and 10,000. Additionally, each lane bid can contain acapacity constraint as well as a budget capacity con-straint covering multiple lanes. The typical numberof lanes is 3000. OptiBidTM does not limit the num-ber of distinct subsets that bidders bid on or the num-ber of items allowed within a package. OptiBidTM

is based on an integer-program solver with a seriesof proprietary formulations and starting heuristicalgorithms.

SAITECH-INC’s bidding software, SBID, is alsobased on integer programming. They report beingable to handle problems of similar size as OptiBidTM.

Exact methods for CAP2 have been proposedby Rothkopf et al. (1998), Fujishima et al. (1999),Sandholm (1999), and Andersson et al. (2000). Thefirst uses straight dynamic programming, while thesecond and third use refinements by substantiallypruning the search tree and introducing additionalbounding heuristics. Andersson et al. use integer pro-gramming. In the second, the method is tested onrandomly generated instances, the largest of whichinvolved 500 items (rows) and 20,000 bids (variables).

The third also tests the method on randomly gen-erated instances, the largest of which involved 400items (rows) and 2000 bids (variables). In these teststhe number of bids examined is far smaller than thenumber of subsets of objects. The last uses integer-programming methods on the test problems gener-ated by the second and third.

By comparison, a straightforward implementationon a commercially available code for solving linearinteger programs (called CPLEX) only runs into dif-ficulties for instances of CAP involving more than 19items if one puts nonzero bids on all subsets. Therewill be more than 219 variables. This already requiresone gigabyte of memory to store. CPLEX can handlein this straightforward approach on the order of 219

variables and 19 constraints before running out of res-ident memory. Notice that this is large enough to han-dle the test problems considered by Sandholm (1999)and Fujishima et al. (1999).

Andersson et al. (2000) (we reported in earlier ver-sions of this survey about our own experiments; thispart became an independent report, de Vries andVohra 2001b) point out that CPLEX dominates (interms of run times) the algorithms of Sandholm (1999)and appear competitive with Fujishima et al. (1999).As pointed out by Andersson et al. (2000) and de Vriesand Vohra (2001b), solution times can be sensitive toproblem structure. For this reason Leyton-Brown et al.(2000b) are compiling a suite of test problems.

3.6. Approximate MethodsOne way of dealing with hard integer programs is togive up on finding the optimal solution. Rather, oneseeks a feasible solution fast and hopes that it is nearoptimal. This raises the obvious question of how closeto optimal the solution is. There have traditionallybeen three ways to assess the accuracy of an approxi-mate solution. The first is by worst-case analysis, thesecond by probabilistic analysis, and the third empir-ically.

Before proceeding it is important to say that prob-ably every heuristic approach for solving generalinteger-programming problems has been applied tothe SPP. Unfortunately, there has not been a compar-ative testing across such methods to determine underwhat circumstances a specific method might perform



best. We think it safe to say that anything one canthink of for approximating the SPP has probably beenthought of. In addition, one can embed approxima-tion algorithms within exact algorithms so that one isattempting to get a sharp approximation to the lowerbound for the problem at the same time that one iter-atively tightens the upper bound.

3.6.1. Worst-Case Analysis. The SPP is difficult toapproximate in a worst-case sense. A major result byHåstad (1999) is that unless ZPP=NP (this assump-tion is actually weaker than P=NP, but as wellbelieved to be unlikely), there is no polynomial-timealgorithm for the SPP that can deliver a worst-caseratio larger than n&−1 for any &>0. (Recall that inCAP1, n would be the number of bids.) On the pos-itive side, polynomial algorithms that have a worst-case ratio of O�n/�logn�2�, see Boppana and Halldórs-son (1992), are known. (In contrast, SCP can beapproximated to within a factor of logn. In this senseSCP is “easier” than SPP.) Bounds that are a func-tion of the data of the underlying input problem arealso known. A recent example of this motivated byCAP1 is given by Akcoglu et al. (2002). The readerinterested in a full account of what is known aboutapproximating the SPP should consult Crescenzi andKann (1995) where an updated list of what is knownabout the worst-case approximation ratio of a wholerange of optimization problems is given.

When interpreting these worst-case results it shouldbe remembered that they shed little light on the “typ-ical” accuracy of an approximation algorithm.

3.6.2. Probabilistic Analysis. Probabilistic analy-sis is an attempt to characterize the typical behavior ofan approximation algorithm. A probability distribu-tion over problem instances is specified. This inducesa distribution over the value of the optimal as wellas approximate solution. The goal is to understandhow close these two distributions might be. Since theresults are asymptotic in nature, attention must bepaid to the convergence results when interpreting theresults. A problematic feature is that the distributionsover instances that are chosen (because of ease ofanalysis) do not necessarily coincide with the distri-butions from which actual instances will be drawn.(Results that assert that “typical” instances are hard

are very rare.) This issue arises also in the empiricaltesting of approximation algorithms.

3.6.3. Empirical Testing. Many approximation al-gorithms will be elaborate enough to defy theoreti-cal analysis. For this reason it is common to resort toempirical testing. Further empirical testing allows oneto consider issues not easily treated analytically.

A good guide to the consumption of an empiricalstudy of approximation algorithms is given by Balland Magazine (1981). They list the following evalua-tion criteria:

1. Proximity to the optimal solution.2. Ease of implementation (coding and data re-

quirements).3. Flexibility; ability to handle changes in the

model.4. Robustness; ability to provide sensitivity analy-

sis and bounds.This is not the forum for an extensive discussionof the issues associated with the empirical test-ing of heuristics. However, some points are worthhighlighting.

The most obvious is the choice of test problems. Arethey realistic? Do they exhibit the features that onethinks one will find in the environment? Interestingly,probabilistic analysis has a role to play here in elim-inating some schemes for randomly generating testproblems. For example it is known that certain gen-eration schemes give rise to problems that are easy tosolve; for example, a randomly generated solution iswith high probability close to optimal. Success on acollection of problems generated in this way conveysno information. Is the accuracy due to the approxima-tion algorithm or the structure of the test problems?

Some approximation algorithms involve a numberof parameters that need to be fine tuned. Comparingtheir performance with heuristics whose parametersare not fine tuned becomes difficult because it is notclear whether one should include the overhead in thetuning stage in the comparison.

4. Iterative AuctionsIterative auctions come in two varieties (with hybridspossible). In the first, bidders submit, in each round,prices on various allocations. The auctioneer makes



a provisional allocation of the items that depends onthe submitted prices. Bidders are allowed to adjusttheir price offers from the previous rounds and theauction continues. Such auctions come equipped withrules to ensure rapid progress and encourage compe-tition. Iterative auctions of this type seem to be mostprevalent in practice.

In the second type, the auctioneer sets the price andbidders announce which bundles they want at theposted prices. The auctioneer observes the requestsand adjusts the prices. The price adjustment is usu-ally governed by the need to balance demand withsupply.

Call auctions of the first type quantity-setting,because the auctioneer sets the allocation or quantityin response to the prices/bids set by bidders. Callthe second price-setting because the auctioneer sets theprice. Quantity-setting auctions are harder to analyzebecause of the freedom they give to bidders. Eachbidder determines the list of bundles as well as theirprices to announce. In price-setting auctions, each bid-der is limited to announcing which bundles meet theirneeds at the announced prices.

In many simple environments, price-setting andquantity-setting auctions can be viewed as being“dual” to one another. The simplest example is theauction of a single object. The popular English ascend-ing auction is an example of a quantity-setting auc-tion. Bidders submit prices in succession, with theobject tentatively assigned to the current highest bid-der. The auction terminates when no one is preparedto top the current high bid. The “dual” version tothis auction has the auctioneer continuously raisingthe price. Bidders signal their willingness to buy atthe current price by keeping their hands raised. Theauction terminates the instant a single bidder remainswith his hand raised. In fact this dual version ofthe English auction is used as a stylized model ofthe English auction itself for the purposes of analy-sis (see for example Klemperer 2002). We believe thatprice-setting auctions are useful stylized models ofquantity-setting auctions and that insights from oneapply to the other.

Our discussion of iterative auctions is motivatedby this “duality.” We will point out that price-settingauctions can be viewed as primal-dual algorithms for

the underlying winner determination problem. Thereverse will also be true. Primal dual algorithms forCAP1 (or CAP2) can be given a price-setting auc-tion interpretation. Dantzig (1963) specifically offersan auction interpretation for the decomposition algo-rithm for linear programming. A more recent exampleis Bertsekas (1991), who has proposed a collection ofdual based algorithms for the class of linear networkoptimization problems. These algorithms he dubs auc-tion algorithms. Auction interpretations of algorithmsfor optimization problems go back at least as far asWalras (see Chapter 17H of the book by Mas-Collelet al. 1995) and all have the same flavor. Dual vari-ables are interpreted as prices, and the updates ontheir value that are executed in these algorithms canbe interpreted as a form of myopic best response onthe part of bidders.

What are the advantages of iterative auctions over,say, single-round sealed-bid auctions? The first is thatthey save bidders from specifying their bids for everypossible combination in advance. Second, such meth-ods can be adapted to dynamic environments wherebidders and objects arrive and depart at differenttimes. Third, in settings where bidders have privateinformation that is relevant to other bidders, suchauctions (with appropriate feedback) allow that infor-mation to be revealed.

Examples of iterative approaches for solving theCAP are given by Fujishima et al. (1999), Rassentiet al. (1982), Parkes and Ungar (2000), and Bikhchan-dani et al. (2002). In the same spirit, Brewer (1999),Wellman et al. (2001) and Kutanoglu and Wu (1999)propose decentralized scheduling procedures in dif-ferent contexts. In their setup the auctioneer choosesa feasible solution and “bidders” are asked to sub-mit improvements to the solution. In return for theseimprovements, the auctioneer agrees to share a por-tion of the revenue gain with the bidder.

In order to understand the behavior of price-settingauctions it is important to identify what propertiesprices must have in order to produce an allocationthat solves CAP1 (or CAP2). Such an understand-ing can be derived from the duality theory of integerprograms.



4.1. Duality in Integer ProgrammingTo describe the dual to SPP let 1 denote the m-vectorof all 1s and aj the jth column of the constraint matrixA. The (superadditive) dual to SPP is the problemof finding a superadditive, non-decreasing functionF � �m→�1 that solves

min F �1�

s.t. F �aj�≥cj ∀j∈VF�0�= 0

We can think of F as being a non-linear price func-tion that assigns a price to each bundle of goods (seeWolsey 1981).

If the primal integer program has the integralityproperty, there is an optimal integer solution to itslinear-programming relaxation, the dual function F

will be linear i.e. F �u�=∑i yiui for some y and all u∈

�m. The dual becomes:

min∑i

yi

s.t.∑i

aijyi≥cj ∀j∈V

yi≥0 ∀i∈MThat is, the superadditive dual reduces to the dual ofthe linear-programming relaxation of SPP. In this casewe can interpret each yi to be the price of object i.Thus, an optimal allocation given by a solution to theCAP can be supported by prices on individual objects.

Optimal objective-function values of SPP and itsdual coincide (when both are well defined). There isalso a complementary slackness condition:

Theorem 4.1 (Nemhauser and Wolsey 1988). If xis an optimal solution to SPP and F an optimal solutionto the superadditive dual then(

F �aj�−cj)xj=0 ∀j�

Solving the superadditive dual problem is as hardas solving the original primal problem. It is possibleto reformulate the superadditive dual problem as alinear program (the number of variables in the for-mulation is exponential in the size of the originalproblem). For small or specially structured problems

this can provide some insight. The interested readeris referred to Nemhauser and Wolsey (1988) for moredetails. In general one relies on the solution to thelinear-programming dual and uses its optimal valueto guide the search for an optimal solution to the orig-inal primal integer program. One way to do it is witha technique known as Lagrangean relaxation.

4.2. Lagrangean RelaxationThe basic idea is to “relax” some of the constraints ofthe original problem by moving them into the objec-tive function with a penalty term. That is, infeasi-ble solutions to the original problem are allowed, butthey are penalized in the objective function in pro-portion to the amount of infeasibility. The constraintsthat are chosen to be relaxed are selected so that theoptimization problem over the remaining set of con-straints is in some sense easy. We describe the barebones of the method first and then give a “market”interpretation of it.

Recall the SPP:

Z = max∑j∈Vcjxj

s.t.∑j∈Vaijxj≤1 ∀i∈M

xj = 0�1 ∀j∈VLet ZLP denote the optimal objective-function value tothe linear-programming relaxation of SPP. Note thatZ≤ZLP . Consider now the following relaxed problem:

Z�.� = max∑j∈Vcjxj+

∑i∈M.i

(1−∑

j∈Vaijxj

)

s.t. 1≥xj≥0 ∀j∈VFor a given ., computing Z�.� is easy. To see whynote that ∑

j∈Vcjxj+

∑i∈M.i

(1−∑

j∈Vaijxj

)

=∑j∈V

(cj−

∑i∈M.iaij

)xj+

∑i∈M.i�

Thus, to find Z�.�, simply set xj=1 if �cj−∑i∈M.iaij�>0 and zero otherwise. It is also easy to

see that Z�.� is piecewise linear and convex. A basicresult that follows from the duality theorem of linear



programming is:

Theorem 4.2.

ZLP=min.≥0

Z�.��

Evaluating Z�.� for each . is a snap. If one can finda fast way to determine the . that solves min.≥0Z�.��

one would have a fast procedure to find ZLP . Theresulting solution (values of the x variables) whileintegral need not be feasible. However it may not be“too infeasible” and so could be fudged into a fea-sible solution without a great reduction in objective-function value.

Finding the . that solves min.≥0Z�.� can be accom-plished using the subgradient algorithm. Suppose thevalue of the Lagrange multiplier . at iteration t is .t .Choose any subgradient of Z�.t� and call it st . Choosethe Lagrange multiplier for iteration t+1 to be .t+1ts

t , where 1t is a positive number called the step size.In fact if xt is the optimal solution associated withZ�.t�,

.t+1=.t+1t�Axt−1��Notice that .t+1

i >.ti for any i such that∑

j aijxtj >1. The

penalty term is increased on any constraint currentlybeing violated.

For an appropriate choice of step size at each itera-tion, this procedure can be shown to converge to theoptimal solution. Specifically, 1t→0 as t→� but

∑t 1t

diverges. Ygge (1999) describes some heuristics fordetermining the multipliers in the context of winnerdetermination.

Here is the auction interpretation. The auctioneerchooses a price vector . for the individual objectsand bidders submit bids. If the highest bid, cj� forthe jth bundle exceeds

∑i∈M aij.i, this bundle is ten-

tatively assigned to that bidder. Notice that the auc-tioneer need not know what cj is ahead of time. Thisis supplied by the bidders after . is announced. Infact, the bidders need not announce bids; they couldsimply state which individual objects are acceptableto them at the announced prices. The auctioneer canrandomly assign objects to bidders in case of ties. Ifthere is a conflict in the assignments, the auctioneeruses the subgradient algorithm to adjust prices andrepeats the process.

Now let us compare this auction interpretation ofLagrangean relaxation with the simultaneous ascend-ing auction (SAA) proposed by P. Milgrom, R. Wilson,and R. P. McAfee (see Milgrom 1995). In the SAA,bidders bid on individual items simultaneously inrounds. To stay in the auction for an item, bids mustbe increased by a specified minimum from one roundto the next just like the step size. Winning bidderspay their bids. The only difference between this andLagrangean relaxation is that the bidders throughtheir bids adjust their prices rather than the auc-tioneer. The adjustment is along a subgradient. Bidsincrease on those items for which there are two ormore bidders competing.

One byproduct of the SAA is called the exposureproblem. Bidders pay too much for individual items orbidders with preferences for certain bundles drop outearly to limit losses. As an illustration, consider anextreme example of a bidder who values the bundleof goods i and j at $100 but each separately at $0. Inthe SAA, this bidder may have to submit high bidson i and j to be able to secure them. Suppose that heloses the bidding on i. Then it is left standing with ahigh bid j� which it values at zero. The presence ofsuch a problem is easily seen within the Lagrangeanrelaxation framework. While Lagrangean relaxationwill yield the optimal objective-function value for thelinear relaxation of the underlying integer program, itis not guaranteed to produce a feasible solution. Thusthe solution generated may not satisfy the comple-mentary slackness conditions. The violation of com-plementary slackness is the exposure problem associ-ated with this auction scheme. To see why, notice thata violation of complementary slackness means∑

i∈Maij.i >cj and xj=1�

Hence the sum of prices exceeds the value of the bun-dle that the agent receives. Notice that any auctionscheme that relies on prices for individual items alonewill face this problem.

In contrast to the SAA outlined above is the adap-tive user selection mechanism (AUSM) proposed byBanks et al. (1989). AUSM is asynchronous in thatbids on subsets can be submitted at any time andso is difficult to connect to the Lagrangean ideas just



described. An important feature of AUSM is an arenathat allows bidders to aggregate bids to exploit syner-gies. DeMartini et al. (1999) propose an iterative auc-tion scheme that is a hybrid of the SAA and AUSMand is easier to connect to the Lagrangean frame-work. In this scheme, bidders submit bids on pack-ages rather than on individual items. Like the SAA,bids on packages must be increased by a specifiedamount from one round to the next. This minimumincrement is a function of the bids submitted in theprevious round. In addition, the number of itemson which a bidder may bid in each round is lim-ited by the number of items he bid on in previousrounds. The particular implementation of this schemeadvanced by DeMartini et al. (1999) can also be givena Lagrangean interpretation. They choose the multi-pliers (which can be interpreted as prices on individ-ual items) so as to try to satisfy the complementary-slackness conditions of linear programming. Giventhe bids in each round, they allocate the objects soas to maximize revenue. Then they solve a linearprogram (that is essentially the dual to CAP1) thatfinds a set of prices/multipliers that approximatelysatisfy the complementary slackness conditions asso-ciated with the allocation.

Wurman and Wellman (2000) propose an iterativeauction that allows bids on subsets but uses anony-mous, non-linear prices to “direct” the auction. Bid-ders submit bids on bundles and using these bids,an instance of CAP2 is formulated and solved. Then,another program is solved to impute prices to thebundles allocated that will satisfy a complementary-slackness condition. In the next round, bidders mustsubmit a bid that is at least as large as the imputedprice of the bundles.

Kelly and Steinberg (2000) also propose an iterativescheme for combinatorial auctions. (The descriptionis tailored to the auction for assigning carrier of lastresort rights in telecommunications.) The auction hastwo phases. The first phase is an SAA where bid-ders bid on individual items. In the second phase anAUSM-like mechanism is used. The important differ-ence is that each bidder submits a (temporary) sug-gestion for an assignment of all the items in the auc-tion. Here a temporary assignment is composed of

previous bids of other players, plus new bids of hisown.

In Parkes (1999) an iterative auction, called iBundle,that allows bidders to bid on combinations of itemsand uses non-linear prices, is proposed. Bidders sub-mit bids for subsets of items. At each iteration theauctioneer announces prices for those subsets of itemsthat receive unsuccessful bids from agents. For a bidon a subset to be “legal” it must exceed the priceposted by the auctioneer. Given the bids, the auction-eer solves an instance of CAP1 and tentatively assignsthe objects. For the next iteration, the prices on eachsubset are either kept the same or adjusted upwards.The upward adjustment is determined by the high-est losing bid for the subset in question, plus a user-specified increment. The auction terminates when thebids from one round to the next do not show suffi-cient change. The scheme can be given a Lagrangianinterpretation as well, but the underlying formula-tion is different from CAP1 or CAP2. We discuss theunderlying formulation in Section 4.4.

By relaxing on a subset of the constraints as opposedto all of them we get different relaxations, some ofwhich give upper bounds on Z that are smaller thanZLP . Details can be found in the book by Nemhauserand Wolsey (1988). Needless to say, there have beenmany applications of Lagrangean relaxation to SPP,SPA, and SCP, and hybrids with exact methods havealso been investigated. See Balas and Carrera (1996)and Beasley (1990) for recent examples.

4.3. Column GenerationColumn generation is a technique for solving linear pro-grams with an exceedingly large number of variables.Each variable gives rise to a column in the constraintmatrix, hence the name “column generation.” A naiveimplementation of a simplex-type algorithm for lin-ear programming would require recording and stor-ing every column of the constraint matrix. However,only a small fraction of those columns would evermake it into an optimal basic feasible solution to thelinear program. Further, of those columns not in thecurrent basis, one only cares about the ones whosereduced cost will be of the appropriate sign. Column



generation exploits this observation in the followingway. First an optimal solution is found using a sub-set of the columns/variables. Next, given the dualvariable implied by this preliminary solution, an opti-mization problem is solved to find a non-basic col-umn/variable that has a reduced cost of appropriatesign. The trick is to design an optimization problemto find this non-basic column without listing all non-basic columns.

Here we propose that the column-generation ideacan be implemented in an auction setting. In the firststep the auctioneer chooses an extreme-point solutionto the CAP. It does not matter which one; any onewill do. Note that this initial solution could involvefractional allocations of objects.

This extreme-point solution together with thereduced costs is reported to all bidders. Each bid-der, looking only at how they value the allocation,proposes a column/variable/subset to enter the basis(along with its value to the bidder). The proposedcolumn and its valuation must satisfy the appropri-ate reduced-cost criterion for inclusion in the basis. Ineffect, each bidder is being used as a subroutine toexecute the column-generation step. In the worst case,the bidder’s pricing problem might be NP-complete,but the bidder knows how his valuation is structured.So to the bidder it might be computationally simplerto solve the pricing problem knowing the underly-ing structure than for the auctioneer who does not.Further, this eases the communication requirementsbetween bidder and auctioneer and permits biddersto reveal only as little of their valuation as is neces-sary to determine whether they win anything.

The auctioneer now gathers up the proposedcolumns (along with their valuations) and usingthese columns and the columns from the initial basisonly (and possibly previously generated nonbasiccolumns), solves a linear program to find a revenue-maximizing (possibly fractional) allocation. The newextreme-point solution generated is handed out tothe bidders who are asked to each identify a newcolumn (if any) to be added to the new basis thatmeets the reduced-cost criterion for inclusion. Theprocess is then repeated until an extreme-point solu-tion is identified that no bidder wishes to modify.To avoid cycling, the auctioneer can always imple-

ment one of the standard anti-cycling rules for linearprogramming.

This auction procedure eliminates the need to trans-mit and process long lists of subsets and their bids.Bids and subsets are generated only as needed.Second, the bidders are provided an opportunity tochallenge an allocation, provided they propose analternative that increases the revenue to the seller. Ifthe bids might lead to a nonintegral allocation, thenthis column generation has to be imbedded into abranch-and-cut/price scheme to produce an integersolution. (We thank Dr. Márta Eso for suggesting thislast refinement. See Eso 1999 for an example of such abranch-and-price scheme. For an approach by branchand price to the FCC Auction #31 see Dietrich andForrest 2002.)

Notice that the ellipsoid method provides a way tosolve the fractional CAP to optimality in polynomialtime while generating only a polynomially boundednumber of columns (provided that the pricing prob-lem is solvable in polynomial time). So if the frac-tional CAP turns out to be integral, CAP itself can besolved in polynomial time. On the other hand, Nisanand Segal (2002) showed, that even for submodularvaluations the computation of the efficient outcomerequires exponential communication.

For a nontrivial efficiently solvable example, con-sider the edges of a tree. Every bidder is promised acertain edge that no one except for himself can win.The bidder has a positive value for this edge and isinterested in buying a subtree of the tree that containshis earmarked edge. A bidder values every subtree bythe sum of his (private) edge values. As the differentbids of the bidder contain the earmarked edge, for-mulation CAP2 suffices to capture the problem. FromSection 3.4.3 the resulting constraint matrix is perfectand so the LP is integral. However, it can have anexponential number of columns. But by starting witha few columns, and then using the bidders as pric-ing oracles, these instances of CAP2 can be solved inpolynomial time. Further, the bidder’s pricing prob-lem is just a maximum-spanning-tree problem.

4.4. Cuts, Extended Formulations,and Nonlinear Prices

The decentralized methods described above work byconveying “price” information to the bidders. Given



a set of bids and an allocation, prices for individualitems that “support” or are “consistent” with the bidsand allocations are derived and communicated to thebidders. Such prices, because they are linear, cannothope to capture fully the interactions between the par-ties. Here we show, with an example, how cutting-plane methods can be used to generate prices thatmore closely reflect the interactions between bids ondifferent sets of objects.

In the example we have six objects with highestbids on various subsets of objects shown below; sub-sets with bids of zero are not shown:

b��1�2�� = b��2�3��=b��3�4��=b��4�5��= b��1�5�6��=2� b��6��=1�

Formulation CAP2 for this example (ignoring theintegrality constraints) is:

max 2x12 + 2x23 + 2x34 + 2x45 + 2x156 + x6

s.t. x12 + x156 ≤1x12 + x23 ≤1

x23 + x34 ≤1x34 + x45 ≤1

x45 + x156 ≤1x156 + x6 ≤1

x12� x23� x34� x45� x156� x6 ≥0

The optimal fractional solution is to set all variablesequal to 1/2� The optimal dual variables are yi=1/2for i=1��5 and y6=1. So, for example, the imputedprice of the set �1�2� is y1+y2=1.

Consider now the inequality

x12+x23+x34+x45+x156≤2�

Every feasible integer solution to the formulationabove satisfies this inequality, but not all fractionalsolutions do. In particular the optimal fractionalsolution above does not satisfy this inequality. Thisinequality is an example of a cut. Classes of cuts forthe SPP are known, the one above belongs to the classof odd-cycle cuts.

Now append this cut to our original formulation:

max 2x12 + 2x23 + 2x34 + 2x45 + 2x156 + x6

s.t. x12 + x156 ≤1x12 + x23 ≤1

x23 + x34 ≤1x34 + x45 ≤1

x45 + x156 ≤1x156 + x6 ≤1

x12 + x23 + x34 + x45 + x156 ≤2x12� x23� x34� x45� x156� x6 ≥0

The optimal solution to this linear program is integral.It is x12=1�x34=1 and x6=1. There are now sevendual variables. One for each of the six objects (yi) andone more for the cut (2). One optimal dual solution isy1=y5=y6=0� y2=y3=y4=1 and 2=1. The imputedprice for the set �1�2� is now y1+y2+2=2. In generalthe price of a set S will be the sum of the item prices,∑

i∈S yi� plus 2 if the “x” variable associated with theset S appears with coefficient 1 in the cut. Notice thatpricing sets of objects in this way means that the pricefunction will be superadditive.

It is instructive to compare the imputed price ofthe set �1�2� in the two formulations. The first formu-lation assigns a price of one to the set. The seconda higher price. The first formulation ignores the factthat if the set �1�2� is assigned to a bidder, the sets�1�5�6� and �2�3� cannot be assigned to anyone else.This fact is captured by the cut. The dual variableassociated with the cut can be interpreted as the asso-ciated opportunity cost of assigning the set �1�2� toa bidder. Thus the actual price of the set �1�2� is thesum of the prices of the objects in it, plus the oppor-tunity cost associated with its sale.

Cuts can be derived in one of two ways. The first isby purely combinatorial reasoning (see Padberg 1973,1975, 1979, Cornuejols and Sassano 1989 and Sassano1989) and the other through an algebraic techniqueintroduced by Ralph Gomory (see Nemhauser andWolsey 1988 for the details). For CAP1 or CAP2, givena fractional extreme point, one can use the Gomorymethod to generate a cut involving only the variablesthat are basic in the current extreme point. This, isuseful for computational purposes as one does nothave to lug all variables around to identify a cut. Sec-ond, the new inequality will be a non-negative linear



combination of the current basic rows; so all coeffi-cients of the new inequality and its right hand sideare non-negative. Thus, the dual variable associatedwith this new constraint will have an additive effecton the prices of various subsets as in the example.

The reader will notice that by picking an extreme-point dual solution, the imputed prices for some setsare zero. Since there is some flexibility in the choiceof dual variables, one can choose an interior (to thefeasible region) dual solution. Other choices are pos-sible. Incentive considerations suggest choosing theone that minimizes the prices bidders pay (see, forexample, Bikhchandani and Ostroy 2001).

Yet another way to get nonlinear prices is by start-ing with a stronger formulation of the underlyingoptimization problem. One formulation is strongerthan another if its set of feasible (fractional) solu-tions is strictly contained in the other. In the exampleabove, the second formulation is stronger than thefirst. Both formulations share the same set of integersolutions, but not fractional solutions. The set of frac-tional solutions to the second formulation is a strictsubset of the fractional solutions to the first one.

Stronger formulations can be obtained, as shownabove, by the addition of inequalities. Yet anotherstandard way of obtaining stronger formulations isthrough the use of additional or auxiliary variables,typically a large number of them. Geometrically, oneis treating the problem formulated in the original setof variables as the projection of a higher-dimensionalbut structurally simpler polyhedron. Formulationsinvolving such additional variables are called extendedformulations and developing these extended formula-tions is called lifting. Using lifting, one can develop ahierarchy of successively stronger formulations of theunderlying integer program.

There is a close connection between lifting, ex-tended formulations, and cutting planes. Perhaps themost accessible introduction to these matters is Balaset al. (1993).

In the auction context, Bikhchandani and Ostroy(2001), propose a number of extended formulationsfor the problem of selecting the winning set of bids. Todescribe the first of their extended formulations, let 3be the set of all possible partitions of the objects in theset M . If 4 is an element of 3, we write S∈4 to mean

that the set S⊆M is a part of the partition 4. Let z4=1if the partition 4 is selected and zero otherwise. Theseare the auxiliary variables. Using them Bikhchandaniand Ostroy (2001) can reformulate CAP1 as follows:

max∑j∈N

∑S⊆M

bj�S�y�S�j�

s.t.∑S⊆M

y�S�j�≤1 ∀j∈N∑j∈Ny�S�j�≤∑

4�Sz4 ∀S⊆M

∑4∈3

z4≤1

y�S�j�=0�1 ∀S⊆M�j∈Nz4=0�1 ∀4∈3

Call this formulation CAP3. In words, CAP3 choosesa partition of M and then assigns the sets of the parti-tion to bidders in such a way as to maximize revenue.It is easy to see that this formulation is stronger thanCAP1 or CAP2. Fix an i∈M and add over all S� i theinequalities

∑j∈Ny�S�j�≤∑

4�Sz4 ∀S⊆M

to obtain ∑S�i

∑j∈Ny�S�j�≤1 ∀i∈M�

which are the inequalities that appear in CAP1. Whilestronger than CAP1, formulation CAP3 still admitsfractional extreme points (Bikhchandani and Ostroy2001).

The dual of the linear relaxation of CAP3 involvesone variable for every constraint of the form

∑S⊆M

y�S�j�≤1 ∀j∈N�

call it sj , which can be interpreted as the surplus thatbidder j obtains. The dual involves one variable forevery constraint of the form

∑j∈Ny�S�j�≤∑

4�Sz4 ∀S⊆M�



which we will denote pS . It can be interpreted as theprice of the subset S. In fact the dual will be

min∑j∈Nsj+2

s.t. sj≥bj�S�−pS ∀j∈N S⊆M2≥∑

S∈4pS ∀4∈3

sj�pS�2≥0

and has the obvious interpretation: minimizing thebidders surplus plus 2. Thus one can obtain nonlinearprices from the extended formulation. These pricesdo not support the optimal allocation since CAP3 isnot integral. Further, they do not depend on the bid-ders; that is, all bidders pay the same price for a givensubset. The catch, of course, is that this formulationinvolves many more variables than CAP1 or CAP2.

In Parkes and Ungar (2000) a condition on bidders’preferences is identified that ensures that the linearrelaxation of CAP3 has an integral solution. The con-dition, called bid safety, is difficult to interpret, buthas the effect of forcing complementary slackness tohold for an integer solution of CAP3. Under this con-dition any algorithm for solving CAP3’s dual (or itsLagrangean relaxation) will generate an optimal solu-tion of CAP3 itself. Since many dual algorithms canbe given an auction interpretation with the iterationsbeing identified as adjustments in bids that a myopicbest-reply agent might execute, one can generate auc-tion schemes that are arguably optimal. This is pre-cisely the tack taken in Parkes and Ungar (2000) tosupport the adoption of the iBundle auction schemeof Parkes (1999).

Bikhchandani and Ostroy (2001) introduce yetanother formulation stronger than CAP3 that is inte-gral. The idea is to use a variable that representsboth a partition of the objects and an allocation, essen-tially one variable for every solution. The dual to thisformulation gives rise to nonlinear prices with thetwist that they are bidder-specific. Different bidderspay different prices for the same subset. Bikhchandaniet al. (2002) propose and investigate another extendedintegral formulation with significantly fewer variablesthan the one in Bikhchandani and Ostroy (2001).

5. Incentive IssuesThus far we have focused on the problem of choos-ing an allocation of the objects so as to maximize theseller’s revenue. The revenue depends on the bidssubmitted but there is no guarantee that the submit-ted bids approximate the actual values that biddersassign to the various subsets. To illustrate how thiscan happen consider three bidders, 1, 2 and 3, andtwo objects �x�y�. Suppose:

v1�x�y� = 100� v1�x�=v1�y�=0�

v2�x� = v2�y�=75� v2�x�y�=0�

v3�x� = v3�y�=40� v3�x�y�=0�

Here vi�·� represents the value to bidder i of a partic-ular subset. Notice that the bid that i submits on theset S, bi�S�� need not equal vi�S�.

If the bidders bid truthfully, the auctioneer shouldaward x to 2 and y to 3, say, to maximize his rev-enue. Notice however that bidder 2 say, under theassumption that bidder 3 continues to bid truthfully,has an incentive to shade his bid down on x and yto, say, 65. Notice that bidders 2 and 3 still win butbidder 2 pays less. This argument applies to bidder 3as well. However, if they both shade their bids down-wards they can end up losing the auction. This fea-ture of combinatorial auctions is called the thresholdproblem (see Bykowsky et al. 2000): a collection of bid-ders whose combined valuation for distinct portionsof a subset of items exceeds the bid submitted onthat subset by some other bidder. It may be difficultfor them to coordinate their bids to outbid the largebidder on that subset. The basic problem is that thebidders 2 and 3 must decide how to divide 75+40−100 between them. Every split can be rationalized asthe equilibrium of an appropriate bargaining game.In linear-programming terms, the threshold problemarises because of a multiplicity of optimal dual solu-tions.

In this section we describe what is known aboutauction mechanisms that give bidders the incentiveto reveal their valuations truthfully. (For more detailson game theory, equilibria, and mechanism design seeFudenberg and Tirole 1992.)

To discuss incentive issues we need a model ofbidders preferences. The simplest conceptual model



endows bidder j∈N with a list �vj�S��S⊆M withvj��=0, abbreviated to vj , that specifies how shevalues (monetarily) each subset of objects. Thus vj�S�represents how much bidder j values the subset S ofobjects. (In the language of mechanism design, thislist of valuations becomes the bidders type. In thiscase, since the type is not a single number it is calledmulti-dimensional. For an introduction to mechanismdesign see Chapter 7 of Fudenberg and Tirole 1992.)

The auction scheme chosen and the bids submittedwill be a function of the beliefs that seller and biddershave about each other. The simplest model of beliefsis the independent-private-values model and it is themodel to which we will restrict ourselves. Each bid-der’s vj is assumed by the seller and all bidders to bean independent draw from a commonly known distri-bution over a compact, convex set. Bidder j knows hervj but not the valuations of the other bidders. Last,bidders and seller are assumed to be risk neutral.

To continue the discussion, it will be useful to dis-tinguish between two popular objectives the auction-eer may have. The first is economic efficiency and thesecond is revenue maximization.

5.1. Economic EfficiencyAn auction is economically efficient if the allocation ofobjects to bidders chosen by the seller solves the fol-lowing:

max∑j∈N

∑S⊆M

vj�S�y�S�j�

s.t.∑S�i

∑j∈Ny�S�j�≤1 ∀i∈M

∑S⊆M

y�S�j�≤1 ∀j∈N

y�S�j�=0�1 ∀S⊆M�j∈NNotice that this is just CAP1 with bj replaced byvj . The optimal objective-function value of this inte-ger program is an upper bound on the revenuethat the seller can achieve if no bidder bids abovetheir valuation. The fact that the seller uses an auc-tion that selects an allocation that solves this inte-ger program does not imply that the seller achievesthis revenue. (In Myerson 1981 it is shown that therevenue-maximizing auction for a single good is not

guaranteed to be efficient. See Jehiel and Moldovanu2001c for a more pronounced version of the same.)

An auctioneer interested in producing an efficientallocation has a puzzle. Since bidders’ valuations areprivate information, he must solve the optimizationproblem above without a knowledge of the objec-tive function! Remarkably, there is a sealed-bid auc-tion that implements the efficient outcome. It doesso because it is a weakly dominant strategy for bid-ders to bid truthfully in the auction. The most generalclass of such auctions was characterized by Clarke(1971) and Groves (1973). A special case was identi-fied earlier by William Vickrey (1961) in an auctionthat bears his name. The version we describe hereis sometimes known as Vickrey-Clarke-Groves (VCG)scheme. It is proved in Krishna and Perry (1998) (seealso Williams 1999 for the same result under slightlydifferent assumptions) that in the independent privatevalues model, amongst all auctions that implementthe efficient allocation, the VCG scheme maximizesthe revenue to the seller.

It works as follows:1. Agent j reports vj . There is nothing to prevent

agent j from misrepresenting themselves. However,given the rules of the auction, it is a weakly dominantstrategy to bid truthfully.

2. The seller chooses the allocation that solves:

V = max∑j∈N

∑S⊆M


s.t.∑S�i

∑j∈Ny�S�j�≤1 ∀i∈M

∑S⊆M

y�S�j�≤1 ∀j∈N

y�S�j�=0�1 ∀S⊆M�j∈NCall this optimal allocation y∗

3. To compute the payment that each bidder mustmake let, for each k∈N ,

V−k= max∑j∈N\k

∑S⊆M


s.t.∑S�i

∑j∈N\k

y�S�j�≤1 ∀i∈M∑S⊆M

y�S�j�≤1 ∀j∈N \k

y�S�j�=0�1 ∀S⊆M�j∈N \k



Denote by yk the optimal solution to this integer pro-gram. Thus yk is the efficient allocation when bidderk is excluded.

4. The payment that bidder k makes is equal to

V−k−[V− ∑

S⊆Mvk�S�y∗�S�k�

]�

Thus bidder k’s payment is the difference in “welfare”of the other bidders without him and the welfare ofothers when he is included in the allocation. Noticethat the payment made by each bidder to the auction-eer is non-negative.If a seller were to adopt the VCG scheme her totalrevenue would be

∑k∈NV−k−∑

k∈N

[V− ∑

S⊆Mvk�S�y∗�S�k�

]

=∑k∈N

∑S⊆N

vk�S�y∗�S�k�+∑k∈N�V−k−V�

=V+∑k∈N�V−k−V��

If there were a large number of agents then no sin-gle agent can have a significant effect, i.e., one wouldexpect that, on average, V is very close in value toV−k. Thus the revenue to the seller would be closeto V , the largest possible revenue that any auctioncould extract. To solidify this intuition we need thatfor all agents k that their valuation vk is superad-ditive, i.e. vk�A�+vk�B�≤vk�A∪B� for all k∈N andA�B⊆M such that A∩B=�. With this assumption wecan find the efficient allocation using CAP2. Thus:

V = max∑S⊆M

{maxj∈N

vj�S�}xS

s.t.∑S�ixS≤1 ∀i∈M

xS=0�1 ∀S⊆M

and

V−k= max∑S⊆M

{maxj∈N\k

vj�S�}xS

s.t.∑S�ixS≤1 ∀i∈M

xS=0�1 ∀S⊆M

Notice now that if the number �N � of bidders is largeand given that the vj ’s live in a compact set, the ran-dom variable maxj∈N vj�S� is very close on averageto maxj∈N\kvj�S�. (In fact, the difference of the two isessentially the difference between the first and secondorder statistic of a large collection of independent ran-dom numbers from a compact set.) Hence the objec-tive function of the program that defines V is essen-tially the same as the objective function of the integerprogram that defines V−k. This argument is made pre-cise in Monderer and Tennenholtz (2000), where it isshown in the model used here that the VCG schemegenerates a revenue for the seller that is asymptoti-cally close to the revenue from the optimal auction.

The VCG scheme is, in general, impractical toimplement, if the number of bidders is very large.However in some circumstances this difficulty canbe avoided. Hershberger and Suri (2001), for exam-ple, show that in the network-routing context at mosttwo optimization problems must be solved to com-pute Vickrey payments. Bikhchandani et al. (2002)show that, in a wide range of situations, the prob-lem of finding the efficient allocation can be for-mulated as a linear program. What is more, opti-mal dual variables in this linear program coincidewith the Vickrey payments. In these instances, two—and in many instances even only one—optimizationproblems must be solved to compute the Vickreypayments.

Another way to overcome the computational diffi-culties is to replace y∗ and yk for all k∈N with approx-imately optimal solutions. Such a modification in thescheme does not in general preserve incentive com-patibility (see Nisan and Ronen 2000). In Lehmannet al. (1999) such a direction is taken. They solvethe embedded optimization problems using a greedy-type algorithm and show that the resulting schemeis not incentive-compatible. However if one is willingto restrict bidders’ valuations drastically it is possibleto generate schemes based on the greedy algorithmthat are incentive-compatible. Lehmann et al. (1999)call this restriction “single mindedness.” Each biddervalues only one subset and no other.

Even if one is willing to relax incentive compat-ibility, an approximate solution to the underlyingoptimization problems in the VCG can lead to other



problems. There can be many different solutions toan optimization problem whose objective functionvalues are within a specified tolerance of the opti-mal objective function value. The payments specifiedby the VCG scheme are very sensitive to the choiceof solution. Thus the choice of approximate solu-tion can have a significant impact on the paymentsmade by bidders. This issue is discussed by Johnsonet al. (1997) in the context of an electricity auctionused to decide the scheduling of short-term electricityneeds. Through simulations they show that variationsin near-optimal schedules that have negligible effecton total system cost can have significant consequenceson the total payments by bidders.

The VCG scheme as described is a single-roundsealed-bid auction. In simple settings the VCG auc-tion has an iterative counterpart that implements theefficient outcome in the sense that it is a Nash equi-librium for bidders to bid truthfully in each round.Such iterative auctions reduce the cognitive burdenon bidders to list their valuations for all bundles.Also, they reduce the amount of information thatbidders must reveal to the auctioneer. An exampleof such can be found in Ausubel (2000) where anascending auction for indivisible heterogenous objectsunder the gross substitutes on preferences assump-tion is proposed that implements the efficient out-come. Bikhchandani and Ostroy (2001) and Bikhchan-dani et al. (2002) show that in many environments theproblem of finding the efficient allocation can be for-mulated as a linear program in such a way that dualvariables correspond to Vickrey payments. (In fact,the environments when this is possible are character-ized in terms of bidders preferences.) The primal-dualalgorithm for these linear programs produce an itera-tive auction that implements the outcome of the VCGauction. That is, bidding truthfully in each round is aNash equilibrium.

Experience with the VCG scheme in field settingsis limited. Isaac and James (1998) report on an experi-ment using the VCG scheme for a combinatorial auc-tion involving three bidders and two objects. On thebasis of their results they argue that the VCG schemecan be operationalized and, in their words, “achievehigh allocative efficiency.” Kagel and Levin (2001)

experiment with iterative auctions for the sale of mul-tiple units of homogeneous goods. They compare theuniform price auction with an iterative auction (dueto Ausubel 1997) that implements the VCG outcome.In their experiments Ausubel’s auction results in out-comes close to the efficient one. Hobbs et al. (2000)explore the possibility that the VCG scheme is vulner-able to collusion. It is pointed out by these authors inenvironments with repeated interactions that not onlyare there many opportunities for collusion among bid-ders but incentive compatibility of the VCG schemecannot be guaranteed. However, this is a weaknessnot unique to VCG.

The efficient auction when bidders values are inter-dependent is more difficult. Jehiel et al. (1999) andJehiel and Moldovanu (2001a) discuss the underlyingmathematical problems. Eso and Maskin (2000) intro-duce a restriction on preferences that they call parti-tion preferences. The collection of subsets to which abidder assigns a positive value form a partition of theset of objects. Note that problem CAP is still NP-hardunder this restriction. For this restricted case theyderive results about the form of the efficient auction.

5.2. Revenue MaximizationThe problem of designing an auction that maxi-mizes the auctioneer’s revenue (optimal auction) ismore difficult. Our discussion of this problem willbe restricted to the case of independent private val-ues. For simplicity we will suppose that each bidderassumes that the other bidders’ value functions areindependent draws from a finite set V of value func-tions with commonly known probability distributionp. The probability that agent j has value function u

will be denoted p�u�. The probability that the n-tuplev= �v1��vn�∈Vn is realized is denoted

∏nj=1p�v

j�.For convenience this will be abbreviated to p�v�.

One can imagine a variety of elaborate andinvolved auction protocols that must be considered.However, the revelation principle (see Myerson 1981)allows one, without loss of generality, to restrict atten-tion to a direct-revelation mechanism that satisfies twoconditions.

In a direct-revelation mechanism, the auctioneerannounces how he will allocate the objects amongstthe bidders and the payments he will extract from



each as a function of the announced value functions.Then bidders are asked to announce their value func-tions only.

The two conditions the mechanism must satisfy arecalled incentive compatibility and individual rationality.The first requires that the expected payoff to a bidderfrom truthfully reporting his value function shouldexceed his expected payoff from misreporting it. Thesecond requires that the expected payoff from truthfulreporting should be at least zero; otherwise, a bidderwill not participate in the auction.

An allocation rule is a mapping from an n-tuple ofvalue functions to an integer solution to CAP1. Thus,an allocation rule specifies how the objects are to bedivided up among the bidders as a function of theirreported value functions. If A is an allocation ruleand v= �v1�v2��vn� an n-tuple of value functions,we will write A�v� to mean the allocation selected bythe rule A.

A payment rule, P, is a mapping from an n-tupleof value functions to an n-tuple of payments, onefor each bidder. If bidder j reports the value func-tion v and the other bidders report v−j, we denoteby Pj�v�v−j� the payment that bidder j makes. Thedependence on j means that we allow payments to bea function of both the reports as well as the identityof the bidder.

With these variables one can formulate the problemof finding a revenue-maximizing auction as a math-ematical program. (This is just a specialization of theusual formulation for optimal auctions with multi-dimensional types. The survey paper by Rochet andStole 2001 has more details on this subject. No closed-form solution to the problem of optimal auctions withmultidimensional types is known and it is unlikelythat any exists.)

There is one decision variable that represents thechoice of allocation rule and another to represent thechoice of payment rule. The objective is

maxA�P

∑v∈Vn

p�v�

[∑j∈NPj�v�

]�

Incentive compatibility requires that for each j∈Nwith value function v and all u �=v:∑

v−j∈Vn−1

�v�A�v�v−j��−Pj�v�v−j��p�v−j�

≥ ∑v−j∈Vn−1

�v�A�u�v−j��−Pj�u�v−j��p�v−j��

That is, the expected utility from truthfully reportingone’s value function should exceed the expected util-ity from reporting some other value function. Givenincentive compatibility we can write the individualrationality constraint as∑

v−j∈Vn−1

�v�A�v�v−j��−Pj�v�v−j��p�v−j�≥0

for all j∈N and value functions v∈V .How might one solve this? First fix a choice for

allocation rule, A. Let

wjA�u�v� =

∑v−j∈Vn−1

v�A�u�v−j��p�v−j�

and8j�u� = ∑

v−j∈Vn−1

Pj�u�v−j�p�v−j��

Thus wjA�u�v� can be interpreted as the expected util-

ity agent j receives under allocation rule A when hereports u given his actual value function is v. Simi-larly, 8j�u� is the expected payment agent j must makeunder allocation rule A when he reports u as his valuefunction. Since the probability that agent j has valuefunction v is independent of j, we can, in the defini-tions of wA and 8 above, suppress the dependence onthe index j. Specifically, the expected utility an agentreceives and expected payment he makes dependsonly on his reported valuation rather than his iden-tity. Thus, for all j�j ′ ∈N we have wj

A�u�v�=wj ′A�u�v�

and 8j�u�=8j ′�u�.Then we can rewrite the incentive compatibility and

individual rationality constraints as follows:

8�v�−8�u� ≤ wA�v�v�−wA�u�v��and

8�v� ≤ wA�v�v��The objective function can be written as:

∑v∈Vn

p�v�

[∑j∈NPj�v�

]=�N �∑

v∈Vp�v�8�v��



Thus the optimization problem becomes:

max �N �∑v∈Vp�v�8�v�

s.t. 8�v�−8�u�≤wA�v�v�−wA�u�v�∀v�u∈V� ∀j∈N

8�v�≤wA�v�v� ∀v∈V� ∀j∈N�

Since the allocation rule A is fixed, the only vari-ables are the 8’s. There are at most two of themper constraint with coefficients of +1 and −1. Hence,given A, the problem of finding a payment rule thatenforces incentive compatibility and individual ratio-nality is a network-flow problem (or, more precisely,the dual to one). (Introduce one vertex for each valuefunction in V . For each ordered pair �v�u�, introducean arc directed from v to u with length wA�v�v�−w�u�v�. Each individual rationality constraint alsointroduces a source node. The dual problem is to findthe shortest-path tree, one for each agent, rooted atthe source node corresponding to that agent.) For eachpossible choice of allocation rule A, we can deter-mine via the duality theorem of linear programmingwhether a payment rule that is incentive-compatibleexists.

To determine the optimal auction one has to solvethis constrained optimization problem for each possi-ble allocation rule. Then pick the allocation rule thatyields the largest revenue. Thus the design of the opti-mal auction may be extremely sensitive to both thechoice of bidders’ value function, as well as distribu-tion of type. Under fairly stringent conditions someresults have been derived as well as comparisons ofrevenue between auctions of different kind. Rochetand Stole (2001) summarize these results.

Levin (1997) identifies the optimal auction undera more restrictive setting. Specifically, all objects,for all bidders, complement each other and biddersare perfectly symmetric. In this case, the revenue-maximizing auction is simply to bundle all the objectstogether and auction the bundle off using an optimalsingle-item auction.

Krishna and Rosenthal (1996), again with a simpli-fied model of preferences, attempt to make revenue

comparisons between different auction schemes. Theyconsider auctions involving two items where biddersare of two kinds. One kind, that they call local, areinterested in receiving a single item. The other, calledglobal, have valuations for the entire set that are super-additive. In this paper they identify equilibria for atwo-item sequential second-price auction, (a particu-lar) simultaneous second-price auction, and a combi-natorial second-price auction. The revenues obtainedfrom each auction (under the identified equilibria) arecompared numerically. They observe that if the syn-ergy is small then the sequential second-price auctiongenerates the largest revenue. For larger synergiesthe simultaneous second-price auction generates thelargest revenue. In their simulations the combinato-rial second-price auction always generates the small-est revenue. It is not known whether this relation is afunction of the equilibrium found or the specificationof the simultaneous auction.

Cybernomics, Inc. (2000) went a step further bycomparing a particular simultaneous multi-roundauction with a particular multi-round combinatorialauction. They performed experiments for additivevalues, and for valuations with synergies of small,medium, or high intensity. Their experiments indi-cate that the combinatorial multi-round auction wasalways superior with respect to efficiency, but the rev-enue was smaller, and it took more rounds of longerduration to finish the auction.

6. SummaryOur survey has had three goals. The first, a pedes-trian one, has been to survey the extant literature. Thesecond, has been to point out “classical” results thatapply directly to the problem of designing combina-torial auctions. The third has been to emphasize theconnections between the duality theory of optimiza-tion problems and the design of auctions.

AcknowledgmentsWe thank Christopher Caplice, Sunil Chopra, Márta Eso, YoshiroIkura, Charles Lee Jackson, John O. Ledyard, David Parkes, MichaelRothkopf, Richard Steinberg, W. David Kelton, and anonymous ref-erees for many useful comments and suggestions.



ReferencesAkcoglu, K., J. Aspnes, B. DasGupta, M. Kao. 2002. An

opportunity-cost algorithm for combinatorial auctions. E. J.Kontoghiorghes, B. Rustem, S. Siokos, eds. Applied Optimiza-tion: Computational Methods in Decision-Making, Economics andFinance. Kluwer, Dordrecht, The Netherlands.

Andersson, A., M. Tenhunen, F. Ygge. 2000. Integer programmingfor combinatorial auction winner determination. Proceedingsof the Fourth International Conference on Multi-Agent Systems(ICMAS00), Boston, MA, 39–46.

Ausubel, L. M. 1997. An efficient ascending-bid auction for multipleobjects. Working Paper No. 97-06, Department of Economics,University of Maryland, College Park, MD.

Ausubel, L. M. 2000. An efficient dynamic auction for heteroge-neous commodities. Working paper, Department of Economics,University of Maryland, College Park, MD.

Ausubel, L. M., P. Cramton. 1998. Demand reduction and ineffi-ciency in multi-unit auctions. Manuscript, Department of Eco-nomics, University of Maryland, College Park, MD.

Balas, E., M. C. Carrera. 1996. A dynamic subgradient-basedbranch-and-bound procedure for set covering. Oper. Res. 44875–890.

Balas, E., S. Ceria, G. Cornuéjols. 1993. A lift-and-project cuttingplane algorithm for mixed 0-1 programs. Math. Programming58 295–324.

Balas, E., M. W. Padberg. 1976. Set partitioning: a survey. SIAM Rev.18 710–760.

Ball, M. O., M. Magazine. 1981. The design and analysis of heuris-tics. Networks 11 215–219.

Banks, J. S., J. O. Ledyard, D. P. Porter. 1989. Allocating uncertainand unresponsive resources: an experimental approach. RANDJ. of Econom. 20 1–25.

Barnhart, C., E. L. Johnson, G. L. Nemhauser, M. W. P. Savelsbergh,P. H. Vance. 1998. Branch and price: column generation forsolving huge integer programs. Oper. Res. 46 316–329.

Beasley, J. E. 1990. A Lagrangian heuristic for set-covering prob-lems. Naval Res. Logist. 37 151–164.

Bertsekas, D. P. 1991. Linear Network Optimization: Algorithms andCodes. MIT Press, Cambridge, MA.

Bikhchandani, S., C.-f. Huang. 1993. The economics of treasurysecurities markets. J. of Econom. Perspectives 7 117–134.

Bikhchandani, S., S. de Vries, J. Schummer, R. V. Vohra. 2002. Linearprogramming and Vickrey auctions. B. Dietrich, R. V. Vohra,eds. Mathematics of the Internet: E-Auction and Markets. The IMAVolumes in Mathematics and Its Applications, Vol. 127. Springer-Verlag, New York, 75–116.

Bikhchandani, S., J. W. Mamer. 1997. Competitive equilibrium in anexchange economy with indivisibilities. J. of Econom. Theory 74385–413.

Bikhchandani, S., J. M. Ostroy. 2001. The package assignmentmodel. Manuscript, The Anderson School, UCLA, Los Angeles,CA.

Binmore, K., P. Klemperer. 2002. The biggest auction ever: the saleof the British 3G telecom licences. Econom. J. 112 C74–C96.

Boppana, R. B., M. M. Halldórsson. 1992. Approximating maximumindependent sets by excluding subgraphs. BIT 32 180–196.

Brewer, P. J. 1999. Decentralized computation procurement andcomputational robustness in a smart market. Econom. Theory 1341–92.

Bykowsky, M. M., R. J. Cull, J. O. Ledyard. 2000. Mutually destruc-tive bidding: the FCC auction design problem. J. of RegulatoryEconom. 17 205–228.

Caplice, C. G. 1996. An optimization based bidding process:A new framework for shipper-carrier relationships. Thesis,Department of Civil and Environmental Engineering, Mas-sachusetts Institute of Technology, Cambridge, MA.

Clarke, E. 1971. Multipart pricing of public goods. Public Choice 819–33.

Cornuéjols, G., A. Sassano. 1989. On the 0-1 facets of the set cover-ing polytope. Math. Programming 43 45–56.

Cramton, P. 2002. Spectrum auctions. In M. Cave, S. Majumdar,I. Vogelsang, eds. Handbook of Telecommunications Economics.Elsevier Science B.V., Amsterdam, The Netherlands.

Crescenzi, P., V. Kann. 1995. A compendium of NP optimiza-tion problems. Technical report SI/RR-95/02, Dipartimento diScienze dell’Informazione, Università di Roma “La Sapienza,”Italy. http://www.nada.kth.se/theory/compendium/. Contin-uously updated.

Cybernomics, Inc. 2000. An experimental comparison of the simul-taneous multi-round auction and the CRA combinatorial auc-tion. http://combin.fcc.gov/98540193.pdf.

Dantzig, G. B. 1963. Linear Programming and Extensions. PrincetonUniversity Press, Princeton, NJ.

Davenport, A., J. Kalagnanam. 2002. Price negotiations for procure-ment of direct inputs. In B. Dietrich, R. V. Vohra, eds. Mathe-matics of the Internet: E-Auction and Markets. The IMA Volumesin Mathematics and Its Applications, Vol. 127. Springer-Verlag,New York, 27–44.

DeMartini, C., A. Kwasnica, J. O. Ledyard, D. Porter. 1999. A newand improved design for multi-object iterative auctions. SocialScience Working Paper No. 1054, Division of the Humani-ties and Social Sciences, California Institute of Technology,Pasadena, CA.

de Vries, S., R. V. Vohra. 2001a. Auctions and the German UMTS-auction. Mitteilungen der DMV No. 1, 31–38.

de Vries, S., R. V. Vohra. 2001b. A solver for winner determinationfor combinatorial auctions. Manuscript, Zentrum Mathematik,TU München, München, Germany.

Dietrich, B., J. J. Forrest. 2002. A column generation approach forcombinatorial auctions. B. Dietrich, R. V. Vohra, eds. Mathe-matics of the Internet: E-Auction and Markets. The IMA Volumesin Mathematics and Its Applications, Vol. 127. Springer-Verlag,New York, 15–26.

Eso, M. 1999. Parallel branch and cut for set partitioning. Ph.D. the-sis, School of Operations Research and Industrial Engineering,Cornell University, Ithaca, NY.

Eso, P., E. Maskin. 2000. Multi-good efficient auctions with multi-dimensional information. Working paper, Kellogg School ofManagement, Northwestern University, Evanston, IL.



Fudenberg, D., J. Tirole. 1992. Game Theory. MIT Press, Cambridge,MA.

Fujishima, Y., K. Leyton-Brown, Y. Shoham. 1999. Taming the com-putational complexity of combinatorial auctions: optimal andapproximate approaches. Proc. of IJCAI-99, Stockholm, Sweden,548–553.

Gilmore, P. C., R. E. Gomory. 1961. A linear programming approachto the cutting stock problem. Oper. Res. 9 849–859.

Gonen, R., D. Lehmann. 2000. Optimal solutions for multi-unitcombinatorial auctions: branch-and-bound heuristics. ACMConf. on Electronic Commerce (EC-00), Minneapolis, MN, 13–20.

Graves, R., J. Sankaran, L. Schrage. 1993. An auction method forcourse registration. Interfaces 23 81–92.

Grimm, V., F. Riedl, E. Wolfstetter. 2001. The third generation(UMTS) spectrum auction in Germany. Manuscript, Instituteof Economic Theory I, Humboldt University of Berlin, Berlin,Germany.

Grötschel, M., L. Lovász, A. Schrijver. 1988. Geometric Algorithms andCombinatorial Optimization. Springer-Verlag, Berlin Heidelberg,Germany.

Groves, T. 1973. Incentives in teams. Econometrica 41 617–631.Golumbic, M. C., P. L. Hammer. 1988. Stability in circular arc

graphs. J. Algorithms 9 314–320.Gul, F., E. Stacchetti. 2000. English and double auctions with differ-

entiated commodities. J. of Econom. Theory 92 66–95.Håstad, J. 1999. Clique is hard to approximate within n1−& . Acta

Mathematica 182 105–142.Hershberger, J., S. Suri. 2001. Vickrey prices and shortest paths:

What is an edge worth? 42nd Annual Sympos. on Foundations ofComput. Sci. (FOCS), Las Vegas, NV, 129–140.

Hobbs, B. F., M. Rothkopf, L. Hyde, R. P. O’Neill. 2000. Evaluationof a truthful revelation auction in the context of energy marketswith nonconcave benefits. J. of Regulatory Econom. 18 5–32.

Hoffman, K., M. W. Padberg. 1993. Solving airline crew schedulingproblems by branch and cut. Management Sci. 39 657–682.

Isaac, R. M., D. James. 1998. Robustness of the incentive compatiblecombinatorial auction. Manuscript, Department of Economics,University of Arizona, Tucson, AZ.

Jackson, C. 1976. Technology for spectrum markets. Ph.D. thesis,Department of Electrical Engineering, Massachusetts Instituteof Technology, Cambridge, MA.

Jehiel, P., B. Moldovanu. 2001a. Efficient design with interdepen-dent valuations. Econometrica 69 1237–1259.

Jehiel, P., B. Moldovanu. 2001b. The European UMTS/IMT-2000license auctions. Discussion paper, Faculty of Economics, Uni-versity of Mannheim, Mannheim, Germany.

Jehiel, P., B. Moldovanu. 2001c. A note on revenue maximizationand efficiency in multi-object auctions. Econom. Bull. 3 1–5.

Jehiel, P., B. Moldovanu, E. Stacchetti. 1999. Multidimensionalmechanism design for auctions with externalities. J. of Econom.Theory 85 258–294.

Johnson, R. B., S. S. Oren, A. J. Svoboda. 1997. Equity and efficiencyof unit commitment in competitive electricity markets. UtilitiesPolicy 6 9–20.

Kagel, J. H., D. Levin. 2001. Behavior in multi-unit demand auc-tions: Experiments with uniform price and dynamic Vickreyauctions. Econometrica 69 413–454.

Keil, J. M. 1992. On the complexity of scheduling tasks with discretestarting times. Oper. Res. Lett. 12 293–295.

Kelso, A. S., V. P. Crawford. 1982. Job matching, coalition formationand gross substitutes. Econometrica 50 1483–1504.

Kelly, F., R. Steinberg. 2000. A combinatorial auction with multiplewinners for universal service. Management Sci. 46 586–596.

Klemperer, P. 2002. What really matters in auction design. J. ofEconom. Perspectives 16 169–190.

Krishna, V., M. Perry. 1998. Efficient mechanism design.Manuscript, Department of Economics, The Pennsylvania StateUniversity, University Park, PA.

Krishna, V., R. W. Rosenthal. 1996. Simultaneous auctions with syn-ergies. Games and Econom. Behavior 17 1–31.

Kutanoglu, E., S. D. Wu. 1999. On combinatorial auction andLagrangean relaxation for distributed resource scheduling. IIETrans. 31 813–826.

Ledyard, J. O., M. Olson, D. Porter, J. A. Swanson, D. P. Torma.2000. The first use of a combined value auction for transporta-tion services. Social Science Working Paper No. 1093, Divisionof the Humanities and Social Sciences, California Institute ofTechnology, Pasadena, CA.

Lehmann, D., L. O’Callaghan, Y. Shoham. 1999. Truth revelationin rapid, approximately efficient combinatorial auctions. Proc.ACM Conf. on Electronic Commerce (EC-99), Denver, CO, 96–102.

Levin, J. 1997. An optimal auction for complements. Games andEconom. Behavior 18 176–192.

Leyton-Brown, K., Y. Shoham, M. Tennenholtz. 2000a. An algorithmfor multi-unit combinatorial auctions. Proc. of the SeventeenthNational Conf. on Artificial Intelligence and Twelfth Conferenceon Innovative Applications of Artificial Intelligence, Austin, TX,56–61.

Leyton-Brown, K., M. Pearson, Y. Shoham. 2000b. Towards a uni-versal test suite for combinatorial auction algorithms. Proc.ACM Conf. on Electronic Commerce (EC-00), Minneapolis, MN,66–76.

Lovász, L., M. D. Plummer. 1986. Matching theory. Annals of Dis-crete Mathematics, 29. North-Holland, Elsevier, Amsterdam, TheNetherlands.

Mas-Collel, A., M. D. Whinston, J. R. Green. 1995. MicroeconomicTheory. Oxford University Press, New York.

Milgrom, P. 1995. Auctioning the radio spectrum. Auction Theoryfor Privatization, Cambridge University Press, Cambridge, Eng-land. Forthcoming.

Monderer, D., M. Tennenholtz. 2000. Asymptotically optimal multi-object auctions for risk-averse agents. Manuscript, Facultyof Industrial Engineering and Management, Technion—IsraelInstitute of Technology, Haifa, Israel.

Murota, K., A. Tamura. 2000. New characterizations of M-convexfunctions and connections to mathematical economics. RIMSPreprint 1307, Research Institute for Mathematical Sciences,Kyoto University, Kyoto, Japan.



Myerson, R. 1981. Optimal auction design. Math. of Oper. Res. 658–73.

Naor, M., B. Pinkas, R. Sumner. 1999. Privacy preserving auctionsand mechanism design. Proc. ACM Conf. on Electronic Commerce(EC-99), Denver, CO, 129–139.

Nemhauser, G. L., L. A. Wolsey. 1988. Integer and Combinatorial Opti-mization. John Wiley, New York.

Nisan, N. 2000. Bidding and allocation in combinatorial auctions.Proc. ACM Conf. on Electronic Commerce (EC-00), Minneapolis,MN, 1–12.

Nisan, N., A. Ronen. 2000. Computationally feasible VCG mecha-nisms. Proc. ACM Conf. on Electronic Commerce (EC-00), Min-neapolis, MN, 242–252.

Nisan, N., I. Segal. 2002. The communications complexity of effi-cient allocation problems. Working paper, Institute of Com-puter Science, Hebrew University, Jerusalem, Israel.

Padberg, M. W. 1973. On the facial structure of set packing polyhe-dra. Math. Programming 5 199–215.

Padberg, M. W. 1975. On the complexity of the set packing polyhe-dra. Ann. of Discrete Math. 1 421–434.

Padberg, M. W. 1979. Covering, packing and knapsack problems.Ann. of Discrete Math. 4 265–287.

Papadimitriou, C. H., K. Steiglitz. 1982. Combinatorial Optimization.Prentice-Hall, Inc., Englewood Cliffs, NJ.

Parkes, D. C. 1999. iBundle: an efficient ascending price bundle auc-tion. Proc. ACM Conf. on Electronic Commerce (EC-99), Denver,CO, 148–157.

Parkes, D. C., L. Ungar. 2000. Iterative combinatorial auctions: the-ory and practice. Proc. 17th National Conf. on Artificial Intelli-gence, Austin, TX, 74–81.

Rassenti, S. J., V. L. Smith, R. L. Bulfin. 1982. A combinatorialauction mechanism for airport time slot allocation. Bell J. ofEconom. 13 402–417.

Rochet, J.-C., L. Stole. 2001. The economics of multidimensionalscreening. Manuscript, Graduate School of Business, Univer-sity of Chicago, Chicago, IL.

Rothkopf, M. H., A. Pekec, R. M. Harstad. 1998. Computation-ally manageable combinatorial auctions. Management Science 441131–1147.

Sandholm, T. 1999. An algorithm for optimal winner determinationin combinatorial auctions. Proc. of IJCAI-99, Stockholm, Swe-den, 542–547.

Sassano, A. 1989. On the facial structure of the set covering poly-tope. Math. Programming 44 181–202.

Schrijver, A. 1986. Theory of Linear and Integer Programming. JohnWiley, New York.

Srinivasan, S., J. Stallert, A. B. Whinston. 1998. Portfolio tradingand electronic networks. Working paper, Center for Researchin Electronic Commerce, The University of Texas at Austin,Austin, Texas.

Strevell, M., P. Chong. 1985. Gambling on vacation. Interfaces 1563–67.

van Hoesel, S., R. Müller. 2001. Optimization in electronic markets:examples in combinatorial auctions. Netnomics 3 23–33.

Vickrey, W. 1961. Counterspeculation, auctions, and competitivesealed tenders. J. of Finance 16 8–37.

Wellman, M. P., W. E. Walsh, P. R. Wurman, J. K. MacKie-Mason.2001. Auction protocols for decentralized scheduling. Gamesand Econom. Behavior 35 271–303.

Williams, S. R. 1999. A characterization of efficient, Bayesian incen-tive compatible mechanisms. Econom. Theory 14 155–180.

Wolsey, L. 1981. Integer programming duality: Price functions andsensitivity analysis. Math. Programming 20 173–195.

Wurman, P. R., M. Wellman. 2000. AkBA: A progressive,anonymous-price combinatorial auction. Proc. Second ACMConf. on Electronic Commerce (EC-00), Minneapolis, MN, 21–29.

Ygge, F. 1999. Improving the computational efficiency of combina-torial auction algorithms. EnerSearch Public Report:EnS 99:08,EnerSearch AB, Göteborg, Sweden.

Yokoo, M., Y. Sakurai, S. Matsubara. 2001. Robust combinatorialauction protocol against false-name bids. Artificial Intelligence J.130 167–181.

Accepted by Ramayya Krishnan; received September 2000; revised June 2001, February 2002; accepted April 2002.


Combinatorial Auctions: A Survey - Department of … Auctions: A Survey Sven de Vries • Rakesh V. Vohra Zentrum Mathematik, TU München, D-80290 München, Germany Department of Managerial

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