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price | Auction Theory | Auction
Volume , Issue   Article 
Lawrence M. Ausubel∗ Paul R. Milgrom†
University of Maryland, ausubel@econ.umd.edu Stanford University, paul@milgrom.net
Copyright c 2002 by the authors. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, bepress. Frontiers of Theoretical Economics is one of The B.E. Journals in Theoretical Economics, produced by The Berkeley Electronic Press (bepress). http://www.bepress.com/bejte.
Lawrence M. Ausubel and Paul R. Milgrom
A family of ascending package auction models is introduced in which bidders may determine their own packages on which to bid. In the proxy auction (revelation game) versions, the outcome is a point in the core of the exchange economy for the reported preferences. When payoﬀs are linear in money and goods are substitutes, sincere reporting constitutes a Nash equilibrium and the outcome coincides with the Vickrey auction outcome. Even when goods are not substitutes, ascending proxy auction equilibria lie in the core with respect to the true preferences. Compared to the Vickrey auction, the proxy auctions generate higher equilibrium revenues, are less vulnerable to shill bidding and collusion, can handle budget constraints much more robustly, and may provide better ex ante investment incentives. KEYWORDS: auction theory, FCC auctions, package bidding, combinatorial bidding, activity rule, bid Improvement rule, e-commerce, electronic commerce
Ausubel and Milgrom: Ascending Package Auctions
Asset sales and procurement auctions often begin with an evaluation of how to structure the lots that are sold or bought. In asset sales, a farm can be sold either as a single entity or it might be sold in pieces such as the house and barn, the arable land, other land, and perhaps water rights; radio spectrum licenses can cover an entire nation or be split among smaller geographic areas and the available set of frequencies in each area sold as a single unit or in smaller pieces; a large company can be sold intact to new owners or broken up into individual product divisions. Similar decisions are made in procurement auctions, in which a manager might decide to buy needed products and services from a single supplier or to break the purchase into smaller parts. The regulators, brokers, investment bankers, auctioneers and industrial buyers who conduct these auctions commonly consult potential bidders in an effort to identify which packages are most attractive to bidders and best serve the auction designer’s revenue, cost or efficiency goals.1 Some auction designs allow bidders a choice of packages on which to bid. For example, Cassady (1967) describes a sale of five buildings of a bankrupt real estate firm in which three buildings were defined to constitute a “complex.” An ascending auction was used for the sale, with bids accepted for either the individual buildings or for the complex.2 More subtly, Dutch flower auctions allow winning bidders to take as many lots as they wish at the winning price.3 Both of these designs encourage direct competition among bidders who seek to buy differently sized packages, which can increase prices compared to auction formats in which the lot size is predetermined. In recent years, there has been growing interest in allowing bidders much greater flexibility to name the packages on which they bid. The London Transportation authority procures bus service from private operators in a sealed-bid auction that allows bids on all combinations of routes within a particular auction, and 46% of winning bids involve such combinations (Cantillon and Pesendorfer, 2001). The ascending auction planned for Federal Communications Commission (FCC) Auction No. 31, which will sell spectrum licenses in the 700 MHz band, permits bids for any of the 4095 possible packages of the twelve licenses on offer. Other examples include proposed auctions for paired airport take-off and landing slots (Rassenti, Smith and Bulfin, 1982) and for industrial procurement, in which individual sellers may offer all or part of a bill of materials and services sought by buyers (Milgrom, 2000b).
The revenue and efficiency criteria can lead to quite different choices; see Palfrey (1983). Milgrom (2000a) reports examples in which the sum of total value and auction revenue is constant across packaging decisions, so that there is a dollar for dollar trade-off between creating value and raising revenue. 2 Sometimes, bidders for large packages are required to specify bids on certain smaller packages as well. An example is an auction one author designed for selling the power portfolio of the Portland General Electric Company (PGE), which was adopted by the company and the Oregon Public Utility Commission. The auction design requires that bidders for the whole package of plants and contracts must also name “decrements” for individual power supply contracts on which there are competing individual bids. 3 Katok and Roth (2001) report experiments to assess the performance of Dutch auctions in the presence of scale economies.
Vol. 1 [2002], No. 1, Article 1
Package auctions can be either static sealed bid auctions, such as the Vickrey (1961) auction or the Bernheim-Whinston (1986) menu auction, or dynamic (multi-round or continuous) auctions. In the latter case, bidders may bid on a multiplicity of packages (including single-item packages) and may improve their bids or add new packages during the course of the auction. The eventual winning bids are the ones that maximize the total selling price of the goods. In Section 2, we review the reasons for the recent interest in ascending package auctions. Subsections discuss advances in technology, changes in spectrum regulation, shortcomings of the generalized Vickrey auction, theory and experience with the FCC’s simultaneous ascending auction (SAA), and evidence from economics laboratories regarding the performance of an ascending package auction design. Our formal analysis begins in Section 3 with descriptions of the package exchange problem and our model of the ascending package auction. That section also introduces a particular myopic strategy, which we call “straightforward bidding,” which may account for the performance of bidders in certain laboratory settings.4 Section 4 explores an associated direct revelation mechanism called the “ascending proxy auction.” We intend this auction to be both a possible account of bidding in certain experiments and a new auction design in its own right. The insight guiding our analysis is that the ascending proxy auction is a new kind of “deferred acceptance algorithm,” having some properties similar to the algorithms studied in two-sided matching theory.5 To develop this idea in detail, we introduce the coalitional game that corresponds to the package economy. We show that the auction algorithm terminates at a core allocation of the package exchange economy relative to the reported preferences. This result, which corresponds to a familiar one in two-sided matching theory, can be extended to models with budget constraints, using the insight that deferred acceptance algorithms apply equally well to cooperative games without transferable utility. In section 5, we study bidding incentives and equilibrium in the proxy auction. We define semi-sincere reports to be reports that reduce the valuation of all packages by an equal amount relative to the truth (except that negative values are reported as zero). We show that for any pure profile of opposing strategies, each bidder always has a best reply in semi-sincere strategies and that equilibria in such strategies exist. If all bidders adopt such strategies and if losing bidders bid sincerely, then the set of possible equilibrium payoff vectors is identical to the set of bidder-Pareto-optimal points in the core of the cooperative game among the bidders and the seller.
An ascending package auction design with “myopic bidding,” similar to our “straightforward bidding,” was studied by Parkes and Ungar (2000a). Their work emphasizes the computational advantages of such a design. The same authors (Parkes and Ungar (2000b)) also study proxy agents with a specification that differs substantially from ours. 5 See Roth and Sotomayor (1990) for an excellent review and survey of two-sided matching theory. Our new algorithm and its analysis differ from all previous matching algorithms and analyses in several important ways. First, our model involves package offers, rather than offers for the individual items. Second, in our algorithm, an offer that is tentatively “rejected” at one round may nevertheless be accepted later as part of the final allocation. Third, much of the analysis of model does not require the assumption of “substitutes,” which has been required for all previous matching analyses. Finally, ours is the first model of many-to-one matching in which Nash equilibrium strategies have been explicitly characterized.
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The adoption in 1993 of legislation authorizing spectrum auctions in the US and the bold decision by the FCC the following year to adopt 6 Some computational aspects of package bidding are surveyed by deVries and Vohra (2001).Ausubel and Milgrom: Ascending Package Auctions 3 Sections 6 and 7 compare the outcomes of the ascending proxy auction and the Vickrey auction. interfaces. however. Produced by The Berkeley Electronic Press. truthful reporting is a Nash equilibrium of the ascending proxy auction and its outcome coincides with the Vickrey outcome. this number of combinations might have overwhelmed users and posed serious computational problems. Thus. Under the same conditions. and research that highlights the potential benefits of package auctions. which enables certain new auction designs. Section 9 concludes. All this must be done quickly. and communications systems that make it practical for users to identify and bid for many combinations. while bidders sit in front of their individual computer terminals. 2004 . suppose that bidders submit bids for overlapping packages. Now. By way of contrast. favorable developments in regulated spectrum markets and unregulated Internet exchange markets. and for all to verify and track the progress of the auction. Background A variety of developments have contributed to the present drive to develop and implement package auctions. One result identifies the conditions on preferences under which the Vickrey outcome is in the core. for auctioneers to compute optimal bid combinations. Provided that all additive valuations are possible. the Vickrey auction outcome entails seller revenues so low that the payoff vector falls outside the core. 2. the most important differences between the designs emerge when goods are not substitutes. there are processors. the Vickrey outcome is guaranteed to be in the core if and only if the goods are substitutes for every possible bidder preference ordering. consider the plan for FCC Auction No.1 Changing Technology and Markets The most important group factors contributing to the new package bidding designs is associated with the rapid advance of technology. Section 8 explores generalizations of the ascending proxy auction that work even when there are limits on monetary transfers. Given these bids.6 Even as technology was advancing. Then. The twelve licenses on offer allow for 4095 distinct combinations involving between one and twelve licenses. To get an idea of the size of the problem. the total bid associated with each such package must be computed and the revenue-maximizing set of “consistent” bids must be found. To understand the technical challenge. In that case. the ascending proxy auction has extensions that preserve many of its desirable properties. algorithms. markets were changing in ways that facilitate the adoption of sophisticated auction designs. 2. the first step of finding the sets of “consistent” bids in which each individual item is included in just one package (“sold just once”) is a hard computational problem. These can be grouped into three general categories: rapid advances in technology. even from remote locations. 31 of spectrum licenses. The Vickrey auction generally loses its desirable properties when such limits are imposed. A decade ago.
which make the buyer’s procurement optimization problem a nonconvex one. can be traced to the seminal paper by William Vickrey (1961). The perceived successes of spectrum auctions have led some to propose even more ambitious designs. These developments and others have inspired new research by economists. then some form of package auction will be needed. Subsequent work by Clarke (1971) and Groves (1973) demonstrated that a generalization of the Vickrey mechanism leads to the same “dominant strategy property” in a much wider range of applications. spectrum regulators eager to “let the market decide” all details of the allocation initiated a serious discussion about the sale of “postage stamp” sized licenses. Article 1 the computerized simultaneous ascending auction (SAA) gave an important boost to advocates of more sophisticated auction designs.com/bejte/frontiers/vol1/iss1/art1 . in the sense of always leading to identical equilibrium outcomes. Each bidder is asked to report to the auctioneer its entire demand schedule for all possible quantities.” The advantages of the VCG mechanism are further confirmed by certain uniqueness and equivalence results. No. Those proposals were shelved because of concerns that the complementarity among the licenses might make the auction and subsequent resale markets perform poorly. like so much of auction theory. In particular. such as a construction project. Vickrey’s mechanism can be described as follows. It then requires each buyer to pay an amount equal to the lowest total bid the buyer could have made to win its part of the final allocation. Multiple suppliers may each supply part of the buyer’s needs on terms that may involve quantity discounts. regardless of the bids made by others. Williams (1999) finds that all Bayesian http://www. operations researchers and computer scientists into the theory and practice of package bidding. Results by Green and Laffont (1979) and Holmstrom (1979) imply that any efficient mechanism with the dominant strategy property and in which losers have zero payoffs is equivalent to the VCG mechanism. These would entail very small geographic areas and narrow slivers of bandwidth to be licensed and ultimately recombined as desired by spectrum buyers.4 Frontiers of Theoretical Economics Vol. Often. The auctioneer uses that information to select the allocation that maximizes the total value. Shortly afterward. and supporting software for these auctions has recently become available. with this payment rule. 2. given the other bids.” This extension has come to be known as the “generalized Vickrey auction” or as the “Vickrey-ClarkeGroves (VCG) mechanism.bepress. industrial buyers seek to purchase not just single components but all the materials and services for a large project. 1 [2002]. provided the original requirement to make bids for “all possible quantities” is replaced by the requirement to make bids on “all possible packages. 1. another area of high-tech applications began to develop as Internetbased businesses raced to develop electronic markets that could serve the needs of business customers. it is in each bidder’s interest to make its “bid” correspond to its actual demand schedule. If such procurements are to be managed by competitive bidding. Vickrey’s conclusion holds even when there are many types of goods. which focused on sales involving a single type of good. Vickrey showed that.2 Vickrey Auctions: Advantages and Disadvantages The theory of package bidding. In Australia.
However. and the Stanford Institute for Economic Policy Research. For example. Bidder 2 bids $2 billion for one item. while his shill bidder 3 separately bids $2 billion for one item. Bidder 2 is willing to pay $1 billion for the pair and $0. A fourth is the incentives that the auction design creates for related choices. it seemed. A third efficiency issue is the cost of the auction to the various parties.7 Efficiency in auction design has several aspects. A quick calculation establishes that the Vickrey outcome allocates the licenses to bidders 2 and 3. 2004 . however. Yokoo and Matsubara (1999) and Yokoo.html. the auctioneer must be wary of whether bidders can profit by submitting additional bids under false identities. Sakurai and Matsubara (2000) refer to bids by a buyer shill as “false-name bids. In auctions of public assets. given the preferences of participants when the auction is run. 7 Produced by The Berkeley Electronic Press.gov/auctions/conferences/combin2000/ papers. for a payment of $1 billion.5 billion for a single license. The second and third disadvantages of the Vickrey design are its vulnerability to shill bidding8 and collusion. These discoveries had wide ramifications. Bidder 1 wants only the package of two licenses. even by losing bidders. since auction revenues can displace distortionary tax revenues. decisive for most practical applications. bidders 2 and 3 together win both items. bidder 1 should win both items. consider an auction of two identical licenses to two bidders. As in the previous example. each at a price of zero! This defect of the VCG mechanism is. then the generalized Vickrey auction would become a practical solution to a wide range of resource allocation problems.fcc. the National Science Foundation. Probably the most important disadvantage of the Vickrey auction is that the revenues it yields can be very low or zero.Ausubel and Milgrom: Ascending Package Auctions 5 mechanisms that yield efficient equilibrium outcomes and in which losers have zero payoffs lead to the same expected equilibrium payments as the VCG mechanism. even when efficiency is the auction designer’s goal. For economists. suppose that bidder 2 misrepresents both his value and his identity by submitting bids under two separate names. while bidders 2 and 3 are each willing to pay $2 billion for a single license. consider a hypothetical auction of two spectrum licenses to three bidders.” They argue that in situations (such as Internet auctions) where the auctioneer cannot completely determine the identities of bidders. they seemed to reduce the economic problem of auction design to a computational problem. As we shall see. If we modify this example to replace bidder 2 by Parts of the following discussion of disadvantages of the Vickrey auction are drawn from a report to the FCC by Charles River Associates and Market Design Inc (1997). For example. For some operations researchers. higher revenues also improve efficiency. the VCG mechanism has profound defects that make it unlikely that it could be useful for sales of publicly owned assets. The shill strategy profitably converts losing bidder 2 into a winning bidder. and bidder 2 should win nothing. by itself. even when the items being sold are quite valuable. paying zero in the Vickrey auction. Vickrey’s findings raised expectations about the possibility of designing effective auctions using economic analysis. but there are other important defects as well. Bidder 1 is willing to pay $2 billion for the pair of licenses and nothing for a single license. In a Vickrey auction. One is efficiency of the allocation itself. The reports to the FCC and related papers were presented at a conference sponsored by the FCC. If only one could describe and compute values and allocations quickly. for which it is willing to pay $2 billion. such as pre-auction investments or choices of technology and organization. See http://wireless. 8 Sakurai.
If instead bidder C bids high. 1. Despite the value created by the merger. if A’s budget limit constrains it to bid no more than $1.2 billion for the package). which makes shill bidding and loser collusion unprofitable. if bidder C bids zero. In the first example above. changing bidder 1’s values in this way forces the others to pay at least $2 billion in total to win. so its budget is always adequate to make its Vickrey payment. Then. In either case.1 billion or it may have access to a substitute license from another source. depending on the last bidder’s decision. Its two competitors. the Vickrey auction creates a significant bias in decisions about technology and organization.6 Frontiers of Theoretical Economics Vol. Since these are inconsistent. each want only one license. the Vickrey outcome was not in the core. because bidder 1 has value only for the entire package. are mitigated in various degrees by the ascending proxy auction. described below. the parties would find this merger 9 This collusion example is based on a suggestion by Jeremy Bulow. and increase the total value of the licenses from $4 billion to $5 billion. 11 Che and Gale (1998) analyze revenue differences among first-price and second-price auctions in the presence of budget constraints. then it has no dominant strategy.2 billion for any package. If we modified the examples to make the licenses substitutes. Article 1 two bidders each of whom is willing to pay $0. The remaining weaknesses. but can spend only $1. A fourth weakness of the Vickrey auction is that its dominant strategy property depends on unlimited bidder budgets. by specifying that bidder 1 was willing to pay $1 billion for each license rather than just $2 billion for the pair.5 billion for one license. The first three weaknesses described above are all remedied by our basic ascending proxy auction.11 To illustrate. For suppose bidders 2 and 3 could merge. by themselves. The distinction between valuations that are substitutes and the remaining cases plays a decisive role in the general theory we develop. say.9 Notice that these first three weaknesses are illustrated in examples in which the licenses are not all substitutes. which is a competitive payment in the sense that the payoff outcome lies in the core. In a Vickrey auction. The coalition consisting of the seller and bidder 1 could “block” the Vickrey outcome since.com/bejte/frontiers/vol1/iss1/art1 . then those bidders could profitably collude by both raising their bids to $2 billion. 10 http://www. consider an auction in which a certain bidder A has values of $1 billion for a single license or $2 billion for two licenses based on the profits it can earn. A’s best reply requires that it bid less than $400 million for a single license (and. bidder A should win either two licenses or one. its total payment will be $800 million. bidding $800 million for a single license. and the fourth is remedied by a variation on that auction.bepress. but the Vickrey outcome gave them only a payoff of zero. In the unmodified example. they earn a coalition payoff of $2 billion. Bidder B is willing to pay $800 million for a license. Yet. in which case it will bid zero. suppose that bidder B plays its dominant strategy. A has no dominant strategy. bidders B and C. for example. then A’s best reply entails bidding more than $800 million for a single license.10 In the shill bidding and collusion examples. adopt a coordinated technology. then the identified problems would vanish. No. The seller would get a total payment of $2 billion in the first example. Bidder C may be willing to pay $1. $1. but they are not completely resolved. becoming winners and paying prices of zero.2 billion to buy licenses. For example. 1 [2002].
See also the discussion of advantages of ascending auctions over sealed-bid auctions in the Introduction of Ausubel (1997a). particularly when the asset valuation process requires substantial human inputs. For package auctions. When the assets in the collection interact in complex ways that affect the optimal business plan.20). 2004 . even when both have made the same bids for those allocations.12. the sheer number of combinations that a bidder must evaluate makes that assumption especially dubious. it will consider labor and contractual constraints. A valuation for a bidder is a triple (x. Another potentially significant issue is the cost of determining valuations. 13 This topic has recently attracted the attention of several researchers.z). When package evaluation is costly. and the package XY. See Parkes. In our experience. suppose the items offered in an auction are {ABCD}. The result is that A and B will each be awarded an item (at an efficient allocation. The final valuation is the result of an optimization over business plans using all this information. A traditional assumption in auction theory analyses is that each bidder knows all its values or can compute them at a zero cost. because the amount they would have to pay in the Vickrey auction would rise from zero to $2 billion. auction design should account for the way those choices are made as well as the evaluation costs that bidders incur. suppose there are two bidders—A and B—and two items—X and Y.12 Potential buyers who find it too expensive to investigate every packaging alternative will instead choose a few packages to evaluate fully. In comparison. In a Vickrey auction or any sealed bid auction. then significant extra costs must be incurred to evaluate each package. For example. specifying how much the bidder would be willing to pay for item X alone.13) and (12. actual and potential transmission capacity. 14 To illustrate the price discrimination problem. Ungar and Foster (1999). a bidder hoping to purchase parts of an electrical generating portfolio might investigate the physical condition of each plant. either bidder may get either item) at prices of 8 and 1 respectively. It is unlikely to be an effective strategy to bid for package AB unless someone else is bidding either on CD or on C and D separately. Suppose the parties report valuations of (12.Ausubel and Milgrom: Ascending Package Auctions 7 unprofitable.y. Total profits would therefore fall from $4 billion to $3 billion. and so on. the availability of land and water for cooling to allow plant expansion. the condition of markets in which power might be sold. bidder valuation costs are relatively less affected by advancing technologies. Produced by The Berkeley Electronic Press. if it is too costly to evaluate all the packages. a multi-round ascending auction economizes on the need to guess because bidders can adapt their plans based on observations made during the auction. valuing significant business assets involves both investigating the asset itself and creating business plans showing how they will be used.12. the interacting choices that bidders make about which packages to evaluate further complicate the analysis. In addition. Compte and Jehiel (2000). even though the items are perfect substitutes and bidders A and B made identical bids for the individual items. item Y alone.14 Such discriminatory pricing may be illegal and is often regarded as “unfair. Ideally. and tempered by human judgment.13 Another characteristic of the Vickrey auction that is sometimes considered a drawback is its use of explicit price discrimination: two bidders may pay different prices for identical allocations. and Rezende (2000). and other physical variables. Compared to most of the other costs involved in conducting combinatorial auctions. partnerships that might enhance the asset value. zoning and other regulatory constraints. For example. then bidders must guess about which packages are most relevant and how to allocate their limited evaluation resources.” 12 The nature of human inputs to the valuation process is a matter of controversy among auction consultants. discouraging the efficient reorganization.
at least for noncomputerized auctions in which secure encryption technologies are not available. Huang and Klemperer (1999)). since the theoretical bid functions are invertible to reveal bidders’ values. the revenues from second price auctions can be nearly zero when there is “common value” or “almost common value” uncertainty (Milgrom (1981). but the auction outcome is not generally “envy free”: a bidder may prefer the price and allocation assigned to another bidder and may complain on that basis. Article 1 Most favorable evaluations of Vickrey auctions depend sensitively on assumptions that each bidder knows its own values (the “private values” model) and that participation is exogenous. Finally. with minor variation. differs from the Vickrey auction in two ways that have made it an attractive practical alternative to the Vickrey auction: it is a pay-as-bid. and conceals the winning bidder’s maximum willingness to pay.S.com/bejte/frontiers/vol1/iss1/art1 . The auction design has an iterative structure. For sales of a single good.bepress. the price discrimination is not so obvious. http://www. and the simultaneous ascending auction with package bidding (SAAPB). the public has sometimes been outraged when bidders for government assets are permitted to pay significantly less than their announced maximum prices in a Vickrey auction (McMillan (1994)). 1. in most other spectrum auctions worldwide). During each round. 15 A similar case can be made against ordinary first-price auctions.15 These various defects of the Vickrey auction have led to increased interest in exploring alternative designs. 1 [2002]. Standard sealed bid auctions do not share this fault. radio spectrum auctions (and. Similarly. which entails no package bidding. The next sections consider two such designs: the simultaneous ascending auction (SAA). The pay-as-bid feature avoids the low revenue outcomes of the Vickrey auction and discourages shill bidding and some kinds of collusive strategies. Initially. Winning bidders may fear that information revealed by their bids will be used by auctioneers to cheat them or by third parties to disadvantage them in subsequent negotiations. such as multiple round pay-as-bid auctions. No. 16 A reserve price may also be used. Klemperer (1998)) or endogenous entry decisions (Bulow.3 Simultaneous Ascending Auctions The simultaneous ascending auction.8 Frontiers of Theoretical Economics Vol. with the “state of the auction” after each round described by the identities of the standing high bidders and the amounts of the standing high bids for each item. the standing high bid for each item is zero16 and the standing high bidder is the seller. which has been employed by the FCC in almost all U. bidders may raise the bid by When the items are not identical. A bidder’s motive to conceal its information can destroy the dominant strategy property that accounts for much of the appeal of the Vickrey auction. 2. economizes on bidder evaluation efforts. the fact that Vickrey auctions require bidders to report their valuations has been regarded as problematic (Rothkopf. The two features combined turn out to also alleviate problems associated with budget constraints. In this respect. Teisberg and Kahn (1990)). multiple round auction design. The multiple rounds feature provides feedback to bidders about relevant packages. ascending auctions are theoretically superior to both kinds of sealed bid auctions because they better conceal the winning bidder’s valuation.
the items for sale are substitutes17 for all the bidders. Bidder 1 values a single license at $3 billion and the pair of licenses at $6 billion. the final allocation is efficient and the final prices are competitive 17 Generally. The process repeats itself until there is a round with no new bids on any item. In such cases. Then. Although early experimental testing of the SAA demonstrated that it performed well in some environments possibly resembling the FCC environment (Plott (1997)). In addition. To explain the role of the substitutes condition in theoretical terms. a condition that may have applied to the radio spectrum auctions (Ausubel. goods are substitutes when increasing the price of one does not reduce demand for the other. which determines new bidders and standing high bids. In particular. Lehmann (private communication) has shown that in models with at least three discrete goods. Since bidders in our auction model (and most others) have quasilinear utilities. The modified terms “gross substitutes” and “net substitutes” are often used to distinguish between substitutes for uncompensated and compensated demand. The associated revenue is high enough that the outcome is a core outcome. As described earlier. Porter and Rangel (1997)). The seller might hope that the SAA would lead to a price of $2 billion (+ ε) for each license. with bidder 1 winning both. suppose bidders bid “straightforwardly” at each round for the items in a package they most prefer at the current prices. This outcome is unsatisfactory both in its inefficiency and low revenues. consider an auction of two identical licenses to two bidders. McAfee and McMillan (1997)).” The substitutes condition has figured prominently in multi-good auction theory since its use by Kelso and Crawford (1982) to evaluate an auction model of the labor market. At that point. Produced by The Berkeley Electronic Press. that establishes that the condition is non-generic. the ascending proxy auction of this paper and the Vickrey auction both avoid the inefficiency of demand reduction in this example by allowing bidder 1 to win the package of two licenses for a price of $2 billion. For example. However. respectively. bidder 1 strategically withholds his demand for the second license and bids for only a single license. 2004 . winning one license at a price of zero (which yields $3 billion in profits) is preferred by bidder 1 to winning two licenses at a price of $2 billion each (which yields $2 billion in profits). Bidder 2 values a single license at $2 billion and has no additional value for the pair. By contrast. In the first situation.Ausubel and Milgrom: Ascending Package Auctions 9 an integer number of increments on any items that they wish. the bid increment is “small” and the initial prices are low enough to attract at least one bid during the auction for every item. bidding on all items is closed and the standing high bids determine the prices. there is no distinction between “gross” and “net” so we refer simply to “substitutes. the substitutes condition holds on a lower-dimensional subspace of the space of possible valuation vectors. the seller cannot do better by reneging on the auction terms and instead negotiating with the losing bidder. it has a variety of theoretical limitations. Another important limitation of the SAA is its degraded performance in laboratory experiments in which the substitutes condition fails (Ledyard. One important issue with the SAA is the incentive that it provides for demand reduction (Ausubel and Cramton (2002)). that is. Gul and Stacchetti (1999) characterize the preference orderings satisfying the condition. there is also an activity rule designed to ensure that bidding activity starts out high and declines during the auction as prices rise far enough to discourage some bidders from continuing. Thus. we compare two different situations. Cramton. causing the auction to conclude at prices of zero.
Second. bids straightforwardly. On the other hand. the information communicated during the course of the SAA is rich enough to allow the auction algorithm to discover equilibrium prices and allocations. Bidder 2’s values are then constructed so that (1) the items are substitutes for bidder 2 and (2) the unique efficient outcome is for bidder 1 to win both licenses. The preceding result entails several interesting conclusions for the case when goods are substitutes. If T is any strict superset of S and provided that there are at least two bidders. Suppose it is so and let pA and pB be the equilibrium prices. Starting from any prices that are below the minimum equilibrium price vector. Suppose that its competitor. For α>½. if the bidders bid straightforwardly from that point forward. if a competitive equilibrium exists.18 Indeed.bepress. there exists a profile of valuations drawn from T such that no competitive equilibrium exists (Milgrom. the auction algorithm can recover from some kinds of anomalous bidding behavior early in the auction. market-clearing prices do exist.and forward-looking price-setting processes. since bidder 1 purchases both goods at equilibrium. 1 [2002].com/bejte/frontiers/vol1/iss1/art1 . http://www. Intuition for this result is provided by Table 1. equilibrium prices for an economy with “almost” the same values as the actual economy. Third. Milgrom and Roberts (1991) showed that the same sort of stability holds over a vast family of discrete and continuous time. First. if bidder 1 does not know whether α>½. differing by at most the relevant bid increment (Milgrom. Consider the situation at any time after the prices exceed a 18 The idea that price formation processes behave drastically differently in the cases of substitutes and complements has a long history in economics. In the example just described. By the first welfare theorem. let S denote the set of valuations in which bidders regard the items as substitutes. Article 1 Table 1: Bidder Values Item A Item B Package AB Bidder 1 a+b+c a b Bidder 2 a+αc a+b b+αc The example uses c>0 and 0<α<1. No. where some items may be complements. In the table. then the sequence of prices and allocations converges to the equilibrium prices and allocation.10 Frontiers of Theoretical Economics Vol. which tabulates bidder values. 2000a). bidder 1 will have to bid more than a for item A and more than b for item B. backward. Since bidder 2 does not demand either good at equilibrium. To win both items. Block and Hurwicz (1959) first established the global stability of tatonnement in the case of “gross” substitutes (substitutes using uncompensated demand functions). Arrow. The preceding conclusions change drastically in the second situation. 2000a). Scarf (1960) provided examples of global instability in the case when the goods are complements. bidder 2. it must be true that a + α c ≤ p A and b + α c ≤ pB . sharply contrasting with the case for substitutes. these inequalities are inconsistent: the contradiction establishes that no market clearing prices exist. bidder 1’s values are any ones for which the two licenses are complements. then the outcome is efficient. synchronous and asynchronous. despite the indivisibility of the items offered for sale. 1. we may infer that a + b + c ≥ p A + pB . then it may face a difficult bidding problem.
Ausubel and Milgrom: Ascending Package Auctions 11 and b and bidder 2 places a bid on. Smith and Bulfin (1982) are often credited with the first experimental study of package auctions. in which payments are first-price in the sense that winning bidders pay their final bids. bidder 1 must always take this risk and always decide in this situation to stop bidding and accept the loss from acquiring a single item. Rassenti. Experimental subjects were assigned valuations for packages of items and participated in these ascending package auctions. has played a crucial role in the growing interest in ascending auctions with package bidding. Banks. more than 90% of the available gains from trade were typically realized in the experiments. as well as theoretical analysis. If bidder 1 stops now. The experimental subjects were assigned valuations for packages of items and were instructed to participate in a sealed-bid auction in which package bids could be submitted. At that point. in which payments are secondprice in the same sense as in the Vickrey auction. While their experiments involved a dynamic mechanism that did not utilize package bids. If α>½ and if it continues to bid. Produced by The Berkeley Electronic Press. one might expect to find inefficient outcomes and bidder losses as common occurrences in circumstances like this one. Their experiments on average realized over 97% of the available efficiencies. and bidders were charged uniform prices intended to approximate shadow prices. the FCC adopted a rule permitting bid withdrawals under some circumstances to mitigate it. and an adaptive user selection mechanism. The problem facing a bidder for whom goods are sometimes complements has come to be called the “exposure problem. it has no hope of avoiding a loss. on average realizing 80% of the available efficiencies.” In 1994. a natural application for package auctions given that landing and takeoff slots are strong complements. as well as some alternative procedures meant to represent administrative processes and markets. the mechanism was similar in flavor (in that it required the auctioneer to calculate the revenue-maximizing feasible allocation at each iteration) and it was also aimed at solving a complex fitting problem. For an efficient outcome always to emerge. it will wind up acquiring only item B for a price greater than its value b to the bidder. say. 1997) suggests that withdrawals do mitigate the problem. These authors developed two dynamic procedures in which bidders submit package bids and the winning bids are selected so as to maximize the sum of the prices bid: an iterative Vickrey-Groves mechanism.4 Experimental Evaluation of Ascending Auction Designs Economic experimentation. There is a venerable line of experimental research suggesting that auctions may enable bidders to solve relatively complex multi-item assignment problems. but they do not solve it completely. Brewer and Plott (1996) investigated a problem of allocating the usage of railroad tracks. Thus. item A. but this is rarely the optimal strategy. Ledyard and Porter (1989) conducted an early and influential study of ascending package auctions. Experimental evidence (Porter. The ascending package auctions outperformed the alternatives. it will eventually find that the total price exceeds a+b+c. They studied the problem of allocating airport time slots. 2. While sincere bidding is not incentive compatible in this mechanism. 2004 . The winning bids were selected so as to maximize the sum of the prices bid.
The remaining three conditions involved increasing amounts of complementarity. 1. 1 [2002].8 Recently. The study was conducted under four experimental conditions.5 9. once for a group of subjects participating in the non-package auction—the SAA—and once for a group participating in the package auction—the SAAPB. the relative simplicity of the package optimization problem might have influenced the experimental outcome. but not in the SAA. In the present case. Efficiency in the study was measured by the ratio of the total value of the allocation resulting from the auction to the maximum of that total over all possible allocations. The experimental results show several prominent features. the SAAPB took roughly three times as many rounds to reach completion. In the experiments. although the reported difference is small. 19 http://www. First. All experiments require making implementation choices that may affect the experimental outcome. Bidder values were drawn at random for each experimental condition and were used twice.12 Frontiers of Theoretical Economics Vol. the ascending package auction generates higher efficiency than the SAA even when there are no complementarities. a bidder’s value for any package was equal to the sum of its values for the individual items in the package. compared to the SAA. labeled low. This condition involves no complementarities. comparing the performance of the SAA to that of a particular combinatorial auction called the simultaneous ascending auction with package bidding (SAAPB).19 Second. revenues are higher in all conditions for the SAA compared to the SAAPB. In addition. the measured efficiency of the SAA falls off markedly as complementarities increase. In the first. This is problematic for two reasons: the selected complementarities appear unlikely to reflect those in the structure of the actual FCC auction and.3 31. a large experimental study was sponsored by the FCC and conducted by Cybernomics (2000). For that reason. the Cybernomics experiment was constructed to involve a simple kind of complementarities that make it relatively easy to compute optimal allocations.3 10 10. medium and high. experimental results are most convincing when similar results are obtained under a variety of relevant conditions. Article 1 Table 2: Findings of the Cybernomics Experiment Complementarity None Low Medium High Condition: Efficiency SAA (No packages) 97% 90% 82% 79% SAAPB 99% 96% 98% 96% Revenues SAA (No packages) 4631 8538 5333 5687 SAAPB 4205 8059 4603 4874 Rounds SAA (No packages) 8.com/bejte/frontiers/vol1/iss1/art1 .9 28 32.5 SAAPB 25. The major findings of that study are summarized in Table 2.bepress. but the efficiency of the package auction is largely unaffected by complementarity. The theoretical analysis for the case of substitutes provides a possible explanation: the straightforward bidding that supports efficient allocations is incentive-compatible in the package auction. viewing the package auction as an optimization algorithm. No.
Third. Finally. Quasilinear utility without externalities: Produced by The Berkeley Electronic Press. Setting Mk >1 reduces the number of possible bids—sometimes drastically so. the history of successes of various package auctions in laboratory settings from Rassenti. In the next several sections. The set of auction participants L consists of the seller. where the stakes are also very much higher. 3. . meaning that the auction treats each item as unique.. z N ) is a vector of integers whose components indicate the number of units of each type in the package.. Long as these sessions were. Each buyer l has a valuation vector vl = ( vl ( z ) : z ∈ [0. subjects in the experiments had no access to expert assistance or to analyses that could pinpoint opportunities for strategic bidding. M ]) . the experimental subjects’ lack of information about other bidders’ values is not typical of FCC spectrum auctions and makes it harder for them to exploit the strategic opportunities that the auction affords. affording subjects little opportunity to evaluate others’ bids and assess the strategic opportunities. designated by l = 1 . unlike bidders in the FCC auction. . then a bidder can completely describe its values by reporting just M1 bids. Compounding this is the fact that rounds were relatively short. M N ) denote the number of items of each type.. is striking. designated by l = 0. there is a practical reason to group together identical items. 2004 .. Although there is no loss of generality in limiting attention to this case. This condition has been common in the FCC auctions. Smith and Bulfin (1982) to Cybernomics (2000).. if there is only one type of item. . 31. we provide a theoretical analysis that seeks to account for the success of the package auction experiments and to explore more generally the strategic opportunities that such auctions create. .Ausubel and Milgrom: Ascending Package Auctions 13 The Cybernomics experimental environment may also have offered less scope for strategic manipulation of the rules than the real-world FCC auction environment.M] denote this set. instead of the 2 M1 bids needed to specify a separate value for each possible package. they fall far short of the preparation undertaken by bidders in the FCC auctions. the relatively long training sessions that subjects required seemed to highlight their difficulty in understanding the rules. An Ascending Package Auction Let there be finite number N of types of items to be sold and let M = ( M 1 . | L | −1 . it does not change its value when it learns about what others are willing to pay.. and the buyers. The relevant packages in the auction are those for which 0 ≤ z ≤ M. An important special case arises when M is a vectors of 1’s.. Despite these limitations. We limit and simplify our analysis by the following assumptions: (i) (ii) Private values: each bidder l knows its own value vector vl . For example. where vl ( z ) specifies the “value” of package z to bidder l. There are several reasons to suspect this. including the planned package Auction No. First. let [0. further limiting their ability to exploit gaps in the rules.. A package or bundle z = ( z1 .
as is standard in auction theory. the relevant factors include technology and demand forecasts. 1. allowing such bids in lieu of mutually exclusive bids could restrict the bidders’ options and prevent the theorems of this paper 20 21 See also Jehiel and Moldovanu (2001) and the references therein. in which the factors that affect value are the same for various bidders but in which bidders have different estimates of those values. A bidder l that acquires package z and pays price bl ( z ) earns a net payoff of vl ( z ) − bl ( z ) . In our analysis. One very important modeling/design issue is the form that package bidding may take.21 The last two assumptions are innocuous and standard. In the spectrum auctions. (iii) (iv) Monotonicity (Free Disposal): For all l and z ≤ z ′ . Zero seller values: v0 ( z ) ≡ 0 . Assumption (ii) limits bidders’ payoffs to be linear in money. but we omit that issue in the present analysis. we abstract from these issues to make progress on other important aspects of the auction design. Allowing such bids in addition to all mutually exclusive bids would have no consequences for our theoretical analysis. Assumption (i) rules out various kinds of “value interdependencies. we assume full mutual exclusivity: bidders are free to make mutually exclusive bids on as many packages as they wish.com/bejte/frontiers/vol1/iss1/art1 . Particularly. b. Article 1 a. In addition. vl ( z ) ≤ vl ( z ′) . supposing that the objective of a bidding language is to express the richest possible set of plausible preferences as succinctly as possible. the generalized ascending package auction formulated in Section 8 may enable the relaxation of Assumption (ii).22 However. however. In practice. which does not depend on what l’s competitors acquire. for allowing such bids merely enriches the language in which the complete bid vector can be expressed. Such a rule imposes no restrictions on what packages the bidder may name or on what amounts it may bid for different packages. 1982).” 22 Nissan (1999) investigates the expressive power of various “languages” for package bidding.” including especially possible “common value” elements (Milgrom and Weber. A bidder l that acquires nothing and pays nothing earns a net payoff of zero: vl (0) = 0 . http://www. it excludes a class of externality issues that was first emphasized by Jehiel and Moldovanu (1996). 1 [2002].14 Frontiers of Theoretical Economics Vol. This is essentially the same rule as is used in the generalized Vickrey auction. in spectrum auctions. buyers do interact after the auction and these interactions could influence bidding behavior. and it is the most flexible form of package bidding.bepress. Rational bidders should often respect common value estimates made by competitors’ analysts as much as or more than the estimates made by their own analysts—a fact that can have a profound impact on bidding strategy. other rules governing packages have sometimes been adopted in which certain bids from a single bidder are not taken to be mutually exclusive. as well as Das Varma (2000a&b).20 For the most part. No. Our auction can be adapted to apply to public decisions in much the same fashion as the BernheimWhinston (1986) “menu auctions.
but bids made in the same round are not. an ability to make mutually exclusive bids in different rounds is sufficient to imply all of our results (provided that there are no activity rules requiring multiple bids within a round). the design of the user interface.27 A final modeling decision is whether bidders are free to bid directly in the auction or whether their bidding is required to be intermediated through “proxy agents” that may place constraints on the form that bids may take. Third. 31 include a combination of exclusive and nonexclusive bids. a bid that was a losing bid at round t can become a provisional winner at a later round. will introduce proxy bidding constraints. the auctioneer announces the full history of winning and losing bids after each round (although straightforward bidders will not utilize all that information). 2004 . and each bidder may be associated with at most one provisionally winning bid. Any new bid a bidder makes on any package z must be positive and must exceed the bidder’s best previous bid on z by some integer number of bid increments. This is 23 The package bidding rules for FCC Auction No. will allow bidders to bid directly. Second. but not both.28 Second. 24 The details omitted in conventional game theoretic analyses include how long each bidder has to submit its bid.Ausubel and Milgrom: Ascending Package Auctions 15 from applying. below. First. might find its options limited. Bids made in different rounds are treated as mutually exclusive. Several differences between the ascending package auction and the SAA merit highlighting.3. Third. as in the FCC design. In our model in this paper. This is illustrated in Table 3 below by the bid of 5 by bidder Y. how much discretion the auctioneer has to make exceptions. subject to two kinds of constraints: at most Mn items of each type n may be sold. We further suppose that. Sections 3. after each round. 25 Some aspects of the ascending package auction technology are described in greater detail in Ausubel (1999. in which there may not exist anonymous prices that support an efficient allocation. the minimum bids can differ among bidders on any item or package. while Section 3. A bidder who prefers its bids made in different rounds to be treated as non-exclusive can accomplish that by “renewing” old bids in the current round. The discussion following Theorem 3 indicates the need for either activity rules or proxy bidders to quicken the pace of ascending package auctions 28 In this respect. 27 In contrast to the rules for both the simultaneous ascending auction and FCC Auction No.26 The auction then ends and the provisionally winning bids at that time become the winning bids in the auction. The rules that figure into our analysis are the following ones.4 et seq. the auctioneer identifies a set of “provisionally winning bids. no “activity rules” are included in the present model. 2000) and Ausubel and Milgrom (2001). such as round t+1.” This is the set of bids that maximizes the total price. 26 The rules for FCC Auction No. all bids are firm offers. the theory of package auctions resembles the theory of public goods. including some that are left unspecified in standard game theoretic analyses. 31 specify that the auction proceeds in a sequence of bidding rounds until two consecutive rounds elapse with no new bids.24 are needed to complete the rules of a practical auction design.23 Many more details. A bidder who wished to buy package A or package B. which becomes a provisional winner in round R+1 even though it was not one in round R. 31.1 – 3. A bidder can never reduce or withdraw a bid it has made on any package. and many more. Produced by The Berkeley Electronic Press. the price of an item or package can decrease from round to round. the auction proceeds in a sequence of bidding rounds until a round elapses in which no new bids are submitted by any bidder.25 First.
Let Blt ( z ) = Bl ( H t . The allocation is “feasible” if: ∑ l∈L \0 N + xl ≤ M . The fact that bids may change from losing to winning explains why. or “history” of bids made by all bidders up to and including round t (t = 1. where xl denotes the package or bundle assigned to bidder l. These may appear to be complicating features: they make the auction less transparent for onlookers and might seem to create certain new strategic bidding issues.16 Frontiers of Theoretical Economics Vol. z ) denote the highest bid made by bidder l for package z up to and including time t and let Bl0 ( z ) ≡ 0 . A bidding strategy bl for any bidder l is a map from histories to new bids that satisfies the minimum bid restriction that. without these features. 3. l ∈ L \ 0 ) . One may equivalently describe l’s strategy in terms of the function Bl. it is necessary to specify whether bids from previous rounds remain binding on bidders. We assume for simplicity that the seller sets reserve prices of zero for all packages. for every package z. for all l ∈ L \ 0 . Nevertheless. z ) is the bid increment applicable to bidder l for package z at round t. in the SAAPB. In the SAA. 2. prices for individual items can never fall. … ).2 Round by Round Optimization An allocation is a vector x = ( xl ∈ N + . 3. as described in the following sub-sections. No. 1 [2002].bepress.com/bejte/frontiers/vol1/iss1/art1 .1 Bidding Strategies Let Ht denote the list. Article 1 Table 3: Sample Rounds in a Package Auction Item A Item B Package AB Round R Bidder X 4 0 0 Bidder Y 5 0 0 Bidder Z 0 0 6 Provisional Winning Bids 6 Round R+1 Bidder X 4 2 0 Provisional Winning Bids 5 2 Round R+2 Bidder Y 5 6 0 Provisional Winning Bids 4 6 illustrated in the table by the fall in the price of Item A from 5 in round R+1 to 4 in round R+2. requiring that Blt ( z ) ≥ Blt −1 ( z ) and that Blt ( z ) > Blt −1 ( z ) ⇒ Blt ( z ) ≥ Blt −1 ( z ) + I lt −1 ( z ) . straightforward bidding would not generally lead to such nearly efficient outcomes. blt ( z ) > Blt −1 ( z ) ⇒ blt ( z ) ≥ Blt −1 ( z ) + I lt −1 ( z ) . xl ∈ (1) http://www. 1. where I lt −1 ( z ) ≡ I l ( H t −1 .
Professor Charles Plott has called this strategy “bidding the gradient” and observed that it is consistent with the behavior of some bidders in his package auction experiments. 2004 . approximately. It holds that. l ∈ L. Properties (i) and (ii) of Theorem 1 characterize the main implications of straightforward bidding and provide the means to evaluate how well the model fits behavior found in experiments. or otherwise that bid plus one increment:  Blt −1 ( xl ) if xl = xl*t −1  B ( xl ) =  t −1 t −1  Bl ( xl ) + I l ( xl ) otherwise  t l (3) ˆ Bidder l’s potential profit from bidding on xl at round t is π lt ( xl ) = vl ( xl ) − Blt ( xl ) and its ˆ “optimal potential profit” is π lt = max(0. max xl π lt ( xl )) . then for all rounds t and packages xl . x∈X (2) Let us assume that in case there are multiple optima in (2). The straightforward bidding ˆ strategy B (i| v ) is the strategy in which l makes new bids only on the packages with the l l optimal potential profit and makes the minimum bid Blt ( xl ) on each such package. Let ε > 0 be an upper bound on the bid increments used during the auction. 3. maximizes the sum of the provisionally accepted bids: x*t ∈ arg max ∑ l∈L \0 Blt ( xl ) . The strategy can be described mathematically by: t  t ˆt ˆ t ( x | v ) =  Bl ( xl ) if π l ( xl ) = π l Bl l l  t −1  Bl ( xl ) otherwise  (4) Theorem 1. one may measure the distance between the strategies actually played and straightforward strategies by finding the smallest ε for Produced by The Berkeley Electronic Press. The provisional winning allocation for round t. xl ∈ [0. M ]) . vl ( xl ) − π lt ) |≤ ε .3 Straightforward Bidding and Limited Straightforward Bidding We now investigate a strategy in which bidders bid “straightforwardly” at each round on the package that has the highest profit potential. The lowest price that l can bid for any package xl at round t is l’s highest bid from the previous round if l was the provisional winner. bidders bid for the same profit on every package within an auction round and that losing bidders stop only when that “target profit” is nearly zero. If bidder l plays the straightforward strategy throughout the auction. | Blt ( xl ) − max(0. there is some fixed tiebreaking rule that depends only on the vector of best bids B t = ( Blt ( xl ). The second constraint says that each bidder’s package is described by a vector of nonnegative integers. Let X = {x|x satisfies (1)} be the set of feasible allocations. x *t .Ausubel and Milgrom: Ascending Package Auctions 17 The first constraint is the usual resource constraint limiting the quantities available. Empirically. and (i) (ii) at the final round T. either l is a provisionally winning bidder or π lT = 0 .
The agent for bidder l accepts as input a valuation profile vl and bids straightforwardly according to that profile. For example. and then use that to make predictions about how well the auction will replicate the other properties derived below. With this specification. shading the proxy instructions accordingly: vl ( xl ) = max {vl ( xl ) − π l . using the equation:  Blt ( xl ) if π lt ( xl ) = max π l . (6) While the use of negligible increments enables clean representations like (6). At the final round.4 The Ascending Proxy Auction In much of the remainder of this paper.bepress. In that case. then this description would entail no restrictions on how a bidder might bid.. bidders l ∉ T will have set π lt = 0 . so: max ∑ x∈X l∈L \0 Blt ( xl ) = max ∑ x∈X l∈L \0 [vl ( xl ) − π lt ] = max ∑ x∈X l∈L \0 vl ( xl ) − ∑ l∈T π lt .” i. 29 http://www. 1 [2002]. in which each buyer l instructs a “proxy agent” that bids on its behalf. 2001) versions of this model in which the instructions to the proxy agent can be changed periodically.. π l ) =  t −1  Bl ( xl ) otherwise . when the auction ends at some round t. We treat ties by augmenting the standard notion of Nash equilibrium in the manner suggested by In this paper.com/bejte/frontiers/vol1/iss1/art1 . the design may be a useful restriction on continuously-changing dynamic strategies. let T denote the set of bidders whose bids maximize total revenue at the final round. if bidders can change their proxy instructions at only limited times. We explore elsewhere (Ausubel and Milgrom. the highest bids made by l’s proxy agent up to any time t are characterized by a number π lt with Blt ( xl ) = max ( 0. each proxy agent can only receive instructions once. It will prove convenient to embed the straightforward strategies in a larger class of limited straightforward bidding strategies—ones in which bidder l bids straightforwardly except that it makes no bids that have a potential profit less than some target amount ˆ π l ≥ 0 . π l ) is described similarly to straightforward bidding. Limited straightforward bidding is implemented by the “semi-sincere strategy” in the proxy auction. we focus attention on a particular version of the model. However. it has the unfortunate byproduct of introducing the possibility of ties into the bidding. 0} . (Straightforward bidding is implemented simply by the “sincere strategy. 29 Observe that a bidder can utilize this proxy agent to implement either a straightforward bidding strategy or a limited straightforward bidding strategy.18 Frontiers of Theoretical Economics Vol. i. Article 1 which statements (i) and (ii) are true. vl ( xl ) − π lt ) . 1. while still maintaining many advantages of a dynamic auction design. No. To see this. We further limit attention to the case in which bid increments are negligibly small.  ( ) (5) 3. max x π lt ( xl ) ˆ ˆ  l ˆ Blt ( xl | vl .e. where vl is the valuation reported to the proxy agent. Observe that if bidders were able to revise their proxy instructions at every round. The limited straightforward bidding strategy Bl (i| vl .) Using a proxy converts the game into a direct revelation mechanism. the winning packages are among those that maximize the total reported value.e. it may allow bidders to reduce their planning costs or to make inferences in a common value setting. truthfully instructing the proxy: vl ( xl ) = vl ( xl ) .
v− l . and rejects all but its most preferred offer. x ) is an augmented Nash equilibrium. The strategy vl is said to be a best reply to v− l at x if the allocation x is consistent with (vl . The allocation x is required to be consistent with the strategy profile {vl }l∈L \0 . while a “Nash” equilibrium consists just of the corresponding strategy profile. (ii) x is not consistent with (vl′. v− l ) .30 Let π l ( v. an augmented equilibrium is a strategy profile and tie-breaking rule such that no bidder can. Matching Theory and the Core Our approach in this section is to analyze the ascending proxy auction as a kind of deferred acceptance algorithm in the sense that term is understood in the theory of twosided matching. The outcome specified in this way is the limit of the discrete increment case.” Each seller compares the offers it has in hand. but also an allocation x of the items. respectively. including any offers it may be holding from the previous round. We shall say that v is a Nash equilibrium if there is some x such that ( v . When the process ends. v− l ) and if there are no alternative strategy vl′ and allocation x′ such that: (i) x′ is consistent with (vl′. With “negligible” bid increments. These reports are processed in a series of rounds that mimic an imaginary market process. 30 Produced by The Berkeley Electronic Press.Ausubel and Milgrom: Ascending Package Auctions 19 Simon and Zame (1990). change the outcome in its favor. x ) is said to be an augmented Nash equilibrium if vl is a best reply to v− l at x for every bidder l. However. The strategy profile and allocation ( v . Package Auctions. bidder 1 bids 5-ε and bidder 2 bids 5. and (iii) π l (vl′. there is a unique pure strategy equilibrium in undominated strategies. any still unrejected offers become finally Readers unfamiliar with the Simon-Zame procedure should reflect on the example of a sealed bid auction for a single item worth 5 and 10 to the two bidders. 2004 . 4. x) . in the sense that x maximizes the total bid: x ∈ arg max y∈X ∑ l∈L \0 vl ( yl ) . Much of the remainder of this paper characterizes Nash equilibria of this ascending proxy auction. an equilibrium comprises not only a vector of strategies {vl }l∈L \0 for every bidder. in the event that there are multiple maximizers of the total bid. players on one side of a market—let us call them the “buyers”—make offers to the “sellers. by changing its bid. With discrete but small bid units ε. At that equilibrium. In our formal treatment of the ascending proxy auction. Informally. All deferred acceptance algorithms begin with a report of ordinal preferences over outcomes by the participants. v− l . x′) > π l (vl . x ) denote the payoff to bidder l from strategy profile v = {vl }l∈L \0 and allocation x. the unique equilibrium in undominated strategies specifies that both bidders bid 5 and that bidder 2 wins. each rejected buyer who wishes to buy more makes the offer that ranks next highest on its preference list. The process iterates until no new offers are made. the tie-breaking rule is specified as part of the solution. v− l ) . At each round. At the next round.
In particular. That is. for any specification of the preferences of the others.w) that is associated with the package economy. A second recurring result in matching models is that the core allocation to which the deferred acceptance algorithm converges is the most preferred point in the core for each buyer and the least preferred point in the core for each seller. However.20 Frontiers of Theoretical Economics Vol. Deferred acceptance algorithms arise both in models where there is no exchange of money—the so-called “marriage problem” and “college admissions problem” are examples—and in ones where money is exchanged. In this section. with buyers shouting offers and the seller holding onto its best offer only until a better offer is made. As part of the development. we find analogues for the results described above and other typical results of matching theory.. Straightforward bidding by the proxy bidders plays an essential role in our analysis: it enforces the condition found in all deferred acceptance algorithms that the buyers first make the offers they prefer most and proceed monotonically to the offers they prefer least. To make formal sense of these claims. A second is sometimes interpreted to mean that. Most closely related to our analysis is one by Kelso and Crawford (1982). we begin by defining a coalitional form game (L. to be discussed below.e. A recurring result in matching models with and without money is that the outcome of the deferred acceptance algorithm is a “stable match” or core allocation of the matching game. the result of the algorithm when a buyer reports truthfully is at least as favorable as when it reports untruthfully. we explore the ascending proxy auction as a kind of deferred acceptance algorithm. and any given untruthful strategy leads to strictly worse outcomes for some profile of preferences of the other players. That is. this implies that such a core point exists.com/bejte/frontiers/vol1/iss1/art1 . Article 1 accepted. the outcome has the property that there is no coalition of players that can match or trade among themselves in a way that coalition members all prefer to the outcome proposed by the algorithm. Each of these results relies on certain assumptions about preferences. from each player’s point of view. No. This class of algorithms acquires its name from the fact that even the best offers are not finally accepted until after the last round. w(S) = 0 if 0∉S) and otherwise is defined by the following expression: http://www. 1. it is not generally a dominant strategy for the sellers to report their preferences truthfully.bepress. stopping at last when no unmade offer is preferred to the no-trade outcome. the parties on the other side of the market are “substitutes” in a limited sense. this result implies that the core of the matching game is non-empty. The ordinary ascending “English” auction is an example of the latter kind. A third result found in some matching models is that it is a dominant strategy for each buyer to report its preferences truthfully. One is that the players on one side of the market—the buyers or the sellers—wish to have just one trading partner. 1 [2002]. The coalitional value function w is zero if the seller is not a member of the coalition (i. In particular. who analyzed a kind of simultaneous ascending auction in which the bidders are firms and the sellers are workers with preferences over both the employer’s identity and income.
π T is unblocked. In the ascending proxy auction. w) = π : w( L) = ∑ l∈L π l . According to the process. for all t. the core is the set of profit allocations that are feasible for the coalition of the whole and unblocked by any coalition S. given the reported preferences. Buyers who are not part of the provisionally winning coalition at t but who have demanded a positive profit then reduce their profit demands at the next round. (ii) the characterization of the { } proxy bidder behavior given above. S ⊂L t The five steps in (9) follow from (i) the definition of π 0 . the final payoff allocation is in the core: π T ∈ Core( L. and (v) the definition of w. Theorem 2. vl ( xl ) − π lt ) = max max ∑ l∈S \0 [vl ( xl ) − π lt ] = max max ∑ l∈S \0 [vl ( xl ) − π lt ] S ⊂L x∈ X S⊂L x∈ X x∈ X x∈ X (9) = max w( S ) − ∑ l∈S \0 π lt . coalition S then offers the seller revenue equal to the coalition value w(S) minus the sum of the members’ profit demands. The provisionally winning coalition is the one that offers the highest price. in particular. An immediate consequence of (9) is that for all coalitions S that include the seller. Produced by The Berkeley Electronic Press. the value of a coalition that includes the seller is the maximum total value the players can create by trading among themselves. This coalitional-bargaining interpretation of the auction process leads to the following characterization of the outcome. We must show that π T is feasible and not blocked by any coalition.Ausubel and Milgrom: Ascending Package Auctions 21 w( S ) = max ∑ l∈S vl ( xl ) . Hence. the payoffs vector π t is unblocked. in which each buyer at round t makes a profit demand π lt ≥ 0 . and the same result holds trivially for coalitions excluding the seller. { } suggests an interpretation of the package auction as a coalitional bargaining process. w( S ) ≤ ∑ l∈S π lt . 2004 . { } (8) Thus. The process ends when all nonwinning buyers are demanding profits of zero and so do not reduce their demands any further. (iv) maximization. the seller’s revenue at any round t of the ascending proxy auction is given by: t ˆ π 0 = max ∑ l∈L \0 Blt ( xl | vl ) = max ∑ l∈L \0 max ( 0. Proof. (iii) the observation that S = { l | vl ( xl ) − π lt ≥ 0} t The characterization that π 0 = max w( S ) − ∑ l∈S \0 π lt S ⊂L achieves the maximum. We then define the core by: Core( L. x∈ X (7) Thus. Given a vector of reported values. w( S ) ≤ ∑ l∈S π l for all S ⊂ L . w) .
v− l ) to be the projection of X * (π l . including the seller.. No.) We claim that vl ( xl ) ≡ max {vl ( xl ) − π l* . If π l* = 0 . However. Then: { } (a) (b) in the ascending proxy auction. meaning that bidder l’s profit cannot exceed π l* . 0} is a best reply for bidder l to v− l at x* .4 above. x* ) + ε . v− l ) < π l (vl . Thus. w( L) ≤ ∑ l∈L π lT = ∑ l∈S * π lT = w( S *) ≤ w( L) (10) Hence. ■ The power of restricting players to bid through proxy agents can be seen immediately in the next theorem. earns l a profit of exactly π l* . and Let π = sup {π l : X l* (π l . no strategy Bl can give bidder l an appreciably better payoff than he earns from vl in the ascending proxy auction. Then by (9) and since π lT = 0 for any “losing bidder” (l ∉ S *) . and let x* ∈ arg max x∈X vl ( xl ) + ∑ k ≠l vk ( xk ) . this provides a justification for restricting players to bid through proxy agents. v−l . and in the ascending package auction with small positive bid increments. If a player’s opponents are allowed to use arbitrary pure strategies in a dynamic package auction. then the player’s best reply may need to be arbitrarily complex. π l* is the largest positive amount that X l* (π l . w( L) = ∑ l∈L π lT .com/bejte/frontiers/vol1/iss1/art1 . v− l ) on bidder bidder l can uniformly shade his valuation while still being efficiently allocated some items or zero if no such positive amount exists. there is a semi-sincere strategy vl that is a best reply for bidder l to v− l at x* . any bid by bidder l that is shaded by more than π l* cannot win. 1 [2002]. which establishes that π T is feasible. then x* awards a package to l which. 1. By Theorem 2. v− l ) ≠ ∅ or π l = 0} . We have: Theorem 3. then the player necessarily has a best reply in limited straightforward strategies. Article 1 Let S* be the final (round T) winning coalition.e. for every ε > 0.22 Frontiers of Theoretical Economics Vol. If π l* > 0 . (Thus. if l reports vl . Suppose that bidders in L \ l use proxy bidders and report the valuation profile v− l . if a player’s opponents are restricted to bidding through proxy agents as described in Section 3. there exists δ > 0 such that if δ is an upper bound on the bid increments used in the auction. and hence yields an allocation x ∈ arg max y∈X ∑ l∈L \0 vl ( yl ) . then the l from every strategy Bl is bounded by: payoff to π l ( Bl . the payoff allocation resulting from any play of the proxy agents is an element of the core (defined with respect to the proxy instructions). http://www. In turn. Proof. v− l ) = arg max x∈X max {vl ( xl ) − π l . the proof is complete.bepress. 0} + ∑ k ≠ l vk ( xk ) define * l { } l. which itself can be implemented with the same form of proxy agent. i. (a) Define X * (π l .
we have already described the incentive for large bidders to distort their reported preferences. the bidder playing σ would place new bids only when the auction would otherwise end with S winning and. In the ascending package auction. if πl > 0. This means that regardless of others’ strategies. which differentiates it from many other existing auction formats.31 We now argue that the temptation is substantial: σ is always a best reply. let V−l (δ ) denote the set of all profiles of valuations for bidder l’s opponents satisfying | vk ( x) − vk ( x) |≤ δ . because bidding does not close until there are two rounds with no new bids. but to know that strategy it may need to know the corresponding profit level πl. there could be “threat equilibria. l would eventually bid just enough so that the payoff profile is unblocked by S ∪ {0} .Ausubel and Milgrom: Ascending Package Auctions 23 (b) Given δ > 0 . An analysis using Theorem 3 also explains the need for either activity rules or proxy bidders to pace the ascending package auction. regardless of which limited straightforward strategies the others may play. this would be feasible in FCC Auction No. According to theorem 3. so strategy σ is a best reply. the bidder could be tempted to adopt the following “maximal delay strategy” σ instead: place no bid until all bidding stops and thereafter bid straightforwardly. it is evident that no lower bids could win against the opposing profile. the ascending proxy auction makes threats such as those impossible: a bidder always has a best reply in which it does not distort its relative valuation among packages. withholding demand in order to reduce prices. and avoid the need for activity rules that are necessary to speed an unstructured auction but can distort the auction outcome. In the FCC’s simultaneous ascending auction. Treating bid increments as negligible. Without that information. 2004 . Produced by The Berkeley Electronic Press. for all k ≠ l and for all x ∈ [0. suppose all bidders except bidder l play limited straightforward strategies. the bidder can bid the full incremental value for any additional package it wishes to acquire.” in which a bidder is deterred from bidding for the packages it most wants to acquire by the threat that its competitors will retaliate to drive up its prices. for example. Suppose that the winning coalition is S when strategy σ calls for l to begin bidding. 31. without fear that doing so will reduce its payoff in the game. The conclusion then follows from the fact that lim π l* (δ ) = π l* . M ] . regardless of the values or profit targets of the other bidders. To illustrate the potential problem. According to theorem 3. Then. no such incentive exists in the ascending proxy or package auctions. v−l ) ≠ ∅ for some v−l ∈ V−l (δ )} . They rule out certain kinds of delay. besides the package it expects to win. bidder l has a best reply that is a limited straightforward strategy as well. ■ Theorem 3 identifies a highly desirable property of the ascending proxy auction. 31 But for the activity rule. allow the use of smaller (more efficient) bid increments. Let π l* (δ ) = sup {π l : π l = 0 or X l* (π l . According to theorem 3. Auctions using proxy bidders thus have a natural speed advantage over less automated alternatives. Bidder l’s payoff from v− l δ →0 any strategy Bl played in the ascending package auction against proxies with reports of is bounded by π l* (δ ) .
π k − π k ) (11) http://www. and others payoffs are as specified by π. using (9): ˆ β ( S ) ≥ w( S ) − ∑ k∈S \0 max(π k . Then there is some player l and some unilateral deviation for that player that ˆ ˆ leads to a winning coalition T (T ∋ l ) and profit outcome vector π . We show that β ( S ) > β (T ) . π and β (T ) . The strict inequality ˆ in the second step follows because l ∈ T \ S and π l > π l .com/bejte/frontiers/vol1/iss1/art1 .bepress. Suppose π ∈ Core( L. Moreover. For bidder l. The third step follows by selection of S. reporting the valuation function of vl (i) ≡ max {vl (i) − π l . π k − π k ) ˆ = π 0 − ∑ k∈T \0 max(0. Article 1 5. We would like to show that the reports vl (i) are a Nash equilibrium of the ascending proxy auction. there exists a Nash equilibrium of the ascending proxy auction with associated payoff vector π. Equilibrium of the Ascending Proxy Auction Let us say that a payoff profile π ∈ Core( L. 1 [2002]. We begin by observing that every such bidder-Pareto-optimal point is the payoff vector associated with a Nash equilibrium of the ascending proxy auction. there would be a point in the core at which l gets π l + ε . the seller gets π 0 − ε . Theorem 4. w) is a bidder-Pareto-optimal point in the core. w) is a bidder-Pareto-optimal point in the core if there is no π ′ ∈ Core( L. contradicting the hypothesis that T is the winning coalition. and the fifth and sixth by the ˆ definitions of T. π l > π l . the fourth because π ∈ Core( L. then π (v ) is a bidder-Pareto-optimal point in Core( L. w) with π ′ ≠ π and π l′ ≥ π l for every bidder l. w) . Since π is bidder Pareto optimal. No. if v is a Nash equilibrium in semi-sincere strategies at which losing bidders bid sincerely.) Let β ( S ) and β (T ) denote the highest total revenue associated with bids by the bidders in coalitions S and T during the proxy auction. π k − π k ) ˆ = w(T ) − ∑ k∈T \0 π k = β (T ) The first step in (11) follows from the proxy rules: any losing bidders in S stop bidding only when their potential profits reach the specified levels. Proof. ˆ and for all bidders k ∈ T . One strategy profile that supports π has each bidder l playing a semisincere strategy. contradicting bidder Pareto optimality. Suppose π is a bidder-Pareto-optimal point in Core( L. w) . Indeed. the proxy strategies imply that π k ≥ π k . π k ) ˆ > w( S ) − ∑ k∈S \0 π k − ∑ k∈T \0 max(0. given the specified deviation by bidder l. 1. w) . for some ε > 0 .24 Frontiers of Theoretical Economics Vol. 0} . Suppose not. Then. there is a coalition S such that l ∉ S and w( S ) = ∑ k∈S π k (Otherwise. ˆ ≥ w(T ) − ∑ k∈T \0 π k − ∑ k∈T \0 max(0.
32 Provided that the others are playing a pure strategy profile. thereby increasing its profits to π l′ . π − l ) ∈ Core( L. A related example arises when there are two identical items and three bidders. A and B.32 there do exist Nash equilibria whose allocations are different from those identified in theorem 4. Then. there is a Nash equilibrium (of both the Vickrey and ascending proxy auctions) in which A bids 12 and B bids 4. For a first example. and another in which A bids 6 and B bids 10. respectively. the incentives for parking is an important reason to prefer proxy implementations over more free-form package auctions. then the semi-sincere strategies described in the theorem are similar: they merely move the no-trade point up in the bidders rank order list without changing the relative ranking of any other pair of outcomes. All of the equilibria described here are supported by the insincere bidding of losing bidders. If one regards the reports vl as specifying ordinal preferences over allocation-payment pairs. In the environment studied here. That paper employs coalition-proofness to single out equilibrium strategy profiles at which payoffs are bidder-Pareto-optimal points in the core. Bidder l can deviate by reporting max {vl − π l + π l′ . suppose there is just one good and two bidders. using the version of the algorithm in which the men make the offers. there is a Nash equilibrium of both the Vickrey and ascending proxy auction in which A and B bid zero for every package and C bids 0 for a single item and 15 for the pair. which is ruled out in the second half of theorem 4. We describe some of those here and compare them with the corresponding Nash equilibrium in the Vickrey auction. in which it delays making serious bids until late in the auction. ■ If π is not bidder-Pareto-optimal in the core. 0} instead of reporting vl . it is the use of package bidding that gives the proxy auction its unique strategic structure. w) . for example. In a previous version of this paper. Indeed. In the matching theory “marriage” model. Although individual bidders suffer no loss by limiting themselves to semi-sincere strategies.Ausubel and Milgrom: Ascending Package Auctions 25 (π l′. Other studies of many-to-one matching have found no equilibrium of this sort. Then. in addition to the familiar equilibrium in which the parties bid 5 and 10. B and C. leading to equilibrium strategies that coincide exactly with the equilibrium reports to proxy bidders identified in theorem 4. Produced by The Berkeley Electronic Press. Parking strategies can sometimes be best implemented by avoiding semi-sincere strategies. then a bidder can benefit from a “parking” strategy. with values of 5 and 10 respectively. Bidders A and B each want one item only and have a value of 10. A. 1990). then there exists l and π l′ > π l such that Theorem 4 is related to theoretical results about Nash equilibrium behavior in some other deferred acceptance algorithms. while bidder C values only the pair and has a value of 15. The theorem is also closely related to the Bernheim-Whinston (1986) equilibrium strategies in their menu auction game. we showed that if the opposing strategy profile is uncertain. the menu auction is simply a sealed-bid package auction in which the auctioneer takes the collection of bids that maximizes its total revenue and allocates the goods accordingly. there is a Nash equilibrium in which all men report truthfully and each woman plays a “semisincere” strategy—moving the no-match outcome up in her rank order list to just below that woman’s most preferred outcome in the core (Roth and Sotomayor. 2004 .
π l = w( L) − w( L \ l ) . Article 1 In models with multiple goods. it is not just losing bidders whose behavior can lead to these odd-seeming outcomes: insincere incremental bids by winning bidders for items and packages they do not win have consequences much like insincere bidding by losing bidders. l . w) . A and B. For bidders (l ∈ L \ 0) . Then there is a Nash equilibrium (of the ascending and Vickrey auctions) in which A and B both bid 12 for one item and 12 for the pair. π l ≤ max {π l | π ∈ Core( L. π l ≥ π l for all π ∈ Core( L. leading to profits of 10 and 9. we focus the second half of theorem 4 on equilibria using semi-sincere strategies. the payoff vector defined by π 0 = w( L \ l ) . if π is in the core.com/bejte/frontiers/vol1/iss1/art1 . while B’s corresponding values are 9 and 18. Theorem 5. ∑ Now suppose that π is a feasible profit allocation with π l > π l for some l ≠ 0. The core contains a bidder Pareto-dominant point if and only if the Vickrey payoff vector π is in the core. π l = w( L) − w( L \ l ) .e. it is bidder Pareto-dominant. 1. then it is bidder Paretodominant. Let π denote the Vickrey payoff vector. 6. A’s values are 10 for a single item or 20 for a pair. the existence is fully characterized by a simple condition relating two of the central concepts of this paper: a bidder-Paretooptimal core outcome exists if and only if the Vickrey outcome is contained in the core. By inspection. No. However. http://www. w) . Then k∈L \ l π k = w( L) − π l < w( L \ l ) . suppose there are two identical items and two bidders. π ∉ Core( L.33 To identify equilibria that we believe are most likely to be played. ■ Theorem 6.bepress. 1 [2002]. upon which the coalition cannot improve. 33 This equilibrium is even coalition proof. satisfies π ∈ Core( L. Proof. In general. in matching models. We have: core.. By Theorem 5. for all l ∈ L \ 0 : π l = w( L) − w( L \ l ) = max {π l | π ∈ Core( L. the analysis can often be further refined to conclude that the outcome is the point in the core that is unanimously most preferred by the buyers and unanimously least preferred by the sellers. i. That is. the payoffs associated with the dominant-strategy equilibrium of the generalized Vickrey auction. w)} . Vickrey Outcomes and the Core We saw in Section 5 that the ascending package auction may be interpreted as a deferred acceptance algorithm and that—similar to deferred acceptance algorithms in two-sided matching models—the outcomes will tend to be core allocations. so L \ l blocks the allocation. A bidder’s Vickrey payoff π l is l’s highest payoff over all points in the Proof. such a point need not exist in the package auction environment. for j ≠ 0. w)} . w) and l ∈ L \ 0 . Hence. Hence. while the seller’s payoff is π 0 = w( L) − ∑ l∈L \0 π l . If π is in the core. So. and π j = 0 .26 Frontiers of Theoretical Economics Vol. For example. but as we shall now see.
Bidders j and k. Once losing bidders have no more profitable bids. under exactly this condition. When Sincere Bidding is an Equilibrium We found in Section 6 that. However. to the extent that bidder j bids more. bidder k can get away with bidding less. this conclusion is not quite right. hoping to avoid making unnecessary concessions.Ausubel and Milgrom: Ascending Package Auctions 27 For the converse. the proof has identified bidders j and k for which there is a nondegenerate bargaining problem. viewed in the space of payoffs. we show that no other core point can be bidder Pareto-dominant. the payoff profile is unblocked. as the following example demonstrates. if the Vickrey payoff vector is an element of the core. it is helpful to depict the licenses as follows: Geographic Space Bandwidth West-20 West-10 East-20 East-10 Produced by The Berkeley Electronic Press. in situations where the condition of Theorem 6 does not hold. 2004 . the profile lies in the core and no bidder can gain by continuing to bid. According to Theorem 4. No opposing bidder k benefits if some bidder j bids a higher amount than would yield a payoff of π j . ■ In situations where the condition of Theorem 6 holds. Using semi-sincere strategies in the proxy auction amounts to deciding how far to concede in a bargaining game. each bidder could simply use the straightforward bidding strategy and this would yield an equilibrium. At every round. In order to understand that the following bidder valuations are sensible. if the condition of Theorem 6 does not hold. bids in this situation are similar to concessions in a bargaining game. ˆ For suppose π ≠ π ∈ Core( L. no bidder has any incentive to manipulate the adjustment path to enter the core at any point other than the bidder-Pareto-preferred point. a bidder in the package auction may bid slowly. 7. if the condition of Theorem 6 holds. Indeed. Conversely. must bid some requisite amount of money in order to block some other bidder. together. there is effectively a coincidence of interests among bidders in the ascending package auction. Just as a bargainer may wish to avoid making concessions to see how much others are willing to concede. then there exists no fundamental bargaining problem among the bidders. Furthermore. the outcome of such bargaining can be any point that is bidder-Pareto-optimal in the core. However. w) . then individual bidders do have an incentive to manipulate where the adjustment path enters the core. It might then seem that. Suppose that there are four spectrum licenses. there is some bidder j for whom ˆ π j < π j and some other core point at which j receives π j . Then by theorem 5. Theorems 4 and 6 together provide some insight into the geometry of the adjustment process of the ascending package auction.
West-10) = 10. Bidders 2 and 3 to bid 15 each. and so there is nothing further to induce Bidder 1 to raise his bid. we develop a stronger condition that renders truthful reporting an equilibrium of the ascending proxy auction. Bidder 5 wants the full 30-MHz of spectrum in the West.28 Frontiers of Theoretical Economics Vol. 5. v4(East-20. and Bidders 4 and 5 to bid 10 apiece. which this paper supercedes. 34 For example. and v5(West-20. by π l ( S ) ≡ w( S ) − w( S \ l ) for l ∈ S \ 0 and π 0 ( S ) ≡ w( S ) − ∑ l∈S \0 π l ( S ) . With starting prices of zero. straightforward bidding would lead Bidder 1 to bid 10. Theorem 7. is an element of the core. Article 1 There are five bidders. he still would have won the West-10 and East-10 licenses. 35 This condition (which is the opposite of the “convexity” condition introduced in Shapley (1971)) and a version of theorem 7 first appeared in Ausubel (1997b). Nevertheless. (20. if each bidder raises his bid by the same bid increment whenever he is not a provisional winner. Observe that the Vickrey payoff vector. reached a bid of 10. l∈S \0 for all coalitions S (0 ∈ S ⊂ L ) .bepress. If Bidder 1 had instead limited his bidding to 0. 1 [2002]. irrevocably. East-20) = 20. 3} is a provisional winner ¼ of the time. 2} is a provisional winner ¼ of the time. 0. No. when his bidder-Pareto-optimal core payment equals 0. 10. 1. Bidders 2 and 3 desire a 20-MHz band of spectrum covering both East and West. Thus: v1(West-10. Coalition {1. For the next theorem.1 Buyer-Submodular Values In this section. East-10) = 10. 0). We will require not only that the Vickrey payoff vector is an element of Core(L. But he has already. The coalitional value function w is buyer-submodular if for all l ∈ L \0 and all coalitions S and S ′ satisfying 0 ∈ S ⊂ S ′ . v3(West-20. That predecessor also included the following necessary condition for the conclusions of theorem 7: w( L ) − w( L \ S ) ≥ ∑ ( w ( L ) − w( L \ l ) ) . East-10) = 10. Bikhchandani and Ostroy (2002) subsequently developed the implications of these conditions for dual problems to the package assignment problem.34 7. Bidder 4 wants the full 30-MHz of spectrum in the East. v2(West-20. straightforward bidding is likely to lead Bidder 1 to pay a positive price. Bidders 4 and 5 drop out of the auction.35 The next two theorems apply to the case in which the coalitional value function is buyer-submodular. East-20) = 25. Bidder 1 desires a 10-MHz band of spectrum covering both East and West. corresponding to Bidder 3 paying 20 for his desired licenses and Bidder 1 paying 0 for his desired licenses. Definition. which we denote π ( S ) . we define the “Vickrey payoff vectors” for the cooperative game restricted to the members of coalition S. Coalition {1. The following three statements are equivalent: (1) The coalitional value function w is buyer-submodular.com/bejte/frontiers/vol1/iss1/art1 . and Coalition {4. At this point. http://www. w( S ∪ {l}) − w( S ) ≥ w( S ′ ∪ {l}) − w( S ′) . we see that (so long as Bidder 2 remains in the auction). 5} is a provisional winner ½ of the time. 0. with all singletons and all other doubletons valued at zero. but that the analogous condition holds for all subcoalitions S that include the seller.w).
. that is... 2004 . the second from π ∈ Π .. the third from the definition of π . For the reverse inclusion. S ′ = {0. It is immediate that (3)⇒(2). w) . This follows because: (1) holds. l}) − w({0. so the payoff allocation π S is blocked by coalition S \ ij . Suppose that suppose π S ∈ Π S and that S = {1.. Hence.Ausubel and Milgrom: Ascending Package Auctions 29 (2) For every coalition S that includes the seller. and the fourth from the condition that bidders are substitutes. (2) also fails. w( S ′) − w( S ′ \ i ) > w( S ′ \ j ) − w( S ′ \ ij ) . there exists a player i such that w( S ) − w( S \ i ) is not weakly decreasing in S.. j ∈ S ′ \ 0 such that. Let S′ be a subcoalition of S including the seller.. We show that the blocking inequality associated with coalition S′ is satisfied.. ∑ l∈S ′ π l = w( S ) − ∑ l = k +1π l |S | |S | | L| | L| ≥ w( S ) − ∑ l = k +1π l ( S ) = w( S ) − ∑ l = k +1 [ w( S ) − w( S \ l )] ≥ w( S ) − ∑ l = k +1 [ w({0. ∑ l∈S ′\ij π l ( S ′) = w( S ′) − π i ( S ′) − π j ( S ′) = w( S ′ \ i ) + w( S ′ \ j ) − w( S ′) < w( S ′ \ ij ) . 0 ≤ π l ≤ π l ( S ) for all l ∈ S \ 0} (12) Proof. We first establish that truthful reporting leads to the Vickrey payoff vector.. that is. For (2)⇒(1). Then. l − 1})] = w( S ) − [ w( S ) − w( S ′)] = w( S ′) The first step in (13) follows from feasibility of π. Define Π S = {π S | ∑ l∈S π l = w( S ). suppose (1) fails. w) = {π S | ∑ l∈S π l = w( S ).. w) ⊂ Π S . there is a unique core point that is Pareto-best for the buyers and. ■ Theorem 8.. (1)⇒(3). truthful reporting is a Nash equilibrium strategy profile of the ascending proxy auction and leads to the generalized Vickrey outcome: π T = π . 0 ≤ π l ≤ π l ( S ) for all l ∈ S \ 0} . k} . (3) For every coalition S that includes the seller. So. indeed: Core( S . Suppose there is some round t at which π lt < π l .| S |} . Suppose that the coalitional value function is buyer-submodular. Hence.. Proof.. Then.. w is not buyer-submodular... Then. the Vickrey payoff vector is in the core: π ( S ) ∈ Core( S . We show that l is necessarily part of the winning coalition at that round. It follows from Theorem 5 that Core( S . say. (13) Produced by The Berkeley Electronic Press. there is a coalition S′ including the seller and players i. Let S be any coalition including the seller but not bidder l.
then it is a dominant strategy for l to report its preferences truthfully. respectively. truthful reporting is always a best reply. l’s profit π lt is at least π l minus one bid increment ε. Taking the bid increment to zero for the ascending proxy auction proves that for l ≠ 0 . Article 1 w( S ) − ∑ k∈S π kt < w( S ) − ∑ k∈S π kt + (π l − π lt ) t = w( S ) − ∑ k∈S ∪{l } π k + w( L) − w( L \ l ) t ≤ w( S ) − ∑ k∈S ∪{l } π k + w( S ∪ l ) − w( S ) t = w( S ∪ {l}) − ∑ k∈S ∪{l }π k (14) So. ■ When there may be multiple goods of each kind. when l’s value is vl . and the reverse inequality follows from theorem 5. we show that truthful reporting is a best response to all other bidders reporting truthfully. 36 http://www. designate the marginal values and maximum desired quantities for each kind of good. which is the payoff that results from truthful reporting. 1. reporting honestly ˆ can possibly pay strictly more than reporting v1 . ■ Theorem 9. l’s payoff to any strategy is bounded above π l = w( L) − w( L \ l ) . Since the total payoff to all players is at most w( L) . Finally.30 Frontiers of Theoretical Economics Vol. Then. theorem 5 implies that the payoff to coalition L \ l is at least w( L \ l ) . Then. For all elements of the package M − z . It remains to show that there is no other uniform best reply. For any bidder l and any report by that bidder. Suppose there is a single good of each type. the corresponding coalitional value function is buyersubmodular. No. j’s 2 value for each other good is zero. Proof. where the vectors λ and z . Consider the restricted ascending proxy auction games in which each buyer l may announce only preferences satisfying vl ∈ Vl . Second. as follows. Hence.bepress. 1 [2002]. we may specify that there is just one other buyer. a similar result can be obtained using the set Vadd of valuations of the form v( z ) = λ i( z ∧ z ) . so we may specify that by specifying a value for each good individually. π lT ≥ π l . j’s value for good ˆ ˆ m is v j ( M − z + 1m ) − v j ( M − z ) = 1 ( vl ( z ) − vl ( z − 1m ) + vl ( z ) − vl ( z − 1m ) ) . say buyer j. If bidder l’s actual preferences are in Vl and there are at least two buyers (including l). Also. By theorem 8.36 Suppose that for all valuation profiles v ∈×l∈L Vl . j’s values are so high that j is assigned at least M − z for either report by buyer l. by our assumptions. ˆ Let l have actual preferences vl and let vl ≠ vl . In particular. l’s report determines only the allocation of good ˆ m.com/bejte/frontiers/vol1/iss1/art1 . there is some package z ˆ ˆ and item m such that vl ( z ) − vl ( z − 1m ) ≠ vl ( z ) − vl ( z − 1m ) . Let Vadd denote the set of additive valuation functions and suppose that Vl ⊃ Vadd for all l ∈ L \ 0 . and reporting honestly pays higher than reporting vl by an amount equal to 1 2 ˆ ˆ ( vl ( z ) − vl ( z − 1m ) + vl ( z ) − vl ( z − 1m ) ) . who has strictly positive values. Bidder j’s values are additive.
for any fixed package z that might be assigned to a new coalition member. So. Theorem 11 holds that if the goods are substitutes for all bidders. Mk = 1. the substitutes condition is satisfied if and only if xlm ( p) is nondecreasing in each pj for j ≠ m. is decreasing in the coalition size. By definition. Goods are “substitutes” for bidder l if and only if the indirect utility function ul (⋅) is submodular. which is the new member’s value of the package it receives minus the coalition’s opportunity cost of that package. that is. For present purposes. other kinds of valuations are generally inconsistent with this conclusion. Indeed. In the present model. Produced by The Berkeley Electronic Press. we shall suppose that the bidders distinguish individual goods so that for all k. Proof. According to theorem 12. the partial derivative is equal to -xlm. Remark. If goods are substitutes for all bidders. the indirect utility function is absolutely continuous and partially differentiable almost everywhere in each good’s price pm. this has implications about the structure of the core. where xlm is the quantity demanded of good m at price vector p.Ausubel and Milgrom: Ascending Package Auctions 31 7. payoffs in the Vickrey auction. Thus. 2004 . the partial derivative exists for precisely those price vectors at which l’s demand for good m is single-valued. (Establishing this with indirect utility functions occupies the main part of the proof. The proof outline reflects the following intuition: When goods are substitutes. Second. the opportunity cost to coalition S of any good is increasing in the size (inclusiveness) of the coalition. ■ Theorem 11.37 the following conclusions are true: First. then there exist valuation profiles such that the coalitional value function is not bidder-submodular. however. Since demand in discrete models is not generally single-valued. assumptions about the coalitional value function are not subject to further analysis. and bidder incentives in the proxy auction. By an envelope theorem.) Consequently.2 When Goods are Substitutes In models for which the coalitional game (L. goods are substitutes if and only if ∂ul / ∂pm is nonincreasing in each pj for j ≠ m. starting from any given package. we shall say that goods are substitutes if the demand for each good is nondecreasing in the prices of other goods when attention is restricted to the set of prices for which demand is single-valued. Theorem 10. the cost of giving away that package is increasing in the coalition size. 37 See Milgrom and Segal (2002). In view of Theorems 7-9. at those prices p. if the set of individual valuations contains (i) the additive valuations and (ii) any valuation for which the goods are not substitutes. This allows express the substitutes condition conveniently using the indirect utility function.w) is the primitive. Corollary 4. if and only if ul is submodular. Third. It is natural to ask: what conditions on bidder valuations imply that bidders are substitutes in the coalition game? We answer that question with two theorems. which expresses the buyer’s maximum utility at a given price vector p: ul ( p ) = max z {vl ( z ) − p i z} . then the coalition value function is bidder-submodular. coalition values are not primitive—they are derived from individual package values. then the coalitional value function is bidder-submodular. the incremental value of an additional member.
vS ( z ) ≥ min p∈[0. ■ http://www. then pm ( S | z ) = vS ( z ) − vS ( z − 1m ) . Hence. The corresponding coalition indirect utility function is given by: uS ( p) = max z {vS ( z ) − pi z} = ∑ l∈S ul ( p ) . To see this. B ]N {uS ( p ) + pi z} . z − 1m ∈ arg max z′ vS ( z ′) − p( S | z )i z ′ ... The feasible set of prices is a closed interval. By inspection. pm ( S | z ) = vL ( M ) .B ]N {uS ( p ) + pi z} . applying the Topkis monotonicity theorem. because each such good is priced at vL ( M ) . Fix any package z and let B be a large number that exceeds the incremental value of any good to any coalition. For all p. for each m such that xm = 0 . vS ( z ) = min p∈[0. Hence. (16) (15) The objective function in (16) is continuous.. By construction. vS ( M ) − vS ( z m ) = ∑ j = m +1 ( vS ( z j ) − vS ( z j −1 ) ) = ∑ j = m +1 p j ( S | z j ) . Then. z ∈ arg max z ′ vS ( z ′) − p( S | z )i z ′ . Let S be a coalition that includes the seller.. Coalition S’s value for a package z is given by vS ( z ) = max x∈X ( z ) ∑ l∈S vl ( xl ) . We claim that if zm = 1 . z − 1m ∈ arg max z′ vS ( z ′) − pε i z′ for all ε>0. Article 1 Proof. The feasible set X(z) incorporates two restrictions: that only the quantity vector z is available and that each buyer may acquire only zero or one good of each type. By the preceding paragraph. 0. and hence pm ( S | z ) = vS ( z ) − vS ( z − 1m ) . note first that by (15) and (16). 1. Hence. Then. 0) denote a vector with n initial 1’s. Notice that w( S ∪ {l}) − w( S ) = max z vS ( z ) + vl ( M − z ) − vS ( M ) . set ′ ′ pε = p( S | z ) + ε 1m ...bepress.32 Frontiers of Theoretical Economics Vol. taking pm = 0 for all m such that zm = 1 and pm = B otherwise leads to uS ( p ) = vS ( z ) − pi z . We may suppose without loss of generality that z = z m for some m. So. demand for good m at price vector pε is zero. the right-hand expression is a maximum of nonincreasing functions of S. so vS ( z ) ≤ min p∈[0. By definition of p ( S | z ) .1. so uS (i) is also submodular. n n (17) This difference is nondecreasing in S since each term is so.. By the theorem of the maximum. uS ( p ) ≥ vS ( z ) − pi z . By the condition of substitutes. the same must hold for ε = 0. which is an isotone (“weakly increasing”) function of S. demand ′ for the remaining goods is undiminished. B ]N {uS ( p ) + pi z} . vS ( z − 1m ) − p( S | z )i( z − 1m ) = vS ( z ) − p ( S | z )i z . that is. Suppose zm = 1 and letting ε>0. 1 [2002]. the set of minimizers has a maximum element p ( S | z ) ..S). So. antitone (“weakly decreasing”) and submodular in p and has weakly decreasing differences in (p. No. demand for goods j for which z j = 0 is zero is zero at price vector ′ pε .com/bejte/frontiers/vol1/iss1/art1 . Let z n = (1. so w( S ∪ {l}) − w( S ) is itself nonincreasing in S.
Proof. p. Then. with the property that ′′′ ′′′ xm = 0 and xn = 1 is strictly suboptimal at price vector p. By Berge’s theorem. v3 ( x) = pm xm + pn xn . Since x′′ is the unique ˆ optimum for buyer 1 at price vector ( p− m . x1m ( pm . contradicting the uniqueness of the optimum at such prices. then for any ε >0. 39 ˆ ˆ v2 ( x) = ∑ k ≠ n . Furthermore. the coalitional value function is bidder-submodular. Suppose that there are at least four possible bidders. w(0123) = w(012) . and Vsub the set of valuation functions for which the goods are substitutes. p− m )i x satisfying xn = xm = 1 . m and n. π ∈ Core( L. Further suppose that there is a single unit of each kind and that Vadd ⊂ V . and a price ˆ vector. (1)⇒(2)⇒(3). such that for 0 ≤ pm < pm . both x′ and x′′ are optimal. Vadd the set of additive valuation functions. x′′′ . there exists pm ∈ ( pm . 2004 . we introduce three sets of goods valuation functions. p− m ) where pm > pm . Suppose that the substitutes condition fails for some valuation v1 ∈ V . x′′′ would also be optimal at any price vector ( pm . p− m ) ≠ x1m ( pm + ε . p− m ) and x1n ( pm + ε . Let V denote the set of valuations from which the bidders in the auction may draw values. w(01234) > w(0124) . and a price vector. w) . the goods-are-substitutes condition is necessary. it follows that at the price vector p. we may take buyers 2. Since x′ is Since Vadd ⊂ V . the change in the price of good m would have no effect on buyer 1’s demand for good n. Then the following three conditions are equivalent: (1) V ⊂ Vsub (2) For every profile of bidder valuations drawn for each bidder from V. pm > 0 . with pn . At the Vickrey payoffs. any alternative bundle. as well as sufficient. (3) For every profile of bidder valuations drawn for each bidder from V.m pk xk . there is a unique maximizer x′ of ′ ′ ˆ ˆ v1 ( x) − ( pm . π 0 + π 1 + π 2 = w(01234) − (π 3 + π 4 ) = w(01234) – ([ w(01234) – 38 The detailed construction begins by noting that. p− m ) .Ausubel and Milgrom: Ascending Package Auctions 33 Given the additional condition that the possible bidder valuations include the additive values. if the substitutes condition fails. which we may take to be the valuation of buyer 1. To state this result clearly. pm + ε ) at which the demand for good n changes from 1 to 0. It remains to show that (3)⇒(1). such that: (i) buyer 1 has unique demands at price vector ( pm . Moreover. m and n. Hence. and v4 ( x) = pm xm where pm > pm . By Theorems 11 and 7. there exist two goods. 3 and 4 to have additive valuations as follows: optimal for buyer 1 at price vector p above. to conclude that the coalitional value function is bidder-submodular. pm ) and since pn > 0 . Theorem 12. by inspection. p− m ) = 1 . p− m ) = 0 . p− m ) . and for pm < pm . p− m ) and x1n ( pm . ( pm . and (ii) buyer 1 has unique demands at price vector ( pm + ε . for otherwise (given the quasilinear preferences). there is a unique maximizer 38 ′′ ′′ x′′ satisfying xn = xm = 0 . there exist two goods. 39 ˆ ˆ Otherwise. Produced by The Berkeley Electronic Press.
we conclude: π l ( L′) ≥ w( L) − ∑ l∈L \{0}π l ( L) = π 0 ( L) . 1. In turn. we have: π l ( L′) = w( L′) − w( L′ \ l ) ≤ w( L) − w( L \ l ) = π l ( L) . Then the following four conditions are equivalent:40 (1) V ⊂ Vsub . No. Hence. http://www. By Theorem 11. Moreover.com/bejte/frontiers/vol1/iss1/art1 . Suppose that there is a single unit of each kind and that Vadd ⊂ V . Bidder monotonicity formalizes the familiar property of ordinary single-item private-values auctions that increasing bidder participation can only benefit the seller.bepress. as required. (4) For every profile of bidder valuations drawn for each bidder from V. L′ ({0} ⊂ L ⊂ L′) be any nested sets of players that include the seller. then the coalitional value function is bidder-submodular. Theorem 13. Indeed. the proof establishes that (1)⇒(2). it should be recognized that these possibilities for manipulation are intimately related to a failure of monotonicity of revenues in the set of bidders. and (4)⇒(1). provided N≥3. provided N≥4. So. possible failures of the substitutes condition are problematic for traditional market mechanisms as well as for the Vickrey mechanism. As before. including the shill bidding and collusion by losers described in the introduction. The failure of the substitutes condition is also closely connected to some extreme possibilities for manipulation in the Vickrey auction. 1 [2002]. any shill bidding is unprofitable in the Vickrey auction. any joint deviation by losing bidders is unprofitable in the Vickrey auction. adding bidders can never reduce the seller’s total revenues in the Vickrey auction. Meanwhile (see the inequality in footnote 35): π0 40 ∑ ( L′) = w( L′) − ∑ l∈L′ \ L π l ( L′) ≤ w( L′) − w( L) . (1)⇒(3).34 Frontiers of Theoretical Economics Vol. coalition 012 blocks the Vickrey payoff allocation and π ∉ Core( L. Proof. The next theorem shows that the substitutes condition is sufficient for bidder monotonicity in the Vickrey auction. (2) For every profile of bidder valuations drawn for each bidder from V. Milgrom (2000a) shows that if Vsub ⊂ V . π ( L) and π ( L′) denote the associated vectors of Vickrey auction payoffs. (2)⇒(1) and (3)⇒(1). and for shill bidding and loser collusion to be unprofitable. (1)⇒(4). Summing these inequalities. ■ Theorem 12 is closely related to theorems about the existence of competitive equilibrium goods prices in models like this one. then a competitive equilibrium exists for every profile of bidder valuations drawn from V if and only if V = Vsub . Article 1 w(0124)] + [w(01234) – w(0123)]) = w(0123) + w(0124) – w(01234) < w(0123) = w(012). (1)⇒(2): Let L. We define an auction to exhibit bidder monotonicity if there is no preference profile such that adding another bidder reduces the seller’s equilibrium revenues. l∈L′ \{0} For models starting with a fixed number of bidders N. w) . the substitutes condition is also necessary for these conclusions if the set of bidder values is otherwise sufficiently inclusive. for every bidder l ∈ L \{0} . Thus. if every bidder has substitutes preferences. (3) For every profile of bidder valuations drawn for each bidder from V.
Ausubel and Milgrom: Ascending Package Auctions 35 (1)⇒(3): By Theorem 11. ■ These failings of the Vickrey auction contrast sharply with the properties of the ascending proxy auction. Following the proof of Theorem 12. Then exactly as in the proof of Theorem 12. which we may take to be the valuation of buyer 1. it illustrates profitable collusion among losing bidders. then the coalitional value function is bidder-submodular. {x′. where xn = xm = 1 and ′′ ′′ ′′ xn = xm = 0 . In that case. (2)⇒(1): The construction in the preceding paragraph also provides a violation of bidder monotonicity. we may take buyers 2 and 3 to have the additive valuations of v2 ( x) = ∑ k ≠ n . However. the shill buyer 4 wins good m for payment of pm . It is obvious that introducing additional bidders in this formulation simply adds constraints to the minimization problem. shill bidding is unprofitable in the Vickrey auction. suppose that buyer 3 can enter an additional bid. a price vector. the equilibrium revenue is min π 0 subject to ∑ l∈S π l ≥ v( S ) for every coalition S ⊂ L . no losing bidder can benefit from such a deviation. to check bidder monotonicity. under substitutes preferences. ′ ′ such that at price vector p. Then. m and n. v is submodular. now a real bidder. increasing π 0 . Also. Observe that. reduces the seller’s revenues. the ascending proxy auction satisfies bidder monotonicity. (4)⇒(1): The preceding construction also applies if buyer 4 is a real bidder but has valuation v4 ( x) = pm xm . Sakurai and Matsubara (2000). With this bid.41 41 The argument in this paragraph together with the earlier characterization. In this sense.m pk xk and v3 ( x) = pm xm + pn xn . under the false name of buyer 4. so the shill bidding is profitable for buyer 3. (1)⇒(4): Let v be the maximum value function for the coalition of winning bidders. (3)⇒(1): Suppose that the substitutes condition fails for some valuation v1 ∈ V . For the latter. p. Let ∆ = v( x− n . Since goods are substitutes for the winning bidders. The right-hand side is the Vickrey price for a lone deviator. that the Vickrey payoffs are the unique core point that is Pareto-best for the buyers (theorems 7 and 11) provide an alternative proof that (1)⇒(2) in Theorem 13. buyer 3 receives a payoff of zero. Adding buyer 4. there exist two goods.” We focus attention on the equilibria of the proxy auction that are consistent with the selection in theorem 4 and minimize revenues. Produced by The Berkeley Electronic Press. x′′} = arg max{v1 ( x) − p ⋅ x} . so a losing bidder’s price for any bundle is less if it deviates alone than if it participates in a joint deviation by coalition S. Hence. Then by Proposition 3 of Yokoo. 2004 .1) − v( x′′) be the incremental value of good n to buyer 1 at x′′ . buyer 3 then wins good n for payment of ∆ < pn . we need to be identify the “equilibrium revenues. Suppose a coalition ˆ S of losing bidders deviates and acquires the bundles ( xl )l∈S . in the sincere bidding equilibrium of the Vickrey auction. The Vickrey price paid by ˆ ˆ any losing bidder l to acquire its bundle is at least v( M − ∑ i∈S \l xi ) − v( M − ∑ i∈S xi ) ≥ ˆ v( M ) − v( M \ xl ) . v4 ( x) = ( pm + pn − ∆ ) xm . and two bundles x′ and x′′ . if the substitutes condition holds for all bidders.
These assumptions can fail in practice for a variety of reasons. π l* . one reason is the possibility of binding budget constraints. attainable by bidder l—singly. A third reason. bidder l does not transact and earns zero utility. we introduce the idea of a generalized ascending package auction. or with additional shill bids. and the auctioneer cares only about the total price offered... the use of fast trains too soon after slow ones or the use of trains heading in opposite directions on the same track. http://www. let FS = { x ∈ F : xl = ∅l for all l ∉ S \ 0} be the set of offer profiles that are feasible for S.. Looking first at bidders. the auctioneer sells rights of passage to Swedish trains. Given the opposing bidders’ strategies. In the analysis of Brewer and Plott (1996). No. despite its preferences. The set of offers that can be made by bidder l is formulated as a general finite set. the proportion of minority suppliers.com/bejte/frontiers/vol1/iss1/art1 . Generalized Ascending Package Auctions Our analysis in this paper has focused on the case in which bidder preferences are “quasi-linear” (utility is representable as the value of packages minus any cash paid). To accommodate all these possibilities and more. In particular. preferences and constraints may also be more complex than represented in our model.. Let F ⊆ X be the set of all feasible offer profiles. or the total number of suppliers used. there exists a maximum profit target. Let X = ×l∈L \0 X l be the set of all possible offer profiles for bidders. at any pure strategy Nash equilibrium. is that bidders may have policies or contracts that limit price discounts but allow other inducements. which includes the “null offer” ∅ l . A buyer. for example. Any strict preference ordering of bidder l over the set X l can be described by a utility function ul . there is no gain to any deviation using shill bids. it might have complex preferences that give weight to such varied factors as the quality of the input. Similar reasoning shows that there are never profitable joint deviations for losing bidders in the proxy auction. ∅|L|−1 ) is assumed always to be feasible and u0 (∅) = 0 . This result follows from the same reasoning as in Theorem 3(a).36 Frontiers of Theoretical Economics Vol. 1 [2002]. relevant for procurement auctions. A second possibility is that a buyer’s preference among packages may depend on the price it pays.bepress. and for any coalition of bidders S. Similarly. A fourth possibility is that some of the bidders may be taking both sides of the market—making offers to sell and to buy— requiring the description of feasible bids to be modified accordingly. At the null offer. when the bid-taker is a buyer procuring inputs. Article 1 The precise statement that shill bidding is unprofitable in the proxy auction is the following: Given any pure strategy profile for opposing bidders –l. there is a best reply for bidder l that does not use shill bids. On the seller/auctioneer’s side. the null offer profile ∅ = ( ∅1 . for example because the price a spectrum buyer pays may affect its ability to make complementary investments in wireless infrastructure. Feasibility and safety constraints limit. 8. X l . may be unable to finance the purchase of the package it wants. The auctioneer’s strict preference ordering over offer profiles in F is represented by the utility function u0 . the constraints are the “packaging constraints” described by (1). ∅ l . 1. the distance of the nearest supplier to a certain facility.
and let the round t–1 winning ˆ coalition be S t −1 = l ∈ L \ 0 : xlt −1 ≠ ∅l . where F t = { x ∈ F : xl ∈ Blt for all l ∈ L \ 0} . Produced by The Berkeley Electronic Press. ˆ auctioneer after round 1. (18) The process continues iteratively. Our equilibrium analysis in the preceding sections employed the transferable utility core and assumed that the actual preferences could also be found in this family. Blt = Blt −1 ∪ {xlt } . u0 ) . The null bid ∅l is also treated as one of bidder l’s offers. and π lt = ul ( xlt ) . Each other bidder makes the next most profitable bid on its list: if Alt ≠ {∅l } and l ∉ S t −1 . One example of a generalized ascending proxy auction is the discretized version of the ascending proxy auction studied earlier. Let Pl be the set of strict preference orderings over X l . terminating after the first round in which no new offers are made. which is determined by the triple ( X . then xlt = arg max x ∈At ul ( xl ) . F . l l • The auctioneer selects its most preferred feasible offer profile according to the objective u0 from the accumulated list of offers: t ˆ ˆ xt = arg max F t {u0 ( x)} and π 0 = u0 ( xt ) . Let Alt identify the individually rational bids not { } yet made by bidder l when round t begins: Alt = {xl | 0 ≤ ul ( xl ) < π lt −1} if π lt −1 > 0 . Preferences can then be completely described by a valuation function and a budget limit. This corresponds to the case in which each bidder’s strategy set is the set of preferences describable by a value function vl (but with offer amounts limited to discrete units). Proceeding iteratively. operates in a series of rounds. let t >1 denote the current round. At the first round. Round t proceeds as follows: • • • Members of the winning coalition make no new offers: If l ∈ S t −1 then Blt = Blt −1 and π lt = π lt −1 . As for the auction without proxy bidders. the proxy agent for bidder l sets xl1 = arg max X l ul ( xl ) . 2004 . and the corresponding profit demand is π l1 = max X l ul ( xl ) . The mechanism chooses x1 = arg max x∈F1 {u0 ( x)} . and S 1 F 1 = {x ∈ F : xl ∈ Bl1 for all l ∈ L \ 0} : this is the feasible set of bids available to the denotes the round 1 winning coalition. The mechanism. and the feasible set F is the set of all compatible profiles of packages. and Alt = {∅l } if π lt −1 = 0 . the auctioneer ranks offer profiles according to total revenues. Each bidder l’s strategy set is a subset of Pl (or a corresponding set of utility functions ul ).Ausubel and Milgrom: Ascending Package Auctions 37 A generalized ascending proxy auction uses this generalized formulation as intermediated through proxy agents. A second example of such a process arises from the earlier environment augmented by a budget constraint that limits bids. so Bl1 = {∅l } ∪ {xl1} . then Blt = Blt −1 and π lt = 0 . Bidders with no remaining profitable bids make no new offers: if Alt = {∅l } .
and Siva Anantham. but also a core allocation of the exchange game.” the auction outcome is not only efficient. Acknowledgements and Sources: We thank Evan Kwerel for inspiring this research. A sequel paper will contain the proof of this theorem and will develop further results concerning generalized ascending package auctions. Parag Pathak. By contrast. but the Vickrey auction is not. Some aspects of the ascending package auction technology are described in greater detail in Ausubel and Milgrom (2001). Article 1 We have proved the following result: Theorem 14. the ascending proxy is neutral about the choice of technology. Lixin Ye and the referees for comments on previous drafts. it applies to models in which bidders are budget-constrained and to procurement problems with their characteristic multi-faceted selection criteria. the Vickrey auction may not even apply. The final outcome of any generalized ascending proxy auction is an NTU-core allocation with respect to the reported preferences. particularly in evaluating the Cybernomics experiment. The ascending proxy auction is a new kind of deferred acceptance algorithm. It avoids the very low revenues possible in the Vickrey auction by always selecting core allocations. This new auction design has significant advantages compared to the Vickrey auction. Hui Li.com/bejte/frontiers/vol1/iss1/art1 . This paper builds upon and supercedes Ausubel (1997b) and Milgrom (2000c). Eva Meyersson Milgrom for surgical jargon-removal. In such circumstances. we have also introduced a generalized version that applies for much more general preferences and constraints than are permitted in the standard Vickrey model. When technology choice affects the organization of the bidders. John Asker for his research assistance. 9. Finally.38 Frontiers of Theoretical Economics Vol. http://www. Peter Cramton. We believe that this family of auction designs holds considerable promise for a variety of practical applications. the Vickrey auction is assured to select such allocations only when goods are substitutes. If bidders bid “straightforwardly. Ilya Segal. No. Besides the basic ascending proxy auction. Conclusion Our analysis of the ascending package auction may explain the efficient outcomes that sometimes emerge in experiments with package bidding and entail new predictions as well. The ascending proxy auction also avoids some of the extreme vulnerability of the Vickrey auction to shill bids and to collusion by coalitions of losing bidders. related to the algorithms studied in matching theory. 1 [2002]. For example. in its multi-stage form. but the generalized ascending proxy auction selects allocations in the NTU-core. Valter Sorana. 1.bepress. or it may lead to inefficient outcomes. the ascending proxy auction economizes on the bidders’ costs of evaluating packages by allowing them to focus their efforts on the packages that they have a reasonable chance to win based on the bids made by competitors earlier in the auction.
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