An Efficient Dynamic Auction for Heterogeneous Commodities 603

An Efficient Dynamic Auction for Heterogeneous Commodities By LAWRENCE M. AUSUBEL* This article proposes a new dynamic design for auctioning multiple heterogeneous commodities. An auctioneer wishes to allocate K types of commodities among n bidders. The auctioneer announces a vector of current prices, bidders report quantities demanded at these prices, and the auctioneer adjusts the prices. Units are credited to bidders at the current prices as their opponents’ demands decline, and the process continues until every commodity market clears. Bidders, rather than being assumed to behave as price-takers, are permitted to strategically exercise their market power. Nevertheless, the proposed auction yields Walrasian equilibrium prices and, as from a Vickrey-Clarke-Groves mechanism, an efficient alloca- tion. (JEL D44) In earlier work (Ausubel, 1997, 2004), I pro- the U.S. Treasury sells in excess of $10 billion posed an efficient ascending auction design for in three-month bills and $10 billion in six- multiple homogeneous items. In environments month bills.1 Current practice is to auction the where bidders have pure private values and three-month and six-month bills separately in diminishing marginal values, this dynamic auc- two independent sealed-bid auctions. In the Eu- tion yields outcomes coinciding with that of the ropean UMTS/IMT-2000 spectrum auctions, (sealed-bid) Vickrey auction (William Vickrey, governments sold both paired and unpaired 3G 1961), but offers advantages of simplicity, spectrum, located at similar frequencies but ap- transparency, and privacy preservation. More- parently exhibiting markedly different values. over, in some environments where bidders have Some governments auctioned these together in interdependent values for the items, this dy- fixed bundles, while other governments auc- namic auction continues to yield efficient out- tioned the paired spectrum followed by the (less comes and thus outperforms even the Vickrey valuable) unpaired spectrum. In the Electricite´ auction. de France (EDF) generation capacity auctions However, situations abound in diverse indus- that have operated quarterly since 2001, as well tries in which heterogeneous (but related) com- as separately in the Electrabel virtual power modities are auctioned. On a typical Monday, plant auctions that have operated quarterly in Belgium since 2003, the companies sell base- load electricity contracts and peak-load electric- * Department of Economics, University of Maryland, ity contracts, of at least five different durations Tydings Hall, Room 3105, College Park, MD 20742 each, simultaneously in one dynamic auction (e-mail: [email protected]). I am grateful to Kathleen procedure (see Ausubel and Peter Cramton, Ausubel, Ken Binmore, Peter Cramton, John Ledyard, Pres- ton McAfee, Paul Milgrom, Thayer Morrill, Ennio Stac- 2004a). chetti, Jeroen Swinkels, Daniel Vincent, three anonymous The current article proposes an efficient dy- referees, and participants in the Heidelberg Conference on namic auction method for heterogeneous items. Auctions and Market Structure, the Stony Brook Multi-Unit The starting point for the new design is a ven- Auctions Workshop, the 2000 World Congress of the erable trading procedure, often associated in Econometric Society, the 2001 NBER General Equilibrium Conference, the 2001 NSF Decentralization Conference, general equilibrium theory with the fictitious and numerous university seminars for helpful comments. Intellectual Property Disclosure: The auction design intro- duced in this article may be subject to issued or pending 1 For example, on 25 July 2005, the U.S. Treasury auc- patents, in particular, U.S. Patent No. 6,026,383 and U.S. tioned $19 billion in 13-week Treasury bills and $17 billion Patent Application No. 09/898,483. Upon request, the au- in 26-week bills (Press Release, Department of the Trea- thor will grant a royalty-free license to the referenced prop- sury, Bureau of the Public Debt, www.publicdebt.treas. erties for noncommercial research on this auction design. gov). 602 VOL. 96 NO. 3 AUSUBEL: AN EFFICIENT DYNAMIC AUCTION FOR HETEROGENEOUS COMMODITIES 603 Walrasian auctioneer, and sometimes imple- One of the objectives of the current article is, mented in modern times as a dynamic clock thus, to provide both a solution to an outstand- auction. An auctioneer wishes to allocate K ing theoretical auction question and a new prac- types of heterogeneous commodities among n tical auction design. In a recent article, Sushil bidders. The auctioneer announces a price vec- Bikhchandani and John W. Mamer (1997, pp. tor, p, and bidders respond by reporting the 405–06) ask: quantity vectors that they wish to transact at these prices. The auctioneer then calculates the “Do there exist simple market mecha- excess demand and increases or decreases each nisms (i.e., mechanisms that assign a coordinate of the price vector according to price to each object) which efficiently al- whether the excess demand is positive or neg- locate multiple indivisible objects when ative (Walrasian taˆtonnement). This iterative market clearing prices exist? ... Whether there are simple incentive compatible process continues until a price vector is reached market mechanisms which converge to a at which excess demand is zero, and trades competitive equilibrium (whenever one occur only at the final price vector. exists) under the more general condition In both the fictitious Walrasian auctioneer that buyers may want to consume more construct and in most real-world dynamic clock than one object is an open question.” auctions, bidders’ payments are linear in the quantities awarded: if bidder i wins quantity Meanwhile, Faruk Gul and Ennio Stacchetti (2000, p. 69) conclude the introduction of their vector qi in an auction with final price vector p, ⅐ recent article by stating: bidder i pays p qi. Unfortunately, a strategic agent who faces linear prices in the auction then possesses an incentive to underreport her true “More importantly, we show that no dy- demand at the announced prices, and this incen- namic auction can reveal sufficient in- tive increases in her market share. This is most formation to implement the Vickrey mechanism if all Gross Substitutes pref- straightforwardly seen in the case of homoge- ϭ erences are allowed. Thus, the unit de- neous goods (i.e., K 1), where there is a mand case of Demange et al. [1986] and growing body of both theoretical arguments and the multiple homogeneous goods case of empirical evidence. (See detailed discussions Ausubel [1997] are the most general en- for sealed-bid auctions in Ausubel and Cram- vironments for which generalizations of ton, 2002, and for ascending auctions in Aus- the English auction can be used to implement efficient, strategy-proof alloca- ubel, 2004.) Consequently, when agents have 3 market power, the Walrasian auction procedure tions.” typically does not result in Walrasian outcomes. In this article, I will put forth an affirmative The current article circumvents this problem by answer to Bikhchandani and Mamer’s question, extending and generalizing an approach intro- while disagreeing with the spirit (but not the duced in Ausubel (1997, 2004): units are cred- letter) of Gul and Stacchetti’s conclusion. A ited to bidders at the current prices whenever simple market mechanism is provided: the auc- the opposing bidders’ demands decline. Speci- fied properly, this nonlinear pricing rule restores the incentive for strategic bidders to bid as this scenario. Second, K simultaneous auctions are effec- price-takers, yielding efficient outcomes even tively conducted, and it is unclear how the progress of one when bidders have market power.2 auction should affect the clinching of units in another. Surprisingly, this article establishes that it suffices to cal- culate independently the crediting of different commodities; the only formal interaction among the K auctions needs to 2 The extension of the efficient ascending auction of occur through the simultaneous bidding and the price ad- Ausubel (1997, 2004) to K Ն 2 commodities poses at least justment rule. With these obstacles resolved, an efficient two significant obstacles. First, unlike in the homogeneous dynamic auction design for heterogeneous commodities goods case, a bidder may now wish to increase her demand emerges. for a given commodity along the path toward equilibrium, 3 They also conclude their article: “Finally, we showed as prices of substitute commodities increase. Thus, units that in general, no efficient, dynamic auction can extract that once appeared to be “clinched” by another bidder may enough information to implement any strategy-proof mech- later be “unclinched,” and the auction rules need to reflect anism” (Gul and Stacchetti, 2000, p. 83). 604 THE AMERICAN ECONOMIC REVIEW JUNE 2006 tioneer announces price vectors, bidders are The current article will also seek to offer a asked to respond with their naı¨ve demands, and modern perspective on the Walrasian auction- there is no benefit to bidders from strategizing eer. Indeed, economists have long been hostile further. Moreover, the dynamic mechanism toward this modeling device. Kenneth J. Arrow economizes on information in the sense that (1959, p. 43) notes at once the motivation for bidders need only report their demands at a the fictitious auctioneer and the logical problem one-dimensional set of price vectors, and it that the auctioneer creates: “It is not explained maintains privacy in the sense that (starting whose decision it is to change prices in accor- from

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