In multiple-object environments where individual bidders may demand more than one object, standard methods of auction generally result in allocative inefficiency. This paper proposes a new ascending-bid method for auctioning homogeneous goods, such as Treasury bills or communications spectrum. The auctioneer announces a current price, bidders report back the quantity demanded at that price, and the auctioneer raises the price. Objects are awarded to bidders at the current price whenever they are “clinched, ” and the process continues until the market clears. With pure private values, the proposed (dynamic) auction yields the same outcome as the (sealed-bid) Vickrey auction, but may be simpler for bidders to understand and has the advantage of assuring the privacy of the upper portions of bidders ’ demand curves. With interdependent values, the proposed auction may still yield efficiency, whereas the Vickrey auction fails due to a problem which could be described as the “Generalized Winner’s Curse.” JEL No.: D44 (Auctions)

This paper considers combinatorial auctions among such submodular buyers. The valuations of such buyers are placed within a hierarchy of valuations that exhibit no complementarities, a hierarchy that includes also OR and XOR combinations of singleton valuations, and valuations satisfying the gross substitutes property. Those last valuations are shown to form a zero-measure subset of the submodular valuations that have positive measure. While we show that the allocation problem among submodular valuations is NP-hard, we present an efficient greedy 2-approximation algorithm for this case and generalize it to the case of limited complementarities. No such approximation algorithm exists in a setting allowing for arbitrary complementarities. Some results about strategic aspects of combinatorial auctions among players with decreasing marginal utilities are also presented.

We show that any communication finding a Pareto efficient allocation in a private-information economy must also discover supporting Lindahl prices. In particular, efficient allocation of L indivisible objects requires naming a price for each of the 2 L ¡1 bundles. Furthermore, exponential communication in L is needed just to ensure a higher share of surplus than that realized by auctioning all items as a bundle, or even a higher expected surplus (for some probability distribution over valuations). When the valuations are submodular, efficiency still requires exponential communication (and fully polynomial approximation is impossible). When the objects are homogeneous, arbitrarily good approximation is obtained using exponentially less communication than that needed for exact efficiency.

The Vickrey sealed bid auction occupies a central place in auction theory because of its efficiency and incentive properties. Implementing the auction requires the auctioneer to solve n + 1 optimization problems,where n is the number of bidders. In this paper we survey various environments (some old and some new) where the payments bidders make under the Vickrey auction correspond to dual variables in certain linear programs. Thus,in these environments,at most two optimization problems must be solved to determine the Vickrey outcome. Furthermore,primal-dual algorithms for some of these linear programs suggest ascending auctions that implement the Vickrey outcome.

This paper analyzes individually-rational ex post equilibrium in the VC (Vickrey-Clarke) combinatorial auctions. If \Sigma is a family of bundles of goods, the organizer may restrict the participants by requiring them to submit their bids only for bundles in \Sigma. The \Sigma-VC combinatorial auctions (multi-good auctions) obtained in this way are known to be individually-rational truthtelling mechanisms. In contrast, this paper deals with non-restricted VC auctions, in which the buyers restrict themselves to bids on bundles in \Sigma, because it is rational for them to do so. That is, it may be that when the buyers report their valuation of the bundles in \Sigma, they are in an equilibrium. We fully characterize those \Sigma that induce individually rational equilibrium in every VC auction, and we refer to the associated equilibrium as a bundling equilibrium. The number of bundles in \Sigma represents the communication complexity of the equilibrium. A special case of bundling equilibrium is partition-based equilibrium, in which \Sigma is a field, that is, it is generated by a partition. We analyze the tradeoff between communication complexity and economic efficiency of bundling equilibrium, focusing in particular on partition-based equilibrium.

We propose the clock-proxy auction as a practical means for auctioning many related items. A clock auction phase is followed by a last-and-final proxy round. The approach combines the simple and transparent price discovery of the clock auction with the efficiency of the proxy auction. Linear pricing is maintained as long as possible, but then is abandoned in the proxy round to improve efficiency and enhance seller revenues. The approach has many advantages over the simultaneous ascending auction. In particular, the clock-proxy auction has no exposure problem, eliminates incentives for demand reduction, and prevents most collusive bidding strategies. 1

William Vickrey’s (1961) inquiry into auctions and “counterspeculation ” marked the first serious attempt by an economist to analyze the details of market rules and to design new rules to achieve superior performance. He demonstrated that a particular pricing rule makes it a dominant strategy for bidders to report their values truthfully, even when they

Combinatorial auctions allow bidders to express complex valuations on bundles of items, and have been proposed in settings as diverse as the allocation of floor space in a new condominium building to individual units (Wired 2000) and the allocation of take-off and landing slots at airports (Smith, Forward). Many applications are described in Part V of this book. The promise of combinatorial auctions (CAs) is that they can allow bidders to better express their private information about preferences for different outcomes and thus enhance competition and market efficiency. Much effort has been spent on developing algorithms for the hard problem of winner determination once bids have been received (Sandholm, Chapter 14). Yet, preference elicitation has emerged as perhaps the key bottleneck in the real-world deployment of combinatorial auctions. Advanced clearing algorithms are worthless if one cannot simplify the bidding problem facing bidders. Preference elicitation is a p

We construct an ascending auction for heterogeneous objects by applying a primal-dual algorithm to a linear program that represents the efficient-allocation problem for this setting. The auction assigns personalized prices to bundles, and asks bidders to report their preferred bundles in each round. A bidder’s prices are increased when he belongs to a “minimally undersupplied ” set of bidders. This concept generalizes the notion of “overdemanded” sets of objects introduced by Demange et al. (1986) for the one-to-one assignment problem. Under a submodularity condition, the auction implements the Vickrey–Clarke–Groves outcome; we show that this type of condition is somewhat necessary to do so. When classifying the ascending-auction literature in terms of their underlying algorithms, our auction fills a gap in that literature. We relate our results to various ascending auctions in the literature.

We analyze the communication complexity of surplus-maximizing allocations. We study both the continuous and discrete models of communication, measuring its complexity with the dimensionality of the message space and the number of transmitted bits, respectively. In both cases, we offer a lower bound on the amount of communication. This bound is applied to the problem of allocating L heterogeneous objects among N agents, whose valuations are (i) unrestricted, (ii) submodular, or (iii) homogeneous in objects. In cases (i) and (ii), efficiency requires exponential communication in L. Furthermore, in case (i), polynomial communication in L cannot achieve a higher surplus than selling all objects as a bundle. On the other hand, in case (iii), exact efficiency requires the transmission of L numbers, but arbitrarily close approximation is achieved with only O(log L) bits. When a Walrasian equilibrium with per-item prices exists, efficiency is achieved with deterministic communication that is polynomial in L.