Truthful approximation mechanisms for restricted combinatorial auctions

When attempting to design a truthful mechanism for a computationally hard problem such as combinatorial auctions, one is faced with the problem that most efficiently computable heuristics can not be embedded in any truthful mechanism (e.g. VCG-like payment rules will not ensure truthfulness).We develop a set of techniques that allow constructing efficiently computable truthful mechanisms for combinatorial auctions in the special case where each bidder desires a specific known subset of items and only the valuation is unknown by the mechanism (the single parameter case). For this case we extend the work of Lehmann, O'Callaghan, and Shoham, who presented greedy heuristics. We show how to use If-Then-Else constructs, perform a partial search, and use the LP relaxation. We apply these techniques for several canonical types of combinatorial auctions, obtaining truthful mechanisms with provable approximation ratios.     

Ascending prices and package bidding: Further experimental analysis

We explore the performance of multi-round, price-guided combinatorial auctions for a previously untested class of value profiles in which synergies arise from shared fixed costs. We find that, in many cases, a simulator that bids straightforwardly does well in predicting auction performance, but exceptions arise because human bidders sometimes rely on cues besides prices to guide their package selection and because they sometimes bid aggressively on items for which they have no value in order to increase payments by bidders seeking complementary packages. In our experiments, this latter behavior not only raises prices, but can also improve efficiency by mitigating the threshold problem. Comparisons between a combinatorial clock auction (CCA) and a simultaneous ascending auction (SAA) are reported.     

Liquidity provision vs. deposit insurance: preventing bank panics without moral hazard

Equilibrium bidding strategies under most multi-unit auction rules cannot be obtained as closed form expressions. Research in multi-unit auctions has, therefore, depended on implicit characterization of equilibrium strategies using the first-order conditions of the bidders’ expected payoff maximization problem. In this paper we consider the pay-as-bid auction with diminishing marginal values for two units and show that any symmetric equilibrium in continuous strategies has the necessary properties to allow such a characterization. Moreover, any increasing solution to the system of differential equations that is used to characterize the equilibrium strategies describes an equilibrium strategy.Generous suggestions and comments from Richard Engelbrecht-Wiggans, Charles M. Kahn, and an anonymous referee are gratefully acknowledged.     

Ascending auctions for multiple objects: the case for the Japanese design

Summary. We consider two ascending auctions for multiple objects, namely, an English and a Japanese auction, and derive a perfect Bayesian equilibrium of the Japanese auction by exploiting its strategic equivalence with the survival auction, which consists of a finite sequence of sealed-bid auctions. Thus an equilibrium of a continuous time game is derived by means of backward induction in finitely many steps. We then show that all equilibria of the Japanese auction induce equilibria of the English auction, but that many collusive or signaling equilibria of the English auction do not have a counterpart in the Japanese auction.Received: 2 September 2004, Revised: 20 April 2005, JEL Classification Numbers: C72, D44.Fabrizio Germano: Correspondence toWe are indebted to Philippe Jehiel for useful discussions and to Nicolas Vieille for suggestions in the proof of Proposition 2; we also thank seminar audiences in Athens, Basel, Beer-Sheva, Berlin, Brussels, Cambridge, Edinburgh, Exeter, Lausanne, Lisbon, London, Louvain-la-Neuve, Namur and Tel Aviv. Germano acknowledges financial support from Euopean Commission, TMR Network Grant ERBFMRXCT0055, “Cooperation and Information” as well from the Spanish Ministry of Science and Technology, Grants SEC2001-0792, SEJ2004-06319, and in form of a Ramon y Cajal Fellowship. Lovo is member of GREGHEC, unité CNRS, FRE-2810. The support of the Economic and Social Research Council (ESRC) is also gratefully acknowledged. The work was part of the programme of the ESRC Research for Economic Learning and Social Evolution.     

Combinatorial auctions with decreasing marginal utilities

In most of microeconomic theory, consumers are assumed to exhibit decreasing marginal utilities. 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.     

Asymptotic Revenue Equivalence in Auctions

We study the revenue in auctions of a single object when the number of bidders becomes large. We show that all sequences of auctions belonging to a class of mechanisms have limiting expected revenue equal to the expected best-use value of the object. An important special case that is covered is common value auctions, but more generally not even affiliation of values is assumed. This provides an asymptotic revenue equivalence result for settings beyond that of private values. Journal of Economic Literature Classification Numbers: D44, C72.     

Robustness of the Incentive Compatible Combinatorial Auction

Goods are said to be combinatorial when the value of a bundle of goods is not equal to the sum of the values of the same goods unbundled. Investigations of combinatorial allocation problems should recognize that there are two separate aspects of such problems: anenvironmental distinction between multiple-unit allocation problems which involve combinatorial goods and those which do not do so, and aninstitutional distinction between auctions in which combinatorial values can be expressed as part of the bidding rules and those in which they cannot. Forsythe and Isaac (Research in Experimental Economics, Vol. 2 (1982). Greenwich, Conn.: JAI Press, Inc.) reports the extension of the Vickrey auction into a demand-revealing, multiple unit, private goods auction that can incorporate combinatorial values. This current paper places that theoretically demand-revealing institution in a series of experimental environments in order to generate results (e.g. efficiencies) which may serve as a benchmark for other auctions (combinatorial or otherwise) whose implementation characteristics may be more favorable. To aid in interpretation of our Vickrey experimental results, we also provide results of alternatives to Vickrey allocations from both institutional and heuristic sources, as well as a discussion of the source of the Vickrey auctions high efficiencies even in the presence of misrevelation.     

Vickrey auctions with reserve pricing

Summary. We generalize the Vickrey auction to allow for reserve pricing in a multi-unit auction with interdependent values. In the Vickrey auction with reserve pricing, the seller determines the quantity to be made available as a function of the bidders' reports of private information, and then efficiently allocates this quantity among the bidders. Truthful bidding is a dominant strategy with private values and an ex post equilibrium with interdependent values. If the auction is followed by resale, then truthful bidding remains an equilibrium in the auction-plus-resale game. In settings with perfect resale, the Vickrey auction with reserve pricing maximizes seller revenues.Received: 31 December 2002, Revised: 5 May 2003, JEL Classification Numbers: D44, C78, D82.Correspondence to: Lawrence M. AusubelThe authors gratefully acknowledge the generous support of National Science Foundation Grants SES-97-31025, SES-01-12906 and IIS-02-05489. We appreciate valuable comments from Ilya Segal. Special thanks go to Mordecai Kurz, who served as Larry's dissertation advisor and who introduced both authors to the economics profession back at IMSSS at Stanford. Congratulations and best wishes are extended to Mordecai and his family on the happy occasion of the publication of Assets, Beliefs, and Equilibria in Economic Dynamics: Essays in Honor of Mordecai Kurz, in which this article also appears.     

Allocating multiple units

Summary. This paper studies the allocation and rent distribution in multi-unit, combinatorial-bid auctions under complete information. We focus on the natural multi-unit analogue of the first-price auction, where buyers bid total payments, pay their bids, and where the seller allocates goods to maximize his revenue. While there are many equilibria in this auction, only efficient equilibria remain when the truthful equilibrium restriction of the menu-auction literature is used. Focusing on these equilibria we first show that the first-price auction just described is revenue and outcome equivalent to a Vickrey auction, which is the multi unit analogue of a second-price auction. Furthermore, we characterize these equilibria when valuations take a number of different forms: diminishing marginal valuations, increasing average valuations, and marginal valuations with single turning points. Received: December 23, 1999; revised version: December 10, 2001     

Sequential vs. single-round uniform-price auctions

We study sequential and single-round uniform-price auctions with affiliated values. We derive symmetric equilibrium for the auction in which k₁ objects are sold in the first round and k₂ in the second round, with and without revelation of the first-round winning bids. We demonstrate that auctioning objects in sequence generates a lowballing effect that reduces the first-round price. Total revenue is greater in a single-round, uniform auction for k=k₁+k₂ objects than in a sequential uniform auction with no bid announcement. When the first-round winning bids are announced, we also identify a positive informational effect on the second-round price. Total expected revenue in a sequential uniform auction with winning-bids announcement may be greater or smaller than in a single-round uniform auction, depending on the model's parameters.     

Auctions with online supply

Online advertising auctions present settings in which there is uncertainty about the number of items for sale. We study mechanisms for selling identical items when the total supply is unknown but is drawn from a known distribution. Items arrive dynamically, and the seller must make immediate allocation and payment decisions with the goal of maximizing social welfare. We devise a simple incentive-compatible mechanism that guarantees some constant fraction of the first-best solution. A surprising feature of our mechanism is that it artificially limits supply, and we show that limiting the supply is essential for obtaining high social welfare. Although common when maximizing revenue, commitment to limit the supply is less intuitive when maximizing social welfare. The performance guarantee of our mechanism is in expectation over the supply distribution; We show that obtaining similar performance guarantee for every realization of supply is impossible.     

Competitive auctions

We study a class of single-round, sealed-bid auctions for an item in unlimited supply, such as a digital good. We introduce the notion of competitive auctions. A competitive auction is truthful (i.e. encourages bidders to bid their true valuations) and on all inputs yields profit that is within a constant factor of the profit of the optimal single sale price. We justify the use of optimal single price profit as a benchmark for evaluating a competitive auctions profit. We exhibit several randomized competitive auctions and show that there is no symmetric deterministic competitive auction. Our results extend to bounded supply markets, for which we also give competitive auctions.     

Informational limitations of ascending combinatorial auctions

We study the inherent limitations of natural widely-used classes of ascending combinatorial auctions. Specifically, we show that ascending combinatorial auctions that do not use both non-linear prices and personalized prices cannot achieve social efficiency with general bidder valuations. We also show that the loss of efficiency can be severe and that only a diminishing fraction of the social welfare may be captured. This justifies the added complexity in the auctions suggested by, e.g., Parkes and Ungar (2000) [29] and Ausubel and Milgrom (2002) [2].     

Expected revenue of all-pay auctions and first-price sealed-bid auctions with budget constraints

We show that all-pay auctions dominate first-price sealed-bid auctions when bidders face budget constraints. This ranking is explained by the fact that budget constraints bind less frequently in the all-pay auctions, which leads to more aggressive bidding in that format.     

Exploring pricing rules in combinatorial sealed-bid auctions

The allocation of public goods such as the radio spectrum is a difficult task that the government must face. Currently, auctions are becoming an important tool to deal with this duty. In this context, the rules that the auctioneer establishes are particularly relevant, as the final outcome depends on them. When auctioning many related items, such as spectrum licenses, the bidders’ values for one item may depend on the number of items already obtained (complements and substitutes items). In such circumstances, combinatorial auctions are the most appropriate alternative for allocating lots. This paper analyzes the implications of selecting a particular pricing mechanism on the final result in a combinatorial sealed-bid auction. The following pricing rules are selected: the first-price mechanism, the Vickrey–Clarke–Groves (VCG) mechanism, and the bidder–Pareto–optimal (BPO) core mechanism, a core-selecting auction. To test these pricing rules, a simulator of the auction model has been developed. Then, to tackle the complex problem of simulating bidders’ behavior, a co-evolutionary system has been designed to identify improved strategies. The results revealed that the first-price mechanism yields inefficient outcomes and a notable reduction in the seller's revenues. Both the VCG and BPO mechanisms yield outcomes that are closer to the efficient allocation, and differences in revenues are affected by the presence of asymmetries.     

A comparison of first price multi-object auctions

We compare simultaneous multi-object English-type ascending price auctions with first price sealed bid auctions in private values environments with multi-object demands. Special attention is paid to the effect of closing rules on ascending auctions’ outcomes. We find that simultaneous ascending auctions with the soft closing rule are the most efficient, while the sealed bid auctions generate the highest revenue. Ascending auctions with the hard closing rule display a significant amount of late bidding, resulting in the lowest among the three institutions revenue and efficiency.

---

Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.      

Late and multiple bidding in second price Internet auctions: Theory and evidence concerning different rules for ending an auction

In second price Internet auctions with a fixed end time, such as those on eBay, many bidders submit their bids in the closing minutes or seconds of an auction. We propose an internet auction model, in which very late bids have a positive probability of not being successfully submitted, and show that late bidding in a fixed deadline auction can occur at equilibrium in auctions both with private values and with uncertain, dependent values. Late bidding may also arise out of equilibrium, as a best reply to incremental bidding. However, the strategic advantages of late bidding are severely attenuated in auctions that apply an automatic extension rule such as auctions conducted on Amazon. Field data show that there is more late bidding on eBay than on Amazon, and this difference grows with experience. We also study the incidence of multiple bidding, and its relation to late bidding.     

Second chance offers versus sequential auctions: theory and behavior

Second chance offers in online marketplaces involve a seller conducting an auction for a single object and then using information from the auction to offer a losing bidder a take-it-or-leave-it price for another unit. We theoretically and experimentally investigate this practice and compare it to two sequential auctions. We show that the equilibrium bidding strategy in the second chance offer mechanism only exists in mixed strategies, and we observe that this mechanism generates more profit for the auctioneer than two sequential auctions. We also observe virtually no rejections of profitable offers in the ultimatum bargaining stage.      

The (in) effectiveness of procurement auctions in the public sector

This article studies procurement auctions in the public sector using game theoretical tools. The article shows that when participants in an auction are agents with low abilities (low type), as is common in the public sector, they place the same contract request. As a result, the auction mechanism will rarely produce real competition, thus making this mechanism, which limits bidders to those already within the system, ineffective for use in the public sector job market.     

Ex-post full surplus extraction,straightforwardly

Consider an estimate of the common value of an auctioned asset that is symmetric in the bidders’ types. Such an estimate can be represented solely in terms of the order statistics of those types. This representation forms the basis for a pricing rule yielding truthful bidding as an equilibrium, whether bidders’ types are affiliated or independent. We highlight the link between the estimator and full surplus extraction, providing a necessary and sufficient condition for ex-post full surplus extraction, including the possibility of independent types. The results offer sharp insights into the strengths and limits of simple auctions by identifying the source of informational rents in such environments. Harstad acknowledges hospitable accommodation by the Fuqua School of Business, Duke University, and the Olin School of Business, Washington University in St. Louis, during parts of this research. We are grateful for comments and suggestions from Richard McLean and Jeroen Swinkels.