Strategic price discounting and rationing in uniform price auctions

I modify the uniform-price auction rules in allowing the seller to ration bidders. This allows me to provide a strategic foundation for underpricing when the seller has an interest in ownership dispersion. Moreover, many of the so-called “collusive-seeming” equilibria disappear.     

Credit contracting and bidding under wealth constraints

Summary. We model credit contracting and bidding in a first-price sealed-bid auction when bidder valuation and wealth are private information. An efficient separating equilibrium exists only if the wealth levels of both bidder types are sufficiently different. If not, high-valuation bidders signal by borrowing more and using less of their wealth – this is inefficient as wealth is a cheaper source of funds. An increase in the amount of borrowing required to signal does not necessarily decrease seller expected revenue. In contrast to separating equilibria, high-valuation bidders adopt pure strategy bids in pooling equilibria. Conditions are identified under which the lower bound on winning bids is higher in pooling than separating equilibria. Received: January 22, 2001; revised version: August 28, 2001     

Multi-unit auctions with private information: an indivisible unit continuous price model

We construct a model of multi-unit auctions in which I bidders bid for two indivisible units of a common value good. Using a first-order approach, we find that there are equilibria in which bidders bid the same price for both units in the discriminatory auction, but not in the uniform auction. When there are only two bidders, under certain conditions, there are linear equilibria for both the discriminatory and the uniform auction formats. In all equilibria, bidders equalize the expected marginal benefit of bidding to the marginal costs of bidding. We show that comparison of the seller??s expected revenue across auction formats depends only on the ratio of the precision of private information to the precision of public information.     

On the informational inefficiency of discriminatory price auctions

We analyze bidding behavior in large discriminatory-price auctions in a common value setting where the number of objects is a non-trivial proportion of the number of bidders. We show that the average price paid in the auction is biased downward from the expected value of the objects, even in the competitive limit. We show that conditional on a signal that falls below a threshold, a bidder bids no more than the expected value of an object conditional on the signal and winning; while conditional on any signal that lies above the threshold the bid is strictly lower than the expected value conditional on the signal and winning.     

Multiple-Object Auctions with Budget Constrained Bidders

A seller with two objects faces a group of bidders who are subject to budget constraints. The objects have common values to all bidders but need not be identical, and may be either complements or substitutes. In a simple complete information setting we show: (1) if the objects are sold by means of a sequence of open ascending auctions, then it is always optimal to sell the more valuable object first; (2) the sequential auction yields more revenue than the simultaneous ascending auction used recently by the FCC if the discrepancy in the values is large, or if there are significant complementarities; (3) a hybrid simultaneous-sequential form is revenue superior to the sequential auction; and (4) budget constraints arise endogenously.     

The Relevance of a Choice of Auction Format in a Competitive Environment

We examine the relevance of an auction format in a competitive environment by comparing uniform and discriminatory price auctions with many bidders in a private values setting. We show that if the number of objects for sale is small relative to the number of bidders, then all equilibria of both auctions are approximately efficient and lead to approximately the same revenue. When the number of objects for sale is proportional to the number of bidders, then the particulars of the auction format matter. All equilibria of the uniform auction are efficient, while all of the equilibria of the discriminatory auction are inefficient. The relative revenue rankings of the auction formats can go in either direction, depending on the specifics of the environment. These conclusions regarding the efficiency and revenue ranking are in contrast to the previous literature, which focused on the case of independent information across agents.     

Non-existence of monotone equilibria in games with correlated signals

Participation in economic games such as auctions is typically costly, which means that players (potential bidders) must consider whether to participate. Such decisions may be no less crucial than how to bid, and yet the literature has been mostly concerned with bidding, assuming an exogenously given number of bidders; see Krishna [Auction Theory, Academic Press, New York, 2002]. Landsberger and Tsirelson [Correlated signals against monotone equilibria, preprint SSRN 222308, Social Science Research Network Electronic Library, May 2000. Available online: http://papers.ssrn.com/abstract=222308] have shown that fundamental results established in symmetric auction theory with correlated signals, such as the existence of a monotone equilibrium, may not hold if participation decisions are part of equilibrium. The major goal of this paper is to illustrate and explain this result that may be considered counter intuitive given the emphasis placed on monotone equilibria in the auction literature.     

Bidding for the future: signaling in auctions with an aftermarket

This paper considers auctions where bidders compete for an advantage in future strategic interactions. When bidders wish to exaggerate their private information, equilibrium bidding functions are biased upwards as bidders attempt to signal via the winning bid. Signaling is most prominent in second-price auctions where equilibrium bids are “above value.” In English and first-price auctions, signaling is less extreme since the winner incurs the cost of her signaling choice. The opportunity to signal lowers bidders’ payoffs and raises revenue. When bidders understate their private information, separating equilibria need not exist and the auction may not be efficient.     

The effectiveness of English auctions

We study the performance of the English auction under different assumptions about the seller's degree of “Bayesian sophistication.” We define the effectiveness of an auction as the ratio between the expected revenue it generates for the seller and the expected valuation of the object to the bidder with the highest valuation (total surplus). We identify tight lower bounds on the effectiveness of the English auction for general private-values environments, and for private-values environments where bidders' valuations are non-negatively correlated. For example, when the seller faces 12 bidders who the seller believes have non-negatively correlated valuations whose expectations are at least as high as 60% of the maximal possible valuation, an English auction with no reserve price generates an expected price that is more than 80% of the value of the object to the bidder with the highest valuation.     

Simultaneous use of auctions and posted prices

I consider a model in which several identical objects are sold simultaneously via an auction and a posted price mechanism. The model explains several empirical regularities regarding bidding behavior in eBay auctions such as the finding that some bidders bid multiple times over the course of the auction, and that bidders tend to bid with greater frequency near the end of the auction than the beginning. I also show that sellers prefer to simultaneously use auctions and posted prices than to use either mechanism individually.     

Congestion games with malicious players

We study the equilibria of non-atomic congestion games in which there are two types of players: rational players, who seek to minimize their own delay, and malicious players, who seek to maximize the average delay experienced by the rational players. We study the existence of pure and mixed Nash equilibria for these games, and we seek to quantify the impact of the malicious players on the equilibrium. One counterintuitive phenomenon which we demonstrate is the “windfall of malice”: paradoxically, when a myopically malicious player gains control of a fraction of the flow, the new equilibrium may be more favorable for the remaining rational players than the previous equilibrium.     

Bidding behavior in competing auctions: Evidence from eBay

Much of the existing auction literature treats auctions as running independently of one another, with each bidder choosing to participate in only one auction. However, in many online auctions, a number of substitutable goods are auctioned concurrently and bidders can bid on several auctions at the same time. Recent theoretical research shows how bidders can gain from the existence of competing auctions, the current paper providing the first empirical evidence in support of competing auctions theory using online auctions data from eBay. Our results indicate that a significant proportion of bidders do bid across competing auctions and that bidders tend to submit bids on auctions with the lowest standing bid, as the theory predicts. The paper also shows that winning bidders who cross-bid pay lower prices on average than winning bidders who do not.     

The all-pay auction with complete information

Summary In a (first price) all-pay auction, bidders simultaneously submit bids for an item. All players forfeit their bids, and the high bidder receives the item. This auction is widely used in economics to model rent seeking, R&D races, political contests, and job promotion tournaments. We fully characterize equilibrium for this class of games, and show that the set of equilibria is much larger than has been recognized in the literature. When there are more than two players, for instance, we show that even when the auction is symmetric there exists a continuum of asymmetric equilibria. Moreover, for economically important configurations of valuations, there is no revenue equivalence across the equilibria; asymmetric equilibria imply higher expected revenues than the symmetric equilibrium.We are grateful to Jacques Crémer, Chuangyin Dang, Jürgen Dennert, Chaim Fershtman, Martin Hellwig, Arthur Robson, Heinrich Ursprung, Eric van Damme, Ton Vorst, and the referees for helpful comments. We benefitted from presentations at the World Congress of the Econometric Society in Barcelona, the European Meeting on the Economics of Information at Tilburg University, the Midwest Mathematical Economics Meetings at the University of Illinois, and seminars at Texas A&M University, The Pennsylvania State University, Tilburg University and the University of Montreal. Baye is grateful for support from the CentER for Economic Research at Tilburg University and the Tinbergen Institute where earlier versions of this paper were completed. Kovenock acknowledges support from Erasmus Universiteit Rotterdam, the Tinbergen Institute, the Center for Economic Studies at the University of Munich, the Institut d'Analisi Economica CSIC at the Universitat Autonoma de Barcelona, the Krannert School of Management, and the Jay N. Ross Young Faculty Scholar Award.     

All equilibria of the multi-unit Vickrey auction

This paper completely characterizes the set of equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there are at least three potential buyers. There are two types of equilibria: In the first class of equilibria there are positive bids below the maximum valuation. In this class, above a threshold value all bidders bid truthfully on all units. One of the bidders bids at the threshold for any unit for which his valuation is below the threshold; the other bidders bid zero in this range. In the second class of equilibria there are as many bids at or above the maximum valuation as there are units. The allocation of these bids is arbitrary across bidders. All the remaining bids equal zero. With any positive reserve price equilibrium becomes unique: Bidders bid truthfully on all units for which their valuation exceeds the reserve price.     

Auctioning divisible goods

Summary. We derive equilibrium bidding strategies in divisible good auctions for asymmetrically informed risk neutral and risk averse bidders when there is random noncompetitive demand. The equilibrium bid schedules contain both strategic considerations and explicit allowances for the winner's curse. When the bidders' information is symmetric, the strategic aspects of bidding imply that there always exist equilibria of a uniform-price auction with lower expected revenue than provided by a discriminatory auction. When bidders are risk averse, there may exist equilibria of the uniform-price auction that provide higher expected revenue than a discriminatory auction. Received: November 4, 1999; revised version: March 9, 2001     

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.     

Optimal auctions with asymmetrically informed bidders

Summary. The paper analyzes a problem of optimal auction design when the seller faces asymmetrically informed bidders. Specifically, we consider a continuum of risk-neutral uninformed bidders taking part into the auction along with n risk-averse informed bidders. The contribution of the paper is threefold. First, we fully characterize the optimal auction in this non standard environment and in a very general set-up. We find that when informed bidders reveal “bad news” about the value of the good, the seller optimally awards the object to the uninformed bidders. Secondly, we show that the seller is better off in presence of uninformed bidders because this allows to lower the informational rents paid to the informed bidders. Last, we find that, with bi-lateral risk neutrality, the seller always awards the good to the uninformed bidders thereby keeping all the surplus.Received: 22 October 2004, Revised: 21 April 2005, JEL Classification Numbers: D44, D82.We are very grateful to two anonymous referees, Jacques Crémer, Patrick François, Angel Hernando-Veciana and seminar participants at the 2004 SCSE conference (Quebec, Canada) for their valuable comments. Moez Bennouri acknowledges the financial support by the Initiative of the New Economy (INE) program of SSHRC (Canada).     

Information aggregation in auctions with an unknown number of bidders

Information aggregation, a key concern for uniform-price, common-value auctions with many bidders, has been characterized in models where bidders know exactly how many rivals they face. A model allowing for uncertainty over the number of bidders is essential for capturing a critical condition for information to aggregate: as the numbers of winning and losing bidders grow large, information aggregates if and only if uncertainty about the fraction of winning bidders vanishes. It may be possible for the seller to impart this information by precommitting to a specified fraction of winning bidders, via a proportional selling policy. Intuitively, this could make the proportion of winners known, and thus provide all the information that bidders need to make winner's curse corrections.     

Ascending price Vickrey auctions

We show that an ascending price auction for multiple units of a homogeneous object proposed by Ausubel (i) raises prices for packages until they reach those nonlinear and non-anonymous market clearing prices at which bidders get their marginal products and (ii) the auction is a primal–dual algorithm applied to an appropriate linear programming formulation in which the dual solution yields those same market clearing prices. We emphasize the similarities with efficient incentive compatible ascending price auctions to implement Vickrey payments when there is a single object or when objects are heterogeneous but each buyer does not desire more than one unit. A potential benefit of these common threads is that it helps to establish the principles upon which Vickrey payments may be implemented through decentralized, incentive compatible procedures.     

Multi-unit auctions with uniform prices

Summary. Auctions in which individuals can purchase more than one unit of the good being sold differ in striking ways from multi-unit auctions in which individuals may purchase only one unit. The uniform price auction in particular frequently yields Nash equilibria in which bidders underbid for their second unit and therefore pay very low prices for the good. This paper characterizes equilibria for the uniform price auction. Received: July 31, 1995; revised version: May 28, 1997