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.     

The insider's curse

The paper shows that in an open-ascending bid auction with multi-dimensional uncertainty about private and common value components, private information about the common value has negative value for a bidder if there are sufficiently many bidders. We discuss the role of the visibility of bids and the multi-dimensionality of private information for this result.     

Information acquisition in auctions: Sealed bids vs. open bids

This paper studies the incentives of a bidder to acquire information in an auction when her information acquisition decision is observed by the other bidders before bidding. Our results show that the sealed bid (second price) auction may induce more information acquisition about a common component of the value than the open (English) auction but less about the private component of the value. Moreover, we show that more information about the private value and less information about the common value may improve efficiency and revenue. Consequently, our results suggest new arguments in favor of the open auction.     

Auctions with variable supply: Uniform price versus discriminatory

We examine an auction in which the seller determines the supply after observing the bids. We compare the uniform price and the discriminatory auction in a setting of supply uncertainty, where uncertainty is caused by the interplay of two factors: the seller's private information about marginal cost and the seller's incentive to sell the profit-maximizing quantity, given the received bids. In every symmetric mixed strategy equilibrium, bidders submit higher bids in the uniform price auction than in the discriminatory auction. In the two-bidder case, this result extends to the set of rationalizable strategies. As a consequence, we find that the uniform price auction generates a higher expected revenue for the seller and a higher trade volume.     

Optimal auctions with information acquisition

This paper studies optimal auction design in a private value setting with endogenous information gathering. We develop a general framework for modeling information acquisition when a seller wants to sell an object to one of several potential buyers, who can each gather information about their valuations prior to participation in the auction. We first demonstrate that the optimal monopoly price is always lower than the standard monopoly price. We then show that standard auctions with a reserve price remain optimal among symmetric mechanisms, but the optimal reserve price lies between the ex ante mean valuation of bidders and the standard reserve price in Myerson (1981). Finally, we show that the optimal asymmetric mechanism softens the price discrimination against “strong” bidders.     

Auction fever: Rising revenue in second-price auction formats

The prevalent term “auction fever” visualizes that ascending auctions – inconsistent with theory – are likely to provoke higher bids than one-shot auctions. To explore and isolate causes of auction fever experimentally, we design four different strategy-proof auction formats and order these according to expected rising bids based on pseudo-endowment effect arguments (psychological ownership and disparity between willingness to pay and willingness to accept). Observed revenues in the experiment in the four formats rank as expected if bidders have private uncertain values (the private information of a bidder is the distribution of her value). A control treatment supports our view that the traditional private certain values approach prevents auction fever in the laboratory. Another control treatment with a procurement auction relates the auction fever bids to bids in a one-shot auction with real endowments. We conclude that, when bidders are uncertain about their valuations, auctions that foster pseudo-endowment may raise bids and revenues.     

Multidimensional private value auctions

We consider parametric examples of symmetric two-bidder private value auctions in which each bidder observes her own private valuation as well as noisy signals about her opponent's private valuation. We show that, in such environments, the revenue equivalence between the first and second price auctions (SPAs) breaks down and there is no definite revenue ranking; while the SPA is always efficient allocatively, the first price auction (FPA) may be inefficient; equilibria may fail to exist for the FPA. We also show that auction mechanisms provide different incentives for bidders to acquire costly information about opponents’ valuation.     

Allotment in first-price auctions: an experimental investigation

We experimentally study the effects of allotment—the division of an item into homogeneous units—in independent private value auctions. We compare a bundling first-price auction with two equivalent treatments where allotment is implemented: a two-unit discriminatory auction and two simultaneous single-unit first-price auctions. We find that allotment in the form of a discriminatory auction generates a loss of efficiency with respect to bundling. In the allotment treatments, we observe large and persistent bid spread, and the discriminatory auction is less efficient than simultaneous auctions. We provide a unified interpretation of our results that is based on both a non-equilibrium response to the coordination problem characterizing the simultaneous auction format and a general class of behavioral preferences that includes risk aversion, joy of winning and loser’s regret as specific cases.     

Single Object Auctions With Externalities: A Tractable Model

We consider the single object auction model with allocative externalities in a private valuation and quasi‐linear setting. We model externalities by assuming that every agent has a private valuation (for the object) and a strict ranking of other agents. The utility for an agent when another agent receives the object is the product of his own valuation and a real number that depends on the rank of this agent in his ranking. When the only private information is the valuation of the agents, we characterise the implementable allocation rules and use these to derive the optimal auction. The optimal auction collects payments from agents who do not receive the object.     

Valuation structure in first-price and least-revenue auctions: an experimental investigation

In many auctions the valuation structure involves both private and common value elements. Existing experimental evidence (e.g. Goeree and Offerman in Am. Econ. Rev. 92(3):625–643, 2002) demonstrates that first-price auctions with this valuation structure tend to be inefficient, and inexperienced subjects tend to bid above the break-even bidding threshold. In this paper, we compare first-price auctions with an alternative auction mechanism: the least-revenue auction. This auction mechanism shifts the risk regarding the common value of the good to the auctioneer. Such a shift is desirable when ex post negative payoffs for the winning bidder results in unfulfilled contracts, as is often the case in infrastructure concessions contracts. We directly compare these two auction formats within two valuation structures: (1) pure common value and (2) common value with a private cost. We find that, relative to first-price auctions, bidding above the break-even bidding threshold is significantly less prevalent in least-revenue auctions regardless of valuation structure. As a result, revenue in first-price auctions is higher than in least-revenue auctions, contrary to theory. Further, when there are private and common value components, least-revenue auctions are significantly more efficient than first-price auctions.     

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     

Researching Preferences,Valuation and Hypothetical Bias

A number of recent papers in environmental economics have focused on the process of researching preferences – agents are uncertain about preferences but with effort may narrow their uncertainty. This issue has arisen in formulating bids in contingent valuation (CV) as well as the debate over the divergence between WTP and WTA. In the context of CV, it has been suggested that the hypothetical nature of the preference elicitation process biases responses. This paper provides both a theoretical model and experimental evidence to contribute to this debate. The model is a model of competitive bidding for a private good with two components that are particularly relevant to the debate. The first component is that bidders are unsure of their own value for the private good but may purchase information about their own value (researching preferences). The second component is that there is a probability that the auction is hypothetical – that the winning bidder will not get the private good and will not pay the winning bid. The experiment tests this theoretical model of bidding equilibrium and analyzes the effects of variations in the parameters (hypotheticalness, information costs and number of agents) on the endogenous variables (such as the proportion of bidders who become informed and the winning bid). Experimental results suggest that an increase in the hypotheticalness of an auction tends to decrease the likelihood that bidders pay for information on their valuation with an ambiguous effect on the winning bid.      

Auctions with cross-shareholdings

Summary. We study the effect of cross-shareholding among two competing firms on their bidding behavior and the expected sales revenue for the seller in an auction environment. The bidders private signals are independent, and the model encompasses the private values model and a particular common value model as special cases. When cross-shareholding is symmetric, the bids decrease towards the collusive level as the degree of cross-shareholding increases. The Revenue Equivalence result no longer holds: the first-price auction generates higher expected revenue for the seller than the second-price auction.With asymmetric cross-shareholding, revenue comparisons are only possible in the common value setting. Expected revenue for the seller is again higher in the first-price than in the second price auction. Bidding behavior in the second-price auction is more sensitive to changes in cross-shareholding and the value environment than in the first-price auction.Received: 18 September 2000, Revised: 27 May 2003, JEL Classification Numbers: C72, D44.Correspondence to: Sudipto DasguptaWe thank Sugato Bhattacharyya, Paul Klemperer, Kunal Sengupta and Guofu Tan for helpful discussions, and an anonymous referee for suggestions that improved the paper. The usual disclaimer, of course, applies.     

Endogenous entry and self-selection in private value auctions: An experimental study

This paper presents the results of an experimental study of endogenous entry in first-price independent private value auctions. N potential bidders simultaneously decide whether to participate in an auction or receive a known outside option. In the second stage, entrants submit bids after learning their own private values and the number of entrants. An equilibrium model of heterogeneous risk averse bidders implies a self-selection effect, where bidding in the auction is lower with endogenous entry because only less risk averse bidders enter. This effect is confirmed by the experiment. We also observe excessive entry relative to the theoretical model.     

Auctions for social lending: A theoretical analysis

Prosper, today the second largest social lending marketplace with nearly 1.5 million members and $380 million in funded loans, employed an auction mechanism amongst lenders to finance each borrower's loan until 2010. Given that a basic premise of social lending is cheap loans for borrowers, how does the Prosper auction do in terms of the borrower's payment, when lenders are strategic agents with private true interest rates? We first analyze the Prosper auction as a game of complete information and fully characterize its Nash equilibria, and show that the uniform-price Prosper mechanism, while simple, can lead to much larger payments for the borrower than the VCG mechanism. We next compare the Prosper mechanism against the borrower-optimal auction in an incomplete information setting, and conclude by examining the Prosper mechanism when modeled as a dynamic auction, and provide tight bounds on the price for a general class of bidding strategies.     

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.     

Information Provision in Procurement Auctions

We study the optimal provision of information in a procurement auction with horizontally differentiated goods. The buyer has private information about her preferred location on the product space and has access to a costless communication device. A seller who pays the entry cost may submit a bid comprising a location and a minimum price. We characterize the optimal information structure and show that the buyer prefers to attract only two bids. Further, additional sellers are inefficient since they reduce total and consumer surplus, gross of entry costs. We show that the buyer will not find it optimal to send public information to all sellers. On the other hand, she may profit from setting a minimum price and that a severe hold‐up problem arises if she lacks commitment to set up the rules of the auction ex ante.     

Non-monotoniticies and the all-pay auction tie-breaking rule

Discontinuous games, such as auctions, may require special tie-breaking rules to guarantee equilibrium existence. The best results available ensure equilibrium existence only in mixed strategy with endogenously defined tie-breaking rules and communication of private information. We show that an all-pay auction tie-breaking rule is sufficient for the existence of pure strategy equilibrium in a class of auctions. The rule is explicitly defined and does not require communication of private information. We also characterize when special tie-breaking rules are really needed.      

Regulating an externality-generating public utility: A multi-dimensional screening approach

In this paper, I present a monopoly screening model in which a firm holds private information on fixed abatement costs. Because of the nature of the optimal contracts, bunching is never optimal even with ‘n’ distinct types. A characterization of full and limited information solutions is offered and it is shown that the design standards alone determine the value of an information rent. The variable charge is completely free of any marginal information costs. This allows the variable charge to track marginal damages and thereby ensures that marginal social costs are covered. A fixed charge implements the optimal solution.     

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.