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.     

Auctions in which losers set the price

We study auctions of a single asset among symmetric bidders with affiliated values. We show that the second-price auction minimizes revenue among all efficient auction mechanisms in which only the winner pays, and the price only depends on the losers' bids. In particular, we show that the kth price auction generates higher revenue than the second-price auction, for all k>2. If rationing is allowed, with shares of the asset rationed among the t highest bidders, then the (t+1)st price auction yields the lowest revenue among all auctions with rationing in which only the winners pay and the unit price only depends on the losers' bids. Finally, we compute bidding functions and revenue of the kth price auction, with and without rationing, for an illustrative example much used in the experimental literature to study first-price, second-price and English auctions.     

Bidding ‘as if’ risk neutral in experimental first price auctions without information feedback

Experimental research on first price sealed bid auctions has usually involved repeated settings with information feedback on winning bids and payoffs after each auction round. Relative to the risk neutral Nash equilibrium, significantly higher bidding has been reported. The present paper reports the results of experimental first price auctions with n=7 where feedback on payoffs and winning bids is withheld. Under these conditions, average bidding is below the risk neutral Nash equilibrium prediction but converges to it with repetition.

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Asymptotic and small sample analysis of the stochastic properties and certainty equivalents of winning bids in independent private values auctions$\!$

Summary. This paper investigates the stochastic properties of the first price and second price winning bids in auctions with risk neutral bidders with independent and identically distributed valuations. In such an environment, the winning first price bids second order stochastically dominate the second price winning bids. A key result of this paper is that the ratio of the variance of the winning first price bid to that of the winning second price bid is strictly less than one even as the number of bidders goes to infinity. This suggests that the stochastic dominance of the winning first price bids does not vanish even in the limit. Both the asymptotic and small sample properties of winning bids are investigated. In particular, the small sample results suggest that the identification power of econometric procedures that rely on winning bids can decline rapidly as the number of bidders increases. This paper also includes a systematic evaluation of the difference in certainty equivalents between the two auction formats for several common distributions and number of bidders for the case of auctioneers with constant coefficient of absolute risk aversion. These results are presented in a form that makes it easy to apply them to a specific auction of interest.Received: 15 July 2002, Revised: 7 April 2003, JEL Classification Numbers: D44.I would like to thank David Pearce for helpful comments and discussion and a referee for suggestions that have greatly improved this paper.     

Multiple-unit English auctions

This paper examines the outcome of an ascending-price multiple-unit auction. Two bidders, facing continuous, downward-sloping demand functions, participate in the auction of some divisible objects. The auctioneer starts the process by announcing an initial price and asking both bidders to submit sealed-bids of desired quantities. The auctioneer increases the price until the total amount bid for is less than the total supply. We compute the outcome of this auction game under full information and suggest that this mechanism may not maximize expected revenue.     

Hybrid allocation mechanisms for publicly provided goods

Motivated by efficiency and equity concerns, public resource managers have increasingly utilized hybrid allocation mechanisms that combine features of commonly used price (e.g., auction) and non-price (e.g., lottery) mechanisms. This study serves as an initial investigation of these hybrid mechanisms, exploring theoretically and experimentally how the opportunity to obtain a homogeneous good in a subsequent lottery affects Nash equilibrium bids in discriminative and uniform price auctions. The lottery imposes an opportunity cost to winning the auction, systematically reducing equilibrium auction bids. In contrast to the uniform price auction, equilibrium bids in the uniform price hybrid mechanism vary with bidder risk preferences. Experimental evidence suggests that the presence of the lottery and risk attitudes (elicited through a preceding experiment) impact auction bids in the directions predicted by theory. Finally, we find that theoretically and experimentally, the subsequent lottery does not compromise the efficiency of the auction component of the hybrid mechanisms.     

Budget-constrained sequential auctions with incomplete information

I study a budget-constrained, private-valuation, sealed-bid sequential auction with two incompletely-informed, risk-neutral bidders in which the valuations and income may be non-monotonic functions of a bidder's type. Multiple equilibrium symmetric bidding functions may exist that differ in allocation, efficiency and revenue. The sequence of sale affects the competition for a good and therefore also affects revenue and the prices of each good in a systematic way that depends on the relationship among the valuations and incomes of bidders. The sequence of sale may affect prices and revenue even when the number of bidders is large relative to the number of goods. If a particular good, say α, is allocated to a strong bidder independent of the sequence of sale, then auction revenue and the price of good α are higher when good α is sold first.     

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     

Second chance offers in auctions

This paper examines situations in which a seller might make a second chance (take-it-or-leave-it) offer to a non-winning bidder at a price equal to their bid at auction. This study is motivated by the take-it-or-leave-it second chance offer rules used by eBay and a number of state procurement agencies. Equilibrium bidder behavior is determined for IPV sealed bid first price, second price, English, and Vickrey auctions when a second chance offer will be made with an exogenous probability $p$ . In all but the Vickrey auction (which elicits the dominant strategy of bidding one’s value) equilibrium bids are lower than if there were no possibility of a second chance offer and higher than if a second chance offer will be made for certain. Further, the possibility of a second chance offer erodes the strategic equivalence between second price bids and English auction drop out levels. If bidders are risk averse (with CRRA preferences), this difference leads to expected revenue dominance of the second price over the English auction, both of which dominate the Vickrey auction. The first price auction is also shown to revenue dominate the Vickrey auction, and moreover, numerical results and intuition from existing literature suggest that the first price auction revenue dominates the second price auction.     

Reserve Prices without Commitment

When potential bidders in an auction have to incur a cost to prepare their bids and thus to learn their valuations, imposing a reserve price and announcing that in case no bid is submitted there will be another auction without a reserve price is both revenue and welfare improving. Reserve prices that induce less than maximum entry in the first auction may be optimal. Also, entry fees are not necessarily better instruments than reserve prices.Journal of Economic LiteratureClassification Number: D44.     

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.     

Sealed bid auctions with ambiguity: Theory and experiments

This study presents a laboratory experiment of the first and second price sealed bid auctions with independent private values, where the distribution of bidder valuations may be unknown. In our experimental setting, in first price auctions, bids are lower with the presence of ambiguity. This result is consistent with ambiguity loving in a model that allows for different ambiguity attitudes. We also find that the first price auction generates significantly higher revenue than the second price auction with and without ambiguity.     

Explaining Overbidding in First Price Auctions Using Controlled Lotteries

In this paper, we study the behavior of individuals when facing two different, but incentive-wise identical, institutions. We pair the first price auction with an equivalent lottery. Once a subject is assigned a value for the auctioned object, the first price auction can be modeled as a lottery in which the individual faces a given probability of winning a certain payoff. This set up allows us to explore to what extent the misperception of the probability of winning in the auction is responsible for bidders in a first price auction to bidding above the risk neutral Nash equilibrium prediction. The first result we obtain is that individuals, even though facing the same choice over probability/payoff pairs, behave differently depending on the type of choice they are called to make. When facing an auction, subjects with high values tend to bid significantly above the bid they choose in the corresponding lottery environment. We further find that in both the lottery and the auction environments, subjects tend to bid in excess of the bid predicted by the risk neutral model, at least for intermediate range values. Finally, we find that the difference between the lottery behavior and the auction behavior is substantially, but not totally, eliminated by showing the subjects the probability of winning the auction.     

A Sealed-Bid Auction That Matches the English Auction

This paper analyzes a two-stage sealed-bid auction that is frequently employed in privatization, takeover, and merger and acquisition contests. This auction format yields the same expected revenue as the open ascending (English) auction, yet is less susceptible to preemptive bidding and collusion. Journal of Economic Literature Classification Number: D44.     

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.     

A Bluff-Bidding Exercise

Consider an auction in which one potential buyer wishes to participate, but the other potential buyer would rather the bidding not start. However, once bidding starts, the reluctant firm participates (submits “bluff bids”) simply to make the eventual winner pay more. This incentive exists when the marginal effect of the winning bid is to increase a rival's profit. In 2004, AT&T Wireless placed itself for sale in an English auction. Some predicted Vodafone would make bluff bids (to make Cingular pay more. Students experience this sort of activity in the game that this article describes. Students also learn that bluff bidding affects profits of the firms involved and therefore has important implications for stock prices of participating firms.     

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.     

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.     

Competition in the economic crisis: Analysis of procurement auctions

We study the effects of the recent economic crisis on firms׳ bidding behavior and markups in sealed bid auctions. Using data from Austrian construction procurements, we estimate bidders׳ construction costs within a private value auction model. We find that markups of all bids submitted decrease by 1.5 percentage points in the recent economic crisis, markups of winning bids decrease by 3.3 percentage points. We also find that without the government stimulus package this decrease would have been larger. These two pieces of evidence point to pro-cyclical markups.     

Bid Functions in Auctions and Fair Division Games: Experimental Evidence

In auctions a seller offers a commodity for sale and collects the revenue. In fair division games the object is collectively owned by the group of bidders who equally share the revenue. We run an experiment in which the participants face four types of allocation games (auctions and fair division game under two price rules, first– versus second–price rule). We collect entire bid functions rather than bids for single values and investigate price and efficiency of the different trading institutions. We find that the first–price auction is more efficient than the second–price auction, whereas economic rationality assuming heterogeneous bidders suggests the opposite. Furthermore, we study the structure of individual bid functions.