CHARITY AUCTIONS*

In a charity auction the public‐goods nature of auction revenue affects bidding incentives. We compare equilibrium bidding and revenue in first‐price, second‐price, and all‐pay charity auctions. Bidding revenue typically varies by selling format. First‐price auctions are less lucrative than second‐price and all‐pay auctions, and with sufficiently many bidders the all‐pay auction has the highest bidding revenue. However, revenue equivalence applies when the auctioneer can set a reserve price and fees plus threaten to cancel the auction. If the auctioneer cannot threaten cancellation, a reserve and bidding fee can augment revenue but again revenue varies by auction format     

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.     

BUNDLING IN AUCTIONS WITH NON‐LINEAR VALUATIONS*

We examine when a revenue‐maximising auctioneer prefers to auction a homogenous product in one bundle (a single‐object auction) than to sell the item in two or more shares. When the items are super‐additive the auctioneer always prefers a single‐object auction. When the product valuations are sub‐additive, the auctioneer is more likely to choose a share auction when there are a large number of potential bidders. When there are a small number of bidders the auctioneer will tend to prefer a single‐object auction.     

Profit Maximizing in Auctions of Public Goods

A profit-maximizing auctioneer can provide a public good to a group of agents. Each group member has a private value for the good being provided to the group. We investigate an auction mechanism where the auctioneer provides the good to the group only if the sum of their bids exceeds a reserve price declared previously by the auctioneer. For the two-bidder case with private values drawn from a uniform distribution we characterize the continuously differentiable symmetric equilibrium bidding functions for the agents, and we find the optimal reserve price for the auctioneer when such functions are used by the bidders. We also examine another interesting family of equilibrium bidding functions for this case, with a discrete number of possible bids, and show the relation (in the limit) to the differentiable bidding functions.     

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.     

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.      

Large common value auctions with risk averse bidders

We analyze large symmetric auctions with conditionally i.i.d. common values and risk averse bidders. Our main result characterizes the asymptotic equilibrium price distribution for the first- and second-price auctions. As an implication, we show that with constant absolute risk aversion (CARA), the second-price auction raises significantly more revenue than the first-price auction. While this ranking seems robust in numerical analysis also outside the CARA specification, we show by counterexamples that the result does not generalize to all risk averse utility functions.     

Unraveling in first-price auction

In a private values first-price auction with random number of bidders, the auctioneer has to decide whether to announce or not the actual number of bidders. If the auctioneer cannot commit to any particular announcement policy, then he would like to reveal when the number of bidders is high in order to increase bidding competition. By the same logic, the auctioneer should like to conceal if the number turns out to be small. When bidders' types are independent, full revelation by the auctioneer is shown to be a unique equilibrium with ‘skeptical belief’ as the unique equilibrium belief.     

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.     

Revealing information in auctions: the allocation effect

When there are two bidders, releasing independent information in an English auction with private values makes the seller worse off. However, this is no longer true with more bidders: when there is enough competition, revelation benefits the auctioneer. In three examples the dividing case is shown to be three bidders. This allocation effect applies to other standard auctions and parallels the bundling decision in a multi-unit auction. I would like to thank Jeremy Bulow, Willie Fuchs, Paul Klemperer, Rob McMillan, Phil Reny, Jeff Zwiebel and especially Andy Skrzypacz. I also grateful to two referees and an Associate Editor for a number of helpful comments.     

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.     

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.     

An Efficient Multi-Unit Ascending Auction

We provide an ascending auction that yields an efficient outcome when there are many identical units for sale and bidders have interdependent values and downward-sloping demand. Our ascending auction both extends and generalizes Ausubel 's (2004) and yields the same outcome as Perry and Reny 's (2002) generalization of Vickrey 's (1961) sealed-bid auction. There are two key features of our auction. Bidders are permitted both to express different demands against different bidders, as well as to increase their demands. The equilibrium strategies are closely related to the familiar „drop out when price equals value” strategy of the English auction.     

Communication,competition and social gift exchange in an auction for public good provision

Reverse auctions are an established policy instrument for allocating conservation contracts. While the auction mechanism has been the subject of a number of studies, less attention has been paid to the post-bidding contract phase. As contracts involving natural resource management are usually incomplete, trust becomes crucial for the effectiveness of the programme. We test the effect of communication between auctioneer and bidders on bidding behaviour and contract fulfilment using experimental economics. We combine a repeated reverse auction with an effort-level game and use a bilateral chatting tool as treatment variable. Without communication, auctioneers tended to select the lowest-priced bidders, who invested substantially less than the socially optimal level of effort when fulfilling their contract to provide the public good. Relational contracting proved important, with effort levels and profits tending to be higher when auctioneers and bidders entered into consecutive contract relationships. In the communication treatment there was no evidence of price competition, as auctioneers were more likely to accept high-priced bids. However, an overall higher price level did not lead to efficiency losses, since contractors realised higher effort levels in return, establishing a ‘social gift exchange’. Our results demonstrate the importance of trust-based relationships between the auctioneering institution and landholders.     

The Impact of Discounting on an Auction with a Buyout Option: a Theoretical Analysis Motivated by eBay’s Buy-It-Now Feature

An auction with a buyout option occurring over continuous time with rules similar to eBays buy it now option is analyzed. When auction participants make no distinction as to when a transaction occurs, the seller optimally chooses a buyout option price so high that bidders never exercise the option. However, time impatience on either side of the transaction can motivate the seller to offer an option price low enough so that the option is exercised with positive probability. Further, allowing a time impatient seller to offer such an option results in an ex ante Pareto improvement when bidders do not discount future transactions.     

The high/low divide: Self-selection by values in auction choice

Most prior theoretical and experimental work involving auction choice has assumed bidders find out their value after making a choice of which auction to enter. We examine whether or not bidders knowing their value prior to making a choice of which among multiple alternative auction formats to enter impacts their choice decision and/or the outcome of the auctions. The results show a strong impact on auction choice. Subjects with low values choose the first price sealed bid auction more often while subjects with high values choose the ascending auction more often. The number of bidders in each auction, revenue, efficiency and average bidder surplus all end up equalized.     

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     

密封价格拍卖或招标中的有限腐败

本文研究了密封价格拍卖或招投标中的有限腐败问题,探讨了当行贿者通过行贿招标主持人获得多次出标机会,而其他竞标者不知道这种有限腐败行为时,对拍卖结果所产生的影响。在第二价格拍卖机制下,由于竞标者按真实估价报价总是弱占优策略,该有限腐败行为在此拍卖机制下不会产生影响。但对于密封的第一价格拍卖机制来说,由于行贿者的多个标价中的最高标价比其他竞标者的标价更强势(aggressive),导致其他竞标者获胜的概率减少,行贿者获胜的概率增加。而行贿者的其他出标机会所用的出标策略比其他竞标者的出标策略要弱势,所以在保证期望收益增加的情况下所付出的期望支付有可能比在没有腐败情形下的期望支付要低。特别地,当行贿者获得更多的出标机会时所采用的出标策略与我们的直觉是有差异的。有限腐败对于物品所有者来说是不利的,他的期望收益随着行贿者的特权的增强而减少。     

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.     

Charitable asymmetric bidders

Recent papers show that the all‐pay auction is better at raising money for charity than the first‐price auction with symmetric bidders under incomplete information. Yet, this result is lost with sufficiently asymmetric bidders under complete information. In this paper, we consider a framework on charity auctions with asymmetric bidders under some incomplete information. We find that the all‐pay auction still raises more money than the first‐price auction. Thus, the all‐pay auction should be seriously considered when one wants to organize a charity auction.