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.     

Auctions with uncertain numbers of bidders

We investigate bidders’ and seller's responses to ambiguity about the number of bidders in the first price auction (FPA) and the second price auction (SPA) with independent private valuations. We model ambiguity aversion using the maxmin expected utility model. We find that bidders prefer the number of bidders to be revealed in the FPA, are indifferent between revealing and concealing in the SPA, and prefer the SPA to the FPA. If bidders are more pessimistic than the seller then the seller prefers to conceal the number of bidders in the FPA, and prefers the FPA to the SPA.     

Almost common values auctions revisited

In almost common-value auctions one bidder (the advantaged bidder) has a valuation advantage over all other (regular) bidders. It is well known that in second-price auctions with two bidders, even a slight private-value advantage can have an explosive effect on auction outcomes as the advantaged bidder wins all the time and auction revenue is substantially lower than in a pure second-price common-value auction. We explore the robustness of these results to the addition of more regular bidders in second-price auctions, and the extent to which these results generalize to ascending-price English auctions in an effort to provide insight into when and why one ought to be concerned about such slight asymmetries.     

不同一口价组合拍卖机制下的卖方收益比较研究

随着电子商务的发展,作为网上拍卖方式之一的一口价拍卖逐渐盛行起来。网上一口价拍卖打破了时间和空间的限制,因而被广泛应用于组合拍卖中。笔者针对目前拍卖网站所采用的两种不同拍卖规则:固定一口价和持久一口价,在独立私人估价模型下分析比较了上述两种规则下的卖方期望收益。     

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.     

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.     

Asymmetric information about rivals'' types in standard auctions

We study auctions in which bidders may know the types of some rival bidders but not others. This asymmetry in bidders' knowledge about rivals' types has different effects on the two standard auction formats. In a second-price auction, it is weakly dominant to bid one's valuation, so the knowledge of rivals' types has no effect, and the good is allocated efficiently. In a first-price auction, bidders refine their bidding strategies based on their knowledge of rivals' types, which yields an inefficient allocation. We show that the inefficient allocation in the first-price auction translates into a poor revenue performance. Given a standard regularity condition, the seller earns higher expected revenue from the second-price auction than from the first-price auction, whereas the bidders are better off from the latter.     

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.     

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.     

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.     

Sealed bid auctions with uncertainty averse bidders

Summary. Traditional analysis of auctions assumes that each bidder's beliefs about opponents' valuations are represented by a probability measure. Motivated by experimental findings such as the Ellsberg Paradox, this paper examines the consequences of relaxing this assumption in the first and second price sealed bid auctions with independent private values. The multiple priors model of Gilboa and Schmeidler [Journal of Mathematical Economics, 18 (1989), 141–153] is adopted specifically to represent the bidders' (and the auctioneer's) preferences. The unique equilibrium bidding strategy in the first price auction is derived. Moreover, under an interesting parametric specialization of the model, it is shown that the first price auction Pareto dominates the second price auction. Received: December 15, 1995; revised version: February 19, 1997     

Risk aversion and optimal reserve prices in first- and second-price auctions

We analyze the effects of buyer and seller risk aversion in first- and second-price auctions in the classic setting of symmetric and independent private values. We show that the seller's optimal reserve price decreases in his own risk aversion, and more so in the first-price auction. The reserve price also decreases in the buyers' risk aversion in the first-price auction. Thus, greater risk aversion increases ex post efficiency in both auctions - especially that of the first-price auction. At the interim stage, the first-price auction is preferred by all buyer types in a lower interval, as well as by the seller.     

Mixed bundling auctions

We study multi-object auctions where agents have private and additive valuations for heterogeneous objects. We focus on the revenue properties of a class of dominant strategy mechanisms where a weight is assigned to each partition of objects. The weights influence the probability with which partitions are chosen in the mechanism. This class contains efficient auctions, pure bundling auctions, mixed bundling auctions, auctions with reserve prices and auctions with pre-packaged bundles. For any number of objects and bidders, both the pure bundling auction and separate, efficient auctions for the single objects are revenue-inferior to an auction that involves mixed bundling.     

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.     

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 almost homogeneous bidders

We prove that, around the symmetric case, where the values are identically distributed, the equilibrium of the first price auction is jointly differentiable with respect to general bidder-specific parameters of the value distributions. We show that the revenue equivalence between the first-price and the second-price auctions to the first-order in the size of the parameters is an immediate consequence of this differentiability and the Revenue Equivalence Theorem; thereby formally establishing the first-order equivalence Fibich et al. [G. Fibich, A. Gavious, A. Sela, Revenue equivalence in asymmetric auctions, J. Econ. Theory 115 (2004) 309-321] noticed for their particular perturbation.     

Bundling and Optimal Auctions of Multiple Products

We study the optimal ( i.e. revenue maximizing) auction of multiple products. We make three major points. First, we extend the relationship between price discrimination and optimal auctions from the single-product case to the multiple-product case. A monopolist setting prices for multiple products may offer discounts on purchases of bundles of products; similarly, the optimal auction of multiple products facilitates price discrimination by allocating products inefficiently to customers who are willing to purchase both products. Second, we demonstrate that optimal auctions are qualitatively distinct from monopoly sales of multiple products. Because of uncertainty about the values of other consumers, two products are bundled probabilistically in an optimal auction for a customer who is willing to buy both of them. A customer may then receive a discount on a lower-valued product without receiving a higher-valued product. Third, we show that in an optimal auction of two products the allocation of one product may vary with the amount of competition for the other product.     

Auctions with Financial Externalities

We study auctions with financial externalities, i.e., auctions in which losers care about how much the winner pays. In the first-price auction, larger financial externalities result in a lower expected price; in the second-price auction, the effect is ambiguous. Although the expected price in the second-price auction may increase if financial externalities increase, the seller is not able to gain more revenue by guaranteeing the losers a fraction of the auction revenue. With a reserve price, we find that both auctions may have pooling at the reserve price. This finding suggests that identical bids need not be a signal of collusion, in contrast to what is sometimes argued in anti-trust cases. We gratefully acknowledge financial support from the Dutch National Science Foundation (NWO 510.010.501 and NWO-VICI 453.03.606). For valuable discussions and comments, we would like to thank Eric van Damme, Jacob Goeree, Thomas Kittsteiner, Marta Kolodziejczyk, seminar participants at Tilburg University, Humboldt University Berlin, and National University of Singapore, and audiences at ESEM 2001 in Lausanne, and the FEEM 2002 conference in Milan on auctions and market design. The suggestions of an anonymous referee of this Journal greatly improved the article. The usual disclaimer applies.     

Efficient collusion in repeated auctions with communication

This paper studies collusion in repeated auctions when bidders communicate prior to each stage auction. For independent and correlated private signals and general interdependent values, the paper identifies conditions under which an equilibrium collusion scheme is fully efficient in the sense that the bidders’ payoff is close to what they get when the object is allocated to the highest valuation bidder at the reserve price in every period.     

Multiple unit auctions with strategic price-quantity decisions

Summary. I study a multiple unit auction where symmetric risk-neutral bidders choose prices and quantities endogenously. In the model, bidders (a) may place non-linear valuations on the auctioned units, and (b) bid for several units at the same price (“lumpy” bids). I characterize quantity-symmetric and strictly monotone-increasing price equilibria for discriminatory and competitive auctions, and show that (i) if quantity strategy profiles are equal across auctions revenue- equivalence holds, (ii) expected revenue is higher if bidders bid for the entire supply rather than for shares of it, and (iii) equilibrium allocations may fail to be Pareto-optimal. Received: April 14, 1995; revised version: September 3, 1997