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.     

Standard Auctions with Financially Constrained Bidders

We develop a methodology for analyzing the revenue and efficiency performance of auctions when buyers have private information about their willingness to pay and ability to pay. We then apply the framework to scenarios involving standard auction mechanisms. In the simplest case, where bidders face absolute spending limits, first-price auctions yield higher expected revenue and social surplus than second-price auctions. The revenue dominance of first-price auctions over second-price auctions carries over to the case where bidders have access to credit. These rankings are explained by differences in the extent to which financial constraints bind in different auction formats.     

Creating competition out of thin air: An experimental study of right-to-choose auctions

This paper presents an experimental study of a mechanism that is commonly used to sell multiple heterogeneous goods. The novel feature of this procedure is that instead of selling each good in a separate auction, the seller executes a single auction in which buyers, who may be interested in completely different goods, compete for the right to choose a good. We provide experimental evidence that a Right-to-Choose (RTC) auction can generate more revenue than the theoretically optimal auction. Moreover, in contrast to the “optimal” auction, the RTC auction is approximately efficient in the sense that the surplus it generates is close to the maximal one. Furthermore, a seller who would like to retain some of his goods can generate more revenue with a restricted RTC auction in which not all rights-to-choose are sold, than with the theoretically optimal auction.     

Optimal Information Disclosure in Auctions and the Handicap Auction

We analyse a situation where a monopolist is selling an indivisible good to risk-neutral buyers who only have an estimate of their private valuations. The seller can release, without observing, certain additional signals that affect the buyers' valuations. Our main result is that in the expected revenue-maximizing mechanism, the seller makes available all the information that she can, and her expected revenue is the same as it would be if she could observe the part of the information that is "new" to the buyers. We also show that this mechanism can be implemented by what we call a handicap auction in interesting applications. In the first round of this auction, each buyer picks a price premium from a menu offered by the seller (a smaller premium costs more). Then the seller releases the additional signals. In the second round, the buyers bid in a second-price auction where the winner pays the sum of his premium and the second highest non-negative bid. In the case of a single buyer, this mechanism simplifies to a menu of European call options.     

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     

Uniqueness of equilibrium in sealed high-bid auctions

For the case of two buyers we show that equilibrium in the sealed high-bid auction is unique when (i) buyers' reservations prices are drawn independently from distributions with finite support and positive mass at the lower endpoint; (ii) buyers have private values; and (iii) buyers' preferences are log supermodular. For more than two buyers, we obtain the same result under the additional assumptions that (iv) buyers with the same reservation price have the same preferences; (v) buyers are risk neutral or risk averse with non-increasing absolute risk aversion; and (vi) the supports of the different buyers' distributions of reservation prices have the same upper endpoint.     

An optimal auction with correlated values and risk aversion

We consider an auction setting where the buyers are risk averse with correlated private valuations (CARA preferences, binary types), and characterize the optimal mechanism for a risk-neutral seller. We show that the optimal auction extracts all buyer surplus whenever the correlation is sufficiently strong (greater than 1/3 in absolute value), no matter how risk averse the buyers are. In contrast, we note that a sufficiently risk-averse seller would not use a full rent extracting mechanism for any positive correlation of the valuations even if the buyers were risk neutral.     

An experimental study of auctions with a buy price under private and common values

eBayʼs Buy It Now format allows a seller to list an auction with a “buy price” at which a bidder may purchase the item immediately and end the auction. When bidders are risk averse, then theoretically a buy price can raise seller revenue when values are private (but not when values are common). We report the results of laboratory experiments designed to determine whether in practice a buy price is advantageous to the seller. We find that a suitably chosen buy price yields a substantial increase in seller revenue when values are private, and a small (but statistically insignificant) increase in revenue when values are common. In both cases a buy price reduces the variance of seller revenue. A behavioral model which incorporates the winnerʼs curse and the overweighting by bidders of their own signal explains the common value auction data better than the rational model.     

Imperfect Competition in Auction Designs

We study the competition between two owners of identical goods who wish to sell them to a pool of potential buyers. The sellers compete simultaneously setting reserve prices for their second price sealed bid auctions. Upon observing the set reserve prices, the buyers decide simultaneously in which auction to bid. We show that this game has (at least) one equilibrium and that all equilibria are inefficient: reserve prices are not driven to zero (cost). We also discuss where and why the parallel between optimal auction design and optimal pricing in the case of monopoly breaks down for oligopoly.     

A Comparison of Multiple‐Unit All‐Pay and Winner‐Pay Auctions Under Incomplete Information*

This article examines the properties of independent‐private‐value all‐pay and winner‐pay auctions when there are multiple units sold. We study bidding behavior, efficiency, and revenue in a set of 23 experimental sessions. Our data show that the all‐pay auction and the winner‐pay auction are empirically revenue equivalent, though both yield higher revenue than the risk neutral Bayesian equilibrium. Our evidence also suggests that the winner‐pay auction is more likely to lead to a Pareto‐efficient allocation than the all‐pay auction.     

Premium auctions and risk preferences

In a premium auction, the seller offers some “payback”, called premium, to a set of high bidders at the end of the auction. This paper investigates how the performance of such premium tactics is related to the bidders? risk preferences. We analyze a two-stage English premium auction model with symmetric interdependent values, in which the bidders may be risk averse or risk preferring. Upon establishing the existence and uniqueness of a symmetric equilibrium, we show that the premium causes the expected revenue to increase in the bidders? risk tolerance. A “net-premium effect” is key to this result.     

Competing auctions with endogenous quantities

Two sellers decide on their discrete supply of a homogenous good. There is a finite number of buyers with unit demand and privately known valuations. In the first model, there is a centralized market place where a uniform auction takes place. In the second, there are two distinct auction sites, each with one seller, and buyers decide where to bid. Using the theory of potential games, we show that in the one-site auction model there is always an equilibrium in pure-strategies. In contrast, if the distribution of buyers values has an increasing failure rate, and if the marginal cost of production is relatively low, there is no pure-strategy equilibrium where both sellers make positive profits in the competing sites model. We also identify conditions under which an equilibrium with a unique active site exists. We deal with the finite and discrete models by using several results about order statistics developed by Richard Barlow and Frank Proschan [R. Barlow, F. Proschan, Mathematical Theory of Reliability, Wiley, New York, 1965; R. Barlow, F. Proschan, Inequalities for linear combinations of order statistics from restricted families, Ann. Math. Statist. 37 (1966) 1593-1601; R. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, McArdle Press, Silver Spring, 1975].     

A choice of auction format in seller cheating: a signaling game analysis

In this paper, we examine which auction format, first-price or second-price, a seller will choose when he can profitably cheat in a second price auction by observing all bids by possible buyers and submitting a shill bid as pretending to be a buyer. We model this choice of auction format in seller cheating as a signaling game in which the buyers may regard the selection of a second price auction by the seller as a signal that he is a shill bidder. By introducing trembling-hand perfectness as a refinement of signaling equilibrium, we find two possible strictly perfect signaling equilibria. One is a separating equilibrium in which a noncheating honest seller selects a first price auction and a cheating seller does a second price auction. In another pooling equilibrium, however, both cheating and non-cheating sellers select a second price auction. The conclusion that a seller chooses a second price auction even if he cannot cheat is in contrast to the previous literature, which focused on the case of independent values. We thank an anonymous referee for useful comments that have improved the paper. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B) 15310023 and (C) 18530139.     

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.     

A simple model of coalitional bidding

Summary. We analyze a model of coalitional bidding in which coalitions form endogenously and compete with each other. Since the nature of this competition influences the way in which agents organize themselves into coalitions, our main aim is to characterize the equilibrium coalition structure and the resulting bids. We do so in a simple model in which the seller may have good reason to allow joint bidding. In particular, we study a model in which the agents are budget constrained, and are allowed to form coalitions to pool their finances before engaging in the first price auction. We show that if the budget constraint is very severe, the equilibrium coalition structure consists of two coalitions, one slightly larger than the other; interestingly, it is not the grand coalition. This equilibrium coalition structure is one which yields (approximately) the maximum expected revenue. Thus the seller can induce the optimal (revenue maximizing) degree of cooperation among budget constrained buyers simply by permitting them to collude. Received: June 25, 1999; revised version: November 13, 2000     

A nearly optimal auction for an uninformed seller

This paper describes a nearly optimal auction mechanism that does not require previous knowledge of the distribution of values of potential buyers. The mechanism we propose builds on the new literature on the elicitation of information from experts. We extend the latter to the case where the secret information shared by the experts–potential buyers in our model–can be used against them if it becomes public knowledge.     

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.     

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.     

Information acquisition during a descending auction

We compare the effects of information acquisition during a descending auction with its static counterpart, the first-price sealed-bid auction. In a framework with heterogeneous prior information, we show that an equilibrium with information acquisition exists in both auction formats. We show that everything else equal information acquisition is more desirable in the dynamic auction. Moreover, we characterize a set of parameter values where more information is acquired in the dynamic auction in equilibrium. If the costs of information acquisition are sufficiently low, the sealed-bid auction generates more revenue although the descending auction is more efficient.     

三种国债拍卖方式的拍卖收入比较研究

国债拍卖主要有美国式拍卖、荷兰式拍卖与混合式拍卖三种,但究竟哪一种拍卖方式的拍卖收入最大?学术界迄今尚无定论。文章对W-Z模型进行扩展,并在此基础上通过结合不同市场利率环境下投标者的不同风险偏好进行比较研究。结果表明:(1)当市场利率趋升时,若非竞争性投标量较大,则混合式拍卖收入最大;若非竞争性投标量较小,但竞争性投标者较少,且市场利率上升趋势极强,则同样是混合式拍卖收入最大,反之则荷兰式拍卖收入最大。(2)当市场利率趋降时,美国式拍卖收入最大。(3)当市场利率平稳时,何种拍卖方式收入最大无法确定。