Regret in auctions: theory and evidence

The sealed-bid first-price auction of a single object in the case of independent privately-known values is the simplest auction setting and understanding it is important for understanding more complex mechanisms. But bidders bid above the risk-neutral Nash equilibrium theory prediction. The reasons for this “over bidding” remain an unsolved puzzle. Several explanations have been offered, including risk aversion, social comparisons, and learning. We present a new explanation based on regret and a model that explains not only the observed over bidding in sealed-bid first-price auctions, but also behavior in several other settings that is inconsistent with risk aversion. The authors gratefully acknowledge support from the National Science Foundation.     

On the continuous equilibria of affiliated-value,all-pay auctions with private budget constraints

We consider all-pay auctions in the presence of interdependent, affiliated valuations and private budget constraints. For the sealed-bid, all-pay auction we characterize a symmetric equilibrium in continuous strategies for the case of N bidders. Budget constraints encourage more aggressive bidding among participants with large endowments and intermediate valuations. We extend our results to the war of attrition where we show that budget constraints lead to a uniform amplification of equilibrium bids among bidders with sufficient endowments. An example shows that with both interdependent valuations and private budget constraints, a revenue ranking between the two auction formats is generally not possible. Equilibria with discontinuous bidding strategies are discussed.     

Beyond chopsticks: Symmetric equilibria in majority auction games

Majority auction games are simultaneous sealed-bid auctions of identical objects among identical bidders who each want to win a specified fraction (more than a half) of the objects. Each bidder receives no benefit from winning less than the specified fraction and no additional benefit from winning more than it. Symmetric equilibria having simple, intuitive forms are shown to exist in first-price, second-price and all-pay versions of such games when the number of bidders is sufficiently large. This contrasts with earlier results for the two-bidder “pure chopstick” majority auction games where the only known equilibria are more complicated.     

Multi-unit demand auctions with synergies: behavior in sealed-bid versus ascending-bid uniform-price auctions

We construct a relatively simple model of bidding with synergies and solve it for both open outcry and sealed-bid uniform-price auctions. The essential behavioral forces involved in these auctions are: (1) A demand reduction force resulting from the monopsony power that bidders with multiple-unit demands have when synergies are relatively inconsequential, and (2) Bidding above stand-alone values in order to capture significant complementarities between units. The latter creates a potential “exposure problem,” as bidders may win only parts of a package and earn negative profits. Bidding outcomes are closer to equilibrium in clock compared to sealed-bid auctions. However, there are substantial and systematic deviations from equilibrium, with patterns of out-of-equilibrium play differing systematically between the two auction formats. These patterns of out-of-equilibrium play are analyzed, along with their effects on revenue and efficiency.     

Behavior in second-price auctions by highly experienced eBay buyers and sellers

We report on sealed-bid second-price auctions that we conducted on the Internet using subjects with substantial prior experience: they were highly experienced participants in eBay auctions. Unlike the novice bidders in previous (laboratory) experiments, the experienced bidders exhibited no greater tendency to overbid than to underbid. However, even subjects with substantial prior experience tended not to bid their values, suggesting that the non-optimal bidding of novice subjects is robust to substantial experience in non-experimental auctions. We found that auction revenue was not significantly different from the expected revenue the auction would generate if bidders bid their values. Auction efficiency, as measured by the percentage of surplus captured, was substantially lower in our SPAs than in previous laboratory experiments.     

Multi-unit demand auctions with synergies: behavior in sealed-bid versus ascending-bid uniform-price auctions

We construct a relatively simple model of bidding with synergies and solve it for both open outcry and sealed-bid uniform-price auctions. The essential behavioral forces involved in these auctions are: (1) A demand reduction force resulting from the monopsony power that bidders with multiple-unit demands have when synergies are relatively inconsequential, and (2) Bidding above stand-alone values in order to capture significant complementarities between units. The latter creates a potential “exposure problem,” as bidders may win only parts of a package and earn negative profits. Bidding outcomes are closer to equilibrium in clock compared to sealed-bid auctions. However, there are substantial and systematic deviations from equilibrium, with patterns of out-of-equilibrium play differing systematically between the two auction formats. These patterns of out-of-equilibrium play are analyzed, along with their effects on revenue and efficiency.     

Time is money: The effect of clock speed on seller’s revenue in Dutch auctions

We study the role of timing in auctions under the premise that time is a valuable resource. When one object is for sale, Dutch and first-price sealed bid auctions are strategically equivalent in standard models, and therefore, they should yield the same revenue for the auctioneer. We study Dutch and first-price sealed bid auctions in the laboratory, with a specific emphasis on the speed of the clock in the Dutch auction. At fast clock speeds, revenue in the Dutch auction is significantly lower than it is in the sealed bid auction. When the clock is sufficiently slow, however, revenue in the Dutch auction is higher than the revenue in the sealed bid auction. We develop and test a simple model of auctions with impatient bidders that is consistent with these laboratory findings.

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Auctions with endogenous participation and an uncertain number of bidders: experimental evidence

Attracting bidders to an auction is a key factor in determining revenue. We experimentally investigate entry and bidding behavior in first-price and English clock auctions to determine the revenue implications of entry. Potential bidders observe their value and then decide whether or not to incur a cost to enter. We also vary whether or not bidders are informed regarding the number of entrants prior to placing their bids. Revenue equivalence is predicted in all four environments. We find that, regardless of whether or not bidders are informed, first-price auctions generate more revenue than English clock auctions. Within a given auction format, the effect of informing bidders differs. In first-price auctions, revenue is higher when bidders are informed, while the opposite is true in English clock auctions. The optimal choice for an auction designer who wishes to maximize revenue is a first-price auction with uninformed bidders.     

Risky procurement with an insider bidder

Procurement auctions carry substantial risk when the value of the project is highly uncertain and known only to insiders. This paper reports the results from a series of experiments comparing the performance of three auction formats in such complex and risky settings. In the experiment, every bidder knows the private value for the project but only a single insider bidder knows the common-value part. In addition to the standard second-price and English auctions we test the “qualifying auction,” a two-stage format commonly used in the sale of complex and risky assets. The qualifying auction has a fully “revealing” equilibrium that implements the revenue-maximizing outcome but it also has an uninformative “babbling” equilibrium in which bidders place arbitrarily high bids in the first stage. In the experiments, the latter equilibrium has more drawing power, which causes the qualifying auction to perform worse than the English auction and only slightly better than a sealed-bid second-price auction. Compared to the two other formats, the English auction is roughly 40% more efficient, yields 50% more revenues, avoids windfall profits for the insider, while protecting uninformed bidders from losses.     

Bidding behavior in competing auctions: Evidence from eBay

Much of the existing auction literature treats auctions as running independently of one another, with each bidder choosing to participate in only one auction. However, in many online auctions, a number of substitutable goods are auctioned concurrently and bidders can bid on several auctions at the same time. Recent theoretical research shows how bidders can gain from the existence of competing auctions, the current paper providing the first empirical evidence in support of competing auctions theory using online auctions data from eBay. Our results indicate that a significant proportion of bidders do bid across competing auctions and that bidders tend to submit bids on auctions with the lowest standing bid, as the theory predicts. The paper also shows that winning bidders who cross-bid pay lower prices on average than winning bidders who do not.     

Auction design with opportunity cost

This paper studies revenue-maximizing auctions in an independent private value setting where potential bidders have known positive opportunity cost of bidding. The main findings are as follows. Firstly, there is no loss of generality in deriving the revenue-maximizing auctions within the class of threshold-entry mechanisms. Secondly, for any given set of entry thresholds, a second-price sealed-bid auction with properly set reserve prices and entry subsidy is revenue-maximizing. Thirdly, a variety of auctions are revenue-maximizing within the symmetric threshold-entry class. Two of them involve no entry subsidy (fee). Fourthly, we identify sufficient conditions under which it is in the seller’s interest to limit the number of potential bidders even if the revenue-maximizing symmetric threshold-entry auction is adopted. Lastly, the revenue-maximizing auction implements asymmetric entry across symmetric bidders in many cases. I am grateful to Jean-Jacques Laffont for his advice. I thank Isabelle Brocas, Juan Carrillo, Indranil Chakraborty, Jeff Ely, Jinwoo Kim, Vijay Krishna, Preston McAfee, Isabelle Perrigne, Eric Rasmusen, Guofu Tan, Quang Vuong, Joseph Wang for helpful comments and suggestions. Insightful comments and suggestions from the associate editor and two anonymous referees have greatly improved the paper. Earlier versions have been presented at the 2004 International Industrial Organization Conference, the 2005 Pan Pacific Game Theory Conference and the 2007 International Game Theory Conference. All errors are mine. Financial support from National University of Singapore (R-122-000-088-112) is gratefully acknowledged.     

Collusion Without Conspiracy: An Experimental Study of One-Sided Auctions

The study investigates collusion-nonfacilitating features of one-sided auctions. We report the results of oral nondescending bid auction experiments in which the strict bid improvement rule was absent, and compare them with the results of sealed bid auction experiments. In the sealed bid experiments the outcomes converged to the competitive equilibrium. In the oral auctions, collusive outcomes emerged and were sustained with bidders using bid matching strategies. We conclude that oral auctions provide opportunities for tacit coordination and collusion enforcement that do not exist under the sealed bid. Therefore, the strict bid improvement rule becomes critical for breaking collusion.     

The multiple unit auction with variable supply

Summary. The theory of multiple unit auctions traditionally assumes that the offered quantity is fixed. I argue that this assumption is not appropriate for many applications because the seller may be able and willing to adjust the supply as a function of the bidding. In this paper I address this shortcoming by analyzing a multi-unit auction game between a monopolistic seller who can produce arbitrary quantities at constant unit cost, and oligopolistic bidders. I establish the existence of a subgame-perfect equilibrium for price discriminating and for uniform price auctions. I also show that bidders have an incentive to misreport their true demand in both auction formats, but they do that in different ways and for different reasons. Furthermore, both auction formats are inefficient, but there is no unambiguous ordering among them. Finally, the more competitive the bidders are, the more likely the seller is to prefer uniform pricing over price discrimination, yet increased competition among bidders may or may not enhance efficiency. Received: June 18, 1998; revised version: January 13, 1999     

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.     

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.     

Private value perturbations and informational advantage in common value auctions

We analyze the value of being better informed than one's rival in a two bidder, second price common value auction. Standard models of these auctions do not pin down relative bidding postures, but we show that by adding small amounts of private value information, a unique equilibrium can be restored. Additional common value information affects a bidder's payoff both directly, by increasing his information rent, and indirectly, by shifting the relative bidding posture of his opponent. Although the latter effect need not be positive, we establish broad conditions under which bidders with more information do better than their rivals. We turn to information acquisition and release and show that the desire to influence relative bidding postures can lead sellers to create new information rents (by releasing information privately to only one bidder) and bidders to forego information rents (instead choosing to gather information that a rival already has).     

Competitive auctions

We study a class of single-round, sealed-bid auctions for an item in unlimited supply, such as a digital good. We introduce the notion of competitive auctions. A competitive auction is truthful (i.e. encourages bidders to bid their true valuations) and on all inputs yields profit that is within a constant factor of the profit of the optimal single sale price. We justify the use of optimal single price profit as a benchmark for evaluating a competitive auctions profit. We exhibit several randomized competitive auctions and show that there is no symmetric deterministic competitive auction. Our results extend to bounded supply markets, for which we also give competitive auctions.     

Auctions with Price-Proportional Benefits to Bidders

After some auctions, including Amish estate sales and buyer rings′ knockout auctions, each bidder receives a share of the revenue generated by the auction. We show that in the symmetric case, equilibrium bids in both first-price sealed-bid auctions and oral auctions increase as each bidder′s share increases. In the case of independent signals, oral auctions result in higher expected equilibrium prices than do first-price sealed-bid auctions. Journal of Economic, Literature classification number: D44.     

Comparing competition and collusion: a numerical approach

Summary. Collusion is a serious problem in many procurement auctions. In this research, I study a model of first price sealed bid procurement auctions with asymmetric bidders. I demonstrate that the equilibrium to the model is unique and describe three algorithms that can be used to compute the inverse equilibrium bid functions. I then use the computational algorithms to compare competitive and collusive bidding. The algorithms are useful for structural estimation of auction models and for assessing the damages from bid-rigging. Received: January 14, 2000; revised version: February 28, 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。