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.     

An experiment on auctions with endogenous budget constraints

We perform laboratory experiments comparing auctions with endogenous budget constraints. A principal imposes a budget limit on a bidder (an agent) in response to a principal-agent problem. In contrast to the existing literature where budget constraints are exogenous, this theory predicts that tighter constraints will be imposed in first-price auctions than in second-price auctions, tending to offset any advantages attributable to the lower bidding strategy of the first-price auction. Our experimental findings support this theory: principals are found to set significantly lower budgets in first-price auctions. The result holds robustly, whether the principal chooses a budget for human bidders or computerized bidders. We further show that the empirical revenue difference between first- and second-price formats persists with and without budget constraints.     

First-price auctions with resale: the case of many bidders

We study first-price auctions with resale when there are many bidders and derive existence and characterization results under the assumption that the winner of the initial auction runs a second-price auction with an optimal reserve price. The fact that symmetrization fails when there are more than two bidders has been observed before, but we also provide the direction: weaker bidders are less likely to win than stronger ones. For a special class of distributions and three bidders, we prove that the bid distributions are more symmetric with resale than without. Numerical simulations suggest that the more bidders there are, the more similar the allocation is to the case without resale, and thus, the more asymmetric the bid distributions are between strong and weak bidders. We also show in an example that the revenue advantage of first-price auctions over second-price auctions is positive, but decreasing in the number of bidders.     

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 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.     

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.     

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.     

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.     

Outside options: Another reason to choose the first-price auction

In this paper we study equilibrium and experimental bidding behaviour in first-price and second-price auctions with outside options.We find that bidders do respond to outside options and to variations of common knowledge about competitors’ outside options. However, overbidding in first-price auctions is significantly higher with outside options than without. First-price auctions yield more revenue than second-price auctions. This revenue-premium is significantly higher with outside options. In second-price auctions the introduction of outside options has only a small effect.     

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     

Revenue ranking of first-price auctions with resale

When the price setter in post-auction resale is chosen according to exogenous probabilities, Hafalir and Krishna (2008) [2] showed that the first-price auction brings more expected revenues than the second-price auction with truth-bidding bidders. We complete their revenue ranking by proving that the first-price auction produces higher expected revenues the higher the probability the auction winner sets the resale price.     

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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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.     

Bidder collusion

We analyze bidder collusion at first-price and second-price auctions. Our focus is on less than all-inclusive cartels and collusive mechanisms that do not rely on auction outcomes. We show that cartels that cannot control the bids of their members can eliminate all ring competition at second-price auctions, but not at first-price auctions. At first-price auctions, when the cartel cannot control members’ bids, cartel behavior involves multiple cartel bids. Cartels that can control bids of their members can suppress all ring competition at both second-price and first-price auctions; however, shill bidding reduces the profitability of collusion at first-price auctions.     

A common value auction model with endogenous entry and information acquisition

Summary Much of the auction literature assumes both a fixed number of bidders and a fixed information setting. This sidesteps the important and often costly decisions a potential bidder must make prior to an auction: Should I enter and, if I do, what level of resources should I expend evaluating the good prior to bidding? We answer these questions for a stylized information model of a common value auction. The expected selling price is shown to be the expected value of the good minus the expected aggregate entry and information costs of the bidders. Thus, the seller indirectly pays for these costs to the bidders. There are auctions where the seller seemingly restricts the bidders' information expenditures. While this restriction does influence the entry decision, we demonstrate that the overall effect can be to improve the selling price. Finally, the probability of entry and the chosen accuracy of the information are never more in the second-price auction than in the first-price auction, and the seller prefers the second-price auction.We are grateful for the comments and suggestions of seminar participants at the University of British Columbia, Dartmouth College, the University of Wisconsin, Yale University, and the International Conference on Game Theory and Economics at SUNY Stony Brook.     

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.     

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.     

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.     

Multi-dimensional reference-dependent preferences in sealed-bid auctions – How (most) laboratory experiments differ from the field

We study bidding behavior in first- and second-price sealed-bid auctions with loss-averse agents. Our model predicts overbidding in first-price induced-value auctions consistent with evidence from most laboratory experiments. Substantially different bidding behavior could result in commodity auctions where money and auction item are consumed along different dimensions of the consumption space. Differences also result in second-price auctions. Our study thereby indicates that transferring qualitative behavioral findings from induced-value laboratory experiments to the field may be problematic if subjects are loss-averse.     

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