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.     

Auctioning Risky Conservation Contracts

Conservation auctions have the potential to increase the efficiency of payments to farmers to adopt conservation-friendly management practices by fostering competition among them. The literature considers bidders that have complete information about the costs of adoption and optimal bidding behavior reflects this information advantage. Farmers seek information rents and bids decrease when risk aversion increases because farmers are more averse to losing the auction. We contribute to the literature by allowing for cost risk. Our paper shows that farmers must balance the risk of losing the auction (thus foregoing information rent) with the risk of submitting a bid that is not high enough to pay the costs of adopting conservation practices (thus incurring losses). We design an experiment to trade off these two risks and examine how risk aversion affects bidding behavior when participants face different sources and levels of risk. Our experiment contributes to a small literature on experimental auctions with risky product valuations. We find that participants decrease their bids as risk aversion increases, even in auctions with cost risk, suggesting that the risk of losing the auction dominates. These findings uncover new challenges for the practical implementation of conservation auctions as an efficient policy instrument.     

Empirical Implications of Equilibrium Bidding in First-Price, Symmetric, Common Value Auctions

This paper studies federal auctions for wildcat leases on the Outer Continental Shelf from 1954 to 1970. These are leases where bidders privately acquire (at some cost) noisy, but equally informative, signals about the amount of oil and gas that may be present. We develop tests of rational and equilibrium bidding in a common values model that are implemented using data on bids and ex post values. We also use data on tract location and ex post values to test the comparative static prediction that bidders may bid less aggressively in common value auctions when they expect more competition. We find that bidders are aware of the "winner's curse" and their bidding is largely consistent with equilibrium.     

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     

DEMAND REDUCTION IN MULTI‐UNIT AUCTIONS WITH VARYING NUMBERS OF BIDDERS: THEORY AND EVIDENCE FROM A FIELD EXPERIMENT*

Recent auction theory and experimental results document strategic demand reduction by bidders in uniform‐price auctions. The present article extends this area of research to consider the effects of varying the number of bidders. Our theoretical model predicts that demand reduction should decrease with an increase in the number of bidders. Considerable demand reduction remains even in the asymptotic limit, although truthful bidding yields profits very close to those of equilibrium play. We experimentally confirm several of our predictions by examining bidding behavior of subjects in an actual marketplace, auctioning dozens of sportscards using both uniform‐price and Vickrey auction formats.     

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.     

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.     

Competition among auctioneers in large markets

We analyse a multistage game of competition among auctioneers. First, the auctioneers commit to some reserve prices; second, the bidders enter one auction, if any; and finally, the auctions take place. We show that for any finite set of feasible reserve prices, each auctioneer announces a reserve price equal to his production cost if the numbers of auctioneers and bidders are sufficiently large, though finite. Our result supports the idea that optimal auctions may be quite simple. Our model also confirms previous results for some “limit” versions of the model by McAfee (Econometrica 61 (1993) 1281-1312), Peters (Rev. Econ. Stud. 64 (1997) 97-123), and Peters and Severinov (J. Econ. Theory 75 (1997) 141-179).     

Collusive Bidding: Lessons from the FCC Spectrum Auctions

The Federal Communications Commission (FCC) spectrum auctions use a simultaneous ascending auction design. Bidders bid on numerous communication licenses simultaneously, with bidding remaining open on all licenses until no bidder is willing to bid higher on any license. With full revelation of bidding information, simultaneous open bidding allows bidders to send messages to their rivals, telling them on which licenses to bid and which to avoid. These strategies can help bidders coordinate a division of the licenses, and enforce the proposed division by directed punishments. We examine solutions to mitigate collusive bidding in the spectrum auctions, and then apply these ideas to the design of daily electricity auctions.     

Auction fever: Rising revenue in second-price auction formats

The prevalent term “auction fever” visualizes that ascending auctions – inconsistent with theory – are likely to provoke higher bids than one-shot auctions. To explore and isolate causes of auction fever experimentally, we design four different strategy-proof auction formats and order these according to expected rising bids based on pseudo-endowment effect arguments (psychological ownership and disparity between willingness to pay and willingness to accept). Observed revenues in the experiment in the four formats rank as expected if bidders have private uncertain values (the private information of a bidder is the distribution of her value). A control treatment supports our view that the traditional private certain values approach prevents auction fever in the laboratory. Another control treatment with a procurement auction relates the auction fever bids to bids in a one-shot auction with real endowments. We conclude that, when bidders are uncertain about their valuations, auctions that foster pseudo-endowment may raise bids and revenues.     

REPUTATIONAL BIDDING

We consider auctions where bidders care about the reputational effects of their bidding and argue that the amount of information disclosed at the end of the auction will influence bidding. We focus on bid disclosure rules that capture all of the realistic cases. We show that bidders distort their bidding in a way that conforms to stylized facts about takeovers/licence auctions. We rank the disclosure rules in terms of their expected revenues and find that, under certain conditions, full disclosure will not be optimal. First‐price and second‐price auctions with price disclosure are not revenue equivalent and we rank them.     

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.     

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.     

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.     

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.     

Learning to bid: The design of auctions under uncertainty and adaptation

We examine auction design in a context where symmetrically informed adaptive agents with common valuations learn to bid for a good. Despite the absence of private valuations, asymmetric information, or risk aversion, bidder strategies do not converge to the Bertrand–Nash equilibrium strategies even in the long run. Deviations from equilibrium strategies depend on uncertainty regarding the value of the good, auction structure, the agents? learning model, and the number of bidders. Although individual agents learn Nash bidding strategies in isolation, the learning of each agent, by flattening the best-reply correspondence of other agents, blocks common learning. These negative externalities are more severe in second-price auctions, auctions with many bidders, and auctions where the good has an uncertain value ex post.     

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.     

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.     

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.     

Multiunit Pay‐Your‐Bid Auction with One‐Dimensional Multiunit Demands*

An arbitrary number of units of a good is sold to two bidders through a discriminatory auction. The bidders are homogeneous ex ante and their demand functions are two‐step functions that depend on a single parameter. We characterize the symmetric Bayesian equilibrium and prove its existence and uniqueness. We compare this equilibrium with the equilibrium of the multiunit Vickrey auction and with the equilibria of the single‐unit first price and second price auctions. We examine the consequences of bundling all units into one package. We study the impacts that variations of the “relative” supply have on the equilibrium, on the bidders' average payoffs per unit, and on the efficiency of the equilibrium allocation.