Tie-breaks and bid-caps in all-pay auctions

We revisit the two bidder complete information all-pay auction with bid-caps introduced by Che and Gale (1998), dropping their assumption that tie-breaking must be symmetric. Any choice of tie-breaking rule leads to a different set of Nash equilibria. Compared to the optimal bid-cap of Che and Gale we obtain that in order to maximize the sum of bids, the designer prefers to set a less restrictive bid-cap combined with a tie-breaking rule which slightly favors the weaker bidder. Moreover, the designer is better off breaking ties deterministically in favor of the weak bidder than symmetrically.     

The all-pay auction with complete information

Summary In a (first price) all-pay auction, bidders simultaneously submit bids for an item. All players forfeit their bids, and the high bidder receives the item. This auction is widely used in economics to model rent seeking, R&D races, political contests, and job promotion tournaments. We fully characterize equilibrium for this class of games, and show that the set of equilibria is much larger than has been recognized in the literature. When there are more than two players, for instance, we show that even when the auction is symmetric there exists a continuum of asymmetric equilibria. Moreover, for economically important configurations of valuations, there is no revenue equivalence across the equilibria; asymmetric equilibria imply higher expected revenues than the symmetric equilibrium.We are grateful to Jacques Crémer, Chuangyin Dang, Jürgen Dennert, Chaim Fershtman, Martin Hellwig, Arthur Robson, Heinrich Ursprung, Eric van Damme, Ton Vorst, and the referees for helpful comments. We benefitted from presentations at the World Congress of the Econometric Society in Barcelona, the European Meeting on the Economics of Information at Tilburg University, the Midwest Mathematical Economics Meetings at the University of Illinois, and seminars at Texas A&M University, The Pennsylvania State University, Tilburg University and the University of Montreal. Baye is grateful for support from the CentER for Economic Research at Tilburg University and the Tinbergen Institute where earlier versions of this paper were completed. Kovenock acknowledges support from Erasmus Universiteit Rotterdam, the Tinbergen Institute, the Center for Economic Studies at the University of Munich, the Institut d'Analisi Economica CSIC at the Universitat Autonoma de Barcelona, the Krannert School of Management, and the Jay N. Ross Young Faculty Scholar Award.     

Multi-stage sequential all-pay auctions

We study multi-stage sequential all-pay contests (auctions) where heterogeneous contestants are privately informed about a parameter (ability) that affects their cost of effort. We characterize the perfect Bayesian equilibrium of these contests and analyze the effect of the number of contestants and their types on the contestants׳ expected highest effort.     

Expected revenue of all-pay auctions and first-price sealed-bid auctions with budget constraints

We show that all-pay auctions dominate first-price sealed-bid auctions when bidders face budget constraints. This ranking is explained by the fact that budget constraints bind less frequently in the all-pay auctions, which leads to more aggressive bidding in that format.     

The best-shot all-pay (group) auction with complete information

We analyze an all-pay group contest in which individual members’ efforts are aggregated via the best-shot technology and the prize is a public good for the winning group. The interplay of within-group free-riding and across-group competition allows for a wide variety of equilibria, according to how well groups overcome internal free-riding. In contrast with the existing literature, we derive equilibria of a symmetric model in which multiple agents per group are active. Our findings differ qualitatively from those of the individualistic all-pay auction: rents are not necessarily dissipated in equilibrium, total expected efforts vary across equilibria, and participation is expected to be greater. Moreover, equilibria with greater symmetry of behavior within a group are shown to have more “wasted” effort but also greater payoffs as overall efforts are lower. In contrast to standard economic intuition, free-riding can be beneficial for players as it reduces competition among groups. Examples of asymmetric group contests are also studied.     

The value of information in asymmetric all-pay auctions

This note analyzes a two-player all-pay auction with incomplete information. More precisely, one bidder is uncertain about the size of the initial advantage of his rival modeled as a head start in the auction.I derive the unique Bayesian Nash equilibrium outcome for a large class of cumulative distribution functions of the head start. The stronger player generates an informational rent if and only if his head start distribution is not stochastically dominated by a uniform distribution. I identify why my results for an uncertain head start differ qualitatively from uncertainty about a cost function or a valuation.     

Lottery versus all-pay auction contests: A revenue dominance theorem

We allow a contest organizer to bias a contest in a discriminatory way; i.e., she can favor specific contestants by designing the contest rule in order to maximize total equilibrium effort (resp. revenue). The two predominant contest regimes are considered, all-pay auctions and lottery contests. For all-pay auctions the optimal bias is derived in closed form: It implies extreme competitive pressure among active contestants and low endogenous participation rates. Moreover, the exclusion principle advanced by Baye et al. (1993) becomes obsolete in this case. In contrast, the optimally biased lottery induces a higher number of actively participating contestants due to softer competition. Our main result regarding total revenue comparison under the optimal biases reveals that the all-pay auction revenue-dominates the lottery contest for all levels of heterogeneity among contestants. The incentive effect due to a strongly discriminating contest rule (all-pay auction) dominates the participation effect due to a weakly discriminating contest rule (lottery).     

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.     

Asymmetric information contests

We consider how asymmetries in information affect contest behavior. We find two effects drive behavior-risk and perceptions of an opponent's value. Our results also suggest that the more tractable one-sided asymmetric information contest might be sufficient to capture contest behavior under uncertainty. However, the efficiency of a contest is sensitive to asymmetric information and the distribution a players' values because players with relatively low values can have an increased probability of success. Thus, it is important (a) to account for information asymmetries to avoid biasing predicted efficiency, and (b) to recognize that this bias is unsystematic.     

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     

Asymmetric contests with general technologies

Summary. We investigate the pure-strategy Nash equilibria of asymmetric, winner-take-all, imperfectly discriminating contests, focussing on existence, uniqueness and rent dissipation. When the contest success function is determined by a production function with decreasing returns for each contestant, there is a unique pure-strategy equilibrium. If marginal product is also bounded, limiting total expenditure is equal to the value of the prize in large contests even if contestants differ. Partial dissipation occurs only when infinite marginal products are permitted. Our analysis relies heavily on the use of share functions and we discuss their theory and application.Received: 28 May 2003, Revised: 26 April 2004, JEL Classification Numbers: C72, D72. Correspondence to: Richard CornesMuch of the research in this paper was undertaken while the first author was a Visiting Scholar at the Centre for Economic Studies, University of Munich. The support of the Centre is gratefully acknowledged.     

Late and multiple bidding in second price Internet auctions: Theory and evidence concerning different rules for ending an auction

In second price Internet auctions with a fixed end time, such as those on eBay, many bidders submit their bids in the closing minutes or seconds of an auction. We propose an internet auction model, in which very late bids have a positive probability of not being successfully submitted, and show that late bidding in a fixed deadline auction can occur at equilibrium in auctions both with private values and with uncertain, dependent values. Late bidding may also arise out of equilibrium, as a best reply to incremental bidding. However, the strategic advantages of late bidding are severely attenuated in auctions that apply an automatic extension rule such as auctions conducted on Amazon. Field data show that there is more late bidding on eBay than on Amazon, and this difference grows with experience. We also study the incidence of multiple bidding, and its relation to late bidding.     

The value of commitment in contests and tournaments when observation is costly

We study the value of commitment in sequential contests when the follower faces small costs to observe the leader's effort. We show that the value of commitment vanishes entirely in this class of games. By contrast, in sequential tournaments—games where, at a cost, the follower can observe the effectiveness of the leader's effort—the value of commitment is preserved completely provided that the observation costs are sufficiently small.     

Outsourcing via reverse auction with a built-in menu of change orders

Many firms invent and design products while outsourcing their production to independent contractors. We consider a dominant strategy mechanism that selects a contractor using a reverse auction, combined with a menu of permitted change orders from which the contractor can choose after updated cost information has become available. That mechanism maximizes the gain from trade, allows the firm to extract the second highest surplus, and induces the contractor to make efficient adjustments to output after updated cost information has emerged.     

Are auction revenues affected by rising art buyers’ premia? The case of early American art

The steady rise in the premiums charged to art buyers at auction (above hammer price) has been underway since 1992. This article, using a stable and bounded sample of repeat purchase of American works created before 1950, reveals that this tact has reduced hammer prices for that art. However, renewed and hyper-competitive efforts to bring more and higher quality art to market by the two main houses, Sotheby’s and Christie’s, have resulted in general profitability. Nevertheless, we calculate that a rise in buyers’ premia at Sotheby’s, a publically traded company, has reduced revenues and profits below their potential in the absence of such increases.     

Budget-constrained sequential auctions with incomplete information

I study a budget-constrained, private-valuation, sealed-bid sequential auction with two incompletely-informed, risk-neutral bidders in which the valuations and income may be non-monotonic functions of a bidder's type. Multiple equilibrium symmetric bidding functions may exist that differ in allocation, efficiency and revenue. The sequence of sale affects the competition for a good and therefore also affects revenue and the prices of each good in a systematic way that depends on the relationship among the valuations and incomes of bidders. The sequence of sale may affect prices and revenue even when the number of bidders is large relative to the number of goods. If a particular good, say α, is allocated to a strong bidder independent of the sequence of sale, then auction revenue and the price of good α are higher when good α is sold first.     

Entry in contests with incomplete information: Theory and experiments

This paper studies entry decisions in contests with private values. Potential contestants observe their value and the common opportunity cost of entry, and make entry decisions simultaneously. Theory predicts that whether or not contestants are informed of the number of entrants prior to choosing their expenditures has no effect on entry or aggregate expenditures. We test these assertions in our experiments. We find substantial over-entry in both information structures. However, entry is higher when contestants are informed. Since expenditures do not, on average, differ across information structures, aggregate expenditure is also higher when contestants are informed. Contestants earn on average less than the opportunity cost of entry.     

Unmediated communication in games with complete and incomplete information

We study the effects of adding unmediated communication to static, finite games of complete and incomplete information. We characterize S^U(G), the set of outcomes of a game G, that are induced by sequential equilibria of cheap talk extensions. A cheap talk extension of G is an extensive-form game in which players communicate before playing G. A reliable mediator is not available and players exchange private or public messages that do not affect directly their payoffs. We first show that if G is a game of complete information with five or more players and rational parameters, then S^U(G) coincides with the set of correlated equilibria of G. Next, we demonstrate that if G is a game of incomplete information with at least five players, rational parameters and full support (i.e., all profiles of types have positive probability), then S^U(G) is equal to the set of communication equilibria of G.     

Cheap talk with multiple audiences: An experimental analysis

We examine strategic information transmission in a controlled laboratory experiment of a cheap talk game with one sender and multiple receivers. We study the change in equilibrium behavior from the addition of another audience as well as from varying the degree of conflict between the senderʼs and receiversʼ preferences. We find that, as in cheap talk games with just one receiver, information transmission is higher in games with a separating equilibrium, than in games with only a babbling equilibrium. More interestingly, we find clear evidence that the addition of another audience alters the communication between the sender and the receiver in a way consistent with the theoretical predictions. There is evidence of the presence of agents that are systematically truthful as senders and trusting as receivers: deviations from the theoretical predictions, however, tend to disappear with experience, and learning is faster precisely in the games where deviations are more pronounced.