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.     

Compensating for the winner's curse: Experimental evidence

We conduct experiments on common value auctions with rationing. In each auction, the good is randomly allocated to one of the k highest bidders, at the (k+1)st highest price. When k>1, bidders are rationed. As the degree of rationing increases, the equilibrium bid function increases. Consistent with prior literature, we find that bidders suffer from the winner's curse and lose money on average. However, bids in the experiments do adjust in the appropriate direction as the degree of rationing changes, providing support for the comparative statics implications of the theory. Our results are consistent with subjects having an intuitive understanding of the winner's curse, but being unable to compute the equilibrium bid levels.     

Chores

We analyze situations where the provision of each of c public goods must be voluntarily assumed by exactly one of n private agents in the absence of transfer schemes or binding contracts. We model this problem as a complete information, potentially infinite horizon game where n agents simultaneously wage c wars of attrition. Providing a public good commits an agent not to take on the provision of another public good for a fixed period. We explore the strategic trade-offs that this commitment ability and the multiplicity of tasks provide. Subgame perfect equilibria (SPEs) are characterized completely for games with two agents and two public goods. For games with two identical agents and c > 1 identical public goods, we establish that an equilibrium that yields a surplus-maximizing outcome always exists and we provide sufficient conditions under which it is the unique equilibrium outcome. We show that under mild conditions, the surplus-maximizing SPE is the unique symmetric SPE. Journal of Economic Literature Classification Number: H41, C72, D13.     

Bidding ‘as if’ risk neutral in experimental first price auctions without information feedback

Experimental research on first price sealed bid auctions has usually involved repeated settings with information feedback on winning bids and payoffs after each auction round. Relative to the risk neutral Nash equilibrium, significantly higher bidding has been reported. The present paper reports the results of experimental first price auctions with n=7 where feedback on payoffs and winning bids is withheld. Under these conditions, average bidding is below the risk neutral Nash equilibrium prediction but converges to it with repetition.

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Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.      

Sequential vs. single-round uniform-price auctions

We study sequential and single-round uniform-price auctions with affiliated values. We derive symmetric equilibrium for the auction in which k₁ objects are sold in the first round and k₂ in the second round, with and without revelation of the first-round winning bids. We demonstrate that auctioning objects in sequence generates a lowballing effect that reduces the first-round price. Total revenue is greater in a single-round, uniform auction for k=k₁+k₂ objects than in a sequential uniform auction with no bid announcement. When the first-round winning bids are announced, we also identify a positive informational effect on the second-round price. Total expected revenue in a sequential uniform auction with winning-bids announcement may be greater or smaller than in a single-round uniform auction, depending on the model's parameters.     

Unique Implementation of the Full Surplus Extraction Outcome in Auctions with Correlated Types

In auctions with correlated types it is possible to design mechanisms such that full surplus extraction can be obtained as the outcome of an equilibrium in which agents use (weakly) dominant strategies. However, it is not assured that the outcome is unique. We present an example in which no mechanism can yield the full surplus extraction outcome as the unique Bayesian equilibrium outcome. Next we show that in the standard auction model the multiplicity problem can be fully resolved using sequential mechanisms, i.e., we show that it is possible to obtain the full surplus extraction outcome as the unique perfect Bayesian equilibrium outcome.Journal of Economic LiteratureClassification Numbers: D44; D70.     

Non-monotoniticies and the all-pay auction tie-breaking rule

Discontinuous games, such as auctions, may require special tie-breaking rules to guarantee equilibrium existence. The best results available ensure equilibrium existence only in mixed strategy with endogenously defined tie-breaking rules and communication of private information. We show that an all-pay auction tie-breaking rule is sufficient for the existence of pure strategy equilibrium in a class of auctions. The rule is explicitly defined and does not require communication of private information. We also characterize when special tie-breaking rules are really needed.      

Equilibrium selection in common-value second-price auctions

This note considers equilibrium selection in common-value second-price auctions with two bidders. We show that for each ex post equilibrium in continuous and undominated strategies, a sequence of “almost common-value” auctions can be constructed such that each of them possesses a unique undominated and continuous equilibrium and the corresponding sequence of equilibria converges to that ex post equilibrium. As an implication, no equilibrium selection of this model based on perturbations seems to be more convincing than others.     

On optimality in intergenerational risk sharing

Summary. This paper defines and studies optimality in a dynamic stochastic economy with finitely lived agents, and investigates the optimality properties of an equilibrium with or without sequentially complete markets. Various Pareto optimality concepts are considered, including interim and ex ante optimality. We show that, at an equilibrium with a productive asset (land) and sequentially complete markets, the intervention of a government may be justified, but only to improve risk sharing between generations. If markets are incomplete, constrained interim optimality is investigated in two-period lived OLG economies. We extend the optimality properties of an equilibrium with land and give conditions under which introducing a pay-as-you-go system at an equilibrium would not lead to any Pareto improvement. Received: October 5, 1998; revised version: April 3, 2001     

Private Value Allocations in Large Economies with Differential Information

We consider a Radner-type (e.g., Radner, 1968, Econometrica36, 31–58) pure exchange economy with differential information and a continuum of agents. We show that under appropriate assumptions the set of Aumann–Shapley private value allocations in such an economy coincides with the set of Radner competitive equilibrium allocations. Journal of Economic Literature Classification Numbers: C71, D51, D82.     

Private monitoring in auctions

We study infinitely repeated first-price auctions in which a bidder only learns whether or not he won the object. While repetition of the stage-game equilibrium is the unique Nash equilibrium in public strategies, with patient bidders there are simple Nash equilibria in private strategies that improve on bid rotation. Sequential rationality is appropriately captured by essentially perfect Bayesian equilibrium (EPBE), which ignores behavior after irrelevant histories. Our main result is the construction of EPBEa that improve upon bid rotation. Assuming symmetry, the exclusionary schemes of Skrzypacz and Hopenhayn [Tacit collusion in repeated auctions, J. Econ. Theory 114 (2004), 153–169], including asymptotically efficient ones, are supported as EPBEa.     

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.     

The Evolutionary Stability of Auctions over Bargaining

This paper considers equilibrium in transaction mechanisms. In an environment with homogeneous buyers and sellers, which eliminates the advantage auctions possess of matching buyers and sellers, both auctions and bargaining are equilibria. However, only auctions are evolutionarily stable. This identifies a new advantage of auctions over bargaining, arising from the division of the gains from trade.Journal of Economic LiteratureClassification Numbers: C78, C73, D44.     

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.     

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.     

Experimental internet auctions with random information retrieval

We run an experiment where 97 subjects could retrieve records of completed past auctions before placing their bids in current one-bid, two-bid, and auction-selection games. Each subject was asked to participate in 3 current auctions; but could retrieve up to 60 records of completed (past) auctions. The results reveal a positive relation between the payoffs earned by the subjects and their history-inspection effort. Subjects act as if responding to the average bidding-ratios of the winners in the samples that they have retrieved. They apply intuitive signal-dependent stopping rules like “sample until observing a winner-value close to my won” or “find a close winner-value and try one more history” when sampling the databases. History-inspection directs bidders with relatively high private-valuations to moderate bidding which increases their realized payoffs. (JEL C9 D4 D8) Electronic Supplementary Material Supplementary material is available in the online version of this article at . JEL Classification C93, D44, D83     

Are sequential round‐robin tournaments discriminatory?

I examine sequential round‐robin tournaments with three and four symmetric players. Each player is matched once with each other player and each match is organized as a Tullock contest. A single prize is allocated to the player with the most matches won. I show that, depending on the position of their matches in the sequence of the tournament, the players' ex ante winning probabilities and expected payoffs differ. Compared to tournaments with matches organized as all‐pay auctions, however, discrimination is weaker and partially reversed. Moreover, aggregate effort in tournaments with suitable Tullock contests is larger than in tournaments with all‐pay auctions. The fairness of round‐robin tournaments may be improved by an endogenous sequence of matches or the requirement that players fix their effort ex ante.     

Stochastic stability in a double auction

In a k-double auction, a buyer and a seller must simultaneously announce a bid and an ask price respectively. Exchange of the indivisible good takes place if and only if the bid is at least as high as the ask, the trading price being the bid price with probability k and the ask price with probability (1−k). We show that the stable equilibria of a complete information k-double approximate an asymmetric Nash bargaining solution with the seller's bargaining power decreasing in k.Note that ceteris paribus, the payoffs of the seller of the one-shot game increase in k. Nevertheless, as the stochastically stable equilibrium price decreases in k, choosing the seller's favorite price with a relatively higher probability in individual encounters makes him worse off in the long run.     

The symmetric multiple prize all-pay auction with complete information

This paper extends the analysis of the n-player all-pay auction with complete information to cover the case of m≤n prizes, valued in weakly decreasing order, but symmetrically across players. We provide a complete characterization of the Nash equilibrium distributions for this class of auctions and provide an exact expression for the expected revenue generated.