The Impact of Discounting on an Auction with a Buyout Option: a Theoretical Analysis Motivated by eBay’s Buy-It-Now Feature

An auction with a buyout option occurring over continuous time with rules similar to eBays buy it now option is analyzed. When auction participants make no distinction as to when a transaction occurs, the seller optimally chooses a buyout option price so high that bidders never exercise the option. However, time impatience on either side of the transaction can motivate the seller to offer an option price low enough so that the option is exercised with positive probability. Further, allowing a time impatient seller to offer such an option results in an ex ante Pareto improvement when bidders do not discount future transactions.     

An experimental study of auctions with a buy price under private and common values

eBayʼs Buy It Now format allows a seller to list an auction with a “buy price” at which a bidder may purchase the item immediately and end the auction. When bidders are risk averse, then theoretically a buy price can raise seller revenue when values are private (but not when values are common). We report the results of laboratory experiments designed to determine whether in practice a buy price is advantageous to the seller. We find that a suitably chosen buy price yields a substantial increase in seller revenue when values are private, and a small (but statistically insignificant) increase in revenue when values are common. In both cases a buy price reduces the variance of seller revenue. A behavioral model which incorporates the winnerʼs curse and the overweighting by bidders of their own signal explains the common value auction data better than the rational model.     

Auctions with a buy price

eBay and Yahoo allow sellers to list their auctions with a buy price at which a bidder may purchase the item immediately. On eBay, the buy-now option disappears once a bid is placed, while on Yahoo the buy-now option remains in effect throughout the auction. We show that when bidders are risk averse, both types of auctions raise seller revenue for a wide range of buy prices. The Yahoo format raises more revenue than the eBay format when bidders have either CARA or DARA. Bidders with DARA prefer the eBay auction, while bidders with CARA are indifferent between the two. Part of this work was completed while Reynolds was a visitor at Instituto de Analisys Economico in Barcelona and while Wooders was a visitor at Hong Kong University of Science and Technology. We are grateful to these institutions for their hospitality.     

Auctioning divisible goods

Summary. We derive equilibrium bidding strategies in divisible good auctions for asymmetrically informed risk neutral and risk averse bidders when there is random noncompetitive demand. The equilibrium bid schedules contain both strategic considerations and explicit allowances for the winner's curse. When the bidders' information is symmetric, the strategic aspects of bidding imply that there always exist equilibria of a uniform-price auction with lower expected revenue than provided by a discriminatory auction. When bidders are risk averse, there may exist equilibria of the uniform-price auction that provide higher expected revenue than a discriminatory auction. Received: November 4, 1999; revised version: March 9, 2001     

Second chance offers in auctions

This paper examines situations in which a seller might make a second chance (take-it-or-leave-it) offer to a non-winning bidder at a price equal to their bid at auction. This study is motivated by the take-it-or-leave-it second chance offer rules used by eBay and a number of state procurement agencies. Equilibrium bidder behavior is determined for IPV sealed bid first price, second price, English, and Vickrey auctions when a second chance offer will be made with an exogenous probability $p$ . In all but the Vickrey auction (which elicits the dominant strategy of bidding one’s value) equilibrium bids are lower than if there were no possibility of a second chance offer and higher than if a second chance offer will be made for certain. Further, the possibility of a second chance offer erodes the strategic equivalence between second price bids and English auction drop out levels. If bidders are risk averse (with CRRA preferences), this difference leads to expected revenue dominance of the second price over the English auction, both of which dominate the Vickrey auction. The first price auction is also shown to revenue dominate the Vickrey auction, and moreover, numerical results and intuition from existing literature suggest that the first price auction revenue dominates the second price auction.     

Sequential negotiations with costly information acquisition

Negotiations about a merger or acquisition are often sequential and only partially disclose to bidders information about each otherʼs bids. This paper explains the seller optimality of partial disclosure in a single-item private-value auction with two bidders. Each bidder can inspect the item at a nonprohibitive cost. If a revenue-maximizing seller cannot charge bidders for the information about the otherʼs bid, then the seller optimally runs a sequential second-price auction with a reserve price and a buy-now price. The seller prefers to keep the bids confidential and, sometimes, to hide the order in which he approaches the bidders.     

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.     

Premium auctions and risk preferences

In a premium auction, the seller offers some “payback”, called premium, to a set of high bidders at the end of the auction. This paper investigates how the performance of such premium tactics is related to the bidders? risk preferences. We analyze a two-stage English premium auction model with symmetric interdependent values, in which the bidders may be risk averse or risk preferring. Upon establishing the existence and uniqueness of a symmetric equilibrium, we show that the premium causes the expected revenue to increase in the bidders? risk tolerance. A “net-premium effect” is key to this result.     

Auctions with a buy price: The case of reference-dependent preferences

In an auction with a buy price, the seller provides bidders with an option to end the auction early by accepting a transaction at a posted price. This paper develops a model of an auction with a buy price in which bidders use the auction's reserve price and buy price to formulate a reference price. The model both explains why a revenue-maximizing seller would want to augment her auction with a buy price and demonstrates that the seller sets a higher reserve price when she can affect the bidders' reference price through the auction's reserve price and buy price than when she can affect the bidders' reference price through the auction's reserve price only. The comparative statics properties of bidding behavior are in sharp contrast to equilibrium behavior in other models where the existence and size of the auction's buy price have no effect on bidding behavior.     

Commitment in sequential auctioning: advance listings and threshold prices

This paper models sequential auctioning of two perfect substitutes by a strategic seller, who learns about demand from the first-auction price. The seller holds the second auction only when the remaining demand is strong enough to cover her opportunity cost. Bidding in anticipation of such a contingent future auction is characterized, including a sufficient condition for existence of an invertible (increasing symmetric pure-strategy) bidding equilibrium that facilitates the seller’s learning. A unique invertible bidding equilibrium exists for the Dutch auction format, but only when the second auction is sufficiently discounted by the bidders. In the equilibrium, high-valuation bidders shade their bids down as if the second auction were guaranteed. To counter such strategic bidding, the seller would value ex-ante commitment to hold the second auction less often. Three forms of such commitment are analyzed: commitment to list future auctions in advance, commitment to not hold the second auction unless the first price exceeds a publicly announced threshold, and commitment to a reserve-price in the second auction. I would like to thank Georgios Katsenos, Thomas Jeitschko, Miguel Villas-Boas, George Deltas, and an anonymous referee for thorough and insightful feedback.