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.     

Multi-unit auctions with private information: an indivisible unit continuous price model

We construct a model of multi-unit auctions in which I bidders bid for two indivisible units of a common value good. Using a first-order approach, we find that there are equilibria in which bidders bid the same price for both units in the discriminatory auction, but not in the uniform auction. When there are only two bidders, under certain conditions, there are linear equilibria for both the discriminatory and the uniform auction formats. In all equilibria, bidders equalize the expected marginal benefit of bidding to the marginal costs of bidding. We show that comparison of the seller??s expected revenue across auction formats depends only on the ratio of the precision of private information to the precision of public information.     

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.     

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.     

Auctions with variable supply: Uniform price versus discriminatory

We examine an auction in which the seller determines the supply after observing the bids. We compare the uniform price and the discriminatory auction in a setting of supply uncertainty, where uncertainty is caused by the interplay of two factors: the seller's private information about marginal cost and the seller's incentive to sell the profit-maximizing quantity, given the received bids. In every symmetric mixed strategy equilibrium, bidders submit higher bids in the uniform price auction than in the discriminatory auction. In the two-bidder case, this result extends to the set of rationalizable strategies. As a consequence, we find that the uniform price auction generates a higher expected revenue for the seller and a higher trade volume.     

Equilibrium reserve prices in sequential ascending auctions

We study a model where bidders have perfectly correlated valuations for two goods sold sequentially in two ascending-price auctions. The seller sets a reserve price before the beginning of each auction. Despite the lack of commitment by the seller, we characterize an equilibrium and study its properties. Strategic non-disclosure of information takes the form of non-participation in the early auction by low-valuation bidders, while high-valuation bidders bid up to their true valuations. Some buyers who would profitably buy at the reserve price refrain from participating in order to decrease the second-auction reserve price.     

Revenue in first-price auctions with a buy-out price and risk-averse bidders

Journal of Economics - We model the “Buy-It-Now or Best Offer” eBay listing option as a first-price auction with a buy-out price and consider seller revenue when bidders are...     

Auctions with uncertain numbers of bidders

We investigate bidders’ and seller's responses to ambiguity about the number of bidders in the first price auction (FPA) and the second price auction (SPA) with independent private valuations. We model ambiguity aversion using the maxmin expected utility model. We find that bidders prefer the number of bidders to be revealed in the FPA, are indifferent between revealing and concealing in the SPA, and prefer the SPA to the FPA. If bidders are more pessimistic than the seller then the seller prefers to conceal the number of bidders in the FPA, and prefers the FPA to the SPA.     

On the effects of joint bidding in independent private value auctions: An experimental study

This article explores the effect of a subset of symmetric bidders joining to bid together. Possible applications include mergers, collusion and joint-bidding arrangements. The change produces a “strong” party with a more advantageous value distribution than the remaining “weak” bidder(s). The predicted effects include inefficiency, a decrease in the sellerʼs revenue, and higher biddersʼ payoffs. Under risk neutrality, the members of the strong party benefit less than the weak bidders. The prediction is reversed when the bidders are sufficiently risk-averse. These hypotheses are tested experimentally. Contrary to the theory, joint bidding increases efficiency and the sellerʼs revenue decreases by less than expected. Strong bidders benefit more than weak bidders indicating that incentives to bid jointly may be greater than hypothesized. Additionally, the experiment assesses the effect of group decision-making. A Nash equilibrium prediction for individual–group differences based on differences in risk attitudes is not supported by the data.     

Reserve price signaling

In a general auction model in which bidders’ signals are affiliated, we characterize the unique separating equilibrium in which the seller can use reserve prices to credibly signal her private information. When the buyers’ signals are independent, the optimal reserve price is shown to be increasing in the number of bidders under certain conditions. We also demonstrate that the probability that the item is sold at the reserve price can increase as the number of bidders increases, which indicates a more central role for reserve prices than perceived in the standard auction models.     

Bid Functions in Auctions and Fair Division Games: Experimental Evidence

In auctions a seller offers a commodity for sale and collects the revenue. In fair division games the object is collectively owned by the group of bidders who equally share the revenue. We run an experiment in which the participants face four types of allocation games (auctions and fair division game under two price rules, first– versus second–price rule). We collect entire bid functions rather than bids for single values and investigate price and efficiency of the different trading institutions. We find that the first–price auction is more efficient than the second–price auction, whereas economic rationality assuming heterogeneous bidders suggests the opposite. Furthermore, we study the structure of individual bid functions.     

Auctions with cross-shareholdings

Summary. We study the effect of cross-shareholding among two competing firms on their bidding behavior and the expected sales revenue for the seller in an auction environment. The bidders private signals are independent, and the model encompasses the private values model and a particular common value model as special cases. When cross-shareholding is symmetric, the bids decrease towards the collusive level as the degree of cross-shareholding increases. The Revenue Equivalence result no longer holds: the first-price auction generates higher expected revenue for the seller than the second-price auction.With asymmetric cross-shareholding, revenue comparisons are only possible in the common value setting. Expected revenue for the seller is again higher in the first-price than in the second price auction. Bidding behavior in the second-price auction is more sensitive to changes in cross-shareholding and the value environment than in the first-price auction.Received: 18 September 2000, Revised: 27 May 2003, JEL Classification Numbers: C72, D44.Correspondence to: Sudipto DasguptaWe thank Sugato Bhattacharyya, Paul Klemperer, Kunal Sengupta and Guofu Tan for helpful discussions, and an anonymous referee for suggestions that improved the paper. The usual disclaimer, of course, applies.     

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.     

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.     

Sequentially Optimal Auctions

In auctions where a seller can post a reserve price but if the object fails to sell cannot commit never to attempt to resell it, revenue equivalence between repeated first price and second price auctions without commitment results. When the time between auctions goes to zero, seller expected revenues converge to those of a static auction with no reserve price. With many bidders, the seller equilibrium reserve price approaches the reserve price in an optimal static auction. An auction in which the simple equilibrium reserve price policy of the seller mirrors a policy commonly used by many auctioneers is computed.Journal of Economic LiteratureClassification Numbers: C78, D44, D82.     

Optimal auctions with information acquisition

This paper studies optimal auction design in a private value setting with endogenous information gathering. We develop a general framework for modeling information acquisition when a seller wants to sell an object to one of several potential buyers, who can each gather information about their valuations prior to participation in the auction. We first demonstrate that the optimal monopoly price is always lower than the standard monopoly price. We then show that standard auctions with a reserve price remain optimal among symmetric mechanisms, but the optimal reserve price lies between the ex ante mean valuation of bidders and the standard reserve price in Myerson (1981). Finally, we show that the optimal asymmetric mechanism softens the price discrimination against “strong” bidders.     

The effectiveness of English auctions

We study the performance of the English auction under different assumptions about the seller's degree of “Bayesian sophistication.” We define the effectiveness of an auction as the ratio between the expected revenue it generates for the seller and the expected valuation of the object to the bidder with the highest valuation (total surplus). We identify tight lower bounds on the effectiveness of the English auction for general private-values environments, and for private-values environments where bidders' valuations are non-negatively correlated. For example, when the seller faces 12 bidders who the seller believes have non-negatively correlated valuations whose expectations are at least as high as 60% of the maximal possible valuation, an English auction with no reserve price generates an expected price that is more than 80% of the value of the object to the bidder with the highest valuation.     

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.     

Optimal advertising of auctions

We study a symmetric independent private values auction model where the revenue-maximizing seller faces a cost cn of attracting n bidders to the auction. If the distribution of valuations possesses an increasing failure rate (IFR), the seller overinvests in attracting bidders compared to the social optimum. Conversely, if the distribution is DFR, the seller underinvests compared to the social optimum. If the distribution of valuations becomes more dispersed, both, a revenue- and a welfare-maximizing seller, attract more bidders.     

Credit contracting and bidding under wealth constraints

Summary. We model credit contracting and bidding in a first-price sealed-bid auction when bidder valuation and wealth are private information. An efficient separating equilibrium exists only if the wealth levels of both bidder types are sufficiently different. If not, high-valuation bidders signal by borrowing more and using less of their wealth – this is inefficient as wealth is a cheaper source of funds. An increase in the amount of borrowing required to signal does not necessarily decrease seller expected revenue. In contrast to separating equilibria, high-valuation bidders adopt pure strategy bids in pooling equilibria. Conditions are identified under which the lower bound on winning bids is higher in pooling than separating equilibria. Received: January 22, 2001; revised version: August 28, 2001