An optimal auction with correlated values and risk aversion

We consider an auction setting where the buyers are risk averse with correlated private valuations (CARA preferences, binary types), and characterize the optimal mechanism for a risk-neutral seller. We show that the optimal auction extracts all buyer surplus whenever the correlation is sufficiently strong (greater than 1/3 in absolute value), no matter how risk averse the buyers are. In contrast, we note that a sufficiently risk-averse seller would not use a full rent extracting mechanism for any positive correlation of the valuations even if the buyers were risk neutral.     

Optimal Information Disclosure in Auctions and the Handicap Auction

We analyse a situation where a monopolist is selling an indivisible good to risk-neutral buyers who only have an estimate of their private valuations. The seller can release, without observing, certain additional signals that affect the buyers' valuations. Our main result is that in the expected revenue-maximizing mechanism, the seller makes available all the information that she can, and her expected revenue is the same as it would be if she could observe the part of the information that is "new" to the buyers. We also show that this mechanism can be implemented by what we call a handicap auction in interesting applications. In the first round of this auction, each buyer picks a price premium from a menu offered by the seller (a smaller premium costs more). Then the seller releases the additional signals. In the second round, the buyers bid in a second-price auction where the winner pays the sum of his premium and the second highest non-negative bid. In the case of a single buyer, this mechanism simplifies to a menu of European call options.     

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.     

Optimal mechanism design with resale via bargaining

In this paper, we examine the optimal mechanism design of selling an indivisible object to one regular buyer and one publicly known buyer, where inter-buyer resale cannot be prohibited. The resale market is modeled as a stochastic ultimatum bargaining game between the two buyers. We fully characterize an optimal mechanism under general conditions. Surprisingly, in this optimal mechanism, the seller never allocates the object to the regular buyer regardless of his bargaining power in the resale market. The seller sells only to the publicly known buyer, and reveals no additional information to the resale market. The possibility of resale causes the seller to sometimes hold back the object, which under our setup is never optimal if resale is prohibited. We find that the seller?s revenue is increasing in the publicly known buyer?s bargaining power in the resale market. When the publicly known buyer has full bargaining power, Myerson?s optimal revenue is achieved; when the publicly known buyer has no bargaining power, a conditionally efficient mechanism prevails.     

Sequential selling mechanisms

Summary In this paper we attempt to formalize the idea that a mechanism that involves multilateral communication between buyers and sellers may be dominated by one that involves simple bilateral communication. To do this we consider the well known problem in which a seller tries to sell a single unit of output to a group ofN buyers who have independently distributed private valuations. Our arguments hinge on two considerations. First, buyers communicate their willingness to negotiate with the seller sequentially, and second, buyers have the option of purchasing the good from some alternative supplier. It is shown that the seller cannot improve upon a procedure in which she offers the good to each buyer in turn at a fixed price. The seller reverts to multilateral communication if possible, only when no buyer is willing to pay the fixed price. In reasonable environments buyers will be too impatient to wait for the outcome of a multilateral negotiation and all communications will be bilateral.In many problems in mechanism design, informed traders have no alternative to participation in the mechanism that is offered by its designer. The best mechanism from the designer's point of view is then the one that is most efficient at extracting informational rents, that is, a simple auction. In a competitive environment it is likely to be costly for buyers to participate in an auction or any other multilateral selling scheme in which the seller must process information from many different buyers because alternative trading opportunities will be disappearing during the time that the seller is collecting this information. Buyers might be willing to participate in an auction, but only if they could be guaranteed that the competition that they face will not eliminate too much of their surplus.At the other extreme to the auction is a simple fixed price selling scheme 1. The seller simply waits until he meets a buyer whose valuation is high enough, given the opportunities that exist in the rest of the market, for him to be willing to pay this price. The seller extracts the minimum of the buyer's informational rents since the price that a buyer pays is independent of his valuation. Yet the seller might like this scheme if adding a second bidder to the process makes it very difficult for him to find a buyer with a valuation high enough to want to participate.In the presence of opportunity costs, the seller faces a trade-off between his ability to extract buyers informational rents and his ability to find buyers who are willing to participate in any competitive process. In practice this trade-off will impose structure on the method that is used to determine a price. In markets where there are auctions, limits are put on buyer participation. In tobacco auctions bids are submitted at a distinct point in time from buyers who are present at that time. In real estate auctions time limits are put on the amount of time the seller will wait before making a decision. These restrictions on participation are presumably endogenously selected by the seller (possibly in competition with other mechanism designers) with this trade-off in mind.On the other hand, markets in which objects appear to trade at a fixed price are rarely so simple. A baker with a fixed supply of fresh bagels is unlikely to collect bids from buyers and award the bagels to the high bidder at the end of the day. Buyers are unlikely to be willing to participate in such a scheme since they can buy fresh bagels from a competitor down the street. Yet despite the fact that bagels sell at a fixed price throughout the day, most bakers are more than willing to let it be known that they will discount price at the end of the day on any bagels that they have not yet sold. Selling used cars presents a similar problem. Each potential buyer for the used car is likely to have inspected a number of alternatives, and is likely to know the prices at which these alternative can be obtained. A seller who suggests that buyers submit a bid, then wait until the seller is sure that no higher offer will be submitted is asking buyers to forgo these alternative opportunities with no gain to themselves. To avoid the rigidity of the pure fixed price scheme most used cars are sold for a fixed price or best offer. These examples suggest that the best selling mechanism may involve a complex interplay between participation and surplus extraction considerations.The purpose of this paper is to provide a simple formalism within which the factors that determine the best contract can be evaluated. We consider the best known environment from the point of view of auction design in which there are a large number of buyers with independent private valuations for a unit of an indivisible commodity that is being sold by a single supplier who acts as the mechanism designer. We modify this standard problem in two critical ways. First, we assume that the seller meets the potential buyers sequentially rather than all at once. Secondly we assume that buyers have a valuable alternative that yields them a sure surplus. This creates a simple bidding cost that is effectively the expected loss in surplus (created by the disappearance of outside alternatives) that the buyer faces during the time that he spends negotiating with the seller.These simple assumptions allow us to calculate the impact of competition and communication costs using completely standard arguments from the mechanism design literature. We are able to show that with these assumptions the seller's expected surplus will be highest if the object is sold according to the following modified fixed price scheme: the seller contacts each of the potential buyers in turn and either offers to negotiate or announces that he no longer wishes to trade. If he offers to negotiate and the buyer agrees, the buyer immediately has the option of trading for sure with the seller at a fixed price set ex ante. If the buyer does not wish to pay this fixed price, he may submit an alternative bid. The seller will then continue to contact new buyers, returning to trade with the buyer only if no buyer wishes to pay the fixed price and no higher bid is submitted.It will be clear that in our environment, both the simple fixed price scheme and the simple auction are feasible. The simple auction prevails when the fixed price is set equal to the maximum possible valuation, while the simple fixed price scheme occurs when the fixed price is set so that buyers are willing to participate if and only if they are willing to pay the fixed price. Our results will show that a simple auction in never optimal for the seller. The seller can always strictly improve his payoff by moving to a scheme in which there is some strictly positive probability that trade will occur at the fixed price. On the other hand, there are reasonable circumstances in which the seller cannot achieve a higher payoff than the one she gets by selling at a fixed price. It is shown that for any positive participation cost, there is a large, but finite, number of potential buyers so that the seller cannot achieve a higher payoff than what she gets by selling at a fixed price. Two simple, but important continuity results are also illustrated. As the cost of participation in the mechanism increases (decreases), the probability with which the seller's unit of output is sold at a fixed price goes to one (zero) in the best modified fixed price mechanism for the seller.Our paper is not the first to generate such a modified fixed price scheme. Both McAfee and McMillan (1988) and Riley and Zeckhauser (1983) come up with similar schemes for the case in which the seller must bear a fixed cost for each new buyer that she contacts. There are two essential differences between our model and theirs. First, as the cost is interpreted as the opportunity cost of participation in the mechanism, it is reasonable to imagine that the seller advertises the mechanism ex ante. Another way of putting this is that the seller pays a fixed rather than a variable cost to communicate the mechanism to buyers. This makes it possible to assume that the mechanism is common knowledge to the seller and all the buyers at the beginning of the communication process. For this reason we can make our case using completely standard arguments. Secondly, the mechanism in the opportunity cost case plays a different allocative role than it does in the case when the seller bears a cost. The mechanism must decide whether buyers should communicate with the seller or pursue their alternative activities, as well as who should trade and at what price. It is this allocative role that makes bilateral communication superior to multilateral communication in a competitive environment. These differences allow us to show, for example, that a simple fixed price scheme is undominated for the seller when the number of buyers is finite. As shown by McAfee and McMillan, this is only possible when the number of potential buyers is infinite when the seller bears the cost of communication.Remarkably, the existence of opportunity costs to buyer participation is not, by itself, sufficient to explain why sellers might prefer bilateral communications mechanisms. Samuelson (1983) and McAfee and McMillan (1987) show that when buyers must pay a fixed cost to submit a bid, which is equivalent to giving up a valuable alternative, a seller cannot expect to earn more than she does in a second price auction (though Samuelson shows that the reserve price may depend on the number of potential buyers). One of the contributions of this paper is to show that the assumption that buyers make their participation decisions simultaneously is critical to this result. Simultaneous entry decisions means that whether or not any particular buyer is assigned to the alternative activity is independent of any other buyer's valuation. With sequential communication the seller is able to relax this constraint. It is precisely the enlargement of the class of feasible mechanism that breaks down the optimality of the simple auction.The second author acknowledges the support of the Social Sciences and Humanities Research Council of Canada and the CRDE at the Université de Montreal.     

How do selling mechanisms affect profits,surplus, capacity and prices with unknown demand?

I analyze a model in which a seller wishes to sell multiple homogeneous goods to a large group of buyers with unknown demand. The seller may either sell objects via a posted‐price mechanism, a discriminatory‐price auction, a uniform‐price auction, their open‐bid analogs, or a revelation mechanism in which the seller first asks all potential buyers to report their valuations and then sets a reserve price. I show that the revelation mechanism leads to the greatest profits, the auction mechanisms result in identical expected profits and the posted‐price mechanism results in the smallest profits. However, the more profitable mechanisms impose stronger informational requirements that may make these mechanisms infeasible in practice, and the posted‐price mechanism also results in the greatest total surplus. I also find the seller chooses a lower capacity and reserve price in an auction than in the posted‐price mechanism.     

Search, common knowledge, and competition

This paper analyzes the effects of buyer search costs and seller private and common knowledge on seller competition. It shows that lack of common knowledge results in the equilibrium price continuously decreasing to the perfectly competitive one as buyer search costs for price decrease from positive for all buyers to zero for all buyers, even if each market agent's uncertainty (in the private knowledge) is small. At the same time, if the uncertainty of each seller about buyer valuations is small, the effects of a small change in the search costs or of information structure on pricing may be large (but continuous).     

Fake Sales: A Dynamic Pricing Perspective

Some sellers display high “regular” prices, but mark down these prices the vast majority of the time, advertising the good as “on sale” or “discounted”. This note suggests a framework for understanding the practice, emphasising the role of buyer uncertainty about their future valuations for the good. We argue that so‐called “regular” prices set buyers’ expectations regarding future prices, expectations that need not be tethered to the prices actually set. By manipulating upwards buyers’ expectations of future prices, the seller can increase demand for the good at the current “sale” price, increasing profits.     

Reputation for quality and adverse selection

We analyze a dynamic market with a seller who can make a one-time investment that affects the returns of tradable assets. The potential buyers of the assets cannot observe the seller׳s investment prior to the trade or verify it in any way after the trade. The market faces two types of inefficiency: the ex-ante inefficiency, i.e., the seller׳s moral hazard problem, and the ex-post inefficiency, i.e., inefficient ex-post allocations due to the adverse selection problem. We analyze how the observability of information by future buyers, through which the seller builds a reputation, affects the two types of inefficiency as well as the interplay between them.     

Auctions with costly information acquisition

We characterize optimal selling mechanisms in auction environments where bidders must incur a cost to learn their valuations. These mechanisms specify for each period, as a function of the bids in previous periods, which new potential buyers should be asked to bid. In addition, these mechanisms must induce the bidders to acquire information about their valuations and to reveal this information truthfully. Using a generalized Groves principle, we prove a very general “full extraction of the surplus” result: the seller can obtain the same profit as if he had full control over the bidders’ acquisition of information and could have observed directly their valuations once they are informed. We also present appealing implementations of the optimal mechanism in special cases. For helpful comments we thank George Deltas, David Martimort, an anonymous referee, and seminar participants in Mannheim, Rutgers, Tel Aviv, Toulouse, the Society for Economic Design 2002 conference in New York, and the 2003 North American Summer Meetings of the Econometric Society in Evanston, IL. Yossi Spiegel thanks the IIBR for financial assistance and Charles Zheng thanks the NSF for grant SES-0214471.     

The role of optimal threats in auction design

This paper studies revenue-maximizing auctions when buyers' outside options depend on their private information and are endogenously chosen by the seller. We show that the revenue-maximizing assignment of the object can depend crucially on the outside options that the seller can choose as threats. Depending on the shape of outside options, sometimes an optimal mechanism allocates the object in an ex-post efficient way, and, other times, buyers obtain the object more often than is efficient.     

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.     

Dynamic bilateral bargaining under private information with a sequence of potential buyers

A seller owning a single, indivisible asset faces the random arrival of privately informed buyers, with whom he can bargain sequentially. Our key result is that despite the arrival of alternative buyers the Coase conjecture continues to hold under stationary strategies if the distribution of buyer valuations has convex support: Negotiations end almost immediately and the asset is sold almost at the minimum of the seller's own reservation value and the lowest possible valuation of a buyer. We also show existence of multiple stationary equilibria, though, in the special case where the support of buyers' valuations exhibits a sufficiently large “interior gap”. Taken together, our findings thus also point to a potential pitfall when analyzing only two-type distributions in more applied work.     

Sustainable reputations with rating systems

In a product choice game played between a long lived seller and an infinite sequence of buyers, we assume that buyers cannot observe past signals. To facilitate the analysis of applications such as online auctions (e.g. eBay), online shopping search engines (e.g. BizRate.com) and consumer reports, we assume that a central mechanism observes all past signals, and makes public announcements every period. The set of announcements and the mapping from observed signals to the set of announcements is called a rating system. We show that, absent reputation effects, information censoring cannot improve attainable payoffs. However, if there is an initial probability that the seller is a commitment type that plays a particular strategy every period, then there exists a finite rating system and an equilibrium of the resulting game such that, the expected present discounted payoff of the seller is almost his Stackelberg payoff after every history. This is in contrast to Cripps, Mailath and Samuelson (2004) [5], where it is shown that reputation effects do not last forever in such games if buyers can observe all past signals. We also construct finite rating systems that increase payoffs of almost all buyers, while decreasing the seller?s payoff.     

ENDOGENOUS ROLE ASSIGNMENT AND TEAM PERFORMANCE

We study how the mechanism used for assigning roles within teams affects team performance. Subjects play the takeover game in buyer–seller teams. Understanding optimal play is demanding for buyers and trivial for sellers, so teams should perform better if the buyer is the abler teammate. When teammates are allowed to jointly choose their roles, abler teammates tend to become buyers, but this is more than offset by disruptions to the learning process. We examine two potential sources for the latter effect and find that endogenous role assignment has a negative psychological and emotional effect on buyers.     

Seller competition by mechanism design

This paper analyzes a market game in which sellers offer trading mechanisms to buyers and buyers decide which seller to go to depending on the trading mechanisms offered. In a (subgame perfect) equilibrium of this market, sellers hold auctions with an efficient reserve price but charge an entry fee. The entry fee depends on the number of buyers and sellers, the distribution of buyer valuations, and the buyer cost of entering the market. As the size of the market increases, the entry fee decreases and converges to zero in the limit. We study how the surplus of buyers and sellers depends on the number of agents on each side of the market in this decentralized trading environment.     

Allocating multiple units

Summary. This paper studies the allocation and rent distribution in multi-unit, combinatorial-bid auctions under complete information. We focus on the natural multi-unit analogue of the first-price auction, where buyers bid total payments, pay their bids, and where the seller allocates goods to maximize his revenue. While there are many equilibria in this auction, only efficient equilibria remain when the truthful equilibrium restriction of the menu-auction literature is used. Focusing on these equilibria we first show that the first-price auction just described is revenue and outcome equivalent to a Vickrey auction, which is the multi unit analogue of a second-price auction. Furthermore, we characterize these equilibria when valuations take a number of different forms: diminishing marginal valuations, increasing average valuations, and marginal valuations with single turning points. Received: December 23, 1999; revised version: December 10, 2001     

Learning about the arrival of sales

We analyse optimal stopping when the economic environment changes because of learning. A primary application is optimal selling of an asset when demand is uncertain. The seller learns about the arrival rate of buyers. As time passes without a sale, the seller becomes more pessimistic about the arrival rate. When the arrival of buyers is not observed, the rate at which the seller revises her beliefs is affected by the price she sets. Learning leads to a higher posted price by the seller. When the seller does observe the arrival of buyers, she sets an even higher price.     

Dynamic market for lemons with endogenous quality choice by the seller

We analyze a dynamic market for lemons in which the quality of the good is endogenously determined by the seller. Potential buyers sequentially submit offers to one seller. The seller can make an investment that determines the quality of the item at the beginning of the game, which is unobservable to buyers. At the interim stage of the game, the information and payoff structures are the same as in the market for lemons. Our main result is that the possibility of trade does not create any efficiency gain if (i) the common discounting is low, and (ii) the static incentive constraints preclude the mutually agreeable ex-ante contract under which the trade happens with probability one. Our result does not depend on whether the offers by buyers are private or public.     

Pricing and signaling with frictions

We study a market where each seller chooses the quality and price of goods and the number of selling sites. Observing sellers? choices of prices and sites, but not quality, buyers choose which site to visit. A seller?s choices of prices can direct buyers? search and signal quality. A unique equilibrium exists and is separating. When the quality differential is large, the equilibrium implements the efficient allocation with public information. Otherwise, the quality of goods and/or the number of sites created is inefficient, due to a conflict between the search-directing and signaling roles of prices.