The relationship between the allocation of goods and a seller’s revenue

Abstract

We examine a seller auctioning off a set of objects to a large number of bidders, and restrict attention to auctions that are self-enforcing in that bidders do not want to walk away from the mechanism after they see the price that they must pay. We show that the seller’s ability to extract the full possible revenue depends on whether the efficient allocation is concentrated among a few bidders or is dispersed so that a non-negligible fraction of bidders obtain objects. If it is concentrated then any sequence of mechanisms that achieve the efficient allocation (and there are many) asymptotically extracts the full surplus, and so it is in the seller’s interest to efficiently allocate objects. In contrast, when the efficient allocation is dispersed then no sequence of mechanisms asymptotically extracts the full surplus. Moreover, in the dispersed case the seller may benefit by inefficiently bundling objects for sale.

Introduction

Consider a monopolist facing a large number of potential buyers, who may differ in their valuations for the goods for sale and/or in the information that they hold regarding the quality of the goods. The central question that we ask is whether the high degree of competition among the large number of buyers enables the seller to extract the full possible surplus from sale of the objects, and whether the seller has an incentive to allocate the objects in an efficient manner.

It is well known, that with a limited number of bidders, a seller may gain from inefficiently allocating goods, for instance by setting a reserve price; and that bidders enjoy some rents for their private information (see Myerson, 1981, Riley and Samuelson, 1981, and Milgrom and Weber (1982), among others). It is also known, however, that as the number of bidders grows a seller of a single object can extract the full possible surplus from the sale of the object; and moreover, this is true regardless of whether the valuations are private or common. Any mechanism that efficiently allocates the object leads to the same full revenue (see Bali and Jackson (2002)). Thus, a seller of a single object to a large number of bidders has an incentive to efficiently allocate the object, extracts the full revenue, and is indifferent between a large number of auction formats. Here we examine whether the same is true when a seller is selling more than one object, and find that the answer depends on the structure of the efficient allocation of the objects.

In order to study this question, we impose a constraint on the auction formats that the seller may choose. In particular, Crémer and McLean, 1988 (see also McAfee and Reny (1992)) have shown that when there is even a minimal amount of correlation among buyers’ types, then a seller can extract full rents through an elaborate payment scheme that depends in careful ways on the underlying distribution of types, and this is true even with small numbers of agents. Thus, there exist auction formats that implement an efficient allocation and extract the entire surplus. However, such mechanisms are extreme and are not used in practice, and might be thought as a theoretical benchmark rather than a practical device. There are several reasons why these might not be practical, and an important one is that agents are sometimes charged prices that they know to exceed the value of the good they obtain. Upon observing their own allocation and payment, agents often prefer to walk away, even without knowing anything about the information of the other agents. This means that the seller would face the unpleasant task of forcing the mechanism outcome on its participants.

The condition we impose is that a mechanism be self-enforcing, which effectively means that this never happens. The condition requires that an agent does not wish to leave after he or she observes his or her allocation and price—but not necessarily anything about the information of other agents. This is a condition that is slightly stronger than an interim individual rationality constraint, but substantially weaker than an ex post constraint. Under this condition, it is still possible that agents eventually regret having purchased an object, say in the case of common values. Nevertheless, this condition is strong enough to rule out the Crémer and McLean types of constructions, and is still satisfied by most standard auction formats that one can think of.¹

With a restriction to self-enforcing mechanisms, our results may be summarized as follows. If the efficient allocation is concentrated in the hands of a few agents, then any mechanism that achieves the efficient allocation also extract the entire surplus. Moreover, such a mechanism always exists. Thus, with concentrated allocations, the seller has an incentive to efficiently allocate the objects for sale. In contrast, if the efficient allocation is dispersed among a non-negligible number of bidders, then no mechanism can extract the entire surplus. Moreover, in this case there is often tension between efficiency and revenue maximization even in the limit. In some situations a mechanism that inefficiently bundles objects for sale may lead to higher revenue than any mechanism that leads to an efficient allocation. Thus, with a large number of objects, a seller may have an incentive to inefficiently allocate objects, even when there are arbitrarily many potential buyers.

Finally, we note that for the case of concentrated allocations, we can pinpoint how fast the revenue of the seller grows with the number of buyers.