A Competitive Distribution of Auctions

Abstract

In this paper a competitive distribution of auctions is described for an economy consisting of an infinite number of buyers and sellers, all of whom differ according to their valuation for the single indivisible object being traded. A competitive distribution of auctions is such that no seller can improve his profits by deviating to any alternative direct mechanism. It is shown that the competitive distribution of auctions will have the property that each buyer and seller's best reply is independent of his beliefs about the tastes of other buyers and sellers on the market. The purpose of this paper is to provide an equilibrium for an economy in which sellers compete by offering different direct mechanisms to buyers. No attempt is made to give a complete characterization of all possible equilibria, rather we focus on one equilibrium in which the contracts that are offered by sellers in equilibrium, and buyer's behaviour under these contracts, are both independent of traders' beliefs about one another. The equilibrium illustrates one of the reasons why contracts that are employed in practice may not be as finely tuned to beliefs as the theory of mechanism design says that they should be. The answer is simply that contracts are not offered in a vacuum. Sellers in the model studied in this paper need to compete with one another in order to attract buyers. Highly-tuned mechanisms tend to extract buyer surplus. In such an environment, sellers are able to attract many buyers by offering to use cruder mechanisms that leave some surplus for buyers. In this paper, sellers are allowed to have different costs for the commodity that is being traded. Nonetheless, the equilibrium (for the limit economy in which there are infinitely many buyers and sellers) is such that every seller offers a second-price auction and publicizes a reserve price equal to his cost. In a standard monopoly auction the optimal reserve price for the seller exceeds his cost by an amount that depends on his beliefs about buyers' valuations (Myerson (1981)). A seller who raises his reserve price has his profits affected in two distinct ways. By raising his reserve price the seller makes it unprofitable for some buyer types to participate in his auction. The seller therefore loses any surplus that he might have earned by trading with these buyers. The buyers who still find it profitable to participate in the seller's auction will end up having to pay a higher price when they win because of this increase in the seller's reserve price. This means that the seller will be able to extract a bit more surplus from buyers who continue to participate. The seller raises the reserve price until the marginal cost associated with raising the reserve price is just equal to the marginal gain associated with the ability to extract more surplus from remaining buyers. Competition mitigates this argument since there is a limit to the amount of surplus that the seller can extract from remaining buyers before they decide that they should instead choose one of the alternative auctions that are being held on the market. With