Competing Auctions with Endogenous Quantities

We study models where two sellers simultaneously decide on their discrete supply of a homogenous good. There is a finite, not necessarily large, number of buyers who have unit demand and privately known valuations. In the first model, there is a centralized market place where a uniform auction takes place. In the second model, there are two distinct auction sites, each with one seller, and buyers decide where to bid. Our results shed some light on the conditions leading to either the emergence of dominant marketplaces or to the coexistence of several competing sites. Using the theory of potential games, we show that in the one-site auction model there is always an (almost symmetric) equilibrium in pure strategies. This equilibrium approximates the Cournot outcome as the number of buyers becomes large. In contrast, if the distribution of buyers values has an increasing failure rate, and if the marginal cost of production is relatively low, there is no pure strategy equilibrium where both sellers make positive profits in the competing sites model. We also identify conditions under which an equilibrium with a unique active site exists. Technically, we are able to deal with the finite and discrete models by using several results about order statistics developed by Richard Barlow and Frank Proschan (1965, 1966, 1975).