Competing auctions: Finite markets and convergence

The literature on competing auctions offers a model where sellers compete for buyers by setting reserve prices. An outstanding conjecture (e.g., Peters and Severinov 1997) is that the sellers post prices close to their marginal costs when the market becomes large. This conjecture is confirmed in this paper: we show that if all sellers have zero costs, then the equilibrium reserve price converges to 0 in distribution. Under further conditions there is a symmetric pure strategy equilibrium. In this equilibrium, if the ratio of buyers to sellers increases, then the equilibrium reserve price increases, and the reserve price is decreasing in the size of the market. Convergence of reserve prices occurs at the fast rate of 1/n if the ratio of buyers to sellers is held constant.