Competing auctions: finite markets and convergence

The literature on competing auctions offers a model where sellers compete for buyers by setting reserve prices freely. An important 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. More precisely, we show that if all sellers have zero costs, then the equilibrium reserve price converges to 0 in distribution. I also show that if there is a high enough lower bound on the buyers’ valuations, then there is a symmetric pure strategy equilibrium. In this equilibrium, if the number of buyers (sellers) increases, then the equilibrium reserve price increases (decreases) and the reserve price is decreasing in the size of the market.