Revenue and Reserve Prices in a Probabilistic Single Item Auction

We investigate the effect of limiting the number of reserve prices on the revenue in a probabilistic single item auction. In the model considered, bidders compete for an impression drawn from a known distribution of possible types. The auction mechanism sets up to ℓ reserve prices, and each impression type is assigned the highest reserve price lower than the valuation of some bidder for it. The bidder proposing the highest bid for an arriving impression gets it provided his bid is at least the corresponding reserve price, and pays the maximum between the reserve price and the second highest bid. Since the number of impression types may be huge, we consider the revenue R_ℓ that can be ensured using only ℓ reserve prices. Our main results are tight lower bounds on R_ℓ for the cases where the impressions are drawn from the uniform or a general probability distribution.