Revenue-superior variants of the second-price auction

Abstract

With two bidders, one strong and one weak, the introduction of at least some degree of anonymous “pay-your-bid” in the payment rule of the second-price auction behoves any risk-neutral seller who, while possibly efficiency minded, cares about revenues. This can be achieved by adding to the winner’s payment a uniform proportion of his own bid, as in Guth and van Damme’s auction, or by having bidders receive a uniform proportion of the losing bid, as in Goeree and Offerman’s Amsterdam auction, or even by selling uniform toeholds to the bidders prior to the auction. We demonstrate one-to-one relations between the equilibria of these auctions and of first-price auctions. By assuming a power relation between the bidders’ value cumulative or decumulative functions, we obtain explicit expressions for the first-order effects of the pay-your-bid rule.