Sealed bid second price auctions with discrete bids

A single item is sold to two bidders by way of a sealed bid second price auction in which bids are restricted to a set of discrete values. Restricting attention to symmetric pure strategy behavior on the part of bidders, a unique equilibrium exists. When following these equilibrium strategies bidders may bid strictly above or below their valuation, implying that the item may be awarded to a bidder other than the high valuation bidder. In an auction with two acceptable bids, the expected revenue of the seller may be maximized by a high bid level not equal to the highest possible bidder valuation and may exceed the expected revenue from an analogous second price auction with continuous bidding (and no reserve price). With three acceptable bids, a revenue maximizing seller may choose unevenly spaced bids. With an arbitrary number of evenly spaced bids, as the number of acceptable bids is increased, the expected revenue of the seller and the probability of ex post inefficiency both may either increase or decrease.