Bidding in Common‐Value Auctions With an Unknown Number of Competitors

Abstract

This paper studies a first‐price common‐value auction in which bidders do not know the number of their competitors. In contrast to the case of common‐value auctions with a known number of rival bidders, the inference from winning is not monotone, and a “winner's blessing” emerges at low bids. As a result, bidding strategies may not be strictly increasing, but instead may contain atoms. Moreover, an equilibrium fails to exist when the expected number of competitors is large and the bid space is continuous. Therefore, we consider auctions on a grid. On a fine grid, high‐signal bidders follow an essentially strictly increasing strategy, whereas low‐signal bidders pool on two adjacent bids on the grid. The solutions of a “communication extension” based on Jackson, Simon, Swinkels, and Zame (2002) capture the equilibrium bidding behavior in the limit, as the grid becomes arbitrarily fine.