Bidding in Common-Value Auctions With an Uncertain Number of Competitors

This paper studies a first-price common-value auction in which bidders are uncertain about the number of their competitors. It shows that this uncertainty invalidates classic findings for common-value auctions with a known number of rival bidders (Milgrom and Weber 1982). In particular, the inference from winning is no longer monotonic, and a “winner’s blessing” emerges at low bids. As a result, bidding strategies may not be strictly increasing but instead contain atoms. The location of the atoms is indeterminate, implying equilibrium multiplicity. 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. For the equilibrium characterization, we utilize a “communication extension” based on Jackson et al. (2002).