  We consider the question of whether collusion among bidders (a "bidding
ring") can be supported in equilibrium of unrepeated first-price auctions.
Unlike previous work on the topic such as that by McAfee and McMillan [1992]
and Marshall and Marx [2007], we do not assume that non-colluding agents have
perfect knowledge about the number of colluding agents whose bids are
suppressed by the bidding ring, and indeed even allow for the existence of
multiple cartels. Furthermore, while we treat the association of bidders with
bidding rings as exogenous, we allow bidders to make strategic decisions about
whether to join bidding rings when invited. We identify a bidding ring protocol
that results in an efficient allocation in Bayes{Nash equilibrium, under which
non-colluding agents bid straightforwardly, and colluding agents join bidding
rings when invited and truthfully declare their valuations to the ring center.
We show that bidding rings benefit ring centers and all agents, both members
and non-members of bidding rings, at the auctioneer's expense. The techniques
we introduce in this paper may also be useful for reasoning about other
problems in which agents have asymmetric information about a setting.
