In a common value auction in which the information partitions of the bidders are connected, all rings are core-stable. More precisely, the ex ante expected utilities of rings, at the (noncooperative) sophisticated equilibrium proposed by Einy, Haimanko, Orzach and Sela (Journal of Mathematical Economics, 2002), describe a cooperative game, in characteristic function form, in spite of the underlying strategic externalities. A ring is core-stable if the core of this characteristic function is not empty. Furthermore, every ring can implement its sophisticated equilibrium strategy by means of an incentive compatible mechanism.
Françoise Forges and Ram Orzach,
"Core-stable Rings in Second Price Auctions with Common Values"
(November 19, 2010).
Fondazione Eni Enrico Mattei Working Papers.
Working Paper 507.