Implementation with coalition formation: A complete characterization

The basic difference between Nash and strong (Nash) implementations originates from the possibility of coalition formation. The results of Nash implementation are applicable when agents in no group can cooperate (or form a coalition) and those of strong implementation are applicable when agents in every group can cooperate. In the present paper we consider an environment where agents in some groups can cooperate and agents in other groups cannot. We completely characterize the set of correspondences that are implementable in such a general environment, and we provide an algorithm which enables us to verify whether a correspondence satisfies the condition that appears in the characterization results. By applying the algorithm in an exchange economy, we show that the core, the definition of which depends on the possibility of cooperation, is implementable in the equilibrium corresponding to the cooperation possibility.