Coalitional Bayesian Nash implementation in differential information economies

A mechanism coalitionally implements a social choice set if any outcome of the social choice set can be achieved as a coalitional Bayesian Nash equili brium of a mechanism and vice versa. We say that a social choice set is coa litionally implementable if there is a mechanism which coalitionally implem ents it. Our main theorem proves that a social choice set is coalitionally implementable if and only if it is interim individually rational, interim e fficient, coalitional Bayesian incentive compatible, and satisfies a coalit ional Bayesian monotonicity condition as well as a closure condition. As an application of our main result, we show that the private core and the priv ate Shapley value of an economy with differential information are coalition ally implementable.