Constrained Mediation: Bayesian Implementability of Joint Posteriors

We examine information structures in settings with privately informed agents and an informationally constrained mediator who supplies additional public signals. Our focus is on characterizing the set of posteriors that the mediator can induce. To this end, we employ a graph-theoretic framework: states are represented as vertices, information sets correspond to edges, and a likelihood ratio function on edges encodes the posterior beliefs. Within this framework, we derive necessary and sufficient conditions, internal and external consistency, for the rationalization of posteriors. Finally, we identify conditions under which a single mediator can implement multiple posteriors, effectively serving as a generator of Blackwell experiments.