Sequential Information Design
The literature on implementation has focused mainly on cases in which agents have both agency over their actions and over the time at which they commit to their actions. We take as given the set of agents, their set of actions, and their payoffs. We ask what distributions over actions are consistent with the players playing according to some extensive form. The main result of the paper is to show that a distribution over outcomes is implementable as a Perfect Bayesian equilibrium (PBE) of an admissible extensive form if, and only if, it is implementable as a PBE of a canonical extensive form. The latter is an admissible extensive form, in which there is a randomization device that not only sends (private) recommendations to the agents, but also selects the order in which the agents move; moreover, subsequent recommendations can be made conditional on actions already taken. This result strictly generalizes Aumann’s notion of correlated equilibrium, and Bergemann and Morris’  notion of Bayes’ correlated equilibrium.