Dynamically Eliciting Unobservable Information

We answer the following question: At t = 1, an expert has (probabilistic) information about a random outcome X. In addition, the expert will obtain further information about X as time passes, up to some time t = T + 1 at which X will be publicly revealed. (How) Can a protocol be devised that induces the expert, as a strict best response, to reveal at the outset his prior assessment of both X and the information flows he anticipates and, subsequently, what information he privately receives? (The protocol can provide the expert with payoffs that depend only on the realization of X, as well as any decisions he may take.) We show that this can be done with the following sort of protocol: At the penultimate time t = T, the expert chooses a payoff function from a menu of such functions, where the menu available to him was chosen by him at time t = T − 1 from a menu of such menus, and so forth. We show that any protocol that affirmatively answers our question can be approximated by a protocol of the form described. We show how these results can be extended from discrete time to continuous time problems of this sort.