Strategic Communication with Minimal Verification

A receiver wants to learn multidimensional information from a sender, and she has the capacity to verify just one dimension. The sender's payoff depends on the belief he induces, via an exogenously given monotone function. We show that by using a randomized verification strategy, the receiver can learn the sender's information fully in many cases. We characterize exactly when it is possible to do so. In particular, when the exogenous payoff function is submodular, we can explicitly describe a full-learning mechanism; when it is (strictly) supermodular, full learning is not possible. In leading cases where full learning is possible, it can be attained using an indirect mechanism in which the sender chooses the probability of verifying each dimension.