Abstract: A receiver wants to learn multidimensional information from a sender, but she has capacity to verify only 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 if the exogenous payoff function is submodular. If it is (strictly) supermodular, then full learning is not possible. In a variant of the model that allows for severe punishments when the sender is found to have lied, we can give a complete characterization of when full learning is possible. Our full learning result does not critically rely on perfect verifiability of one dimension: in an example with noisy verification, the receiver's ex-post perceived distribution of information converges in distribution to the true value as the noise vanishes.
