Information design with endogenous state verification

We study a persuasion problem in which the receiver has the ability to probabilistically verify the state at a cost. The sender wants to convince the receiver to accept a project, but the receiver is only willing to accept the project when its quality is above a threshold. The optimal disclosure policy strikes a balance between influencing the receiver’s decisions to accept and to verify the quality. It is deterministic and involves at most three messages, each consisting of an acceptance recommendation and a verification recommendation. Under the optimal disclosure policy, the acceptance recommendation exhibits a cutoff structure, while the verification recommendation exhibits a negative assortative structure. Specifically, the optimal disclosure policy recommends acceptance when the quality exceeds a certain threshold. When the quality falls below this threshold, rejection without verification is recommended. For qualities above the threshold, verification is not recommended if the quality lies within the middle range of the interval. Using the characterization of the optimal disclosure policy, we conduct comparative statics and show that verification is recommended when the probability of successful verification is moderate.