Procuring unverifiable information

We study settings where information in the form of Bayesian signals is acquired by an expert on behalf of a principal. Information acquisition is costly for the expert, and crucially not verifiable by the principal. The expert is compensated by the principal with a menu of state-contingent payments. We provide a full characterization of the set of all menus that implement (resp., strictly implement) each signal. Moreover, we provide a closed-form characterization for the expected cost for the cheapest such menu, which we call proxy cost of the signal. Surprisingly, in general, the proxy cost is neither increasing in the Blackwell order, nor posterior-separable, even when the expert’s cost function is posterior-separable itself. Subsequently, we study the full agency problem (by introducing a downstream decision), thus endogenizing the signal. We show that there is always an optimal signal that can be strictly implemented, meaning that it is without loss of generality to exogenously restrict attention to strict implementation. As a result, similarly to Bayesian persuasion, the complexity of the principal’s optimal signal is bounded by the cardinality of the state space. Finally, we present some applications of interest.