Optimal Mechanism Design with Post-Allocation Signal

We study the design of the optimal mechanism in a setting with one buyer and one item in which the seller receives a binary signal correlated with the buyer's type after the allocation decision and can condition the payment on the signal realization. The optimal mechanism depends on the correlation between the signal and the buyer's type through the conditional probability of the signal given the type, which we call the \emph{buyer's belief function}. Under the standard monotone likelihood property (MLRP), we establish that the curvature of the buyer's belief function plays a central role: full surplus extraction is achievable if and only if the belief function is convex in the buyer's type. In contrast, when beliefs are concave, it is optimal to fully allocate to the buyer if and only if her type is outside an intermediary range of types (corresponding to types with a higher correlation with the signal). We further establish that, in the case of concave beliefs,
the optimal mechanism can be implemented as a \emph{state-contingent posted payment} offered on a take-it-or-leave-it basis. We further provide a structural characterization of the optimal deterministic mechanism under general belief functions: the state space partitions into several regions, depending on the curvature of the belief function, and in each region, we either have full surplus extraction or a state-contingent posted payment. Finally, we show how our results can be extended to a setting with multiple buyers.