Optimal Auctions with Restricted Allocations

We study the problem of designing optimal auctions under restrictions on the set of permissible allocations.
In addition to allowing us to restrict to deterministic mechanisms, we can also indirectly model non-additive
valuations. We prove a strong duality result, extending a result due to Daskalakis et al. [2015], that guarantees
the existence of a certificate of optimality for optimal restricted mechanisms. As a corollary of our
result, we provide a new characterization of the set of allocations that the optimal mechanism may actually
use. To illustrate our result we find and certify optimal mechanisms for four settings where previous frameworks
do not apply, and provide new economic intuition about some of the tools that have previously been
used to find optimal mechanisms.