We study when and how explanations of complex models can aid a decision maker (DM) whose payoff depends on a state of the world described by inputs and outputs. The DM cannot directly understand the true model linking inputs to outputs and must instead rely on an explanation from a class of simpler intelligible models. We analyze mappings from the infinite-dimensional space of possible true models to the finite-dimensional space of intelligible ones — what we call explainers — and focus on those that yield explanations which are robustly useful: that is, they allow the DM to improve their worst-case payoff across all models that are consistent with the explanation received.

When firms and statistical agencies disseminate information about the data they collect, they are often constrained by the need to protect the privacy of the individuals in their sample. In many cases, they do so by committing to a publication mechanism --- that is, a stochastic map from possible datasets to published outputs --- so as to satisfy a differential privacy [1] requirement.