We study the set of multidimensional monotone functions from [0, 1]n to [0, 1] as well as their one-dimensional marginals. We characterize the extreme points of these convex sets subject to finitely many linear constraints. These characterizations lead to new results in various mechanism design and information design problems, including public good provision with interdependent values; interim efficient bilateral trade mechanisms; asymmetric reduced form auctions; and optimal private private information structure. As another application, we also present a mechanism anti-equivalence theorem for two-agent, two-alternative social choice problems: A mechanism is payoff-equivalent to a deterministic DIC mechanism if and only if they are ex-post equivalent.

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.

We characterize the extreme points of first-order stochastic dominance (FOSD) intervals and show how these intervals are at the heart of many topics in economics. An FOSD interval is a set of distributions that dominate a distribution and are simultaneously dominated by another distribution, in the sense of FOSD. The convexity of FOSD intervals means that their extreme points are fundamental to understanding their properties. We show that a distribution is an extreme point of an FOSD interval if and only if the distribution either coincides with one of the FOSD bounds or is flat. Wherever the distribution is flat, at least one end of the flat portion must be attached to one of the FOSD bounds.