We characterize the extreme points of multidimensional monotone functions from [0,1]^n to [0,1], as well as the extreme points of the set of one-dimensional marginals of these functions. 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 consider the problem of explaining models to a decision maker (DM) whose payoff depends on a state of the world described by inputs and outputs. A true model specifies the relationship between these inputs and outputs, but is not intelligible to the DM. Instead, the true model must be explained via a simpler model from a finite-dimensional set. If the DM maximizes their average payoff, then an explanation using ordinary least squares is as good as understanding the true model itself. However, if the DM maximizes their worst-case payoff, then any explanation is no better than no explanation at all. We discuss how these results apply to policy evaluation and explainable AI.

An intermediary serves as a platform through which a producer with a private cost sells a product to a unit mass of consumers. The intermediary has information about the match values between consumers and the product, which can be used to inform both the consumers and the producer. This paper studies the revenue-maximizing mechanisms for the intermediary under two critical business models: the retail model and the marketplace model. We show that the market outcomes are equivalent across the two business models. Furthermore, it is optimal for the intermediary to either (i) provide upper-censored partial information to consumers and no information to the producer, or (ii) fully inform consumers about their values and partially disclose that information to the producer and induce quasi-perfect price discrimination. This result suggests that product recommendation and price discrimination are outcome-equivalent mechanisms for an intermediary.