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.

Leveraging an established exercise in negotiation education, we build a novel dataset for studying how the use of language shapes bilateral bargaining.Our dataset extends existing work in two ways: 1) we recruit participants via behavioral labs instead of crowdsourcing platforms and allow participants to negotiate through audio, enabling more naturalistic interactions; 2) we add a control setting where participants negotiate only through alternating, written numeric offers.Despite the two contrasting forms of communication, we find that the average agreed prices of the two treatments are identical.But when subjects can talk, fewer offers are exchanged, negotiations finish faster, the likelihood of reaching agreement rises, and the variance of prices at which subjects agree drops substantially.We further propose a taxonomy of speech acts in negotiation and enrich the dataset with annotated speech acts.We set up prediction tasks to predict negotiation success and find that being reactive to the arguments of the other party is advantageous over driving the negotiation.

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.