Single-Crossing Differences in Convex Environments

TL;DR

This paper characterizes utility functions with single-crossing differences in convex environments, extending the concept to various preference types without assuming order on choices, and explores applications in economics.

Contribution

It provides a comprehensive characterization of SCD in convex environments, including unordered cases, and links them to interval choice and applications in economic models.

Findings

SCD characterizes preferences over lotteries and bundles in convex environments.

Unordered SCD is necessary and sufficient for interval choice.

Applications include cheap talk, observational learning, and collective choice.

Abstract

An agent's preferences depend on an ordered parameter or type. We characterize the set of utility functions with single-crossing differences (SCD) in convex environments. These include preferences over lotteries, both in expected utility and rank-dependent utility frameworks, and preferences over bundles of goods and over consumption streams. Our notion of SCD does not presume an order on the choice space. This unordered SCD is necessary and sufficient for ''interval choice'' comparative statics. We present applications to cheap talk, observational learning, and collective choice, showing how convex environments arise in these problems and how SCD/interval choice are useful. Methodologically, our main characterization stems from a result on linear aggregations of single-crossing functions.