Quasi-Concavity, Convexity of Optimal Actions, and the Local Single-Crossing Property

TL;DR

This paper establishes conditions under which decision problems are quasi-concave and demonstrates that quasi-concavity implies a local single crossing property after relabeling states, enhancing understanding of decision structure.

Contribution

It introduces two novel results linking convexity of optimal actions to quasi-concavity and the local single crossing property in decision problems.

Findings

Decision problems are quasi-concave if optimal actions form convex sets under all beliefs.

Quasi-concavity implies the local single crossing property after relabeling states.

Provides conditions under which these properties hold.

Abstract

This note presents two results. First, it shows that under mild conditions, a decision problem is quasi-concave if the set of optimal actions is convex under every belief. Second, it shows that if a decision problem is quasi-concave, then it satisfies the local single crossing property after relabeling the states.