Representations of cones and applications to decision theory

TL;DR

This paper characterizes cones in topological vector spaces via dual set families, linking geometric structures to decision theory and solving an open problem related to the independence axiom.

Contribution

It introduces a unique representation of cones using families of subsets in the dual space, connecting geometric and decision-theoretic concepts.

Findings

Characterization of cones through dual set families.

Identification of properties distinguishing cone types.

Application to decision theory and solving an open problem.

Abstract

Let $C$ be a cone in a locally convex Hausdorff topological vector space $X$ containing $0$. We show that there exists a (essentially unique) nonempty family $\mathscr{K}$ of nonempty subsets of the topological dual $X^\prime$ such that $$ C=\{x \in X: \forall K \in \mathscr{K}, \exists f \in K, \,\, f(x) \ge 0\}. $$ Then, we identify the additional properties on the family $\mathscr{K}$ which characterize, among others, closed convex cones, open convex cones, closed cones, and convex cones. For instance, if $X$ is a Banach space, then $C$ is a closed cone if and only if the family $\mathscr{K}$ can be chosen with nonempty convex compact sets. These representations provide abstract versions of several recent results in decision theory and give us the proper framework to obtain new ones. This allows us to characterize preorders which satisfy the independence axiom over certain probability measures, answering an open question in [Econometrica~\textbf{87} (2019), no. 3, 933--980].