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Representations of cones and applications to decision theory

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  • Paolo Leonetti
  • Giulio Principi

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].

Suggested Citation

  • Paolo Leonetti & Giulio Principi, 2022. "Representations of cones and applications to decision theory," Papers 2209.06310, arXiv.org, revised Jan 2023.
  • Handle: RePEc:arx:papers:2209.06310
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    References listed on IDEAS

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    Cited by:

    1. Paolo Leonetti, 2022. "Expected multi-utility representations of preferences over lotteries," Papers 2210.04739, arXiv.org, revised Jan 2024.

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