Results about sets of desirable gamble sets

TL;DR

This work develops a rigorous framework for coherent sets of desirable gamble sets as a model of uncertainty and choice. It introduces the natural extension Ext as the minimal coherent extension and proves a key coherence property that characterizes Ext, tying finite and (where possible) infinite addition into a unified theory. A central result shows that every coherent gamble-set family can be represented as a filter-like structure of coherent gambles $D$, providing a powerful structural link to classical representations via gambles and probability measures. The paper also situates its approach relative to Debock (2018), clarifies finite vs infinite settings, and presents both D-based and probability-filter representations, offering a versatile foundation for decision-making under uncertainty. Overall, it advances the mathematical understanding of coherence, extension, and representation in sets of desirable gamble sets with potential implications for robust decision theory and imprecise probabilities.

Abstract

Coherent sets of desirable gamble sets is used as a model for representing an agents opinions and choice preferences under uncertainty. In this paper we provide some results about the axioms required for coherence and the natural extension of a given set of desirable gamble sets. We also show that coherent sets of desirable gamble sets can be represented by a proper filter of coherent sets of desirable gambles.

Figures (3)

• Figure 1: Suppose $A_1=\set{g_1^1,g_1^2}\in\mathcal{K}$ and $A_2=\set{g_2^1,g_2^2}\in\mathcal{K}$. Then this reasoning (described in \ref{['property:1']}) gets us that $\set{f_{\langle g_1^1,g_2^1\rangle},f_{\langle g_1^1,g_2^2\rangle},f_{\langle g_1^2,g_2^1\rangle},f_{\langle g_1^2,g_2^2\rangle}}\in\mathcal{K}$.
• Figure :
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Theorems & Definitions (41)

• Definition 1.2
• Proposition 1.3
• Proposition 1.4
• Definition 1.5
• Theorem 2.1
• proof
• proof
• Definition 2.2
• Theorem 2.3
• proof