Interpreting, axiomatising and representing coherent choice functions in terms of desirability
This work addresses foundational issues in imprecise-probabilistic decision making for researchers in AI and decision theory, but it is incremental as it builds on existing frameworks.
The paper tackled the problem of modeling choice under uncertainty with choice functions by providing a desirability interpretation, deriving coherence axioms, and showing representation via sets of strict preference orders, with additional properties leading to representations like lexicographic probability systems.
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that appear in imprecise-probabilistic decision making. We provide these choice functions with a clear interpretation in terms of desirability, use this interpretation to derive a set of basic coherence axioms, and show that this notion of coherence leads to a representation in terms of sets of strict preference orders. By imposing additional properties such as totality, the mixing property and Archimedeanity, we obtain representation in terms of sets of strict total orders, lexicographic probability systems, coherent lower previsions or linear previsions.