Resolutions of Probabilistic Choice Spaces

Resolutions of choice spaces capture hierarchical processes of choice. Cantone, Giar-lotta and Watson (2021) formalize the concept in the setting of deterministic choice. We extend the notion of a resolution to probabilistic choices. Our main result is about the random utility model, also known as the multiple choice model (MCM, characterized by Falmagne in 1978). It states that any resolution of probabilistic choices satisfying the MCM also satisfies the MCM. We provide three proofs, each offering specific insights. In contrast, Luce model (1959) of probabilistic choice is not preserved by resolutions. A probabilistic choice is hierarchical if it is equal to the outcome of some nontrivial resolution. Our second result is a simpler characterization of such choices, as those admitting a shrinkable subset. Here a shrinkable subset is a collection of elements that are, intuitively, indistinguishable from outside the collection. Our final result establishes that special ‘single-point’ resolutions suffice to generate all resolutions.