  Recently the Local Friendliness (LF) no-go theorem has gained a lot of
attention, owing to its deep foundational implications. This no-go theorem
applies to scenarios which combine Bell experiments with Wigner's friend-type
set ups, containing space-like separated superobservers who are assumed to be
capable of performing quantum operations on a local observer, also known as
their "friend". Analogously to the hypothesis of local hidden variables in Bell
scenarios, a set of assumptions termed "Local Friendliness" constrains the
space of probabilistic behaviours accessible to the superobservers to be a
particular subset of the no-signalling polytope in such scenarios. It has
additionally been shown, that there are scenarios where the set of behaviours
compatible with Local Friendliness is strictly larger than the Bell-local
polytope, while in some scenarios those sets are equal. In this work, we
complete the picture by identifying all the canonical Local Friendliness
scenarios, with arbitrary but finite numbers of superobservers, friends,
measurements and outcomes, where the set of LF correlations admits a local
hidden variable model, and where they do not. Our proof is constructive in the
sense that we also demonstrate how a local hidden variable model can be
constructed, given a behaviour compatible with LF in the appropriate scenarios.
While our principal motivation is the foundational question of better
understanding the constraints from Local Friendliness, the same inequalities
constraining LF polytopes have been shown to arise in a priori unrelated
contexts of device-independent information processing. Our results may thus
find use in those research areas as well.
