  In the well-studied (2,2,2) Bell experiment consisting of two parties, two
measurement settings per party, and two possible outcomes per setting, it is
known that if the experiment obeys no-signaling constraints, then the set of
admissible experimental probability distributions is fully characterized as the
convex hull of 24 distributions: 8 Popescu-Rohrlich (PR) boxes and 16 local
deterministic distributions. Here, we refine this result to show that in the
(2,2,2) case, any nonlocal nonsignaling distribution can always be uniquely
expressed as a convex combination of exactly one PR box and (up to) eight local
deterministic distributions. In this representation each PR box will always
occur only with a fixed set of eight local deterministic distributions with
which it is affiliated. This decomposition has multiple applications: we
demonstrate an analytical proof that the minimum detection efficiency for which
nonlocality can be observed is $\eta>2/3$ even for theories constrained only by
the no-signaling principle, and we develop new algorithms that speed the
calculation of important statistical functions of Bell test data. Finally, we
enumerate the vertices of the no-signaling polytope for the (2, n, 2) "chained
Bell" scenario and find that similar decomposition results are possible in this
general case. Here, our results allow us to prove the optimality of a bound,
derived in Barrett et al. [1], on the proportion of local theories in a
local/nonlocal mixture that can be inferred from the experimental violation of
a chained Bell inequality.
