The notion of a θ -winning set provides a natural way to define two families of more decisive voting correspondences using the same approach: First, we can pick a θ ∈ (0, 1] and output all candidates that belong to some minimal (or minimum-size) θ-winning set. Alternatively, we can fix the desired size of the winning set (say, k), find the largest value of θ such that there exists a θ-winning set of size k, and output all candidates that belong to some such set. A variant of the latter approach is to pick one θ -winning set of size k according to some tie-breaking rule, thus obtaining a committee selection r ule (see Sect. 7)