Multiwinner voting
Multiwinner,[1] at-large, or committee[2][3] voting refers to electoral systems that elect several candidates at once. Such methods can be used to elect parliaments or committees.
Excellence means that the committee should contain the "best" candidates. Excellence-based voting rules are often called screening rules.[18] They are often used as a first step in a selection of a single best candidate, that is, a method for creating a shortlist. A basic property that should be satisfied by such a rule is committee monotonicity (also called house monotonicity, a variant of resource monotonicity): if some k candidates are elected by a rule, and then the committee size increases to k+1 and the rule is re-applied, then the first k candidates should still be elected. Some families of committee-monotone rules are:
The property of committee monotonicity is incompatible with the property of stability (a particular adaptation of Condorcet's criterion): there exists a single voting profile that admits a unique Condorcet set of size 2, and a unique Condorcet set of size 3, and they are disjoint (the set of size 2 is not contained in the set of size 3).[18]
On the other hand, there exists a family of positional scoring rules - the separable positional scoring rules - that are committee-monotone. These rules are also computable in polynomial time (if their underlying single-winner scoring functions are).[1] For example, k-Borda is separable while multiple non-transferable vote is not.
Diversity means that the committee should contain the top-ranked candidates of as many voters as possible. Formally, the following axioms are reasonable for diversity-centered applications: