Perfect representation

A winner set W provides perfect representation for a group of n voters and a total seat size k if n = ks for some positive integer s and the voters can be split into k pairwise disjoint groups ''N1,. . ., Nk of size s each in such a way that there is a one-to-one mapping μ : W → {N1,. . ., Nk} such that for each candidate a ∈ W all voters in μ(a) approve a''.

In other words, in a multi-winner approval-voting election, for a given set of ballots cast, if there is a possible election result where candidates could each be assigned an equal number of voters where each voter has approved their assigned candidate and no voter is left without a candidate, then for a method to pass the perfect representation criterion, such a result must be the actual result.

Consider the following election with two winners, where A, B, C and D are candidates, and the number of voters approving each candidate are as follows:

100 voters: A, B, C

100 voters: A, B, D

1 voter: C

1 voter: D

A method passing the perfect representation criterion must elect candidates C and D despite near universal support for candidates A and B. This could be seen as an argument against perfect representation as a useful criterion.

Every winner set that provides perfect representation also provides Proportional Justified Representation. In contrast, Extended Justified Representation may rule out all winner sets that provide perfect representation.