Justified representation: Difference between revisions

Given a matrix of approval votes A where each column represents a candidate c ∈ C and each row represents a voter v ∈ V. A winner set of candidates, W, of size |W| = k provides '''<math> \ell</math> -justified representation''' for (A, k) if there does not exist a subset of voters N∗ ⊆ N with |N∗| ≥ <math> \ell</math>n/k for a positive integer <math> \ell</math> such that there are at least <math> \ell</math> non-winnerscandidates they all approve and they all approve fewer than <math> \ell</math> winners. W provides '''extended justified representation''' (EJR) for (A, k) if it provides <math> \ell</math>-JR for (A, k) for all <math> \ell</math>, 1 ≤ <math> \ell</math> ≤ k. We say that an approval-based voting rule satisfies '''<math> \ell</math>-justified representation''' (<math> \ell</math>-JR) if for every matrix A and every target committee size k it outputs a committee that provides <math> \ell</math>-JR for (A, k). Finally, we say that a voting system satisfies '''extended justified representation''' (EJR) if it satisfies <math> \ell</math>-JR for all ℓ, 1 ≤ ℓ ≤ k.

Given a matrix of approval votes A where each column represents a candidate c ∈ C and each row represents a voter v ∈ V. A winner set of candidates, W, of size |W| = k provides '''proportional''' '''justified representation''' for (A, k) if there does not exist a positive integer <math> \ell</math> and a subset of voters N∗ ⊆ N with |N∗| ≥ <math> \ell</math>n/k such that there are at least <math> \ell</math> non-winnerscandidates they all approve and ''fewer than <math> \ell</math> winners any of them approve'' (emphasis to distinguish from extended justified representation).