Justified representation

Justified representation is a term advocated by people applying approval voting to proportional representation.

Justified representation is an an extension to the Hare Quota rule for multi-member approval voting systems. There are at least three variants: "justified representation", "extended justified representation" and "proportional justified representation" (though it seems more can be found on the English Wikipedia article: Justified representation). These alternative definitions can be thought of as an alternative to the definition of "proportional representation" for dealing with a representative systems and not a "partisan system".

Justified representation establishes requirements on when a large enough group of voters is justified to have at least one of the candidates they approve elected. Similarly, Extended Justified representation establishes requirements on when a large enough group of agents justified to have to have several of the candidates approved by them elected. Proportional Justified representation is a fix to the earlier definition of Extended Justified representation to be consistent with Perfect representation in the limiting case. The formal definitions can be found in the image. and will be reviewed later in this paper. These definitions are interesting because voting rules that satisfy them guarantee that a large enough group of agents (even if it is a minority of the total agents) will receive at least one (for JR) or at least x (for EJR and PJR) representatives that they approve regardless of any strategic vote followed by the reminder voters.



Justified representation
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 justified representation for (A, k) if there does not exist a subset of voters N∗ ⊆ N with |N∗| ≥ n/k such that the voters all approve of the same candidate not in the winner set and none of the candidates in the winner set. We say that an approval-based voting system satisfies justified representation (JR) if for every profile A = (A1, . . ., An) and every target committee size k it outputs a winning set that provides justified representation for (A, k).

Extended justified representation
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 $$ \ell$$ -justified representation for (A, k) if there does not exist a subset of voters N∗ ⊆ N with |N∗| ≥ $$ \ell$$n/k for a positive integer $$ \ell$$ such that there are at least $$ \ell$$ candidates they all approve and they all approve fewer than $$ \ell$$ winners. W provides extended justified representation (EJR) for (A, k) if it provides $$ \ell$$-JR for (A, k) for all $$ \ell$$, 1 ≤ $$ \ell$$ ≤ k. We say that an approval-based voting rule satisfies $$ \ell$$-justified representation ($$ \ell$$-JR) if for every matrix A and every target committee size k it outputs a committee that provides $$ \ell$$-JR for (A, k). Finally, we say that a voting system satisfies extended justified representation (EJR) if it satisfies $$ \ell$$-JR for all ℓ, 1 ≤ ℓ ≤ k.

Proportional justified representation
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 $$ \ell$$ and a subset of voters N∗ ⊆ N with |N∗| ≥ $$ \ell$$n/k such that there are at least $$ \ell$$ candidates they all approve and fewer than $$ \ell$$ winners any of them approve (emphasis to distinguish from extended justified representation).

Full justified representation
In approval elections, 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 full justified representation for (A, k) if there does not exist a subset of voters N∗ ⊆ N with |N∗| ≥ $$ \ell$$n/k for a positive integer $$ \ell$$ such that there is a subset T ⊆ C of no more than $$ \ell$$ candidates such that for some β ≤ $$ \ell$$, each voter in N∗ approves at least β candidates in T and they all approve fewer than β winners.

In score elections, "each voter in N∗ approves at least β candidates in T" is replaced by "each voter in N∗ gives the candidates in T a total of at least β points (with scores normalized to [0,1])."

Full Justified Representation implies Extended Justified Representation since the requirements are the same for β = $$ \ell$$.

Implications
While approval voting is strategy-proof for voters with dichotomous preferences, every method passing justified representation is susceptible to strategic voting, even in this setting.

Comparison
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. It is easily seen that PJR is a weaker requirement than EJR, and a stronger one than JR. A method satisfying EJR also satisfies PJR, and that a method satisfying PJR also satisfies JR.

Even though Justified representation may appear to be similar to core stability, it is, in fact, a strictly weaker condition. Indeed, the core stability condition appears to be too demanding, as no known voting system is guaranteed to produce a core stable outcome, even when the core is known to be non-empty.

Extension to score systems
Simply applying the Kotze-Pereira transformation will allow for a generalization to cardinal voting systems with greater than 2 gradations.