Justified Representation (COMSOC)

Definition

Justified Representation ( COMSOC)

Let $W\subseteq C$ be a committee of size $k$. The committee $W$ satisfies justified representation (JR) if every sufficiently large group $V^{\prime }\subseteq V$ that agrees on at least one candidate (a cohesive voter group), i.e.,

$$|V^{\prime }|\ge \frac{|V|}{k}\text{and}\bigcap _{v_i\in V^{\prime }}{A}_i\ne \varnothing ,$$

has at least one of its voters represented by committee, i.e.,

$$W\cap \bigcup _{v_i\in V^{\prime }}{A}_i\ne \varnothing .$$

Such a group $V^{\prime }$ is called a cohesive voter group.

Intuition

The condition

$$\bigcap _{v_i\in V^{\prime }}{A}_i\ne \varnothing$$

establishes that the group is sufficiently cohesive: all voters in $V^{\prime }$ agree on at least one candidate.

Unlike strong justified representation, JR does not require one of these commonly approved candidates to be elected. Instead, it only requires

$$W\cap \bigcup _{v_i\in V^{\prime }}{A}_i\ne \varnothing .$$

Hence, at least one voter in the cohesive group must approve at least one elected candidate.

Equivalently,

$$\exists v_i\in V^{\prime }:W\cap A_i\ne \varnothing .$$

Thus, JR guarantees one unit of representation to every sufficiently large cohesive group without requiring the committee to elect a candidate unanimously approved by the entire group