Strong Justified Representation (COMSOC)

Definition

Strong Justified Representation ( COMSOC)

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

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

must have one of its unanimously approved candidated elected, i.e.,

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

Why It Can Be Too Strong

Strong justified representation requires every sufficiently large cohesive group to have a commonly approved candidate in the committee.

A group $V^{\prime }\subseteq V$ is relevant whenever

$$|V^{\prime }|\ge \frac{|V|}{k}$$

and

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

For every such group, SJR requires

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

This can be impossible to satisfy: different sufficiently large groups may overlap while having different sets of commonly approved candidates. They can therefore impose more distinct representation requirements than there are $k$ available seats.

Hence, a committee satisfying strong justified representation need not exist.

This motivates weakening the requirement to justified representation, where a cohesive group only needs at least one of its voters to be represented, rather than requiring a candidate approved by every voter in the group.