I've been on the EndFPTP Reddit posting on this subject and I got ChatGPT to formalise the PRIL criterion. I'll copy my post from there wholesale:
On the formalisation of PRIL, I put to ChatGPT my idea of using the var-Phragmén metric and the concept of loads and told it to put the criterion in formal notation. PRIL is method agnostic though and could be used in conjunction with other methods that aim for Perfect Representation like leximax-Phragmén or Monroe. Anyway, here is what ChatGPT came up with. I make no guarantees an error hasn't slipped in:
Formal Definition of PRIL Using var-Phragmén:
Consider an election with: A set of voters N = {1, 2, . . . , n}.
A set of candidates C = {c1, c2, . . . , cm}.
Each voter i ∈ N has an approval ballot Ai ⊆ C.
A target number of winners k.
In the context of the var-Phragmén method:
Each voter i is assigned a "load" ℓi representing their share in the election of the chosen committee.
The PRIL criterion can be formalized as follows:
For any arbitrarily small positive number ϵ>0, there exists a number of winners k such that for all k′ ≥ k, the normalized variance of the voter loads satisfies:
Var (ℓ) / k′ < ϵ
This condition ensures that as the number of elected candidates k′ increases, the distribution of voter loads becomes increasingly uniform, approaching perfect representation in the limit. In essence, PRIL requires that for sufficiently large committees, the method should allocate representation so evenly among voters that the per-candidate variance of their loads becomes arbitrarily small, reflecting an ideal proportional representation.
