@poppeacock here is another toy example, which shows that while seating may be pluralistic/consensus based, the ultimate power-based representation may be strongly majoritarian.

Example: A “universally approved” compromise seat has ~0 power.

N = 3 seats, 5 candidates, 100 voters
• 40: A > M | B C D (approve {A,M})
• 35: B > M | A C D (approve {B,M})
• 25: C > B > M > A | D (approve {C,B,M,A})

Approvals: M=100, A=65, B=60, C=25 → Seated: {M, A, B}.
Strength:
• 40 A-first → A
• 35 B-first → B
• 25 C-first → among seated {M,A,B} they rank B > M > A → B
Final: A=40, B=60, M=0.

The only universally approved candidate M is seated, but powerless, and the majority coalition has monopolized all legitimate majority-based bargaining power through representative B.

Maybe that’s OK with so few seats? I’m just trying to think through the implications of the system. My guess is that this same kind of power-based majoritarianism may still cause issues even with more seats available. I could be wrong.

However, it’s important to note that the proportionality/plurality/consensus aspect hinges strongly and primarily on the initial seating method. Using approval, for example, makes the seats susceptible to strategic nomination of clones, where a majority faction could dominate all seats by approving many clone candidates.

So the method would need to rely on other more specialized PR-based seating algorithms. In that case, the majoritarian power allocation aspect becomes a questionable design choice, since it seems to potentially undermine the very pluralism that the PR-based seating is meant to achieve.

The majoritarianism becomes less of an issue though if decisions require supermajorities of power.

This is out of my wheelhouse, but possibly this could be addressed by removing candidates with zero power, and enabling runners-up to take their seats. And in the approval case, perhaps also allocating approval to the top-seated candidate. This is a suggested direction for using approval:

Select provisional open seat allocations by approval. If there are approval ties, resolve them by a head-to-head match if possible, otherwise by a predetermined sort order of the candidates. If any voter has their top-ranked candidate provisionally seated, remove all other approval support they give to other candidates, and discount any rankings from their ballot in subsequent head-to-head matches. Allocate provisional power. If there are any filled seats with zero power, completely remove the provisionally seated candidates with zero power from the election and vacate their seats. Repeat from step (1) using the updated approval and ranking profiles, until there is no change in the seating or power allocation.

I think also that one could “smooth” the final power allocations to prevent majoritarianism. If the (max-min)+1 power is added everywhere, then any single representative can swing any power gap between two other disagreeing representatives. It also removes the possibility of a filled seat with zero power.