When the STAR-PR methods disagree

Allocated Score, SSS and RRV usually elect the same slate — which makes it easy to assume the choice between them is academic. It isn't. Two elections of three voters each are enough to split all three apart, and the seat turns on one thing: what a winner costs the voters who backed them.

→ The methods themselves: Allocated Score · SSS · RRV · the hub: the three STAR-PR methods · the theory: the math behind proportional STAR

Level: 301 · deep dive

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The two elections

Both are 3 candidates, 3 voters, 2 seats — the smallest size at which any of this can happen. In both, seat 1 is unanimous across the methods and the entire argument is about seat 2.

Election	Allocated	SSS	RRV	What it separates
Three neighbors	Bo, Cleo	Ana, Cleo	Ana, Cleo	The two quota methods, from each other
Two officers	Dana, Finn	Dana, Finn	Dana, Eli	The quota family, from the divisor one

Between them every pair disagrees somewhere: Allocated ≠ SSS and Allocated ≠ RRV in the first, RRV ≠ SSS and RRV ≠ Allocated in the second.

Why they can disagree at all

All three fill seat 1 the same way — elect the highest scorer. They differ only in what happens next, and there are exactly two decisions to make:

1. Do you spend ballot weight, or divide it? Allocated Score and SSS are quota methods: they charge a seat's cost against a quota's worth of ballot weight and use those voters up. RRV is a divisor method: nobody is ever used up, every ballot is simply divided by 1 + (score given to winners / max score). Two officers is that difference, isolated — the same two voters land at weight 0.25 under the quota methods and 0.556 under RRV. Exhausted versus merely turned down, and it changes the seat.

2. If you spend, do you charge by tier or in proportion? Allocated Score sorts a winner's supporters into score groups and allocates them group by group — every 5-star backer first, then every 4-star backer — until the quota is full, splitting only the last group. SSS charges every supporter at once, each paying in proportion to the score they gave. Three neighbors is that difference, isolated: a lukewarm 2-star supporter is never reached by the tiers, so Allocated Score leaves her at full weight and she picks seat 2 by herself.

That is the whole taxonomy, and it is why the classification in what "proportional" actually means is load-bearing rather than bookkeeping: quota methods pass the Hare Quota Criterion and pay for it with the non-monotonicity that produces the Alabama paradox; RRV is coherent and monotone and pays for that by not guaranteeing a quota.

How these were found

Both were located by exhaustive smallest-first search over random 0–5 profiles, walking up from 3 ballots, and neither involves a tie or a lot. Each candidate profile was tabulated under five different tiebreaker seeds and discarded unless all five agreed — so what you are looking at is a property of the methods, not a coin toss. The hand-arithmetic in each page reproduces the engine's weights exactly.

Worth stating plainly: these are constructed minimal examples, not observed elections. They prove the disagreement is real and show its mechanism at a size you can check by hand; they say nothing about how often it happens on realistic ballots. On this library's larger PR cases the three methods usually agree — the 63-ballot baseline has all three electing Alice, Ben and Dan.

Bigger, and not minimal

One more divergence, on a real academic profile rather than a constructed one — the Lackner & Skowron shadow election, 7 candidates, 12 ballots, 4 seats:

Method	Winners
Allocated Score	A, B, C, D
RRV	A, B, C, F

Three seats agree, the fourth doesn't — the same quota-versus-divisor split as Two officers, at a size where hand-checking stops being practical.

The cases

Case	Method	Winners	Read · run
Three neighbors	allocated	Bo, Cleo	count · yaml
Three neighbors	sss	Ana, Cleo	count · yaml
Three neighbors	rrv	Ana, Cleo	count · yaml
Two officers	allocated	Dana, Finn	count · yaml
Two officers	sss	Dana, Finn	count · yaml
Two officers	rrv	Dana, Eli	count · yaml