Scale granularity can flip the winner (a 301 case)¶
A score ballot's resolution — how many rungs the scale has — is a modeling choice, not a neutral detail. Compressing a 0–9 ballot down to STAR's 0–5 can change who becomes a finalist, and through that, who wins. This page works one real example where it does exactly that.
→ Companions: the fidelity ladder · scores vs. ranks · Black Curtain lesson #5 (the 0–9 → 0–5 problem) in The Black Curtain — one electorate, four "identical" landslides · Curriculum: 301.1 (RRV/proportional) and 301.7 (when Score/Runoff/Condorcet disagree)
Files: rrv_sample_c15_b13_three-parties.yaml · full report …_tabulated.txt
The one idea¶
The fidelity ladder is about converting between scores and ranks. This is a subtler cousin: staying entirely within scores, but changing the scale's granularity — 0–9 down to 0–5. Rescaling is monotone (it never reorders one voter's own preferences), so you'd expect the winner to be safe. It usually is. But STAR picks its two finalists by summed score, and when several candidates are bunched in a near-tie for that second seat, quantization can nudge a different one into the runoff — and a different finalist can win.
The case¶
The example is BetterVoting.org's Reweighted Range Voting sample election — three "parties" (Purple, Orange, Yellow), five candidates each, 13 voters mostly loyal to a party (3 Purple, 6 Orange, 4 Yellow), grading on 0–9. RRV is a proportional multi-winner method; here we instead run the same ballots through single-winner STAR to ask who one seat would go to. Orange has the most loyal voters (6 of 13), so — no surprise — an Orange candidate wins. The surprise is which one.
On STAR's native 0–5 scale → Orange5¶
Mapping the source 0–9 scores to 0–5 with round(x·5/9), the LH engine reports:
[Divergence from STAR]
STAR = Orange5
Choose-One (Plurality) = Orange1 (differs from STAR)
RCV-IRV = Orange1 (differs from STAR)
Approval = Orange1 (differs from STAR)
Note: 13 of 13 ballots (100%) had equal non-zero scores, so their ranks
were decided by candidate priority order. The RCV-IRV result may be
an artifact of score-to-rank tie-breaking rather than a deep
difference.
Note: Ranked Robin (RCV-RR) agrees with STAR, so RCV-IRV is the lone
outlier — the classic center-squeeze signature.
Full round-by-round reports (generated for review):
RCV-IRV rounds: cases_tabulated/rrv_sample_c15_b13_three-parties_RCV-IRV_tabulated.txt
[Runoff Reversal]
- Score Round Winner(s) = (Orange1)
- Runoff Round Winner = (Orange5)
Candidate Orange1 earned the highest total score, but
Candidate Orange5 won the automatic runoff — not a malfunction,
STAR working as designed: the runoff elects the finalist preferred
by the majority (of voters with a preference).
--- STAR Voting Method (single winner) ---
[STAR Voting]
Tabulating 13 ballots.
Purple1,Purple2,Purple3,Purple4,Purple5,Orange1,Orange2,Orange3,Orange4,Orange5,Yellow1,Yellow2,Yellow3,Yellow4,Yellow5
5, 4, 5, 4, 5, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0
5, 4, 4, 5, 5, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0
4, 5, 5, 2, 4, 1, 1, 1, 1, 2, 1, 0, 0, 1, 0
0, 0, 0, 0, 0, 5, 5, 4, 5, 5, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 5, 1, 4, 4, 4, 0, 0, 1, 0, 0
0, 0, 0, 0, 0, 5, 3, 4, 4, 2, 0, 1, 0, 0, 1
0, 0, 0, 0, 0, 5, 4, 4, 4, 4, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 4, 5, 5, 5, 5, 0, 1, 0, 0, 1
0, 0, 0, 0, 0, 4, 5, 5, 5, 5, 0, 0, 0, 0, 0
1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 5, 5, 2, 5, 5
1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 4, 4, 4, 4, 4
1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 5, 4, 4, 5, 4
2, 0, 2, 2, 1, 0, 0, 0, 0, 0, 5, 5, 4, 4, 5
[STAR Voting: Scoring Round]
The two highest-scoring candidates advance to the next round.
Orange1 -- 31 -- First place
Orange5 -- 30 -- Second place
Orange3 -- 29
Orange4 -- 29
Orange2 -- 25
Yellow2 -- 21
Yellow1 -- 20
Yellow4 -- 20
Yellow5 -- 20
Purple1 -- 19
Purple3 -- 19
Purple5 -- 17
Purple2 -- 16
Purple4 -- 16
Yellow3 -- 15
Orange1 and Orange5 advance.
[STAR Voting: Automatic Runoff Round]
The candidate preferred in the most head-to-head matchups wins.
Orange5 -- 4 -- First place
Orange1 -- 3
Equal Support -- 6
Orange5 wins.
Runoff math:
13 ballots cast
− 6 Equal Support (no preference between the two finalists)
──
7 voters with a preference (majority = 4)
Orange5 4 (57%) · Orange1 3 (43%)
[STAR Voting: Winner — STAR Voting Method (single winner)]
Orange5
Note how tight the score round is: Orange1 31, Orange5 30, Orange3 29, Orange4 29 — four co-partisans inside a two-point band. Orange5 edges into the second finalist slot, then wins the runoff 4–3.
On the original 0–9 scale → Orange1¶
Summing the raw 0–9 ballots instead (the engine only accepts 0–5, so this is counted directly from the source scores):
| Orange candidate | 0–9 total |
|---|---|
| Orange1 | 54 |
| Orange3 | 53 |
| Orange5 | 53 |
| Orange4 | 52 |
| Orange2 | 45 |
Now Orange3 and Orange5 are exactly tied at 53 for the second seat, and the tie-break sends Orange3 to the runoff — where Orange1 beats Orange3, 4–3. So on the source scale the winner is Orange1, not Orange5.
Why the winner moved¶
Nothing about any voter's order changed — rescaling is monotone. What changed is who came second in the score round:
- On 0–9, Orange3 and Orange5 tie at 53; the second finalist is decided by a tiebreak (→ Orange3).
- On 0–5, the
round(x·5/9)compression separates them — Orange5 lands at 30, Orange3 at 29 — so Orange5 takes the slot outright.
And the two candidates behave differently in the runoff: Orange5 beats Orange1, but Orange3 loses to Orange1. Swap the finalist and you swap the champion. The runoff round is not where the divergence is born — it's the finalist selection upstream, made fragile by a cluster of near-tied co-partisans.
The lesson (and the honest caveats)¶
- Resolution is a design decision. How many rungs your ballot offers (0–5, 0–9, 0–10…) is part of the method. When the top contenders are bunched, that choice can pick the winner. This is the concrete, winner-flipping version of Black Curtain lesson #5, where a finer scale merely preserved a pure-Score near-tie; here it moves the STAR result.
- It lives on a knife's edge — say so. The 0–9 outcome rests on a 2nd-place tie (53–53) resolved by tiebreak; the 0–5 outcome rests on a one-point gap. This is a fragile divergence, not a robust law — exactly the kind of case where you should report the assumptions, never just the winner (the methodological habit of 301.7).
- It's mapping-dependent. A different 0–9 → 0–5 rule (floor, ceiling, a custom table) could land differently. The fragility is the point: when a race is this close, small modeling choices are decisive.
- Both winners are Orange. The majority party still wins either way — STAR faithfully hands the seat to the 6-voter bloc. What's unstable is which co-partisan, which is precisely the internal choice a proportional method like the source RRV would instead resolve by seating several of them.
Takeaway¶
When you port a real election onto a coarser score scale to teach or tabulate it, check whether the finalists are bunched. If they are, treat the winner as scale-sensitive and present both counts — as we do here — rather than quoting one number as if the scale were free.
Terminology: the source method is RRV (Reweighted Range Voting), a proportional multi-winner method; this page runs its ballots through single-winner STAR purely to expose the scale effect. See GLOSSARY.md.