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How often do STAR and Approval disagree?

A common question with a deceptively honest answer: there is no single number — and the reason why is the actual lesson. This page measures the divergence rate (with a runnable, seeded simulation), explains why the two methods split, and links the worked elections where you can watch it happen on real ballots.

Level: 301 · deep dive Companion: What makes a good winner? · election simulation models · the worked cases in Black Curtain.

Why there's no single rate

STAR has a canonical sincere ballot: min-max your feelings onto 0–5. Approval does not — every voter must choose a 0/1 cutoff, and where they draw it changes the winner. So the divergence rate depends on two modelling choices, not one:

  1. the electorate model (how voters' preferences are generated), and
  2. the approval-cutoff rule (how a voter turns feelings into approvals).

Quote a divergence percentage without both and it's meaningless — the same house rule the repo applies to every simulated number (301.7/301.10; simulations README).

The measured rates — the approval cutoff is the knob

Because Approval has no canonical sincere ballot, the honest way to report this is to sweep the cutoff. Read each voter's 0–5 STAR ballot as approvals at threshold N (approve everything scored ≥ N): ge5 = approve only your top (near-bullet), ge1 = approve anyone but your worst. From star_vs_approval_divergence.py (20,000 elections, 51 voters, seed 12345, sincere):

3 candidates — divergence by where the approval line is drawn:

The two columns are two assumptions about the electorate. Spatial (realistic) is the issue-space model — voters and candidates are points on a map, you prefer whoever's closest; real electorates have this structure (a center), so paradoxes are relatively rare, and this column ≈ what you'd actually expect. Impartial (stress) is impartial culture — every preference independent and random, no structure at all; it manufactures far more disagreement than reality and is a deliberate worst-case ceiling, not a prediction. The truth for a real electorate sits between the two, much closer to spatial — which is why both are shown rather than one number.

Approve scores ≥ Spatial (realistic) Impartial (stress)
5 — only your top (near-bullet) 14% 22%
4 10% ← most STAR-like 22%
3 — top half of the scale 12% 23%
2 20% 24%
1 — anyone but your worst 27% 27%

Four findings:

  • The cutoff is the dominant knob under a realistic electorate. Just moving the approval line swings divergence from ~10% to ~27% (3 candidates); at 4 candidates it's ~15% to ~39%. So "how often do STAR and Approval disagree?" has no answer until you say where voters approve — which is the whole point.
  • It's non-monotonic, with a sweet spot. Agreement is highest at a moderate cutoff (approve your top ~half, ge3–ge4) and worst at the generous extreme (ge1): approving everyone but your worst discards nearly all intensity and collapses Approval toward "least-hated" — exactly where it parts from STAR's majority-preferred runoff winner.
  • More candidates → more divergence, at every cutoff.
  • The electorate model still matters (impartial > spatial), but under the realistic model the cutoff moves the number more than the model does. (An earlier version of this page tested only two similar cutoffs and wrongly concluded the cutoff barely mattered — the sweep corrects that.)

A fair one-liner: with a sensible sincere cutoff (approve your top half) in realistic 3–5 candidate elections, STAR and Approval agree ~80–90% of the time; loosen the cutoff toward "approve almost everyone" and they part far more often — because that's when Approval stops measuring intensity. When they split, it's the intensity-vs-breadth disagreement below.

Why they diverge (when they do)

Both methods reward broad support — that's why they agree most of the time. They split on the intensity-vs-breadth axis:

  • STAR reads intensity and runs a runoff. Its 0–5 ballot records how much, and the automatic runoff then asks how many prefer A to B among the two strongest. So STAR leans toward the candidate a majority actively prefers head-to-head.
  • Approval is a coarse 0/1 and has no runoff. It elects whoever is acceptable to the most voters — the broadest common denominator — and can't distinguish "approve enthusiastically" from "approve as a tolerable compromise."

So the classic split is: a broadly-tolerable consensus candidate (Approval's pick) vs. an intensely-preferred majority favorite (STAR's pick, surfaced by the runoff). Neither is "wrong" — it's the good-winner question (utilitarian-broad vs. majoritarian) showing up as a method disagreement.

Watch it on real ballots — worked examples

The Black Curtain set is exactly this, made countable: the same small electorate counted by Approval, STAR (and RCV-IRV / Range), so you can see where they agree and where they part:

  • Hidden consensus — a broadly-liked compromise that Approval elects and STAR's runoff can pass over.
  • Near-clones and polarized variants — how the split appears (or doesn't) as the preference structure changes.
  • Range vs the rest — the same ballots under Score, for contrast.

For the single-election view in the engine, any Approval or STAR file prints a [Divergence from STAR] block when the methods disagree (divergence review ledger).

Run it yourself

python3 06_Other/simulations/star_vs_approval_divergence.py                       # default: sweep score cutoffs ge5..ge1
python3 06_Other/simulations/star_vs_approval_divergence.py --candidates 5 --voters 101
python3 06_Other/simulations/star_vs_approval_divergence.py --cutoffs mean midpoint   # utility cutoffs instead
python3 06_Other/simulations/star_vs_approval_divergence.py --selftest            # known-answer checks

The --cutoffs parameter is the one that matters: geN reads the STAR ballot as approvals at threshold N (the sweep above); mean/midpoint are utility-based cutoffs. Sweeping it is how you see that the "divergence rate" is a function of the approval rule, not a fixed fact.

It reuses the electorate models and STAR tabulator from fbc_simulation.py, is fully seeded (reproducible), and self-tests its tabulators (a known Approval count and a constructed STAR≠Approval divergence) before every run. Everything it reports carries its model and cutoff — because, as above, the number is meaningless without them.

The honest caveats

  • Sincere only. This measures sincere ballots. Real Approval voters adjust their cutoff to the race (polls, frontrunners), which is a different study.
  • Model-bounded. Spatial ≈ realistic; impartial culture ≈ adversarial upper bound. The truth for any real electorate sits between and depends on its actual preference structure.
  • Tie rules fixed. STAR and Approval ties are broken deterministically (documented in the script); a different tie rule would nudge the edges.