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Teaching Runoff Reversal — a step-by-step guide

The single question STAR raises most often: "Why didn't the candidate with the most stars win?" This is the presenter's playbook for answering it — the order to teach it in, the examples to use, why a reversal is a good thing, and the hard ("devil's advocate") questions with answers.

The one-sentence answer (lead with this)

Leading the Scoring Round makes you a finalist, not the winner. The winner is whichever of the top two more voters actually prefer — and that's the whole point of the runoff.

Why a reversal is a good thing (the core message)

A big star total can come from intensity or breadth — not from being the candidate the most people prefer head-to-head. The Automatic Runoff asks the majority-protecting question that raw scores can't: of the two finalists, which one do more voters prefer? When those agree (the usual case) the leader wins. When they disagree, majority preference is exactly what should win — otherwise a passionate minority could outvote the majority just by scoring harder. The reversal is the safeguard doing its job, not a glitch.

A bonus: because your runoff vote is set by the scores you already gave, you never have to lowball a rival to protect your favorite. Honest scoring is also optimal strategy.

Two reassurances worth saying out loud. First, when the second-highest scorer wins the runoff it's because they had the stronger base of preference — and in practice that only happens when the two finalists' totals are close, so both are broadly-supported candidates who'd make good winners; the runoff just picks the majority-preferred one. Second, the deeper framing: the Scoring Round is utilitarian (total support) and the runoff is majoritarian (whom more voters prefer). Rather than force a choice between those values, STAR answers "both" — a winner with broad support and a head-to-head majority. (Notably, STAR tends to score better on utility in simulations than plain Score voting, which has only the utilitarian half.)

The two voter patterns that cause a reversal

Almost every reversal is one of these (both are the same lesson — stars measure support, the runoff measures preference):

  1. The narrow-but-deep leader (polarizing). A candidate piles up stars from an enthusiastic minority (lots of 5s) while the majority rates them low or 0. Big total, but loses the head-to-head. (See 02, 05.)
  2. The broadly-liked-but-second-choice leader. A candidate everyone rates well (a high consensus total) but who is most voters' second pick; the majority's actual favorite wins the runoff. (See 01a below.)

Step by step

Step 1 — the atom (3 voters): 01a

Show the ballots, then the result. Everyone likes Almond — but two of the three prefer Brownie:

Almond, Brownie, Cocoa
     5,       1,     2
     4,       5,     0
     4,       5,     0

[Divergence from STAR]
  STAR     = Brownie
  Approval = Almond   (differs from STAR)

[Runoff Reversal]
 - Score Round Winner(s) = (Almond)
 - Runoff Round Winner   = (Brownie)
  Candidate Almond earned the highest total score, but
  Candidate Brownie 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 3 ballots.
Count × Almond,Brownie,Cocoa
    2 ×      4,      5,    0
    1 ×      5,      1,    2

[STAR Voting: Scoring Round]
 The two highest-scoring candidates advance to the next round.
   Almond        -- 13 -- First place
   Brownie       -- 11 -- Second place
   Cocoa         --  2
 Almond and Brownie advance.

[STAR Voting: Automatic Runoff Round]
 The candidate preferred in the most head-to-head matchups wins.
   Brownie       -- 2 -- First place
   Almond        -- 1
   Equal Support -- 0
 Brownie wins.
   Runoff math:
     3  ballots cast
   − 0  Equal Support (no preference between the two finalists)
     ─
     3  voters with a preference  (majority = 2)
           Brownie 2 (67%)  ·  Almond 1 (33%)

[STAR Voting: Winner — STAR Voting Method (single winner)]
 Brownie

Talking point: Almond's 13 is real — it is broadly liked. But when each voter must pick between the two finalists, the majority chose Brownie. Stars found the finalists; the runoff picked the winner.

Step 2 — the control (same machinery, leader CONFIRMED): 04

Immediately defuse "the runoff is rigged against the leader" by showing the same process keep a leader who is most-preferred:

Amber, Blue, Coral, Dune
    2,    5,     1,    0
    2,    5,     0,    1
    5,    4,     0,    0

--- STAR Voting Method (single winner) ---

[STAR Voting]
 Tabulating 3 ballots.
Amber,Blue,Coral,Dune
    2,   5,    1,   0
    2,   5,    0,   1
    5,   4,    0,   0

[STAR Voting: Scoring Round]
 The two highest-scoring candidates advance to the next round.
   Blue          -- 14 -- First place
   Amber         --  9 -- Second place
   Coral         --  1
   Dune          --  1
 Blue and Amber advance.

[STAR Voting: Automatic Runoff Round]
 The candidate preferred in the most head-to-head matchups wins.
   Blue          -- 2 -- First place
   Amber         -- 1
   Equal Support -- 0
 Blue wins.
   Runoff math:
     3  ballots cast
   − 0  Equal Support (no preference between the two finalists)
     ─
     3  voters with a preference  (majority = 2)
           Blue 2 (67%)  ·  Amber 1 (33%)

[STAR Voting: Winner — STAR Voting Method (single winner)]
 Blue

Talking point: the runoff isn't biased against the leader — it just checks the leader. Here Blue leads and is preferred, so Blue wins. The reversal only happens when intensity and majority preference point at different candidates.

Step 3 — it holds at scale, and it's real

  • 01b — the same election with more voters: 67% / 33%. The atom wasn't a small-numbers fluke.
  • 02 — 5 candidates: a high-ceiling favorite (the narrow-but-deep pattern) loses to the broad compromise.
  • 03 (real BV 4c7kp9) and 05 (real BV BV1265) — actual recorded elections; 05 is the one that triggers BetterVoting's own "why is the top scorer different?" popover.

Devil's-advocate questions (with answers)

"Isn't it undemocratic to deny the candidate with the most points?" Points aren't votes for winning — they're how voters describe how much they like each candidate, and they're used to pick the two finalists. The win goes to whichever finalist more voters prefer. Denying that would let a minority's high scores override a majority's preference — that would be undemocratic.

"Doesn't this punish popular candidates / reward being everyone's bland second choice?" No — the runoff is a head-to-head majority. A "bland" candidate only wins if, one on one, more voters genuinely prefer them to the other finalist. If the popular candidate is truly most-preferred, they win (that's the control case).

"Couldn't I game it by scoring strategically?" Your runoff vote is whatever your own scores already imply, so there's no gain from lowballing a rival or burying your second choice. Honest scoring is the best strategy — a property pure-score methods lack.

"Why only the top two? What about third place?" The top two carry the most support, so the decisive question is which of them the majority prefers. If you worry the "right" candidate was a finalist at all, check the Condorcet winner (the candidate who beats everyone head-to-head) in the full report — STAR's runoff winner usually matches it. → three notions of "winner"

"Is the reversal random, or tied to a tie-break?" Neither. It's fully deterministic from the ballots, and it has nothing to do with tie-breaking — a reversal isn't a tie. (Tie-breaking only runs on an exact runoff split; see the flow.)

"Voters will be confused/angry when their leader loses." That's the real risk — so name it and explain it in one breath, the way Explaining Runoff Reversal to Voters does. Confusion comes from an unexplained reversal, not the reversal itself.

Two views of the real cases (BV + LH) — assets to add

For the recorded races (03 = BV 4c7kp9, 05 = BV BV1265), the strongest proof is the same election shown both ways — BetterVoting's screenshot beside the LH report. To complete this (mirrors the pattern in pet_real_bv_election/):

  1. Screenshot each BV result and save under runoff_overturns_leader/img/ (e.g. img/REPLACE_4c7kp9_bv_result.png, img/bv1265_bv_result.png).
  2. Export each BV JSON, drop it here, and convert it to YAML (01_convert_json_yaml.py) — the BV JSON → YAML → LH report flow — so the ballots and both tabulations sit side by side.

See also: the full audit report for each case in runoff_overturns_leader_tabulated/, and how often reversals occur in 06_Other/simulations/runoff_reversal_simulation.py.