Three Notions of "Winner" — Condorcet, Score, and Runoff¶

One line: the word winner isn't one thing. In a single election, three reasonable definitions can point at three different candidates — and STAR's job is to choose among them on purpose, not by accident.

→ Related: the simpler two-way case, Runoff Reversal (score leader ≠ STAR winner). Glossary: Condorcet winner.

The three notions¶

Condorcet winner — the candidate who beats every other candidate one-on-one. The "no one would rather have someone else, head-to-head" winner.
Score (utility) winner — the candidate with the most total stars. The "highest overall enthusiasm" winner.
Runoff winner (= the STAR winner) — the candidate the majority prefers between the two finalists. STAR uses the Scoring Round to pick the two finalists, then the Automatic Runoff to choose between them.

When these three agree — which is common — STAR, score, and Condorcet all name the same person and there's nothing to discuss. The lesson is what happens when they don't.

One election, three winners¶

→ three_winners_cw_score_runoff.yaml — five voters, three candidates:

  Ann  Bob  Carl
   5    4    3      ×3   (love Ann, like Bob, ok Carl)
   0    3    5      ×2   (love Carl, ok Bob, reject Ann)

  Total stars:  Ann 15   Bob 18   Carl 19

Score winner — Carl (19 stars). His two devotees max him out; he edges Bob and Ann on raw totals.
Condorcet winner — Ann. Head-to-head she beats Bob 3–2 and beats Carl 3–2. More voters prefer Ann than any rival, one-on-one.
STAR winner — Bob. The Scoring Round sends the top two — Carl (19) and Bob (18) — to the Automatic Runoff. There, 3 of 5 prefer Bob to Carl, so Bob wins 3–2. Bob is the broad compromise the majority actually lands on once the field is narrowed to two.

(All three are confirmed by the engine, which prints Condorcet = Ann (differs from STAR) and Carl earned the highest total score, but Bob won the automatic runoff.)

Why isn't this a bug?¶

Each method optimizes a different, defensible goal, so disagreement is information, not error:

Score rewards intensity — but intensity alone can let a candidate adored by a minority outrank a candidate liked by everyone.
Condorcet rewards pairwise dominance — but a Condorcet winner can be almost nobody's favorite, and the criterion ignores how much voters prefer one to another.
STAR splits the difference on purpose: scores find the finalists, a majority runoff picks the winner. It captures strength of support and applies a majority check — at the deliberate cost of not always electing the Condorcet winner.

Here Ann (Condorcet) is shut out only because she's nudged into third by total score and never reaches the runoff. That's the design tradeoff in one picture: STAR is not a Condorcet method, and it doesn't pretend to be — it's a score-then-majority method, and Bob is exactly who that procedure is meant to elect.

This is exactly where STAR and Ranked Robin part ways. On these same ballots, Ranked Robin — a Condorcet method — elects Ann (the head-to-head winner), while STAR elects Bob (the score-then-runoff winner). "RR = Condorcet winner, STAR = the runoff winner" is the whole STAR-vs-RR fault line, and it's the preference-vs-support split: RR rewards order, STAR rewards how much support each candidate has. A gallery of 30 more elections where this happens (cycles and dark-horse Condorcet winners, each with its cause spelled out): STAR vs Ranked Robin — divergence samples. The full list of criteria STAR gives up (with receipts): Every criterion STAR fails.

The takeaway for a debate¶

When someone says "but STAR didn't pick the Condorcet winner!", the honest answer isn't a dodge — it's: correct, and on purpose. STAR trades guaranteed Condorcet compliance for expressiveness plus a majority runoff. This little five-voter election is the cleanest way to show all three notions of "winner" at once, and to be precise about which one STAR targets.

Sources: Condorcet criterion (Wikipedia), Utilitarian vs. majoritarian social choice (Wikipedia).