STAR's runoff is spoiler-sensitive under a Condorcet cycle¶

Method: STAR (single winner) · 1 seat · Expected winner: Alice · full count →

▶ Live on BetterVoting: vote · results ↗ (election g3f7r2, Test ID BV2212).

One line: when the head-to-head results form a Condorcet cycle, a candidate who cannot win can still decide who does — because their presence changes which two candidates meet in STAR's automatic runoff. This is STAR's honest Independence-of-Irrelevant-Alternatives (IIA) limit, shown mechanically, on perfectly sincere ballots.

This is the sharpest fair criticism of STAR from the r/EndFPTP thread that prompted it (user Excellent_Air8235) — not a strawman, and worth teaching honestly alongside STAR's strengths. See also the broader STAR — Honest Limits.

The setup — 23 voters, three rotating factions, all sincere¶

Three candidates (Alice, Ben, Carla) and three blocs, each voting its true preference — favorite 5, compromise 3, last choice 0. No strategy anywhere.

Count	Alice	Ben	Carla	Sincere order
10	5	3	0	Alice > Ben > Carla
6	0	5	3	Ben > Carla > Alice
7	3	0	5	Carla > Alice > Ben

The three factions rotate, so every candidate beats one rival head-to-head and loses to the other — a Condorcet cycle with no Condorcet winner:

Alice ▸ Ben (17–6)
Ben ▸ Carla (16–7)
Carla ▸ Alice (13–10)

What STAR does — Alice wins¶

Scoring round totals Alice 71, Ben 60, Carla 53, so the finalists are Alice and Ben, and Alice takes the runoff 17–6. STAR elects Alice — cleanly, with no tie-break (BetterVoting reports tieBreakType: none, and BV's own count agrees: elected = Alice).

The IIA failure — delete Ben, and Carla wins¶

Ben finishes second and loses the runoff — he never had a path to victory. Remove him from the ballot and change nothing else:

Count	Alice	Carla
10	5	0
6	0	3
7	3	5

Now the finalists are Alice (71) and Carla (53), and the runoff flips: Carla beats Alice 13–10, so STAR elects Carla.

A losing candidate (Ben) determined whether Alice or Carla wins — with not one voter changing a single score. STAR's ballot is cardinal, but its finish is ordinal (a top-two runoff), and inside a Condorcet cycle the choice of finalists is itself sensitive to a third candidate who can't win. That's the IIA failure, located precisely in the runoff stage.

It is a real, honest cost — and note what it is not: it needs no dishonest ballots (these are fully sincere), and it only bites when the electorate has no Condorcet winner at all (a genuine three-way cycle). When a Condorcet winner exists, STAR's runoff can't be spoiled this way.

Bonus — one election, three different winners¶

The same 23 sincere ballots, counted three ways, split every method apart:

Method	Winner	Why
STAR	Alice	Finalists Alice (71) & Ben (60); Alice wins the runoff 17–6
RCV-IRV	Carla	Ben has the fewest first choices, is eliminated, and his 6 ballots transfer to Carla (13 vs Alice's 10)
Ranked Robin (Copeland)	Alice	Every candidate is 1–1 in the cycle; LH breaks the Copeland tie by margin → Alice's +11 win is largest

Because the Ranked Robin result is a tie-break, it is not freezable on BetterVoting — LH breaks the Copeland tie deterministically (by margin), but BetterVoting breaks it randomly, so a live BV RR race here couldn't be reproduced. This case therefore ships STAR-only on BetterVoting (same reasoning as BV830). The RR angle stays LH-side commentary.

View 1 — BetterVoting¶

Live results: bettervoting.com/g3f7r2/results. BetterVoting's STAR count agrees with LH exactly — elected: Alice, finalists Alice (71) & Ben (60), runoff 17–6, tieBreakType: none. The frozen export (Election + Ballots + Results) is bv2212_g3f7r2_cycle_spoiler_bv_export.json; its winsAgainst data encodes the same cycle (Alice▸Ben, Ben▸Carla, Carla▸Alice).

View 2 — the LH tabulation (on-screen report)¶

[Divergence from STAR]
  STAR    = Alice
  RCV-IRV = Carla   (differs from STAR)
  Note: no ballots had tied scores, so RCV-IRV vs STAR here is a genuine
        method difference, not a tie-breaking artifact.
  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/bv2212_g3f7r2_cycle_spoiler_RCV-IRV_tabulated.txt

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

[STAR Voting]
 Tabulating 23 ballots.
Count × Alice,Ben,Carla
   10 ×     5,  3,    0
    7 ×     3,  0,    5
    6 ×     0,  5,    3

[STAR Voting: Scoring Round]
 The two highest-scoring candidates advance to the next round.
   Alice         -- 71 -- First place
   Ben           -- 60 -- Second place
   Carla         -- 53
 Alice and Ben advance.

[STAR Voting: Automatic Runoff Round]
 The candidate preferred in the most head-to-head matchups wins.
   Alice         -- 17 -- First place
   Ben           --  6
   Equal Support --  0
 Alice wins.
   Runoff math:
     23  ballots cast
   −  0  Equal Support (no preference between the two finalists)
     ──
     23  voters with a preference  (majority = 12)
           Alice 17 (74%)  ·  Ben 6 (26%)

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

Full engine detail: iia_cycle_spoiler_tabulated/bv2212_g3f7r2_cycle_spoiler_tabulated.txt · run it: bv2212_g3f7r2_cycle_spoiler.yaml.