Ranked Robin tiebreaks — LH vs. BetterVoting (a documented divergence)¶

Ranked Robin (RCV-RR / Copeland) almost always just elects the Condorcet winner, and every engine agrees. But when Copeland scores tie, the engines part ways in how they break it — and one of them is non-deterministic. This note pins the difference down, with two tested cases: one where everyone agrees, one where they can't.

→ The method: Ranked Robin · cycles in depth: cycle_resolution.md · same-animal-until-a-cycle: ranked_robin_vs_condorcet.md

The two ladders¶

Both engines score Copeland the same way — win = 1, tie = ½ — and elect the highest. They differ only in the tiebreak when the top Copeland score is shared:

Fixed 2026-07-26. LH's rung 1 used to sort on the raw win count while printing the wins + ½·ties column beside it, so the two disagreed whenever a pairwise draw existed — the report could contradict its own table and elect a candidate it had ranked third. Rung 1 is now the Copeland score, as this table always claimed. Ranking by wins + ½·ties and by wins − losses are affine transforms of each other and always agree; the raw win count was the outlier.

Rung	LH `run_ranked_robin` (`starvote_larry_hastings.py`)	BetterVoting `RankedRobin.ts`
1	highest Copeland score (wins + ½·ties)	highest Copeland score (wins + ½·ties)
2	total margin (sum of For − Against)	head-to-head — only if exactly 2 are tied
3	lot order (pre-published `lot_numbers`)	"random" — a seeded shuffle

Consequence: LH is fully deterministic at every rung (margin, then a pre-published lot). BetterVoting is deterministic only for a clean 2-way tie that the head-to-head resolves — otherwise (3+ tied, or a 2-way tie whose head-to-head is itself a tie) it falls through to its rung of last resort.

"Random" is the wrong word for BV's rung 3 — read this before relying on the row above. BetterVoting's tiebreak is not a coin flip. shuffleCandidatesForRandomTiebreak.ts says outright that the protocol is written to be deterministic so that re-running the tabulator gives the same result; it seeds TinyRand with (rawVoteCount + hash(raceId)) >>> 0, shuffles once, and stores each candidate's index as tieBreakOrder. The shuffled order ships in the results JSON as perm, so the export publishes the entire tiebreak sequence — winner and runners-up — and a re-tally reproduces it. Confirmed on a live election built for the purpose: BV2261 y2fbpc (two races, both re-fetched byte-identical), and BV2141 has been quietly relying on it all along by pinning lot_numbers to a recorded perm.

What stays true is the weaker, sharper claim: BV's order is recorded but not derivable. It is a function of the ballot count and the race id, never of how anyone voted — so an independent engine can replay it from the export but can never predict it from the ballots. And because rawVoteCount is in the seed, every new ballot re-rolls the order, so a perm frozen mid-election describes that moment only. Both properties are why a tie-deciding case is still best kept LH-only: LH's lot is the only rung reproducible from published inputs alone.

Check it yourself rather than taking the above on trust. tools_adam/bv_replay_tiebreak.py ports TinyRand + the BV shuffle to Python; hand it any frozen _bv_export.json and it recomputes each race's perm from (rawVoteCount, raceId) and diffs it against what BV recorded. It is also the cleanest demonstration of the "not derivable" half — no ballot content is passed in, and the order still comes out right.

Neither engine is wrong — and the divergence was unavoidable. A small impossibility theorem (Moulin, 1983) proves that no anonymous, neutral, Pareto rule can always name exactly one winner; on those profiles a tie is forced and the rule must reach outside the ballots for something to decide with. The literature offers four ways to do that, and the two engines simply took different ones: LH takes the fixed-order approach (a pre-published lot) and pays in neutrality — permute the candidates' names and the winner can change. BetterVoting takes the randomized approach and pays in predictability — its winner is not a function of the ballots. (Not in determinism, note: the shuffle is seeded and reproducible, and it is recorded in the export — see the box above. What it can't be is derived from how people voted, which is why the dead-heat case below stays LH-only.) (pref_voting, consulted as the third opinion, takes a third approach: it returns the whole tied leader set and declines to choose.) Read that way, this page documents three defensible answers to a forced choice rather than a discrepancy to be resolved. Theory: Ties Are Forced.

Where they agree — a clean Condorcet winner¶

When a Condorcet winner exists there's no tie to break, so every engine agrees. The BV-backed Tennessee case (BV2131) is the worked example: Nashville is 3–0 (Copeland 3), tieBreakType: none, and LH-native, BetterVoting (RankedRobin.ts), and pref_voting's independent Copeland all elect Nashville. Agreement is the common case — this note is about the exception.

Where they diverge — a dead heat¶

The LH-only dead-heat case is engineered to tie every deterministic rung. Ada and Ben each go 1–0–1 (Copeland 1.5), both beat Cara, and their margins are identical (+4). What each engine does:

LH: Copeland tie (1.5 = 1.5) → margin tie (+4 = +4) → lot order [Ada, Ben, Cara] → Ada. Reproducible every run.
BetterVoting: Copeland tie → tries the 2-way head-to-head… but Ada vs Ben is itself a tie (1–1, with 2 Equal Support) → falls through to its seeded shuffle. Whatever it draws would be recorded and stable — but it is not predictable from these ballots, so the case can't demonstrate a rule, only one draw.

That's why the dead-heat case has no BetterVoting election: BV's answer here would be an accident of the seed rather than a consequence of the ballots, so there'd be nothing to teach from it. It documents the LH ladder specifically. (BetterVoting would agree Ada and Ben are co-leaders; it just wouldn't choose between them on the evidence of the votes.) If you want to see what BV records when it does land on that rung, BV2261 is the case built to show it.

Where they diverge — live (BV2176, and the bench version BV2270)¶

The Post-it RCV example (BV2176, p8dp28) is the first live BetterVoting election to land on the divergence — a clean 2-way tie whose head-to-head is decisive, so both ladders stay deterministic and simply disagree. (The same race runs live a second time inside the seven-method sweep BV2177 v8r66y; and the companion BV2178 8kg698 shows how knife-edge the tie is — two flipped ballots dissolve the cycle into a clean Condorcet winner and both engines snap back to agreement.) Green and Blue each go 2–1 inside a genuine cycle. BetterVoting's rung 2 (head-to-head between the two tied) elects Green, who beats Blue 7–4 — confirmed on the live results page and freezable. LH's rung 2 (total margin) elects Blue (+5 vs Green's +4). Same ballots, same Copeland tie, two published rules, two winners — pref_voting's independent Copeland reports the leader set {Blue, Green}, and each engine tie-broke inside it consistently with its own ladder.

BV2270 8h4bvh is the same divergence built deliberately, and stripped to the bone — four candidates, nine ballots, six matchups you can check by hand. Alder and Birch both finish on Copeland 2; BV's rung 2 elects Alder (who beat Birch 5–4), LH's elects Birch (margin +3 to Alder's +1), and pref_voting again returns the leader set and declines to choose. Use BV2176 when you want the divergence as it actually turned up; use BV2270 when you want to show someone the mechanism in a minute. (BV2270 was minted for an unrelated BetterVoting display bug — its results page stars the wrong candidate, bettervoting#1480 — so read its winner off the heading, not off the star.)

Practical guidance¶

For teaching a clean outcome (a Condorcet winner, or any un-tied Copeland ranking), any engine is fine and they agree — cross-check freely.

Confirmed by Equal Vote — and scheduled to change (as of 2026-07-29). On electowiki's Ranked Robin talk page in 2025, Sara Wolk wrote that BetterVoting "currently has a random tiebreaker in place, but the plan is to allow users to select from a handful of tiebreaker options (Smith-Minimax, Copeland-Margins, Simple Favorite) before it goes to random." So the divergence this page documents is real and acknowledged by the people who built the other engine — and it has an expiry date. If BetterVoting ships a selectable Copeland-Margins option, its rung 1–2 could line up with LH's and the dead-heat case below may stop being LH-only. Re-check before relying on the table above.
For a case that turns on the tiebreak, state which engine's rule you're relying on. Only the LH rule (margin → lot) is reproducible from the ballots plus a published lot. If a real BetterVoting election lands on a Copeland tie, its winner is stable and recorded — but it is decided by the seed, not a function of the ballots, so no argument about how people voted can rest on it.
When you want determinism from a tie, pin lot_numbers in the YAML and use the LH tally; the printed winner line names the rung that decided it.

Engine wording (fixed)¶

The winner line now distinguishes a dead heat from a real cycle. run_ranked_robin tests whether the tied leaders draw their head-to-heads (dead heat) or beat around a loop (cycle):

# co-top dead heat (leaders draw each other, both beat the rest):
*** 2 candidates tie on the highest Copeland score (1.5): Ada, Ben — a dead heat (they draw head-to-head, not a cycle). Resolved by total margin, then lot order.

# genuine rock-paper-scissors cycle (directed loop, no Condorcet winner):
*** 3 candidates tie for the most wins (Rock, Scissors, Paper) — a Condorcet cycle (no candidate beats all others). Resolved by total margin, then lot order. (… Minimax / Ranked Pairs / Schulze differ — see cycle_resolution.md.)

Both lead with the tie, not a verdict; "cycle" is reserved for a genuine loop. The lead phrasing adapts: with no draws among the leaders it says "tie for the most wins" (then tying on Copeland is tying on wins), and once a draw is in play — as in the dead heat above, where Ada and Ben draw each other — it names the Copeland score instead, because "most wins" would no longer be true. Locked by tests/test_ranked_robin.py (the RPS case asserts "Condorcet cycle"; the dead-heat case asserts "dead heat" and not "Condorcet cycle").

Tested cases¶

Case	Method	Outcome	Engines
BV2131 — Tennessee	RankedRobin	Nashville (Condorcet, no tiebreak)	LH = BV = pref_voting ✓
Dead heat → lot	RankedRobin	Ada (LH lot); BV random	LH deterministic; BV non-deterministic
BV2176 — Post-it RCV example	RankedRobin	2-way tie, decisive head-to-head: Green (BV) vs Blue (LH)	both deterministic — the ladder divergence, live
BV2261 — the tiebreak is recorded	RankedRobin ×2	3-way tie twice (by draws, then by cycle) → Anika both times	BV's `perm` replayed as LH's lot: winner and full order match; `pref_voting` returns the leader set
BV2262 — nine-way dead heat	RankedRobin	9 candidates all 4–4–0, margin +0 → Boris (6th in list order)	the scale check: nine-deep `perm`, LH matches all nine positions, shuffle independently recomputed, `pref_voting` consistent
BV2270 — rung 2, on a bench	RankedRobin	2-way tie, decisive head-to-head: Alder (BV) vs Birch (LH)	the same divergence as BV2176, built minimal — 4 candidates, 9 ballots, no lot on either side

The tie-break itself can't be made strategyproof — Brandt, Saile & Stricker, "Strategyproof social choice when preferences and outcomes may contain ties" (Journal of Economic Theory 202, 2022, 105447) prove that no anonymous, Pareto-optimal tie-breaking rule — a fixed order (LH's lot) or a coin flip (BV's random) — escapes manipulability once voters may express ties, so the LH-vs-BV choice above is a design trade-off, not a solved problem.
Ranked Robin (the method) · cycle resolution · ranked_robin_vs_condorcet.md
This same Copeland-margin logic is offered as an optional STAR tiebreaker — Equal Vote's Condorcet Tiebreaker runs a mini round-robin (most head-to-head wins → margin) among candidates STAR's own rounds left tied. Ranked Robin is this arithmetic as a whole method; there it is a subroutine firing only on an exact STAR tie.
BetterVoting tabulation engine notes: tabulation_engines/BV/tabulation_engine/README.md (the RankedRobin.ts row: "2-way tie → head-to-head; else random")
Glossary

file: rr_tiebreak_lh_vs_bv.md¶