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The ranked-ballot method zoo — many ways to count one ranked ballot

Robert LeGrand's ranked-ballot voting calculator offers fifteen-plus methods — Baldwin, Black, Borda, Bucklin, Carey, Coombs, Copeland, Dodgson, Hare, Nanson, Raynaud, Schulze, Simpson, Small, Tideman — all fed the same ranked ballots. That single screen is the clearest possible demonstration of this repo's core terminology point: RCV names a ballot (ranked); it does not name a tabulation. IRV is just one way to count a ranked ballot; here are a dozen more, and they routinely pick different winners on identical votes.

→ Level: 301 · deep dive — Curriculum 301.11. Companions: Ranked Robin vs. the Condorcet winner · Cycle resolution — why Minimax, Ranked Pairs, Schulze exist · Scoring vs. ranked methods · terminology canon: TIPS_terminology.

Live tool: calculator · authoritative method write-ups: descriptions (LeGrand). Paste ballots like 14:Alan>Beth>Carl (14 identical ballots; = for equal ranks) and it runs all methods at once — a great sandbox for the examples below.

The one idea to hold onto

Every method on this page takes the identical ranked ballot and differs only in how it counts. They split into a few families by what information they look at:

  • Positional methods score a candidate by where it sits on each ballot (Borda and its relatives).
  • Sequential-elimination methods drop a loser and recount, round by round (Hare/IRV, Coombs, Carey).
  • Graduated-majority methods widen the net until someone crosses 50% (Bucklin).
  • Pairwise (Condorcet) methods treat the election as every candidate-vs-candidate duel at once (Copeland, Simpson, Schulze, Tideman, and the rest).

With only two candidates they all agree (they all elect the majority winner). With three or more they diverge — and that divergence is the whole subject of voting theory.

Family 1 — Positional (Borda-based)

First, the unification that makes this whole family one method. A scoring rule is defined by a single score vector w = (w₁, w₂, …, w_m): each voter awards w₁ points to their top-ranked candidate, w₂ to their second, and so on; highest total wins. The vector is proper when w₁ ≥ w₂ ≥ … ≥ w_m and w₁ > w_m. Change the numbers and you change the method — and several rules you'd never file together turn out to be the same machine on a different setting:

Rule Score vector w
Plurality / Choose-One (1, 0, 0, …, 0)
Anti-plurality (veto) (1, 1, 1, …, 1, 0)
k-Approval (1, …, 1, 0, …, 0) — k ones
Borda (m−1, m−2, …, 1, 0)
Formula One World Championship (25, 18, 15, 12, 10, 8, 6, 4, 2, 1, 0, …)

That last row is not a joke, and it's the best hook in this section: since 2010 an F1 driver scores 25 points for a win, 18 for second, and so on, and the championship goes to the highest season total. The races are the voters and the drivers are the candidates — the World Championship is a positional voting rule, and every criticism on this page applies to it. It's the cleanest proof that voting rules are not a niche of politics.

The teaching payoff: Plurality is not a different kind of thing from Borda. It is the most extreme proper score vector there is — all the weight on first place, nothing anywhere else — which is precisely why it discards so much of the ballot. "Choose-One vs ranked" is really a question about how flat the weights are, and that reframes the whole debate as a dial rather than a wall. (Same-ballot, different-count is the theme of Same opinions, every method.)

Borda gives each candidate points for its rank position: in LeGrand's form, (times ranked over another) − (times ranked under another). It's often the best method when everyone votes sincerely — and the most easily manipulated when they don't. Its signature flaw is clone dependence: a party can win simply by running extra similar candidates (LeGrand's Eric/Fran/Gary example — Eric has 63% of first places but Borda hands it to Fran because Gary pads her totals). Borda is a ranked method but not Condorcet-consistent.

Three methods bolt recursive elimination onto Borda and, in doing so, become Condorcet-consistent:

  • Nanson — eliminate every candidate with a negative Borda score, recompute, repeat.
  • Baldwin — eliminate only the single lowest Borda score, recompute, repeat.
  • Rouse — Baldwin with an extra layer of recursion (repeatedly excuse the highest Borda candidate from elimination). Present in LeGrand's descriptions; a rarity in practice.

On the same three ballots these three can each pick a different winner (LeGrand's Jana/Kurt/Lisa example: Nanson→Jana, Baldwin→Lisa, Rouse→Jana) — a compact lesson that "Borda-based" is not one method.

Black is the simplest hybrid: elect the Condorcet winner if one exists, otherwise the Borda winner. It is the most decisive method in the whole list (it almost never needs a tiebreaker).

Family 2 — Sequential elimination

  • Hare is the original name for Instant-Runoff Voting (IRV) — in this repo, RCV-IRV. Count first choices, eliminate the candidate with the fewest, repeat until someone has a majority. Intuitive, but not Condorcet-consistent, and the source of the repo's recurring critiques: center squeeze, non-monotonicity, and exhausted ballots. It looks at the least ballot information at any one step.
  • Coombs is Hare in reverse: each round eliminate the candidate with the most last-place votes. Often elects a strong compromise candidate, but also not Condorcet-consistent.
  • Carey (a generalization of Craig Carey's three-candidate IFPP) eliminates all below-average first-rank candidates each step, not just one. Like Hare, it can punish sincere voting — LeGrand's Katy/Luke/Mary example shows both Hare and Carey rewarding strategic ballots.

Family 3 — Graduated majority

Bucklin counts first-place votes; if nobody has a majority it adds in every second-place vote, then thirds, and so on until someone crosses 50%. A "median-rank" idea. Notably it can crown a candidate who would lose head-to-head to another (LeGrand's Mark/Nell/Owen example: Nell wins on added seconds even though Mark beats her one-on-one) — so Bucklin, too, is not Condorcet-consistent.

Family 4 — Pairwise (Condorcet) methods

These build a pairwise matrix in one pass — for every pair, how many voters ranked A over B (a tie counts half each way) — then reason over those duels. When some candidate wins all its duels (the Condorcet winner), every method below elects it (as do Nanson, Baldwin, Rouse, and Black above). They differ only when there's no Condorcet winner — a cycle (A beats B beats C beats A). How each resolves the cycle is its personality:

  • Copeland — most pairwise wins (ties = ½). This is exactly the core of the repo's Ranked Robin (RCV-RR) (win–loss record). Transparent, but indecisive — it ties often, ignoring how big each win was.
  • Small — Copeland, then break Copeland ties by eliminating the non-top-scorers and recomputing until it can't. A more decisive Copeland.
  • Dodgson (Lewis Carroll) — the candidate needing the fewest ballot-swaps to become a Condorcet winner. Rarely, it can pick a Condorcet loser (LeGrand's Cora, who loses every duel but each by a hair). Not the same as "smallest sum of defeat margins": that gloss describes a different, simpler rule (LeGrand's simplification), and the difference matters — true Dodgson needs the ballots themselves, not just the pairwise matrix, which is why it sits in C3 alongside Young rather than with the margin-reading C2 methods (the tiers).
  • Simpson (a.k.a. Minimax / Simpson–Kramer) — smallest single worst pairwise defeat. Also rarely picks a Condorcet loser.
  • Raynaud — elimination on Simpson: repeatedly drop the candidate with the largest single defeat until one remains.
  • Schulze (beatpaths / CSSD) — resolves cycles via strongest indirect paths of victories. Clone-independent, monotonic, and never picks a Condorcet loser.
  • Tideman (Ranked Pairs) — lock in pairwise victories strongest-first, skipping any that would contradict a stronger, already-locked one. Also clone-independent, monotonic, and never a Condorcet loser.

Minimax (Simpson), Ranked Pairs (Tideman), and Schulze are the three cycle-resolvers the repo treats in depth in cycle_resolution.md — Ranked Robin runs the round-robin and, if a cycle appears, hands off to one of these.

Beyond single winners — the set-valued three

Smith, Schwartz, and Landau are usually treated as producing a set of candidates, not one winner — they're too indecisive to serve as a method on their own. The Smith set (smallest set that beats everyone outside it) is the important one: a method is "Smith-efficient" if it always elects from it. It's the formal version of "the group of candidates any of whom has a legitimate claim."

"Landau" is the graph theorist's name for the uncovered set (also seen as the Fishburn set) — worth flagging, because the name gives no hint of that. Despite sitting in this trio it is the sharpest of the three, not the vaguest: uncovered ⊆ Smith, always. It's also the one with a direct bearing on our methods — Ranked Robin can never elect a candidate outside it, and STAR demonstrably can.

At a glance

Method Family Condorcet winner? Never elects a Condorcet loser? Known as / repo tie-in
Borda positional ✗ ✗ classic point-count; clone-vulnerable
Nanson positional+elim ✓ ✓ below-average Borda cut
Baldwin positional+elim ✓ ✓ lowest-Borda cut
Rouse positional+elim ✓ ✓ Baldwin variant (rare)
Black hybrid ✓ ✓ Condorcet-else-Borda; most decisive
Hare elimination ✗ ✗ IRV / RCV-IRV
Coombs elimination ✗ ✗ Hare in reverse (most last-place out)
Carey elimination ✗ ✗ below-average first-place cut (IFPP)
Bucklin graduated majority ✗ ✗ add later ranks until a majority
Copeland pairwise ✓ ✓ Ranked Robin (RCV-RR) core
Small pairwise ✓ ✓ decisive Copeland
Dodgson pairwise ✓ ✗ (rarely) fewest swaps to Condorcet
Simpson pairwise ✓ ✗ (rarely) Minimax / Simpson–Kramer
Raynaud pairwise ✓ ✓ eliminate largest defeat
Schulze pairwise ✓ ✓ beatpaths; clone-proof, monotonic
Tideman pairwise ✓ ✓ Ranked Pairs; clone-proof, monotonic
Smith / Schwartz / Landau set-valued — — sets, not single winners; Landau = uncovered set

("Condorcet winner? ✓" means the method always elects the candidate who beats all others head-to-head, when one exists. The last two columns are exactly why the repo prefers Condorcet methods like Ranked Robin that never seat a candidate who would lose every duel — a guarantee IRV/Hare, Borda, and Bucklin do not give.)

Why this belongs in a STAR repo

Three payoffs for our teaching:

  1. It nails the terminology. One ranked ballot, fifteen tabulations, fifteen possible winners. "RCV" cannot mean a method — it names the ballot. IRV/Hare is one child of that ballot; Ranked Robin (Copeland), Ranked Pairs (Tideman), and Schulze are siblings. (House canon: TIPS_terminology, GLOSSARY.)
  2. It maps the repo's own engines onto the wider field. The vendored IRV engine = Hare. The Ranked Robin engine = Copeland (with the house margin→lot tiebreak). The cycle-resolution page's Minimax/Ranked Pairs/Schulze = Simpson/Tideman/Schulze here.
  3. It shows why STAR isn't on this list. Every method here counts a ranked ballot. STAR counts a scored (0–5) ballot — a different, more expressive input — so it can't appear on a ranked-only calculator at all. That contrast is the point of scoring vs. ranked methods: the ballot comes first, the tabulation second.

Worked live on BetterVoting. Two of LeGrand's examples are reproduced end-to-end in this repo — same ballots, tabulated by IRV / Ranked Robin / STV / STAR on BetterVoting and cross-checked against pref_voting and LeGrand's calculator: the center squeeze (BV2137) where 13 of 15 methods elect the centrist Condorcet winner that IRV discards, and the no-Condorcet-winner five-way (BV2138) where the winner genuinely depends on the method.

A caution, stated plainly: none of this ranks the methods for you. LeGrand's own descriptions note real trade-offs on every one — Borda is best-when-sincere but most manipulable; Copeland is transparent but indecisive; Dodgson and Simpson can (rarely) crown a Condorcet loser; Schulze and Tideman have the strongest criterion-compliance but are the hardest to hand-count and explain. Which properties matter is the judgment call laid out in what makes a voting method good? — this page is the catalog, not the verdict.

References