Skip to content

One Ranked Electorate, Many Tabulations — the winner depends on the method (BV2138)

▶ Live on BetterVoting: vote · results ↗ (election cxrf8v).

Robert LeGrand's flagship "the method decides everything" example, from his ranked-ballot calculator. 921 voters rank five candidates, and there is no Condorcet winner — a top cycle. Across the ~15 ranked methods the win splits five ways. Run through the four tabulations BetterVoting supports, one electorate yields three different winners — and one of them is a documented LH-vs-BetterVoting tiebreak divergence.

→ Level: 301 · deep dive — Curriculum 301.11. See also: the ranked-ballot method zoo · cycle resolution · RR tiebreak: LH vs BV.

The electorate

921 voters, five candidates (Abby, Brad, Cora, Dave, Erin):

 98:Abby>Cora>Erin>Dave>Brad     124:Cora>Abby>Erin>Dave>Brad
 64:Brad>Abby>Erin>Cora>Dave      76:Cora>Erin>Abby>Dave>Brad
 12:Brad>Abby>Erin>Dave>Cora      21:Dave>Abby>Brad>Erin>Cora
 98:Brad>Erin>Abby>Cora>Dave      30:Dave>Brad>Abby>Erin>Cora
 13:Brad>Erin>Abby>Dave>Cora      98:Dave>Brad>Erin>Cora>Abby
125:Brad>Erin>Dave>Abby>Cora     139:Dave>Cora>Abby>Brad>Erin
                                   23:Dave>Cora>Brad>Abby>Erin

There is no Condorcet winner: the pairwise contests form a cycle (the Smith set — the smallest set that beats everyone outside it — is {Abby, Brad, Dave, Erin}). With no candidate beating all others, each method's cycle-handling personality picks a different winner.

The result: five methods, five winners (four on BetterVoting)

Method Winner Engines
IRV (Hare) Dave LeGrand · pref_voting · LH · BV
STV, 1 seat Dave LeGrand · LH · BV (= IRV single-winner)
Ranked Robin (Copeland) Abby (LH) / Brad (BV) see divergence below
STAR (ranks→scores) Brad LH · BV
Borda, Coombs, Baldwin, Raynaud, Schulze Abby LeGrand · pref_voting
Nanson, Tideman (Ranked Pairs), Small Brad LeGrand · pref_voting
Dodgson, Simpson (Minimax) Cora LeGrand · pref_voting
Bucklin Erin LeGrand · pref_voting

Every candidate in the Smith set wins under some method, and even the Condorcet loser Cora wins under Dodgson/Simpson (she loses every duel, but only barely). This is the strongest possible statement of the repo's thesis: with no Condorcet winner, "who won?" has no method-independent answer.

The Ranked Robin divergence: LH ≠ BetterVoting

Copeland (Ranked Robin's core) ties Abby and Brad — each wins three head-to-heads and loses two. The two engines break that tie by different rules, and here the rules disagree:

   Abby  beats Cora   461 – 460      Brad  beats Abby   463 – 458
   Abby  beats Erin   511 – 410      Brad  beats Cora   461 – 460
   Abby  beats Dave   485 – 436      ...
  • LH breaks the Copeland tie by total margin, then lot → Abby (her wins are by larger margins).
  • BetterVoting breaks it head-to-head → Brad (Brad beats Abby directly, 463–458).

Both are deterministic here; they simply encode different notions of "who deserves the tie." This is the same LH-vs-BV Ranked Robin tiebreak split documented in rr_tiebreak_lh_vs_bv.md. The .yaml's expected_winners is LH's Abby (that's what the LH engine computes and the test suite checks); BetterVoting's live result is Brad. Note that STAR's automatic runoff also resolves Abby-vs-Brad head-to-head → Brad, so BV's Ranked Robin and STAR agree, and LH's Ranked Robin is the lone Abby.

The rank→score conversion (STAR race)

As in BV2137, the STAR race maps each ranking to 0–5 scores linearly, top → 5, bottom → 1:

score(rank) = round( 1 + 4·(N − rank) / (N − 1) ) → for N = 5 candidates: 5, 4, 3, 2, 1.

STAR then gives Abby the top score total (2836) but the automatic runoff flips to Brad, who beats Abby head-to-head 463–458 — a compact demonstration that STAR's runoff, not its score round, decides.

Which methods are on BetterVoting — and which aren't

BetterVoting natively runs four of the ~15 methods (IRV, Ranked Robin, STV, STAR-via-scores). The other eleven — Borda, Bucklin, Coombs, Dodgson, Simpson, Schulze, Tideman, Nanson, Baldwin, Raynaud, Small — have no BetterVoting equivalent and are verified with pref_voting and LeGrand's calculator. That's why the table above lists more winners than BetterVoting alone can show: it's a limit of BV's method menu, not of the election.

Sources