Condorcet's 1788 rebuttal to Borda — where the Condorcet criterion comes from¶
Borda's pitch (1770) was that his rank-points rule fixed plurality's embarrassment: the plurality winner can lose a direct majority contest to someone else. Condorcet's reply was that Borda's rule has the same disease — and he produced the election that proves it. Both plurality and Borda elect Paul here, yet Peter beats every opponent head-to-head. This is the argument that became the Condorcet criterion, and it is also the cleanest demonstration of what STAR's second round is for.
▶ Live on BetterVoting: vote · results ↗ (election khcwm4, Test ID BV2250 — three races on the same 11 ballots: STAR, Ranked Robin, RCV-IRV).
→ Related: Borda count · Ranked Robin · the dark horse (Borda's strategic pathology, as opposed to this sincere one)
The election¶
Three candidates and 11 voters. The names are Condorcet's own:
Each row is one voter's ranking, most-preferred first (N: prefix = N identical ballots).
4:Peter>Paul>James
3:Paul>James>Peter
2:Paul>Peter>James
2:James>Peter>Paul
Plurality counts first choices: Paul 5, Peter 4, James 2 → Paul.
Borda (2 / 1 / 0 points) → Paul:
Paul = 4·1 + 3·2 + 2·2 + 2·0 = 14 ← Borda winner
Peter = 4·2 + 3·0 + 2·1 + 2·1 = 12
James = 4·0 + 3·1 + 2·0 + 2·2 = 7
But every head-to-head contest says Peter:
Peter beats Paul 6 – 5
Peter beats James 6 – 5
Paul beats James 9 – 2
Peter is the Condorcet winner — and neither of the two methods on trial elects him. That is Condorcet's whole point: Borda diagnosed plurality's disease correctly and then caught it himself.
Where each method lands¶
| Method | Winner | Elects the Condorcet winner? |
|---|---|---|
| Choose-One (Plurality) | Paul | ✗ |
| Borda | Paul | ✗ — Condorcet's target |
| Approval | Paul | ✗ |
| RCV-IRV | Peter | ✓ |
| Ranked Robin | Peter | ✓ |
| STAR | Peter | ✓ |
Keep it fair — this is not an IRV failure case. RCV-IRV gets it right: James is eliminated with 2 first choices, both his ballots transfer to Peter, and Peter wins 6–5 (condorcet_1788_irv.md). The methods that miss the Condorcet winner in this particular election are plurality, Borda and approval. (IRV's own Condorcet failures need a center squeeze, which this profile doesn't contain.) The IRV race is on the live BetterVoting election deliberately, so nobody can accuse the demo of hiding it.
BetterVoting agrees with the LH engine on all three races — STAR, Ranked Robin and RCV-IRV each elect Peter, with the same pairwise counts (Peter over Paul 6–5, over James 6–5) and the same first-round IRV elimination of James. The frozen export is condorcet_1788_star_bv_export.json.
The STAR lesson: the scoring round is Borda, and then the runoff checks it¶
Ranks carry no intensity, so running this ranked profile under STAR means converting rank to score — exactly the fabrication the Borda page warns about. We use an even 5 / 3 / 0 spacing, and the outcome is robust to the choice (5/4/0, 5/2/0, 5/1/0 and 4/2/0 all behave identically).
That conversion makes the point sharper rather than weaker: under a uniform spacing, STAR's scoring round is a Borda count. So STAR's first round reproduces Borda's answer exactly — Paul first — and then the automatic runoff runs the direct majority contest Condorcet demanded:
Scoring Round
Paul -- 37 -- First place ← Borda's winner
Peter -- 32 -- Second place
James -- 19
Paul and Peter advance.
Automatic Runoff Round
Peter -- 6 -- First place
Paul -- 5
Equal Support -- 0
Peter wins.
Voters with a preference: 11 of 11 (no Equal Support).
Peter 6 (55%) vs Paul 5 (45%); majority = 6.
[Runoff Reversal]
- Score Round Winner(s) = (Paul)
- Runoff Round Winner = (Peter)
[Condorcet Winner]
Condorcet Winner: Peter — matches the STAR winner
[Divergence from STAR]
STAR = Peter
Choose-One (Plurality) = Paul (differs from STAR)
Approval = Paul (differs from STAR)
Want the whole count? See the full LH reports → condorcet_1788_star.md · condorcet_1788_ranked_robin.md.
One more thing the pairwise table shows: James is the Condorcet loser — he loses every matchup. No method here elects him, but it is worth noticing that Borda ranks him above nobody while plurality's 2 first-choice votes are all he has.
Reproduce it¶
.venv/bin/python STARVote_LH_tabulation_engine/starvote_larry_hastings.py method_comparisons/borda_condorcet_1788/cases/condorcet_1788_star.yaml
The LH engine has no Borda tabulator, so the Borda figures are cross-checked with pref_voting:
uv run python -c "
from pref_voting.profiles import Profile
from pref_voting.scoring_methods import borda, plurality
n={'Peter':0,'Paul':1,'James':2}; names=list(n)
rows=[(4,['Peter','Paul','James']),(3,['Paul','James','Peter']),(2,['Paul','Peter','James']),(2,['James','Peter','Paul'])]
P=Profile([tuple(n[c] for c in o) for _,o in rows], rcounts=[c for c,_ in rows])
print('borda scores ->', P.borda_scores())
print('borda ->', [names[w] for w in borda(P)])
print('plurality ->', [names[w] for w in plurality(P)])
print('condorcet ->', names[P.condorcet_winner()])"
borda scores -> {0: 12, 1: 14, 2: 7}
borda -> ['Paul']
plurality -> ['Paul']
condorcet -> Peter
Notes on the source¶
The profile is the simplified 11-voter version of an example Condorcet described in 1788, the form standard textbooks use when they introduce the Condorcet criterion. Condorcet formalized the head-to-head idea in 1785; Borda had proposed his rank-points rule in 1770, arguing from an example much like Pliny the Younger's.
The candidate names are Condorcet's own (Peter / Paul / James) and are kept for fidelity to the source, even though Peter and Paul share an initial — which this repo's naming rule would otherwise avoid.
Why Borda misses Peter — the mechanism, not just the result¶
Condorcet's objection isn't that Borda "got it wrong" here; it's that Borda reads the pairwise numbers through a sum, and a sum can be dominated by one lopsided pairing.
Peter's margins: +1 over Paul, +1 over James — he wins both, narrowly. Sum: +2. Paul's margins: −1 to Peter, +7 over James. Sum: +6.
Paul loses the only matchup that decides a head-to-head champion, and still finishes ahead on Borda — because crushing James by 7 more than repays losing to Peter by 1. That is the whole disagreement in two lines of arithmetic. Borda asks how much did you beat people by, on aggregate; Condorcet asks did you beat each of them.
(The margins are exactly what a Borda score is made of: Borda(x) = ½·Σ M(x,y) + n(m−1)/2, and the constant is the same for everyone.)
Which is right is a values question, not a math question — and it's the same fork Copeland vs Borda margins makes you choose at, with the structural version underneath. See also what a method reads.