Skip to content

Borda Count — Manufacturing Scores from Ranks

If a score ballot can be reduced to a ranking (drop the intensity), can you go the other way — turn a ranking back into scores? You can. The one principled way to do it is the Borda count, and the act of doing it is exactly inventing the intensity the voter never gave.

→ The reverse direction: scores vs. ranks · strict vs. weak ranks · Borda-elimination methods: Baldwin & Nanson · Glossary: Borda

What Borda does

Assign points by a candidate's position in the ranking: with N candidates, 1st place = N−1 points, 2nd = N−2, …, last = 0. Then add the points across all ballots; highest total wins. That's the whole method — positional scoring.

Conveniently, the even-spaced points for a 6-candidate ballot land on 0–5, so a Borda'd ranking looks exactly like a STAR ballot:

ranked:   E > A > B > F > D > C
Borda  →  E=5  A=4  B=3  F=2  D=1  C=0

Tied ranks get the average of the positions they span (a "midrank"). So E > A=B=F > D > C → E=5, then A,B,F share (4+3+2)/3 = 3 each, D=1, C=0.

The catch: Borda assumes equal gaps

Those manufactured scores are scores under an assumption — never the voter's scores. A ranking carries order only; it is silent on how much you prefer one rank over the next. Borda fills that silence by assuming every gap is the same size. But the voter might have felt a cliff, not a ramp:

ranked:        E   >  A  >  B  >  F  >  D  >  C
the voter felt:  E ≈ A ≈ B  (all great)   ≫   F, D, C  (all bad)
real scores:     5,  5,  5,                    1,  0,  0
Borda's guess:   5,  4,  3,                    2,  1,  0   ← flattens the cliff into a ramp

Borda has no way to know the difference — the intensity was never on the ballot. So any scores it produces are an artifact of the spacing rule, not the electorate.

Different spacings → different "scores"

There isn't one "right" way to invent intensity; each spacing tells a different story:

Spacing Points (top → bottom) Assumes…
Borda (even) 5, 4, 3, 2, 1, 0 every gap is equal
Dowdall 1, ½, ⅓, ¼, … the top ranks matter far more
Convex / concave your choice a steeper or flatter intensity curve

All are defensible conventions; none is the answer. (This is also why Borda is strategy-prone — burying a rival or running clones shifts everyone's positional points.)

The fidelity ladder — how much each conversion invents

This is the key idea. Converting between scores and ranks is never free; what matters is how much you fabricate:

Conversion Used by Drops Invents
score → weak rank Ranked Robin intensity nothing — ties stay ties (A=B)
score → strict rank RCV-IRV intensity a tie-break order (A=B forced to A>B)
rank → score Borda — the whole intensity (the spacing)

Read top to bottom and the fabrication grows. Ranked Robin's reduction is the honest one — precisely because weak ranks let it keep the ties the voter expressed, so it invents nothing. IRV can't represent a tie, so it has to make one up (that's its fragility). Borda sits at the far end: it conjures the entire intensity dimension out of pure order.

That asymmetry is why STAR → ranks works but ranks → STAR doesn't recover anything: a score ballot contains its ranking, but a ranking never contained scores.

Is Borda "bad," then?

No — Borda is a real, century-old, sometimes-used method (it sits in the Positional branch of the methods family tree). It's even the engine inside the Condorcet-safe IRV variants Baldwin and Nanson, which eliminate by Borda score. The honest rule is just about labeling:

Manufacturing scores from ranks is fine — call them "Borda / positional points," never "the voter's scores." The only real error is printing invented numbers next to the word scores as if the voter had typed them.

Two facts that keep the criticism calibrated. Borda's failures are usually cited without their bounds, so here are both directions:

  • Borda can deny a majority favorite. Five voters: three rank a ≻ b ≻ c, two rank b ≻ c ≻ a. With weights (2,1,0), b wins 7 to 6 — even though a is top-ranked by an outright majority and is the Condorcet winner. This is Borda failing the majority criterion in the smallest possible way, and it's why "Borda isn't Condorcet" isn't a technicality. Note also the sympathetic reading, which is genuinely held: some describe Borda as a compromise that takes minority views into account, even when a majority completely agrees with one another — whether that's a bug depends on what you think an election is for.
  • But Borda can never elect a Condorcet loser. A Condorcet winner's Borda score is always strictly above the average, and a Condorcet loser's is always strictly below it — so a Borda winner, being at or above average, can never be the candidate who loses every head-to-head. That's a real guarantee, and one Choose-One plurality does not have.

In this repo

  • The pref_voting engine reports the Borda winner as a bonus column beside Copeland on any example election — clearly labeled as Borda / positional points (not scores, and not what Ranked Robin actually counts — RR uses pairwise).
  • Borda is not Condorcet — and that's not an abstract criterion note, it's the objection Condorcet himself raised in 1788, with a worked 11-voter election you can run: Condorcet's rebuttal to Borda. Borda's Condorcet-fixing cousins are Baldwin & Nanson.

Borda is the pairwise table, added up

A Borda score is not a separate thing from the head-to-head numbers — it is them, summed. Writing M(x,y) for the margin of x over y, n for ballots and m for candidates:

Borda(x) = ½ · Σ M(x,y) + n(m−1)/2

The right-hand term is the same for every candidate, so the pairwise margins alone fix the entire Borda ranking. Equivalently and more simply: add up x's head-to-head vote totals against each opponent, and that sum is x's Borda score.

This is why Borda sits in Fishburn's C2 tier — it reads the weighted tournament, exactly like Minimax and Ranked Pairs do, despite not being a Condorcet method. Three conditions: standard equally-spaced points; ties handled by splitting points evenly; and truncation breaks it — under the usual "unranked get 0" rule Borda is no longer margin-determined, so real truncated-ballot Borda is not C2.

The structural payoff: because Borda reads the table through a sum while Condorcet methods read it through signs, their disagreement isn't random — it lives entirely in the circulating component of the margins. Worked on twelve ballots in Copeland vs Borda margins, and framed in what a method reads.