Copeland score — why a drawn matchup is worth half a win¶

Ranked Robin is usually explained as "whoever wins the most head-to-head matchups wins." That is a shorthand, and it is very nearly always right. This page is about the case where it isn't — where the actual rule, the Copeland score, elects someone the shorthand would not.

The rule¶

Every candidate plays every other candidate head-to-head. Score the results like a chess tournament:

Result	Credit
win	1
draw	½
loss	0

Copeland score = wins + ½·ties. Highest score wins. That half-point for a draw is the entire subject of this page.

You will also see Copeland written as wins − losses, and the repo's own glossary uses that form. The two are not in conflict: with every candidate playing the same number of matchups, wins − losses = 2·(wins + ½·ties) − (n−1). That is an affine transform — multiply by two, shift by a constant — so the two formulas always produce the same ranking. Pick whichever you find easier to explain; the winner never changes.

What is different is the raw win count. Ignore draws entirely and you get a third ordering, one that can disagree with both. Raw wins is the odd one out — and since a report that prints "W–L–T" invites you to read the W column as the answer, it is worth seeing exactly how that goes wrong.

When the shorthand breaks¶

If every matchup has a winner, there are no draws, so wins + ½·ties is the win count and the shorthand is exact. This is the common case, which is why the shorthand survives.

The moment one matchup is drawn, the two part company — and a single half-point is enough to decide an election.

The case: the chess club elects a president¶

Thirty club members rank five candidates.

Each row is one voter's ranking, most-preferred first (N: prefix = N identical ballots).

9:Alice>Elena>Dmitri>Bruno>Carmen
8:Carmen>Bruno>Alice>Elena>Dmitri
7:Dmitri>Bruno>Carmen>Alice>Elena
6:Carmen>Alice>Dmitri>Elena>Bruno

Every pair meets head-to-head:

Alice  beats Dmitri  23 – 7        Bruno  beats Carmen  16 – 14
Alice  beats Elena   30 – 0        Dmitri beats Bruno   22 – 8
Alice  ties  Bruno   15 – 15       Dmitri beats Carmen  16 – 14
Carmen beats Alice   21 – 9        Carmen beats Elena   21 – 9
Bruno  ties  Elena   15 – 15       Elena  beats Dmitri  17 – 13

Which gives the round-robin table:

Win–loss record — Copeland score = wins + ½·ties (highest score wins; ties broken by total margin, then lot order):
    #  Candidate  W–L–T  Copeland  Margin  Beats
    1  Alice      2–1–1       2.5     +34  Dmitri, Elena
    2  Carmen     2–2–0         2     +20  Alice, Elena
    3  Dmitri     2–2–0         2      -4  Carmen, Bruno
    4  Bruno      1–1–2         2     -12  Carmen
    5  Elena      1–2–1       1.5     -38  Dmitri

Winner — Ranked Robin (RCV-RR): Alice
   the highest Copeland score (2.5 = wins + ½·ties).

Why Alice wins¶

Read the W column alone and Alice looks unremarkable: two wins, exactly like Carmen and Dmitri. On raw wins this is a three-way tie at the top.

Now count the draw. Alice's tie with Bruno is worth ½, so Alice sits on 2.5 while Carmen, Dmitri and Bruno are all on exactly 2.0. Nobody else has 2 wins and a draw. The half-point is not a tiebreaker here — it is the whole margin of victory.

And note what Alice's record actually contains: a loss. Carmen beat Alice 21–9, a thumping. Ranked Robin still elects Alice, because nobody in this election beat everybody — there is no Condorcet winner at all. The Smith set is all five candidates: the whole field is one big cycle. When no one beats everyone, Ranked Robin falls back to the best overall record — and the best overall record can belong to someone who lost a match.

What the half-point costs¶

Here is the part worth being honest about. Run this same profile through the other Condorcet methods and every one of them elects Carmen, not Alice:

Method	Winner
Copeland (= Ranked Robin)	Alice
Minimax	Carmen
Ranked Pairs	Carmen
Schulze	Carmen
Split Cycle	Carmen
Stable Voting	Carmen

Copeland stands alone against the field. That is not a rounding artifact — it follows directly from what Copeland is allowed to look at.

Copeland reads only who beat whom. A win is a win; the size of it is invisible. So compare the two candidates' losses:

Alice lost once — to Carmen, by 12 votes.
Carmen lost twice — to Dmitri by 2, and to Bruno by 2.

Copeland sees "one loss" versus "two losses" and prefers Alice. Every method in the right-hand column can see margins, and they see a candidate who was barely nudged out of two matchups versus one who was decisively beaten in another — and they prefer Carmen.

That is the C1 / C2 distinction exactly: Copeland is a tournament solution, reading only the arrows; the others read the numbers on the arrows. Neither is obviously correct — discarding margins is a deliberate choice that buys resistance to certain manipulations — but a profile like this one is where the choice becomes visible, and it is a fair thing for a critic to raise.

Cross-checks¶

Verified two ways, per the repo's Ranked Robin practice:

LH engine (this repo's tabulator) → Alice, Copeland 2.5
pref_voting (Eric Pacuit's independent library, Copeland computed as wins − losses) → Alice, AGREE ✓ (unique Copeland winner)

Two different formulas, one winner — the affine-equivalence claim above, confirmed in practice rather than asserted.

There is no BetterVoting election behind this case; it is a pure engine-mechanics illustration. The result is fully deterministic (Alice is the unique Copeland leader, so no tiebreak rung is ever reached), so nothing here depends on a lot draw.

Run it yourself¶

.venv/bin/python STARVote_LH_tabulation_engine/starvote_larry_hastings.py 05_Ranked_Robin/02_Examples/copeland_score/cases/copeland_half_credit_decides.yaml

Want the whole count — the full pairwise grid, the Smith-set audit, the ballot listing? See the full LH report → copeland_half_credit_decides, or the raw _tabulated mirror. Source: copeland_half_credit_decides.yaml.