Ranked Robin Is Summable — the Pairwise Matrix Adds

One line: RCV-RR (Ranked Robin / Copeland) is precinct-summable. Its whole tally is the pairwise matrix — for every pair of candidates, how many ballots rank A over B, B over A, or neither — and those matrices add across precincts. So the same ranked ballot IRV must count centrally, Ranked Robin can tally locally and add up.

→ Cross-method topic hub: Summability · the method itself: Ranked Robin · the contrast — IRV is not summable · the score-method counterpart: STAR is summable · Glossary: summability.

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Why it's summable

Ranked Robin's winner is read entirely from the pairwise (For / Against / No-preference) matrix: most head-to-head wins takes it. (How that matrix is built, one ballot at a time: pairwise counting — each ballot is a tiny matrix, and the election's matrix is the ballots' sum; a precinct's matrix is just that same sum stopped partway.) That matrix is a small fixed-size table (C×C), and each precinct can build its own and publish it — the statewide matrix is just the sum. No elimination order, no transfers, nothing that depends on the other precincts' ballots. (This is the very property IRV lacks: summability is about the count, not the ballot.)

Worked example — the same ballots IRV couldn't combine

Take the two districts IRV can't sum (B wins both, but is eliminated when merged). Here are the actual ranked ballots — 13 voters in each district (summability_demo/):

District A (13 voters)        District B (13 voters)
  6 × A                         6 × C
  4 × B                         4 × B
  3 × C > B > A                 3 × A > B > C

Each precinct turns its own ballots into a pairwise For – Against – No-preference table. In District A, the 6 A ballots rank A over B, while the 4 B and 3 C>B>A ballots rank B over A — so the A-vs-B cell is 6 – 7 – 0. Do that for every pair, in each district, and the matrices add cell by cell:

                A vs B        A vs C        B vs C
District A     6 – 7 – 0     6 – 3 – 4     4 – 3 – 6
District B     3 – 4 – 6     3 – 6 – 4     7 – 6 – 0
─────────────────────────────────────────────────────
Combined       9 – 11 – 6    9 – 9 – 8    11 – 9 – 6     (= the sum)

From the summed matrix: B beats A (11–9) and beats C (11–9) → B is the Ranked Robin / Condorcet winner. The candidate who won both districts wins the merge too — reached by adding precinct tables, never pooling ballots. (Same ranked ballots that broke IRV's count.)

Tabulate & verify it

The LH engine computes the matrix (calculate_preference_matrix); the pref_voting engine reports the Copeland = Ranked Robin winner and is itself cross-checked against the LH engine — so the summable artifact is independently confirmed.

The nuance — same as STAR, opposite of IRV

	summable?	the artifact that adds
STAR	✅	score totals + pairwise matrix
Ranked Robin / RCV-RR	✅	the pairwise matrix
RCV-IRV	❌	none — needs every ballot centrally

Because Ranked Robin keeps a fixed-size, additive table, it's precinct-auditable and supports meaningful partial counts — two things IRV gives up.