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IRV Is Not Summable — Every Ballot Must Be Counted Centrally

One line: RCV-IRV cannot be tallied by adding up precinct totals. Because the winner depends on the elimination order, and the elimination order depends on every ballot, you need the full ballot set in one place to run the count.

Applies to: all sequential-elimination variants — Hare, BTR, Coombs, Baldwin/Nanson, and STV — since every one of them depends on elimination order. The summable alternatives are the pairwise Ranked Robin (precinct matrices just add up) and positional methods. See Which RCV-IRV?.

Why this page matters: summability sounds like a dry administrative detail, but it drives real things voters care about — how elections are audited, whether a precinct can verify its own result, and whether early/partial counts mean anything. IRV quietly gives all of that up.

→ Cross-method topic hub: Summability (STAR / Ranked Robin / IRV side by side). STAR keeps this property — STAR is summable. Glossary: summability.

Why IRV isn't summable

IRV is a sequence of eliminations. Which candidate is eliminated in round 2 depends on the combined round-1 result of every precinct — you cannot add up precinct winners, because a candidate who leads several precincts can still be eliminated statewide. To know who transfers to whom, you need each individual ballot's full ranking, centrally.

There's no small fixed-size table a precinct can publish that sums to the whole. The only "summable" object is the entire pile of ballots.

Worked example — two districts both won by B, merged, B loses

The classic demonstration (after rangevoting.org, verified on the engine — run summability_demo/):

District A          District B          Combined (A + B)
  6 : A               6 : C               6 : A   6 : C
  4 : B               4 : B               4 : B   4 : B
  3 : C>B>A           3 : A>B>C           3 : C>B>A   3 : A>B>C

IRV: C out → B 7, A 6   IRV: A out → B 7, C 6   first-choices: A 9, C 9, B 8
  → B wins                → B wins              → B (fewest) ELIMINATED → A/C tie

B wins each district outright, yet is eliminated when the districts are combined. That's the whole problem in one example: the district winners (B, B) tell you nothing about the combined winner, and there is no precinct subtotal you could have published that would add up to it. To get the real answer you must pool every ballot and recount from scratch. (Contrast STAR below, where the precinct subtotals do add up.)

What that costs

  • Central tabulation — ballots (or full cast-vote records) must be gathered in one place, a single point of failure and a heavier, slower audit. (Expanded on its own page: Maine's courier operation, the NYC 2021 and Alameda 2022 central-count errors, and why IRV audits need heavier machinery.)
  • No meaningful precinct subtotals — a precinct can't certify its own contribution to the outcome the way it can under summable methods.
  • Partial counts can mislead — the candidate "ahead" in first-choices partway through can lose once transfers run, so early IRV numbers are easy to misread.

(This is one row on the method scorecard: Summable / local tally? — STAR: yes, RCV-IRV: no. See the flagship Segment 6 in What's So Good About STAR Voting?.)

The nuance — it's IRV's count, not ranked ballots

Summability is about the tabulation, not the ballot. The same ranked ballot, counted by a Condorcet method (Ranked Robin), is summable — via a pairwise matrix that adds across precincts. So "ranked ballots can't be summed" is wrong; it's IRV's elimination count specifically that can't. (See Tips — Terminology: RCV vs IRV vs RCV-IRV (and friends) and the STAR counterpart, STAR is summable.)

Same ballots, summed — Ranked Robin on the example above

Take the exact two districts IRV couldn't combine and build each one's pairwise matrix (For–Against–NoPref for every pair). 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 that summed matrix: B beats A (11–9) and beats C (11–9) → B is the Ranked Robin / Condorcet winner. So the candidate who won both districts wins the merged election too — and you reach it by adding precinct tables, never pooling ballots. Same ranked ballots that broke IRV's count; Ranked Robin sums them and gets the sensible answer. (Produced by the LH engine's calculate_preference_matrix; the pref_voting engine reports the same Copeland = Ranked Robin winner. Files: summability_demo/.) Full RR-side treatment: Ranked Robin is summable.