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Pairwise Counting — Every Ballot Is a Tiny Matrix

How the preference matrix actually gets built: one ballot at a time. Once you see that each ballot is already a little head-to-head table, the matrix — and why it's precinct-summable — stops being mysterious.

→ Level: 201 · deep dive — Curriculum 201.1 (reading the results) · Glossary: preference matrix · the summability payoff: Summability topic hub

Two terms, one thing (a verb and a noun)

  • Pairwise counting is the process: for every pair of candidates, tally each ballot's verdict — does this ballot support A over B, B over A, or neither?
  • The preference matrix is the artifact that process produces: the table holding those tallies for every pair.

They are the same idea seen as verb and noun. The artifact goes by many names — preference matrix, pairwise matrix, pairwise-comparison matrix, head-to-head table, and, on Wikipedia, beats matrix, tournament matrix, or outranking matrix. When it holds the summed tallies for the whole electorate it's the sum matrix. The LH engine prints it as the "Runoff (Preference) Matrix" with the legend For – Equal Support – Against. All the same table. (electowiki calls the process "pairwise counting"; we use its mechanics pages here while noting it's an advocacy-adjacent wiki.)

Seen the "strange" Wikipedia matrix? That article shows a single ballot written as a 0/1 grid (1 = the row candidate beats the column candidate), then adds the ballots into a sum matrix. That two-step — one ballot is a matrix, and matrices add — is exactly what this page builds below, and it's why the count is precinct-summable.

The one idea: a ballot is already a matrix

You don't need the whole electorate to start a preference matrix — a single ballot is one. Take the first voter of the canonical Ann, Bob, Cal election (3 voters × 3 candidates, registered canonical #1 — live on BetterVoting). She scored Ann 5, Bob 4, Cal 0. Written as her own tiny matrix (For – Equal Support – Against, row vs column):

Voter 1 (Ann 5, Bob 4, Cal 0)
        |    Ann    |    Bob    |    Cal    |
  Ann > |    ---    | 1 - 0 - 0 | 1 - 0 - 0 |
  Bob > | 0 - 0 - 1 |    ---    | 1 - 0 - 0 |
  Cal > | 0 - 0 - 1 | 0 - 0 - 1 |    ---    |

Her 5 beats the 4, so this ballot is one vote For Ann over Bob — even though she likes Bob too. Here's the one thing not to misread: that 1 is a count, not a margin — it is not 5 − 4. Look at Ann over Cal (5 vs 0), which is also just 1: the cell records only whether one candidate is higher, never by how much. A 5-vs-4 and a 5-vs-0 are each exactly one vote For. (That intensity is thrown away here on purpose — which is why STAR keeps the score totals too, right alongside this matrix.) Every cell is just "which did this ballot put higher?"

Counting the election = adding the ballots' matrices. Here are all three ballots' pair verdicts, and their sum:

Ballot (Ann, Bob, Cal) Ann vs Bob Ann vs Cal Bob vs Cal
Voter 1 — 5, 4, 0 Ann (5>4) Ann (5>0) Bob (4>0)
Voter 2 — 3, 5, 2 Bob (5>3) Ann (3>2) Bob (5>2)
Voter 3 — 0, 3, 5 Bob (3>0) Cal (5>0) Cal (5>3)
Sum (For – Equal Support – Against) Ann: 1 – 0 – 2 Ann: 2 – 0 – 1 Bob: 2 – 0 – 1

That bottom row is the preference matrix. Here it is straight from the LH engine (the full report of bv2187_qrw6wb_ann-bob-cal.yaml):

[Divergence from STAR]
  STAR                   = Bob
  Choose-One (Plurality) = Ann   (differs from STAR)

--- STAR Voting Method (single winner) ---

[STAR Voting]
 Tabulating 3 ballots.
Ann,Bob,Cal
  5,  4,  0
  3,  5,  2
  0,  3,  5

[STAR Voting: Scoring Round]
 The two highest-scoring candidates advance to the next round.
   Bob           -- 12 -- First place
   Ann           --  8 -- Second place
   Cal           --  7
 Bob and Ann advance.

[STAR Voting: Automatic Runoff Round]
 The candidate preferred in the most head-to-head matchups wins.
   Bob           -- 2 -- First place
   Ann           -- 1
   Equal Support -- 0
 Bob wins.
   Runoff math:
     3  ballots cast
   − 0  Equal Support (no preference between the two finalists)
     ─
     3  voters with a preference  (majority = 2)
           Bob 2 (67%)  ·  Ann 1 (33%)

[STAR Voting: Winner — STAR Voting Method (single winner)]
 Bob

Reading it: Bob beats Ann 2–1 and beats Cal 2–1, so Bob wins every head-to-head — the Condorcet winner. And look at the Bob-vs-Ann cell: 2 – 0 – 1 is exactly STAR's Automatic Runoff (Bob preferred on 2 ballots, Ann on 1). The runoff isn't a separate computation — it's one cell of this matrix.

If a ballot puts a pair equal — the same score, or both blank — that ballot lands in the middle Equal Support bucket for that pair (none happen to occur here). No verdict is ever lost: every ballot counts For, Against, or Equal Support on every pair, which is what makes the matrix self-reconciling in an audit.

Ranked or scored — same construction

The matrix doesn't care which ballot style fed it, only order:

  • Scored ballot (0–5): higher score = For. Strength is deliberately thrown away — a 5-vs-4 and a 5-vs-0 are each one vote For (that's why STAR keeps the score totals too).
  • Ranked ballot (A>B>C): higher rank = For. An unranked candidate counts below every ranked one; two unranked (or equal-ranked, where allowed) candidates are Equal Support for that pair.

Why this one idea earns its keep

  1. It's the auditable heart of the count. The matrix is a small fixed-size table anyone can recompute from the ballots — the annotated tour of a full report is How to Read a STAR Result Report, and the display demo is 04b_c4_b3_display-options-all.
  2. Ranked Robin reads its whole result off it — most head-to-head wins takes the seat. The matrix isn't a supporting exhibit there; it is the tally.
  3. It's why these counts are summable. The election's matrix is the sum of the ballots' matrices — so a precinct's matrix is just a partial sum, and precinct tables add to the statewide result. That's the worked two-district demo in Ranked Robin is summable and STAR is summable, and the very thing IRV's count can't do: IRV has no per-ballot artifact that adds — its rounds depend on everyone else's ballots.

Further afield (301): write-ins and bookkeeping variants

Real election offices hit a wrinkle: a write-in candidate discovered mid-count has no row or column yet. electowiki's pairwise counting page works through the administrative fixes — retrofitting the new candidate's cells, and a "negative counting" variant that records only the upsets to cut bookkeeping. Clever, but it changes nothing conceptually: every variant still ends at the same summed matrix. File under election administration, not voting theory.

Cross-references