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What a method reads — the informational basis of a voting rule

Every tabulation is a two-step act: compress the ballots into some summary, then decide from the summary. Change the second step and you have a different method; change the first and you have changed what the method is even capable of noticing. This page is about the first step — which statistic a rule actually reads — the question Peter Fishburn turned into a classification in 1977. It answers a debate question worth having ("how much of my ballot does your method look at?") and defuses two claims that sound right and aren't.

→ Related: the C1/C2/C3 tiers in the Condorcet reading list · summability — how big the summary is · the cycle–cocycle decomposition — how the pairwise summary splits · Borda · pairwise counting.

Runnable: Same matrix, different plurality — three electorates, one pairwise table, three plurality winners.

Three different questions, routinely confused

Question Asks Answered by
Which statistic does the rule read? can it even see margins? first choices? Fishburn's tiers, below
How big is that statistic? what must a precinct publish? summability
How hard is it to compute from? can you do it by hand? complexity

These are independent, and every pairing of them comes apart somewhere. Keeping them separate is most of the value on this page.

Fishburn's tiers

Peter Fishburn (1977) sorted rules by how much of the pairwise data they need:

Tier Reads Members you'll meet
C1 the tournament — who beat whom, and which pairs tied. Direction only, no sizes Copeland (the core of Ranked Robin), Smith set, Top Cycle, uncovered set — the tournament solutions
C2 the weighted tournament — the same graph with the margins on it Minimax/Simpson, Ranked Pairs, Schulze, Kemeny, Split Cycle — and Borda
C3 more than the pairwise matrix contains Dodgson, Young, plurality, RCV-IRV

The one-line test for C1 vs C2: hand the rule two elections whose head-to-head directions match but whose margins differ. A C1 rule must return the same answer; a C2 rule may not. Copeland genuinely cannot tell a 50-vote blowout from a 1-vote squeaker. Minimax and Kemeny can, and do.

Scope note, stated once so nobody has to catch us on it. Fishburn's 1977 paper classified Condorcet social choice functions. Extending the labels to Borda, plurality and IRV is later convention — universal in the literature, but convention. It is not a direct citation of Fishburn.

The claim that gets it wrong: "plurality needs more information"

Because plurality is C3 and Borda is C2, it is tempting to say plurality needs more information than Borda. That is false, and it inverts badly for a lay reader — a plurality ballot carries far less than a Borda ballot, and readers will hear the sentence as the opposite of the truth.

The two statistics are incomparable, not nested. First-place counts don't determine the pairwise table; the pairwise table doesn't determine first-place counts. C3 isn't a higher rung — it's the residual class, everything the pairwise matrix fails to capture. The precise claim is narrow:

Plurality's winner is not a function of the pairwise matrix.

And it is demonstrable. Three 12-ballot electorates produce the identical pairwise table — Ben beats Ada 7–5, Ada ties Cal 6–6, Ben beats Cal 7–5, hence the same Condorcet winner, the same Borda scores, the same Ranked Robin / Minimax / Ranked Pairs / Kemeny result — and three different plurality winners, one per candidate.

The mechanism is worth carrying, because it explains the whole tier: a ballot and its exact mirror cancel pairwise but not in the first-choice tally. Ada>Ben>Cal plus Cal>Ben>Ada puts one vote on each side of every head-to-head, leaving every margin untouched, while handing out two different first preferences. Swap mirror pairs in and out and the plurality winner roams while the matrix sits still.

Borda: the C2 rule that isn't a Condorcet method

Borda belongs in C2 because a Borda score is a row of the pairwise table added up. With M(x,y) the margin, n ballots and m candidates:

Borda(x) = ½ · Σ M(x,y) + n(m−1)/2

The second term is identical for everyone, so it can't move anybody: the margins alone fix the entire Borda ranking. (They fix the raw scores only if you also supply n and m — a margin matrix doesn't remember how many people voted. If you want exact scores from a matrix alone, sum the pairwise support counts instead: Borda(x) = Σ N(x,y). That is all a Borda count ever was.)

Three conditions on that identity, because it is not unconditional:

  1. Equal spacing — standard Borda points (m−1, m−2, …, 0).
  2. Complete ballots, or ties handled by splitting points evenly.
  3. Truncation breaks it. Under the common "unranked candidates get 0 points" rule, Borda is not margin-determined — so real truncated-ballot Borda is not C2. Worth knowing, since real ranked ballots are truncated.

Why the identity matters beyond bookkeeping: it is the reason Borda and the Condorcet family diverge in a structured way rather than randomly. Borda reads the pairwise table through a sum, which is blind to circulation; Condorcet methods read the same table through its signs, which circulation can flip.

STAR, Score and Approval have no Fishburn class

Not "an unusual one" — none, and the reason is sharper than the obvious one.

The obvious reason is that Fishburn classifies rules whose input is a ranked profile. A critic answers that easily: map the score ballots to the rankings they induce, then classify. The decisive reason is that this cannot be done, because STAR is not a function of the ranked profile at all.

Two score profiles can induce exactly the same ranked profile — hence exactly the same pairwise matrix — and elect different STAR winners. Six voters ranking Ada>Ben>Cara, five Ben>Cara>Ada, four Cara>Ada>Ben, scored 5,4,0 / 0,5,1 / 4,0,5, elect Ada; the same rankings scored 5,4,0 / 0,5,4 / 1,0,5 elect Ben. Plurality, RCV-IRV and Ranked Robin return Ada on both. Only STAR moves.

A function must return the same answer on the same input. There is simply nothing here for C1/C2/C3 to classify. In any tier column, STAR / Score / Approval get n/a — not a ranked-ballot rule — never a hedged class like "C3" or "beyond C2," because readers strip hedges and would come away thinking STAR counts like Dodgson.

This isn't STAR dodging a question. It's the same type distinction as the SWF/SCF one: a classification applies to a domain, and STAR is not in it. What STAR's tabulation actually needs is a score-count matrix plus a pairwise matrix — richer than either alone, and still perfectly summable.

(Footnote for completeness: on the Brams–Fishburn dichotomous-preference domain, Approval is a genuine SCF on preference profiles and is determined by the majority tournament — C1-like. That's a theoretical domain, not real approval ballots, but a critic will find it, so here it is.)

Why C3 is not a demerit

C3 is a bag, not a basement. It holds plurality — the cheapest, most summable method in this library, one number per candidate — and RCV-IRV, the one method here whose count doesn't summarize into precinct subtotals at all. Those two have nothing in common except that the pairwise matrix doesn't determine them.

Which is exactly why the tiers must not be read as a ladder:

  • Not a summability ladder. Plurality is C3 and first-order summable; Kemeny is C2 and NP-hard.
  • Not a difficulty ladder. See the previous line.
  • Not a quality ladder. C1 vs C2 is invisible to a precinct: Copeland and Schulze publish the identical C×C table. The tier says which part of it they look at.

The debate use

The version that travels is not the taxonomy — it's the question:

How much of my ballot does your method actually look at?

It is positive rather than an attack: it says what pairwise and score counting see, not what any method gets wrong. It makes vote-splitting precise instead of rhetorical — in two of the three electorates above, the plurality winner isn't the candidate a majority prefers head-to-head, and you can watch the ballots that do it. And it cuts symmetrically, which is what makes it usable: pressed honestly, the same framing forces the admission that STAR sits outside the ladder entirely — which is the pro-STAR point restated, not dodged.

Sources

  • Peter C. Fishburn, "Condorcet Social Choice Functions," SIAM Journal on Applied Mathematics 33(3), 1977, pp. 469–489 — the classification, for Condorcet SCFs. Lean: neutral; taxonomy.
  • William S. Zwicker, "Introduction to the Theory of Voting," in Handbook of Computational Social Choice (2016), §2.5 — the modern restatement, including the observation that one should balk at "plurality needs more information than Borda." Lean: neutral.
  • The tier assignments used across this repo, and the [C1]/[C2] tags printed by cycle_resolution_report.py, follow pref_voting's own module organization.