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Voting 301 — proportional, criteria, debate theory

Audience: skeptics, academics, RCV advocates, deep self-study. Concede limits honestly — that candor is the credibility.

Prereq: Voting 101 + 201. · Up: Curriculum hub.

The two proportional rungs come first and are deliberately separate: 301.1 is the score-ballot family, 301.2 the ranked-ballot one. They answer the same question with different machinery and different ballots — read them in order, but don't fuse them.

301.1 — Proportional STAR

  • Objective: give a cohesive minority the representation it earned — on the ordinary 0–5 STAR ballot, counted by reweighting instead of by transfers.
  • Key terms: proportional representation, Droop quota, Allocated Score, Sequentially Spent Score (SSS), Reweighted Range Voting (RRV), ballot weight, vote unitarity.
  • See it: the proportional trio in 03_STAR_PR/ — allocated / SSS / RRV — one ballot file, three tabulations, switched with voting_method:. Contrast with Bloc STAR. Exercise: Two seats, one neighborhood (ex12) — the Bloc sweep vs the Allocated share on ten ballots.
  • The point: proportional methods seat coalitions in proportion; Bloc STAR lets a majority sweep every seat. Same ballots, opposite goal — so the majoritarian-or-proportional choice is made before the count, not by it (electing more than one).
  • The criteria, demonstrated: vote unitarity — influence is spent only in exchange for representation gained: a seven-voter SSS election where two bullet voters' unspent star budgets decide the second seat (also the profile that caught a real engine defect, upstream starvote #19). And the quota family's structural price: the Alabama paradox — the committee grows by a seat and a sitting member loses hers.
  • Deeper math: the math behind proportional STAR — quotas, D'Hondt reweighting, JR/EJR, fair division.
  • ABC rules & the utilitarian–egalitarian spectrum: ABC rules — the approval-committee formalism (AV / PAV / Chamberlin–Courant / Phragmén) and the shadow-STAR bridge (Bloc STAR = AV, RRV = PAV), verified with abcvoting.
  • Thiele methods: Thiele methods — AV/PAV/CC as one parameterised family; RRV is the score-ballot cousin of seq-PAV, while Allocated Score / SSS follow the quota-and-spend lineage of 301.2.
  • Keep it honest: what "proportional" actually means — exact proportionality has one definition and almost no real election meets it; quotas are a guarantee, not a price; and without parties there is no obvious thing to be proportional to (proportional to what?).

301.2 — STV

  • Objective: learn the established ranked-ballot proportional method on its own terms — quota, surplus, transfer. It has a century of real-world use (Ireland, Malta, the Australian Senate, Scottish councils, Cambridge MA), it is the one your audience has heard of, and it is the one STAR-PR will always be measured against.
  • Key terms: Single Transferable Vote, Droop quota, surplus transfer, elimination, district magnitude (M), Hare vs Droop.
  • See it: the method's door is 06_Other/STV/ — three seats, seven candidates, quota and transfers end to end. Exercise: The transfer machine (ex14) — one surplus, one elimination, counted by hand. In the wild: Australia's Senate, the multiparty chamber sitting next to an IRV lower house.
  • The point: STV is the proportional multi-winner cousin of IRV — it is not IRV, and it is not "RCV." Same ranked ballot, a different count, a different job (terminology). Folding it into "RCV" is the single most common error in this whole area.
  • Head to head with 301.1: STV vs STAR-PR counts one shared 100-voter, 3-seat electorate both ways. The result worth carrying: all four proportional methods elect the identical slate, and only the majoritarian one differs — the ballot argument and the proportionality argument are separate arguments.
  • Keep it honest: there is no settled metric that ranks proportional methods against each other — Quinn shelved AVEC unfinished — so the defensible comparisons are mechanical and administrative (summability, expressiveness, auditability), never a satisfaction score. Anyone quoting a precise "proportional accuracy" figure is ahead of the evidence, and that applies to STAR-PR exactly as much as to STV. The one sharp mechanical difference: summability — STV's is worse than IRV's, because eliminating never-elected candidates to free their votes defeats even the seat-capping workaround that does rescue the STAR-PR family at 3-seat districts. And engines are not theory: a live BetterVoting STV crash, bisected with five public elections.

301.3 — Favorite betrayal: does only "RCV" avoid it?

  • Key terms: Favorite-Betrayal Criterion, Later-No-Harm, center squeeze, the incompatibility theorem.
  • Material: favorite betrayal (301). Companion exercise: Later-no-harm, both readings (ex10) — the criterion IRV keeps and STAR trades, live in one election.
  • The point: neither STAR nor RCV-IRV is FBC-proof; RCV-IRV fails it structurally (Alaska '22), STAR only in lab constructions.

301.4 — "Are equal-score votes discounted?"

301.5 — The honest limits & theory

  • Key terms: Gibbard / Gibbard–Satterthwaite, strategy resistance vs proofness, Condorcet efficiency, Test of Balance.
  • Material: STAR's honest limits; "resistant, not proof." Worked gamble: Bullet voting backfires (ex06).
  • The Equal Vote / Test of Balance: The Equally Weighted Vote (why STAR passes) and RCV-IRV Fails the Equal Vote Criterion (why RCV-IRV fails — stated fairly, with the honest caveats).
  • Deeper math: the math behind Condorcet (tournaments, Smith/Schwartz, Arrow & Gibbard–Satterthwaite).
  • Approval's limits, from the literature rather than from STAR advocates: Approval in the theory literature — the six standard arguments and five standard criticisms (two of the five don't survive scrutiny — "unfair to voters who approve more" dies on a symmetry argument), the three incompatible readings of what "approve" means, and why the strategy question is downstream of that choice rather than of any simulation. Ends on Approval = Borda = Condorcet under dichotomous preferences — true, elegant, and shown in five runnable ballots not to transfer to an election where voters compress their own rankings.
  • The generalized Condorcet winner: The Smith set — the smallest club that beats everyone outside it — when a cycle erases the Condorcet winner, the smallest set that beats everyone outside it is the principled "still in contention" list; the Smith-efficiency scorecard (Ranked Robin ✅, Minimax ❌, STAR ❌ by design), the Smith//X constructions, and ISDA — worked on a runnable 4-candidate cycle (04_smith_set_c4_b7).

301.6 — The vote-splitting formula (blocs)

  • Objective: confirm a spoiler in numbers for a declared bloc.
  • Material: the blocs: field + [Vote-splitting check] (see the engine); the split-voting set.

301.7 — When Condorcet, Score, and Runoff disagree (and how often)

301.8 — "Exhausted ballots": what FairVote's word hides

  • Objective: untangle the overloaded RCV-IRV term — separate the fair, voter-side cases from the method-caused ones a fully and correctly ranked ballot still suffers.
  • Key terms: ballot exhaustion, inactive ballot, ranking limit, majority-of-remaining-ballots, forced vs voluntary exhaustion.
  • Material: exhausted ballots · forced vs voluntary exhaustion; contrast are equal-score votes discounted?.
  • The point: exhaustion is IRV-specific (Ranked Robin reads every rank); STAR counts every ballot in both rounds, so nothing exhausts.

301.9 — Scale granularity can flip the winner

  • Objective: a score scale's resolution (0–5 vs 0–9…) is a modeling choice; when the top contenders are bunched, compressing the scale can move a finalist and flip the STAR winner — even though rescaling never reorders any voter's own preferences.
  • Key terms: score resolution / granularity, finalist selection, near-tie, quantization.
  • Material: Scale granularity can flip the winner; case rrv_sample_c15_b13_three-parties (0–5 → Orange5, 0–9 → Orange1).
  • Companion — "unorthodox STAR": Running STAR on a scale wider than 0–5 — the 0–5 range is a convention, not a rule; which round the scale touches (finalists, never the runoff), the honest tradeoff, and how to make LH tabulate a 0–10 STAR election (maximum_score=N).
  • The point: a fragile, mapping-dependent divergence — present both counts, never quote one.

301.10 — How the simulations are built (VSE's foundation)

  • Objective: understand the models that generate synthetic electorates for VSE / Bayesian-Regret studies — and why every simulation result is conditional on the model.
  • Key terms: Impartial Culture (IC) / Impartial Anonymous Culture (IAC), spatial model, Euclidean distance, Mallows (Kendall-Tau, φ), Plackett-Luce, Pólya urn, Yee diagram.
  • Page: Election simulation models — the neutral menu, the math prerequisites, and the standing caveat. Prerequisite: Simulate utilities, not ballots — the methodology point that you sample preferences and derive ballots, never sample random ballots directly (and why [0,1] utilities beat random {0..5} scores).
  • The point: pairs with 301.7's habit — never quote a rate without the model — and with 201.6's VSE overview.
  • A worked frequency study: How often do STAR and Approval disagree? — a seeded, self-testing simulation (~12% at 3 candidates / ~25% at 5, realistic model) that makes the "no single number" lesson concrete: divergence depends on the electorate model and the approval cutoff, because Approval has no canonical sincere ballot. The cutoff's outcome-power as a paper drill: Where do you draw the line? (ex13) — one electorate, three thresholds, three Approval winners.

301.11 — The ranked-ballot method zoo (many tabulations of one ballot)

  • Objective: a single ranked ballot supports a dozen-plus tabulations that pick different winners — the definitive proof that "RCV" names a ballot, not a method — and locate the repo's engines (Hare = IRV, Copeland = Ranked Robin) inside the wider field.
  • Page: The ranked-ballot method zoo — grouped by family, an at-a-glance criterion table, and a live calculator sandbox.
  • The point: the catalog, not the verdict — which properties matter is 201.6's question.

301.12 — Reading advocacy claims: a claim-check, worked

301.13 — Participation: can showing up ever hurt you?

  • Objective: the Participation criterion and the no-show / Twin paradoxes — "vote, it can only help your side" is a promise some methods structurally can't make.
  • Material: the Participation topic hub + the live pair method_comparisons/participation_no_show/; catalog: no-show paradox; the rare STAR-side failure, as a predict-then-peek exercise: The tenth ballot.
  • The point: the honest scorecard — only pure summation (Score/Approval) and Choose-One are immune; STAR's runoff costs it the formal guarantee too, just far more rarely than IRV's elimination machinery. Frame per reading these fairly.
  • The three-candidate twist: Moulin's theorem (every Condorcet method fails no-show) needs ≥ 4 candidates. At exactly three, Brandt, Dong & Peters (2024) prove there is a no-show-immune family — the refinements of maximin (leximin, Nanson), and uniquely so. A rare case where restricting the problem buys a clean escape.

301.14 — Consistency: winning every district ≠ winning the whole

  • Objective: the consistency criterion (join-consistency / reinforcement) — if every district separately elects X, must the combined electorate? Point-summing rules must, by arithmetic; STAR (like RCV-IRV, top-two, and Condorcet methods) need not, because "who advances to the runoff" is not additive.
  • Key terms: consistency / reinforcement, runoff pairing, precinct-summable (the contrast).
  • Material: the predict-then-peek exercise Two districts, one mayor (live on BetterVoting as BV2188–90); catalog: the multiple-districts (reinforcement) paradox, whose Felsenthal trio (BV2147–49) is the IRV-side sibling; contrast STAR Is Summable — the tallies add across precincts even though the winner inference doesn't.
  • Run it across every method: Reinforcement paradox — both halves pick Ada, the whole picks Cara — the same 9 voters counted five ways: Score/Approval/Plurality keep the promise (additive), Ranked Robin can't, and STAR's scoring round keeps it but the runoff breaks it (a live Runoff Reversal). The concrete instance of a theorem: Brandt, Dong & Peters (2024) prove every Condorcet method must show this paradox once there are ≥ 8 voters.
  • The point: "she carried both districts and still lost" is a headline risk, not a tabulation risk — no one tallies a citywide seat district-by-district. Carry the worked answer.

301.15 — Distortion: putting a number on what a ranking throws away

  • Objective: meet the mainstream-academic formalization of this library's central question — if a ballot records order but never degree, what does that cost? — and learn to quote it without overclaiming in either direction.
  • Key terms: distortion, misrepresentation, social welfare, unit-sum (normalized) vs metric model, uncovered set, cardinal queries.
  • Page: Distortion — the two models and why the model decides the verdict, the scoreboard (Copeland/Ranked Robin ≤ 5 constant · STV/IRV O(log m), not constant · Plurality & Borda linear in the field · approval-only input unbounded), and the result that most supports score ballots: rankings plus a few numeric queries collapse the worst case from Θ(m²) to constant.
  • The point: distortion is VSE's academic sibling — same premise (utility is the target, a ballot is a lossy channel), opposite instrument (proved worst case vs simulated average case). Useful when someone dismisses VSE as reform-movement math: the peer-reviewed literature made the identical modeling choice.
  • Keep it honest: the metric bound is only 3, so rankings are not hopeless; the best strategyproof mechanisms are essentially ordinal; and no published distortion bound for STAR exists that we could find — an open gap, not a result to cite. Frame per reading these fairly.
  • How a bound of 3 is even possible (same page, second half): the three-line Condorcet-winner ≤ 3 proof (triangle inequality + "a majority said so"), the midpoint-voters lower bound (near-indifference is the one thing a ranking cannot say), and where STAR's rounds sit: the automatic runoff is pairwise majority, so the ≤ 3 lemma plugs straight in between the finalists — a conditional guarantee, since the scoring round's worst case lies outside the ordinal theory entirely.
  • Run it (the founding impossibility): same ranks, different utilities — Procaccia & Rosenschein's Proposition 1 on 3 voters and 2 candidates: two files with identical rankings and opposite welfare-optimal winners, so every ranked rule is wrong on one of them. Watch STAR's scoring round report the gap (B 9, A 6) and its runoff overrule it — at two candidates STAR is majority rule, which is where this collides with May's theorem.
  • The escape hatch: misrepresentation — restrict utilities to rank positions and the impossibilities lift (Plurality and top-two runoff both m−1, Ranked Robin ≤ m−1, RCV-IRV ≤ 1.5(m−1), Veto unbounded). Borda scores a perfect 1 — because the measure is the Borda count, a peer-reviewed instance of a criterion built to fit the method. Also where Veto satisfies participation + monotonicity + consistency and still has unbounded loss: criteria are pass/fail on edge cases, bounds are magnitudes.
  • Run it: the valuable Condorcet loser, counted — the runnable companion to Ebadian–Latifian–Shah (AAMAS 2023), the closest published relative to STAR's shape and the unit-sum counterweight to the runoff-as-insurance reading: there a majority runoff raises approval voting's distortion (Θ(m) → Θ(m²)) because it structurally blocks the highest-welfare candidate; their pivotal profile on nine ballots (vcl_c4_b9) — Score elects Amy (20 vs 11), STAR's runoff rejects her 4:5, welfare ratio 1.8. Both verdicts on the runoff are theorems; the model decides.

301.16 — Grading as a rival primitive (Balinski & Laraki)

  • Objective: see the strongest academic argument that the preference order is the wrong primitive — not merely lossy — and learn why it is not an ally of score-summing.
  • Key terms: common language of grades, majority grade (median), majority-gauge, strategy-proof in grading, absolute vs relative evaluation.
  • Page: Grading as a rival primitive; the case against, worked: Majority Judgment's paradoxes; the reading-list entry: rated & score methods.
  • The point: "preference = ordering" is a contested modeling choice with a serious academic rival, not the neutral default — which is the citable backing for preference vs. support and for refusing to cede the word "preference" to ranked ballots.
  • Keep it honest: Balinski & Laraki argue the median over the sum, so they cut against STAR's Scoring Round as much as against IRV; MJ fails the majority and Condorcet criteria by design; and a 2026 experiment (Delemazure, Brunetti, Baujard & Bouveret, N = 1955) rejects the common-language premise that the comparability argument rests on.

301.17 — The social welfare function (what Arrow's theorem is about)

  • Objective: learn the type distinction that most impossibility debate skips — a rule that outputs a ranking (SWF) vs one that outputs a winner (SCF) — and stop both over- and under-applying Arrow.
  • Key terms: social welfare function f : L(A)ⁿ → R(A), social choice function, social preference order, weak vs strong Pareto, IIA, decisive coalition, Contagion / Splitting Lemma, welfarist social welfare.
  • Page: Social welfare function — the two objects, both axioms stated at both levels, who actually fails Pareto, and the two senses of "social welfare."
  • The point: majority rule is both Paretian and IIA, and escapes Arrow only because it is not an SWF — its output can cycle, so it isn't a ranking at all. That reframes Condorcet methods correctly: a completion like Ranked Robin is an SWF, so Arrow applies in full, and the cycle-resolution rule is what costs it IIA. Patching the cycle doesn't buy an exemption.
  • Keep it honest: Pareto forbids, it does not require — plurality is Paretian and still elects poorly; and a dictatorship is Paretian, which is exactly why Arrow's conclusion bites. Pairs with 301.5's Arrow/Gibbard framing and with Does Arrow apply to STAR? — that page's ordinal/cardinal escape is a claim about the SWF's domain, and this page supplies the type it escapes from.

301.18 — May's Theorem: why the hard part starts at three

  • Objective: meet the positive result the impossibility theorems are measured against, and get the fairest available framing of Choose-One voting.
  • Key terms: May's theorem, anonymity, neutrality, positive responsiveness, decisiveness, the two-alternative case.
  • Page: May's Theorem — the four conditions, what giving up each one buys you, and the hinge into Arrow.
  • The point: with exactly two alternatives, majority rule is provably the unique reasonable rule — so FPTP is majority rule at two candidates and cannot be improved on there. Every spoiler, vote-split and wasted-vote pathology comes from running a two-candidate rule on three or more. That's a stronger and fairer critique than "FPTP is broken," and it reframes two-party dominance as the system being pushed back toward the only domain where it works.
  • Read as a pair with 301.17: majority rule is uniquely right for two alternatives (May), and cannot be coherently extended to three (Condorcet's paradox, then Arrow). Every method in this library is a proposal for what to do about that gap.
  • Keep it honest: May's theorem says nothing about which multi-candidate method is better — its content evaporates at three. Citing it to defend a particular reform is overreach. Useful sibling fact: at two candidates STAR, RCV-IRV, Ranked Robin and Approval all collapse to majority rule.
  • Also new here: the Duggan–Schwartz theorem — allowing tied winners does not escape Gibbard–Satterthwaite. You cannot tiebreak your way out of manipulability.

301.19 — Ties are forced (the small impossibility theorem)

  • Objective: stop treating ties as an engineering wart. Learn the theorem that makes them unavoidable, the arithmetic that says exactly when, and the four-way menu every real system has to pick from.
  • Key terms: resolute / irresolute SCF, nonimposition, reverse Borda, Moulin's proposition, the P₄ rotation profile, set extension principle.
  • Page: Ties Are Forced — the theorem, the witness profile (which this repo already runs), the four ways out and what each costs.
  • The point: anonymity + neutrality + Pareto are incompatible with always naming one winner whenever the voter count n has a divisor r with 1 < r ≤ m candidates (Moulin, 1983). Restated: ties are forced exactly when the smallest prime factor of n is at most m — so every even electorate has a forced tie, for any number of candidates. The design question was never "how do we avoid ties," it's "which axiom do we spend."
  • The repo payoff: the three engines here have each already answered it differently — LH breaks ties by a pre-published lot order (spends neutrality), BetterVoting breaks them at random (spends determinism, which is why a random BV tie can't be frozen into an export), and pref_voting returns the whole tied leader set (spends decisiveness, and by Duggan–Schwartz buys no strategyproofness for it). The LH-vs-BV divergence is three defensible answers to a forced choice, not a bug. And the "contrived" tie probes turn out to be the theorem's own witness profile.
  • Keep it honest: "forced" means some profile ties, not that ties are likely — exact ties stay astronomically rare in large electorates, and the theorem ranks no method above another (all are caught). It also costs only resoluteness, which has four known workarounds; do not cite it as though it were Arrow. Note too that the proposition is stated for ordinal rules — what carries over to STAR's score ballot is the symmetry argument, not the proposition itself.
  • Read with 301.18: May's positive responsiveness breaks every tie that isn't forced, which is why majority rule is resolute on an odd electorate and cannot be on an even one. The 50/50 split is this theorem's two-candidate face.

301.20 — Campbell–Kelly: May's theorem for three or more alternatives

  • Objective: finish the arc 301.17–301.19 with the one positive characterization result at three-plus candidates — and learn to state it at full strength and name its price, which is the whole skill.
  • Key terms: Pairwise Majority Rule (PMR), Condorcet domain, Condorcet extension, strategyproofness on a restricted domain, group strategyproofness, nonimposition.
  • Page: The Campbell–Kelly theorem — the statement, the May comparison, the one-line strategyproofness proof, and four ways it gets overstated.
  • The point: on the full domain there is no satisfactory extension of May's theorem to 3+ alternatives. Restrict to the Condorcet domain (profiles where a Condorcet winner exists) and uniqueness returns: PMR is resolute, anonymous, neutral and strategyproof there, uniquely so for odd n. The axiom swap isn't a swap — at two candidates monotonicity is strategyproofness, so Campbell–Kelly keeps May's fourth condition in the form that still has teeth at three.
  • The lesson that generalizes: it does not contradict Gibbard–Satterthwaite, because restricted-domain strategyproofness requires the sincere and manipulated profiles to stay inside the domain. So — G–S's bite comes from the full domain, not from having three or more candidates. Same shape as 301.19: insisting on a total function is what costs you the axiom.
  • The price tag: 301.22 states what Condorcet consistency costs on the other axis — every Condorcet extension provably fails reinforcement. Campbell–Kelly and that result are the two halves of one bargain.
  • Keep it honest: the restriction does enormous work — 𝒟_Condorcet is exactly the set of easy profiles, so the theorem is silent on cycles, which is where every real disagreement lives. It also cannot rank Ranked Robin above Minimax above Schulze (all agree on that domain by definition). And it lands against us: STAR is not a Condorcet extension, so a Condorcet advocate citing this theorem is citing it correctly — say so, then argue the axis rather than the fact.

301.21 — The cycle–cocycle decomposition: quality signal vs. circulation

  • Objective: learn the one linear-algebra theorem that explains why Borda-style and Condorcet-style methods can disagree — every margin graph is uniquely a quality signal plus a rock-paper-scissors circulation, and different methods read different parts.
  • Key terms: margin graph as a flow, basic cycle, basic cocycle, star / cut / potential difference, orthogonal decomposition, cycle strength λ, net outflow = symmetric Borda score.
  • Page: The cycle–cocycle decomposition — the two atoms of Zwicker's Figure 2.2, the theorem, which part each method reads, and the gelato loop decomposed to exact fractions.
  • The point: Borda is the cocycle part — its score is the margin graph's net outflow, and circulations have none, so they are invisible to it by construction — while Condorcet methods read the sign of each summed margin, which a strong enough circulation can point against the quality order. The gelato case in numbers: quality signal Berry > Almond > Cocoa (at most 4/3 of a vote on any pair), circulation 8/3 — every head-to-head sign follows the loop, Copeland ties all three, Borda alone reads the signal.
  • Keep it honest: the cycle part is not voter irrationality — even a single transitive ballot carries a half-strength circulation, and real electorates' circulations are usually too weak to flip any sign (Condorcet winners exist in the overwhelming majority of real ranked elections). And the decomposition crowns no method: treating each pairwise majority as a verdict (Copeland) and trusting margin sizes (Borda) are both principled readings — Borda's purity about the signal is also exactly the surface Condorcet attacked in 1788.
  • Read with 301.19: the forced-tie rotation profiles are the far end of this spectrum — quality signal exactly zero, pure circulation. And with cycle resolution: resolution rules are what Condorcet methods do about the cycle part once it has flipped a sign.

301.22 — The price of Condorcet: reinforcement, and what monotonicity really says

  • Objective: close the bargain 301.20 opened — see what Condorcet consistency costs, and learn to read a criterion claim sceptically enough to ask which version of the criterion is meant.
  • Key terms: reinforcement / consistency, voting situation, homogeneity, compound scoring rule, continuity (Archimedean property), lifting simply, resolute vs irresolute monotonicity, Peleg monotonicity, scoring run-off rule.
  • Pages: the reinforcement paradox, worked — the theorems and all three branches, runnable · monotonicity hub — the theorems behind the table.
  • The two theorems, as a matched pair:
  • Smith (1973) / Young (1975): the anonymous, neutral and reinforcing SCFs are exactly the compound scoring rules (a cascade of score vectors — ties under w₁ broken by w₂, and so on). Add continuity / the Archimedean property and the class narrows to the simple, one-vector scoring rules. An iff, not the usual one-directional "scoring rules pass."
  • Zwicker, Prop. 2.5: every Condorcet extension for three or more candidates violates reinforcement. No ceiling on the candidate count — it generalizes the tight three-candidate result of Brandt, Dong & Peters that this repo already ran.
  • The sharp consequence: since STAR fails reinforcement, STAR is provably not a compound scoring rule — no cascade of score vectors reproduces it, and the automatic runoff is exactly where it leaves the class. Its scoring round is a scoring rule and keeps the promise; the runoff forfeits it. A far more precise concession than "STAR isn't consistent."
  • The runnable payoff: the reinforcement folder's ballots are the profiles Zwicker's proof uses, and the proof's case analysis is now built out: whichever candidate the symmetric cycle resolves to, there is a South district agreeing with it and a merged electorate overturning it. Note why that mattered — LH is resolute, so it spends neutrality on a lot order and lands on one branch; a single pair would have been a lucky draw.
  • Monotonicity, said properly: "raise the winner" means lifting simply — x moves up while the relative order of every pair not involving x is unchanged. Smith (1980): every scoring run-off rule violates monotonicity — the general theorem behind Alaska 2022 and SF D7. STAR sits outside that class: Smith's family re-tallies scores after each elimination; STAR scores once, then compares two candidates pairwise.
  • Keep it honest — the criterion-scepticism lesson: the standard resolute monotonicity definition can be satisfied vacuously, by modifying any rule at all to add one tied alternative to every outcome. Peleg (1981) repairs it (x stays a winner and no new winners appear). So "method M satisfies criterion C" is only ever as strong as C's definition — ask which version is claimed. Files next to ordered majority rule as a second, literature-sourced case of a criterion that flatters by construction.

301.23 — What a method reads (the informational basis)

  • Objective: separate three questions that get conflated — which statistic a rule reads, how big that statistic is, and how hard it is to compute from.
  • Key terms: informational basis, tournament, weighted tournament, Fishburn C1/C2/C3, sufficient statistic.
  • Page: What a method reads. Runnable: Same matrix, different plurality — three 12-ballot electorates with the identical pairwise table and three different plurality winners, one per candidate.
  • The point: the debate-usable question is "how much of my ballot does your method actually look at?" — positive rather than an attack, and it makes vote-splitting precise instead of rhetorical.
  • Keep it honest: do not say plurality "needs more information" than Borda — the two statistics are incomparable, not nested, and C3 is a residual class, not a higher rung. The tiers are not a summability, difficulty, or quality ladder: plurality is C3 and the cheapest summable method there is; Kemeny is C2 and NP-hard; Copeland and Schulze publish the identical matrix. And STAR/Score/Approval have no Fishburn class at all — not a hedged one — because STAR is not a function of the ranked profile (two score profiles inducing the same pairwise matrix elect different STAR winners).

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