Topic: Monotonicity (more support shouldn't hurt you)¶
Topic hub — a cross-method view. A method is monotonic if ranking or scoring the eventual winner higher can never cause them to lose (and moving a loser down can never make them win). It's the property that makes "vote your honest favorite first" safe.
The one idea to take away: non-monotonicity comes from sequential elimination, not from ranked ballots. RCV-IRV (Hare) — and the other eliminate-and-transfer variants — can punish a candidate for gaining support, because added first-choices change who is eliminated when. Methods that read the whole ballot at once (Ranked Robin, STAR) don't have this hole.
Which methods are monotonic — and where each is treated¶
| Method | Monotonic? | Why | Full page |
|---|---|---|---|
| STAR | ✅ | scores are added, not eliminated — raising a candidate only helps them | STAR monotonicity |
| Ranked Robin / Condorcet | ✅ | pairwise wins only improve when you rank someone higher | Ranked Robin |
| Approval / Score | ✅ | more approval/points can't hurt | scoring methods |
| RCV-IRV (Hare) | ❌ | added first-choices can change the elimination order and flip the winner | IRV non-monotonicity |
| Other IRV variants (BTR, Coombs, Baldwin, Nanson) | ❌ | same cause — they still eliminate round by round | Which RCV-IRV? |
So unlike center squeeze (which is Hare-specific), non-monotonicity is shared by all the sequential-elimination methods — only the non-eliminating methods (STAR, Ranked Robin) escape it.
The theorems behind that table¶
Everything above is stated as observed behaviour. It is all theorem-backed, and knowing the theorems tells you where the boundary actually runs.
Saying it precisely first — "lifting simply." The loose phrasing "raise the winner on some ballots" hides a condition that has to be there. The precise move (Fishburn, 1982) is that a voter lifts x simply: x moves from below one or more candidates to above them, and the relative order of every pair that doesn't involve x is left untouched. A resolute SCF is monotonic if lifting the winner x simply always leaves x the winner. The italicised clause is what makes the criterion mean anything — without it a "lift" could smuggle in arbitrary other reordering, and a flipped result would prove nothing about support for x.
Why the eliminating methods fail, as a theorem, not a collection of examples. Smith (1980) proved that every scoring run-off rule violates monotonicity — the whole family at once, not RCV-IRV in particular. That is the general statement behind this hub's "shared by all the sequential-elimination methods," and it covers plurality run-off, STV, and the Borda-elimination variants (Baldwin, Nanson) as well as Hare. The Alaska 2022 and San Francisco D7 cases are not unlucky elections; they are instances of a rule the theorem says must have them.
Why the others pass, also as a theorem. There's a clean sufficient condition for score-maximising rules: if lifting x simply can never lower x's score and never raise anyone else's, the rule is monotonic — and stays monotonic if a fixed ordering breaks its ties. That single argument covers Copeland (= Ranked Robin), Simpson/minimax, and every proper scoring rule (Plurality, Borda, k-approval and the rest). It also settles the boundary question for STAR: STAR is not a scoring run-off rule in Smith's sense. Smith's family eliminates a candidate and re-tallies the scores, round after round; STAR scores once, takes the top two, and finishes with a pairwise comparison. It never re-scores, so it sits outside the class the theorem condemns — which is the formal version of this hub's "scores are added, not eliminated."
And a warning about the criterion itself. The resolute definition above can be satisfied vacuously: take any SCF at all, modify it to add one tied alternative to its outcome on every profile, and the "the winner must not change" test can no longer bite. A criterion that a trivial cosmetic change can buy is not measuring what it appears to. Peleg (1981) proposed the repair, and it is the version worth quoting: after any simple lift of a winning x, x remains a winner and no new winners are added. Sanver and Zwicker (2012) argue for exactly this form — it resists the trick, and it is also a better fit for what monotonicity is for, since insisting the winning set not change at all is stricter than "the output should move in the same direction as the input." Copeland, Simpson, the proper scoring rules, sequential majority comparison and Top Cycle all satisfy Peleg's stronger version too.
This is worth filing next to the repo's other criterion-scepticism material: "method M satisfies criterion C" is only as strong as C's definition, and here is a case, straight from the literature, where the standard definition is gameable by construction. Ask which version of a criterion is being claimed before crediting it.
Worked real examples — both flavours, both real: - Upward ("more is less" — raise the winner, she loses): Alaska 2022. Ranking the winner Peltola higher would have made her lose. - Downward ("less is more" — lower a loser, they win): San Francisco D7 2020. Ranking the loser Engardio lower would have made him win.
Both are reproduced on the real ballots, and in both, Ranked Robin elects the Condorcet winner — unmoved — while RCV-IRV flips.
Glossary: monotonicity · lifting simply · Peleg monotonicity.
Sources¶
- J. H. Smith, "Aggregation of preferences with variable electorate," Econometrica 41 (1973); and the 1980 result that every scoring run-off rule violates monotonicity. Lean: neutral.
- P. C. Fishburn (1982) — the standard "lifting simply" formulation. · B. Peleg (1981) — the irresolute strengthening. · M. R. Sanver & W. S. Zwicker (2012) — the argument for Peleg's version. Lean: neutral.
- William S. Zwicker, "Introduction to the Theory of Voting," in Handbook of Computational Social Choice (CUP 2016), §2.6 — Definition 2.10 and the surrounding discussion, including the vacuous-satisfaction problem and the score-based sufficient condition. Lean: neutral; the standard academic reference.
This is a topic hub (cross-method index). The authoritative write-ups live in the per-method folders linked above. See the topics index for the other topic hubs.