Reversal symmetry — when a method's "best" candidate is also its "worst"¶
A voting method has reversal symmetry if flipping every voter's ballot — so everyone now expresses the exact opposite preference, as if choosing the worst candidate — can never re-elect the original winner. RCV-IRV fails it: here's a 24-voter election where IRV elects A whether the voters are picking the best candidate or the worst. It's a real, IRV-specific defect — conceded — though the example is a deliberately hard one (a Condorcet cycle), and the argument comes from a Range-advocacy source whose lean we disclose.
→ Related: IRV non-monotonicity · the participation / no-show paradox · Condorcet cycles · glossary reversal symmetry.
The example¶
24 voters, three candidates, forming a Condorcet cycle (A beats B, B beats C, C beats A — nobody beats everyone):
Each row is one voter's ranking, most-preferred first (N: prefix = N identical ballots).
9:B>C>A
8:A>B>C
7:C>A>B
Original — RCV-IRV elects A (reversal_irv_original): C has the fewest first-choices, is eliminated, and C's ballots flow to A → A wins 15–9.
Now reverse every ballot (each voter expresses the exact opposite — as if electing the worst candidate) (reversal_irv_reversed): you'd expect A to finish last. Instead B is eliminated first, B's ballots flow to A, and A wins again, 16–8. IRV's "best" candidate and its "worst" candidate are the same one. That's a reversal symmetry failure.
Which methods fail it¶
| Method | Reversal symmetry | On this example |
|---|---|---|
| RCV-IRV, Plurality | ❌ fail | IRV elects A both ways (winner = loser) |
| Range / Score, Borda, Approval | ✅ pass | additive — reversing the ballots reverses the totals |
| Ranked Pairs, Schulze (Condorcet) | ✅ pass | — |
| STAR | avoids it here | original → B, reversed → A (star original · reversed) — a different winner each way, so no winner=loser |
| Ranked Robin | avoids it here | flags the cycle and resolves by margin |
A careful, honest note on STAR: the cases above show STAR avoiding the winner=loser on this electorate — not a proof that STAR satisfies reversal symmetry as a criterion. STAR is a hybrid (an additive scoring round plus a pairwise runoff), and the rank→score mapping used here (5/3/0) is a modeling choice. We claim only what we ran: STAR gives B then A, so its best and worst here differ.
Keep it in proportion — the fair reading¶
- The failure is real and IRV-specific — worth knowing, and a clean symptom of the same root cause as center squeeze and non-monotonicity: IRV eliminates on first-choice counts, so reversing preferences doesn't cleanly reverse the outcome.
- But the electorate is a deliberate Condorcet cycle — there is no "correct" winner (Gibbard territory), so this is an engineered worst case, not a "STAR right, IRV wrong" story. Every method is on hard ground in a cycle.
- Two of the source's other "disasters" are things we already cover. The page's "strategic reversal" twist (raising A from bottom to top makes A lose) is plain non-monotonicity; its "no-show" twist is the participation paradox. Nothing new there.
- Disclose the lean. The source is rangevoting.org — Warren Smith / the Center for Range Voting, an aggressively pro-Range advocacy site (advocacy-adjacent: strong for mechanics, weak for verdicts). Its framing ("psychotic," "insane," "silly myth") is polemic, and the criterion it prizes — like self-consistency — is one additive methods have and elimination methods lack, so the argument is really a case for pure Score (it would fault STAR's runoff too, in principle). We include it because the library checks pro-Range anti-IRV arguments by the same standard it checks the pro-RCV ones. (The page also cites "Peru 2006" as a real instance; we haven't verified that and don't repeat it as fact.)
Reproduce it¶
python3 STARVote_LH_tabulation_engine/starvote_larry_hastings.py method_comparisons/reversal_symmetry/cases/reversal_irv_original.yaml
python3 STARVote_LH_tabulation_engine/starvote_larry_hastings.py method_comparisons/reversal_symmetry/cases/reversal_irv_reversed.yaml