Voting paradoxes — Felsenthal's catalog, demonstrated¶

This folder teaches the classic voting paradoxes one page at a time, each backed by a runnable repo election and (where possible) a live BetterVoting election. Cases are tagged in their YAML (paradoxes: [tag, …]) and the auto-generated registry groups every tagged case by paradox: the paradox registry.

Primary source: Dan S. Felsenthal, "Review of Paradoxes Afflicting Various Voting Procedures Where One Out of m Candidates (m ≥ 2) Must Be Elected", University of Haifa / Centre for Philosophy of Natural and Social Science, LSE; revised 26 May 2010. Presented at the Leverhulme Trust Voting Power in Practice workshop, Château du Baffy, Normandy, 30 July – 2 August 2010.

Felsenthal's two kinds of paradox¶

Felsenthal splits paradoxes into two families, and the split is worth learning before any individual paradox:

Simple (or "straightforward") paradoxes — the relevant data as given leads to a surprising, arguably undesirable outcome. The "relevant data" include the number of voters and candidates, how many must be elected, every voter's preference ordering, what voters know about each other's preferences, the order of voting, whether voting is open or secret, and how ties are broken. Nothing changes; the result is startling on its own. Example: a candidate whom a majority ranks last wins.

Conditional paradoxes — changing one relevant datum while holding everything else constant changes the outcome in a surprising way. Examples: a hopeless candidate drops out and the winner changes; a candidate gains support and thereby loses.

Paradox pages (covered)¶

Paradox	Also known as	Kind	Page	Demonstrated by
Condorcet winner paradox	Thwarted majorities	Simple	Condorcet winner paradox	BV2144 Felsenthal Ex.1 · BV2137 center squeeze
Condorcet loser paradox	Borda's paradox	Simple	Condorcet loser paradox	BV2144 Felsenthal Ex.1
Absolute loser paradox	—	Simple	Absolute loser paradox	BV2144 Felsenthal Ex.1
SCC / spoiler	Subset choice condition; IIA failure (informal)	Conditional	SCC / spoiler	BV2144 Felsenthal Ex.1 · BV2145 Felsenthal Ex.2 · the split-voting set
Non-monotonicity	More-is-less; additional-support; lack of monotonicity	Conditional	Non-monotonicity	BV2145→BV2146 Felsenthal Ex.2 pair · the monotonicity set
Reinforcement / multiple-districts	Inconsistency paradox	Conditional	Reinforcement / multiple-districts	BV2147+BV2148→BV2149 Felsenthal Ex.3 trio
No-Show + Twin	Participation failure	Conditional	No-Show + Twin	BV2150→BV2151 Felsenthal Ex.4 pair · BV2165→BV2166 (STAR flips too) · BV2159 (Brams)
Truncation paradox	Preference-truncation	Conditional	Truncation paradox	BV2162→BV2163 Nurmi pair · BV2160 Borda flavor

Covered elsewhere in the repo (tagged, no dedicated page yet)¶

Paradox	Also known as	Kind	Repo home
Condorcet's paradox (cycle)	Cyclical majorities	Simple	BV2157 (rock-paper-scissors) · cycle resolution
Clone dependence / teaming	Cloning paradox	Conditional	Ranked Robin & clone independence
Favorite betrayal	—	Conditional	Favorite betrayal
Absolute Majority paradox	Majority criterion failure	Simple	BV2153 Felsenthal Ex.7
Pareto-dominated winner	Pareto violation	Simple	Felsenthal Ex.6 (LH-only)

Deliberately not listed: the runoff reversal¶

The most-asked-about omission. STAR's automatic runoff overturning the scoring leader — "the top-scorer lost!" — is not in this catalog or the registry, on purpose. In Felsenthal's terms nothing paradoxical has happened: no datum changed, and the elected candidate is the finalist a majority of deciding voters prefers — which is exactly the job the "AR" in STAR exists to do. It makes a fine attack line (the mudroom collection keeps it honest), but a designed mechanism doing its designed job is a feature demonstration, and it is taught as one: when the runoff overturns the scoring leader.

The same bar keeps strategy backfires out. Tactical maximization — hedged 5s silencing the hedgers' own voice in the runoff — is a criteria lesson, not a paradox: the backfire is even monotone (the raised score helps exactly the candidate it was given to). STAR earns genuine registry entries where it genuinely fails; the no-show pair above catches STAR flipping too, and those cases are tagged like any other. The paradoxes: vocabulary itself is controlled by build_paradox_index.py, which fails the build on an unknown tag rather than minting a new paradox by accident — extend its VOCAB only for a Felsenthal-style pathology, never for a feature demonstration or a strategy backfire (settled 2026-08-09, when a stray runoff-reversal tag was removed rather than adopted).

Minimal instances and how our examples compare¶

Our worked examples come from Felsenthal (2010) and are chosen for teaching clarity — small, memorable, named casts — not for provable minimality. Brandt, Matthäus & Saile, "Minimal voting paradoxes", Journal of Theoretical Politics 34(4), 2022 (DOI · open PDF) settle the companion question with integer-linear-programming: the smallest electorate — (candidates m, voters n) — that forces each paradox, for 13 ranked rules. Cross-checking our pages against their Table 3:

Our example	Rule · paradox	Ours (m, n)	Paper minimal	Status
BV2144 (Felsenthal Ex.1)	Plurality · Condorcet winner	3, 7	3, 7	✅ provably minimal
BV2144	Plurality · Condorcet loser	3, 7	3, 7	✅ provably minimal
BV2144	Plurality · absolute loser	3, 7	3, 7	✅ provably minimal
BV2144	Plurality · SCC / subset choice	3, 7	3, 7	✅ provably minimal
BV2145→46 (Ex.2)	IRV · non-monotonicity (add'l support)	3, 17	3, 17	✅ provably minimal (at 3 cand.)
BV2150→51 (Ex.4)	IRV · no-show	3, 11	3, 11	✅ provably minimal
Coombs Ex.19 (BV2165/66)	Coombs · no-show	3, 15	3, 15	✅ provably minimal (at 3 cand.)
BV2147/48→49 (Ex.3)	Plurality-runoff · reinforcement	3, 32	3, 15	larger (Felsenthal's electorate)
Coombs Ex.17 (BV2164)	Coombs · Condorcet winner	4, 33	3, 13 / 4, 9	larger — and refutes a conjecture (below)
Coombs Ex.20	Coombs · reinforcement	3, 41	3, 13	larger
Coombs Ex.22	Coombs · SCC	4, 29	4, 7	larger
Minimax Ex.29 (BV2167)	Maximin · Condorcet & absolute loser	4, 11	4, 9	slightly larger
Minimax Ex.30	Maximin · no-show	4, 19	4, 9	larger
Minimax Ex.32	Maximin · reinforcement	4, 14	3, 15 / 5, 9	comparable
Minimax Ex.33	Maximin · SCC	4, 7	3, 7	one candidate more

Two takeaways. (1) The BV2144 quartet and the monotonicity/no-show pairs are already the mathematically smallest possible — a nice, citable validation that the sharpest teaching cases are also the tightest. (2) The larger cases are Felsenthal's own historical electorates; we keep them for their named casts and live BV backing, but now have a reference for how far they could shrink. The paper studies only ranked rules and Felsenthal's nine paradoxes, so our truncation pages (score/ranked, and 103-voter Nurmi), range voting and majority judgment (score methods), and successive-elimination examples fall outside its scope — no minimal to compare against.

Planned (from the wider literature)¶

Ostrogorski's paradox, Anscombe's paradox, the paradox of multiple elections, dedicated pages for the Absolute-Majority and Pareto-dominated paradoxes, and Simpson's paradox — each gets a page + a tagged case as examples are built.

Every worked-tables page is now runnable. Four procedures that no engine here could count — Coombs (§A7), Minimax (§A10), successive elimination (§A4), and the two grade methods, Range Voting (§A8) and Majority Judgment (§A9) — have tabulators in tools_adam/pref_voting_tabulation_engine/, each cross-checked against pref_voting where a counterpart exists:

Page	Tool	Levers it exposes
coombs.md	`coombs_report.py`	`--drop` (SCC)
minimax.md	`minimax_report.py`	`--drop`, `--equal-prob` (truncation convention)
successive_elimination.md	`successive_elimination_report.py`	`--agenda` (path independence), `--tiebreak`, `--drop`
range_voting.md · majority_judgment.md	`grade_methods_report.py`	`--ungrade` (truncation), `--abstain` (no-show)

These pages are the case against a method — start with the method. Each of the two grade procedures has a front door that shows its ballot, its count worked on a small election, and its pros and cons: Range / Score Voting and Majority Judgment. Landing on a paradox first is landing in the middle of an argument.

Every example on those pages is a case file, ballots or grades in YAML rather than in a parenthetical. All 27 counts reproduce Felsenthal's published results, with three documented exceptions that the pages state rather than smooth over: Minimax Ex.31's winner depends on a truncation convention; successive elimination Ex.10's amalgamated round 1 is a 2:2 tie, not the win the page used to assert, so its reinforcement failure holds only under one tie-break reading; and MJ Ex.26 has no case file at all, because its grade tables live in the source and were never reproduced here.

The summability demo is the operational cousin of reinforcement (district-by-district counting) and is cross-linked from multiple_districts.md.