Paradoxes & whoopses â when methods disagree đ¶
A small gallery of low-stakes elections with surprising results â the kind where the same ballots hand the win to a different candidate depending on the method. They're fun ("ha! three winners from one set of ballots!"), they're great teaching moments, and â read with care â they keep us honest.
The even-handedness pledge. These examples are not a stick for beating one method. Every voting method fails some fairness criterion (that's Arrow & Gibbard â no method escapes), so this folder makes a point of letting each camp take a turn as the punchline. If you catch yourself only collecting IRV's faults, that's the bias to resist â add the matching STAR, Approval, and Condorcet whoopses too. Read Reading these fairly â the test for an honest "whoops" before adding any case.
The cases¶
| Case | Lesson | The whoops | Who takes the hit | Live |
|---|---|---|---|---|
| BV2155 | Tennessee â four ways, three winners | one ballot set â Plurality=Memphis, IRV=Knoxville, STAR & Ranked Robin=Nashville | Plurality & IRV | results â |
| BV2156 | STAR misses the Condorcet winner | the head-to-head winner is too low-scored to reach the runoff | STAR | results â |
| BV2157 | a Condorcet cycle (rock-paper-scissors) | majority rule is intransitive â no Condorcet winner exists | Condorcet / Ranked Robin | results â |
| BV2158 | IRV buries the centrist (Ossipoff) | the plurality and Condorcet winner is eliminated; IRV elects D | IRV | results â |
| BV2159 | many IRV pathologies in one (Brams) | Condorcet failure + no-show + truncation + non-monotonicity | IRV | results â |
| BV2183 | Forced Exhaustion Ceiling | (constructed ceiling) a 2-rank cap force-exhausts 21 of 50 ballots â more than the winner's own votes | IRV | results â |
Cases 01â03 are STAR/score files (engine-verified, in the test library); 04â05 are ranked-ballot RCV-IRV cases (IRV rounds engine-verified, Condorcet winners checked by pairwise tally). Each carries a fairness box.
Balance ledger¶
The even-handedness pledge isn't a vibe â it's a count. Keep this honest, and when it tilts, the next case should hit the over-represented direction's opposite.
| Method | Times it's the punchline | Cases |
|---|---|---|
| Plurality | 1 | 01 |
| RCV-IRV | 3 | 01, 04, 05 |
| STAR / score | 1 | 02 |
| Condorcet / Ranked Robin | 1 | 03 |
| Approval | 0 | â |
Balance owed (as of cases 01â05): the folder leans IRV (3). The next additions should embarrass the score family â Approval (0 so far) and STAR â not IRV. Candidate follow-ups: an Approval "bland-winner" / chicken-dilemma case, or a STAR non-monotonicity construction. Don't add another IRV whoops until this evens out.
How to read these (the short version)¶
A "whoops" is only worth teaching if it's an honest one. The full test is in Reading these fairly â the test for an honest "whoops", but the gist:
- Structural, not contrived â happens across a region of realistic electorates, not at one knife-edge with absurd weights.
- Sincere, not strategic â failures under honest voting are fair game; "but you can game it" applies to every method (Gibbard), so it's the weakest, muddiest angle.
- Realistic electorate â spatial / natural distributions, not alien voter behavior.
- Bonus: it really happened â Burlington 2009, Alaska 2022 aren't constructions.
All three cases here pass that test (Tennessee is canonical; the STAR miss and the Condorcet cycle are foundational, sincere-vote results) â which is itself the lesson: these aren't cheap gotchas.
The opposite of this page, if you want it: the Mudroom is a per-method criterion-failure gallery written the unfair way on purpose â deliberately mean and one-sided hit-reels (IRV, STAR, Ranked Robin, more), the cheap gotchas that don't pass the honest test above, curated for sport. It's kept low-profile and clearly labeled for exactly that reason; every fact in it is still real, and every reel links back here to the fair version. Fun, not for citing.
Live on BetterVoting (BV2155âBV2159)¶
Every case in this gallery is now a real BetterVoting election (created 2026-07; STAR is race 1 in each, per house style, with the comparison methods as further races on the same ballots). All 16 races match an independent count from Larry Hastings' starvote, extended in this repo (the "LH" engine), exactly â the contrast here is method vs method, and BetterVoting and the engine agree on every count. Frozen exports sit beside each case as bv215N_<bvid>_bv_export.json.
Run them yourself¶
cd STARVote_LH_tabulation_engine
python starvote_larry_hastings.py "../method_comparisons/paradoxes_and_whoops/cases/bv2155_cphxpt_tennessee_four_ways.yaml"
Each writes a full audit copy to paradoxes_and_whoops_tabulated/. All three also live as flat-schema positive test cases in YAML_library/1_positive/ (each verifies the STAR winner â the gallery is about disagreement, so the test pins STAR's answer, and the lesson narrates the others).
Books on voting paradoxes¶
Voting theory is rife with mathematical anomalies, and several books catalogue them:
- Hannu Nurmi â Voting Paradoxes and How to Deal with Them (1999). The reference manual: systematically catalogues the no-show paradox, monotonicity failures, Condorcet cycles, etc., and rates how vulnerable each method is to each.
- Donald G. Saari â Disposing Dictators, Demystifying Voting Paradoxes (2008). A geometric take on social choice â argues Arrow's theorem and the paradoxes reflect positional systems discarding information, not democracy being broken.
- William Poundstone â Gaming the Vote: Why Elections Aren't Fair (2008). The general-audience pick: history of paradoxes and strategic voting, arguing independent- score methods (Approval, Score) sidestep paradoxes inherent to strict ranking like IRV.
- Michel Balinski & Rida Laraki â Majority Judgment (2011). Argues that turning individual rankings into a collective ranking makes Arrow/Condorcet paradoxes inevitable, and proposes grading candidates on a common scale instead.
More whoopses to mine (sources)¶
- rangevoting.org (Warren Smith, the Center for Range Voting) â a deep trove of method-failure constructions and paradoxes.
- Burlington 2009, Alaska 2022 (real center-squeeze elections â see Center Squeeze).
- Classic social-choice paradoxes (Condorcet, Arrow, no-show, monotonicity).
Adding one? Run it through the engine, write the fairness box, and make sure the camp it embarrasses isn't always the same one.