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Paradox index — test cases grouped by voting paradox

Auto-generated by tools_adam/scripts/build_paradox_index.py from each case YAML's paradoxes: tags — do not edit by hand. Taxonomy and teaching pages: voting_paradoxes/README.md.

192 tagged case(s), 14 paradox(es).

Condorcet winner paradox

Simple paradox (Felsenthal). The candidate who beats every rival head-to-head is not elected (a.k.a. thwarted majorities; center squeeze is the IRV route).

Case Method Expected winner(s) BV YAML
Favorite betrayal in STAR (1 of 2) — honest ballots: the tepid consensus misses the runoff STAR Clover BV2206 bv2206_7mckyg_fbc_honest_tepid_consensus.yaml
STAR vs RR divergence -- 3 cands, 15 voters, cycle (STAR A, RR B) STAR A cycle_C03_fewV15_noise_1.yaml
STAR vs RR divergence -- 3 cands, 15 voters, cycle (STAR A, RR C) STAR A cycle_C03_fewV15_noise_2.yaml
STAR vs RR divergence -- 3 cands, 45 voters, cycle (STAR A, RR B) STAR A cycle_C03_medV45_noise_1.yaml
STAR vs RR divergence -- 5 cands, 15 voters, cycle (STAR A, RR E) STAR A cycle_C05_fewV15_noise_2.yaml
STAR vs RR divergence -- 5 cands, 28 voters, cycle (STAR A, RR E) STAR A cycle_C05_fewV28_bloc_1.yaml
STAR vs RR divergence -- 5 cands, 45 voters, cycle (STAR A, RR B) STAR A cycle_C05_medV45_noise_1.yaml
STAR vs RR divergence -- 5 cands, 45 voters, cycle (STAR D, RR B) STAR D cycle_C05_medV45_noise_2.yaml
STAR vs RR divergence -- 7 cands, 15 voters, cycle (STAR D, RR A) STAR D cycle_C07_fewV15_noise_1.yaml
STAR vs RR divergence -- 7 cands, 598 voters, cycle (STAR C, RR E) STAR C cycle_C07_largeV598_bloc_1.yaml
STAR vs RR divergence -- 7 cands, 149 voters, cycle (STAR F, RR C) STAR F cycle_C07_medV149_bloc_2.yaml
STAR vs RR divergence -- 10 cands, 15 voters, cycle (STAR A, RR C) STAR A cycle_C10_fewV15_noise_1.yaml
STAR vs RR divergence -- 10 cands, 15 voters, cycle (STAR J, RR F) STAR J cycle_C10_fewV15_noise_2.yaml
STAR vs RR divergence -- 10 cands, 28 voters, cycle (STAR C, RR F) STAR C cycle_C10_fewV28_bloc_1.yaml
STAR vs RR divergence -- 10 cands, 29 voters, cycle (STAR C, RR B) STAR C cycle_C10_fewV29_bloc_2.yaml
STAR vs RR divergence -- 10 cands, 149 voters, cycle (STAR I, RR H) STAR I cycle_C10_medV149_bloc_2.yaml
STAR vs RR divergence -- 10 cands, 45 voters, cycle (STAR E, RR G) STAR E cycle_C10_medV45_noise_1.yaml
STAR vs RR divergence -- 10 cands, 45 voters, cycle (STAR A, RR I) STAR A cycle_C10_medV45_noise_2.yaml
STAR vs RR divergence -- 3 cands, 15 voters, darkhorse (STAR A, RR C) STAR A darkhorse_C03_fewV15_noise_1.yaml
STAR vs RR divergence -- 5 cands, 599 voters, darkhorse (STAR A, RR E) STAR A darkhorse_C05_largeV599_bloc_1.yaml
STAR vs RR divergence -- 7 cands, 30 voters, darkhorse (STAR D, RR C) STAR D darkhorse_C07_fewV30_bloc_1.yaml
STAR vs RR divergence -- 7 cands, 597 voters, darkhorse (STAR D, RR E) STAR D darkhorse_C07_largeV597_bloc_1.yaml
STAR vs RR divergence -- 7 cands, 598 voters, darkhorse (STAR E, RR G) STAR E darkhorse_C07_largeV598_bloc_2.yaml
STAR vs RR divergence -- 7 cands, 147 voters, darkhorse (STAR F, RR D) STAR F darkhorse_C07_medV147_bloc_1.yaml
STAR vs RR divergence -- 7 cands, 45 voters, darkhorse (STAR E, RR A) STAR E darkhorse_C07_medV45_noise_1.yaml
STAR vs RR divergence -- 10 cands, 598 voters, darkhorse (STAR G, RR F) STAR G darkhorse_C10_largeV598_bloc_1.yaml
STAR vs RR divergence -- 10 cands, 599 voters, darkhorse (STAR B, RR E) STAR B darkhorse_C10_largeV599_bloc_2.yaml
Alaska 2022 US House special (reduced model) — STAR & Ranked Robin elect Begich, the Condorcet winner RCV-IRV cut STAR Begich BV2213 bv2213_k3fmwv_alaska_2022.yaml
Alaska 2022 US House general (reduced model) — IRV got it right: all four counts elect the Condorcet winner, Peltola STAR Peltola BV2214 bv2214_m3hb6y_alaska_2022_general.yaml
Hamlin & Hua §4.1 — the approval count as printed: B wins with 100% Approval B hh41_01_approval_as_printed.yaml
Hamlin & Hua §4.1 — the assumed preferences, counted pairwise: A is the Condorcet winner RankedRobin A hh41_02_preferences_ranked_robin.yaml
Hamlin & Hua §4.1 — the same marks read pairwise: 60 of 100 voters express no preference STAR B hh41_03_marks_read_pairwise.yaml
Hamlin & Hua §4.1 — their own utility stipulation, on a 0-5 ballot: A 380, B 370, A wins STAR A hh41_04_stipulated_utilities_star.yaml
Burlington 2009 mayor — RCV-IRV: the real center squeeze (Kiss wins, the Condorcet winner is third) RCV_IRV Kiss burlington_2009_irv.yaml
Center squeeze (RCV-IRV) — minimal 27-voter case (the moderate is eliminated) RCV_IRV Left center_squeeze_irv.yaml
Center squeeze — STAR elects the consensus (Center) STAR Center center_squeeze_star.yaml
Center squeeze — the voteline 1D spectrum (Red / Green / Yellow) STAR Green center_squeeze_voteline_1d.yaml
Center Squeeze — IRV (Hare): the center gets squeezed out IRV Carter BV2137 bv2137_ywckmg_irv.yaml
Center Squeeze — Ranked Robin (Copeland): the Condorcet winner survives RankedRobin Anderson BV2137 bv2137_ywckmg_ranked_robin.yaml
Center Squeeze — STAR (ranks→scores): agrees with Condorcet, not IRV STAR Anderson BV2137 bv2137_ywckmg_star.yaml
Center Squeeze — STV (1 seat = IRV single-winner): same squeeze STV Carter BV2137 bv2137_ywckmg_stv.yaml
Chicken / Burr dilemma — STAR resolves it (allies A & B beat C; A wins honestly) STAR A chicken_star.yaml
Dark Horse — STAR elects the honest winner A (Borda would elect the dark horse D) STAR A dark_horse_star.yaml
Edelman's 'Myth of the Condorcet Winner' 81 voters — the score count says Ben, the majorities say Ada STAR Ada BV2173 bv2173_gmfv4c_edelman_saari_cancellation.yaml
FairVote's own hypothetical (45/12/43) — the moderate IS the majority's pairwise choice STAR Moderate BV2168 bv2168_6w2gq7_fairvote_40_15_40_moderate_cw.yaml
FairVote white paper — French 2017 (HONEST): STAR elects the centrist Macron STAR Macron BV2229 bv2229_7j2bqf_french_2017_honest.yaml
FairVote white paper — French 2017 (BURIAL): coordinated strategy squeezes Macron out STAR Melenchon BV2230 bv2230_2hqmrd_french_2017_burial.yaml
FairVote white paper — Washington 2010 (HONEST): STAR elects the moderate Berkey STAR Berkey BV2231 bv2231_b4yr3v_wa_2010_honest.yaml
FairVote white paper — Washington 2010 (BURIAL): the squeeze works on STAR, but IRV resists it STAR Harper BV2232 bv2232_24b623_wa_2010_burial.yaml
Favorite betrayal (RCV-IRV) — HONEST: Left voters rank their favorite first, and their WORST wins RCV_IRV Right BV2227 bv2227_3xgkck_honest_irv.yaml
Favorite betrayal — the fix: STAR & Ranked Robin elect Center from the HONEST ballots STAR Center BV2227 bv2227_3xgkck_honest_star.yaml
Favorite betrayal (RCV-IRV) — BETRAY: 2 Left voters rank their favorite SECOND, and win RCV_IRV Center BV2228 bv2228_bgcmxx_betray_irv.yaml
Felsenthal Ex.1 — Choose-One (Plurality): the absolute loser wins Plurality Ana BV2144 bv2144_mxfmhm_plurality.yaml
Felsenthal Ex.1 — STAR (ranks→scores): elects the Condorcet winner STAR Bo BV2144 bv2144_mxfmhm_star.yaml
Felsenthal Ex.2 — Runoff/IRV: the Condorcet winner is eliminated first IRV Ben BV2145 bv2145_6fj2kg_irv.yaml
Felsenthal Ex.2 — Ranked Robin: the Condorcet winner survives RankedRobin Ada BV2145 bv2145_6fj2kg_ranked_robin.yaml
Felsenthal Ex.2 — STAR (ranks→scores): elects the Condorcet winner STAR Ada BV2145 bv2145_6fj2kg_star.yaml
Felsenthal Ex.4 full electorate — Runoff/IRV: their worst choice wins IRV Carl BV2150 bv2150_dxg8pb_irv.yaml
Felsenthal Ex.4 full electorate — Ranked Robin: the Condorcet winner Beth RankedRobin Beth BV2150 bv2150_dxg8pb_ranked_robin.yaml
Felsenthal Ex.4 full electorate — STAR: elects the Condorcet winner Beth STAR Beth BV2150 bv2150_dxg8pb_star.yaml
Felsenthal Ex.5 — Approval: the Condorcet winner loses the approval count Approval Anna BV2152 bv2152_r6ctvy_approval.yaml
Felsenthal Ex.5 — Ranked Robin: the Condorcet winner Bert RankedRobin Bert BV2152 bv2152_r6ctvy_ranked_robin.yaml
Felsenthal Ex.8 — IRV: a third winner, Clara IRV Clara BV2154 bv2154_wq6yv7_irv.yaml
Nurmi Ex.16 sincere — RCV-IRV: full rankings elect A, not the Condorcet winner IRV A BV2162 bv2162_4htk44_irv.yaml
Nurmi Ex.16 sincere — Ranked Robin: the Condorcet winner B RankedRobin B BV2162 bv2162_4htk44_ranked_robin.yaml
Nurmi Ex.16 sincere — STAR: elects the Condorcet winner B STAR B BV2162 bv2162_4htk44_star.yaml
Coombs Ex.17 — Choose-One: Bree on first choices Plurality Bree BV2164 bv2164_xbqq8t_plurality.yaml
Coombs Ex.17 — Ranked Robin: the Condorcet winner Arlo RankedRobin Arlo BV2164 bv2164_xbqq8t_ranked_robin.yaml
Coombs Ex.17 — STAR: Bree tops the scores, the Condorcet winner Arlo wins the runoff STAR Arlo BV2164 bv2164_xbqq8t_star.yaml
Coombs Ex.18 — Bree is raised on four ballots and loses because of it RCV_IRV Arlo coombs_ex18_monotonicity.yaml
Felsenthal Ex.6 — Approval can elect a Pareto-dominated candidate (LH-only) Approval Beau felsenthal_ex6_pareto_approval.yaml
MJ Ex.25 — the median elects y though a majority grades x higher ? mj_ex25_absolute_winner_loses.yaml
Range Ex.23 — four paradoxes with two candidates: one enthusiast outvotes three majorities ? range_ex23_mean_overrules_majority.yaml
Successive elimination Ex.11 — after: a twin joins and costs their own side the win RankedRobin B succ_elim_ex11_twin_after.yaml
Successive elimination Ex.9 — two D voters stay home and get a better result RankedRobin A succ_elim_ex9_noshow.yaml
Hillinger Table 4 — three methods, three winners (EV-3 scale) STAR Ana hillinger_t4_ev3.yaml
Minority winner — 34% wins Choose-One, but STAR & Ranked Robin elect the majority's real choice (BV2215, 2p33qq) STAR Cleo BV2215 bv2215_2p33qq_minority_winner.yaml
Tennessee capital, four ways — same ballots; Memphis, Knoxville or Nashville by count (BV2155) STAR Nashville BV2155 bv2155_cphxpt_tennessee_four_ways.yaml
STAR misses the Condorcet winner — STAR's own signature failure (BV2156) STAR Ada BV2156 bv2156_3grpbb_star_misses_condorcet.yaml
IRV buries the centrist — Ossipoff's 303-voter one-dimensional example (BV2158) RCV_IRV BV2158 bv2158_gr72hd_ossipoff_centrist_irv.yaml
Many IRV pathologies in one election — Brams' 21-voter sampler (BV2159) RCV_IRV BV2159 bv2159_f4cjpy_brams_irv_pathologies.yaml
The Post-it switch, made real — RCV-IRV: Green eliminated, Blue 10, Purple 9 IRV Blue BV2178 bv2178_8kg698_irv.yaml
The Post-it switch, made real — Ranked Robin: Blue, now the outright Condorcet winner RankedRobin Blue BV2178 bv2178_8kg698_ranked_robin.yaml
The Post-it switch, made real — STAR: Blue, now also the score co-leader STAR Blue BV2178 bv2178_8kg698_star.yaml
Preference vs Support — the center TOLERATED (wings score Blair 1) STAR Alex BV2225 bv2225_ywx39y_center_tolerated.yaml
Preference vs Support — the center SUPPORTED (wings score Blair 4) STAR Blair BV2226 bv2226_82gg36_center_supported.yaml
Condorcet centrist, minimal form (8 voters) — STAR elects Casey STAR Casey BV2171 bv2171_h93tm4_star.yaml
Condorcet centrist, full form (100 voters, 47/47/3/3) — STAR elects Casey STAR Casey BV2172 bv2172_bkwfjr_star.yaml
Symmetric centrist (47/47/3/3) — RCV-IRV: the centrist is squeezed, the poles deadlock IRV Avery BV2170 bv2170_pp2q4q_irv.yaml
Symmetric centrist (47/47/3/3) — Choose-One: the poles tie, the centrist gets 6 Plurality Blake BV2170 bv2170_pp2q4q_plurality.yaml
Symmetric centrist (47/47/3/3) — Ranked Robin: elects Casey, beats everyone head-to-head RankedRobin Casey BV2170 bv2170_pp2q4q_ranked_robin.yaml
Symmetric centrist (47/47/3/3) — STAR: elects Casey, the Condorcet winner STAR Casey BV2170 bv2170_pp2q4q_star.yaml

Condorcet loser paradox

Simple paradox (Felsenthal). The candidate who loses to every rival head-to-head is elected (a.k.a. Borda's paradox).

Case Method Expected winner(s) BV YAML
Felsenthal Ex.1 — Choose-One (Plurality): the absolute loser wins Plurality Ana BV2144 bv2144_mxfmhm_plurality.yaml
Felsenthal Ex.1 — STAR (ranks→scores): elects the Condorcet winner STAR Bo BV2144 bv2144_mxfmhm_star.yaml
Felsenthal Ex.8 — Approval: the absolute loser wins on one strategic ballot Approval April BV2154 bv2154_wq6yv7_approval.yaml
Felsenthal Ex.8 — Ranked Robin: the Condorcet winner Bruce RankedRobin Bruce BV2154 bv2154_wq6yv7_ranked_robin.yaml
Minimax Ex.29 — Choose-One: agrees with Minimax, elects the absolute loser D Plurality D BV2167 bv2167_f3dxq9_plurality.yaml
Minimax Ex.29 — STAR: elects B, a top-cycle member, not the universal loser STAR B BV2167 bv2167_f3dxq9_star.yaml
MJ Ex.25 — the median elects y though a majority grades x higher ? mj_ex25_absolute_winner_loses.yaml
Range Ex.23 — four paradoxes with two candidates: one enthusiast outvotes three majorities ? range_ex23_mean_overrules_majority.yaml

Absolute loser paradox

Simple paradox (Felsenthal). A candidate ranked last by an outright majority is elected.

Case Method Expected winner(s) BV YAML
Felsenthal Ex.1 — Choose-One (Plurality): the absolute loser wins Plurality Ana BV2144 bv2144_mxfmhm_plurality.yaml
Felsenthal Ex.1 — STAR (ranks→scores): elects the Condorcet winner STAR Bo BV2144 bv2144_mxfmhm_star.yaml
Felsenthal Ex.8 — Approval: the absolute loser wins on one strategic ballot Approval April BV2154 bv2154_wq6yv7_approval.yaml
Felsenthal Ex.8 — Ranked Robin: the Condorcet winner Bruce RankedRobin Bruce BV2154 bv2154_wq6yv7_ranked_robin.yaml
Minimax Ex.29 — Choose-One: agrees with Minimax, elects the absolute loser D Plurality D BV2167 bv2167_f3dxq9_plurality.yaml
Minimax Ex.29 — STAR: elects B, a top-cycle member, not the universal loser STAR B BV2167 bv2167_f3dxq9_star.yaml
Range Ex.23 — four paradoxes with two candidates: one enthusiast outvotes three majorities ? range_ex23_mean_overrules_majority.yaml

SCC / spoiler

Conditional paradox (Felsenthal). A candidate who cannot win enters or leaves the race and the winner changes.

Case Method Expected winner(s) BV YAML
BV2267 — Committee spoiler (1 of 2): three candidates, before a fourth runs Bloc STAR Cyrus, Ari BV2267 bv2267_my9jd9_council_before_dane.yaml
BV2268 — Committee spoiler (2 of 2): a fourth candidate wins nothing and changes who does Bloc STAR Cyrus, Bea BV2268 bv2268_6m3gxq_spoiler_changes_council.yaml
Felsenthal Ex.1 — Choose-One (Plurality): the absolute loser wins Plurality Ana BV2144 bv2144_mxfmhm_plurality.yaml
Felsenthal Ex.1 — STAR (ranks→scores): elects the Condorcet winner STAR Bo BV2144 bv2144_mxfmhm_star.yaml
Felsenthal Ex.2 — Runoff/IRV: the Condorcet winner is eliminated first IRV Ben BV2145 bv2145_6fj2kg_irv.yaml
Borda SCC Ex.15 — Choose-One: C wins on first choices Plurality C BV2161 bv2161_q3h4fk_plurality.yaml
Borda SCC Ex.15 — STAR: C wins (and stays won when a loser exits) STAR C BV2161 bv2161_q3h4fk_star.yaml
Coombs Ex.22 — SCC: C drops out and A wins instead of B RCV_IRV A coombs_ex22_scc.yaml
Minimax Ex.33 — SCC: drop a loser and the winner changes RankedRobin D minimax_ex33_scc.yaml
Successive elimination Ex.9 — the agenda elects B, whom every voter ranks below A RankedRobin A succ_elim_ex9_pareto.yaml

Condorcet's paradox (cycle)

Simple paradox (Felsenthal). Pairwise majorities form a cycle; no Condorcet winner exists.

Case Method Expected winner(s) BV YAML
BV2142 — Ranked Robin clone independence (1/2): a no-Condorcet cycle, LH vs BV tiebreak RankedRobin A BV2142 bv2142_4gfwdq_clone_cycle_pre.yaml
Clone independence (1/2) — before cloning: A, B, C tie in a cycle RankedRobin A clone_teaming_01_pre.yaml
A cycle Copeland can't break — three trails tie 1-1, and the refined rules all rescue Alder RankedRobin Alder cycle_copeland_ties_c4_b21.yaml
The whole Condorcet family splits — Minimax & Schulze pick Ava, Ranked Pairs picks Ben, on one set of ballots RankedRobin Ava cycle_family_splits_c5_b77.yaml
Same ballots, different Condorcet rule — Schulze says Ana, Ranked Pairs says Bruno, Split Cycle says both RankedRobin Ana cycle_schulze_vs_ranked_pairs_c4_b40.yaml
A perfect 'Condorcet component' (30 voters) — every count ties; any winner is an artifact STAR Ada edelman_perfect_component_c3_b30.yaml
Felsenthal Ex.3 District I — Runoff/IRV: Bruno wins the runoff IRV Bruno BV2147 bv2147_9gdrqg_irv.yaml
Felsenthal Ex.3 Combined — Runoff/IRV: Bruno won both districts, Alma wins the whole IRV Alma BV2149 bv2149_byk9v2_irv.yaml
Fishburn Ex.14 — Choose-One: A wins on first choices Plurality A BV2160 bv2160_r6qc8h_plurality.yaml
Fishburn Ex.14 — STAR: B wins the electorate whose Borda count is truncation-unstable STAR B BV2160 bv2160_r6qc8h_star.yaml
Borda SCC Ex.15 — Choose-One: C wins on first choices Plurality C BV2161 bv2161_q3h4fk_plurality.yaml
Borda SCC Ex.15 — STAR: C wins (and stays won when a loser exits) STAR C BV2161 bv2161_q3h4fk_star.yaml
Coombs Ex.19 full — Choose-One: Cass on first choices Plurality Cass BV2165 bv2165_9vxcj7_plurality.yaml
Coombs Ex.19 full — STAR: Boone wins the runoff 8–7 STAR Boone BV2165 bv2165_9vxcj7_star.yaml
Minimax Ex.29 — Choose-One: agrees with Minimax, elects the absolute loser D Plurality D BV2167 bv2167_f3dxq9_plurality.yaml
Minimax Ex.29 — STAR: elects B, a top-cycle member, not the universal loser STAR B BV2167 bv2167_f3dxq9_star.yaml
Minimax Ex.30 — after: three C>A>B>D voters stay home, Minimax elects A RankedRobin D minimax_ex30_noshow_after.yaml
Minimax Ex.30 — before: all 19 vote, Minimax elects B RankedRobin A minimax_ex30_noshow_before.yaml
Minimax Ex.31 — truncation: the same 19 voters, four ballots cut to their top two RankedRobin A minimax_ex31_truncation.yaml
Minimax Ex.32 — amalgamated: both districts elected D, together they cannot RankedRobin B minimax_ex32_amalgamated.yaml
Minimax Ex.33 — SCC: drop a loser and the winner changes RankedRobin D minimax_ex33_scc.yaml
Successive elimination Ex.10 — amalgamated: every round ties, and the tie-break picks the winner RankedRobin C succ_elim_ex10_amalgamated.yaml
Successive elimination Ex.11 — before: six voters, the agenda elects C RankedRobin B succ_elim_ex11_twin_before.yaml
Successive elimination Ex.12 — sincere: the A>B>C>D voter gets their last choice RankedRobin C succ_elim_ex12_sincere.yaml
Successive elimination Ex.12 — truncated: naming only A turns last place into first RankedRobin C succ_elim_ex12_truncated.yaml
Successive elimination Ex.9 — the agenda elects B, whom every voter ranks below A RankedRobin A succ_elim_ex9_pareto.yaml
Minimal tilted cycle — 5 voters, margins 3–1–1 (RCV-IRV) RCV-IRV Cara tilted_cycle_c3_b5_irv.yaml
Minimal tilted cycle — 5 voters, margins 3–1–1 (Ranked Robin) RankedRobin Ada tilted_cycle_c3_b5_rr.yaml
A Condorcet cycle — rock, paper, scissors: no pairwise winner exists (BV2157) STAR Rock BV2157 bv2157_mmcmpy_condorcet_cycle_rps.yaml
No-show paradox (1 of 2) — 8 April fans stay home; RCV-IRV elects Bruno STAR Bruno BV2174 bv2174_yyhr66_noshow_baseline.yaml
No-show paradox (2 of 2) — the 8 April fans vote; RCV-IRV hands them their LAST choice STAR April BV2175 bv2175_9dhv8y_noshow_showup.yaml
The Post-it RCV example (20 voters) — RCV-IRV: Purple wins the video's whiteboard count IRV Purple BV2176 bv2176_p8dp28_irv.yaml
The Post-it RCV example (20 voters) — Ranked Robin: a cycle, a 2-1 tie, and two ladders RankedRobin Blue BV2176 bv2176_p8dp28_ranked_robin.yaml
The Post-it RCV example (20 voters) — STAR: Blue wins the runoff RCV-IRV never held STAR Blue BV2176 bv2176_p8dp28_star.yaml
The Post-it election, seven ways — Approval: Pink, if approving means any support Approval Pink BV2177 bv2177_v8r66y_approval.yaml
The Post-it election, seven ways — Choose-One: Purple on 7 first choices Plurality Purple BV2177 bv2177_v8r66y_plurality.yaml
Reinforcement — North district alone (6 voters, a perfect cycle) RankedRobin Ada reinf_north_c3_b6_rr.yaml
Reversal symmetry — RCV-IRV, original: A wins (best) RCV_IRV A reversal_irv_original.yaml
Reversal symmetry — RCV-IRV, reversed: A wins AGAIN (worst = best) RCV_IRV A reversal_irv_reversed.yaml
Reversal symmetry — STAR, original: B (STAR does not winner=loser here) STAR B reversal_star_original.yaml
Reversal symmetry — STAR, reversed: A (differs from original B — no winner=loser) STAR A reversal_star_reversed.yaml
A candidate nobody prefers still flips the winner — Schulze's spoiler, and Split Cycle's immunity RankedRobin Cascade split_cycle_schulze_spoiler_c5_b40.yaml

Non-monotonicity

Conditional paradox (Felsenthal). Ranking/scoring a candidate HIGHER makes them lose (or lower makes them win); a.k.a. more-is-less, additional-support paradox.

Case Method Expected winner(s) BV YAML
Burlington 2009, the raise — 750 Wright voters rank Kiss first, and Kiss loses RCV_IRV Montroll burlington_2009_raise_kiss_nonmono.yaml
Felsenthal Ex.2 — Runoff/IRV: the Condorcet winner is eliminated first IRV Ben BV2145 bv2145_6fj2kg_irv.yaml
Felsenthal Ex.2 after the raise — Runoff/IRV: Ben loses by GAINING support IRV Ada BV2146 bv2146_krk2px_irv.yaml
Felsenthal Ex.2 after the raise — Ranked Robin: unmoved, still Ada RankedRobin Ada BV2146 bv2146_krk2px_ranked_robin.yaml
Felsenthal Ex.2 after the raise — STAR: unmoved, still Ada STAR Ada BV2146 bv2146_krk2px_star.yaml
Coombs Ex.18 — Bree is raised on four ballots and loses because of it RCV_IRV Arlo coombs_ex18_monotonicity.yaml
Upward monotonicity (Alaska 2022) — AFTER: raise the winner, she loses RCV_IRV Begich alaska_upward_after.yaml
Upward monotonicity (Alaska 2022) — BEFORE: Peltola wins RCV_IRV Peltola alaska_upward_before.yaml
STAR mono-raise-delete — part 2: raise X, delete Y-below-X, X loses STAR Z mono_raise_delete_after.yaml
STAR mono-raise-delete — part 1: baseline, X wins STAR X mono_raise_delete_before.yaml
Non-monotonicity (RCV-IRV) — part 2: raising X makes X lose RCV_IRV Z monotonicity_irv_after.yaml
Non-monotonicity (RCV-IRV) — part 1: baseline, X wins RCV_IRV X monotonicity_irv_before.yaml
Monotonicity — STAR counterpart (AFTER — X still wins) STAR X monotonicity_star_after.yaml
Monotonicity — STAR counterpart (BEFORE — X wins) STAR X monotonicity_star_before.yaml
Downward monotonicity (San Francisco D7 2020) — AFTER: rank the loser lower, he wins RCV_IRV Engardio sf_d7_downward_after.yaml
Downward monotonicity (San Francisco D7 2020) — BEFORE: Melgar wins RCV_IRV Melgar sf_d7_downward_before.yaml
Many IRV pathologies in one election — Brams' 21-voter sampler (BV2159) RCV_IRV BV2159 bv2159_f4cjpy_brams_irv_pathologies.yaml

No-show paradox

Conditional paradox (Felsenthal). Voters do better by staying home: adding ballots ranking X last can make X win.

Case Method Expected winner(s) BV YAML
BV2264 — Participation (1 of 2): the council six voters elect Bloc STAR Ada, Bruno BV2264 bv2264_j3hqvb_council_before_joiner.yaml
BV2265 — Participation (2 of 2): one more honest voter, and the council gets worse for them Bloc STAR Ada, Dov BV2265 bv2265_th3pbp_joiner_council_worse.yaml
Burlington 2009, the raise — 750 Wright voters rank Kiss first, and Kiss loses RCV_IRV Montroll burlington_2009_raise_kiss_nonmono.yaml
Felsenthal Ex.4 full electorate — Runoff/IRV: their worst choice wins IRV Carl BV2150 bv2150_dxg8pb_irv.yaml
Felsenthal Ex.4 full electorate — Ranked Robin: the Condorcet winner Beth RankedRobin Beth BV2150 bv2150_dxg8pb_ranked_robin.yaml
Felsenthal Ex.4 full electorate — STAR: elects the Condorcet winner Beth STAR Beth BV2150 bv2150_dxg8pb_star.yaml
Felsenthal Ex.4 after two no-shows — Runoff/IRV: the abstainers do better IRV Beth BV2151 bv2151_97hbpw_irv.yaml
Felsenthal Ex.4 after two no-shows — Ranked Robin: unmoved, still Beth RankedRobin Beth BV2151 bv2151_97hbpw_ranked_robin.yaml
Felsenthal Ex.4 after two no-shows — STAR: unmoved, still Beth STAR Beth BV2151 bv2151_97hbpw_star.yaml
Coombs Ex.19 full — Choose-One: Cass on first choices Plurality Cass BV2165 bv2165_9vxcj7_plurality.yaml
Coombs Ex.19 full — STAR: Boone wins the runoff 8–7 STAR Boone BV2165 bv2165_9vxcj7_star.yaml
Coombs Ex.19 no-show — Choose-One: still Cass, unmoved Plurality Cass BV2166 bv2166_b7b8dv_plurality.yaml
Coombs Ex.19 no-show — STAR: the two abstainers get their favorite (STAR flips too) STAR Cass BV2166 bv2166_b7b8dv_star.yaml
Minimax Ex.30 — after: three C>A>B>D voters stay home, Minimax elects A RankedRobin D minimax_ex30_noshow_after.yaml
Minimax Ex.30 — before: all 19 vote, Minimax elects B RankedRobin A minimax_ex30_noshow_before.yaml
MJ Ex.27 — two voters do better by staying home ? mj_ex27_noshow_twin.yaml
Successive elimination Ex.9 — two D voters stay home and get a better result RankedRobin A succ_elim_ex9_noshow.yaml
Successive elimination Ex.9 — the agenda elects B, whom every voter ranks below A RankedRobin A succ_elim_ex9_pareto.yaml
Many IRV pathologies in one election — Brams' 21-voter sampler (BV2159) RCV_IRV BV2159 bv2159_f4cjpy_brams_irv_pathologies.yaml
No-show paradox (1 of 2) — 8 April fans stay home; RCV-IRV elects Bruno STAR Bruno BV2174 bv2174_yyhr66_noshow_baseline.yaml
No-show paradox (2 of 2) — the 8 April fans vote; RCV-IRV hands them their LAST choice STAR April BV2175 bv2175_9dhv8y_noshow_showup.yaml

Twin paradox (weak form)

Conditional paradox (Felsenthal). Voters are made worse off by the arrival of 'twins' — added voters with an identical preference ordering.

Case Method Expected winner(s) BV YAML
Felsenthal Ex.4 full electorate — Runoff/IRV: their worst choice wins IRV Carl BV2150 bv2150_dxg8pb_irv.yaml
Felsenthal Ex.4 after two no-shows — Runoff/IRV: the abstainers do better IRV Beth BV2151 bv2151_97hbpw_irv.yaml
Coombs Ex.21 — after: two twins join for B, and B's win becomes a coin flip RCV_IRV B coombs_ex21_twin_after.yaml
Coombs Ex.21 — before: 20 voters, Coombs elects B outright RCV_IRV B coombs_ex21_twin_before.yaml
Minimax Ex.30 — after: three C>A>B>D voters stay home, Minimax elects A RankedRobin D minimax_ex30_noshow_after.yaml
Minimax Ex.30 — before: all 19 vote, Minimax elects B RankedRobin A minimax_ex30_noshow_before.yaml
MJ Ex.27 — two voters do better by staying home ? mj_ex27_noshow_twin.yaml
Successive elimination Ex.11 — after: a twin joins and costs their own side the win RankedRobin B succ_elim_ex11_twin_after.yaml
Successive elimination Ex.11 — before: six voters, the agenda elects C RankedRobin B succ_elim_ex11_twin_before.yaml
No-show paradox (2 of 2) — the 8 April fans vote; RCV-IRV hands them their LAST choice STAR April BV2175 bv2175_9dhv8y_noshow_showup.yaml

Truncation paradox

Conditional paradox (Felsenthal). Ranking fewer candidates gives a voter a better outcome than ranking all.

Case Method Expected winner(s) BV YAML
Fishburn Ex.14 — Choose-One: A wins on first choices Plurality A BV2160 bv2160_r6qc8h_plurality.yaml
Fishburn Ex.14 — STAR: B wins the electorate whose Borda count is truncation-unstable STAR B BV2160 bv2160_r6qc8h_star.yaml
Nurmi Ex.16 sincere — RCV-IRV: full rankings elect A, not the Condorcet winner IRV A BV2162 bv2162_4htk44_irv.yaml
Nurmi Ex.16 sincere — STAR: elects the Condorcet winner B STAR B BV2162 bv2162_4htk44_star.yaml
Nurmi Ex.16 truncated — RCV-IRV: ranking ONLY their favorite serves 17 voters better IRV B BV2163 bv2163_74j6vv_irv.yaml
Nurmi Ex.16 truncated — Ranked Robin: unmoved, still B RankedRobin B BV2163 bv2163_74j6vv_ranked_robin.yaml
Nurmi Ex.16 truncated — STAR: unmoved, still B STAR B BV2163 bv2163_74j6vv_star.yaml
Minimax Ex.31 — truncation: the same 19 voters, four ballots cut to their top two RankedRobin A minimax_ex31_truncation.yaml
MJ Ex.28 — V6 grades only y, and x's median collapses from D to A ? mj_ex28_truncation.yaml
Range Ex.24 — V4 does better by not grading y at all ? range_ex24_truncation.yaml
Successive elimination Ex.12 — sincere: the A>B>C>D voter gets their last choice RankedRobin C succ_elim_ex12_sincere.yaml
Successive elimination Ex.12 — truncated: naming only A turns last place into first RankedRobin C succ_elim_ex12_truncated.yaml
Many IRV pathologies in one election — Brams' 21-voter sampler (BV2159) RCV_IRV BV2159 bv2159_f4cjpy_brams_irv_pathologies.yaml

Clone dependence / teaming

Conditional paradox (Felsenthal). Adding a near-copy of a candidate changes who wins (teaming = doing it on purpose).

Case Method Expected winner(s) BV YAML
BV2142 — Ranked Robin clone independence (1/2): a no-Condorcet cycle, LH vs BV tiebreak RankedRobin A BV2142 bv2142_4gfwdq_clone_cycle_pre.yaml
BV2143 — Ranked Robin clone independence (2/2): teaming succeeds on LH, FAILS on BV RankedRobin A1 BV2143 bv2143_9pr3wr_teaming_fails.yaml
Clone independence (1/2) — before cloning: A, B, C tie in a cycle RankedRobin A clone_teaming_01_pre.yaml
Clone independence (2/2) — teaming: A runs clones and turns a coin flip into a win RankedRobin A1 clone_teaming_02_post.yaml

Favorite betrayal

Conditional paradox (Felsenthal). A voter gets a better result by ranking/scoring their true favorite below another candidate.

Case Method Expected winner(s) BV YAML
Favorite betrayal in STAR (1 of 2) — honest ballots: the tepid consensus misses the runoff STAR Clover BV2206 bv2206_7mckyg_fbc_honest_tepid_consensus.yaml
Favorite betrayal in STAR (2 of 2) — nine voters demote their favorite and it pays STAR Bluebell BV2207 bv2207_b6xrdr_fbc_betrayal_pays.yaml
Many IRV pathologies in one election — Brams' 21-voter sampler (BV2159) RCV_IRV BV2159 bv2159_f4cjpy_brams_irv_pathologies.yaml

Reinforcement / multiple-districts paradox

Conditional paradox (Felsenthal). A candidate wins in every district separately but loses the combined election (a.k.a. inconsistency paradox).

Case Method Expected winner(s) BV YAML
Felsenthal Ex.3 District I — Runoff/IRV: Bruno wins the runoff IRV Bruno BV2147 bv2147_9gdrqg_irv.yaml
Felsenthal Ex.3 District I — STAR: also Bruno STAR Bruno BV2147 bv2147_9gdrqg_star.yaml
Felsenthal Ex.3 District II — Runoff/IRV: Bruno wins in round one IRV Bruno BV2148 bv2148_h87k6v_irv.yaml
Felsenthal Ex.3 District II — STAR: also Bruno STAR Bruno BV2148 bv2148_h87k6v_star.yaml
Felsenthal Ex.3 Combined — Runoff/IRV: Bruno won both districts, Alma wins the whole IRV Alma BV2149 bv2149_byk9v2_irv.yaml
Felsenthal Ex.3 Combined — STAR: Bruno, consistent with both districts STAR Bruno BV2149 bv2149_byk9v2_star.yaml
Coombs Ex.20 — amalgamated: both districts chose B, their union chooses A RCV_IRV B coombs_ex20_amalgamated.yaml
Coombs Ex.20 — District I: 34 voters, Coombs elects B RCV_IRV B coombs_ex20_district1.yaml
Coombs Ex.20 — District II: 7 voters, B wins outright RCV_IRV B coombs_ex20_district2.yaml
Minimax Ex.32 — amalgamated: both districts elected D, together they cannot RankedRobin B minimax_ex32_amalgamated.yaml
Minimax Ex.32 — District II: three voters, D wins outright RankedRobin D minimax_ex32_district2.yaml
Successive elimination Ex.10 — amalgamated: every round ties, and the tie-break picks the winner RankedRobin C succ_elim_ex10_amalgamated.yaml
Successive elimination Ex.10 — District I: three voters, the agenda elects C RankedRobin B succ_elim_ex10_district1.yaml
Successive elimination Ex.10 — District II: a single voter, C wins RankedRobin C succ_elim_ex10_district2.yaml
Reinforcement — Combined, Ben branch (9 voters; both halves say Ben, the whole says Ada) RankedRobin Ada reinf_combined_ben_c3_b9_rr.yaml
Reinforcement — combined 9 voters, counted by Ranked Robin (Cara wins) RankedRobin Cara BV2254 reinf_combined_c3_b9_rr.yaml
Reinforcement — combined 9 voters, counted by STAR (scoring round → Ada, runoff → Cara) STAR Cara BV2254 reinf_combined_c3_b9_star.yaml
Reinforcement — Combined, Cara branch (9 voters; both halves say Cara, the whole says Ben) RankedRobin Ben reinf_combined_cara_c3_b9_rr.yaml
Reinforcement — North district alone (6 voters, a perfect cycle) RankedRobin Ada reinf_north_c3_b6_rr.yaml
Reinforcement — South district, Ben branch (3 voters, Ben is the clear Condorcet winner) RankedRobin Ben reinf_south_ben_c3_b3_rr.yaml
Reinforcement — South district alone (3 voters, Ada is the clear Condorcet winner) RankedRobin Ada reinf_south_c3_b3_rr.yaml
Reinforcement — South district, Cara branch (3 voters, Cara is the clear Condorcet winner) RankedRobin Cara reinf_south_cara_c3_b3_rr.yaml

Majority criterion failure

Simple paradox (Felsenthal). A candidate who is the first choice of an absolute majority is not elected (Felsenthal's Absolute Majority paradox).

Case Method Expected winner(s) BV YAML
Hamlin & Hua §4.1 — the approval count as printed: B wins with 100% Approval B hh41_01_approval_as_printed.yaml
Hamlin & Hua §4.1 — the same marks read pairwise: 60 of 100 voters express no preference STAR B hh41_03_marks_read_pairwise.yaml
Felsenthal Ex.7 — Approval: a majority's first choice loses Approval Bella BV2153 bv2153_pcttmr_approval.yaml
Felsenthal Ex.7 — IRV: the majority favorite wins round one IRV Amos BV2153 bv2153_pcttmr_irv.yaml
Felsenthal Ex.7 — Ranked Robin: the Condorcet winner Amos RankedRobin Amos BV2153 bv2153_pcttmr_ranked_robin.yaml
MJ Ex.25 — the median elects y though a majority grades x higher ? mj_ex25_absolute_winner_loses.yaml
Range Ex.23 — four paradoxes with two candidates: one enthusiast outvotes three majorities ? range_ex23_mean_overrules_majority.yaml

Pareto-dominated winner

Simple paradox (Felsenthal). A candidate is (or can be) elected although EVERY voter prefers some other candidate.

Case Method Expected winner(s) BV YAML
Felsenthal Ex.6 — Approval can elect a Pareto-dominated candidate (LH-only) Approval Beau felsenthal_ex6_pareto_approval.yaml
Felsenthal Ex.6 — Ranked Robin: the Pareto-dominant Condorcet winner Aria (LH-only) RankedRobin Aria felsenthal_ex6_ranked_robin.yaml
Successive elimination Ex.9 — the agenda elects B, whom every voter ranks below A RankedRobin A succ_elim_ex9_pareto.yaml