Approval and the majority criterion — Hamlin & Hua's own example, counted five ways¶
The tabulatable evidence behind Claim check — Hamlin & Hua (2023), The case for approval voting. One electorate — the worked example printed in that paper's §4.1 — read five ways, so you can watch a majority criterion failure happen, see exactly where the deciding information goes, and test the paper's own defence of it.
Until now this repo asserted that Approval fails the majority criterion (it's a row in the criterion table) but only ever worked STAR's version of the failure. This set fills that gap — using the example the approval side chose for itself.
The electorate. 100 voters, three candidates, kept at the paper's own labels because the point is line-by-line correspondence with the printed example:
| voters | preference | the approval ballot §4.1 assumes they cast |
|---|---|---|
| 60 | A > B > C | A + B |
| 30 | B > C > A | B + C |
| 10 | C > B > A | C + B |
A is the first choice of 60%. B is approved on all 100 ballots. Approval elects B.
| Page (read this) | What it shows | src |
|---|---|---|
| 01 — the approval count as printed | B wins, 100 approvals (100%), A 60, C 40 — the failure exactly as the paper prints it. Note the 30 and the 10 cast identical ballots and collapse to one 40-voter row | .yaml |
| 02 — the same preferences, counted pairwise | A is the Condorcet winner — beats B 60–40 and C 60–40. The paper opens §4.1 noting a Condorcet winner needn't exist; in its own example one does, and Approval doesn't elect it | .yaml |
| 03 — the same marks read pairwise | Would a runoff on approval ballots fix it? No: "Voters with a preference: 40 of 100 (60 Equal Support)." The 60 who prefer A approved both, so they say nothing in the head-to-head | .yaml |
| 04 — the paper's own utility stipulation, on a 0–5 ballot | §4.1 says a real utility gap "would require certain assumptions." Written down, those assumptions are three ballot rows: score round A 380, B 370 — the gap really is tiny, and STAR elects A anyway, 60–40 in the runoff | .yaml |
| 05 — the majority bullet-votes instead | Same voters, same opinions, one threshold moved: A wins 60–40. The violation lives in where the line is drawn, not in the electorate | .yaml |
The one table that carries the lesson¶
Same 100 voters, same opinions, every row an engine result:
| what the ballot recorded | who wins | by how much |
|---|---|---|
| full preferences, pairwise (02) | A | beats B 60–40, C 60–40 |
| first choices only (Choose-One / RCV-IRV) | A | 60% in round one |
| the paper's stipulated utilities, 0–5 (04) | A | score 380–370, runoff 60–40 |
| approve / don't approve (01) | B | 100 – 60 |
| the same marks, pairwise (03) | B | 40 – 0, with 60 Equal Support |
Read the last two rows against the third. On the full-resolution ballot A and B are ten points apart in five hundred — a coin flip, exactly as the paper argues. Compressed to checkmarks, the same electorate reports B over A by 100 to 60. The compression doesn't merely lose the gap; it manufactures a landslide in the opposite direction. That is a sharper statement of Approval's honest limits §1 than the limits page itself makes.
Fairness notes¶
- The paper's framework is the right one, and this repo already uses it. Hamlin & Hua argue a criterion violation should be judged on frequency × severity, not as a pass/fail checkbox. Agreed — the same standard is what stops this library treating "STAR fails the majority criterion" as a knockout. Case 04 concedes their severity point outright.
- It cuts against STAR too. STAR fails the majority criterion as well — worked at BV95a / BV95b. The difference is the trigger: STAR's failure needs the majority to support two rivals, Approval's needs one. That is the Relaxed Majority Criterion, and it is a difference of degree, not of kind.
- Case 05 is a counterfactual and is labelled one. The paper's assumed ballots are case 01; case 05 changes them to isolate what the example depends on.
Concept hubs: majority criterion · Approval Voting · Approval in the theory literature · Black Curtain — the same compression, five ballots · up: method_comparisons — same ballots, different methods