Worked Examples — The Impossibility Floor, With Arithmetic

Eight machine-checked ballot profiles that make the social-choice pathologies concrete, built for the CGP Grey STV refutation. Two separate attacks, deliberately not blended: (A) impossibility — no method satisfies every desideratum, so “the fair system” names a corner the theorems forbid; and (B) underdetermination — “STV” is not one rule, and the implementation choices the cartoon skips decide the winner. Attack B is the stronger one for a video and the one nobody makes. Headline result: identical ballots, four ordinary rule-sets, three different councils. Every number here is computed by tools/voting-paradoxes.py and re-derived on every --selftest run — none of it is asserted.

Tool: tools/voting-paradoxes.py — --selftest re-derives all eight claims from the raw ballots and exits non-zero if any stops holding. --report prints the tables below. Links: CGP Grey — “Too Good for Politicians to Allow” (STV) (the parent review — this page is the arithmetic its Open Questions asked for), Government Formation (Duverger + the impossibility floor), Aggregation vs. Sorting, Game Theory as Normative, Not Descriptive, Scope Confusion, The Load-Bearing Word

Method page. Not a transcript review — a reusable evidence bank. Built for a planned refutation video, so each example is stated in the form a hostile viewer would have to answer.

0. Precision first — don’t fight looseness with looseness

The charge against the video is that it is loose about the rules that let a candidate advance. That charge is only survivable if the refutation is tighter, so every example below is labelled with which result it actually demonstrates. Bundling all of these under “Arrow’s law” is the single most common way an otherwise-correct critique gets dismantled on camera.

The rhetorical order that works: lead with §6 (the rules decide the winner — concrete, no theory needed), then §1 (five methods, five winners — no theory needed either), and only then name the theorems as the explanation for why this is unavoidable rather than a fixable bug. Theory last.

1. One electorate. Six methods. Five different winners.

111 voters, one fixed set of sincere preferences. Nobody changes their mind; only the counting rule changes.

Five of the five candidates win under some perfectly ordinary rule. And a Condorcet winner does exist here (B), so the excuse “there was no majority will to find” is unavailable — there was a pairwise-dominant candidate and four of six methods missed him.

The line this example buys: “the electorate did not decide this election — the rulebook did.” That is the whole impossibility argument in one table, before a single theorem is named.

2. Arrow proper — an irrelevant alternative flips the winner

96 voters. This one profile is a double: an IIA violation and a Condorcet failure, which is the Alaska 2022 center-squeeze shape in miniature.

IRV rounds: A=40, B=19, C=37 → B eliminated → A=40, C=56. IRV elects C.

Now delete A — a loser, who won nothing — and re-count the same ballots:

IRV elects B.

Not one voter changed their B-vs-C ranking. The winner still flipped, because A’s presence determined who got eliminated first. That is exactly the independence condition Arrow proved cannot be kept.

And in the same profile: B beats A head-to-head, and B beats C head-to-head — B is the Condorcet winner — while A is the Condorcet loser yet survives to the final round. The squeezed centrist dies first because first preferences, not pairwise strength, decide elimination order.

3. Non-monotonicity — helping your candidate hurts your candidate

67 voters. C wins.

Rounds: B=18, C=20, A=29 → B eliminated → C=38, A=29. C wins.

Now 12 of the A-first voters change their minds and promote C to first (A > B > C becomes C > A > B). Nothing else changes; the electorate is the same size; C is strictly more popular than before.

Rounds: B=18, C=32, A=17 → A is now eliminated instead of B → A’s 17 flow to B → B=35, C=32. C loses.

Gaining 12 first-preference votes cost C the election. The selftest verifies mechanically that the only difference between the two profiles is C moving up on 12 ballots.

Why it matters beyond the curiosity: this is the formal death of “there’s no point in strategizing about how everyone else votes.” If raising a candidate can defeat them, then sincere ranking is not weakly dominant, and a voter who knows the polls has something to compute.

4. The no-show paradox — voting makes you worse off

118 voters elect C. Then 9 more voters turn up whose sincere ranking is B > D > C > A.

• They abstain → C wins. They rank C third.
• They vote → A wins. They rank A last.

By casting sincere ballots they replaced their 3rd choice with their 4th. Staying home was strictly better. (Verified tie-break-free in both counts, so it is not an artefact of a coin-flip rule — a detail worth stating pre-emptively, because it is the first thing a critic will reach for.)

This is Moulin’s theorem, not Arrow: no Condorcet-consistent rule can guarantee that participating never harms you, once there are four or more candidates.

5. Gibbard–Satterthwaite — burying your own favourite pays

100 voters, three candidates: Left, Centre, Right.

Everyone sincere → Centre is eliminated first (32, fewest) → Centre’s votes flow to Left → Left wins. The Right bloc gets its worst outcome.

Now 6 Right voters lie, ranking Centre first (Centre > Right > Left):

Right (29) is now eliminated first, its votes flow to Centre, and Centre wins — which those 6 voters prefer to Left. Betraying their own favourite got them a better result. Electorate unchanged; only the reported rankings moved.

This is the concrete form of the claim the video denies outright. Gibbard–Satterthwaite says such a profile exists for every non-dictatorial deterministic method — so this is not a flaw of IRV that a better ranked system fixes.

6. THE HEADLINE — same ballots, same voters, three different councils

This is the direct answer to “the video is loose about the rules that let a candidate advance.” Not a paradox, not a theorem: just four combinations of two implementation choices that real jurisdictions actually make, applied to one identical set of ballots.

• Quota — Droop (⌊V/(S+1)⌋+1) or Hare (V/S). Both are in real use.
• Surplus transfer — Gregory (fractional: all of the winner’s ballots move at reduced weight) or a whole-ballot sample (a subset of physical ballots moves at full weight). Both are in real use; several jurisdictions have drawn the sample at random.

172 voters, 3 seats:

Three different three-member councils from one set of ballots. Every voter’s preferences are identical in all four counts. What changed is a paragraph of statute that the cartoon covers with the phrase “their surplus votes get transferred.”

6a. The quota alone decides it

Surplus rule held fixed at Gregory; only the quota changes:

6b. The surplus-transfer rule alone decides it

Quota held fixed at Droop; only the transfer method changes:

Why this is the best material for a video. It needs no theorem, no counterintuitive paradox, and no social-choice literacy from the audience. It says: you cannot advocate “STV” — there is no such single thing. Name your quota and name your surplus rule, because those choices, not the voters, picked the third seat. And it lands precisely on the gap the review already identified — the mechanism the cartoon glosses fastest is the most consequential one.

The random-sample variant sharpens it further: where the surplus is drawn by lot, a recount of the same ballots can seat a different person. That is not a paradox about voters. It is a property of the statute.

7. Majority-criterion failure — Borda overrides an outright majority

Found by the selftest itself, as a failed sanity check — which is the best provenance an example can have.

A holds an outright 60% first-preference majority. Plurality, runoff, IRV, Coombs and Condorcet all elect A. Borda elects B (A: 120, B: 140).

Useful as the fairness-pluralism example: Borda is not broken, it is optimising something else — mean rank rather than majority support. Which is the point. “Fair” has no single referent, so “the fairest system” is a category error, and every method is an answer to a question the advocate usually declines to state.

8. How to deploy these

• Lead with §6, not Arrow. The rules-decide-the-winner table needs no theory and cannot be answered by “well, no system is perfect” — because it is not a claim about systems, it is a claim about this system being underspecified.
• Then §1. Five winners, one electorate. Still no theory.
• Only then the theorems, and name them correctly (§0). The theorems explain why §1 and §6 are permanent rather than fixable — that is their job in the argument, and it is the last job, not the first.
• Pre-empt the tie-break dodge. §4 is verified tie-break-free precisely because “that’s just an artefact of your tie-breaking rule” is the standard escape.
• Concede what’s true. FPTP’s pathologies are real; STV is a genuine improvement on several axes. The target is the oversell — “maximizes happiness,” “no point in strategizing,” “too good to allow” — not the mechanism. The parent review already establishes that the strongest version of this critique is scope-bounded, and overreaching here would hand back the advantage.
• Keep IRV and STV distinct. Single-winner IRV and multi-winner STV share an engine but not a failure profile; §2–§5 are single-winner results and §6 is multi-winner. Blurring them is the same looseness being criticised.

9. Open questions

• §6 uses a deterministic stand-in for the random draw. The whole transfer rule here takes the last-arriving ballots rather than a genuine random sample. That is enough to prove the rule matters, but a true Monte-Carlo over random draws would let the page state a distribution of councils from one ballot set — a strictly stronger and more filmable claim. Worth doing before a video.
• Batch elimination is unmodelled. Some jurisdictions eliminate several trailing candidates at once to save rounds. That is a third implementation lever and almost certainly changes winners too; it is not yet in the tool.
• Ballot truncation. All profiles here are fully ranked. Real ballots are truncated and exhaust, which is where much real-world RCV controversy lives (exhausted ballots mean winners with less than a true majority of ballots cast). Not yet modelled.
• Is there a profile where all six methods give six different winners? The search found five. Six should exist with more candidates; not yet located.

Tags

politics · philosophy · debates