Matpic
mathematics, in pictures
Concept

Voting rule

Any recipe turning a collection of submitted rankings into a verdict — a winner, or an ordering of the candidates. Every one of them fails at least one condition that looks indispensable, which is what the impossibility theorems establish.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

A majority cycle over 3 candidates, and how often 3 voters produce one. The majority tournament as a directed polygon with each arc's margin, beside one cell for every profile of the stated size, filled where no Condorcet winner exists.

The majority that goes in a circle

Every voter hands in a ranking, and a ranking is transitive by construction. Compare the candidates two at a time and let the majority decide each pair, and the verdicts need not fit together into a ranking at all.

applied · Voting rules Five rules on one profile of 27 ballots, and 5 different winners. The ballot groups as columns beside a table of five voting rules with the winner each returns and the count that decided it.

Five rules and five winners

Twenty-seven ranked ballots, five entirely reasonable ways of counting them, and five different candidates declared the winner. Every count is correct, every rule is defensible, and the answer turns out to be a property of the rule rather than of the ballots.

applied · Voting rules Independence of irrelevant alternatives, broken by Borda. Two profiles that agree on every voter's ranking of two candidates and differ only in where the others sit, with the rule's verdict between the two reversed.

Four conditions, and no rule that has all of them

The rung below shows five reasonable rules returning five different winners, which invites the obvious question of which one is right. The answer is that the conditions anybody would write down cannot all hold at once — and here each named rule's own violation is found by search rather than quoted.

applied · Voting rules Every ballot one voter could submit under instant runoff. One voter's true ranking beside every ranking that voter could submit instead, with the winner each produces and the profitable misreports marked.

A lie that pays

Three rungs of this ladder have read a ballot as a report of a preference. This one reads it as a move, and walks every move one voter has — all six rankings, the winner each produces, and the ones that beat honesty.

applied · Voting rules 3 consistent judges, and a majority that is not. A table of judges against three questions, every judge's row internally consistent, with the majority answer to each question underneath forming a combination no judge holds.

The court that contradicts itself

Three judges each answer three questions, and each answers them consistently. Take the majority on each question separately and the answers no longer hang together — the body as a whole asserts a combination no member of it holds, and no rearrangement of the procedure removes the problem.

applied · Judgement aggregation 16 rules, and none that survives. A table of every systematic anonymous aggregation rule for 3 judges: one row per rule, showing the verdict it gives at each count of yes-votes, whether it decides every proposition, and whether it is consistent. No row has both.

No rule escapes the doctrinal paradox

A court whose members each hold a consistent position can reach an inconsistent verdict by majority. The anchor's first rung exhibits one such case, which invites the hope that a better rule would avoid it — and every rule that responds to the votes at all fails somewhere.

applied · Judgement aggregation Where the two procedures part company. A table of 3 judges' verdicts on two premises and the conclusion each is committed to, with the two majorities at the foot disagreeing about the conclusion.

Deciding the premises or the conclusion

A body that cannot be both decisive and coherent has to choose which. The two live options are to vote on the reasons and let the verdict follow, or to vote on the verdict and let the reasons look after themselves — and they reach opposite answers on exactly the profiles the impossibility identifies.

applied · Judgement aggregation

Named alongside it

The objects these essays reach for when they reach for this one.

Preference profileCondorcet cyclePairwise majorityAggregationConsistencyCounterexampleMajorityImpossibilityIndependence of irrelevant alternativesJudgement aggregationLogicArrows theorem

All concepts