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Concept

Majority rule

The rule that settles a yes-or-no question by giving it to the side with more votes. It is the only rule that treats voters alike, treats the two answers alike and never punishes a candidate for gaining support — May's theorem — and it can still produce a set of verdicts that contradict one another when several questions are decided at once.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Five rules, and the one condition each of them gives up. A table with one row per aggregation rule and one column per condition, marking which conditions each rule satisfies when run over every profile of the agenda.

Four ways out, and what each costs

An impossibility theorem lists conditions and says no rule has them all. That leaves exactly as many escapes as there are conditions, each of them a real institution — a dictator, a two-stage procedure, a supermajority, a restricted agenda — and each escape's price can be counted rather than argued about.

applied · Judgement aggregation The nearest consistent verdict: a 3-way tie at distance 4. A table of the 4 consistent judgement sets on the agenda p, q, and p and q, each with its number of disagreements with each of 3 judges and the total; the smallest total is marked.

The nearest consistent verdict

When a court's majorities contradict each other, one repair is to announce the consistent verdict that disagrees with the judges least. It treats the premises and the conclusion alike, which neither of the two standard procedures does. On the classic case it returns a three-way tie; on five judges, with every question weighted equally, it never returns a single answer on a troubled profile at all — and what breaks the tie is a decision about which question matters more.

applied · Judgement aggregation Which agendas majority can vote on safely. A table of 7 agendas with the size of their largest minimally inconsistent set and the count of inconsistent majority outcomes over all profiles of three and five judges.

Agendas that cannot contradict themselves

A court voting on two unconnected questions never contradicts itself, and neither does one voting on a chain of thresholds. A court voting on two premises and their conjunction sometimes does. What separates them is the size of the smallest sets of judgements that cannot all be true: pairs are harmless, because two majorities always share a judge, and triples are not. The same count says exactly how large a supermajority has to be to stay consistent on any agenda.

applied · Judgement aggregation Weighted votes: two functions with a cut, and parity without one. For three functions of three letters, the eight assignments placed on a line by a weighted count, with true assignments filled and the threshold marked where one exists.

A plane through the cube

Some truth functions are weighted votes: give each letter a weight, add the weights of the true letters, and say yes when the total passes a threshold. On the cube of assignments, such a function is a plane cutting the true corners from the false. Majority is one. Exclusive-or is not, and never can be — and of the 65,536 functions of four letters, only 1,882 are. The ones that are are exactly what a single artificial neuron can compute.

logic · Truth functions

Named alongside it

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

Exhaustive searchImpossibility theoremJudgement aggregationCondorcet cycleHypercubeQuota ruleBoolean functionDictatorshipDomain restrictionExclusive-orHamming distanceMedian voter

All concepts