Matpic
mathematics, in pictures
Concept

Impossibility theorem

A theorem showing that no procedure can satisfy a given list of requirements at once, rather than that nobody has yet found one. Arrow's and Gibbard's are the famous examples in voting; each derives a contradiction from the requirements, which is why the response is always to give one of them up.

Named by 4 essays across one field — 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 What survives on each agenda, 3 judges. A table of four agendas with, for each, the size of its largest inconsistent set, the number of independent unanimous rules for 3 judges, how many are consistent on every profile, and how many of those are dictatorships, oligarchies and other rules: two unconnected questions 324 of 324; a chain of two thresholds 129 of 324; two premises and their conjunction 7 of 5,832; a ranking of three options 3 of 5,832.

What the agenda leaves standing

Ask for a rule that settles each question from the votes on that question, follows a unanimous court and never contradicts itself, and search every such rule for three judges. On a ranking of three options, three survive: one dictator per judge. On two premises and their conjunction, seven survive: every rule in which a fixed set of judges must all agree. On a chain of thresholds, a hundred and twenty-nine, majority among them. The difference is not in the rules. It is in which answers force which, and whether that forcing ever runs back.

applied · Judgement aggregation

Named alongside it

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

Exhaustive searchJudgement aggregationMajority ruleCondorcet cycleQuota ruleDictatorshipDomain restrictionHamming distanceHypercubeMedian voterOligarchy

All concepts