Dictatorship
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
An electorate too large for a dictator
Arrow proved that a voting rule with his four properties must have a dictator, and his proof uses the finiteness of the electorate exactly once. With infinitely many voters the dictator can vanish: the sets of voters who get their way form an ultrafilter, and every free ultrafilter is a rule with all four properties and nobody in charge. The catch is that no such rule can be written down. Its existence is a fragment of the axiom of choice.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchImpossibility theoremJudgement aggregationMajority ruleQuota ruleArrow's theoremAxiom of choiceCondorcet cycleDecisive setDomain restrictionIndependence of irrelevant alternativesInfinity