Majority rule
Named by 12 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.
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.
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.
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.
How few voters any majority needs
Any pattern of head-to-head majorities whatever — cycles within cycles, a candidate who beats the winner of every other contest and loses to its loser — can be produced by voters who each rank the candidates sensibly. McGarvey's recipe needs n(n − 1) of them for n candidates. The truth is far fewer: every pattern on five candidates takes three voters at most, a counting argument shows the number must eventually grow, and it grows only like n divided by its logarithm.
The fewest swaps to a winner
When no candidate beats every other head to head, Charles Dodgson proposed in 1876 to elect the one that is closest to doing so — the candidate that the fewest swaps of neighbouring names on the ballots would turn into a winner of every contest. The rule is easy to state and hard to compute: the count needs a search, and deciding the winner is provably among the hardest problems of its kind. A much simpler count, the votes still to be won, usually agrees, more often the larger the electorate.
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.
Which vote finds the truth more often
A court that must decide two premises and their conjunction can vote on the premises or on the conclusion, and the two votes sometimes disagree. Suppose there is a right answer and each judge is more often right than wrong. Then the question has an exact answer: for three judges the conclusion vote is more often right when competence is below √½ and the premise vote above it, and in a large court the premise vote wins at any competence — while the conclusion vote turns out to be a built-in standard of proof.
Every premise makes the verdict vote stricter
A court that must find several things true before it finds liability can vote on each of them or on the verdict. With two elements, judges right seven times in ten are enough for either. With five, the verdict vote needs judges right eighty-seven times in a hundred, or a larger court makes it worse — while the premise vote still needs only better than even. And when any one ground is enough instead of all, the two procedures swap their failures.
A party that loses every issue can win
Two parties take opposite sides on three issues, and every voter backs the party they agree with on more of them. The party that wins the election can be on the losing side of all three issues — in a fifth of random electorates it loses most of them, and in about one in a hundred it loses every one. A majority of voters can even find themselves outvoted on most of the questions decided. Only when issues are settled by three-to-one majorities on average is that ruled out.
The element the judges are worst at
A court can vote element by element or on the verdict, and the comparison so far assumed judges equally good at every element. Real judges are surer about the law than about a contested fact. Make one element hard and both procedures sink towards the majority on that element, but they sink differently: the verdict vote sees only the product of the judges' competences and cannot tell an even court from a lopsided one, while the premise vote is ruled by its weakest majority. And when a judge's errors on two elements come from the same misreading of the facts, the gap between the procedures closes.
A vote worth a square root
Give each country in a council votes in proportion to its people, and a citizen of the largest has about fifteen times the say of a citizen of the smallest. Give each country one vote and the same ratio runs the other way. Lionel Penrose saw in 1946 that the fair weight is the square root of the population, because one voter's chance of deciding an election falls as one over the square root of the crowd. Computed for twenty-seven states, the best quota is 61.5%, and it holds the ratio to 1.006.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchJudgement aggregationImpossibility theoremCondorcet cycleCondorcet jury theoremQuota ruleAggregationBinomial distributionCondorcet paradoxDictatorshipHypercubeMedian voter