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Concept

Probability model

A precise statement of which outcomes are possible and how likely each is, chosen so that a question about chance becomes a calculation. Different models of one situation can give different answers, so the model is part of the claim rather than a detail of it.

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

How often three candidates' majorities go in a circle. The exact probability that pairwise majority among three candidates is cyclic, for every odd number of voters from 1 to 41, under two models of how ballots are drawn, with their limits of about 8.77% and 6.25%.

How often the majority goes in a circle

Three voters and three candidates give 216 profiles, and 12 of them are cycles. Count every electorate up to 41 voters exactly and the share climbs towards 8.77%, a number Guilbaud found in 1952 as the solid angle where three half-spaces at the tetrahedral angle overlap. Add candidates and a winner goes missing half the time; let voters share one axis and cycles vanish. The number is always a property of the model of how ballots are drawn.

applied · Voting rules A majority of independent voters, more often right than any of them. The probability that a simple majority of n independent voters is right, for n from 1 to 201, when each voter is right with probability 0.45, 0.51, 0.55, 0.6, 0.7.

A majority wiser than its members

Condorcet's other theorem turns voting round: the voters no longer have preferences but judgements about a single fact, each a little more likely right than wrong. Then a simple majority of many of them is almost certainly right — 6,763 voters who are each right 51% of the time make a majority right 95% of the time. The theorem survives voters worse than a coin, if the average is better. It does not survive voters who share their mistakes, and when their skills differ the right rule weighs votes rather than counting them.

applied · Voting rules

Named alongside it

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

Pairwise majorityBayes' theoremBinomial distributionCondorcet cycleCondorcet jury theoremCorrelationExhaustive searchIndependenceLaw of large numbersLimitNormal distributionPreference profile

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