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Concept

Tournament

A contest in which every pair of players meets once and one of the two wins, drawn as a complete graph with every edge given a direction. A tournament can contain cycles in which each player beats the next, so winning more games does not by itself decide who is strongest.

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

A majority cycle over 3 candidates, and how often 3 voters produce one. The majority tournament as a directed polygon with each arc's margin, beside one cell for every profile of the stated size, filled where no Condorcet winner exists.

The majority that goes in a circle

Every voter hands in a ranking, and a ranking is transitive by construction. Compare the candidates two at a time and let the majority decide each pair, and the verdicts need not fit together into a ranking at all.

applied · Voting rules 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

Named alongside it

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

Condorcet cyclePairwise majorityPreference profileCounterexampleExhaustive searchGraphLimitNormal distributionProbability modelSingle-peaked preferencesTransitivityVoting rule

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