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Concept

Pairwise majority

The relation formed by taking candidates two at a time and letting the larger side of the electorate settle each pair. It need not be transitive, and the cycles it can produce are why a majority verdict may fail to assemble into a ranking.

Named by 6 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 Five rules on one profile of 27 ballots, and 5 different winners. The ballot groups as columns beside a table of five voting rules with the winner each returns and the count that decided it.

Five rules and five winners

Twenty-seven ranked ballots, five entirely reasonable ways of counting them, and five different candidates declared the winner. Every count is correct, every rule is defensible, and the answer turns out to be a property of the rule rather than of the ballots.

applied · Voting rules Independence of irrelevant alternatives, broken by Borda. Two profiles that agree on every voter's ranking of two candidates and differ only in where the others sit, with the rule's verdict between the two reversed.

Four conditions, and no rule that has all of them

Five reasonable rules can return five different winners on one set of ballots, which invites the obvious question of which one is right. The answer is that the conditions anybody would write down cannot all hold at once — and here each named rule's own violation is found by search rather than quoted.

applied · Voting rules Every ballot one voter could submit under instant runoff. One voter's true ranking beside every ranking that voter could submit instead, with the winner each produces and the profitable misreports marked.

A lie that pays

A ballot is usually read as a report of a preference. This one reads it as a move, and walks every move one voter has — all six rankings, the winner each produces, and the ones that beat honesty.

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 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.

Condorcet cyclePreference profileVoting ruleCounterexampleIndependence of irrelevant alternativesProbability modelTournamentArrows theoremBayes' theoremBinomial distributionBorda countCondorcet jury theorem

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