Matpic
mathematics, in pictures
Concept

Transitivity

The property that if A beats B and B beats C, then A beats C. Rankings by a single number always have it; majority votes, dice and races between coin patterns can lack it, and then there may be no best option at all.

Named by 2 essays across 2 fields — 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 Three patterns that beat one another in a circle: HHHT, TTHH, HTTH. Three coin-toss patterns at the corners of a triangle with arrows showing which beats which in two-way races, and each pattern's chance of winning when all three race.

Three patterns in a circle

Race three coin patterns at once and the gamblers' accounting still gives each one's chance of arriving first — one fairness equation per pattern. What it does not give is any way to read the three-way result off the two-way ones. HHHT, TTHH and HTTH beat one another in a circle, and HHH loses both its head-to-head races and still finishes ahead of one of the patterns that beat it.

probability · Expectation

Named alongside it

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

Condorcet cycleCounterexampleExhaustive searchExpectationFinite automatonGraphLinear systemMarkov chainMartingalePairwise majorityPreference profileTournament

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