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Concept

Martingale

A sequence of fortunes in a fair game: whatever has happened so far, the expected next value equals the current one. Stopped at a sensible moment it still averages its starting value, which turns waiting times and winning chances into bookkeeping about fair bets.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

A walk with a barrier at each end. Three games played to absorption on a table of 12, beside the chance of ruin from each starting stake — a straight line, because the walk is fair.

Two barriers and a fair game

A fair walk between two absorbing barriers is ruined with a probability that is a straight line in the starting stake, and lasts for a number of steps that is the product of what each side can lose. Both facts come from the same two-line recurrence, and both are bad news for the smaller player.

probability · Random walk How much the first player can guarantee, as the coin's bias moves. A plot of the first player's best guaranteed winning chance in Penney's game against the probability of heads, a third on a fair coin and rising past one half only when the coin is heavily biased.

A coin that lets the first player win

On a fair coin the second player in Penney's game always has a better pattern than the first, and the first can hold them to no worse than two to one. Bend the coin and every overlap is paid for in the letters it uses: the replies change, the first player's share swings between a third and a half, and past a heads chance of 1/∛2 the first player simply names HHH and wins.

probability · Expectation 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.

ExpectationMarkov chainFinite automatonAbsorbing stateConditional probabilityExhaustive searchGamblers ruinLinear systemMinimaxRandom walkRecurrence relationTransitivity

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