Concept

Gamblers ruin

The chance that a gambler betting fixed stakes loses everything before reaching a target. For a fair game it is the distance to the target as a share of the whole range, and an unfavourable game makes ruin nearly certain.

Named by 2 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 $Four fractions staked on one run of tosses. p=0.6, 1000 tosses, 604 heads; final log10 wealth: Kelly 9.449, half Kelly 6.881, twice Kelly 0.409, everything ruined at toss 10.$

The best average is the worst bet

A coin lands heads six times in ten and pays even money. Stake everything on every toss and the expected wealth grows by a fifth a toss — and the bettor is ruined at the first tail, which is certain to come. Stake a fifth of wealth each time instead and the average grows far more slowly, while the typical bettor grows faster than with any other fraction. The expectation is maximised by the strategy almost nobody should use.

probability · Expectation

Named alongside it

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

Expectation Random walk Absorbing state Entropy Law of large numbers Logarithm Markov chain Martingale Median Recurrence relation

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