Gambler's ruin

Gambler's ruin studies a nearest-neighbour random walk stopped on reaching either endpoint of a finite interval.

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Expected duration of biased gambler's ruin ( $m_{i} = [N (1 - ρ^{i}) / (1 - ρ^{N}) - i] / (p - q)$ )

For a nearest-neighbour fortune on

{0, \dots, N}

, with win probability

p

, loss probability

q = 1 - p

and

p \neq = q

, put

ρ = q / p

. First-step analysis gives

m_{i} = 1 + p m_{i + 1} + q m_{i - 1}

, with zero endpoint values. Solving the recurrence gives the displayed mean duration. Its symmetric limit is

i (N - i)

.

Biased gambler's ruin probability

For a walk that steps right with probability

p > 1/2

, put

λ = (1 - p) / p

. If

a < 0 < b

and

T_{x}

is the first hitting time of

x

, then the martingale

λ^{S_{n}}

and the optional stopping theorem give

P (T_{a} < T_{b}) = \frac{λ ^{b} - 1}{λ ^{b} - λ ^{a}} .

Letting the opposite boundary tend to infinity gives

P (T_{a} < \infty) = λ^{- a}, P (T_{b} < \infty) = 1.

Expected duration of symmetric gambler's ruin

For a symmetric random walk started at

i \in {- a, \dots, b}

and stopped on first reaching

- a

or

b

, the mean duration is

E_{i} T = (i + a) (b - i) .

In particular,

E_{0} T = ab

. This follows either from the first-step recurrence or by stopping the martingale

S_{n}^{2} - n

.

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Gambler's ruin is a concept from probability theory and statistics that models a gambling scenario where a gambler continues to gamble until either they lose all their money or reach a predetermined target amount. It is often used to illustrate the principles of random walks and the behavior of stochastic processes. In a typical setup, a gambler starts with a certain amount of capital and bets on a game with a fixed probability of winning or losing.

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