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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Duration of gambler's ruin conditioned on a chosen boundary ( $E_{i} [τ_{0} ∣ τ_{0} < τ_{r}] = \frac{i ( 2 r - i )}{3}$ )

For a simple symmetric random walk absorbed at

0, r

, the probability of absorption at

0

is

h_{i} = (r - i) / r

. Conditioning on that absorption gives the Doob h-transform with transition probabilities

(r - i + 1) / [2 (r - i)]

downwards and

(r - i - 1) / [2 (r - i)]

upwards. If

g_{i} = E_{i} [τ 1_{{τ_{0} < τ_{r}}}]

, first-step analysis gives

g_{i} = h_{i} + (g_{i - 1} + g_{i + 1}) /2

,

g_{0} = g_{r} = 0

. Its solution is

g_{i} = i (r - i) (2 r - i) / (3 r)

, and dividing by

h_{i}

gives the conditional duration.

Discounted gambler's ruin hitting probability ( $u_{k} = E_{k} [s^{τ} 1_{{X_{τ} = 0}}]$ )

For a nearest-neighbour random walk on

{0, \dots, N}

with right-step probability

p

, left-step probability

q = 1 - p

, and absorption time

τ

, first-step conditioning gives

u_{k} = s (p u_{k + 1} + q u_{k - 1})

,

u_{0} = 1

,

u_{N} = 0

. For

0 < s < 1

, the two positive distinct characteristic roots are

r_{\pm} = (1 \pm 1 - 4 pq s^{2}) / (2 p s)

. Solving the boundary equations gives

u_{k} = (r_{+}^{N} r_{-}^{k} - r_{-}^{N} r_{+}^{k}) / (r_{+}^{N} - r_{-}^{N})

. The maximum principle

∥ u - v ∥_{\infty} \leq s ∥ u - v ∥_{\infty}

proves uniqueness for equal boundary data. This is the joint generating function of the absorption time and the lower-end absorption event.

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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