Section 16.2

Gambler's Ruin

Absorption between two barriers: ruin probabilities, expected durations, matches decided by a lead of N.

Toolkit

Model the fortune as a random walk on $\{0,1,\dots,N\}$ with absorbing barriers at $0$ and $N$ , and use martingale/optional-stopping arguments to extract absorption probabilities and expected durations without solving difference equations by hand.

Let $p$ be the probability of a $+1$ step, $q=1-p$ , and let $X_0=i$ .

Symmetric case $p=q=1/2$ : $S_n$ is a martingale, so
$\mathbb{P}(\text{hit } N \text{ before } 0) = \frac{i}{N}.$
Biased case $p\neq q$ : $\left(\frac{q}{p}\right)^{S_n}$ is a martingale, giving
$\mathbb{P}(\text{hit } N \text{ before } 0) = \frac{1-(q/p)^i}{1-(q/p)^N}.$
Symmetric expected duration: $S_n^2-n$ is a martingale, so starting at $i$ between barriers $0$ and $N$ ,
$\mathbb{E}[\text{time to absorption}] = i(N-i).$
For a match decided by a lead of $N$ (steps $\pm1$ with probabilities $p,q$ ), treat each pair of non-tying rolls as an effective trial and apply the above with the reduced step probabilities $p/(p+q)$ , $q/(p+q)$ where ties are discarded.
Number of visits to an intermediate level before absorption is geometric, built from the probability of returning to that level before being absorbed.

Problems