Gambler's Ruin Problem

Topics: Random Walk - Stochastic Process

---

Say we’re studying a game between two players A and B. In it, there’s a probability $p$ of winning and $q$ of losing; there can’t be a draw.

The two players agree on giving each other a monetary unit upon losing. Player A starts with $K$ units, while B starts with $N-K$, adding up to a total of $N$.

(theorem)

Let $\left\{X_n\mid n\in \mathbb{N}\right\}$ be the stochastic process that represents the wealth of player A at the time $n$.

This stochastic process is a random walk that starts at $K$ and finishes either at $0$ (player A is ruined) or at $N$ (player B is ruined). Hence, $0$ and $N$ are both absorbing states.

We can obtain the probability of ruin of a given player, as well as the expected amount of bets before a given player gets ruined.