Gambler's ruin - problem 6

Probability Level 3

You walk into Martin Gale's Betting Room with an initial budget of $k$

As usual, you can play the following game any number of times (if you still have what it costs)

You pay Martin a dollar.
Martin tosses a fair coin
If the coin comes up heads Martin pays you two dollars. If it comes up tails, you get nothing.

Your plan is to just keep playing indefinitely, if you can.

Which of the following is true?

1) You will lose all your money within $O(k^2)$ games.

2) You will lose all your money within $O(2^k)$ games.

3) There is an integer $n = n(k)$ such that it is impossible to increase your wealth to $n$

4) For each integer $n > k$ , you have a positive probability of increasing your wealth to $n$ , hence you have a positive probability of being able to play indefinitely.

5) For each integer $n > k$ , you have a positive probability of increasing your wealth to $n$ , but you will inevitably eventually lose all your money.

Note: You may assume that Martin's budget is infinite. In fact this is implicit in the fact that you are even allowed, in principle, to play indefinitely. If Martin had a finite budget $B$ then situation would be equivalent to your deciding to play up to threshold $k+B$ .

The answer is 5.

1 solution

Mark Hennings
Feb 14, 2019

Suppose that $p_{n,k}$ is the probability of reaching a wealth of $n$ without going bust, given that you start with $k$ . Then $p_{n,0}=0$ and $p_{n,n} = 1$ , while $p_{n,k} \; = \; \tfrac12(p_{n,k-1} + p_{n,k+1}) \hspace{2cm} 1 \le k \le n-1$ Thus $p_{n,k} = \tfrac{k}{n}$ for all $0 \le k \le n$ . Thus you have probability $\tfrac{k}{n}$ of increasing your wealth from $k$ to $n$ .

Let $q_k$ be the probability that you eventually go bust, given that you start with $k$ . Then $q_0 = 1$ and $q_k \; = \; \tfrac12(q_{k+1} + q_{k-1}) \hspace{2cm} k \ge 1$ Thus $q_k = 1 + Ak$ for $k \ge 0$ for some constant $A$ . Since $0 \le q_k \le 1$ for all $k$ , we deduce that $A=0$ and so $q_k=1$ for all $k$ , so you are sure to go bust.

These results tell us that $\boxed{(5)}$ is true, which certainly makes (3) & (4) false. There is a nonzero probability of winning a streak of any length, so you can never be certain of going bust in a fixed finite number of moves, which makes (1) & (2) false.

Gambler's ruin - problem 6

The answer is 5.

1 solution

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