From now on, we will assume that the gambler's stopping rule is a very simple and standard one: she will bet on the games until she either loses her entire fortune and is ruined or reaches a fixed tar...From now on, we will assume that the gambler's stopping rule is a very simple and standard one: she will bet on the games until she either loses her entire fortune and is ruined or reaches a fixed target fortune \(a\): \[ N = \min\{n \in \N: X_n = 0 \text{ or } X_n = a\} \] Thus, any strategy (betting function) \(S\) must satisfy \(s(x) \le \min\{x, a - x\}\) for \(0 \le x \le a\): the gambler cannot bet what she does not have, and will not bet more than is necessary to reach the target \(a\).
