Improving on bold play when the gambler is restricted

Suppose a gambler starts with a fortune in (0,1) and wishes to attain a fortune of 1 by making a sequence of bets. Assume thay whenever the gambler stakes the amount s, the gambler's fortune increases by s with probability w and decreases by s with probability 1 - w, where w < 1/2. Dubins and Savage showed that the optimal strategy, which they called "bold play", is always to stake min{f, 1-f}, where f is the gambler's current fortune. Here we consider the problem in which the gambler may stake no more than l at one time. We show that the bold strategy of always betting min{l, f, 1-f} is not optimal if l is irrational, extending a result of Heath, Pruitt, and Sudderth.