Suppose that a gambler starts with a fortune in (0,1) and wishes to attain a fortune of 1 by making a sequence of bets. Assume that whenever the gambler stakes an amount s, the gambler's fortune increases by s with probability w and decreases by s with probability 1 − w, where w < ½. Dubins and Savage showed that the optimal strategy, which they called ‘bold play’, is always to bet 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.