On discounted subfair primitive casino

As is well known, in a subfair primitive casino a gambler with an initial fortune f, 0<f<1, desiring to reach 1 (his goal) should play boldly since there is no other strategy that can provide him with a higher utility (the probability of reaching his goal). Now suppose the game is modified by adding a discount factor which is used to motivate the gambler to recognize the time value of his goal and complete the game as quickly as is reasonably consistent with reaching his goal. Then one would intuitively suspect that again the bold play would be optimal. We will show in this paper that for certain subfair or fair primitive casinos the bold play is always optimal regardless of the discount factor; however, for some subfair or fair primitive casinos, there exist some discount factors for which the bold play is no longer optimal.