Bold Play

Say you are playing blackjack (or any other unfair game). You have $x and wish to play until you increase your worth to $L or go broke, whichever comes first. (So L > x.) What betting strategy maximizes the probability of reaching L and minimizes the probability of going broke?

A famous result from Dubins and Savage ("How to Gamble if You Must") is that bold play is an optimal strategy (but not the only 0ne). That is, bet min(x, L - x). So if you have $100 and need to make $1000 to pay a loan shark, bet all $100 on the first hand. If you only need to make $20, just bet $20.

This more recent paper also gives a good survey of the literature in the introduction. This survey of gambling problems also covers the problem. Interestingly, if you are play a fair game (positive expected value), timid play is the optimal strategy.