Thursday, July 20, 2006

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.

Tuesday, July 18, 2006

The Gambler's Ruin

A game of blackjack is a good example of a random walk. There is some good theoretical material on this available online (see p. 487).

An example is illuminating. Assume you start with $10. Each game, you have a 49% chance of winning $1 and a 51% chance of losing a dollar. (Real blackjack odds are better.) Say you decide to play until you have $11 or go broke, whichever comes first. In this case, you only have an 11% chance of going broke and an 89% chance of winning the $11. But if you play until $15, you have a 40% chance of going broke. Play to $20 and it increases to 60%. The longer you play, the lesser your expected value. Play forever--greedy you!--and you have a 100% chance of going broke. This is known as the gambler's ruin.