For example, suppose you have a bet that has a 50% chance of losing everything and a 50% chance of quadrupling your investment, the Kelly criterion says not to take it, since losing everything has infinite disutility.
A bet where you quadruple your investment has a b of 3, and p is .5. The Kelly criterion says you should bet (b*p-q)/b, which is (3*.5-.5)/3, which is one third of your bankroll every time. The expected value after n times is (4/3)^n.
The assumption of the Kelly criterion is that you get to decide the scale of your investment, and that the investment scales with your bankroll.
If you take it n times, you have a 2^(-n) chance of having 4^n times as much as you started with, which gives an expected value of 2^n.
Indeed, but the probability that the Kelly better does better than that better is 1-2^(-n)!
A bet where you quadruple your investment has a b of 3, and p is .5. The Kelly criterion says you should bet (b*p-q)/b, which is (3*.5-.5)/3, which is one third of your bankroll every time. The expected value after n times is (4/3)^n.
The assumption of the Kelly criterion is that you get to decide the scale of your investment, and that the investment scales with your bankroll.
Indeed, but the probability that the Kelly better does better than that better is 1-2^(-n)!