The Kelly criterion doesn’t maximize expected wealth, it maximizes expected log wealth, as the article you linked mentions
I thought the log function operating on real positive numbers was real and monotonically increasing with wealth.
I thought wealth for the purposes of the wikipedia article and Kelly criterion calculations was real and positive.
So how can something which is said to maximize log(wealth) not also be said to maximize wealth with identical meaning?
Seriously, if there is some meaningful sense in which something that maximizes log(wealth) does not also maximize wealth, I am at a loss to even guess what it is and would appreciate being enlightened.
I gave several examples in my comment, but here’s another with explicitly calculated logs:
1) $100 with certainty. Log base 10 is 2. So expected log wealth 2, expected wealth $100.
2) 50% chance of $1, for a log of 0. 50% chance of $1,000 with a log of 3. Expected log wealth is therefore ((0+3)/2)=1.5, and expected wealth is ($1+$1000)/2=$500.5.
1) has higher expected log wealth, but 2) has higher expected wealth.
Please enlighten a poor Physicist. You write:
I thought the log function operating on real positive numbers was real and monotonically increasing with wealth.
I thought wealth for the purposes of the wikipedia article and Kelly criterion calculations was real and positive.
So how can something which is said to maximize log(wealth) not also be said to maximize wealth with identical meaning?
Seriously, if there is some meaningful sense in which something that maximizes log(wealth) does not also maximize wealth, I am at a loss to even guess what it is and would appreciate being enlightened.
I gave several examples in my comment, but here’s another with explicitly calculated logs:
1) $100 with certainty. Log base 10 is 2. So expected log wealth 2, expected wealth $100.
2) 50% chance of $1, for a log of 0. 50% chance of $1,000 with a log of 3. Expected log wealth is therefore ((0+3)/2)=1.5, and expected wealth is ($1+$1000)/2=$500.5.
1) has higher expected log wealth, but 2) has higher expected wealth.