Actually, no, it does not. The Kelly Rule aims to maximize the expected bankroll (not the log of it) after many bets. It’s a real-world rule and in the real world people want money, not log(money).
I think there are a couple of points of confusion here. The first is between maximizing what and maximizing how. We want to maximize the amount that you have on hand after a long series of bets. Figuring out what bets to accept and which to decline involves logs, but in the end you just want to have max(money). The second is between maximizing over a long series of bets and maximizing in one single bet (trial).
If your current bankroll is $1000 and you can choose between (1) a gamble that gives you $100 with probability 3⁄4 and $1000 with probability 1⁄4 and (2) sticking with what you’ve got, maximising expected bankroll will tell you to choose #1 but maximising expected log bankroll will tell you to choose #2.
Your example is screwed up. First, no one would ever choose #1, and second, the Kelly Rule isn’t about picking between two bets, it’s about how much of your bankroll to invest in a particular bet.
In any case, this subthread seems to have fallen into the negative-karma hole.
Actually, no, it does not. The Kelly Rule aims to maximize the expected bankroll (not the log of it) after many bets. It’s a real-world rule and in the real world people want money, not log(money).
I think there are a couple of points of confusion here. The first is between maximizing what and maximizing how. We want to maximize the amount that you have on hand after a long series of bets. Figuring out what bets to accept and which to decline involves logs, but in the end you just want to have max(money). The second is between maximizing over a long series of bets and maximizing in one single bet (trial).
Your example is screwed up. First, no one would ever choose #1, and second, the Kelly Rule isn’t about picking between two bets, it’s about how much of your bankroll to invest in a particular bet.
In any case, this subthread seems to have fallen into the negative-karma hole.