Average of 1 and 100 is 50.5. Average of log(1) and log(100) is log(10). If you’re offered a choice of a 50:50 chance of 1 vs 10 and a sure amount between 10 and 50.5, then you’re better off with the bet for maximizing expected value, but you’re better off with the sure thing for maximizing the expected log value.
You are confused between maximizing the log of the expected value of your bets and maximizing the expected value of the log of your bets. These are, of course, not the same.
In Kelly Rule bets you do not get paid the log of the outcomes.
>>> from sympy import *
>>> x,b,p = symbols('x b p')
>>> y = p*log(1+b*x) + (1-p)*log(1-x)
>>> solve(diff(y,x), x)
[-(1 - p - b*p)/b]
It is maximizing the function y = p*log(1+b*x) + (1-p)*log(1-x). Maximizing the function y = p*(1+b*x) + (1-p)*(1-x) with the restriction that p is between 0 and 1 will give p as 0 or 1, since it’s a linear function.
If you take any of the bets, your bankroll is a probability distribution. Probability distributions have no standard ordering, and cannot be maximized.
The Kelly criterion maximizes the expected value of the logarithmic bankroll. Not the expected value of the bankroll.
And we have come full circle to noting again that because log is a monotonic function maximizing the expected value of the log of the bankroll is exactly the same thing as maximizing the expected value of the bankroll.
The expected value of the bankroll is a simple number, a positive scalar. Maximizing this value is equivalent to maximizing the log of this value.
But the Kelly criterion does not maximize the log of the expected value of the bankroll. It maximizes the expected value of the log of the bankroll. The way to maximize the log of the expected value of the bankroll is to take all of the bets with all of your money.
I’m not really saying anything DanielLC hasn’t said already, but perhaps it will be easier for you to take seriously when it’s not just one other person disagreeing with you.
DanielLC is right and you are wrong.
Kelly says: maximise expected log bankroll. This is not the same as maximising expected bankroll, and it is not true in general that maximising E(X) and maximising E(f(X)) are equivalent when f is monotonic increasing.
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.
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.
Average of 1 and 100 is 50.5. Average of log(1) and log(100) is log(10). If you’re offered a choice of a 50:50 chance of 1 vs 10 and a sure amount between 10 and 50.5, then you’re better off with the bet for maximizing expected value, but you’re better off with the sure thing for maximizing the expected log value.
You are confused between maximizing the log of the expected value of your bets and maximizing the expected value of the log of your bets. These are, of course, not the same.
In Kelly Rule bets you do not get paid the log of the outcomes.
The Kelly rule maximizes the expected value of the log of your bets. You get paid the outcome, but you presumably value the log of what you get paid.
The proof section on the Wikipedia article gives the derivation using Python and SymPy:
It is maximizing the function y = p*log(1+b*x) + (1-p)*log(1-x). Maximizing the function y = p*(1+b*x) + (1-p)*(1-x) with the restriction that p is between 0 and 1 will give p as 0 or 1, since it’s a linear function.
The Kelly rule maximizes the log of your bankroll as the number of trials goes to infinity. Note that Wikipedia says:
You’re maximizing “the expected value of the logarithmic bankroll y(x)”.
If you take any of the bets, your bankroll is a probability distribution. Probability distributions have no standard ordering, and cannot be maximized.
Yes, that’s why you’re maximizing the expected value and not an entire probability distribution.
I seem to have misread the second thing you said, which is more helpful.
Yes. The Kelly criterion maximizes the expected value of the logarithmic bankroll. Not the expected value of the bankroll.
And we have come full circle to noting again that because log is a monotonic function maximizing the expected value of the log of the bankroll is exactly the same thing as maximizing the expected value of the bankroll.
The expected value of the bankroll is a simple number, a positive scalar. Maximizing this value is equivalent to maximizing the log of this value.
But the Kelly criterion does not maximize the log of the expected value of the bankroll. It maximizes the expected value of the log of the bankroll. The way to maximize the log of the expected value of the bankroll is to take all of the bets with all of your money.
I’m not really saying anything DanielLC hasn’t said already, but perhaps it will be easier for you to take seriously when it’s not just one other person disagreeing with you.
DanielLC is right and you are wrong.
Kelly says: maximise expected log bankroll. This is not the same as maximising expected bankroll, and it is not true in general that maximising E(X) and maximising E(f(X)) are equivalent when f is monotonic increasing.
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.
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.