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Lumifer comments on An investment analogy for Pascal's Mugging - Less Wrong Discussion

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Comment author: Lumifer 10 December 2014 10:20:37PM * -1 points [-]

The Kelly rule maximizes the expected value of the log of your bets.

The Kelly rule maximizes the log of your bankroll as the number of trials goes to infinity. Note that Wikipedia says:

one would set the derivative y'(x) of the expected value of the logarithmic bankroll y(x) to 0 and solve for x:

You're maximizing "the expected value of the logarithmic bankroll y(x)".

Comment author: DanielLC 10 December 2014 10:34:28PM 1 point [-]

If you take any of the bets, your bankroll is a probability distribution. Probability distributions have no standard ordering, and cannot be maximized.

Comment author: Lumifer 11 December 2014 01:05:13AM -1 points [-]

Yes, that's why you're maximizing the expected value and not an entire probability distribution.

Comment author: DanielLC 11 December 2014 06:31:47AM 2 points [-]

The Kelly rule maximizes the log of your bankroll as the number of trials goes to infinity.

I seem to have misread the second thing you said, which is more helpful.

You're maximizing "the expected value of the logarithmic bankroll y(x)".

Yes. The Kelly criterion maximizes the expected value of the logarithmic bankroll. Not the expected value of the bankroll.

Comment author: Lumifer comment score below threshold [+] (3 children)

Comment author: DanielLC 11 December 2014 07:41:52AM 5 points [-]

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.

Comment author: gjm 11 December 2014 10:01:47AM 2 points [-]

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.

Comment author: Lumifer 11 December 2014 03:46:39PM * 0 points [-]

Kelly says: maximise expected log bankroll.

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.

Lumifer comments on An investment analogy for Pascal's Mugging - Less Wrong Discussion

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