“I don’t know, I recall something called the Kelly criterion which says you shouldn’t scale your willingness to make risky bets proportionally with available capital—that is, you shouldn’t be just as eager to bet your capital away when you have a lot as when you have very little, or you’ll go into the red much faster.
I think I’m misunderstanding something here. Let’s say you have n dollars and are looking for the optimum number of dollars to bet on something that causes you to gain α dollars with probability p and lose β dollars with probability 1−p. The optimum number of dollars you should bet via the Kelly criterion seems to be
n(αp+β(1−p)αβ)
(assuming positive expectation; i.e. the numerator is positive), which does scale linearly with n. And this seems fundamental to this post.
I think you’re right, I had misunderstood! Kind of an egregious misunderstanding, too.
I’m curious how it seems to you to be fundamental to the post, maybe I missed something on that count. I’m planning on replacing ‘shouldn’t scale . . . proportionally’ with ‘shouldn’t scale more than proportionally’, and I don’t see how that substantially changes anything, given that Jill is right, and the concept of a Kelly fraction isn’t applicable when you’re starting out with zero capital of your own to gamble.
I think even the scaling thing doesn’t apply here because they’re not insuring bigger trips: they’re insuring more trips (which makes things strictly better). I’m having some trouble understanding Dennis’ point.
0 trips → 1 trip is an addition to the number of games played, but it’s also an addition to the percentage of income bet on that one game—right?
Dennis is also having trouble understanding his own point, FWIW. That’s how the dialogue came out; both people in that part are thinking in loose/sketchy terms and missing important points.
The thing Dennis was trying to get at by bringing up the concrete example of an optimal Kelly fraction is that it doesn’t make sense for willingness to make a risky bet to have no dependence on available capital; he perceives Jill as suggesting that this is the case.
I think I’m misunderstanding something here. Let’s say you have n dollars and are looking for the optimum number of dollars to bet on something that causes you to gain α dollars with probability p and lose β dollars with probability 1−p. The optimum number of dollars you should bet via the Kelly criterion seems to be
n(αp+β(1−p)αβ)(assuming positive expectation; i.e. the numerator is positive), which does scale linearly with n. And this seems fundamental to this post.
I think you’re right, I had misunderstood! Kind of an egregious misunderstanding, too.
I’m curious how it seems to you to be fundamental to the post, maybe I missed something on that count. I’m planning on replacing ‘shouldn’t scale . . . proportionally’ with ‘shouldn’t scale more than proportionally’, and I don’t see how that substantially changes anything, given that Jill is right, and the concept of a Kelly fraction isn’t applicable when you’re starting out with zero capital of your own to gamble.
I think even the scaling thing doesn’t apply here because they’re not insuring bigger trips: they’re insuring more trips (which makes things strictly better). I’m having some trouble understanding Dennis’ point.
0 trips → 1 trip is an addition to the number of games played, but it’s also an addition to the percentage of income bet on that one game—right?
Dennis is also having trouble understanding his own point, FWIW. That’s how the dialogue came out; both people in that part are thinking in loose/sketchy terms and missing important points.
The thing Dennis was trying to get at by bringing up the concrete example of an optimal Kelly fraction is that it doesn’t make sense for willingness to make a risky bet to have no dependence on available capital; he perceives Jill as suggesting that this is the case.