I think the key thing to note here is that “maximizing expected growth” looks the same whether the thing you’re trying to grow is money or log-money or sqrt-money or what. It “just happens” that (at least in this framework) the way one maximizes expected growth is the same as the way one maximizes expected log-money.
I’ve recently written about this myself. My goal was partly to clarify this, though I don’t know if I succeeded.
I think the post confuses things by motivating the Kelly bet as the thing that maximizes expected log-money, and also has other neat properties. To my mind, if you want to maximize expected log-money, you just… do the arithmetic to figure out what that means. It’s not quite trivial, but it’s stats-101 stuff. I don’t think it seems more interesting to do the arithmetic that maximizes expected log-money compared to expected money or expected sqrt-money. Kelly certainly didn’t introduce the criterion as “hey guys, here’s a way to maximize expected log-money”. (Admittedly, I don’t much care about his framing either. The original paper is information-theoretic in a way that seems to be mostly forgotten about these days.)
To my mind, the important thing about the Kelly bet is the “almost certainly win more money than anyone using a different strategy, over a long enough time period” thing. (Which is the same as maximizing expected growth rate, when growth is exponential. If growth is linear you still might care if you’re earning $2/day or $1/day, but the “growth rate” of both is 0 as defined here.) So I prefer to motivate the Kelly bet as being the thing that does that, and then say “and incidentally, turns out this also maximizes expected log-wealth, which is neat because...”.
I think the key thing to note here is that “maximizing expected growth” looks the same whether the thing you’re trying to grow is money or log-money or sqrt-money or what. It “just happens” that (at least in this framework) the way one maximizes expected growth is the same as the way one maximizes expected log-money.
I’ve recently written about this myself. My goal was partly to clarify this, though I don’t know if I succeeded.
I think the post confuses things by motivating the Kelly bet as the thing that maximizes expected log-money, and also has other neat properties. To my mind, if you want to maximize expected log-money, you just… do the arithmetic to figure out what that means. It’s not quite trivial, but it’s stats-101 stuff. I don’t think it seems more interesting to do the arithmetic that maximizes expected log-money compared to expected money or expected sqrt-money. Kelly certainly didn’t introduce the criterion as “hey guys, here’s a way to maximize expected log-money”. (Admittedly, I don’t much care about his framing either. The original paper is information-theoretic in a way that seems to be mostly forgotten about these days.)
To my mind, the important thing about the Kelly bet is the “almost certainly win more money than anyone using a different strategy, over a long enough time period” thing. (Which is the same as maximizing expected growth rate, when growth is exponential. If growth is linear you still might care if you’re earning $2/day or $1/day, but the “growth rate” of both is 0 as defined here.) So I prefer to motivate the Kelly bet as being the thing that does that, and then say “and incidentally, turns out this also maximizes expected log-wealth, which is neat because...”.