Sure, but the point of the Kelly calculation is to PICK the amount, relative to the potential gain and risk of ruin. Which ends up equivalent to logarithmic utility.
For the final bet (or the induction base for a finite sequence), one cannot pick an amount without knowing the zero-point on the utility curve.
What my comment about small differences amounts to is if you can’t bet the whole bankroll; for example if your utility is (altruistically) logarithmic in the resources of all humanity, but you can only control how to gamble with a small fraction of that.
This might justify EAs behaving like their utility is linear in resources, even if it’s ultimately logarithmic.
The problem is in the “I have” statement in the setup. After your final bet, you will be dead (or at least not have any ability for further decisions about the money). You have to specify what “have” means, in terms of your utility. Perhaps you’ve willed it all to a home for cats, in that case the home has 500,100 +/- x. Perhaps you care about humanity as a whole, in which case your wager has no impact—any that you add or remove from “your” $100 comes out of someone else’s. Or if the wager is making something worth x, or destroying x value as your final act, then humanity as a whole has $90T +/- x.
Sure, but the point of the Kelly calculation is to PICK the amount, relative to the potential gain and risk of ruin. Which ends up equivalent to logarithmic utility.
For the final bet (or the induction base for a finite sequence), one cannot pick an amount without knowing the zero-point on the utility curve.
Agreed.
What my comment about small differences amounts to is if you can’t bet the whole bankroll; for example if your utility is (altruistically) logarithmic in the resources of all humanity, but you can only control how to gamble with a small fraction of that.
This might justify EAs behaving like their utility is linear in resources, even if it’s ultimately logarithmic.
I’m a little confused about what you mean sorry -
What’s wrong with this example?:
It’s time for the final bet, I have $100 and my utility is U=ln($)
I have the opportunity to bet on a coin which lands heads with probability 34, at 1:1 odds.
If I bet $x on heads, then my expected utility is E(U)=34ln(100+x)+14ln(100−x), which is maximized when x=50.
So I decide to bet 50 dollars.
What am I missing here?
The problem is in the “I have” statement in the setup. After your final bet, you will be dead (or at least not have any ability for further decisions about the money). You have to specify what “have” means, in terms of your utility. Perhaps you’ve willed it all to a home for cats, in that case the home has 500,100 +/- x. Perhaps you care about humanity as a whole, in which case your wager has no impact—any that you add or remove from “your” $100 comes out of someone else’s. Or if the wager is making something worth x, or destroying x value as your final act, then humanity as a whole has $90T +/- x.