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.
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.