The tax man's dilemma, an infinite decision tree grounded in reality:
Assume you're the anthropomorphization of government. And you have a decision to make: You need to decide the ideal tax rate for businesses.
In your society, corporations reliably make 5% returns on investments, accounting for inflation. That money is reliably reinvested, although not necessarily in the same corporation.
How should you tax those returns in order to maximize total utility? You may change taxes at any point. Also, you're the anthropomorphic representation of government - you are, for all intents and purposes, immortal.
Assume a future utility discount rate of less than the investment return rate, and assume you don't know the money-utility relationship - you can say you weigh the possibility of future disasters which require immense funds against the possibility that money has declining utility over time to produce a constant relationship for simplicity, if you wish. Assume that your returns will be less than corporate returns, and corporate utility will be less than your utility. (Simplified, you produce no investment returns, corporations produce no utility.)
Let's say you have a box that has a token in it that can be redeemed for 1 utilon. Every day, its contents double. There is no limit on how many utilons you can buy with these tokens. You are immortal. It is sealed, and if you open it, it becomes an ordinary box. You get the tokens it has created, but the box does not double its contents anymore. There are no other ways to get utilons.
How long do you wait before opening it? If you never open it, you get nothing (you lose! Good day, sir or madam!) and whenever you take it, taking it one day later would have been twice as good.
I hope this doesn't sound like a reductio ad absurdum against unbounded utility functions or not discounting the future, because if it does you are in danger of amputating the wrong limb to save yourself from paradox-gangrene.
What if instead of growing exponentially without bound, it decays exponentially to the bound of your utility function? If your utility function is bounded at 10, what if the first day it is 5, the second 7.5, the third 8.75, etc. Assume all the little details, like remembering about the box, trading in the tokens, etc, are free.
If you discount the future using any function that doesn't ever hit 0, then the growth rate of the tokens can be chosen to more than make up for your discounting.
If it does hit 0 at time T, what if instead of doubling, it just increases by however many utilons will be adjusted to 1 by your discounting at that point every time of growth, but the intervals of growth shrink to nothing? You get an adjusted 1 utilon at time T - 1s, and another adjusted 1 utilon at T - 0.5s, and another at T - 0.25s, etc? Suppose you can think as fast as you want, and open the box at arbitrary speed. Also, that whatever solution your present self precommits to will be followed by the future self. (Their decision won't be changed by any change in what times they care about)
EDIT: People in the comments have suggested using a utility function that is both bounded and discounting. If your utility function isn't so strongly discounting that it drops to 0 right after the present, then you can find some time interval very close to the present where the discounting is all nonzero. And if it's nonzero, you can have a box that disappears, taking all possible utility with it at the end of that interval, and that, leading up to that interval, grows the utility in intervals that shrink to nothing as you approach the end of the interval, and increasing the utility-worth of tokens in the box such that it compensates for whatever your discounting function is exactly enough to asymptotically approach your bound.
Here is my solution. You can't assume that your future self will make the optimal decision, or even a good decision. You have to treat your future self as a physical object that your choices affect, and take the probability distribution of what decisions your future self will make, and how much utility they will net you into account.
Think if yourself as a Turing machine. If you do not halt and open the box, you lose and get nothing. No matter how complicated your brain, you have a finite number of states. You want to be a busy beaver and take the most possible time to halt, but still halt.
If, at the end, you say to yourself "I just counted to the highest number I could, counting once per day, and then made a small mark on my skin, and repeated, and when my skin was full of marks, that I was constantly refreshing to make sure they didn't go away...
...but I could let it double one more time, for more utility!"
If you return to a state you have already been at, you know you are going to be waiting forever and lose and get nothing. So it is in your best interest to open the box.
So there is not a universal optimal solution to this problem, but there is an optimal solution for a finite mind.
I remember reading a while ago about a paradox where you start with $1, and can trade that for a 50% chance of $2.01, which you can trade for a 25% chance of $4.03, which you can trade for a 12.5% chance of $8.07, etc (can't remember where I read it).
This is the same paradox with one of the traps for wannabe Captain Kirks (using dollars instead of utilons) removed and one of the unnecessary variables (uncertainty) cut out.
My solution also works on that. Every trade is analogous to a day waited to open the box.