There are many paradoxes with unbounded utility functions. For instance, consider whether it's rational to spend eternity in Hell:
Suppose that you die, and God offers you a deal. You can spend 1 day in Hell, and he will give you 2 days in Heaven, and then you will spend the rest of eternity in Purgatory (which is positioned exactly midway in utility between heaven and hell). You decide that it's a good deal, and accept. At the end of your first day in Hell, God offers you the same deal: 1 extra day in Hell, and you will get 2 more days in Heaven. Again you accept. The same deal is offered at the end of the second day.
And the result is... that you spend eternity in Hell. There is never a rational moment to leave for Heaven - that decision is always dominated by the decision to stay in Hell.
Or consider a simpler paradox:
You're immortal. Tell Omega any natural number, and he will give you that much utility. On top of that, he will give you any utility you may have lost in the decision process (such as the time wasted choosing and specifying your number). Then he departs. What number will you choose?
Again, there's no good answer to this problem - any number you name, you could have got more by naming a higher one. And since Omega compensates you for extra effort, there's never any reason to not name a higher number.
It seems that these are problems caused by unbounded utility. But that's not the case, in fact! Consider:
You're immortal. Tell Omega any real number r > 0, and he'll give you 1-r utility. On top of that, he will give you any utility you may have lost in the decision process (such as the time wasted choosing and specifying your number). Then he departs. What number will you choose?
Again, there is not best answer - for any r, r/2 would have been better. So these problems arise not because of unbounded utility, but because of unbounded options. You have infinitely many options to choose from (sequentially in the Heaven and Hell problem, all at once in the other two) and the set of possible utilities from your choices does not possess a maximum - so there is no best choice.
What should you do? In the Heaven and Hell problem, you end up worse off if you make the locally dominant decision at each decision node - if you always choose to add an extra day in Hell, you'll never get out of it. At some point (maybe at the very beginning), you're going to have to give up an advantageous deal. In fact, since giving up once means you'll never be offered the deal again, you're going to have to give up arbitrarily much utility. Is there a way out of this conundrum?
Assume first that you're a deterministic agent, and imagine that you're sitting down for an hour to think about this (don't worry, Satan can wait, he's just warming up the pokers). Since you're deterministic, and you know it, then your ultimate life future will be entirely determined by what you decide right now (in fact your life history is already determined, you just don't know it yet - still, by the Markov property, your current decision also determines the future). Now, you don't have to reach any grand decision now - you're just deciding what you'll do for the next hour or so. Some possible options are:
- Ignore everything, sing songs to yourself.
- Think about this some more, thinking of yourself as an algorithm.
- Think about this some more, thinking of yourself as a collection of arguing agents.
- Pick a number N, and accept all of God's deals until day N.
- Promise yourself you'll reject all of God's deals.
- Accept God's deal for today, hope something turns up.
- Defer any decision until another hour has passed.
There are many other options - in fact, there are precisely as many options as you've considered during that hour. And, crucially, you can put an estimated expected utility to each one. For instance, you might know yourself, and suspect that you'll always do the same thing (you have no self discipline where cake and Heaven are concerned), so any decision apart from immediately rejecting all of God's deals will give you -∞ utility. Or maybe you know yourself, and have great self discipline and perfect precommitments- therefore if you pick a number N in the coming hour, you'll stick to it. Thinking some more may have a certain expected utility - which may differ depending on what directions you direct your thoughts. And if you know that you can't direct your thoughts - well then they'll all have the same expected utility.
But notice what's happening here: you've reduced the expected utility calculation over infinitely many options, to one over finitely many options - namely, all the interim decisions that you can consider in the course of an hour. Since you are deterministic, the infinitely many options don't have an impact: whatever interim decision you follow, will uniquely determine how much utility you actually get out of this. And given finitely many options, each with expected utility, choosing one doesn't give any paradoxes.
And note that you don't need determinism - adding stochastic components to yourself doesn't change anything, as you're already using expected utility anyway. So all you need is an assumption of naturalism - that you're subject to the laws of nature, that your decision will be the result of deterministic or stochastic processes. In other words, you don't have 'spooky' free will that contradicts the laws of physics.
Of course, you might be wrong about your estimates - maybe you have more/less willpower than you initially thought. That doesn't invalidate the model - at every hour, at every interim decision, you need to choose the option that will, in your estimation, ultimately result in the most utility (not just for the next few moments or days).
If we want to be more formal, we can say that you're deciding on a decision policy - choosing among the different agents that you could be, the one most likely to reach high expected utility. Here are some policies you could choose from (the challenge is to find a policy that gets you the most days in Hell/Heaven, without getting stuck and going on forever):
- Decide to count the days, and reject God's deal as soon as you lose count.
- Fix a probability distribution over future days, and reject God's deal with a certain probability.
- Model yourself as a finite state machine. Figure out the Busy Beaver number of that finite state machine. Reject the deal when the number of days climbs close to that.
- Realise that you probably can't compute the Busy Beaver number for yourself, and instead use some very fast growing function like the Ackermann functions instead.
- Use the Ackermann function to count down the days during which you formulate a policy; after that, implement it.
- Estimate that there is a non-zero probability of falling into a loop (which would give you -∞ utility), so reject God's deal as soon as possible.
- Estimate that there is a non-zero probability of accidentally telling God the wrong thing, so commit to accepting all of God's deals (and count on accidents to rescue you from -∞ utility).
But why spend a whole hour thinking about it? Surely the same applies for half an hour, a minute, a second, a microsecond? That's entirely a convenience choice - if you think about things in one second increments, then the interim decision "think some more" is nearly always going to be the dominant one.
The mention of the Busy Beaver number hints at a truth - given the limitations of your mind and decision abilities, there is one policy, among all possible policies that you could implement, that gives you the most utility. More complicated policies you can't implement (which generally means you'd hit a loop and get -∞ utility), and simpler policies would give you less utility. Of course, you likely won't find that policy, or anything close to it. It all really depends on how good your policy finding policy is (and your policy finding policy finding policy...).
That's maybe the most important aspect of these problems: some agents are just better than others. Unlike finite cases where any agent can simply list all the options, take their time, and choose the best one, here an agent with a better decision algorithm will outperform another. Even if they start with the same resources (memory capacity, cognitive shortcuts, etc...) one may be a lot better than another. If the agents don't acquire more resources during their time in Hell, then their maximal possible utility is related to their Busy Beaver number - basically the maximal length that a finite-state agent can survive without falling into an infinite loop. Busy Beaver numbers are extremely uncomputable, so some agents, by pure chance, may be capable of acquiring much greater utility than others. And agents that start with more resources have a much larger theoretical maximum - not fair, but deal with it. Hence it's not really an infinite option scenario, but an infinite agent scenario, with each agent having a different maximal expected utility that they can extract from the setup.
It should be noted that God, or any being capable of hypercomputation, has real problems in these situations: they actually have infinite options (not a finite options of choosing their future policy), and so don't have any solution available.
This is also related to theoretical maximally optimum agent that is AIXI: for any computable agent that approximates AIXI, there will be other agents that approximate it better (and hence get higher expected utility). Again, it's not fair, but not unexpected either: smarter agents are smarter.
What to do?
This analysis doesn't solve the vexing question of what to do - what is the right answer to these kind of problems? These depend on what type of agent you are, but what you need to do is estimate the maximal integer you are capable of computing (and storing), and endure for that many days. Certain probabilistic strategies may improve your performance further, but you have to put the effort into finding them.