Infinite t does not necessarily deliver infinite utility.
Perhaps it would be simpler if I instead let t be in (0, 1], and U(t) = {t if t < 1; 0 if t = 1}.
It’s the same problem, with 1 replacing infinity. I have edited the question with this example instead.
(It’s not a particularly weird utility function—consider, e.g. if the agent needs to expend a resource such that the utility from expending the resource at time t is some fast-growing function f(t). But never expending the resource gives zero utility. In any case, an adverserial agent can always create this situation.)
I’m not personally aware of any “best” or “correct” solutions, and I would be quite surprised if there were one (mathematically, at least, we know there’s no single maximizer). But I think concretely speaking, you can restrict the choice set of t to a compact set of size (0, 1 - \epsilon] and develop the appropriate bounds for the analysis you’re interested in. Maybe not the most satisfying answer, but I guess that’s Analysis in a nutshell.
Infinite t does not necessarily deliver infinite utility.
Perhaps it would be simpler if I instead let t be in (0, 1], and U(t) = {t if t < 1; 0 if t = 1}.
It’s the same problem, with 1 replacing infinity. I have edited the question with this example instead.
(It’s not a particularly weird utility function—consider, e.g. if the agent needs to expend a resource such that the utility from expending the resource at time t is some fast-growing function f(t). But never expending the resource gives zero utility. In any case, an adverserial agent can always create this situation.)
I see what you mean now, thanks for clarifying.
I’m not personally aware of any “best” or “correct” solutions, and I would be quite surprised if there were one (mathematically, at least, we know there’s no single maximizer). But I think concretely speaking, you can restrict the choice set of t to a compact set of size (0, 1 - \epsilon] and develop the appropriate bounds for the analysis you’re interested in. Maybe not the most satisfying answer, but I guess that’s Analysis in a nutshell.