This problem makes more sense if you strip out time and the doubling, and look at this one:
Choose an integer N. Receive N utilons.
This problem has no optimal solution (because there is no largest integer). You can compare any two strategies to each other, but you cannot find a supremum; the closest thing available is an infinite series of successively better strategies, which eventually passes any single strategy.
In the original problem, the options are “don’t open the box” or “wait N days, then open the box”. The former can be crossed off; the latter has the same infinite series of successively better strategies. (The apparent time-symmetry is a false one, because there are only two time-invariant strategies, and they both lose.)
The way to solve this in decision theory is to either introduce finiteness somewhere that caps the number of possible strategies, or to output an ordering over choices instead of a single choice. The latter seems right; if you define and prove an infinite sequence of successively better options, you still have to pick one; and lattices seem like a good way to represent the results of partial reasoning.
This seems like a helpful simplification of the problem. Note that it also works if you receive 1-1/N utilons, so as with the original post this isn’t an unbounded utility issue as such.
Just one point though—in the original problem specification it’s obvious what “choose an integer N” means: opening a physical box on day n corresponds to choosing 2^n. But how does your problem get embedded in reality? Do you need to write your chosen number down? Assuming there’s no time limit to writing it down then this becomes very similar to the original problem except you’re multiplying by 10 instead of 2 and the time interval is the time taken to write an extra digit instead of a day.
This problem makes more sense if you strip out time and the doubling, and look at this one:
This problem has no optimal solution (because there is no largest integer). You can compare any two strategies to each other, but you cannot find a supremum; the closest thing available is an infinite series of successively better strategies, which eventually passes any single strategy.
In the original problem, the options are “don’t open the box” or “wait N days, then open the box”. The former can be crossed off; the latter has the same infinite series of successively better strategies. (The apparent time-symmetry is a false one, because there are only two time-invariant strategies, and they both lose.)
The way to solve this in decision theory is to either introduce finiteness somewhere that caps the number of possible strategies, or to output an ordering over choices instead of a single choice. The latter seems right; if you define and prove an infinite sequence of successively better options, you still have to pick one; and lattices seem like a good way to represent the results of partial reasoning.
This is pretty much the only comment in the entire thread that doesn’t fight the hypothetical. Well done, I guess?
This seems like a helpful simplification of the problem. Note that it also works if you receive 1-1/N utilons, so as with the original post this isn’t an unbounded utility issue as such.
Just one point though—in the original problem specification it’s obvious what “choose an integer N” means: opening a physical box on day n corresponds to choosing 2^n. But how does your problem get embedded in reality? Do you need to write your chosen number down? Assuming there’s no time limit to writing it down then this becomes very similar to the original problem except you’re multiplying by 10 instead of 2 and the time interval is the time taken to write an extra digit instead of a day.
It doesn’t. It gets embedded in something with infinite time and in which infinite utility can be given out (in infinite different denominations).
Writing decimal digits isn’t the optimal way to write big numbers. (Of course this doesn’t invalidate your point.)
It kind of is if you have to be able to write down any number.