If you can use mixed strategies (i.e. are not required to be deterministically predictable), you can use the following strategy for the doubling-utility case: every day, toss a coin; if it comes up heads, open the box, otherwise wait another day. Expected utility of each day is constant 1⁄2, since the probability of getting heads on a particular day halves with each subsequent day, and utility doubles, so the series diverges and you get infinite total expected utility.
This suggests a strategy; tile the universe with coins and flip each of them every day. If they all come up heads, open the box (presumably it’s full of even more coins).
better yet, every day count one more integer toward the highest number you can think of, when you reach it, flip the coins. If they don’t all come up heads, start over again.
There are meaningful ways to compare two outcomes which both have infinite expected utility. For example, suppose X is your favorite infinite-expected-utility outcome. Then a 20% chance of X (and 80% chance of nothing) is better than a 10% chance of X. Something similar happens with tossing two coins instead of one, although it’s more subtle.
Actually what you get is another divergent infinite series that grows faster. They both grow arbitrarily large, but the one with p=0.25 grows arbitrarily larger than the series with p=0.5, as you compute more terms. So there is no sense in which the second series is twice as big, although there is a sense in which it is infinitely larger. (I know your point is that they’re both technically the same size, but I think this is worth noting.)
This is what I was going to say; it’s consistent with the apparent time symmetry, and is the only solution that makes sense if we accept the problem as stated. But it seems like the wrong answer intuitively, because it means that every strategy is equal, as long as the probability of opening the box on a given day is in the half-open interval (0,0.5]. I’d certainly be happier with, say, p=0.01 than p=0.5, (and so would everyone else, apparently) which suggests that I don’t actually have a real-valued utility function. This might be a good argument against real-valued utility functions in general (bounded or not). Especially since a lot of the proposed solutions here “fight the hypothetical” by pointing out that real agents can only choose from a finite set of strategies.
If you can use mixed strategies (i.e. are not required to be deterministically predictable), you can use the following strategy for the doubling-utility case: every day, toss a coin; if it comes up heads, open the box, otherwise wait another day. Expected utility of each day is constant 1⁄2, since the probability of getting heads on a particular day halves with each subsequent day, and utility doubles, so the series diverges and you get infinite total expected utility.
Even better, however, would be to toss two coins every day, and only open the box if both come up heads :)
This suggests a strategy; tile the universe with coins and flip each of them every day. If they all come up heads, open the box (presumably it’s full of even more coins).
better yet, every day count one more integer toward the highest number you can think of, when you reach it, flip the coins. If they don’t all come up heads, start over again.
That way your expected utility becomes INFINITY TIMES TWO! :)
There are meaningful ways to compare two outcomes which both have infinite expected utility. For example, suppose X is your favorite infinite-expected-utility outcome. Then a 20% chance of X (and 80% chance of nothing) is better than a 10% chance of X. Something similar happens with tossing two coins instead of one, although it’s more subtle.
Actually what you get is another divergent infinite series that grows faster. They both grow arbitrarily large, but the one with p=0.25 grows arbitrarily larger than the series with p=0.5, as you compute more terms. So there is no sense in which the second series is twice as big, although there is a sense in which it is infinitely larger. (I know your point is that they’re both technically the same size, but I think this is worth noting.)
This is what I was going to say; it’s consistent with the apparent time symmetry, and is the only solution that makes sense if we accept the problem as stated. But it seems like the wrong answer intuitively, because it means that every strategy is equal, as long as the probability of opening the box on a given day is in the half-open interval (0,0.5]. I’d certainly be happier with, say, p=0.01 than p=0.5, (and so would everyone else, apparently) which suggests that I don’t actually have a real-valued utility function. This might be a good argument against real-valued utility functions in general (bounded or not). Especially since a lot of the proposed solutions here “fight the hypothetical” by pointing out that real agents can only choose from a finite set of strategies.
So you can have infinite expected utility, but be guaranteed to have finite utility? That is weird.