I figured that the right answer is (and that FDT would also reason):
If I choose to take the big box only, I only get $1M.
If I I don’t take the big box only, then that number is composite so I get $2M.
One way to not take the big box only is to take both boxes thus netting $2M+$1,000.
Separately, there’s the option of factoring/primality testing the number. (I may be unsure of it’s primality, but for less than $1,000 I should be able to get more sure.) (If there’s enough time to decide, I could take the small box, use the money in it to get more info about that number, and then go back and decide if I’m going to take the other box.)
Edited to add:
If the two numbers weren’t the same, then you could (as a quick primality/composability check):
The difference between your reasoning and the reasoning of FDT, is that your reasoning acts like the equality of the number in the big box and the number chosen by Omicron is robust, whereas the setup of the problem indicates that while the number in the big box is sensitive to your action, the number chosen by Omicron is not. As such, FDT says you shouldn’t imagine them covarying; when you imagine changing your action you should imagine the number in the big box changing while the number chosen by Omicron stays fixed. And indeed, as illustrated in the expected utility calculation in the OP, FDT’s reasoning is “correct” in the sense of winning more utility (in all cases, and in expectation).
The consequences of not having enough time to think.
winning more utility
more money.
EDIT: It’s not clear what effects the amount of time restriction has. ‘Not enough time to factor this number’ could still be a lot of time, or it could be very little.
I figured that the right answer is (and that FDT would also reason):
If I choose to take the big box only, I only get $1M.
If I I don’t take the big box only, then that number is composite so I get $2M.
One way to not take the big box only is to take both boxes thus netting $2M+$1,000.
Separately, there’s the option of factoring/primality testing the number. (I may be unsure of it’s primality, but for less than $1,000 I should be able to get more sure.) (If there’s enough time to decide, I could take the small box, use the money in it to get more info about that number, and then go back and decide if I’m going to take the other box.)
Edited to add:
If the two numbers weren’t the same, then you could (as a quick primality/composability check):
divide the larger by the smaller
find the greatest common factor
The difference between your reasoning and the reasoning of FDT, is that your reasoning acts like the equality of the number in the big box and the number chosen by Omicron is robust, whereas the setup of the problem indicates that while the number in the big box is sensitive to your action, the number chosen by Omicron is not. As such, FDT says you shouldn’t imagine them covarying; when you imagine changing your action you should imagine the number in the big box changing while the number chosen by Omicron stays fixed. And indeed, as illustrated in the expected utility calculation in the OP, FDT’s reasoning is “correct” in the sense of winning more utility (in all cases, and in expectation).
The consequences of not having enough time to think.
more money.
EDIT: It’s not clear what effects the amount of time restriction has. ‘Not enough time to factor this number’ could still be a lot of time, or it could be very little.