This scenario seems impossible, as in contradictory / not self-consistent. I cannot say exactly why it breaks, but at least the two statements here seem to be not consistent:
today they [Omicron] happen to have selected the number X
and
[Omega puts] a prime number in that box iff they predicted you will take only the big box
Both of these statements have implications for X and cannot both be always true. The number cannot both, be random, and be chosen by Omega/you, can it?
From another angle, the statement
FDT will always see a prime number
demonstrates that something fishy is going on. The “random” number X that Omicron has chosen—and is in the box—and seen my FDT—is “always prime”. Then it is not a random number?
Edit: See my reply below, the contradiction is that Omega cannot predict EDT’s behaviour when Omicron chose a prime number. EDT’s decision depends on Omega’s decision, and EDT’s decision depends on Omega’s decision (via the “do the numbers coincide” link). On days where Omicron chooses a prime number this cyclic dependence leads to a contradiction / Omega cannot predict correctly.
Yes that was my reasoning too. The situation presumably goes:
Omicron chooses a random number X, either prime or composite
Omega simulates you, makes its prediction, and decides whether X’s primality is consistent with its prediction
If it is, then:
Omega puts X into the box
Omega teleports you into the room with the boxes and has you make your choice
If it’s not, then...? I think the correct solution depends on what Omega does in this case.
Maybe it just quietly waits until tomorrow and tries again? In which case no one is ever shown a case where the box does not contain Omicron’s number. If this is how Omega is acting, then I think you can act as though your choice affects Omircon’s number, even though that number is technically random on this particular day.
Maybe it just picks its own number, and shows you the problem anyway. I believe this was the assumption in the post.
I think I found the problem: Omega is unable to predict your action in this scenario, i.e. the assumption “Omega is good at predicting your behaviour” is wrong / impossible / inconsistent.
Consider a day where Omicron (randomly) chose a prime number (Omega knows this). Now an EDT is on their way to the room with the boxes, and Omega has to put a prime or non-prime (composite) number into the box, predicting EDT’s action.
If Omega makes X prime (i.e. coincides) then EDT two-boxes and therefore Omega has failed in predicting.
If Omega makes X non-prime (i.e. numbers don’t coincide) then EDT one-boxes and therefore Omega has failed in predicting.
Edit: To clarify, EDT’s policy is two-box if Omega and Omicron’s numbers coincide, one-box if they don’t.
If the agent is EDT and Omicron chooses a prime number, then Omega has to choose a different prime number. Fortunately, for every prime number there exists a distinct prime number.
EDT’s policy is not “two-box if both numbers are prime or both numbers are composite”, it’s “two-box if both numbers are equal”. EDT can’t (by hypothesis) figure out in the allotted time whether the number in the box (or the number that Omicron chose) is prime. (It can readily verify the equality of the two numbers, though, and this equality is what causes it—erroneously, in my view—to believe it has control over whether it gets paid by Omicron.)
This scenario seems impossible, as in contradictory / not self-consistent. I cannot say exactly why it breaks, but at least the two statements here seem to be not consistent:
and
Both of these statements have implications for X and cannot both be always true. The number cannot both, be random, and be chosen by Omega/you, can it?
From another angle, the statement
demonstrates that something fishy is going on. The “random” number X that Omicron has chosen—and is in the box—and seen my FDT—is “always prime”. Then it is not a random number?
Edit: See my reply below, the contradiction is that Omega cannot predict EDT’s behaviour when Omicron chose a prime number. EDT’s decision depends on Omega’s decision, and EDT’s decision depends on Omega’s decision (via the “do the numbers coincide” link). On days where Omicron chooses a prime number this cyclic dependence leads to a contradiction / Omega cannot predict correctly.
The fact that the 2 numbers are equal is not always true, it is randomly true on this day.
Yes that was my reasoning too. The situation presumably goes:
Omicron chooses a random number X, either prime or composite
Omega simulates you, makes its prediction, and decides whether X’s primality is consistent with its prediction
If it is, then:
Omega puts X into the box
Omega teleports you into the room with the boxes and has you make your choice
If it’s not, then...? I think the correct solution depends on what Omega does in this case.
Maybe it just quietly waits until tomorrow and tries again? In which case no one is ever shown a case where the box does not contain Omicron’s number. If this is how Omega is acting, then I think you can act as though your choice affects Omircon’s number, even though that number is technically random on this particular day.
Maybe it just picks its own number, and shows you the problem anyway. I believe this was the assumption in the post.
I think I found the problem: Omega is unable to predict your action in this scenario, i.e. the assumption “Omega is good at predicting your behaviour” is wrong / impossible / inconsistent.
Consider a day where Omicron (randomly) chose a prime number (Omega knows this). Now an EDT is on their way to the room with the boxes, and Omega has to put a prime or non-prime (composite) number into the box, predicting EDT’s action.
If Omega makes X prime (i.e. coincides) then EDT two-boxes and therefore Omega has failed in predicting.
If Omega makes X non-prime (i.e. numbers don’t coincide) then EDT one-boxes and therefore Omega has failed in predicting.
Edit: To clarify, EDT’s policy is two-box if Omega and Omicron’s numbers coincide, one-box if they don’t.
If the agent is EDT and Omicron chooses a prime number, then Omega has to choose a different prime number. Fortunately, for every prime number there exists a distinct prime number.
EDT’s policy is not “two-box if both numbers are prime or both numbers are composite”, it’s “two-box if both numbers are equal”. EDT can’t (by hypothesis) figure out in the allotted time whether the number in the box (or the number that Omicron chose) is prime. (It can readily verify the equality of the two numbers, though, and this equality is what causes it—erroneously, in my view—to believe it has control over whether it gets paid by Omicron.)