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.)
The way the scenario is given, player is informed that Omega and Omicron’s numbers coincide, but needs to decide for themselves what that implies for whether that number is prime or composite. So if the player is EDT, that player will always two box in this scenario.
I think the sequence of events goes like this:
Omega knows it is about to encounter an EDT, and so starts to simulate them.
The simulated EDT reasons “I should take both boxes, because this makes X composite, which pays more than making X prime by taking one box (the extra $1k being inconsequential)”
Omega, seeing that EDT would take both boxes, thus decides put a composite number in the box.
Omicron selects a random number. It just so happens that the number it selected was coincidentally X.
EDT arrives at the scenario and, as predicted by Omega, takes both boxes.
Omega does not pay 1M (but EDT gets to keep the 1K).
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.)
The way the scenario is given, player is informed that Omega and Omicron’s numbers coincide, but needs to decide for themselves what that implies for whether that number is prime or composite. So if the player is EDT, that player will always two box in this scenario.
I think the sequence of events goes like this:
Omega knows it is about to encounter an EDT, and so starts to simulate them.
The simulated EDT reasons “I should take both boxes, because this makes X composite, which pays more than making X prime by taking one box (the extra $1k being inconsequential)”
Omega, seeing that EDT would take both boxes, thus decides put a composite number in the box.
Omicron selects a random number. It just so happens that the number it selected was coincidentally X.
EDT arrives at the scenario and, as predicted by Omega, takes both boxes.
Omega does not pay 1M (but EDT gets to keep the 1K).
Omicron pays 2M.