This is not a well formed problem. If my strategy is to 1-box iff the numbers match, then omega, choosing his number independently of the lottery number must choose a composite number, since I will 2-box 99.9% of the time. Therefore if omega is correct 99.9% of the time when the numbers match, then the lottery number must be composite at least 99.9% of the time.
However, if my strategy is to 2-box iff the numbers match, then omega, choosing his number independently of the lottery number must choose a prime number, since I will 1-box 99.9% of the time. Therefore if omega is correct 99.9% of the time when the numbers match, then the lottery number must be prime at least 99.9% of the time.
I don’t know the odds of the lottery, but it cannot be both prime at least 99.9% of the time and composite at least 99.9%, so one of these two strategies will consistently make omega when the boxes match.
I realized this is slightly wrong as written, because Omega doesn’t have to be correct 100% of the time when they don’t match, so he could do a little better using a random algorithm, but this just means that some of the 99.9%s need to be replaced with 99.8%.
Written before reading comments:
This is not a well formed problem. If my strategy is to 1-box iff the numbers match, then omega, choosing his number independently of the lottery number must choose a composite number, since I will 2-box 99.9% of the time. Therefore if omega is correct 99.9% of the time when the numbers match, then the lottery number must be composite at least 99.9% of the time.
However, if my strategy is to 2-box iff the numbers match, then omega, choosing his number independently of the lottery number must choose a prime number, since I will 1-box 99.9% of the time. Therefore if omega is correct 99.9% of the time when the numbers match, then the lottery number must be prime at least 99.9% of the time.
I don’t know the odds of the lottery, but it cannot be both prime at least 99.9% of the time and composite at least 99.9%, so one of these two strategies will consistently make omega when the boxes match.
I realized this is slightly wrong as written, because Omega doesn’t have to be correct 100% of the time when they don’t match, so he could do a little better using a random algorithm, but this just means that some of the 99.9%s need to be replaced with 99.8%.