I’ll disregard my earlier comment and assume the latter interpretation for now.
So here are the things that can (and can’t) happen:
Alpha puts £1 000 000 in the envelope.
Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope.
You refuse the £10, and find £1 000 000 in the envelope.
You accept the £10 — contradiction, won’t happen.
Omega has predicted you will accept the £10 and that the envelope is empty — contradiction, won’t happen.
Alpha puts nothing in the envelope.
Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope — contradiction, won’t happen.
Omega has predicted you will accept the £10 and that the envelope is empty.
You refuse the £10 — contradiction, won’t happen.
You accept the £10, and find nothing in the envelope.
So, starting with Alpha’s coin flip, here is what can happen:
Alpha puts £1 000 000 in the envelope. Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope. You refuse the £10, and find £1 000 000 in the envelope.
Alpha puts nothing in the envelope. Omega has predicted you will accept the £10 and that the envelope is empty. You accept the £10, and find nothing in the envelope.
Alpha puts £1 000 000 in the envelope. Omega predicts that if it were to offer you this puzzle (while remaining completely honest about its own prediction), you would accept the £10, so it has no choice but to leave you alone. You find £1 000 000 in the envelope.
Alpha puts nothing in the envelope. Omega predicts that if it were to offer you this puzzle (while remaining completely honest about its own prediction), you would reject the £10, so it has no choice but to leave you alone. You find nothing in the envelope.
If you happen to get as far as having Omega present you with this conundrum as stated, and you find yourself able to convince yourself to refuse the £10, then do so. If the envelope were empty and you were the sort of person who would make this choice, then Omega would not have made the prediction in the first place, as it would have been wrong. It would have left you alone with your empty envelope. Same if the envelope contained the £1 000 000 and you could not convince yourself to refuse the £10, as Omega would have known that in advance.
I’ll disregard my earlier comment and assume the latter interpretation for now.
So here are the things that can (and can’t) happen:
Alpha puts £1 000 000 in the envelope.
Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope.
You refuse the £10, and find £1 000 000 in the envelope.
You accept the £10 — contradiction, won’t happen.
Omega has predicted you will accept the £10 and that the envelope is empty — contradiction, won’t happen.
Alpha puts nothing in the envelope.
Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope — contradiction, won’t happen.
Omega has predicted you will accept the £10 and that the envelope is empty.
You refuse the £10 — contradiction, won’t happen.
You accept the £10, and find nothing in the envelope.
So, starting with Alpha’s coin flip, here is what can happen:
Alpha puts £1 000 000 in the envelope. Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope. You refuse the £10, and find £1 000 000 in the envelope.
Alpha puts nothing in the envelope. Omega has predicted you will accept the £10 and that the envelope is empty. You accept the £10, and find nothing in the envelope.
Alpha puts £1 000 000 in the envelope. Omega predicts that if it were to offer you this puzzle (while remaining completely honest about its own prediction), you would accept the £10, so it has no choice but to leave you alone. You find £1 000 000 in the envelope.
Alpha puts nothing in the envelope. Omega predicts that if it were to offer you this puzzle (while remaining completely honest about its own prediction), you would reject the £10, so it has no choice but to leave you alone. You find nothing in the envelope.
If you happen to get as far as having Omega present you with this conundrum as stated, and you find yourself able to convince yourself to refuse the £10, then do so. If the envelope were empty and you were the sort of person who would make this choice, then Omega would not have made the prediction in the first place, as it would have been wrong. It would have left you alone with your empty envelope. Same if the envelope contained the £1 000 000 and you could not convince yourself to refuse the £10, as Omega would have known that in advance.