Take the £10, my reasoning goes as follows: if I precommit to refuse it, either I get the £1,000,000 and refuse £10, or I get £0 and omega doesn’t even show up; if I precommit to accept it, either I get the £1,000,000 and omega doesn’t even show up, or I get £10 from omega showing up and me accepting (the respective expected utilities being £500,000 and £500,005). I do better by precommitting to take it, so to be reflectively consistent (and win), I must now take it.
Take the £10, my reasoning goes as follows: if I precommit to refuse it, either I get the £1,000,000 and refuse £10, or I get £0 and omega doesn’t even show up; if I precommit to accept it, either I get the £1,000,000 and omega doesn’t even show up, or I get £10 from omega showing up and me accepting (the respective expected utilities being £500,000 and £500,005). I do better by precommitting to take it, so to be reflectively consistent (and win), I must now take it.