My chances of gaining or losing if I switch are not determined by my knowledge or, more to the point, lack of knowledge, of the values of the cheques but by their actual values. In the above example, the cheques have values $4 and $8. If I have chosen the $4 cheque, I don't know whether the other cheque is $2 or $8 but if I switch I will not get the $2 cheque because it does not exist. I will get the $8 cheque with absolute certainty and be $4 better off.

If I had first chosen the $8 cheque, I would not know whether the other cheque was $4 or $16 but if I switch, I will not get the £16 cheque because it does not exist. I will get the $4 cheque with absolute certainty and be $4 worse off.

The way the problem is stated turns "I don't know whether the other cheque is half or double the one I've got" into "There is an equal chance that the other cheque is half or double the one I've got". Sorry, not so.

Incidentally, the 2ⁿ and 2ⁿ⁺¹ business is just a smoke screen to add further confusion. The cheques don't have to have values of powers of 2. It is sufficient that one is twice the other. $5 and $10 would have worked just as well.