Inserting A=60 into the original line of reasoning (just to show that it doesn't immediately resolve the paradox):
1. Denote by A=60 the amount in the player's selected envelope.
2. The probability that A=60 is the smaller amount is 1/2, and that it is the larger amount is also 1/2.
3. The other envelope may contain either 2A=120 or A/2=30.
4. If A=60 is the smaller amount, then the other envelope contains 2A=120.
5. If A=60 is the larger amount, then the other envelope contains A/2=30.
6. Thus the other envelope contains 2A=120 with probability 1/2 and A/2=30 with probability 1/2.
7. So the expected value of the money in the other envelope is: 1/2(120) + 1/2(30) = 75
8. This is greater than A=60 so, on average, the person reasons that they stand to gain by swapping.
9. ...
edit: If anything, to me this obscures things further, because it makes it non-obvious that we are actually talking about ratios (i.e. 2A, 1/2A) rather than absolute numbers, and so the expectation calculated in step 7 is inappropriate - a geometric mean recovers the correct answer.
1. Denote by A=60 the amount in the player's selected envelope.
2. The probability that A=60 is the smaller amount is 1/2, and that it is the larger amount is also 1/2.
3. The other envelope may contain either 2A=120 or A/2=30.
4. If A=60 is the smaller amount, then the other envelope contains 2A=120.
5. If A=60 is the larger amount, then the other envelope contains A/2=30.
6. Thus the other envelope contains 2A=120 with probability 1/2 and A/2=30 with probability 1/2.
7. So the expected value of the money in the other envelope is: 1/2(120) + 1/2(30) = 75
8. This is greater than A=60 so, on average, the person reasons that they stand to gain by swapping.
9. ...
edit: If anything, to me this obscures things further, because it makes it non-obvious that we are actually talking about ratios (i.e. 2A, 1/2A) rather than absolute numbers, and so the expectation calculated in step 7 is inappropriate - a geometric mean recovers the correct answer.