As they reduce the number of boxes in the game they are losing some of the different amounts of money in each box. At various points in the game they will be made a deal based on the amounts that are left. If they don’t accept the deal then they will eventually end up with two boxes, the one that they have and the one other remaining box. They know the two amounts of money that the boxes contain, but not which of the two amounts is in their box.
At this point they are offered the chance to swap their box for the other one. They will do this if they think that the other box contains a larger amount of money.
So should they swap boxes?
It seems obvious that there is no logical reason to swap boxes, the player has no information about which box will contain the larger amount.
Let’s do the calculation just to check.
To keep it simple, let’s assume that one amount is double the other. Let’s denote the amount in the player’s box by the letter A.
If the player has the smaller amount then the amount in the other box is 2A. There is a 50% chance of this.
If the player has the larger amount then the amount in the other box is A/2. There is also a 50% chance of this.
This means that, on average, the amount that is expected to be in the other box is (2A x 1/2) + (A/2 x 1/2). This is (A + A/4) or 5A/4.
This is more than the amount that the player has in their own box, which is A. So they should swap boxes.
Now that they have swapped boxes they can go through the same logic again about the amount in their new box and they will decide to swap again. Something must be wrong!
What is wrong with the seemingly straightforward logic above?
The key thing to notice is that the player first assumes that he has the smaller amount in his box, and then during the same calculation he assumes that he has the larger amount. He can’t have both the smaller and the larger amount at the same time so this is what leads him to the false conclusion that he should swap boxes.