Luco wins the internets.

[1] [2] [3]
The probability is 1/3 to choose the box with the money the first time, and 2/3 that the money is in a non-chosen box.
( [1/3] ) ( [1/3] [1/3] )
The revealed box is always empty, so the 2/3's chance of containing the money has collapsed into one box.
( [1/3] ) ( [2/3] [0] )

1) The EV of not switching is $100
2) The EV of switching is $200
3) Always switch.