I think part of the reason people are
confused with this is that they didn't read the problem closely:
I give you the first envelope (with $100); you may either keep it or switch.
You know you have $100, which means you can think
objectively about the game. The way I think of the
game is in terms of a
gambling game -
tossing a coin. The
rules (taken from the original problem) are simple:
1: You must always bet half, and only half of your money.
2: Heads you win 2:1 (ie $50 wins you an extra $100)
3: Tails you lose
Looking at it like this, we can see two things:
1: You will never lose all your
money - you could end up with a minute fraction of a cent, but you can still play with half of it.
2: The game favours the
player - the odds are
50/50, yet the game is paying
2:1. 50/50 odds normally (in a
casino) pay 1:1 (except nothing in a casino is actually 50/50).
If you were
lucky enough to be playing this game, the strategy is simple:
1: If you have less than $100 you should play. The $100 was
given to you, so you have nothing to loose.
2: Only ever quit when you are ahead.
Rule 2 is the catch - since the odds are 50/50, playing an
infinite number of games should leave you with $100 (roughly 50% wins and 50% losses). You have to decide when you feel you have enough money.
The game is a
free ride. You only need to win 4 times in a row (from $100) to leave the game with $1200 dollars.