The Wallet Paradox

Another classic game theory puzzle, this time with a different twist. Here's the basic setup:

Bill Gates meets Warren Buffett at a dinner party and the host tells them to play a game. Each person will place his wallet on the table. The person with less money in his wallet wins all the money. 

Is anyone favored to win this game? 

...Afterwards, it turns out both Bill Gates and Warren Buffett loved the game.
They add a twist as follows. They will play the game every day for a month. To manage the stakes, they agree to carry an average (mean) of $100. Is this game fair? What is the best possible strategy, given you know what your opponent is doing?

Think about this game for a while. The solution is here.