Guessing Each Others' Coins

Source

I got this problem from Rustan Leino, who got it from Raphael Reischuk, who also has a little puzzle collection.

I solved it and wrote up my solution.

Problem    

You and a friend each have a fair coin. You can decide on a strategy and then play the following game, without any further communication with each other. You flip your coin and then write down a guess as to what your friend's coin will say. Meanwhile, your friend flips her coin and writes down a guess as to what your coin says. There's a third person involved: The third person collects your guesses and inspects your coins. If both you and your friend correctly guessed each other's coins, then your team (you and your friend) receive \$2 from the third person. But if either you or your friend (or both) gets the guess wrong, then your team has to pay \$1 to the third person. This procedure is repeated all day. Assuming your object is to win money, are you happy to be on your team or would you rather trade places with the third person?

Solution     Reveal