[antimagic]

I personally just worked through a concrete example. So Alice flips her coin and gets heads. I then just assume that she is going to guess the opposite of what she flipped, so she guesses tails.

For Alice to lose, Bob would have to have flipped heads. Now Bob knows, because he has discussed strategy with Alice prior to doing the flips that either Alice flipped tails, in which case she has won, or she has flipped heads, in which case she has lost. But Bob can then bet on heads, in which case he knows that he wins if Alice loses.

That was my way of reasoning things anyhow. As you can see from the other responses, there are many different ways of arriving at the right answer.