Think of it like this: What is the probability that two coins come up with the same face value? 0.5 If we use the guessing scheme where each player guesses hi

Think of it like this: What is the probability that two coins come up with the same face value? 0.5 If we use the guessing scheme where each player guesses his own guess as the result of the other player's coin, then players A and B win if their coins are the same. Their coins are the same with probability 0.5. Thus, Expectation[game] = Expectation[Game|A & B Win] * P(A & B win) + Expectation[Game|A & B Lose] * P(A & B Lose) = 1 * .5 + -2 * .5 = -1, so C should not play. If the players guessed randomly then the expectation of the game would be as you say it is.