2 (20 points) Consider a 2-player game where the types of the players are 0, 1 or 2. Nature decides the type of the players by flipping three fair coins, where heads is counted as a 1 and tails as a 0. To be precise: let's call the outcome of the coins Xo, X, and X's. Then each of these is a Bernoulli(?) random variable, and they are independent. The type of player 1 is now No + X's, and the type of player 2 is No + Xz. (a) (4 points) What is the probability distribution of the type of player 1? (b) (4 points) If player 1 sees her type is 0, what is her belief about the probability distribution of the type of player 2? (c) (4 points) If player I sees her type is 1, what is her belief about the probability distribution of the type of player 2? After seeing their type, the players play a Battle of the Sexes-like game: they have to choose between playing O or F. Their payoff depends on their type and on whether the other player chooses the same action. To be precise, for each player i = 1, 2 the payoffs are (0, 0;0) = 3 (F, F:0) = 1 (0, 0; 1) = 2 (F. F 1) - 2 (F, F: 2) = 3, and the payoffs for any outcome where they choose different actions is (. (d) (4 points) Suppose player 1 knows player 2 is playing the strategy so = OOF, where the first letter indicates the action if player 2 is type O, the second if player 2 is type 1, and the last if player 2 is type 2. What is the best response to sa? (e) (4 points) What is the expected payoff for player 1 when she plays the strategy in (d) against S = OOF