Consider the following game, played between two Players. First, Player 1 chooses C or D. Second, without observing Player 1's choice, Player 2 chooses C or D as well. Finally, Nature moves as follows: If both players chose C, Nature flips a coin (50%-50% odds), with the winner of the coin toss getting 40 utility and the loser 0 utility. Player 1 wins if the coin comes up heads, and Player 2 wins if the coin comes up tails. If both players chose D, Nature flips a coin (50%-50% odds), with the winner of the coin toss getting 10 utility and the loser 0 utility. Player 1 wins if the coin comes up heads, and Player 2 wins if the coin comes up tails. If one Player chooses D and the other Player chooses C, Nature flips another coin (50%-50% odds), and awards the D-choosing player 40 utility if the coin comes up heads, and 0 utility if the coin comes up tails. It awards the C-choosing player -20 utility if the coin comes up tails, and 20 utility if the coin comes up heads. Now, analyse this game using the following steps. (a) (5 points) Draw the game tree described above, being careful to label the payoffs and probabilities for Nature's moves. (b) (3 points) Suppose Player 2 believes Player 1 is choosing C; what is their best response? What is Player 1's best response to Player 2's best response? Is this an equilibrium? Explain. (c) (3 points) Suppose Player 2 believes Player 1 is choosing D; what is their best response? What is Player 1's best response to Player 2's best response? Is this an equilibrium?

(d) (5 points) You did a bunch of expected utility calculations in the above analysis. Replace the "Nature" branches with the expected utility values associated with those branches. For example, replace the branch under C, C with (20, 20). What type of game is this? Draw it in a matrix and find its Nash equilibrium. (e) (4 points) Explain how the NE in (d) relate to the equilibria in (b) and (c). Why does the uncertainty matter (or not) in this game? Relate to the ideas of asymmetric information developed in class.