5. [20 points] Consider a game between two players, P1 and P2. Players bargain over how to divide a pie between themselves. The game lasts for three periods. A player's utility is given by MC) = c, where c 6 [0,1] is the amount of pie he consumes. Both players discount utility across periods, where 61- E (U, l) is the discount factor for player i = 1,2. The timing of the game is as follows. In period 1, rst P1 makes an offer (3:,1 3:), where .r. 6 {0,1}. That is, P1 consumes a: and P2 consumes 1 3:. Then given Pl's o'er, P2 decides whether to accept the offer. If P2 accepts the offer, the game ends and the players divide the pie according to Pl's offer. If P2 rejects the offer, both players move On to period 2. In period 2, rst P2 makes a counteroffer (y,1 y) with y E {0, 1] and P1 responds to the offer. Similarly, in period 3, P1 makes another oeunteroffer (2,1 z) with z 6 [0,1] and P2 raspOnds. If P2 rejects the offer, the game ends and both players receive nothing. Assume that if a player is indifferent between two strategies, he/she will accept the one that is most preferred by his/ her opponent. (a) Draw the game tree. Then describe the best responses of both players. (b) Find the subgame perfect equilibrium. Describe the outcome of the game and the payoffs to each player