Consider a bargaining problem with two agents 1 and 2. There is a prize of $1 to be divided. Each agent has a common discount factor 0 < < 1. There are two periods, i.e., t {0, 1}. This is a two period but random symmetric bargaining model. At any date t {0, 1} we toss a fair coin. If it comes out "Head" ( with probability p = 1 2 ) player 1 is selected. If it comes out "Tail", (again with probability 1 p = 1 2 ), player 2 is selected. The selected player makes an offer (x, y) where x, y 0 and x + y 1. After observing the offer, the other player can either accept or reject the offer. If the offer is accepted the game ends yielding payoffs ( tx, t y). If the offer is rejected there are two possibilities:

if t = 0, then the game moves to period t = 1, when the same procedure is repeated.

if t = 1, the game ends and the pay-off vector (0, 0) realizes, i.e., each player gets 0.

(a) Suppose that there is only one period,i.e., t = 0. Compute the Subgame perfect Equilibrium (SPE). What is the expected utility of each player before the coin toss, given that they will play the SPE.

(b) Suppose now there are two periods i.e., t = 0, 1. Compute the Subgame perfect Equilibrium (SPE). What is the expected utility of each player before the first coin toss, given that they will play the SPE.