Consider a bargaining problem with two agents 1 and 2. There isa prize of $1 to be divided. Each agent has a common discountfactor 0 < δ < 1. There are two periods, i.e., t ∈ {0, 1}.This is a two period but random symmetric bargaining model. At anydate t ∈ {0, 1} we toss a fair coin. If it comes out “Head†( withprobability p = 1 2 ) player 1 is selected. If it comes out “Tailâ€,(again with probability 1 −p = 1 2 ), player 2 is selected. Theselected player makes an offer (x, y) where x, y ≥ 0 and x + y ≤ 1.After observing the offer, the other player can either accept orreject the offer. If the offer is accepted the game ends yieldingpayoffs (δ tx, δt y). If the offer is rejected there are twopossibilities:
• if t = 0, then the game moves to period t = 1, when the sameprocedure 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. Computethe Subgame perfect Equilibrium (SPE). What is the expected utilityof each player before the coin toss, given that they will play theSPE.
(b) Suppose now there are two periods i.e., t = 0, 1. Computethe Subgame perfect Equilibrium (SPE). What is the expected utilityof each player before the first coin toss, given that they willplay the SPE.
