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 2 f0; 1g. This is a two period

but random symmetric bargaining model. At any date t 2 f0; 1g 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 1p = 1

2 ),

player 2 is selected. The selected player makes an oer (x; y) where

x; y 0 and x + y 1. After observing the oer, the other player

can either accept or reject the oer. If the oer is accepted the game

ends yielding payos (tx; ty). If the oer 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-o 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 rst coin toss, given that they will play

the SPE.