Question 6 (20 points) Their rich uncle left 100 pounds of gold to Todd and Steven. The negotiating process for allocating the treasure between them was also laid out in their uncle's will. They have three rounds by which to come to an agreement. In an odd (even) round, Todd (Steven) is required to propose an allocation. In response to a proposal, the other nephew can accept or reject it. If he accepts the proposal, the process is ended and the proposed allocation is made. If he rejects the proposal, the game moves to the next round. Failure to agree by the end of the third round means that all of the gold goes to charity, so none of it lands in the pockets of Todd and Steven. Furthermore, at the end of each round in which an agreement has not been reached, a fraction of 1/ 10 the allotment of gold is given to charity. Thus, there are 9/ 10*100=90 pounds of gold at the beginning of round 2 (after an agreement was not reached in the rst round) and only 9/ 10*90=81 pounds of gold at the beginning of round 3 (after an agreement was not reached in the rst two rounds). In other words, there is a cost to delaying agreement and, of course, a cost to ever failing to agree. Each nephew's payoff equals the number of pounds of gold he ends up with (there is no discounting), so neither cares about the other or about their uncle's favorite charity. For notational purposes, assume that a proposal in round t is a value for xt, where xt is the amount of gold for Todd. 1) Suppose that no agreement has been reached by the beginning of the third round. Describe the optimal strategies of Todd (the proposer) and Steve in round 3, where the amount to be divided is 81 and failure to reach an agreement leads to payoff zero for each player. 2) Find the rollback equilibrium for the entire three-period game. Make sure to write down the complete equilibrium strategies, that is, the optimal proposal and acceptance rule for each player. 3) According to the equilibrium, in what day the agreement is reached