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. (Isn’t it clever how Todd moves in odd rounds and Steven moves in even rounds?) 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 1 d of the allotment of gold is given to charity, where 0 d 1.

Thus, there are 100d pounds of gold at the beginning of round 2 (after an agreement was not reached in the first round) and only 100d2pounds of gold at the beginning of round 3 (after an agreement was not reached in the first 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, 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 xi, where xiis the share of the remaining amount of gold for Todd and, therefore, Steven’s share is 1 xi. Note that 0 xi1 and thus is any number between 0 and 1 inclusive. Find an SPNE.