Suppose the president of the local teachers' union bargains with the superintendent of schools over teachers' salaries. Assume the salary is a number between 0 and 1, 1 being the teachers' preferred amount and 0 being the superintendent's preferred amount.

(a) Model this bargaining problem by using a simple ultimatum game. The superintendent picks a number x, between 0 and 1, which we interpret as his offer. After observing this offer, the president of the union says "yes" or "no." If she says "yes," then an agreement is reached; in this case, the superintendent (and the administration that she represents) receives 1 - x and the president (and the union) receives x. If the president says "no," then both parties receive 0. Using the concept of backward induction, what would you predict in this game?

(b) Let us enrich the model. Suppose that before the negotiation takes place, the president of the union meets with the teachers and promises to hold out for an agreement of at least a salary of z. Suppose also that both the superintendent and the president of the union understand that the president will be fired as union leader if she accepts an offer x.

(c) Given your answer to part (b), what is the outcome of the game? (What is the superintendent's offer and what is the president's response?) Comment on how the union's final salary depends on y.

(d) Given your answer to part (b), what kind of promise should the president make?