Problem Set #7
- PlSc 349
- Due: Tuesday, 26 October 2010
- [4 points] Problem U18.1
- [13] Problem U18.2
- [4] Problem U18.3
- [9] [from Watson 2007]
Consider a three-player bargaining game where the players are negotiating
over a surplus of one unit of utility. The game begins with player 1 proposing a
three-way split of the surplus. Then player 2 must decide whether to accept the
proposal or to substitute for player 1's proposal his own alternative proposal.
Finally, player 3 must decide whether to accept or reject the current proposal
(whether it is player 1's or player 2's). If he accepts, then the players obtain
specified shares of the surplus. If player 3 rejects, then the players each get
0. Draw the extensive form of this perfect-information game and determine the
subgame perfect equilibria. Discuss where this model might be applicable in politics.
- [11] [from Watson 2007]
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, 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. What is the rollback [backwards induction, subgame perfect Nash] equilibrium?
(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 < z. They also understand that the
president of the union values both the salary and her job as the leader of the
union and that she will suffer a great personal cost if she is dismissed as president.
To be precise, suppose that the president suffers a cost of y utility units if
she is fired (this is subtracted from whatever salary amount is reached through
negotiation, 0 if the president rejects the salary offer). That is, if the president
accepts an offer of x, then she receives x - y in the event that x < z, and x in
the event that x ≥ z. If the president rejects the offer, then she obtains a payoff
of 0. In the bargaining game, what offers of the superintendent will the president
accept? (For what values of x will the president say "yes"? Your answer
should be a function of y and z.)
(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?
- [9] [Watson 2007]
In experimental tests of the ultimatum bargaining game, subjects who propose
the split rarely offer a tiny share of the surplus to the o.ther party. Furthermore,
sometimes subjects reject positive offers. These findings seem to contradict
the standard analysis of the ultimatum game. Many scholars conclude that the
payoffs specified in the basic model do not represent the actual preferences
of the people who participate in the experiments. In reality, people care about
more than their own monetary rewards. For example, people also act on feelings
of spite and the ideal of fairness. Suppose that, in the ultimatum game, the
responder's payoff is given by y+a(y-z), where y is the responder's monetary
reward, z is the offerer's monetary take, and a is a positive constant. That is,
the responder cares about how much money he gets and he cares about relative
monetary amounts (the difference between the money he gets and the money
the other player gets). Assume that the offerer's payoff is as in the basic model.
(a) Represent this game in the extensive form, writing the payoffs in terms of
m, the monetary offer of the proposer, and the parameter a.
(b) Find and report the subgame perfect equilibrium. Note how equilibrium
behavior depends on a.
(c) What is the equilibrium monetary split as a becomes large? Explain why
this is the case.
- [for future planning] Which problem was most useful in learning
the concepts? Which problem was least useful?
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