Econ 521 - Week 7:

1. Extensive game with perfect information - The political figures Rosa and Ernesto each has to take a position on an issue. The options are Berlin (B) or Havana (H). They choose sequentially. A third person, Karl, determines who chooses first. Both Rosa and Ernesto care only about the action they choose, not about who chooses first. Rosa prefers the outcome in which both she and Ernesto choose B to that in which they both choose H, and prefers this outcome to either of the ones in which she and Ernesto choose different actions; she is indifferent between these last two outcomes. Ernesto's preferences differ from Rosas in that the roles of B and H are reversed. Karl's preferences are the same as Ernesto's. Based on this,

(a) Model this situation as an extensive game with perfect information. Specify the components of the game.

(b) Represent the game in a diagram and find the outcome of this game using backward induction.

2. Extensive game with three players - Consider the case of two employees who start with wealth 2 each and that have been assigned the task of building a widget, and their manager. Each of the two employees must choose between sleeping and working. Workings imply an effort equivalent to 1 in terms of wealth. The quality of the widget depends on the number of employees who work. If at least one employee works, the manager can readily observe that the widget is of high quality (payoff 4). If neither employee works the widget is of unacceptable quality (payoff -4). Each employee likes sleeping better than working, and since the manager yells at them both when he opts to perform an inspection (which cost 1), the employees dislike this very much and is equivalent to a reduction in 4 units of wealth. After the manager performs an inspection, the widget is of acceptable quality.

(a) Model this situation as an extensive game with perfect information when workers choose first sequentially and then the manager decides whether or not to perform an inspection. Specify the components of the game.

(b) Represent the game in a diagram. Find the outcome of this game using backward induction.

3. Bargaining over two indivisible objects - Consider a variant of the ultimatum game, with indivisible units. Two people use the following procedure to allocate two desirable identical indivisible objects. One person proposes an allocation (both objects go to person 1, both go to person 2, one goes to each person), which the other person then either accepts or rejects. In the event of rejection, neither person receives either object. Each person cares only about the number of objects she obtains.

(a) Construct an extensive game that models this situation and find its outcomes.