Assignment: Games and Economic Behavior

1. Fred is a 92 year old man who sits by a pond in the park all day, snacking on a sandwich his wife made him. Every once in awhile, a duck will swim by Fred and do one of two things, Quack(Q) or Not Quack(NQ). If the duck chooses NQ, the duck just swims on by and gets a payoff of 0, while Fred gets a payoff of 4 since he is uninterrupted in his thoughts. If the duck chooses Q, this distracts Fred and alerts him of the duck's existence. Fred thinks the duck would like some bread from his sandwich and Fred would get some joy from giving the duck some bread. Specifically, Fred can choose to Feed(F) or Not Feed(NF) the duck. If he chooses F, the duck gets 1 and Fred gets 3. If he chooses NF, the duck gets -1 and Fred gets 2. Thus, Fred is tempted to Feed the ducks, but the other ducks will see this and Quack at him in the future.

A) Draw the game in Extensive form

B) Suppose that each duck plays the following strategy:

Choose NQ if Fred never played F in the past. Otherwise play Q; and that Fred plays the following strategy:

Choose NF if Fred never played F in the past. Otherwise choose F.

Show that the above strategies make a SPNE if Fred cares sufficiently about the fu- ture. What's the smallest discount factor Fred can have to keep him playing the SPNE?

2. Consider a three-period sequential bargaining model where 2 players have to split a pie of size 1 but each player has his or her own discount factor: δ₁ ∈ [0, 1) and δ₂ ∈ [0, 1).

A) Compute the outcome of the unique SPNE.

B) Show that when δ₁ = δ₂, player 1 has an advantage.