Problem 1:

Part a: Draw two extensive form games of perfect information that satisfy all of the following properties:

1. They both have the same normal form representation;

2. Both have unique backward induction solutions;

3. But the resulting payoffs of the backward induction solutions are different in the two games.

Justify that the two extensive form games that you drew satisfy each of the above properties.

Part b: Consider the following extensive form game. First, player 1 chooses either to give a gift to player 2 or not. Then after observing whether or not player 1 gave player 2 a gift, player 2 chooses whether or not to give a gift to player 1. Receiving a gift generates an additional benefit of 5 utils to the recipient. However, giving a gift is costly incurs a cost of 1 util. Thus, if player 1 both receives and gives a gift, then she obtains 4 utils. If player 1 neither received nor gave a gift, then player 1 obtains 0 utils.

Write the extensive form representation of this game. What is the backward induction solution of this game? What is the backward induction outcome?