Question 1 Consider the following sequential game. First, Evan decides between staying home and going hiking. If he decides to stay in, the game ends. If he decides to go hiking, then he wants to meet with his friend Seth who is definitely going hiking. There are two trails that Seth and Evan can hike, the Billy Goat Trail and the Cascade Falls Trail. However, Seth and Evan do not use cellphones so they cannot communicate with each other and ask which trail they should go to. So if Evan decides to go hiking, then Seth and Evan independently and simultaneously choose which hiking trail to go to, without knowing what the other person has decided. Evan prefers hiking the Cascade Falls Trail with Seth to hiking the Billy Goat Trail with Seth and he prefers hiking any of the two trails to staying at home as long as he chooses the same trail with Seth. However, he would rather stay at home than hike any of the two trails by himself. In particular, he prefers staying home to hiking the Billy Goat Trail by himself which he prefers over hiking the Cascade Falls Trail by himself. Seth prefers hiking the Billy Goat Trail with Evan to hiking the Cascade Falls trail with Evan. He is indifferent between hiking either of the trails alone. So Seth gets the same payoff when Evan decides to stay home or they end up going to different hikes. Seth prefers hiking any of the trails with Evan to hiking alone. 1

a) (5 points) Draw a game tree (i.e. use extensive form representation) that represents the sequential game describe above. Remember that the exact number you are using to describe payoffs does not matter, but if a player likes an outcome more than the other, he should be getting more payoff in his preferred outcome compared to the other one.

b) (5 points) Write down the strategy sets of Evan and Seth.

c) (5 points) Convert the game tree into a game table (i.e. use strategic/normal form representation.)

d) (5 points) Find all pure strategy Nash equilibria of the game you draw in part c.

e) (5 points) Find all pure strategy subgame perfect Nash equilibria of the game you draw in part a.