To give Mom a day of rest, Dad plans to take his two children, Bart and Cassie, on an outing on Sunday. Bart prefers to go to the amusement park (A), whereas Cassie prefers to go to the science museum (S). Each child gets 3 units of utility from his/her more preferred activity and only 2 units of utility from his/her less preferred activity. Dad gets 2 units of utility for either of the two activities. To choose their activity, Dad plans rest to ask Bart for his preference, then to ask Cassie after she hears Bart's choice. Each child can choose either the amusement park (A) or the science museum (S). If both children choose the same activity, then that is what they will all do. If the children choose dierent activities, Dad will make a tie-breaking decision. As the parent, Dad has an additional option: he can choose the amusement park, the science museum, or his personal favorite, the mountain hike (M). Bart and Cassie each get 1 unit of utility from the mountain hike, and Dad gets 3 units of utility from the mountain hike. Because Dad wants his children to cooperate with one another, he gets 2 extra units of utility if the children choose the same activity (no matter which one of the two it is).
a) Draw the game tree, with payos, for this three-person game.
b) What is the rollback equilibrium to this game? Make sure to describe the strategies, not just the payos.
c) How many deferent complete strategies does Bart have? Explain.
d) How many complete strategies does Cassie have? Explain.