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 amusment park (A), whereas Cassie prefers to go to the science musuem (S). Each child gets 3 units of utility from his/her more preferred activity and only 2 units of utlility from his/her less prefered activity. Dad gets 2 units of utility for either of the two activities. To Choose their activity, Dad plans firtst to ask Bart for his preference, then to ask Cassie after she hears Bart's choice. Each child can chose either the amusment 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 different activities, Dad will make a tie-breaking descion. As the parent, Dad has an additional option: he can choose the amusment park, the science musuem, 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 utilitiy if the children choose the same activity(no matter which one of the two it is)
a) Draw the game tree, with payoffs, for the this three person game.
b) What is the rollback equilibrium to this game? Make sure to describe the stratagies, not just the payoffs.
c) How many different complete stratagies dose Bart have? Explain
d) How many complete stratagies does Cassie have? Explain