# To give Mom a day of rest, Dad plans to take his two children, Bart and Cassie, on an outing on…

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 first 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 different 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).

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(a) Draw the game tree, with payoffs, for this
three-person game.

(b) What is the rollback equilibrium to this game? Make
sure to describe the strategies, not just the payoffs.

(c) How many different complete strategies does Bart
have? Explain.

(d) How many complete strategies does Cassie have?
Explain.