# In the classic film Mary Poppins, the Banks children are players in a strategic game with a number..

In the classic film Mary Poppins, the Banks children are
players in a strategic game with a number of different nannies. In their view
of the world, nannies are inherently harsh, and playing tricks on nannies is
great fun. That is, they view themselves as playing a game in which the nanny
moves first, showing herself to be either Harsh or Nice, and the children move
second, choosing to be either Good or Mischievous. The nanny prefers to have
Good children to take care of but is also inherently harsh, and so she gets her
highest payoff of 4 from (Harsh, Good) and her lowest payoff of 1 from (Nice,
Mischievous), with (Nice, Good) yielding 3 and (Harsh, Mischievous) yielding 2.
The children similarly most prefer to have a Nice nanny and then to be
Mischievous; they get their highest two payoffs when the nanny is Nice (4 if
Mischievous, 3 if Good) and their lowest two payoffs when the nanny is Harsh (2
if Mischievous, 1 if Good).

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(a) Draw the game tree for this game and find the
subgame-perfect equilibrium in the absence of any strategic moves.

(b) In the film, before the arrival of Mary Poppins, the
children write their own ad for a new nanny in which they state: “If you won’t
scold and dominate us, we will never give you cause to hate us; we won’t hide
your tea.” Use the tree from part (a) to argue that this statement constitutes
a promise. What would the outcome of the game be if the children keep their
promise?

(c) What is the implied threat that goes with the promise
in part (b)? Is that implied threat automatically credible? Explain your