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).
(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 spectacles so you can’t see, put toads in your bed, or pep‑ per in 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 answer.
(d) How could the children make the promise in part (b) credible? (e) Is the promise in part (b) compellent or deterrent? Explain your answer by referring to the status quo in the game—namely, what would happen in the absence of the strategic move.