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, but playing tricks on such nannies is great fun. That is, they view themselves as playing a game in which the nanny moves first, show­ ing herself to be either Harsh or Nice, and the children move second, choosing to be 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 1from (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 tvvo payoffs when the nanny is Harsh (2 if Mischievous, 1if Good).

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

"William C. Charron, "Greeks and Games: Forerunners of Modern Game Theory," Forum for So­ cial Economics, vol. 29, no. 2 (Spring 2000), pp. 1-32.

(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 pepper in your tea." Use the tree from part a to argue that this statement consti­ tutes a promise. What would the outcome of the game be if the promise works?

(c) What is the implied threat that goes with the promise in part b? Is that promise 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.