The introduction to this chapter of Workouts, recounted the sad tale of roommates Victoria and Albert and their dirty room. The payoff matrix for their relationship was given as follows.
Suppose that we add a second stage to this game in which Victoria and Albert each have a chance to punish the other. Imagine that at the end of the day, Victoria and Albert are each able to see whether the other has done any housecleaning. After seeing what the other has done, each has the option of starting a quarrel. A quarrel hurts both of them, regardless of who started it. Thus we will assume that if either or both of them starts a quarrel, the day’s payoff for each of them is reduced by 2. (For example if Victoria cleans and Albert doesn’t clean and if Victoria, on seeing this result, starts a quarrel, Albert’s payoff will be 6 − 2 = 4 and Victoria’s will be 2 − 2 = 0.)
(a) Suppose that it is evening and Victoria sees that Albert has chosen not to clean and she thinks that he will not start a quarrel. Which strategy will give her a higher payoff for the whole day, Quarrel or Not Quarrel? _________.
(b) Suppose that Victoria and Albert each believe that the other will try to take the actions that will maximize his or her total payoff for the day. Does either believe the other will start a quarrel? _________ Assuming that each is trying to maximize his or her own payoff, given the actions of the other, what would you expect each of them to do in the first stage of the game, clean or not Clean? _________.
(c) Suppose that Victoria and Albert are governed by emotions that they cannot control. Neither can avoid getting angry if the other does not clean. And if either one is angry, they will quarrel so that the payoff of each is diminished by 2. Given that there is certain to be a quarrel if either does not clean, the payoff matrix for the game between Victoria and Albert becomes:
(d) If the other player cleans, is it better to clean or not clean? Clean If the other player does not clean, is it better to clean or not clean. ________ Explain _________.
(e) Does this game have a dominant strategy? _______. Explain _______.
(f) This game has two Nash equilibria. What are they? Both clean the room and both don’t clean the room.
(g) Explain how it could happen that Albert and Victoria would both be better off if both are easy to anger than if they are rational about when to get angry, but it might also happen that they would both be worse off. ____________.
(h) Suppose that Albert and Victoria are both aware that Albert will get angry and start a quarrel if Victoria does not clean, but that Victoria is level-headed and will not start a quarrel. What would be the equilibrium outcome? _______.