Microeconomics question (Game theory)

Exercise 8.]. John and Mary are discussing whether to go to the ballet, to the boxing match, or to stay home tonight. They have a random- number generator, and they can agree to let their decision depend on its output in any way, so as to create a lottery with any probability distribution over the three possible outcomes (go to the ballet, go to boxing, or stay home). If they cannot agree, they will stay home. John wants to go to the boxing match, and he is indifferent between going to the ballet and staying home. For Mary, going to the ballet would be best, staying home would be worst, and she would be indifferent be- tween going to the boxing match or taking a lottery with probability V4 Exercises 413 of going to the ballet and probability 5% of staying home. They both satisfy the axioms of von Neumann-Morgenstern utility theory. a. Describe this situation by a two-person bargaining problem (F ,v) and compute its Nash bargaining solution. How should they implement the Nash bargaining solution? b. The above description of this situation does not specify a specic two-person bargaining problem, as dened in Section 8.2. Which axi- omatic properties of the Nash bargaining solution permitted you to answer part (a) anyway? c. Find natural utility scales for John and Mary in which the egali- tarian and utilitarian solutions both coincide with the Nash bargaining solution. d. If the television set were broken, the prospect of staying home would become much worse for John. To be specic, John would be indifferent between going to the ballet and a lottery with probability W3 of going to boxing and probability 1/3 of staying home with a broken television set. A broken television set would not affect Mary's prefer- ences. If Mary breaks the television set, her brother (who is a television repairman, and who is coming for breakfast tomorrow) will x it for free. How would breaking the television set change the probability of going to the ballet in the Nash bargaining solution