### Question 1.

Ada and Bella are flatmates. Bella is social and dynamic, and often makes plans for herself and her flatmate. Ada is more of a follower, but sometimes opts out of Bella’s plans. Suppose that they have the following preferences: Bella always prefers to be with Ada than alone. For Bella, there’s nothing better than partying at a club all night with Ada, but when she’s not with Ada, she’d rather be at home with a book than out alone. Ada is much more laid-back – she’s equally happy alone, or at home with Bella, but doesn’t really like clubbing with Bella because Bella does embarrassing things when drunk. For Bella, the choice is whether to stay in for the evening or go out partying; while for Ada, the choice is whether to go along with whatever Bella has planned, or avoid her after class and be alone in her room for the evening.

**(a)** Draw up a matrix describing the outcomes for the two flatmates. Use ordinal values. As Ada and Bella go through this process every evening, you can assume through the question that this is an iterated game, and that their strategies will reflect this.

**(b)** Assuming each flatmate is only interested in satisfying their own preferences, and both are rational, what will they end up doing in the evenings? Be careful to explicitly state the assumptions about rationality that lead to your outcomes(s). In your explanation, refer to the matrix above.

**(c)** Suppose that Ada is weakly altruistic, so that she prefers to maximise Bella’s value when selecting between two options for which she is indifferent. Give an argument why it might be rational for Ada to always avoid Bella after class, and an argument why it might be rational to always stick with Bella in the evenings. In each case, be careful to explicitly state the assumptions of rationality (including Ada’s weak altruism) that lead to the differing strategies. Again, refer to the matrix above.

**(d)** Suppose that Bella knows that Ada is weakly altruistic, and so Ada will choose to always stick with Bella in the evenings, for the reasons you have given above. Give an argument why Bella, being rational, would choose to stay at home in the flat, and an argument why Bella rationally would choose to go out clubbing. In each case, be careful to explicitly state the assumptions of rationality (including Ada’s weak altruism) that lead to the differing strategies.

**(e)** One evening, the flatties ring up their ex-flatmate Caro, and ask her to decide what Ada and Bella should do for the evening. Caro is yet another rational agent (which is why they were such great flatmates), is equally concerned with both her friends having a good time, and knows both their preferences very well. What does Caro advise, and why? Explicitly state Caro’s reasoning, including your assumptions of rationality. Is asking Caro a better way to decide what to do in the evening than simply each flatmate trying to satisfy their own preferences? Is either flatmate disadvantaged compared with the other?

**(f)** Propose a probabilistic strategy for at least one of the flatmates, and explain how it could lead to a better outcome than having a deterministic strategy. State your assumptions carefully.

**(g)** Describe what you think is the right – that is, ethical – choice for each of Ada and Bella. Describe the assumptions of both rationality and ethical decision making that will lead to this choice, and justify your choice of rational principles. If you claim that a standard game-theory technique such as selecting a Pareto optimal or Nash equilibrium strategy always produces a correct ethical decision, you will need to justify this.

### Question 2.

John claims that he has more chance of being killed by a mass-extinction event (like the Chicxulub asteroid which probably killed the dinosaurs 66 million years ago) than of being killed by sharks.

**(a)** Suppose John has one chance in a twenty of dying this year given that he is killed by sharks, one chance in 80 of dying this year, and one chance in a million of being killed by sharks. What is the probability that he is killed by sharks given that he dies this year?

**(b)** Explain why Bayes’ Theorem shouldn’t be used to calculate the conditional probability of John being killed by sharks given that he dies in the next hundred years.

**(c)** Re-state John’s claim for an epistemic interpretation of probability, and discuss what evidence would be relevant to determining whether his claim is accurate.

**(d)** Repeat Q2(c) for a frequentist interpretation of probability.

**(e)** Repeat Q2(c) for a logical interpretation of probability.

**(f)** Which of the interpretations in Q2(c)-(e) seems to be most appropriate for John’s claim? Why?

**(g)** Restate John’s absolute probability claim as a conditional probability claim; include the substantive assumptions that you think John may have made. Include at least one condition that heavily influences your preference for the best interpretation of John’s probability claim, and explain why it affects your preference.

**(h)** Melissa believes that John is at least twice as likely to be killed by his hobby of chainsaw juggling as by a mass-extinction event. Isaac believes that chainsaw juggling is safe, and so John is much less likely to die juggling than during his regular job as a shark dental hygenist. Design a Dutch Book so that you will profit exactly $1000 from Melissa, Isaac’s, and John’s beliefs regardless of the correct probabilities.

**(i)** How would the Dutch Book from the previous question change if Melissa only believes that John’s hobby is somewhat more likely to be fatal than a mass-extinction event?

**(j)** If a Dutch Book an appropriate tool for assessing rationality in this case, given that one of the outcomes will involve all participants’ certain death?

### Question 3.

Professor Oldbrush likes to give her students a choice of assignments. Each student is given an envelope containing an assignment. They may also choose to take a copy of a hard assignment (which they may look through before deciding whether to take). If they take both assignments, they can attempt both of them, and the marks for each assignment will be added together. But there is a catch. If Professor Oldbrush believes the student will take the hard assignment, the envelope will contain a fiendishly difficult assignment, while if she believes they will not take the extra assignment, the envelope will contain a straightforward assignment. Her students are aware of this, and they also know that Professor Oldbrush is an excellent reader of students, and is usually right in predicting whether they’ll take the hard assignment or not. Moreover, she is just as good at predicting if they will choose randomly, and in this case the envelope will contain the fiendishly difficult assignment.

**(a)** Would you take the hard assignment as well as the assignment in the envelope if you were in Prof Old brushes class?

**(b)** Give a cogent argument why a rational student should take the hard assignment as well as the assignment in the envelope

**(c)** Give a cogent argument why a rational student should only take the assignment in the envelope.

**(d)** Explain what assumptions about rationality differ between the two cases above. Explain why you prefer the assumptions about rationality that led to your choice in the first part of this question.

**(e)** Suppose that a student believes that they will get an average of 5 marks on the hard assignment, 1 mark on the fiendish assignment, and 15 marks on the easy assignment, and that Professor Oldbrush is 90% accurate in predicting whether a student will take the hard assignment as well. You can then calculate an expected mark for each of the two strategies. What are the expected marks? What parts of the problem does this help with, and what parts does it not?

**(f)** One week Professor Oldbrush is away, and her tutor Sean Black distributes the assignments instead. However, Sean only accurately reads students around three-quarters of the time. Does this make a difference to the dilemma? Why? What is the critical level of predictive accuracy required to create this dilemma?

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