Q3. (Sophistication vs Naivete) Suppose there are three periods: bedtime, 8am and 9am. The agent has an exam at 9am the next day, and an alarm clock that can be turned off or can be set / reset only for Sam or 9am. The decision tree describes the actions available in each period: At bedtime she can either set the alarm for 8 or 9 or she can choose not to set an alarm. At 8am her possible actions are getting out of bed at 8am, resetting the alarm for 9am, or unplugging the alarm clock. At 9am her actions are getting out of bed at 9am or to unplug the alarm clock. [You should draw the decision tree]. In this setting, a plan is a sequence of actions. For instance, (reset for 9, wake up at 9) is a possible plan available to the 8am self, which described an action for her, and an action for the 9am self. Suppose the agent's preferences (over plans) at each point in time are fully dictated by her preferences over when to wake up: at bedtime, she prefers getting out of bed at 9am to getting out of bed at 8am to skipping the exam altogether; at 8am, she prefers getting out of bed at 9am to getting out of bed at 8am to skipping the exam; at 9am, she prefers to skip the exam over getting out of bed. (a) What plan will the agent choose in each period if she is (i) naive in every period, (ii) sephisticated in every period? (Hint: for the sophisticated case, the answer is NOT that she skips the exam eventually). (b) In each case, determine if the bed time self would be interested in com mitment devices. (c) Consider the sophisticated agent's choice again but now suppose that the option to skip the exam at 9am is no longer feasible (for instance, suppose the agent tells her friend to come get her at 9am unless she cancels at 8am). Write down the sophisticated agent's chosen plans in each period. (d) Suppose an outside observer observed the sophisticated agent's choice (out of objective menus) at bed time in the cases given in parts (a) and (c) Would the observer nd the agent's choices to be consistent with preference maximization (in the sense of chapter 3 of the lecture notes)