Bill injured his knee. His doctor gives him the following options: 1. No physical activity for 2 months; in 45% of the cases there is complete recovery; in the remaining 55% of the cases there is only some pain reduction. 2. Take strong doses of Ibuprofen for 2 months; in 70% of the cases there is complete recovery, without side effects; in 20% of the cases there is complete recovery, but also the development of a stomach ulcer; in the remaining 10% of the cases there is only some pain reduction, with no side effects. 3. Surgery. This type of surgery has been performed many times in the past and it led to complete recovery in 95% of the patients. For the remaining 5%, however, there was damage to the knee during the procedure, which led to permanent inability to perform a range of physical activities (running, walking, etc.). (a) Draw a decision tree to represent Bill's decision problem. The basic outcomes are: (A) complete recovery, (B) partial pain reduction, (C) complete recovery plus ulcer, (D) permanent damage of the knee. Bill ranks these outcomes as follows: (A) (B) (C) (D). He is indifferent between (B) for sure and a lottery that gives (A) with probability and (D) with probability Furthermore, he is 2 2 indifferent between (C) for sure and a lottery that gives (A) with probability probability Bill satisfies the axioms of expected utility. 3 4 . 4 and (D) with (b) Re-draw the decision tree replacing outcomes with the corresponding utility (use a utility function whose lowest value is 0 and highest value is 100). (c) Use backward-induction to compute the optimal decision for Bill.