(15 points) Joe plays a game with three possible prizes: a high prize of $100, a medium prize of $40, and a low prize of $0. Joe's utility for the high prize is 120, and his utility for the low prize is 30. For notation, let the probabilities for any lottery over these prizes be denoted q = (Pr(low), Pr(medium), Pr(high)). (a) Suppose Joe is an expected utility maximizer, and his utility of the middle prize is v. What is v if Joe is risk averse? (b) Suppose we don't know if Joe is risk averse, but he strictly prefers higher prizes to lower ones. He is offered two different lotteries over these three possible prizes: lottery q = (0.4, 0.4, 0.2) and lottery r = (0.25, 0.5, 0.25). Which lottery does Joe prefer? (c) Suppose Joe has preferences over lotteries over these prizes, such that (0.1, 0.7, 0.2) > (0.15, 0.6, 0.25) (0.8075, 0.13, 0.0625) > (0.805, 0.135, 0.06) Are these preferences consistent with expected utility maximization