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[6 marks] 3. Decision Making A rational agent R must choose between two lotteries, the outcomes of which depends on which colour ball is picked from a jar: Lottery A: If the chosen ball is red, then you win £1 with probability p and £0 with probability (1 - p). If, on the other hand, the chosen ball isn't red, then you are certain to get £0. Lottery B: If the chosen ball is red, then you are certain to win £0. If, on the other hand, the chosen ball isn't red, then you win £1 with probability a and £0 with probability (1-4). (a) Suppose that R doesn't know how many balls are in the jar, nor which colours they are. Define the expected utility of lottery A and lottery B. (b) Suppose that p and q are such that the agent R is indifferent between lot- teries A and B and suppose further more that R always chooses an action that maximises R's expected utility. Then using the definition in part (a), define what R must believe the probability of picking a red ball from the jar must be in terms of (only) p and q. Be sure to show all the workings in your answer. (c) Now suppose that there are 10 balls in the jar; 4 are red and 6 aren't. Furthermore, suppose R is indifferent between lotteries A and B. Then define p in terms of . (d) Now consider an irrational agent I. I has a choice between 3 lotteries: lotteries A and B as defined above with p=9= 1, and lottery C: Lottery C: You win £1 with probability 0.5 and £0 with probability 0.5 I prefers lottery C to both lotteries A and B. Show that I cannot be max- imising expected utility, whatever the values of U(£1) and U(£0). (10 marks] [3 marks [6 marks]