Question 3. (20 marks) a) The following axioms are central in Subjective Expected Utility (SEU) theory: Axiom 1: For all events A and B, and all outcomes x > y and x' > y', we have: [( A, x); (-A, y)] ~ [(B, x); (-B, y)] > [(A, x'); (-A, y)] > [(B, x'); (-B, y)]. Axiom 2: For all events C and all outcomes z, z' we have: [( A] , x1 ); ... ; (A;, x;); (C, z)] z [(B, VI); ... ; (BK, VK); (C, z) ] [(A1, x1 ); ... ; (A;, x;); (C, z')] z [(B,, VI); ... ; (BK, VK); (C, z')]. Explain these axioms and show that they are necessary for SEU maximisation. b) Define an order _* over events so that A _* B if and only if, for some x and y, we have [(A, x); (-A, y)] Z [(B, x); (-B, y)]. Explain how such an order can be interpreted if axioms Al and A2 hold. Assuming these axioms hold, show that the following must be true: For all events A, B, C with AnC = BOC = 0 we have A _* B if and only if AUC _* BUC. c) An urn contains 200 balls, each of which are either red, black, green, or yellow (R, B, G, Y). There are 50 green and 50 yellow balls. The remaining 100 balls are either red or black, but nothing more is known. One ball is drawn and bets are placed on its colour. Consider the following acts: a = [(R, f100); (B, f100); (G, fo); (Y , fo)] b = [(R, f100); (B, fo); (G, f100); (Y, fo)] c = [(R, fO); (B, f100); (G, fO); (Y, f100)] d = [(R, fo); (B, fo); (G, f100); (Y, f100)] The modal pattern of preferences is a > b and c