Let X = {X1, 22, 23} be the set of prizes. In each of the following questions, do the following. Check which of the three conditions: rationality, independence, and solvability, the preference R satisfies. Find an expected utility representation if R satisfies all three conditions, otherwise give a counterexample for each condition that it violates. (a) Worst-case scenario: pRq min i > min i p(xi)>0 9(2)>0 P, q E A(X). That is, our decision-maker compares lotteries by looking at the lowest in- dexed prize which has positive probability under that lottery. (b) Let Q1, Q2, Q3 > 0) and: pRq ap(+1) . (22) og(+3) > aq(21) a (22), (23) P, qe A(X). p,qe A(X). (c) pRq p(x1)2 +p(x2)2 +p(23)2 > q(21)? +9(x2)2 +9(23)2 (d) Lexicographic: pRq p(x3) > q(X3), or [p(x3) = 9(x3) and p(x2) > 9(x2)] P, q E A(X). Let X = {X1, 22, 23} be the set of prizes. In each of the following questions, do the following. Check which of the three conditions: rationality, independence, and solvability, the preference R satisfies. Find an expected utility representation if R satisfies all three conditions, otherwise give a counterexample for each condition that it violates. (a) Worst-case scenario: pRq min i > min i p(xi)>0 9(2)>0 P, q E A(X). That is, our decision-maker compares lotteries by looking at the lowest in- dexed prize which has positive probability under that lottery. (b) Let Q1, Q2, Q3 > 0) and: pRq ap(+1) . (22) og(+3) > aq(21) a (22), (23) P, qe A(X). p,qe A(X). (c) pRq p(x1)2 +p(x2)2 +p(23)2 > q(21)? +9(x2)2 +9(23)2 (d) Lexicographic: pRq p(x3) > q(X3), or [p(x3) = 9(x3) and p(x2) > 9(x2)] P, q E A(X)