2. Jill is contemplating her preferences concerning tonight's activities. She strictly prefers: - going out with...

 

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2. Jill is contemplating her preferences concerning tonight's activities. She strictly prefers: - going out with 50% chance and working with 50% chance, over - watching TV for sure, over - going out with 40% chance, watching TV with 20% chance and working with 40% chance. a) Show that Jill's preferences violate the independence axiom. [Hint: Start by decomposing the third lottery into a combination of the first two. That is, if the first lottery is denoted A, the second B, and the third C, find x € [0,1] such that C = xA + (1x)B. Then show that A > B> xA+ (1 - x)B violates the independence axiom.] b) Since Jill's preferences violate the independence axiom, we know that they do not admit an expected utility representation. Show directly that it is impossible to assign utilities to the outcomes so that the ranking of the expected utilities of the three lotteries matches Jill's preference ranking. [Hint: To do this, first assume that the outcomes generate utilities u₁, U₂, U3. Then compute the expected utilities of the three lotteries above, and derive a contradiction between the inequalities corresponding to Jill's preferences.] 2. Jill is contemplating her preferences concerning tonight's activities. She strictly prefers: - going out with 50% chance and working with 50% chance, over - watching TV for sure, over - going out with 40% chance, watching TV with 20% chance and working with 40% chance. a) Show that Jill's preferences violate the independence axiom. [Hint: Start by decomposing the third lottery into a combination of the first two. That is, if the first lottery is denoted A, the second B, and the third C, find x € [0,1] such that C = xA + (1x)B. Then show that A > B> xA+ (1 - x)B violates the independence axiom.] b) Since Jill's preferences violate the independence axiom, we know that they do not admit an expected utility representation. Show directly that it is impossible to assign utilities to the outcomes so that the ranking of the expected utilities of the three lotteries matches Jill's preference ranking. [Hint: To do this, first assume that the outcomes generate utilities u₁, U₂, U3. Then compute the expected utilities of the three lotteries above, and derive a contradiction between the inequalities corresponding to Jill's preferences.]

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a Let a a a denote the activities going out watching TV and working respectively The options faced b... View the full answer