Fairness Preferences. Suppose you can award exactly one person with a prize. Person A and person B both equally deserve the prize and you are indifferent between giving the prize to A or B. However, you strictly prefer to toss a fair coin to decide who gets the prize over giving the prize to either of the two individuals directly.

(a) What are three different lotteries for which a preference ranking is specified in the question? Write down the preference ranking between these three lotteries.

(b) Show that such preferences violate one of the vNM axioms and hence are not compatible with expected utility representation.

(c) Now assume by contradiction that the preferences are consistent with expected utility representation, state what the preference ranking implies for the utility values and derive a contradiction.