5. (10 points) The version of Arrow's theorem we discussed in class states that, for A...

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5. (10 points) The version of Arrow's theorem we discussed in class states that, for A 3 alternatives, any ranking rule that satisfies unanimity and independence of irrelevant alternatives (IIA) is dictatorial, i.e., there is an agent i such that the outcome of the rule equals >; for any profile of preferences (>1,...,n) of preferences. What if we dropped the assumption of unanimity? We would get many boring rules, such as those that do not depend on preference profile at all. To exclude non-interesting answers, let's replace unanimity with the following weaker requirement. A ranking rule is onto if for any ranking > there is a preference profile (>1,...,n) such that the outcome of the rule on this profile equals >. Construct an example of a ranking rule that is onto and satisfies IIA but is not a dictatorship. Hint 1: The outcome of a rule may still be determined by a preference >i of a single agent i but not be equal to >i. Hint 2: The answer is simple. Remark: There is a version of Arrow's theorem which states that all rules that are onto and satisfy IIA are either dictatorships or the rules you've come up with. 5. (10 points) The version of Arrow's theorem we discussed in class states that, for A 3 alternatives, any ranking rule that satisfies unanimity and independence of irrelevant alternatives (IIA) is dictatorial, i.e., there is an agent i such that the outcome of the rule equals >; for any profile of preferences (>1,...,n) of preferences. What if we dropped the assumption of unanimity? We would get many boring rules, such as those that do not depend on preference profile at all. To exclude non-interesting answers, let's replace unanimity with the following weaker requirement. A ranking rule is onto if for any ranking > there is a preference profile (>1,...,n) such that the outcome of the rule on this profile equals >. Construct an example of a ranking rule that is onto and satisfies IIA but is not a dictatorship. Hint 1: The outcome of a rule may still be determined by a preference >i of a single agent i but not be equal to >i. Hint 2: The answer is simple. Remark: There is a version of Arrow's theorem which states that all rules that are onto and satisfy IIA are either dictatorships or the rules you've come up with.

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