Exercise 22.20. * (1) Consider a society with two individuals 1 and 2 and three choices,

a, b and

c. For the purposes of this exercise, only consider strict individual and social orderings (i.e., no indifference allowed). Suppose that the preferences of the first agent are given by abc (short for a  b Â

c, i.e., a strictly preferred to

b, strictly preferred to c). Consider the six possible preference orderings of the second individual, i.e., s2 ∈ {abc, acb, bac, ...}, etc.. Define a social ordering as a mapping from the preferences of the second agent (given the preferences of the first) into a social ranking of the three outcomes, i.e., some function φ such that the social ranking is s = φ (s2). Illustrate the Arrow impossibility theorem using this example [Hint: start as follows: abc = φ (abc), i.e., when the second agent’s ordering is abc, the social ranking must be abc; next, φ (acb) = abc or acb (why?); then if φ (acb) = abc, we must also have φ (cab) = abc (why?); and proceeding this way to show that the social ordering is either dictatorial or it violates one of the axioms]. (2) Now suppose we have the following aggregation rule: individual 1 will (sincerely) rank the three outcomes, his first choice will get 6 votes, the second 3 votes, the third 1 vote. Individual 2 will do the same, his first choice will get 8 votes, the second 4 votes, and the third 0 vote. The three choices are ranked according to the total number of votes. Which of the axioms of the Arrow’s Theorem does this aggregation rule violate? (3) With the above voting rule, show that for a certain configuration of preferences, either agent has an incentive to distort his true ranking (i.e., not vote sincerely). (4) Now consider a society consisting of three individuals, with preferences given by: 1 a  b  c 2 c  a  b 3 b  c  a Consider a series of pairwise votes between the alternatives. Show that when agents vote sincerely, the resulting social ordering will be “intransitive”. Relate this to the Theorem 22.1. (5) Show that if the preferences of the second agent are changed to b  a Â

c, the social ordering is no longer intransitive. Relate this to “single-peaked preferences”. (6) Explain intuitively why single-peaked preferences are sufficient to ensure that there will not be intransitive social orderings. How does this relate to Theorem 22.1?