Study how to aggregate individual preferences

35pts) We want to study how to aggregate individual preferences. Environment is the ollowing I : a set of agents i E I = {1,2,3}, X: a set of alternatives X = {x, y, z} Assume that each agent i has a rational preference relation R' on X, called a preference ordering. Here we only consider a strict preference relation Pi on X derived from Ri. 1) Consider the set of strict preference ordering, P. How many number of elements does the set have? (1pts) 2) Consider the set of strict preference profiles, P3 = (P, P, P). How many number of elements does the set have? (1pts) 3) We learned the pairwise majority voting in the class. Provide an example of preference profile, p E P3 that shows that the preference relation created by the pairwise majority voting is not transitive. (4pts) Let S C R3 be given. A social welfare functional is a map R: S - R. Here we can consider S = P3 4) We learned how to define the Rank-order vote social welfare functional in the class