One possible method of determining a group preference relation is the Borda count. The way it works is that each voter is asked to rank all of the alternatives. If there are 10 alternatives, you give your first choice a 10, your second choice a 9, and so on. The voters scores for each alternative are then added over all voters. The total score for an alternative is called its Borda count. For any two alternatives, x and y, if the Borda count of x is larger than or the same as the Borda count for y, then x is at least as good as y for the group. Suppose that there are a finite number of alternatives to choose from and that every individual has complete, reflexive, and transitive preferences. For the time being, let us also suppose that individuals are never indifferent between any two different alternatives but always prefer one to the other.
(a) Is the social preference ordering defined in this way complete? Is it reflexive? Is it transitive?
(b) If everyone prefers x to y, will the Borda count rank x as as preferred to y for that group? Explain your answer.
(c) Suppose that there are two voters and three candidates, x, y, and z. Suppose that Voter 1 ranks the candidates, x first, z second, and y third. Suppose that Voter 2 ranks the candidates, y first, x second, and z third. What is the Borda count for x, y and z?
(d) Now suppose that for the same three voters and candidates above, that it is discovered that candidate z once put ketchup on his eggs. Voter 1, is appalled and changes his ranking to x first, y second, z third. Voter 2, has always secretly ketchuped' his eggs, is impressed by z's courage and changes his ranking to y first, z second, x third. Now what is the Borda count for x, y and z now?
(e) Does the social preference relation defined by the Borda count satisfy Arrow's condition of "independence of irrelevant alternatives" Explain.