One possible method of determining a social preference relation is the Borda count, also known as rank-order voting. Each voter is asked to rank all of the alternatives. If there are 10 alternatives, you give your first choice a 1, your second choice a 2, 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 smaller than or the same as the Borda count for y, then x is “socially at least as good as” y. 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? ________ Reflexive? __________ Transitive? __________
(b) If everyone prefers x to y, will the Borda count rank x as socially preferred to y? 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? _________ For y? __________ For z? Now suppose that it is discovered that candidate z once lifted a beagle by the ears. Voter 1, who has rather large ears himself, is appalled and changes his ranking to x first, y second, z third. Voter 2, who picks up his own children by the ears, is favorably impressed and changes his ranking to y first, z second, x third. Now what is the Borda count for x? _________ For y? _______ For z? __________
(d) Does the social preference relation defined by the Borda count have the property that social preferences between x and y depend only on how people rank x versus y and not on how they rank other alternatives? Explain.