Question

4. (a) Consider the following group of people (labelled A to G) who have the rankings shown of candidates for election X, Y and Z. The symbol > means “is preferred to.”

A: X>Y>Z B: X>Y>Z C: Y>X>Z D: Y>X>Z
E: Z>X>Y F: Z>X>Y G: Z>X>Y


(i) Explain which candidate will win if voting is on the first past the post basis. (2)

(ii) Suppose, instead, that voting is conducted by running off candidates in pairs; that is X vs Y, Y vs Z, and Z vs X. Explain who will win in each of these runoffs and comment, in the light of Condorcet’s paradox, whether transitivity is violated in this case. (5)

(iii) Given your answers in (i) and (ii), it should be clear that one of the candidates appears to be favoured, if not by a majority, at least by a plurality of voters yet, in a runoff against any one of the other candidates, that candidate would lose. In the light of Arrow’s impossibility theorem, what comment would you make on the two voting systems presented in this question? (3)

(iv) Suppose you were asked to justify candidate X as the most appropriate to be elected given the preferences of these voters. Explain on what basis you could justify this outcome. (3)

Answer 1

(i) In a first past the post basis, the candidate who wins is the one with the most first-place votes. Based on the given rankings, candidate Z would win because they are ranked first by all voters.

(ii) In the pair runoffs, comparing X and Y, X is preferred by A, B, E, and F, while Y is preferred by C and D. Thus, X would win the X vs Y runoff. Comparing Y and Z, Y is preferred by A, B, C, and D, while Z is preferred by E, F, and G. Therefore, Y would win the Y vs Z runoff. Comparing Z and X, Z is preferred by all voters. Hence, Z would win the Z vs X runoff.

Condorcet's paradox refers to a situation where a cyclic pattern of preferences occurs among three or more candidates. In this case, there is no Condorcet winner because each candidate can lose in a pairwise comparison. Transitivity is violated since it is not possible to establish a clear overall preference order among the candidates.

(iii) Arrow's impossibility theorem states that no voting system that meets certain reasonable criteria can avoid all possible paradoxes and inconsistencies. In this case, the voting systems presented in (i) and (ii) both fail to identify a clear winner. They demonstrate the limitations of voting systems in accurately reflecting the collective preferences of the voters and avoiding paradoxes.

(iv) Justifying candidate X as the most appropriate based on the preferences of these voters could be done by considering the concept of the Borda count. The Borda count assigns points to candidates based on their ranking. In this case, X would receive 5 points (2nd choice for A, B, C, D, and 3rd choice for E, F, G), while Y and Z would receive 4 points each. By this count, X would have the highest score and could be justified as the most appropriate candidate.