91Ó°ÊÓ

Table 27 shows the preference schedule for an election. Rewrite Table 27 using the alternative preference schedule format. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Number of voters } & 14 & 10 & 8 & 7 & 4 \\ \hline 1 \text { st } & B & B & A & D & E \\ \hline \text { 2nd } & A & D & B & C & B \\ \hline \text { 3rd } & E & A & E & B & A \\ \hline \text { 4th } & D & E & D & E & C \\ \hline \text { 5th } & C & C & C & A & D \\ \hline \end{array} $$

Table 31 shows the preference schedule for an election with four candidates \((A, B, C,\) and \(D)\). Use the plurality method to (a) find the winner of the election. (b) find the complete ranking of the candidates. $$ \begin{array}{|l|r|r|r|r|r|l|} \hline \text { Number of voters } & \mathbf{2 7} & \mathbf{1 5} & \mathbf{1 1} & \mathbf{9} & \mathbf{8} & \mathbf{1} \\ \hline \text { 1st } & C & A & B & D & B & B \\ \hline \text { 2nd } & D & B & D & A & A & A \\ \hline \text { 3rd } & B & D & A & B & C & D \\ \hline \text { 4th } & A & C & C & C & D & C \\ \hline \end{array} $$

Table 32 shows the preference schedule for an election with four candidates \((A, B, C,\) and \(D) .\) Use the plurality method to (a) find the winner of the election. (b) find the complete ranking of the candidates. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Number of voters } & \mathbf{2 9} & \mathbf{2 1} & \mathbf{1 8} & \mathbf{1 0} & \mathbf{1} \\ \hline \text { 1st } & D & A & B & C & C \\ \hline \text { 2nd } & C & C & A & B & B \\ \hline \text { 3rd } & A & B & C & A & D \\ \hline \text { 4th } & B & D & D & D & A \\ \hline \end{array} $$

Use Table 43 to illustrate why the plurality-with-elimination method violates the IIA criterion. (Hint: Find the winner, then eliminate \(C\) and see what happens.) $$ \begin{array}{l|c|c|c|c|c|c} \text { Number of voters } & 5 & 5 & 3 & 3 & 3 & 2 \\ \hline 1 \text { st } & A & C & A & D & B & D \\ \hline \text { 2nd } & B & E & D & C & E & C \\ \hline \text { 3rd } & C & D & B & B & A & B \\ \hline \text { 4th } & D & B & C & E & C & A \\ \hline \text { 5th } & E & A & E & A & D & E \end{array} $$

The following fairness criterion was proposed by Italian economist Vilfredo Pareto (1848-1923): If every voter prefers candidate \(X\) to candidate \(Y\), then \(X\) should be ranked above \(Y\). (a) Explain why the Borda count method satisfies the Pareto criterion. (b) Explain why the pairwise-comparisons method satisfies the Pareto criterion.

If there is a candidate who loses in a one-to-one comparison to each of the other candidates, then that candidate should not be the winner of the election. (This fairness criterion is a sort of mirror image of the regular Condorcet criterion.) (a) Give an example that illustrates why the plurality method violates the Condorcet loser criterion. (b) Give an example that illustrates why the plurality-withelimination method violates the Condorcet loser criterion. (c) Explain why the Borda count method satisfies the Condorcet loser criterion.

Consider the following fairness criterion: If a majority of the voters have candidate \(X\) ranked last, then candidate \(X\) should not be a winner of the election. (a) Give an example to illustrate why the plurality method violates this criterion. (b) Give an example to illustrate why the plurality-withelimination method violates this criterion. (c) Explain why the method of pairwise comparisons satisfies this criterion. (d) Explain why the Borda count method satisfies this criterion.